diff options
Diffstat (limited to 'src/share')
| -rw-r--r-- | src/share/algebra/browse.daase | 3436 | ||||
| -rw-r--r-- | src/share/algebra/category.daase | 6968 | ||||
| -rw-r--r-- | src/share/algebra/compress.daase | 139 | ||||
| -rw-r--r-- | src/share/algebra/interp.daase | 8035 | ||||
| -rw-r--r-- | src/share/algebra/operation.daase | 21551 |
5 files changed, 20105 insertions, 20024 deletions
diff --git a/src/share/algebra/browse.daase b/src/share/algebra/browse.daase index 4b2d1519..52405adb 100644 --- a/src/share/algebra/browse.daase +++ b/src/share/algebra/browse.daase @@ -1,12 +1,12 @@ -(1962894 . 3538630437) +(1965419 . 3539125283) (-18 A S) ((|constructor| (NIL "One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \\spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently grow or shrink on demand as can list structures (see however \\spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result."))) NIL NIL (-19 S) ((|constructor| (NIL "One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \\spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently grow or shrink on demand as can list structures (see however \\spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL (-20 S) ((|constructor| (NIL "The class of abelian groups,{} \\spadignore{i.e.} additive monoids where each element has an additive inverse. \\blankline")) (- (($ $ $) "\\spad{x-y} is the difference of \\spad{x} and \\spad{y} \\spadignore{i.e.} \\spad{x + (-y)}.") (($ $) "\\spad{-x} is the additive inverse of \\spad{x}"))) @@ -38,7 +38,7 @@ NIL NIL (-27) ((|constructor| (NIL "Model for algebraically closed fields.")) (|zerosOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{zerosOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) (|Polynomial| $)) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible. Otherwise they are implicit algebraic quantities. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|zeroOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{zeroOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}; if possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity which displays as \\spad{'y}.") (($ (|SparseUnivariatePolynomial| $)) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}; if possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity.") (($ (|Polynomial| $)) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. If possible,{} \\spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootsOf| (((|List| $) (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\spad{rootsOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) (|Polynomial| $)) "\\spad{rootsOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ (|SparseUnivariatePolynomial| $) (|Symbol|)) "\\spad{rootOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ (|SparseUnivariatePolynomial| $)) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}.") (($ (|Polynomial| $)) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL (-28 S R) ((|constructor| (NIL "Model for algebraically closed function spaces.")) (|zerosOf| (((|List| $) $ (|Symbol|)) "\\spad{zerosOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable.")) (|zeroOf| (($ $ (|Symbol|)) "\\spad{zeroOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \\spad{'y}.") (($ $) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \\spad{p} has more than one variable.")) (|rootsOf| (((|List| $) $ (|Symbol|)) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ $ (|Symbol|)) "\\spad{rootOf(p,y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ $) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) @@ -46,7 +46,7 @@ NIL NIL (-29 R) ((|constructor| (NIL "Model for algebraically closed function spaces.")) (|zerosOf| (((|List| $) $ (|Symbol|)) "\\spad{zerosOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities which display as \\spad{'yi}. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{zerosOf(p)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}. The \\spad{yi}'s are expressed in radicals if possible. The returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable.")) (|zeroOf| (($ $ (|Symbol|)) "\\spad{zeroOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \\spad{'y}.") (($ $) "\\spad{zeroOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. The value \\spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \\spad{p} has more than one variable.")) (|rootsOf| (((|List| $) $ (|Symbol|)) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; The returned roots display as \\spad{'y1},{}...,{}\\spad{'yn}. Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values.") (((|List| $) $) "\\spad{rootsOf(p, y)} returns \\spad{[y1,...,yn]} such that \\spad{p(yi) = 0}; Note: the returned symbols \\spad{y1},{}...,{}yn are bound in the interpreter to respective root values. Error: if \\spad{p} has more than one variable \\spad{y}.")) (|rootOf| (($ $ (|Symbol|)) "\\spad{rootOf(p,y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}.") (($ $) "\\spad{rootOf(p)} returns \\spad{y} such that \\spad{p(y) = 0}. Error: if \\spad{p} has more than one variable \\spad{y}."))) -((-3975 . T) (-3973 . T) (-3972 . T) ((-3980 "*") . T) (-3971 . T) (-3976 . T) (-3970 . T)) +((-3986 . T) (-3984 . T) (-3983 . T) ((-3991 "*") . T) (-3982 . T) (-3987 . T) (-3981 . T)) NIL (-30) ((|constructor| (NIL "\\indented{1}{Plot a NON-SINGULAR plane algebraic curve \\spad{p}(\\spad{x},{}\\spad{y}) = 0.} Author: Clifton \\spad{J}. Williamson Date Created: Fall 1988 Date Last Updated: 27 April 1990 Keywords: algebraic curve,{} non-singular,{} plot Examples: References:")) (|refine| (($ $ (|DoubleFloat|)) "\\spad{refine(p,x)} \\undocumented{}")) (|makeSketch| (($ (|Polynomial| (|Integer|)) (|Symbol|) (|Symbol|) (|Segment| (|Fraction| (|Integer|))) (|Segment| (|Fraction| (|Integer|)))) "\\spad{makeSketch(p,x,y,a..b,c..d)} creates an ACPLOT of the curve \\spad{p = 0} in the region {\\em a <= x <= b, c <= y <= d}. More specifically,{} 'makeSketch' plots a non-singular algebraic curve \\spad{p = 0} in an rectangular region {\\em xMin <= x <= xMax},{} {\\em yMin <= y <= yMax}. The user inputs \\spad{makeSketch(p,x,y,xMin..xMax,yMin..yMax)}. Here \\spad{p} is a polynomial in the variables \\spad{x} and \\spad{y} with integer coefficients (\\spad{p} belongs to the domain \\spad{Polynomial Integer}). The case where \\spad{p} is a polynomial in only one of the variables is allowed. The variables \\spad{x} and \\spad{y} are input to specify the the coordinate axes. The horizontal axis is the \\spad{x}-axis and the vertical axis is the \\spad{y}-axis. The rational numbers xMin,{}...,{}yMax specify the boundaries of the region in which the curve is to be plotted."))) @@ -56,14 +56,14 @@ NIL ((|constructor| (NIL "This domain represents the syntax for an add-expression.")) (|body| (((|SpadAst|) $) "base(\\spad{d}) returns the actual body of the add-domain expression `d'.")) (|base| (((|SpadAst|) $) "\\spad{base(d)} returns the base domain(\\spad{s}) of the add-domain expression."))) NIL NIL -(-32 R -3078) +(-32 R -3088) ((|constructor| (NIL "This package provides algebraic functions over an integral domain.")) (|iroot| ((|#2| |#1| (|Integer|)) "\\spad{iroot(p, n)} should be a non-exported function.")) (|definingPolynomial| ((|#2| |#2|) "\\spad{definingPolynomial(f)} returns the defining polynomial of \\spad{f} as an element of \\spad{F}. Error: if \\spad{f} is not a kernel.")) (|minPoly| (((|SparseUnivariatePolynomial| |#2|) (|Kernel| |#2|)) "\\spad{minPoly(k)} returns the defining polynomial of \\spad{k}.")) (** ((|#2| |#2| (|Fraction| (|Integer|))) "\\spad{x ** q} is \\spad{x} raised to the rational power \\spad{q}.")) (|droot| (((|OutputForm|) (|List| |#2|)) "\\spad{droot(l)} should be a non-exported function.")) (|inrootof| ((|#2| (|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{inrootof(p, x)} should be a non-exported function.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}. Error: if \\spad{op} is not an algebraic operator,{} that is,{} an \\spad{n}th root or implicit algebraic operator.")) (|rootOf| ((|#2| (|SparseUnivariatePolynomial| |#2|) (|Symbol|)) "\\spad{rootOf(p, y)} returns \\spad{y} such that \\spad{p(y) = 0}. The object returned displays as \\spad{'y}."))) NIL -((|HasCategory| |#1| (QUOTE (-945 (-480))))) +((|HasCategory| |#1| (QUOTE (-950 (-483))))) (-33 S) ((|constructor| (NIL "The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as \"the set of \\spad{x} satisfying relation {\\em r(x)}\" An attribute \\spadatt{finiteAggregate} is used to assert that a domain has a finite number of elements.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# u} returns the number of items in \\spad{u}.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|empty?| (((|Boolean|) $) "\\spad{empty?(u)} tests if \\spad{u} has 0 elements.")) (|empty| (($) "\\spad{empty()}\\$\\spad{D} creates an aggregate of type \\spad{D} with 0 elements. Note: The {\\em \\$D} can be dropped if understood by context,{} \\spadignore{e.g.} \\axiom{u: \\spad{D} := empty()}.")) (|copy| (($ $) "\\spad{copy(u)} returns a top-level (non-recursive) copy of \\spad{u}. Note: for collections,{} \\axiom{copy(\\spad{u}) == [\\spad{x} for \\spad{x} in \\spad{u}]}.")) (|eq?| (((|Boolean|) $ $) "\\spad{eq?(u,v)} tests if \\spad{u} and \\spad{v} are same objects."))) NIL -((|HasAttribute| |#1| (QUOTE -3978))) +((|HasAttribute| |#1| (QUOTE -3989))) (-34) ((|constructor| (NIL "The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as \"the set of \\spad{x} satisfying relation {\\em r(x)}\" An attribute \\spadatt{finiteAggregate} is used to assert that a domain has a finite number of elements.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# u} returns the number of items in \\spad{u}.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|empty?| (((|Boolean|) $) "\\spad{empty?(u)} tests if \\spad{u} has 0 elements.")) (|empty| (($) "\\spad{empty()}\\$\\spad{D} creates an aggregate of type \\spad{D} with 0 elements. Note: The {\\em \\$D} can be dropped if understood by context,{} \\spadignore{e.g.} \\axiom{u: \\spad{D} := empty()}.")) (|copy| (($ $) "\\spad{copy(u)} returns a top-level (non-recursive) copy of \\spad{u}. Note: for collections,{} \\axiom{copy(\\spad{u}) == [\\spad{x} for \\spad{x} in \\spad{u}]}.")) (|eq?| (((|Boolean|) $ $) "\\spad{eq?(u,v)} tests if \\spad{u} and \\spad{v} are same objects."))) NIL @@ -74,7 +74,7 @@ NIL NIL (-36 |Key| |Entry|) ((|constructor| (NIL "An association list is a list of key entry pairs which may be viewed as a table. It is a poor mans version of a table: searching for a key is a linear operation.")) (|assoc| (((|Union| (|Record| (|:| |key| |#1|) (|:| |entry| |#2|)) "failed") |#1| $) "\\spad{assoc(k,u)} returns the element \\spad{x} in association list \\spad{u} stored with key \\spad{k},{} or \"failed\" if \\spad{u} has no key \\spad{k}."))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL (-37 S R) ((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline"))) @@ -82,20 +82,20 @@ NIL NIL (-38 R) ((|constructor| (NIL "The category of associative algebras (modules which are themselves rings). \\blankline"))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL (-39 UP) ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in \\spadtype{AlgebraicNumber}.")) (|doublyTransitive?| (((|Boolean|) |#1|) "\\spad{doublyTransitive?(p)} is \\spad{true} if \\spad{p} is irreducible over over the field \\spad{K} generated by its coefficients,{} and if \\spad{p(X) / (X - a)} is irreducible over \\spad{K(a)} where \\spad{p(a) = 0}.")) (|split| (((|Factored| |#1|) |#1|) "\\spad{split(p)} returns a prime factorisation of \\spad{p} over its splitting field.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p} over the field generated by its coefficients.") (((|Factored| |#1|) |#1| (|List| (|AlgebraicNumber|))) "\\spad{factor(p, [a1,...,an])} returns a prime factorisation of \\spad{p} over the field generated by its coefficients and \\spad{a1},{}...,{}an."))) NIL NIL -(-40 -3078 UP UPUP -2600) +(-40 -3088 UP UPUP -2610) ((|constructor| (NIL "Function field defined by \\spad{f}(\\spad{x},{} \\spad{y}) = 0.")) (|knownInfBasis| (((|Void|) (|NonNegativeInteger|)) "\\spad{knownInfBasis(n)} \\undocumented{}"))) -((-3971 |has| (-345 |#2|) (-309)) (-3976 |has| (-345 |#2|) (-309)) (-3970 |has| (-345 |#2|) (-309)) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-345 |#2|) (QUOTE (-116))) (|HasCategory| (-345 |#2|) (QUOTE (-118))) (|HasCategory| (-345 |#2|) (QUOTE (-296))) (OR (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-296)))) (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-315))) (OR (-12 (|HasCategory| (-345 |#2|) (QUOTE (-188))) (|HasCategory| (-345 |#2|) (QUOTE (-309)))) (|HasCategory| (-345 |#2|) (QUOTE (-296)))) (OR (-12 (|HasCategory| (-345 |#2|) (QUOTE (-188))) (|HasCategory| (-345 |#2|) (QUOTE (-309)))) (-12 (|HasCategory| (-345 |#2|) (QUOTE (-187))) (|HasCategory| (-345 |#2|) (QUOTE (-309)))) (|HasCategory| (-345 |#2|) (QUOTE (-296)))) (OR (-12 (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-804 (-1081))))) (-12 (|HasCategory| (-345 |#2|) (QUOTE (-296))) (|HasCategory| (-345 |#2|) (QUOTE (-804 (-1081)))))) (OR (-12 (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-804 (-1081))))) (-12 (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-806 (-1081)))))) (|HasCategory| (-345 |#2|) (QUOTE (-577 (-480)))) (OR (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-945 (-345 (-480)))))) (|HasCategory| (-345 |#2|) (QUOTE (-945 (-345 (-480))))) (|HasCategory| (-345 |#2|) (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-315))) (-12 (|HasCategory| (-345 |#2|) (QUOTE (-187))) (|HasCategory| (-345 |#2|) (QUOTE (-309)))) (-12 (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-806 (-1081))))) (-12 (|HasCategory| (-345 |#2|) (QUOTE (-188))) (|HasCategory| (-345 |#2|) (QUOTE (-309)))) (-12 (|HasCategory| (-345 |#2|) (QUOTE (-309))) (|HasCategory| (-345 |#2|) (QUOTE (-804 (-1081)))))) -(-41 R -3078) +((-3982 |has| (-347 |#2|) (-311)) (-3987 |has| (-347 |#2|) (-311)) (-3981 |has| (-347 |#2|) (-311)) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-347 |#2|) (QUOTE (-118))) (|HasCategory| (-347 |#2|) (QUOTE (-120))) (|HasCategory| (-347 |#2|) (QUOTE (-298))) (OR (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-298)))) (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-317))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-190))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (|HasCategory| (-347 |#2|) (QUOTE (-298)))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-190))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-189))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (|HasCategory| (-347 |#2|) (QUOTE (-298)))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088))))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-298))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088)))))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088))))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-811 (-1088)))))) (|HasCategory| (-347 |#2|) (QUOTE (-580 (-483)))) (OR (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-950 (-347 (-483)))))) (|HasCategory| (-347 |#2|) (QUOTE (-950 (-347 (-483))))) (|HasCategory| (-347 |#2|) (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-317))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-189))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-811 (-1088))))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-190))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088)))))) +(-41 R -3088) ((|constructor| (NIL "AlgebraicManipulations provides functions to simplify and expand expressions involving algebraic operators.")) (|rootKerSimp| ((|#2| (|BasicOperator|) |#2| (|NonNegativeInteger|)) "\\spad{rootKerSimp(op,f,n)} should be local but conditional.")) (|rootSimp| ((|#2| |#2|) "\\spad{rootSimp(f)} transforms every radical of the form \\spad{(a * b**(q*n+r))**(1/n)} appearing in \\spad{f} into \\spad{b**q * (a * b**r)**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{b}.")) (|rootProduct| ((|#2| |#2|) "\\spad{rootProduct(f)} combines every product of the form \\spad{(a**(1/n))**m * (a**(1/s))**t} into a single power of a root of \\spad{a},{} and transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form.")) (|rootPower| ((|#2| |#2|) "\\spad{rootPower(f)} transforms every radical power of the form \\spad{(a**(1/n))**m} into a simpler form if \\spad{m} and \\spad{n} have a common factor.")) (|ratPoly| (((|SparseUnivariatePolynomial| |#2|) |#2|) "\\spad{ratPoly(f)} returns a polynomial \\spad{p} such that \\spad{p} has no algebraic coefficients,{} and \\spad{p(f) = 0}.")) (|ratDenom| ((|#2| |#2| (|List| (|Kernel| |#2|))) "\\spad{ratDenom(f, [a1,...,an])} removes the \\spad{ai}'s which are algebraic from the denominators in \\spad{f}.") ((|#2| |#2| (|List| |#2|)) "\\spad{ratDenom(f, [a1,...,an])} removes the \\spad{ai}'s which are algebraic kernels from the denominators in \\spad{f}.") ((|#2| |#2| |#2|) "\\spad{ratDenom(f, a)} removes \\spad{a} from the denominators in \\spad{f} if \\spad{a} is an algebraic kernel.") ((|#2| |#2|) "\\spad{ratDenom(f)} rationalizes the denominators appearing in \\spad{f} by moving all the algebraic quantities into the numerators.")) (|rootSplit| ((|#2| |#2|) "\\spad{rootSplit(f)} transforms every radical of the form \\spad{(a/b)**(1/n)} appearing in \\spad{f} into \\spad{a**(1/n) / b**(1/n)}. This transformation is not in general valid for all complex numbers \\spad{a} and \\spad{b}.")) (|coerce| (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{coerce(x)} \\undocumented")) (|denom| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{denom(x)} \\undocumented")) (|numer| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{numer(x)} \\undocumented"))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#2| (|%list| (QUOTE -359) (|devaluate| |#1|))))) +((-12 (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#2| (|%list| (QUOTE -361) (|devaluate| |#1|))))) (-42 OV E P) ((|constructor| (NIL "This package factors multivariate polynomials over the domain of \\spadtype{AlgebraicNumber} by allowing the user to specify a list of algebraic numbers generating the particular extension to factor over.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#3|)) (|SparseUnivariatePolynomial| |#3|) (|List| (|AlgebraicNumber|))) "\\spad{factor(p,lan)} factors the polynomial \\spad{p} over the extension generated by the algebraic numbers given by the list \\spad{lan}. \\spad{p} is presented as a univariate polynomial with multivariate coefficients.") (((|Factored| |#3|) |#3| (|List| (|AlgebraicNumber|))) "\\spad{factor(p,lan)} factors the polynomial \\spad{p} over the extension generated by the algebraic numbers given by the list \\spad{lan}."))) NIL @@ -103,34 +103,34 @@ NIL (-43 R A) ((|constructor| (NIL "AlgebraPackage assembles a variety of useful functions for general algebras.")) (|basis| (((|Vector| |#2|) (|Vector| |#2|)) "\\spad{basis(va)} selects a basis from the elements of \\spad{va}.")) (|radicalOfLeftTraceForm| (((|List| |#2|)) "\\spad{radicalOfLeftTraceForm()} returns basis for null space of \\spad{leftTraceMatrix()},{} if the algebra is associative,{} alternative or a Jordan algebra,{} then this space equals the radical (maximal nil ideal) of the algebra.")) (|basisOfCentroid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfCentroid()} returns a basis of the centroid,{} \\spadignore{i.e.} the endomorphism ring of \\spad{A} considered as \\spad{(A,A)}-bimodule.")) (|basisOfRightNucloid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfRightNucloid()} returns a basis of the space of endomorphisms of \\spad{A} as left module. Note: right nucloid coincides with right nucleus if \\spad{A} has a unit.")) (|basisOfLeftNucloid| (((|List| (|Matrix| |#1|))) "\\spad{basisOfLeftNucloid()} returns a basis of the space of endomorphisms of \\spad{A} as right module. Note: left nucloid coincides with left nucleus if \\spad{A} has a unit.")) (|basisOfCenter| (((|List| |#2|)) "\\spad{basisOfCenter()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{commutator(x,a) = 0} and \\spad{associator(x,a,b) = associator(a,x,b) = associator(a,b,x) = 0} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfNucleus| (((|List| |#2|)) "\\spad{basisOfNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{associator(x,a,b) = associator(a,x,b) = associator(a,b,x) = 0} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfMiddleNucleus| (((|List| |#2|)) "\\spad{basisOfMiddleNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(a,x,b)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfRightNucleus| (((|List| |#2|)) "\\spad{basisOfRightNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(a,b,x)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfLeftNucleus| (((|List| |#2|)) "\\spad{basisOfLeftNucleus()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = associator(x,a,b)} for all \\spad{a},{}\\spad{b} in \\spad{A}.")) (|basisOfRightAnnihilator| (((|List| |#2|) |#2|) "\\spad{basisOfRightAnnihilator(a)} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = a*x}.")) (|basisOfLeftAnnihilator| (((|List| |#2|) |#2|) "\\spad{basisOfLeftAnnihilator(a)} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = x*a}.")) (|basisOfCommutingElements| (((|List| |#2|)) "\\spad{basisOfCommutingElements()} returns a basis of the space of all \\spad{x} of \\spad{A} satisfying \\spad{0 = commutator(x,a)} for all \\spad{a} in \\spad{A}.")) (|biRank| (((|NonNegativeInteger|) |#2|) "\\spad{biRank(x)} determines the number of linearly independent elements in \\spad{x},{} \\spad{x*bi},{} \\spad{bi*x},{} \\spad{bi*x*bj},{} \\spad{i,j=1,...,n},{} where \\spad{b=[b1,...,bn]} is a basis. Note: if \\spad{A} has a unit,{} then \\spadfunFrom{doubleRank}{AlgebraPackage},{} \\spadfunFrom{weakBiRank}{AlgebraPackage} and \\spadfunFrom{biRank}{AlgebraPackage} coincide.")) (|weakBiRank| (((|NonNegativeInteger|) |#2|) "\\spad{weakBiRank(x)} determines the number of linearly independent elements in the \\spad{bi*x*bj},{} \\spad{i,j=1,...,n},{} where \\spad{b=[b1,...,bn]} is a basis.")) (|doubleRank| (((|NonNegativeInteger|) |#2|) "\\spad{doubleRank(x)} determines the number of linearly independent elements in \\spad{b1*x},{}...,{}\\spad{x*bn},{} where \\spad{b=[b1,...,bn]} is a basis.")) (|rightRank| (((|NonNegativeInteger|) |#2|) "\\spad{rightRank(x)} determines the number of linearly independent elements in \\spad{b1*x},{}...,{}\\spad{bn*x},{} where \\spad{b=[b1,...,bn]} is a basis.")) (|leftRank| (((|NonNegativeInteger|) |#2|) "\\spad{leftRank(x)} determines the number of linearly independent elements in \\spad{x*b1},{}...,{}\\spad{x*bn},{} where \\spad{b=[b1,...,bn]} is a basis."))) NIL -((|HasCategory| |#1| (QUOTE (-255)))) +((|HasCategory| |#1| (QUOTE (-257)))) (-44 R |n| |ls| |gamma|) ((|constructor| (NIL "AlgebraGivenByStructuralConstants implements finite rank algebras over a commutative ring,{} given by the structural constants \\spad{gamma} with respect to a fixed basis \\spad{[a1,..,an]},{} where \\spad{gamma} is an \\spad{n}-vector of \\spad{n} by \\spad{n} matrices \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{ai * aj = gammaij1 * a1 + ... + gammaijn * an}. The symbols for the fixed basis have to be given as a list of symbols.")) (|coerce| (($ (|Vector| |#1|)) "\\spad{coerce(v)} converts a vector to a member of the algebra by forming a linear combination with the basis element. Note: the vector is assumed to have length equal to the dimension of the algebra."))) -((-3975 |has| |#1| (-491)) (-3973 . T) (-3972 . T)) -((|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) +((-3986 |has| |#1| (-494)) (-3984 . T) (-3983 . T)) +((|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (-45 |Key| |Entry|) ((|constructor| (NIL "\\spadtype{AssociationList} implements association lists. These may be viewed as lists of pairs where the first part is a key and the second is the stored value. For example,{} the key might be a string with a persons employee identification number and the value might be a record with personnel data."))) -((-3978 . T) (-3979 . T)) -((OR (-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-751)))) (-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007))))) (OR (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-751))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-751))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-751))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007))))) +((-3989 . 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T)) +((OR (-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-756)))) (-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012))))) (OR (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-756))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-756))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-756))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012))))) (-46 S R E) ((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#2|) "\\spad{p/c} divides \\spad{p} by the coefficient \\spad{c}.")) (|coefficient| ((|#2| $ |#3|) "\\spad{coefficient(p,e)} extracts the coefficient of the monomial with exponent \\spad{e} from polynomial \\spad{p},{} or returns zero if exponent is not present.")) (|reductum| (($ $) "\\spad{reductum(u)} returns \\spad{u} minus its leading monomial returns zero if handed the zero element.")) (|monomial| (($ |#2| |#3|) "\\spad{monomial(r,e)} makes a term from a coefficient \\spad{r} and an exponent \\spad{e}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(p)} tests if \\spad{p} is a single monomial.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|degree| ((|#3| $) "\\spad{degree(p)} returns the maximum of the exponents of the terms of \\spad{p}.")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(p)} returns the monomial of \\spad{p} with the highest degree.")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) NIL -((|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-309)))) +((|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-311)))) (-47 R E) ((|constructor| (NIL "Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} and with the coefficients (which themselves may or may not be commutative). See \\spadtype{FiniteAbelianMonoidRing} for the case of finite support a useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.")) (/ (($ $ |#1|) "\\spad{p/c} divides \\spad{p} by the coefficient \\spad{c}.")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(p,e)} extracts the coefficient of the monomial with exponent \\spad{e} from polynomial \\spad{p},{} or returns zero if exponent is not present.")) (|reductum| (($ $) "\\spad{reductum(u)} returns \\spad{u} minus its leading monomial returns zero if handed the zero element.")) (|monomial| (($ |#1| |#2|) "\\spad{monomial(r,e)} makes a term from a coefficient \\spad{r} and an exponent \\spad{e}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(p)} tests if \\spad{p} is a single monomial.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|degree| ((|#2| $) "\\spad{degree(p)} returns the maximum of the exponents of the terms of \\spad{p}.")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(p)} returns the monomial of \\spad{p} with the highest degree.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \\spad{p}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL (-48) ((|constructor| (NIL "Algebraic closure of the rational numbers,{} with mathematical =")) (|norm| (($ $ (|List| (|Kernel| $))) "\\spad{norm(f,l)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernels \\spad{l}") (($ $ (|Kernel| $)) "\\spad{norm(f,k)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernel \\spad{k}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|List| (|Kernel| $))) "\\spad{norm(p,l)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernels \\spad{l}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{norm(p,k)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernel \\spad{k}")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \\spad{f} by applying their defining relations.")) (|denom| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|numer| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| $ (QUOTE (-956))) (|HasCategory| $ (QUOTE (-945 (-480))))) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| $ (QUOTE (-961))) (|HasCategory| $ (QUOTE (-950 (-483))))) (-49) ((|constructor| (NIL "This domain implements anonymous functions")) (|body| (((|Syntax|) $) "\\spad{body(f)} returns the body of the unnamed function `f'.")) (|parameters| (((|List| (|Identifier|)) $) "\\spad{parameters(f)} returns the list of parameters bound by `f'."))) NIL NIL (-50 R |lVar|) ((|constructor| (NIL "The domain of antisymmetric polynomials.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,p)} changes each coefficient of \\spad{p} by the application of \\spad{f}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} returns the homogeneous degree of \\spad{p}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?(p)} tests if \\spad{p} is a 0-form,{} \\spadignore{i.e.} if degree(\\spad{p}) = 0.")) (|homogeneous?| (((|Boolean|) $) "\\spad{homogeneous?(p)} tests if all of the terms of \\spad{p} have the same degree.")) (|exp| (($ (|List| (|Integer|))) "\\spad{exp([i1,...in])} returns \\spad{u_1\\^{i_1} ... u_n\\^{i_n}}")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(n)} returns the \\spad{n}th multiplicative generator,{} a basis term.")) (|coefficient| ((|#1| $ $) "\\spad{coefficient(p,u)} returns the coefficient of the term in \\spad{p} containing the basis term \\spad{u} if such a term exists,{} and 0 otherwise. Error: if the second argument \\spad{u} is not a basis element.")) (|reductum| (($ $) "\\spad{reductum(p)},{} where \\spad{p} is an antisymmetric polynomial,{} returns \\spad{p} minus the leading term of \\spad{p} if \\spad{p} has at least two terms,{} and 0 otherwise.")) (|leadingBasisTerm| (($ $) "\\spad{leadingBasisTerm(p)} returns the leading basis term of antisymmetric polynomial \\spad{p}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(p)} returns the leading coefficient of antisymmetric polynomial \\spad{p}."))) -((-3975 . T)) +((-3986 . T)) NIL (-51) ((|constructor| (NIL "\\spadtype{Any} implements a type that packages up objects and their types in objects of \\spadtype{Any}. Roughly speaking that means that if \\spad{s : S} then when converted to \\spadtype{Any},{} the new object will include both the original object and its type. This is a way of converting arbitrary objects into a single type without losing any of the original information. Any object can be converted to one of \\spadtype{Any}. The original object can be recovered by `is-case' pattern matching as exemplified here and \\spad{AnyFunctions1}.")) (|obj| (((|None|) $) "\\spad{obj(a)} essentially returns the original object that was converted to \\spadtype{Any} except that the type is forced to be \\spadtype{None}.")) (|dom| (((|SExpression|) $) "\\spad{dom(a)} returns a \\spadgloss{LISP} form of the type of the original object that was converted to \\spadtype{Any}.")) (|any| (($ (|SExpression|) (|None|)) "\\spad{any(type,object)} is a technical function for creating an \\spad{object} of \\spadtype{Any}. Arugment \\spad{type} is a \\spadgloss{LISP} form for the \\spad{type} of \\spad{object}."))) @@ -144,7 +144,7 @@ NIL ((|constructor| (NIL "\\spad{ApplyUnivariateSkewPolynomial} (internal) allows univariate skew polynomials to be applied to appropriate modules.")) (|apply| ((|#2| |#3| (|Mapping| |#2| |#2|) |#2|) "\\spad{apply(p, f, m)} returns \\spad{p(m)} where the action is given by \\spad{x m = f(m)}. \\spad{f} must be an \\spad{R}-pseudo linear map on \\spad{M}."))) NIL NIL -(-54 |Base| R -3078) +(-54 |Base| R -3088) ((|constructor| (NIL "This package apply rewrite rules to expressions,{} calling the pattern matcher.")) (|localUnquote| ((|#3| |#3| (|List| (|Symbol|))) "\\spad{localUnquote(f,ls)} is a local function.")) (|applyRules| ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3| (|PositiveInteger|)) "\\spad{applyRules([r1,...,rn], expr, n)} applies the rules \\spad{r1},{}...,{}rn to \\spad{f} a most \\spad{n} times.") ((|#3| (|List| (|RewriteRule| |#1| |#2| |#3|)) |#3|) "\\spad{applyRules([r1,...,rn], expr)} applies the rules \\spad{r1},{}...,{}rn to \\spad{f} an unlimited number of times,{} \\spadignore{i.e.} until none of \\spad{r1},{}...,{}rn is applicable to the expression."))) NIL NIL @@ -158,28 +158,28 @@ NIL NIL (-57 R |Row| |Col|) ((|constructor| (NIL "\\indented{1}{TwoDimensionalArrayCategory is a general array category which} allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and columns returned as objects of type Col. The index of the 'first' row may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.")) (|map!| (($ (|Mapping| |#1| |#1|) $) "\\spad{map!(f,a)} assign \\spad{a(i,j)} to \\spad{f(a(i,j))} for all \\spad{i, j}")) (|map| (($ (|Mapping| |#1| |#1| |#1|) $ $ |#1|) "\\spad{map(f,a,b,r)} returns \\spad{c},{} where \\spad{c(i,j) = f(a(i,j),b(i,j))} when both \\spad{a(i,j)} and \\spad{b(i,j)} exist; else \\spad{c(i,j) = f(r, b(i,j))} when \\spad{a(i,j)} does not exist; else \\spad{c(i,j) = f(a(i,j),r)} when \\spad{b(i,j)} does not exist; otherwise \\spad{c(i,j) = f(r,r)}.") (($ (|Mapping| |#1| |#1| |#1|) $ $) "\\spad{map(f,a,b)} returns \\spad{c},{} where \\spad{c(i,j) = f(a(i,j),b(i,j))} for all \\spad{i, j}") (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,a)} returns \\spad{b},{} where \\spad{b(i,j) = f(a(i,j))} for all \\spad{i, j}")) (|setColumn!| (($ $ (|Integer|) |#3|) "\\spad{setColumn!(m,j,v)} sets to \\spad{j}th column of \\spad{m} to \\spad{v}")) (|setRow!| (($ $ (|Integer|) |#2|) "\\spad{setRow!(m,i,v)} sets to \\spad{i}th row of \\spad{m} to \\spad{v}")) (|qsetelt!| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\spad{qsetelt!(m,i,j,r)} sets the element in the \\spad{i}th row and \\spad{j}th column of \\spad{m} to \\spad{r} NO error check to determine if indices are in proper ranges")) (|setelt| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\spad{setelt(m,i,j,r)} sets the element in the \\spad{i}th row and \\spad{j}th column of \\spad{m} to \\spad{r} error check to determine if indices are in proper ranges")) (|parts| (((|List| |#1|) $) "\\spad{parts(m)} returns a list of the elements of \\spad{m} in row major order")) (|column| ((|#3| $ (|Integer|)) "\\spad{column(m,j)} returns the \\spad{j}th column of \\spad{m} error check to determine if index is in proper ranges")) (|row| ((|#2| $ (|Integer|)) "\\spad{row(m,i)} returns the \\spad{i}th row of \\spad{m} error check to determine if index is in proper ranges")) (|qelt| ((|#1| $ (|Integer|) (|Integer|)) "\\spad{qelt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the array \\spad{m} NO error check to determine if indices are in proper ranges")) (|elt| ((|#1| $ (|Integer|) (|Integer|) |#1|) "\\spad{elt(m,i,j,r)} returns the element in the \\spad{i}th row and \\spad{j}th column of the array \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{} and returns \\spad{r} otherwise") ((|#1| $ (|Integer|) (|Integer|)) "\\spad{elt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the array \\spad{m} error check to determine if indices are in proper ranges")) (|ncols| (((|NonNegativeInteger|) $) "\\spad{ncols(m)} returns the number of columns in the array \\spad{m}")) (|nrows| (((|NonNegativeInteger|) $) "\\spad{nrows(m)} returns the number of rows in the array \\spad{m}")) (|maxColIndex| (((|Integer|) $) "\\spad{maxColIndex(m)} returns the index of the 'last' column of the array \\spad{m}")) (|minColIndex| (((|Integer|) $) "\\spad{minColIndex(m)} returns the index of the 'first' column of the array \\spad{m}")) (|maxRowIndex| (((|Integer|) $) "\\spad{maxRowIndex(m)} returns the index of the 'last' row of the array \\spad{m}")) (|minRowIndex| (((|Integer|) $) "\\spad{minRowIndex(m)} returns the index of the 'first' row of the array \\spad{m}")) (|fill!| (($ $ |#1|) "\\spad{fill!(m,r)} fills \\spad{m} with \\spad{r}'s")) (|new| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) |#1|) "\\spad{new(m,n,r)} is an \\spad{m}-by-\\spad{n} array all of whose entries are \\spad{r}")) (|finiteAggregate| ((|attribute|) "two-dimensional arrays are finite")) (|shallowlyMutable| ((|attribute|) "one may destructively alter arrays"))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL (-58 S) ((|constructor| (NIL "This is the domain of 1-based one dimensional arrays")) (|oneDimensionalArray| (($ (|NonNegativeInteger|) |#1|) "\\spad{oneDimensionalArray(n,s)} creates an array from \\spad{n} copies of element \\spad{s}") (($ (|List| |#1|)) "\\spad{oneDimensionalArray(l)} creates an array from a list of elements \\spad{l}"))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (-59 A B) ((|constructor| (NIL "\\indented{1}{This package provides tools for operating on one-dimensional arrays} with unary and binary functions involving different underlying types")) (|map| (((|OneDimensionalArray| |#2|) (|Mapping| |#2| |#1|) (|OneDimensionalArray| |#1|)) "\\spad{map(f,a)} applies function \\spad{f} to each member of one-dimensional array \\spad{a} resulting in a new one-dimensional array over a possibly different underlying domain.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|OneDimensionalArray| |#1|) |#2|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the one-dimensional array \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as the identity element for the function \\spad{f}.")) (|scan| (((|OneDimensionalArray| |#2|) (|Mapping| |#2| |#1| |#2|) (|OneDimensionalArray| |#1|) |#2|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-arrays \\spad{x} of one-dimensional array \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}."))) NIL NIL (-60 R) ((|constructor| (NIL "\\indented{1}{A TwoDimensionalArray is a two dimensional array with} 1-based indexing for both rows and columns.")) (|shallowlyMutable| ((|attribute|) "One may destructively alter TwoDimensionalArray's."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) (-61 R L) ((|constructor| (NIL "\\spadtype{AssociatedEquations} provides functions to compute the associated equations needed for factoring operators")) (|associatedEquations| (((|Record| (|:| |minor| (|List| (|PositiveInteger|))) (|:| |eq| |#2|) (|:| |minors| (|List| (|List| (|PositiveInteger|)))) (|:| |ops| (|List| |#2|))) |#2| (|PositiveInteger|)) "\\spad{associatedEquations(op, m)} returns \\spad{[w, eq, lw, lop]} such that \\spad{eq(w) = 0} where \\spad{w} is the given minor,{} and \\spad{lw_i = lop_i(w)} for all the other minors.")) (|uncouplingMatrices| (((|Vector| (|Matrix| |#1|)) (|Matrix| |#1|)) "\\spad{uncouplingMatrices(M)} returns \\spad{[A_1,...,A_n]} such that if \\spad{y = [y_1,...,y_n]} is a solution of \\spad{y' = M y},{} then \\spad{[\\$y_j',y_j'',...,y_j^{(n)}\\$] = \\$A_j y\\$} for all \\spad{j}'s.")) (|associatedSystem| (((|Record| (|:| |mat| (|Matrix| |#1|)) (|:| |vec| (|Vector| (|List| (|PositiveInteger|))))) |#2| (|PositiveInteger|)) "\\spad{associatedSystem(op, m)} returns \\spad{[M,w]} such that the \\spad{m}-th associated equation system to \\spad{L} is \\spad{w' = M w}."))) NIL -((|HasCategory| |#1| (QUOTE (-309)))) +((|HasCategory| |#1| (QUOTE (-311)))) (-62 S) ((|constructor| (NIL "A stack represented as a flexible array.")) (|arrayStack| (($ (|List| |#1|)) "\\spad{arrayStack([x,y,...,z])} creates an array stack with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last element \\spad{z}."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) (-63 S) ((|constructor| (NIL "This is the category of Spad abstract syntax trees."))) NIL @@ -202,11 +202,11 @@ NIL NIL (-68) ((|constructor| (NIL "This category exports the attributes in the AXIOM Library")) (|canonical| ((|attribute|) "\\spad{canonical} is \\spad{true} if and only if distinct elements have distinct data structures. For example,{} a domain of mathematical objects which has the \\spad{canonical} attribute means that two objects are mathematically equal if and only if their data structures are equal.")) (|multiplicativeValuation| ((|attribute|) "\\spad{multiplicativeValuation} implies \\spad{euclideanSize(a*b)=euclideanSize(a)*euclideanSize(b)}.")) (|additiveValuation| ((|attribute|) "\\spad{additiveValuation} implies \\spad{euclideanSize(a*b)=euclideanSize(a)+euclideanSize(b)}.")) (|noetherian| ((|attribute|) "\\spad{noetherian} is \\spad{true} if all of its ideals are finitely generated.")) (|central| ((|attribute|) "\\spad{central} is \\spad{true} if,{} given an algebra over a ring \\spad{R},{} the image of \\spad{R} is the center of the algebra,{} \\spadignore{i.e.} the set of members of the algebra which commute with all others is precisely the image of \\spad{R} in the algebra.")) (|partiallyOrderedSet| ((|attribute|) "\\spad{partiallyOrderedSet} is \\spad{true} if a set with \\spadop{<} which is transitive,{} but \\spad{not(a < b or a = b)} does not necessarily imply \\spad{b<a}.")) (|arbitraryPrecision| ((|attribute|) "\\spad{arbitraryPrecision} means the user can set the precision for subsequent calculations.")) (|canonicalsClosed| ((|attribute|) "\\spad{canonicalsClosed} is \\spad{true} if \\spad{unitCanonical(a)*unitCanonical(b) = unitCanonical(a*b)}.")) (|canonicalUnitNormal| ((|attribute|) "\\spad{canonicalUnitNormal} is \\spad{true} if we can choose a canonical representative for each class of associate elements,{} that is \\spad{associates?(a,b)} returns \\spad{true} if and only if \\spad{unitCanonical(a) = unitCanonical(b)}.")) (|noZeroDivisors| ((|attribute|) "\\spad{noZeroDivisors} is \\spad{true} if \\spad{x * y \\~~= 0} implies both \\spad{x} and \\spad{y} are non-zero.")) (|rightUnitary| ((|attribute|) "\\spad{rightUnitary} is \\spad{true} if \\spad{x * 1 = x} for all \\spad{x}.")) (|leftUnitary| ((|attribute|) "\\spad{leftUnitary} is \\spad{true} if \\spad{1 * x = x} for all \\spad{x}.")) (|unitsKnown| ((|attribute|) "\\spad{unitsKnown} is \\spad{true} if a monoid (a multiplicative semigroup with a 1) has \\spad{unitsKnown} means that the operation \\spadfun{recip} can only return \"failed\" if its argument is not a unit.")) (|shallowlyMutable| ((|attribute|) "\\spad{shallowlyMutable} is \\spad{true} if its values have immediate components that are updateable (mutable). Note: the properties of any component domain are irrevelant to the \\spad{shallowlyMutable} proper.")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} is \\spad{true} if it has an operation \\spad{\"*\": (D,D) -> D} which is commutative.")) (|finiteAggregate| ((|attribute|) "\\spad{finiteAggregate} is \\spad{true} if it is an aggregate with a finite number of elements."))) -((-3978 . T) ((-3980 "*") . T) (-3979 . T) (-3975 . T) (-3973 . T) (-3972 . T) (-3971 . T) (-3976 . T) (-3970 . T) (-3969 . T) (-3968 . T) (-3967 . T) (-3966 . T) (-3974 . T) (-3977 . T) (|NullSquare| . T) (|JacobiIdentity| . T) (-3965 . T)) +((-3989 . T) ((-3991 "*") . T) (-3990 . T) (-3986 . T) (-3984 . T) (-3983 . T) (-3982 . T) (-3987 . T) (-3981 . T) (-3980 . T) (-3979 . T) (-3978 . T) (-3977 . T) (-3985 . T) (-3988 . T) (|NullSquare| . T) (|JacobiIdentity| . T) (-3976 . T)) NIL (-69 R) ((|constructor| (NIL "Automorphism \\spad{R} is the multiplicative group of automorphisms of \\spad{R}.")) (|morphism| (($ (|Mapping| |#1| |#1| (|Integer|))) "\\spad{morphism(f)} returns the morphism given by \\spad{f^n(x) = f(x,n)}.") (($ (|Mapping| |#1| |#1|) (|Mapping| |#1| |#1|)) "\\spad{morphism(f, g)} returns the invertible morphism given by \\spad{f},{} where \\spad{g} is the inverse of \\spad{f}..") (($ (|Mapping| |#1| |#1|)) "\\spad{morphism(f)} returns the non-invertible morphism given by \\spad{f}."))) -((-3975 . T)) +((-3986 . T)) NIL (-70 R UP) ((|constructor| (NIL "This package provides balanced factorisations of polynomials.")) (|balancedFactorisation| (((|Factored| |#2|) |#2| (|List| |#2|)) "\\spad{balancedFactorisation(a, [b1,...,bn])} returns a factorisation \\spad{a = p1^e1 ... pm^em} such that each \\spad{pi} is balanced with respect to \\spad{[b1,...,bm]}.") (((|Factored| |#2|) |#2| |#2|) "\\spad{balancedFactorisation(a, b)} returns a factorisation \\spad{a = p1^e1 ... pm^em} such that each \\spad{pi} is balanced with respect to \\spad{b}."))) @@ -222,4515 +222,4543 @@ NIL NIL (-73 S) ((|constructor| (NIL "\\spadtype{BalancedBinaryTree(S)} is the domain of balanced binary trees (bbtree). A balanced binary tree of \\spad{2**k} leaves,{} for some \\spad{k > 0},{} is symmetric,{} that is,{} the left and right subtree of each interior node have identical shape. In general,{} the left and right subtree of a given node can differ by at most leaf node.")) (|mapDown!| (($ $ |#1| (|Mapping| (|List| |#1|) |#1| |#1| |#1|)) "\\spad{mapDown!(t,p,f)} returns \\spad{t} after traversing \\spad{t} in \"preorder\" (node then left then right) fashion replacing the successive interior nodes as follows. Let \\spad{l} and \\spad{r} denote the left and right subtrees of \\spad{t}. The root value \\spad{x} of \\spad{t} is replaced by \\spad{p}. Then \\spad{f}(value \\spad{l},{} value \\spad{r},{} \\spad{p}),{} where \\spad{l} and \\spad{r} denote the left and right subtrees of \\spad{t},{} is evaluated producing two values pl and pr. Then \\spad{mapDown!(l,pl,f)} and \\spad{mapDown!(l,pr,f)} are evaluated.") (($ $ |#1| (|Mapping| |#1| |#1| |#1|)) "\\spad{mapDown!(t,p,f)} returns \\spad{t} after traversing \\spad{t} in \"preorder\" (node then left then right) fashion replacing the successive interior nodes as follows. The root value \\spad{x} is replaced by \\spad{q} := \\spad{f}(\\spad{p},{}\\spad{x}). The mapDown!(\\spad{l},{}\\spad{q},{}\\spad{f}) and mapDown!(\\spad{r},{}\\spad{q},{}\\spad{f}) are evaluated for the left and right subtrees \\spad{l} and \\spad{r} of \\spad{t}.")) (|mapUp!| (($ $ $ (|Mapping| |#1| |#1| |#1| |#1| |#1|)) "\\spad{mapUp!(t,t1,f)} traverses \\spad{t} in an \"endorder\" (left then right then node) fashion returning \\spad{t} with the value at each successive interior node of \\spad{t} replaced by \\spad{f}(\\spad{l},{}\\spad{r},{}\\spad{l1},{}\\spad{r1}) where \\spad{l} and \\spad{r} are the values at the immediate left and right nodes. Values \\spad{l1} and \\spad{r1} are values at the corresponding nodes of a balanced binary tree \\spad{t1},{} of identical shape at \\spad{t}.") ((|#1| $ (|Mapping| |#1| |#1| |#1|)) "\\spad{mapUp!(t,f)} traverses balanced binary tree \\spad{t} in an \"endorder\" (left then right then node) fashion returning \\spad{t} with the value at each successive interior node of \\spad{t} replaced by \\spad{f}(\\spad{l},{}\\spad{r}) where \\spad{l} and \\spad{r} are the values at the immediate left and right nodes.")) (|setleaves!| (($ $ (|List| |#1|)) "\\spad{setleaves!(t, ls)} sets the leaves of \\spad{t} in left-to-right order to the elements of ls.")) (|balancedBinaryTree| (($ (|NonNegativeInteger|) |#1|) "\\spad{balancedBinaryTree(n, s)} creates a balanced binary tree with \\spad{n} nodes each with value \\spad{s}."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) (-74 R UP M |Row| |Col|) ((|constructor| (NIL "\\spadtype{BezoutMatrix} contains functions for computing resultants and discriminants using Bezout matrices.")) (|bezoutDiscriminant| ((|#1| |#2|) "\\spad{bezoutDiscriminant(p)} computes the discriminant of a polynomial \\spad{p} by computing the determinant of a Bezout matrix.")) (|bezoutResultant| ((|#1| |#2| |#2|) "\\spad{bezoutResultant(p,q)} computes the resultant of the two polynomials \\spad{p} and \\spad{q} by computing the determinant of a Bezout matrix.")) (|bezoutMatrix| ((|#3| |#2| |#2|) "\\spad{bezoutMatrix(p,q)} returns the Bezout matrix for the two polynomials \\spad{p} and \\spad{q}.")) (|sylvesterMatrix| ((|#3| |#2| |#2|) "\\spad{sylvesterMatrix(p,q)} returns the Sylvester matrix for the two polynomials \\spad{p} and \\spad{q}."))) NIL -((|HasAttribute| |#1| (QUOTE (-3980 "*")))) +((|HasAttribute| |#1| (QUOTE (-3991 "*")))) (-75 A S) ((|constructor| (NIL "A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.")) (|inspect| ((|#2| $) "\\spad{inspect(u)} returns an (random) element from a bag.")) (|insert!| (($ |#2| $) "\\spad{insert!(x,u)} inserts item \\spad{x} into bag \\spad{u}.")) (|extract!| ((|#2| $) "\\spad{extract!(u)} destructively removes a (random) item from bag \\spad{u}.")) (|bag| (($ (|List| |#2|)) "\\spad{bag([x,y,...,z])} creates a bag with elements \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.")) (|shallowlyMutable| ((|attribute|) "shallowlyMutable means that elements of bags may be destructively changed."))) NIL NIL (-76 S) ((|constructor| (NIL "A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.")) (|inspect| ((|#1| $) "\\spad{inspect(u)} returns an (random) element from a bag.")) (|insert!| (($ |#1| $) "\\spad{insert!(x,u)} inserts item \\spad{x} into bag \\spad{u}.")) (|extract!| ((|#1| $) "\\spad{extract!(u)} destructively removes a (random) item from bag \\spad{u}.")) (|bag| (($ (|List| |#1|)) "\\spad{bag([x,y,...,z])} creates a bag with elements \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.")) (|shallowlyMutable| ((|attribute|) "shallowlyMutable means that elements of bags may be destructively changed."))) -((-3979 . T)) +((-3990 . T)) NIL (-77) ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating binary expansions.")) (|binary| (($ (|Fraction| (|Integer|))) "\\spad{binary(r)} converts a rational number to a binary expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(b)} returns the fractional part of a binary expansion."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-480) (QUOTE (-816))) (|HasCategory| (-480) (QUOTE (-945 (-1081)))) (|HasCategory| (-480) (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-118))) (|HasCategory| (-480) (QUOTE (-550 (-469)))) (|HasCategory| (-480) (QUOTE (-928))) (|HasCategory| (-480) (QUOTE (-735))) (|HasCategory| (-480) (QUOTE (-751))) (OR (|HasCategory| (-480) (QUOTE (-735))) (|HasCategory| (-480) (QUOTE (-751)))) (|HasCategory| (-480) (QUOTE (-945 (-480)))) (|HasCategory| (-480) (QUOTE (-1057))) (|HasCategory| (-480) (QUOTE (-791 (-325)))) (|HasCategory| (-480) (QUOTE (-791 (-480)))) (|HasCategory| (-480) (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-480) (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-480) (QUOTE (-187))) (|HasCategory| (-480) (QUOTE (-806 (-1081)))) (|HasCategory| (-480) (QUOTE (-188))) (|HasCategory| (-480) (QUOTE (-804 (-1081)))) (|HasCategory| (-480) (QUOTE (-449 (-1081) (-480)))) (|HasCategory| (-480) (QUOTE (-257 (-480)))) (|HasCategory| (-480) (QUOTE (-239 (-480) (-480)))) (|HasCategory| (-480) (QUOTE (-255))) (|HasCategory| (-480) (QUOTE (-479))) (|HasCategory| (-480) (QUOTE (-577 (-480)))) (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-816)))) (OR (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-816)))) (|HasCategory| (-480) (QUOTE (-116))))) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-483) (QUOTE (-821))) (|HasCategory| (-483) (QUOTE (-950 (-1088)))) (|HasCategory| (-483) (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-120))) (|HasCategory| (-483) (QUOTE (-553 (-472)))) (|HasCategory| (-483) (QUOTE (-933))) (|HasCategory| (-483) (QUOTE (-740))) (|HasCategory| (-483) (QUOTE (-756))) (OR (|HasCategory| (-483) (QUOTE (-740))) (|HasCategory| (-483) (QUOTE (-756)))) (|HasCategory| (-483) (QUOTE (-950 (-483)))) (|HasCategory| (-483) (QUOTE (-1064))) (|HasCategory| (-483) (QUOTE (-796 (-327)))) (|HasCategory| (-483) (QUOTE (-796 (-483)))) (|HasCategory| (-483) (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-483) (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-483) (QUOTE (-189))) (|HasCategory| (-483) (QUOTE (-811 (-1088)))) (|HasCategory| (-483) (QUOTE (-190))) (|HasCategory| (-483) (QUOTE (-809 (-1088)))) (|HasCategory| (-483) (QUOTE (-452 (-1088) (-483)))) (|HasCategory| (-483) (QUOTE (-259 (-483)))) (|HasCategory| (-483) (QUOTE (-241 (-483) (-483)))) (|HasCategory| (-483) (QUOTE (-257))) (|HasCategory| (-483) (QUOTE (-482))) (|HasCategory| (-483) (QUOTE (-580 (-483)))) (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-821)))) (OR (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-821)))) (|HasCategory| (-483) (QUOTE (-118))))) (-78) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. A `Binding' is a name asosciated with a collection of properties.")) (|binding| (($ (|Identifier|) (|List| (|Property|))) "\\spad{binding(n,props)} constructs a binding with name `n' and property list `props'.")) (|properties| (((|List| (|Property|)) $) "\\spad{properties(b)} returns the properties associated with binding \\spad{b}.")) (|name| (((|Identifier|) $) "\\spad{name(b)} returns the name of binding \\spad{b}"))) NIL NIL -(-79) +(-79 T$) +((|constructor| (NIL "This domain implements binary operations.")) (|binaryOperation| (($ (|Mapping| |#1| |#1| |#1|)) "\\spad{binaryOperation f} constructs a binary operation value out of any homogeneous mapping of arity 2."))) +NIL +NIL +(-80 T$) +((|constructor| (NIL "This is the category of all domains that implement binary operations."))) +NIL +NIL +(-81) ((|constructor| (NIL "\\spadtype{Bits} provides logical functions for Indexed Bits.")) (|bits| (($ (|NonNegativeInteger|) (|Boolean|)) "\\spad{bits(n,b)} creates bits with \\spad{n} values of \\spad{b}"))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| (-83) (QUOTE (-257 (-83)))) (|HasCategory| (-83) (QUOTE (-1007)))) (|HasCategory| (-83) (QUOTE (-550 (-469)))) (|HasCategory| (-83) (QUOTE (-751))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| (-83) (QUOTE (-1007))) (|HasCategory| (-83) (QUOTE (-549 (-767)))) (|HasCategory| (-83) (QUOTE (-72)))) -(-80 R S) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| (-85) (QUOTE (-259 (-85)))) (|HasCategory| (-85) (QUOTE (-1012)))) (|HasCategory| (-85) (QUOTE (-553 (-472)))) (|HasCategory| (-85) (QUOTE (-756))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| (-85) (QUOTE (-1012))) (|HasCategory| (-85) (QUOTE (-552 (-772)))) (|HasCategory| (-85) (QUOTE (-72)))) +(-82 R S) ((|constructor| (NIL "A \\spadtype{BiModule} is both a left and right module with respect to potentially different rings. \\blankline")) (|rightUnitary| ((|attribute|) "\\spad{x * 1 = x}")) (|leftUnitary| ((|attribute|) "\\spad{1 * x = x}"))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-81 S) +(-83 S) ((|constructor| (NIL "This is the category of Boolean logic structures.")) (|or| (($ $ $) "\\spad{x or y} returns the disjunction of \\spad{x} and \\spad{y}.")) (|and| (($ $ $) "\\spad{x and y} returns the conjunction of \\spad{x} and \\spad{y}.")) (|not| (($ $) "\\spad{not x} returns the complement or negation of \\spad{x}."))) NIL NIL -(-82) +(-84) ((|constructor| (NIL "This is the category of Boolean logic structures.")) (|or| (($ $ $) "\\spad{x or y} returns the disjunction of \\spad{x} and \\spad{y}.")) (|and| (($ $ $) "\\spad{x and y} returns the conjunction of \\spad{x} and \\spad{y}.")) (|not| (($ $) "\\spad{not x} returns the complement or negation of \\spad{x}."))) NIL NIL -(-83) +(-85) ((|constructor| (NIL "\\indented{1}{\\spadtype{Boolean} is the elementary logic with 2 values:} \\spad{true} and \\spad{false}")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical negation of \\spad{a} or \\spad{b}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical negation of \\spad{a} and \\spad{b}.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical exclusive {\\em or} of Boolean \\spad{a} and \\spad{b}."))) NIL NIL -(-84) +(-86) ((|constructor| (NIL "A basic operator is an object that can be applied to a list of arguments from a set,{} the result being a kernel over that set.")) (|setProperties| (($ $ (|AssociationList| (|String|) (|None|))) "\\spad{setProperties(op, l)} sets the property list of \\spad{op} to \\spad{l}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|setProperty| (($ $ (|Identifier|) (|None|)) "\\spad{setProperty(op, p, v)} attaches property \\spad{p} to \\spad{op},{} and sets its value to \\spad{v}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.") (($ $ (|String|) (|None|)) "\\spad{setProperty(op, s, v)} attaches property \\spad{s} to \\spad{op},{} and sets its value to \\spad{v}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|property| (((|Maybe| (|None|)) $ (|Identifier|)) "\\spad{property(op, p)} returns the value of property \\spad{p} if it is attached to \\spad{op},{} otherwise \\spad{nothing}.") (((|Union| (|None|) "failed") $ (|String|)) "\\spad{property(op, s)} returns the value of property \\spad{s} if it is attached to \\spad{op},{} and \"failed\" otherwise.")) (|deleteProperty!| (($ $ (|Identifier|)) "\\spad{deleteProperty!(op, p)} unattaches property \\spad{p} from \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.") (($ $ (|String|)) "\\spad{deleteProperty!(op, s)} unattaches property \\spad{s} from \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|assert| (($ $ (|Identifier|)) "\\spad{assert(op, p)} attaches property \\spad{p} to \\spad{op}. Argument \\spad{op} is modified \"in place\",{} \\spadignore{i.e.} no copy is made.")) (|has?| (((|Boolean|) $ (|Identifier|)) "\\spad{has?(op,p)} tests if property \\spad{s} is attached to \\spad{op}.")) (|input| (((|Maybe| (|Mapping| (|InputForm|) (|List| (|InputForm|)))) $) "\\spad{input(op)} returns the \"\\%input\" property of \\spad{op} if it has one attached,{} \\spad{nothing} otherwise.") (($ $ (|Mapping| (|InputForm|) (|List| (|InputForm|)))) "\\spad{input(op, foo)} attaches foo as the \"\\%input\" property of \\spad{op}. If \\spad{op} has a \"\\%input\" property \\spad{f},{} then \\spad{op(a1,...,an)} gets converted to InputForm as \\spad{f(a1,...,an)}.")) (|display| (($ $ (|Mapping| (|OutputForm|) (|OutputForm|))) "\\spad{display(op, foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a)} gets converted to OutputForm as \\spad{f(a)}. Argument \\spad{op} must be unary.") (($ $ (|Mapping| (|OutputForm|) (|List| (|OutputForm|)))) "\\spad{display(op, foo)} attaches foo as the \"\\%display\" property of \\spad{op}. If \\spad{op} has a \"\\%display\" property \\spad{f},{} then \\spad{op(a1,...,an)} gets converted to OutputForm as \\spad{f(a1,...,an)}.") (((|Maybe| (|Mapping| (|OutputForm|) (|List| (|OutputForm|)))) $) "\\spad{display(op)} returns the \"\\%display\" property of \\spad{op} if it has one attached,{} and \\spad{nothing} otherwise.")) (|comparison| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{comparison(op, foo?)} attaches foo? as the \"\\%less?\" property to \\spad{op}. If \\spad{op1} and \\spad{op2} have the same name,{} and one of them has a \"\\%less?\" property \\spad{f},{} then \\spad{f(op1, op2)} is called to decide whether \\spad{op1 < op2}.")) (|equality| (($ $ (|Mapping| (|Boolean|) $ $)) "\\spad{equality(op, foo?)} attaches foo? as the \"\\%equal?\" property to \\spad{op}. If \\spad{op1} and \\spad{op2} have the same name,{} and one of them has an \"\\%equal?\" property \\spad{f},{} then \\spad{f(op1, op2)} is called to decide whether \\spad{op1} and \\spad{op2} should be considered equal.")) (|weight| (($ $ (|NonNegativeInteger|)) "\\spad{weight(op, n)} attaches the weight \\spad{n} to \\spad{op}.") (((|NonNegativeInteger|) $) "\\spad{weight(op)} returns the weight attached to \\spad{op}.")) (|nary?| (((|Boolean|) $) "\\spad{nary?(op)} tests if \\spad{op} has arbitrary arity.")) (|unary?| (((|Boolean|) $) "\\spad{unary?(op)} tests if \\spad{op} is unary.")) (|nullary?| (((|Boolean|) $) "\\spad{nullary?(op)} tests if \\spad{op} is nullary.")) (|operator| (($ (|Symbol|) (|Arity|)) "\\spad{operator(f, a)} makes \\spad{f} into an operator of arity \\spad{a}.") (($ (|Symbol|) (|NonNegativeInteger|)) "\\spad{operator(f, n)} makes \\spad{f} into an \\spad{n}-ary operator.") (($ (|Symbol|)) "\\spad{operator(f)} makes \\spad{f} into an operator with arbitrary arity.")) (|copy| (($ $) "\\spad{copy(op)} returns a copy of \\spad{op}.")) (|properties| (((|AssociationList| (|String|) (|None|)) $) "\\spad{properties(op)} returns the list of all the properties currently attached to \\spad{op}."))) NIL NIL -(-85 A) +(-87 A) ((|constructor| (NIL "This package exports functions to set some commonly used properties of operators,{} including properties which contain functions.")) (|constantOpIfCan| (((|Union| |#1| "failed") (|BasicOperator|)) "\\spad{constantOpIfCan(op)} returns \\spad{a} if \\spad{op} is the constant nullary operator always returning \\spad{a},{} \"failed\" otherwise.")) (|constantOperator| (((|BasicOperator|) |#1|) "\\spad{constantOperator(a)} returns a nullary operator op such that \\spad{op()} always evaluate to \\spad{a}.")) (|derivative| (((|Union| (|List| (|Mapping| |#1| (|List| |#1|))) "failed") (|BasicOperator|)) "\\spad{derivative(op)} returns the value of the \"\\%diff\" property of \\spad{op} if it has one,{} and \"failed\" otherwise.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| |#1|)) "\\spad{derivative(op, foo)} attaches foo as the \"\\%diff\" property of \\spad{op}. If \\spad{op} has an \"\\%diff\" property \\spad{f},{} then applying a derivation \\spad{D} to \\spad{op}(a) returns \\spad{f(a) * D(a)}. Argument \\spad{op} must be unary.") (((|BasicOperator|) (|BasicOperator|) (|List| (|Mapping| |#1| (|List| |#1|)))) "\\spad{derivative(op, [foo1,...,foon])} attaches [\\spad{foo1},{}...,{}foon] as the \"\\%diff\" property of \\spad{op}. If \\spad{op} has an \"\\%diff\" property \\spad{[f1,...,fn]} then applying a derivation \\spad{D} to \\spad{op(a1,...,an)} returns \\spad{f1(a1,...,an) * D(a1) + ... + fn(a1,...,an) * D(an)}.")) (|evaluate| (((|Union| (|Mapping| |#1| (|List| |#1|)) "failed") (|BasicOperator|)) "\\spad{evaluate(op)} returns the value of the \"\\%eval\" property of \\spad{op} if it has one,{} and \"failed\" otherwise.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| |#1|)) "\\spad{evaluate(op, foo)} attaches foo as the \"\\%eval\" property of \\spad{op}. If \\spad{op} has an \"\\%eval\" property \\spad{f},{} then applying \\spad{op} to a returns the result of \\spad{f(a)}. Argument \\spad{op} must be unary.") (((|BasicOperator|) (|BasicOperator|) (|Mapping| |#1| (|List| |#1|))) "\\spad{evaluate(op, foo)} attaches foo as the \"\\%eval\" property of \\spad{op}. If \\spad{op} has an \"\\%eval\" property \\spad{f},{} then applying \\spad{op} to \\spad{(a1,...,an)} returns the result of \\spad{f(a1,...,an)}.") (((|Union| |#1| "failed") (|BasicOperator|) (|List| |#1|)) "\\spad{evaluate(op, [a1,...,an])} checks if \\spad{op} has an \"\\%eval\" property \\spad{f}. If it has,{} then \\spad{f(a1,...,an)} is returned,{} and \"failed\" otherwise."))) NIL NIL -(-86 -3078 UP) +(-88 -3088 UP) ((|constructor| (NIL "\\spadtype{BoundIntegerRoots} provides functions to find lower bounds on the integer roots of a polynomial.")) (|integerBound| (((|Integer|) |#2|) "\\spad{integerBound(p)} returns a lower bound on the negative integer roots of \\spad{p},{} and 0 if \\spad{p} has no negative integer roots."))) NIL NIL -(-87 |p|) +(-89 |p|) ((|constructor| (NIL "Stream-based implementation of Zp: \\spad{p}-adic numbers are represented as sum(\\spad{i} = 0..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in -(\\spad{p} - 1)\\spad{/2},{}...,{}(\\spad{p} - 1)\\spad{/2}."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-88 |p|) +(-90 |p|) ((|constructor| (NIL "Stream-based implementation of Qp: numbers are represented as sum(\\spad{i} = \\spad{k}..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in -(\\spad{p} - 1)\\spad{/2},{}...,{}(\\spad{p} - 1)\\spad{/2}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-87 |#1|) (QUOTE (-816))) (|HasCategory| (-87 |#1|) (QUOTE (-945 (-1081)))) (|HasCategory| (-87 |#1|) (QUOTE (-116))) (|HasCategory| (-87 |#1|) (QUOTE (-118))) (|HasCategory| (-87 |#1|) (QUOTE (-550 (-469)))) (|HasCategory| (-87 |#1|) (QUOTE (-928))) (|HasCategory| (-87 |#1|) (QUOTE (-735))) (|HasCategory| (-87 |#1|) (QUOTE (-751))) (OR (|HasCategory| (-87 |#1|) (QUOTE (-735))) (|HasCategory| (-87 |#1|) (QUOTE (-751)))) (|HasCategory| (-87 |#1|) (QUOTE (-945 (-480)))) (|HasCategory| (-87 |#1|) (QUOTE (-1057))) (|HasCategory| (-87 |#1|) (QUOTE (-791 (-325)))) (|HasCategory| (-87 |#1|) (QUOTE (-791 (-480)))) (|HasCategory| (-87 |#1|) (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-87 |#1|) (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-87 |#1|) (QUOTE (-577 (-480)))) (|HasCategory| (-87 |#1|) (QUOTE (-187))) (|HasCategory| (-87 |#1|) (QUOTE (-806 (-1081)))) (|HasCategory| (-87 |#1|) (QUOTE (-188))) (|HasCategory| (-87 |#1|) (QUOTE (-804 (-1081)))) (|HasCategory| (-87 |#1|) (|%list| (QUOTE -449) (QUOTE (-1081)) (|%list| (QUOTE -87) (|devaluate| |#1|)))) (|HasCategory| (-87 |#1|) (|%list| (QUOTE -257) (|%list| (QUOTE -87) (|devaluate| |#1|)))) (|HasCategory| (-87 |#1|) (|%list| (QUOTE -239) (|%list| (QUOTE -87) (|devaluate| |#1|)) (|%list| (QUOTE -87) (|devaluate| |#1|)))) (|HasCategory| (-87 |#1|) (QUOTE (-255))) (|HasCategory| (-87 |#1|) (QUOTE (-479))) (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-87 |#1|) (QUOTE (-816)))) (OR (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-87 |#1|) (QUOTE (-816)))) (|HasCategory| (-87 |#1|) (QUOTE (-116))))) -(-89 A S) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-89 |#1|) (QUOTE (-821))) (|HasCategory| (-89 |#1|) (QUOTE (-950 (-1088)))) (|HasCategory| (-89 |#1|) (QUOTE (-118))) (|HasCategory| (-89 |#1|) (QUOTE (-120))) (|HasCategory| (-89 |#1|) (QUOTE (-553 (-472)))) (|HasCategory| (-89 |#1|) (QUOTE (-933))) (|HasCategory| (-89 |#1|) (QUOTE (-740))) (|HasCategory| (-89 |#1|) (QUOTE (-756))) (OR (|HasCategory| (-89 |#1|) (QUOTE (-740))) (|HasCategory| (-89 |#1|) (QUOTE (-756)))) (|HasCategory| (-89 |#1|) (QUOTE (-950 (-483)))) (|HasCategory| (-89 |#1|) (QUOTE (-1064))) (|HasCategory| (-89 |#1|) (QUOTE (-796 (-327)))) (|HasCategory| (-89 |#1|) (QUOTE (-796 (-483)))) (|HasCategory| (-89 |#1|) (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-89 |#1|) (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-89 |#1|) (QUOTE (-580 (-483)))) (|HasCategory| (-89 |#1|) (QUOTE (-189))) (|HasCategory| (-89 |#1|) (QUOTE (-811 (-1088)))) (|HasCategory| (-89 |#1|) (QUOTE (-190))) (|HasCategory| (-89 |#1|) (QUOTE (-809 (-1088)))) (|HasCategory| (-89 |#1|) (|%list| (QUOTE -452) (QUOTE (-1088)) (|%list| (QUOTE -89) (|devaluate| |#1|)))) (|HasCategory| (-89 |#1|) (|%list| (QUOTE -259) (|%list| (QUOTE -89) (|devaluate| |#1|)))) (|HasCategory| (-89 |#1|) (|%list| (QUOTE -241) (|%list| (QUOTE -89) (|devaluate| |#1|)) (|%list| (QUOTE -89) (|devaluate| |#1|)))) (|HasCategory| (-89 |#1|) (QUOTE (-257))) (|HasCategory| (-89 |#1|) (QUOTE (-482))) (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-89 |#1|) (QUOTE (-821)))) (OR (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-89 |#1|) (QUOTE (-821)))) (|HasCategory| (-89 |#1|) (QUOTE (-118))))) +(-91 A S) ((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "right" $) "\\spad{setelt(a,\"right\",b)} (also written \\axiom{\\spad{b} . right := \\spad{b}}) is equivalent to \\axiom{setright!(a,{}\\spad{b})}.") (($ $ "left" $) "\\spad{setelt(a,\"left\",b)} (also written \\axiom{a . left := \\spad{b}}) is equivalent to \\axiom{setleft!(a,{}\\spad{b})}.")) (|right| (($ $) "\\spad{right(a)} returns the right child.")) (|elt| (($ $ "right") "\\spad{elt(a,\"right\")} (also written: \\axiom{a . right}) is equivalent to \\axiom{right(a)}.") (($ $ "left") "\\spad{elt(u,\"left\")} (also written: \\axiom{a . left}) is equivalent to \\axiom{left(a)}.")) (|left| (($ $) "\\spad{left(u)} returns the left child."))) NIL -((|HasAttribute| |#1| (QUOTE -3979))) -(-90 S) +((|HasAttribute| |#1| (QUOTE -3990))) +(-92 S) ((|constructor| (NIL "A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure")) (|setright!| (($ $ $) "\\spad{setright!(a,x)} sets the right child of \\spad{t} to be \\spad{x}.")) (|setleft!| (($ $ $) "\\spad{setleft!(a,b)} sets the left child of \\axiom{a} to be \\spad{b}.")) (|setelt| (($ $ "right" $) "\\spad{setelt(a,\"right\",b)} (also written \\axiom{\\spad{b} . right := \\spad{b}}) is equivalent to \\axiom{setright!(a,{}\\spad{b})}.") (($ $ "left" $) "\\spad{setelt(a,\"left\",b)} (also written \\axiom{a . left := \\spad{b}}) is equivalent to \\axiom{setleft!(a,{}\\spad{b})}.")) (|right| (($ $) "\\spad{right(a)} returns the right child.")) (|elt| (($ $ "right") "\\spad{elt(a,\"right\")} (also written: \\axiom{a . right}) is equivalent to \\axiom{right(a)}.") (($ $ "left") "\\spad{elt(u,\"left\")} (also written: \\axiom{a . left}) is equivalent to \\axiom{left(a)}.")) (|left| (($ $) "\\spad{left(u)} returns the left child."))) NIL NIL -(-91 UP) +(-93 UP) ((|constructor| (NIL "\\indented{1}{Author: Frederic Lehobey,{} James \\spad{H}. Davenport} Date Created: 28 June 1994 Date Last Updated: 11 July 1997 Basic Operations: brillhartIrreducible? Related Domains: Also See: AMS Classifications: Keywords: factorization Examples: References: [1] John Brillhart,{} Note on Irreducibility Testing,{} Mathematics of Computation,{} vol. 35,{} num. 35,{} Oct. 1980,{} 1379-1381 [2] James Davenport,{} On Brillhart Irreducibility. To appear. [3] John Brillhart,{} On the Euler and Bernoulli polynomials,{} \\spad{J}. Reine Angew. Math.,{} \\spad{v}. 234,{} (1969),{} pp. 45-64")) (|noLinearFactor?| (((|Boolean|) |#1|) "\\spad{noLinearFactor?(p)} returns \\spad{true} if \\spad{p} can be shown to have no linear factor by a theorem of Lehmer,{} \\spad{false} else. \\spad{I} insist on the fact that \\spad{false} does not mean that \\spad{p} has a linear factor.")) (|brillhartTrials| (((|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{brillhartTrials(n)} sets to \\spad{n} the number of tests in \\spadfun{brillhartIrreducible?} and returns the previous value.") (((|NonNegativeInteger|)) "\\spad{brillhartTrials()} returns the number of tests in \\spadfun{brillhartIrreducible?}.")) (|brillhartIrreducible?| (((|Boolean|) |#1| (|Boolean|)) "\\spad{brillhartIrreducible?(p,noLinears)} returns \\spad{true} if \\spad{p} can be shown to be irreducible by a remark of Brillhart,{} \\spad{false} else. If \\spad{noLinears} is \\spad{true},{} we are being told \\spad{p} has no linear factors \\spad{false} does not mean that \\spad{p} is reducible.") (((|Boolean|) |#1|) "\\spad{brillhartIrreducible?(p)} returns \\spad{true} if \\spad{p} can be shown to be irreducible by a remark of Brillhart,{} \\spad{false} is inconclusive."))) NIL NIL -(-92 S) +(-94 S) ((|constructor| (NIL "BinarySearchTree(\\spad{S}) is the domain of a binary trees where elements are ordered across the tree. A binary search tree is either empty or has a value which is an \\spad{S},{} and a right and left which are both BinaryTree(\\spad{S}) Elements are ordered across the tree.")) (|split| (((|Record| (|:| |less| $) (|:| |greater| $)) |#1| $) "\\spad{split(x,b)} splits binary tree \\spad{b} into two trees,{} one with elements greater than \\spad{x},{} the other with elements less than \\spad{x}.")) (|insertRoot!| (($ |#1| $) "\\spad{insertRoot!(x,b)} inserts element \\spad{x} as a root of binary search tree \\spad{b}.")) (|insert!| (($ |#1| $) "\\spad{insert!(x,b)} inserts element \\spad{x} as leaves into binary search tree \\spad{b}.")) (|binarySearchTree| (($ (|List| |#1|)) "\\spad{binarySearchTree(l)} \\undocumented"))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-93 S) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-95 S) ((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical {\\em exclusive-or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical {\\em nor} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical {\\em nand} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}."))) NIL NIL -(-94) +(-96) ((|constructor| (NIL "The bit aggregate category models aggregates representing large quantities of Boolean data.")) (|xor| (($ $ $) "\\spad{xor(a,b)} returns the logical {\\em exclusive-or} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nor| (($ $ $) "\\spad{nor(a,b)} returns the logical {\\em nor} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}.")) (|nand| (($ $ $) "\\spad{nand(a,b)} returns the logical {\\em nand} of bit aggregates \\axiom{a} and \\axiom{\\spad{b}}."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-95 A S) +(-97 A S) ((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#2| $) "\\spad{node(left,v,right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}.")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) NIL NIL -(-96 S) +(-98 S) ((|constructor| (NIL "\\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \\spadfun{node} consisting of a value and a \\spadfun{left} and \\spadfun{right},{} both binary trees.")) (|node| (($ $ |#1| $) "\\spad{node(left,v,right)} creates a binary tree with value \\spad{v},{} a binary tree \\spad{left},{} and a binary tree \\spad{right}.")) (|finiteAggregate| ((|attribute|) "Binary trees have a finite number of components")) (|shallowlyMutable| ((|attribute|) "Binary trees have updateable components"))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL -(-97 S) +(-99 S) ((|constructor| (NIL "\\spadtype{BinaryTournament(S)} is the domain of binary trees where elements are ordered down the tree. A binary search tree is either empty or is a node containing a \\spadfun{value} of type \\spad{S},{} and a \\spadfun{right} and a \\spadfun{left} which are both \\spadtype{BinaryTree(S)}")) (|insert!| (($ |#1| $) "\\spad{insert!(x,b)} inserts element \\spad{x} as leaves into binary tournament \\spad{b}.")) (|binaryTournament| (($ (|List| |#1|)) "\\spad{binaryTournament(ls)} creates a binary tournament with the elements of \\spad{ls} as values at the nodes."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-98 S) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-100 S) ((|constructor| (NIL "\\spadtype{BinaryTree(S)} is the domain of all binary trees. A binary tree over \\spad{S} is either empty or has a \\spadfun{value} which is an \\spad{S} and a \\spadfun{right} and \\spadfun{left} which are both binary trees.")) (|binaryTree| (($ $ |#1| $) "\\spad{binaryTree(l,v,r)} creates a binary tree with value \\spad{v} with left subtree \\spad{l} and right subtree \\spad{r}.") (($ |#1|) "\\spad{binaryTree(v)} is an non-empty binary tree with value \\spad{v},{} and left and right empty."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-99) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-101) ((|constructor| (NIL "Byte is the datatype of 8-bit sized unsigned integer values.")) (|sample| (($) "\\spad{sample} gives a sample datum of type Byte.")) (|bitior| (($ $ $) "bitor(\\spad{x},{}\\spad{y}) returns the bitwise `inclusive or' of `x' and `y'.")) (|bitand| (($ $ $) "\\spad{bitand(x,y)} returns the bitwise `and' of `x' and `y'.")) (|byte| (($ (|NonNegativeInteger|)) "\\spad{byte(x)} injects the unsigned integer value `v' into the Byte algebra. `v' must be non-negative and less than 256."))) NIL NIL -(-100) +(-102) ((|constructor| (NIL "ByteBuffer provides datatype for buffers of bytes. This domain differs from PrimitiveArray Byte in that it is not as rigid as PrimitiveArray Byte. That is,{} the typical use of ByteBuffer is to pre-allocate a vector of Byte of some capacity `n'. The array can then store up to `n' bytes. The actual interesting bytes count (the length of the buffer) is therefore different from the capacity. The length is no more than the capacity,{} but it can be set dynamically as needed. This functionality is used for example when reading bytes from input/output devices where we use buffers to transfer data in and out of the system. Note: a value of type ByteBuffer is 0-based indexed,{} as opposed \\indented{6}{Vector,{} but not unlike PrimitiveArray Byte.}")) (|finiteAggregate| ((|attribute|) "A ByteBuffer object is a finite aggregate")) (|setLength!| (((|NonNegativeInteger|) $ (|NonNegativeInteger|)) "\\spad{setLength!(buf,n)} sets the number of active bytes in the `buf'. Error if `n' is more than the capacity.")) (|capacity| (((|NonNegativeInteger|) $) "\\spad{capacity(buf)} returns the pre-allocated maximum size of `buf'.")) (|byteBuffer| (($ (|NonNegativeInteger|)) "\\spad{byteBuffer(n)} creates a buffer of capacity \\spad{n},{} and length 0."))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| (-99) (QUOTE (-257 (-99)))) (|HasCategory| (-99) (QUOTE (-751)))) (-12 (|HasCategory| (-99) (QUOTE (-257 (-99)))) (|HasCategory| (-99) (QUOTE (-1007))))) (|HasCategory| (-99) (QUOTE (-549 (-767)))) (|HasCategory| (-99) (QUOTE (-550 (-469)))) (OR (|HasCategory| (-99) (QUOTE (-751))) (|HasCategory| (-99) (QUOTE (-1007)))) (|HasCategory| (-99) (QUOTE (-751))) (OR (|HasCategory| (-99) (QUOTE (-72))) (|HasCategory| (-99) (QUOTE (-751))) (|HasCategory| (-99) (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| (-99) (QUOTE (-1007))) (|HasCategory| (-99) (QUOTE (-72))) (-12 (|HasCategory| (-99) (QUOTE (-257 (-99)))) (|HasCategory| (-99) (QUOTE (-1007))))) -(-101) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| (-101) (QUOTE (-259 (-101)))) (|HasCategory| (-101) (QUOTE (-756)))) (-12 (|HasCategory| (-101) (QUOTE (-259 (-101)))) (|HasCategory| (-101) (QUOTE (-1012))))) (|HasCategory| (-101) (QUOTE (-552 (-772)))) (|HasCategory| (-101) (QUOTE (-553 (-472)))) (OR (|HasCategory| (-101) (QUOTE (-756))) (|HasCategory| (-101) (QUOTE (-1012)))) (|HasCategory| (-101) (QUOTE (-756))) (OR (|HasCategory| (-101) (QUOTE (-72))) (|HasCategory| (-101) (QUOTE (-756))) (|HasCategory| (-101) (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| (-101) (QUOTE (-1012))) (|HasCategory| (-101) (QUOTE (-72))) (-12 (|HasCategory| (-101) (QUOTE (-259 (-101)))) (|HasCategory| (-101) (QUOTE (-1012))))) +(-103) ((|constructor| (NIL "This datatype describes byte order of machine values stored memory.")) (|unknownEndian| (($) "\\spad{unknownEndian} for none of the above.")) (|bigEndian| (($) "\\spad{bigEndian} describes big endian host")) (|littleEndian| (($) "\\spad{littleEndian} describes little endian host"))) NIL NIL -(-102) +(-104) ((|constructor| (NIL "This is an \\spadtype{AbelianMonoid} with the cancellation property,{} \\spadignore{i.e.} \\spad{ a+b = a+c => b=c }. This is formalised by the partial subtraction operator,{} which satisfies the axioms listed below: \\blankline")) (|subtractIfCan| (((|Union| $ "failed") $ $) "\\spad{subtractIfCan(x, y)} returns an element \\spad{z} such that \\spad{z+y=x} or \"failed\" if no such element exists."))) NIL NIL -(-103) +(-105) ((|constructor| (NIL "A cachable set is a set whose elements keep an integer as part of their structure.")) (|setPosition| (((|Void|) $ (|NonNegativeInteger|)) "\\spad{setPosition(x, n)} associates the integer \\spad{n} to \\spad{x}.")) (|position| (((|NonNegativeInteger|) $) "\\spad{position(x)} returns the integer \\spad{n} associated to \\spad{x}."))) NIL NIL -(-104) +(-106) ((|constructor| (NIL "This domain represents the capsule of a domain definition.")) (|body| (((|List| (|SpadAst|)) $) "\\spad{body(c)} returns the list of top level expressions appearing in `c'."))) NIL NIL -(-105) +(-107) ((|constructor| (NIL "Members of the domain CardinalNumber are values indicating the cardinality of sets,{} both finite and infinite. Arithmetic operations are defined on cardinal numbers as follows. \\blankline If \\spad{x = \\#X} and \\spad{y = \\#Y} then \\indented{2}{\\spad{x+y\\space{2}= \\#(X+Y)}\\space{3}\\tab{30}disjoint union} \\indented{2}{\\spad{x-y\\space{2}= \\#(X-Y)}\\space{3}\\tab{30}relative complement} \\indented{2}{\\spad{x*y\\space{2}= \\#(X*Y)}\\space{3}\\tab{30}cartesian product} \\indented{2}{\\spad{x**y = \\#(X**Y)}\\space{2}\\tab{30}\\spad{X**Y = \\{g| g:Y->X\\}}} \\blankline The non-negative integers have a natural construction as cardinals \\indented{2}{\\spad{0 = \\#\\{\\}},{} \\spad{1 = \\{0\\}},{} \\spad{2 = \\{0, 1\\}},{} ...,{} \\spad{n = \\{i| 0 <= i < n\\}}.} \\blankline That \\spad{0} acts as a zero for the multiplication of cardinals is equivalent to the axiom of choice. \\blankline The generalized continuum hypothesis asserts \\center{\\spad{2**Aleph i = Aleph(i+1)}} and is independent of the axioms of set theory [Goedel 1940]. \\blankline Three commonly encountered cardinal numbers are \\indented{3}{\\spad{a = \\#Z}\\space{7}\\tab{30}countable infinity} \\indented{3}{\\spad{c = \\#R}\\space{7}\\tab{30}the continuum} \\indented{3}{\\spad{f = \\#\\{g| g:[0,1]->R\\}}} \\blankline In this domain,{} these values are obtained using \\indented{3}{\\spad{a := Aleph 0},{} \\spad{c := 2**a},{} \\spad{f := 2**c}.} \\blankline")) (|generalizedContinuumHypothesisAssumed| (((|Boolean|) (|Boolean|)) "\\spad{generalizedContinuumHypothesisAssumed(bool)} is used to dictate whether the hypothesis is to be assumed.")) (|generalizedContinuumHypothesisAssumed?| (((|Boolean|)) "\\spad{generalizedContinuumHypothesisAssumed?()} tests if the hypothesis is currently assumed.")) (|countable?| (((|Boolean|) $) "\\spad{countable?(\\spad{a})} determines whether \\spad{a} is a countable cardinal,{} \\spadignore{i.e.} an integer or \\spad{Aleph 0}.")) (|finite?| (((|Boolean|) $) "\\spad{finite?(\\spad{a})} determines whether \\spad{a} is a finite cardinal,{} \\spadignore{i.e.} an integer.")) (|Aleph| (($ (|NonNegativeInteger|)) "\\spad{Aleph(n)} provides the named (infinite) cardinal number.")) (** (($ $ $) "\\spad{x**y} returns \\spad{\\#(X**Y)} where \\spad{X**Y} is defined \\indented{1}{as \\spad{\\{g| g:Y->X\\}}.}")) (- (((|Union| $ "failed") $ $) "\\spad{x - y} returns an element \\spad{z} such that \\spad{z+y=x} or \"failed\" if no such element exists.")) (|commutative| ((|attribute| "*") "a domain \\spad{D} has \\spad{commutative(\"*\")} if it has an operation \\spad{\"*\": (D,D) -> D} which is commutative."))) -(((-3980 "*") . T)) +(((-3991 "*") . T)) NIL -(-106 |minix| -2607 R) +(-108 |minix| -2617 R) ((|constructor| (NIL "CartesianTensor(minix,{}dim,{}\\spad{R}) provides Cartesian tensors with components belonging to a commutative ring \\spad{R}. These tensors can have any number of indices. Each index takes values from \\spad{minix} to \\spad{minix + dim - 1}.")) (|sample| (($) "\\spad{sample()} returns an object of type \\%.")) (|unravel| (($ (|List| |#3|)) "\\spad{unravel(t)} produces a tensor from a list of components such that \\indented{2}{\\spad{unravel(ravel(t)) = t}.}")) (|ravel| (((|List| |#3|) $) "\\spad{ravel(t)} produces a list of components from a tensor such that \\indented{2}{\\spad{unravel(ravel(t)) = t}.}")) (|leviCivitaSymbol| (($) "\\spad{leviCivitaSymbol()} is the rank \\spad{dim} tensor defined by \\spad{leviCivitaSymbol()(i1,...idim) = +1/0/-1} if \\spad{i1,...,idim} is an even/is nota /is an odd permutation of \\spad{minix,...,minix+dim-1}.")) (|kroneckerDelta| (($) "\\spad{kroneckerDelta()} is the rank 2 tensor defined by \\indented{3}{\\spad{kroneckerDelta()(i,j)}} \\indented{6}{\\spad{= 1\\space{2}if i = j}} \\indented{6}{\\spad{= 0 if\\space{2}i \\~= j}}")) (|reindex| (($ $ (|List| (|Integer|))) "\\spad{reindex(t,[i1,...,idim])} permutes the indices of \\spad{t}. For example,{} if \\spad{r = reindex(t, [4,1,2,3])} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank for tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(l,i,j,k)}.}")) (|transpose| (($ $ (|Integer|) (|Integer|)) "\\spad{transpose(t,i,j)} exchanges the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th indices of \\spad{t}. For example,{} if \\spad{r = transpose(t,2,3)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(i,k,j,l)}.}") (($ $) "\\spad{transpose(t)} exchanges the first and last indices of \\spad{t}. For example,{} if \\spad{r = transpose(t)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = t(l,j,k,i)}.}")) (|contract| (($ $ (|Integer|) (|Integer|)) "\\spad{contract(t,i,j)} is the contraction of tensor \\spad{t} which sums along the \\spad{i}\\spad{-}th and \\spad{j}\\spad{-}th indices. For example,{} if \\spad{r = contract(t,1,3)} for a rank 4 tensor \\spad{t},{} then \\spad{r} is the rank 2 \\spad{(= 4 - 2)} tensor given by \\indented{4}{\\spad{r(i,j) = sum(h=1..dim,t(h,i,h,j))}.}") (($ $ (|Integer|) $ (|Integer|)) "\\spad{contract(t,i,s,j)} is the inner product of tenors \\spad{s} and \\spad{t} which sums along the \\spad{k1}\\spad{-}th index of \\spad{t} and the \\spad{k2}\\spad{-}th index of \\spad{s}. For example,{} if \\spad{r = contract(s,2,t,1)} for rank 3 tensors rank 3 tensors \\spad{s} and \\spad{t},{} then \\spad{r} is the rank 4 \\spad{(= 3 + 3 - 2)} tensor given by \\indented{4}{\\spad{r(i,j,k,l) = sum(h=1..dim,s(i,h,j)*t(h,k,l))}.}")) (* (($ $ $) "\\spad{s*t} is the inner product of the tensors \\spad{s} and \\spad{t} which contracts the last index of \\spad{s} with the first index of \\spad{t},{} \\spadignore{i.e.} \\indented{4}{\\spad{t*s = contract(t,rank t, s, 1)}} \\indented{4}{\\spad{t*s = sum(k=1..N, t[i1,..,iN,k]*s[k,j1,..,jM])}} This is compatible with the use of \\spad{M*v} to denote the matrix-vector inner product.")) (|product| (($ $ $) "\\spad{product(s,t)} is the outer product of the tensors \\spad{s} and \\spad{t}. For example,{} if \\spad{r = product(s,t)} for rank 2 tensors \\spad{s} and \\spad{t},{} then \\spad{r} is a rank 4 tensor given by \\indented{4}{\\spad{r(i,j,k,l) = s(i,j)*t(k,l)}.}")) (|elt| ((|#3| $ (|List| (|Integer|))) "\\spad{elt(t,[i1,...,iN])} gives a component of a rank \\spad{N} tensor.") ((|#3| $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{elt(t,i,j,k,l)} gives a component of a rank 4 tensor.") ((|#3| $ (|Integer|) (|Integer|) (|Integer|)) "\\spad{elt(t,i,j,k)} gives a component of a rank 3 tensor.") ((|#3| $ (|Integer|) (|Integer|)) "\\spad{elt(t,i,j)} gives a component of a rank 2 tensor.") ((|#3| $) "\\spad{elt(t)} gives the component of a rank 0 tensor.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(t)} returns the tensorial rank of \\spad{t} (that is,{} the number of indices). This is the same as the graded module degree."))) NIL NIL -(-107 |minix| -2607 S T$) +(-109 |minix| -2617 S T$) ((|constructor| (NIL "This package provides functions to enable conversion of tensors given conversion of the components.")) (|map| (((|CartesianTensor| |#1| |#2| |#4|) (|Mapping| |#4| |#3|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{map(f,ts)} does a componentwise conversion of the tensor \\spad{ts} to a tensor with components of type \\spad{T}.")) (|reshape| (((|CartesianTensor| |#1| |#2| |#4|) (|List| |#4|) (|CartesianTensor| |#1| |#2| |#3|)) "\\spad{reshape(lt,ts)} organizes the list of components \\spad{lt} into a tensor with the same shape as \\spad{ts}."))) NIL NIL -(-108) +(-110) ((|constructor| (NIL "This domain represents a `case' expression.")) (|rhs| (((|SpadAst|) $) "\\spad{rhs(e)} returns the right hand side of the case expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the case expression `e'."))) NIL NIL -(-109) +(-111) ((|constructor| (NIL "This domain represents the unnamed category defined \\indented{2}{by a list of exported signatures}")) (|body| (((|List| (|SpadAst|)) $) "\\spad{body(c)} returns the list of exports in category syntax `c'.")) (|kind| (((|ConstructorKind|) $) "\\spad{kind(c)} returns the kind of unnamed category,{} either 'domain' or 'package'."))) NIL NIL -(-110) +(-112) ((|constructor| (NIL "This domain provides representations for category constructors."))) NIL NIL -(-111) +(-113) ((|parents| (((|List| (|ConstructorCall| (|CategoryConstructor|))) $) "\\spad{parents(c)} returns the list of all category forms directly extended by the category `c'.")) (|principalAncestors| (((|List| (|ConstructorCall| (|CategoryConstructor|))) $) "\\spad{principalAncestors(c)} returns the list of all category forms that are principal ancestors of the the category `c'.")) (|exportedOperators| (((|List| (|OperatorSignature|)) $) "\\spad{exportedOperators(c)} returns the list of all operator signatures exported by the category `c',{} along with their predicates.")) (|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Create: December 20,{} 2008. Date Last Updated: February 16,{} 2008. Basic Operations: coerce Related Constructors: Also See: Type") (((|CategoryConstructor|) $) "\\spad{constructor(c)} returns the category constructor used to instantiate the category object `c'."))) NIL NIL -(-112) +(-114) ((|constructor| (NIL "This domain allows classes of characters to be defined and manipulated efficiently.")) (|alphanumeric| (($) "\\spad{alphanumeric()} returns the class of all characters for which \\spadfunFrom{alphanumeric?}{Character} is \\spad{true}.")) (|alphabetic| (($) "\\spad{alphabetic()} returns the class of all characters for which \\spadfunFrom{alphabetic?}{Character} is \\spad{true}.")) (|lowerCase| (($) "\\spad{lowerCase()} returns the class of all characters for which \\spadfunFrom{lowerCase?}{Character} is \\spad{true}.")) (|upperCase| (($) "\\spad{upperCase()} returns the class of all characters for which \\spadfunFrom{upperCase?}{Character} is \\spad{true}.")) (|hexDigit| (($) "\\spad{hexDigit()} returns the class of all characters for which \\spadfunFrom{hexDigit?}{Character} is \\spad{true}.")) (|digit| (($) "\\spad{digit()} returns the class of all characters for which \\spadfunFrom{digit?}{Character} is \\spad{true}.")) (|charClass| (($ (|List| (|Character|))) "\\spad{charClass(l)} creates a character class which contains exactly the characters given in the list \\spad{l}.") (($ (|String|)) "\\spad{charClass(s)} creates a character class which contains exactly the characters given in the string \\spad{s}."))) -((-3978 . T) (-3968 . T) (-3979 . T)) -((OR (-12 (|HasCategory| (-115) (QUOTE (-257 (-115)))) (|HasCategory| (-115) (QUOTE (-315)))) (-12 (|HasCategory| (-115) (QUOTE (-257 (-115)))) (|HasCategory| (-115) (QUOTE (-1007))))) (|HasCategory| (-115) (QUOTE (-550 (-469)))) (|HasCategory| (-115) (QUOTE (-315))) (|HasCategory| (-115) (QUOTE (-751))) (|HasCategory| (-115) (QUOTE (-1007))) (|HasCategory| (-115) (QUOTE (-549 (-767)))) (|HasCategory| (-115) (QUOTE (-72))) (-12 (|HasCategory| (-115) (QUOTE (-257 (-115)))) (|HasCategory| (-115) (QUOTE (-1007))))) -(-113 R Q A) +((-3989 . T) (-3979 . T) (-3990 . T)) +((OR (-12 (|HasCategory| (-117) (QUOTE (-259 (-117)))) (|HasCategory| (-117) (QUOTE (-317)))) (-12 (|HasCategory| (-117) (QUOTE (-259 (-117)))) (|HasCategory| (-117) (QUOTE (-1012))))) (|HasCategory| (-117) (QUOTE (-553 (-472)))) (|HasCategory| (-117) (QUOTE (-317))) (|HasCategory| (-117) (QUOTE (-756))) (|HasCategory| (-117) (QUOTE (-1012))) (|HasCategory| (-117) (QUOTE (-552 (-772)))) (|HasCategory| (-117) (QUOTE (-72))) (-12 (|HasCategory| (-117) (QUOTE (-259 (-117)))) (|HasCategory| (-117) (QUOTE (-1012))))) +(-115 R Q A) ((|constructor| (NIL "CommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#3|) "\\spad{splitDenominator([q1,...,qn])} returns \\spad{[[p1,...,pn], d]} such that \\spad{qi = pi/d} and \\spad{d} is a common denominator for the \\spad{qi}'s.")) (|clearDenominator| ((|#3| |#3|) "\\spad{clearDenominator([q1,...,qn])} returns \\spad{[p1,...,pn]} such that \\spad{qi = pi/d} where \\spad{d} is a common denominator for the \\spad{qi}'s.")) (|commonDenominator| ((|#1| |#3|) "\\spad{commonDenominator([q1,...,qn])} returns a common denominator \\spad{d} for \\spad{q1},{}...,{}qn."))) NIL NIL -(-114) +(-116) ((|constructor| (NIL "Category for the usual combinatorial functions.")) (|permutation| (($ $ $) "\\spad{permutation(n, m)} returns the number of permutations of \\spad{n} objects taken \\spad{m} at a time. Note: \\spad{permutation(n,m) = n!/(n-m)!}.")) (|factorial| (($ $) "\\spad{factorial(n)} computes the factorial of \\spad{n} (denoted in the literature by \\spad{n!}) Note: \\spad{n! = n (n-1)! when n > 0}; also,{} \\spad{0! = 1}.")) (|binomial| (($ $ $) "\\spad{binomial(n,r)} returns the \\spad{(n,r)} binomial coefficient (often denoted in the literature by \\spad{C(n,r)}). Note: \\spad{C(n,r) = n!/(r!(n-r)!)} where \\spad{n >= r >= 0}."))) NIL NIL -(-115) +(-117) ((|constructor| (NIL "This domain provides the basic character data type.")) (|alphanumeric?| (((|Boolean|) $) "\\spad{alphanumeric?(c)} tests if \\spad{c} is either a letter or number,{} \\spadignore{i.e.} one of 0..9,{} a..\\spad{z} or A..\\spad{Z}.")) (|lowerCase?| (((|Boolean|) $) "\\spad{lowerCase?(c)} tests if \\spad{c} is an lower case letter,{} \\spadignore{i.e.} one of a..\\spad{z}.")) (|upperCase?| (((|Boolean|) $) "\\spad{upperCase?(c)} tests if \\spad{c} is an upper case letter,{} \\spadignore{i.e.} one of A..\\spad{Z}.")) (|alphabetic?| (((|Boolean|) $) "\\spad{alphabetic?(c)} tests if \\spad{c} is a letter,{} \\spadignore{i.e.} one of a..\\spad{z} or A..\\spad{Z}.")) (|hexDigit?| (((|Boolean|) $) "\\spad{hexDigit?(c)} tests if \\spad{c} is a hexadecimal numeral,{} \\spadignore{i.e.} one of 0..9,{} a..\\spad{f} or A..\\spad{F}.")) (|digit?| (((|Boolean|) $) "\\spad{digit?(c)} tests if \\spad{c} is a digit character,{} \\spadignore{i.e.} one of 0..9.")) (|lowerCase| (($ $) "\\spad{lowerCase(c)} converts an upper case letter to the corresponding lower case letter. If \\spad{c} is not an upper case letter,{} then it is returned unchanged.")) (|upperCase| (($ $) "\\spad{upperCase(c)} converts a lower case letter to the corresponding upper case letter. If \\spad{c} is not a lower case letter,{} then it is returned unchanged.")) (|escape| (($) "\\spad{escape} designate the escape character.")) (|verticalTab| (($) "\\spad{verticalTab} designates vertical tab.")) (|horizontalTab| (($) "\\spad{horizontalTab} designates horizontal tab.")) (|backspace| (($) "\\spad{backspace} designates the backspace character.")) (|formfeed| (($) "\\spad{formfeed} designates the form feed character.")) (|linefeed| (($) "\\spad{linefeed} designates the line feed character.")) (|carriageReturn| (($) "\\spad{carriageReturn} designates carriage return.")) (|newline| (($) "\\spad{newline} designates the new line character.")) (|underscore| (($) "\\spad{underscore} designates the underbar character.")) (|quote| (($) "\\spad{quote} provides the string quote character,{} \\spad{\"}.")) (|space| (($) "\\spad{space} provides the blank character.")) (|char| (($ (|String|)) "\\spad{char(s)} provides a character from a string \\spad{s} of length one.") (($ (|NonNegativeInteger|)) "\\spad{char(i)} provides a character corresponding to the integer code \\spad{i}. It is always \\spad{true} that \\spad{ord char i = i}.")) (|ord| (((|NonNegativeInteger|) $) "\\spad{ord(c)} provides an integral code corresponding to the character \\spad{c}. It is always \\spad{true} that \\spad{char ord c = c}."))) NIL NIL -(-116) +(-118) ((|constructor| (NIL "Rings of Characteristic Non Zero")) (|charthRoot| (((|Maybe| $) $) "\\spad{charthRoot(x)} returns the \\spad{p}th root of \\spad{x} where \\spad{p} is the characteristic of the ring."))) -((-3975 . T)) +((-3986 . T)) NIL -(-117 R) +(-119 R) ((|constructor| (NIL "This package provides a characteristicPolynomial function for any matrix over a commutative ring.")) (|characteristicPolynomial| ((|#1| (|Matrix| |#1|) |#1|) "\\spad{characteristicPolynomial(m,r)} computes the characteristic polynomial of the matrix \\spad{m} evaluated at the point \\spad{r}. In particular,{} if \\spad{r} is the polynomial 'x,{} then it returns the characteristic polynomial expressed as a polynomial in 'x."))) NIL NIL -(-118) +(-120) ((|constructor| (NIL "Rings of Characteristic Zero."))) -((-3975 . T)) +((-3986 . T)) NIL -(-119 -3078 UP UPUP) +(-121 -3088 UP UPUP) ((|constructor| (NIL "Tools to send a point to infinity on an algebraic curve.")) (|chvar| (((|Record| (|:| |func| |#3|) (|:| |poly| |#3|) (|:| |c1| (|Fraction| |#2|)) (|:| |c2| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) |#3| |#3|) "\\spad{chvar(f(x,y), p(x,y))} returns \\spad{[g(z,t), q(z,t), c1(z), c2(z), n]} such that under the change of variable \\spad{x = c1(z)},{} \\spad{y = t * c2(z)},{} one gets \\spad{f(x,y) = g(z,t)}. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x, y) = 0}. The algebraic relation between \\spad{z} and \\spad{t} is \\spad{q(z, t) = 0}.")) (|eval| ((|#3| |#3| (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{eval(p(x,y), f(x), g(x))} returns \\spad{p(f(x), y * g(x))}.")) (|goodPoint| ((|#1| |#3| |#3|) "\\spad{goodPoint(p, q)} returns an integer a such that a is neither a pole of \\spad{p(x,y)} nor a branch point of \\spad{q(x,y) = 0}.")) (|rootPoly| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| (|Fraction| |#2|)) (|:| |radicand| |#2|)) (|Fraction| |#2|) (|NonNegativeInteger|)) "\\spad{rootPoly(g, n)} returns \\spad{[m, c, P]} such that \\spad{c * g ** (1/n) = P ** (1/m)} thus if \\spad{y**n = g},{} then \\spad{z**m = P} where \\spad{z = c * y}.")) (|radPoly| (((|Union| (|Record| (|:| |radicand| (|Fraction| |#2|)) (|:| |deg| (|NonNegativeInteger|))) "failed") |#3|) "\\spad{radPoly(p(x, y))} returns \\spad{[c(x), n]} if \\spad{p} is of the form \\spad{y**n - c(x)},{} \"failed\" otherwise.")) (|mkIntegral| (((|Record| (|:| |coef| (|Fraction| |#2|)) (|:| |poly| |#3|)) |#3|) "\\spad{mkIntegral(p(x,y))} returns \\spad{[c(x), q(x,z)]} such that \\spad{z = c * y} is integral. The algebraic relation between \\spad{x} and \\spad{y} is \\spad{p(x, y) = 0}. The algebraic relation between \\spad{x} and \\spad{z} is \\spad{q(x, z) = 0}."))) NIL NIL -(-120 R CR) +(-122 R CR) ((|constructor| (NIL "This package provides the generalized euclidean algorithm which is needed as the basic step for factoring polynomials.")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| |#2|)) "failed") (|List| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} where (\\spad{fi} relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g} = sum \\spad{ai} prod fj (\\spad{j} \\= \\spad{i}) or equivalently g/prod fj = sum (ai/fi) or returns \"failed\" if no such list exists"))) NIL NIL -(-121 A S) +(-123 A S) ((|constructor| (NIL "A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \\spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \\spadignore{e.g.} \\spadfun{list} for \\spadtype{List},{} \\spadfun{flexibleArray} for \\spadtype{FlexibleArray},{} and so on.")) (|removeDuplicates| (($ $) "\\spad{removeDuplicates(u)} returns a copy of \\spad{u} with all duplicates removed.")) (|select| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select(p,u)} returns a copy of \\spad{u} containing only those elements such \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{select(\\spad{p},{}\\spad{u}) == [\\spad{x} for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})]}.")) (|remove| (($ |#2| $) "\\spad{remove(x,u)} returns a copy of \\spad{u} with all elements \\axiom{\\spad{y} = \\spad{x}} removed. Note: \\axiom{remove(\\spad{y},{}\\spad{c}) == [\\spad{x} for \\spad{x} in \\spad{c} | \\spad{x} ~= \\spad{y}]}.") (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove(p,u)} returns a copy of \\spad{u} removing all elements \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{remove(\\spad{p},{}\\spad{u}) == [\\spad{x} for \\spad{x} in \\spad{u} | not \\spad{p}(\\spad{x})]}.")) (|reduce| ((|#2| (|Mapping| |#2| |#2| |#2|) $ |#2| |#2|) "\\spad{reduce(f,u,x,z)} reduces the binary operation \\spad{f} across \\spad{u},{} stopping when an \"absorbing element\" \\spad{z} is encountered. As for \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})},{} \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})} when \\spad{u} contains no element \\spad{z}. Thus the third argument \\spad{x} is returned when \\spad{u} is empty.") ((|#2| (|Mapping| |#2| |#2| |#2|) $ |#2|) "\\spad{reduce(f,u,x)} reduces the binary operation \\spad{f} across \\spad{u},{} where \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u})} if \\spad{u} has 2 or more elements. Returns \\axiom{\\spad{f}(\\spad{x},{}\\spad{y})} if \\spad{u} has one element \\spad{y},{} \\spad{x} if \\spad{u} is empty. For example,{} \\axiom{reduce(+,{}\\spad{u},{}0)} returns the sum of the elements of \\spad{u}.") ((|#2| (|Mapping| |#2| |#2| |#2|) $) "\\spad{reduce(f,u)} reduces the binary operation \\spad{f} across \\spad{u}. For example,{} if \\spad{u} is \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]} then \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\axiom{\\spad{f}(..\\spad{f}(\\spad{f}(\\spad{x},{}\\spad{y}),{}...),{}\\spad{z})}. Note: if \\spad{u} has one element \\spad{x},{} \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\spad{x}. Error: if \\spad{u} is empty.")) (|find| (((|Union| |#2| "failed") (|Mapping| (|Boolean|) |#2|) $) "\\spad{find(p,u)} returns the first \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \"failed\" otherwise.")) (|construct| (($ (|List| |#2|)) "\\axiom{construct(\\spad{x},{}\\spad{y},{}...,{}\\spad{z})} returns the collection of elements \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}} ordered as given. Equivalently written as \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]\\$\\spad{D}},{} where \\spad{D} is the domain. \\spad{D} may be omitted for those of type List."))) NIL -((|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasAttribute| |#1| (QUOTE -3978))) -(-122 S) +((|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasAttribute| |#1| (QUOTE -3989))) +(-124 S) ((|constructor| (NIL "A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \\spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \\spadignore{e.g.} \\spadfun{list} for \\spadtype{List},{} \\spadfun{flexibleArray} for \\spadtype{FlexibleArray},{} and so on.")) (|removeDuplicates| (($ $) "\\spad{removeDuplicates(u)} returns a copy of \\spad{u} with all duplicates removed.")) (|select| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select(p,u)} returns a copy of \\spad{u} containing only those elements such \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{select(\\spad{p},{}\\spad{u}) == [\\spad{x} for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})]}.")) (|remove| (($ |#1| $) "\\spad{remove(x,u)} returns a copy of \\spad{u} with all elements \\axiom{\\spad{y} = \\spad{x}} removed. Note: \\axiom{remove(\\spad{y},{}\\spad{c}) == [\\spad{x} for \\spad{x} in \\spad{c} | \\spad{x} ~= \\spad{y}]}.") (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove(p,u)} returns a copy of \\spad{u} removing all elements \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. Note: \\axiom{remove(\\spad{p},{}\\spad{u}) == [\\spad{x} for \\spad{x} in \\spad{u} | not \\spad{p}(\\spad{x})]}.")) (|reduce| ((|#1| (|Mapping| |#1| |#1| |#1|) $ |#1| |#1|) "\\spad{reduce(f,u,x,z)} reduces the binary operation \\spad{f} across \\spad{u},{} stopping when an \"absorbing element\" \\spad{z} is encountered. As for \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})},{} \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u},{}\\spad{x})} when \\spad{u} contains no element \\spad{z}. Thus the third argument \\spad{x} is returned when \\spad{u} is empty.") ((|#1| (|Mapping| |#1| |#1| |#1|) $ |#1|) "\\spad{reduce(f,u,x)} reduces the binary operation \\spad{f} across \\spad{u},{} where \\spad{x} is the identity operation of \\spad{f}. Same as \\axiom{reduce(\\spad{f},{}\\spad{u})} if \\spad{u} has 2 or more elements. Returns \\axiom{\\spad{f}(\\spad{x},{}\\spad{y})} if \\spad{u} has one element \\spad{y},{} \\spad{x} if \\spad{u} is empty. For example,{} \\axiom{reduce(+,{}\\spad{u},{}0)} returns the sum of the elements of \\spad{u}.") ((|#1| (|Mapping| |#1| |#1| |#1|) $) "\\spad{reduce(f,u)} reduces the binary operation \\spad{f} across \\spad{u}. For example,{} if \\spad{u} is \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]} then \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\axiom{\\spad{f}(..\\spad{f}(\\spad{f}(\\spad{x},{}\\spad{y}),{}...),{}\\spad{z})}. Note: if \\spad{u} has one element \\spad{x},{} \\axiom{reduce(\\spad{f},{}\\spad{u})} returns \\spad{x}. Error: if \\spad{u} is empty.")) (|find| (((|Union| |#1| "failed") (|Mapping| (|Boolean|) |#1|) $) "\\spad{find(p,u)} returns the first \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \"failed\" otherwise.")) (|construct| (($ (|List| |#1|)) "\\axiom{construct(\\spad{x},{}\\spad{y},{}...,{}\\spad{z})} returns the collection of elements \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}} ordered as given. Equivalently written as \\axiom{[\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]\\$\\spad{D}},{} where \\spad{D} is the domain. \\spad{D} may be omitted for those of type List."))) NIL NIL -(-123 |n| K Q) +(-125 |n| K Q) ((|constructor| (NIL "CliffordAlgebra(\\spad{n},{} \\spad{K},{} \\spad{Q}) defines a vector space of dimension \\spad{2**n} over \\spad{K},{} given a quadratic form \\spad{Q} on \\spad{K**n}. \\blankline If \\spad{e[i]},{} \\spad{1<=i<=n} is a basis for \\spad{K**n} then \\indented{3}{1,{} \\spad{e[i]} (\\spad{1<=i<=n}),{} \\spad{e[i1]*e[i2]}} (\\spad{1<=i1<i2<=n}),{}...,{}\\spad{e[1]*e[2]*..*e[n]} is a basis for the Clifford Algebra. \\blankline The algebra is defined by the relations \\indented{3}{\\spad{e[i]*e[j] = -e[j]*e[i]}\\space{2}(\\spad{i \\~~= j}),{}} \\indented{3}{\\spad{e[i]*e[i] = Q(e[i])}} \\blankline Examples of Clifford Algebras are: gaussians,{} quaternions,{} exterior algebras and spin algebras.")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} computes the multiplicative inverse of \\spad{x} or \"failed\" if \\spad{x} is not invertible.")) (|coefficient| ((|#2| $ (|List| (|PositiveInteger|))) "\\spad{coefficient(x,[i1,i2,...,iN])} extracts the coefficient of \\spad{e(i1)*e(i2)*...*e(iN)} in \\spad{x}.")) (|monomial| (($ |#2| (|List| (|PositiveInteger|))) "\\spad{monomial(c,[i1,i2,...,iN])} produces the value given by \\spad{c*e(i1)*e(i2)*...*e(iN)}.")) (|e| (($ (|PositiveInteger|)) "\\spad{e(n)} produces the appropriate unit element."))) -((-3973 . T) (-3972 . T) (-3975 . T)) +((-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-124) +(-126) ((|constructor| (NIL "\\indented{1}{The purpose of this package is to provide reasonable plots of} functions with singularities.")) (|clipWithRanges| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|)))) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{clipWithRanges(pointLists,xMin,xMax,yMin,yMax)} performs clipping on a list of lists of points,{} \\spad{pointLists}. Clipping is done within the specified ranges of \\spad{xMin},{} \\spad{xMax} and \\spad{yMin},{} \\spad{yMax}. This function is used internally by the \\fakeAxiomFun{iClipParametric} subroutine in this package.")) (|clipParametric| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|) (|Fraction| (|Integer|)) (|Fraction| (|Integer|))) "\\spad{clipParametric(p,frac,sc)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)}; the fraction parameter is specified by \\spad{frac} and the scale parameter is specified by \\spad{sc} for use in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|)) "\\spad{clipParametric(p)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)}; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.")) (|clip| (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{clip(ll)} performs two-dimensional clipping on a list of lists of points,{} \\spad{ll}; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|List| (|Point| (|DoubleFloat|)))) "\\spad{clip(l)} performs two-dimensional clipping on a curve \\spad{l},{} which is a list of points; the default parameters \\spad{1/2} for the fraction and \\spad{5/1} for the scale are used in the \\fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|) (|Fraction| (|Integer|)) (|Fraction| (|Integer|))) "\\spad{clip(p,frac,sc)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the graph of one variable \\spad{y = f(x)}; the fraction parameter is specified by \\spad{frac} and the scale parameter is specified by \\spad{sc} for use in the \\spadfun{clip} function.") (((|Record| (|:| |brans| (|List| (|List| (|Point| (|DoubleFloat|))))) (|:| |xValues| (|Segment| (|DoubleFloat|))) (|:| |yValues| (|Segment| (|DoubleFloat|)))) (|Plot|)) "\\spad{clip(p)} performs two-dimensional clipping on a plot,{} \\spad{p},{} from the domain \\spadtype{Plot} for the graph of one variable,{} \\spad{y = f(x)}; the default parameters \\spad{1/4} for the fraction and \\spad{5/1} for the scale are used in the \\spadfun{clip} function."))) NIL NIL -(-125) +(-127) ((|constructor| (NIL "This domain represents list comprehension syntax.")) (|body| (((|SpadAst|) $) "\\spad{body(e)} return the expression being collected by the list comprehension `e'.")) (|iterators| (((|List| (|SpadAst|)) $) "\\spad{iterators(e)} returns the list of the iterators of the list comprehension `e'."))) NIL NIL -(-126 UP |Par|) +(-128 UP |Par|) ((|complexZeros| (((|List| (|Complex| |#2|)) |#1| |#2|) "\\spad{complexZeros(poly, eps)} finds the complex zeros of the univariate polynomial \\spad{poly} to precision eps with solutions returned as complex floats or rationals depending on the type of eps."))) NIL NIL -(-127) +(-129) ((|constructor| (NIL "This domain represents type specification \\indented{2}{for an identifier or expression.}")) (|rhs| (((|TypeAst|) $) "\\spad{rhs(e)} returns the right hand side of the colon expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the colon expression `e'."))) NIL NIL -(-128) +(-130) ((|constructor| (NIL "Color() specifies a domain of 27 colors provided in the \\Language{} system (the colors mix additively).")) (|color| (($ (|Integer|)) "\\spad{color(i)} returns a color of the indicated hue \\spad{i}.")) (|numberOfHues| (((|PositiveInteger|)) "\\spad{numberOfHues()} returns the number of total hues,{} set in totalHues.")) (|hue| (((|Integer|) $) "\\spad{hue(c)} returns the hue index of the indicated color \\spad{c}.")) (|blue| (($) "\\spad{blue()} returns the position of the blue hue from total hues.")) (|green| (($) "\\spad{green()} returns the position of the green hue from total hues.")) (|yellow| (($) "\\spad{yellow()} returns the position of the yellow hue from total hues.")) (|red| (($) "\\spad{red()} returns the position of the red hue from total hues.")) (+ (($ $ $) "\\spad{c1 + c2} additively mixes the two colors \\spad{c1} and \\spad{c2}.")) (* (($ (|DoubleFloat|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}.") (($ (|PositiveInteger|) $) "\\spad{s * c},{} returns the color \\spad{c},{} whose weighted shade has been scaled by \\spad{s}."))) NIL NIL -(-129 R -3078) +(-131 R -3088) ((|constructor| (NIL "Provides combinatorial functions over an integral domain.")) (|ipow| ((|#2| (|List| |#2|)) "\\spad{ipow(l)} should be local but conditional.")) (|iidprod| ((|#2| (|List| |#2|)) "\\spad{iidprod(l)} should be local but conditional.")) (|iidsum| ((|#2| (|List| |#2|)) "\\spad{iidsum(l)} should be local but conditional.")) (|iipow| ((|#2| (|List| |#2|)) "\\spad{iipow(l)} should be local but conditional.")) (|iiperm| ((|#2| (|List| |#2|)) "\\spad{iiperm(l)} should be local but conditional.")) (|iibinom| ((|#2| (|List| |#2|)) "\\spad{iibinom(l)} should be local but conditional.")) (|iifact| ((|#2| |#2|) "\\spad{iifact(x)} should be local but conditional.")) (|product| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{product(f(n), n = a..b)} returns \\spad{f}(a) * ... * \\spad{f}(\\spad{b}) as a formal product.") ((|#2| |#2| (|Symbol|)) "\\spad{product(f(n), n)} returns the formal product \\spad{P}(\\spad{n}) which verifies \\spad{P}(\\spad{n+1})/P(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|summation| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{summation(f(n), n = a..b)} returns \\spad{f}(a) + ... + \\spad{f}(\\spad{b}) as a formal sum.") ((|#2| |#2| (|Symbol|)) "\\spad{summation(f(n), n)} returns the formal sum \\spad{S}(\\spad{n}) which verifies \\spad{S}(\\spad{n+1}) - \\spad{S}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|factorials| ((|#2| |#2| (|Symbol|)) "\\spad{factorials(f, x)} rewrites the permutations and binomials in \\spad{f} involving \\spad{x} in terms of factorials.") ((|#2| |#2|) "\\spad{factorials(f)} rewrites the permutations and binomials in \\spad{f} in terms of factorials.")) (|factorial| ((|#2| |#2|) "\\spad{factorial(n)} returns the factorial of \\spad{n},{} \\spadignore{i.e.} n!.")) (|permutation| ((|#2| |#2| |#2|) "\\spad{permutation(n, r)} returns the number of permutations of \\spad{n} objects taken \\spad{r} at a time,{} \\spadignore{i.e.} n!/(\\spad{n}-\\spad{r})!.")) (|binomial| ((|#2| |#2| |#2|) "\\spad{binomial(n, r)} returns the number of subsets of \\spad{r} objects taken among \\spad{n} objects,{} \\spadignore{i.e.} n!/(r! * (\\spad{n}-\\spad{r})!).")) (** ((|#2| |#2| |#2|) "\\spad{a ** b} is the formal exponential a**b.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}; error if \\spad{op} is not a combinatorial operator.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is a combinatorial operator."))) NIL NIL -(-130 I) +(-132 I) ((|stirling2| ((|#1| |#1| |#1|) "\\spad{stirling2(n,m)} returns the Stirling number of the second kind denoted \\spad{SS[n,m]}.")) (|stirling1| ((|#1| |#1| |#1|) "\\spad{stirling1(n,m)} returns the Stirling number of the first kind denoted \\spad{S[n,m]}.")) (|permutation| ((|#1| |#1| |#1|) "\\spad{permutation(n)} returns \\spad{!P(n,r) = n!/(n-r)!}. This is the number of permutations of \\spad{n} objects taken \\spad{r} at a time.")) (|partition| ((|#1| |#1|) "\\spad{partition(n)} returns the number of partitions of the integer \\spad{n}. This is the number of distinct ways that \\spad{n} can be written as a sum of positive integers.")) (|multinomial| ((|#1| |#1| (|List| |#1|)) "\\spad{multinomial(n,[m1,m2,...,mk])} returns the multinomial coefficient \\spad{n!/(m1! m2! ... mk!)}.")) (|factorial| ((|#1| |#1|) "\\spad{factorial(n)} returns \\spad{n!}. this is the product of all integers between 1 and \\spad{n} (inclusive). Note: \\spad{0!} is defined to be 1.")) (|binomial| ((|#1| |#1| |#1|) "\\spad{binomial(n,r)} returns the binomial coefficient \\spad{C(n,r) = n!/(r! (n-r)!)},{} where \\spad{n >= r >= 0}. This is the number of combinations of \\spad{n} objects taken \\spad{r} at a time."))) NIL NIL -(-131) +(-133) ((|constructor| (NIL "CombinatorialOpsCategory is the category obtaining by adjoining summations and products to the usual combinatorial operations.")) (|product| (($ $ (|SegmentBinding| $)) "\\spad{product(f(n), n = a..b)} returns \\spad{f}(a) * ... * \\spad{f}(\\spad{b}) as a formal product.") (($ $ (|Symbol|)) "\\spad{product(f(n), n)} returns the formal product \\spad{P}(\\spad{n}) which verifies \\spad{P}(\\spad{n+1})/P(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|summation| (($ $ (|SegmentBinding| $)) "\\spad{summation(f(n), n = a..b)} returns \\spad{f}(a) + ... + \\spad{f}(\\spad{b}) as a formal sum.") (($ $ (|Symbol|)) "\\spad{summation(f(n), n)} returns the formal sum \\spad{S}(\\spad{n}) which verifies \\spad{S}(\\spad{n+1}) - \\spad{S}(\\spad{n}) = \\spad{f}(\\spad{n}).")) (|factorials| (($ $ (|Symbol|)) "\\spad{factorials(f, x)} rewrites the permutations and binomials in \\spad{f} involving \\spad{x} in terms of factorials.") (($ $) "\\spad{factorials(f)} rewrites the permutations and binomials in \\spad{f} in terms of factorials."))) NIL NIL -(-132) +(-134) ((|constructor| (NIL "A type for basic commutators")) (|mkcomm| (($ $ $) "\\spad{mkcomm(i,j)} \\undocumented{}") (($ (|Integer|)) "\\spad{mkcomm(i)} \\undocumented{}"))) NIL NIL -(-133) +(-135) ((|constructor| (NIL "This domain represents the syntax of a comma-separated \\indented{2}{list of expressions.}")) (|body| (((|List| (|SpadAst|)) $) "\\spad{body(e)} returns the list of expressions making up `e'."))) NIL NIL -(-134) +(-136) ((|constructor| (NIL "This package exports the elementary operators,{} with some semantics already attached to them. The semantics that is attached here is not dependent on the set in which the operators will be applied.")) (|operator| (((|BasicOperator|) (|Symbol|)) "\\spad{operator(s)} returns an operator with name \\spad{s},{} with the appropriate semantics if \\spad{s} is known. If \\spad{s} is not known,{} the result has no semantics."))) NIL NIL -(-135 R UP UPUP) +(-137 R UP UPUP) ((|constructor| (NIL "A package for swapping the order of two variables in a tower of two UnivariatePolynomialCategory extensions.")) (|swap| ((|#3| |#3|) "\\spad{swap(p(x,y))} returns \\spad{p}(\\spad{y},{}\\spad{x})."))) NIL NIL -(-136 S R) +(-138 S R) ((|constructor| (NIL "This category represents the extension of a ring by a square root of \\spad{-1}.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} or \"failed\" if \\spad{x} is not a rational number.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a rational number.")) (|polarCoordinates| (((|Record| (|:| |r| |#2|) (|:| |phi| |#2|)) $) "\\spad{polarCoordinates(x)} returns (\\spad{r},{} phi) such that \\spad{x} = \\spad{r} * exp(\\%\\spad{i} * phi).")) (|argument| ((|#2| $) "\\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\\spad{x}).")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x} = sqrt(norm(\\spad{x})).")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(x, r)} returns the exact quotient of \\spad{x} by \\spad{r},{} or \"failed\" if \\spad{r} does not divide \\spad{x} exactly.")) (|norm| ((|#2| $) "\\spad{norm(x)} returns \\spad{x} * conjugate(\\spad{x})")) (|real| ((|#2| $) "\\spad{real(x)} returns real part of \\spad{x}.")) (|imag| ((|#2| $) "\\spad{imag(x)} returns imaginary part of \\spad{x}.")) (|conjugate| (($ $) "\\spad{conjugate(x + \\%i y)} returns \\spad{x} - \\%\\spad{i} \\spad{y}.")) (|imaginary| (($) "\\spad{imaginary()} = sqrt(\\spad{-1}) = \\%\\spad{i}.")) (|complex| (($ |#2| |#2|) "\\spad{complex(x,y)} constructs \\spad{x} + \\%i*y.") 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(|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} or \"failed\" if \\spad{x} is not a rational number.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a rational number.")) (|polarCoordinates| (((|Record| (|:| |r| |#1|) (|:| |phi| |#1|)) $) "\\spad{polarCoordinates(x)} returns (\\spad{r},{} phi) such that \\spad{x} = \\spad{r} * exp(\\%\\spad{i} * phi).")) (|argument| ((|#1| $) "\\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\\spad{x}).")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x} = sqrt(norm(\\spad{x})).")) 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T)) +((-3982 OR (|has| |#1| (-494)) (-12 (|has| |#1| (-257)) (|has| |#1| (-821)))) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3985 |has| |#1| (-6 -3985)) (-3988 |has| |#1| (-6 -3988)) (-1373 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-138 RR PR) +(-140 RR PR) ((|constructor| (NIL "\\indented{1}{Author:} Date Created: Date Last Updated: Basic Functions: Related Constructors: Complex,{} UnivariatePolynomial Also See: AMS Classifications: Keywords: complex,{} polynomial factorization,{} factor References:")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} factorizes the polynomial \\spad{p} with complex coefficients."))) NIL NIL -(-139) +(-141) ((|constructor| (NIL "This package implements a Spad compiler.")) (|elaborate| (((|Maybe| (|Elaboration|)) (|SpadAst|)) "\\spad{elaborate(s)} returns the elaboration of the syntax object \\spad{s} in the empty environement.")) 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T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . 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(|map| (((|Complex| |#2|) (|Mapping| |#2| |#1|) (|Complex| |#1|)) "\\spad{map(f,u)} maps \\spad{f} onto real and imaginary parts of \\spad{u}."))) NIL NIL -(-142 R S CS) +(-144 R S CS) ((|constructor| (NIL "This package supports converting complex expressions to patterns")) (|convert| (((|Pattern| |#1|) |#3|) "\\spad{convert(cs)} converts the complex expression \\spad{cs} to a pattern"))) NIL NIL -(-143) +(-145) ((|constructor| (NIL "This domain implements some global properties of subspaces.")) (|copy| (($ $) "\\spad{copy(x)} \\undocumented")) (|solid| (((|Boolean|) $ (|Boolean|)) "\\spad{solid(x,b)} \\undocumented")) (|close| (((|Boolean|) $ (|Boolean|)) "\\spad{close(x,b)} \\undocumented")) (|solid?| (((|Boolean|) $) "\\spad{solid?(x)} \\undocumented")) (|closed?| (((|Boolean|) $) "\\spad{closed?(x)} \\undocumented")) (|new| (($) "\\spad{new()} \\undocumented"))) NIL NIL -(-144) +(-146) ((|constructor| (NIL "The category of commutative rings with unity,{} \\spadignore{i.e.} rings where \\spadop{*} is commutative,{} and which have a multiplicative identity. element.")) (|commutative| ((|attribute| "*") "multiplication is commutative."))) -(((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-145) +(-147) ((|constructor| (NIL "This category is the root of the I/O conduits.")) (|close!| (($ $) "\\spad{close!(c)} closes the conduit \\spad{c},{} changing its state to one that is invalid for future read or write operations."))) NIL NIL -(-146 R) +(-148 R) ((|constructor| (NIL "\\spadtype{ContinuedFraction} implements general \\indented{1}{continued fractions.\\space{2}This version is not restricted to simple,{}} \\indented{1}{finite fractions and uses the \\spadtype{Stream} as a} \\indented{1}{representation.\\space{2}The arithmetic functions assume that the} \\indented{1}{approximants alternate below/above the convergence point.} \\indented{1}{This is enforced by ensuring the partial numerators and partial} \\indented{1}{denominators are greater than 0 in the Euclidean domain view of \\spad{R}} \\indented{1}{(\\spadignore{i.e.} \\spad{sizeLess?(0, x)}).}")) (|complete| (($ $) "\\spad{complete(x)} causes all entries in \\spadvar{\\spad{x}} to be computed. Normally entries are only computed as needed. If \\spadvar{\\spad{x}} is an infinite continued fraction,{} a user-initiated interrupt is necessary to stop the computation.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(x,n)} causes the first \\spadvar{\\spad{n}} entries in the continued fraction \\spadvar{\\spad{x}} to be computed. Normally entries are only computed as needed.")) (|denominators| (((|Stream| |#1|) $) "\\spad{denominators(x)} returns the stream of denominators of the approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|numerators| (((|Stream| |#1|) $) "\\spad{numerators(x)} returns the stream of numerators of the approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|convergents| (((|Stream| (|Fraction| |#1|)) $) "\\spad{convergents(x)} returns the stream of the convergents of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.")) (|approximants| (((|Stream| (|Fraction| |#1|)) $) "\\spad{approximants(x)} returns the stream of approximants of the continued fraction \\spadvar{\\spad{x}}. If the continued fraction is finite,{} then the stream will be infinite and periodic with period 1.")) (|reducedForm| (($ $) "\\spad{reducedForm(x)} puts the continued fraction \\spadvar{\\spad{x}} in reduced form,{} \\spadignore{i.e.} the function returns an equivalent continued fraction of the form \\spad{continuedFraction(b0,[1,1,1,...],[b1,b2,b3,...])}.")) (|wholePart| ((|#1| $) "\\spad{wholePart(x)} extracts the whole part of \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{wholePart(x) = b0}.")) (|partialQuotients| (((|Stream| |#1|) $) "\\spad{partialQuotients(x)} extracts the partial quotients in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{partialQuotients(x) = [b0,b1,b2,b3,...]}.")) (|partialDenominators| (((|Stream| |#1|) $) "\\spad{partialDenominators(x)} extracts the denominators in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{partialDenominators(x) = [b1,b2,b3,...]}.")) (|partialNumerators| (((|Stream| |#1|) $) "\\spad{partialNumerators(x)} extracts the numerators in \\spadvar{\\spad{x}}. That is,{} if \\spad{x = continuedFraction(b0, [a1,a2,a3,...], [b1,b2,b3,...])},{} then \\spad{partialNumerators(x) = [a1,a2,a3,...]}.")) (|reducedContinuedFraction| (($ |#1| (|Stream| |#1|)) "\\spad{reducedContinuedFraction(b0,b)} constructs a continued fraction in the following way: if \\spad{b = [b1,b2,...]} then the result is the continued fraction \\spad{b0 + 1/(b1 + 1/(b2 + ...))}. That is,{} the result is the same as \\spad{continuedFraction(b0,[1,1,1,...],[b1,b2,b3,...])}.")) (|continuedFraction| (($ |#1| (|Stream| |#1|) (|Stream| |#1|)) "\\spad{continuedFraction(b0,a,b)} constructs a continued fraction in the following way: if \\spad{a = [a1,a2,...]} and \\spad{b = [b1,b2,...]} then the result is the continued fraction \\spad{b0 + a1/(b1 + a2/(b2 + ...))}.") (($ (|Fraction| |#1|)) "\\spad{continuedFraction(r)} converts the fraction \\spadvar{\\spad{r}} with components of type \\spad{R} to a continued fraction over \\spad{R}."))) -(((-3980 "*") . T) (-3971 . T) (-3976 . T) (-3970 . T) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") . T) (-3982 . T) (-3987 . T) (-3981 . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-147) +(-149) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. A `Contour' a list of bindings making up a `virtual scope'.")) (|findBinding| (((|Maybe| (|Binding|)) (|Identifier|) $) "\\spad{findBinding(c,n)} returns the first binding associated with `n'. Otherwise `nothing.")) (|push| (($ (|Binding|) $) "\\spad{push(c,b)} augments the contour with binding `b'.")) (|bindings| (((|List| (|Binding|)) $) "\\spad{bindings(c)} returns the list of bindings in countour \\spad{c}."))) NIL NIL -(-148 R) +(-150 R) ((|constructor| (NIL "CoordinateSystems provides coordinate transformation functions for plotting. Functions in this package return conversion functions which take points expressed in other coordinate systems and return points with the corresponding Cartesian coordinates.")) (|conical| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1| |#1|) "\\spad{conical(a,b)} transforms from conical coordinates to Cartesian coordinates: \\spad{conical(a,b)} is a function which will map the point \\spad{(lambda,mu,nu)} to \\spad{x = lambda*mu*nu/(a*b)},{} \\spad{y = lambda/a*sqrt((mu**2-a**2)*(nu**2-a**2)/(a**2-b**2))},{} \\spad{z = lambda/b*sqrt((mu**2-b**2)*(nu**2-b**2)/(b**2-a**2))}.")) (|toroidal| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{toroidal(a)} transforms from toroidal coordinates to Cartesian coordinates: \\spad{toroidal(a)} is a function which will map the point \\spad{(u,v,phi)} to \\spad{x = a*sinh(v)*cos(phi)/(cosh(v)-cos(u))},{} \\spad{y = a*sinh(v)*sin(phi)/(cosh(v)-cos(u))},{} \\spad{z = a*sin(u)/(cosh(v)-cos(u))}.")) (|bipolarCylindrical| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{bipolarCylindrical(a)} transforms from bipolar cylindrical coordinates to Cartesian coordinates: \\spad{bipolarCylindrical(a)} is a function which will map the point \\spad{(u,v,z)} to \\spad{x = a*sinh(v)/(cosh(v)-cos(u))},{} \\spad{y = a*sin(u)/(cosh(v)-cos(u))},{} \\spad{z}.")) (|bipolar| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{bipolar(a)} transforms from bipolar coordinates to Cartesian coordinates: \\spad{bipolar(a)} is a function which will map the point \\spad{(u,v)} to \\spad{x = a*sinh(v)/(cosh(v)-cos(u))},{} \\spad{y = a*sin(u)/(cosh(v)-cos(u))}.")) (|oblateSpheroidal| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{oblateSpheroidal(a)} transforms from oblate spheroidal coordinates to Cartesian coordinates: \\spad{oblateSpheroidal(a)} is a function which will map the point \\spad{(xi,eta,phi)} to \\spad{x = a*sinh(xi)*sin(eta)*cos(phi)},{} \\spad{y = a*sinh(xi)*sin(eta)*sin(phi)},{} \\spad{z = a*cosh(xi)*cos(eta)}.")) (|prolateSpheroidal| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{prolateSpheroidal(a)} transforms from prolate spheroidal coordinates to Cartesian coordinates: \\spad{prolateSpheroidal(a)} is a function which will map the point \\spad{(xi,eta,phi)} to \\spad{x = a*sinh(xi)*sin(eta)*cos(phi)},{} \\spad{y = a*sinh(xi)*sin(eta)*sin(phi)},{} \\spad{z = a*cosh(xi)*cos(eta)}.")) (|ellipticCylindrical| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{ellipticCylindrical(a)} transforms from elliptic cylindrical coordinates to Cartesian coordinates: \\spad{ellipticCylindrical(a)} is a function which will map the point \\spad{(u,v,z)} to \\spad{x = a*cosh(u)*cos(v)},{} \\spad{y = a*sinh(u)*sin(v)},{} \\spad{z}.")) (|elliptic| (((|Mapping| (|Point| |#1|) (|Point| |#1|)) |#1|) "\\spad{elliptic(a)} transforms from elliptic coordinates to Cartesian coordinates: \\spad{elliptic(a)} is a function which will map the point \\spad{(u,v)} to \\spad{x = a*cosh(u)*cos(v)},{} \\spad{y = a*sinh(u)*sin(v)}.")) (|paraboloidal| (((|Point| |#1|) (|Point| |#1|)) "\\spad{paraboloidal(pt)} transforms \\spad{pt} from paraboloidal coordinates to Cartesian coordinates: the function produced will map the point \\spad{(u,v,phi)} to \\spad{x = u*v*cos(phi)},{} \\spad{y = u*v*sin(phi)},{} \\spad{z = 1/2 * (u**2 - v**2)}.")) (|parabolicCylindrical| (((|Point| |#1|) (|Point| |#1|)) "\\spad{parabolicCylindrical(pt)} transforms \\spad{pt} from parabolic cylindrical coordinates to Cartesian coordinates: the function produced will map the point \\spad{(u,v,z)} to \\spad{x = 1/2*(u**2 - v**2)},{} \\spad{y = u*v},{} \\spad{z}.")) (|parabolic| (((|Point| |#1|) (|Point| |#1|)) "\\spad{parabolic(pt)} transforms \\spad{pt} from parabolic coordinates to Cartesian coordinates: the function produced will map the point \\spad{(u,v)} to \\spad{x = 1/2*(u**2 - v**2)},{} \\spad{y = u*v}.")) (|spherical| (((|Point| |#1|) (|Point| |#1|)) "\\spad{spherical(pt)} transforms \\spad{pt} from spherical coordinates to Cartesian coordinates: the function produced will map the point \\spad{(r,theta,phi)} to \\spad{x = r*sin(phi)*cos(theta)},{} \\spad{y = r*sin(phi)*sin(theta)},{} \\spad{z = r*cos(phi)}.")) (|cylindrical| (((|Point| |#1|) (|Point| |#1|)) "\\spad{cylindrical(pt)} transforms \\spad{pt} from polar coordinates to Cartesian coordinates: the function produced will map the point \\spad{(r,theta,z)} to \\spad{x = r * cos(theta)},{} \\spad{y = r * sin(theta)},{} \\spad{z}.")) (|polar| (((|Point| |#1|) (|Point| |#1|)) "\\spad{polar(pt)} transforms \\spad{pt} from polar coordinates to Cartesian coordinates: the function produced will map the point \\spad{(r,theta)} to \\spad{x = r * cos(theta)} ,{} \\spad{y = r * sin(theta)}.")) (|cartesian| (((|Point| |#1|) (|Point| |#1|)) "\\spad{cartesian(pt)} returns the Cartesian coordinates of point \\spad{pt}."))) NIL NIL -(-149 R |PolR| E) +(-151 R |PolR| E) ((|constructor| (NIL "This package implements characteristicPolynomials for monogenic algebras using resultants")) (|characteristicPolynomial| ((|#2| |#3|) "\\spad{characteristicPolynomial(e)} returns the characteristic polynomial of \\spad{e} using resultants"))) NIL NIL -(-150 R S CS) +(-152 R S CS) ((|constructor| (NIL "This package supports matching patterns involving complex expressions")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(cexpr, pat, res)} matches the pattern \\spad{pat} to the complex expression \\spad{cexpr}. res contains the variables of \\spad{pat} which are already matched and their matches."))) NIL -((|HasCategory| (-852 |#2|) (|%list| (QUOTE -791) (|devaluate| |#1|)))) -(-151 R) +((|HasCategory| (-857 |#2|) (|%list| (QUOTE -796) (|devaluate| |#1|)))) +(-153 R) ((|constructor| (NIL "This package \\undocumented{}")) (|multiEuclideanTree| (((|List| |#1|) (|List| |#1|) |#1|) "\\spad{multiEuclideanTree(l,r)} \\undocumented{}")) (|chineseRemainder| (((|List| |#1|) (|List| (|List| |#1|)) (|List| |#1|)) "\\spad{chineseRemainder(llv,lm)} returns a list of values,{} each of which corresponds to the Chinese remainder of the associated element of \\axiom{\\spad{llv}} and axiom{\\spad{lm}}. This is more efficient than applying chineseRemainder several times.") ((|#1| (|List| |#1|) (|List| |#1|)) "\\spad{chineseRemainder(lv,lm)} returns a value \\axiom{\\spad{v}} such that,{} if \\spad{x} is \\axiom{\\spad{lv}.\\spad{i}} modulo \\axiom{\\spad{lm}.\\spad{i}} for all \\axiom{\\spad{i}},{} then \\spad{x} is \\axiom{\\spad{v}} modulo \\axiom{\\spad{lm}(1)*lm(2)*...*lm(\\spad{n})}.")) (|modTree| (((|List| |#1|) |#1| (|List| |#1|)) "\\spad{modTree(r,l)} \\undocumented{}"))) NIL NIL -(-152) +(-154) ((|constructor| (NIL "This domain represents `coerce' expressions.")) (|target| (((|TypeAst|) $) "\\spad{target(e)} returns the target type of the conversion..")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression being converted."))) NIL NIL -(-153 R UP) +(-155 R UP) ((|constructor| (NIL "\\spadtype{ComplexRootFindingPackage} provides functions to find all roots of a polynomial \\spad{p} over the complex number by using Plesken's idea to calculate in the polynomial ring modulo \\spad{f} and employing the Chinese Remainder Theorem. In this first version,{} the precision (see \\spadfunFrom{digits}{Float}) is not increased when this is necessary to avoid rounding errors. Hence it is the user's responsibility to increase the precision if necessary. Note also,{} if this package is called with \\spadignore{e.g.} \\spadtype{Fraction Integer},{} the precise calculations could require a lot of time. Also note that evaluating the zeros is not necessarily a good check whether the result is correct: already evaluation can cause rounding errors.")) (|startPolynomial| (((|Record| (|:| |start| |#2|) (|:| |factors| (|Factored| |#2|))) |#2|) "\\spad{startPolynomial(p)} uses the ideas of Schoenhage's variant of Graeffe's method to construct circles which separate roots to get a good start polynomial,{} \\spadignore{i.e.} one whose image under the Chinese Remainder Isomorphism has both entries of norm smaller and greater or equal to 1. In case the roots are found during internal calculations. The corresponding factors are in {\\em factors} which are otherwise 1.")) (|setErrorBound| ((|#1| |#1|) "\\spad{setErrorBound(eps)} changes the internal error bound,{} by default being {\\em 10 ** (-3)} to \\spad{eps},{} if \\spad{R} is a member in the category \\spadtype{QuotientFieldCategory Integer}. The internal {\\em globalDigits} is set to {\\em ceiling(1/r)**2*10} being {\\em 10**7} by default.")) (|schwerpunkt| (((|Complex| |#1|) |#2|) "\\spad{schwerpunkt(p)} determines the 'Schwerpunkt' of the roots of the polynomial \\spad{p} of degree \\spad{n},{} \\spadignore{i.e.} the center of gravity,{} which is {\\em coeffient of \\spad{x**(n-1)}} divided by {\\em n times coefficient of \\spad{x**n}}.")) (|rootRadius| ((|#1| |#2|) "\\spad{rootRadius(p)} calculates the root radius of \\spad{p} with a maximal error quotient of {\\em 1+globalEps},{} where {\\em globalEps} is the internal error bound,{} which can be set by {\\em setErrorBound}.") ((|#1| |#2| |#1|) "\\spad{rootRadius(p,errQuot)} calculates the root radius of \\spad{p} with a maximal error quotient of {\\em errQuot}.")) (|reciprocalPolynomial| ((|#2| |#2|) "\\spad{reciprocalPolynomial(p)} calulates a polynomial which has exactly the inverses of the non-zero roots of \\spad{p} as roots,{} and the same number of 0-roots.")) (|pleskenSplit| (((|Factored| |#2|) |#2| |#1|) "\\spad{pleskenSplit(poly, eps)} determines a start polynomial {\\em start}\\\\ by using \"startPolynomial then it increases the exponent \\spad{n} of {\\em start ** n mod poly} to get an approximate factor of {\\em poly},{} in general of degree \"degree \\spad{poly} -1\". Then a divisor cascade is calculated and the best splitting is chosen,{} as soon as the error is small enough.") (((|Factored| |#2|) |#2| |#1| (|Boolean|)) "\\spad{pleskenSplit(poly,eps,info)} determines a start polynomial {\\em start} by using \"startPolynomial then it increases the exponent \\spad{n} of {\\em start ** n mod poly} to get an approximate factor of {\\em poly},{} in general of degree \"degree \\spad{poly} -1\". Then a divisor cascade is calculated and the best splitting is chosen,{} as soon as the error is small enough. If {\\em info} is {\\em true},{} then information messages are issued.")) (|norm| ((|#1| |#2|) "\\spad{norm(p)} determines sum of absolute values of coefficients Note: this function depends on \\spadfunFrom{abs}{Complex}.")) (|graeffe| ((|#2| |#2|) "\\spad{graeffe p} determines \\spad{q} such that \\spad{q(-z**2) = p(z)*p(-z)}. Note that the roots of \\spad{q} are the squares of the roots of \\spad{p}.")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} tries to factor \\spad{p} into linear factors with error atmost {\\em globalEps},{} the internal error bound,{} which can be set by {\\em setErrorBound}. An overall error bound {\\em eps0} is determined and iterated tree-like calls to {\\em pleskenSplit} are used to get the factorization.") (((|Factored| |#2|) |#2| |#1|) "\\spad{factor(p, eps)} tries to factor \\spad{p} into linear factors with error atmost {\\em eps}. An overall error bound {\\em eps0} is determined and iterated tree-like calls to {\\em pleskenSplit} are used to get the factorization.") (((|Factored| |#2|) |#2| |#1| (|Boolean|)) "\\spad{factor(p, eps, info)} tries to factor \\spad{p} into linear factors with error atmost {\\em eps}. An overall error bound {\\em eps0} is determined and iterated tree-like calls to {\\em pleskenSplit} are used to get the factorization. If {\\em info} is {\\em true},{} then information messages are given.")) (|divisorCascade| (((|List| (|Record| (|:| |factors| (|List| |#2|)) (|:| |error| |#1|))) |#2| |#2|) "\\spad{divisorCascade(p,tp)} assumes that degree of polynomial {\\em tp} is smaller than degree of polynomial \\spad{p},{} both monic. A sequence of divisions is calculated using the remainder,{} made monic,{} as divisor for the the next division. The result contains also the error of the factorizations,{} \\spadignore{i.e.} the norm of the remainder polynomial.") (((|List| (|Record| (|:| |factors| (|List| |#2|)) (|:| |error| |#1|))) |#2| |#2| (|Boolean|)) "\\spad{divisorCascade(p,tp)} assumes that degree of polynomial {\\em tp} is smaller than degree of polynomial \\spad{p},{} both monic. A sequence of divisions are calculated using the remainder,{} made monic,{} as divisor for the the next division. The result contains also the error of the factorizations,{} \\spadignore{i.e.} the norm of the remainder polynomial. If {\\em info} is {\\em true},{} then information messages are issued.")) (|complexZeros| (((|List| (|Complex| |#1|)) |#2| |#1|) "\\spad{complexZeros(p, eps)} tries to determine all complex zeros of the polynomial \\spad{p} with accuracy given by {\\em eps}.") (((|List| (|Complex| |#1|)) |#2|) "\\spad{complexZeros(p)} tries to determine all complex zeros of the polynomial \\spad{p} with accuracy given by the package constant {\\em globalEps} which you may change by {\\em setErrorBound}."))) NIL NIL -(-154 S ST) +(-156 S ST) ((|constructor| (NIL "This package provides tools for working with cyclic streams.")) (|computeCycleEntry| ((|#2| |#2| |#2|) "\\spad{computeCycleEntry(x,cycElt)},{} where \\spad{cycElt} is a pointer to a node in the cyclic part of the cyclic stream \\spad{x},{} returns a pointer to the first node in the cycle")) (|computeCycleLength| (((|NonNegativeInteger|) |#2|) "\\spad{computeCycleLength(s)} returns the length of the cycle of a cyclic stream \\spad{t},{} where \\spad{s} is a pointer to a node in the cyclic part of \\spad{t}.")) (|cycleElt| (((|Union| |#2| "failed") |#2|) "\\spad{cycleElt(s)} returns a pointer to a node in the cycle if the stream \\spad{s} is cyclic and returns \"failed\" if \\spad{s} is not cyclic"))) NIL NIL -(-155) +(-157) ((|constructor| (NIL "This domain provides implementations for constructors.")) (|findConstructor| (((|Maybe| $) (|Identifier|)) "\\spad{findConstructor(s)} attempts to find a constructor named \\spad{s}. If successful,{} returns that constructor; otherwise,{} returns \\spad{nothing}."))) NIL NIL -(-156 C) +(-158 C) ((|arguments| (((|List| (|Syntax|)) $) "\\spad{arguments(t)} returns the list of syntax objects for the arguments used to invoke the constructor.")) (|constructor| (NIL "This domains represents a syntax object that designates a category,{} domain,{} or a package. See Also: Syntax,{} Domain") ((|#1| $) "\\spad{constructor(t)} returns the name of the constructor used to make the call."))) NIL NIL -(-157 S) +(-159 S) ((|constructor| (NIL "This category declares basic operations on all constructors.")) (|operations| (((|List| (|OverloadSet|)) $) "\\spad{operations(c)} returns the list of all operator exported by instantiations of constructor \\spad{c}. The operators are partitioned into overload sets.")) (|dualSignature| (((|List| (|Boolean|)) $) "\\spad{dualSignature(c)} returns a list \\spad{l} of Boolean values with the following meaning: \\indented{2}{\\spad{l}.(\\spad{i+1}) holds when the constructor takes a domain object} \\indented{10}{as the `i'th argument.\\space{2}Otherwise the argument} \\indented{10}{must be a non-domain object.}")) (|kind| (((|ConstructorKind|) $) "\\spad{kind(ctor)} returns the kind of the constructor `ctor'."))) NIL NIL -(-158) +(-160) ((|constructor| (NIL "This category declares basic operations on all constructors.")) (|operations| (((|List| (|OverloadSet|)) $) "\\spad{operations(c)} returns the list of all operator exported by instantiations of constructor \\spad{c}. The operators are partitioned into overload sets.")) (|dualSignature| (((|List| (|Boolean|)) $) "\\spad{dualSignature(c)} returns a list \\spad{l} of Boolean values with the following meaning: \\indented{2}{\\spad{l}.(\\spad{i+1}) holds when the constructor takes a domain object} \\indented{10}{as the `i'th argument.\\space{2}Otherwise the argument} \\indented{10}{must be a non-domain object.}")) (|kind| (((|ConstructorKind|) $) "\\spad{kind(ctor)} returns the kind of the constructor `ctor'."))) NIL NIL -(-159) +(-161) ((|constructor| (NIL "This domain enumerates the three kinds of constructors available in OpenAxiom: category constructors,{} domain constructors,{} and package constructors.")) (|package| (($) "`package' is the kind of package constructors.")) (|domain| (($) "`domain' is the kind of domain constructors")) (|category| (($) "`category' is the kind of category constructors"))) NIL NIL -(-160 R -3078) +(-162 R -3088) ((|constructor| (NIL "\\spadtype{ComplexTrigonometricManipulations} provides function that compute the real and imaginary parts of complex functions.")) (|complexForm| (((|Complex| (|Expression| |#1|)) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f, imag f]}.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| (((|Expression| |#1|) |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| (((|Expression| |#1|) |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f, x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f, x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) NIL NIL -(-161 R) +(-163 R) ((|constructor| (NIL "CoerceVectorMatrixPackage: an unexposed,{} technical package for data conversions")) (|coerce| (((|Vector| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Vector| (|Matrix| |#1|))) "\\spad{coerce(v)} coerces a vector \\spad{v} with entries in \\spadtype{Matrix R} as vector over \\spadtype{Matrix Fraction Polynomial R}")) (|coerceP| (((|Vector| (|Matrix| (|Polynomial| |#1|))) (|Vector| (|Matrix| |#1|))) "\\spad{coerceP(v)} coerces a vector \\spad{v} with entries in \\spadtype{Matrix R} as vector over \\spadtype{Matrix Polynomial R}"))) NIL NIL -(-162) +(-164) ((|constructor| (NIL "Enumeration by cycle indices.")) (|skewSFunction| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{skewSFunction(li1,li2)} is the \\spad{S}-function \\indented{1}{of the partition difference \\spad{li1 - li2}} \\indented{1}{expressed in terms of power sum symmetric functions.}")) (|SFunction| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|List| (|PositiveInteger|))) "\\spad{SFunction(li)} is the \\spad{S}-function of the partition \\spad{li} \\indented{1}{expressed in terms of power sum symmetric functions.}")) (|wreath| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{wreath(s1,s2)} is the cycle index of the wreath product \\indented{1}{of the two groups whose cycle indices are \\spad{s1} and} \\indented{1}{\\spad{s2}.}")) (|eval| (((|Fraction| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{eval s} is the sum of the coefficients of a cycle index.")) (|cup| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{cup(s1,s2)},{} introduced by Redfield,{} \\indented{1}{is the scalar product of two cycle indices,{} in which the} \\indented{1}{power sums are retained to produce a cycle index.}")) (|cap| (((|Fraction| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|))) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{cap(s1,s2)},{} introduced by Redfield,{} \\indented{1}{is the scalar product of two cycle indices.}")) (|graphs| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|PositiveInteger|)) "\\spad{graphs n} is the cycle index of the group induced on \\indented{1}{the edges of a graph by applying the symmetric function to the} \\indented{1}{\\spad{n} nodes.}")) (|dihedral| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|PositiveInteger|)) "\\spad{dihedral n} is the cycle index of the \\indented{1}{dihedral group of degree \\spad{n}.}")) (|cyclic| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|PositiveInteger|)) "\\spad{cyclic n} is the cycle index of the \\indented{1}{cyclic group of degree \\spad{n}.}")) (|alternating| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|PositiveInteger|)) "\\spad{alternating n} is the cycle index of the \\indented{1}{alternating group of degree \\spad{n}.}")) (|elementary| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|PositiveInteger|)) "\\spad{elementary n} is the \\spad{n} th elementary symmetric \\indented{1}{function expressed in terms of power sums.}")) (|powerSum| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|PositiveInteger|)) "\\spad{powerSum n} is the \\spad{n} th power sum symmetric \\indented{1}{function.}")) (|complete| (((|SymmetricPolynomial| (|Fraction| (|Integer|))) (|PositiveInteger|)) "\\spad{complete n} is the \\spad{n} th complete homogeneous \\indented{1}{symmetric function expressed in terms of power sums.} \\indented{1}{Alternatively it is the cycle index of the symmetric} \\indented{1}{group of degree \\spad{n}.}"))) NIL NIL -(-163) +(-165) ((|constructor| (NIL "This package \\undocumented{}")) (|cyclotomicFactorization| (((|Factored| (|SparseUnivariatePolynomial| (|Integer|))) (|Integer|)) "\\spad{cyclotomicFactorization(n)} \\undocumented{}")) (|cyclotomic| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{cyclotomic(n)} \\undocumented{}")) (|cyclotomicDecomposition| (((|List| (|SparseUnivariatePolynomial| (|Integer|))) (|Integer|)) "\\spad{cyclotomicDecomposition(n)} \\undocumented{}"))) NIL NIL -(-164 N T$) +(-166 N T$) ((|constructor| (NIL "This domain provides for a fixed-sized homogeneous data buffer.")) (|qsetelt| ((|#2| $ (|NonNegativeInteger|) |#2|) "setelt(\\spad{b},{}\\spad{i},{}\\spad{x}) sets the \\spad{i}th entry of data buffer `b' to `x'. Indexing is 0-based.")) (|qelt| ((|#2| $ (|NonNegativeInteger|)) "elt(\\spad{b},{}\\spad{i}) returns the \\spad{i}th element in buffer `b'. Indexing is 0-based.")) (|new| (($) "\\spad{new()} returns a fresly allocated data buffer or length \\spad{N}."))) NIL NIL -(-165 S) +(-167 S) ((|constructor| (NIL "\\indented{1}{This domain implements a simple view of a database whose fields are} indexed by symbols")) (- (($ $ $) "\\spad{db1-db2} returns the difference of databases \\spad{db1} and \\spad{db2} \\spadignore{i.e.} consisting of elements in \\spad{db1} but not in \\spad{db2}")) (+ (($ $ $) "\\spad{db1+db2} returns the merge of databases \\spad{db1} and \\spad{db2}")) (|fullDisplay| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{fullDisplay(db,start,end )} prints full details of entries in the range \\axiom{\\spad{start}..end} in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(db)} prints full details of each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{fullDisplay(x)} displays \\spad{x} in detail")) (|display| (((|Void|) $) "\\spad{display(db)} prints a summary line for each entry in \\axiom{\\spad{db}}.") (((|Void|) $) "\\spad{display(x)} displays \\spad{x} in some form")) (|elt| (((|DataList| (|String|)) $ (|Symbol|)) "\\spad{elt(db,s)} returns the \\axiom{\\spad{s}} field of each element of \\axiom{\\spad{db}}.") (($ $ (|QueryEquation|)) "\\spad{elt(db,q)} returns all elements of \\axiom{\\spad{db}} which satisfy \\axiom{\\spad{q}}.") (((|String|) $ (|Symbol|)) "\\spad{elt(x,s)} returns an element of \\spad{x} indexed by \\spad{s}"))) NIL NIL -(-166 |vars|) +(-168 |vars|) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: July 2,{} 2010 Date Last Modified: July 2,{} 2010 Descrption: \\indented{2}{Representation of a dual vector space basis,{} given by symbols.}")) (|dual| (($ (|LinearBasis| |#1|)) "\\spad{dual x} constructs the dual vector of a linear element which is part of a basis."))) NIL NIL -(-167 -3078 UP UPUP R) +(-169 -3088 UP UPUP R) ((|constructor| (NIL "This package provides functions for computing the residues of a function on an algebraic curve.")) (|doubleResultant| ((|#2| |#4| (|Mapping| |#2| |#2|)) "\\spad{doubleResultant(f, ')} returns \\spad{p}(\\spad{x}) whose roots are rational multiples of the residues of \\spad{f} at all its finite poles. Argument ' is the derivation to use."))) NIL NIL -(-168 -3078 FP) +(-170 -3088 FP) ((|constructor| (NIL "Package for the factorization of a univariate polynomial with coefficients in a finite field. The algorithm used is the \"distinct degree\" algorithm of Cantor-Zassenhaus,{} modified to use trace instead of the norm and a table for computing Frobenius as suggested by Naudin and Quitte .")) (|irreducible?| (((|Boolean|) |#2|) "\\spad{irreducible?(p)} tests whether the polynomial \\spad{p} is irreducible.")) (|tracePowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{tracePowMod(u,k,v)} produces the sum of \\spad{u**(q**i)} for \\spad{i} running and q= size \\spad{F}")) (|trace2PowMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{trace2PowMod(u,k,v)} produces the sum of \\spad{u**(2**i)} for \\spad{i} running from 1 to \\spad{k} all computed modulo the polynomial \\spad{v}.")) (|exptMod| ((|#2| |#2| (|NonNegativeInteger|) |#2|) "\\spad{exptMod(u,k,v)} raises the polynomial \\spad{u} to the \\spad{k}th power modulo the polynomial \\spad{v}.")) (|separateFactors| (((|List| |#2|) (|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|)))) "\\spad{separateFactors(lfact)} takes the list produced by \\spadfunFrom{separateDegrees}{DistinctDegreeFactorization} and produces the complete list of factors.")) (|separateDegrees| (((|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |prod| |#2|))) |#2|) "\\spad{separateDegrees(p)} splits the square free polynomial \\spad{p} into factors each of which is a product of irreducibles of the same degree.")) (|distdfact| (((|Record| (|:| |cont| |#1|) (|:| |factors| (|List| (|Record| (|:| |irr| |#2|) (|:| |pow| (|Integer|)))))) |#2| (|Boolean|)) "\\spad{distdfact(p,sqfrflag)} produces the complete factorization of the polynomial \\spad{p} returning an internal data structure. If argument \\spad{sqfrflag} is \\spad{true},{} the polynomial is assumed square free.")) (|factorSquareFree| (((|Factored| |#2|) |#2|) "\\spad{factorSquareFree(p)} produces the complete factorization of the square free polynomial \\spad{p}.")) (|factor| (((|Factored| |#2|) |#2|) "\\spad{factor(p)} produces the complete factorization of the polynomial \\spad{p}."))) NIL NIL -(-169) +(-171) ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating decimal expansions.")) (|decimal| (($ (|Fraction| (|Integer|))) "\\spad{decimal(r)} converts a rational number to a decimal expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(d)} returns the fractional part of a decimal expansion."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-480) (QUOTE (-816))) (|HasCategory| (-480) (QUOTE (-945 (-1081)))) (|HasCategory| (-480) (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-118))) (|HasCategory| (-480) (QUOTE (-550 (-469)))) (|HasCategory| (-480) (QUOTE (-928))) (|HasCategory| (-480) (QUOTE (-735))) (|HasCategory| (-480) (QUOTE (-751))) (OR (|HasCategory| (-480) (QUOTE (-735))) (|HasCategory| (-480) (QUOTE (-751)))) (|HasCategory| (-480) (QUOTE (-945 (-480)))) (|HasCategory| (-480) (QUOTE (-1057))) (|HasCategory| (-480) (QUOTE (-791 (-325)))) (|HasCategory| (-480) (QUOTE (-791 (-480)))) (|HasCategory| (-480) (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-480) (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-480) (QUOTE (-187))) (|HasCategory| (-480) (QUOTE (-806 (-1081)))) (|HasCategory| (-480) (QUOTE (-188))) (|HasCategory| (-480) (QUOTE (-804 (-1081)))) (|HasCategory| (-480) (QUOTE (-449 (-1081) (-480)))) (|HasCategory| (-480) (QUOTE (-257 (-480)))) (|HasCategory| (-480) (QUOTE (-239 (-480) (-480)))) (|HasCategory| (-480) (QUOTE (-255))) (|HasCategory| (-480) (QUOTE (-479))) (|HasCategory| (-480) (QUOTE (-577 (-480)))) (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-816)))) (OR (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-816)))) (|HasCategory| (-480) (QUOTE (-116))))) -(-170) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-483) (QUOTE (-821))) (|HasCategory| (-483) (QUOTE (-950 (-1088)))) (|HasCategory| (-483) (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-120))) (|HasCategory| (-483) (QUOTE (-553 (-472)))) (|HasCategory| (-483) (QUOTE (-933))) (|HasCategory| (-483) (QUOTE (-740))) (|HasCategory| (-483) (QUOTE (-756))) (OR (|HasCategory| (-483) (QUOTE (-740))) (|HasCategory| (-483) (QUOTE (-756)))) (|HasCategory| (-483) (QUOTE (-950 (-483)))) (|HasCategory| (-483) (QUOTE (-1064))) (|HasCategory| (-483) (QUOTE (-796 (-327)))) (|HasCategory| (-483) (QUOTE (-796 (-483)))) (|HasCategory| (-483) (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-483) (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-483) (QUOTE (-189))) (|HasCategory| (-483) (QUOTE (-811 (-1088)))) (|HasCategory| (-483) (QUOTE (-190))) (|HasCategory| (-483) (QUOTE (-809 (-1088)))) (|HasCategory| (-483) (QUOTE (-452 (-1088) (-483)))) (|HasCategory| (-483) (QUOTE (-259 (-483)))) (|HasCategory| (-483) (QUOTE (-241 (-483) (-483)))) (|HasCategory| (-483) (QUOTE (-257))) (|HasCategory| (-483) (QUOTE (-482))) (|HasCategory| (-483) (QUOTE (-580 (-483)))) (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-821)))) (OR (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-821)))) (|HasCategory| (-483) (QUOTE (-118))))) +(-172) ((|constructor| (NIL "This domain represents the syntax of a definition.")) (|body| (((|SpadAst|) $) "\\spad{body(d)} returns the right hand side of the definition `d'.")) (|signature| (((|Signature|) $) "\\spad{signature(d)} returns the signature of the operation being defined. Note that this list may be partial in that it contains only the types actually specified in the definition.")) (|head| (((|HeadAst|) $) "\\spad{head(d)} returns the head of the definition `d'. This is a list of identifiers starting with the name of the operation followed by the name of the parameters,{} if any."))) NIL NIL -(-171 R -3078) +(-173 R -3088) ((|constructor| (NIL "\\spadtype{ElementaryFunctionDefiniteIntegration} provides functions to compute definite integrals of elementary functions.")) (|innerint| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| #1="failed") (|:| |pole| #2="potentialPole")) |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{innerint(f, x, a, b, ignore?)} should be local but conditional")) (|integrate| (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| #1#) (|:| |pole| #2#)) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|)) (|String|)) "\\spad{integrate(f, x = a..b, \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| |#2|)) (|:| |f2| (|List| (|OrderedCompletion| |#2|))) (|:| |fail| #1#) (|:| |pole| #2#)) |#2| (|SegmentBinding| (|OrderedCompletion| |#2|))) "\\spad{integrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}."))) NIL NIL -(-172 R) +(-174 R) ((|constructor| (NIL "\\spadtype{RationalFunctionDefiniteIntegration} provides functions to compute definite integrals of rational functions.")) (|integrate| (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| #1="failed") (|:| |pole| #2="potentialPole")) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|)))) (|String|)) "\\spad{integrate(f, x = a..b, \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| #1#) (|:| |pole| #2#)) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))))) "\\spad{integrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}.") (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| #1#) (|:| |pole| #2#)) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Expression| |#1|))) (|String|)) "\\spad{integrate(f, x = a..b, \"noPole\")} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. If it is not possible to check whether \\spad{f} has a pole for \\spad{x} between a and \\spad{b} (because of parameters),{} then this function will assume that \\spad{f} has no such pole. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b} or if the last argument is not \"noPole\".") (((|Union| (|:| |f1| (|OrderedCompletion| (|Expression| |#1|))) (|:| |f2| (|List| (|OrderedCompletion| (|Expression| |#1|)))) (|:| |fail| #1#) (|:| |pole| #2#)) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|OrderedCompletion| (|Expression| |#1|)))) "\\spad{integrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b}. Error: if \\spad{f} has a pole for \\spad{x} between a and \\spad{b}."))) NIL NIL -(-173 R1 R2) +(-175 R1 R2) ((|constructor| (NIL "This package \\undocumented{}")) (|expand| (((|List| (|Expression| |#2|)) (|Expression| |#2|) (|PositiveInteger|)) "\\spad{expand(f,n)} \\undocumented{}")) (|reduce| (((|Record| (|:| |pol| (|SparseUnivariatePolynomial| |#1|)) (|:| |deg| (|PositiveInteger|))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{reduce(p)} \\undocumented{}"))) NIL NIL -(-174 S) +(-176 S) ((|constructor| (NIL "Linked list implementation of a Dequeue")) (|dequeue| (($ (|List| |#1|)) "\\spad{dequeue([x,y,...,z])} creates a dequeue with first (top or front) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom or back) element \\spad{z}."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-175 |CoefRing| |listIndVar|) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-177 |CoefRing| |listIndVar|) ((|constructor| (NIL "The deRham complex of Euclidean space,{} that is,{} the class of differential forms of arbitary degree over a coefficient ring. See Flanders,{} Harley,{} Differential Forms,{} With Applications to the Physical Sciences,{} New York,{} Academic Press,{} 1963.")) (|exteriorDifferential| (($ $) "\\spad{exteriorDifferential(df)} returns the exterior derivative (gradient,{} curl,{} divergence,{} ...) of the differential form \\spad{df}.")) (|totalDifferential| (($ (|Expression| |#1|)) "\\spad{totalDifferential(x)} returns the total differential (gradient) form for element \\spad{x}.")) (|map| (($ (|Mapping| (|Expression| |#1|) (|Expression| |#1|)) $) "\\spad{map(f,df)} replaces each coefficient \\spad{x} of differential form \\spad{df} by \\spad{f(x)}.")) (|degree| (((|Integer|) $) "\\spad{degree(df)} returns the homogeneous degree of differential form \\spad{df}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?(df)} tests if differential form \\spad{df} is a 0-form,{} \\spadignore{i.e.} if degree(\\spad{df}) = 0.")) (|homogeneous?| (((|Boolean|) $) "\\spad{homogeneous?(df)} tests if all of the terms of differential form \\spad{df} have the same degree.")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(n)} returns the \\spad{n}th basis term for a differential form.")) (|coefficient| (((|Expression| |#1|) $ $) "\\spad{coefficient(df,u)},{} where \\spad{df} is a differential form,{} returns the coefficient of \\spad{df} containing the basis term \\spad{u} if such a term exists,{} and 0 otherwise.")) (|reductum| (($ $) "\\spad{reductum(df)},{} where \\spad{df} is a differential form,{} returns \\spad{df} minus the leading term of \\spad{df} if \\spad{df} has two or more terms,{} and 0 otherwise.")) (|leadingBasisTerm| (($ $) "\\spad{leadingBasisTerm(df)} returns the leading basis term of differential form \\spad{df}.")) (|leadingCoefficient| (((|Expression| |#1|) $) "\\spad{leadingCoefficient(df)} returns the leading coefficient of differential form \\spad{df}."))) -((-3975 . T)) +((-3986 . T)) NIL -(-176 R -3078) +(-178 R -3088) ((|constructor| (NIL "\\spadtype{DefiniteIntegrationTools} provides common tools used by the definite integration of both rational and elementary functions.")) (|checkForZero| (((|Union| (|Boolean|) "failed") (|SparseUnivariatePolynomial| |#2|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p, a, b, incl?)} is \\spad{true} if \\spad{p} has a zero between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.") (((|Union| (|Boolean|) "failed") (|Polynomial| |#1|) (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{checkForZero(p, x, a, b, incl?)} is \\spad{true} if \\spad{p} has a zero for \\spad{x} between a and \\spad{b},{} \\spad{false} otherwise,{} \"failed\" if this cannot be determined. Check for a and \\spad{b} inclusive if incl? is \\spad{true},{} exclusive otherwise.")) (|computeInt| (((|Union| (|OrderedCompletion| |#2|) "failed") (|Kernel| |#2|) |#2| (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|) (|Boolean|)) "\\spad{computeInt(x, g, a, b, eval?)} returns the integral of \\spad{f} for \\spad{x} between a and \\spad{b},{} assuming that \\spad{g} is an indefinite integral of \\spad{f} and \\spad{f} has no pole between a and \\spad{b}. If \\spad{eval?} is \\spad{true},{} then \\spad{g} can be evaluated safely at \\spad{a} and \\spad{b},{} provided that they are finite values. Otherwise,{} limits must be computed.")) (|ignore?| (((|Boolean|) (|String|)) "\\spad{ignore?(s)} is \\spad{true} if \\spad{s} is the string that tells the integrator to assume that the function has no pole in the integration interval."))) NIL NIL -(-177) +(-179) ((|constructor| (NIL "\\indented{1}{\\spadtype{DoubleFloat} is intended to make accessible} hardware floating point arithmetic in \\Language{},{} either native double precision,{} or IEEE. On most machines,{} there will be hardware support for the arithmetic operations: \\spadfunFrom{+}{DoubleFloat},{} \\spadfunFrom{*}{DoubleFloat},{} \\spadfunFrom{/}{DoubleFloat} and possibly also the \\spadfunFrom{sqrt}{DoubleFloat} operation. The operations \\spadfunFrom{exp}{DoubleFloat},{} \\spadfunFrom{log}{DoubleFloat},{} \\spadfunFrom{sin}{DoubleFloat},{} \\spadfunFrom{cos}{DoubleFloat},{} \\spadfunFrom{atan}{DoubleFloat} are normally coded in software based on minimax polynomial/rational approximations. Note that under Lisp/VM,{} \\spadfunFrom{atan}{DoubleFloat} is not available at this time. Some general comments about the accuracy of the operations: the operations \\spadfunFrom{+}{DoubleFloat},{} \\spadfunFrom{*}{DoubleFloat},{} \\spadfunFrom{/}{DoubleFloat} and \\spadfunFrom{sqrt}{DoubleFloat} are expected to be fully accurate. The operations \\spadfunFrom{exp}{DoubleFloat},{} \\spadfunFrom{log}{DoubleFloat},{} \\spadfunFrom{sin}{DoubleFloat},{} \\spadfunFrom{cos}{DoubleFloat} and \\spadfunFrom{atan}{DoubleFloat} are not expected to be fully accurate. In particular,{} \\spadfunFrom{sin}{DoubleFloat} and \\spadfunFrom{cos}{DoubleFloat} will lose all precision for large arguments. \\blankline The \\spadtype{Float} domain provides an alternative to the \\spad{DoubleFloat} domain. It provides an arbitrary precision model of floating point arithmetic. This means that accuracy problems like those above are eliminated by increasing the working precision where necessary. \\spadtype{Float} provides some special functions such as \\spadfunFrom{erf}{DoubleFloat},{} the error function in addition to the elementary functions. The disadvantage of \\spadtype{Float} is that it is much more expensive than small floats when the latter can be used.")) (|nan?| (((|Boolean|) $) "\\spad{nan? x} holds if \\spad{x} is a Not a Number floating point data in the IEEE 754 sense.")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n, b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)} (that is,{} \\spad{|(r-f)/f| < b**(-n)}).") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|Beta| (($ $ $) "\\spad{Beta(x,y)} is \\spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.")) (|Gamma| (($ $) "\\spad{Gamma(x)} is the Euler Gamma function.")) (|atan| (($ $ $) "\\spad{atan(x,y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm with base 10 for \\spad{x}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm with base 2 for \\spad{x}.")) (|exp1| (($) "\\spad{exp1()} returns the natural log base \\spad{2.718281828...}.")) (** (($ $ $) "\\spad{x ** y} returns the \\spad{y}th power of \\spad{x} (equal to \\spad{exp(y log x)}).")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) -((-3753 . T) (-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3764 . T) (-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-178) +(-180) ((|constructor| (NIL "This package provides special functions for double precision real and complex floating point.")) (|hypergeometric0F1| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{hypergeometric0F1(c,z)} is the hypergeometric function \\spad{0F1(; c; z)}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{hypergeometric0F1(c,z)} is the hypergeometric function \\spad{0F1(; c; z)}.")) (|airyBi| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Bi''(x) - x * Bi(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Bi''(x) - x * Bi(x) = 0}.}")) (|airyAi| (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Ai''(x) - x * Ai(x) = 0}.}") (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}. This function satisfies the differential equation: \\indented{2}{\\spad{Ai''(x) - x * Ai(x) = 0}.}")) (|besselK| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselK(v,x)} is the modified Bessel function of the first kind,{} \\spad{K(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{K(v,x) = \\%pi/2*(I(-v,x) - I(v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselK(v,x)} is the modified Bessel function of the first kind,{} \\spad{K(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{K(v,x) = \\%pi/2*(I(-v,x) - I(v,x))/sin(v*\\%pi)}.} so is not valid for integer values of \\spad{v}.")) (|besselI| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselI(v,x)} is the modified Bessel function of the first kind,{} \\spad{I(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselI(v,x)} is the modified Bessel function of the first kind,{} \\spad{I(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}")) (|besselY| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselY(v,x)} is the Bessel function of the second kind,{} \\spad{Y(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{Y(v,x) = (J(v,x) cos(v*\\%pi) - J(-v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselY(v,x)} is the Bessel function of the second kind,{} \\spad{Y(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implmentation uses the relation \\indented{2}{\\spad{Y(v,x) = (J(v,x) cos(v*\\%pi) - J(-v,x))/sin(v*\\%pi)}} so is not valid for integer values of \\spad{v}.")) (|besselJ| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{besselJ(v,x)} is the Bessel function of the first kind,{} \\spad{J(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{besselJ(v,x)} is the Bessel function of the first kind,{} \\spad{J(v,x)}. This function satisfies the differential equation: \\indented{2}{\\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}")) (|polygamma| (((|Complex| (|DoubleFloat|)) (|NonNegativeInteger|) (|Complex| (|DoubleFloat|))) "\\spad{polygamma(n, x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.") (((|DoubleFloat|) (|NonNegativeInteger|) (|DoubleFloat|)) "\\spad{polygamma(n, x)} is the \\spad{n}-th derivative of \\spad{digamma(x)}.")) (|digamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{digamma(x)} is the function,{} \\spad{psi(x)},{} defined by \\indented{2}{\\spad{psi(x) = Gamma'(x)/Gamma(x)}.}")) (|logGamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{logGamma(x)} is the natural log of \\spad{Gamma(x)}. This can often be computed even if \\spad{Gamma(x)} cannot.")) (|Beta| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Beta(x, y)} is the Euler beta function,{} \\spad{B(x,y)},{} defined by \\indented{2}{\\spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}") (((|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{Beta(x, y)} is the Euler beta function,{} \\spad{B(x,y)},{} defined by \\indented{2}{\\spad{Beta(x,y) = integrate(t^(x-1)*(1-t)^(y-1), t=0..1)}.} This is related to \\spad{Gamma(x)} by \\indented{2}{\\spad{Beta(x,y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}")) (|Gamma| (((|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\\%infinity)}.}") (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{Gamma(x)} is the Euler gamma function,{} \\spad{Gamma(x)},{} defined by \\indented{2}{\\spad{Gamma(x) = integrate(t^(x-1)*exp(-t), t=0..\\%infinity)}.}"))) NIL NIL -(-179 R) +(-181 R) ((|constructor| (NIL "\\indented{1}{A Denavit-Hartenberg Matrix is a 4x4 Matrix of the form:} \\indented{1}{\\spad{nx ox ax px}} \\indented{1}{\\spad{ny oy ay py}} \\indented{1}{\\spad{nz oz az pz}} \\indented{2}{\\spad{0\\space{2}0\\space{2}0\\space{2}1}} (\\spad{n},{} \\spad{o},{} and a are the direction cosines)")) (|translate| (($ |#1| |#1| |#1|) "\\spad{translate(X,Y,Z)} returns a dhmatrix for translation by \\spad{X},{} \\spad{Y},{} and \\spad{Z}")) (|scale| (($ |#1| |#1| |#1|) "\\spad{scale(sx,sy,sz)} returns a dhmatrix for scaling in the \\spad{X},{} \\spad{Y} and \\spad{Z} directions")) (|rotatez| (($ |#1|) "\\spad{rotatez(r)} returns a dhmatrix for rotation about axis \\spad{Z} for \\spad{r} degrees")) (|rotatey| (($ |#1|) "\\spad{rotatey(r)} returns a dhmatrix for rotation about axis \\spad{Y} for \\spad{r} degrees")) (|rotatex| (($ |#1|) "\\spad{rotatex(r)} returns a dhmatrix for rotation about axis \\spad{X} for \\spad{r} degrees")) (|identity| (($) "\\spad{identity()} create the identity dhmatrix")) (* (((|Point| |#1|) $ (|Point| |#1|)) "\\spad{t*p} applies the dhmatrix \\spad{t} to point \\spad{p}"))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-255))) (|HasCategory| |#1| (QUOTE (-491))) (|HasAttribute| |#1| (QUOTE (-3980 "*"))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-72)))) -(-180 A S) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-257))) (|HasCategory| |#1| (QUOTE (-494))) (|HasAttribute| |#1| (QUOTE (-3991 "*"))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-72)))) +(-182 A S) ((|constructor| (NIL "A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones."))) NIL NIL -(-181 S) +(-183 S) ((|constructor| (NIL "A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones."))) -((-3979 . T)) +((-3990 . T)) NIL -(-182 R) +(-184 R) ((|constructor| (NIL "Differential extensions of a ring \\spad{R}. Given a differentiation on \\spad{R},{} extend it to a differentiation on \\%."))) -((-3975 . T)) +((-3986 . T)) NIL -(-183 S T$) +(-185 S T$) ((|constructor| (NIL "This category captures the interface of domains with a distinguished operation named \\spad{differentiate}. Usually,{} additional properties are wanted. For example,{} that it obeys the usual Leibniz identity of differentiation of product,{} in case of differential rings. One could also want \\spad{differentiate} to obey the chain rule when considering differential manifolds. The lack of specific requirement in this category is an implicit admission that currently \\Language{} is not expressive enough to express the most general notion of differentiation in an adequate manner,{} suitable for computational purposes.")) (D ((|#2| $) "\\spad{D x} is a shorthand for \\spad{differentiate x}")) (|differentiate| ((|#2| $) "\\spad{differentiate x} compute the derivative of \\spad{x}."))) NIL NIL -(-184 T$) +(-186 T$) ((|constructor| (NIL "This category captures the interface of domains with a distinguished operation named \\spad{differentiate}. Usually,{} additional properties are wanted. For example,{} that it obeys the usual Leibniz identity of differentiation of product,{} in case of differential rings. One could also want \\spad{differentiate} to obey the chain rule when considering differential manifolds. The lack of specific requirement in this category is an implicit admission that currently \\Language{} is not expressive enough to express the most general notion of differentiation in an adequate manner,{} suitable for computational purposes.")) (D ((|#1| $) "\\spad{D x} is a shorthand for \\spad{differentiate x}")) (|differentiate| ((|#1| $) "\\spad{differentiate x} compute the derivative of \\spad{x}."))) NIL NIL -(-185 R) +(-187 R) ((|constructor| (NIL "An \\spad{R}-module equipped with a distinguised differential operator. If \\spad{R} is a differential ring,{} then differentiation on the module should extend differentiation on the differential ring \\spad{R}. The latter can be the null operator. In that case,{} the differentiation operator on the module is just an \\spad{R}-linear operator. For that reason,{} we do not require that the ring \\spad{R} be a DifferentialRing; \\blankline"))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-186 S) +(-188 S) ((|constructor| (NIL "This category is like \\spadtype{DifferentialDomain} where the target of the differentiation operator is the same as its source.")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(x, n)} returns the \\spad{n}\\spad{-}th derivative of \\spad{x}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(x,n)} returns the \\spad{n}\\spad{-}th derivative of \\spad{x}."))) NIL NIL -(-187) +(-189) ((|constructor| (NIL "This category is like \\spadtype{DifferentialDomain} where the target of the differentiation operator is the same as its source.")) (D (($ $ (|NonNegativeInteger|)) "\\spad{D(x, n)} returns the \\spad{n}\\spad{-}th derivative of \\spad{x}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(x,n)} returns the \\spad{n}\\spad{-}th derivative of \\spad{x}."))) NIL NIL -(-188) +(-190) ((|constructor| (NIL "An ordinary differential ring,{} that is,{} a ring with an operation \\spadfun{differentiate}. \\blankline"))) -((-3975 . T)) +((-3986 . T)) NIL -(-189) +(-191) ((|constructor| (NIL "Dioid is the class of semirings where the addition operation induces a canonical order relation."))) NIL NIL -(-190 A S) +(-192 A S) ((|constructor| (NIL "This category is a collection of operations common to both categories \\spadtype{Dictionary} and \\spadtype{MultiDictionary}")) (|select!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select!(p,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is not \\spad{true}.")) (|remove!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove!(p,d)} destructively changes dictionary \\spad{d} by removeing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.") (($ |#2| $) "\\spad{remove!(x,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{y} such that \\axiom{\\spad{y} = \\spad{x}}.")) (|dictionary| (($ (|List| |#2|)) "\\spad{dictionary([x,y,...,z])} creates a dictionary consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{dictionary()}\\$\\spad{D} creates an empty dictionary of type \\spad{D}."))) NIL -((|HasAttribute| |#1| (QUOTE -3978))) -(-191 S) +((|HasAttribute| |#1| (QUOTE -3989))) +(-193 S) ((|constructor| (NIL "This category is a collection of operations common to both categories \\spadtype{Dictionary} and \\spadtype{MultiDictionary}")) (|select!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select!(p,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is not \\spad{true}.")) (|remove!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove!(p,d)} destructively changes dictionary \\spad{d} by removeing all entries \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.") (($ |#1| $) "\\spad{remove!(x,d)} destructively changes dictionary \\spad{d} by removing all entries \\spad{y} such that \\axiom{\\spad{y} = \\spad{x}}.")) (|dictionary| (($ (|List| |#1|)) "\\spad{dictionary([x,y,...,z])} creates a dictionary consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{dictionary()}\\$\\spad{D} creates an empty dictionary of type \\spad{D}."))) -((-3979 . T)) +((-3990 . T)) NIL -(-192) +(-194) ((|constructor| (NIL "any solution of a homogeneous linear Diophantine equation can be represented as a sum of minimal solutions,{} which form a \"basis\" (a minimal solution cannot be represented as a nontrivial sum of solutions) in the case of an inhomogeneous linear Diophantine equation,{} each solution is the sum of a inhomogeneous solution and any number of homogeneous solutions therefore,{} it suffices to compute two sets: \\indented{3}{1. all minimal inhomogeneous solutions} \\indented{3}{2. all minimal homogeneous solutions} the algorithm implemented is a completion procedure,{} which enumerates all solutions in a recursive depth-first-search it can be seen as finding monotone paths in a graph for more details see Reference")) (|dioSolve| (((|Record| (|:| |varOrder| (|List| (|Symbol|))) (|:| |inhom| (|Union| (|List| (|Vector| (|NonNegativeInteger|))) "failed")) (|:| |hom| (|List| (|Vector| (|NonNegativeInteger|))))) (|Equation| (|Polynomial| (|Integer|)))) "\\spad{dioSolve(u)} computes a basis of all minimal solutions for linear homogeneous Diophantine equation \\spad{u},{} then all minimal solutions of inhomogeneous equation"))) NIL NIL -(-193 S -2607 R) +(-195 S -2617 R) ((|constructor| (NIL "\\indented{2}{This category represents a finite cartesian product of a given type.} Many categorical properties are preserved under this construction.")) (|dot| ((|#3| $ $) "\\spad{dot(x,y)} computes the inner product of the vectors \\spad{x} and \\spad{y}.")) (|unitVector| (($ (|PositiveInteger|)) "\\spad{unitVector(n)} produces a vector with 1 in position \\spad{n} and zero elsewhere.")) (|directProduct| (($ (|Vector| |#3|)) "\\spad{directProduct(v)} converts the vector \\spad{v} to become a direct product. Error: if the length of \\spad{v} is different from dim.")) (|finiteAggregate| ((|attribute|) "attribute to indicate an aggregate of finite size"))) NIL -((|HasCategory| |#3| (QUOTE (-309))) (|HasCategory| |#3| (QUOTE (-712))) (|HasCategory| |#3| (QUOTE (-751))) (|HasAttribute| |#3| (QUOTE -3975)) (|HasCategory| |#3| (QUOTE (-144))) (|HasCategory| |#3| (QUOTE (-315))) (|HasCategory| |#3| (QUOTE (-660))) (|HasCategory| |#3| (QUOTE (-21))) (|HasCategory| |#3| (QUOTE (-23))) (|HasCategory| |#3| (QUOTE (-102))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-956))) (|HasCategory| |#3| (QUOTE (-1007)))) -(-194 -2607 R) +((|HasCategory| |#3| (QUOTE (-311))) (|HasCategory| |#3| (QUOTE (-717))) (|HasCategory| |#3| (QUOTE (-756))) (|HasAttribute| |#3| (QUOTE -3986)) (|HasCategory| |#3| (QUOTE (-146))) (|HasCategory| |#3| (QUOTE (-317))) (|HasCategory| |#3| (QUOTE (-663))) (|HasCategory| |#3| (QUOTE (-21))) (|HasCategory| |#3| (QUOTE (-23))) (|HasCategory| |#3| (QUOTE (-104))) (|HasCategory| |#3| (QUOTE (-25))) (|HasCategory| |#3| (QUOTE (-961))) (|HasCategory| |#3| (QUOTE (-1012)))) +(-196 -2617 R) ((|constructor| (NIL "\\indented{2}{This category represents a finite cartesian product of a given type.} Many categorical properties are preserved under this construction.")) 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NIL NIL -(-197) +(-199) ((|constructor| (NIL "DisplayPackage allows one to print strings in a nice manner,{} including highlighting substrings.")) (|sayLength| (((|Integer|) (|List| (|String|))) "\\spad{sayLength(l)} returns the length of a list of strings \\spad{l} as an integer.") (((|Integer|) (|String|)) "\\spad{sayLength(s)} returns the length of a string \\spad{s} as an integer.")) (|say| (((|Void|) (|List| (|String|))) "\\spad{say(l)} sends a list of strings \\spad{l} to output.") (((|Void|) (|String|)) "\\spad{say(s)} sends a string \\spad{s} to output.")) (|center| (((|List| (|String|)) (|List| (|String|)) (|Integer|) (|String|)) "\\spad{center(l,i,s)} takes a list of strings \\spad{l},{} and centers them within a list of strings which is \\spad{i} characters long,{} in which the remaining spaces are filled with strings composed of as many repetitions as possible of the last string parameter \\spad{s}.") (((|String|) (|String|) (|Integer|) (|String|)) "\\spad{center(s,i,s)} takes the first string \\spad{s},{} and centers it within a string of length \\spad{i},{} in which the other elements of the string are composed of as many replications as possible of the second indicated string,{} \\spad{s} which must have a length greater than that of an empty string.")) (|copies| (((|String|) (|Integer|) (|String|)) "\\spad{copies(i,s)} will take a string \\spad{s} and create a new string composed of \\spad{i} copies of \\spad{s}.")) (|newLine| (((|String|)) "\\spad{newLine()} sends a new line command to output.")) (|bright| (((|List| (|String|)) (|List| (|String|))) "\\spad{bright(l)} sets the font property of a list of strings,{} \\spad{l},{} to bold-face type.") (((|List| (|String|)) (|String|)) "\\spad{bright(s)} sets the font property of the string \\spad{s} to bold-face type."))) NIL NIL -(-198 S) +(-200 S) ((|constructor| (NIL "A division ring (sometimes called a skew field),{} \\spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.")) (|inv| (($ $) "\\spad{inv x} returns the multiplicative inverse of \\spad{x}. Error: if \\spad{x} is 0.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}."))) NIL NIL -(-199) +(-201) ((|constructor| (NIL "A division ring (sometimes called a skew field),{} \\spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.")) (|inv| (($ $) "\\spad{inv x} returns the multiplicative inverse of \\spad{x}. Error: if \\spad{x} is 0.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}."))) -((-3971 . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) (-3983 . T) (-3984 . T) (-3986 . 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(|previous| (($ $) "\\spad{previous(l)} returns the doubly-link list beginning with its previous element. Error: if \\spad{l} has no previous element. Note: \\axiom{next(previous(\\spad{l})) = \\spad{l}}.")) (|tail| (($ $) "\\spad{tail(l)} returns the doubly-linked aggregate \\spad{l} starting at its second element. Error: if \\spad{l} is empty.")) (|head| (($ $) "\\spad{head(l)} returns the first element of a doubly-linked aggregate \\spad{l}. Error: if \\spad{l} is empty.")) (|last| ((|#1| $) "\\spad{last(l)} returns the last element of a doubly-linked aggregate \\spad{l}. Error: if \\spad{l} is empty."))) NIL NIL -(-201 S) +(-203 S) ((|constructor| (NIL "This domain provides some nice functions on lists")) (|elt| (((|NonNegativeInteger|) $ "count") "\\axiom{\\spad{l}.\"count\"} returns the number of elements in \\axiom{\\spad{l}}.") (($ $ "sort") "\\axiom{\\spad{l}.sort} returns \\axiom{\\spad{l}} with elements sorted. 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T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-202 M) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-204 M) ((|constructor| (NIL "DiscreteLogarithmPackage implements help functions for discrete logarithms in monoids using small cyclic groups.")) (|shanksDiscLogAlgorithm| (((|Union| (|NonNegativeInteger|) "failed") |#1| |#1| (|NonNegativeInteger|)) "\\spad{shanksDiscLogAlgorithm(b,a,p)} computes \\spad{s} with \\spad{b**s = a} for assuming that \\spad{a} and \\spad{b} are elements in a 'small' cyclic group of order \\spad{p} by Shank's algorithm. Note: this is a subroutine of the function \\spadfun{discreteLog}.")) (** ((|#1| |#1| (|Integer|)) "\\spad{x ** n} returns \\spad{x} raised to the integer power \\spad{n}"))) NIL NIL -(-203 R) +(-205 R) ((|constructor| (NIL "Category of modules that extend differential rings. \\blankline"))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-204 |vl| R) +(-206 |vl| R) ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is lexicographic specified by the variable list parameter with the most significant variable first in the list.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p, perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) -(((-3980 "*") |has| |#2| (-144)) (-3971 |has| |#2| (-491)) (-3976 |has| |#2| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . 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T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-821))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-821)))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-494)))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-327)))) (|HasCategory| (-773 |#1|) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-483)))) (|HasCategory| (-773 |#1|) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-472))))) (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-311))) (|HasAttribute| |#2| (QUOTE -3987)) (|HasCategory| |#2| (QUOTE (-389))) (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#2| (QUOTE (-118))))) +(-207) ((|showSummary| (((|Void|) $) "\\spad{showSummary(d)} prints out implementation detail information of domain `d'.")) 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((|constructor| (NIL "\\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \\spadfun{order} and \\spadfun{weight} are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \\spadfun{leader},{} \\spadfun{initial},{} \\spadfun{separant},{} \\spadfun{differentialVariables},{} and \\spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \\spadfun{eval}. A convenient way of referencing derivatives is provided by the functions \\spadfun{makeVariable}. \\blankline To construct a domain using this constructor,{} one needs to provide a ground ring \\spad{R},{} an ordered set \\spad{S} of differential indeterminates,{} a ranking \\spad{V} on the set of derivatives of the differential indeterminates,{} and a set \\spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \\blankline")) (|separant| (($ $) "\\spad{separant(p)} returns the partial derivative of the differential polynomial \\spad{p} with respect to its leader.")) (|initial| (($ $) "\\spad{initial(p)} returns the leading coefficient when the differential polynomial \\spad{p} is written as a univariate polynomial in its leader.")) (|leader| ((|#4| $) "\\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \\spad{p} Note: an error occurs if \\spad{p} is in the ground ring.")) (|isobaric?| (((|Boolean|) $) "\\spad{isobaric?(p)} returns \\spad{true} if every differential monomial appearing in the differential polynomial \\spad{p} has same weight,{} and returns \\spad{false} otherwise.")) (|weight| (((|NonNegativeInteger|) $ |#3|) "\\spad{weight(p, s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|NonNegativeInteger|) $) "\\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p}.")) (|weights| (((|List| (|NonNegativeInteger|)) $ |#3|) "\\spad{weights(p, s)} returns a list of weights of differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|List| (|NonNegativeInteger|)) $) "\\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $ |#3|) "\\spad{degree(p, s)} returns the maximum degree of the differential polynomial \\spad{p} viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of the differential polynomial \\spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \\spad{p}.") (((|NonNegativeInteger|) $ |#3|) "\\spad{order(p,s)} returns the order of the differential polynomial \\spad{p} in differential indeterminate \\spad{s}.")) (|differentialVariables| (((|List| |#3|) $) "\\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \\spad{p}.")) (|makeVariable| (((|Mapping| $ (|NonNegativeInteger|)) $) "\\spad{makeVariable(p)} views \\spad{p} as an element of a differential ring,{} in such a way that the \\spad{n}-th derivative of \\spad{p} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} := makeVariable(\\spad{p}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored.") (((|Mapping| $ (|NonNegativeInteger|)) |#3|) "\\spad{makeVariable(s)} views \\spad{s} as a differential indeterminate,{} in such a way that the \\spad{n}-th derivative of \\spad{s} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} :=makeVariable(\\spad{s}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored."))) NIL -((|HasCategory| |#2| (QUOTE (-188)))) -(-211 R S V E) +((|HasCategory| |#2| (QUOTE (-190)))) +(-213 R S V E) ((|constructor| (NIL "\\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \\spadfun{order} and \\spadfun{weight} are extended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \\spadfun{leader},{} \\spadfun{initial},{} \\spadfun{separant},{} \\spadfun{differentialVariables},{} and \\spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \\spadfun{eval}. A convenient way of referencing derivatives is provided by the functions \\spadfun{makeVariable}. \\blankline To construct a domain using this constructor,{} one needs to provide a ground ring \\spad{R},{} an ordered set \\spad{S} of differential indeterminates,{} a ranking \\spad{V} on the set of derivatives of the differential indeterminates,{} and a set \\spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \\blankline")) (|separant| (($ $) "\\spad{separant(p)} returns the partial derivative of the differential polynomial \\spad{p} with respect to its leader.")) (|initial| (($ $) "\\spad{initial(p)} returns the leading coefficient when the differential polynomial \\spad{p} is written as a univariate polynomial in its leader.")) (|leader| ((|#3| $) "\\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \\spad{p} Note: an error occurs if \\spad{p} is in the ground ring.")) (|isobaric?| (((|Boolean|) $) "\\spad{isobaric?(p)} returns \\spad{true} if every differential monomial appearing in the differential polynomial \\spad{p} has same weight,{} and returns \\spad{false} otherwise.")) (|weight| (((|NonNegativeInteger|) $ |#2|) "\\spad{weight(p, s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|NonNegativeInteger|) $) "\\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \\spad{p}.")) (|weights| (((|List| (|NonNegativeInteger|)) $ |#2|) "\\spad{weights(p, s)} returns a list of weights of differential monomials appearing in the differential polynomial \\spad{p} when \\spad{p} is viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.") (((|List| (|NonNegativeInteger|)) $) "\\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \\spad{p}.")) (|degree| (((|NonNegativeInteger|) $ |#2|) "\\spad{degree(p, s)} returns the maximum degree of the differential polynomial \\spad{p} viewed as a differential polynomial in the differential indeterminate \\spad{s} alone.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of the differential polynomial \\spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \\spad{p}.") (((|NonNegativeInteger|) $ |#2|) "\\spad{order(p,s)} returns the order of the differential polynomial \\spad{p} in differential indeterminate \\spad{s}.")) (|differentialVariables| (((|List| |#2|) $) "\\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \\spad{p}.")) (|makeVariable| (((|Mapping| $ (|NonNegativeInteger|)) $) "\\spad{makeVariable(p)} views \\spad{p} as an element of a differential ring,{} in such a way that the \\spad{n}-th derivative of \\spad{p} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} := makeVariable(\\spad{p}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored.") (((|Mapping| $ (|NonNegativeInteger|)) |#2|) "\\spad{makeVariable(s)} views \\spad{s} as a differential indeterminate,{} in such a way that the \\spad{n}-th derivative of \\spad{s} may be simply referenced as \\spad{z}.\\spad{n} where \\spad{z} :=makeVariable(\\spad{s}). Note: In the interpreter,{} \\spad{z} is given as an internal map,{} which may be ignored."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-212 S) +(-214 S) ((|constructor| (NIL "A dequeue is a doubly ended stack,{} that is,{} a bag where first items inserted are the first items extracted,{} at either the front or the back end of the data structure.")) (|reverse!| (($ $) "\\spad{reverse!(d)} destructively replaces \\spad{d} by its reverse dequeue,{} \\spadignore{i.e.} the top (front) element is now the bottom (back) element,{} and so on.")) (|extractBottom!| ((|#1| $) "\\spad{extractBottom!(d)} destructively extracts the bottom (back) element from the dequeue \\spad{d}. Error: if \\spad{d} is empty.")) (|extractTop!| ((|#1| $) "\\spad{extractTop!(d)} destructively extracts the top (front) element from the dequeue \\spad{d}. Error: if \\spad{d} is empty.")) (|insertBottom!| ((|#1| |#1| $) "\\spad{insertBottom!(x,d)} destructively inserts \\spad{x} into the dequeue \\spad{d} at the bottom (back) of the dequeue.")) (|insertTop!| ((|#1| |#1| $) "\\spad{insertTop!(x,d)} destructively inserts \\spad{x} into the dequeue \\spad{d},{} that is,{} at the top (front) of the dequeue. The element previously at the top of the dequeue becomes the second in the dequeue,{} and so on.")) (|bottom!| ((|#1| $) "\\spad{bottom!(d)} returns the element at the bottom (back) of the dequeue.")) (|top!| ((|#1| $) "\\spad{top!(d)} returns the element at the top (front) of the dequeue.")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(d)} returns the number of elements in dequeue \\spad{d}. Note: \\axiom{height(\\spad{d}) = \\# \\spad{d}}.")) (|dequeue| (($ (|List| |#1|)) "\\spad{dequeue([x,y,...,z])} creates a dequeue with first (top or front) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom or back) element \\spad{z}.") (($) "\\spad{dequeue()}\\$\\spad{D} creates an empty dequeue of type \\spad{D}."))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL -(-213 |Ex|) +(-215 |Ex|) ((|constructor| (NIL "TopLevelDrawFunctions provides top level functions for drawing graphics of expressions.")) (|makeObject| (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{makeObject(f(x,y),x = a..b,y = c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} appears as the default title.") (((|ThreeSpace| (|DoubleFloat|)) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f(x,y),x = a..b,y = c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{makeObject(curve(f(t),g(t),h(t)),t = a..b)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f(t),g(t),h(t)),t = a..b,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.")) (|draw| (((|ThreeDimensionalViewport|) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d)} draws the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title.") (((|ThreeDimensionalViewport|) (|ParametricSurface| |#1|) (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(surface(f(u,v),g(u,v),h(u,v)),u = a..b,v = c..d,l)} draws the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|))) "\\spad{draw(f(x,y),x = a..b,y = c..d)} draws the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} appears in the title bar.") (((|ThreeDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f(x,y),x = a..b,y = c..d,l)} draws the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; \\spad{f(x,y)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{draw(curve(f(t),g(t),h(t)),t = a..b)} draws the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f(t),g(t),h(t)),t = a..b,l)} draws the graph of the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{h(t)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| |#1|) (|SegmentBinding| (|Float|))) "\\spad{draw(curve(f(t),g(t)),t = a..b)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{(f(t),g(t))} appears in the title bar.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| |#1|) (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f(t),g(t)),t = a..b,l)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{(f(t),g(t))} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) |#1| (|SegmentBinding| (|Float|))) "\\spad{draw(f(x),x = a..b)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{f(x)} appears in the title bar.") (((|TwoDimensionalViewport|) |#1| (|SegmentBinding| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f(x),x = a..b,l)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; \\spad{f(x)} is the default title,{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied."))) NIL NIL -(-214) +(-216) ((|constructor| (NIL "TopLevelDrawFunctionsForCompiledFunctions provides top level functions for drawing graphics of expressions.")) (|recolor| (((|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{recolor()},{} uninteresting to top level user; exported in order to compile package.")) (|makeObject| (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(surface(f,g,h),a..b,c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(surface(f,g,h),a..b,c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(f,a..b,c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{f(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f,a..b,c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric surface \\spad{f(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{makeObject(f,a..b,c..d)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(f,a..b,c..d,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)},{} and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{makeObject(sp,curve(f,g,h),a..b)} returns the space \\spad{sp} of the domain \\spadtype{ThreeSpace} with the addition of the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f,g,h),a..b,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{makeObject(sp,curve(f,g,h),a..b)} returns the space \\spad{sp} of the domain \\spadtype{ThreeSpace} with the addition of the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeSpace| (|DoubleFloat|)) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{makeObject(curve(f,g,h),a..b,l)} returns a space of the domain \\spadtype{ThreeSpace} which contains the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.")) (|draw| (((|ThreeDimensionalViewport|) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(surface(f,g,h),a..b,c..d)} draws the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}.") (((|ThreeDimensionalViewport|) (|ParametricSurface| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(surface(f,g,h),a..b,c..d)} draws the graph of the parametric surface \\spad{x = f(u,v)},{} \\spad{y = g(u,v)},{} \\spad{z = h(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}; The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b,c..d)} draws the graph of the parametric surface \\spad{f(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)} The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,a..b,c..d)} draws the graph of the parametric surface \\spad{f(u,v)} as \\spad{u} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{v} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b,c..d)} draws the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}.") (((|ThreeDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,a..b,c..d,l)} draws the graph of \\spad{z = f(x,y)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)} and \\spad{y} ranges from \\spad{min(c,d)} to \\spad{max(c,d)}. and the options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b,l)} draws the graph of the parametric curve \\spad{f} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeDimensionalViewport|) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,a..b,l)} draws the graph of the parametric curve \\spad{f} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{draw(curve(f,g,h),a..b,l)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|ThreeDimensionalViewport|) (|ParametricSpaceCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f,g,h),a..b,l)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t), z = h(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|))) "\\spad{draw(curve(f,g),a..b)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|TwoDimensionalViewport|) (|ParametricPlaneCurve| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(curve(f,g),a..b,l)} draws the graph of the parametric curve \\spad{x = f(t), y = g(t)} as \\spad{t} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|))) "\\spad{draw(f,a..b)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}.") (((|TwoDimensionalViewport|) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|Float|)) (|List| (|DrawOption|))) "\\spad{draw(f,a..b,l)} draws the graph of \\spad{y = f(x)} as \\spad{x} ranges from \\spad{min(a,b)} to \\spad{max(a,b)}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied."))) NIL NIL -(-215 R |Ex|) +(-217 R |Ex|) ((|constructor| (NIL "TopLevelDrawFunctionsForAlgebraicCurves provides top level functions for drawing non-singular algebraic curves.")) (|draw| (((|TwoDimensionalViewport|) (|Equation| |#2|) (|Symbol|) (|Symbol|) (|List| (|DrawOption|))) "\\spad{draw(f(x,y) = g(x,y),x,y,l)} draws the graph of a polynomial equation. The list \\spad{l} of draw options must specify a region in the plane in which the curve is to sketched."))) NIL NIL -(-216) +(-218) ((|setClipValue| (((|DoubleFloat|) (|DoubleFloat|)) "\\spad{setClipValue(x)} sets to \\spad{x} the maximum value to plot when drawing complex functions. Returns \\spad{x}.")) (|setImagSteps| (((|Integer|) (|Integer|)) "\\spad{setImagSteps(i)} sets to \\spad{i} the number of steps to use in the imaginary direction when drawing complex functions. Returns \\spad{i}.")) (|setRealSteps| (((|Integer|) (|Integer|)) "\\spad{setRealSteps(i)} sets to \\spad{i} the number of steps to use in the real direction when drawing complex functions. Returns \\spad{i}.")) (|drawComplexVectorField| (((|ThreeDimensionalViewport|) (|Mapping| (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{drawComplexVectorField(f,rRange,iRange)} draws a complex vector field using arrows on the \\spad{x--y} plane. These vector fields should be viewed from the top by pressing the \"XY\" translate button on the 3-\\spad{d} viewport control panel.\\newline Sample call: \\indented{3}{\\spad{f z == sin z}} \\indented{3}{\\spad{drawComplexVectorField(f, -2..2, -2..2)}} Parameter descriptions: \\indented{2}{\\spad{f} : the function to draw} \\indented{2}{\\spad{rRange} : the range of the real values} \\indented{2}{\\spad{iRange} : the range of the imaginary values} Call the functions \\axiomFunFrom{setRealSteps}{DrawComplex} and \\axiomFunFrom{setImagSteps}{DrawComplex} to change the number of steps used in each direction.")) (|drawComplex| (((|ThreeDimensionalViewport|) (|Mapping| (|Complex| (|DoubleFloat|)) (|Complex| (|DoubleFloat|))) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Boolean|)) "\\spad{drawComplex(f,rRange,iRange,arrows?)} draws a complex function as a height field. It uses the complex norm as the height and the complex argument as the color. It will optionally draw arrows on the surface indicating the direction of the complex value.\\newline Sample call: \\indented{2}{\\spad{f z == exp(1/z)}} \\indented{2}{\\spad{drawComplex(f, 0.3..3, 0..2*\\%pi, false)}} Parameter descriptions: \\indented{2}{f:\\space{2}the function to draw} \\indented{2}{\\spad{rRange} : the range of the real values} \\indented{2}{\\spad{iRange} : the range of imaginary values} \\indented{2}{\\spad{arrows?} : a flag indicating whether to draw the phase arrows for \\spad{f}} Call the functions \\axiomFunFrom{setRealSteps}{DrawComplex} and \\axiomFunFrom{setImagSteps}{DrawComplex} to change the number of steps used in each direction."))) NIL NIL -(-217 R) +(-219 R) ((|constructor| (NIL "Hack for the draw interface. DrawNumericHack provides a \"coercion\" from something of the form \\spad{x = a..b} where \\spad{a} and \\spad{b} are formal expressions to a binding of the form \\spad{x = c..d} where \\spad{c} and \\spad{d} are the numerical values of \\spad{a} and \\spad{b}. This \"coercion\" fails if \\spad{a} and \\spad{b} contains symbolic variables,{} but is meant for expressions involving \\%\\spad{pi}.")) (|coerce| (((|SegmentBinding| (|Float|)) (|SegmentBinding| (|Expression| |#1|))) "\\spad{coerce(x = a..b)} returns \\spad{x = c..d} where \\spad{c} and \\spad{d} are the numerical values of \\spad{a} and \\spad{b}."))) NIL NIL -(-218) +(-220) ((|constructor| (NIL "TopLevelDrawFunctionsForPoints provides top level functions for drawing curves and surfaces described by sets of points.")) (|draw| (((|ThreeDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{draw(lx,ly,lz,l)} draws the surface constructed by projecting the values in the \\axiom{\\spad{lz}} list onto the rectangular grid formed by the The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|ThreeDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|))) "\\spad{draw(lx,ly,lz)} draws the surface constructed by projecting the values in the \\axiom{\\spad{lz}} list onto the rectangular grid formed by the \\axiom{\\spad{lx} \\spad{X} \\spad{ly}}.") (((|TwoDimensionalViewport|) (|List| (|Point| (|DoubleFloat|))) (|List| (|DrawOption|))) "\\spad{draw(lp,l)} plots the curve constructed from the list of points \\spad{lp}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|List| (|Point| (|DoubleFloat|)))) "\\spad{draw(lp)} plots the curve constructed from the list of points \\spad{lp}.") (((|TwoDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{draw(lx,ly,l)} plots the curve constructed of points (\\spad{x},{}\\spad{y}) for \\spad{x} in \\spad{lx} for \\spad{y} in \\spad{ly}. The options contained in the list \\spad{l} of the domain \\spad{DrawOption} are applied.") (((|TwoDimensionalViewport|) (|List| (|DoubleFloat|)) (|List| (|DoubleFloat|))) "\\spad{draw(lx,ly)} plots the curve constructed of points (\\spad{x},{}\\spad{y}) for \\spad{x} in \\spad{lx} for \\spad{y} in \\spad{ly}."))) NIL NIL -(-219) +(-221) ((|constructor| (NIL "DrawOption allows the user to specify defaults for the creation and rendering of plots.")) (|option?| (((|Boolean|) (|List| $) (|Symbol|)) "\\spad{option?()} is not to be used at the top level; option? internally returns \\spad{true} for drawing options which are indicated in a draw command,{} or \\spad{false} for those which are not.")) (|option| (((|Union| (|Any|) "failed") (|List| $) (|Symbol|)) "\\spad{option()} is not to be used at the top level; option determines internally which drawing options are indicated in a draw command.")) (|unit| (($ (|List| (|Float|))) "\\spad{unit(lf)} will mark off the units according to the indicated list \\spad{lf}. This option is expressed in the form \\spad{unit == [f1,f2]}.")) (|coord| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)))) "\\spad{coord(p)} specifies a change of coordinates of point \\spad{p}. This option is expressed in the form \\spad{coord == p}.")) (|tubePoints| (($ (|PositiveInteger|)) "\\spad{tubePoints(n)} specifies the number of points,{} \\spad{n},{} defining the circle which creates the tube around a 3D curve,{} the default is 6. This option is expressed in the form \\spad{tubePoints == n}.")) (|var2Steps| (($ (|PositiveInteger|)) "\\spad{var2Steps(n)} indicates the number of subdivisions,{} \\spad{n},{} of the second range variable. This option is expressed in the form \\spad{var2Steps == n}.")) (|var1Steps| (($ (|PositiveInteger|)) "\\spad{var1Steps(n)} indicates the number of subdivisions,{} \\spad{n},{} of the first range variable. This option is expressed in the form \\spad{var1Steps == n}.")) (|space| (($ (|ThreeSpace| (|DoubleFloat|))) "\\spad{space specifies} the space into which we will draw. If none is given then a new space is created.")) (|ranges| (($ (|List| (|Segment| (|Float|)))) "\\spad{ranges(l)} provides a list of user-specified ranges \\spad{l}. This option is expressed in the form \\spad{ranges == l}.")) (|range| (($ (|List| (|Segment| (|Fraction| (|Integer|))))) "\\spad{range([i])} provides a user-specified range \\spad{i}. This option is expressed in the form \\spad{range == [i]}.") (($ (|List| (|Segment| (|Float|)))) "\\spad{range([l])} provides a user-specified range \\spad{l}. This option is expressed in the form \\spad{range == [l]}.")) (|tubeRadius| (($ (|Float|)) "\\spad{tubeRadius(r)} specifies a radius,{} \\spad{r},{} for a tube plot around a 3D curve; is expressed in the form \\spad{tubeRadius == 4}.")) (|colorFunction| (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{colorFunction(f(x,y,z))} specifies the color for three dimensional plots as a function of \\spad{x},{} \\spad{y},{} and \\spad{z} coordinates. This option is expressed in the form \\spad{colorFunction == f(x,y,z)}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{colorFunction(f(u,v))} specifies the color for three dimensional plots as a function based upon the two parametric variables. This option is expressed in the form \\spad{colorFunction == f(u,v)}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) "\\spad{colorFunction(f(z))} specifies the color based upon the \\spad{z}-component of three dimensional plots. This option is expressed in the form \\spad{colorFunction == f(z)}.")) (|curveColor| (($ (|Palette|)) "\\spad{curveColor(p)} specifies a color index for 2D graph curves from the spadcolors palette \\spad{p}. This option is expressed in the form \\spad{curveColor ==p}.") (($ (|Float|)) "\\spad{curveColor(v)} specifies a color,{} \\spad{v},{} for 2D graph curves. This option is expressed in the form \\spad{curveColor == v}.")) (|pointColor| (($ (|Palette|)) "\\spad{pointColor(p)} specifies a color index for 2D graph points from the spadcolors palette \\spad{p}. This option is expressed in the form \\spad{pointColor == p}.") (($ (|Float|)) "\\spad{pointColor(v)} specifies a color,{} \\spad{v},{} for 2D graph points. This option is expressed in the form \\spad{pointColor == v}.")) (|coordinates| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)))) "\\spad{coordinates(p)} specifies a change of coordinate systems of point \\spad{p}. This option is expressed in the form \\spad{coordinates == p}.")) (|toScale| (($ (|Boolean|)) "\\spad{toScale(b)} specifies whether or not a plot is to be drawn to scale; if \\spad{b} is \\spad{true} it is drawn to scale,{} if \\spad{b} is \\spad{false} it is not. This option is expressed in the form \\spad{toScale == b}.")) (|style| (($ (|String|)) "\\spad{style(s)} specifies the drawing style in which the graph will be plotted by the indicated string \\spad{s}. This option is expressed in the form \\spad{style == s}.")) (|title| (($ (|String|)) "\\spad{title(s)} specifies a title for a plot by the indicated string \\spad{s}. This option is expressed in the form \\spad{title == s}.")) (|viewpoint| (($ (|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)))) "\\spad{viewpoint(vp)} creates a viewpoint data structure corresponding to the list of values. The values are interpreted as [theta,{} phi,{} scale,{} scaleX,{} scaleY,{} scaleZ,{} deltaX,{} deltaY]. This option is expressed in the form \\spad{viewpoint == ls}.")) (|clip| (($ (|List| (|Segment| (|Float|)))) "\\spad{clip([l])} provides ranges for user-defined clipping as specified in the list \\spad{l}. This option is expressed in the form \\spad{clip == [l]}.") (($ (|Boolean|)) "\\spad{clip(b)} turns 2D clipping on if \\spad{b} is \\spad{true},{} or off if \\spad{b} is \\spad{false}. This option is expressed in the form \\spad{clip == b}.")) (|adaptive| (($ (|Boolean|)) "\\spad{adaptive(b)} turns adaptive 2D plotting on if \\spad{b} is \\spad{true},{} or off if \\spad{b} is \\spad{false}. This option is expressed in the form \\spad{adaptive == b}."))) NIL NIL -(-220) +(-222) ((|constructor| (NIL "This package \\undocumented{}")) (|units| (((|List| (|Float|)) (|List| (|DrawOption|)) (|List| (|Float|))) "\\spad{units(l,u)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{unit}. If the option does not exist the value,{} \\spad{u} is returned.")) (|coord| (((|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) (|List| (|DrawOption|)) (|Mapping| (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)))) "\\spad{coord(l,p)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{coord}. If the option does not exist the value,{} \\spad{p} is returned.")) (|tubeRadius| (((|Float|) (|List| (|DrawOption|)) (|Float|)) "\\spad{tubeRadius(l,n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{tubeRadius}. If the option does not exist the value,{} \\spad{n} is returned.")) (|tubePoints| (((|PositiveInteger|) (|List| (|DrawOption|)) (|PositiveInteger|)) "\\spad{tubePoints(l,n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{tubePoints}. If the option does not exist the value,{} \\spad{n} is returned.")) (|space| (((|ThreeSpace| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{space(l)} takes a list of draw options,{} \\spad{l},{} and checks to see if it contains the option \\spad{space}. If the the option doesn't exist,{} then an empty space is returned.")) (|var2Steps| (((|PositiveInteger|) (|List| (|DrawOption|)) (|PositiveInteger|)) "\\spad{var2Steps(l,n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{var2Steps}. If the option does not exist the value,{} \\spad{n} is returned.")) (|var1Steps| (((|PositiveInteger|) (|List| (|DrawOption|)) (|PositiveInteger|)) "\\spad{var1Steps(l,n)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{var1Steps}. If the option does not exist the value,{} \\spad{n} is returned.")) (|ranges| (((|List| (|Segment| (|Float|))) (|List| (|DrawOption|)) (|List| (|Segment| (|Float|)))) "\\spad{ranges(l,r)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{ranges}. If the option does not exist the value,{} \\spad{r} is returned.")) (|curveColorPalette| (((|Palette|) (|List| (|DrawOption|)) (|Palette|)) "\\spad{curveColorPalette(l,p)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{curveColorPalette}. If the option does not exist the value,{} \\spad{p} is returned.")) (|pointColorPalette| (((|Palette|) (|List| (|DrawOption|)) (|Palette|)) "\\spad{pointColorPalette(l,p)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{pointColorPalette}. If the option does not exist the value,{} \\spad{p} is returned.")) (|toScale| (((|Boolean|) (|List| (|DrawOption|)) (|Boolean|)) "\\spad{toScale(l,b)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{toScale}. If the option does not exist the value,{} \\spad{b} is returned.")) (|style| (((|String|) (|List| (|DrawOption|)) (|String|)) "\\spad{style(l,s)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{style}. If the option does not exist the value,{} \\spad{s} is returned.")) (|title| (((|String|) (|List| (|DrawOption|)) (|String|)) "\\spad{title(l,s)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{title}. If the option does not exist the value,{} \\spad{s} is returned.")) (|viewpoint| (((|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|))) (|List| (|DrawOption|)) (|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)))) "\\spad{viewpoint(l,ls)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{viewpoint}. IF the option does not exist,{} the value \\spad{ls} is returned.")) (|clipBoolean| (((|Boolean|) (|List| (|DrawOption|)) (|Boolean|)) "\\spad{clipBoolean(l,b)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{clipBoolean}. If the option does not exist the value,{} \\spad{b} is returned.")) (|adaptive| (((|Boolean|) (|List| (|DrawOption|)) (|Boolean|)) "\\spad{adaptive(l,b)} takes the list of draw options,{} \\spad{l},{} and checks the list to see if it contains the option \\spad{adaptive}. If the option does not exist the value,{} \\spad{b} is returned."))) NIL NIL -(-221 S) +(-223 S) ((|constructor| (NIL "This package \\undocumented{}")) (|option| (((|Union| |#1| "failed") (|List| (|DrawOption|)) (|Symbol|)) "\\spad{option(l,s)} determines whether the indicated drawing option,{} \\spad{s},{} is contained in the list of drawing options,{} \\spad{l},{} which is defined by the draw command."))) NIL NIL -(-222 S R) +(-224 S R) ((|constructor| (NIL "Extension of a base differential space with a derivation. \\blankline")) (D (($ $ (|Mapping| |#2| |#2|) (|NonNegativeInteger|)) "\\spad{D(x,d,n)} is a shorthand for \\spad{differentiate(x,d,n)}.") (($ $ (|Mapping| |#2| |#2|)) "\\spad{D(x,d)} is a shorthand for \\spad{differentiate(x,d)}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|) (|NonNegativeInteger|)) "\\spad{differentiate(x,d,n)} computes the \\spad{n}\\spad{-}th derivative of \\spad{x} using a derivation extending \\spad{d} on \\spad{R}.") (($ $ (|Mapping| |#2| |#2|)) "\\spad{differentiate(x,d)} computes the derivative of \\spad{x},{} extending differentiation \\spad{d} on \\spad{R}."))) NIL -((|HasCategory| |#2| (QUOTE (-806 (-1081)))) (|HasCategory| |#2| (QUOTE (-187)))) -(-223 R) +((|HasCategory| |#2| (QUOTE (-811 (-1088)))) (|HasCategory| |#2| (QUOTE (-189)))) +(-225 R) ((|constructor| (NIL "Extension of a base differential space with a derivation. \\blankline")) (D (($ $ (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{D(x,d,n)} is a shorthand for \\spad{differentiate(x,d,n)}.") (($ $ (|Mapping| |#1| |#1|)) "\\spad{D(x,d)} is a shorthand for \\spad{differentiate(x,d)}.")) (|differentiate| (($ $ (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{differentiate(x,d,n)} computes the \\spad{n}\\spad{-}th derivative of \\spad{x} using a derivation extending \\spad{d} on \\spad{R}.") (($ $ (|Mapping| |#1| |#1|)) "\\spad{differentiate(x,d)} computes the derivative of \\spad{x},{} extending differentiation \\spad{d} on \\spad{R}."))) NIL NIL -(-224 R S V) +(-226 R S V) ((|constructor| (NIL "\\spadtype{DifferentialSparseMultivariatePolynomial} implements an ordinary differential polynomial ring by combining a domain belonging to the category \\spadtype{DifferentialVariableCategory} with the domain \\spadtype{SparseMultivariatePolynomial}. \\blankline"))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . 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T)) +((|HasCategory| |#1| (QUOTE (-821))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| |#3| (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| |#3| (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| |#3| (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#3| (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#3| (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-227 A S) ((|constructor| (NIL "\\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \\spad{x},{}...,{}\\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \\spad{x},{}\\spad{x'},{}\\spad{x''},{}...,{} \\spad{y},{}\\spad{y'},{}\\spad{y''},{}... (Note: in the interpreter,{} the \\spad{n}-th derivative of \\spad{y} is displayed as \\spad{y} with a subscript \\spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \\blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \\spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \\spadfun{<}. For example,{} one may define one derivative to be less than another by lexicographically comparing first the \\spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \\blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{} this constructor provides a function \\spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \\spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.")) (|weight| (((|NonNegativeInteger|) $) "\\spad{weight(v)} returns the weight of the derivative \\spad{v}.")) (|variable| ((|#2| $) "\\spad{variable(v)} returns \\spad{s} if \\spad{v} is any derivative of the differential indeterminate \\spad{s}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(v)} returns \\spad{n} if \\spad{v} is the \\spad{n}-th derivative of any differential indeterminate.")) (|makeVariable| (($ |#2| (|NonNegativeInteger|)) "\\spad{makeVariable(s, n)} returns the \\spad{n}-th derivative of a differential indeterminate \\spad{s} as an algebraic indeterminate."))) NIL NIL -(-226 S) +(-228 S) ((|constructor| (NIL "\\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \\spad{x},{}...,{}\\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \\spad{x},{}\\spad{x'},{}\\spad{x''},{}...,{} \\spad{y},{}\\spad{y'},{}\\spad{y''},{}... (Note: in the interpreter,{} the \\spad{n}-th derivative of \\spad{y} is displayed as \\spad{y} with a subscript \\spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \\blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \\spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \\spadfun{<}. For example,{} one may define one derivative to be less than another by lexicographically comparing first the \\spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \\blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{} this constructor provides a function \\spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \\spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.")) (|weight| (((|NonNegativeInteger|) $) "\\spad{weight(v)} returns the weight of the derivative \\spad{v}.")) (|variable| ((|#1| $) "\\spad{variable(v)} returns \\spad{s} if \\spad{v} is any derivative of the differential indeterminate \\spad{s}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(v)} returns \\spad{n} if \\spad{v} is the \\spad{n}-th derivative of any differential indeterminate.")) (|makeVariable| (($ |#1| (|NonNegativeInteger|)) "\\spad{makeVariable(s, n)} returns the \\spad{n}-th derivative of a differential indeterminate \\spad{s} as an algebraic indeterminate."))) NIL NIL -(-227) +(-229) ((|constructor| (NIL "A domain used in the construction of the exterior algebra on a set \\spad{X} over a ring \\spad{R}. This domain represents the set of all ordered subsets of the set \\spad{X},{} assumed to be in correspondance with {1,{}2,{}3,{} ...}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements which can only be 0 or 1. \\blankline The multiplicative identity element of the exterior algebra corresponds to the empty subset of \\spad{X}. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.")) (|Nul| (($ (|NonNegativeInteger|)) "\\spad{Nul()} gives the basis element 1 for the algebra generated by \\spad{n} generators.")) (|exponents| (((|List| (|Integer|)) $) "\\spad{exponents(x)} converts a domain element into a list of zeros and ones corresponding to the exponents in the basis element that \\spad{x} represents.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(x)} gives the numbers of 1's in \\spad{x},{} \\spadignore{i.e.} the number of non-zero exponents in the basis element that \\spad{x} represents.")) (|coerce| (($ (|List| (|Integer|))) "\\spad{coerce(l)} converts a list of 0's and 1's into a basis element,{} where 1 (respectively 0) designates that the variable of the corresponding index of \\spad{l} is (respectively,{} is not) present. Error: if an element of \\spad{l} is not 0 or 1."))) NIL NIL -(-228 R -3078) +(-230 R -3088) ((|constructor| (NIL "Provides elementary functions over an integral domain.")) (|localReal?| (((|Boolean|) |#2|) "\\spad{localReal?(x)} should be local but conditional")) (|specialTrigs| (((|Union| |#2| "failed") |#2| (|List| (|Record| (|:| |func| |#2|) (|:| |pole| (|Boolean|))))) "\\spad{specialTrigs(x,l)} should be local but conditional")) (|iiacsch| ((|#2| |#2|) "\\spad{iiacsch(x)} should be local but conditional")) (|iiasech| ((|#2| |#2|) "\\spad{iiasech(x)} should be local but conditional")) (|iiacoth| ((|#2| |#2|) "\\spad{iiacoth(x)} should be local but conditional")) (|iiatanh| ((|#2| |#2|) "\\spad{iiatanh(x)} should be local but conditional")) (|iiacosh| ((|#2| |#2|) "\\spad{iiacosh(x)} should be local but conditional")) (|iiasinh| ((|#2| |#2|) "\\spad{iiasinh(x)} should be local but conditional")) (|iicsch| ((|#2| |#2|) "\\spad{iicsch(x)} should be local but conditional")) (|iisech| ((|#2| |#2|) "\\spad{iisech(x)} should be local but conditional")) (|iicoth| ((|#2| |#2|) "\\spad{iicoth(x)} should be local but conditional")) (|iitanh| ((|#2| |#2|) "\\spad{iitanh(x)} should be local but conditional")) (|iicosh| ((|#2| |#2|) "\\spad{iicosh(x)} should be local but conditional")) (|iisinh| ((|#2| |#2|) "\\spad{iisinh(x)} should be local but conditional")) (|iiacsc| ((|#2| |#2|) "\\spad{iiacsc(x)} should be local but conditional")) (|iiasec| ((|#2| |#2|) "\\spad{iiasec(x)} should be local but conditional")) (|iiacot| ((|#2| |#2|) "\\spad{iiacot(x)} should be local but conditional")) (|iiatan| ((|#2| |#2|) "\\spad{iiatan(x)} should be local but conditional")) (|iiacos| ((|#2| |#2|) "\\spad{iiacos(x)} should be local but conditional")) (|iiasin| ((|#2| |#2|) "\\spad{iiasin(x)} should be local but conditional")) (|iicsc| ((|#2| |#2|) "\\spad{iicsc(x)} should be local but conditional")) (|iisec| ((|#2| |#2|) "\\spad{iisec(x)} should be local but conditional")) (|iicot| ((|#2| |#2|) "\\spad{iicot(x)} should be local but conditional")) (|iitan| ((|#2| |#2|) "\\spad{iitan(x)} should be local but conditional")) (|iicos| ((|#2| |#2|) "\\spad{iicos(x)} should be local but conditional")) (|iisin| ((|#2| |#2|) "\\spad{iisin(x)} should be local but conditional")) (|iilog| ((|#2| |#2|) "\\spad{iilog(x)} should be local but conditional")) (|iiexp| ((|#2| |#2|) "\\spad{iiexp(x)} should be local but conditional")) (|iisqrt3| ((|#2|) "\\spad{iisqrt3()} should be local but conditional")) (|iisqrt2| ((|#2|) "\\spad{iisqrt2()} should be local but conditional")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(p)} returns an elementary operator with the same symbol as \\spad{p}")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(p)} returns \\spad{true} if operator \\spad{p} is elementary")) (|pi| ((|#2|) "\\spad{pi()} returns the \\spad{pi} operator")) (|acsch| ((|#2| |#2|) "\\spad{acsch(x)} applies the inverse hyperbolic cosecant operator to \\spad{x}")) (|asech| ((|#2| |#2|) "\\spad{asech(x)} applies the inverse hyperbolic secant operator to \\spad{x}")) (|acoth| ((|#2| |#2|) "\\spad{acoth(x)} applies the inverse hyperbolic cotangent operator to \\spad{x}")) (|atanh| ((|#2| |#2|) "\\spad{atanh(x)} applies the inverse hyperbolic tangent operator to \\spad{x}")) (|acosh| ((|#2| |#2|) "\\spad{acosh(x)} applies the inverse hyperbolic cosine operator to \\spad{x}")) (|asinh| ((|#2| |#2|) "\\spad{asinh(x)} applies the inverse hyperbolic sine operator to \\spad{x}")) (|csch| ((|#2| |#2|) "\\spad{csch(x)} applies the hyperbolic cosecant operator to \\spad{x}")) (|sech| ((|#2| |#2|) "\\spad{sech(x)} applies the hyperbolic secant operator to \\spad{x}")) (|coth| ((|#2| |#2|) "\\spad{coth(x)} applies the hyperbolic cotangent operator to \\spad{x}")) (|tanh| ((|#2| |#2|) "\\spad{tanh(x)} applies the hyperbolic tangent operator to \\spad{x}")) (|cosh| ((|#2| |#2|) "\\spad{cosh(x)} applies the hyperbolic cosine operator to \\spad{x}")) (|sinh| ((|#2| |#2|) "\\spad{sinh(x)} applies the hyperbolic sine operator to \\spad{x}")) (|acsc| ((|#2| |#2|) "\\spad{acsc(x)} applies the inverse cosecant operator to \\spad{x}")) (|asec| ((|#2| |#2|) "\\spad{asec(x)} applies the inverse secant operator to \\spad{x}")) (|acot| ((|#2| |#2|) "\\spad{acot(x)} applies the inverse cotangent operator to \\spad{x}")) (|atan| ((|#2| |#2|) "\\spad{atan(x)} applies the inverse tangent operator to \\spad{x}")) (|acos| ((|#2| |#2|) "\\spad{acos(x)} applies the inverse cosine operator to \\spad{x}")) (|asin| ((|#2| |#2|) "\\spad{asin(x)} applies the inverse sine operator to \\spad{x}")) (|csc| ((|#2| |#2|) "\\spad{csc(x)} applies the cosecant operator to \\spad{x}")) (|sec| ((|#2| |#2|) "\\spad{sec(x)} applies the secant operator to \\spad{x}")) (|cot| ((|#2| |#2|) "\\spad{cot(x)} applies the cotangent operator to \\spad{x}")) (|tan| ((|#2| |#2|) "\\spad{tan(x)} applies the tangent operator to \\spad{x}")) (|cos| ((|#2| |#2|) "\\spad{cos(x)} applies the cosine operator to \\spad{x}")) (|sin| ((|#2| |#2|) "\\spad{sin(x)} applies the sine operator to \\spad{x}")) (|log| ((|#2| |#2|) "\\spad{log(x)} applies the logarithm operator to \\spad{x}")) (|exp| ((|#2| |#2|) "\\spad{exp(x)} applies the exponential operator to \\spad{x}"))) NIL NIL -(-229 R -3078) +(-231 R -3088) ((|constructor| (NIL "ElementaryFunctionStructurePackage provides functions to test the algebraic independence of various elementary functions,{} using the Risch structure theorem (real and complex versions). It also provides transformations on elementary functions which are not considered simplifications.")) (|tanQ| ((|#2| (|Fraction| (|Integer|)) |#2|) "\\spad{tanQ(q,a)} is a local function with a conditional implementation.")) (|rootNormalize| ((|#2| |#2| (|Kernel| |#2|)) "\\spad{rootNormalize(f, k)} returns \\spad{f} rewriting either \\spad{k} which must be an \\spad{n}th-root in terms of radicals already in \\spad{f},{} or some radicals in \\spad{f} in terms of \\spad{k}.")) (|validExponential| (((|Union| |#2| "failed") (|List| (|Kernel| |#2|)) |#2| (|Symbol|)) "\\spad{validExponential([k1,...,kn],f,x)} returns \\spad{g} if \\spad{exp(f)=g} and \\spad{g} involves only \\spad{k1...kn},{} and \"failed\" otherwise.")) (|realElementary| ((|#2| |#2| (|Symbol|)) "\\spad{realElementary(f,x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log, exp, tan, atan}.") ((|#2| |#2|) "\\spad{realElementary(f)} rewrites \\spad{f} in terms of the 4 fundamental real transcendental elementary functions: \\spad{log, exp, tan, atan}.")) (|rischNormalize| (((|Record| (|:| |func| |#2|) (|:| |kers| (|List| (|Kernel| |#2|))) (|:| |vals| (|List| |#2|))) |#2| (|Symbol|)) "\\spad{rischNormalize(f, x)} returns \\spad{[g, [k1,...,kn], [h1,...,hn]]} such that \\spad{g = normalize(f, x)} and each \\spad{ki} was rewritten as \\spad{hi} during the normalization.")) (|normalize| ((|#2| |#2| (|Symbol|)) "\\spad{normalize(f, x)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{normalize(f)} rewrites \\spad{f} using the least possible number of real algebraically independent kernels."))) NIL NIL -(-230 |Coef| UTS ULS) +(-232 |Coef| UTS ULS) ((|constructor| (NIL "\\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return \"failed\" when this is not possible.")) (|acsch| ((|#3| |#3|) "\\spad{acsch(z)} returns the inverse hyperbolic cosecant of Laurent series \\spad{z}.")) (|asech| ((|#3| |#3|) "\\spad{asech(z)} returns the inverse hyperbolic secant of Laurent series \\spad{z}.")) (|acoth| ((|#3| |#3|) "\\spad{acoth(z)} returns the inverse hyperbolic cotangent of Laurent series \\spad{z}.")) (|atanh| ((|#3| |#3|) "\\spad{atanh(z)} returns the inverse hyperbolic tangent of Laurent series \\spad{z}.")) (|acosh| ((|#3| |#3|) "\\spad{acosh(z)} returns the inverse hyperbolic cosine of Laurent series \\spad{z}.")) (|asinh| ((|#3| |#3|) "\\spad{asinh(z)} returns the inverse hyperbolic sine of Laurent series \\spad{z}.")) (|csch| ((|#3| |#3|) "\\spad{csch(z)} returns the hyperbolic cosecant of Laurent series \\spad{z}.")) (|sech| ((|#3| |#3|) "\\spad{sech(z)} returns the hyperbolic secant of Laurent series \\spad{z}.")) (|coth| ((|#3| |#3|) "\\spad{coth(z)} returns the hyperbolic cotangent of Laurent series \\spad{z}.")) (|tanh| ((|#3| |#3|) "\\spad{tanh(z)} returns the hyperbolic tangent of Laurent series \\spad{z}.")) (|cosh| ((|#3| |#3|) "\\spad{cosh(z)} returns the hyperbolic cosine of Laurent series \\spad{z}.")) (|sinh| ((|#3| |#3|) "\\spad{sinh(z)} returns the hyperbolic sine of Laurent series \\spad{z}.")) (|acsc| ((|#3| |#3|) "\\spad{acsc(z)} returns the arc-cosecant of Laurent series \\spad{z}.")) (|asec| ((|#3| |#3|) "\\spad{asec(z)} returns the arc-secant of Laurent series \\spad{z}.")) (|acot| ((|#3| |#3|) "\\spad{acot(z)} returns the arc-cotangent of Laurent series \\spad{z}.")) (|atan| ((|#3| |#3|) "\\spad{atan(z)} returns the arc-tangent of Laurent series \\spad{z}.")) (|acos| ((|#3| |#3|) "\\spad{acos(z)} returns the arc-cosine of Laurent series \\spad{z}.")) (|asin| ((|#3| |#3|) "\\spad{asin(z)} returns the arc-sine of Laurent series \\spad{z}.")) (|csc| ((|#3| |#3|) "\\spad{csc(z)} returns the cosecant of Laurent series \\spad{z}.")) (|sec| ((|#3| |#3|) "\\spad{sec(z)} returns the secant of Laurent series \\spad{z}.")) (|cot| ((|#3| |#3|) "\\spad{cot(z)} returns the cotangent of Laurent series \\spad{z}.")) (|tan| ((|#3| |#3|) "\\spad{tan(z)} returns the tangent of Laurent series \\spad{z}.")) (|cos| ((|#3| |#3|) "\\spad{cos(z)} returns the cosine of Laurent series \\spad{z}.")) (|sin| ((|#3| |#3|) "\\spad{sin(z)} returns the sine of Laurent series \\spad{z}.")) (|log| ((|#3| |#3|) "\\spad{log(z)} returns the logarithm of Laurent series \\spad{z}.")) (|exp| ((|#3| |#3|) "\\spad{exp(z)} returns the exponential of Laurent series \\spad{z}.")) (** ((|#3| |#3| (|Fraction| (|Integer|))) "\\spad{s ** r} raises a Laurent series \\spad{s} to a rational power \\spad{r}"))) NIL -((|HasCategory| |#1| (QUOTE (-309)))) -(-231 |Coef| ULS UPXS EFULS) +((|HasCategory| |#1| (QUOTE (-311)))) +(-233 |Coef| ULS UPXS EFULS) ((|constructor| (NIL "\\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return \"failed\" when this is not possible.")) (|acsch| ((|#3| |#3|) "\\spad{acsch(z)} returns the inverse hyperbolic cosecant of a Puiseux series \\spad{z}.")) (|asech| ((|#3| |#3|) "\\spad{asech(z)} returns the inverse hyperbolic secant of a Puiseux series \\spad{z}.")) (|acoth| ((|#3| |#3|) "\\spad{acoth(z)} returns the inverse hyperbolic cotangent of a Puiseux series \\spad{z}.")) (|atanh| ((|#3| |#3|) "\\spad{atanh(z)} returns the inverse hyperbolic tangent of a Puiseux series \\spad{z}.")) (|acosh| ((|#3| |#3|) "\\spad{acosh(z)} returns the inverse hyperbolic cosine of a Puiseux series \\spad{z}.")) (|asinh| ((|#3| |#3|) "\\spad{asinh(z)} returns the inverse hyperbolic sine of a Puiseux series \\spad{z}.")) (|csch| ((|#3| |#3|) "\\spad{csch(z)} returns the hyperbolic cosecant of a Puiseux series \\spad{z}.")) (|sech| ((|#3| |#3|) "\\spad{sech(z)} returns the hyperbolic secant of a Puiseux series \\spad{z}.")) (|coth| ((|#3| |#3|) "\\spad{coth(z)} returns the hyperbolic cotangent of a Puiseux series \\spad{z}.")) (|tanh| ((|#3| |#3|) "\\spad{tanh(z)} returns the hyperbolic tangent of a Puiseux series \\spad{z}.")) (|cosh| ((|#3| |#3|) "\\spad{cosh(z)} returns the hyperbolic cosine of a Puiseux series \\spad{z}.")) (|sinh| ((|#3| |#3|) "\\spad{sinh(z)} returns the hyperbolic sine of a Puiseux series \\spad{z}.")) (|acsc| ((|#3| |#3|) "\\spad{acsc(z)} returns the arc-cosecant of a Puiseux series \\spad{z}.")) (|asec| ((|#3| |#3|) "\\spad{asec(z)} returns the arc-secant of a Puiseux series \\spad{z}.")) (|acot| ((|#3| |#3|) "\\spad{acot(z)} returns the arc-cotangent of a Puiseux series \\spad{z}.")) (|atan| ((|#3| |#3|) "\\spad{atan(z)} returns the arc-tangent of a Puiseux series \\spad{z}.")) (|acos| ((|#3| |#3|) "\\spad{acos(z)} returns the arc-cosine of a Puiseux series \\spad{z}.")) (|asin| ((|#3| |#3|) "\\spad{asin(z)} returns the arc-sine of a Puiseux series \\spad{z}.")) (|csc| ((|#3| |#3|) "\\spad{csc(z)} returns the cosecant of a Puiseux series \\spad{z}.")) (|sec| ((|#3| |#3|) "\\spad{sec(z)} returns the secant of a Puiseux series \\spad{z}.")) (|cot| ((|#3| |#3|) "\\spad{cot(z)} returns the cotangent of a Puiseux series \\spad{z}.")) (|tan| ((|#3| |#3|) "\\spad{tan(z)} returns the tangent of a Puiseux series \\spad{z}.")) (|cos| ((|#3| |#3|) "\\spad{cos(z)} returns the cosine of a Puiseux series \\spad{z}.")) (|sin| ((|#3| |#3|) "\\spad{sin(z)} returns the sine of a Puiseux series \\spad{z}.")) (|log| ((|#3| |#3|) "\\spad{log(z)} returns the logarithm of a Puiseux series \\spad{z}.")) (|exp| ((|#3| |#3|) "\\spad{exp(z)} returns the exponential of a Puiseux series \\spad{z}.")) (** ((|#3| |#3| (|Fraction| (|Integer|))) "\\spad{z ** r} raises a Puiseaux series \\spad{z} to a rational power \\spad{r}"))) NIL -((|HasCategory| |#1| (QUOTE (-309)))) -(-232) +((|HasCategory| |#1| (QUOTE (-311)))) +(-234) ((|constructor| (NIL "This domains an expresion as elaborated by the interpreter. See Also:")) (|getOperands| (((|Union| (|List| $) "failed") $) "\\spad{getOperands(e)} returns the list of operands in `e',{} assuming it is a call form.")) (|getOperator| (((|Union| (|Identifier|) "failed") $) "\\spad{getOperator(e)} retrieves the operator being invoked in `e',{} when `e' is an expression.")) (|callForm?| (((|Boolean|) $) "\\spad{callForm?(e)} is \\spad{true} when `e' is a call expression.")) (|getIdentifier| (((|Union| (|Identifier|) "failed") $) "\\spad{getIdentifier(e)} retrieves the name of the variable `e'.")) (|variable?| (((|Boolean|) $) "\\spad{variable?(e)} returns \\spad{true} if `e' is a variable.")) (|getConstant| (((|Union| (|SExpression|) "failed") $) "\\spad{getConstant(e)} retrieves the constant value of `e'e.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(e)} returns \\spad{true} if `e' is a constant.")) (|type| (((|Syntax|) $) "\\spad{type(e)} returns the type of the expression as computed by the interpreter."))) NIL NIL -(-233) +(-235) ((|environment| (((|Environment|) $) "\\spad{environment(x)} returns the environment of the elaboration \\spad{x}.")) (|typeForm| (((|InternalTypeForm|) $) "\\spad{typeForm(x)} returns the type form of the elaboration \\spad{x}.")) (|irForm| (((|InternalRepresentationForm|) $) "\\spad{irForm(x)} returns the internal representation form of the elaboration \\spad{x}.")) (|elaboration| (($ (|InternalRepresentationForm|) (|InternalTypeForm|) (|Environment|)) "\\spad{elaboration(ir,ty,env)} construct an elaboration object for for the internal representation form \\spad{ir},{} with type \\spad{ty},{} and environment \\spad{env}."))) NIL NIL -(-234 A S) +(-236 A S) ((|constructor| (NIL "An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \\spadtype{FlexibleArray} for an exception.")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(u)} destructively removes duplicates from \\spad{u}.")) (|select!| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select!(p,u)} destructively changes \\spad{u} by keeping only values \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})}.")) (|merge!| (($ $ $) "\\spad{merge!(u,v)} destructively merges \\spad{u} and \\spad{v} in ascending order.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $ $) "\\spad{merge!(p,u,v)} destructively merges \\spad{u} and \\spad{v} using predicate \\spad{p}.")) (|insert!| (($ $ $ (|Integer|)) "\\spad{insert!(v,u,i)} destructively inserts aggregate \\spad{v} into \\spad{u} at position \\spad{i}.") (($ |#2| $ (|Integer|)) "\\spad{insert!(x,u,i)} destructively inserts \\spad{x} into \\spad{u} at position \\spad{i}.")) (|remove!| (($ |#2| $) "\\spad{remove!(x,u)} destructively removes all values \\spad{x} from \\spad{u}.") (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove!(p,u)} destructively removes all elements \\spad{x} of \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.")) (|delete!| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete!(u,i..j)} destructively deletes elements \\spad{u}.\\spad{i} through \\spad{u}.\\spad{j}.") (($ $ (|Integer|)) "\\spad{delete!(u,i)} destructively deletes the \\axiom{\\spad{i}}th element of \\spad{u}.")) (|concat!| (($ $ $) "\\spad{concat!(u,v)} destructively appends \\spad{v} to the end of \\spad{u}. \\spad{v} is unchanged") (($ $ |#2|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}."))) NIL -((|HasCategory| |#2| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-1007)))) -(-235 S) +((|HasCategory| |#2| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-1012)))) +(-237 S) ((|constructor| (NIL "An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \\spadtype{FlexibleArray} for an exception.")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(u)} destructively removes duplicates from \\spad{u}.")) (|select!| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select!(p,u)} destructively changes \\spad{u} by keeping only values \\spad{x} such that \\axiom{\\spad{p}(\\spad{x})}.")) (|merge!| (($ $ $) "\\spad{merge!(u,v)} destructively merges \\spad{u} and \\spad{v} in ascending order.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $ $) "\\spad{merge!(p,u,v)} destructively merges \\spad{u} and \\spad{v} using predicate \\spad{p}.")) (|insert!| (($ $ $ (|Integer|)) "\\spad{insert!(v,u,i)} destructively inserts aggregate \\spad{v} into \\spad{u} at position \\spad{i}.") (($ |#1| $ (|Integer|)) "\\spad{insert!(x,u,i)} destructively inserts \\spad{x} into \\spad{u} at position \\spad{i}.")) (|remove!| (($ |#1| $) "\\spad{remove!(x,u)} destructively removes all values \\spad{x} from \\spad{u}.") (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove!(p,u)} destructively removes all elements \\spad{x} of \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}.")) (|delete!| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete!(u,i..j)} destructively deletes elements \\spad{u}.\\spad{i} through \\spad{u}.\\spad{j}.") (($ $ (|Integer|)) "\\spad{delete!(u,i)} destructively deletes the \\axiom{\\spad{i}}th element of \\spad{u}.")) (|concat!| (($ $ $) "\\spad{concat!(u,v)} destructively appends \\spad{v} to the end of \\spad{u}. \\spad{v} is unchanged") (($ $ |#1|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}."))) -((-3979 . T)) +((-3990 . T)) NIL -(-236 S) +(-238 S) ((|constructor| (NIL "Category for the elementary functions.")) (** (($ $ $) "\\spad{x**y} returns \\spad{x} to the power \\spad{y}.")) (|exp| (($ $) "\\spad{exp(x)} returns \\%\\spad{e} to the power \\spad{x}.")) (|log| (($ $) "\\spad{log(x)} returns the natural logarithm of \\spad{x}."))) NIL NIL -(-237) +(-239) ((|constructor| (NIL "Category for the elementary functions.")) (** (($ $ $) "\\spad{x**y} returns \\spad{x} to the power \\spad{y}.")) (|exp| (($ $) "\\spad{exp(x)} returns \\%\\spad{e} to the power \\spad{x}.")) (|log| (($ $) "\\spad{log(x)} returns the natural logarithm of \\spad{x}."))) NIL NIL -(-238 |Coef| UTS) +(-240 |Coef| UTS) ((|constructor| (NIL "The elliptic functions sn,{} sc and dn are expanded as Taylor series.")) (|sncndn| (((|List| (|Stream| |#1|)) (|Stream| |#1|) |#1|) "\\spad{sncndn(s,c)} is used internally.")) (|dn| ((|#2| |#2| |#1|) "\\spad{dn(x,k)} expands the elliptic function dn as a Taylor \\indented{1}{series.}")) (|cn| ((|#2| |#2| |#1|) "\\spad{cn(x,k)} expands the elliptic function cn as a Taylor \\indented{1}{series.}")) (|sn| ((|#2| |#2| |#1|) "\\spad{sn(x,k)} expands the elliptic function sn as a Taylor \\indented{1}{series.}"))) NIL NIL -(-239 S T$) +(-241 S T$) ((|constructor| (NIL "An eltable over domains \\spad{S} and \\spad{T} is a structure which can be viewed as a function from \\spad{S} to \\spad{T}. Examples of eltable structures range from data structures,{} \\spadignore{e.g.} those of type \\spadtype{List},{} to algebraic structures,{} \\spadignore{e.g.} \\spadtype{Polynomial}.")) (|elt| ((|#2| $ |#1|) "\\spad{elt(u,s)} (also written: \\spad{u.s}) returns the value of \\spad{u} at \\spad{s}. Error: if \\spad{u} is not defined at \\spad{s}."))) NIL NIL -(-240 S |Dom| |Im|) +(-242 S |Dom| |Im|) ((|constructor| (NIL "An eltable aggregate is one which can be viewed as a function. For example,{} the list \\axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\\em Dom} to an image domain {\\em Im}.")) (|qsetelt!| ((|#3| $ |#2| |#3|) "\\spad{qsetelt!(u,x,y)} sets the image of \\axiom{\\spad{x}} to be \\axiom{\\spad{y}} under \\axiom{\\spad{u}},{} without checking that \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If such a check is required use the function \\axiom{setelt}.")) (|setelt| ((|#3| $ |#2| |#3|) "\\spad{setelt(u,x,y)} sets the image of \\spad{x} to be \\spad{y} under \\spad{u},{} assuming \\spad{x} is in the domain of \\spad{u}. Error: if \\spad{x} is not in the domain of \\spad{u}.")) (|qelt| ((|#3| $ |#2|) "\\spad{qelt(u, x)} applies \\axiom{\\spad{u}} to \\axiom{\\spad{x}} without checking whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If \\axiom{\\spad{x}} is not in the domain of \\axiom{\\spad{u}} a memory-access violation may occur. If a check on whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}} is required,{} use the function \\axiom{elt}.")) (|elt| ((|#3| $ |#2| |#3|) "\\spad{elt(u, x, y)} applies \\spad{u} to \\spad{x} if \\spad{x} is in the domain of \\spad{u},{} and returns \\spad{y} otherwise. For example,{} if \\spad{u} is a polynomial in \\axiom{\\spad{x}} over the rationals,{} \\axiom{elt(\\spad{u},{}\\spad{n},{}0)} may define the coefficient of \\axiom{\\spad{x}} to the power \\spad{n},{} returning 0 when \\spad{n} is out of range."))) NIL -((|HasAttribute| |#1| (QUOTE -3979))) -(-241 |Dom| |Im|) +((|HasAttribute| |#1| (QUOTE -3990))) +(-243 |Dom| |Im|) ((|constructor| (NIL "An eltable aggregate is one which can be viewed as a function. For example,{} the list \\axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\\em Dom} to an image domain {\\em Im}.")) (|qsetelt!| ((|#2| $ |#1| |#2|) "\\spad{qsetelt!(u,x,y)} sets the image of \\axiom{\\spad{x}} to be \\axiom{\\spad{y}} under \\axiom{\\spad{u}},{} without checking that \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If such a check is required use the function \\axiom{setelt}.")) (|setelt| ((|#2| $ |#1| |#2|) "\\spad{setelt(u,x,y)} sets the image of \\spad{x} to be \\spad{y} under \\spad{u},{} assuming \\spad{x} is in the domain of \\spad{u}. Error: if \\spad{x} is not in the domain of \\spad{u}.")) (|qelt| ((|#2| $ |#1|) "\\spad{qelt(u, x)} applies \\axiom{\\spad{u}} to \\axiom{\\spad{x}} without checking whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}}. If \\axiom{\\spad{x}} is not in the domain of \\axiom{\\spad{u}} a memory-access violation may occur. If a check on whether \\axiom{\\spad{x}} is in the domain of \\axiom{\\spad{u}} is required,{} use the function \\axiom{elt}.")) (|elt| ((|#2| $ |#1| |#2|) "\\spad{elt(u, x, y)} applies \\spad{u} to \\spad{x} if \\spad{x} is in the domain of \\spad{u},{} and returns \\spad{y} otherwise. For example,{} if \\spad{u} is a polynomial in \\axiom{\\spad{x}} over the rationals,{} \\axiom{elt(\\spad{u},{}\\spad{n},{}0)} may define the coefficient of \\axiom{\\spad{x}} to the power \\spad{n},{} returning 0 when \\spad{n} is out of range."))) NIL NIL -(-242 S R |Mod| -2025 -3501 |exactQuo|) +(-244 S R |Mod| -2033 -3512 |exactQuo|) ((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{ModularRing},{} \\spadtype{ModularField}")) (|inv| (($ $) "\\spad{inv(x)} \\undocumented")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} \\undocumented")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#2| |#3|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#2| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#3| $) "\\spad{modulus(x)} \\undocumented"))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-243) +(-245) ((|constructor| (NIL "Entire Rings (non-commutative Integral Domains),{} \\spadignore{i.e.} a ring not necessarily commutative which has no zero divisors. \\blankline")) (|noZeroDivisors| ((|attribute|) "if a product is zero then one of the factors must be zero."))) -((-3971 . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-244) +(-246) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: March 18,{} 2010. An `Environment' is a stack of scope.")) (|categoryFrame| (($) "the current category environment in the interpreter.")) (|interactiveEnv| (($) "the current interactive environment in effect.")) (|currentEnv| (($) "the current normal environment in effect.")) (|putProperties| (($ (|Identifier|) (|List| (|Property|)) $) "\\spad{putProperties(n,props,e)} set the list of properties of \\spad{n} to \\spad{props} in \\spad{e}.")) (|getProperties| (((|List| (|Property|)) (|Identifier|) $) "\\spad{getBinding(n,e)} returns the list of properties of \\spad{n} in \\spad{e}.")) (|putProperty| (($ (|Identifier|) (|Identifier|) (|SExpression|) $) "\\spad{putProperty(n,p,v,e)} binds the property \\spad{(p,v)} to \\spad{n} in the topmost scope of \\spad{e}.")) (|getProperty| (((|Maybe| (|SExpression|)) (|Identifier|) (|Identifier|) $) "\\spad{getProperty(n,p,e)} returns the value of property with name \\spad{p} for the symbol \\spad{n} in environment \\spad{e}. Otherwise,{} \\spad{nothing}.")) (|scopes| (((|List| (|Scope|)) $) "\\spad{scopes(e)} returns the stack of scopes in environment \\spad{e}.")) (|empty| (($) "\\spad{empty()} constructs an empty environment"))) NIL NIL -(-245 R) +(-247 R) ((|constructor| (NIL "This is a package for the exact computation of eigenvalues and eigenvectors. This package can be made to work for matrices with coefficients which are rational functions over a ring where we can factor polynomials. Rational eigenvalues are always explicitly computed while the non-rational ones are expressed in terms of their minimal polynomial.")) (|eigenvectors| (((|List| (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |eigmult| (|NonNegativeInteger|)) (|:| |eigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|))))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvectors(m)} returns the eigenvalues and eigenvectors for the matrix \\spad{m}. The rational eigenvalues and the correspondent eigenvectors are explicitely computed,{} while the non rational ones are given via their minimal polynomial and the corresponding eigenvectors are expressed in terms of a \"generic\" root of such a polynomial.")) (|generalizedEigenvectors| (((|List| (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |geneigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|))))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{generalizedEigenvectors(m)} returns the generalized eigenvectors of the matrix \\spad{m}.")) (|generalizedEigenvector| (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Record| (|:| |eigval| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|:| |eigmult| (|NonNegativeInteger|)) (|:| |eigvec| (|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{generalizedEigenvector(eigen,m)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \\spad{eigen},{} as returned by the function eigenvectors.") (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|))) (|Matrix| (|Fraction| (|Polynomial| |#1|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generalizedEigenvector(alpha,m,k,g)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \\spad{alpha}. The integers \\spad{k} and \\spad{g} are respectively the algebraic and the geometric multiplicity of tye eigenvalue \\spad{alpha}. \\spad{alpha} can be either rational or not. In the seconda case apha is the minimal polynomial of the eigenvalue.")) (|eigenvector| (((|List| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvector(eigval,m)} returns the eigenvectors belonging to the eigenvalue \\spad{eigval} for the matrix \\spad{m}.")) (|eigenvalues| (((|List| (|Union| (|Fraction| (|Polynomial| |#1|)) (|SuchThat| (|Symbol|) (|Polynomial| |#1|)))) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eigenvalues(m)} returns the eigenvalues of the matrix \\spad{m} which are expressible as rational functions over the rational numbers.")) (|characteristicPolynomial| (((|Polynomial| |#1|) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{characteristicPolynomial(m)} returns the characteristicPolynomial of the matrix \\spad{m} using a new generated symbol symbol as the main variable.") (((|Polynomial| |#1|) (|Matrix| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{characteristicPolynomial(m,var)} returns the characteristicPolynomial of the matrix \\spad{m} using the symbol \\spad{var} as the main variable."))) NIL NIL -(-246 S) +(-248 S) ((|constructor| (NIL "Equations as mathematical objects. All properties of the basis domain,{} \\spadignore{e.g.} being an abelian group are carried over the equation domain,{} by performing the structural operations on the left and on the right hand side.")) (|subst| (($ $ $) "\\spad{subst(eq1,eq2)} substitutes \\spad{eq2} into both sides of \\spad{eq1} the lhs of \\spad{eq2} should be a kernel")) (|inv| (($ $) "\\spad{inv(x)} returns the multiplicative inverse of \\spad{x}.")) (/ (($ $ $) "\\spad{e1/e2} produces a new equation by dividing the left and right hand sides of equations \\spad{e1} and \\spad{e2}.")) (|factorAndSplit| (((|List| $) $) "\\spad{factorAndSplit(eq)} make the right hand side 0 and factors the new left hand side. Each factor is equated to 0 and put into the resulting list without repetitions.")) (|rightOne| (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side.") (((|Union| $ "failed") $) "\\spad{rightOne(eq)} divides by the right hand side,{} if possible.")) (|leftOne| (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side.") (((|Union| $ "failed") $) "\\spad{leftOne(eq)} divides by the left hand side,{} if possible.")) (* (($ $ |#1|) "\\spad{eqn*x} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.") (($ |#1| $) "\\spad{x*eqn} produces a new equation by multiplying both sides of equation eqn by \\spad{x}.")) (- (($ $ |#1|) "\\spad{eqn-x} produces a new equation by subtracting \\spad{x} from both sides of equation eqn.") (($ |#1| $) "\\spad{x-eqn} produces a new equation by subtracting both sides of equation eqn from \\spad{x}.")) (|rightZero| (($ $) "\\spad{rightZero(eq)} subtracts the right hand side.")) (|leftZero| (($ $) "\\spad{leftZero(eq)} subtracts the left hand side.")) (+ (($ $ |#1|) "\\spad{eqn+x} produces a new equation by adding \\spad{x} to both sides of equation eqn.") (($ |#1| $) "\\spad{x+eqn} produces a new equation by adding \\spad{x} to both sides of equation eqn.")) (|eval| (($ $ (|List| $)) "\\spad{eval(eqn, [x1=v1, ... xn=vn])} replaces \\spad{xi} by \\spad{vi} in equation \\spad{eqn}.") (($ $ $) "\\spad{eval(eqn, x=f)} replaces \\spad{x} by \\spad{f} in equation \\spad{eqn}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,eqn)} constructs a new equation by applying \\spad{f} to both sides of \\spad{eqn}.")) (|rhs| ((|#1| $) "\\spad{rhs(eqn)} returns the right hand side of equation \\spad{eqn}.")) (|lhs| ((|#1| $) "\\spad{lhs(eqn)} returns the left hand side of equation \\spad{eqn}.")) (|swap| (($ $) "\\spad{swap(eq)} interchanges left and right hand side of equation \\spad{eq}.")) (|equation| (($ |#1| |#1|) "\\spad{equation(a,b)} creates an equation.")) (= (($ |#1| |#1|) "\\spad{a=b} creates an equation."))) -((-3975 OR (|has| |#1| (-956)) (|has| |#1| (-408))) (-3972 |has| |#1| (-956)) (-3973 |has| |#1| (-956))) -((|HasCategory| |#1| (QUOTE (-309))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-956)))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-956))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (OR (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-956)))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-956)))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-956)))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-956)))) (OR (|HasCategory| |#1| (QUOTE (-408))) (|HasCategory| |#1| (QUOTE (-660)))) (|HasCategory| |#1| (QUOTE (-408))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-408))) (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-956))) (|HasCategory| |#1| (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-1007)))) (OR (|HasCategory| |#1| (QUOTE (-408))) (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#1| (QUOTE (-1017)))) (|HasCategory| |#1| (|%list| (QUOTE -449) (QUOTE (-1081)) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-251))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-408)))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-660)))) (OR (|HasCategory| |#1| (QUOTE (-408))) (|HasCategory| |#1| (QUOTE (-956)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-660)))) -(-247 S R) +((-3986 OR (|has| |#1| (-961)) (|has| |#1| (-410))) (-3983 |has| |#1| (-961)) (-3984 |has| |#1| (-961))) +((|HasCategory| |#1| (QUOTE (-311))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311)))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-961))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (OR (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#1| (QUOTE (-663)))) (|HasCategory| |#1| (QUOTE (-410))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-961))) (|HasCategory| |#1| (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-1012)))) (OR (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#1| (QUOTE (-1024)))) (|HasCategory| |#1| (|%list| (QUOTE -452) (QUOTE (-1088)) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-253))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-410)))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-663)))) (OR (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-663)))) +(-249 S R) ((|constructor| (NIL "This package provides operations for mapping the sides of equations.")) (|map| (((|Equation| |#2|) (|Mapping| |#2| |#1|) (|Equation| |#1|)) "\\spad{map(f,eq)} returns an equation where \\spad{f} is applied to the sides of \\spad{eq}"))) NIL NIL -(-248 |Key| |Entry|) +(-250 |Key| |Entry|) ((|constructor| (NIL "This domain provides tables where the keys are compared using \\spadfun{eq?}. Thus keys are considered equal only if they are the same instance of a structure."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-1007))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) -(-249) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-1012))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) +(-251) ((|constructor| (NIL "ErrorFunctions implements error functions callable from the system interpreter. Typically,{} these functions would be called in user functions. The simple forms of the functions take one argument which is either a string (an error message) or a list of strings which all together make up a message. The list can contain formatting codes (see below). The more sophisticated versions takes two arguments where the first argument is the name of the function from which the error was invoked and the second argument is either a string or a list of strings,{} as above. When you use the one argument version in an interpreter function,{} the system will automatically insert the name of the function as the new first argument. Thus in the user interpreter function \\indented{2}{\\spad{f x == if x < 0 then error \"negative argument\" else x}} the call to error will actually be of the form \\indented{2}{\\spad{error(\"f\",\"negative argument\")}} because the interpreter will have created a new first argument. \\blankline Formatting codes: error messages may contain the following formatting codes (they should either start or end a string or else have blanks around them): \\indented{3}{\\spad{\\%l}\\space{6}start a new line} \\indented{3}{\\spad{\\%b}\\space{6}start printing in a bold font (where available)} \\indented{3}{\\spad{\\%d}\\space{6}stop\\space{2}printing in a bold font (where available)} \\indented{3}{\\spad{ \\%ceon}\\space{2}start centering message lines} \\indented{3}{\\spad{\\%ceoff}\\space{2}stop\\space{2}centering message lines} \\indented{3}{\\spad{\\%rjon}\\space{3}start displaying lines \"ragged left\"} \\indented{3}{\\spad{\\%rjoff}\\space{2}stop\\space{2}displaying lines \"ragged left\"} \\indented{3}{\\spad{\\%i}\\space{6}indent\\space{3}following lines 3 additional spaces} \\indented{3}{\\spad{\\%u}\\space{6}unindent following lines 3 additional spaces} \\indented{3}{\\spad{\\%xN}\\space{5}insert \\spad{N} blanks (eg,{} \\spad{\\%x10} inserts 10 blanks)} \\blankline")) (|error| (((|Exit|) (|String|) (|List| (|String|))) "\\spad{error(nam,lmsg)} displays error messages \\spad{lmsg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|String|) (|String|)) "\\spad{error(nam,msg)} displays error message \\spad{msg} preceded by a message containing the name \\spad{nam} of the function in which the error is contained.") (((|Exit|) (|List| (|String|))) "\\spad{error(lmsg)} displays error message \\spad{lmsg} and terminates.") (((|Exit|) (|String|)) "\\spad{error(msg)} displays error message \\spad{msg} and terminates."))) NIL NIL -(-250 S) +(-252 S) ((|constructor| (NIL "An expression space is a set which is closed under certain operators.")) (|odd?| (((|Boolean|) $) "\\spad{odd? x} is \\spad{true} if \\spad{x} is an odd integer.")) (|even?| (((|Boolean|) $) "\\spad{even? x} is \\spad{true} if \\spad{x} is an even integer.")) (|definingPolynomial| (($ $) "\\spad{definingPolynomial(x)} returns an expression \\spad{p} such that \\spad{p(x) = 0}.")) (|minPoly| (((|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{minPoly(k)} returns \\spad{p} such that \\spad{p(k) = 0}.")) (|eval| (($ $ (|BasicOperator|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|BasicOperator|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.")) (|freeOf?| (((|Boolean|) $ (|Symbol|)) "\\spad{freeOf?(x, s)} tests if \\spad{x} does not contain any operator whose name is \\spad{s}.") (((|Boolean|) $ $) "\\spad{freeOf?(x, y)} tests if \\spad{x} does not contain any occurrence of \\spad{y},{} where \\spad{y} is a single kernel.")) (|map| (($ (|Mapping| $ $) (|Kernel| $)) "\\spad{map(f, k)} returns \\spad{op(f(x1),...,f(xn))} where \\spad{k = op(x1,...,xn)}.")) (|kernel| (($ (|BasicOperator|) (|List| $)) "\\spad{kernel(op, [f1,...,fn])} constructs \\spad{op(f1,...,fn)} without evaluating it.") (($ (|BasicOperator|) $) "\\spad{kernel(op, x)} constructs \\spad{op}(\\spad{x}) without evaluating it.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(x, s)} tests if \\spad{x} is a kernel and is the name of its operator is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(x, op)} tests if \\spad{x} is a kernel and is its operator is op.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} tests if \\% accepts \\spad{op} as applicable to its elements.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\%.")) (|operators| (((|List| (|BasicOperator|)) $) "\\spad{operators(f)} returns all the basic operators appearing in \\spad{f},{} no matter what their levels are.")) (|tower| (((|List| (|Kernel| $)) $) "\\spad{tower(f)} returns all the kernels appearing in \\spad{f},{} no matter what their levels are.")) (|kernels| (((|List| (|Kernel| $)) $) "\\spad{kernels(f)} returns the list of all the top-level kernels appearing in \\spad{f},{} but not the ones appearing in the arguments of the top-level kernels.")) (|mainKernel| (((|Union| (|Kernel| $) "failed") $) "\\spad{mainKernel(f)} returns a kernel of \\spad{f} with maximum nesting level,{} or if \\spad{f} has no kernels (\\spadignore{i.e.} \\spad{f} is a constant).")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(f)} returns the highest nesting level appearing in \\spad{f}. Constants have height 0. Symbols have height 1. For any operator op and expressions \\spad{f1},{}...,{}fn,{} \\spad{op(f1,...,fn)} has height equal to \\spad{1 + max(height(f1),...,height(fn))}.")) (|distribute| (($ $ $) "\\spad{distribute(f, g)} expands all the kernels in \\spad{f} that contain \\spad{g} in their arguments and that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or a \\spadfunFrom{paren}{ExpressionSpace} expression.") (($ $) "\\spad{distribute(f)} expands all the kernels in \\spad{f} that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or \\spadfunFrom{paren}{ExpressionSpace} expression.")) (|paren| (($ (|List| $)) "\\spad{paren([f1,...,fn])} returns \\spad{(f1,...,fn)}. This prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(paren [x, 2])} returns the formal kernel \\spad{atan((x, 2))}.") (($ $) "\\spad{paren(f)} returns (\\spad{f}). This prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(paren 1)} returns the formal kernel log((1)).")) (|box| (($ (|List| $)) "\\spad{box([f1,...,fn])} returns \\spad{(f1,...,fn)} with a 'box' around them that prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(box [x, 2])} returns the formal kernel \\spad{atan(x, 2)}.") (($ $) "\\spad{box(f)} returns \\spad{f} with a 'box' around it that prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(box 1)} returns the formal kernel log(1).")) (|subst| (($ $ (|List| (|Kernel| $)) (|List| $)) "\\spad{subst(f, [k1...,kn], [g1,...,gn])} replaces the kernels \\spad{k1},{}...,{}kn by \\spad{g1},{}...,{}gn formally in \\spad{f}.") (($ $ (|List| (|Equation| $))) "\\spad{subst(f, [k1 = g1,...,kn = gn])} replaces the kernels \\spad{k1},{}...,{}kn by \\spad{g1},{}...,{}gn formally in \\spad{f}.") (($ $ (|Equation| $)) "\\spad{subst(f, k = g)} replaces the kernel \\spad{k} by \\spad{g} formally in \\spad{f}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op,[x1,...,xn])} or \\spad{op}([\\spad{x1},{}...,{}xn]) applies the \\spad{n}-ary operator \\spad{op} to \\spad{x1},{}...,{}xn.") (($ (|BasicOperator|) $ $ $ $) "\\spad{elt(op,x,y,z,t)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z},{} \\spad{t}) applies the 4-ary operator \\spad{op} to \\spad{x},{} \\spad{y},{} \\spad{z} and \\spad{t}.") (($ (|BasicOperator|) $ $ $) "\\spad{elt(op,x,y,z)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z}) applies the ternary operator \\spad{op} to \\spad{x},{} \\spad{y} and \\spad{z}.") (($ (|BasicOperator|) $ $) "\\spad{elt(op,x,y)} or \\spad{op}(\\spad{x},{} \\spad{y}) applies the binary operator \\spad{op} to \\spad{x} and \\spad{y}.") (($ (|BasicOperator|) $) "\\spad{elt(op,x)} or \\spad{op}(\\spad{x}) applies the unary operator \\spad{op} to \\spad{x}."))) NIL -((|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-956)))) -(-251) +((|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-961)))) +(-253) ((|constructor| (NIL "An expression space is a set which is closed under certain operators.")) (|odd?| (((|Boolean|) $) "\\spad{odd? x} is \\spad{true} if \\spad{x} is an odd integer.")) (|even?| (((|Boolean|) $) "\\spad{even? x} is \\spad{true} if \\spad{x} is an even integer.")) (|definingPolynomial| (($ $) "\\spad{definingPolynomial(x)} returns an expression \\spad{p} such that \\spad{p(x) = 0}.")) (|minPoly| (((|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{minPoly(k)} returns \\spad{p} such that \\spad{p(k) = 0}.")) (|eval| (($ $ (|BasicOperator|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|BasicOperator|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|BasicOperator|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ $)) "\\spad{eval(x, s, f)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, f)} replaces every \\spad{s(a1,..,am)} in \\spad{x} by \\spad{f(a1,..,am)} for any \\spad{a1},{}...,{}\\spad{am}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}\\spad{an}.") (($ $ (|List| (|Symbol|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [f1,...,fm])} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.")) (|freeOf?| (((|Boolean|) $ (|Symbol|)) "\\spad{freeOf?(x, s)} tests if \\spad{x} does not contain any operator whose name is \\spad{s}.") (((|Boolean|) $ $) "\\spad{freeOf?(x, y)} tests if \\spad{x} does not contain any occurrence of \\spad{y},{} where \\spad{y} is a single kernel.")) (|map| (($ (|Mapping| $ $) (|Kernel| $)) "\\spad{map(f, k)} returns \\spad{op(f(x1),...,f(xn))} where \\spad{k = op(x1,...,xn)}.")) (|kernel| (($ (|BasicOperator|) (|List| $)) "\\spad{kernel(op, [f1,...,fn])} constructs \\spad{op(f1,...,fn)} without evaluating it.") (($ (|BasicOperator|) $) "\\spad{kernel(op, x)} constructs \\spad{op}(\\spad{x}) without evaluating it.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(x, s)} tests if \\spad{x} is a kernel and is the name of its operator is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(x, op)} tests if \\spad{x} is a kernel and is its operator is op.")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} tests if \\% accepts \\spad{op} as applicable to its elements.")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\%.")) (|operators| (((|List| (|BasicOperator|)) $) "\\spad{operators(f)} returns all the basic operators appearing in \\spad{f},{} no matter what their levels are.")) (|tower| (((|List| (|Kernel| $)) $) "\\spad{tower(f)} returns all the kernels appearing in \\spad{f},{} no matter what their levels are.")) (|kernels| (((|List| (|Kernel| $)) $) "\\spad{kernels(f)} returns the list of all the top-level kernels appearing in \\spad{f},{} but not the ones appearing in the arguments of the top-level kernels.")) (|mainKernel| (((|Union| (|Kernel| $) "failed") $) "\\spad{mainKernel(f)} returns a kernel of \\spad{f} with maximum nesting level,{} or if \\spad{f} has no kernels (\\spadignore{i.e.} \\spad{f} is a constant).")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(f)} returns the highest nesting level appearing in \\spad{f}. Constants have height 0. Symbols have height 1. For any operator op and expressions \\spad{f1},{}...,{}fn,{} \\spad{op(f1,...,fn)} has height equal to \\spad{1 + max(height(f1),...,height(fn))}.")) (|distribute| (($ $ $) "\\spad{distribute(f, g)} expands all the kernels in \\spad{f} that contain \\spad{g} in their arguments and that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or a \\spadfunFrom{paren}{ExpressionSpace} expression.") (($ $) "\\spad{distribute(f)} expands all the kernels in \\spad{f} that are formally enclosed by a \\spadfunFrom{box}{ExpressionSpace} or \\spadfunFrom{paren}{ExpressionSpace} expression.")) (|paren| (($ (|List| $)) "\\spad{paren([f1,...,fn])} returns \\spad{(f1,...,fn)}. This prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(paren [x, 2])} returns the formal kernel \\spad{atan((x, 2))}.") (($ $) "\\spad{paren(f)} returns (\\spad{f}). This prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(paren 1)} returns the formal kernel log((1)).")) (|box| (($ (|List| $)) "\\spad{box([f1,...,fn])} returns \\spad{(f1,...,fn)} with a 'box' around them that prevents the \\spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \\spad{atan(box [x, 2])} returns the formal kernel \\spad{atan(x, 2)}.") (($ $) "\\spad{box(f)} returns \\spad{f} with a 'box' around it that prevents \\spad{f} from being evaluated when operators are applied to it. For example,{} \\spad{log(1)} returns 0,{} but \\spad{log(box 1)} returns the formal kernel log(1).")) (|subst| (($ $ (|List| (|Kernel| $)) (|List| $)) "\\spad{subst(f, [k1...,kn], [g1,...,gn])} replaces the kernels \\spad{k1},{}...,{}kn by \\spad{g1},{}...,{}gn formally in \\spad{f}.") (($ $ (|List| (|Equation| $))) "\\spad{subst(f, [k1 = g1,...,kn = gn])} replaces the kernels \\spad{k1},{}...,{}kn by \\spad{g1},{}...,{}gn formally in \\spad{f}.") (($ $ (|Equation| $)) "\\spad{subst(f, k = g)} replaces the kernel \\spad{k} by \\spad{g} formally in \\spad{f}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op,[x1,...,xn])} or \\spad{op}([\\spad{x1},{}...,{}xn]) applies the \\spad{n}-ary operator \\spad{op} to \\spad{x1},{}...,{}xn.") (($ (|BasicOperator|) $ $ $ $) "\\spad{elt(op,x,y,z,t)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z},{} \\spad{t}) applies the 4-ary operator \\spad{op} to \\spad{x},{} \\spad{y},{} \\spad{z} and \\spad{t}.") (($ (|BasicOperator|) $ $ $) "\\spad{elt(op,x,y,z)} or \\spad{op}(\\spad{x},{} \\spad{y},{} \\spad{z}) applies the ternary operator \\spad{op} to \\spad{x},{} \\spad{y} and \\spad{z}.") (($ (|BasicOperator|) $ $) "\\spad{elt(op,x,y)} or \\spad{op}(\\spad{x},{} \\spad{y}) applies the binary operator \\spad{op} to \\spad{x} and \\spad{y}.") (($ (|BasicOperator|) $) "\\spad{elt(op,x)} or \\spad{op}(\\spad{x}) applies the unary operator \\spad{op} to \\spad{x}."))) NIL NIL -(-252 -3078 S) +(-254 -3088 S) ((|constructor| (NIL "This package allows a map from any expression space into any object to be lifted to a kernel over the expression set,{} using a given property of the operator of the kernel.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|String|) (|Kernel| |#1|)) "\\spad{map(f, p, k)} uses the property \\spad{p} of the operator of \\spad{k},{} in order to lift \\spad{f} and apply it to \\spad{k}."))) NIL NIL -(-253 E -3078) +(-255 E -3088) ((|constructor| (NIL "This package allows a mapping \\spad{E} -> \\spad{F} to be lifted to a kernel over \\spad{E}; This lifting can fail if the operator of the kernel cannot be applied in \\spad{F}; Do not use this package with \\spad{E} = \\spad{F},{} since this may drop some properties of the operators.")) (|map| ((|#2| (|Mapping| |#2| |#1|) (|Kernel| |#1|)) "\\spad{map(f, k)} returns \\spad{g = op(f(a1),...,f(an))} where \\spad{k = op(a1,...,an)}."))) NIL NIL -(-254 S) +(-256 S) ((|constructor| (NIL "A constructive euclidean domain,{} \\spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\\spadfun{euclideanSize}) than the divisor. \\blankline Conditional attributes: \\indented{2}{multiplicativeValuation\\tab{25}\\spad{Size(a*b)=Size(a)*Size(b)}} \\indented{2}{additiveValuation\\tab{25}\\spad{Size(a*b)=Size(a)+Size(b)}}")) (|multiEuclidean| (((|Union| (|List| $) "failed") (|List| $) $) "\\spad{multiEuclidean([f1,...,fn],z)} returns a list of coefficients \\spad{[a1, ..., an]} such that \\spad{ z / prod fi = sum aj/fj}. If no such list of coefficients exists,{} \"failed\" is returned.")) (|extendedEuclidean| (((|Union| (|Record| (|:| |coef1| $) (|:| |coef2| $)) "failed") $ $ $) "\\spad{extendedEuclidean(x,y,z)} either returns a record rec where \\spad{rec.coef1*x+rec.coef2*y=z} or returns \"failed\" if \\spad{z} cannot be expressed as a linear combination of \\spad{x} and \\spad{y}.") (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{extendedEuclidean(x,y)} returns a record rec where \\spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a gcd of \\spad{x} and \\spad{y}. The gcd is unique only up to associates if \\spadatt{canonicalUnitNormal} is not asserted. \\spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.")) (|rem| (($ $ $) "\\spad{x rem y} is the same as \\spad{divide(x,y).remainder}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|quo| (($ $ $) "\\spad{x quo y} is the same as \\spad{divide(x,y).quotient}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(x,y)} divides \\spad{x} by \\spad{y} producing a record containing a \\spad{quotient} and \\spad{remainder},{} where the remainder is smaller (see \\spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \\spad{y}.")) (|euclideanSize| (((|NonNegativeInteger|) $) "\\spad{euclideanSize(x)} returns the euclidean size of the element \\spad{x}. Error: if \\spad{x} is zero.")) (|sizeLess?| (((|Boolean|) $ $) "\\spad{sizeLess?(x,y)} tests whether \\spad{x} is strictly smaller than \\spad{y} with respect to the \\spadfunFrom{euclideanSize}{EuclideanDomain}."))) NIL NIL -(-255) +(-257) ((|constructor| (NIL "A constructive euclidean domain,{} \\spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\\spadfun{euclideanSize}) than the divisor. \\blankline Conditional attributes: \\indented{2}{multiplicativeValuation\\tab{25}\\spad{Size(a*b)=Size(a)*Size(b)}} \\indented{2}{additiveValuation\\tab{25}\\spad{Size(a*b)=Size(a)+Size(b)}}")) (|multiEuclidean| (((|Union| (|List| $) "failed") (|List| $) $) "\\spad{multiEuclidean([f1,...,fn],z)} returns a list of coefficients \\spad{[a1, ..., an]} such that \\spad{ z / prod fi = sum aj/fj}. If no such list of coefficients exists,{} \"failed\" is returned.")) (|extendedEuclidean| (((|Union| (|Record| (|:| |coef1| $) (|:| |coef2| $)) "failed") $ $ $) "\\spad{extendedEuclidean(x,y,z)} either returns a record rec where \\spad{rec.coef1*x+rec.coef2*y=z} or returns \"failed\" if \\spad{z} cannot be expressed as a linear combination of \\spad{x} and \\spad{y}.") (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{extendedEuclidean(x,y)} returns a record rec where \\spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a gcd of \\spad{x} and \\spad{y}. The gcd is unique only up to associates if \\spadatt{canonicalUnitNormal} is not asserted. \\spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.")) (|rem| (($ $ $) "\\spad{x rem y} is the same as \\spad{divide(x,y).remainder}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|quo| (($ $ $) "\\spad{x quo y} is the same as \\spad{divide(x,y).quotient}. See \\spadfunFrom{divide}{EuclideanDomain}.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(x,y)} divides \\spad{x} by \\spad{y} producing a record containing a \\spad{quotient} and \\spad{remainder},{} where the remainder is smaller (see \\spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \\spad{y}.")) (|euclideanSize| (((|NonNegativeInteger|) $) "\\spad{euclideanSize(x)} returns the euclidean size of the element \\spad{x}. Error: if \\spad{x} is zero.")) (|sizeLess?| (((|Boolean|) $ $) "\\spad{sizeLess?(x,y)} tests whether \\spad{x} is strictly smaller than \\spad{y} with respect to the \\spadfunFrom{euclideanSize}{EuclideanDomain}."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-256 S R) +(-258 S R) ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation'' substitutions.")) (|eval| (($ $ (|List| (|Equation| |#2|))) "\\spad{eval(f, [x1 = v1,...,xn = vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ (|Equation| |#2|)) "\\spad{eval(f,x = v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) NIL NIL -(-257 R) +(-259 R) ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation'' substitutions.")) (|eval| (($ $ (|List| (|Equation| |#1|))) "\\spad{eval(f, [x1 = v1,...,xn = vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ (|Equation| |#1|)) "\\spad{eval(f,x = v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) NIL NIL -(-258 -3078) +(-260 -3088) ((|constructor| (NIL "This package is to be used in conjuction with \\indented{12}{the CycleIndicators package. It provides an evaluation} \\indented{12}{function for SymmetricPolynomials.}")) (|eval| ((|#1| (|Mapping| |#1| (|Integer|)) (|SymmetricPolynomial| (|Fraction| (|Integer|)))) "\\spad{eval(f,s)} evaluates the cycle index \\spad{s} by applying \\indented{1}{the function \\spad{f} to each integer in a monomial partition,{}} \\indented{1}{forms their product and sums the results over all monomials.}"))) NIL NIL -(-259) +(-261) ((|constructor| (NIL "A function which does not return directly to its caller should have Exit as its return type. \\blankline Note: It is convenient to have a formal \\spad{coerce} into each type from type Exit. This allows,{} for example,{} errors to be raised in one half of a type-balanced \\spad{if}."))) NIL NIL -(-260) +(-262) ((|constructor| (NIL "This domain represents exit expressions.")) (|level| (((|Integer|) $) "\\spad{level(e)} returns the nesting exit level of `e'")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the exit expression of `e'."))) NIL NIL -(-261 R FE |var| |cen|) +(-263 R FE |var| |cen|) ((|constructor| (NIL "UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent essential singularities of functions. Objects in this domain are quotients of sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) "\\spad{coerce(f)} converts a \\spadtype{UnivariatePuiseuxSeries} to an \\spadtype{ExponentialExpansion}.")) (|limitPlus| (((|Union| (|OrderedCompletion| |#2|) "failed") $) "\\spad{limitPlus(f(var))} returns \\spad{limit(var -> a+,f(var))}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-816))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-945 (-1081)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-116))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-118))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-550 (-469)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-928))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-735))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-751))) (OR (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-735))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-751)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-945 (-480)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-1057))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-791 (-325)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-791 (-480)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-577 (-480)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-187))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-806 (-1081)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-188))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-804 (-1081)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (|%list| (QUOTE -449) (QUOTE (-1081)) (|%list| (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (|%list| (QUOTE -257) (|%list| (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (|%list| (QUOTE -239) (|%list| (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)) (|%list| (QUOTE -1157) (|devaluate| |#1|) (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#4|)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-255))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-479))) (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-816)))) (OR (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-816)))) (|HasCategory| (-1157 |#1| |#2| |#3| |#4|) (QUOTE (-116))))) -(-262 R) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . 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(|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} \\undocumented{}")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} \\undocumented{}")) (|simplifyPower| (($ $ (|Integer|)) "simplifyPower?(\\spad{f},{}\\spad{n}) \\undocumented{}")) (|number?| (((|Boolean|) $) "\\spad{number?(f)} tests if \\spad{f} is rational")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic quantities present in \\spad{f} by applying their defining relations."))) -((-3975 OR (-12 (|has| |#1| (-491)) (OR (|has| |#1| (-956)) (|has| |#1| (-408)))) (|has| |#1| (-956)) (|has| |#1| (-408))) (-3973 |has| |#1| (-144)) (-3972 |has| |#1| (-144)) ((-3980 "*") |has| |#1| (-491)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-491)) (-3970 |has| |#1| (-491))) -((OR (-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-945 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-491))) (OR (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-956)))) (|HasCategory| |#1| (QUOTE (-956))) (|HasCategory| |#1| (QUOTE (-21))) (OR (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-956)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-577 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-577 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-577 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-577 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-956))))) (OR (|HasCategory| |#1| (QUOTE (-408))) 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(-483)))))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-580 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-580 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-580 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-580 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-961))))) (OR (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#1| (QUOTE (-1024)))) (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| |#1| (QUOTE (-796 (-483)))) 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(-12 (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-21)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1024)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-25)))) (OR (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#1| (QUOTE (-961)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-950 (-483)))))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| $ (QUOTE (-961))) (|HasCategory| $ (QUOTE (-950 (-483))))) +(-265 R S) ((|constructor| (NIL "Lifting of maps to Expressions. Date Created: 16 Jan 1989 Date Last Updated: 22 Jan 1990")) (|map| (((|Expression| |#2|) (|Mapping| |#2| |#1|) (|Expression| |#1|)) "\\spad{map(f, e)} applies \\spad{f} to all the constants appearing in \\spad{e}."))) NIL NIL -(-264 R FE) +(-266 R FE) ((|constructor| (NIL "This package provides functions to convert functional expressions to power series.")) (|series| (((|Any|) |#2| (|Equation| |#2|) (|Fraction| (|Integer|))) "\\spad{series(f,x = a,n)} expands the expression \\spad{f} as a series in powers of (\\spad{x} - a); terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{series(f,x = a)} expands the expression \\spad{f} as a series in powers of (\\spad{x} - a).") (((|Any|) |#2| (|Fraction| (|Integer|))) "\\spad{series(f,n)} returns a series expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{series(f)} returns a series expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{series(x)} returns \\spad{x} viewed as a series.")) (|puiseux| (((|Any|) |#2| (|Equation| |#2|) (|Fraction| (|Integer|))) "\\spad{puiseux(f,x = a,n)} expands the expression \\spad{f} as a Puiseux series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{puiseux(f,x = a)} expands the expression \\spad{f} as a Puiseux series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|Fraction| (|Integer|))) "\\spad{puiseux(f,n)} returns a Puiseux expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{puiseux(f)} returns a Puiseux expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{puiseux(x)} returns \\spad{x} viewed as a Puiseux series.")) (|laurent| (((|Any|) |#2| (|Equation| |#2|) (|Integer|)) "\\spad{laurent(f,x = a,n)} expands the expression \\spad{f} as a Laurent series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{laurent(f,x = a)} expands the expression \\spad{f} as a Laurent series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|Integer|)) "\\spad{laurent(f,n)} returns a Laurent expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{laurent(f)} returns a Laurent expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{laurent(x)} returns \\spad{x} viewed as a Laurent series.")) (|taylor| (((|Any|) |#2| (|Equation| |#2|) (|NonNegativeInteger|)) "\\spad{taylor(f,x = a)} expands the expression \\spad{f} as a Taylor series in powers of \\spad{(x - a)}; terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2| (|Equation| |#2|)) "\\spad{taylor(f,x = a)} expands the expression \\spad{f} as a Taylor series in powers of \\spad{(x - a)}.") (((|Any|) |#2| (|NonNegativeInteger|)) "\\spad{taylor(f,n)} returns a Taylor expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \\spad{n}.") (((|Any|) |#2|) "\\spad{taylor(f)} returns a Taylor expansion of the expression \\spad{f}. Note: \\spad{f} should have only one variable; the series will be expanded in powers of that variable.") (((|Any|) (|Symbol|)) "\\spad{taylor(x)} returns \\spad{x} viewed as a Taylor series."))) NIL NIL -(-265 R -3078) +(-267 R -3088) ((|constructor| (NIL "Taylor series solutions of explicit ODE's.")) (|seriesSolve| (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq, y, x = a, [b0,...,bn])} is equivalent to \\spad{seriesSolve(eq = 0, y, x = a, [b0,...,b(n-1)])}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq, y, x = a, y a = b)} is equivalent to \\spad{seriesSolve(eq=0, y, x=a, y a = b)}.") (((|Any|) |#2| (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq, y, x = a, b)} is equivalent to \\spad{seriesSolve(eq = 0, y, x = a, y a = b)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) |#2|) "\\spad{seriesSolve(eq,y, x=a, b)} is equivalent to \\spad{seriesSolve(eq, y, x=a, y a = b)}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,[y1 a = b1,..., yn a = bn])} is equivalent to \\spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x = a, [y1 a = b1,..., yn a = bn])}.") (((|Any|) (|List| |#2|) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])} is equivalent to \\spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x=a, [b1,...,bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])} is equivalent to \\spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a, [y1 a = b1,..., yn a = bn])}.") (((|Any|) (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Equation| |#2|) (|List| (|Equation| |#2|))) "\\spad{seriesSolve([eq1,...,eqn],[y1,...,yn],x = a,[y1 a = b1,...,yn a = bn])} returns a taylor series solution of \\spad{[eq1,...,eqn]} around \\spad{x = a} with initial conditions \\spad{yi(a) = bi}. Note: eqi must be of the form \\spad{fi(x, y1 x, y2 x,..., yn x) y1'(x) + gi(x, y1 x, y2 x,..., yn x) = h(x, y1 x, y2 x,..., yn x)}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{seriesSolve(eq,y,x=a,[b0,...,b(n-1)])} returns a Taylor series solution of \\spad{eq} around \\spad{x = a} with initial conditions \\spad{y(a) = b0},{} \\spad{y'(a) = b1},{} \\spad{y''(a) = b2},{} ...,{}\\spad{y(n-1)(a) = b(n-1)} \\spad{eq} must be of the form \\spad{f(x, y x, y'(x),..., y(n-1)(x)) y(n)(x) + g(x,y x,y'(x),...,y(n-1)(x)) = h(x,y x, y'(x),..., y(n-1)(x))}.") (((|Any|) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|Equation| |#2|)) "\\spad{seriesSolve(eq,y,x=a, y a = b)} returns a Taylor series solution of \\spad{eq} around \\spad{x} = a with initial condition \\spad{y(a) = b}. Note: \\spad{eq} must be of the form \\spad{f(x, y x) y'(x) + g(x, y x) = h(x, y x)}."))) NIL NIL -(-266) +(-268) ((|constructor| (NIL "\\indented{1}{Author: Clifton \\spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.")) (|tubePlot| (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|) (|String|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n,s)} puts a tube of radius \\spad{r} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. If \\spad{s} = \"closed\",{} the tube is considered to be closed; if \\spad{s} = \"open\",{} the tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|DoubleFloat|) (|Integer|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n)} puts a tube of radius \\spad{r} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. The tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Integer|) (|String|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n,s)} puts a tube of radius \\spad{r(t)} with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. If \\spad{s} = \"closed\",{} the tube is considered to be closed; if \\spad{s} = \"open\",{} the tube is considered to be open.") (((|TubePlot| (|Plot3D|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Integer|)) "\\spad{tubePlot(f,g,h,colorFcn,a..b,r,n)} puts a tube of radius \\spad{r}(\\spad{t}) with \\spad{n} points on each circle about the curve \\spad{x = f(t)},{} \\spad{y = g(t)},{} \\spad{z = h(t)} for \\spad{t} in \\spad{[a,b]}. The tube is considered to be open.")) (|constantToUnaryFunction| (((|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|DoubleFloat|)) "\\spad{constantToUnaryFunction(s)} is a local function which takes the value of \\spad{s},{} which may be a function of a constant,{} and returns a function which always returns the value \\spadtype{DoubleFloat} \\spad{s}."))) NIL NIL -(-267 FE |var| |cen|) +(-269 FE |var| |cen|) ((|constructor| (NIL "ExponentialOfUnivariatePuiseuxSeries is a domain used to represent essential singularities of functions. An object in this domain is a function of the form \\spad{exp(f(x))},{} where \\spad{f(x)} is a Puiseux series with no terms of non-negative degree. Objects are ordered according to order of singularity,{} with functions which tend more rapidly to zero or infinity considered to be larger. Thus,{} if \\spad{order(f(x)) < order(g(x))},{} \\spadignore{i.e.} the first non-zero term of \\spad{f(x)} has lower degree than the first non-zero term of \\spad{g(x)},{} then \\spad{exp(f(x)) > exp(g(x))}. If \\spad{order(f(x)) = order(g(x))},{} then the ordering is essentially random. This domain is used in computing limits involving functions with essential singularities.")) (|exponentialOrder| (((|Fraction| (|Integer|)) $) "\\spad{exponentialOrder(exp(c * x **(-n) + ...))} returns \\spad{-n}. exponentialOrder(0) returns \\spad{0}.")) (|exponent| (((|UnivariatePuiseuxSeries| |#1| |#2| |#3|) $) "\\spad{exponent(exp(f(x)))} returns \\spad{f(x)}")) (|exponential| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{exponential(f(x))} returns \\spad{exp(f(x))}. Note: the function does NOT check that \\spad{f(x)} has no non-negative terms."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|)))) (|HasCategory| (-345 (-480)) (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-309))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-29 (-480)))) (|HasCategory| |#1| (QUOTE (-866))) (|HasCategory| |#1| (QUOTE (-1106)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1081))))) (|HasSignature| |#1| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#1|))))))) -(-268 M) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|)))) (|HasCategory| (-347 (-483)) (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-311))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-29 (-483)))) (|HasCategory| |#1| (QUOTE (-871))) (|HasCategory| |#1| (QUOTE (-1113)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1088))))) (|HasSignature| |#1| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#1|))))))) +(-270 M) ((|constructor| (NIL "computes various functions on factored arguments.")) (|log| (((|List| (|Record| (|:| |coef| (|NonNegativeInteger|)) (|:| |logand| |#1|))) (|Factored| |#1|)) "\\spad{log(f)} returns \\spad{[(a1,b1),...,(am,bm)]} such that the logarithm of \\spad{f} is equal to \\spad{a1*log(b1) + ... + am*log(bm)}.")) (|nthRoot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#1|) (|:| |radicand| (|List| |#1|))) (|Factored| |#1|) (|NonNegativeInteger|)) "\\spad{nthRoot(f, n)} returns \\spad{(p, r, [r1,...,rm])} such that the \\spad{n}th-root of \\spad{f} is equal to \\spad{r * \\spad{p}th-root(r1 * ... * rm)},{} where \\spad{r1},{}...,{}rm are distinct factors of \\spad{f},{} each of which has an exponent smaller than \\spad{p} in \\spad{f}."))) NIL NIL -(-269 E OV R P) +(-271 E OV R P) ((|constructor| (NIL "This package provides utilities used by the factorizers which operate on polynomials represented as univariate polynomials with multivariate coefficients.")) (|ran| ((|#3| (|Integer|)) "\\spad{ran(k)} computes a random integer between -k and \\spad{k} as a member of \\spad{R}.")) (|normalDeriv| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|Integer|)) "\\spad{normalDeriv(poly,i)} computes the \\spad{i}th derivative of \\spad{poly} divided by i!.")) (|raisePolynomial| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#3|)) "\\spad{raisePolynomial(rpoly)} converts \\spad{rpoly} from a univariate polynomial over \\spad{r} to be a univariate polynomial with polynomial coefficients.")) (|lowerPolynomial| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{lowerPolynomial(upoly)} converts \\spad{upoly} to be a univariate polynomial over \\spad{R}. An error if the coefficients contain variables.")) (|variables| (((|List| |#2|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{variables(upoly)} returns the list of variables for the coefficients of \\spad{upoly}.")) (|degree| (((|List| (|NonNegativeInteger|)) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|)) "\\spad{degree(upoly, lvar)} returns a list containing the maximum degree for each variable in lvar.")) (|completeEval| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| |#3|)) "\\spad{completeEval(upoly, lvar, lval)} evaluates the polynomial \\spad{upoly} with each variable in \\spad{lvar} replaced by the corresponding value in lval. Substitutions are done for all variables in \\spad{upoly} producing a univariate polynomial over \\spad{R}."))) NIL NIL -(-270 S) +(-272 S) ((|constructor| (NIL "The free abelian group on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,[ni * si])} where the \\spad{si}'s are in \\spad{S},{} and the \\spad{ni}'s are integers. The operation is commutative."))) -((-3973 . T) (-3972 . T)) -((|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| (-480) (QUOTE (-711)))) -(-271 S E) +((-3984 . T) (-3983 . T)) +((|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| (-483) (QUOTE (-716)))) +(-273 S E) ((|constructor| (NIL "A free abelian monoid on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,[ni * si])} where the \\spad{si}'s are in \\spad{S},{} and the \\spad{ni}'s are in a given abelian monoid. The operation is commutative.")) (|highCommonTerms| (($ $ $) "\\spad{highCommonTerms(e1 a1 + ... + en an, f1 b1 + ... + fm bm)} returns \\indented{2}{\\spad{reduce(+,[max(ei, fi) ci])}} where \\spad{ci} ranges in the intersection of \\spad{{a1,...,an}} and \\spad{{b1,...,bm}}.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, e1 a1 +...+ en an)} returns \\spad{e1 f(a1) +...+ en f(an)}.")) (|mapCoef| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapCoef(f, e1 a1 +...+ en an)} returns \\spad{f(e1) a1 +...+ f(en) an}.")) (|coefficient| ((|#2| |#1| $) "\\spad{coefficient(s, e1 a1 + ... + en an)} returns \\spad{ei} such that \\spad{ai} = \\spad{s},{} or 0 if \\spad{s} is not one of the \\spad{ai}'s.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x, n)} returns the factor of the n^th term of \\spad{x}.")) (|nthCoef| ((|#2| $ (|Integer|)) "\\spad{nthCoef(x, n)} returns the coefficient of the n^th term of \\spad{x}.")) (|terms| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| |#2|))) $) "\\spad{terms(e1 a1 + ... + en an)} returns \\spad{[[a1, e1],...,[an, en]]}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of terms in \\spad{x}. mapGen(\\spad{f},{} \\spad{a1}\\^\\spad{e1} ... an\\^en) returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (* (($ |#2| |#1|) "\\spad{e * s} returns \\spad{e} times \\spad{s}.")) (+ (($ |#1| $) "\\spad{s + x} returns the sum of \\spad{s} and \\spad{x}."))) NIL NIL -(-272 S) +(-274 S) ((|constructor| (NIL "The free abelian monoid on a set \\spad{S} is the monoid of finite sums of the form \\spad{reduce(+,[ni * si])} where the \\spad{si}'s are in \\spad{S},{} and the \\spad{ni}'s are non-negative integers. The operation is commutative."))) NIL -((|HasCategory| (-689) (QUOTE (-711)))) -(-273 S R E) +((|HasCategory| (-694) (QUOTE (-716)))) +(-275 S R E) ((|constructor| (NIL "This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.")) (|primitivePart| (($ $) "\\spad{primitivePart(p)} returns the unit normalized form of polynomial \\spad{p} divided by the content of \\spad{p}.")) (|content| ((|#2| $) "\\spad{content(p)} gives the gcd of the coefficients of polynomial \\spad{p}.")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(p,r)} returns the exact quotient of polynomial \\spad{p} by \\spad{r},{} or \"failed\" if none exists.")) (|binomThmExpt| (($ $ $ (|NonNegativeInteger|)) "\\spad{binomThmExpt(p,q,n)} returns \\spad{(x+y)^n} by means of the binomial theorem trick.")) (|pomopo!| (($ $ |#2| |#3| $) "\\spad{pomopo!(p1,r,e,p2)} returns \\spad{p1 + monomial(e,r) * p2} and may use \\spad{p1} as workspace. The constaant \\spad{r} is assumed to be nonzero.")) (|mapExponents| (($ (|Mapping| |#3| |#3|) $) "\\spad{mapExponents(fn,u)} maps function \\spad{fn} onto the exponents of the non-zero monomials of polynomial \\spad{u}.")) (|minimumDegree| ((|#3| $) "\\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \\spad{p}. Error: if applied to 0.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \\spad{p}.")) (|coefficients| (((|List| |#2|) $) "\\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \\spad{p}.")) (|ground| ((|#2| $) "\\spad{ground(p)} retracts polynomial \\spad{p} to the coefficient ring.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(p)} tests if polynomial \\spad{p} is a member of the coefficient ring."))) NIL -((|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-144)))) -(-274 R E) +((|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146)))) +(-276 R E) ((|constructor| (NIL "This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.")) (|primitivePart| (($ $) "\\spad{primitivePart(p)} returns the unit normalized form of polynomial \\spad{p} divided by the content of \\spad{p}.")) (|content| ((|#1| $) "\\spad{content(p)} gives the gcd of the coefficients of polynomial \\spad{p}.")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(p,r)} returns the exact quotient of polynomial \\spad{p} by \\spad{r},{} or \"failed\" if none exists.")) (|binomThmExpt| (($ $ $ (|NonNegativeInteger|)) "\\spad{binomThmExpt(p,q,n)} returns \\spad{(x+y)^n} by means of the binomial theorem trick.")) (|pomopo!| (($ $ |#1| |#2| $) "\\spad{pomopo!(p1,r,e,p2)} returns \\spad{p1 + monomial(e,r) * p2} and may use \\spad{p1} as workspace. The constaant \\spad{r} is assumed to be nonzero.")) (|mapExponents| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapExponents(fn,u)} maps function \\spad{fn} onto the exponents of the non-zero monomials of polynomial \\spad{u}.")) (|minimumDegree| ((|#2| $) "\\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \\spad{p}. Error: if applied to 0.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \\spad{p}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \\spad{p}.")) (|ground| ((|#1| $) "\\spad{ground(p)} retracts polynomial \\spad{p} to the coefficient ring.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(p)} tests if polynomial \\spad{p} is a member of the coefficient ring."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-275 S) +(-277 S) ((|constructor| (NIL "\\indented{1}{A FlexibleArray is the notion of an array intended to allow for growth} at the end only. Hence the following efficient operations \\indented{2}{\\spad{append(x,a)} meaning append item \\spad{x} at the end of the array \\spad{a}} \\indented{2}{\\spad{delete(a,n)} meaning delete the last item from the array \\spad{a}} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets."))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-276 S -3078) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-278 S -3088) ((|constructor| (NIL "FiniteAlgebraicExtensionField {\\em F} is the category of fields which are finite algebraic extensions of the field {\\em F}. If {\\em F} is finite then any finite algebraic extension of {\\em F} is finite,{} too. Let {\\em K} be a finite algebraic extension of the finite field {\\em F}. The exponentiation of elements of {\\em K} defines a \\spad{Z}-module structure on the multiplicative group of {\\em K}. The additive group of {\\em K} becomes a module over the ring of polynomials over {\\em F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em K},{} {\\em c,d} from {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\$SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)} where {\\em q=size()\\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#2|) "failed") $ $) "\\spad{linearAssociatedLog(b,a)} returns a polynomial {\\em g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals {\\em a}. If there is no such polynomial {\\em g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial {\\em g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals {\\em a}.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\\em g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#2|)) "\\spad{linearAssociatedExp(a,f)} is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em \\$},{} {\\em c,d} form {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\$SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)},{} where {\\em q=size()\\$F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: \\spad{trace(a,d) = reduce(+,[a**(q**(d*i)) for i in 0..n/d])}.") ((|#2| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: norm(a,{}\\spad{d}) = reduce(*,{}[a**(q**(d*i)) for \\spad{i} in 0..n/d])") ((|#2| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#2|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}vn are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}'s with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\$ as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\$ as \\spad{F}-vectorspace."))) NIL -((|HasCategory| |#2| (QUOTE (-315)))) -(-277 -3078) +((|HasCategory| |#2| (QUOTE (-317)))) +(-279 -3088) ((|constructor| (NIL "FiniteAlgebraicExtensionField {\\em F} is the category of fields which are finite algebraic extensions of the field {\\em F}. If {\\em F} is finite then any finite algebraic extension of {\\em F} is finite,{} too. Let {\\em K} be a finite algebraic extension of the finite field {\\em F}. The exponentiation of elements of {\\em K} defines a \\spad{Z}-module structure on the multiplicative group of {\\em K}. The additive group of {\\em K} becomes a module over the ring of polynomials over {\\em F} via the operation \\spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \\spad{F}) which is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em K},{} {\\em c,d} from {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\$SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)} where {\\em q=size()\\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \\spadfun{linearAssociatedExp}. These are the functions \\spadfun{linearAssociatedOrder} and \\spadfun{linearAssociatedLog},{} respectively.")) (|linearAssociatedLog| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") $ $) "\\spad{linearAssociatedLog(b,a)} returns a polynomial {\\em g},{} such that the \\spadfun{linearAssociatedExp}(\\spad{b},{}\\spad{g}) equals {\\em a}. If there is no such polynomial {\\em g},{} then \\spadfun{linearAssociatedLog} fails.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedLog(a)} returns a polynomial {\\em g},{} such that \\spadfun{linearAssociatedExp}(normalElement(),{}\\spad{g}) equals {\\em a}.")) (|linearAssociatedOrder| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\\em g} of least degree,{} such that \\spadfun{linearAssociatedExp}(a,{}\\spad{g}) is 0.")) (|linearAssociatedExp| (($ $ (|SparseUnivariatePolynomial| |#1|)) "\\spad{linearAssociatedExp(a,f)} is linear over {\\em F},{} \\spadignore{i.e.} for elements {\\em a} from {\\em \\$},{} {\\em c,d} form {\\em F} and {\\em f,g} univariate polynomials over {\\em F} we have \\spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\\em c} times \\spadfun{linearAssociatedExp}(a,{}\\spad{f}) plus {\\em d} times \\spadfun{linearAssociatedExp}(a,{}\\spad{g}). Therefore \\spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\\em F[X]}: \\spadfun{linearAssociatedExp}(a,{}monomial(1,{}\\spad{k})\\$SUP(\\spad{F})) is defined to be \\spadfun{Frobenius}(a,{}\\spad{k}) which is {\\em a**(q**k)},{} where {\\em q=size()\\$F}.")) (|generator| (($) "\\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.")) (|normal?| (((|Boolean|) $) "\\spad{normal?(a)} tests whether the element \\spad{a} is normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i <= extensionDegree()-1} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.")) (|normalElement| (($) "\\spad{normalElement()} returns a element,{} normal over the ground field \\spad{F},{} \\spadignore{i.e.} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. At the first call,{} the element is computed by \\spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global variable.")) (|createNormalElement| (($) "\\spad{createNormalElement()} computes a normal element over the ground field \\spad{F},{} that is,{} \\spad{a**(q**i), 0 <= i < extensionDegree()} is an \\spad{F}-basis,{} where \\spad{q = size()\\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.")) (|trace| (($ $ (|PositiveInteger|)) "\\spad{trace(a,d)} computes the trace of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size \\spad{q}. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: \\spad{trace(a,d) = reduce(+,[a**(q**(d*i)) for i in 0..n/d])}.") ((|#1| $) "\\spad{trace(a)} computes the trace of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|norm| (($ $ (|PositiveInteger|)) "\\spad{norm(a,d)} computes the norm of \\spad{a} with respect to the field of extension degree \\spad{d} over the ground field of size. Error: if \\spad{d} does not divide the extension degree of \\spad{a}. Note: norm(a,{}\\spad{d}) = reduce(*,{}[a**(q**(d*i)) for \\spad{i} in 0..n/d])") ((|#1| $) "\\spad{norm(a)} computes the norm of \\spad{a} with respect to the field considered as an algebra with 1 over the ground field \\spad{F}.")) (|degree| (((|PositiveInteger|) $) "\\spad{degree(a)} returns the degree of the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|extensionDegree| (((|PositiveInteger|)) "\\spad{extensionDegree()} returns the degree of field extension.")) (|definingPolynomial| (((|SparseUnivariatePolynomial| |#1|)) "\\spad{definingPolynomial()} returns the polynomial used to define the field extension.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| $) $ (|PositiveInteger|)) "\\spad{minimalPolynomial(x,n)} computes the minimal polynomial of \\spad{x} over the field of extension degree \\spad{n} over the ground field \\spad{F}.") (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \\spad{a} over the ground field \\spad{F}.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{}...,{}vn are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}'s with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{F}-vectorspace basis.")) (|basis| (((|Vector| $) (|PositiveInteger|)) "\\spad{basis(n)} returns a fixed basis of a subfield of \\$ as \\spad{F}-vectorspace.") (((|Vector| $)) "\\spad{basis()} returns a fixed basis of \\$ as \\spad{F}-vectorspace."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-278 E) +(-280 E) ((|constructor| (NIL "\\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: 12 June 1992 Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:")) (|argument| ((|#1| $) "\\spad{argument(x)} returns the argument of a given sin/cos expressions")) (|sin?| (((|Boolean|) $) "\\spad{sin?(x)} returns \\spad{true} if term is a sin,{} otherwise \\spad{false}")) (|cos| (($ |#1|) "\\spad{cos(x)} makes a cos kernel for use in Fourier series")) (|sin| (($ |#1|) "\\spad{sin(x)} makes a sin kernel for use in Fourier series"))) NIL NIL -(-279) +(-281) ((|constructor| (NIL "Represntation of data needed to instantiate a domain constructor.")) (|lookupFunction| (((|Identifier|) $) "\\spad{lookupFunction x} returns the name of the lookup function associated with the functor data \\spad{x}.")) (|categories| (((|PrimitiveArray| (|ConstructorCall| (|CategoryConstructor|))) $) "\\spad{categories x} returns the list of categories forms each domain object obtained from the domain data \\spad{x} belongs to.")) (|encodingDirectory| (((|PrimitiveArray| (|NonNegativeInteger|)) $) "\\spad{encodintDirectory x} returns the directory of domain-wide entity description.")) (|attributeData| (((|List| (|Pair| (|Syntax|) (|NonNegativeInteger|))) $) "\\spad{attributeData x} returns the list of attribute-predicate bit vector index pair associated with the functor data \\spad{x}.")) (|domainTemplate| (((|DomainTemplate|) $) "\\spad{domainTemplate x} returns the domain template vector associated with the functor data \\spad{x}."))) NIL NIL -(-280 -3078 UP UPUP R) +(-282 -3088 UP UPUP R) ((|constructor| (NIL "This domains implements finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}'s are integers and the \\spad{P}'s are finite rational points on the curve.")) (|lSpaceBasis| (((|Vector| |#4|) $) "\\spad{lSpaceBasis(d)} returns a basis for \\spad{L(d) = {f | (f) >= -d}} as a module over \\spad{K[x]}.")) (|finiteBasis| (((|Vector| |#4|) $) "\\spad{finiteBasis(d)} returns a basis for \\spad{d} as a module over {\\em K[x]}."))) NIL NIL -(-281 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) +(-283 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) ((|constructor| (NIL "\\indented{1}{Lift a map to finite divisors.} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 19 May 1993")) (|map| (((|FiniteDivisor| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{map(f,d)} \\undocumented{}"))) NIL NIL -(-282 S -3078 UP UPUP R) +(-284 S -3088 UP UPUP R) ((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}'s are integers and the \\spad{P}'s are finite rational points on the curve.")) (|generator| (((|Union| |#5| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) (|:| |principalPart| |#5|)) $) "\\spad{decompose(d)} returns \\spad{[id, f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#5| |#3| |#3| |#3| |#2|) "\\spad{divisor(h, d, d', g, r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,discriminant)} contains the ramified zeros of \\spad{d}") (($ |#2| |#2| (|Integer|)) "\\spad{divisor(a, b, n)} makes the divisor \\spad{nP} where P: \\spad{(x = a, y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#2| |#2|) "\\spad{divisor(a, b)} makes the divisor P: \\spad{(x = a, y = b)}. Error: if \\spad{P} is singular.") (($ |#5|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#3| (|Fraction| |#3|) |#4| |#5|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) NIL NIL -(-283 -3078 UP UPUP R) +(-285 -3088 UP UPUP R) ((|constructor| (NIL "This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\\spad{n} * \\spad{P}) where the \\spad{n}'s are integers and the \\spad{P}'s are finite rational points on the curve.")) (|generator| (((|Union| |#4| "failed") $) "\\spad{generator(d)} returns \\spad{f} if \\spad{(f) = d},{} \"failed\" if \\spad{d} is not principal.")) (|principal?| (((|Boolean|) $) "\\spad{principal?(D)} tests if the argument is the divisor of a function.")) (|reduce| (($ $) "\\spad{reduce(D)} converts \\spad{D} to some reduced form (the reduced forms can be differents in different implementations).")) (|decompose| (((|Record| (|:| |id| (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) (|:| |principalPart| |#4|)) $) "\\spad{decompose(d)} returns \\spad{[id, f]} where \\spad{d = (id) + div(f)}.")) (|divisor| (($ |#4| |#2| |#2| |#2| |#1|) "\\spad{divisor(h, d, d', g, r)} returns the sum of all the finite points where \\spad{h/d} has residue \\spad{r}. \\spad{h} must be integral. \\spad{d} must be squarefree. \\spad{d'} is some derivative of \\spad{d} (not necessarily dd/dx). \\spad{g = gcd(d,discriminant)} contains the ramified zeros of \\spad{d}") (($ |#1| |#1| (|Integer|)) "\\spad{divisor(a, b, n)} makes the divisor \\spad{nP} where P: \\spad{(x = a, y = b)}. \\spad{P} is allowed to be singular if \\spad{n} is a multiple of the rank.") (($ |#1| |#1|) "\\spad{divisor(a, b)} makes the divisor P: \\spad{(x = a, y = b)}. Error: if \\spad{P} is singular.") (($ |#4|) "\\spad{divisor(g)} returns the divisor of the function \\spad{g}.") (($ (|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|)) "\\spad{divisor(I)} makes a divisor \\spad{D} from an ideal \\spad{I}.")) (|ideal| (((|FractionalIdeal| |#2| (|Fraction| |#2|) |#3| |#4|) $) "\\spad{ideal(D)} returns the ideal corresponding to a divisor \\spad{D}."))) NIL NIL -(-284 S R) +(-286 S R) ((|constructor| (NIL "This category provides a selection of evaluation operations depending on what the argument type \\spad{R} provides.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(f, ex)} evaluates ex,{} applying \\spad{f} to values of type \\spad{R} in ex."))) NIL -((|HasCategory| |#2| (|%list| (QUOTE -449) (QUOTE (-1081)) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -239) (|devaluate| |#2|) (|devaluate| |#2|)))) -(-285 R) +((|HasCategory| |#2| (|%list| (QUOTE -452) (QUOTE (-1088)) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -241) (|devaluate| |#2|) (|devaluate| |#2|)))) +(-287 R) ((|constructor| (NIL "This category provides a selection of evaluation operations depending on what the argument type \\spad{R} provides.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f, ex)} evaluates ex,{} applying \\spad{f} to values of type \\spad{R} in ex."))) NIL NIL -(-286 |p| |n|) +(-288 |p| |n|) ((|constructor| (NIL "FiniteField(\\spad{p},{}\\spad{n}) implements finite fields with p**n elements. This packages checks that \\spad{p} is prime. For a non-checking version,{} see \\spadtype{InnerFiniteField}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| (-812 |#1|) (QUOTE (-116))) (|HasCategory| (-812 |#1|) (QUOTE (-315)))) (|HasCategory| (-812 |#1|) (QUOTE (-118))) (|HasCategory| (-812 |#1|) (QUOTE (-315))) (|HasCategory| (-812 |#1|) (QUOTE (-116)))) -(-287 S -3078 UP UPUP) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| (-817 |#1|) (QUOTE (-118))) (|HasCategory| (-817 |#1|) (QUOTE (-317)))) (|HasCategory| (-817 |#1|) (QUOTE (-120))) (|HasCategory| (-817 |#1|) (QUOTE (-317))) (|HasCategory| (-817 |#1|) (QUOTE (-118)))) +(-289 S -3088 UP UPUP) ((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#2|))) "\\spad{rationalPoints()} returns the list of all the affine rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#2|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in \\spad{u1},{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#3|) (|:| |derivden| |#3|) (|:| |gd| |#3|)) $ (|Mapping| |#3| |#3|)) "\\spad{algSplitSimple(f, D)} returns \\spad{[h,d,d',g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d, discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#3| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#3| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#2| $ |#2| |#2|) "\\spad{elt(f,a,b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a, y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#3| |#3|)) "\\spad{differentiate(x, d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#3|)) (|:| |den| |#3|)) (|Mapping| |#3| |#3|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(wi)} with respect to \\spad{(w1,...,wn)} where \\spad{(w1,...,wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#3|) |#3|) "\\spad{integralRepresents([A1,...,An], D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,...,wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,...,wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#3|) |#3|) "\\spad{represents([A0,...,A(n-1)],D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#3|)) (|:| |den| |#3|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\spad{inverseIntegralMatrixAtInfinity()} returns \\spad{M} such that \\spad{M (v1,...,vn) = (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#3|))) "\\spad{integralMatrixAtInfinity()} returns \\spad{M} such that \\spad{(v1,...,vn) = M (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\spad{inverseIntegralMatrix()} returns \\spad{M} such that \\spad{M (w1,...,wn) = (1, y, ..., y**(n-1))} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|integralMatrix| (((|Matrix| (|Fraction| |#3|))) "\\spad{integralMatrix()} returns \\spad{M} such that \\spad{(w1,...,wn) = M (1, y, ..., y**(n-1))},{} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,...,bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,...,bn)} returns the complementary basis \\spad{(b1',...,bn')} of \\spad{(b1,...,bn)}.")) (|integral?| (((|Boolean|) $ |#3|) "\\spad{integral?(f, p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}.") (((|Boolean|) $ |#2|) "\\spad{integral?(f, a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.")) (|integralBasis| (((|Vector| $)) "\\spad{integralBasis()} returns the integral basis for the curve.")) (|ramified?| (((|Boolean|) |#3|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#2|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#3|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#2|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#3|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#2|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.")) (|rationalPoint?| (((|Boolean|) |#2| |#2|) "\\spad{rationalPoint?(a, b)} tests if \\spad{(x=a,y=b)} is on the curve.")) (|absolutelyIrreducible?| (((|Boolean|)) "\\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?")) (|genus| (((|NonNegativeInteger|)) "\\spad{genus()} returns the genus of one absolutely irreducible component")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\spad{numberOfComponents()} returns the number of absolutely irreducible components."))) NIL -((|HasCategory| |#2| (QUOTE (-315))) (|HasCategory| |#2| (QUOTE (-309)))) -(-288 -3078 UP UPUP) +((|HasCategory| |#2| (QUOTE (-317))) (|HasCategory| |#2| (QUOTE (-311)))) +(-290 -3088 UP UPUP) ((|constructor| (NIL "This category is a model for the function field of a plane algebraic curve.")) (|rationalPoints| (((|List| (|List| |#1|))) "\\spad{rationalPoints()} returns the list of all the affine rational points.")) (|nonSingularModel| (((|List| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{nonSingularModel(u)} returns the equations in \\spad{u1},{}...,{}un of an affine non-singular model for the curve.")) (|algSplitSimple| (((|Record| (|:| |num| $) (|:| |den| |#2|) (|:| |derivden| |#2|) (|:| |gd| |#2|)) $ (|Mapping| |#2| |#2|)) "\\spad{algSplitSimple(f, D)} returns \\spad{[h,d,d',g]} such that \\spad{f=h/d},{} \\spad{h} is integral at all the normal places \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{d' = Dd},{} \\spad{g = gcd(d, discriminant())} and \\spad{D} is the derivation to use. \\spad{f} must have at most simple finite poles.")) (|hyperelliptic| (((|Union| |#2| "failed")) "\\spad{hyperelliptic()} returns \\spad{p(x)} if the curve is the hyperelliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elliptic| (((|Union| |#2| "failed")) "\\spad{elliptic()} returns \\spad{p(x)} if the curve is the elliptic defined by \\spad{y**2 = p(x)},{} \"failed\" otherwise.")) (|elt| ((|#1| $ |#1| |#1|) "\\spad{elt(f,a,b)} or \\spad{f}(a,{} \\spad{b}) returns the value of \\spad{f} at the point \\spad{(x = a, y = b)} if it is not singular.")) (|primitivePart| (($ $) "\\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \\spad{f}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|)) "\\spad{differentiate(x, d)} extends the derivation \\spad{d} from UP to \\$ and applies it to \\spad{x}.")) (|integralDerivationMatrix| (((|Record| (|:| |num| (|Matrix| |#2|)) (|:| |den| |#2|)) (|Mapping| |#2| |#2|)) "\\spad{integralDerivationMatrix(d)} extends the derivation \\spad{d} from UP to \\$ and returns (\\spad{M},{} \\spad{Q}) such that the i^th row of \\spad{M} divided by \\spad{Q} form the coordinates of \\spad{d(wi)} with respect to \\spad{(w1,...,wn)} where \\spad{(w1,...,wn)} is the integral basis returned by integralBasis().")) (|integralRepresents| (($ (|Vector| |#2|) |#2|) "\\spad{integralRepresents([A1,...,An], D)} returns \\spad{(A1 w1+...+An wn)/D} where \\spad{(w1,...,wn)} is the integral basis of \\spad{integralBasis()}.")) (|integralCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{integralCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 w1 +...+ An wn) / D} where \\spad{(w1,...,wn)} is the integral basis returned by \\spad{integralBasis()}.")) (|represents| (($ (|Vector| |#2|) |#2|) "\\spad{represents([A0,...,A(n-1)],D)} returns \\spad{(A0 + A1 y +...+ A(n-1)*y**(n-1))/D}.")) (|yCoordinates| (((|Record| (|:| |num| (|Vector| |#2|)) (|:| |den| |#2|)) $) "\\spad{yCoordinates(f)} returns \\spad{[[A1,...,An], D]} such that \\spad{f = (A1 + A2 y +...+ An y**(n-1)) / D}.")) (|inverseIntegralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\spad{inverseIntegralMatrixAtInfinity()} returns \\spad{M} such that \\spad{M (v1,...,vn) = (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|integralMatrixAtInfinity| (((|Matrix| (|Fraction| |#2|))) "\\spad{integralMatrixAtInfinity()} returns \\spad{M} such that \\spad{(v1,...,vn) = M (1, y, ..., y**(n-1))} where \\spad{(v1,...,vn)} is the local integral basis at infinity returned by \\spad{infIntBasis()}.")) (|inverseIntegralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\spad{inverseIntegralMatrix()} returns \\spad{M} such that \\spad{M (w1,...,wn) = (1, y, ..., y**(n-1))} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|integralMatrix| (((|Matrix| (|Fraction| |#2|))) "\\spad{integralMatrix()} returns \\spad{M} such that \\spad{(w1,...,wn) = M (1, y, ..., y**(n-1))},{} where \\spad{(w1,...,wn)} is the integral basis of \\spadfunFrom{integralBasis}{FunctionFieldCategory}.")) (|reduceBasisAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{reduceBasisAtInfinity(b1,...,bn)} returns \\spad{(x**i * bj)} for all \\spad{i},{}\\spad{j} such that \\spad{x**i*bj} is locally integral at infinity.")) (|normalizeAtInfinity| (((|Vector| $) (|Vector| $)) "\\spad{normalizeAtInfinity(v)} makes \\spad{v} normal at infinity.")) (|complementaryBasis| (((|Vector| $) (|Vector| $)) "\\spad{complementaryBasis(b1,...,bn)} returns the complementary basis \\spad{(b1',...,bn')} of \\spad{(b1,...,bn)}.")) (|integral?| (((|Boolean|) $ |#2|) "\\spad{integral?(f, p)} tests whether \\spad{f} is locally integral at \\spad{p(x) = 0}.") (((|Boolean|) $ |#1|) "\\spad{integral?(f, a)} tests whether \\spad{f} is locally integral at \\spad{x = a}.") (((|Boolean|) $) "\\spad{integral?()} tests if \\spad{f} is integral over \\spad{k[x]}.")) (|integralAtInfinity?| (((|Boolean|) $) "\\spad{integralAtInfinity?()} tests if \\spad{f} is locally integral at infinity.")) (|integralBasisAtInfinity| (((|Vector| $)) "\\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.")) (|integralBasis| (((|Vector| $)) "\\spad{integralBasis()} returns the integral basis for the curve.")) (|ramified?| (((|Boolean|) |#2|) "\\spad{ramified?(p)} tests whether \\spad{p(x) = 0} is ramified.") (((|Boolean|) |#1|) "\\spad{ramified?(a)} tests whether \\spad{x = a} is ramified.")) (|ramifiedAtInfinity?| (((|Boolean|)) "\\spad{ramifiedAtInfinity?()} tests if infinity is ramified.")) (|singular?| (((|Boolean|) |#2|) "\\spad{singular?(p)} tests whether \\spad{p(x) = 0} is singular.") (((|Boolean|) |#1|) "\\spad{singular?(a)} tests whether \\spad{x = a} is singular.")) (|singularAtInfinity?| (((|Boolean|)) "\\spad{singularAtInfinity?()} tests if there is a singularity at infinity.")) (|branchPoint?| (((|Boolean|) |#2|) "\\spad{branchPoint?(p)} tests whether \\spad{p(x) = 0} is a branch point.") (((|Boolean|) |#1|) "\\spad{branchPoint?(a)} tests whether \\spad{x = a} is a branch point.")) (|branchPointAtInfinity?| (((|Boolean|)) "\\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.")) (|rationalPoint?| (((|Boolean|) |#1| |#1|) "\\spad{rationalPoint?(a, b)} tests if \\spad{(x=a,y=b)} is on the curve.")) (|absolutelyIrreducible?| (((|Boolean|)) "\\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?")) (|genus| (((|NonNegativeInteger|)) "\\spad{genus()} returns the genus of one absolutely irreducible component")) (|numberOfComponents| (((|NonNegativeInteger|)) "\\spad{numberOfComponents()} returns the number of absolutely irreducible components."))) -((-3971 |has| (-345 |#2|) (-309)) (-3976 |has| (-345 |#2|) (-309)) (-3970 |has| (-345 |#2|) (-309)) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 |has| (-347 |#2|) (-311)) (-3987 |has| (-347 |#2|) (-311)) (-3981 |has| (-347 |#2|) (-311)) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-289 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) +(-291 R1 UP1 UPUP1 F1 R2 UP2 UPUP2 F2) ((|constructor| (NIL "Lifts a map from rings to function fields over them.")) (|map| ((|#8| (|Mapping| |#5| |#1|) |#4|) "\\spad{map(f, p)} lifts \\spad{f} to \\spad{F1} and applies it to \\spad{p}."))) NIL NIL -(-290 |p| |extdeg|) +(-292 |p| |extdeg|) ((|constructor| (NIL "FiniteFieldCyclicGroup(\\spad{p},{}\\spad{n}) implements a finite field extension of degee \\spad{n} over the prime field with \\spad{p} elements. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\\em createPrimitivePoly} from \\spadtype{FiniteFieldPolynomialPackage}. The Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| (-812 |#1|) (QUOTE (-116))) (|HasCategory| (-812 |#1|) (QUOTE (-315)))) (|HasCategory| (-812 |#1|) (QUOTE (-118))) (|HasCategory| (-812 |#1|) (QUOTE (-315))) (|HasCategory| (-812 |#1|) (QUOTE (-116)))) -(-291 GF |defpol|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| (-817 |#1|) (QUOTE (-118))) (|HasCategory| (-817 |#1|) (QUOTE (-317)))) (|HasCategory| (-817 |#1|) (QUOTE (-120))) (|HasCategory| (-817 |#1|) (QUOTE (-317))) (|HasCategory| (-817 |#1|) (QUOTE (-118)))) +(-293 GF |defpol|) ((|constructor| (NIL "FiniteFieldCyclicGroupExtensionByPolynomial(GF,{}defpol) implements a finite extension field of the ground field {\\em GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial {\\em defpol},{} which MUST be primitive (user responsibility). Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field it is used to perform additions in the field quickly."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-116)))) -(-292 GF |extdeg|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-118)))) +(-294 GF |extdeg|) ((|constructor| (NIL "FiniteFieldCyclicGroupExtension(GF,{}\\spad{n}) implements a extension of degree \\spad{n} over the ground field {\\em GF}. Its elements are represented by powers of a primitive element,{} \\spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\\em createPrimitivePoly} from \\spadtype{FiniteFieldPolynomialPackage}. Zech logarithms are stored in a table of size half of the field size,{} and use \\spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.")) (|getZechTable| (((|PrimitiveArray| (|SingleInteger|))) "\\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-116)))) -(-293 GF) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-118)))) +(-295 GF) ((|constructor| (NIL "FiniteFieldFunctions(GF) is a package with functions concerning finite extension fields of the finite ground field {\\em GF},{} \\spadignore{e.g.} Zech logarithms.")) (|createLowComplexityNormalBasis| (((|Union| (|SparseUnivariatePolynomial| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) (|PositiveInteger|)) "\\spad{createLowComplexityNormalBasis(n)} tries to find a a low complexity normal basis of degree {\\em n} over {\\em GF} and returns its multiplication matrix If no low complexity basis is found it calls \\axiomFunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}(\\spad{n}) to produce a normal polynomial of degree {\\em n} over {\\em GF}")) (|createLowComplexityTable| (((|Union| (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) "failed") (|PositiveInteger|)) "\\spad{createLowComplexityTable(n)} tries to find a low complexity normal basis of degree {\\em n} over {\\em GF} and returns its multiplication matrix Fails,{} if it does not find a low complexity basis")) (|sizeMultiplication| (((|NonNegativeInteger|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{sizeMultiplication(m)} returns the number of entries of the multiplication table {\\em m}.")) (|createMultiplicationMatrix| (((|Matrix| |#1|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{createMultiplicationMatrix(m)} forms the multiplication table {\\em m} into a matrix over the ground field.")) (|createMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createMultiplicationTable(f)} generates a multiplication table for the normal basis of the field extension determined by {\\em f}. This is needed to perform multiplications between elements represented as coordinate vectors to this basis. See \\spadtype{FFNBP},{} \\spadtype{FFNBX}.")) (|createZechTable| (((|PrimitiveArray| (|SingleInteger|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{createZechTable(f)} generates a Zech logarithm table for the cyclic group representation of a extension of the ground field by the primitive polynomial {\\em f(x)},{} \\spadignore{i.e.} \\spad{Z(i)},{} defined by {\\em x**Z(i) = 1+x**i} is stored at index \\spad{i}. This is needed in particular to perform addition of field elements in finite fields represented in this way. See \\spadtype{FFCGP},{} \\spadtype{FFCGX}."))) NIL NIL -(-294 F1 GF F2) +(-296 F1 GF F2) ((|constructor| (NIL "FiniteFieldHomomorphisms(\\spad{F1},{}GF,{}\\spad{F2}) exports coercion functions of elements between the fields {\\em F1} and {\\em F2},{} which both must be finite simple algebraic extensions of the finite ground field {\\em GF}.")) (|coerce| ((|#1| |#3|) "\\spad{coerce(x)} is the homomorphic image of \\spad{x} from {\\em F2} in {\\em F1},{} where {\\em coerce} is a field homomorphism between the fields extensions {\\em F2} and {\\em F1} both over ground field {\\em GF} (the second argument to the package). Error: if the extension degree of {\\em F2} doesn't divide the extension degree of {\\em F1}. Note that the other coercion function in the \\spadtype{FiniteFieldHomomorphisms} is a left inverse.") ((|#3| |#1|) "\\spad{coerce(x)} is the homomorphic image of \\spad{x} from {\\em F1} in {\\em F2}. Thus {\\em coerce} is a field homomorphism between the fields extensions {\\em F1} and {\\em F2} both over ground field {\\em GF} (the second argument to the package). Error: if the extension degree of {\\em F1} doesn't divide the extension degree of {\\em F2}. Note that the other coercion function in the \\spadtype{FiniteFieldHomomorphisms} is a left inverse."))) NIL NIL -(-295 S) +(-297 S) ((|constructor| (NIL "FiniteFieldCategory is the category of finite fields")) (|representationType| (((|Union| "prime" "polynomial" "normal" "cyclic")) "\\spad{representationType()} returns the type of the representation,{} one of: \\spad{prime},{} \\spad{polynomial},{} \\spad{normal},{} or \\spad{cyclic}.")) (|order| (((|PositiveInteger|) $) "\\spad{order(b)} computes the order of an element \\spad{b} in the multiplicative group of the field. Error: if \\spad{b} equals 0.")) (|discreteLog| (((|NonNegativeInteger|) $) "\\spad{discreteLog(a)} computes the discrete logarithm of \\spad{a} with respect to \\spad{primitiveElement()} of the field.")) (|primitive?| (((|Boolean|) $) "\\spad{primitive?(b)} tests whether the element \\spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \\spadignore{i.e.} is a primitive element. Implementation Note: see ch.IX.1.3,{} th.2 in \\spad{D}. Lipson.")) (|primitiveElement| (($) "\\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \\spadfun{createPrimitiveElement}.")) (|createPrimitiveElement| (($) "\\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.")) (|tableForDiscreteLogarithm| (((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|)) "\\spad{tableForDiscreteLogarithm(a,n)} returns a table of the discrete logarithms of \\spad{a**0} up to \\spad{a**(n-1)} which,{} called with key \\spad{lookup(a**i)} returns \\spad{i} for \\spad{i} in \\spad{0..n-1}. Error: if not called for prime divisors of order of \\indented{7}{multiplicative group.}")) (|factorsOfCyclicGroupSize| (((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|))))) "\\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\\spad{-1}")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or \"failed\" if no such vector exists.")) (|charthRoot| (($ $) "\\spad{charthRoot(a)} takes the characteristic'th root of {\\em a}. Note: such a root is alway defined in finite fields."))) NIL NIL -(-296) +(-298) ((|constructor| (NIL "FiniteFieldCategory is the category of finite fields")) (|representationType| (((|Union| "prime" "polynomial" "normal" "cyclic")) "\\spad{representationType()} returns the type of the representation,{} one of: \\spad{prime},{} \\spad{polynomial},{} \\spad{normal},{} or \\spad{cyclic}.")) (|order| (((|PositiveInteger|) $) "\\spad{order(b)} computes the order of an element \\spad{b} in the multiplicative group of the field. Error: if \\spad{b} equals 0.")) (|discreteLog| (((|NonNegativeInteger|) $) "\\spad{discreteLog(a)} computes the discrete logarithm of \\spad{a} with respect to \\spad{primitiveElement()} of the field.")) (|primitive?| (((|Boolean|) $) "\\spad{primitive?(b)} tests whether the element \\spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \\spadignore{i.e.} is a primitive element. Implementation Note: see ch.IX.1.3,{} th.2 in \\spad{D}. Lipson.")) (|primitiveElement| (($) "\\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \\spadfun{createPrimitiveElement}.")) (|createPrimitiveElement| (($) "\\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.")) (|tableForDiscreteLogarithm| (((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|)) "\\spad{tableForDiscreteLogarithm(a,n)} returns a table of the discrete logarithms of \\spad{a**0} up to \\spad{a**(n-1)} which,{} called with key \\spad{lookup(a**i)} returns \\spad{i} for \\spad{i} in \\spad{0..n-1}. Error: if not called for prime divisors of order of \\indented{7}{multiplicative group.}")) (|factorsOfCyclicGroupSize| (((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|))))) "\\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\\spad{-1}")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or \"failed\" if no such vector exists.")) (|charthRoot| (($ $) "\\spad{charthRoot(a)} takes the characteristic'th root of {\\em a}. Note: such a root is alway defined in finite fields."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-297 R UP -3078) +(-299 R UP -3088) ((|constructor| (NIL "In this package \\spad{R} is a Euclidean domain and \\spad{F} is a framed algebra over \\spad{R}. The package provides functions to compute the integral closure of \\spad{R} in the quotient field of \\spad{F}. It is assumed that \\spad{char(R/P) = char(R)} for any prime \\spad{P} of \\spad{R}. A typical instance of this is when \\spad{R = K[x]} and \\spad{F} is a function field over \\spad{R}.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) |#1|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{R} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) NIL NIL -(-298 |p| |extdeg|) +(-300 |p| |extdeg|) ((|constructor| (NIL "FiniteFieldNormalBasis(\\spad{p},{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the prime field with \\spad{p} elements. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial created by \\spadfunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}.")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: The time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| (|PrimeField| |#1|))) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| (|PrimeField| |#1|)) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| (-812 |#1|) (QUOTE (-116))) (|HasCategory| (-812 |#1|) (QUOTE (-315)))) (|HasCategory| (-812 |#1|) (QUOTE (-118))) (|HasCategory| (-812 |#1|) (QUOTE (-315))) (|HasCategory| (-812 |#1|) (QUOTE (-116)))) -(-299 GF |uni|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| (-817 |#1|) (QUOTE (-118))) (|HasCategory| (-817 |#1|) (QUOTE (-317)))) (|HasCategory| (-817 |#1|) (QUOTE (-120))) (|HasCategory| (-817 |#1|) (QUOTE (-317))) (|HasCategory| (-817 |#1|) (QUOTE (-118)))) +(-301 GF |uni|) ((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(GF,{}uni) implements a finite extension of the ground field {\\em GF}. The elements are represented by coordinate vectors with respect to. a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element,{} where \\spad{q} is the size of {\\em GF}. The normal element is chosen as a root of the extension polynomial,{} which MUST be normal over {\\em GF} (user responsibility)")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-116)))) -(-300 GF |extdeg|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-118)))) +(-302 GF |extdeg|) ((|constructor| (NIL "FiniteFieldNormalBasisExtensionByPolynomial(GF,{}\\spad{n}) implements a finite extension field of degree \\spad{n} over the ground field {\\em GF}. The elements are represented by coordinate vectors with respect to a normal basis,{} \\spadignore{i.e.} a basis consisting of the conjugates (\\spad{q}-powers) of an element,{} in this case called normal element. This is chosen as a root of the extension polynomial,{} created by {\\em createNormalPoly} from \\spadtype{FiniteFieldPolynomialPackage}")) (|sizeMultiplication| (((|NonNegativeInteger|)) "\\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.")) (|getMultiplicationMatrix| (((|Matrix| |#1|)) "\\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.")) (|getMultiplicationTable| (((|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|)))))) "\\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-116)))) -(-301 GF |defpol|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-118)))) +(-303 GF |defpol|) ((|constructor| (NIL "FiniteFieldExtensionByPolynomial(GF,{} defpol) implements the extension of the finite field {\\em GF} generated by the extension polynomial {\\em defpol} which MUST be irreducible. Note: the user has the responsibility to ensure that {\\em defpol} is irreducible."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-116)))) -(-302 GF) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-118)))) +(-304 GF) ((|constructor| (NIL "This package provides a number of functions for generating,{} counting and testing irreducible,{} normal,{} primitive,{} random polynomials over finite fields.")) (|reducedQPowers| (((|PrimitiveArray| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{reducedQPowers(f)} generates \\spad{[x,x**q,x**(q**2),...,x**(q**(n-1))]} reduced modulo \\spad{f} where \\spad{q = size()\\$GF} and \\spad{n = degree f}.")) (|leastAffineMultiple| (((|SparseUnivariatePolynomial| |#1|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{leastAffineMultiple(f)} computes the least affine polynomial which is divisible by the polynomial \\spad{f} over the finite field {\\em GF},{} \\spadignore{i.e.} a polynomial whose exponents are 0 or a power of \\spad{q},{} the size of {\\em GF}.")) (|random| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{random(m,n)}\\$FFPOLY(GF) generates a random monic polynomial of degree \\spad{d} over the finite field {\\em GF},{} \\spad{d} between \\spad{m} and \\spad{n}.") (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{random(n)}\\$FFPOLY(GF) generates a random monic polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|nextPrimitiveNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitiveNormalPoly(f)} yields the next primitive normal polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g} or,{} in case these numbers are equal,{} if the {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than this number for \\spad{g}. If these numbers are equals,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g},{} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are coefficients according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}. This operation is equivalent to nextNormalPrimitivePoly(\\spad{f}).")) (|nextNormalPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPrimitivePoly(f)} yields the next normal primitive polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g} or if {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than this number for \\spad{g}. Otherwise,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents for \\spad{f} are lexicographically less than those for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}. This operation is equivalent to nextPrimitiveNormalPoly(\\spad{f}).")) (|nextNormalPoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextNormalPoly(f)} yields the next normal polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the coefficient of the term of degree {\\em n-1} of \\spad{f} is less than that for \\spad{g}. In case these numbers are equal,{} \\spad{f < g} if if the number of monomials of \\spad{f} is less that for \\spad{g} or if the list of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|nextPrimitivePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextPrimitivePoly(f)} yields the next primitive polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the {\\em lookup} of the constant term of \\spad{f} is less than this number for \\spad{g}. If these values are equal,{} then \\spad{f < g} if if the number of monomials of \\spad{f} is less than that for \\spad{g} or if the lists of exponents of \\spad{f} are lexicographically less than the corresponding list for \\spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|nextIrreduciblePoly| (((|Union| (|SparseUnivariatePolynomial| |#1|) "failed") (|SparseUnivariatePolynomial| |#1|)) "\\spad{nextIrreduciblePoly(f)} yields the next monic irreducible polynomial over a finite field {\\em GF} of the same degree as \\spad{f} in the following order,{} or \"failed\" if there are no greater ones. Error: if \\spad{f} has degree 0. Note: the input polynomial \\spad{f} is made monic. Also,{} \\spad{f < g} if the number of monomials of \\spad{f} is less than this number for \\spad{g}. If \\spad{f} and \\spad{g} have the same number of monomials,{} the lists of exponents are compared lexicographically. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\\em GF} given by {\\em lookup}.")) (|createPrimitiveNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitiveNormalPoly(n)}\\$FFPOLY(GF) generates a normal and primitive polynomial of degree \\spad{n} over the field {\\em GF}. polynomial of degree \\spad{n} over the field {\\em GF}.")) (|createNormalPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPrimitivePoly(n)}\\$FFPOLY(GF) generates a normal and primitive polynomial of degree \\spad{n} over the field {\\em GF}. Note: this function is equivalent to createPrimitiveNormalPoly(\\spad{n})")) (|createNormalPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createNormalPoly(n)}\\$FFPOLY(GF) generates a normal polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|createPrimitivePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createPrimitivePoly(n)}\\$FFPOLY(GF) generates a primitive polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|createIrreduciblePoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{createIrreduciblePoly(n)}\\$FFPOLY(GF) generates a monic irreducible univariate polynomial of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfNormalPoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfNormalPoly(n)}\\$FFPOLY(GF) yields the number of normal polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfPrimitivePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfPrimitivePoly(n)}\\$FFPOLY(GF) yields the number of primitive polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|numberOfIrreduciblePoly| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{numberOfIrreduciblePoly(n)}\\$FFPOLY(GF) yields the number of monic irreducible univariate polynomials of degree \\spad{n} over the finite field {\\em GF}.")) (|normal?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{normal?(f)} tests whether the polynomial \\spad{f} over a finite field is normal,{} \\spadignore{i.e.} its roots are linearly independent over the field.")) (|primitive?| (((|Boolean|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{primitive?(f)} tests whether the polynomial \\spad{f} over a finite field is primitive,{} \\spadignore{i.e.} all its roots are primitive."))) NIL NIL -(-303 -3078 GF) +(-305 -3088 GF) ((|constructor| (NIL "\\spad{FiniteFieldPolynomialPackage2}(\\spad{F},{}GF) exports some functions concerning finite fields,{} which depend on a finite field {\\em GF} and an algebraic extension \\spad{F} of {\\em GF},{} \\spadignore{e.g.} a zero of a polynomial over {\\em GF} in \\spad{F}.")) (|rootOfIrreduciblePoly| ((|#1| (|SparseUnivariatePolynomial| |#2|)) "\\spad{rootOfIrreduciblePoly(f)} computes one root of the monic,{} irreducible polynomial \\spad{f},{} which degree must divide the extension degree of {\\em F} over {\\em GF},{} \\spadignore{i.e.} \\spad{f} splits into linear factors over {\\em F}.")) (|Frobenius| ((|#1| |#1|) "\\spad{Frobenius(x)} \\undocumented{}")) (|basis| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{basis(n)} \\undocumented{}")) (|lookup| (((|PositiveInteger|) |#1|) "\\spad{lookup(x)} \\undocumented{}")) (|coerce| ((|#1| |#2|) "\\spad{coerce(x)} \\undocumented{}"))) NIL NIL -(-304 -3078 FP FPP) +(-306 -3088 FP FPP) ((|constructor| (NIL "This package solves linear diophantine equations for Bivariate polynomials over finite fields")) (|solveLinearPolynomialEquation| (((|Union| (|List| |#3|) "failed") (|List| |#3|) |#3|) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}'s exists."))) NIL NIL -(-305 GF |n|) +(-307 GF |n|) ((|constructor| (NIL "FiniteFieldExtensionByPolynomial(GF,{} \\spad{n}) implements an extension of the finite field {\\em GF} of degree \\spad{n} generated by the extension polynomial constructed by \\spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage} from \\spadtype{FiniteFieldPolynomialPackage}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-116)))) -(-306 R |ls|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-118)))) +(-308 R |ls|) ((|constructor| (NIL "This is just an interface between several packages and domains. The goal is to compute lexicographical Groebner bases of sets of polynomial with type \\spadtype{Polynomial R} by the {\\em FGLM} algorithm if this is possible (\\spadignore{i.e.} if the input system generates a zero-dimensional ideal).")) (|groebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|))) "\\axiom{groebner(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}}. If \\axiom{\\spad{lq1}} generates a zero-dimensional ideal then the {\\em FGLM} strategy is used,{} otherwise the {\\em Sugar} strategy is used.")) (|fglmIfCan| (((|Union| (|List| (|Polynomial| |#1|)) "failed") (|List| (|Polynomial| |#1|))) "\\axiom{fglmIfCan(\\spad{lq1})} returns the lexicographical Groebner basis of \\axiom{\\spad{lq1}} by using the {\\em FGLM} strategy,{} if \\axiom{zeroDimensional?(\\spad{lq1})} holds.")) (|zeroDimensional?| (((|Boolean|) (|List| (|Polynomial| |#1|))) "\\axiom{zeroDimensional?(\\spad{lq1})} returns \\spad{true} iff \\axiom{\\spad{lq1}} generates a zero-dimensional ideal \\spad{w}.\\spad{r}.\\spad{t}. the variables of \\axiom{ls}."))) NIL NIL -(-307 S) +(-309 S) ((|constructor| (NIL "The free group on a set \\spad{S} is the group of finite products of the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}'s are in \\spad{S},{} and the \\spad{ni}'s are integers. The multiplication is not commutative.")) (|factors| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| (|Integer|)))) $) "\\spad{factors(a1\\^e1,...,an\\^en)} returns \\spad{[[a1, e1],...,[an, en]]}.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| (|Integer|) (|Integer|)) $) "\\spad{mapExpon(f, a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x, n)} returns the factor of the n^th monomial of \\spad{x}.")) (|nthExpon| (((|Integer|) $ (|Integer|)) "\\spad{nthExpon(x, n)} returns the exponent of the n^th monomial of \\spad{x}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of monomials in \\spad{x}.")) (** (($ |#1| (|Integer|)) "\\spad{s ** n} returns the product of \\spad{s} by itself \\spad{n} times.")) (* (($ $ |#1|) "\\spad{x * s} returns the product of \\spad{x} by \\spad{s} on the right.") (($ |#1| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) -((-3975 . T)) +((-3986 . T)) NIL -(-308 S) +(-310 S) ((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) NIL NIL -(-309) +(-311) ((|constructor| (NIL "The category of commutative fields,{} \\spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \\spadfun{factor} operation while trivial is useful to have defined. \\blankline")) (|canonicalsClosed| ((|attribute|) "since \\spad{0*0=0},{} \\spad{1*1=1}")) (|canonicalUnitNormal| ((|attribute|) "either 0 or 1.")) (/ (($ $ $) "\\spad{x/y} divides the element \\spad{x} by the element \\spad{y}. Error: if \\spad{y} is 0."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-310 S) +(-312 S) ((|constructor| (NIL "This domain provides a basic model of files to save arbitrary values. The operations provide sequential access to the contents.")) (|readIfCan!| (((|Union| |#1| "failed") $) "\\spad{readIfCan!(f)} returns a value from the file \\spad{f},{} if possible. If \\spad{f} is not open for reading,{} or if \\spad{f} is at the end of file then \\spad{\"failed\"} is the result."))) NIL NIL -(-311 |Name| S) +(-313 |Name| S) ((|constructor| (NIL "This category provides an interface to operate on files in the computer's file system. The precise method of naming files is determined by the Name parameter. The type of the contents of the file is determined by \\spad{S}.")) (|write!| ((|#2| $ |#2|) "\\spad{write!(f,s)} puts the value \\spad{s} into the file \\spad{f}. The state of \\spad{f} is modified so subsequents call to \\spad{write!} will append one after another.")) (|read!| ((|#2| $) "\\spad{read!(f)} extracts a value from file \\spad{f}. The state of \\spad{f} is modified so a subsequent call to \\spadfun{read!} will return the next element.")) (|iomode| (((|String|) $) "\\spad{iomode(f)} returns the status of the file \\spad{f}. The input/output status of \\spad{f} may be \"input\",{} \"output\" or \"closed\" mode.")) (|name| ((|#1| $) "\\spad{name(f)} returns the external name of the file \\spad{f}.")) (|close!| (($ $) "\\spad{close!(f)} returns the file \\spad{f} closed to input and output.")) (|reopen!| (($ $ (|String|)) "\\spad{reopen!(f,mode)} returns a file \\spad{f} reopened for operation in the indicated mode: \"input\" or \"output\". \\spad{reopen!(f,\"input\")} will reopen the file \\spad{f} for input.")) (|open| (($ |#1| (|String|)) "\\spad{open(s,mode)} returns a file \\spad{s} open for operation in the indicated mode: \"input\" or \"output\".") (($ |#1|) "\\spad{open(s)} returns the file \\spad{s} open for input."))) NIL NIL -(-312 S R) +(-314 S R) ((|constructor| (NIL "A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|unitsKnown| ((|attribute|) "unitsKnown means that \\spadfun{recip} truly yields reciprocal or \\spad{\"failed\"} if not a unit,{} similarly for \\spadfun{leftRecip} and \\spadfun{rightRecip}. The reason is that we use left,{} respectively right,{} minimal polynomials to decide this question.")) (|unit| (((|Union| $ "failed")) "\\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnit| (((|Union| $ "failed")) "\\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|leftUnit| (((|Union| $ "failed")) "\\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|rightMinimalPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|leftMinimalPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|associatorDependence| (((|List| (|Vector| |#2|))) "\\spad{associatorDependence()} looks for the associator identities,{} \\spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \\spad{associator(a,b,c)} which yield 0,{} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. The order of the permutations is \\spad{123 231 312 132 321 213}.")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|lieAlgebra?| (((|Boolean|)) "\\spad{lieAlgebra?()} tests if the algebra is anticommutative and \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Lie algebra \\spad{(A,+,*)},{} where \\spad{a*b := a@b-b@a}.")) (|jordanAlgebra?| (((|Boolean|)) "\\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \\spad{(a*b)*a**2 - a*(b*a**2) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Jordan algebra \\spad{(A,+,*)},{} where \\spad{a*b := (a@b+b@a)/2}.")) (|noncommutativeJordanAlgebra?| (((|Boolean|)) "\\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.")) (|jordanAdmissible?| (((|Boolean|)) "\\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \\spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \\spadignore{i.e.} satisfies the Jordan identity. The property of \\spadatt{commutative(\"*\")} follows from by definition.")) (|lieAdmissible?| (((|Boolean|)) "\\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \\spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.")) (|jacobiIdentity?| (((|Boolean|)) "\\spad{jacobiIdentity?()} tests if \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.")) (|powerAssociative?| (((|Boolean|)) "\\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.")) (|alternative?| (((|Boolean|)) "\\spad{alternative?()} tests if \\spad{2*associator(a,a,b) = 0 = 2*associator(a,b,b)} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|flexible?| (((|Boolean|)) "\\spad{flexible?()} tests if \\spad{2*associator(a,b,a) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|rightAlternative?| (((|Boolean|)) "\\spad{rightAlternative?()} tests if \\spad{2*associator(a,b,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|leftAlternative?| (((|Boolean|)) "\\spad{leftAlternative?()} tests if \\spad{2*associator(a,a,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|antiAssociative?| (((|Boolean|)) "\\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \\spadignore{i.e.} \\spad{(a*b)*c + a*(b*c) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra.")) (|associative?| (((|Boolean|)) "\\spad{associative?()} tests if multiplication in algebra is associative.")) (|antiCommutative?| (((|Boolean|)) "\\spad{antiCommutative?()} tests if \\spad{a*a = 0} for all \\spad{a} in the algebra. Note: this implies \\spad{a*b + b*a = 0} for all \\spad{a} and \\spad{b}.")) (|commutative?| (((|Boolean|)) "\\spad{commutative?()} tests if multiplication in the algebra is commutative.")) (|rightCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \\spad{a} with respect to any basis.")) (|leftCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \\spad{a} with respect to any basis.")) (|rightTraceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{rightTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}.")) (|leftTraceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{leftTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}.")) (|rightDiscriminant| ((|#2| (|Vector| $)) "\\spad{rightDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(rightTraceMatrix([v1,...,vn]))}.")) (|leftDiscriminant| ((|#2| (|Vector| $)) "\\spad{leftDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(leftTraceMatrix([v1,...,vn]))}.")) (|represents| (($ (|Vector| |#2|) (|Vector| $)) "\\spad{represents([a1,...,am],[v1,...,vm])} returns the linear combination \\spad{a1*vm + ... + an*vm}.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([a1,...,am],[v1,...,vn])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.") (((|Vector| |#2|) $ (|Vector| $)) "\\spad{coordinates(a,[v1,...,vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rightNorm| ((|#2| $) "\\spad{rightNorm(a)} returns the determinant of the right regular representation of \\spad{a}.")) (|leftNorm| ((|#2| $) "\\spad{leftNorm(a)} returns the determinant of the left regular representation of \\spad{a}.")) (|rightTrace| ((|#2| $) "\\spad{rightTrace(a)} returns the trace of the right regular representation of \\spad{a}.")) (|leftTrace| ((|#2| $) "\\spad{leftTrace(a)} returns the trace of the left regular representation of \\spad{a}.")) (|rightRegularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{rightRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|leftRegularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{leftRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|structuralConstants| (((|Vector| (|Matrix| |#2|)) (|Vector| $)) "\\spad{structuralConstants([v1,v2,...,vm])} calculates the structural constants \\spad{[(gammaijk) for k in 1..m]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \\spad{[v1,...,vm]} is an \\spad{R}-module basis of a subalgebra.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#2|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,...,vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra as \\spad{R}-module.")) (|someBasis| (((|Vector| $)) "\\spad{someBasis()} returns some \\spad{R}-module basis."))) NIL -((|HasCategory| |#2| (QUOTE (-491)))) -(-313 R) +((|HasCategory| |#2| (QUOTE (-494)))) +(-315 R) ((|constructor| (NIL "A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|unitsKnown| ((|attribute|) "unitsKnown means that \\spadfun{recip} truly yields reciprocal or \\spad{\"failed\"} if not a unit,{} similarly for \\spadfun{leftRecip} and \\spadfun{rightRecip}. The reason is that we use left,{} respectively right,{} minimal polynomials to decide this question.")) (|unit| (((|Union| $ "failed")) "\\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnit| (((|Union| $ "failed")) "\\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|leftUnit| (((|Union| $ "failed")) "\\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \\spad{\"failed\"} if there is none.")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none.")) (|rightMinimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|leftMinimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \\spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.")) (|associatorDependence| (((|List| (|Vector| |#1|))) "\\spad{associatorDependence()} looks for the associator identities,{} \\spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \\spad{associator(a,b,c)} which yield 0,{} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. The order of the permutations is \\spad{123 231 312 132 321 213}.")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if there is no unit element,{} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|lieAlgebra?| (((|Boolean|)) "\\spad{lieAlgebra?()} tests if the algebra is anticommutative and \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Lie algebra \\spad{(A,+,*)},{} where \\spad{a*b := a@b-b@a}.")) (|jordanAlgebra?| (((|Boolean|)) "\\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \\spad{(a*b)*a**2 - a*(b*a**2) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \\spad{(A,+,@)} we can construct a Jordan algebra \\spad{(A,+,*)},{} where \\spad{a*b := (a@b+b@a)/2}.")) (|noncommutativeJordanAlgebra?| (((|Boolean|)) "\\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.")) (|jordanAdmissible?| (((|Boolean|)) "\\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \\spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \\spadignore{i.e.} satisfies the Jordan identity. The property of \\spadatt{commutative(\"*\")} follows from by definition.")) (|lieAdmissible?| (((|Boolean|)) "\\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \\spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.")) (|jacobiIdentity?| (((|Boolean|)) "\\spad{jacobiIdentity?()} tests if \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.")) (|powerAssociative?| (((|Boolean|)) "\\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.")) (|alternative?| (((|Boolean|)) "\\spad{alternative?()} tests if \\spad{2*associator(a,a,b) = 0 = 2*associator(a,b,b)} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|flexible?| (((|Boolean|)) "\\spad{flexible?()} tests if \\spad{2*associator(a,b,a) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|rightAlternative?| (((|Boolean|)) "\\spad{rightAlternative?()} tests if \\spad{2*associator(a,b,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|leftAlternative?| (((|Boolean|)) "\\spad{leftAlternative?()} tests if \\spad{2*associator(a,a,b) = 0} for all \\spad{a},{} \\spad{b} in the algebra. Note: we only can test this; in general we don't know whether \\spad{2*a=0} implies \\spad{a=0}.")) (|antiAssociative?| (((|Boolean|)) "\\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \\spadignore{i.e.} \\spad{(a*b)*c + a*(b*c) = 0} for all \\spad{a},{}\\spad{b},{}\\spad{c} in the algebra.")) (|associative?| (((|Boolean|)) "\\spad{associative?()} tests if multiplication in algebra is associative.")) (|antiCommutative?| (((|Boolean|)) "\\spad{antiCommutative?()} tests if \\spad{a*a = 0} for all \\spad{a} in the algebra. Note: this implies \\spad{a*b + b*a = 0} for all \\spad{a} and \\spad{b}.")) (|commutative?| (((|Boolean|)) "\\spad{commutative?()} tests if multiplication in the algebra is commutative.")) (|rightCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \\spad{a} with respect to any basis.")) (|leftCharacteristicPolynomial| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \\spad{a} with respect to any basis.")) (|rightTraceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{rightTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}.")) (|leftTraceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{leftTraceMatrix([v1,...,vn])} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}.")) (|rightDiscriminant| ((|#1| (|Vector| $)) "\\spad{rightDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(rightTraceMatrix([v1,...,vn]))}.")) (|leftDiscriminant| ((|#1| (|Vector| $)) "\\spad{leftDiscriminant([v1,...,vn])} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj}. Note: the same as \\spad{determinant(leftTraceMatrix([v1,...,vn]))}.")) (|represents| (($ (|Vector| |#1|) (|Vector| $)) "\\spad{represents([a1,...,am],[v1,...,vm])} returns the linear combination \\spad{a1*vm + ... + an*vm}.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([a1,...,am],[v1,...,vn])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.") (((|Vector| |#1|) $ (|Vector| $)) "\\spad{coordinates(a,[v1,...,vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rightNorm| ((|#1| $) "\\spad{rightNorm(a)} returns the determinant of the right regular representation of \\spad{a}.")) (|leftNorm| ((|#1| $) "\\spad{leftNorm(a)} returns the determinant of the left regular representation of \\spad{a}.")) (|rightTrace| ((|#1| $) "\\spad{rightTrace(a)} returns the trace of the right regular representation of \\spad{a}.")) (|leftTrace| ((|#1| $) "\\spad{leftTrace(a)} returns the trace of the left regular representation of \\spad{a}.")) (|rightRegularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{rightRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|leftRegularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{leftRegularRepresentation(a,[v1,...,vn])} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{R}-module basis \\spad{[v1,...,vn]}.")) (|structuralConstants| (((|Vector| (|Matrix| |#1|)) (|Vector| $)) "\\spad{structuralConstants([v1,v2,...,vm])} calculates the structural constants \\spad{[(gammaijk) for k in 1..m]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \\spad{[v1,...,vm]} is an \\spad{R}-module basis of a subalgebra.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#1|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,...,vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra as \\spad{R}-module.")) (|someBasis| (((|Vector| $)) "\\spad{someBasis()} returns some \\spad{R}-module basis."))) -((-3975 |has| |#1| (-491)) (-3973 . T) (-3972 . T)) +((-3986 |has| |#1| (-494)) (-3984 . T) (-3983 . T)) NIL -(-314 S) +(-316 S) ((|constructor| (NIL "The category of domains composed of a finite set of elements. We include the functions \\spadfun{lookup} and \\spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \\blankline")) (|random| (($) "\\spad{random()} returns a random element from the set.")) (|lookup| (((|PositiveInteger|) $) "\\spad{lookup(x)} returns a positive integer such that \\spad{x = index lookup x}.")) (|index| (($ (|PositiveInteger|)) "\\spad{index(i)} takes a positive integer \\spad{i} less than or equal to \\spad{size()} and returns the \\spad{i}\\spad{-}th element of the set. This operation establishs a bijection between the elements of the finite set and \\spad{1..size()}.")) (|size| (((|NonNegativeInteger|)) "\\spad{size()} returns the number of elements in the set."))) NIL NIL -(-315) +(-317) ((|constructor| (NIL "The category of domains composed of a finite set of elements. We include the functions \\spadfun{lookup} and \\spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \\blankline")) (|random| (($) "\\spad{random()} returns a random element from the set.")) (|lookup| (((|PositiveInteger|) $) "\\spad{lookup(x)} returns a positive integer such that \\spad{x = index lookup x}.")) (|index| (($ (|PositiveInteger|)) "\\spad{index(i)} takes a positive integer \\spad{i} less than or equal to \\spad{size()} and returns the \\spad{i}\\spad{-}th element of the set. This operation establishs a bijection between the elements of the finite set and \\spad{1..size()}.")) (|size| (((|NonNegativeInteger|)) "\\spad{size()} returns the number of elements in the set."))) NIL NIL -(-316 S R UP) +(-318 S R UP) ((|constructor| (NIL "A FiniteRankAlgebra is an algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|minimalPolynomial| ((|#3| $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of \\spad{a}.")) (|characteristicPolynomial| ((|#3| $) "\\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \\spad{a} with respect to any basis.")) (|traceMatrix| (((|Matrix| |#2|) (|Vector| $)) "\\spad{traceMatrix([v1,..,vn])} is the \\spad{n}-by-\\spad{n} matrix ( Tr(\\spad{vi} * vj) )")) (|discriminant| ((|#2| (|Vector| $)) "\\spad{discriminant([v1,..,vn])} returns \\spad{determinant(traceMatrix([v1,..,vn]))}.")) (|represents| (($ (|Vector| |#2|) (|Vector| $)) "\\spad{represents([a1,..,an],[v1,..,vn])} returns \\spad{a1*v1 + ... + an*vn}.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([v1,...,vm], basis)} returns the coordinates of the \\spad{vi}'s with to the basis \\spad{basis}. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $ (|Vector| $)) "\\spad{coordinates(a,basis)} returns the coordinates of \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|norm| ((|#2| $) "\\spad{norm(a)} returns the determinant of the regular representation of \\spad{a} with respect to any basis.")) (|trace| ((|#2| $) "\\spad{trace(a)} returns the trace of the regular representation of \\spad{a} with respect to any basis.")) (|regularRepresentation| (((|Matrix| |#2|) $ (|Vector| $)) "\\spad{regularRepresentation(a,basis)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra."))) NIL -((|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-309)))) -(-317 R UP) +((|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-311)))) +(-319 R UP) ((|constructor| (NIL "A FiniteRankAlgebra is an algebra over a commutative ring \\spad{R} which is a free \\spad{R}-module of finite rank.")) (|minimalPolynomial| ((|#2| $) "\\spad{minimalPolynomial(a)} returns the minimal polynomial of \\spad{a}.")) (|characteristicPolynomial| ((|#2| $) "\\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \\spad{a} with respect to any basis.")) (|traceMatrix| (((|Matrix| |#1|) (|Vector| $)) "\\spad{traceMatrix([v1,..,vn])} is the \\spad{n}-by-\\spad{n} matrix ( Tr(\\spad{vi} * vj) )")) (|discriminant| ((|#1| (|Vector| $)) "\\spad{discriminant([v1,..,vn])} returns \\spad{determinant(traceMatrix([v1,..,vn]))}.")) (|represents| (($ (|Vector| |#1|) (|Vector| $)) "\\spad{represents([a1,..,an],[v1,..,vn])} returns \\spad{a1*v1 + ... + an*vn}.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $) (|Vector| $)) "\\spad{coordinates([v1,...,vm], basis)} returns the coordinates of the \\spad{vi}'s with to the basis \\spad{basis}. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $ (|Vector| $)) "\\spad{coordinates(a,basis)} returns the coordinates of \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|norm| ((|#1| $) "\\spad{norm(a)} returns the determinant of the regular representation of \\spad{a} with respect to any basis.")) (|trace| ((|#1| $) "\\spad{trace(a)} returns the trace of the regular representation of \\spad{a} with respect to any basis.")) (|regularRepresentation| (((|Matrix| |#1|) $ (|Vector| $)) "\\spad{regularRepresentation(a,basis)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the \\spad{basis} \\spad{basis}.")) (|rank| (((|PositiveInteger|)) "\\spad{rank()} returns the rank of the algebra."))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-318 A S) +(-320 A S) ((|constructor| (NIL "A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \\spadtype{LinearAggregate} such as \\spadfun{reverse},{} \\spadfun{sort},{} and so on.")) (|sort!| (($ $) "\\spad{sort!(u)} returns \\spad{u} with its elements in ascending order.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sort!(p,u)} returns \\spad{u} with its elements ordered by \\spad{p}.")) (|reverse!| (($ $) "\\spad{reverse!(u)} returns \\spad{u} with its elements in reverse order.")) (|copyInto!| (($ $ $ (|Integer|)) "\\spad{copyInto!(u,v,i)} returns aggregate \\spad{u} containing a copy of \\spad{v} inserted at element \\spad{i}.")) (|position| (((|Integer|) |#2| $ (|Integer|)) "\\spad{position(x,a,n)} returns the index \\spad{i} of the first occurrence of \\spad{x} in \\axiom{a} where \\axiom{\\spad{i} >= \\spad{n}},{} and \\axiom{minIndex(a) - 1} if no such \\spad{x} is found.") (((|Integer|) |#2| $) "\\spad{position(x,a)} returns the index \\spad{i} of the first occurrence of \\spad{x} in a,{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.") (((|Integer|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{position(p,a)} returns the index \\spad{i} of the first \\spad{x} in \\axiom{a} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.")) (|sorted?| (((|Boolean|) $) "\\spad{sorted?(u)} tests if the elements of \\spad{u} are in ascending order.") (((|Boolean|) (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sorted?(p,a)} tests if \\axiom{a} is sorted according to predicate \\spad{p}.")) (|sort| (($ $) "\\spad{sort(u)} returns an \\spad{u} with elements in ascending order. Note: \\axiom{sort(\\spad{u}) = sort(<=,{}\\spad{u})}.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $) "\\spad{sort(p,a)} returns a copy of \\axiom{a} sorted using total ordering predicate \\spad{p}.")) (|reverse| (($ $) "\\spad{reverse(a)} returns a copy of \\axiom{a} with elements in reverse order.")) (|merge| (($ $ $) "\\spad{merge(u,v)} merges \\spad{u} and \\spad{v} in ascending order. Note: \\axiom{merge(\\spad{u},{}\\spad{v}) = merge(<=,{}\\spad{u},{}\\spad{v})}.") (($ (|Mapping| (|Boolean|) |#2| |#2|) $ $) "\\spad{merge(p,a,b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) NIL -((|HasAttribute| |#1| (QUOTE -3979)) (|HasCategory| |#2| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-1007)))) -(-319 S) +((|HasAttribute| |#1| (QUOTE -3990)) (|HasCategory| |#2| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-1012)))) +(-321 S) ((|constructor| (NIL "A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \\spadtype{LinearAggregate} such as \\spadfun{reverse},{} \\spadfun{sort},{} and so on.")) (|sort!| (($ $) "\\spad{sort!(u)} returns \\spad{u} with its elements in ascending order.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sort!(p,u)} returns \\spad{u} with its elements ordered by \\spad{p}.")) (|reverse!| (($ $) "\\spad{reverse!(u)} returns \\spad{u} with its elements in reverse order.")) (|copyInto!| (($ $ $ (|Integer|)) "\\spad{copyInto!(u,v,i)} returns aggregate \\spad{u} containing a copy of \\spad{v} inserted at element \\spad{i}.")) (|position| (((|Integer|) |#1| $ (|Integer|)) "\\spad{position(x,a,n)} returns the index \\spad{i} of the first occurrence of \\spad{x} in \\axiom{a} where \\axiom{\\spad{i} >= \\spad{n}},{} and \\axiom{minIndex(a) - 1} if no such \\spad{x} is found.") (((|Integer|) |#1| $) "\\spad{position(x,a)} returns the index \\spad{i} of the first occurrence of \\spad{x} in a,{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.") (((|Integer|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{position(p,a)} returns the index \\spad{i} of the first \\spad{x} in \\axiom{a} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true},{} and \\axiom{minIndex(a) - 1} if there is no such \\spad{x}.")) (|sorted?| (((|Boolean|) $) "\\spad{sorted?(u)} tests if the elements of \\spad{u} are in ascending order.") (((|Boolean|) (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sorted?(p,a)} tests if \\axiom{a} is sorted according to predicate \\spad{p}.")) (|sort| (($ $) "\\spad{sort(u)} returns an \\spad{u} with elements in ascending order. Note: \\axiom{sort(\\spad{u}) = sort(<=,{}\\spad{u})}.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $) "\\spad{sort(p,a)} returns a copy of \\axiom{a} sorted using total ordering predicate \\spad{p}.")) (|reverse| (($ $) "\\spad{reverse(a)} returns a copy of \\axiom{a} with elements in reverse order.")) (|merge| (($ $ $) "\\spad{merge(u,v)} merges \\spad{u} and \\spad{v} in ascending order. Note: \\axiom{merge(\\spad{u},{}\\spad{v}) = merge(<=,{}\\spad{u},{}\\spad{v})}.") (($ (|Mapping| (|Boolean|) |#1| |#1|) $ $) "\\spad{merge(p,a,b)} returns an aggregate \\spad{c} which merges \\axiom{a} and \\spad{b}. The result is produced by examining each element \\spad{x} of \\axiom{a} and \\spad{y} of \\spad{b} successively. If \\axiom{\\spad{p}(\\spad{x},{}\\spad{y})} is \\spad{true},{} then \\spad{x} is inserted into the result; otherwise \\spad{y} is inserted. If \\spad{x} is chosen,{} the next element of \\axiom{a} is examined,{} and so on. When all the elements of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \\axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \\axiom{[1,{}2,{}3,{}7,{}5]}."))) -((-3978 . T)) +((-3989 . T)) NIL -(-320 S A R B) +(-322 S A R B) ((|constructor| (NIL "\\spad{FiniteLinearAggregateFunctions2} provides functions involving two FiniteLinearAggregates where the underlying domains might be different. An example of this might be creating a list of rational numbers by mapping a function across a list of integers where the function divides each integer by 1000.")) (|scan| ((|#4| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-aggregates \\spad{x} of aggregrate \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}.")) (|reduce| ((|#3| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the aggregate \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as the identity element for the function \\spad{f}.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f,a)} applies function \\spad{f} to each member of aggregate \\spad{a} resulting in a new aggregate over a possibly different underlying domain."))) NIL NIL -(-321 |VarSet| R) +(-323 |VarSet| R) ((|constructor| (NIL "The category of free Lie algebras. It is used by domains of non-commutative algebra: \\spadtype{LiePolynomial} and \\spadtype{XPBWPolynomial}. \\newline Author: Michel Petitot (petitot@lifl.fr)")) (|eval| (($ $ (|List| |#1|) (|List| $)) "\\axiom{eval(\\spad{p},{} [\\spad{x1},{}...,{}xn],{} [\\spad{v1},{}...,{}vn])} replaces \\axiom{\\spad{xi}} by \\axiom{\\spad{vi}} in \\axiom{\\spad{p}}.") (($ $ |#1| $) "\\axiom{eval(\\spad{p},{} \\spad{x},{} \\spad{v})} replaces \\axiom{\\spad{x}} by \\axiom{\\spad{v}} in \\axiom{\\spad{p}}.")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{x})} returns the list of distinct entries of \\axiom{\\spad{x}}.")) (|trunc| (($ $ (|NonNegativeInteger|)) "\\axiom{trunc(\\spad{p},{}\\spad{n})} returns the polynomial \\axiom{\\spad{p}} truncated at order \\axiom{\\spad{n}}.")) (|mirror| (($ $) "\\axiom{mirror(\\spad{x})} returns \\axiom{Sum(r_i mirror(w_i))} if \\axiom{\\spad{x}} is \\axiom{Sum(r_i w_i)}.")) (|LiePoly| (($ (|LyndonWord| |#1|)) "\\axiom{LiePoly(\\spad{l})} returns the bracketed form of \\axiom{\\spad{l}} as a Lie polynomial.")) (|rquo| (((|XRecursivePolynomial| |#1| |#2|) (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{rquo(\\spad{x},{}\\spad{y})} returns the right simplification of \\axiom{\\spad{x}} by \\axiom{\\spad{y}}.")) (|lquo| (((|XRecursivePolynomial| |#1| |#2|) (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{lquo(\\spad{x},{}\\spad{y})} returns the left simplification of \\axiom{\\spad{x}} by \\axiom{\\spad{y}}.")) (|degree| (((|NonNegativeInteger|) $) "\\axiom{degree(\\spad{x})} returns the greatest length of a word in the support of \\axiom{\\spad{x}}.")) (|coerce| (((|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as a recursive polynomial.") (((|XDistributedPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as distributed polynomial.") (($ |#1|) "\\axiom{coerce(\\spad{x})} returns \\axiom{\\spad{x}} as a Lie polynomial.")) (|coef| ((|#2| (|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coef(\\spad{x},{}\\spad{y})} returns the scalar product of \\axiom{\\spad{x}} by \\axiom{\\spad{y}},{} the set of words being regarded as an orthogonal basis."))) -((|JacobiIdentity| . T) (|NullSquare| . T) (-3973 . T) (-3972 . T)) +((|JacobiIdentity| . T) (|NullSquare| . T) (-3984 . T) (-3983 . T)) NIL -(-322 S V) +(-324 S V) ((|constructor| (NIL "This package exports 3 sorting algorithms which work over FiniteLinearAggregates.")) (|shellSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{shellSort(f, agg)} sorts the aggregate agg with the ordering function \\spad{f} using the shellSort algorithm.")) (|heapSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{heapSort(f, agg)} sorts the aggregate agg with the ordering function \\spad{f} using the heapsort algorithm.")) (|quickSort| ((|#2| (|Mapping| (|Boolean|) |#1| |#1|) |#2|) "\\spad{quickSort(f, agg)} sorts the aggregate agg with the ordering function \\spad{f} using the quicksort algorithm."))) NIL NIL -(-323 S R) +(-325 S R) ((|constructor| (NIL "\\spad{S} is \\spadtype{FullyLinearlyExplicitRingOver R} means that \\spad{S} is a \\spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \\spad{R} is a \\spadtype{LinearlyExplicitRingOver Integer},{} then so is \\spad{S}"))) NIL -((|HasCategory| |#2| (QUOTE (-577 (-480))))) -(-324 R) +((|HasCategory| |#2| (QUOTE (-580 (-483))))) +(-326 R) ((|constructor| (NIL "\\spad{S} is \\spadtype{FullyLinearlyExplicitRingOver R} means that \\spad{S} is a \\spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \\spad{R} is a \\spadtype{LinearlyExplicitRingOver Integer},{} then so is \\spad{S}"))) NIL NIL -(-325) +(-327) ((|constructor| (NIL "\\spadtype{Float} implements arbitrary precision floating point arithmetic. The number of significant digits of each operation can be set to an arbitrary value (the default is 20 decimal digits). The operation \\spad{float(mantissa,exponent,\\spadfunFrom{base}{FloatingPointSystem})} for integer \\spad{mantissa},{} \\spad{exponent} specifies the number \\spad{mantissa * \\spadfunFrom{base}{FloatingPointSystem} ** exponent} The underlying representation for floats is binary not decimal. The implications of this are described below. \\blankline The model adopted is that arithmetic operations are rounded to to nearest unit in the last place,{} that is,{} accurate to within \\spad{2**(-\\spadfunFrom{bits}{FloatingPointSystem})}. Also,{} the elementary functions and constants are accurate to one unit in the last place. A float is represented as a record of two integers,{} the mantissa and the exponent. The \\spadfunFrom{base}{FloatingPointSystem} of the representation is binary,{} hence a \\spad{Record(m:mantissa,e:exponent)} represents the number \\spad{m * 2 ** e}. Though it is not assumed that the underlying integers are represented with a binary \\spadfunFrom{base}{FloatingPointSystem},{} the code will be most efficient when this is the the case (this is \\spad{true} in most implementations of Lisp). The decision to choose the \\spadfunFrom{base}{FloatingPointSystem} to be binary has some unfortunate consequences. First,{} decimal numbers like 0.3 cannot be represented exactly. Second,{} there is a further loss of accuracy during conversion to decimal for output. To compensate for this,{} if \\spad{d} digits of precision are specified,{} \\spad{1 + ceiling(log2 d)} bits are used. Two numbers that are displayed identically may therefore be not equal. On the other hand,{} a significant efficiency loss would be incurred if we chose to use a decimal \\spadfunFrom{base}{FloatingPointSystem} when the underlying integer base is binary. \\blankline Algorithms used: For the elementary functions,{} the general approach is to apply identities so that the taylor series can be used,{} and,{} so that it will converge within \\spad{O( sqrt n )} steps. For example,{} using the identity \\spad{exp(x) = exp(x/2)**2},{} we can compute \\spad{exp(1/3)} to \\spad{n} digits of precision as follows. We have \\spad{exp(1/3) = exp(2 ** (-sqrt s) / 3) ** (2 ** sqrt s)}. The taylor series will converge in less than sqrt \\spad{n} steps and the exponentiation requires sqrt \\spad{n} multiplications for a total of \\spad{2 sqrt n} multiplications. Assuming integer multiplication costs \\spad{O( n**2 )} the overall running time is \\spad{O( sqrt(n) n**2 )}. This approach is the best known approach for precisions up to about 10,{}000 digits at which point the methods of Brent which are \\spad{O( log(n) n**2 )} become competitive. Note also that summing the terms of the taylor series for the elementary functions is done using integer operations. This avoids the overhead of floating point operations and results in efficient code at low precisions. This implementation makes no attempt to reuse storage,{} relying on the underlying system to do \\spadgloss{garbage collection}. \\spad{I} estimate that the efficiency of this package at low precisions could be improved by a factor of 2 if in-place operations were available. \\blankline Running times: in the following,{} \\spad{n} is the number of bits of precision \\indented{5}{\\spad{*},{} \\spad{/},{} \\spad{sqrt},{} \\spad{pi},{} \\spad{exp1},{} \\spad{log2},{} \\spad{log10}: \\spad{ O( n**2 )}} \\indented{5}{\\spad{exp},{} \\spad{log},{} \\spad{sin},{} \\spad{atan}:\\space{2}\\spad{ O( sqrt(n) n**2 )}} The other elementary functions are coded in terms of the ones above.")) (|outputSpacing| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputSpacing(n)} inserts a space after \\spad{n} (default 10) digits on output; outputSpacing(0) means no spaces are inserted.")) (|outputGeneral| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputGeneral(n)} sets the output mode to general notation with \\spad{n} significant digits displayed.") (((|Void|)) "\\spad{outputGeneral()} sets the output mode (default mode) to general notation; numbers will be displayed in either fixed or floating (scientific) notation depending on the magnitude.")) (|outputFixed| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFixed(n)} sets the output mode to fixed point notation,{} with \\spad{n} digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFixed()} sets the output mode to fixed point notation; the output will contain a decimal point.")) (|outputFloating| (((|Void|) (|NonNegativeInteger|)) "\\spad{outputFloating(n)} sets the output mode to floating (scientific) notation with \\spad{n} significant digits displayed after the decimal point.") (((|Void|)) "\\spad{outputFloating()} sets the output mode to floating (scientific) notation,{} \\spadignore{i.e.} \\spad{mantissa * 10 exponent} is displayed as \\spad{0.mantissa E exponent}.")) (|atan| (($ $ $) "\\spad{atan(x,y)} computes the arc tangent from \\spad{x} with phase \\spad{y}.")) (|exp1| (($) "\\spad{exp1()} returns exp 1: \\spad{2.7182818284...}.")) (|log10| (($ $) "\\spad{log10(x)} computes the logarithm for \\spad{x} to base 10.") (($) "\\spad{log10()} returns \\spad{ln 10}: \\spad{2.3025809299...}.")) (|log2| (($ $) "\\spad{log2(x)} computes the logarithm for \\spad{x} to base 2.") (($) "\\spad{log2()} returns \\spad{ln 2},{} \\spadignore{i.e.} \\spad{0.6931471805...}.")) (|rationalApproximation| (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n, b)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< b**(-n)},{} that is \\spad{|(r-f)/f| < b**(-n)}.") (((|Fraction| (|Integer|)) $ (|NonNegativeInteger|)) "\\spad{rationalApproximation(f, n)} computes a rational approximation \\spad{r} to \\spad{f} with relative error \\spad{< 10**(-n)}.")) (|shift| (($ $ (|Integer|)) "\\spad{shift(x,n)} adds \\spad{n} to the exponent of float \\spad{x}.")) (|relerror| (((|Integer|) $ $) "\\spad{relerror(x,y)} computes the absolute value of \\spad{x - y} divided by \\spad{y},{} when \\spad{y \\~= 0}.")) (|normalize| (($ $) "\\spad{normalize(x)} normalizes \\spad{x} at current precision.")) (** (($ $ $) "\\spad{x ** y} computes \\spad{exp(y log x)} where \\spad{x >= 0}.")) (/ (($ $ (|Integer|)) "\\spad{x / i} computes the division from \\spad{x} by an integer \\spad{i}."))) -((-3961 . T) (-3969 . T) (-3753 . T) (-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3972 . T) (-3980 . T) (-3764 . T) (-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-326 |Par|) +(-328 |Par|) ((|constructor| (NIL "\\indented{3}{This is a package for the approximation of complex solutions for} systems of equations of rational functions with complex rational coefficients. The results are expressed as either complex rational numbers or complex floats depending on the type of the precision parameter which can be either a rational number or a floating point number.")) (|complexRoots| (((|List| (|List| (|Complex| |#1|))) (|List| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) (|List| (|Symbol|)) |#1|) "\\spad{complexRoots(lrf, lv, eps)} finds all the complex solutions of a list of rational functions with rational number coefficients with respect the the variables appearing in \\spad{lv}. Each solution is computed to precision eps and returned as list corresponding to the order of variables in \\spad{lv}.") (((|List| (|Complex| |#1|)) (|Fraction| (|Polynomial| (|Complex| (|Integer|)))) |#1|) "\\spad{complexRoots(rf, eps)} finds all the complex solutions of a univariate rational function with rational number coefficients. The solutions are computed to precision eps.")) (|complexSolve| (((|List| (|Equation| (|Polynomial| (|Complex| |#1|)))) (|Equation| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) |#1|) "\\spad{complexSolve(eq,eps)} finds all the complex solutions of the equation \\spad{eq} of rational functions with rational rational coefficients with respect to all the variables appearing in \\spad{eq},{} with precision \\spad{eps}.") (((|List| (|Equation| (|Polynomial| (|Complex| |#1|)))) (|Fraction| (|Polynomial| (|Complex| (|Integer|)))) |#1|) "\\spad{complexSolve(p,eps)} find all the complex solutions of the rational function \\spad{p} with complex rational coefficients with respect to all the variables appearing in \\spad{p},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| (|Complex| |#1|))))) (|List| (|Equation| (|Fraction| (|Polynomial| (|Complex| (|Integer|)))))) |#1|) "\\spad{complexSolve(leq,eps)} finds all the complex solutions to precision \\spad{eps} of the system \\spad{leq} of equations of rational functions over complex rationals with respect to all the variables appearing in lp.") (((|List| (|List| (|Equation| (|Polynomial| (|Complex| |#1|))))) (|List| (|Fraction| (|Polynomial| (|Complex| (|Integer|))))) |#1|) "\\spad{complexSolve(lp,eps)} finds all the complex solutions to precision \\spad{eps} of the system \\spad{lp} of rational functions over the complex rationals with respect to all the variables appearing in \\spad{lp}."))) NIL NIL -(-327 |Par|) +(-329 |Par|) ((|constructor| (NIL "\\indented{3}{This is a package for the approximation of real solutions for} systems of polynomial equations over the rational numbers. The results are expressed as either rational numbers or floats depending on the type of the precision parameter which can be either a rational number or a floating point number.")) (|realRoots| (((|List| |#1|) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{realRoots(rf, eps)} finds the real zeros of a univariate rational function with precision given by eps.") (((|List| (|List| |#1|)) (|List| (|Fraction| (|Polynomial| (|Integer|)))) (|List| (|Symbol|)) |#1|) "\\spad{realRoots(lp,lv,eps)} computes the list of the real solutions of the list \\spad{lp} of rational functions with rational coefficients with respect to the variables in \\spad{lv},{} with precision \\spad{eps}. Each solution is expressed as a list of numbers in order corresponding to the variables in \\spad{lv}.")) (|solve| (((|List| (|Equation| (|Polynomial| |#1|))) (|Equation| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(eq,eps)} finds all of the real solutions of the univariate equation \\spad{eq} of rational functions with respect to the unique variables appearing in \\spad{eq},{} with precision \\spad{eps}.") (((|List| (|Equation| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| (|Integer|))) |#1|) "\\spad{solve(p,eps)} finds all of the real solutions of the univariate rational function \\spad{p} with rational coefficients with respect to the unique variable appearing in \\spad{p},{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| (|Integer|))))) |#1|) "\\spad{solve(leq,eps)} finds all of the real solutions of the system \\spad{leq} of equationas of rational functions with respect to all the variables appearing in lp,{} with precision \\spad{eps}.") (((|List| (|List| (|Equation| (|Polynomial| |#1|)))) (|List| (|Fraction| (|Polynomial| (|Integer|)))) |#1|) "\\spad{solve(lp,eps)} finds all of the real solutions of the system \\spad{lp} of rational functions over the rational numbers with respect to all the variables appearing in \\spad{lp},{} with precision \\spad{eps}."))) NIL NIL -(-328 R S) +(-330 R S) ((|constructor| (NIL "A \\spad{bi}-module is a free module over a ring with generators indexed by an ordered set. Each element can be expressed as a finite linear combination of generators. Only non-zero terms are stored."))) -((-3973 . T) (-3972 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007))))) -(-329 R S) +((-3984 . T) (-3983 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012))))) +(-331 R S) ((|constructor| (NIL "This domain implements linear combinations of elements from the domain \\spad{S} with coefficients in the domain \\spad{R} where \\spad{S} is an ordered set and \\spad{R} is a ring (which may be non-commutative). This domain is used by domains of non-commutative algebra such as: \\indented{4}{\\spadtype{XDistributedPolynomial},{}} \\indented{4}{\\spadtype{XRecursivePolynomial}.} Author: Michel Petitot (petitot@lifl.fr)")) (* (($ |#2| |#1|) "\\spad{s*r} returns the product \\spad{r*s} used by \\spadtype{XRecursivePolynomial}"))) -((-3973 . T) (-3972 . T)) -((|HasCategory| |#1| (QUOTE (-144)))) -(-330 R |Basis|) +((-3984 . T) (-3983 . T)) +((|HasCategory| |#1| (QUOTE (-146)))) +(-332 R |Basis|) ((|constructor| (NIL "A domain of this category implements formal linear combinations of elements from a domain \\spad{Basis} with coefficients in a domain \\spad{R}. The domain \\spad{Basis} needs only to belong to the category \\spadtype{SetCategory} and \\spad{R} to the category \\spadtype{Ring}. Thus the coefficient ring may be non-commutative. See the \\spadtype{XDistributedPolynomial} constructor for examples of domains built with the \\spadtype{FreeModuleCat} category constructor. Author: Michel Petitot (petitot@lifl.fr)")) (|reductum| (($ $) "\\spad{reductum(x)} returns \\spad{x} minus its leading term.")) (|leadingTerm| (((|Record| (|:| |k| |#2|) (|:| |c| |#1|)) $) "\\spad{leadingTerm(x)} returns the first term which appears in \\spad{ListOfTerms(x)}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(x)} returns the first coefficient which appears in \\spad{ListOfTerms(x)}.")) (|leadingMonomial| ((|#2| $) "\\spad{leadingMonomial(x)} returns the first element from \\spad{Basis} which appears in \\spad{ListOfTerms(x)}.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(x)} returns the number of monomials of \\spad{x}.")) (|monomials| (((|List| $) $) "\\spad{monomials(x)} returns the list of \\spad{r_i*b_i} whose sum is \\spad{x}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(x)} returns the list of coefficients of \\spad{x}.")) (|ListOfTerms| (((|List| (|Record| (|:| |k| |#2|) (|:| |c| |#1|))) $) "\\spad{ListOfTerms(x)} returns a list \\spad{lt} of terms with type \\spad{Record(k: Basis, c: R)} such that \\spad{x} equals \\spad{reduce(+, map(x +-> monom(x.k, x.c), lt))}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} returns \\spad{true} if \\spad{x} contains a single monomial.")) (|monom| (($ |#2| |#1|) "\\spad{monom(b,r)} returns the element with the single monomial \\indented{1}{\\spad{b} and coefficient \\spad{r}.}")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients \\indented{1}{of the non-zero monomials of \\spad{u}.}")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(x,b)} returns the coefficient of \\spad{b} in \\spad{x}.")) (* (($ |#1| |#2|) "\\spad{r*b} returns the product of \\spad{r} by \\spad{b}."))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-331 S) +(-333 S) ((|constructor| (NIL "A free monoid on a set \\spad{S} is the monoid of finite products of the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}'s are in \\spad{S},{} and the \\spad{ni}'s are nonnegative integers. The multiplication is not commutative.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| (|NonNegativeInteger|) (|NonNegativeInteger|)) $) "\\spad{mapExpon(f, a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(x, n)} returns the factor of the n^th monomial of \\spad{x}.")) (|nthExpon| (((|NonNegativeInteger|) $ (|Integer|)) "\\spad{nthExpon(x, n)} returns the exponent of the n^th monomial of \\spad{x}.")) (|factors| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| (|NonNegativeInteger|)))) $) "\\spad{factors(a1\\^e1,...,an\\^en)} returns \\spad{[[a1, e1],...,[an, en]]}.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(x)} returns the number of monomials in \\spad{x}.")) (|overlap| (((|Record| (|:| |lm| $) (|:| |mm| $) (|:| |rm| $)) $ $) "\\spad{overlap(x, y)} returns \\spad{[l, m, r]} such that \\spad{x = l * m},{} \\spad{y = m * r} and \\spad{l} and \\spad{r} have no overlap,{} \\spadignore{i.e.} \\spad{overlap(l, r) = [l, 1, r]}.")) (|divide| (((|Union| (|Record| (|:| |lm| $) (|:| |rm| $)) "failed") $ $) "\\spad{divide(x, y)} returns the left and right exact quotients of \\spad{x} by \\spad{y},{} \\spadignore{i.e.} \\spad{[l, r]} such that \\spad{x = l * y * r},{} \"failed\" if \\spad{x} is not of the form \\spad{l * y * r}.")) (|rquo| (((|Union| $ "failed") $ $) "\\spad{rquo(x, y)} returns the exact right quotient of \\spad{x} by \\spad{y} \\spadignore{i.e.} \\spad{q} such that \\spad{x = q * y},{} \"failed\" if \\spad{x} is not of the form \\spad{q * y}.")) (|lquo| (((|Union| $ "failed") $ $) "\\spad{lquo(x, y)} returns the exact left quotient of \\spad{x} by \\spad{y} \\spadignore{i.e.} \\spad{q} such that \\spad{x = y * q},{} \"failed\" if \\spad{x} is not of the form \\spad{y * q}.")) (|hcrf| (($ $ $) "\\spad{hcrf(x, y)} returns the highest common right factor of \\spad{x} and \\spad{y},{} \\spadignore{i.e.} the largest \\spad{d} such that \\spad{x = a d} and \\spad{y = b d}.")) (|hclf| (($ $ $) "\\spad{hclf(x, y)} returns the highest common left factor of \\spad{x} and \\spad{y},{} \\spadignore{i.e.} the largest \\spad{d} such that \\spad{x = d a} and \\spad{y = d b}.")) (** (($ |#1| (|NonNegativeInteger|)) "\\spad{s ** n} returns the product of \\spad{s} by itself \\spad{n} times.")) (* (($ $ |#1|) "\\spad{x * s} returns the product of \\spad{x} by \\spad{s} on the right.") (($ |#1| $) "\\spad{s * x} returns the product of \\spad{x} by \\spad{s} on the left."))) NIL NIL -(-332 S) +(-334 S) ((|constructor| (NIL "The free monoid on a set \\spad{S} is the monoid of finite products of the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}'s are in \\spad{S},{} and the \\spad{ni}'s are nonnegative integers. The multiplication is not commutative."))) NIL -((|HasCategory| |#1| (QUOTE (-751)))) -(-333) +((|HasCategory| |#1| (QUOTE (-756)))) +(-335) ((|constructor| (NIL "This domain provides an interface to names in the file system."))) NIL NIL -(-334) +(-336) ((|constructor| (NIL "This category provides an interface to names in the file system.")) (|new| (($ (|String|) (|String|) (|String|)) "\\spad{new(d,pref,e)} constructs the name of a new writable file with \\spad{d} as its directory,{} \\spad{pref} as a prefix of its name and \\spad{e} as its extension. When \\spad{d} or \\spad{t} is the empty string,{} a default is used. An error occurs if a new file cannot be written in the given directory.")) (|writable?| (((|Boolean|) $) "\\spad{writable?(f)} tests if the named file be opened for writing. The named file need not already exist.")) (|readable?| (((|Boolean|) $) "\\spad{readable?(f)} tests if the named file exist and can it be opened for reading.")) (|exists?| (((|Boolean|) $) "\\spad{exists?(f)} tests if the file exists in the file system.")) (|extension| (((|String|) $) "\\spad{extension(f)} returns the type part of the file name.")) (|name| (((|String|) $) "\\spad{name(f)} returns the name part of the file name.")) (|directory| (((|String|) $) "\\spad{directory(f)} returns the directory part of the file name.")) (|filename| (($ (|String|) (|String|) (|String|)) "\\spad{filename(d,n,e)} creates a file name with \\spad{d} as its directory,{} \\spad{n} as its name and \\spad{e} as its extension. This is a portable way to create file names. When \\spad{d} or \\spad{t} is the empty string,{} a default is used."))) NIL NIL -(-335 |n| |class| R) +(-337 |n| |class| R) ((|constructor| (NIL "Generate the Free Lie Algebra over a ring \\spad{R} with identity; A \\spad{P}. Hall basis is generated by a package call to HallBasis.")) (|generator| (($ (|NonNegativeInteger|)) "\\spad{generator(i)} is the \\spad{i}th Hall Basis element")) (|shallowExpand| (((|OutputForm|) $) "\\spad{shallowExpand(x)} \\undocumented{}")) (|deepExpand| (((|OutputForm|) $) "\\spad{deepExpand(x)} \\undocumented{}")) (|dimension| (((|NonNegativeInteger|)) "\\spad{dimension()} is the rank of this Lie algebra"))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-336 -3078 UP UPUP R) +(-338 -3088 UP UPUP R) ((|constructor| (NIL "\\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 11 Jul 1990")) (|order| (((|NonNegativeInteger|) (|FiniteDivisor| |#1| |#2| |#3| |#4|)) "\\spad{order(x)} \\undocumented"))) NIL NIL -(-337 -3078 UP) +(-339 -3088 UP) ((|constructor| (NIL "\\indented{1}{Full partial fraction expansion of rational functions} Author: Manuel Bronstein Date Created: 9 December 1992 Date Last Updated: June 18,{} 2010 References: \\spad{M}.Bronstein & \\spad{B}.Salvy,{} \\indented{12}{Full Partial Fraction Decomposition of Rational Functions,{}} \\indented{12}{in Proceedings of \\spad{ISSAC'93},{} Kiev,{} ACM Press.}")) (|construct| (($ (|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|)))) "\\spad{construct(l)} is the inverse of fracPart.")) (|fracPart| (((|List| (|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |center| |#2|) (|:| |num| |#2|))) $) "\\spad{fracPart(f)} returns the list of summands of the fractional part of \\spad{f}.")) (|polyPart| ((|#2| $) "\\spad{polyPart(f)} returns the polynomial part of \\spad{f}.")) (|fullPartialFraction| (($ (|Fraction| |#2|)) "\\spad{fullPartialFraction(f)} returns \\spad{[p, [[j, Dj, Hj]...]]} such that \\spad{f = p(x) + \\sum_{[j,Dj,Hj] in l} \\sum_{Dj(a)=0} Hj(a)/(x - a)\\^j}.")) (+ (($ |#2| $) "\\spad{p + x} returns the sum of \\spad{p} and \\spad{x}"))) NIL NIL -(-338 R) +(-340 R) ((|constructor| (NIL "A set \\spad{S} is PatternMatchable over \\spad{R} if \\spad{S} can lift the pattern-matching functions of \\spad{S} over the integers and float to itself (necessary for matching in towers)."))) NIL NIL -(-339 S) +(-341 S) ((|constructor| (NIL "FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \\spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.")) (|primeFrobenius| (($ $ (|NonNegativeInteger|)) "\\spad{primeFrobenius(a,s)} returns \\spad{a**(p**s)} where \\spad{p} is the characteristic.") (($ $) "\\spad{primeFrobenius(a)} returns \\spad{a ** p} where \\spad{p} is the characteristic.")) (|discreteLog| (((|Union| (|NonNegativeInteger|) "failed") $ $) "\\spad{discreteLog(b,a)} computes \\spad{s} with \\spad{b**s = a} if such an \\spad{s} exists.")) (|order| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \\spad{a} is 0."))) NIL NIL -(-340) +(-342) ((|constructor| (NIL "FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \\spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.")) (|primeFrobenius| (($ $ (|NonNegativeInteger|)) "\\spad{primeFrobenius(a,s)} returns \\spad{a**(p**s)} where \\spad{p} is the characteristic.") (($ $) "\\spad{primeFrobenius(a)} returns \\spad{a ** p} where \\spad{p} is the characteristic.")) (|discreteLog| (((|Union| (|NonNegativeInteger|) "failed") $ $) "\\spad{discreteLog(b,a)} computes \\spad{s} with \\spad{b**s = a} if such an \\spad{s} exists.")) (|order| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \\spad{a} is 0."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-341 S) +(-343 S) ((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline \\indented{2}{1: \\spadfunFrom{base}{FloatingPointSystem} of the \\spadfunFrom{exponent}{FloatingPointSystem}.} \\indented{9}{(actual implemenations are usually binary or decimal)} \\indented{2}{2: \\spadfunFrom{precision}{FloatingPointSystem} of the \\spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \\indented{2}{3: rounding error for operations} \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\"+\") does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\spad{max()} returns the maximum floating point number.")) (|min| (($) "\\spad{min()} returns the minimum floating point number.")) (|decreasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{decreasePrecision(n)} decreases the current \\spadfunFrom{precision}{FloatingPointSystem} precision by \\spad{n} decimal digits.")) (|increasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{increasePrecision(n)} increases the current \\spadfunFrom{precision}{FloatingPointSystem} by \\spad{n} decimal digits.")) (|precision| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(n)} set the precision in the base to \\spad{n} decimal digits.") (((|PositiveInteger|)) "\\spad{precision()} returns the precision in digits base.")) (|digits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{digits(d)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{d} digits.") (((|PositiveInteger|)) "\\spad{digits()} returns ceiling's precision in decimal digits.")) (|bits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{bits(n)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{n} bits.") (((|PositiveInteger|)) "\\spad{bits()} returns ceiling's precision in bits.")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(x)} returns the mantissa part of \\spad{x}.")) (|exponent| (((|Integer|) $) "\\spad{exponent(x)} returns the \\spadfunFrom{exponent}{FloatingPointSystem} part of \\spad{x}.")) (|base| (((|PositiveInteger|)) "\\spad{base()} returns the base of the \\spadfunFrom{exponent}{FloatingPointSystem}.")) (|order| (((|Integer|) $) "\\spad{order x} is the order of magnitude of \\spad{x}. Note: \\spad{base ** order x <= |x| < base ** (1 + order x)}.")) (|float| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{float(a,e,b)} returns \\spad{a * b ** e}.") (($ (|Integer|) (|Integer|)) "\\spad{float(a,e)} returns \\spad{a * base() ** e}.")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\"."))) NIL -((|HasAttribute| |#1| (QUOTE -3961)) (|HasAttribute| |#1| (QUOTE -3969))) -(-342) +((|HasAttribute| |#1| (QUOTE -3972)) (|HasAttribute| |#1| (QUOTE -3980))) +(-344) ((|constructor| (NIL "This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \\blankline \\indented{2}{1: \\spadfunFrom{base}{FloatingPointSystem} of the \\spadfunFrom{exponent}{FloatingPointSystem}.} \\indented{9}{(actual implemenations are usually binary or decimal)} \\indented{2}{2: \\spadfunFrom{precision}{FloatingPointSystem} of the \\spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \\indented{2}{3: rounding error for operations} \\blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\"+\") does not hold. Algorithms defined over a field need special considerations when the field is a floating point system.")) (|max| (($) "\\spad{max()} returns the maximum floating point number.")) (|min| (($) "\\spad{min()} returns the minimum floating point number.")) (|decreasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{decreasePrecision(n)} decreases the current \\spadfunFrom{precision}{FloatingPointSystem} precision by \\spad{n} decimal digits.")) (|increasePrecision| (((|PositiveInteger|) (|Integer|)) "\\spad{increasePrecision(n)} increases the current \\spadfunFrom{precision}{FloatingPointSystem} by \\spad{n} decimal digits.")) (|precision| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{precision(n)} set the precision in the base to \\spad{n} decimal digits.") (((|PositiveInteger|)) "\\spad{precision()} returns the precision in digits base.")) (|digits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{digits(d)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{d} digits.") (((|PositiveInteger|)) "\\spad{digits()} returns ceiling's precision in decimal digits.")) (|bits| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{bits(n)} set the \\spadfunFrom{precision}{FloatingPointSystem} to \\spad{n} bits.") (((|PositiveInteger|)) "\\spad{bits()} returns ceiling's precision in bits.")) (|mantissa| (((|Integer|) $) "\\spad{mantissa(x)} returns the mantissa part of \\spad{x}.")) (|exponent| (((|Integer|) $) "\\spad{exponent(x)} returns the \\spadfunFrom{exponent}{FloatingPointSystem} part of \\spad{x}.")) (|base| (((|PositiveInteger|)) "\\spad{base()} returns the base of the \\spadfunFrom{exponent}{FloatingPointSystem}.")) (|order| (((|Integer|) $) "\\spad{order x} is the order of magnitude of \\spad{x}. Note: \\spad{base ** order x <= |x| < base ** (1 + order x)}.")) (|float| (($ (|Integer|) (|Integer|) (|PositiveInteger|)) "\\spad{float(a,e,b)} returns \\spad{a * b ** e}.") (($ (|Integer|) (|Integer|)) "\\spad{float(a,e)} returns \\spad{a * base() ** e}.")) (|approximate| ((|attribute|) "\\spad{approximate} means \"is an approximation to the real numbers\"."))) -((-3753 . T) (-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3764 . T) (-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-343 R) +(-345 R) ((|constructor| (NIL "\\spadtype{Factored} creates a domain whose objects are kept in factored form as long as possible. Thus certain operations like multiplication and gcd are relatively easy to do. Others,{} like addition require somewhat more work,{} and unless the argument domain provides a factor function,{} the result may not be completely factored. Each object consists of a unit and a list of factors,{} where a factor has a member of \\spad{R} (the \"base\"),{} and exponent and a flag indicating what is known about the base. A flag may be one of \"nil\",{} \"sqfr\",{} \"irred\" or \"prime\",{} which respectively mean that nothing is known about the base,{} it is square-free,{} it is irreducible,{} or it is prime. The current restriction to integral domains allows simplification to be performed without worrying about multiplication order.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(u)} returns a rational number if \\spad{u} really is one,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(u)} assumes spadvar{\\spad{u}} is actually a rational number and does the conversion to rational number (see \\spadtype{Fraction Integer}).")) (|rational?| (((|Boolean|) $) "\\spad{rational?(u)} tests if \\spadvar{\\spad{u}} is actually a rational number (see \\spadtype{Fraction Integer}).")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps the function \\userfun{\\spad{fn}} across the factors of \\spadvar{\\spad{u}} and creates a new factored object. Note: this clears the information flags (sets them to \"nil\") because the effect of \\userfun{\\spad{fn}} is clearly not known in general.")) (|unitNormalize| (($ $) "\\spad{unitNormalize(u)} normalizes the unit part of the factorization. For example,{} when working with factored integers,{} this operation will ensure that the bases are all positive integers.")) (|unit| ((|#1| $) "\\spad{unit(u)} extracts the unit part of the factorization.")) (|flagFactor| (($ |#1| (|Integer|) (|Union| #1="nil" #2="sqfr" #3="irred" #4="prime")) "\\spad{flagFactor(base,exponent,flag)} creates a factored object with a single factor whose \\spad{base} is asserted to be properly described by the information \\spad{flag}.")) (|sqfrFactor| (($ |#1| (|Integer|)) "\\spad{sqfrFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be square-free (flag = \"sqfr\").")) (|primeFactor| (($ |#1| (|Integer|)) "\\spad{primeFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be prime (flag = \"prime\").")) (|numberOfFactors| (((|NonNegativeInteger|) $) "\\spad{numberOfFactors(u)} returns the number of factors in \\spadvar{\\spad{u}}.")) (|nthFlag| (((|Union| #1# #2# #3# #4#) $ (|Integer|)) "\\spad{nthFlag(u,n)} returns the information flag of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} \"nil\" is returned.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(u,n)} returns the base of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 1 is returned. If \\spadvar{\\spad{u}} consists only of a unit,{} the unit is returned.")) (|nthExponent| (((|Integer|) $ (|Integer|)) "\\spad{nthExponent(u,n)} returns the exponent of the \\spad{n}th factor of \\spadvar{\\spad{u}}. If \\spadvar{\\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 0 is returned.")) (|irreducibleFactor| (($ |#1| (|Integer|)) "\\spad{irreducibleFactor(base,exponent)} creates a factored object with a single factor whose \\spad{base} is asserted to be irreducible (flag = \"irred\").")) (|factors| (((|List| (|Record| (|:| |factor| |#1|) (|:| |exponent| (|Integer|)))) $) "\\spad{factors(u)} returns a list of the factors in a form suitable for iteration. That is,{} it returns a list where each element is a record containing a base and exponent. The original object is the product of all the factors and the unit (which can be extracted by \\axiom{unit(\\spad{u})}).")) (|nilFactor| (($ |#1| (|Integer|)) "\\spad{nilFactor(base,exponent)} creates a factored object with a single factor with no information about the kind of \\spad{base} (flag = \"nil\").")) (|factorList| (((|List| (|Record| (|:| |flg| (|Union| #1# #2# #3# #4#)) (|:| |fctr| |#1|) (|:| |xpnt| (|Integer|)))) $) "\\spad{factorList(u)} returns the list of factors with flags (for use by factoring code).")) 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T)) -((|HasCategory| |#1| (QUOTE (-449 (-1081) $))) (|HasCategory| |#1| (QUOTE (-257 $))) (|HasCategory| |#1| (QUOTE (-239 $ $))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-1125))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-1125)))) (|HasCategory| |#1| (QUOTE (-928))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (|%list| (QUOTE -449) (QUOTE (-1081)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -239) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-387)))) -(-344 R S) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-452 (-1088) $))) (|HasCategory| |#1| (QUOTE (-259 $))) (|HasCategory| |#1| (QUOTE (-241 $ $))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-1132))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-1132)))) (|HasCategory| |#1| (QUOTE (-933))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (|%list| (QUOTE -452) (QUOTE (-1088)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -241) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-482))) (|HasCategory| |#1| (QUOTE (-389)))) +(-346 R S) ((|constructor| (NIL "\\spadtype{FactoredFunctions2} contains functions that involve factored objects whose underlying domains may not be the same. For example,{} \\spadfun{map} might be used to coerce an object of type \\spadtype{Factored(Integer)} to \\spadtype{Factored(Complex(Integer))}.")) (|map| (((|Factored| |#2|) (|Mapping| |#2| |#1|) (|Factored| |#1|)) "\\spad{map(fn,u)} is used to apply the function \\userfun{\\spad{fn}} to every factor of \\spadvar{\\spad{u}}. The new factored object will have all its information flags set to \"nil\". This function is used,{} for example,{} to coerce every factor base to another type."))) NIL NIL -(-345 S) +(-347 S) ((|constructor| (NIL "Fraction takes an IntegralDomain \\spad{S} and produces the domain of Fractions with numerators and denominators from \\spad{S}. If \\spad{S} is also a GcdDomain,{} then gcd's between numerator and denominator will be cancelled during all operations.")) (|canonical| ((|attribute|) "\\spad{canonical} means that equal elements are in fact identical."))) -((-3965 -12 (|has| |#1| (-6 -3976)) (|has| |#1| (-387)) (|has| |#1| (-6 -3965))) (-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| |#1| (QUOTE (-945 (-1081)))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-928))) (|HasCategory| |#1| (QUOTE (-735))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-735))) (|HasCategory| |#1| (QUOTE (-751)))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-1057))) (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (|%list| (QUOTE -449) (QUOTE (-1081)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -239) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-255))) (|HasCategory| |#1| (QUOTE (-479))) (-12 (|HasAttribute| |#1| (QUOTE -3965)) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387)))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-346 A B) +((-3976 -12 (|has| |#1| (-6 -3987)) (|has| |#1| (-389)) (|has| |#1| (-6 -3976))) (-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| |#1| (QUOTE (-950 (-1088)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-933))) (|HasCategory| |#1| (QUOTE (-740))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-740))) (|HasCategory| |#1| (QUOTE (-756)))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-1064))) (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (|%list| (QUOTE -452) (QUOTE (-1088)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -241) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-257))) (|HasCategory| |#1| (QUOTE (-482))) (-12 (|HasAttribute| |#1| (QUOTE -3976)) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389)))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-348 A B) ((|constructor| (NIL "This package extends a map between integral domains to a map between Fractions over those domains by applying the map to the numerators and denominators.")) (|map| (((|Fraction| |#2|) (|Mapping| |#2| |#1|) (|Fraction| |#1|)) "\\spad{map(func,frac)} applies the function \\spad{func} to the numerator and denominator of the fraction \\spad{frac}."))) NIL NIL -(-347 S R UP) +(-349 S R UP) ((|constructor| (NIL "A \\spadtype{FramedAlgebra} is a \\spadtype{FiniteRankAlgebra} together with a fixed \\spad{R}-module basis.")) (|regularRepresentation| (((|Matrix| |#2|) $) "\\spad{regularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed basis.")) (|discriminant| ((|#2|) "\\spad{discriminant()} = determinant(traceMatrix()).")) (|traceMatrix| (((|Matrix| |#2|)) "\\spad{traceMatrix()} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr(vi * vj)} ),{} where \\spad{v1},{} ...,{} vn are the elements of the fixed basis.")) (|convert| (($ (|Vector| |#2|)) "\\spad{convert([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} vn are the elements of the fixed basis.") (((|Vector| |#2|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} vn are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}'s with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) NIL NIL -(-348 R UP) +(-350 R UP) ((|constructor| (NIL "A \\spadtype{FramedAlgebra} is a \\spadtype{FiniteRankAlgebra} together with a fixed \\spad{R}-module basis.")) (|regularRepresentation| (((|Matrix| |#1|) $) "\\spad{regularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed basis.")) (|discriminant| ((|#1|) "\\spad{discriminant()} = determinant(traceMatrix()).")) (|traceMatrix| (((|Matrix| |#1|)) "\\spad{traceMatrix()} is the \\spad{n}-by-\\spad{n} matrix ( \\spad{Tr(vi * vj)} ),{} where \\spad{v1},{} ...,{} vn are the elements of the fixed basis.")) (|convert| (($ (|Vector| |#1|)) "\\spad{convert([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} vn are the elements of the fixed basis.") (((|Vector| |#1|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,..,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} vn are the elements of the fixed basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([v1,...,vm])} returns the coordinates of the \\spad{vi}'s with to the fixed basis. The coordinates of \\spad{vi} are contained in the \\spad{i}th row of the matrix returned by this function.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-349 A S) +(-351 A S) ((|constructor| (NIL "\\indented{2}{A is fully retractable to \\spad{B} means that A is retractable to \\spad{B},{} and,{}} \\indented{2}{in addition,{} if \\spad{B} is retractable to the integers or rational} \\indented{2}{numbers then so is A.} \\indented{2}{In particular,{} what we are asserting is that there are no integers} \\indented{2}{(rationals) in A which don't retract into \\spad{B}.} Date Created: March 1990 Date Last Updated: 9 April 1991"))) NIL -((|HasCategory| |#2| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-480))))) -(-350 S) +((|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483))))) +(-352 S) ((|constructor| (NIL "\\indented{2}{A is fully retractable to \\spad{B} means that A is retractable to \\spad{B},{} and,{}} \\indented{2}{in addition,{} if \\spad{B} is retractable to the integers or rational} \\indented{2}{numbers then so is A.} \\indented{2}{In particular,{} what we are asserting is that there are no integers} \\indented{2}{(rationals) in A which don't retract into \\spad{B}.} Date Created: March 1990 Date Last Updated: 9 April 1991"))) NIL NIL -(-351 R -3078 UP A) +(-353 R -3088 UP A) ((|constructor| (NIL "Fractional ideals in a framed algebra.")) (|randomLC| ((|#4| (|NonNegativeInteger|) (|Vector| |#4|)) "\\spad{randomLC(n,x)} should be local but conditional.")) (|minimize| (($ $) "\\spad{minimize(I)} returns a reduced set of generators for \\spad{I}.")) (|denom| ((|#1| $) "\\spad{denom(1/d * (f1,...,fn))} returns \\spad{d}.")) (|numer| (((|Vector| |#4|) $) "\\spad{numer(1/d * (f1,...,fn))} = the vector \\spad{[f1,...,fn]}.")) (|norm| ((|#2| $) "\\spad{norm(I)} returns the norm of the ideal \\spad{I}.")) (|basis| (((|Vector| |#4|) $) "\\spad{basis((f1,...,fn))} returns the vector \\spad{[f1,...,fn]}.")) (|ideal| (($ (|Vector| |#4|)) "\\spad{ideal([f1,...,fn])} returns the ideal \\spad{(f1,...,fn)}."))) -((-3975 . T)) +((-3986 . T)) NIL -(-352 R1 F1 U1 A1 R2 F2 U2 A2) +(-354 R1 F1 U1 A1 R2 F2 U2 A2) ((|constructor| (NIL "\\indented{1}{Lifting of morphisms to fractional ideals.} Author: Manuel Bronstein Date Created: 1 Feb 1989 Date Last Updated: 27 Feb 1990 Keywords: ideal,{} algebra,{} module.")) (|map| (((|FractionalIdeal| |#5| |#6| |#7| |#8|) (|Mapping| |#5| |#1|) (|FractionalIdeal| |#1| |#2| |#3| |#4|)) "\\spad{map(f,i)} \\undocumented{}"))) NIL NIL -(-353 R -3078 UP A |ibasis|) +(-355 R -3088 UP A |ibasis|) ((|constructor| (NIL "Module representation of fractional ideals.")) (|module| (($ (|FractionalIdeal| |#1| |#2| |#3| |#4|)) "\\spad{module(I)} returns \\spad{I} viewed has a module over \\spad{R}.") (($ (|Vector| |#4|)) "\\spad{module([f1,...,fn])} = the module generated by \\spad{(f1,...,fn)} over \\spad{R}.")) (|norm| ((|#2| $) "\\spad{norm(f)} returns the norm of the module \\spad{f}.")) (|basis| (((|Vector| |#4|) $) "\\spad{basis((f1,...,fn))} = the vector \\spad{[f1,...,fn]}."))) NIL -((|HasCategory| |#4| (|%list| (QUOTE -945) (|devaluate| |#2|)))) -(-354 AR R AS S) +((|HasCategory| |#4| (|%list| (QUOTE -950) (|devaluate| |#2|)))) +(-356 AR R AS S) ((|constructor| (NIL "\\spad{FramedNonAssociativeAlgebraFunctions2} implements functions between two framed non associative algebra domains defined over different rings. The function map is used to coerce between algebras over different domains having the same structural constants.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the coordinates of \\spad{u} to get an element in \\spad{AS} via identification of the basis of \\spad{AR} as beginning part of the basis of \\spad{AS}."))) NIL NIL -(-355 S R) +(-357 S R) ((|constructor| (NIL "FramedNonAssociativeAlgebra(\\spad{R}) is a \\spadtype{FiniteRankNonAssociativeAlgebra} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank) over a commutative ring \\spad{R} together with a fixed \\spad{R}-module basis.")) (|apply| (($ (|Matrix| |#2|) $) "\\spad{apply(m,a)} defines a left operation of \\spad{n} by \\spad{n} matrices where \\spad{n} is the rank of the algebra in terms of matrix-vector multiplication,{} this is a substitute for a left module structure. Error: if shape of matrix doesn't fit.")) (|rightRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#2|))) "\\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|leftRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#2|))) "\\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|rightRegularRepresentation| (((|Matrix| |#2|) $) "\\spad{rightRegularRepresentation(a)} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|leftRegularRepresentation| (((|Matrix| |#2|) $) "\\spad{leftRegularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|rightTraceMatrix| (((|Matrix| |#2|)) "\\spad{rightTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|leftTraceMatrix| (((|Matrix| |#2|)) "\\spad{leftTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|rightDiscriminant| ((|#2|) "\\spad{rightDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(rightTraceMatrix())}.")) (|leftDiscriminant| ((|#2|) "\\spad{leftDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(leftTraceMatrix())}.")) (|convert| (($ (|Vector| |#2|)) "\\spad{convert([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.") (((|Vector| |#2|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#2|)) "\\spad{represents([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#2|))) "\\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \\spad{R}-module basis.")) (|structuralConstants| (((|Vector| (|Matrix| |#2|))) "\\spad{structuralConstants()} calculates the structural constants \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \\spad{v1},{}...,{}\\spad{vn} is the fixed \\spad{R}-module basis.")) (|coordinates| (((|Matrix| |#2|) (|Vector| $)) "\\spad{coordinates([a1,...,am])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the fixed \\spad{R}-module basis.") (((|Vector| |#2|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) NIL -((|HasCategory| |#2| (QUOTE (-309)))) -(-356 R) +((|HasCategory| |#2| (QUOTE (-311)))) +(-358 R) ((|constructor| (NIL "FramedNonAssociativeAlgebra(\\spad{R}) is a \\spadtype{FiniteRankNonAssociativeAlgebra} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank) over a commutative ring \\spad{R} together with a fixed \\spad{R}-module basis.")) (|apply| (($ (|Matrix| |#1|) $) "\\spad{apply(m,a)} defines a left operation of \\spad{n} by \\spad{n} matrices where \\spad{n} is the rank of the algebra in terms of matrix-vector multiplication,{} this is a substitute for a left module structure. Error: if shape of matrix doesn't fit.")) (|rightRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#1|))) "\\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|leftRankPolynomial| (((|SparseUnivariatePolynomial| (|Polynomial| |#1|))) "\\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.")) (|rightRegularRepresentation| (((|Matrix| |#1|) $) "\\spad{rightRegularRepresentation(a)} returns the matrix of the linear map defined by right multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|leftRegularRepresentation| (((|Matrix| |#1|) $) "\\spad{leftRegularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|rightTraceMatrix| (((|Matrix| |#1|)) "\\spad{rightTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|leftTraceMatrix| (((|Matrix| |#1|)) "\\spad{leftTraceMatrix()} is the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|rightDiscriminant| ((|#1|) "\\spad{rightDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the right trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(rightTraceMatrix())}.")) (|leftDiscriminant| ((|#1|) "\\spad{leftDiscriminant()} returns the determinant of the \\spad{n}-by-\\spad{n} matrix whose element at the \\spad{i}\\spad{-}th row and \\spad{j}\\spad{-}th column is given by the left trace of the product \\spad{vi*vj},{} where \\spad{v1},{}...,{}\\spad{vn} are the elements of the fixed \\spad{R}-module basis. Note: the same as \\spad{determinant(leftTraceMatrix())}.")) (|convert| (($ (|Vector| |#1|)) "\\spad{convert([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.") (((|Vector| |#1|) $) "\\spad{convert(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|represents| (($ (|Vector| |#1|)) "\\spad{represents([a1,...,an])} returns \\spad{a1*v1 + ... + an*vn},{} where \\spad{v1},{} ...,{} \\spad{vn} are the elements of the fixed \\spad{R}-module basis.")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#1|))) "\\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \\spad{R}-module basis.")) (|structuralConstants| (((|Vector| (|Matrix| |#1|))) "\\spad{structuralConstants()} calculates the structural constants \\spad{[(gammaijk) for k in 1..rank()]} defined by \\spad{vi * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \\spad{v1},{}...,{}\\spad{vn} is the fixed \\spad{R}-module basis.")) (|coordinates| (((|Matrix| |#1|) (|Vector| $)) "\\spad{coordinates([a1,...,am])} returns a matrix whose \\spad{i}-th row is formed by the coordinates of \\spad{ai} with respect to the fixed \\spad{R}-module basis.") (((|Vector| |#1|) $) "\\spad{coordinates(a)} returns the coordinates of \\spad{a} with respect to the fixed \\spad{R}-module basis.")) (|basis| (((|Vector| $)) "\\spad{basis()} returns the fixed \\spad{R}-module basis."))) -((-3975 |has| |#1| (-491)) (-3973 . T) (-3972 . T)) +((-3986 |has| |#1| (-494)) (-3984 . T) (-3983 . T)) NIL -(-357 R) +(-359 R) ((|constructor| (NIL "\\spadtype{FactoredFunctionUtilities} implements some utility functions for manipulating factored objects.")) (|mergeFactors| (((|Factored| |#1|) (|Factored| |#1|) (|Factored| |#1|)) "\\spad{mergeFactors(u,v)} is used when the factorizations of \\spadvar{\\spad{u}} and \\spadvar{\\spad{v}} are known to be disjoint,{} \\spadignore{e.g.} resulting from a content/primitive part split. Essentially,{} it creates a new factored object by multiplying the units together and appending the lists of factors.")) (|refine| (((|Factored| |#1|) (|Factored| |#1|) (|Mapping| (|Factored| |#1|) |#1|)) "\\spad{refine(u,fn)} is used to apply the function \\userfun{\\spad{fn}} to each factor of \\spadvar{\\spad{u}} and then build a new factored object from the results. For example,{} if \\spadvar{\\spad{u}} were created by calling \\spad{nilFactor(10,2)} then \\spad{refine(u,factor)} would create a factored object equal to that created by \\spad{factor(100)} or \\spad{primeFactor(2,2) * primeFactor(5,2)}."))) NIL NIL -(-358 S R) +(-360 S R) ((|constructor| (NIL "A space of formal functions with arguments in an arbitrary ordered set.")) (|univariate| (((|Fraction| (|SparseUnivariatePolynomial| $)) $ (|Kernel| $)) "\\spad{univariate(f, k)} returns \\spad{f} viewed as a univariate fraction in \\spad{k}.")) (/ (($ (|SparseMultivariatePolynomial| |#2| (|Kernel| $)) (|SparseMultivariatePolynomial| |#2| (|Kernel| $))) "\\spad{p1/p2} returns the quotient of \\spad{p1} and \\spad{p2} as an element of \\%.")) (|denominator| (($ $) "\\spad{denominator(f)} returns the denominator of \\spad{f} converted to \\%.")) (|denom| (((|SparseMultivariatePolynomial| |#2| (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|convert| (($ (|Factored| $)) "\\spad{convert(f1\\^e1 ... fm\\^em)} returns \\spad{(f1)\\^e1 ... (fm)\\^em} as an element of \\%,{} using formal kernels created using a \\spadfunFrom{paren}{ExpressionSpace}.")) (|isPower| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|numerator| (($ $) "\\spad{numerator(f)} returns the numerator of \\spad{f} converted to \\%.")) (|numer| (((|SparseMultivariatePolynomial| |#2| (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R} if \\spad{R} is an integral domain. If not,{} then numer(\\spad{f}) = \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|coerce| (($ (|Fraction| (|Polynomial| (|Fraction| |#2|)))) "\\spad{coerce(f)} returns \\spad{f} as an element of \\%.") (($ (|Polynomial| (|Fraction| |#2|))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.") (($ (|Fraction| |#2|)) "\\spad{coerce(q)} returns \\spad{q} as an element of \\%.") (($ (|SparseMultivariatePolynomial| |#2| (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.")) (|isMult| (((|Union| (|Record| (|:| |coef| (|Integer|)) (|:| |var| (|Kernel| $))) "failed") $) "\\spad{isMult(p)} returns \\spad{[n, x]} if \\spad{p = n * x} and \\spad{n <> 0}.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if \\spad{p = m1 +...+ mn} and \\spad{n > 1}.")) (|isExpt| (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|Symbol|)) "\\spad{isExpt(p,f)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = f(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|BasicOperator|)) "\\spad{isExpt(p,op)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = op(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1*...*an} and \\spad{n > 1}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns \\spad{x} * \\spad{x} * \\spad{x} * ... * \\spad{x} (\\spad{n} times).")) (|eval| (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ $)) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a)**n} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a1,...,am)**n} in \\spad{x} by \\spad{f(a1,...,am)} for any \\spad{a1},{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)**ni} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a)**ni} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|List| (|BasicOperator|)) (|List| $) (|Symbol|)) "\\spad{eval(x, [s1,...,sm], [f1,...,fm], y)} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $ (|BasicOperator|) $ (|Symbol|)) "\\spad{eval(x, s, f, y)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $) "\\spad{eval(f)} unquotes all the quoted operators in \\spad{f}.") (($ $ (|List| (|Symbol|))) "\\spad{eval(f, [foo1,...,foon])} unquotes all the \\spad{fooi}'s in \\spad{f}.") (($ $ (|Symbol|)) "\\spad{eval(f, foo)} unquotes all the foo's in \\spad{f}.")) (|applyQuote| (($ (|Symbol|) (|List| $)) "\\spad{applyQuote(foo, [x1,...,xn])} returns \\spad{'foo(x1,...,xn)}.") (($ (|Symbol|) $ $ $ $) "\\spad{applyQuote(foo, x, y, z, t)} returns \\spad{'foo(x,y,z,t)}.") (($ (|Symbol|) $ $ $) "\\spad{applyQuote(foo, x, y, z)} returns \\spad{'foo(x,y,z)}.") (($ (|Symbol|) $ $) "\\spad{applyQuote(foo, x, y)} returns \\spad{'foo(x,y)}.") (($ (|Symbol|) $) "\\spad{applyQuote(foo, x)} returns \\spad{'foo(x)}.")) (|variables| (((|List| (|Symbol|)) $) "\\spad{variables(f)} returns the list of all the variables of \\spad{f}.")) (|ground| ((|#2| $) "\\spad{ground(f)} returns \\spad{f} as an element of \\spad{R}. An error occurs if \\spad{f} is not an element of \\spad{R}.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(f)} tests if \\spad{f} is an element of \\spad{R}."))) NIL -((|HasCategory| |#2| (QUOTE (-945 (-480)))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-956))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-408))) (|HasCategory| |#2| (QUOTE (-1017))) (|HasCategory| |#2| (QUOTE (-550 (-469))))) -(-359 R) +((|HasCategory| |#2| (QUOTE (-950 (-483)))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-961))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-25))) (|HasCategory| |#2| (QUOTE (-410))) (|HasCategory| |#2| (QUOTE (-1024))) (|HasCategory| |#2| (QUOTE (-553 (-472))))) +(-361 R) ((|constructor| (NIL "A space of formal functions with arguments in an arbitrary ordered set.")) (|univariate| (((|Fraction| (|SparseUnivariatePolynomial| $)) $ (|Kernel| $)) "\\spad{univariate(f, k)} returns \\spad{f} viewed as a univariate fraction in \\spad{k}.")) (/ (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $)) (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{p1/p2} returns the quotient of \\spad{p1} and \\spad{p2} as an element of \\%.")) (|denominator| (($ $) "\\spad{denominator(f)} returns the denominator of \\spad{f} converted to \\%.")) (|denom| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|convert| (($ (|Factored| $)) "\\spad{convert(f1\\^e1 ... fm\\^em)} returns \\spad{(f1)\\^e1 ... (fm)\\^em} as an element of \\%,{} using formal kernels created using a \\spadfunFrom{paren}{ExpressionSpace}.")) (|isPower| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|numerator| (($ $) "\\spad{numerator(f)} returns the numerator of \\spad{f} converted to \\%.")) (|numer| (((|SparseMultivariatePolynomial| |#1| (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{R} if \\spad{R} is an integral domain. If not,{} then numer(\\spad{f}) = \\spad{f} viewed as a polynomial in the kernels over \\spad{R}.")) (|coerce| (($ (|Fraction| (|Polynomial| (|Fraction| |#1|)))) "\\spad{coerce(f)} returns \\spad{f} as an element of \\%.") (($ (|Polynomial| (|Fraction| |#1|))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.") (($ (|Fraction| |#1|)) "\\spad{coerce(q)} returns \\spad{q} as an element of \\%.") (($ (|SparseMultivariatePolynomial| |#1| (|Kernel| $))) "\\spad{coerce(p)} returns \\spad{p} as an element of \\%.")) (|isMult| (((|Union| (|Record| (|:| |coef| (|Integer|)) (|:| |var| (|Kernel| $))) "failed") $) "\\spad{isMult(p)} returns \\spad{[n, x]} if \\spad{p = n * x} and \\spad{n <> 0}.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if \\spad{p = m1 +...+ mn} and \\spad{n > 1}.")) (|isExpt| (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|Symbol|)) "\\spad{isExpt(p,f)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = f(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $ (|BasicOperator|)) "\\spad{isExpt(p,op)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0} and \\spad{x = op(a)}.") (((|Union| (|Record| (|:| |var| (|Kernel| $)) (|:| |exponent| (|Integer|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1*...*an} and \\spad{n > 1}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns \\spad{x} * \\spad{x} * \\spad{x} * ... * \\spad{x} (\\spad{n} times).")) (|eval| (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ $)) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a)**n} in \\spad{x} by \\spad{f(a)} for any \\spad{a}.") (($ $ (|Symbol|) (|NonNegativeInteger|) (|Mapping| $ (|List| $))) "\\spad{eval(x, s, n, f)} replaces every \\spad{s(a1,...,am)**n} in \\spad{x} by \\spad{f(a1,...,am)} for any \\spad{a1},{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ (|List| $)))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a1,...,an)**ni} in \\spad{x} by \\spad{fi(a1,...,an)} for any \\spad{a1},{}...,{}am.") (($ $ (|List| (|Symbol|)) (|List| (|NonNegativeInteger|)) (|List| (|Mapping| $ $))) "\\spad{eval(x, [s1,...,sm], [n1,...,nm], [f1,...,fm])} replaces every \\spad{si(a)**ni} in \\spad{x} by \\spad{fi(a)} for any \\spad{a}.") (($ $ (|List| (|BasicOperator|)) (|List| $) (|Symbol|)) "\\spad{eval(x, [s1,...,sm], [f1,...,fm], y)} replaces every \\spad{si(a)} in \\spad{x} by \\spad{fi(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $ (|BasicOperator|) $ (|Symbol|)) "\\spad{eval(x, s, f, y)} replaces every \\spad{s(a)} in \\spad{x} by \\spad{f(y)} with \\spad{y} replaced by \\spad{a} for any \\spad{a}.") (($ $) "\\spad{eval(f)} unquotes all the quoted operators in \\spad{f}.") (($ $ (|List| (|Symbol|))) "\\spad{eval(f, [foo1,...,foon])} unquotes all the \\spad{fooi}'s in \\spad{f}.") (($ $ (|Symbol|)) "\\spad{eval(f, foo)} unquotes all the foo's in \\spad{f}.")) (|applyQuote| (($ (|Symbol|) (|List| $)) "\\spad{applyQuote(foo, [x1,...,xn])} returns \\spad{'foo(x1,...,xn)}.") (($ (|Symbol|) $ $ $ $) "\\spad{applyQuote(foo, x, y, z, t)} returns \\spad{'foo(x,y,z,t)}.") (($ (|Symbol|) $ $ $) "\\spad{applyQuote(foo, x, y, z)} returns \\spad{'foo(x,y,z)}.") (($ (|Symbol|) $ $) "\\spad{applyQuote(foo, x, y)} returns \\spad{'foo(x,y)}.") (($ (|Symbol|) $) "\\spad{applyQuote(foo, x)} returns \\spad{'foo(x)}.")) (|variables| (((|List| (|Symbol|)) $) "\\spad{variables(f)} returns the list of all the variables of \\spad{f}.")) (|ground| ((|#1| $) "\\spad{ground(f)} returns \\spad{f} as an element of \\spad{R}. An error occurs if \\spad{f} is not an element of \\spad{R}.")) (|ground?| (((|Boolean|) $) "\\spad{ground?(f)} tests if \\spad{f} is an element of \\spad{R}."))) -((-3975 OR (|has| |#1| (-956)) (|has| |#1| (-408))) (-3973 |has| |#1| (-144)) (-3972 |has| |#1| (-144)) ((-3980 "*") |has| |#1| (-491)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-491)) (-3970 |has| |#1| (-491))) +((-3986 OR (|has| |#1| (-961)) (|has| |#1| (-410))) (-3984 |has| |#1| (-146)) (-3983 |has| |#1| (-146)) ((-3991 "*") |has| |#1| (-494)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-494)) (-3981 |has| |#1| (-494))) NIL -(-360 R A S B) +(-362 R A S B) ((|constructor| (NIL "This package allows a mapping \\spad{R} -> \\spad{S} to be lifted to a mapping from a function space over \\spad{R} to a function space over \\spad{S}.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f, a)} applies \\spad{f} to all the constants in \\spad{R} appearing in \\spad{a}."))) NIL NIL -(-361 R FE |x| |cen|) +(-363 R FE |x| |cen|) ((|constructor| (NIL "This package converts expressions in some function space to exponential expansions.")) (|localAbs| ((|#2| |#2|) "\\spad{localAbs(fcn)} = \\spad{abs(fcn)} or \\spad{sqrt(fcn**2)} depending on whether or not FE has a function \\spad{abs}. This should be a local function,{} but the compiler won't allow it.")) (|exprToXXP| (((|Union| (|:| |%expansion| (|ExponentialExpansion| |#1| |#2| |#3| |#4|)) (|:| |%problem| (|Record| (|:| |func| (|String|)) (|:| |prob| (|String|))))) |#2| (|Boolean|)) "\\spad{exprToXXP(fcn,posCheck?)} converts the expression \\spad{fcn} to an exponential expansion. If \\spad{posCheck?} is \\spad{true},{} log's of negative numbers are not allowed nor are \\spad{n}th roots of negative numbers with \\spad{n} even. If \\spad{posCheck?} is \\spad{false},{} these are allowed."))) NIL NIL -(-362 R FE |Expon| UPS TRAN |x|) +(-364 R FE |Expon| UPS TRAN |x|) ((|constructor| (NIL "This package converts expressions in some function space to power series in a variable \\spad{x} with coefficients in that function space. The function \\spadfun{exprToUPS} converts expressions to power series whose coefficients do not contain the variable \\spad{x}. The function \\spadfun{exprToGenUPS} converts functional expressions to power series whose coefficients may involve functions of \\spad{log(x)}.")) (|localAbs| ((|#2| |#2|) "\\spad{localAbs(fcn)} = \\spad{abs(fcn)} or \\spad{sqrt(fcn**2)} depending on whether or not FE has a function \\spad{abs}. This should be a local function,{} but the compiler won't allow it.")) (|exprToGenUPS| (((|Union| (|:| |%series| |#4|) (|:| |%problem| (|Record| (|:| |func| (|String|)) (|:| |prob| (|String|))))) |#2| (|Boolean|) (|String|)) "\\spad{exprToGenUPS(fcn,posCheck?,atanFlag)} converts the expression \\spad{fcn} to a generalized power series. If \\spad{posCheck?} is \\spad{true},{} log's of negative numbers are not allowed nor are \\spad{n}th roots of negative numbers with \\spad{n} even. If \\spad{posCheck?} is \\spad{false},{} these are allowed. \\spad{atanFlag} determines how the case \\spad{atan(f(x))},{} where \\spad{f(x)} has a pole,{} will be treated. The possible values of \\spad{atanFlag} are \\spad{\"complex\"},{} \\spad{\"real: two sides\"},{} \\spad{\"real: left side\"},{} \\spad{\"real: right side\"},{} and \\spad{\"just do it\"}. If \\spad{atanFlag} is \\spad{\"complex\"},{} then no series expansion will be computed because,{} viewed as a function of a complex variable,{} \\spad{atan(f(x))} has an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \\spad{f(x)} determines the constant coefficient in the series expansion of \\spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \\spad{atanFlag} is \\spad{\"just do it\"}. When the leading term in the series expansion of \\spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficient in the series expansion of \\spad{atan(f(x))} for values to the left differs from that for values to the right. If \\spad{atanFlag} is \\spad{\"real: two sides\"},{} no series expansion will be computed. If \\spad{atanFlag} is \\spad{\"real: left side\"} the constant coefficient for values to the left will be used and if \\spad{atanFlag} \\spad{\"real: right side\"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} we return a record containing the name of the function that caused the problem and a brief description of the problem. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \\spad{x -> x + a} before calling this function.")) (|exprToUPS| (((|Union| (|:| |%series| |#4|) (|:| |%problem| (|Record| (|:| |func| (|String|)) (|:| |prob| (|String|))))) |#2| (|Boolean|) (|String|)) "\\spad{exprToUPS(fcn,posCheck?,atanFlag)} converts the expression \\spad{fcn} to a power series. If \\spad{posCheck?} is \\spad{true},{} log's of negative numbers are not allowed nor are \\spad{n}th roots of negative numbers with \\spad{n} even. If \\spad{posCheck?} is \\spad{false},{} these are allowed. \\spad{atanFlag} determines how the case \\spad{atan(f(x))},{} where \\spad{f(x)} has a pole,{} will be treated. The possible values of \\spad{atanFlag} are \\spad{\"complex\"},{} \\spad{\"real: two sides\"},{} \\spad{\"real: left side\"},{} \\spad{\"real: right side\"},{} and \\spad{\"just do it\"}. If \\spad{atanFlag} is \\spad{\"complex\"},{} then no series expansion will be computed because,{} viewed as a function of a complex variable,{} \\spad{atan(f(x))} has an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \\spad{f(x)} determines the constant coefficient in the series expansion of \\spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \\spad{atanFlag} is \\spad{\"just do it\"}. When the leading term in the series expansion of \\spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficient in the series expansion of \\spad{atan(f(x))} for values to the left differs from that for values to the right. If \\spad{atanFlag} is \\spad{\"real: two sides\"},{} no series expansion will be computed. If \\spad{atanFlag} is \\spad{\"real: left side\"} the constant coefficient for values to the left will be used and if \\spad{atanFlag} \\spad{\"real: right side\"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} a record containing the name of the function that caused the problem and a brief description of the problem is returned. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \\spad{x -> x + a} before calling this function.")) (|integrate| (($ $) "\\spad{integrate(x)} returns the integral of \\spad{x} since we need to be able to integrate a power series")) (|differentiate| (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x} since we need to be able to differentiate a power series"))) NIL NIL -(-363 A S) +(-365 A S) ((|constructor| (NIL "A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \\spadtype{Set} for an example.")) (|min| ((|#2| $) "\\spad{min(u)} returns the smallest element of aggregate \\spad{u}.")) (|max| ((|#2| $) "\\spad{max(u)} returns the largest element of aggregate \\spad{u}.")) (|universe| (($) "\\spad{universe()}\\$\\spad{D} returns the universal set for finite set aggregate \\spad{D}.")) (|complement| (($ $) "\\spad{complement(u)} returns the complement of the set \\spad{u},{} \\spadignore{i.e.} the set of all values not in \\spad{u}.")) (|cardinality| (((|NonNegativeInteger|) $) "\\spad{cardinality(u)} returns the number of elements of \\spad{u}. Note: \\axiom{cardinality(\\spad{u}) = \\#u}."))) NIL -((|HasCategory| |#2| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-315)))) -(-364 S) +((|HasCategory| |#2| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-317)))) +(-366 S) ((|constructor| (NIL "A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \\spadtype{Set} for an example.")) (|min| ((|#1| $) "\\spad{min(u)} returns the smallest element of aggregate \\spad{u}.")) (|max| ((|#1| $) "\\spad{max(u)} returns the largest element of aggregate \\spad{u}.")) (|universe| (($) "\\spad{universe()}\\$\\spad{D} returns the universal set for finite set aggregate \\spad{D}.")) (|complement| (($ $) "\\spad{complement(u)} returns the complement of the set \\spad{u},{} \\spadignore{i.e.} the set of all values not in \\spad{u}.")) (|cardinality| (((|NonNegativeInteger|) $) "\\spad{cardinality(u)} returns the number of elements of \\spad{u}. Note: \\axiom{cardinality(\\spad{u}) = \\#u}."))) -((-3978 . T) (-3968 . T) (-3979 . T)) +((-3989 . T) (-3979 . T) (-3990 . T)) NIL -(-365 S A R B) +(-367 S A R B) ((|constructor| (NIL "\\spad{FiniteSetAggregateFunctions2} provides functions involving two finite set aggregates where the underlying domains might be different. An example of this is to create a set of rational numbers by mapping a function across a set of integers,{} where the function divides each integer by 1000.")) (|scan| ((|#4| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-aggregates \\spad{x} of aggregate \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad {[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}.")) (|reduce| ((|#3| (|Mapping| |#3| |#1| |#3|) |#2| |#3|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the aggregate \\spad{a} and an accumulant initialised to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does a \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as an identity element for the function.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f,a)} applies function \\spad{f} to each member of aggregate \\spad{a},{} creating a new aggregate with a possibly different underlying domain."))) NIL NIL -(-366 R -3078) +(-368 R -3088) ((|constructor| (NIL "\\spadtype{FunctionSpaceComplexIntegration} provides functions for the indefinite integration of complex-valued functions.")) (|complexIntegrate| ((|#2| |#2| (|Symbol|)) "\\spad{complexIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable.")) (|internalIntegrate0| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate0 should} be a local function,{} but is conditional.")) (|internalIntegrate| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{internalIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable."))) NIL NIL -(-367 R E) +(-369 R E) ((|constructor| (NIL "\\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:")) (|makeCos| (($ |#2| |#1|) "\\spad{makeCos(e,r)} makes a sin expression with given argument and coefficient")) (|makeSin| (($ |#2| |#1|) "\\spad{makeSin(e,r)} makes a sin expression with given argument and coefficient")) (|coerce| (($ (|FourierComponent| |#2|)) "\\spad{coerce(c)} converts sin/cos terms into Fourier Series") (($ |#1|) "\\spad{coerce(r)} converts coefficients into Fourier Series"))) -((-3965 -12 (|has| |#1| (-6 -3965)) (|has| |#2| (-6 -3965))) (-3972 . T) (-3973 . T) (-3975 . T)) -((-12 (|HasAttribute| |#1| (QUOTE -3965)) (|HasAttribute| |#2| (QUOTE -3965)))) -(-368 R -3078) +((-3976 -12 (|has| |#1| (-6 -3976)) (|has| |#2| (-6 -3976))) (-3983 . T) (-3984 . T) (-3986 . T)) +((-12 (|HasAttribute| |#1| (QUOTE -3976)) (|HasAttribute| |#2| (QUOTE -3976)))) +(-370 R -3088) ((|constructor| (NIL "\\spadtype{FunctionSpaceIntegration} provides functions for the indefinite integration of real-valued functions.")) (|integrate| (((|Union| |#2| (|List| |#2|)) |#2| (|Symbol|)) "\\spad{integrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a real variable."))) NIL NIL -(-369 R -3078) +(-371 R -3088) ((|constructor| (NIL "Provides some special functions over an integral domain.")) (|iiabs| ((|#2| |#2|) "\\spad{iiabs(x)} should be local but conditional.")) (|iiGamma| ((|#2| |#2|) "\\spad{iiGamma(x)} should be local but conditional.")) (|airyBi| ((|#2| |#2|) "\\spad{airyBi(x)} returns the airybi function applied to \\spad{x}")) (|airyAi| ((|#2| |#2|) "\\spad{airyAi(x)} returns the airyai function applied to \\spad{x}")) (|besselK| ((|#2| |#2| |#2|) "\\spad{besselK(x,y)} returns the besselk function applied to \\spad{x} and \\spad{y}")) (|besselI| ((|#2| |#2| |#2|) "\\spad{besselI(x,y)} returns the besseli function applied to \\spad{x} and \\spad{y}")) (|besselY| ((|#2| |#2| |#2|) "\\spad{besselY(x,y)} returns the bessely function applied to \\spad{x} and \\spad{y}")) (|besselJ| ((|#2| |#2| |#2|) "\\spad{besselJ(x,y)} returns the besselj function applied to \\spad{x} and \\spad{y}")) (|polygamma| ((|#2| |#2| |#2|) "\\spad{polygamma(x,y)} returns the polygamma function applied to \\spad{x} and \\spad{y}")) (|digamma| ((|#2| |#2|) "\\spad{digamma(x)} returns the digamma function applied to \\spad{x}")) (|Beta| ((|#2| |#2| |#2|) "\\spad{Beta(x,y)} returns the beta function applied to \\spad{x} and \\spad{y}")) (|Gamma| ((|#2| |#2| |#2|) "\\spad{Gamma(a,x)} returns the incomplete Gamma function applied to a and \\spad{x}") ((|#2| |#2|) "\\spad{Gamma(f)} returns the formal Gamma function applied to \\spad{f}")) (|abs| ((|#2| |#2|) "\\spad{abs(f)} returns the absolute value operator applied to \\spad{f}")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns a copy of \\spad{op} with the domain-dependent properties appropriate for \\spad{F}; error if \\spad{op} is not a special function operator")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} is \\spad{true} if \\spad{op} is a special function operator."))) NIL NIL -(-370 R -3078) +(-372 R -3088) ((|constructor| (NIL "FunctionsSpacePrimitiveElement provides functions to compute primitive elements in functions spaces.")) (|primitiveElement| (((|Record| (|:| |primelt| |#2|) (|:| |pol1| (|SparseUnivariatePolynomial| |#2|)) (|:| |pol2| (|SparseUnivariatePolynomial| |#2|)) (|:| |prim| (|SparseUnivariatePolynomial| |#2|))) |#2| |#2|) "\\spad{primitiveElement(a1, a2)} returns \\spad{[a, q1, q2, q]} such that \\spad{k(a1, a2) = k(a)},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The minimal polynomial for \\spad{a2} may involve \\spad{a1},{} but the minimal polynomial for \\spad{a1} may not involve \\spad{a2}; This operations uses \\spadfun{resultant}.") (((|Record| (|:| |primelt| |#2|) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#2|))) (|:| |prim| (|SparseUnivariatePolynomial| |#2|))) (|List| |#2|)) "\\spad{primitiveElement([a1,...,an])} returns \\spad{[a, [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}."))) NIL ((|HasCategory| |#2| (QUOTE (-27)))) -(-371 R -3078) +(-373 R -3088) ((|constructor| (NIL "This package provides function which replaces transcendental kernels in a function space by random integers. The correspondence between the kernels and the integers is fixed between calls to new().")) (|newReduc| (((|Void|)) "\\spad{newReduc()} \\undocumented")) (|bringDown| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) |#2| (|Kernel| |#2|)) "\\spad{bringDown(f,k)} \\undocumented") (((|Fraction| (|Integer|)) |#2|) "\\spad{bringDown(f)} \\undocumented"))) NIL NIL -(-372) +(-374) ((|constructor| (NIL "Creates and manipulates objects which correspond to the basic FORTRAN data types: REAL,{} INTEGER,{} COMPLEX,{} LOGICAL and CHARACTER")) (= (((|Boolean|) $ $) "\\spad{x=y} tests for equality")) (|logical?| (((|Boolean|) $) "\\spad{logical?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type LOGICAL.")) (|character?| (((|Boolean|) $) "\\spad{character?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type CHARACTER.")) (|doubleComplex?| (((|Boolean|) $) "\\spad{doubleComplex?(t)} tests whether \\spad{t} is equivalent to the (non-standard) FORTRAN type DOUBLE COMPLEX.")) (|complex?| (((|Boolean|) $) "\\spad{complex?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type COMPLEX.")) (|integer?| (((|Boolean|) $) "\\spad{integer?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type INTEGER.")) (|double?| (((|Boolean|) $) "\\spad{double?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type DOUBLE PRECISION")) (|real?| (((|Boolean|) $) "\\spad{real?(t)} tests whether \\spad{t} is equivalent to the FORTRAN type REAL.")) (|coerce| (((|SExpression|) $) "\\spad{coerce(x)} returns the \\spad{s}-expression associated with \\spad{x}") (((|Symbol|) $) "\\spad{coerce(x)} returns the symbol associated with \\spad{x}") (($ (|Symbol|)) "\\spad{coerce(s)} transforms the symbol \\spad{s} into an element of FortranScalarType provided \\spad{s} is one of real,{} complex,{}double precision,{} logical,{} integer,{} character,{} REAL,{} COMPLEX,{} LOGICAL,{} INTEGER,{} CHARACTER,{} DOUBLE PRECISION") (($ (|String|)) "\\spad{coerce(s)} transforms the string \\spad{s} into an element of FortranScalarType provided \\spad{s} is one of \"real\",{} \"double precision\",{} \"complex\",{} \"logical\",{} \"integer\",{} \"character\",{} \"REAL\",{} \"COMPLEX\",{} \"LOGICAL\",{} \"INTEGER\",{} \"CHARACTER\",{} \"DOUBLE PRECISION\""))) NIL NIL -(-373 R -3078 UP) +(-375 R -3088 UP) ((|constructor| (NIL "\\indented{1}{Used internally by IR2F} Author: Manuel Bronstein Date Created: 12 May 1988 Date Last Updated: 22 September 1993 Keywords: function,{} space,{} polynomial,{} factoring")) (|anfactor| (((|Union| (|Factored| (|SparseUnivariatePolynomial| (|AlgebraicNumber|))) "failed") |#3|) "\\spad{anfactor(p)} tries to factor \\spad{p} over algebraic numbers,{} returning \"failed\" if it cannot")) (|UP2ifCan| (((|Union| (|:| |overq| (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) (|:| |overan| (|SparseUnivariatePolynomial| (|AlgebraicNumber|))) (|:| |failed| (|Boolean|))) |#3|) "\\spad{UP2ifCan(x)} should be local but conditional.")) (|qfactor| (((|Union| (|Factored| (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "failed") |#3|) "\\spad{qfactor(p)} tries to factor \\spad{p} over fractions of integers,{} returning \"failed\" if it cannot")) (|ffactor| (((|Factored| |#3|) |#3|) "\\spad{ffactor(p)} tries to factor a univariate polynomial \\spad{p} over \\spad{F}"))) NIL -((|HasCategory| |#2| (QUOTE (-945 (-48))))) -(-374) +((|HasCategory| |#2| (QUOTE (-950 (-48))))) +(-376) ((|constructor| (NIL "Creates and manipulates objects which correspond to FORTRAN data types,{} including array dimensions.")) (|fortranCharacter| (($) "\\spad{fortranCharacter()} returns CHARACTER,{} an element of FortranType")) (|fortranDoubleComplex| (($) "\\spad{fortranDoubleComplex()} returns DOUBLE COMPLEX,{} an element of FortranType")) (|fortranComplex| (($) "\\spad{fortranComplex()} returns COMPLEX,{} an element of FortranType")) (|fortranLogical| (($) "\\spad{fortranLogical()} returns LOGICAL,{} an element of FortranType")) (|fortranInteger| (($) "\\spad{fortranInteger()} returns INTEGER,{} an element of FortranType")) (|fortranDouble| (($) "\\spad{fortranDouble()} returns DOUBLE PRECISION,{} an element of FortranType")) (|fortranReal| (($) "\\spad{fortranReal()} returns REAL,{} an element of FortranType")) (|construct| (($ (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1="void")) (|List| (|Polynomial| (|Integer|))) (|Boolean|)) "\\spad{construct(type,dims)} creates an element of FortranType") (($ (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#)) (|List| (|Symbol|)) (|Boolean|)) "\\spad{construct(type,dims)} creates an element of FortranType")) (|external?| (((|Boolean|) $) "\\spad{external?(u)} returns \\spad{true} if \\spad{u} is declared to be EXTERNAL")) (|dimensionsOf| (((|List| (|Polynomial| (|Integer|))) $) "\\spad{dimensionsOf(t)} returns the dimensions of \\spad{t}")) (|scalarTypeOf| (((|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#)) $) "\\spad{scalarTypeOf(t)} returns the FORTRAN data type of \\spad{t}")) (|coerce| (($ (|FortranScalarType|)) "\\spad{coerce(t)} creates an element from a scalar type"))) NIL NIL -(-375 |f|) +(-377 |f|) ((|constructor| (NIL "This domain implements named functions")) (|name| (((|Symbol|) $) "\\spad{name(x)} returns the symbol"))) NIL NIL -(-376) +(-378) ((|constructor| (NIL "This is the datatype for exported function descriptor. A function descriptor consists of: (1) a signature; (2) a predicate; and (3) a slot into the scope object.")) (|signature| (((|Signature|) $) "\\spad{signature(x)} returns the signature of function described by \\spad{x}."))) NIL NIL -(-377 UP) +(-379 UP) ((|constructor| (NIL "\\spadtype{GaloisGroupFactorizer} provides functions to factor resolvents.")) (|btwFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|) (|Set| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{btwFact(p,sqf,pd,r)} returns the factorization of \\spad{p},{} the result is a Record such that \\spad{contp=}content \\spad{p},{} \\spad{factors=}List of irreducible factors of \\spad{p} with exponent. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors). \\spad{pd} is the \\spadtype{Set} of possible degrees. \\spad{r} is a lower bound for the number of factors of \\spad{p}. Please do not use this function in your code because its design may change.")) (|henselFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|)) "\\spad{henselFact(p,sqf)} returns the factorization of \\spad{p},{} the result is a Record such that \\spad{contp=}content \\spad{p},{} \\spad{factors=}List of irreducible factors of \\spad{p} with exponent. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors).")) (|factorOfDegree| (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|) (|Boolean|)) "\\spad{factorOfDegree(d,p,listOfDegrees,r,sqf)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees},{} and that \\spad{p} has at least \\spad{r} factors. If \\spad{sqf=true} the polynomial is assumed to be square free (\\spadignore{i.e.} without repeated factors).") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factorOfDegree(d,p,listOfDegrees,r)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees},{} and that \\spad{p} has at least \\spad{r} factors.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factorOfDegree(d,p,listOfDegrees)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1| (|NonNegativeInteger|)) "\\spad{factorOfDegree(d,p,r)} returns a factor of \\spad{p} of degree \\spad{d} knowing that \\spad{p} has at least \\spad{r} factors.") (((|Union| |#1| "failed") (|PositiveInteger|) |#1|) "\\spad{factorOfDegree(d,p)} returns a factor of \\spad{p} of degree \\spad{d}.")) (|factorSquareFree| (((|Factored| |#1|) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,d,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{d} divides the degree of all factors of \\spad{p} and that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,listOfDegrees,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees} and that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factorSquareFree(p,listOfDegrees)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{factorSquareFree(p,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has at least \\spad{r} factors. \\spad{f} is supposed not having any repeated factor (this is not checked).") (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(p)} returns the factorization of \\spad{p} which is supposed not having any repeated factor (this is not checked).")) (|factor| (((|Factored| |#1|) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factor(p,d,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{d} divides the degree of all factors of \\spad{p} and that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{factor(p,listOfDegrees,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm,{} knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees} and that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1| (|List| (|NonNegativeInteger|))) "\\spad{factor(p,listOfDegrees)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has for possible splitting of its degree \\spad{listOfDegrees}.") (((|Factored| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{factor(p,r)} factorizes the polynomial \\spad{p} using the single factor bound algorithm and knowing that \\spad{p} has at least \\spad{r} factors.") (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns the factorization of \\spad{p} over the integers.")) (|tryFunctionalDecomposition| (((|Boolean|) (|Boolean|)) "\\spad{tryFunctionalDecomposition(b)} chooses whether factorizers have to look for functional decomposition of polynomials (\\spad{true}) or not (\\spad{false}). Returns the previous value.")) (|tryFunctionalDecomposition?| (((|Boolean|)) "\\spad{tryFunctionalDecomposition?()} returns \\spad{true} if factorizers try functional decomposition of polynomials before factoring them.")) (|eisensteinIrreducible?| (((|Boolean|) |#1|) "\\spad{eisensteinIrreducible?(p)} returns \\spad{true} if \\spad{p} can be shown to be irreducible by Eisenstein's criterion,{} \\spad{false} is inconclusive.")) (|useEisensteinCriterion| (((|Boolean|) (|Boolean|)) "\\spad{useEisensteinCriterion(b)} chooses whether factorizers check Eisenstein's criterion before factoring: \\spad{true} for using it,{} \\spad{false} else. Returns the previous value.")) (|useEisensteinCriterion?| (((|Boolean|)) "\\spad{useEisensteinCriterion?()} returns \\spad{true} if factorizers check Eisenstein's criterion before factoring.")) (|useSingleFactorBound| (((|Boolean|) (|Boolean|)) "\\spad{useSingleFactorBound(b)} chooses the algorithm to be used by the factorizers: \\spad{true} for algorithm with single factor bound,{} \\spad{false} for algorithm with overall bound. Returns the previous value.")) (|useSingleFactorBound?| (((|Boolean|)) "\\spad{useSingleFactorBound?()} returns \\spad{true} if algorithm with single factor bound is used for factorization,{} \\spad{false} for algorithm with overall bound.")) (|modularFactor| (((|Record| (|:| |prime| (|Integer|)) (|:| |factors| (|List| |#1|))) |#1|) "\\spad{modularFactor(f)} chooses a \"good\" prime and returns the factorization of \\spad{f} modulo this prime in a form that may be used by \\spadfunFrom{completeHensel}{GeneralHenselPackage}. If prime is zero it means that \\spad{f} has been proved to be irreducible over the integers or that \\spad{f} is a unit (\\spadignore{i.e.} 1 or \\spad{-1}). \\spad{f} shall be primitive (\\spadignore{i.e.} content(\\spad{p})\\spad{=1}) and square free (\\spadignore{i.e.} without repeated factors).")) (|numberOfFactors| (((|NonNegativeInteger|) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|))))) "\\spad{numberOfFactors(ddfactorization)} returns the number of factors of the polynomial \\spad{f} modulo \\spad{p} where \\spad{ddfactorization} is the distinct degree factorization of \\spad{f} computed by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \\spad{p}.")) (|stopMusserTrials| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{stopMusserTrials(n)} sets to \\spad{n} the bound on the number of factors for which \\spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \\spad{2**n} trials. Returns the previous value.") (((|PositiveInteger|)) "\\spad{stopMusserTrials()} returns the bound on the number of factors for which \\spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \\spad{2**stopMusserTrials()} trials.")) (|musserTrials| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{musserTrials(n)} sets to \\spad{n} the number of primes to be tried in \\spadfun{modularFactor} and returns the previous value.") (((|PositiveInteger|)) "\\spad{musserTrials()} returns the number of primes that are tried in \\spadfun{modularFactor}.")) (|degreePartition| (((|Multiset| (|NonNegativeInteger|)) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|))))) "\\spad{degreePartition(ddfactorization)} returns the degree partition of the polynomial \\spad{f} modulo \\spad{p} where \\spad{ddfactorization} is the distinct degree factorization of \\spad{f} computed by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \\spad{p}.")) (|makeFR| (((|Factored| |#1|) (|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|))))))) "\\spad{makeFR(flist)} turns the final factorization of henselFact into a \\spadtype{Factored} object."))) NIL NIL -(-378 R UP -3078) +(-380 R UP -3088) ((|constructor| (NIL "\\spadtype{GaloisGroupFactorizationUtilities} provides functions that will be used by the factorizer.")) (|length| ((|#3| |#2|) "\\spad{length(p)} returns the sum of the absolute values of the coefficients of the polynomial \\spad{p}.")) (|height| ((|#3| |#2|) "\\spad{height(p)} returns the maximal absolute value of the coefficients of the polynomial \\spad{p}.")) (|infinityNorm| ((|#3| |#2|) "\\spad{infinityNorm(f)} returns the maximal absolute value of the coefficients of the polynomial \\spad{f}.")) (|quadraticNorm| ((|#3| |#2|) "\\spad{quadraticNorm(f)} returns the \\spad{l2} norm of the polynomial \\spad{f}.")) (|norm| ((|#3| |#2| (|PositiveInteger|)) "\\spad{norm(f,p)} returns the lp norm of the polynomial \\spad{f}.")) (|singleFactorBound| (((|Integer|) |#2|) "\\spad{singleFactorBound(p,r)} returns a bound on the infinite norm of the factor of \\spad{p} with smallest Bombieri's norm. \\spad{p} shall be of degree higher or equal to 2.") (((|Integer|) |#2| (|NonNegativeInteger|)) "\\spad{singleFactorBound(p,r)} returns a bound on the infinite norm of the factor of \\spad{p} with smallest Bombieri's norm. \\spad{r} is a lower bound for the number of factors of \\spad{p}. \\spad{p} shall be of degree higher or equal to 2.")) (|rootBound| (((|Integer|) |#2|) "\\spad{rootBound(p)} returns a bound on the largest norm of the complex roots of \\spad{p}.")) (|bombieriNorm| ((|#3| |#2| (|PositiveInteger|)) "\\spad{bombieriNorm(p,n)} returns the \\spad{n}th Bombieri's norm of \\spad{p}.") ((|#3| |#2|) "\\spad{bombieriNorm(p)} returns quadratic Bombieri's norm of \\spad{p}.")) (|beauzamyBound| (((|Integer|) |#2|) "\\spad{beauzamyBound(p)} returns a bound on the larger coefficient of any factor of \\spad{p}."))) NIL NIL -(-379 R UP) +(-381 R UP) ((|constructor| (NIL "\\spadtype{GaloisGroupPolynomialUtilities} provides useful functions for univariate polynomials which should be added to \\spadtype{UnivariatePolynomialCategory} or to \\spadtype{Factored} (July 1994).")) (|factorsOfDegree| (((|List| |#2|) (|PositiveInteger|) (|Factored| |#2|)) "\\spad{factorsOfDegree(d,f)} returns the factors of degree \\spad{d} of the factored polynomial \\spad{f}.")) (|factorOfDegree| ((|#2| (|PositiveInteger|) (|Factored| |#2|)) "\\spad{factorOfDegree(d,f)} returns a factor of degree \\spad{d} of the factored polynomial \\spad{f}. Such a factor shall exist.")) (|degreePartition| (((|Multiset| (|NonNegativeInteger|)) (|Factored| |#2|)) "\\spad{degreePartition(f)} returns the degree partition (\\spadignore{i.e.} the multiset of the degrees of the irreducible factors) of the polynomial \\spad{f}.")) (|shiftRoots| ((|#2| |#2| |#1|) "\\spad{shiftRoots(p,c)} returns the polynomial which has for roots \\spad{c} added to the roots of \\spad{p}.")) (|scaleRoots| ((|#2| |#2| |#1|) "\\spad{scaleRoots(p,c)} returns the polynomial which has \\spad{c} times the roots of \\spad{p}.")) (|reverse| ((|#2| |#2|) "\\spad{reverse(p)} returns the reverse polynomial of \\spad{p}.")) (|unvectorise| ((|#2| (|Vector| |#1|)) "\\spad{unvectorise(v)} returns the polynomial which has for coefficients the entries of \\spad{v} in the increasing order.")) (|monic?| (((|Boolean|) |#2|) "\\spad{monic?(p)} tests if \\spad{p} is monic (\\spadignore{i.e.} leading coefficient equal to 1)."))) NIL NIL -(-380 R) +(-382 R) ((|constructor| (NIL "\\spadtype{GaloisGroupUtilities} provides several useful functions.")) (|safetyMargin| (((|NonNegativeInteger|)) "\\spad{safetyMargin()} returns the number of low weight digits we do not trust in the floating point representation (used by \\spadfun{safeCeiling}).") (((|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{safetyMargin(n)} sets to \\spad{n} the number of low weight digits we do not trust in the floating point representation and returns the previous value (for use by \\spadfun{safeCeiling}).")) (|safeFloor| (((|Integer|) |#1|) "\\spad{safeFloor(x)} returns the integer which is lower or equal to the largest integer which has the same floating point number representation.")) (|safeCeiling| (((|Integer|) |#1|) "\\spad{safeCeiling(x)} returns the integer which is greater than any integer with the same floating point number representation.")) (|fillPascalTriangle| (((|Void|)) "\\spad{fillPascalTriangle()} fills the stored table.")) (|sizePascalTriangle| (((|NonNegativeInteger|)) "\\spad{sizePascalTriangle()} returns the number of entries currently stored in the table.")) (|rangePascalTriangle| (((|NonNegativeInteger|)) "\\spad{rangePascalTriangle()} returns the maximal number of lines stored.") (((|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{rangePascalTriangle(n)} sets the maximal number of lines which are stored and returns the previous value.")) (|pascalTriangle| ((|#1| (|NonNegativeInteger|) (|Integer|)) "\\spad{pascalTriangle(n,r)} returns the binomial coefficient \\spad{C(n,r)=n!/(r! (n-r)!)} and stores it in a table to prevent recomputation."))) NIL -((|HasCategory| |#1| (QUOTE (-342)))) -(-381) +((|HasCategory| |#1| (QUOTE (-344)))) +(-383) ((|constructor| (NIL "Package for the factorization of complex or gaussian integers.")) (|prime?| (((|Boolean|) (|Complex| (|Integer|))) "\\spad{prime?(zi)} tests if the complex integer \\spad{zi} is prime.")) (|sumSquares| (((|List| (|Integer|)) (|Integer|)) "\\spad{sumSquares(p)} construct \\spad{a} and \\spad{b} such that \\spad{a**2+b**2} is equal to the integer prime \\spad{p},{} and otherwise returns an error. It will succeed if the prime number \\spad{p} is 2 or congruent to 1 mod 4.")) (|factor| (((|Factored| (|Complex| (|Integer|))) (|Complex| (|Integer|))) "\\spad{factor(zi)} produces the complete factorization of the complex integer \\spad{zi}."))) NIL NIL -(-382 |Dom| |Expon| |VarSet| |Dpol|) +(-384 |Dom| |Expon| |VarSet| |Dpol|) ((|constructor| (NIL "\\spadtype{GroebnerPackage} computes groebner bases for polynomial ideals. The basic computation provides a distinguished set of generators for polynomial ideals over fields. This basis allows an easy test for membership: the operation \\spadfun{normalForm} returns zero on ideal members. When the provided coefficient domain,{} Dom,{} is not a field,{} the result is equivalent to considering the extended ideal with \\spadtype{Fraction(Dom)} as coefficients,{} but considerably more efficient since all calculations are performed in Dom. Additional argument \"info\" and \"redcrit\" can be given to provide incremental information during computation. Argument \"info\" produces a computational summary for each \\spad{s}-polynomial. Argument \"redcrit\" prints out the reduced critical pairs. The term ordering is determined by the polynomial type used. Suggested types include \\spadtype{DistributedMultivariatePolynomial},{} \\spadtype{HomogeneousDistributedMultivariatePolynomial},{} \\spadtype{GeneralDistributedMultivariatePolynomial}.")) (|normalForm| ((|#4| |#4| (|List| |#4|)) "\\spad{normalForm(poly,gb)} reduces the polynomial \\spad{poly} modulo the precomputed groebner basis \\spad{gb} giving a canonical representative of the residue class.")) (|groebner| (((|List| |#4|) (|List| |#4|) (|String|) (|String|)) "\\spad{groebner(lp, \"info\", \"redcrit\")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp},{} displaying both a summary of the critical pairs considered (\\spad{\"info\"}) and the result of reducing each critical pair (\"redcrit\"). If the second or third arguments have any other string value,{} the indicated information is suppressed.") (((|List| |#4|) (|List| |#4|) (|String|)) "\\spad{groebner(lp, infoflag)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp}. Argument infoflag is used to get information on the computation. If infoflag is \"info\",{} then summary information is displayed for each \\spad{s}-polynomial generated. If infoflag is \"redcrit\",{} the reduced critical pairs are displayed. If infoflag is any other string,{} no information is printed during computation.") (((|List| |#4|) (|List| |#4|)) "\\spad{groebner(lp)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp}."))) NIL -((|HasCategory| |#1| (QUOTE (-309)))) -(-383 |Dom| |Expon| |VarSet| |Dpol|) +((|HasCategory| |#1| (QUOTE (-311)))) +(-385 |Dom| |Expon| |VarSet| |Dpol|) ((|constructor| (NIL "\\spadtype{EuclideanGroebnerBasisPackage} computes groebner bases for polynomial ideals over euclidean domains. The basic computation provides a distinguished set of generators for these ideals. This basis allows an easy test for membership: the operation \\spadfun{euclideanNormalForm} returns zero on ideal members. The string \"info\" and \"redcrit\" can be given as additional args to provide incremental information during the computation. If \"info\" is given,{} \\indented{1}{a computational summary is given for each \\spad{s}-polynomial. If \"redcrit\"} is given,{} the reduced critical pairs are printed. The term ordering is determined by the polynomial type used. Suggested types include \\spadtype{DistributedMultivariatePolynomial},{} \\spadtype{HomogeneousDistributedMultivariatePolynomial},{} \\spadtype{GeneralDistributedMultivariatePolynomial}.")) (|euclideanGroebner| (((|List| |#4|) (|List| |#4|) (|String|) (|String|)) "\\spad{euclideanGroebner(lp, \"info\", \"redcrit\")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \\spad{lp}. If the second argument is \\spad{\"info\"},{} a summary is given of the critical pairs. If the third argument is \"redcrit\",{} critical pairs are printed.") (((|List| |#4|) (|List| |#4|) (|String|)) "\\spad{euclideanGroebner(lp, infoflag)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \\spad{lp}. During computation,{} additional information is printed out if infoflag is given as either \"info\" (for summary information) or \"redcrit\" (for reduced critical pairs)") (((|List| |#4|) (|List| |#4|)) "\\spad{euclideanGroebner(lp)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \\spad{lp}.")) (|euclideanNormalForm| ((|#4| |#4| (|List| |#4|)) "\\spad{euclideanNormalForm(poly,gb)} reduces the polynomial \\spad{poly} modulo the precomputed groebner basis \\spad{gb} giving a canonical representative of the residue class."))) NIL NIL -(-384 |Dom| |Expon| |VarSet| |Dpol|) +(-386 |Dom| |Expon| |VarSet| |Dpol|) ((|constructor| (NIL "\\spadtype{GroebnerFactorizationPackage} provides the function groebnerFactor\" which uses the factorization routines of \\Language{} to factor each polynomial under consideration while doing the groebner basis algorithm. Then it writes the ideal as an intersection of ideals determined by the irreducible factors. Note that the whole ring may occur as well as other redundancies. We also use the fact,{} that from the second factor on we can assume that the preceding factors are not equal to 0 and we divide all polynomials under considerations by the elements of this list of \"nonZeroRestrictions\". The result is a list of groebner bases,{} whose union of solutions of the corresponding systems of equations is the solution of the system of equation corresponding to the input list. The term ordering is determined by the polynomial type used. Suggested types include \\spadtype{DistributedMultivariatePolynomial},{} \\spadtype{HomogeneousDistributedMultivariatePolynomial},{} \\spadtype{GeneralDistributedMultivariatePolynomial}.")) (|groebnerFactorize| (((|List| (|List| |#4|)) (|List| |#4|) (|Boolean|)) "\\spad{groebnerFactorize(listOfPolys, info)} returns a list of groebner bases. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys}. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p},{} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}. If {\\em info} is \\spad{true},{} information is printed about partial results.") (((|List| (|List| |#4|)) (|List| |#4|)) "\\spad{groebnerFactorize(listOfPolys)} returns a list of groebner bases. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys}. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p},{} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}.") (((|List| (|List| |#4|)) (|List| |#4|) (|List| |#4|) (|Boolean|)) "\\spad{groebnerFactorize(listOfPolys, nonZeroRestrictions, info)} returns a list of groebner basis. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys} under the restriction that the polynomials of {\\em nonZeroRestrictions} don't vanish. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}. If argument {\\em info} is \\spad{true},{} information is printed about partial results.") (((|List| (|List| |#4|)) (|List| |#4|) (|List| |#4|)) "\\spad{groebnerFactorize(listOfPolys, nonZeroRestrictions)} returns a list of groebner basis. The union of their solutions is the solution of the system of equations given by {\\em listOfPolys} under the restriction that the polynomials of {\\em nonZeroRestrictions} don't vanish. At each stage the polynomial \\spad{p} under consideration (either from the given basis or obtained from a reduction of the next \\spad{S}-polynomial) is factorized. For each irreducible factors of \\spad{p},{} a new {\\em createGroebnerBasis} is started doing the usual updates with the factor in place of \\spad{p}.")) (|factorGroebnerBasis| (((|List| (|List| |#4|)) (|List| |#4|) (|Boolean|)) "\\spad{factorGroebnerBasis(basis,info)} checks whether the \\spad{basis} contains reducible polynomials and uses these to split the \\spad{basis}. If argument {\\em info} is \\spad{true},{} information is printed about partial results.") (((|List| (|List| |#4|)) (|List| |#4|)) "\\spad{factorGroebnerBasis(basis)} checks whether the \\spad{basis} contains reducible polynomials and uses these to split the \\spad{basis}."))) NIL NIL -(-385 |Dom| |Expon| |VarSet| |Dpol|) +(-387 |Dom| |Expon| |VarSet| |Dpol|) ((|constructor| (NIL "\\indented{1}{Author:} Date Created: Date Last Updated: Keywords: Description This package provides low level tools for Groebner basis computations")) (|virtualDegree| (((|NonNegativeInteger|) |#4|) "\\spad{virtualDegree }\\undocumented")) (|makeCrit| (((|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) (|Record| (|:| |totdeg| (|NonNegativeInteger|)) (|:| |pol| |#4|)) |#4| (|NonNegativeInteger|)) "\\spad{makeCrit }\\undocumented")) (|critpOrder| (((|Boolean|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) "\\spad{critpOrder }\\undocumented")) (|prinb| (((|Void|) (|Integer|)) "\\spad{prinb }\\undocumented")) (|prinpolINFO| (((|Void|) (|List| |#4|)) "\\spad{prinpolINFO }\\undocumented")) (|fprindINFO| (((|Integer|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) |#4| |#4| (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{fprindINFO }\\undocumented")) (|prindINFO| (((|Integer|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)) |#4| |#4| (|Integer|) (|Integer|) (|Integer|)) "\\spad{prindINFO }\\undocumented")) (|prinshINFO| (((|Void|) |#4|) "\\spad{prinshINFO }\\undocumented")) (|lepol| (((|Integer|) |#4|) "\\spad{lepol }\\undocumented")) (|minGbasis| (((|List| |#4|) (|List| |#4|)) "\\spad{minGbasis }\\undocumented")) (|updatD| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{updatD }\\undocumented")) (|sPol| ((|#4| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) "\\spad{sPol }\\undocumented")) (|updatF| (((|List| (|Record| (|:| |totdeg| (|NonNegativeInteger|)) (|:| |pol| |#4|))) |#4| (|NonNegativeInteger|) (|List| (|Record| (|:| |totdeg| (|NonNegativeInteger|)) (|:| |pol| |#4|)))) "\\spad{updatF }\\undocumented")) (|hMonic| ((|#4| |#4|) "\\spad{hMonic }\\undocumented")) (|redPo| (((|Record| (|:| |poly| |#4|) (|:| |mult| |#1|)) |#4| (|List| |#4|)) "\\spad{redPo }\\undocumented")) (|critMonD1| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) |#2| (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{critMonD1 }\\undocumented")) (|critMTonD1| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{critMTonD1 }\\undocumented")) (|critBonD| (((|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) |#4| (|List| (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|)))) "\\spad{critBonD }\\undocumented")) (|critB| (((|Boolean|) |#2| |#2| |#2| |#2|) "\\spad{critB }\\undocumented")) (|critM| (((|Boolean|) |#2| |#2|) "\\spad{critM }\\undocumented")) (|critT| (((|Boolean|) (|Record| (|:| |lcmfij| |#2|) (|:| |totdeg| (|NonNegativeInteger|)) (|:| |poli| |#4|) (|:| |polj| |#4|))) "\\spad{critT }\\undocumented")) (|gbasis| (((|List| |#4|) (|List| |#4|) (|Integer|) (|Integer|)) "\\spad{gbasis }\\undocumented")) (|redPol| ((|#4| |#4| (|List| |#4|)) "\\spad{redPol }\\undocumented")) (|credPol| ((|#4| |#4| (|List| |#4|)) "\\spad{credPol }\\undocumented"))) NIL NIL -(-386 S) +(-388 S) ((|constructor| (NIL "This category describes domains where \\spadfun{gcd} can be computed but where there is no guarantee of the existence of \\spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \\spadfun{factor} operation exist,{} factorization will be unique up to order and units.")) (|lcm| (($ (|List| $)) "\\spad{lcm(l)} returns the least common multiple of the elements of the list \\spad{l}.") (($ $ $) "\\spad{lcm(x,y)} returns the least common multiple of \\spad{x} and \\spad{y}.")) (|gcd| (($ (|List| $)) "\\spad{gcd(l)} returns the common gcd of the elements in the list \\spad{l}.") (($ $ $) "\\spad{gcd(x,y)} returns the greatest common divisor of \\spad{x} and \\spad{y}."))) NIL NIL -(-387) +(-389) ((|constructor| (NIL "This category describes domains where \\spadfun{gcd} can be computed but where there is no guarantee of the existence of \\spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \\spadfun{factor} operation exist,{} factorization will be unique up to order and units.")) (|lcm| (($ (|List| $)) "\\spad{lcm(l)} returns the least common multiple of the elements of the list \\spad{l}.") (($ $ $) "\\spad{lcm(x,y)} returns the least common multiple of \\spad{x} and \\spad{y}.")) (|gcd| (($ (|List| $)) "\\spad{gcd(l)} returns the common gcd of the elements in the list \\spad{l}.") (($ $ $) "\\spad{gcd(x,y)} returns the greatest common divisor of \\spad{x} and \\spad{y}."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-388 R |n| |ls| |gamma|) +(-390 R |n| |ls| |gamma|) ((|constructor| (NIL "AlgebraGenericElementPackage allows you to create generic elements of an algebra,{} \\spadignore{i.e.} the scalars are extended to include symbolic coefficients")) (|conditionsForIdempotents| (((|List| (|Polynomial| |#1|))) "\\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \\spad{R}-module basis") (((|List| (|Polynomial| |#1|)) (|Vector| $)) "\\spad{conditionsForIdempotents([v1,...,vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}")) (|genericRightDiscriminant| (((|Fraction| (|Polynomial| |#1|))) "\\spad{genericRightDiscriminant()} is the determinant of the generic left trace forms of all products of basis element,{} if the generic left trace form is associative,{} an algebra is separable if the generic left discriminant is invertible,{} if it is non-zero,{} there is some ring extension which makes the algebra separable")) (|genericRightTraceForm| (((|Fraction| (|Polynomial| |#1|)) $ $) "\\spad{genericRightTraceForm (a,b)} is defined to be \\spadfun{genericRightTrace (a*b)},{} this defines a symmetric bilinear form on the algebra")) (|genericLeftDiscriminant| (((|Fraction| (|Polynomial| |#1|))) "\\spad{genericLeftDiscriminant()} is the determinant of the generic left trace forms of all products of basis element,{} if the generic left trace form is associative,{} an algebra is separable if the generic left discriminant is invertible,{} if it is non-zero,{} there is some ring extension which makes the algebra separable")) (|genericLeftTraceForm| (((|Fraction| (|Polynomial| |#1|)) $ $) "\\spad{genericLeftTraceForm (a,b)} is defined to be \\spad{genericLeftTrace (a*b)},{} this defines a symmetric bilinear form on the algebra")) (|genericRightNorm| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericRightNorm(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the constant term in \\spadfun{rightRankPolynomial} and changes the sign if the degree of this polynomial is odd")) (|genericRightTrace| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericRightTrace(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the second highest term in \\spadfun{rightRankPolynomial} and changes the sign")) (|genericRightMinimalPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|))) $) "\\spad{genericRightMinimalPolynomial(a)} substitutes the coefficients of \\spad{a} for the generic coefficients in \\spadfun{rightRankPolynomial}")) (|rightRankPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) "\\spad{rightRankPolynomial()} returns the right minimimal polynomial of the generic element")) (|genericLeftNorm| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericLeftNorm(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the constant term in \\spadfun{leftRankPolynomial} and changes the sign if the degree of this polynomial is odd. This is a form of degree \\spad{k}")) (|genericLeftTrace| (((|Fraction| (|Polynomial| |#1|)) $) "\\spad{genericLeftTrace(a)} substitutes the coefficients of \\spad{a} for the generic coefficients into the coefficient of the second highest term in \\spadfun{leftRankPolynomial} and changes the sign. \\indented{1}{This is a linear form}")) (|genericLeftMinimalPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|))) $) "\\spad{genericLeftMinimalPolynomial(a)} substitutes the coefficients of {em a} for the generic coefficients in \\spad{leftRankPolynomial()}")) (|leftRankPolynomial| (((|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) "\\spad{leftRankPolynomial()} returns the left minimimal polynomial of the generic element")) (|generic| (($ (|Vector| (|Symbol|)) (|Vector| $)) "\\spad{generic(vs,ve)} returns a generic element,{} \\spadignore{i.e.} the linear combination of \\spad{ve} with the symbolic coefficients \\spad{vs} error,{} if the vector of symbols is shorter than the vector of elements") (($ (|Symbol|) (|Vector| $)) "\\spad{generic(s,v)} returns a generic element,{} \\spadignore{i.e.} the linear combination of \\spad{v} with the symbolic coefficients \\spad{s1,s2,..}") (($ (|Vector| $)) "\\spad{generic(ve)} returns a generic element,{} \\spadignore{i.e.} the linear combination of \\spad{ve} basis with the symbolic coefficients \\spad{\\%x1,\\%x2,..}") (($ (|Vector| (|Symbol|))) "\\spad{generic(vs)} returns a generic element,{} \\spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \\spad{vs}; error,{} if the vector of symbols is too short") (($ (|Symbol|)) "\\spad{generic(s)} returns a generic element,{} \\spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \\spad{s1,s2,..}") (($) "\\spad{generic()} returns a generic element,{} \\spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \\spad{\\%x1,\\%x2,..}")) (|rightUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \\spad{\"failed\"} if there is none")) (|leftUnits| (((|Union| (|Record| (|:| |particular| $) (|:| |basis| (|List| $))) "failed")) "\\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \\spad{\"failed\"} if there is none")) (|coerce| (($ (|Vector| (|Fraction| (|Polynomial| |#1|)))) "\\spad{coerce(v)} assumes that it is called with a vector of length equal to the dimension of the algebra,{} then a linear combination with the basis element is formed"))) -((-3975 |has| (-345 (-852 |#1|)) (-491)) (-3973 . T) (-3972 . T)) -((|HasCategory| (-345 (-852 |#1|)) (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| (-345 (-852 |#1|)) (QUOTE (-491)))) -(-389 |vl| R E) +((-3986 |has| (-347 (-857 |#1|)) (-494)) (-3984 . T) (-3983 . T)) +((|HasCategory| (-347 (-857 |#1|)) (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| (-347 (-857 |#1|)) (QUOTE (-494)))) +(-391 |vl| R E) ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is specified by its third parameter. Suggested types which define term orderings include: \\spadtype{DirectProduct},{} \\spadtype{HomogeneousDirectProduct},{} \\spadtype{SplitHomogeneousDirectProduct} and finally \\spadtype{OrderedDirectProduct} which accepts an arbitrary user function to define a term ordering.")) (|reorder| (($ $ (|List| (|Integer|))) "\\spad{reorder(p, perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial"))) -(((-3980 "*") |has| |#2| (-144)) (-3971 |has| |#2| (-491)) (-3976 |has| |#2| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-816))) (OR (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-816)))) (OR (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-816)))) (OR (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-816)))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-144))) (OR (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-491)))) (-12 (|HasCategory| |#2| (QUOTE (-791 (-325)))) (|HasCategory| (-768 |#1|) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#2| (QUOTE (-791 (-480)))) (|HasCategory| (-768 |#1|) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#2| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-768 |#1|) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#2| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-768 |#1|) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| (-768 |#1|) (QUOTE (-550 (-469))))) (|HasCategory| |#2| (QUOTE (-577 (-480)))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#2| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-309))) (|HasAttribute| |#2| (QUOTE -3976)) (|HasCategory| |#2| (QUOTE (-387))) (-12 (|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#2| (QUOTE (-116))))) -(-390 R BP) +(((-3991 "*") |has| |#2| (-146)) (-3982 |has| |#2| (-494)) (-3987 |has| |#2| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-821))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-821)))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-494)))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-327)))) (|HasCategory| (-773 |#1|) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-483)))) (|HasCategory| (-773 |#1|) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-472))))) (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-311))) (|HasAttribute| |#2| (QUOTE -3987)) (|HasCategory| |#2| (QUOTE (-389))) (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#2| (QUOTE (-118))))) +(-392 R BP) ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni.} January 1990 The equation \\spad{Af+Bg=h} and its generalization to \\spad{n} polynomials is solved for solutions over the \\spad{R},{} euclidean domain. A table containing the solutions of \\spad{Af+Bg=x**k} is used. The operations are performed modulus a prime which are in principle big enough,{} but the solutions are tested and,{} in case of failure,{} a hensel lifting process is used to get to the right solutions. It will be used in the factorization of multivariate polynomials over finite field,{} with \\spad{R=F[x]}.")) (|testModulus| (((|Boolean|) |#1| (|List| |#2|)) "\\spad{testModulus(p,lp)} returns \\spad{true} if the the prime \\spad{p} is valid for the list of polynomials \\spad{lp},{} \\spadignore{i.e.} preserves the degree and they remain relatively prime.")) (|solveid| (((|Union| (|List| |#2|) "failed") |#2| |#1| (|Vector| (|List| |#2|))) "\\spad{solveid(h,table)} computes the coefficients of the extended euclidean algorithm for a list of polynomials whose tablePow is \\spad{table} and with right side \\spad{h}.")) (|tablePow| (((|Union| (|Vector| (|List| |#2|)) "failed") (|NonNegativeInteger|) |#1| (|List| |#2|)) "\\spad{tablePow(maxdeg,prime,lpol)} constructs the table with the coefficients of the Extended Euclidean Algorithm for \\spad{lpol}. Here the right side is \\spad{x**k},{} for \\spad{k} less or equal to \\spad{maxdeg}. The operation returns \"failed\" when the elements are not coprime modulo \\spad{prime}.")) (|compBound| (((|NonNegativeInteger|) |#2| (|List| |#2|)) "\\spad{compBound(p,lp)} computes a bound for the coefficients of the solution polynomials. Given a polynomial right hand side \\spad{p},{} and a list \\spad{lp} of left hand side polynomials. Exported because it depends on the valuation.")) (|reduction| ((|#2| |#2| |#1|) "\\spad{reduction(p,prime)} reduces the polynomial \\spad{p} modulo \\spad{prime} of \\spad{R}. Note: this function is exported only because it's conditional."))) NIL NIL -(-391 OV E S R P) +(-393 OV E S R P) ((|constructor| (NIL "\\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \\spadtype{Integer} or \\spadtype{Fraction Integer}.")) (|factor| (((|Factored| |#5|) |#5|) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} over its coefficient domain")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} makes an element from symbol \\spad{s} or fails.")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) NIL NIL -(-392 E OV R P) +(-394 E OV R P) ((|constructor| (NIL "This package provides operations for GCD computations on polynomials")) (|randomR| ((|#3|) "\\spad{randomR()} should be local but conditional")) (|gcdPolynomial| (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{gcdPolynomial(p,q)} returns the GCD of \\spad{p} and \\spad{q}"))) NIL NIL -(-393 R) +(-395 R) ((|constructor| (NIL "\\indented{1}{Description} This package provides operations for the factorization of univariate polynomials with integer coefficients. The factorization is done by \"lifting\" the finite \"berlekamp's\" factorization")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{factor(p)} returns the factorisation of \\spad{p}"))) NIL NIL -(-394 R FE) +(-396 R FE) ((|constructor| (NIL "\\spadtype{GenerateUnivariatePowerSeries} provides functions that create power series from explicit formulas for their \\spad{n}th coefficient.")) (|series| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{series(a(n),n,x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{series(a(n),n,x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Fraction| (|Integer|))) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{series(n +-> a(n),x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{series(n +-> a(n),x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.") (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{series(a(n),n,x=a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{series(a(n),n,x=a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{series(n +-> a(n),x = a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{series(n +-> a(n),x = a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.") (((|Any|) |#2| (|Symbol|) (|Equation| |#2|)) "\\spad{series(a(n),n,x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|)) "\\spad{series(n +-> a(n),x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}.")) (|puiseux| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{puiseux(a(n),n,x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{puiseux(a(n),n,x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Fraction| (|Integer|))) (|Equation| |#2|) (|UniversalSegment| (|Fraction| (|Integer|))) (|Fraction| (|Integer|))) "\\spad{puiseux(n +-> a(n),x = a,r0..,r)} returns \\spad{sum(n = r0,r0 + r,r0 + 2*r..., a(n) * (x - a)**n)}; \\spad{puiseux(n +-> a(n),x = a,r0..r1,r)} returns \\spad{sum(n = r0 + k*r while n <= r1, a(n) * (x - a)**n)}.")) (|laurent| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{laurent(a(n),n,x=a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{laurent(a(n),n,x=a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|) (|UniversalSegment| (|Integer|))) "\\spad{laurent(n +-> a(n),x = a,n0..)} returns \\spad{sum(n = n0..,a(n) * (x - a)**n)}; \\spad{laurent(n +-> a(n),x = a,n0..n1)} returns \\spad{sum(n = n0..n1,a(n) * (x - a)**n)}.")) (|taylor| (((|Any|) |#2| (|Symbol|) (|Equation| |#2|) (|UniversalSegment| (|NonNegativeInteger|))) "\\spad{taylor(a(n),n,x = a,n0..)} returns \\spad{sum(n = n0..,a(n)*(x-a)**n)}; \\spad{taylor(a(n),n,x = a,n0..n1)} returns \\spad{sum(n = n0..,a(n)*(x-a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|) (|UniversalSegment| (|NonNegativeInteger|))) "\\spad{taylor(n +-> a(n),x = a,n0..)} returns \\spad{sum(n=n0..,a(n)*(x-a)**n)}; \\spad{taylor(n +-> a(n),x = a,n0..n1)} returns \\spad{sum(n = n0..,a(n)*(x-a)**n)}.") (((|Any|) |#2| (|Symbol|) (|Equation| |#2|)) "\\spad{taylor(a(n),n,x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}.") (((|Any|) (|Mapping| |#2| (|Integer|)) (|Equation| |#2|)) "\\spad{taylor(n +-> a(n),x = a)} returns \\spad{sum(n = 0..,a(n)*(x-a)**n)}."))) NIL NIL -(-395 RP TP) +(-397 RP TP) ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni} General Hensel Lifting Used for Factorization of bivariate polynomials over a finite field.")) (|reduction| ((|#2| |#2| |#1|) "\\spad{reduction(u,pol)} computes the symmetric reduction of \\spad{u} mod \\spad{pol}")) (|completeHensel| (((|List| |#2|) |#2| (|List| |#2|) |#1| (|PositiveInteger|)) "\\spad{completeHensel(pol,lfact,prime,bound)} lifts \\spad{lfact},{} the factorization mod \\spad{prime} of \\spad{pol},{} to the factorization mod prime**k>bound. Factors are recombined on the way.")) (|HenselLift| (((|Record| (|:| |plist| (|List| |#2|)) (|:| |modulo| |#1|)) |#2| (|List| |#2|) |#1| (|PositiveInteger|)) "\\spad{HenselLift(pol,lfacts,prime,bound)} lifts \\spad{lfacts},{} that are the factors of \\spad{pol} mod \\spad{prime},{} to factors of \\spad{pol} mod prime**k > \\spad{bound}. No recombining is done ."))) NIL NIL -(-396 |vl| R IS E |ff| P) +(-398 |vl| R IS E |ff| P) ((|constructor| (NIL "This package \\undocumented")) (* (($ |#6| $) "\\spad{p*x} \\undocumented")) (|multMonom| (($ |#2| |#4| $) "\\spad{multMonom(r,e,x)} \\undocumented")) (|build| (($ |#2| |#3| |#4|) "\\spad{build(r,i,e)} \\undocumented")) (|unitVector| (($ |#3|) "\\spad{unitVector(x)} \\undocumented")) (|monomial| (($ |#2| (|ModuleMonomial| |#3| |#4| |#5|)) "\\spad{monomial(r,x)} \\undocumented")) (|reductum| (($ $) "\\spad{reductum(x)} \\undocumented")) (|leadingIndex| ((|#3| $) "\\spad{leadingIndex(x)} \\undocumented")) (|leadingExponent| ((|#4| $) "\\spad{leadingExponent(x)} \\undocumented")) (|leadingMonomial| (((|ModuleMonomial| |#3| |#4| |#5|) $) "\\spad{leadingMonomial(x)} \\undocumented")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(x)} \\undocumented"))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-397 E V R P Q) +(-399 E V R P Q) ((|constructor| (NIL "Gosper's summation algorithm.")) (|GospersMethod| (((|Union| |#5| "failed") |#5| |#2| (|Mapping| |#2|)) "\\spad{GospersMethod(b, n, new)} returns a rational function \\spad{rf(n)} such that \\spad{a(n) * rf(n)} is the indefinite sum of \\spad{a(n)} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{a(n+1) * rf(n+1) - a(n) * rf(n) = a(n)},{} where \\spad{b(n) = a(n)/a(n-1)} is a rational function. Returns \"failed\" if no such rational function \\spad{rf(n)} exists. Note: \\spad{new} is a nullary function returning a new \\spad{V} every time. The condition on \\spad{a(n)} is that \\spad{a(n)/a(n-1)} is a rational function of \\spad{n}."))) NIL NIL -(-398 R E |VarSet| P) +(-400 R E |VarSet| P) ((|constructor| (NIL "A domain for polynomial sets.")) (|convert| (($ (|List| |#4|)) "\\axiom{convert(lp)} returns the polynomial set whose members are the polynomials of \\axiom{lp}."))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#4| (|%list| (QUOTE -257) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-550 (-469)))) (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#4| (QUOTE (-549 (-767)))) (|HasCategory| |#4| (QUOTE (-72)))) -(-399 S R E) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#4| (|%list| (QUOTE -259) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-553 (-472)))) (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#4| (QUOTE (-552 (-772)))) (|HasCategory| |#4| (QUOTE (-72)))) +(-401 S R E) ((|constructor| (NIL "GradedAlgebra(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-algebra''. A graded algebra is a graded module together with a degree preserving \\spad{R}-linear map,{} called the {\\em product}. \\blankline The name ``product'' is written out in full so inner and outer products with the same mapping type can be distinguished by name.")) (|product| (($ $ $) "\\spad{product(a,b)} is the degree-preserving \\spad{R}-linear product: \\blankline \\indented{2}{\\spad{degree product(a,b) = degree a + degree b}} \\indented{2}{\\spad{product(a1+a2,b) = product(a1,b) + product(a2,b)}} \\indented{2}{\\spad{product(a,b1+b2) = product(a,b1) + product(a,b2)}} \\indented{2}{\\spad{product(r*a,b) = product(a,r*b) = r*product(a,b)}} \\indented{2}{\\spad{product(a,product(b,c)) = product(product(a,b),c)}}")) (|One| (($) "1 is the identity for \\spad{product}."))) NIL NIL -(-400 R E) +(-402 R E) ((|constructor| (NIL "GradedAlgebra(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-algebra''. A graded algebra is a graded module together with a degree preserving \\spad{R}-linear map,{} called the {\\em product}. \\blankline The name ``product'' is written out in full so inner and outer products with the same mapping type can be distinguished by name.")) (|product| (($ $ $) "\\spad{product(a,b)} is the degree-preserving \\spad{R}-linear product: \\blankline \\indented{2}{\\spad{degree product(a,b) = degree a + degree b}} \\indented{2}{\\spad{product(a1+a2,b) = product(a1,b) + product(a2,b)}} \\indented{2}{\\spad{product(a,b1+b2) = product(a,b1) + product(a,b2)}} \\indented{2}{\\spad{product(r*a,b) = product(a,r*b) = r*product(a,b)}} \\indented{2}{\\spad{product(a,product(b,c)) = product(product(a,b),c)}}")) (|One| (($) "1 is the identity for \\spad{product}."))) NIL NIL -(-401) +(-403) ((|constructor| (NIL "GrayCode provides a function for efficiently running through all subsets of a finite set,{} only changing one element by another one.")) (|firstSubsetGray| (((|Vector| (|Vector| (|Integer|))) (|PositiveInteger|)) "\\spad{firstSubsetGray(n)} creates the first vector {\\em ww} to start a loop using {\\em nextSubsetGray(ww,n)}")) (|nextSubsetGray| (((|Vector| (|Vector| (|Integer|))) (|Vector| (|Vector| (|Integer|))) (|PositiveInteger|)) "\\spad{nextSubsetGray(ww,n)} returns a vector {\\em vv} whose components have the following meanings:\\begin{items} \\item {\\em vv.1}: a vector of length \\spad{n} whose entries are 0 or 1. This \\indented{3}{can be interpreted as a code for a subset of the set 1,{}...,{}\\spad{n};} \\indented{3}{{\\em vv.1} differs from {\\em ww.1} by exactly one entry;} \\item {\\em vv.2.1} is the number of the entry of {\\em vv.1} which \\indented{3}{will be changed next time;} \\item {\\em vv.2.1 = n+1} means that {\\em vv.1} is the last subset; \\indented{3}{trying to compute nextSubsetGray(vv) if {\\em vv.2.1 = n+1}} \\indented{3}{will produce an error!} \\end{items} The other components of {\\em vv.2} are needed to compute nextSubsetGray efficiently. Note: this is an implementation of [Williamson,{} Topic II,{} 3.54,{} \\spad{p}. 112] for the special case {\\em r1 = r2 = ... = rn = 2}; Note: nextSubsetGray produces a side-effect,{} \\spadignore{i.e.} {\\em nextSubsetGray(vv)} and {\\em vv := nextSubsetGray(vv)} will have the same effect."))) NIL NIL -(-402) +(-404) ((|constructor| (NIL "TwoDimensionalPlotSettings sets global flags and constants for 2-dimensional plotting.")) (|screenResolution| (((|Integer|) (|Integer|)) "\\spad{screenResolution(n)} sets the screen resolution to \\spad{n}.") (((|Integer|)) "\\spad{screenResolution()} returns the screen resolution \\spad{n}.")) (|minPoints| (((|Integer|) (|Integer|)) "\\spad{minPoints()} sets the minimum number of points in a plot.") (((|Integer|)) "\\spad{minPoints()} returns the minimum number of points in a plot.")) (|maxPoints| (((|Integer|) (|Integer|)) "\\spad{maxPoints()} sets the maximum number of points in a plot.") (((|Integer|)) "\\spad{maxPoints()} returns the maximum number of points in a plot.")) (|adaptive| (((|Boolean|) (|Boolean|)) "\\spad{adaptive(true)} turns adaptive plotting on; \\spad{adaptive(false)} turns adaptive plotting off.") (((|Boolean|)) "\\spad{adaptive()} determines whether plotting will be done adaptively.")) (|drawToScale| (((|Boolean|) (|Boolean|)) "\\spad{drawToScale(true)} causes plots to be drawn to scale. \\spad{drawToScale(false)} causes plots to be drawn so that they fill up the viewport window. The default setting is \\spad{false}.") (((|Boolean|)) "\\spad{drawToScale()} determines whether or not plots are to be drawn to scale.")) (|clipPointsDefault| (((|Boolean|) (|Boolean|)) "\\spad{clipPointsDefault(true)} turns on automatic clipping; \\spad{clipPointsDefault(false)} turns off automatic clipping. The default setting is \\spad{true}.") (((|Boolean|)) "\\spad{clipPointsDefault()} determines whether or not automatic clipping is to be done."))) NIL NIL -(-403) +(-405) ((|constructor| (NIL "TwoDimensionalGraph creates virtual two dimensional graphs (to be displayed on TwoDimensionalViewports).")) (|putColorInfo| (((|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|Palette|))) "\\spad{putColorInfo(llp,lpal)} takes a list of list of points,{} \\spad{llp},{} and returns the points with their hue and shade components set according to the list of palette colors,{} \\spad{lpal}.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(gi)} returns the indicated graph,{} \\spad{gi},{} of domain \\spadtype{GraphImage} as output of the domain \\spadtype{OutputForm}.") (($ (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{coerce(llp)} component(\\spad{gi},{}pt) creates and returns a graph of the domain \\spadtype{GraphImage} which is composed of the list of list of points given by \\spad{llp},{} and whose point colors,{} line colors and point sizes are determined by the default functions \\spadfun{pointColorDefault},{} \\spadfun{lineColorDefault},{} and \\spadfun{pointSizeDefault}. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.")) (|point| (((|Void|) $ (|Point| (|DoubleFloat|)) (|Palette|)) "\\spad{point(gi,pt,pal)} modifies the graph \\spad{gi} of the domain \\spadtype{GraphImage} to contain one point component,{} \\spad{pt} whose point color is set to be the palette color \\spad{pal},{} and whose line color and point size are determined by the default functions \\spadfun{lineColorDefault} and \\spadfun{pointSizeDefault}.")) (|appendPoint| (((|Void|) $ (|Point| (|DoubleFloat|))) "\\spad{appendPoint(gi,pt)} appends the point \\spad{pt} to the end of the list of points component for the graph,{} \\spad{gi},{} which is of the domain \\spadtype{GraphImage}.")) (|component| (((|Void|) $ (|Point| (|DoubleFloat|)) (|Palette|) (|Palette|) (|PositiveInteger|)) "\\spad{component(gi,pt,pal1,pal2,ps)} modifies the graph \\spad{gi} of the domain \\spadtype{GraphImage} to contain one point component,{} \\spad{pt} whose point color is set to the palette color \\spad{pal1},{} line color is set to the palette color \\spad{pal2},{} and point size is set to the positive integer \\spad{ps}.") (((|Void|) $ (|Point| (|DoubleFloat|))) "\\spad{component(gi,pt)} modifies the graph \\spad{gi} of the domain \\spadtype{GraphImage} to contain one point component,{} \\spad{pt} whose point color,{} line color and point size are determined by the default functions \\spadfun{pointColorDefault},{} \\spadfun{lineColorDefault},{} and \\spadfun{pointSizeDefault}.") (((|Void|) $ (|List| (|Point| (|DoubleFloat|))) (|Palette|) (|Palette|) (|PositiveInteger|)) "\\spad{component(gi,lp,pal1,pal2,p)} sets the components of the graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} to the values given. The point list for \\spad{gi} is set to the list \\spad{lp},{} the color of the points in \\spad{lp} is set to the palette color \\spad{pal1},{} the color of the lines which connect the points \\spad{lp} is set to the palette color \\spad{pal2},{} and the size of the points in \\spad{lp} is given by the integer \\spad{p}.")) (|units| (((|List| (|Float|)) $ (|List| (|Float|))) "\\spad{units(gi,lu)} modifies the list of unit increments for the \\spad{x} and \\spad{y} axes of the given graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} to be that of the list of unit increments,{} \\spad{lu},{} and returns the new list of units for \\spad{gi}.") (((|List| (|Float|)) $) "\\spad{units(gi)} returns the list of unit increments for the \\spad{x} and \\spad{y} axes of the indicated graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|ranges| (((|List| (|Segment| (|Float|))) $ (|List| (|Segment| (|Float|)))) "\\spad{ranges(gi,lr)} modifies the list of ranges for the given graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} to be that of the list of range segments,{} \\spad{lr},{} and returns the new range list for \\spad{gi}.") (((|List| (|Segment| (|Float|))) $) "\\spad{ranges(gi)} returns the list of ranges of the point components from the indicated graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|key| (((|Integer|) $) "\\spad{key(gi)} returns the process ID of the given graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|pointLists| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{pointLists(gi)} returns the list of lists of points which compose the given graph,{} \\spad{gi},{} of the domain \\spadtype{GraphImage}.")) (|makeGraphImage| (($ (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|Palette|)) (|List| (|Palette|)) (|List| (|PositiveInteger|)) (|List| (|DrawOption|))) "\\spad{makeGraphImage(llp,lpal1,lpal2,lp,lopt)} returns a graph of the domain \\spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \\spad{llp},{} whose point colors are indicated by the list of palette colors,{} \\spad{lpal1},{} and whose lines are colored according to the list of palette colors,{} \\spad{lpal2}. The paramater \\spad{lp} is a list of integers which denote the size of the data points,{} and \\spad{lopt} is the list of draw command options. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.") (($ (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|Palette|)) (|List| (|Palette|)) (|List| (|PositiveInteger|))) "\\spad{makeGraphImage(llp,lpal1,lpal2,lp)} returns a graph of the domain \\spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \\spad{llp},{} whose point colors are indicated by the list of palette colors,{} \\spad{lpal1},{} and whose lines are colored according to the list of palette colors,{} \\spad{lpal2}. The paramater \\spad{lp} is a list of integers which denote the size of the data points. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.") (($ (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{makeGraphImage(llp)} returns a graph of the domain \\spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \\spad{llp},{} with default point size and default point and line colours. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.") (($ $) "\\spad{makeGraphImage(gi)} takes the given graph,{} \\spad{gi} of the domain \\spadtype{GraphImage},{} and sends it's data to the viewport manager where it waits to be included in a two-dimensional viewport window. \\spad{gi} cannot be an empty graph,{} and it's elements must have been created using the \\spadfun{point} or \\spadfun{component} functions,{} not by a previous \\spadfun{makeGraphImage}.")) (|graphImage| (($) "\\spad{graphImage()} returns an empty graph with 0 point lists of the domain \\spadtype{GraphImage}. A graph image contains the graph data component of a two dimensional viewport."))) NIL NIL -(-404 S R E) +(-406 S R E) ((|constructor| (NIL "GradedModule(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-module'',{} \\spadignore{i.e.} collection of \\spad{R}-modules indexed by an abelian monoid \\spad{E}. An element \\spad{g} of \\spad{G[s]} for some specific \\spad{s} in \\spad{E} is said to be an element of \\spad{G} with {\\em degree} \\spad{s}. Sums are defined in each module \\spad{G[s]} so two elements of \\spad{G} have a sum if they have the same degree. \\blankline Morphisms can be defined and composed by degree to give the mathematical category of graded modules.")) (+ (($ $ $) "\\spad{g+h} is the sum of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.")) (- (($ $ $) "\\spad{g-h} is the difference of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.") (($ $) "\\spad{-g} is the additive inverse of \\spad{g} in the module of elements of the same grade as \\spad{g}.")) (* (($ $ |#2|) "\\spad{g*r} is right module multiplication.") (($ |#2| $) "\\spad{r*g} is left module multiplication.")) (|Zero| (($) "0 denotes the zero of degree 0.")) (|degree| ((|#3| $) "\\spad{degree(g)} names the degree of \\spad{g}. The set of all elements of a given degree form an \\spad{R}-module."))) NIL NIL -(-405 R E) +(-407 R E) ((|constructor| (NIL "GradedModule(\\spad{R},{}\\spad{E}) denotes ``E-graded \\spad{R}-module'',{} \\spadignore{i.e.} collection of \\spad{R}-modules indexed by an abelian monoid \\spad{E}. An element \\spad{g} of \\spad{G[s]} for some specific \\spad{s} in \\spad{E} is said to be an element of \\spad{G} with {\\em degree} \\spad{s}. Sums are defined in each module \\spad{G[s]} so two elements of \\spad{G} have a sum if they have the same degree. \\blankline Morphisms can be defined and composed by degree to give the mathematical category of graded modules.")) (+ (($ $ $) "\\spad{g+h} is the sum of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.")) (- (($ $ $) "\\spad{g-h} is the difference of \\spad{g} and \\spad{h} in the module of elements of the same degree as \\spad{g} and \\spad{h}. Error: if \\spad{g} and \\spad{h} have different degrees.") (($ $) "\\spad{-g} is the additive inverse of \\spad{g} in the module of elements of the same grade as \\spad{g}.")) (* (($ $ |#1|) "\\spad{g*r} is right module multiplication.") (($ |#1| $) "\\spad{r*g} is left module multiplication.")) (|Zero| (($) "0 denotes the zero of degree 0.")) (|degree| ((|#2| $) "\\spad{degree(g)} names the degree of \\spad{g}. The set of all elements of a given degree form an \\spad{R}-module."))) NIL NIL -(-406 |lv| -3078 R) +(-408 |lv| -3088 R) ((|constructor| (NIL "\\indented{1}{Author : \\spad{P}.Gianni,{} Summer \\spad{'88},{} revised November \\spad{'89}} Solve systems of polynomial equations using Groebner bases Total order Groebner bases are computed and then converted to lex ones This package is mostly intended for internal use.")) (|genericPosition| (((|Record| (|:| |dpolys| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |coords| (|List| (|Integer|)))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{genericPosition(lp,lv)} puts a radical zero dimensional ideal in general position,{} for system \\spad{lp} in variables \\spad{lv}.")) (|testDim| (((|Union| (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "failed") (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{testDim(lp,lv)} tests if the polynomial system \\spad{lp} in variables \\spad{lv} is zero dimensional.")) (|groebSolve| (((|List| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|OrderedVariableList| |#1|))) "\\spad{groebSolve(lp,lv)} reduces the polynomial system \\spad{lp} in variables \\spad{lv} to triangular form. Algorithm based on groebner bases algorithm with linear algebra for change of ordering. Preprocessing for the general solver. The polynomials in input are of type \\spadtype{DMP}."))) NIL NIL -(-407 S) +(-409 S) ((|constructor| (NIL "The class of multiplicative groups,{} \\spadignore{i.e.} monoids with multiplicative inverses. \\blankline")) (|commutator| (($ $ $) "\\spad{commutator(p,q)} computes \\spad{inv(p) * inv(q) * p * q}.")) (|conjugate| (($ $ $) "\\spad{conjugate(p,q)} computes \\spad{inv(q) * p * q}; this is 'right action by conjugation'.")) (|unitsKnown| ((|attribute|) "unitsKnown asserts that recip only returns \"failed\" for non-units.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}.")) (/ (($ $ $) "\\spad{x/y} is the same as \\spad{x} times the inverse of \\spad{y}.")) (|inv| (($ $) "\\spad{inv(x)} returns the inverse of \\spad{x}."))) NIL NIL -(-408) +(-410) ((|constructor| (NIL "The class of multiplicative groups,{} \\spadignore{i.e.} monoids with multiplicative inverses. \\blankline")) (|commutator| (($ $ $) "\\spad{commutator(p,q)} computes \\spad{inv(p) * inv(q) * p * q}.")) (|conjugate| (($ $ $) "\\spad{conjugate(p,q)} computes \\spad{inv(q) * p * q}; this is 'right action by conjugation'.")) (|unitsKnown| ((|attribute|) "unitsKnown asserts that recip only returns \"failed\" for non-units.")) (** (($ $ (|Integer|)) "\\spad{x**n} returns \\spad{x} raised to the integer power \\spad{n}.")) (/ (($ $ $) "\\spad{x/y} is the same as \\spad{x} times the inverse of \\spad{y}.")) (|inv| (($ $) "\\spad{inv(x)} returns the inverse of \\spad{x}."))) -((-3975 . T)) +((-3986 . T)) NIL -(-409 |Coef| |var| |cen|) +(-411 |Coef| |var| |cen|) ((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x\\^r)}.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|coerce| (($ (|UnivariatePuiseuxSeries| |#1| |#2| |#3|)) "\\spad{coerce(f)} converts a Puiseux series to a general power series.") (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a Puiseux series."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|)))) (|HasCategory| (-345 (-480)) (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-309))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-29 (-480)))) (|HasCategory| |#1| (QUOTE (-866))) (|HasCategory| |#1| (QUOTE (-1106)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1081))))) (|HasSignature| |#1| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#1|))))))) -(-410 |Key| |Entry| |Tbl| |dent|) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|)))) (|HasCategory| (-347 (-483)) (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-311))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-29 (-483)))) (|HasCategory| |#1| (QUOTE (-871))) (|HasCategory| |#1| (QUOTE (-1113)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1088))))) (|HasSignature| |#1| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#1|))))))) +(-412 |Key| |Entry| |Tbl| |dent|) ((|constructor| (NIL "A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key."))) -((-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) -(-411 R E V P) +((-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) +(-413 R E V P) ((|constructor| (NIL "A domain constructor of the category \\axiomType{TriangularSetCategory}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. Triangular sets are stored as sorted lists \\spad{w}.\\spad{r}.\\spad{t}. the main variables of their members but they are displayed in reverse order.\\newline References : \\indented{1}{[1] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)}"))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#4| (|%list| (QUOTE -257) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-550 (-469)))) (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#3| (QUOTE (-315))) (|HasCategory| |#4| (QUOTE (-549 (-767)))) (|HasCategory| |#4| (QUOTE (-72)))) -(-412) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#4| (|%list| (QUOTE -259) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-553 (-472)))) (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#3| (QUOTE (-317))) (|HasCategory| |#4| (QUOTE (-552 (-772)))) (|HasCategory| |#4| (QUOTE (-72)))) +(-414) ((|constructor| (NIL "\\indented{1}{Symbolic fractions in \\%\\spad{pi} with integer coefficients;} \\indented{1}{The point for using \\spad{Pi} as the default domain for those fractions} \\indented{1}{is that \\spad{Pi} is coercible to the float types,{} and not Expression.} Date Created: 21 Feb 1990 Date Last Updated: 12 Mai 1992")) (|pi| (($) "\\spad{pi()} returns the symbolic \\%\\spad{pi}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-413) +(-415) ((|constructor| (NIL "This domain represents a `has' expression.")) (|rhs| (((|SpadAst|) $) "\\spad{rhs(e)} returns the right hand side of the case expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the has expression `e'."))) NIL NIL -(-414 |Key| |Entry| |hashfn|) +(-416 |Key| |Entry| |hashfn|) ((|constructor| (NIL "This domain provides access to the underlying Lisp hash tables. By varying the hashfn parameter,{} tables suited for different purposes can be obtained."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-1007))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) -(-415) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-1012))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) +(-417) ((|constructor| (NIL "\\indented{1}{Author : Larry Lambe} Date Created : August 1988 Date Last Updated : March 9 1990 Related Constructors: OrderedSetInts,{} Commutator,{} FreeNilpotentLie AMS Classification: Primary 17B05,{} 17B30; Secondary 17A50 Keywords: free Lie algebra,{} Hall basis,{} basic commutators Description : Generate a basis for the free Lie algebra on \\spad{n} generators over a ring \\spad{R} with identity up to basic commutators of length \\spad{c} using the algorithm of \\spad{P}. Hall as given in Serre's book Lie Groups -- Lie Algebras")) (|generate| (((|Vector| (|List| (|Integer|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{generate(numberOfGens, maximalWeight)} generates a vector of elements of the form [left,{}weight,{}right] which represents a \\spad{P}. Hall basis element for the free lie algebra on \\spad{numberOfGens} generators. We only generate those basis elements of weight less than or equal to maximalWeight")) (|inHallBasis?| (((|Boolean|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{inHallBasis?(numberOfGens, leftCandidate, rightCandidate, left)} tests to see if a new element should be added to the \\spad{P}. Hall basis being constructed. 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(|any?| (((|Boolean|) (|Mapping| (|Boolean|) |#2|) $) "\\spad{any?(p,u)} tests if \\axiom{\\spad{p}(\\spad{x})} is \\spad{true} for any element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{any?(\\spad{p},{}\\spad{u}) = reduce(or,{}map(\\spad{f},{}\\spad{u}),{}\\spad{false},{}\\spad{true})}.")) (|map!| (($ (|Mapping| |#2| |#2|) $) "\\spad{map!(f,u)} destructively replaces each element \\spad{x} of \\spad{u} by \\axiom{\\spad{f}(\\spad{x})}.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(f,u)} returns a copy of \\spad{u} with each element \\spad{x} replaced by \\spad{f}(\\spad{x}). For collections,{} \\axiom{map(\\spad{f},{}\\spad{u}) = [\\spad{f}(\\spad{x}) for \\spad{x} in \\spad{u}]}."))) NIL -((|HasAttribute| |#1| (QUOTE -3978)) (|HasAttribute| |#1| (QUOTE -3979)) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) -(-424 S) +((|HasAttribute| |#1| (QUOTE -3989)) (|HasAttribute| |#1| (QUOTE -3990)) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) +(-426 S) ((|constructor| (NIL "A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \\spadatt{finiteAggregate} have a finite number of members. Those with attribute \\spadatt{shallowlyMutable} allow an element to be modified or updated without changing its overall value.")) (|member?| (((|Boolean|) |#1| $) "\\spad{member?(x,u)} tests if \\spad{x} is a member of \\spad{u}. For collections,{} \\axiom{member?(\\spad{x},{}\\spad{u}) = reduce(or,{}[x=y for \\spad{y} in \\spad{u}],{}\\spad{false})}.")) (|members| (((|List| |#1|) $) "\\spad{members(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|parts| (((|List| |#1|) $) "\\spad{parts(u)} returns a list of the consecutive elements of \\spad{u}. For collections,{} \\axiom{parts([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = (\\spad{x},{}\\spad{y},{}...,{}\\spad{z})}.")) (|count| (((|NonNegativeInteger|) |#1| $) "\\spad{count(x,u)} returns the number of occurrences of \\spad{x} in \\spad{u}. For collections,{} \\axiom{count(\\spad{x},{}\\spad{u}) = reduce(+,{}[x=y for \\spad{y} in \\spad{u}],{}0)}.") (((|NonNegativeInteger|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{count(p,u)} returns the number of elements \\spad{x} in \\spad{u} such that \\axiom{\\spad{p}(\\spad{x})} is \\spad{true}. For collections,{} \\axiom{count(\\spad{p},{}\\spad{u}) = reduce(+,{}[1 for \\spad{x} in \\spad{u} | \\spad{p}(\\spad{x})],{}0)}.")) (|every?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{every?(f,u)} tests if \\spad{p}(\\spad{x}) is \\spad{true} for all elements \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{every?(\\spad{p},{}\\spad{u}) = reduce(and,{}map(\\spad{f},{}\\spad{u}),{}\\spad{true},{}\\spad{false})}.")) (|any?| (((|Boolean|) (|Mapping| (|Boolean|) |#1|) $) "\\spad{any?(p,u)} tests if \\axiom{\\spad{p}(\\spad{x})} is \\spad{true} for any element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{any?(\\spad{p},{}\\spad{u}) = reduce(or,{}map(\\spad{f},{}\\spad{u}),{}\\spad{false},{}\\spad{true})}.")) (|map!| (($ (|Mapping| |#1| |#1|) $) "\\spad{map!(f,u)} destructively replaces each element \\spad{x} of \\spad{u} by \\axiom{\\spad{f}(\\spad{x})}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,u)} returns a copy of \\spad{u} with each element \\spad{x} replaced by \\spad{f}(\\spad{x}). For collections,{} \\axiom{map(\\spad{f},{}\\spad{u}) = [\\spad{f}(\\spad{x}) for \\spad{x} in \\spad{u}]}."))) NIL NIL -(-425 S) +(-427 S) ((|constructor| (NIL "A is homotopic to \\spad{B} iff any element of domain \\spad{B} can be automically converted into an element of domain \\spad{B},{} and nay element of domain \\spad{B} can be automatically converted into an A."))) NIL NIL -(-426) +(-428) ((|constructor| (NIL "This domain represents hostnames on computer network.")) (|host| (($ (|String|)) "\\spad{host(n)} constructs a Hostname from the name `n'."))) NIL NIL -(-427 S) +(-429 S) ((|constructor| (NIL "Category for the hyperbolic trigonometric functions.")) (|tanh| (($ $) "\\spad{tanh(x)} returns the hyperbolic tangent of \\spad{x}.")) (|sinh| (($ $) "\\spad{sinh(x)} returns the hyperbolic sine of \\spad{x}.")) (|sech| (($ $) "\\spad{sech(x)} returns the hyperbolic secant of \\spad{x}.")) (|csch| (($ $) "\\spad{csch(x)} returns the hyperbolic cosecant of \\spad{x}.")) (|coth| (($ $) "\\spad{coth(x)} returns the hyperbolic cotangent of \\spad{x}.")) (|cosh| (($ $) "\\spad{cosh(x)} returns the hyperbolic cosine of \\spad{x}."))) NIL NIL -(-428) +(-430) ((|constructor| (NIL "Category for the hyperbolic trigonometric functions.")) (|tanh| (($ $) "\\spad{tanh(x)} returns the hyperbolic tangent of \\spad{x}.")) (|sinh| (($ $) "\\spad{sinh(x)} returns the hyperbolic sine of \\spad{x}.")) (|sech| (($ $) "\\spad{sech(x)} returns the hyperbolic secant of \\spad{x}.")) (|csch| (($ $) "\\spad{csch(x)} returns the hyperbolic cosecant of \\spad{x}.")) (|coth| (($ $) "\\spad{coth(x)} returns the hyperbolic cotangent of \\spad{x}.")) (|cosh| (($ $) "\\spad{cosh(x)} returns the hyperbolic cosine of \\spad{x}."))) NIL NIL -(-429 -3078 UP |AlExt| |AlPol|) +(-431 -3088 UP |AlExt| |AlPol|) ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of a field over which we can factor UP's.")) (|factor| (((|Factored| |#4|) |#4| (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{factor(p, f)} returns a prime factorisation of \\spad{p}; \\spad{f} is a factorisation map for elements of UP."))) NIL NIL -(-430) +(-432) ((|constructor| (NIL "Algebraic closure of the rational numbers.")) (|norm| (($ $ (|List| (|Kernel| $))) "\\spad{norm(f,l)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernels \\spad{l}") (($ $ (|Kernel| $)) "\\spad{norm(f,k)} computes the norm of the algebraic number \\spad{f} with respect to the extension generated by kernel \\spad{k}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|List| (|Kernel| $))) "\\spad{norm(p,l)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernels \\spad{l}") (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|Kernel| $)) "\\spad{norm(p,k)} computes the norm of the polynomial \\spad{p} with respect to the extension generated by kernel \\spad{k}")) (|trueEqual| (((|Boolean|) $ $) "\\spad{trueEqual(x,y)} tries to determine if the two numbers are equal")) (|reduce| (($ $) "\\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \\spad{f} by applying their defining relations.")) (|denom| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{denom(f)} returns the denominator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}.")) (|numer| (((|SparseMultivariatePolynomial| (|Integer|) (|Kernel| $)) $) "\\spad{numer(f)} returns the numerator of \\spad{f} viewed as a polynomial in the kernels over \\spad{Z}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| $ (QUOTE (-956))) (|HasCategory| $ (QUOTE (-945 (-480))))) -(-431 S |mn|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| $ (QUOTE (-961))) (|HasCategory| $ (QUOTE (-950 (-483))))) +(-433 S |mn|) ((|constructor| (NIL "\\indented{1}{Author Micheal Monagan \\spad{Aug/87}} This is the basic one dimensional array data type."))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-432 R |mnRow| |mnCol|) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-434 R |mnRow| |mnCol|) ((|constructor| (NIL "\\indented{1}{An IndexedTwoDimensionalArray is a 2-dimensional array where} the minimal row and column indices are parameters of the type. Rows and columns are returned as IndexedOneDimensionalArray's with minimal indices matching those of the IndexedTwoDimensionalArray. The index of the 'first' row may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-433 K R UP) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-435 K R UP) ((|constructor| (NIL "\\indented{1}{Author: Clifton Williamson} Date Created: 9 August 1993 Date Last Updated: 3 December 1993 Basic Operations: chineseRemainder,{} factorList Related Domains: PAdicWildFunctionFieldIntegralBasis(\\spad{K},{}\\spad{R},{}UP,{}\\spad{F}) Also See: WildFunctionFieldIntegralBasis,{} FunctionFieldIntegralBasis AMS Classifications: Keywords: function field,{} finite field,{} integral basis Examples: References: Description:")) (|chineseRemainder| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) (|List| |#3|) (|List| (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) (|NonNegativeInteger|)) "\\spad{chineseRemainder(lu,lr,n)} \\undocumented")) (|listConjugateBases| (((|List| (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) (|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{listConjugateBases(bas,q,n)} returns the list \\spad{[bas,bas^Frob,bas^(Frob^2),...bas^(Frob^(n-1))]},{} where \\spad{Frob} raises the coefficients of all polynomials appearing in the basis \\spad{bas} to the \\spad{q}th power.")) (|factorList| (((|List| (|SparseUnivariatePolynomial| |#1|)) |#1| (|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{factorList(k,n,m,j)} \\undocumented"))) NIL NIL -(-434 R UP -3078) +(-436 R UP -3088) ((|constructor| (NIL "This package contains functions used in the packages FunctionFieldIntegralBasis and NumberFieldIntegralBasis.")) (|moduleSum| (((|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) (|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|))) (|Record| (|:| |basis| (|Matrix| |#1|)) (|:| |basisDen| |#1|) (|:| |basisInv| (|Matrix| |#1|)))) "\\spad{moduleSum(m1,m2)} returns the sum of two modules in the framed algebra \\spad{F}. Each module \\spad{mi} is represented as follows: \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn} and \\spad{mi} is a record \\spad{[basis,basisDen,basisInv]}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then a basis \\spad{v1,...,vn} for \\spad{mi} is given by \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|idealiserMatrix| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{idealiserMatrix(m1, m2)} returns the matrix representing the linear conditions on the Ring associatied with an ideal defined by \\spad{m1} and \\spad{m2}.")) (|idealiser| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{idealiser(m1,m2,d)} computes the order of an ideal defined by \\spad{m1} and \\spad{m2} where \\spad{d} is the known part of the denominator") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{idealiser(m1,m2)} computes the order of an ideal defined by \\spad{m1} and \\spad{m2}")) (|leastPower| (((|NonNegativeInteger|) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{leastPower(p,n)} returns \\spad{e},{} where \\spad{e} is the smallest integer such that \\spad{p **e >= n}")) (|divideIfCan!| ((|#1| (|Matrix| |#1|) (|Matrix| |#1|) |#1| (|Integer|)) "\\spad{divideIfCan!(matrix,matrixOut,prime,n)} attempts to divide the entries of \\spad{matrix} by \\spad{prime} and store the result in \\spad{matrixOut}. If it is successful,{} 1 is returned and if not,{} \\spad{prime} is returned. Here both \\spad{matrix} and \\spad{matrixOut} are \\spad{n}-by-\\spad{n} upper triangular matrices.")) (|matrixGcd| ((|#1| (|Matrix| |#1|) |#1| (|NonNegativeInteger|)) "\\spad{matrixGcd(mat,sing,n)} is \\spad{gcd(sing,g)} where \\spad{g} is the gcd of the entries of the \\spad{n}-by-\\spad{n} upper-triangular matrix \\spad{mat}.")) (|diagonalProduct| ((|#1| (|Matrix| |#1|)) "\\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \\spad{m}")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(x)} returns a square-free factorisation of \\spad{x}"))) NIL NIL -(-435 |mn|) +(-437 |mn|) ((|constructor| (NIL "\\spadtype{IndexedBits} is a domain to compactly represent large quantities of Boolean data."))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| (-83) (QUOTE (-257 (-83)))) (|HasCategory| (-83) (QUOTE (-1007)))) (|HasCategory| (-83) (QUOTE (-550 (-469)))) (|HasCategory| (-83) (QUOTE (-751))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| (-83) (QUOTE (-1007))) (|HasCategory| (-83) (QUOTE (-549 (-767)))) (|HasCategory| (-83) (QUOTE (-72)))) -(-436 K R UP L) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| (-85) (QUOTE (-259 (-85)))) (|HasCategory| (-85) (QUOTE (-1012)))) (|HasCategory| (-85) (QUOTE (-553 (-472)))) (|HasCategory| (-85) (QUOTE (-756))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| (-85) (QUOTE (-1012))) (|HasCategory| (-85) (QUOTE (-552 (-772)))) (|HasCategory| (-85) (QUOTE (-72)))) +(-438 K R UP L) ((|constructor| (NIL "IntegralBasisPolynomialTools provides functions for \\indented{1}{mapping functions on the coefficients of univariate and bivariate} \\indented{1}{polynomials.}")) (|mapBivariate| (((|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#4|)) (|Mapping| |#4| |#1|) |#3|) "\\spad{mapBivariate(f,p(x,y))} applies the function \\spad{f} to the coefficients of \\spad{p(x,y)}.")) (|mapMatrixIfCan| (((|Union| (|Matrix| |#2|) "failed") (|Mapping| (|Union| |#1| "failed") |#4|) (|Matrix| (|SparseUnivariatePolynomial| |#4|))) "\\spad{mapMatrixIfCan(f,mat)} applies the function \\spad{f} to the coefficients of the entries of \\spad{mat} if possible,{} and returns \\spad{\"failed\"} otherwise.")) (|mapUnivariateIfCan| (((|Union| |#2| "failed") (|Mapping| (|Union| |#1| "failed") |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{mapUnivariateIfCan(f,p(x))} applies the function \\spad{f} to the coefficients of \\spad{p(x)},{} if possible,{} and returns \\spad{\"failed\"} otherwise.")) (|mapUnivariate| (((|SparseUnivariatePolynomial| |#4|) (|Mapping| |#4| |#1|) |#2|) "\\spad{mapUnivariate(f,p(x))} applies the function \\spad{f} to the coefficients of \\spad{p(x)}.") ((|#2| (|Mapping| |#1| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{mapUnivariate(f,p(x))} applies the function \\spad{f} to the coefficients of \\spad{p(x)}."))) NIL NIL -(-437) +(-439) ((|constructor| (NIL "\\indented{1}{This domain implements a container of information} about the AXIOM library")) (|fullDisplay| (((|Void|) $) "\\spad{fullDisplay(ic)} prints all of the information contained in \\axiom{\\spad{ic}}.")) (|display| (((|Void|) $) "\\spad{display(ic)} prints a summary of the information contained in \\axiom{\\spad{ic}}.")) (|elt| (((|String|) $ (|Symbol|)) "\\spad{elt(ic,s)} selects a particular field from \\axiom{\\spad{ic}}. Valid fields are \\axiom{name,{} nargs,{} exposed,{} type,{} abbreviation,{} kind,{} origin,{} params,{} condition,{} doc}."))) NIL NIL -(-438 R Q A B) +(-440 R Q A B) ((|constructor| (NIL "InnerCommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#4|) "\\spad{splitDenominator([q1,...,qn])} returns \\spad{[[p1,...,pn], d]} such that \\spad{qi = pi/d} and \\spad{d} is a common denominator for the \\spad{qi}'s.")) (|clearDenominator| ((|#3| |#4|) "\\spad{clearDenominator([q1,...,qn])} returns \\spad{[p1,...,pn]} such that \\spad{qi = pi/d} where \\spad{d} is a common denominator for the \\spad{qi}'s.")) (|commonDenominator| ((|#1| |#4|) "\\spad{commonDenominator([q1,...,qn])} returns a common denominator \\spad{d} for \\spad{q1},{}...,{}qn."))) NIL NIL -(-439 -3078 |Expon| |VarSet| |DPoly|) +(-441 -3088 |Expon| |VarSet| |DPoly|) ((|constructor| (NIL "This domain represents polynomial ideals with coefficients in any field and supports the basic ideal operations,{} including intersection sum and quotient. An ideal is represented by a list of polynomials (the generators of the ideal) and a boolean that is \\spad{true} if the generators are a Groebner basis. The algorithms used are based on Groebner basis computations. The ordering is determined by the datatype of the input polynomials. Users may use refinements of total degree orderings.")) (|relationsIdeal| (((|SuchThat| (|List| (|Polynomial| |#1|)) (|List| (|Equation| (|Polynomial| |#1|)))) (|List| |#4|)) "\\spad{relationsIdeal(polyList)} returns the ideal of relations among the polynomials in \\spad{polyList}.")) (|saturate| (($ $ |#4| (|List| |#3|)) "\\spad{saturate(I,f,lvar)} is the saturation with respect to the prime principal ideal which is generated by \\spad{f} in the polynomial ring \\spad{F[lvar]}.") (($ $ |#4|) "\\spad{saturate(I,f)} is the saturation of the ideal \\spad{I} with respect to the multiplicative set generated by the polynomial \\spad{f}.")) (|coerce| (($ (|List| |#4|)) "\\spad{coerce(polyList)} converts the list of polynomials \\spad{polyList} to an ideal.")) (|generators| (((|List| |#4|) $) "\\spad{generators(I)} returns a list of generators for the ideal \\spad{I}.")) (|groebner?| (((|Boolean|) $) "\\spad{groebner?(I)} tests if the generators of the ideal \\spad{I} are a Groebner basis.")) (|groebnerIdeal| (($ (|List| |#4|)) "\\spad{groebnerIdeal(polyList)} constructs the ideal generated by the list of polynomials \\spad{polyList} which are assumed to be a Groebner basis. Note: this operation avoids a Groebner basis computation.")) (|ideal| (($ (|List| |#4|)) "\\spad{ideal(polyList)} constructs the ideal generated by the list of polynomials \\spad{polyList}.")) (|leadingIdeal| (($ $) "\\spad{leadingIdeal(I)} is the ideal generated by the leading terms of the elements of the ideal \\spad{I}.")) (|dimension| (((|Integer|) $) "\\spad{dimension(I)} gives the dimension of the ideal \\spad{I}. in the ring \\spad{F[lvar]},{} where lvar are the variables appearing in \\spad{I}") (((|Integer|) $ (|List| |#3|)) "\\spad{dimension(I,lvar)} gives the dimension of the ideal \\spad{I},{} in the ring \\spad{F[lvar]}")) (|backOldPos| (($ (|Record| (|:| |mval| (|Matrix| |#1|)) (|:| |invmval| (|Matrix| |#1|)) (|:| |genIdeal| $))) "\\spad{backOldPos(genPos)} takes the result produced by \\spadfunFrom{generalPosition}{PolynomialIdeals} and performs the inverse transformation,{} returning the original ideal \\spad{backOldPos(generalPosition(I,listvar))} = \\spad{I}.")) (|generalPosition| (((|Record| (|:| |mval| (|Matrix| |#1|)) (|:| |invmval| (|Matrix| |#1|)) (|:| |genIdeal| $)) $ (|List| |#3|)) "\\spad{generalPosition(I,listvar)} perform a random linear transformation on the variables in \\spad{listvar} and returns the transformed ideal along with the change of basis matrix.")) (|groebner| (($ $) "\\spad{groebner(I)} returns a set of generators of \\spad{I} that are a Groebner basis for \\spad{I}.")) (|quotient| (($ $ |#4|) "\\spad{quotient(I,f)} computes the quotient of the ideal \\spad{I} by the principal ideal generated by the polynomial \\spad{f},{} \\spad{(I:(f))}.") (($ $ $) "\\spad{quotient(I,J)} computes the quotient of the ideals \\spad{I} and \\spad{J},{} \\spad{(I:J)}.")) (|intersect| (($ (|List| $)) "\\spad{intersect(LI)} computes the intersection of the list of ideals \\spad{LI}.") (($ $ $) "\\spad{intersect(I,J)} computes the intersection of the ideals \\spad{I} and \\spad{J}.")) (|zeroDim?| (((|Boolean|) $) "\\spad{zeroDim?(I)} tests if the ideal \\spad{I} is zero dimensional,{} \\spadignore{i.e.} all its associated primes are maximal,{} in the ring \\spad{F[lvar]},{} where lvar are the variables appearing in \\spad{I}") (((|Boolean|) $ (|List| |#3|)) "\\spad{zeroDim?(I,lvar)} tests if the ideal \\spad{I} is zero dimensional,{} \\spadignore{i.e.} all its associated primes are maximal,{} in the ring \\spad{F[lvar]}")) (|inRadical?| (((|Boolean|) |#4| $) "\\spad{inRadical?(f,I)} tests if some power of the polynomial \\spad{f} belongs to the ideal \\spad{I}.")) (|in?| (((|Boolean|) $ $) "\\spad{in?(I,J)} tests if the ideal \\spad{I} is contained in the ideal \\spad{J}.")) (|element?| (((|Boolean|) |#4| $) "\\spad{element?(f,I)} tests whether the polynomial \\spad{f} belongs to the ideal \\spad{I}.")) (|zero?| (((|Boolean|) $) "\\spad{zero?(I)} tests whether the ideal \\spad{I} is the zero ideal")) (|one?| (((|Boolean|) $) "\\spad{one?(I)} tests whether the ideal \\spad{I} is the unit ideal,{} \\spadignore{i.e.} contains 1.")) (+ (($ $ $) "\\spad{I+J} computes the ideal generated by the union of \\spad{I} and \\spad{J}.")) (** (($ $ (|NonNegativeInteger|)) "\\spad{I**n} computes the \\spad{n}th power of the ideal \\spad{I}.")) (* (($ $ $) "\\spad{I*J} computes the product of the ideal \\spad{I} and \\spad{J}."))) NIL -((|HasCategory| |#3| (QUOTE (-550 (-1081))))) -(-440 |vl| |nv|) +((|HasCategory| |#3| (QUOTE (-553 (-1088))))) +(-442 |vl| |nv|) ((|constructor| (NIL "\\indented{2}{This package provides functions for the primary decomposition of} polynomial ideals over the rational numbers. The ideals are members of the \\spadtype{PolynomialIdeals} domain,{} and the polynomial generators are required to be from the \\spadtype{DistributedMultivariatePolynomial} domain.")) (|contract| (((|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|List| (|OrderedVariableList| |#1|))) "\\spad{contract(I,lvar)} contracts the ideal \\spad{I} to the polynomial ring \\spad{F[lvar]}.")) (|primaryDecomp| (((|List| (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{primaryDecomp(I)} returns a list of primary ideals such that their intersection is the ideal \\spad{I}.")) (|radical| (((|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{radical(I)} returns the radical of the ideal \\spad{I}.")) (|prime?| (((|Boolean|) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{prime?(I)} tests if the ideal \\spad{I} is prime.")) (|zeroDimPrimary?| (((|Boolean|) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{zeroDimPrimary?(I)} tests if the ideal \\spad{I} is 0-dimensional primary.")) (|zeroDimPrime?| (((|Boolean|) (|PolynomialIdeals| (|Fraction| (|Integer|)) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|OrderedVariableList| |#1|) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{zeroDimPrime?(I)} tests if the ideal \\spad{I} is a 0-dimensional prime."))) NIL NIL -(-441) +(-443 T$) +((|constructor| (NIL "This is the category of all domains that implement idempotent operations."))) +(((|%Rule| |idempotence| (|%Forall| (|%Sequence| (|:| |f| $) (|:| |x| |#1|)) (-3052 (|f| |x| |x|) |x|))) . T)) +NIL +(-444) ((|constructor| (NIL "This domain provides representation for plain identifiers. It differs from Symbol in that it does not support any form of scripting. It is a plain basic data structure. \\blankline")) (|gensym| (($) "\\spad{gensym()} returns a new identifier,{} different from any other identifier in the running system"))) NIL NIL -(-442 A S) +(-445 A S) ((|constructor| (NIL "\\indented{1}{Indexed direct products of abelian groups over an abelian group \\spad{A} of} generators indexed by the ordered set \\spad{S}. All items have finite support: only non-zero terms are stored."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007))))) -(-443 A S) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012))))) +(-446 A S) ((|constructor| (NIL "\\indented{1}{Indexed direct products of abelian monoids over an abelian monoid \\spad{A} of} generators indexed by the ordered set \\spad{S}. All items have finite support. Only non-zero terms are stored."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007))))) -(-444 A S) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012))))) +(-447 A S) ((|constructor| (NIL "This category represents the direct product of some set with respect to an ordered indexing set.")) (|terms| (((|List| (|Pair| |#2| |#1|)) $) "\\spad{terms x} returns the list of terms in \\spad{x}. Each term is a pair of a support (the first component) and the corresponding value (the second component).")) (|reductum| (($ $) "\\spad{reductum(z)} returns a new element created by removing the leading coefficient/support pair from the element \\spad{z}. Error: if \\spad{z} has no support.")) (|leadingSupport| ((|#2| $) "\\spad{leadingSupport(z)} returns the index of leading (with respect to the ordering on the indexing set) monomial of \\spad{z}. Error: if \\spad{z} has no support.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(z)} returns the coefficient of the leading (with respect to the ordering on the indexing set) monomial of \\spad{z}. Error: if \\spad{z} has no support.")) (|monomial| (($ |#1| |#2|) "\\spad{monomial(a,s)} constructs a direct product element with the \\spad{s} component set to \\spad{a}")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,z)} returns the new element created by applying the function \\spad{f} to each component of the direct product element \\spad{z}."))) NIL NIL -(-445 A S) +(-448 A S) ((|constructor| (NIL "Indexed direct products of objects over a set \\spad{A} of generators indexed by an ordered set \\spad{S}. All items have finite support."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007))))) -(-446 A S) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012))))) +(-449 A S) ((|constructor| (NIL "\\indented{1}{Indexed direct products of ordered abelian monoids \\spad{A} of} generators indexed by the ordered set \\spad{S}. The inherited order is lexicographical. All items have finite support: only non-zero terms are stored."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007))))) -(-447 A S) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012))))) +(-450 A S) ((|constructor| (NIL "\\indented{1}{Indexed direct products of ordered abelian monoid sups \\spad{A},{}} generators indexed by the ordered set \\spad{S}. All items have finite support: only non-zero terms are stored."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007))))) -(-448 S A B) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012))))) +(-451 S A B) ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation'' substitutions. The difference between this and \\spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.")) (|eval| (($ $ (|List| |#2|) (|List| |#3|)) "\\spad{eval(f, [x1,...,xn], [v1,...,vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ |#2| |#3|) "\\spad{eval(f, x, v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) NIL NIL -(-449 A B) +(-452 A B) ((|constructor| (NIL "This category provides \\spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation'' substitutions. The difference between this and \\spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.")) (|eval| (($ $ (|List| |#1|) (|List| |#2|)) "\\spad{eval(f, [x1,...,xn], [v1,...,vn])} replaces \\spad{xi} by \\spad{vi} in \\spad{f}.") (($ $ |#1| |#2|) "\\spad{eval(f, x, v)} replaces \\spad{x} by \\spad{v} in \\spad{f}."))) NIL NIL -(-450 S E |un|) +(-453 S E |un|) ((|constructor| (NIL "Internal implementation of a free abelian monoid."))) NIL -((|HasCategory| |#2| (QUOTE (-711)))) -(-451 S |mn|) +((|HasCategory| |#2| (QUOTE (-716)))) +(-454 S |mn|) ((|constructor| (NIL "\\indented{1}{Author: Michael Monagan \\spad{July/87},{} modified SMW \\spad{June/91}} A FlexibleArray is the notion of an array intended to allow for growth at the end only. Hence the following efficient operations \\indented{2}{\\spad{append(x,a)} meaning append item \\spad{x} at the end of the array \\spad{a}} \\indented{2}{\\spad{delete(a,n)} meaning delete the last item from the array \\spad{a}} Flexible arrays support the other operations inherited from \\spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \\spad{O(1)} access time property of arrays with growing and shrinking at the end in \\spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets.")) (|shrinkable| (((|Boolean|) (|Boolean|)) "\\spad{shrinkable(b)} sets the shrinkable attribute of flexible arrays to \\spad{b} and returns the previous value")) (|physicalLength!| (($ $ (|Integer|)) "\\spad{physicalLength!(x,n)} changes the physical length of \\spad{x} to be \\spad{n} and returns the new array.")) (|physicalLength| (((|NonNegativeInteger|) $) "\\spad{physicalLength(x)} returns the number of elements \\spad{x} can accomodate before growing")) (|flexibleArray| (($ (|List| |#1|)) "\\spad{flexibleArray(l)} creates a flexible array from the list of elements \\spad{l}"))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-452) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-455) ((|constructor| (NIL "This domain represents AST for conditional expressions.")) (|elseBranch| (((|SpadAst|) $) "thenBranch(\\spad{e}) returns the `else-branch' of `e'.")) (|thenBranch| (((|SpadAst|) $) "\\spad{thenBranch(e)} returns the `then-branch' of `e'.")) (|condition| (((|SpadAst|) $) "\\spad{condition(e)} returns the condition of the if-expression `e'."))) NIL NIL -(-453 |p| |n|) +(-456 |p| |n|) ((|constructor| (NIL "InnerFiniteField(\\spad{p},{}\\spad{n}) implements finite fields with \\spad{p**n} elements where \\spad{p} is assumed prime but does not check. For a version which checks that \\spad{p} is prime,{} see \\spadtype{FiniteField}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((OR (|HasCategory| (-513 |#1|) (QUOTE (-116))) (|HasCategory| (-513 |#1|) (QUOTE (-315)))) (|HasCategory| (-513 |#1|) (QUOTE (-118))) (|HasCategory| (-513 |#1|) (QUOTE (-315))) (|HasCategory| (-513 |#1|) (QUOTE (-116)))) -(-454 R |mnRow| |mnCol| |Row| |Col|) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((OR (|HasCategory| (-516 |#1|) (QUOTE (-118))) (|HasCategory| (-516 |#1|) (QUOTE (-317)))) (|HasCategory| (-516 |#1|) (QUOTE (-120))) (|HasCategory| (-516 |#1|) (QUOTE (-317))) (|HasCategory| (-516 |#1|) (QUOTE (-118)))) +(-457 R |mnRow| |mnCol| |Row| |Col|) ((|constructor| (NIL "\\indented{1}{This is an internal type which provides an implementation of} 2-dimensional arrays as PrimitiveArray's of PrimitiveArray's."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-455 R |Row| |Col| M) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-458 R |Row| |Col| M) ((|constructor| (NIL "\\spadtype{InnerMatrixLinearAlgebraFunctions} is an internal package which provides standard linear algebra functions on domains in \\spad{MatrixCategory}")) (|inverse| (((|Union| |#4| "failed") |#4|) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|generalizedInverse| ((|#4| |#4|) "\\spad{generalizedInverse(m)} returns the generalized (Moore--Penrose) inverse of the matrix \\spad{m},{} \\spadignore{i.e.} the matrix \\spad{h} such that m*h*m=h,{} h*m*h=m,{} m*h and h*m are both symmetric matrices.")) (|determinant| ((|#1| |#4|) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. an error message is returned if the matrix is not square.")) (|nullSpace| (((|List| |#3|) |#4|) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) |#4|) "\\spad{nullity(m)} returns the mullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) |#4|) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| ((|#4| |#4|) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}."))) NIL -((|HasAttribute| |#3| (QUOTE -3979))) -(-456 R |Row| |Col| M QF |Row2| |Col2| M2) +((|HasAttribute| |#3| (QUOTE -3990))) +(-459 R |Row| |Col| M QF |Row2| |Col2| M2) ((|constructor| (NIL "\\spadtype{InnerMatrixQuotientFieldFunctions} provides functions on matrices over an integral domain which involve the quotient field of that integral domain. The functions rowEchelon and inverse return matrices with entries in the quotient field.")) (|nullSpace| (((|List| |#3|) |#4|) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|inverse| (((|Union| |#8| "failed") |#4|) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square. Note: the result will have entries in the quotient field.")) (|rowEchelon| ((|#8| |#4|) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}. the result will have entries in the quotient field."))) NIL -((|HasAttribute| |#7| (QUOTE -3979))) -(-457 R |mnRow| |mnCol|) +((|HasAttribute| |#7| (QUOTE -3990))) +(-460 R |mnRow| |mnCol|) ((|constructor| (NIL "An \\spad{IndexedMatrix} is a matrix where the minimal row and column indices are parameters of the type. The domains Row and Col are both IndexedVectors. The index of the 'first' row may be obtained by calling the function \\spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \\spadfun{minColIndex}. The index of the first element of a 'Row' is the same as the index of the first column in a matrix and vice versa."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-255))) (|HasCategory| |#1| (QUOTE (-491))) (|HasAttribute| |#1| (QUOTE (-3980 "*"))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-72)))) -(-458) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-257))) (|HasCategory| |#1| (QUOTE (-494))) (|HasAttribute| |#1| (QUOTE (-3991 "*"))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-72)))) +(-461) ((|constructor| (NIL "This domain represents an `import' of types.")) (|imports| (((|List| (|TypeAst|)) $) "\\spad{imports(x)} returns the list of imported types.")) (|coerce| (($ (|List| (|TypeAst|))) "ts::ImportAst constructs an ImportAst for the list if types `ts'."))) NIL NIL -(-459) +(-462) ((|constructor| (NIL "This domain represents the `in' iterator syntax.")) (|sequence| (((|SpadAst|) $) "\\spad{sequence(i)} returns the sequence expression being iterated over by `i'.")) (|iterationVar| (((|Identifier|) $) "\\spad{iterationVar(i)} returns the name of the iterating variable of the `in' iterator 'i'"))) NIL NIL -(-460 S) +(-463 S) ((|constructor| (NIL "This category describes input byte stream conduits.")) (|readBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{readBytes!(c,b)} reads byte sequences from conduit `c' into the byte buffer `b'. The actual number of bytes written is returned,{} and the length of `b' is set to that amount.")) (|readUInt32!| (((|Maybe| (|UInt32|)) $) "\\spad{readUInt32!(cond)} attempts to read a \\spad{UInt32} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt32!| (((|Maybe| (|Int32|)) $) "\\spad{readInt32!(cond)} attempts to read an \\spad{Int32} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt16!| (((|Maybe| (|UInt16|)) $) "\\spad{readUInt16!(cond)} attempts to read a \\spad{UInt16} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt16!| (((|Maybe| (|Int16|)) $) "\\spad{readInt16!(cond)} attempts to read an \\spad{Int16} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt8!| (((|Maybe| (|UInt8|)) $) "\\spad{readUInt8!(cond)} attempts to read a \\spad{UInt8} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt8!| (((|Maybe| (|Int8|)) $) "\\spad{readInt8!(cond)} attempts to read an \\spad{Int8} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readByte!| (((|Maybe| (|Byte|)) $) "\\spad{readByte!(cond)} attempts to read a byte from the input conduit `cond'. Returns the read byte if successful,{} otherwise \\spad{nothing}."))) NIL NIL -(-461) +(-464) ((|constructor| (NIL "This category describes input byte stream conduits.")) (|readBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{readBytes!(c,b)} reads byte sequences from conduit `c' into the byte buffer `b'. The actual number of bytes written is returned,{} and the length of `b' is set to that amount.")) (|readUInt32!| (((|Maybe| (|UInt32|)) $) "\\spad{readUInt32!(cond)} attempts to read a \\spad{UInt32} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt32!| (((|Maybe| (|Int32|)) $) "\\spad{readInt32!(cond)} attempts to read an \\spad{Int32} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt16!| (((|Maybe| (|UInt16|)) $) "\\spad{readUInt16!(cond)} attempts to read a \\spad{UInt16} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt16!| (((|Maybe| (|Int16|)) $) "\\spad{readInt16!(cond)} attempts to read an \\spad{Int16} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readUInt8!| (((|Maybe| (|UInt8|)) $) "\\spad{readUInt8!(cond)} attempts to read a \\spad{UInt8} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readInt8!| (((|Maybe| (|Int8|)) $) "\\spad{readInt8!(cond)} attempts to read an \\spad{Int8} value from the input conduit `cond'. Returns the value if successful,{} otherwise \\spad{nothing}.")) (|readByte!| (((|Maybe| (|Byte|)) $) "\\spad{readByte!(cond)} attempts to read a byte from the input conduit `cond'. Returns the read byte if successful,{} otherwise \\spad{nothing}."))) NIL NIL -(-462 GF) +(-465 GF) ((|constructor| (NIL "InnerNormalBasisFieldFunctions(GF) (unexposed): This package has functions used by every normal basis finite field extension domain.")) (|minimalPolynomial| (((|SparseUnivariatePolynomial| |#1|) (|Vector| |#1|)) "\\spad{minimalPolynomial(x)} \\undocumented{} See \\axiomFunFrom{minimalPolynomial}{FiniteAlgebraicExtensionField}")) (|normalElement| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{normalElement(n)} \\undocumented{} See \\axiomFunFrom{normalElement}{FiniteAlgebraicExtensionField}")) (|basis| (((|Vector| (|Vector| |#1|)) (|PositiveInteger|)) "\\spad{basis(n)} \\undocumented{} See \\axiomFunFrom{basis}{FiniteAlgebraicExtensionField}")) (|normal?| (((|Boolean|) (|Vector| |#1|)) "\\spad{normal?(x)} \\undocumented{} See \\axiomFunFrom{normal?}{FiniteAlgebraicExtensionField}")) (|lookup| (((|PositiveInteger|) (|Vector| |#1|)) "\\spad{lookup(x)} \\undocumented{} See \\axiomFunFrom{lookup}{Finite}")) (|inv| (((|Vector| |#1|) (|Vector| |#1|)) "\\spad{inv x} \\undocumented{} See \\axiomFunFrom{inv}{DivisionRing}")) (|trace| (((|Vector| |#1|) (|Vector| |#1|) (|PositiveInteger|)) "\\spad{trace(x,n)} \\undocumented{} See \\axiomFunFrom{trace}{FiniteAlgebraicExtensionField}")) (|norm| (((|Vector| |#1|) (|Vector| |#1|) (|PositiveInteger|)) "\\spad{norm(x,n)} \\undocumented{} See \\axiomFunFrom{norm}{FiniteAlgebraicExtensionField}")) (/ (((|Vector| |#1|) (|Vector| |#1|) (|Vector| |#1|)) "\\spad{x/y} \\undocumented{} See \\axiomFunFrom{/}{Field}")) (* (((|Vector| |#1|) (|Vector| |#1|) (|Vector| |#1|)) "\\spad{x*y} \\undocumented{} See \\axiomFunFrom{*}{SemiGroup}")) (** (((|Vector| |#1|) (|Vector| |#1|) (|Integer|)) "\\spad{x**n} \\undocumented{} See \\axiomFunFrom{**}{DivisionRing}")) (|qPot| (((|Vector| |#1|) (|Vector| |#1|) (|Integer|)) "\\spad{qPot(v,e)} computes \\spad{v**(q**e)},{} interpreting \\spad{v} as an element of normal basis field,{} \\spad{q} the size of the ground field. This is done by a cyclic \\spad{e}-shift of the vector \\spad{v}.")) (|expPot| (((|Vector| |#1|) (|Vector| |#1|) (|SingleInteger|) (|SingleInteger|)) "\\spad{expPot(v,e,d)} returns the sum from \\spad{i = 0} to \\spad{e - 1} of \\spad{v**(q**i*d)},{} interpreting \\spad{v} as an element of a normal basis field and where \\spad{q} is the size of the ground field. Note: for a description of the algorithm,{} see \\spad{T}.Itoh and \\spad{S}.Tsujii,{} \"A fast algorithm for computing multiplicative inverses in GF(2^m) using normal bases\",{} Information and Computation 78,{} pp.171-177,{} 1988.")) (|repSq| (((|Vector| |#1|) (|Vector| |#1|) (|NonNegativeInteger|)) "\\spad{repSq(v,e)} computes \\spad{v**e} by repeated squaring,{} interpreting \\spad{v} as an element of a normal basis field.")) (|dAndcExp| (((|Vector| |#1|) (|Vector| |#1|) (|NonNegativeInteger|) (|SingleInteger|)) "\\spad{dAndcExp(v,n,k)} computes \\spad{v**e} interpreting \\spad{v} as an element of normal basis field. A divide and conquer algorithm similar to the one from \\spad{D}.\\spad{R}.Stinson,{} \"Some observations on parallel Algorithms for fast exponentiation in GF(2^n)\",{} Siam \\spad{J}. Computation,{} Vol.19,{} No.4,{} pp.711-717,{} August 1990 is used. Argument \\spad{k} is a parameter of this algorithm.")) (|xn| (((|SparseUnivariatePolynomial| |#1|) (|NonNegativeInteger|)) "\\spad{xn(n)} returns the polynomial \\spad{x**n-1}.")) (|pol| (((|SparseUnivariatePolynomial| |#1|) (|Vector| |#1|)) "\\spad{pol(v)} turns the vector \\spad{[v0,...,vn]} into the polynomial \\spad{v0+v1*x+ ... + vn*x**n}.")) (|index| (((|Vector| |#1|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{index(n,m)} is a index function for vectors of length \\spad{n} over the ground field.")) (|random| (((|Vector| |#1|) (|PositiveInteger|)) "\\spad{random(n)} creates a vector over the ground field with random entries.")) (|setFieldInfo| (((|Void|) (|Vector| (|List| (|Record| (|:| |value| |#1|) (|:| |index| (|SingleInteger|))))) |#1|) "\\spad{setFieldInfo(m,p)} initializes the field arithmetic,{} where \\spad{m} is the multiplication table and \\spad{p} is the respective normal element of the ground field GF."))) NIL NIL -(-463) +(-466) ((|constructor| (NIL "This domain provides representation for binary files open for input operations. `Binary' here means that the conduits do not interpret their contents.")) (|position!| (((|SingleInteger|) $ (|SingleInteger|)) "position(\\spad{f},{}\\spad{p}) sets the current byte-position to `i'.")) (|position| (((|SingleInteger|) $) "\\spad{position(f)} returns the current byte-position in the file `f'.")) (|isOpen?| (((|Boolean|) $) "\\spad{isOpen?(ifile)} holds if `ifile' is in open state.")) (|eof?| (((|Boolean|) $) "\\spad{eof?(ifile)} holds when the last read reached end of file.")) (|inputBinaryFile| (($ (|String|)) "\\spad{inputBinaryFile(f)} returns an input conduit obtained by opening the file named by `f' as a binary file.") (($ (|FileName|)) "\\spad{inputBinaryFile(f)} returns an input conduit obtained by opening the file named by `f' as a binary file."))) NIL NIL -(-464 R) +(-467 R) ((|constructor| (NIL "This package provides operations to create incrementing functions.")) (|incrementBy| (((|Mapping| |#1| |#1|) |#1|) "\\spad{incrementBy(n)} produces a function which adds \\spad{n} to whatever argument it is given. For example,{} if {\\spad{f} := increment(\\spad{n})} then \\spad{f x} is \\spad{x+n}.")) (|increment| (((|Mapping| |#1| |#1|)) "\\spad{increment()} produces a function which adds \\spad{1} to whatever argument it is given. For example,{} if {\\spad{f} := increment()} then \\spad{f x} is \\spad{x+1}."))) NIL NIL -(-465 |Varset|) +(-468 |Varset|) ((|constructor| (NIL "\\indented{2}{IndexedExponents of an ordered set of variables gives a representation} for the degree of polynomials in commuting variables. It gives an ordered pairing of non negative integer exponents with variables"))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| (-689) (QUOTE (-1007))))) -(-466 K -3078 |Par|) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| (-694) (QUOTE (-1012))))) +(-469 K -3088 |Par|) ((|constructor| (NIL "This package is the inner package to be used by NumericRealEigenPackage and NumericComplexEigenPackage for the computation of numeric eigenvalues and eigenvectors.")) (|innerEigenvectors| (((|List| (|Record| (|:| |outval| |#2|) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| |#2|))))) (|Matrix| |#1|) |#3| (|Mapping| (|Factored| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|))) "\\spad{innerEigenvectors(m,eps,factor)} computes explicitly the eigenvalues and the correspondent eigenvectors of the matrix \\spad{m}. The parameter \\spad{eps} determines the type of the output,{} \\spad{factor} is the univariate factorizer to br used to reduce the characteristic polynomial into irreducible factors.")) (|solve1| (((|List| |#2|) (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{solve1(pol, eps)} finds the roots of the univariate polynomial polynomial \\spad{pol} to precision eps. If \\spad{K} is \\spad{Fraction Integer} then only the real roots are returned,{} if \\spad{K} is \\spad{Complex Fraction Integer} then all roots are found.")) (|charpol| (((|SparseUnivariatePolynomial| |#1|) (|Matrix| |#1|)) "\\spad{charpol(m)} computes the characteristic polynomial of a matrix \\spad{m} with entries in \\spad{K}. This function returns a polynomial over \\spad{K},{} while the general one (that is in EiegenPackage) returns Fraction \\spad{P} \\spad{K}"))) NIL NIL -(-467) +(-470) NIL NIL NIL -(-468) +(-471) ((|constructor| (NIL "Default infinity signatures for the interpreter; Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989")) (|minusInfinity| (((|OrderedCompletion| (|Integer|))) "\\spad{minusInfinity()} returns minusInfinity.")) (|plusInfinity| (((|OrderedCompletion| (|Integer|))) "\\spad{plusInfinity()} returns plusIinfinity.")) (|infinity| (((|OnePointCompletion| (|Integer|))) "\\spad{infinity()} returns infinity."))) NIL NIL -(-469) +(-472) ((|constructor| (NIL "Domain of parsed forms which can be passed to the interpreter. This is also the interface between algebra code and facilities in the interpreter.")) (|compile| (((|Symbol|) (|Symbol|) (|List| $)) "\\spad{compile(f, [t1,...,tn])} forces the interpreter to compile the function \\spad{f} with signature \\spad{(t1,...,tn) -> ?}. returns the symbol \\spad{f} if successful. Error: if \\spad{f} was not defined beforehand in the interpreter,{} or if the \\spad{ti}'s are not valid types,{} or if the compiler fails.")) (|declare| (((|Symbol|) (|List| $)) "\\spad{declare(t)} returns a name \\spad{f} such that \\spad{f} has been declared to the interpreter to be of type \\spad{t},{} but has not been assigned a value yet. Note: \\spad{t} should be created as \\spad{devaluate(T)\\$Lisp} where \\spad{T} is the actual type of \\spad{f} (this hack is required for the case where \\spad{T} is a mapping type).")) (|parseString| (($ (|String|)) "parseString is the inverse of unparse. It parses a string to InputForm.")) (|unparse| (((|String|) $) "\\spad{unparse(f)} returns a string \\spad{s} such that the parser would transform \\spad{s} to \\spad{f}. Error: if \\spad{f} is not the parsed form of a string.")) (|flatten| (($ $) "\\spad{flatten(s)} returns an input form corresponding to \\spad{s} with all the nested operations flattened to triples using new local variables. If \\spad{s} is a piece of code,{} this speeds up the compilation tremendously later on.")) (|One| (($) "\\spad{1} returns the input form corresponding to 1.")) (|Zero| (($) "\\spad{0} returns the input form corresponding to 0.")) (** (($ $ (|Integer|)) "\\spad{a ** b} returns the input form corresponding to \\spad{a ** b}.") (($ $ (|NonNegativeInteger|)) "\\spad{a ** b} returns the input form corresponding to \\spad{a ** b}.")) (/ (($ $ $) "\\spad{a / b} returns the input form corresponding to \\spad{a / b}.")) (* (($ $ $) "\\spad{a * b} returns the input form corresponding to \\spad{a * b}.")) (+ (($ $ $) "\\spad{a + b} returns the input form corresponding to \\spad{a + b}.")) (|lambda| (($ $ (|List| (|Symbol|))) "\\spad{lambda(code, [x1,...,xn])} returns the input form corresponding to \\spad{(x1,...,xn) +-> code} if \\spad{n > 1},{} or to \\spad{x1 +-> code} if \\spad{n = 1}.")) (|function| (($ $ (|List| (|Symbol|)) (|Symbol|)) "\\spad{function(code, [x1,...,xn], f)} returns the input form corresponding to \\spad{f(x1,...,xn) == code}.")) (|binary| (($ $ (|List| $)) "\\spad{binary(op, [a1,...,an])} returns the input form corresponding to \\spad{a1 op a2 op ... op an}.")) (|convert| (($ (|SExpression|)) "\\spad{convert(s)} makes \\spad{s} into an input form.")) (|interpret| (((|Any|) $) "\\spad{interpret(f)} passes \\spad{f} to the interpreter."))) NIL NIL -(-470 R) +(-473 R) ((|constructor| (NIL "Tools for manipulating input forms.")) (|interpret| ((|#1| (|InputForm|)) "\\spad{interpret(f)} passes \\spad{f} to the interpreter,{} and transforms the result into an object of type \\spad{R}.")) (|packageCall| (((|InputForm|) (|Symbol|)) "\\spad{packageCall(f)} returns the input form corresponding to \\spad{f}\\$\\spad{R}."))) NIL NIL -(-471 |Coef| UTS) +(-474 |Coef| UTS) ((|constructor| (NIL "This package computes infinite products of univariate Taylor series over an integral domain of characteristic 0.")) (|generalInfiniteProduct| ((|#2| |#2| (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| ((|#2| |#2|) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| ((|#2| |#2|) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| ((|#2| |#2|) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) NIL NIL -(-472 K -3078 |Par|) +(-475 K -3088 |Par|) ((|constructor| (NIL "This is an internal package for computing approximate solutions to systems of polynomial equations. The parameter \\spad{K} specifies the coefficient field of the input polynomials and must be either \\spad{Fraction(Integer)} or \\spad{Complex(Fraction Integer)}. The parameter \\spad{F} specifies where the solutions must lie and can be one of the following: \\spad{Float},{} \\spad{Fraction(Integer)},{} \\spad{Complex(Float)},{} \\spad{Complex(Fraction Integer)}. The last parameter specifies the type of the precision operand and must be either \\spad{Fraction(Integer)} or \\spad{Float}.")) (|makeEq| (((|List| (|Equation| (|Polynomial| |#2|))) (|List| |#2|) (|List| (|Symbol|))) "\\spad{makeEq(lsol,lvar)} returns a list of equations formed by corresponding members of \\spad{lvar} and \\spad{lsol}.")) (|innerSolve| (((|List| (|List| |#2|)) (|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|)) |#3|) "\\spad{innerSolve(lnum,lden,lvar,eps)} returns a list of solutions of the system of polynomials \\spad{lnum},{} with the side condition that none of the members of \\spad{lden} vanish identically on any solution. Each solution is expressed as a list corresponding to the list of variables in \\spad{lvar} and with precision specified by \\spad{eps}.")) (|innerSolve1| (((|List| |#2|) (|Polynomial| |#1|) |#3|) "\\spad{innerSolve1(p,eps)} returns the list of the zeros of the polynomial \\spad{p} with precision \\spad{eps}.") (((|List| |#2|) (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{innerSolve1(up,eps)} returns the list of the zeros of the univariate polynomial \\spad{up} with precision \\spad{eps}."))) NIL NIL -(-473 R BP |pMod| |nextMod|) +(-476 R BP |pMod| |nextMod|) ((|reduction| ((|#2| |#2| |#1|) "\\spad{reduction(f,p)} reduces the coefficients of the polynomial \\spad{f} modulo the prime \\spad{p}.")) (|modularGcd| ((|#2| (|List| |#2|)) "\\spad{modularGcd(listf)} computes the gcd of the list of polynomials \\spad{listf} by modular methods.")) (|modularGcdPrimitive| ((|#2| (|List| |#2|)) "\\spad{modularGcdPrimitive(f1,f2)} computes the gcd of the two polynomials \\spad{f1} and \\spad{f2} by modular methods."))) NIL NIL -(-474 OV E R P) +(-477 OV E R P) ((|constructor| (NIL "\\indented{2}{This is an inner package for factoring multivariate polynomials} over various coefficient domains in characteristic 0. The univariate factor operation is passed as a parameter. Multivariate hensel lifting is used to lift the univariate factorization")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|) (|Mapping| (|Factored| (|SparseUnivariatePolynomial| |#3|)) (|SparseUnivariatePolynomial| |#3|))) "\\spad{factor(p,ufact)} factors the multivariate polynomial \\spad{p} by specializing variables and calling the univariate factorizer \\spad{ufact}. \\spad{p} is represented as a univariate polynomial with multivariate coefficients.") (((|Factored| |#4|) |#4| (|Mapping| (|Factored| (|SparseUnivariatePolynomial| |#3|)) (|SparseUnivariatePolynomial| |#3|))) "\\spad{factor(p,ufact)} factors the multivariate polynomial \\spad{p} by specializing variables and calling the univariate factorizer \\spad{ufact}."))) NIL NIL -(-475 K UP |Coef| UTS) +(-478 K UP |Coef| UTS) ((|constructor| (NIL "This package computes infinite products of univariate Taylor series over an arbitrary finite field.")) (|generalInfiniteProduct| ((|#4| |#4| (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| ((|#4| |#4|) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| ((|#4| |#4|) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| ((|#4| |#4|) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) NIL NIL -(-476 |Coef| UTS) +(-479 |Coef| UTS) ((|constructor| (NIL "This package computes infinite products of univariate Taylor series over a field of prime order.")) (|generalInfiniteProduct| ((|#2| |#2| (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| ((|#2| |#2|) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| ((|#2| |#2|) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| ((|#2| |#2|) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) NIL NIL -(-477 R UP) +(-480 R UP) ((|constructor| (NIL "Find the sign of a polynomial around a point or infinity.")) (|signAround| (((|Union| (|Integer|) #1="failed") |#2| |#1| (|Mapping| (|Union| (|Integer|) #1#) |#1|)) "\\spad{signAround(u,r,f)} \\undocumented") (((|Union| (|Integer|) #1#) |#2| |#1| (|Integer|) (|Mapping| (|Union| (|Integer|) #1#) |#1|)) "\\spad{signAround(u,r,i,f)} \\undocumented") (((|Union| (|Integer|) #1#) |#2| (|Integer|) (|Mapping| (|Union| (|Integer|) #1#) |#1|)) "\\spad{signAround(u,i,f)} \\undocumented"))) NIL NIL -(-478 S) +(-481 S) ((|constructor| (NIL "An \\spad{IntegerNumberSystem} is a model for the integers.")) (|invmod| (($ $ $) "\\spad{invmod(a,b)},{} \\spad{0<=a<b>1},{} \\spad{(a,b)=1} means \\spad{1/a mod b}.")) (|powmod| (($ $ $ $) "\\spad{powmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a**b mod p}.")) (|mulmod| (($ $ $ $) "\\spad{mulmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a*b mod p}.")) (|submod| (($ $ $ $) "\\spad{submod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a-b mod p}.")) (|addmod| (($ $ $ $) "\\spad{addmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a+b mod p}.")) (|mask| (($ $) "\\spad{mask(n)} returns \\spad{2**n-1} (an \\spad{n} bit mask).")) (|dec| (($ $) "\\spad{dec(x)} returns \\spad{x - 1}.")) (|inc| (($ $) "\\spad{inc(x)} returns \\spad{x + 1}.")) (|copy| (($ $) "\\spad{copy(n)} gives a copy of \\spad{n}.")) (|random| (($ $) "\\spad{random(a)} creates a random element from 0 to \\spad{a-1}.") (($) "\\spad{random()} creates a random element.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(n)} creates a rational number,{} or returns \"failed\" if this is not possible.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(n)} creates a rational number (see \\spadtype{Fraction Integer})..")) (|rational?| (((|Boolean|) $) "\\spad{rational?(n)} tests if \\spad{n} is a rational number (see \\spadtype{Fraction Integer}).")) (|symmetricRemainder| (($ $ $) "\\spad{symmetricRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{ -b/2 <= r < b/2 }.")) (|positiveRemainder| (($ $ $) "\\spad{positiveRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{0 <= r < b} and \\spad{r == a rem b}.")) (|bit?| (((|Boolean|) $ $) "\\spad{bit?(n,i)} returns \\spad{true} if and only if \\spad{i}-th bit of \\spad{n} is a 1.")) (|shift| (($ $ $) "\\spad{shift(a,i)} shift \\spad{a} by \\spad{i} digits.")) (|length| (($ $) "\\spad{length(a)} length of \\spad{a} in digits.")) (|base| (($) "\\spad{base()} returns the base for the operations of \\spad{IntegerNumberSystem}.")) (|multiplicativeValuation| ((|attribute|) "euclideanSize(a*b) returns \\spad{euclideanSize(a)*euclideanSize(b)}.")) (|even?| (((|Boolean|) $) "\\spad{even?(n)} returns \\spad{true} if and only if \\spad{n} is even.")) (|odd?| (((|Boolean|) $) "\\spad{odd?(n)} returns \\spad{true} if and only if \\spad{n} is odd."))) NIL NIL -(-479) +(-482) ((|constructor| (NIL "An \\spad{IntegerNumberSystem} is a model for the integers.")) (|invmod| (($ $ $) "\\spad{invmod(a,b)},{} \\spad{0<=a<b>1},{} \\spad{(a,b)=1} means \\spad{1/a mod b}.")) (|powmod| (($ $ $ $) "\\spad{powmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a**b mod p}.")) (|mulmod| (($ $ $ $) "\\spad{mulmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a*b mod p}.")) (|submod| (($ $ $ $) "\\spad{submod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a-b mod p}.")) (|addmod| (($ $ $ $) "\\spad{addmod(a,b,p)},{} \\spad{0<=a,b<p>1},{} means \\spad{a+b mod p}.")) (|mask| (($ $) "\\spad{mask(n)} returns \\spad{2**n-1} (an \\spad{n} bit mask).")) (|dec| (($ $) "\\spad{dec(x)} returns \\spad{x - 1}.")) (|inc| (($ $) "\\spad{inc(x)} returns \\spad{x + 1}.")) (|copy| (($ $) "\\spad{copy(n)} gives a copy of \\spad{n}.")) (|random| (($ $) "\\spad{random(a)} creates a random element from 0 to \\spad{a-1}.") (($) "\\spad{random()} creates a random element.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(n)} creates a rational number,{} or returns \"failed\" if this is not possible.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(n)} creates a rational number (see \\spadtype{Fraction Integer})..")) (|rational?| (((|Boolean|) $) "\\spad{rational?(n)} tests if \\spad{n} is a rational number (see \\spadtype{Fraction Integer}).")) (|symmetricRemainder| (($ $ $) "\\spad{symmetricRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{ -b/2 <= r < b/2 }.")) (|positiveRemainder| (($ $ $) "\\spad{positiveRemainder(a,b)} (where \\spad{b > 1}) yields \\spad{r} where \\spad{0 <= r < b} and \\spad{r == a rem b}.")) (|bit?| (((|Boolean|) $ $) "\\spad{bit?(n,i)} returns \\spad{true} if and only if \\spad{i}-th bit of \\spad{n} is a 1.")) (|shift| (($ $ $) "\\spad{shift(a,i)} shift \\spad{a} by \\spad{i} digits.")) (|length| (($ $) "\\spad{length(a)} length of \\spad{a} in digits.")) (|base| (($) "\\spad{base()} returns the base for the operations of \\spad{IntegerNumberSystem}.")) (|multiplicativeValuation| ((|attribute|) "euclideanSize(a*b) returns \\spad{euclideanSize(a)*euclideanSize(b)}.")) (|even?| (((|Boolean|) $) "\\spad{even?(n)} returns \\spad{true} if and only if \\spad{n} is even.")) (|odd?| (((|Boolean|) $) "\\spad{odd?(n)} returns \\spad{true} if and only if \\spad{n} is odd."))) -((-3976 . T) (-3977 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3987 . T) (-3988 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-480) +(-483) ((|constructor| (NIL "\\spadtype{Integer} provides the domain of arbitrary precision integers.")) (|noetherian| ((|attribute|) "ascending chain condition on ideals.")) (|canonicalsClosed| ((|attribute|) "two positives multiply to give positive.")) (|canonical| ((|attribute|) "mathematical equality is data structure equality."))) -((-3966 . T) (-3970 . T) (-3965 . T) (-3976 . T) (-3977 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3977 . T) (-3981 . T) (-3976 . T) (-3987 . T) (-3988 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-481) +(-484) ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 16 bits."))) NIL NIL -(-482) +(-485) ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 32 bits."))) NIL NIL -(-483) +(-486) ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 64 bits."))) NIL NIL -(-484) +(-487) ((|constructor| (NIL "This domain is a datatype for (signed) integer values of precision 8 bits."))) NIL NIL -(-485 |Key| |Entry| |addDom|) +(-488 |Key| |Entry| |addDom|) ((|constructor| (NIL "This domain is used to provide a conditional \"add\" domain for the implementation of \\spadtype{Table}."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-1007))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) -(-486 R -3078) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-1012))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) +(-489 R -3088) ((|constructor| (NIL "This package provides functions for the integration of algebraic integrands over transcendental functions.")) (|algint| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|SparseUnivariatePolynomial| |#2|) (|SparseUnivariatePolynomial| |#2|))) "\\spad{algint(f, x, y, d)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x}; \\spad{d} is the derivation to use on \\spad{k[x]}."))) NIL NIL -(-487 R0 -3078 UP UPUP R) +(-490 R0 -3088 UP UPUP R) ((|constructor| (NIL "This package provides functions for integrating a function on an algebraic curve.")) (|palginfieldint| (((|Union| |#5| "failed") |#5| (|Mapping| |#3| |#3|)) "\\spad{palginfieldint(f, d)} returns an algebraic function \\spad{g} such that \\spad{dg = f} if such a \\spad{g} exists,{} \"failed\" otherwise. Argument \\spad{f} must be a pure algebraic function.")) (|palgintegrate| (((|IntegrationResult| |#5|) |#5| (|Mapping| |#3| |#3|)) "\\spad{palgintegrate(f, d)} integrates \\spad{f} with respect to the derivation \\spad{d}. Argument \\spad{f} must be a pure algebraic function.")) (|algintegrate| (((|IntegrationResult| |#5|) |#5| (|Mapping| |#3| |#3|)) "\\spad{algintegrate(f, d)} integrates \\spad{f} with respect to the derivation \\spad{d}."))) NIL NIL -(-488) +(-491) ((|constructor| (NIL "This package provides functions to lookup bits in integers")) (|bitTruth| (((|Boolean|) (|Integer|) (|Integer|)) "\\spad{bitTruth(n,m)} returns \\spad{true} if coefficient of 2**m in abs(\\spad{n}) is 1")) (|bitCoef| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{bitCoef(n,m)} returns the coefficient of 2**m in abs(\\spad{n})")) (|bitLength| (((|Integer|) (|Integer|)) "\\spad{bitLength(n)} returns the number of bits to represent abs(\\spad{n})"))) NIL NIL -(-489 R) +(-492 R) ((|constructor| (NIL "\\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This category implements of interval arithmetic and transcendental + functions over intervals.")) (|contains?| (((|Boolean|) $ |#1|) "\\spad{contains?(i,f)} returns \\spad{true} if \\axiom{\\spad{f}} is contained within the interval \\axiom{\\spad{i}},{} \\spad{false} otherwise.")) (|negative?| (((|Boolean|) $) "\\spad{negative?(u)} returns \\axiom{\\spad{true}} if every element of \\spad{u} is negative,{} \\axiom{\\spad{false}} otherwise.")) (|positive?| (((|Boolean|) $) "\\spad{positive?(u)} returns \\axiom{\\spad{true}} if every element of \\spad{u} is positive,{} \\axiom{\\spad{false}} otherwise.")) (|width| ((|#1| $) "\\spad{width(u)} returns \\axiom{sup(\\spad{u}) - inf(\\spad{u})}.")) (|sup| ((|#1| $) "\\spad{sup(u)} returns the supremum of \\axiom{\\spad{u}}.")) (|inf| ((|#1| $) "\\spad{inf(u)} returns the infinum of \\axiom{\\spad{u}}.")) (|qinterval| (($ |#1| |#1|) "\\spad{qinterval(inf,sup)} creates a new interval \\axiom{[\\spad{inf},{}\\spad{sup}]},{} without checking the ordering on the elements.")) (|interval| (($ (|Fraction| (|Integer|))) "\\spad{interval(f)} creates a new interval around \\spad{f}.") (($ |#1|) "\\spad{interval(f)} creates a new interval around \\spad{f}.") (($ |#1| |#1|) "\\spad{interval(inf,sup)} creates a new interval,{} either \\axiom{[\\spad{inf},{}\\spad{sup}]} if \\axiom{\\spad{inf} <= \\spad{sup}} or \\axiom{[\\spad{sup},{}in]} otherwise."))) -((-3753 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3764 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-490 S) +(-493 S) ((|constructor| (NIL "The category of commutative integral domains,{} \\spadignore{i.e.} commutative rings with no zero divisors. \\blankline Conditional attributes: \\indented{2}{canonicalUnitNormal\\tab{20}the canonical field is the same for all associates} \\indented{2}{canonicalsClosed\\tab{20}the product of two canonicals is itself canonical}")) (|unit?| (((|Boolean|) $) "\\spad{unit?(x)} tests whether \\spad{x} is a unit,{} \\spadignore{i.e.} is invertible.")) (|associates?| (((|Boolean|) $ $) "\\spad{associates?(x,y)} tests whether \\spad{x} and \\spad{y} are associates,{} \\spadignore{i.e.} differ by a unit factor.")) (|unitCanonical| (($ $) "\\spad{unitCanonical(x)} returns \\spad{unitNormal(x).canonical}.")) (|unitNormal| (((|Record| (|:| |unit| $) (|:| |canonical| $) (|:| |associate| $)) $) "\\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \\spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the \"canonical\" element is the same across all associates of \\spad{x} if \\spad{unitNormal(x) = [u,c,a]} then \\spad{u*c = x},{} \\spad{a*u = 1}.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} either returns an element \\spad{c} such that \\spad{c*b=a} or \"failed\" if no such element can be found."))) NIL NIL -(-491) +(-494) ((|constructor| (NIL "The category of commutative integral domains,{} \\spadignore{i.e.} commutative rings with no zero divisors. \\blankline Conditional attributes: \\indented{2}{canonicalUnitNormal\\tab{20}the canonical field is the same for all associates} \\indented{2}{canonicalsClosed\\tab{20}the product of two canonicals is itself canonical}")) (|unit?| (((|Boolean|) $) "\\spad{unit?(x)} tests whether \\spad{x} is a unit,{} \\spadignore{i.e.} is invertible.")) (|associates?| (((|Boolean|) $ $) "\\spad{associates?(x,y)} tests whether \\spad{x} and \\spad{y} are associates,{} \\spadignore{i.e.} differ by a unit factor.")) (|unitCanonical| (($ $) "\\spad{unitCanonical(x)} returns \\spad{unitNormal(x).canonical}.")) (|unitNormal| (((|Record| (|:| |unit| $) (|:| |canonical| $) (|:| |associate| $)) $) "\\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \\spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the \"canonical\" element is the same across all associates of \\spad{x} if \\spad{unitNormal(x) = [u,c,a]} then \\spad{u*c = x},{} \\spad{a*u = 1}.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} either returns an element \\spad{c} such that \\spad{c*b=a} or \"failed\" if no such element can be found."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-492 R -3078) +(-495 R -3088) ((|constructor| (NIL "This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for elemntary functions.")) (|lfextlimint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #1="failed") |#2| (|Symbol|) (|Kernel| |#2|) (|List| (|Kernel| |#2|))) "\\spad{lfextlimint(f,x,k,[k1,...,kn])} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f - c dk/dx}. Value \\spad{h} is looked for in a field containing \\spad{f} and \\spad{k1},{}...,{}kn (the \\spad{ki}'s must be logs).")) (|lfintegrate| (((|IntegrationResult| |#2|) |#2| (|Symbol|)) "\\spad{lfintegrate(f, x)} = \\spad{g} such that \\spad{dg/dx = f}.")) (|lfinfieldint| (((|Union| |#2| "failed") |#2| (|Symbol|)) "\\spad{lfinfieldint(f, x)} returns a function \\spad{g} such that \\spad{dg/dx = f} if \\spad{g} exists,{} \"failed\" otherwise.")) (|lflimitedint| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Symbol|) (|List| |#2|)) "\\spad{lflimitedint(f,x,[g1,...,gn])} returns functions \\spad{[h,[[ci, gi]]]} such that the \\spad{gi}'s are among \\spad{[g1,...,gn]},{} and \\spad{d(h+sum(ci log(gi)))/dx = f},{} if possible,{} \"failed\" otherwise.")) (|lfextendedint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #1#) |#2| (|Symbol|) |#2|) "\\spad{lfextendedint(f, x, g)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f - cg},{} if (\\spad{h},{} \\spad{c}) exist,{} \"failed\" otherwise."))) NIL NIL -(-493 I) +(-496 I) ((|constructor| (NIL "\\indented{1}{This Package contains basic methods for integer factorization.} The factor operation employs trial division up to 10,{}000. It then tests to see if \\spad{n} is a perfect power before using Pollards rho method. Because Pollards method may fail,{} the result of factor may contain composite factors. We should also employ Lenstra's eliptic curve method.")) (|PollardSmallFactor| (((|Union| |#1| "failed") |#1|) "\\spad{PollardSmallFactor(n)} returns a factor of \\spad{n} or \"failed\" if no one is found")) (|BasicMethod| (((|Factored| |#1|) |#1|) "\\spad{BasicMethod(n)} returns the factorization of integer \\spad{n} by trial division")) (|squareFree| (((|Factored| |#1|) |#1|) "\\spad{squareFree(n)} returns the square free factorization of integer \\spad{n}")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(n)} returns the full factorization of integer \\spad{n}"))) NIL NIL -(-494 R -3078 L) +(-497 R -3088 L) ((|constructor| (NIL "This internal package rationalises integrands on curves of the form: \\indented{2}{\\spad{y\\^2 = a x\\^2 + b x + c}} \\indented{2}{\\spad{y\\^2 = (a x + b) / (c x + d)}} \\indented{2}{\\spad{f(x, y) = 0} where \\spad{f} has degree 1 in \\spad{x}} The rationalization is done for integration,{} limited integration,{} extended integration and the risch differential equation.")) (|palgLODE0| (((|Record| (|:| |particular| (|Union| |#2| #1="failed")) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgLODE0(op,g,x,y,z,t,c)} returns the solution of \\spad{op f = g} Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}.") (((|Record| (|:| |particular| (|Union| |#2| #1#)) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgLODE0(op, g, x, y, d, p)} returns the solution of \\spad{op f = g}. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}.")) (|lift| (((|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) (|SparseUnivariatePolynomial| |#2|) (|Kernel| |#2|)) "\\spad{lift(u,k)} \\undocumented")) (|multivariate| ((|#2| (|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) (|Kernel| |#2|) |#2|) "\\spad{multivariate(u,k,f)} \\undocumented")) (|univariate| (((|SparseUnivariatePolynomial| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|SparseUnivariatePolynomial| |#2|)) "\\spad{univariate(f,k,k,p)} \\undocumented")) (|palgRDE0| (((|Union| |#2| #2="failed") |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| #2#) |#2| |#2| (|Symbol|)) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgRDE0(f, g, x, y, foo, t, c)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{foo},{} called by \\spad{foo(a, b, x)},{} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}.") (((|Union| |#2| #2#) |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| #2#) |#2| |#2| (|Symbol|)) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgRDE0(f, g, x, y, foo, d, p)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}. Argument \\spad{foo},{} called by \\spad{foo(a, b, x)},{} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}.")) (|palglimint0| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) #3="failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palglimint0(f, x, y, [u1,...,un], z, t, c)} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}'s are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}.") (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) #3#) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palglimint0(f, x, y, [u1,...,un], d, p)} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}'s are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} and \"failed\" otherwise. Argument \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2y(x)\\^2 = P(x)}.")) (|palgextint0| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #4="failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgextint0(f, x, y, g, z, t, c)} returns functions \\spad{[h, d]} such that \\spad{dh/dx = f(x,y) - d g},{} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy},{} and \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{z} is a dummy variable not appearing in \\spad{f(x,y)}. The operation returns \"failed\" if no such functions exist.") (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) #4#) |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgextint0(f, x, y, g, d, p)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f(x,y) - c g},{} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2 y(x)\\^2 = P(x)},{} or \"failed\" if no such functions exist.")) (|palgint0| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|Fraction| (|SparseUnivariatePolynomial| |#2|))) "\\spad{palgint0(f, x, y, z, t, c)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{f(x,y)dx = c f(t,y) dy}; \\spad{c} and \\spad{t} are rational functions of \\spad{y}. Argument \\spad{z} is a dummy variable not appearing in \\spad{f(x,y)}.") (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) "\\spad{palgint0(f, x, y, d, p)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x} satisfying \\spad{d(x)\\^2 y(x)\\^2 = P(x)}."))) NIL -((|HasCategory| |#3| (|%list| (QUOTE -597) (|devaluate| |#2|)))) -(-495) +((|HasCategory| |#3| (|%list| (QUOTE -600) (|devaluate| |#2|)))) +(-498) ((|constructor| (NIL "This package provides various number theoretic functions on the integers.")) (|sumOfKthPowerDivisors| (((|Integer|) (|Integer|) (|NonNegativeInteger|)) "\\spad{sumOfKthPowerDivisors(n,k)} returns the sum of the \\spad{k}th powers of the integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. the sum of the \\spad{k}th powers of the divisors of \\spad{n} is often denoted by \\spad{sigma_k(n)}.")) (|sumOfDivisors| (((|Integer|) (|Integer|)) "\\spad{sumOfDivisors(n)} returns the sum of the integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. The sum of the divisors of \\spad{n} is often denoted by \\spad{sigma(n)}.")) (|numberOfDivisors| (((|Integer|) (|Integer|)) "\\spad{numberOfDivisors(n)} returns the number of integers between 1 and \\spad{n} (inclusive) which divide \\spad{n}. The number of divisors of \\spad{n} is often denoted by \\spad{tau(n)}.")) (|moebiusMu| (((|Integer|) (|Integer|)) "\\spad{moebiusMu(n)} returns the Moebius function \\spad{mu(n)}. \\spad{mu(n)} is either \\spad{-1},{}0 or 1 as follows: \\spad{mu(n) = 0} if \\spad{n} is divisible by a square > 1,{} \\spad{mu(n) = (-1)^k} if \\spad{n} is square-free and has \\spad{k} distinct prime divisors.")) (|legendre| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{legendre(a,p)} returns the Legendre symbol \\spad{L(a/p)}. \\spad{L(a/p) = (-1)**((p-1)/2) mod p} (\\spad{p} prime),{} which is 0 if \\spad{a} is 0,{} 1 if \\spad{a} is a quadratic residue \\spad{mod p} and \\spad{-1} otherwise. Note: because the primality test is expensive,{} if it is known that \\spad{p} is prime then use \\spad{jacobi(a,p)}.")) (|jacobi| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{jacobi(a,b)} returns the Jacobi symbol \\spad{J(a/b)}. When \\spad{b} is odd,{} \\spad{J(a/b) = product(L(a/p) for p in factor b )}. Note: by convention,{} 0 is returned if \\spad{gcd(a,b) ~= 1}. Iterative \\spad{O(log(b)^2)} version coded by Michael Monagan June 1987.")) (|harmonic| (((|Fraction| (|Integer|)) (|Integer|)) "\\spad{harmonic(n)} returns the \\spad{n}th harmonic number. This is \\spad{H[n] = sum(1/k,k=1..n)}.")) (|fibonacci| (((|Integer|) (|Integer|)) "\\spad{fibonacci(n)} returns the \\spad{n}th Fibonacci number. the Fibonacci numbers \\spad{F[n]} are defined by \\spad{F[0] = F[1] = 1} and \\spad{F[n] = F[n-1] + F[n-2]}. The algorithm has running time \\spad{O(log(n)^3)}. Reference: Knuth,{} The Art of Computer Programming Vol 2,{} Semi-Numerical Algorithms.")) (|eulerPhi| (((|Integer|) (|Integer|)) "\\spad{eulerPhi(n)} returns the number of integers between 1 and \\spad{n} (including 1) which are relatively prime to \\spad{n}. This is the Euler phi function \\spad{\\phi(n)} is also called the totient function.")) (|euler| (((|Integer|) (|Integer|)) "\\spad{euler(n)} returns the \\spad{n}th Euler number. This is \\spad{2^n E(n,1/2)},{} where \\spad{E(n,x)} is the \\spad{n}th Euler polynomial.")) (|divisors| (((|List| (|Integer|)) (|Integer|)) "\\spad{divisors(n)} returns a list of the divisors of \\spad{n}.")) (|chineseRemainder| (((|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{chineseRemainder(x1,m1,x2,m2)} returns \\spad{w},{} where \\spad{w} is such that \\spad{w = x1 mod m1} and \\spad{w = x2 mod m2}. Note: \\spad{m1} and \\spad{m2} must be relatively prime.")) (|bernoulli| (((|Fraction| (|Integer|)) (|Integer|)) "\\spad{bernoulli(n)} returns the \\spad{n}th Bernoulli number. this is \\spad{B(n,0)},{} where \\spad{B(n,x)} is the \\spad{n}th Bernoulli polynomial."))) NIL NIL -(-496 -3078 UP UPUP R) +(-499 -3088 UP UPUP R) ((|constructor| (NIL "algebraic Hermite redution.")) (|HermiteIntegrate| (((|Record| (|:| |answer| |#4|) (|:| |logpart| |#4|)) |#4| (|Mapping| |#2| |#2|)) "\\spad{HermiteIntegrate(f, ')} returns \\spad{[g,h]} such that \\spad{f = g' + h} and \\spad{h} has a only simple finite normal poles."))) NIL NIL -(-497 -3078 UP) +(-500 -3088 UP) ((|constructor| (NIL "Hermite integration,{} transcendental case.")) (|HermiteIntegrate| (((|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |logpart| (|Fraction| |#2|)) (|:| |specpart| (|Fraction| |#2|)) (|:| |polypart| |#2|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{HermiteIntegrate(f, D)} returns \\spad{[g, h, s, p]} such that \\spad{f = Dg + h + s + p},{} \\spad{h} has a squarefree denominator normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and all the squarefree factors of the denominator of \\spad{s} are special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D}. Furthermore,{} \\spad{h} and \\spad{s} have no polynomial parts. \\spad{D} is the derivation to use on \\spadtype{UP}."))) NIL NIL -(-498 R -3078 L) +(-501 R -3088 L) ((|constructor| (NIL "This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for pure algebraic integrands.")) (|palgLODE| (((|Record| (|:| |particular| (|Union| |#2| #1="failed")) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Symbol|)) "\\spad{palgLODE(op, g, kx, y, x)} returns the solution of \\spad{op f = g}. \\spad{y} is an algebraic function of \\spad{x}.")) (|palgRDE| (((|Union| |#2| #1#) |#2| |#2| |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|Mapping| (|Union| |#2| #1#) |#2| |#2| (|Symbol|))) "\\spad{palgRDE(nfp, f, g, x, y, foo)} returns a function \\spad{z(x,y)} such that \\spad{dz/dx + n * df/dx z(x,y) = g(x,y)} if such a \\spad{z} exists,{} \"failed\" otherwise; \\spad{y} is an algebraic function of \\spad{x}; \\spad{foo(a, b, x)} is a function that solves \\spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \\spad{u(x)} not involving \\spad{y}. \\spad{nfp} is \\spad{n * df/dx}.")) (|palglimint| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) (|List| |#2|)) "\\spad{palglimint(f, x, y, [u1,...,un])} returns functions \\spad{[h,[[ci, ui]]]} such that the \\spad{ui}'s are among \\spad{[u1,...,un]} and \\spad{d(h + sum(ci log(ui)))/dx = f(x,y)} if such functions exist,{} \"failed\" otherwise; \\spad{y} is an algebraic function of \\spad{x}.")) (|palgextint| (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| (|Kernel| |#2|) (|Kernel| |#2|) |#2|) "\\spad{palgextint(f, x, y, g)} returns functions \\spad{[h, c]} such that \\spad{dh/dx = f(x,y) - c g},{} where \\spad{y} is an algebraic function of \\spad{x}; returns \"failed\" if no such functions exist.")) (|palgint| (((|IntegrationResult| |#2|) |#2| (|Kernel| |#2|) (|Kernel| |#2|)) "\\spad{palgint(f, x, y)} returns the integral of \\spad{f(x,y)dx} where \\spad{y} is an algebraic function of \\spad{x}."))) NIL -((|HasCategory| |#3| (|%list| (QUOTE -597) (|devaluate| |#2|)))) -(-499 R -3078) +((|HasCategory| |#3| (|%list| (QUOTE -600) (|devaluate| |#2|)))) +(-502 R -3088) ((|constructor| (NIL "\\spadtype{PatternMatchIntegration} provides functions that use the pattern matcher to find some indefinite and definite integrals involving special functions and found in the litterature.")) (|pmintegrate| (((|Union| |#2| "failed") |#2| (|Symbol|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|)) "\\spad{pmintegrate(f, x = a..b)} returns the integral of \\spad{f(x)dx} from a to \\spad{b} if it can be found by the built-in pattern matching rules.") (((|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|)) "\\spad{pmintegrate(f, x)} returns either \"failed\" or \\spad{[g,h]} such that \\spad{integrate(f,x) = g + integrate(h,x)}.")) (|pmComplexintegrate| (((|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|)) "\\spad{pmComplexintegrate(f, x)} returns either \"failed\" or \\spad{[g,h]} such that \\spad{integrate(f,x) = g + integrate(h,x)}. It only looks for special complex integrals that pmintegrate does not return.")) (|splitConstant| (((|Record| (|:| |const| |#2|) (|:| |nconst| |#2|)) |#2| (|Symbol|)) "\\spad{splitConstant(f, x)} returns \\spad{[c, g]} such that \\spad{f = c * g} and \\spad{c} does not involve \\spad{t}."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| |#2| (QUOTE (-1044)))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| |#2| (QUOTE (-566))))) -(-500 -3078 UP) +((-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| |#2| (QUOTE (-1051)))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| |#2| (QUOTE (-569))))) +(-503 -3088 UP) ((|constructor| (NIL "This package provides functions for the base case of the Risch algorithm.")) (|limitedint| (((|Union| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|)))))) "failed") (|Fraction| |#2|) (|List| (|Fraction| |#2|))) "\\spad{limitedint(f, [g1,...,gn])} returns fractions \\spad{[h,[[ci, gi]]]} such that the \\spad{gi}'s are among \\spad{[g1,...,gn]},{} \\spad{ci' = 0},{} and \\spad{(h+sum(ci log(gi)))' = f},{} if possible,{} \"failed\" otherwise.")) (|extendedint| (((|Union| (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{extendedint(f, g)} returns fractions \\spad{[h, c]} such that \\spad{c' = 0} and \\spad{h' = f - cg},{} if \\spad{(h, c)} exist,{} \"failed\" otherwise.")) (|infieldint| (((|Union| (|Fraction| |#2|) "failed") (|Fraction| |#2|)) "\\spad{infieldint(f)} returns \\spad{g} such that \\spad{g' = f} or \"failed\" if the integral of \\spad{f} is not a rational function.")) (|integrate| (((|IntegrationResult| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{integrate(f)} returns \\spad{g} such that \\spad{g' = f}."))) NIL NIL -(-501 S) +(-504 S) ((|constructor| (NIL "Provides integer testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991")) (|integerIfCan| (((|Union| (|Integer|) "failed") |#1|) "\\spad{integerIfCan(x)} returns \\spad{x} as an integer,{} \"failed\" if \\spad{x} is not an integer.")) (|integer?| (((|Boolean|) |#1|) "\\spad{integer?(x)} is \\spad{true} if \\spad{x} is an integer,{} \\spad{false} otherwise.")) (|integer| (((|Integer|) |#1|) "\\spad{integer(x)} returns \\spad{x} as an integer; error if \\spad{x} is not an integer."))) NIL NIL -(-502 -3078) +(-505 -3088) ((|constructor| (NIL "This package provides functions for the integration of rational functions.")) (|extendedIntegrate| (((|Union| (|Record| (|:| |ratpart| (|Fraction| (|Polynomial| |#1|))) (|:| |coeff| (|Fraction| (|Polynomial| |#1|)))) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|Fraction| (|Polynomial| |#1|))) "\\spad{extendedIntegrate(f, x, g)} returns fractions \\spad{[h, c]} such that \\spad{dc/dx = 0} and \\spad{dh/dx = f - cg},{} if \\spad{(h, c)} exist,{} \"failed\" otherwise.")) (|limitedIntegrate| (((|Union| (|Record| (|:| |mainpart| (|Fraction| (|Polynomial| |#1|))) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| (|Polynomial| |#1|))) (|:| |logand| (|Fraction| (|Polynomial| |#1|))))))) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{limitedIntegrate(f, x, [g1,...,gn])} returns fractions \\spad{[h, [[ci,gi]]]} such that the \\spad{gi}'s are among \\spad{[g1,...,gn]},{} \\spad{dci/dx = 0},{} and \\spad{d(h + sum(ci log(gi)))/dx = f} if possible,{} \"failed\" otherwise.")) (|infieldIntegrate| (((|Union| (|Fraction| (|Polynomial| |#1|)) "failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{infieldIntegrate(f, x)} returns a fraction \\spad{g} such that \\spad{dg/dx = f} if \\spad{g} exists,{} \"failed\" otherwise.")) (|internalIntegrate| (((|IntegrationResult| (|Fraction| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{internalIntegrate(f, x)} returns \\spad{g} such that \\spad{dg/dx = f}."))) NIL NIL -(-503 R) +(-506 R) ((|constructor| (NIL "\\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This domain is an implementation of interval arithmetic and transcendental + functions over intervals."))) -((-3753 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3764 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-504) +(-507) ((|constructor| (NIL "This package provides the implementation for the \\spadfun{solveLinearPolynomialEquation} operation over the integers. It uses a lifting technique from the package GenExEuclid")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| (|Integer|))) "failed") (|List| (|SparseUnivariatePolynomial| (|Integer|))) (|SparseUnivariatePolynomial| (|Integer|))) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}'s exists."))) NIL NIL -(-505 R -3078) +(-508 R -3088) ((|constructor| (NIL "\\indented{1}{Tools for the integrator} Author: Manuel Bronstein Date Created: 25 April 1990 Date Last Updated: 9 June 1993 Keywords: elementary,{} function,{} integration.")) (|intPatternMatch| (((|IntegrationResult| |#2|) |#2| (|Symbol|) (|Mapping| (|IntegrationResult| |#2|) |#2| (|Symbol|)) (|Mapping| (|Union| (|Record| (|:| |special| |#2|) (|:| |integrand| |#2|)) "failed") |#2| (|Symbol|))) "\\spad{intPatternMatch(f, x, int, pmint)} tries to integrate \\spad{f} first by using the integration function \\spad{int},{} and then by using the pattern match intetgration function \\spad{pmint} on any remaining unintegrable part.")) (|mkPrim| ((|#2| |#2| (|Symbol|)) "\\spad{mkPrim(f, x)} makes the logs in \\spad{f} which are linear in \\spad{x} primitive with respect to \\spad{x}.")) (|removeConstantTerm| ((|#2| |#2| (|Symbol|)) "\\spad{removeConstantTerm(f, x)} returns \\spad{f} minus any additive constant with respect to \\spad{x}.")) (|vark| (((|List| (|Kernel| |#2|)) (|List| |#2|) (|Symbol|)) "\\spad{vark([f1,...,fn],x)} returns the set-theoretic union of \\spad{(varselect(f1,x),...,varselect(fn,x))}.")) (|union| (((|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|))) "\\spad{union(l1, l2)} returns set-theoretic union of \\spad{l1} and \\spad{l2}.")) (|ksec| (((|Kernel| |#2|) (|Kernel| |#2|) (|List| (|Kernel| |#2|)) (|Symbol|)) "\\spad{ksec(k, [k1,...,kn], x)} returns the second top-level \\spad{ki} after \\spad{k} involving \\spad{x}.")) (|kmax| (((|Kernel| |#2|) (|List| (|Kernel| |#2|))) "\\spad{kmax([k1,...,kn])} returns the top-level \\spad{ki} for integration.")) (|varselect| (((|List| (|Kernel| |#2|)) (|List| (|Kernel| |#2|)) (|Symbol|)) "\\spad{varselect([k1,...,kn], x)} returns the \\spad{ki} which involve \\spad{x}."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| |#2| (QUOTE (-237))) (|HasCategory| |#2| (QUOTE (-566))) (|HasCategory| |#2| (QUOTE (-945 (-1081))))) (-12 (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-237)))) (|HasCategory| |#1| (QUOTE (-491)))) -(-506 -3078 UP) +((-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| |#2| (QUOTE (-239))) (|HasCategory| |#2| (QUOTE (-569))) (|HasCategory| |#2| (QUOTE (-950 (-1088))))) (-12 (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-239)))) (|HasCategory| |#1| (QUOTE (-494)))) +(-509 -3088 UP) ((|constructor| (NIL "This package provides functions for the transcendental case of the Risch algorithm.")) (|monomialIntPoly| (((|Record| (|:| |answer| |#2|) (|:| |polypart| |#2|)) |#2| (|Mapping| |#2| |#2|)) "\\spad{monomialIntPoly(p, ')} returns [\\spad{q},{} \\spad{r}] such that \\spad{p = q' + r} and \\spad{degree(r) < degree(t')}. Error if \\spad{degree(t') < 2}.")) (|monomialIntegrate| (((|Record| (|:| |ir| (|IntegrationResult| (|Fraction| |#2|))) (|:| |specpart| (|Fraction| |#2|)) (|:| |polypart| |#2|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomialIntegrate(f, ')} returns \\spad{[ir, s, p]} such that \\spad{f = ir' + s + p} and all the squarefree factors of the denominator of \\spad{s} are special \\spad{w}.\\spad{r}.\\spad{t} the derivation '.")) (|expintfldpoly| (((|Union| (|LaurentPolynomial| |#1| |#2|) "failed") (|LaurentPolynomial| |#1| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|)) "\\spad{expintfldpoly(p, foo)} returns \\spad{q} such that \\spad{p' = q} or \"failed\" if no such \\spad{q} exists. Argument foo is a Risch differential equation function on \\spad{F}.")) (|primintfldpoly| (((|Union| |#2| "failed") |#2| (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1="failed") |#1|) |#1|) "\\spad{primintfldpoly(p, ', t')} returns \\spad{q} such that \\spad{p' = q} or \"failed\" if no such \\spad{q} exists. Argument \\spad{t'} is the derivative of the primitive generating the extension.")) (|primlimintfrac| (((|Union| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|)))))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|List| (|Fraction| |#2|))) "\\spad{primlimintfrac(f, ', [u1,...,un])} returns \\spad{[v, [c1,...,cn]]} such that \\spad{ci' = 0} and \\spad{f = v' + +/[ci * ui'/ui]}. Error: if \\spad{degree numer f >= degree denom f}.")) (|primextintfrac| (((|Union| (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Fraction| |#2|)) "\\spad{primextintfrac(f, ', g)} returns \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0}. Error: if \\spad{degree numer f >= degree denom f} or if \\spad{degree numer g >= degree denom g} or if \\spad{denom g} is not squarefree.")) (|explimitedint| (((|Union| (|Record| (|:| |answer| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|))))))) (|:| |a0| |#1|)) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|) (|List| (|Fraction| |#2|))) "\\spad{explimitedint(f, ', foo, [u1,...,un])} returns \\spad{[v, [c1,...,cn], a]} such that \\spad{ci' = 0},{} \\spad{f = v' + a + reduce(+,[ci * ui'/ui])},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}. Returns \"failed\" if no such \\spad{v},{} \\spad{ci},{} a exist. Argument \\spad{foo} is a Risch differential equation function on \\spad{F}.")) (|primlimitedint| (((|Union| (|Record| (|:| |answer| (|Record| (|:| |mainpart| (|Fraction| |#2|)) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| (|Fraction| |#2|)) (|:| |logand| (|Fraction| |#2|))))))) (|:| |a0| |#1|)) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1#) |#1|) (|List| (|Fraction| |#2|))) "\\spad{primlimitedint(f, ', foo, [u1,...,un])} returns \\spad{[v, [c1,...,cn], a]} such that \\spad{ci' = 0},{} \\spad{f = v' + a + reduce(+,[ci * ui'/ui])},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Returns \"failed\" if no such \\spad{v},{} \\spad{ci},{} a exist. Argument \\spad{foo} is an extended integration function on \\spad{F}.")) (|expextendedint| (((|Union| (|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |a0| |#1|)) (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|) (|Fraction| |#2|)) "\\spad{expextendedint(f, ', foo, g)} returns either \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0},{} or \\spad{[v, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}. Returns \"failed\" if neither case can hold. Argument \\spad{foo} is a Risch differential equation function on \\spad{F}.")) (|primextendedint| (((|Union| (|Record| (|:| |answer| (|Fraction| |#2|)) (|:| |a0| |#1|)) (|Record| (|:| |ratpart| (|Fraction| |#2|)) (|:| |coeff| (|Fraction| |#2|))) "failed") (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1#) |#1|) (|Fraction| |#2|)) "\\spad{primextendedint(f, ', foo, g)} returns either \\spad{[v, c]} such that \\spad{f = v' + c g} and \\spad{c' = 0},{} or \\spad{[v, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Returns \"failed\" if neither case can hold. Argument \\spad{foo} is an extended integration function on \\spad{F}.")) (|tanintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|List| |#1|) "failed") (|Integer|) |#1| |#1|)) "\\spad{tanintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}; Argument foo is a Risch differential system solver on \\spad{F}.")) (|expintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Record| (|:| |ans| |#1|) (|:| |right| |#1|) (|:| |sol?| (|Boolean|))) (|Integer|) |#1|)) "\\spad{expintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in \\spad{F}; Argument foo is a Risch differential equation solver on \\spad{F}.")) (|primintegrate| (((|Record| (|:| |answer| (|IntegrationResult| (|Fraction| |#2|))) (|:| |a0| |#1|)) (|Fraction| |#2|) (|Mapping| |#2| |#2|) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) #1#) |#1|)) "\\spad{primintegrate(f, ', foo)} returns \\spad{[g, a]} such that \\spad{f = g' + a},{} and \\spad{a = 0} or \\spad{a} has no integral in UP. Argument foo is an extended integration function on \\spad{F}."))) NIL NIL -(-507 R -3078) +(-510 R -3088) ((|constructor| (NIL "This package computes the inverse Laplace Transform.")) (|inverseLaplace| (((|Union| |#2| "failed") |#2| (|Symbol|) (|Symbol|)) "\\spad{inverseLaplace(f, s, t)} returns the Inverse Laplace transform of \\spad{f(s)} using \\spad{t} as the new variable or \"failed\" if unable to find a closed form."))) NIL NIL -(-508) +(-511) ((|constructor| (NIL "This category describes byte stream conduits supporting both input and output operations."))) NIL NIL -(-509) +(-512) ((|constructor| (NIL "\\indented{2}{This domain provides representation for binary files open} \\indented{2}{for input and output operations.} See Also: InputBinaryFile,{} OutputBinaryFile")) (|isOpen?| (((|Boolean|) $) "\\spad{isOpen?(f)} holds if `f' is in open state.")) (|inputOutputBinaryFile| (($ (|String|)) "\\spad{inputOutputBinaryFile(f)} returns an input/output conduit obtained by opening the file named by `f' as a binary file.") (($ (|FileName|)) "\\spad{inputOutputBinaryFile(f)} returns an input/output conduit obtained by opening the file designated by `f' as a binary file."))) NIL NIL -(-510) +(-513) ((|constructor| (NIL "This domain provides constants to describe directions of IO conduits (file,{} etc) mode of operations.")) (|closed| (($) "\\spad{closed} indicates that the IO conduit has been closed.")) (|bothWays| (($) "\\spad{bothWays} indicates that an IO conduit is for both input and output.")) (|output| (($) "\\spad{output} indicates that an IO conduit is for output")) (|input| (($) "\\spad{input} indicates that an IO conduit is for input."))) NIL NIL -(-511) +(-514) ((|constructor| (NIL "This domain provides representation for ARPA Internet \\spad{IP4} addresses.")) (|resolve| (((|Maybe| $) (|Hostname|)) "\\spad{resolve(h)} returns the \\spad{IP4} address of host `h'.")) (|bytes| (((|DataArray| 4 (|Byte|)) $) "\\spad{bytes(x)} returns the bytes of the numeric address `x'.")) (|ip4Address| (($ (|String|)) "\\spad{ip4Address(a)} builds a numeric address out of the ASCII form `a'."))) NIL NIL -(-512 |p| |unBalanced?|) +(-515 |p| |unBalanced?|) ((|constructor| (NIL "This domain implements Zp,{} the \\spad{p}-adic completion of the integers. This is an internal domain."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-513 |p|) +(-516 |p|) ((|constructor| (NIL "InnerPrimeField(\\spad{p}) implements the field with \\spad{p} elements. Note: argument \\spad{p} MUST be a prime (this domain does not check). See \\spadtype{PrimeField} for a domain that does check."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| $ (QUOTE (-118))) (|HasCategory| $ (QUOTE (-116))) (|HasCategory| $ (QUOTE (-315)))) -(-514) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| $ (QUOTE (-120))) (|HasCategory| $ (QUOTE (-118))) (|HasCategory| $ (QUOTE (-317)))) +(-517) ((|constructor| (NIL "A package to print strings without line-feed nor carriage-return.")) (|iprint| (((|Void|) (|String|)) "\\axiom{iprint(\\spad{s})} prints \\axiom{\\spad{s}} at the current position of the cursor."))) NIL NIL -(-515 -3078) +(-518 -3088) ((|constructor| (NIL "If a function \\spad{f} has an elementary integral \\spad{g},{} then \\spad{g} can be written in the form \\spad{g = h + c1 log(u1) + c2 log(u2) + ... + cn log(un)} where \\spad{h},{} which is in the same field than \\spad{f},{} is called the rational part of the integral,{} and \\spad{c1 log(u1) + ... cn log(un)} is called the logarithmic part of the integral. This domain manipulates integrals represented in that form,{} by keeping both parts separately. The logs are not explicitly computed.")) (|differentiate| ((|#1| $ (|Symbol|)) "\\spad{differentiate(ir,x)} differentiates \\spad{ir} with respect to \\spad{x}") ((|#1| $ (|Mapping| |#1| |#1|)) "\\spad{differentiate(ir,D)} differentiates \\spad{ir} with respect to the derivation \\spad{D}.")) (|integral| (($ |#1| (|Symbol|)) "\\spad{integral(f,x)} returns the formal integral of \\spad{f} with respect to \\spad{x}") (($ |#1| |#1|) "\\spad{integral(f,x)} returns the formal integral of \\spad{f} with respect to \\spad{x}")) (|elem?| (((|Boolean|) $) "\\spad{elem?(ir)} tests if an integration result is elementary over F?")) (|notelem| (((|List| (|Record| (|:| |integrand| |#1|) (|:| |intvar| |#1|))) $) "\\spad{notelem(ir)} returns the non-elementary part of an integration result")) (|logpart| (((|List| (|Record| (|:| |scalar| (|Fraction| (|Integer|))) (|:| |coeff| (|SparseUnivariatePolynomial| |#1|)) (|:| |logand| (|SparseUnivariatePolynomial| |#1|)))) $) "\\spad{logpart(ir)} returns the logarithmic part of an integration result")) (|ratpart| ((|#1| $) "\\spad{ratpart(ir)} returns the rational part of an integration result")) (|mkAnswer| (($ |#1| (|List| (|Record| (|:| |scalar| (|Fraction| (|Integer|))) (|:| |coeff| (|SparseUnivariatePolynomial| |#1|)) (|:| |logand| (|SparseUnivariatePolynomial| |#1|)))) (|List| (|Record| (|:| |integrand| |#1|) (|:| |intvar| |#1|)))) "\\spad{mkAnswer(r,l,ne)} creates an integration result from a rational part \\spad{r},{} a logarithmic part \\spad{l},{} and a non-elementary part \\spad{ne}."))) -((-3973 . T) (-3972 . T)) -((|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-945 (-1081))))) -(-516 E -3078) +((-3984 . T) (-3983 . T)) +((|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-950 (-1088))))) +(-519 E -3088) ((|constructor| (NIL "\\indented{1}{Internally used by the integration packages} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 12 August 1992 Keywords: integration.")) (|map| (((|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") (|Mapping| |#2| |#1|) (|Union| (|Record| (|:| |mainpart| |#1|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#1|) (|:| |logand| |#1|))))) "failed")) "\\spad{map(f,ufe)} \\undocumented") (((|Union| |#2| "failed") (|Mapping| |#2| |#1|) (|Union| |#1| "failed")) "\\spad{map(f,ue)} \\undocumented") (((|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") (|Mapping| |#2| |#1|) (|Union| (|Record| (|:| |ratpart| |#1|) (|:| |coeff| |#1|)) "failed")) "\\spad{map(f,ure)} \\undocumented") (((|IntegrationResult| |#2|) (|Mapping| |#2| |#1|) (|IntegrationResult| |#1|)) "\\spad{map(f,ire)} \\undocumented"))) NIL NIL -(-517 R -3078) +(-520 R -3088) ((|constructor| (NIL "This package allows a sum of logs over the roots of a polynomial to be expressed as explicit logarithms and arc tangents,{} provided that the indexing polynomial can be factored into quadratics.")) (|complexExpand| ((|#2| (|IntegrationResult| |#2|)) "\\spad{complexExpand(i)} returns the expanded complex function corresponding to \\spad{i}.")) (|expand| (((|List| |#2|) (|IntegrationResult| |#2|)) "\\spad{expand(i)} returns the list of possible real functions corresponding to \\spad{i}.")) (|split| (((|IntegrationResult| |#2|) (|IntegrationResult| |#2|)) "\\spad{split(u(x) + sum_{P(a)=0} Q(a,x))} returns \\spad{u(x) + sum_{P1(a)=0} Q(a,x) + ... + sum_{Pn(a)=0} Q(a,x)} where \\spad{P1},{}...,{}Pn are the factors of \\spad{P}."))) NIL NIL -(-518) +(-521) ((|constructor| (NIL "This domain provides representations for the intermediate form data structure used by the Spad elaborator.")) (|irDef| (($ (|Identifier|) (|InternalTypeForm|) $) "\\spad{irDef(f,ts,e)} returns an IR representation for a definition of a function named \\spad{f},{} with signature \\spad{ts} and body \\spad{e}.")) (|irCtor| (($ (|Identifier|) (|InternalTypeForm|)) "\\spad{irCtor(n,t)} returns an IR for a constructor reference of type designated by the type form \\spad{t}")) (|irVar| (($ (|Identifier|) (|InternalTypeForm|)) "\\spad{irVar(x,t)} returns an IR for a variable reference of type designated by the type form \\spad{t}"))) NIL NIL -(-519 I) +(-522 I) ((|constructor| (NIL "The \\spadtype{IntegerRoots} package computes square roots and \\indented{2}{\\spad{n}th roots of integers efficiently.}")) (|approxSqrt| ((|#1| |#1|) "\\spad{approxSqrt(n)} returns an approximation \\spad{x} to \\spad{sqrt(n)} such that \\spad{-1 < x - sqrt(n) < 1}. Compute an approximation \\spad{s} to \\spad{sqrt(n)} such that \\indented{10}{\\spad{-1 < s - sqrt(n) < 1}} A variable precision Newton iteration is used. The running time is \\spad{O( log(n)**2 )}.")) (|perfectSqrt| (((|Union| |#1| "failed") |#1|) "\\spad{perfectSqrt(n)} returns the square root of \\spad{n} if \\spad{n} is a perfect square and returns \"failed\" otherwise")) (|perfectSquare?| (((|Boolean|) |#1|) "\\spad{perfectSquare?(n)} returns \\spad{true} if \\spad{n} is a perfect square and \\spad{false} otherwise")) (|approxNthRoot| ((|#1| |#1| (|NonNegativeInteger|)) "\\spad{approxRoot(n,r)} returns an approximation \\spad{x} to \\spad{n**(1/r)} such that \\spad{-1 < x - n**(1/r) < 1}")) (|perfectNthRoot| (((|Record| (|:| |base| |#1|) (|:| |exponent| (|NonNegativeInteger|))) |#1|) "\\spad{perfectNthRoot(n)} returns \\spad{[x,r]},{} where \\spad{n = x\\^r} and \\spad{r} is the largest integer such that \\spad{n} is a perfect \\spad{r}th power") (((|Union| |#1| "failed") |#1| (|NonNegativeInteger|)) "\\spad{perfectNthRoot(n,r)} returns the \\spad{r}th root of \\spad{n} if \\spad{n} is an \\spad{r}th power and returns \"failed\" otherwise")) (|perfectNthPower?| (((|Boolean|) |#1| (|NonNegativeInteger|)) "\\spad{perfectNthPower?(n,r)} returns \\spad{true} if \\spad{n} is an \\spad{r}th power and \\spad{false} otherwise"))) NIL NIL -(-520 GF) +(-523 GF) ((|constructor| (NIL "This package exports the function generateIrredPoly that computes a monic irreducible polynomial of degree \\spad{n} over a finite field.")) (|generateIrredPoly| (((|SparseUnivariatePolynomial| |#1|) (|PositiveInteger|)) "\\spad{generateIrredPoly(n)} generates an irreducible univariate polynomial of the given degree \\spad{n} over the finite field."))) NIL NIL -(-521 R) +(-524 R) ((|constructor| (NIL "\\indented{2}{This package allows a sum of logs over the roots of a polynomial} \\indented{2}{to be expressed as explicit logarithms and arc tangents,{} provided} \\indented{2}{that the indexing polynomial can be factored into quadratics.} Date Created: 21 August 1988 Date Last Updated: 4 October 1993")) (|complexIntegrate| (((|Expression| |#1|) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{complexIntegrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a complex variable.")) (|integrate| (((|Union| (|Expression| |#1|) (|List| (|Expression| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{integrate(f, x)} returns the integral of \\spad{f(x)dx} where \\spad{x} is viewed as a real variable..")) (|complexExpand| (((|Expression| |#1|) (|IntegrationResult| (|Fraction| (|Polynomial| |#1|)))) "\\spad{complexExpand(i)} returns the expanded complex function corresponding to \\spad{i}.")) (|expand| (((|List| (|Expression| |#1|)) (|IntegrationResult| (|Fraction| (|Polynomial| |#1|)))) "\\spad{expand(i)} returns the list of possible real functions corresponding to \\spad{i}.")) (|split| (((|IntegrationResult| (|Fraction| (|Polynomial| |#1|))) (|IntegrationResult| (|Fraction| (|Polynomial| |#1|)))) "\\spad{split(u(x) + sum_{P(a)=0} Q(a,x))} returns \\spad{u(x) + sum_{P1(a)=0} Q(a,x) + ... + sum_{Pn(a)=0} Q(a,x)} where \\spad{P1},{}...,{}Pn are the factors of \\spad{P}."))) NIL -((|HasCategory| |#1| (QUOTE (-118)))) -(-522) +((|HasCategory| |#1| (QUOTE (-120)))) +(-525) ((|constructor| (NIL "IrrRepSymNatPackage contains functions for computing the ordinary irreducible representations of symmetric groups on \\spad{n} letters {\\em {1,2,...,n}} in Young's natural form and their dimensions. These representations can be labelled by number partitions of \\spad{n},{} \\spadignore{i.e.} a weakly decreasing sequence of integers summing up to \\spad{n},{} \\spadignore{e.g.} {\\em [3,3,3,1]} labels an irreducible representation for \\spad{n} equals 10. Note: whenever a \\spadtype{List Integer} appears in a signature,{} a partition required.")) (|irreducibleRepresentation| (((|List| (|Matrix| (|Integer|))) (|List| (|PositiveInteger|)) (|List| (|Permutation| (|Integer|)))) "\\spad{irreducibleRepresentation(lambda,listOfPerm)} is the list of the irreducible representations corresponding to {\\em lambda} in Young's natural form for the list of permutations given by {\\em listOfPerm}.") (((|List| (|Matrix| (|Integer|))) (|List| (|PositiveInteger|))) "\\spad{irreducibleRepresentation(lambda)} is the list of the two irreducible representations corresponding to the partition {\\em lambda} in Young's natural form for the following two generators of the symmetric group,{} whose elements permute {\\em {1,2,...,n}},{} namely {\\em (1 2)} (2-cycle) and {\\em (1 2 ... n)} (\\spad{n}-cycle).") (((|Matrix| (|Integer|)) (|List| (|PositiveInteger|)) (|Permutation| (|Integer|))) "\\spad{irreducibleRepresentation(lambda,pi)} is the irreducible representation corresponding to partition {\\em lambda} in Young's natural form of the permutation {\\em pi} in the symmetric group,{} whose elements permute {\\em {1,2,...,n}}.")) (|dimensionOfIrreducibleRepresentation| (((|NonNegativeInteger|) (|List| (|PositiveInteger|))) "\\spad{dimensionOfIrreducibleRepresentation(lambda)} is the dimension of the ordinary irreducible representation of the symmetric group corresponding to {\\em lambda}. Note: the Robinson-Thrall hook formula is implemented."))) NIL NIL -(-523 R E V P TS) +(-526 R E V P TS) ((|constructor| (NIL "\\indented{1}{An internal package for computing the rational univariate representation} \\indented{1}{of a zero-dimensional algebraic variety given by a square-free} \\indented{1}{triangular set.} \\indented{1}{The main operation is \\axiomOpFrom{rur}{InternalRationalUnivariateRepresentationPackage}.} \\indented{1}{It is based on the {\\em generic} algorithm description in [1]. \\newline References:} [1] \\spad{D}. LAZARD \"Solving Zero-dimensional Algebraic Systems\" \\indented{4}{Journal of Symbolic Computation,{} 1992,{} 13,{} 117-131}")) (|checkRur| (((|Boolean|) |#5| (|List| |#5|)) "\\spad{checkRur(ts,lus)} returns \\spad{true} if \\spad{lus} is a rational univariate representation of \\spad{ts}.")) (|rur| (((|List| |#5|) |#5| (|Boolean|)) "\\spad{rur(ts,univ?)} returns a rational univariate representation of \\spad{ts}. This assumes that the lowest polynomial in \\spad{ts} is a variable \\spad{v} which does not occur in the other polynomials of \\spad{ts}. This variable will be used to define the simple algebraic extension over which these other polynomials will be rewritten as univariate polynomials with degree one. If \\spad{univ?} is \\spad{true} then these polynomials will have a constant initial."))) NIL NIL -(-524) +(-527) ((|constructor| (NIL "This domain represents a `has' expression.")) (|rhs| (((|SpadAst|) $) "\\spad{rhs(e)} returns the right hand side of the is expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the is expression `e'."))) NIL NIL -(-525 E V R P) +(-528 E V R P) ((|constructor| (NIL "tools for the summation packages.")) (|sum| (((|Record| (|:| |num| |#4|) (|:| |den| (|Integer|))) |#4| |#2|) "\\spad{sum(p(n), n)} returns \\spad{P(n)},{} the indefinite sum of \\spad{p(n)} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{P(n+1) - P(n) = a(n)}.") (((|Record| (|:| |num| |#4|) (|:| |den| (|Integer|))) |#4| |#2| (|Segment| |#4|)) "\\spad{sum(p(n), n = a..b)} returns \\spad{p(a) + p(a+1) + ... + p(b)}."))) NIL NIL -(-526 |Coef|) +(-529 |Coef|) ((|constructor| (NIL "InnerSparseUnivariatePowerSeries is an internal domain \\indented{2}{used for creating sparse Taylor and Laurent series.}")) (|cAcsch| (($ $) "\\spad{cAcsch(f)} computes the inverse hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsech| (($ $) "\\spad{cAsech(f)} computes the inverse hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcoth| (($ $) "\\spad{cAcoth(f)} computes the inverse hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtanh| (($ $) "\\spad{cAtanh(f)} computes the inverse hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcosh| (($ $) "\\spad{cAcosh(f)} computes the inverse hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsinh| (($ $) "\\spad{cAsinh(f)} computes the inverse hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsch| (($ $) "\\spad{cCsch(f)} computes the hyperbolic cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSech| (($ $) "\\spad{cSech(f)} computes the hyperbolic secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCoth| (($ $) "\\spad{cCoth(f)} computes the hyperbolic cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTanh| (($ $) "\\spad{cTanh(f)} computes the hyperbolic tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCosh| (($ $) "\\spad{cCosh(f)} computes the hyperbolic cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSinh| (($ $) "\\spad{cSinh(f)} computes the hyperbolic sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcsc| (($ $) "\\spad{cAcsc(f)} computes the arccosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsec| (($ $) "\\spad{cAsec(f)} computes the arcsecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcot| (($ $) "\\spad{cAcot(f)} computes the arccotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAtan| (($ $) "\\spad{cAtan(f)} computes the arctangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAcos| (($ $) "\\spad{cAcos(f)} computes the arccosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cAsin| (($ $) "\\spad{cAsin(f)} computes the arcsine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCsc| (($ $) "\\spad{cCsc(f)} computes the cosecant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSec| (($ $) "\\spad{cSec(f)} computes the secant of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCot| (($ $) "\\spad{cCot(f)} computes the cotangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cTan| (($ $) "\\spad{cTan(f)} computes the tangent of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cCos| (($ $) "\\spad{cCos(f)} computes the cosine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cSin| (($ $) "\\spad{cSin(f)} computes the sine of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cLog| (($ $) "\\spad{cLog(f)} computes the logarithm of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cExp| (($ $) "\\spad{cExp(f)} computes the exponential of the power series \\spad{f}. For use when the coefficient ring is commutative.")) (|cRationalPower| (($ $ (|Fraction| (|Integer|))) "\\spad{cRationalPower(f,r)} computes \\spad{f^r}. For use when the coefficient ring is commutative.")) (|cPower| (($ $ |#1|) "\\spad{cPower(f,r)} computes \\spad{f^r},{} where \\spad{f} has constant coefficient 1. For use when the coefficient ring is commutative.")) (|integrate| (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. Warning: function does not check for a term of degree \\spad{-1}.")) (|seriesToOutputForm| (((|OutputForm|) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) (|Reference| (|OrderedCompletion| (|Integer|))) (|Symbol|) |#1| (|Fraction| (|Integer|))) "\\spad{seriesToOutputForm(st,refer,var,cen,r)} prints the series \\spad{f((var - cen)^r)}.")) (|iCompose| (($ $ $) "\\spad{iCompose(f,g)} returns \\spad{f(g(x))}. This is an internal function which should only be called for Taylor series \\spad{f(x)} and \\spad{g(x)} such that the constant coefficient of \\spad{g(x)} is zero.")) (|taylorQuoByVar| (($ $) "\\spad{taylorQuoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...}")) (|iExquo| (((|Union| $ "failed") $ $ (|Boolean|)) "\\spad{iExquo(f,g,taylor?)} is the quotient of the power series \\spad{f} and \\spad{g}. If \\spad{taylor?} is \\spad{true},{} then we must have \\spad{order(f) >= order(g)}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(fn,f)} returns the series \\spad{sum(fn(n) * an * x^n,n = n0..)},{} where \\spad{f} is the series \\spad{sum(an * x^n,n = n0..)}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(f)} tests if \\spad{f} is a single monomial.")) (|series| (($ (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.")) (|getStream| (((|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|))) $) "\\spad{getStream(f)} returns the stream of terms representing the series \\spad{f}.")) (|getRef| (((|Reference| (|OrderedCompletion| (|Integer|))) $) "\\spad{getRef(f)} returns a reference containing the order to which the terms of \\spad{f} have been computed.")) (|makeSeries| (($ (|Reference| (|OrderedCompletion| (|Integer|))) (|Stream| (|Record| (|:| |k| (|Integer|)) (|:| |c| |#1|)))) "\\spad{makeSeries(refer,str)} creates a power series from the reference \\spad{refer} and the stream \\spad{str}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-480)) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-480)) (|devaluate| |#1|)))) (|HasCategory| (-480) (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-309))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-480)))))) -(-527 |Coef|) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-483)) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-483)) (|devaluate| |#1|)))) (|HasCategory| (-483) (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-311))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-483)))))) +(-530 |Coef|) ((|constructor| (NIL "Internal package for dense Taylor series. This is an internal Taylor series type in which Taylor series are represented by a \\spadtype{Stream} of \\spadtype{Ring} elements. For univariate series,{} the \\spad{Stream} elements are the Taylor coefficients. For multivariate series,{} the \\spad{n}th Stream element is a form of degree \\spad{n} in the power series variables.")) (* (($ $ (|Integer|)) "\\spad{x*i} returns the product of integer \\spad{i} and the series \\spad{x}.")) (|order| (((|NonNegativeInteger|) $ (|NonNegativeInteger|)) "\\spad{order(x,n)} returns the minimum of \\spad{n} and the order of \\spad{x}.") (((|NonNegativeInteger|) $) "\\spad{order(x)} returns the order of a power series \\spad{x},{} \\indented{1}{\\spadignore{i.e.} the degree of the first non-zero term of the series.}")) (|pole?| (((|Boolean|) $) "\\spad{pole?(x)} tests if the series \\spad{x} has a pole. \\indented{1}{Note: this is \\spad{false} when \\spad{x} is a Taylor series.}")) (|series| (($ (|Stream| |#1|)) "\\spad{series(s)} creates a power series from a stream of \\indented{1}{ring elements.} \\indented{1}{For univariate series types,{} the stream \\spad{s} should be a stream} \\indented{1}{of Taylor coefficients. For multivariate series types,{} the} \\indented{1}{stream \\spad{s} should be a stream of forms the \\spad{n}th element} \\indented{1}{of which is a} \\indented{1}{form of degree \\spad{n} in the power series variables.}")) (|coefficients| (((|Stream| |#1|) $) "\\spad{coefficients(x)} returns a stream of ring elements. \\indented{1}{When \\spad{x} is a univariate series,{} this is a stream of Taylor} \\indented{1}{coefficients. When \\spad{x} is a multivariate series,{} the} \\indented{1}{\\spad{n}th element of the stream is a form of} \\indented{1}{degree \\spad{n} in the power series variables.}"))) -(((-3980 "*") |has| |#1| (-491)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-491)))) -(-528) +(((-3991 "*") |has| |#1| (-494)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-494)))) +(-531) ((|constructor| (NIL "This domain provides representations for internal type form.")) (|mappingMode| (($ $ (|List| $)) "\\spad{mappingMode(r,ts)} returns a mapping mode with return mode \\spad{r},{} and parameter modes \\spad{ts}.")) (|categoryMode| (($) "\\spad{categoryMode} is a constant mode denoting Category.")) (|voidMode| (($) "\\spad{voidMode} is a constant mode denoting Void.")) (|noValueMode| (($) "\\spad{noValueMode} is a constant mode that indicates that the value of an expression is to be ignored.")) (|jokerMode| (($) "\\spad{jokerMode} is a constant that stands for any mode in a type inference context"))) NIL NIL -(-529 A B) +(-532 A B) ((|constructor| (NIL "Functions defined on streams with entries in two sets.")) (|map| (((|InfiniteTuple| |#2|) (|Mapping| |#2| |#1|) (|InfiniteTuple| |#1|)) "\\spad{map(f,[x0,x1,x2,...])} returns \\spad{[f(x0),f(x1),f(x2),..]}."))) NIL NIL -(-530 A B C) +(-533 A B C) ((|constructor| (NIL "Functions defined on streams with entries in two sets.")) (|map| (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|InfiniteTuple| |#1|) (|Stream| |#2|)) "\\spad{map(f,a,b)} \\undocumented") (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|Stream| |#1|) (|InfiniteTuple| |#2|)) "\\spad{map(f,a,b)} \\undocumented") (((|InfiniteTuple| |#3|) (|Mapping| |#3| |#1| |#2|) (|InfiniteTuple| |#1|) (|InfiniteTuple| |#2|)) "\\spad{map(f,a,b)} \\undocumented"))) NIL NIL -(-531 R -3078 FG) +(-534 R -3088 FG) ((|constructor| (NIL "This package provides transformations from trigonometric functions to exponentials and logarithms,{} and back. \\spad{F} and FG should be the same type of function space.")) (|trigs2explogs| ((|#3| |#3| (|List| (|Kernel| |#3|)) (|List| (|Symbol|))) "\\spad{trigs2explogs(f, [k1,...,kn], [x1,...,xm])} rewrites all the trigonometric functions appearing in \\spad{f} and involving one of the \\spad{xi's} in terms of complex logarithms and exponentials. A kernel of the form \\spad{tan(u)} is expressed using \\spad{exp(u)**2} if it is one of the \\spad{ki's},{} in terms of \\spad{exp(2*u)} otherwise.")) (|explogs2trigs| (((|Complex| |#2|) |#3|) "\\spad{explogs2trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (F2FG ((|#3| |#2|) "\\spad{F2FG(a + sqrt(-1) b)} returns \\spad{a + i b}.")) (FG2F ((|#2| |#3|) "\\spad{FG2F(a + i b)} returns \\spad{a + sqrt(-1) b}.")) (GF2FG ((|#3| (|Complex| |#2|)) "\\spad{GF2FG(a + i b)} returns \\spad{a + i b} viewed as a function with the \\spad{i} pushed down into the coefficient domain."))) NIL NIL -(-532 S) +(-535 S) ((|constructor| (NIL "This package implements 'infinite tuples' for the interpreter. The representation is a stream.")) (|construct| (((|Stream| |#1|) $) "\\spad{construct(t)} converts an infinite tuple to a stream.")) (|generate| (($ (|Mapping| |#1| |#1|) |#1|) "\\spad{generate(f,s)} returns \\spad{[s,f(s),f(f(s)),...]}.")) (|select| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select(p,t)} returns \\spad{[x for x in t | p(x)]}.")) (|filterUntil| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterUntil(p,t)} returns \\spad{[x for x in t while not p(x)]}.")) (|filterWhile| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterWhile(p,t)} returns \\spad{[x for x in t while p(x)]}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(f,t)} replaces the tuple \\spad{t} by \\spad{[f(x) for x in t]}."))) NIL NIL -(-533 R |mn|) +(-536 R |mn|) ((|constructor| (NIL "\\indented{2}{This type represents vector like objects with varying lengths} and a user-specified initial index."))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#1| (QUOTE (-956))) (-12 (|HasCategory| |#1| (QUOTE (-910))) (|HasCategory| |#1| (QUOTE (-956)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-534 S |Index| |Entry|) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#1| (QUOTE (-961))) (-12 (|HasCategory| |#1| (QUOTE (-915))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-537 S |Index| |Entry|) ((|constructor| (NIL "An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.")) (|swap!| (((|Void|) $ |#2| |#2|) "\\spad{swap!(u,i,j)} interchanges elements \\spad{i} and \\spad{j} of aggregate \\spad{u}. No meaningful value is returned.")) (|fill!| (($ $ |#3|) "\\spad{fill!(u,x)} replaces each entry in aggregate \\spad{u} by \\spad{x}. The modified \\spad{u} is returned as value.")) (|first| ((|#3| $) "\\spad{first(u)} returns the first element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{first([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = \\spad{x}}. Error: if \\spad{u} is empty.")) (|minIndex| ((|#2| $) "\\spad{minIndex(u)} returns the minimum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{minIndex(a) = reduce(min,{}[\\spad{i} for \\spad{i} in indices a])}; for lists,{} \\axiom{minIndex(a) = 1}.")) (|maxIndex| ((|#2| $) "\\spad{maxIndex(u)} returns the maximum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{maxIndex(\\spad{u}) = reduce(max,{}[\\spad{i} for \\spad{i} in indices \\spad{u}])}; if \\spad{u} is a list,{} \\axiom{maxIndex(\\spad{u}) = \\#u}.")) (|entry?| (((|Boolean|) |#3| $) "\\spad{entry?(x,u)} tests if \\spad{x} equals \\axiom{\\spad{u} . \\spad{i}} for some index \\spad{i}.")) (|indices| (((|List| |#2|) $) "\\spad{indices(u)} returns a list of indices of aggregate \\spad{u} in no particular order.")) (|index?| (((|Boolean|) |#2| $) "\\spad{index?(i,u)} tests if \\spad{i} is an index of aggregate \\spad{u}.")) (|entries| (((|List| |#3|) $) "\\spad{entries(u)} returns a list of all the entries of aggregate \\spad{u} in no assumed order."))) NIL -((|HasAttribute| |#1| (QUOTE -3979)) (|HasCategory| |#2| (QUOTE (-751))) (|HasAttribute| |#1| (QUOTE -3978)) (|HasCategory| |#3| (QUOTE (-1007)))) -(-535 |Index| |Entry|) +((|HasAttribute| |#1| (QUOTE -3990)) (|HasCategory| |#2| (QUOTE (-756))) (|HasAttribute| |#1| (QUOTE -3989)) (|HasCategory| |#3| (QUOTE (-1012)))) +(-538 |Index| |Entry|) ((|constructor| (NIL "An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.")) (|swap!| (((|Void|) $ |#1| |#1|) "\\spad{swap!(u,i,j)} interchanges elements \\spad{i} and \\spad{j} of aggregate \\spad{u}. No meaningful value is returned.")) (|fill!| (($ $ |#2|) "\\spad{fill!(u,x)} replaces each entry in aggregate \\spad{u} by \\spad{x}. The modified \\spad{u} is returned as value.")) (|first| ((|#2| $) "\\spad{first(u)} returns the first element \\spad{x} of \\spad{u}. Note: for collections,{} \\axiom{first([\\spad{x},{}\\spad{y},{}...,{}\\spad{z}]) = \\spad{x}}. Error: if \\spad{u} is empty.")) (|minIndex| ((|#1| $) "\\spad{minIndex(u)} returns the minimum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{minIndex(a) = reduce(min,{}[\\spad{i} for \\spad{i} in indices a])}; for lists,{} \\axiom{minIndex(a) = 1}.")) (|maxIndex| ((|#1| $) "\\spad{maxIndex(u)} returns the maximum index \\spad{i} of aggregate \\spad{u}. Note: in general,{} \\axiom{maxIndex(\\spad{u}) = reduce(max,{}[\\spad{i} for \\spad{i} in indices \\spad{u}])}; if \\spad{u} is a list,{} \\axiom{maxIndex(\\spad{u}) = \\#u}.")) (|entry?| (((|Boolean|) |#2| $) "\\spad{entry?(x,u)} tests if \\spad{x} equals \\axiom{\\spad{u} . \\spad{i}} for some index \\spad{i}.")) (|indices| (((|List| |#1|) $) "\\spad{indices(u)} returns a list of indices of aggregate \\spad{u} in no particular order.")) (|index?| (((|Boolean|) |#1| $) "\\spad{index?(i,u)} tests if \\spad{i} is an index of aggregate \\spad{u}.")) (|entries| (((|List| |#2|) $) "\\spad{entries(u)} returns a list of all the entries of aggregate \\spad{u} in no assumed order."))) NIL NIL -(-536) +(-539) ((|constructor| (NIL "This domain represents the join of categories ASTs.")) (|categories| (((|List| (|TypeAst|)) $) "catehories(\\spad{x}) returns the types in the join `x'.")) (|coerce| (($ (|List| (|TypeAst|))) "ts::JoinAst construct the AST for a join of the types `ts'."))) NIL NIL -(-537 R A) +(-540 R A) ((|constructor| (NIL "\\indented{1}{AssociatedJordanAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$A}} \\indented{1}{to define the new multiplications \\spad{a*b := (a *\\$A b + b *\\$A a)/2}} \\indented{1}{(anticommutator).} \\indented{1}{The usual notation \\spad{{a,b}_+} cannot be used due to} \\indented{1}{restrictions in the current language.} \\indented{1}{This domain only gives a Jordan algebra if the} \\indented{1}{Jordan-identity \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds} \\indented{1}{for all \\spad{a},{}\\spad{b},{}\\spad{c} in \\spad{A}.} \\indented{1}{This relation can be checked by} \\indented{1}{\\spadfun{jordanAdmissible?()\\$A}.} \\blankline If the underlying algebra is of type \\spadtype{FramedNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank,{} together with a fixed \\spad{R}-module basis),{} then the same is \\spad{true} for the associated Jordan algebra. Moreover,{} if the underlying algebra is of type \\spadtype{FiniteRankNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank),{} then the same \\spad{true} for the associated Jordan algebra.")) (|coerce| (($ |#2|) "\\spad{coerce(a)} coerces the element \\spad{a} of the algebra \\spad{A} to an element of the Jordan algebra \\spadtype{AssociatedJordanAlgebra}(\\spad{R},{}A)."))) -((-3975 OR (-2548 (|has| |#2| (-313 |#1|)) (|has| |#1| (-491))) (-12 (|has| |#2| (-356 |#1|)) (|has| |#1| (-491)))) (-3973 . T) (-3972 . T)) -((OR (|HasCategory| |#2| (|%list| (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|)))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#2| (|%list| (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|))))) (|HasCategory| |#2| (|%list| (QUOTE -313) (|devaluate| |#1|)))) -(-538) +((-3986 OR (-2558 (|has| |#2| (-315 |#1|)) (|has| |#1| (-494))) (-12 (|has| |#2| (-358 |#1|)) (|has| |#1| (-494)))) (-3984 . T) (-3983 . T)) +((OR (|HasCategory| |#2| (|%list| (QUOTE -315) (|devaluate| |#1|))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|)))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#2| (|%list| (QUOTE -315) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|))))) (|HasCategory| |#2| (|%list| (QUOTE -315) (|devaluate| |#1|)))) +(-541) ((|constructor| (NIL "This is the datatype for the JVM bytecodes."))) NIL NIL -(-539) +(-542) ((|constructor| (NIL "JVM class file access bitmask and values.")) (|jvmAbstract| (($) "The class was declared abstract; therefore object of this class may not be created.")) (|jvmInterface| (($) "The class file represents an interface,{} not a class.")) (|jvmSuper| (($) "Instruct the JVM to treat base clss method invokation specially.")) (|jvmFinal| (($) "The class was declared final; therefore no derived class allowed.")) (|jvmPublic| (($) "The class was declared public,{} therefore may be accessed from outside its package"))) NIL NIL -(-540) +(-543) ((|constructor| (NIL "JVM class file constant pool tags.")) (|jvmNameAndTypeConstantTag| (($) "The correspondong constant pool entry represents the name and type of a field or method info.")) (|jvmInterfaceMethodConstantTag| (($) "The correspondong constant pool entry represents an interface method info.")) (|jvmMethodrefConstantTag| (($) "The correspondong constant pool entry represents a class method info.")) (|jvmFieldrefConstantTag| (($) "The corresponding constant pool entry represents a class field info.")) (|jvmStringConstantTag| (($) "The corresponding constant pool entry is a string constant info.")) (|jvmClassConstantTag| (($) "The corresponding constant pool entry represents a class or and interface.")) (|jvmDoubleConstantTag| (($) "The corresponding constant pool entry is a double constant info.")) (|jvmLongConstantTag| (($) "The corresponding constant pool entry is a long constant info.")) (|jvmFloatConstantTag| (($) "The corresponding constant pool entry is a float constant info.")) (|jvmIntegerConstantTag| (($) "The corresponding constant pool entry is an integer constant info.")) (|jvmUTF8ConstantTag| (($) "The corresponding constant pool entry is sequence of bytes representing Java \\spad{UTF8} string constant."))) NIL NIL -(-541) +(-544) ((|constructor| (NIL "JVM class field access bitmask and values.")) (|jvmTransient| (($) "The field was declared transient.")) (|jvmVolatile| (($) "The field was declared volatile.")) (|jvmFinal| (($) "The field was declared final; therefore may not be modified after initialization.")) (|jvmStatic| (($) "The field was declared static.")) (|jvmProtected| (($) "The field was declared protected; therefore may be accessed withing derived classes.")) (|jvmPrivate| (($) "The field was declared private; threfore can be accessed only within the defining class.")) (|jvmPublic| (($) "The field was declared public; therefore mey accessed from outside its package."))) NIL NIL -(-542) +(-545) ((|constructor| (NIL "JVM class method access bitmask and values.")) (|jvmStrict| (($) "The method was declared fpstrict; therefore floating-point mode is FP-strict.")) (|jvmAbstract| (($) "The method was declared abstract; therefore no implementation is provided.")) (|jvmNative| (($) "The method was declared native; therefore implemented in a language other than Java.")) (|jvmSynchronized| (($) "The method was declared synchronized.")) (|jvmFinal| (($) "The method was declared final; therefore may not be overriden. in derived classes.")) (|jvmStatic| (($) "The method was declared static.")) (|jvmProtected| (($) "The method was declared protected; therefore may be accessed withing derived classes.")) (|jvmPrivate| (($) "The method was declared private; threfore can be accessed only within the defining class.")) (|jvmPublic| (($) "The method was declared public; therefore mey accessed from outside its package."))) NIL NIL -(-543) +(-546) ((|constructor| (NIL "This is the datatype for the JVM opcodes."))) NIL NIL -(-544 |Entry|) +(-547 |Entry|) ((|constructor| (NIL "This domain allows a random access file to be viewed both as a table and as a file object.")) (|pack!| (($ $) "\\spad{pack!(f)} reorganizes the file \\spad{f} on disk to recover unused space."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (QUOTE (|:| -3843 (-1064))) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-1007)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| (-1064) (QUOTE (-751))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-72)))) -(-545 S |Key| |Entry|) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (QUOTE (|:| -3854 (-1071))) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-1012)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| (-1071) (QUOTE (-756))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-72)))) +(-548 S |Key| |Entry|) ((|constructor| (NIL "A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.")) (|search| (((|Union| |#3| "failed") |#2| $) "\\spad{search(k,t)} searches the table \\spad{t} for the key \\spad{k},{} returning the entry stored in \\spad{t} for key \\spad{k}. If \\spad{t} has no such key,{} \\axiom{search(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|remove!| (((|Union| |#3| "failed") |#2| $) "\\spad{remove!(k,t)} searches the table \\spad{t} for the key \\spad{k} removing (and return) the entry if there. If \\spad{t} has no such key,{} \\axiom{remove!(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|keys| (((|List| |#2|) $) "\\spad{keys(t)} returns the list the keys in table \\spad{t}.")) (|key?| (((|Boolean|) |#2| $) "\\spad{key?(k,t)} tests if \\spad{k} is a key in table \\spad{t}."))) NIL NIL -(-546 |Key| |Entry|) +(-549 |Key| |Entry|) ((|constructor| (NIL "A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.")) (|search| (((|Union| |#2| "failed") |#1| $) "\\spad{search(k,t)} searches the table \\spad{t} for the key \\spad{k},{} returning the entry stored in \\spad{t} for key \\spad{k}. If \\spad{t} has no such key,{} \\axiom{search(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|remove!| (((|Union| |#2| "failed") |#1| $) "\\spad{remove!(k,t)} searches the table \\spad{t} for the key \\spad{k} removing (and return) the entry if there. If \\spad{t} has no such key,{} \\axiom{remove!(\\spad{k},{}\\spad{t})} returns \"failed\".")) (|keys| (((|List| |#1|) $) "\\spad{keys(t)} returns the list the keys in table \\spad{t}.")) (|key?| (((|Boolean|) |#1| $) "\\spad{key?(k,t)} tests if \\spad{k} is a key in table \\spad{t}."))) -((-3979 . T)) +((-3990 . T)) NIL -(-547 S) +(-550 S) ((|constructor| (NIL "A kernel over a set \\spad{S} is an operator applied to a given list of arguments from \\spad{S}.")) (|is?| (((|Boolean|) $ (|Symbol|)) "\\spad{is?(op(a1,...,an), s)} tests if the name of op is \\spad{s}.") (((|Boolean|) $ (|BasicOperator|)) "\\spad{is?(op(a1,...,an), f)} tests if op = \\spad{f}.")) (|symbolIfCan| (((|Union| (|Symbol|) "failed") $) "\\spad{symbolIfCan(k)} returns \\spad{k} viewed as a symbol if \\spad{k} is a symbol,{} and \"failed\" otherwise.")) (|kernel| (($ (|Symbol|)) "\\spad{kernel(x)} returns \\spad{x} viewed as a kernel.") (($ (|BasicOperator|) (|List| |#1|) (|NonNegativeInteger|)) "\\spad{kernel(op, [a1,...,an], m)} returns the kernel \\spad{op(a1,...,an)} of nesting level \\spad{m}. Error: if \\spad{op} is \\spad{k}-ary for some \\spad{k} not equal to \\spad{m}.")) (|height| (((|NonNegativeInteger|) $) "\\spad{height(k)} returns the nesting level of \\spad{k}.")) (|argument| (((|List| |#1|) $) "\\spad{argument(op(a1,...,an))} returns \\spad{[a1,...,an]}.")) (|operator| (((|BasicOperator|) $) "\\spad{operator(op(a1,...,an))} returns the operator op."))) NIL -((|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| |#1| (QUOTE (-550 (-795 (-480)))))) -(-548 R S) +((|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| |#1| (QUOTE (-553 (-800 (-483)))))) +(-551 R S) ((|constructor| (NIL "This package exports some auxiliary functions on kernels")) (|constantIfCan| (((|Union| |#1| "failed") (|Kernel| |#2|)) "\\spad{constantIfCan(k)} \\undocumented")) (|constantKernel| (((|Kernel| |#2|) |#1|) "\\spad{constantKernel(r)} \\undocumented"))) NIL NIL -(-549 S) +(-552 S) ((|constructor| (NIL "A is coercible to \\spad{B} means any element of A can automatically be converted into an element of \\spad{B} by the interpreter.")) (|coerce| ((|#1| $) "\\spad{coerce(a)} transforms a into an element of \\spad{S}."))) NIL NIL -(-550 S) +(-553 S) ((|constructor| (NIL "A is convertible to \\spad{B} means any element of A can be converted into an element of \\spad{B},{} but not automatically by the interpreter.")) (|convert| ((|#1| $) "\\spad{convert(a)} transforms a into an element of \\spad{S}."))) NIL NIL -(-551 -3078 UP) +(-554 -3088 UP) ((|constructor| (NIL "\\spadtype{Kovacic} provides a modified Kovacic's algorithm for solving explicitely irreducible 2nd order linear ordinary differential equations.")) (|kovacic| (((|Union| (|SparseUnivariatePolynomial| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{kovacic(a_0,a_1,a_2,ezfactor)} returns either \"failed\" or \\spad{P}(\\spad{u}) such that \\spad{\\$e^{\\int(-a_1/2a_2)} e^{\\int u}\\$} is a solution of \\indented{5}{\\spad{\\$a_2 y'' + a_1 y' + a0 y = 0\\$}} whenever \\spad{u} is a solution of \\spad{P u = 0}. The equation must be already irreducible over the rational functions. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|Union| (|SparseUnivariatePolynomial| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{kovacic(a_0,a_1,a_2)} returns either \"failed\" or \\spad{P}(\\spad{u}) such that \\spad{\\$e^{\\int(-a_1/2a_2)} e^{\\int u}\\$} is a solution of \\indented{5}{\\spad{a_2 y'' + a_1 y' + a0 y = 0}} whenever \\spad{u} is a solution of \\spad{P u = 0}. The equation must be already irreducible over the rational functions."))) NIL NIL -(-552 S) +(-555 S) ((|constructor| (NIL "A is coercible from \\spad{B} iff any element of domain \\spad{B} can be automically converted into an element of domain A.")) (|coerce| (($ |#1|) "\\spad{coerce(s)} transforms `s' into an element of `\\%'."))) NIL NIL -(-553) +(-556) ((|constructor| (NIL "This domain implements Kleene's 3-valued propositional logic.")) (|case| (((|Boolean|) $ (|[\|\|]| |true|)) "\\spad{s case true} holds if the value of `x' is `true'.") (((|Boolean|) $ (|[\|\|]| |unknown|)) "\\spad{x case unknown} holds if the value of `x' is `unknown'") (((|Boolean|) $ (|[\|\|]| |false|)) "\\spad{x case false} holds if the value of `x' is `false'")) (|unknown| (($) "the indefinite `unknown'"))) NIL NIL -(-554 S) +(-557 S) ((|constructor| (NIL "A is convertible from \\spad{B} iff any element of domain \\spad{B} can be explicitly converted into an element of domain A.")) (|convert| (($ |#1|) "\\spad{convert(s)} transforms `s' into an element of `\\%'."))) NIL NIL -(-555 A R S) +(-558 A R S) ((|constructor| (NIL "LocalAlgebra produces the localization of an algebra,{} \\spadignore{i.e.} fractions whose numerators come from some \\spad{R} algebra.")) (|denom| ((|#3| $) "\\spad{denom x} returns the denominator of \\spad{x}.")) (|numer| ((|#1| $) "\\spad{numer x} returns the numerator of \\spad{x}.")) (/ (($ |#1| |#3|) "\\spad{a / d} divides the element \\spad{a} by \\spad{d}.") (($ $ |#3|) "\\spad{x / d} divides the element \\spad{x} by \\spad{d}."))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-750)))) -(-556 S R) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-755)))) +(-559 S R) ((|constructor| (NIL "The category of all left algebras over an arbitrary ring.")) (|coerce| (($ |#2|) "\\spad{coerce(r)} returns \\spad{r} * 1 where 1 is the identity of the left algebra."))) NIL NIL -(-557 R) +(-560 R) ((|constructor| (NIL "The category of all left algebras over an arbitrary ring.")) (|coerce| (($ |#1|) "\\spad{coerce(r)} returns \\spad{r} * 1 where 1 is the identity of the left algebra."))) -((-3975 . T)) +((-3986 . T)) NIL -(-558 R -3078) +(-561 R -3088) ((|constructor| (NIL "This package computes the forward Laplace Transform.")) (|laplace| ((|#2| |#2| (|Symbol|) (|Symbol|)) "\\spad{laplace(f, t, s)} returns the Laplace transform of \\spad{f(t)} using \\spad{s} as the new variable. This is \\spad{integral(exp(-s*t)*f(t), t = 0..\\%plusInfinity)}. Returns the formal object \\spad{laplace(f, t, s)} if it cannot compute the transform."))) NIL NIL -(-559 R UP) +(-562 R UP) ((|constructor| (NIL "\\indented{1}{Univariate polynomials with negative and positive exponents.} Author: Manuel Bronstein Date Created: May 1988 Date Last Updated: 26 Apr 1990")) (|separate| (((|Record| (|:| |polyPart| $) (|:| |fracPart| (|Fraction| |#2|))) (|Fraction| |#2|)) "\\spad{separate(x)} \\undocumented")) (|monomial| (($ |#1| (|Integer|)) "\\spad{monomial(x,n)} \\undocumented")) (|coefficient| ((|#1| $ (|Integer|)) "\\spad{coefficient(x,n)} \\undocumented")) (|trailingCoefficient| ((|#1| $) "\\spad{trailingCoefficient }\\undocumented")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient }\\undocumented")) (|reductum| (($ $) "\\spad{reductum(x)} \\undocumented")) (|order| (((|Integer|) $) "\\spad{order(x)} \\undocumented")) (|degree| (((|Integer|) $) "\\spad{degree(x)} \\undocumented")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} \\undocumented"))) -((-3973 . T) (-3972 . T) ((-3980 "*") . T) (-3971 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-804 (-1081)))) (|HasCategory| |#2| (QUOTE (-806 (-1081)))) (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480))))) -(-560 R E V P TS ST) +((-3984 . T) (-3983 . T) ((-3991 "*") . T) (-3982 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasCategory| |#2| (QUOTE (-811 (-1088)))) (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483))))) +(-563 R E V P TS ST) ((|constructor| (NIL "A package for solving polynomial systems by means of Lazard triangular sets [1]. This package provides two operations. One for solving in the sense of the regular zeros,{} and the other for solving in the sense of the Zariski closure. Both produce square-free regular sets. Moreover,{} the decompositions do not contain any redundant component. However,{} only zero-dimensional regular sets are normalized,{} since normalization may be time consumming in positive dimension. The decomposition process is that of [2].\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|zeroSetSplit| (((|List| |#6|) (|List| |#4|) (|Boolean|)) "\\axiom{zeroSetSplit(lp,{}clos?)} has the same specifications as \\axiomOpFrom{zeroSetSplit(lp,{}clos?)}{RegularTriangularSetCategory}.")) (|normalizeIfCan| ((|#6| |#6|) "\\axiom{normalizeIfCan(ts)} returns \\axiom{ts} in an normalized shape if \\axiom{ts} is zero-dimensional."))) NIL NIL -(-561 OV E Z P) +(-564 OV E Z P) ((|constructor| (NIL "Package for leading coefficient determination in the lifting step. Package working for every \\spad{R} euclidean with property \"F\".")) (|distFact| (((|Union| (|Record| (|:| |polfac| (|List| |#4|)) (|:| |correct| |#3|) (|:| |corrfact| (|List| (|SparseUnivariatePolynomial| |#3|)))) "failed") |#3| (|List| (|SparseUnivariatePolynomial| |#3|)) (|Record| (|:| |contp| |#3|) (|:| |factors| (|List| (|Record| (|:| |irr| |#4|) (|:| |pow| (|Integer|)))))) (|List| |#3|) (|List| |#1|) (|List| |#3|)) "\\spad{distFact(contm,unilist,plead,vl,lvar,lval)},{} where \\spad{contm} is the content of the evaluated polynomial,{} \\spad{unilist} is the list of factors of the evaluated polynomial,{} \\spad{plead} is the complete factorization of the leading coefficient,{} \\spad{vl} is the list of factors of the leading coefficient evaluated,{} \\spad{lvar} is the list of variables,{} \\spad{lval} is the list of values,{} returns a record giving the list of leading coefficients to impose on the univariate factors,{}")) (|polCase| (((|Boolean|) |#3| (|NonNegativeInteger|) (|List| |#3|)) "\\spad{polCase(contprod, numFacts, evallcs)},{} where \\spad{contprod} is the product of the content of the leading coefficient of the polynomial to be factored with the content of the evaluated polynomial,{} \\spad{numFacts} is the number of factors of the leadingCoefficient,{} and evallcs is the list of the evaluated factors of the leadingCoefficient,{} returns \\spad{true} if the factors of the leading Coefficient can be distributed with this valuation."))) NIL NIL -(-562) +(-565) ((|constructor| (NIL "This domain represents assignment expressions.")) (|rhs| (((|SpadAst|) $) "\\spad{rhs(e)} returns the right hand side of the assignment expression `e'.")) (|lhs| (((|SpadAst|) $) "\\spad{lhs(e)} returns the left hand side of the assignment expression `e'."))) NIL NIL -(-563 |VarSet| R |Order|) +(-566 |VarSet| R |Order|) ((|constructor| (NIL "Management of the Lie Group associated with a free nilpotent Lie algebra. Every Lie bracket with length greater than \\axiom{Order} are assumed to be null. The implementation inherits from the \\spadtype{XPBWPolynomial} domain constructor: Lyndon coordinates are exponential coordinates of the second kind. \\newline Author: Michel Petitot (petitot@lifl.fr).")) (|identification| (((|List| (|Equation| |#2|)) $ $) "\\axiom{identification(\\spad{g},{}\\spad{h})} returns the list of equations \\axiom{g_i = h_i},{} where \\axiom{g_i} (resp. \\axiom{h_i}) are exponential coordinates of \\axiom{\\spad{g}} (resp. \\axiom{\\spad{h}}).")) (|LyndonCoordinates| (((|List| (|Record| (|:| |k| (|LyndonWord| |#1|)) (|:| |c| |#2|))) $) "\\axiom{LyndonCoordinates(\\spad{g})} returns the exponential coordinates of \\axiom{\\spad{g}}.")) (|LyndonBasis| (((|List| (|LiePolynomial| |#1| |#2|)) (|List| |#1|)) "\\axiom{LyndonBasis(lv)} returns the Lyndon basis of the nilpotent free Lie algebra.")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{g})} returns the list of variables of \\axiom{\\spad{g}}.")) (|mirror| (($ $) "\\axiom{mirror(\\spad{g})} is the mirror of the internal representation of \\axiom{\\spad{g}}.")) (|coerce| (((|XPBWPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{g})} returns the internal representation of \\axiom{\\spad{g}}.") (((|XDistributedPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{g})} returns the internal representation of \\axiom{\\spad{g}}.")) (|ListOfTerms| (((|List| (|Record| (|:| |k| (|PoincareBirkhoffWittLyndonBasis| |#1|)) (|:| |c| |#2|))) $) "\\axiom{ListOfTerms(\\spad{p})} returns the internal representation of \\axiom{\\spad{p}}.")) (|log| (((|LiePolynomial| |#1| |#2|) $) "\\axiom{log(\\spad{p})} returns the logarithm of \\axiom{\\spad{p}}.")) (|exp| (($ (|LiePolynomial| |#1| |#2|)) "\\axiom{exp(\\spad{p})} returns the exponential of \\axiom{\\spad{p}}."))) -((-3975 . T)) +((-3986 . T)) NIL -(-564 R |ls|) +(-567 R |ls|) ((|constructor| (NIL "A package for solving polynomial systems with finitely many solutions. The decompositions are given by means of regular triangular sets. The computations use lexicographical Groebner bases. The main operations are \\axiomOpFrom{lexTriangular}{LexTriangularPackage} and \\axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}. The second one provide decompositions by means of square-free regular triangular sets. Both are based on the {\\em lexTriangular} method described in [1]. They differ from the algorithm described in [2] by the fact that multiciplities of the roots are not kept. With the \\axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} operation all multiciplities are removed. With the other operation some multiciplities may remain. Both operations admit an optional argument to produce normalized triangular sets. \\newline")) (|zeroSetSplit| (((|List| (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#2|)) (|OrderedVariableList| |#2|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{zeroSetSplit(lp,{} norm?)} decomposes the variety associated with \\axiom{lp} into square-free regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{lp} needs to generate a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.") (((|List| (|RegularChain| |#1| |#2|)) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{zeroSetSplit(lp,{} norm?)} decomposes the variety associated with \\axiom{lp} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{lp} needs to generate a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.")) (|squareFreeLexTriangular| (((|List| (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#2|)) (|OrderedVariableList| |#2|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{squareFreeLexTriangular(base,{} norm?)} decomposes the variety associated with \\axiom{base} into square-free regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{base} needs to be a lexicographical Groebner basis of a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.")) (|lexTriangular| (((|List| (|RegularChain| |#1| |#2|)) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|)) "\\axiom{lexTriangular(base,{} norm?)} decomposes the variety associated with \\axiom{base} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{base} needs to be a lexicographical Groebner basis of a zero-dimensional ideal. If \\axiom{norm?} is \\axiom{\\spad{true}} then the regular sets are normalized.")) (|groebner| (((|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) "\\axiom{groebner(lp)} returns the lexicographical Groebner basis of \\axiom{lp}. If \\axiom{lp} generates a zero-dimensional ideal then the {\\em FGLM} strategy is used,{} otherwise the {\\em Sugar} strategy is used.")) (|fglmIfCan| (((|Union| (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) "failed") (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) "\\axiom{fglmIfCan(lp)} returns the lexicographical Groebner basis of \\axiom{lp} by using the {\\em FGLM} strategy,{} if \\axiom{zeroDimensional?(lp)} holds .")) (|zeroDimensional?| (((|Boolean|) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|)))) "\\axiom{zeroDimensional?(lp)} returns \\spad{true} iff \\axiom{lp} generates a zero-dimensional ideal \\spad{w}.\\spad{r}.\\spad{t}. the variables involved in \\axiom{lp}."))) NIL NIL -(-565 R -3078) +(-568 R -3088) ((|constructor| (NIL "This package provides liouvillian functions over an integral domain.")) (|integral| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{integral(f,x = a..b)} denotes the definite integral of \\spad{f} with respect to \\spad{x} from \\spad{a} to \\spad{b}.") ((|#2| |#2| (|Symbol|)) "\\spad{integral(f,x)} indefinite integral of \\spad{f} with respect to \\spad{x}.")) (|dilog| ((|#2| |#2|) "\\spad{dilog(f)} denotes the dilogarithm")) (|erf| ((|#2| |#2|) "\\spad{erf(f)} denotes the error function")) (|li| ((|#2| |#2|) "\\spad{li(f)} denotes the logarithmic integral")) (|Ci| ((|#2| |#2|) "\\spad{Ci(f)} denotes the cosine integral")) (|Si| ((|#2| |#2|) "\\spad{Si(f)} denotes the sine integral")) (|Ei| ((|#2| |#2|) "\\spad{Ei(f)} denotes the exponential integral")) (|operator| (((|BasicOperator|) (|BasicOperator|)) "\\spad{operator(op)} returns the Liouvillian operator based on \\spad{op}")) (|belong?| (((|Boolean|) (|BasicOperator|)) "\\spad{belong?(op)} checks if \\spad{op} is Liouvillian"))) NIL NIL -(-566) +(-569) ((|constructor| (NIL "Category for the transcendental Liouvillian functions.")) (|erf| (($ $) "\\spad{erf(x)} returns the error function of \\spad{x},{} \\spadignore{i.e.} \\spad{2 / sqrt(\\%pi)} times the integral of \\spad{exp(-x**2) dx}.")) (|dilog| (($ $) "\\spad{dilog(x)} returns the dilogarithm of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{log(x) / (1 - x) dx}.")) (|li| (($ $) "\\spad{li(x)} returns the logarithmic integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{dx / log(x)}.")) (|Ci| (($ $) "\\spad{Ci(x)} returns the cosine integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{cos(x) / x dx}.")) (|Si| (($ $) "\\spad{Si(x)} returns the sine integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{sin(x) / x dx}.")) (|Ei| (($ $) "\\spad{Ei(x)} returns the exponential integral of \\spad{x},{} \\spadignore{i.e.} the integral of \\spad{exp(x)/x dx}."))) NIL NIL -(-567 |lv| -3078) +(-570 |lv| -3088) ((|constructor| (NIL "\\indented{1}{Given a Groebner basis \\spad{B} with respect to the total degree ordering for} a zero-dimensional ideal \\spad{I},{} compute a Groebner basis with respect to the lexicographical ordering by using linear algebra.")) (|transform| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{transform }\\undocumented")) (|choosemon| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{choosemon }\\undocumented")) (|intcompBasis| (((|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{intcompBasis }\\undocumented")) (|anticoord| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|List| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{anticoord }\\undocumented")) (|coord| (((|Vector| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{coord }\\undocumented")) (|computeBasis| (((|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{computeBasis }\\undocumented")) (|minPol| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|)) "\\spad{minPol }\\undocumented") (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) (|OrderedVariableList| |#1|)) "\\spad{minPol }\\undocumented")) (|totolex| (((|List| (|DistributedMultivariatePolynomial| |#1| |#2|)) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{totolex }\\undocumented")) (|groebgen| (((|Record| (|:| |glbase| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |glval| (|List| (|Integer|)))) (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{groebgen }\\undocumented")) (|linGenPos| (((|Record| (|:| |gblist| (|List| (|DistributedMultivariatePolynomial| |#1| |#2|))) (|:| |gvlist| (|List| (|Integer|)))) (|List| (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|))) "\\spad{linGenPos }\\undocumented"))) NIL NIL -(-568) +(-571) ((|constructor| (NIL "This domain provides a simple way to save values in files.")) (|setelt| (((|Any|) $ (|Symbol|) (|Any|)) "\\spad{lib.k := v} saves the value \\spad{v} in the library \\spad{lib}. It can later be extracted using the key \\spad{k}.")) (|pack!| (($ $) "\\spad{pack!(f)} reorganizes the file \\spad{f} on disk to recover unused space.")) (|library| (($ (|FileName|)) "\\spad{library(ln)} creates a new library file."))) -((-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-257 (-2 (|:| -3843 (-1064)) (|:| |entry| (-51)))))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-1007)))) (OR (|HasCategory| (-51) (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-1007)))) (OR (|HasCategory| (-51) (QUOTE (-72))) (|HasCategory| (-51) (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-549 (-767)))) (|HasCategory| (-51) (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-550 (-469)))) (-12 (|HasCategory| (-51) (QUOTE (-257 (-51)))) (|HasCategory| (-51) (QUOTE (-1007)))) (|HasCategory| (-1064) (QUOTE (-751))) (OR (|HasCategory| (-51) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-72)))) (|HasCategory| (-51) (QUOTE (-1007))) (|HasCategory| (-51) (QUOTE (-72))) (|HasCategory| (-51) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| (-51))) (QUOTE (-1007)))) -(-569 R A) +((-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-259 (-2 (|:| -3854 (-1071)) (|:| |entry| (-51)))))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-1012)))) (OR (|HasCategory| (-51) (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-1012)))) (OR (|HasCategory| (-51) (QUOTE (-72))) (|HasCategory| (-51) (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-552 (-772)))) (|HasCategory| (-51) (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-553 (-472)))) (-12 (|HasCategory| (-51) (QUOTE (-259 (-51)))) (|HasCategory| (-51) (QUOTE (-1012)))) (|HasCategory| (-1071) (QUOTE (-756))) (OR (|HasCategory| (-51) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-72)))) (|HasCategory| (-51) (QUOTE (-1012))) (|HasCategory| (-51) (QUOTE (-72))) (|HasCategory| (-51) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| (-51))) (QUOTE (-1012)))) +(-572 R A) ((|constructor| (NIL "AssociatedLieAlgebra takes an algebra \\spad{A} and uses \\spadfun{*\\$A} to define the Lie bracket \\spad{a*b := (a *\\$A b - b *\\$A a)} (commutator). Note that the notation \\spad{[a,b]} cannot be used due to restrictions of the current compiler. This domain only gives a Lie algebra if the Jacobi-identity \\spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds for all \\spad{a},{}\\spad{b},{}\\spad{c} in \\spad{A}. This relation can be checked by \\spad{lieAdmissible?()\\$A}. \\blankline If the underlying algebra is of type \\spadtype{FramedNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank,{} together with a fixed \\spad{R}-module basis),{} then the same is \\spad{true} for the associated Lie algebra. Also,{} if the underlying algebra is of type \\spadtype{FiniteRankNonAssociativeAlgebra(R)} (\\spadignore{i.e.} a non associative algebra over \\spad{R} which is a free \\spad{R}-module of finite rank),{} then the same is \\spad{true} for the associated Lie algebra.")) (|coerce| (($ |#2|) "\\spad{coerce(a)} coerces the element \\spad{a} of the algebra \\spad{A} to an element of the Lie algebra \\spadtype{AssociatedLieAlgebra}(\\spad{R},{}A)."))) -((-3975 OR (-2548 (|has| |#2| (-313 |#1|)) (|has| |#1| (-491))) (-12 (|has| |#2| (-356 |#1|)) (|has| |#1| (-491)))) (-3973 . T) (-3972 . T)) -((OR (|HasCategory| |#2| (|%list| (QUOTE -313) (|devaluate| |#1|))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|)))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#2| (|%list| (QUOTE -313) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#2| (|%list| (QUOTE -356) (|devaluate| |#1|))))) (|HasCategory| |#2| (|%list| (QUOTE -313) (|devaluate| |#1|)))) -(-570 S R) +((-3986 OR (-2558 (|has| |#2| (-315 |#1|)) (|has| |#1| (-494))) (-12 (|has| |#2| (-358 |#1|)) (|has| |#1| (-494)))) (-3984 . T) (-3983 . T)) +((OR (|HasCategory| |#2| (|%list| (QUOTE -315) (|devaluate| |#1|))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|)))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#2| (|%list| (QUOTE -315) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#2| (|%list| (QUOTE -358) (|devaluate| |#1|))))) (|HasCategory| |#2| (|%list| (QUOTE -315) (|devaluate| |#1|)))) +(-573 S R) ((|constructor| (NIL "\\axiom{JacobiIdentity} means that \\axiom{[\\spad{x},{}[\\spad{y},{}\\spad{z}]]+[\\spad{y},{}[\\spad{z},{}\\spad{x}]]+[\\spad{z},{}[\\spad{x},{}\\spad{y}]] = 0} holds.")) (/ (($ $ |#2|) "\\axiom{x/r} returns the division of \\axiom{\\spad{x}} by \\axiom{\\spad{r}}.")) (|construct| (($ $ $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket of \\axiom{\\spad{x}} and \\axiom{\\spad{y}}."))) NIL -((|HasCategory| |#2| (QUOTE (-309)))) -(-571 R) +((|HasCategory| |#2| (QUOTE (-311)))) +(-574 R) ((|constructor| (NIL "\\axiom{JacobiIdentity} means that \\axiom{[\\spad{x},{}[\\spad{y},{}\\spad{z}]]+[\\spad{y},{}[\\spad{z},{}\\spad{x}]]+[\\spad{z},{}[\\spad{x},{}\\spad{y}]] = 0} holds.")) (/ (($ $ |#1|) "\\axiom{x/r} returns the division of \\axiom{\\spad{x}} by \\axiom{\\spad{r}}.")) (|construct| (($ $ $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket of \\axiom{\\spad{x}} and \\axiom{\\spad{y}}."))) -((|JacobiIdentity| . T) (|NullSquare| . T) (-3973 . T) (-3972 . T)) +((|JacobiIdentity| . T) (|NullSquare| . T) (-3984 . T) (-3983 . T)) NIL -(-572 R FE) +(-575 R FE) ((|constructor| (NIL "PowerSeriesLimitPackage implements limits of expressions in one or more variables as one of the variables approaches a limiting value. Included are two-sided limits,{} left- and right- hand limits,{} and limits at plus or minus infinity.")) (|complexLimit| (((|Union| (|OnePointCompletion| |#2|) "failed") |#2| (|Equation| (|OnePointCompletion| |#2|))) "\\spad{complexLimit(f(x),x = a)} computes the complex limit \\spad{lim(x -> a,f(x))}.")) (|limit| (((|Union| (|OrderedCompletion| |#2|) #1="failed") |#2| (|Equation| |#2|) (|String|)) "\\spad{limit(f(x),x=a,\"left\")} computes the left hand real limit \\spad{lim(x -> a-,f(x))}; \\spad{limit(f(x),x=a,\"right\")} computes the right hand real limit \\spad{lim(x -> a+,f(x))}.") (((|Union| (|OrderedCompletion| |#2|) (|Record| (|:| |leftHandLimit| (|Union| (|OrderedCompletion| |#2|) #1#)) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| |#2|) #1#))) "failed") |#2| (|Equation| (|OrderedCompletion| |#2|))) "\\spad{limit(f(x),x = a)} computes the real limit \\spad{lim(x -> a,f(x))}."))) NIL NIL -(-573 R) +(-576 R) ((|constructor| (NIL "Computation of limits for rational functions.")) (|complexLimit| (((|OnePointCompletion| (|Fraction| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{complexLimit(f(x),x = a)} computes the complex limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}.") (((|OnePointCompletion| (|Fraction| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|OnePointCompletion| (|Polynomial| |#1|)))) "\\spad{complexLimit(f(x),x = a)} computes the complex limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}.")) (|limit| (((|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1="failed") (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|String|)) "\\spad{limit(f(x),x,a,\"left\")} computes the real limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a} from the left; limit(\\spad{f}(\\spad{x}),{}\\spad{x},{}a,{}\"right\") computes the corresponding limit as \\spad{x} approaches \\spad{a} from the right.") (((|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) (|Record| (|:| |leftHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#)) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#))) #2="failed") (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{limit(f(x),x = a)} computes the real two-sided limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}.") (((|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) (|Record| (|:| |leftHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#)) (|:| |rightHandLimit| (|Union| (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|))) #1#))) #2#) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|OrderedCompletion| (|Polynomial| |#1|)))) "\\spad{limit(f(x),x = a)} computes the real two-sided limit of \\spad{f} as its argument \\spad{x} approaches \\spad{a}."))) NIL NIL -(-574 |vars|) +(-577 |vars|) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: July 2,{} 2010 Date Last Modified: July 2,{} 2010 Descrption: \\indented{2}{Representation of a vector space basis,{} given by symbols.}")) (|dual| (($ (|DualBasis| |#1|)) "\\spad{dual f} constructs the dual vector of a linear form which is part of a basis."))) NIL NIL -(-575 S R) +(-578 S R) ((|constructor| (NIL "Test for linear dependence.")) (|solveLinear| (((|Union| (|Vector| (|Fraction| |#1|)) "failed") (|Vector| |#2|) |#2|) "\\spad{solveLinear([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such \\spad{ci}'s exist in the quotient field of \\spad{S}.") (((|Union| (|Vector| |#1|) "failed") (|Vector| |#2|) |#2|) "\\spad{solveLinear([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such \\spad{ci}'s exist in \\spad{S}.")) (|linearDependence| (((|Union| (|Vector| |#1|) "failed") (|Vector| |#2|)) "\\spad{linearDependence([v1,...,vn])} returns \\spad{[c1,...,cn]} if \\spad{c1*v1 + ... + cn*vn = 0} and not all the \\spad{ci}'s are 0,{} \"failed\" if the \\spad{vi}'s are linearly independent over \\spad{S}.")) (|linearlyDependent?| (((|Boolean|) (|Vector| |#2|)) "\\spad{linearlyDependent?([v1,...,vn])} returns \\spad{true} if the \\spad{vi}'s are linearly dependent over \\spad{S},{} \\spad{false} otherwise."))) NIL -((-2546 (|HasCategory| |#1| (QUOTE (-309)))) (|HasCategory| |#1| (QUOTE (-309)))) -(-576 K B) +((-2556 (|HasCategory| |#1| (QUOTE (-311)))) (|HasCategory| |#1| (QUOTE (-311)))) +(-579 K B) ((|constructor| (NIL "A simple data structure for elements that form a vector space of finite dimension over a given field,{} with a given symbolic basis.")) (|coordinates| (((|Vector| |#1|) $) "\\spad{coordinates x} returns the coordinates of the linear element with respect to the basis \\spad{B}.")) (|linearElement| (($ (|List| |#1|)) "\\spad{linearElement [x1,..,xn]} returns a linear element \\indented{1}{with coordinates \\spad{[x1,..,xn]} with respect to} the basis elements \\spad{B}."))) -((-3973 . T) (-3972 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| (-574 |#2|) (QUOTE (-1007))))) -(-577 R) +((-3984 . T) (-3983 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| (-577 |#2|) (QUOTE (-1012))))) +(-580 R) ((|constructor| (NIL "An extension of left-module with an explicit linear dependence test.")) (|reducedSystem| (((|Record| (|:| |mat| (|Matrix| |#1|)) (|:| |vec| (|Vector| |#1|))) (|Matrix| $) (|Vector| $)) "\\spad{reducedSystem(A, v)} returns a matrix \\spad{B} and a vector \\spad{w} such that \\spad{A x = v} and \\spad{B x = w} have the same solutions in \\spad{R}.") (((|Matrix| |#1|) (|Matrix| $)) "\\spad{reducedSystem(A)} returns a matrix \\spad{B} such that \\spad{A x = 0} and \\spad{B x = 0} have the same solutions in \\spad{R}.")) (|leftReducedSystem| (((|Record| (|:| |mat| (|Matrix| |#1|)) (|:| |vec| (|Vector| |#1|))) (|Vector| $) $) "\\spad{reducedSystem([v1,...,vn],u)} returns a matrix \\spad{M} with coefficients in \\spad{R} and a vector \\spad{w} such that the system of equations \\spad{c1*v1 + ... + cn*vn = u} has the same solution as \\spad{c * M = w} where \\spad{c} is the row vector \\spad{[c1,...cn]}.") (((|Matrix| |#1|) (|Vector| $)) "\\spad{leftReducedSystem [v1,...,vn]} returns a matrix \\spad{M} with coefficients in \\spad{R} such that the system of equations \\spad{c1*v1 + ... + cn*vn = 0\\$\\%} has the same solution as \\spad{c * M = 0} where \\spad{c} is the row vector \\spad{[c1,...cn]}."))) NIL NIL -(-578 K B) +(-581 K B) ((|constructor| (NIL "A simple data structure for linear forms on a vector space of finite dimension over a given field,{} with a given symbolic basis.")) (|coordinates| (((|Vector| |#1|) $) "\\spad{coordinates x} returns the coordinates of the linear form with respect to the basis \\spad{DualBasis B}.")) (|linearForm| (($ (|List| |#1|)) "\\spad{linearForm [x1,..,xn]} constructs a linear form with coordinates \\spad{[x1,..,xn]} with respect to the basis elements \\spad{DualBasis B}."))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-579 S) +(-582 S) ((|constructor| (NIL "\\indented{2}{A set is an \\spad{S}-linear set if it is stable by dilation} \\indented{2}{by elements in the semigroup \\spad{S}.} See Also: LeftLinearSet,{} RightLinearSet."))) NIL NIL -(-580 S) +(-583 S) ((|constructor| (NIL "\\spadtype{List} implements singly-linked lists that are addressable by indices; the index of the first element is 1. this constructor provides some LISP-like functions such as \\spadfun{null} and \\spadfun{cons}.")) (|setDifference| (($ $ $) "\\spad{setDifference(u1,u2)} returns a list of the elements of \\spad{u1} that are not also in \\spad{u2}. The order of elements in the resulting list is unspecified.")) (|setIntersection| (($ $ $) "\\spad{setIntersection(u1,u2)} returns a list of the elements that lists \\spad{u1} and \\spad{u2} have in common. The order of elements in the resulting list is unspecified.")) (|setUnion| (($ $ $) "\\spad{setUnion(u1,u2)} appends the two lists \\spad{u1} and \\spad{u2},{} then removes all duplicates. The order of elements in the resulting list is unspecified.")) (|append| (($ $ $) "\\spad{append(u1,u2)} appends the elements of list \\spad{u1} onto the front of list \\spad{u2}. This new list and \\spad{u2} will share some structure.")) (|cons| (($ |#1| $) "\\spad{cons(element,u)} appends \\spad{element} onto the front of list \\spad{u} and returns the new list. This new list and the old one will share some structure.")) (|null| (((|Boolean|) $) "\\spad{null(u)} tests if list \\spad{u} is the empty list.")) (|nil| (($) "\\spad{nil} is the empty list."))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-581 A B) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-584 A B) ((|constructor| (NIL "\\spadtype{ListFunctions2} implements utility functions that operate on two kinds of lists,{} each with a possibly different type of element.")) (|map| (((|List| |#2|) (|Mapping| |#2| |#1|) (|List| |#1|)) "\\spad{map(fn,u)} applies \\spad{fn} to each element of list \\spad{u} and returns a new list with the results. For example \\spad{map(square,[1,2,3]) = [1,4,9]}.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|List| |#1|) |#2|) "\\spad{reduce(fn,u,ident)} successively uses the binary function \\spad{fn} on the elements of list \\spad{u} and the result of previous applications. \\spad{ident} is returned if the \\spad{u} is empty. Note the order of application in the following examples: \\spad{reduce(fn,[1,2,3],0) = fn(3,fn(2,fn(1,0)))} and \\spad{reduce(*,[2,3],1) = 3 * (2 * 1)}.")) (|scan| (((|List| |#2|) (|Mapping| |#2| |#1| |#2|) (|List| |#1|) |#2|) "\\spad{scan(fn,u,ident)} successively uses the binary function \\spad{fn} to reduce more and more of list \\spad{u}. \\spad{ident} is returned if the \\spad{u} is empty. The result is a list of the reductions at each step. See \\spadfun{reduce} for more information. Examples: \\spad{scan(fn,[1,2],0) = [fn(2,fn(1,0)),fn(1,0)]} and \\spad{scan(*,[2,3],1) = [2 * 1, 3 * (2 * 1)]}."))) NIL NIL -(-582 A B) +(-585 A B) ((|constructor| (NIL "\\spadtype{ListToMap} allows mappings to be described by a pair of lists of equal lengths. The image of an element \\spad{x},{} which appears in position \\spad{n} in the first list,{} is then the \\spad{n}th element of the second list. A default value or default function can be specified to be used when \\spad{x} does not appear in the first list. In the absence of defaults,{} an error will occur in that case.")) (|match| ((|#2| (|List| |#1|) (|List| |#2|) |#1| (|Mapping| |#2| |#1|)) "\\spad{match(la, lb, a, f)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length. and applies this map to a. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Argument \\spad{f} is a default function to call if a is not in \\spad{la}. The value returned is then obtained by applying \\spad{f} to argument a.") (((|Mapping| |#2| |#1|) (|List| |#1|) (|List| |#2|) (|Mapping| |#2| |#1|)) "\\spad{match(la, lb, f)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Argument \\spad{f} is used as the function to call when the given function argument is not in \\spad{la}. The value returned is \\spad{f} applied to that argument.") ((|#2| (|List| |#1|) (|List| |#2|) |#1| |#2|) "\\spad{match(la, lb, a, b)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length. and applies this map to a. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Argument \\spad{b} is the default target value if a is not in \\spad{la}. Error: if \\spad{la} and \\spad{lb} are not of equal length.") (((|Mapping| |#2| |#1|) (|List| |#1|) (|List| |#2|) |#2|) "\\spad{match(la, lb, b)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length,{} where \\spad{b} is used as the default target value if the given function argument is not in \\spad{la}. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Error: if \\spad{la} and \\spad{lb} are not of equal length.") ((|#2| (|List| |#1|) (|List| |#2|) |#1|) "\\spad{match(la, lb, a)} creates a map defined by lists \\spad{la} and \\spad{lb} of equal length,{} where \\spad{a} is used as the default source value if the given one is not in \\spad{la}. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index \\spad{lb}. Error: if \\spad{la} and \\spad{lb} are not of equal length.") (((|Mapping| |#2| |#1|) (|List| |#1|) (|List| |#2|)) "\\spad{match(la, lb)} creates a map with no default source or target values defined by lists \\spad{la} and lb of equal length. The target of a source value \\spad{x} in \\spad{la} is the value \\spad{y} with the same index lb. Error: if \\spad{la} and lb are not of equal length. Note: when this map is applied,{} an error occurs when applied to a value missing from \\spad{la}."))) NIL NIL -(-583 A B C) +(-586 A B C) ((|constructor| (NIL "\\spadtype{ListFunctions3} implements utility functions that operate on three kinds of lists,{} each with a possibly different type of element.")) (|map| (((|List| |#3|) (|Mapping| |#3| |#1| |#2|) (|List| |#1|) (|List| |#2|)) "\\spad{map(fn,list1, u2)} applies the binary function \\spad{fn} to corresponding elements of lists \\spad{u1} and \\spad{u2} and returns a list of the results (in the same order). Thus \\spad{map(/,[1,2,3],[4,5,6]) = [1/4,2/4,1/2]}. The computation terminates when the end of either list is reached. That is,{} the length of the result list is equal to the minimum of the lengths of \\spad{u1} and \\spad{u2}."))) NIL NIL -(-584 T$) +(-587 T$) ((|constructor| (NIL "This domain represents AST for Spad literals."))) NIL NIL -(-585 S) +(-588 S) ((|constructor| (NIL "\\indented{2}{A set is an \\spad{S}-left linear set if it is stable by left-dilation} \\indented{2}{by elements in the semigroup \\spad{S}.} See Also: RightLinearSet.")) (* (($ |#1| $) "\\spad{s*x} is the left-dilation of \\spad{x} by \\spad{s}."))) NIL NIL -(-586 S) +(-589 S) ((|substitute| (($ |#1| |#1| $) "\\spad{substitute(x,y,d)} replace \\spad{x}'s with \\spad{y}'s in dictionary \\spad{d}.")) (|duplicates?| (((|Boolean|) $) "\\spad{duplicates?(d)} tests if dictionary \\spad{d} has duplicate entries."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-587 R) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-590 R) ((|constructor| (NIL "The category of left modules over an rng (ring not necessarily with unit). This is an abelian group which supports left multiplation by elements of the rng. \\blankline"))) NIL NIL -(-588 S E |un|) +(-591 S E |un|) ((|constructor| (NIL "This internal package represents monoid (abelian or not,{} with or without inverses) as lists and provides some common operations to the various flavors of monoids.")) (|mapGen| (($ (|Mapping| |#1| |#1|) $) "\\spad{mapGen(f, a1\\^e1 ... an\\^en)} returns \\spad{f(a1)\\^e1 ... f(an)\\^en}.")) (|mapExpon| (($ (|Mapping| |#2| |#2|) $) "\\spad{mapExpon(f, a1\\^e1 ... an\\^en)} returns \\spad{a1\\^f(e1) ... an\\^f(en)}.")) (|commutativeEquality| (((|Boolean|) $ $) "\\spad{commutativeEquality(x,y)} returns \\spad{true} if \\spad{x} and \\spad{y} are equal assuming commutativity")) (|plus| (($ $ $) "\\spad{plus(x, y)} returns \\spad{x + y} where \\spad{+} is the monoid operation,{} which is assumed commutative.") (($ |#1| |#2| $) "\\spad{plus(s, e, x)} returns \\spad{e * s + x} where \\spad{+} is the monoid operation,{} which is assumed commutative.")) (|leftMult| (($ |#1| $) "\\spad{leftMult(s, a)} returns \\spad{s * a} where \\spad{*} is the monoid operation,{} which is assumed non-commutative.")) (|rightMult| (($ $ |#1|) "\\spad{rightMult(a, s)} returns \\spad{a * s} where \\spad{*} is the monoid operation,{} which is assumed non-commutative.")) (|makeUnit| (($) "\\spad{makeUnit()} returns the unit element of the monomial.")) (|size| (((|NonNegativeInteger|) $) "\\spad{size(l)} returns the number of monomials forming \\spad{l}.")) (|reverse!| (($ $) "\\spad{reverse!(l)} reverses the list of monomials forming \\spad{l},{} destroying the element \\spad{l}.")) (|reverse| (($ $) "\\spad{reverse(l)} reverses the list of monomials forming \\spad{l}. This has some effect if the monoid is non-abelian,{} \\spadignore{i.e.} \\spad{reverse(a1\\^e1 ... an\\^en) = an\\^en ... a1\\^e1} which is different.")) (|nthFactor| ((|#1| $ (|Integer|)) "\\spad{nthFactor(l, n)} returns the factor of the n^th monomial of \\spad{l}.")) (|nthExpon| ((|#2| $ (|Integer|)) "\\spad{nthExpon(l, n)} returns the exponent of the n^th monomial of \\spad{l}.")) (|makeMulti| (($ (|List| (|Record| (|:| |gen| |#1|) (|:| |exp| |#2|)))) "\\spad{makeMulti(l)} returns the element whose list of monomials is \\spad{l}.")) (|makeTerm| (($ |#1| |#2|) "\\spad{makeTerm(s, e)} returns the monomial \\spad{s} exponentiated by \\spad{e} (\\spadignore{e.g.} s^e or \\spad{e} * \\spad{s}).")) (|listOfMonoms| (((|List| (|Record| (|:| |gen| |#1|) (|:| |exp| |#2|))) $) "\\spad{listOfMonoms(l)} returns the list of the monomials forming \\spad{l}.")) (|outputForm| (((|OutputForm|) $ (|Mapping| (|OutputForm|) (|OutputForm|) (|OutputForm|)) (|Mapping| (|OutputForm|) (|OutputForm|) (|OutputForm|)) (|Integer|)) "\\spad{outputForm(l, fop, fexp, unit)} converts the monoid element represented by \\spad{l} to an \\spadtype{OutputForm}. Argument unit is the output form for the \\spadignore{unit} of the monoid (\\spadignore{e.g.} 0 or 1),{} \\spad{fop(a, b)} is the output form for the monoid operation applied to \\spad{a} and \\spad{b} (\\spadignore{e.g.} \\spad{a + b},{} \\spad{a * b},{} \\spad{ab}),{} and \\spad{fexp(a, n)} is the output form for the exponentiation operation applied to \\spad{a} and \\spad{n} (\\spadignore{e.g.} \\spad{n a},{} \\spad{n * a},{} \\spad{a ** n},{} \\spad{a\\^n})."))) NIL NIL -(-589 A S) +(-592 A S) ((|constructor| (NIL "A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \\spadfun{concat} of two lists needs only to copy its first argument,{} whereas \\spadfun{concat} of two arrays needs to copy both arguments. Most of the operations exported here apply to infinite objects (\\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \\spadtype{FiniteLinearAggregate}.")) (|setelt| ((|#2| $ (|UniversalSegment| (|Integer|)) |#2|) "\\spad{setelt(u,i..j,x)} (also written: \\axiom{\\spad{u}(\\spad{i}..\\spad{j}) := \\spad{x}}) destructively replaces each element in the segment \\axiom{\\spad{u}(\\spad{i}..\\spad{j})} by \\spad{x}. The value \\spad{x} is returned. Note: \\spad{u} is destructively change so that \\axiom{\\spad{u}.\\spad{k} := \\spad{x} for \\spad{k} in \\spad{i}..\\spad{j}}; its length remains unchanged.")) (|insert| (($ $ $ (|Integer|)) "\\spad{insert(v,u,k)} returns a copy of \\spad{u} having \\spad{v} inserted beginning at the \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{v},{}\\spad{u},{}\\spad{k}) = concat( \\spad{u}(0..\\spad{k}-1),{} \\spad{v},{} \\spad{u}(\\spad{k}..) )}.") (($ |#2| $ (|Integer|)) "\\spad{insert(x,u,i)} returns a copy of \\spad{u} having \\spad{x} as its \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{x},{}a,{}\\spad{k}) = concat(concat(a(0..\\spad{k}-1),{}\\spad{x}),{}a(\\spad{k}..))}.")) (|delete| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete(u,i..j)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th through \\axiom{\\spad{j}}th element deleted. Note: \\axiom{delete(a,{}\\spad{i}..\\spad{j}) = concat(a(0..\\spad{i}-1),{}a(\\spad{j+1}..))}.") (($ $ (|Integer|)) "\\spad{delete(u,i)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th element deleted. Note: for lists,{} \\axiom{delete(a,{}\\spad{i}) == concat(a(0..\\spad{i} - 1),{}a(\\spad{i} + 1,{}..))}.")) (|map| (($ (|Mapping| |#2| |#2| |#2|) $ $) "\\spad{map(f,u,v)} returns a new collection \\spad{w} with elements \\axiom{\\spad{z} = \\spad{f}(\\spad{x},{}\\spad{y})} for corresponding elements \\spad{x} and \\spad{y} from \\spad{u} and \\spad{v}. Note: for linear aggregates,{} \\axiom{\\spad{w}.\\spad{i} = \\spad{f}(\\spad{u}.\\spad{i},{}\\spad{v}.\\spad{i})}.")) (|concat| (($ (|List| $)) "\\spad{concat(u)},{} where \\spad{u} is a lists of aggregates \\axiom{[a,{}\\spad{b},{}...,{}\\spad{c}]},{} returns a single aggregate consisting of the elements of \\axiom{a} followed by those of \\spad{b} followed ... by the elements of \\spad{c}. Note: \\axiom{concat(a,{}\\spad{b},{}...,{}\\spad{c}) = concat(a,{}concat(\\spad{b},{}...,{}\\spad{c}))}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} then \\axiom{\\spad{w}.\\spad{i} = \\spad{u}.\\spad{i} for \\spad{i} in indices \\spad{u}} and \\axiom{\\spad{w}.(\\spad{j} + maxIndex \\spad{u}) = \\spad{v}.\\spad{j} for \\spad{j} in indices \\spad{v}}.") (($ |#2| $) "\\spad{concat(x,u)} returns aggregate \\spad{u} with additional element at the front. Note: for lists: \\axiom{concat(\\spad{x},{}\\spad{u}) == concat([\\spad{x}],{}\\spad{u})}.") (($ $ |#2|) "\\spad{concat(u,x)} returns aggregate \\spad{u} with additional element \\spad{x} at the end. Note: for lists,{} \\axiom{concat(\\spad{u},{}\\spad{x}) == concat(\\spad{u},{}[\\spad{x}])}")) (|new| (($ (|NonNegativeInteger|) |#2|) "\\spad{new(n,x)} returns \\axiom{fill!(new \\spad{n},{}\\spad{x})}."))) NIL -((|HasAttribute| |#1| (QUOTE -3979))) -(-590 S) +((|HasAttribute| |#1| (QUOTE -3990))) +(-593 S) ((|constructor| (NIL "A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \\spadfun{concat} of two lists needs only to copy its first argument,{} whereas \\spadfun{concat} of two arrays needs to copy both arguments. Most of the operations exported here apply to infinite objects (\\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \\spadtype{FiniteLinearAggregate}.")) (|setelt| ((|#1| $ (|UniversalSegment| (|Integer|)) |#1|) "\\spad{setelt(u,i..j,x)} (also written: \\axiom{\\spad{u}(\\spad{i}..\\spad{j}) := \\spad{x}}) destructively replaces each element in the segment \\axiom{\\spad{u}(\\spad{i}..\\spad{j})} by \\spad{x}. The value \\spad{x} is returned. Note: \\spad{u} is destructively change so that \\axiom{\\spad{u}.\\spad{k} := \\spad{x} for \\spad{k} in \\spad{i}..\\spad{j}}; its length remains unchanged.")) (|insert| (($ $ $ (|Integer|)) "\\spad{insert(v,u,k)} returns a copy of \\spad{u} having \\spad{v} inserted beginning at the \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{v},{}\\spad{u},{}\\spad{k}) = concat( \\spad{u}(0..\\spad{k}-1),{} \\spad{v},{} \\spad{u}(\\spad{k}..) )}.") (($ |#1| $ (|Integer|)) "\\spad{insert(x,u,i)} returns a copy of \\spad{u} having \\spad{x} as its \\axiom{\\spad{i}}th element. Note: \\axiom{insert(\\spad{x},{}a,{}\\spad{k}) = concat(concat(a(0..\\spad{k}-1),{}\\spad{x}),{}a(\\spad{k}..))}.")) (|delete| (($ $ (|UniversalSegment| (|Integer|))) "\\spad{delete(u,i..j)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th through \\axiom{\\spad{j}}th element deleted. Note: \\axiom{delete(a,{}\\spad{i}..\\spad{j}) = concat(a(0..\\spad{i}-1),{}a(\\spad{j+1}..))}.") (($ $ (|Integer|)) "\\spad{delete(u,i)} returns a copy of \\spad{u} with the \\axiom{\\spad{i}}th element deleted. Note: for lists,{} \\axiom{delete(a,{}\\spad{i}) == concat(a(0..\\spad{i} - 1),{}a(\\spad{i} + 1,{}..))}.")) (|map| (($ (|Mapping| |#1| |#1| |#1|) $ $) "\\spad{map(f,u,v)} returns a new collection \\spad{w} with elements \\axiom{\\spad{z} = \\spad{f}(\\spad{x},{}\\spad{y})} for corresponding elements \\spad{x} and \\spad{y} from \\spad{u} and \\spad{v}. Note: for linear aggregates,{} \\axiom{\\spad{w}.\\spad{i} = \\spad{f}(\\spad{u}.\\spad{i},{}\\spad{v}.\\spad{i})}.")) (|concat| (($ (|List| $)) "\\spad{concat(u)},{} where \\spad{u} is a lists of aggregates \\axiom{[a,{}\\spad{b},{}...,{}\\spad{c}]},{} returns a single aggregate consisting of the elements of \\axiom{a} followed by those of \\spad{b} followed ... by the elements of \\spad{c}. Note: \\axiom{concat(a,{}\\spad{b},{}...,{}\\spad{c}) = concat(a,{}concat(\\spad{b},{}...,{}\\spad{c}))}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} then \\axiom{\\spad{w}.\\spad{i} = \\spad{u}.\\spad{i} for \\spad{i} in indices \\spad{u}} and \\axiom{\\spad{w}.(\\spad{j} + maxIndex \\spad{u}) = \\spad{v}.\\spad{j} for \\spad{j} in indices \\spad{v}}.") (($ |#1| $) "\\spad{concat(x,u)} returns aggregate \\spad{u} with additional element at the front. Note: for lists: \\axiom{concat(\\spad{x},{}\\spad{u}) == concat([\\spad{x}],{}\\spad{u})}.") (($ $ |#1|) "\\spad{concat(u,x)} returns aggregate \\spad{u} with additional element \\spad{x} at the end. Note: for lists,{} \\axiom{concat(\\spad{u},{}\\spad{x}) == concat(\\spad{u},{}[\\spad{x}])}")) (|new| (($ (|NonNegativeInteger|) |#1|) "\\spad{new(n,x)} returns \\axiom{fill!(new \\spad{n},{}\\spad{x})}."))) NIL NIL -(-591 M R S) +(-594 M R S) ((|constructor| (NIL "Localize(\\spad{M},{}\\spad{R},{}\\spad{S}) produces fractions with numerators from an \\spad{R} module \\spad{M} and denominators from some multiplicative subset \\spad{D} of \\spad{R}.")) (|denom| ((|#3| $) "\\spad{denom x} returns the denominator of \\spad{x}.")) (|numer| ((|#1| $) "\\spad{numer x} returns the numerator of \\spad{x}.")) (/ (($ |#1| |#3|) "\\spad{m / d} divides the element \\spad{m} by \\spad{d}.") (($ $ |#3|) "\\spad{x / d} divides the element \\spad{x} by \\spad{d}."))) -((-3973 . T) (-3972 . T)) -((|HasCategory| |#1| (QUOTE (-709)))) -(-592 R -3078 L) +((-3984 . T) (-3983 . T)) +((|HasCategory| |#1| (QUOTE (-714)))) +(-595 R -3088 L) ((|constructor| (NIL "\\spad{ElementaryFunctionLODESolver} provides the top-level functions for finding closed form solutions of linear ordinary differential equations and initial value problems.")) (|solve| (((|Union| |#2| "failed") |#3| |#2| (|Symbol|) |#2| (|List| |#2|)) "\\spad{solve(op, g, x, a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{op y = g, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; \\spad{x} is the dependent variable.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) "failed") |#3| |#2| (|Symbol|)) "\\spad{solve(op, g, x)} returns either a solution of the ordinary differential equation \\spad{op y = g} or \"failed\" if no non-trivial solution can be found; When found,{} the solution is returned in the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{op y = 0}. A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; \\spad{x} is the dependent variable."))) NIL NIL -(-593 A -2478) +(-596 A -2488) ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperator} defines a ring of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}"))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-309)))) -(-594 A) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-311)))) +(-597 A) ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperator1} defines a ring of differential operators with coefficients in a differential ring A. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}"))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-309)))) -(-595 A M) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-311)))) +(-598 A M) ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperator2} defines a ring of differential operators with coefficients in a differential ring A and acting on an A-module \\spad{M}. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|differentiate| (($ $) "\\spad{differentiate(x)} returns the derivative of \\spad{x}"))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-309)))) -(-596 S A) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-311)))) +(-599 S A) ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|directSum| (($ $ $) "\\spad{directSum(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}.")) (|symmetricSquare| (($ $) "\\spad{symmetricSquare(a)} computes \\spad{symmetricProduct(a,a)} using a more efficient method.")) (|symmetricPower| (($ $ (|NonNegativeInteger|)) "\\spad{symmetricPower(a,n)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of \\spad{n} solutions of \\spad{a}.")) (|symmetricProduct| (($ $ $) "\\spad{symmetricProduct(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of a solution of \\spad{a} by a solution of \\spad{b}.")) (|adjoint| (($ $) "\\spad{adjoint(a)} returns the adjoint operator of a.")) (D (($) "\\spad{D()} provides the operator corresponding to a derivation in the ring \\spad{A}."))) NIL -((|HasCategory| |#2| (QUOTE (-309)))) -(-597 A) +((|HasCategory| |#2| (QUOTE (-311)))) +(-600 A) ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \\indented{4}{\\spad{(L1 * L2).(f) = L1 L2 f}}")) (|directSum| (($ $ $) "\\spad{directSum(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}.")) (|symmetricSquare| (($ $) "\\spad{symmetricSquare(a)} computes \\spad{symmetricProduct(a,a)} using a more efficient method.")) (|symmetricPower| (($ $ (|NonNegativeInteger|)) "\\spad{symmetricPower(a,n)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of \\spad{n} solutions of \\spad{a}.")) (|symmetricProduct| (($ $ $) "\\spad{symmetricProduct(a,b)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of a solution of \\spad{a} by a solution of \\spad{b}.")) (|adjoint| (($ $) "\\spad{adjoint(a)} returns the adjoint operator of a.")) (D (($) "\\spad{D()} provides the operator corresponding to a derivation in the ring \\spad{A}."))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-598 -3078 UP) +(-601 -3088 UP) ((|constructor| (NIL "\\spadtype{LinearOrdinaryDifferentialOperatorFactorizer} provides a factorizer for linear ordinary differential operators whose coefficients are rational functions.")) (|factor1| (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{factor1(a)} returns the factorisation of a,{} assuming that a has no first-order right factor.")) (|factor| (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{factor(a)} returns the factorisation of a.") (((|List| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{factor(a, zeros)} returns the factorisation of a. \\spad{zeros} is a zero finder in \\spad{UP}."))) NIL ((|HasCategory| |#1| (QUOTE (-27)))) -(-599 A L) +(-602 A L) ((|constructor| (NIL "\\spad{LinearOrdinaryDifferentialOperatorsOps} provides symmetric products and sums for linear ordinary differential operators.")) (|directSum| ((|#2| |#2| |#2| (|Mapping| |#1| |#1|)) "\\spad{directSum(a,b,D)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the sums of a solution of \\spad{a} by a solution of \\spad{b}. \\spad{D} is the derivation to use.")) (|symmetricPower| ((|#2| |#2| (|NonNegativeInteger|) (|Mapping| |#1| |#1|)) "\\spad{symmetricPower(a,n,D)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of \\spad{n} solutions of \\spad{a}. \\spad{D} is the derivation to use.")) (|symmetricProduct| ((|#2| |#2| |#2| (|Mapping| |#1| |#1|)) "\\spad{symmetricProduct(a,b,D)} computes an operator \\spad{c} of minimal order such that the nullspace of \\spad{c} is generated by all the products of a solution of \\spad{a} by a solution of \\spad{b}. \\spad{D} is the derivation to use."))) NIL NIL -(-600 S) +(-603 S) ((|constructor| (NIL "`Logic' provides the basic operations for lattices,{} \\spadignore{e.g.} boolean algebra.")) (|\\/| (($ $ $) "\\spadignore{ \\/ } returns the logical `join',{} \\spadignore{e.g.} `or'.")) (|/\\| (($ $ $) "\\spadignore { /\\ }returns the logical `meet',{} \\spadignore{e.g.} `and'.")) (~ (($ $) "\\spad{~(x)} returns the logical complement of \\spad{x}."))) NIL NIL -(-601) +(-604) ((|constructor| (NIL "`Logic' provides the basic operations for lattices,{} \\spadignore{e.g.} boolean algebra.")) (|\\/| (($ $ $) "\\spadignore{ \\/ } returns the logical `join',{} \\spadignore{e.g.} `or'.")) (|/\\| (($ $ $) "\\spadignore { /\\ }returns the logical `meet',{} \\spadignore{e.g.} `and'.")) (~ (($ $) "\\spad{~(x)} returns the logical complement of \\spad{x}."))) NIL NIL -(-602 R) +(-605 R) ((|constructor| (NIL "Given a PolynomialFactorizationExplicit ring,{} this package provides a defaulting rule for the \\spad{solveLinearPolynomialEquation} operation,{} by moving into the field of fractions,{} and solving it there via the \\spad{multiEuclidean} operation.")) (|solveLinearPolynomialEquationByFractions| (((|Union| (|List| (|SparseUnivariatePolynomial| |#1|)) "failed") (|List| (|SparseUnivariatePolynomial| |#1|)) (|SparseUnivariatePolynomial| |#1|)) "\\spad{solveLinearPolynomialEquationByFractions([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such exists."))) NIL NIL -(-603 |VarSet| R) +(-606 |VarSet| R) ((|constructor| (NIL "This type supports Lie polynomials in Lyndon basis see Free Lie Algebras by \\spad{C}. Reutenauer (Oxford science publications). \\newline Author: Michel Petitot (petitot@lifl.fr).")) (|construct| (($ $ (|LyndonWord| |#1|)) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket \\axiom{[\\spad{x},{}\\spad{y}]}.") (($ (|LyndonWord| |#1|) $) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket \\axiom{[\\spad{x},{}\\spad{y}]}.") (($ (|LyndonWord| |#1|) (|LyndonWord| |#1|)) "\\axiom{construct(\\spad{x},{}\\spad{y})} returns the Lie bracket \\axiom{[\\spad{x},{}\\spad{y}]}.")) (|LiePolyIfCan| (((|Union| $ "failed") (|XDistributedPolynomial| |#1| |#2|)) "\\axiom{LiePolyIfCan(\\spad{p})} returns \\axiom{\\spad{p}} in Lyndon basis if \\axiom{\\spad{p}} is a Lie polynomial,{} otherwise \\axiom{\"failed\"} is returned."))) -((|JacobiIdentity| . T) (|NullSquare| . T) (-3973 . T) (-3972 . T)) -((|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-144)))) -(-604 A S) +((|JacobiIdentity| . T) (|NullSquare| . T) (-3984 . T) (-3983 . T)) +((|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-146)))) +(-607 A S) ((|constructor| (NIL "A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.")) (|list| (($ |#2|) "\\spad{list(x)} returns the list of one element \\spad{x}."))) NIL NIL -(-605 S) +(-608 S) ((|constructor| (NIL "A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.")) (|list| (($ |#1|) "\\spad{list(x)} returns the list of one element \\spad{x}."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-606 -3078 |Row| |Col| M) +(-609 -3088 |Row| |Col| M) ((|constructor| (NIL "This package solves linear system in the matrix form \\spad{AX = B}.")) (|rank| (((|NonNegativeInteger|) |#4| |#3|) "\\spad{rank(A,B)} computes the rank of the complete matrix \\spad{(A|B)} of the linear system \\spad{AX = B}.")) (|hasSolution?| (((|Boolean|) |#4| |#3|) "\\spad{hasSolution?(A,B)} tests if the linear system \\spad{AX = B} has a solution.")) (|particularSolution| (((|Union| |#3| #1="failed") |#4| |#3|) "\\spad{particularSolution(A,B)} finds a particular solution of the linear system \\spad{AX = B}.")) (|solve| (((|List| (|Record| (|:| |particular| (|Union| |#3| #1#)) (|:| |basis| (|List| |#3|)))) |#4| (|List| |#3|)) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|Record| (|:| |particular| (|Union| |#3| #1#)) (|:| |basis| (|List| |#3|))) |#4| |#3|) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}."))) NIL NIL -(-607 -3078) +(-610 -3088) ((|constructor| (NIL "This package solves linear system in the matrix form \\spad{AX = B}. It is essentially a particular instantiation of the package \\spadtype{LinearSystemMatrixPackage} for Matrix and Vector. This package's existence makes it easier to use \\spadfun{solve} in the AXIOM interpreter.")) (|rank| (((|NonNegativeInteger|) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{rank(A,B)} computes the rank of the complete matrix \\spad{(A|B)} of the linear system \\spad{AX = B}.")) (|hasSolution?| (((|Boolean|) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{hasSolution?(A,B)} tests if the linear system \\spad{AX = B} has a solution.")) (|particularSolution| (((|Union| (|Vector| |#1|) #1="failed") (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{particularSolution(A,B)} finds a particular solution of the linear system \\spad{AX = B}.")) (|solve| (((|List| (|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|))))) (|List| (|List| |#1|)) (|List| (|Vector| |#1|))) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|List| (|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|))))) (|Matrix| |#1|) (|List| (|Vector| |#1|))) "\\spad{solve(A,LB)} finds a particular soln of the systems \\spad{AX = B} and a basis of the associated homogeneous systems \\spad{AX = 0} where \\spad{B} varies in the list of column vectors \\spad{LB}.") (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|)))) (|List| (|List| |#1|)) (|Vector| |#1|)) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}.") (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) #1#)) (|:| |basis| (|List| (|Vector| |#1|)))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{solve(A,B)} finds a particular solution of the system \\spad{AX = B} and a basis of the associated homogeneous system \\spad{AX = 0}."))) NIL NIL -(-608 R E OV P) +(-611 R E OV P) ((|constructor| (NIL "this package finds the solutions of linear systems presented as a list of polynomials.")) (|linSolve| (((|Record| (|:| |particular| (|Union| (|Vector| (|Fraction| |#4|)) "failed")) (|:| |basis| (|List| (|Vector| (|Fraction| |#4|))))) (|List| |#4|) (|List| |#3|)) "\\spad{linSolve(lp,lvar)} finds the solutions of the linear system of polynomials \\spad{lp} = 0 with respect to the list of symbols \\spad{lvar}."))) NIL NIL -(-609 |n| R) +(-612 |n| R) ((|constructor| (NIL "LieSquareMatrix(\\spad{n},{}\\spad{R}) implements the Lie algebra of the \\spad{n} by \\spad{n} matrices over the commutative ring \\spad{R}. The Lie bracket (commutator) of the algebra is given by \\spad{a*b := (a *\\$SQMATRIX(n,R) b - b *\\$SQMATRIX(n,R) a)},{} where \\spadfun{*\\$SQMATRIX(\\spad{n},{}\\spad{R})} is the usual matrix multiplication."))) -((-3975 . T) (-3978 . T) (-3972 . T) (-3973 . T)) -((|HasCategory| |#2| (QUOTE (-804 (-1081)))) (|HasCategory| |#2| (QUOTE (-806 (-1081)))) (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (QUOTE (-187))) (|HasAttribute| |#2| (QUOTE (-3980 #1="*"))) (|HasCategory| |#2| (QUOTE (-577 (-480)))) (|HasCategory| |#2| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-577 (-480)))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-804 (-1081)))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|))))) (|HasCategory| |#2| (QUOTE (-255))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-491))) (OR (|HasAttribute| |#2| (QUOTE (-3980 #1#))) (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (QUOTE (-804 (-1081))))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-72))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-144)))) -(-610) +((-3986 . T) (-3989 . T) (-3983 . T) (-3984 . T)) +((|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasCategory| |#2| (QUOTE (-811 (-1088)))) (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (QUOTE (-189))) (|HasAttribute| |#2| (QUOTE (-3991 #1="*"))) (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|))))) (|HasCategory| |#2| (QUOTE (-257))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-494))) (OR (|HasAttribute| |#2| (QUOTE (-3991 #1#))) (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (QUOTE (-809 (-1088))))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-72))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-146)))) +(-613) ((|constructor| (NIL "This domain represents `literal sequence' syntax.")) (|elements| (((|List| (|SpadAst|)) $) "\\spad{elements(e)} returns the list of expressions in the `literal' list `e'."))) NIL NIL -(-611 |VarSet|) +(-614 |VarSet|) ((|constructor| (NIL "Lyndon words over arbitrary (ordered) symbols: see Free Lie Algebras by \\spad{C}. Reutenauer (Oxford science publications). A Lyndon word is a word which is smaller than any of its right factors \\spad{w}.\\spad{r}.\\spad{t}. the pure lexicographical ordering. If \\axiom{a} and \\axiom{\\spad{b}} are two Lyndon words such that \\axiom{a < \\spad{b}} holds \\spad{w}.\\spad{r}.\\spad{t} lexicographical ordering then \\axiom{a*b} is a Lyndon word. Parenthesized Lyndon words can be generated from symbols by using the following rule: \\axiom{[[a,{}\\spad{b}],{}\\spad{c}]} is a Lyndon word iff \\axiom{a*b < \\spad{c} <= \\spad{b}} holds. Lyndon words are internally represented by binary trees using the \\spadtype{Magma} domain constructor. Two ordering are provided: lexicographic and length-lexicographic. \\newline Author : Michel Petitot (petitot@lifl.fr).")) (|LyndonWordsList| (((|List| $) (|List| |#1|) (|PositiveInteger|)) "\\axiom{LyndonWordsList(vl,{} \\spad{n})} returns the list of Lyndon words over the alphabet \\axiom{vl},{} up to order \\axiom{\\spad{n}}.")) (|LyndonWordsList1| (((|OneDimensionalArray| (|List| $)) (|List| |#1|) (|PositiveInteger|)) "\\axiom{\\spad{LyndonWordsList1}(vl,{} \\spad{n})} returns an array of lists of Lyndon words over the alphabet \\axiom{vl},{} up to order \\axiom{\\spad{n}}.")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{x})} returns the list of distinct entries of \\axiom{\\spad{x}}.")) (|lyndonIfCan| (((|Union| $ "failed") (|OrderedFreeMonoid| |#1|)) "\\axiom{lyndonIfCan(\\spad{w})} convert \\axiom{\\spad{w}} into a Lyndon word.")) (|lyndon| (($ (|OrderedFreeMonoid| |#1|)) "\\axiom{lyndon(\\spad{w})} convert \\axiom{\\spad{w}} into a Lyndon word,{} error if \\axiom{\\spad{w}} is not a Lyndon word.")) (|lyndon?| (((|Boolean|) (|OrderedFreeMonoid| |#1|)) "\\axiom{lyndon?(\\spad{w})} test if \\axiom{\\spad{w}} is a Lyndon word.")) (|factor| (((|List| $) (|OrderedFreeMonoid| |#1|)) "\\axiom{factor(\\spad{x})} returns the decreasing factorization into Lyndon words.")) (|coerce| (((|Magma| |#1|) $) "\\axiom{coerce(\\spad{x})} returns the element of \\axiomType{Magma}(VarSet) corresponding to \\axiom{\\spad{x}}.") (((|OrderedFreeMonoid| |#1|) $) "\\axiom{coerce(\\spad{x})} returns the element of \\axiomType{OrderedFreeMonoid}(VarSet) corresponding to \\axiom{\\spad{x}}.")) (|lexico| (((|Boolean|) $ $) "\\axiom{lexico(\\spad{x},{}\\spad{y})} returns \\axiom{\\spad{true}} iff \\axiom{\\spad{x}} is smaller than \\axiom{\\spad{y}} \\spad{w}.\\spad{r}.\\spad{t}. the lexicographical ordering induced by \\axiom{VarSet}.")) (|length| (((|PositiveInteger|) $) "\\axiom{length(\\spad{x})} returns the number of entries in \\axiom{\\spad{x}}.")) (|right| (($ $) "\\axiom{right(\\spad{x})} returns right subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{LyndonWord}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|left| (($ $) "\\axiom{left(\\spad{x})} returns left subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{LyndonWord}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|retractable?| (((|Boolean|) $) "\\axiom{retractable?(\\spad{x})} tests if \\axiom{\\spad{x}} is a tree with only one entry."))) NIL NIL -(-612 A S) +(-615 A S) ((|constructor| (NIL "LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \\spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.")) (|complete| (($ $) "\\spad{complete(st)} causes all entries of 'st' to be computed. this function should only be called on streams which are known to be finite.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(st,n)} causes entries to be computed,{} if necessary,{} so that 'st' will have at least 'n' explicit entries or so that all entries of 'st' will be computed if 'st' is finite with length <= \\spad{n}.")) (|numberOfComputedEntries| (((|NonNegativeInteger|) $) "\\spad{numberOfComputedEntries(st)} returns the number of explicitly computed entries of stream \\spad{st} which exist immediately prior to the time this function is called.")) (|rst| (($ $) "\\spad{rst(s)} returns a pointer to the next node of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|frst| ((|#2| $) "\\spad{frst(s)} returns the first element of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|lazyEvaluate| (($ $) "\\spad{lazyEvaluate(s)} causes one lazy evaluation of stream \\spad{s}. Caution: the first node must be a lazy evaluation mechanism (satisfies \\spad{lazy?(s) = true}) as there is no error check. Note: a call to this function may or may not produce an explicit first entry")) (|lazy?| (((|Boolean|) $) "\\spad{lazy?(s)} returns \\spad{true} if the first node of the stream \\spad{s} is a lazy evaluation mechanism which could produce an additional entry to \\spad{s}.")) (|explicitlyEmpty?| (((|Boolean|) $) "\\spad{explicitlyEmpty?(s)} returns \\spad{true} if the stream is an (explicitly) empty stream. Note: this is a null test which will not cause lazy evaluation.")) (|explicitEntries?| (((|Boolean|) $) "\\spad{explicitEntries?(s)} returns \\spad{true} if the stream \\spad{s} has explicitly computed entries,{} and \\spad{false} otherwise.")) (|select| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{select(f,st)} returns a stream consisting of those elements of stream \\spad{st} satisfying the predicate \\spad{f}. Note: \\spad{select(f,st) = [x for x in st | f(x)]}.")) (|remove| (($ (|Mapping| (|Boolean|) |#2|) $) "\\spad{remove(f,st)} returns a stream consisting of those elements of stream \\spad{st} which do not satisfy the predicate \\spad{f}. Note: \\spad{remove(f,st) = [x for x in st | not f(x)]}."))) NIL NIL -(-613 S) +(-616 S) ((|constructor| (NIL "LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \\spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.")) (|complete| (($ $) "\\spad{complete(st)} causes all entries of 'st' to be computed. this function should only be called on streams which are known to be finite.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(st,n)} causes entries to be computed,{} if necessary,{} so that 'st' will have at least 'n' explicit entries or so that all entries of 'st' will be computed if 'st' is finite with length <= \\spad{n}.")) (|numberOfComputedEntries| (((|NonNegativeInteger|) $) "\\spad{numberOfComputedEntries(st)} returns the number of explicitly computed entries of stream \\spad{st} which exist immediately prior to the time this function is called.")) (|rst| (($ $) "\\spad{rst(s)} returns a pointer to the next node of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|frst| ((|#1| $) "\\spad{frst(s)} returns the first element of stream \\spad{s}. Caution: this function should only be called after a \\spad{empty?} test has been made since there no error check.")) (|lazyEvaluate| (($ $) "\\spad{lazyEvaluate(s)} causes one lazy evaluation of stream \\spad{s}. Caution: the first node must be a lazy evaluation mechanism (satisfies \\spad{lazy?(s) = true}) as there is no error check. Note: a call to this function may or may not produce an explicit first entry")) (|lazy?| (((|Boolean|) $) "\\spad{lazy?(s)} returns \\spad{true} if the first node of the stream \\spad{s} is a lazy evaluation mechanism which could produce an additional entry to \\spad{s}.")) (|explicitlyEmpty?| (((|Boolean|) $) "\\spad{explicitlyEmpty?(s)} returns \\spad{true} if the stream is an (explicitly) empty stream. Note: this is a null test which will not cause lazy evaluation.")) (|explicitEntries?| (((|Boolean|) $) "\\spad{explicitEntries?(s)} returns \\spad{true} if the stream \\spad{s} has explicitly computed entries,{} and \\spad{false} otherwise.")) (|select| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{select(f,st)} returns a stream consisting of those elements of stream \\spad{st} satisfying the predicate \\spad{f}. Note: \\spad{select(f,st) = [x for x in st | f(x)]}.")) (|remove| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{remove(f,st)} returns a stream consisting of those elements of stream \\spad{st} which do not satisfy the predicate \\spad{f}. Note: \\spad{remove(f,st) = [x for x in st | not f(x)]}."))) NIL NIL -(-614) +(-617) ((|constructor| (NIL "This domain represents the syntax of a macro definition.")) (|body| (((|SpadAst|) $) "\\spad{body(m)} returns the right hand side of the definition `m'.")) (|head| (((|HeadAst|) $) "\\spad{head(m)} returns the head of the macro definition `m'. This is a list of identifiers starting with the name of the macro followed by the name of the parameters,{} if any."))) NIL NIL -(-615 |VarSet|) +(-618 |VarSet|) ((|constructor| (NIL "This type is the basic representation of parenthesized words (binary trees over arbitrary symbols) useful in \\spadtype{LiePolynomial}. \\newline Author: Michel Petitot (petitot@lifl.fr).")) (|varList| (((|List| |#1|) $) "\\axiom{varList(\\spad{x})} returns the list of distinct entries of \\axiom{\\spad{x}}.")) (|right| (($ $) "\\axiom{right(\\spad{x})} returns right subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|retractable?| (((|Boolean|) $) "\\axiom{retractable?(\\spad{x})} tests if \\axiom{\\spad{x}} is a tree with only one entry.")) (|rest| (($ $) "\\axiom{rest(\\spad{x})} return \\axiom{\\spad{x}} without the first entry or error if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|mirror| (($ $) "\\axiom{mirror(\\spad{x})} returns the reversed word of \\axiom{\\spad{x}}. That is \\axiom{\\spad{x}} itself if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true} and \\axiom{mirror(\\spad{z}) * mirror(\\spad{y})} if \\axiom{\\spad{x}} is \\axiom{y*z}.")) (|lexico| (((|Boolean|) $ $) "\\axiom{lexico(\\spad{x},{}\\spad{y})} returns \\axiom{\\spad{true}} iff \\axiom{\\spad{x}} is smaller than \\axiom{\\spad{y}} \\spad{w}.\\spad{r}.\\spad{t}. the lexicographical ordering induced by \\axiom{VarSet}. \\spad{N}.\\spad{B}. This operation does not take into account the tree structure of its arguments. Thus this is not a total ordering.")) (|length| (((|PositiveInteger|) $) "\\axiom{length(\\spad{x})} returns the number of entries in \\axiom{\\spad{x}}.")) (|left| (($ $) "\\axiom{left(\\spad{x})} returns left subtree of \\axiom{\\spad{x}} or error if \\axiomOpFrom{retractable?}{Magma}(\\axiom{\\spad{x}}) is \\spad{true}.")) (|first| ((|#1| $) "\\axiom{first(\\spad{x})} returns the first entry of the tree \\axiom{\\spad{x}}.")) (|coerce| (((|OrderedFreeMonoid| |#1|) $) "\\axiom{coerce(\\spad{x})} returns the element of \\axiomType{OrderedFreeMonoid}(VarSet) corresponding to \\axiom{\\spad{x}} by removing parentheses.")) (* (($ $ $) "\\axiom{x*y} returns the tree \\axiom{[\\spad{x},{}\\spad{y}]}."))) NIL NIL -(-616 A) +(-619 A) ((|constructor| (NIL "various Currying operations.")) (|recur| ((|#1| (|Mapping| |#1| (|NonNegativeInteger|) |#1|) (|NonNegativeInteger|) |#1|) "\\spad{recur(n,g,x)} is \\spad{g(n,g(n-1,..g(1,x)..))}.")) (|iter| ((|#1| (|Mapping| |#1| |#1|) (|NonNegativeInteger|) |#1|) "\\spad{iter(f,n,x)} applies \\spad{f n} times to \\spad{x}."))) NIL NIL -(-617 A C) +(-620 A C) ((|constructor| (NIL "various Currying operations.")) (|arg2| ((|#2| |#1| |#2|) "\\spad{arg2(a,c)} selects its second argument.")) (|arg1| ((|#1| |#1| |#2|) "\\spad{arg1(a,c)} selects its first argument."))) NIL NIL -(-618 A B C) +(-621 A B C) ((|constructor| (NIL "various Currying operations.")) (|comp| ((|#3| (|Mapping| |#3| |#2|) (|Mapping| |#2| |#1|) |#1|) "\\spad{comp(f,g,x)} is \\spad{f(g x)}."))) NIL NIL -(-619) +(-622) ((|constructor| (NIL "This domain represents a mapping type AST. A mapping AST \\indented{2}{is a syntactic description of a function type,{} \\spadignore{e.g.} its result} \\indented{2}{type and the list of its argument types.}")) (|target| (((|TypeAst|) $) "\\spad{target(s)} returns the result type AST for `s'.")) (|source| (((|List| (|TypeAst|)) $) "\\spad{source(s)} returns the parameter type AST list of `s'.")) (|mappingAst| (($ (|List| (|TypeAst|)) (|TypeAst|)) "\\spad{mappingAst(s,t)} builds the mapping AST \\spad{s} -> \\spad{t}")) (|coerce| (($ (|Signature|)) "sig::MappingAst builds a MappingAst from the Signature `sig'."))) NIL NIL -(-620 A) +(-623 A) ((|constructor| (NIL "various Currying operations.")) (|recur| (((|Mapping| |#1| (|NonNegativeInteger|) |#1|) (|Mapping| |#1| (|NonNegativeInteger|) |#1|)) "\\spad{recur(g)} is the function \\spad{h} such that \\indented{1}{\\spad{h(n,x)= g(n,g(n-1,..g(1,x)..))}.}")) (** (((|Mapping| |#1| |#1|) (|Mapping| |#1| |#1|) (|NonNegativeInteger|)) "\\spad{f**n} is the function which is the \\spad{n}-fold application \\indented{1}{of \\spad{f}.}")) (|id| ((|#1| |#1|) "\\spad{id x} is \\spad{x}.")) (|fixedPoint| (((|List| |#1|) (|Mapping| (|List| |#1|) (|List| |#1|)) (|Integer|)) "\\spad{fixedPoint(f,n)} is the fixed point of function \\indented{1}{\\spad{f} which is assumed to transform a list of length} \\indented{1}{\\spad{n}.}") ((|#1| (|Mapping| |#1| |#1|)) "\\spad{fixedPoint f} is the fixed point of function \\spad{f}. \\indented{1}{\\spadignore{i.e.} such that \\spad{fixedPoint f = f(fixedPoint f)}.}")) (|coerce| (((|Mapping| |#1|) |#1|) "\\spad{coerce A} changes its argument into a \\indented{1}{nullary function.}")) (|nullary| (((|Mapping| |#1|) |#1|) "\\spad{nullary A} changes its argument into a \\indented{1}{nullary function.}"))) NIL NIL -(-621 A C) +(-624 A C) ((|constructor| (NIL "various Currying operations.")) (|diag| (((|Mapping| |#2| |#1|) (|Mapping| |#2| |#1| |#1|)) "\\spad{diag(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g a = f(a,a)}.}")) (|constant| (((|Mapping| |#2| |#1|) (|Mapping| |#2|)) "\\spad{vu(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g a= f ()}.}")) (|curry| (((|Mapping| |#2|) (|Mapping| |#2| |#1|) |#1|) "\\spad{cu(f,a)} is the function \\spad{g} \\indented{1}{such that \\spad{g ()= f a}.}")) (|const| (((|Mapping| |#2| |#1|) |#2|) "\\spad{const c} is a function which produces \\spad{c} when \\indented{1}{applied to its argument.}"))) NIL NIL -(-622 A B C) +(-625 A B C) ((|constructor| (NIL "various Currying operations.")) (* (((|Mapping| |#3| |#1|) (|Mapping| |#3| |#2|) (|Mapping| |#2| |#1|)) "\\spad{f*g} is the function \\spad{h} \\indented{1}{such that \\spad{h x= f(g x)}.}")) (|twist| (((|Mapping| |#3| |#2| |#1|) (|Mapping| |#3| |#1| |#2|)) "\\spad{twist(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g (a,b)= f(b,a)}.}")) (|constantLeft| (((|Mapping| |#3| |#1| |#2|) (|Mapping| |#3| |#2|)) "\\spad{constantLeft(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g (a,b)= f b}.}")) (|constantRight| (((|Mapping| |#3| |#1| |#2|) (|Mapping| |#3| |#1|)) "\\spad{constantRight(f)} is the function \\spad{g} \\indented{1}{such that \\spad{g (a,b)= f a}.}")) (|curryLeft| (((|Mapping| |#3| |#2|) (|Mapping| |#3| |#1| |#2|) |#1|) "\\spad{curryLeft(f,a)} is the function \\spad{g} \\indented{1}{such that \\spad{g b = f(a,b)}.}")) (|curryRight| (((|Mapping| |#3| |#1|) (|Mapping| |#3| |#1| |#2|) |#2|) "\\spad{curryRight(f,b)} is the function \\spad{g} such that \\indented{1}{\\spad{g a = f(a,b)}.}"))) NIL NIL -(-623 S R |Row| |Col|) +(-626 S R |Row| |Col|) ((|constructor| (NIL "\\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by calling the function \\spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \\spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|minordet| ((|#2| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors. Error: if the matrix is not square.")) (|determinant| ((|#2| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. Error: if the matrix is not square.")) (|nullSpace| (((|List| |#4|) $) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#2|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if matrix is not square or if the matrix is square but not invertible.") (($ $ (|NonNegativeInteger|)) "\\spad{x ** n} computes a non-negative integral power of the matrix \\spad{x}. Error: if the matrix is not square.")) (* ((|#3| |#3| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#4| $ |#4|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.") (($ (|Integer|) $) "\\spad{n * x} is an integer multiple.") (($ $ |#2|) "\\spad{x * r} is the right scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ |#2| $) "\\spad{r*x} is the left scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ $ $) "\\spad{x * y} is the product of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (- (($ $) "\\spad{-x} returns the negative of the matrix \\spad{x}.") (($ $ $) "\\spad{x - y} is the difference of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (+ (($ $ $) "\\spad{x + y} is the sum of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (|setsubMatrix!| (($ $ (|Integer|) (|Integer|) $) "\\spad{setsubMatrix(x,i1,j1,y)} destructively alters the matrix \\spad{x}. Here \\spad{x(i,j)} is set to \\spad{y(i-i1+1,j-j1+1)} for \\spad{i = i1,...,i1-1+nrows y} and \\spad{j = j1,...,j1-1+ncols y}.")) (|subMatrix| (($ $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{subMatrix(x,i1,i2,j1,j2)} extracts the submatrix \\spad{[x(i,j)]} where the index \\spad{i} ranges from \\spad{i1} to \\spad{i2} and the index \\spad{j} ranges from \\spad{j1} to \\spad{j2}.")) (|swapColumns!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapColumns!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th columns of \\spad{m}. This destructively alters the matrix.")) (|swapRows!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapRows!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th rows of \\spad{m}. This destructively alters the matrix.")) (|setelt| (($ $ (|List| (|Integer|)) (|List| (|Integer|)) $) "\\spad{setelt(x,rowList,colList,y)} destructively alters the matrix \\spad{x}. If \\spad{y} is \\spad{m}-by-\\spad{n},{} \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then \\spad{x(i<k>,j<l>)} is set to \\spad{y(k,l)} for \\spad{k = 1,...,m} and \\spad{l = 1,...,n}.")) (|elt| (($ $ (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{elt(x,rowList,colList)} returns an \\spad{m}-by-\\spad{n} matrix consisting of elements of \\spad{x},{} where \\spad{m = \\# rowList} and \\spad{n = \\# colList}. If \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then the \\spad{(k,l)}th entry of \\spad{elt(x,rowList,colList)} is \\spad{x(i<k>,j<l>)}.")) (|listOfLists| (((|List| (|List| |#2|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|vertConcat| (($ $ $) "\\spad{vertConcat(x,y)} vertically concatenates two matrices with an equal number of columns. The entries of \\spad{y} appear below of the entries of \\spad{x}. Error: if the matrices do not have the same number of columns.")) (|horizConcat| (($ $ $) "\\spad{horizConcat(x,y)} horizontally concatenates two matrices with an equal number of rows. The entries of \\spad{y} appear to the right of the entries of \\spad{x}. Error: if the matrices do not have the same number of rows.")) (|squareTop| (($ $) "\\spad{squareTop(m)} returns an \\spad{n}-by-\\spad{n} matrix consisting of the first \\spad{n} rows of the \\spad{m}-by-\\spad{n} matrix \\spad{m}. Error: if \\spad{m < n}.")) (|transpose| (($ $) "\\spad{transpose(m)} returns the transpose of the matrix \\spad{m}.") (($ |#3|) "\\spad{transpose(r)} converts the row \\spad{r} to a row matrix.")) (|coerce| (($ |#4|) "\\spad{coerce(col)} converts the column \\spad{col} to a column matrix.")) (|diagonalMatrix| (($ (|List| $)) "\\spad{diagonalMatrix([m1,...,mk])} creates a block diagonal matrix \\spad{M} with block matrices {\\em m1},{}...,{}{\\em mk} down the diagonal,{} with 0 block matrices elsewhere. More precisly: if \\spad{ri := nrows mi},{} \\spad{ci := ncols mi},{} then \\spad{m} is an (r1+..+rk) by (c1+..+ck) - matrix with entries \\spad{m.i.j = ml.(i-r1-..-r(l-1)).(j-n1-..-n(l-1))},{} if \\spad{(r1+..+r(l-1)) < i <= r1+..+rl} and \\spad{(c1+..+c(l-1)) < i <= c1+..+cl},{} \\spad{m.i.j} = 0 otherwise.") (($ (|List| |#2|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ (|NonNegativeInteger|) |#2|) "\\spad{scalarMatrix(n,r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}'s on the diagonal and zeroes elsewhere.")) (|matrix| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) (|Mapping| |#2| (|Integer|) (|Integer|))) "\\spad{matrix(n,m,f)} construcys and \\spad{n * m} matrix with the \\spad{(i,j)} entry equal to \\spad{f(i,j)}.") (($ (|List| (|List| |#2|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|zero| (($ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{zero(m,n)} returns an \\spad{m}-by-\\spad{n} zero matrix.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|finiteAggregate| ((|attribute|) "matrices are finite")) (|shallowlyMutable| ((|attribute|) "One may destructively alter matrices"))) NIL -((|HasAttribute| |#2| (QUOTE (-3980 "*"))) (|HasCategory| |#2| (QUOTE (-255))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-491)))) -(-624 R |Row| |Col|) +((|HasAttribute| |#2| (QUOTE (-3991 "*"))) (|HasCategory| |#2| (QUOTE (-257))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-494)))) +(-627 R |Row| |Col|) ((|constructor| (NIL "\\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by calling the function \\spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \\spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|minordet| ((|#1| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors. Error: if the matrix is not square.")) (|determinant| ((|#1| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. Error: if the matrix is not square.")) (|nullSpace| (((|List| |#3|) $) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#1|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if matrix is not square or if the matrix is square but not invertible.") (($ $ (|NonNegativeInteger|)) "\\spad{x ** n} computes a non-negative integral power of the matrix \\spad{x}. Error: if the matrix is not square.")) (* ((|#2| |#2| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#3| $ |#3|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.") (($ (|Integer|) $) "\\spad{n * x} is an integer multiple.") (($ $ |#1|) "\\spad{x * r} is the right scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ |#1| $) "\\spad{r*x} is the left scalar multiple of the scalar \\spad{r} and the matrix \\spad{x}.") (($ $ $) "\\spad{x * y} is the product of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (- (($ $) "\\spad{-x} returns the negative of the matrix \\spad{x}.") (($ $ $) "\\spad{x - y} is the difference of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (+ (($ $ $) "\\spad{x + y} is the sum of the matrices \\spad{x} and \\spad{y}. Error: if the dimensions are incompatible.")) (|setsubMatrix!| (($ $ (|Integer|) (|Integer|) $) "\\spad{setsubMatrix(x,i1,j1,y)} destructively alters the matrix \\spad{x}. Here \\spad{x(i,j)} is set to \\spad{y(i-i1+1,j-j1+1)} for \\spad{i = i1,...,i1-1+nrows y} and \\spad{j = j1,...,j1-1+ncols y}.")) (|subMatrix| (($ $ (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{subMatrix(x,i1,i2,j1,j2)} extracts the submatrix \\spad{[x(i,j)]} where the index \\spad{i} ranges from \\spad{i1} to \\spad{i2} and the index \\spad{j} ranges from \\spad{j1} to \\spad{j2}.")) (|swapColumns!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapColumns!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th columns of \\spad{m}. This destructively alters the matrix.")) (|swapRows!| (($ $ (|Integer|) (|Integer|)) "\\spad{swapRows!(m,i,j)} interchanges the \\spad{i}th and \\spad{j}th rows of \\spad{m}. This destructively alters the matrix.")) (|setelt| (($ $ (|List| (|Integer|)) (|List| (|Integer|)) $) "\\spad{setelt(x,rowList,colList,y)} destructively alters the matrix \\spad{x}. If \\spad{y} is \\spad{m}-by-\\spad{n},{} \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then \\spad{x(i<k>,j<l>)} is set to \\spad{y(k,l)} for \\spad{k = 1,...,m} and \\spad{l = 1,...,n}.")) (|elt| (($ $ (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{elt(x,rowList,colList)} returns an \\spad{m}-by-\\spad{n} matrix consisting of elements of \\spad{x},{} where \\spad{m = \\# rowList} and \\spad{n = \\# colList}. If \\spad{rowList = [i<1>,i<2>,...,i<m>]} and \\spad{colList = [j<1>,j<2>,...,j<n>]},{} then the \\spad{(k,l)}th entry of \\spad{elt(x,rowList,colList)} is \\spad{x(i<k>,j<l>)}.")) (|listOfLists| (((|List| (|List| |#1|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|vertConcat| (($ $ $) "\\spad{vertConcat(x,y)} vertically concatenates two matrices with an equal number of columns. The entries of \\spad{y} appear below of the entries of \\spad{x}. Error: if the matrices do not have the same number of columns.")) (|horizConcat| (($ $ $) "\\spad{horizConcat(x,y)} horizontally concatenates two matrices with an equal number of rows. The entries of \\spad{y} appear to the right of the entries of \\spad{x}. Error: if the matrices do not have the same number of rows.")) (|squareTop| (($ $) "\\spad{squareTop(m)} returns an \\spad{n}-by-\\spad{n} matrix consisting of the first \\spad{n} rows of the \\spad{m}-by-\\spad{n} matrix \\spad{m}. Error: if \\spad{m < n}.")) (|transpose| (($ $) "\\spad{transpose(m)} returns the transpose of the matrix \\spad{m}.") (($ |#2|) "\\spad{transpose(r)} converts the row \\spad{r} to a row matrix.")) (|coerce| (($ |#3|) "\\spad{coerce(col)} converts the column \\spad{col} to a column matrix.")) (|diagonalMatrix| (($ (|List| $)) "\\spad{diagonalMatrix([m1,...,mk])} creates a block diagonal matrix \\spad{M} with block matrices {\\em m1},{}...,{}{\\em mk} down the diagonal,{} with 0 block matrices elsewhere. More precisly: if \\spad{ri := nrows mi},{} \\spad{ci := ncols mi},{} then \\spad{m} is an (r1+..+rk) by (c1+..+ck) - matrix with entries \\spad{m.i.j = ml.(i-r1-..-r(l-1)).(j-n1-..-n(l-1))},{} if \\spad{(r1+..+r(l-1)) < i <= r1+..+rl} and \\spad{(c1+..+c(l-1)) < i <= c1+..+cl},{} \\spad{m.i.j} = 0 otherwise.") (($ (|List| |#1|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ (|NonNegativeInteger|) |#1|) "\\spad{scalarMatrix(n,r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}'s on the diagonal and zeroes elsewhere.")) (|matrix| (($ (|NonNegativeInteger|) (|NonNegativeInteger|) (|Mapping| |#1| (|Integer|) (|Integer|))) "\\spad{matrix(n,m,f)} construcys and \\spad{n * m} matrix with the \\spad{(i,j)} entry equal to \\spad{f(i,j)}.") (($ (|List| (|List| |#1|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|zero| (($ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{zero(m,n)} returns an \\spad{m}-by-\\spad{n} zero matrix.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|finiteAggregate| ((|attribute|) "matrices are finite")) (|shallowlyMutable| ((|attribute|) "One may destructively alter matrices"))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL -(-625 R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) +(-628 R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) ((|constructor| (NIL "\\spadtype{MatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \\spadfun{map} and \\spadfun{reduce}.")) (|reduce| ((|#5| (|Mapping| |#5| |#1| |#5|) |#4| |#5|) "\\spad{reduce(f,m,r)} returns a matrix \\spad{n} where \\spad{n[i,j] = f(m[i,j],r)} for all indices \\spad{i} and \\spad{j}.")) (|map| (((|Union| |#8| "failed") (|Mapping| (|Union| |#5| "failed") |#1|) |#4|) "\\spad{map(f,m)} applies the function \\spad{f} to the elements of the matrix \\spad{m}.") ((|#8| (|Mapping| |#5| |#1|) |#4|) "\\spad{map(f,m)} applies the function \\spad{f} to the elements of the matrix \\spad{m}."))) NIL NIL -(-626 R |Row| |Col| M) +(-629 R |Row| |Col| M) ((|constructor| (NIL "\\spadtype{MatrixLinearAlgebraFunctions} provides functions to compute inverses and canonical forms.")) (|inverse| (((|Union| |#4| "failed") |#4|) "\\spad{inverse(m)} returns the inverse of the matrix. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|normalizedDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{normalizedDivide(n,d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \\spadignore{e.g.} positive remainders")) (|rowEchelon| ((|#4| |#4|) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (|adjoint| (((|Record| (|:| |adjMat| |#4|) (|:| |detMat| |#1|)) |#4|) "\\spad{adjoint(m)} returns the ajoint matrix of \\spad{m} (\\spadignore{i.e.} the matrix \\spad{n} such that m*n = determinant(\\spad{m})*id) and the detrminant of \\spad{m}.")) (|invertIfCan| (((|Union| |#4| "failed") |#4|) "\\spad{invertIfCan(m)} returns the inverse of \\spad{m} over \\spad{R}")) (|fractionFreeGauss!| ((|#4| |#4|) "\\spad{fractionFreeGauss(m)} performs the fraction free gaussian elimination on the matrix \\spad{m}.")) (|nullSpace| (((|List| |#3|) |#4|) "\\spad{nullSpace(m)} returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) |#4|) "\\spad{nullity(m)} returns the mullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) |#4|) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|elColumn2!| ((|#4| |#4| |#1| (|Integer|) (|Integer|)) "\\spad{elColumn2!(m,a,i,j)} adds to column \\spad{i} a*column(\\spad{m},{}\\spad{j}) : elementary operation of second kind. (\\spad{i} ~=j)")) (|elRow2!| ((|#4| |#4| |#1| (|Integer|) (|Integer|)) "\\spad{elRow2!(m,a,i,j)} adds to row \\spad{i} a*row(\\spad{m},{}\\spad{j}) : elementary operation of second kind. (\\spad{i} ~=j)")) (|elRow1!| ((|#4| |#4| (|Integer|) (|Integer|)) "\\spad{elRow1!(m,i,j)} swaps rows \\spad{i} and \\spad{j} of matrix \\spad{m} : elementary operation of first kind")) (|minordet| ((|#1| |#4|) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors. Error: if the matrix is not square.")) (|determinant| ((|#1| |#4|) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}. an error message is returned if the matrix is not square."))) NIL -((|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-255))) (|HasCategory| |#1| (QUOTE (-491)))) -(-627 R) +((|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-257))) (|HasCategory| |#1| (QUOTE (-494)))) +(-630 R) ((|constructor| (NIL "\\spadtype{Matrix} is a matrix domain where 1-based indexing is used for both rows and columns.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m}. If the matrix is not invertible,{} \"failed\" is returned. Error: if the matrix is not square.")) (|diagonalMatrix| (($ (|Vector| |#1|)) "\\spad{diagonalMatrix(v)} returns a diagonal matrix where the elements of \\spad{v} appear on the diagonal."))) -((-3978 . T) (-3979 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-255))) (|HasCategory| |#1| (QUOTE (-491))) (|HasAttribute| |#1| (QUOTE (-3980 "*"))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-628 R) +((-3989 . T) (-3990 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-257))) (|HasCategory| |#1| (QUOTE (-494))) (|HasAttribute| |#1| (QUOTE (-3991 "*"))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-631 R) ((|constructor| (NIL "This package provides standard arithmetic operations on matrices. The functions in this package store the results of computations in existing matrices,{} rather than creating new matrices. This package works only for matrices of type Matrix and uses the internal representation of this type.")) (** (((|Matrix| |#1|) (|Matrix| |#1|) (|NonNegativeInteger|)) "\\spad{x ** n} computes the \\spad{n}-th power of a square matrix. The power \\spad{n} is assumed greater than 1.")) (|power!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|NonNegativeInteger|)) "\\spad{power!(a,b,c,m,n)} computes \\spad{m} ** \\spad{n} and stores the result in \\spad{a}. The matrices \\spad{b} and \\spad{c} are used to store intermediate results. Error: if \\spad{a},{} \\spad{b},{} \\spad{c},{} and \\spad{m} are not square and of the same dimensions.")) (|times!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{times!(c,a,b)} computes the matrix product \\spad{a * b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have compatible dimensions.")) (|rightScalarTimes!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rightScalarTimes!(c,a,r)} computes the scalar product \\spad{a * r} and stores the result in the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions.")) (|leftScalarTimes!| (((|Matrix| |#1|) (|Matrix| |#1|) |#1| (|Matrix| |#1|)) "\\spad{leftScalarTimes!(c,r,a)} computes the scalar product \\spad{r * a} and stores the result in the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions.")) (|minus!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{!minus!(c,a,b)} computes the matrix difference \\spad{a - b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have the same dimensions.") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{minus!(c,a)} computes \\spad{-a} and stores the result in the matrix \\spad{c}. Error: if a and \\spad{c} do not have the same dimensions.")) (|plus!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{plus!(c,a,b)} computes the matrix sum \\spad{a + b} and stores the result in the matrix \\spad{c}. Error: if \\spad{a},{} \\spad{b},{} and \\spad{c} do not have the same dimensions.")) (|copy!| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{copy!(c,a)} copies the matrix \\spad{a} into the matrix \\spad{c}. Error: if \\spad{a} and \\spad{c} do not have the same dimensions."))) NIL NIL -(-629 T$) +(-632 T$) ((|constructor| (NIL "This domain implements the notion of optional value,{} where a computation may fail to produce expected value.")) (|nothing| (($) "\\spad{nothing} represents failure or absence of value.")) (|autoCoerce| ((|#1| $) "\\spad{autoCoerce} is a courtesy coercion function used by the compiler in case it knows that `x' really is a \\spadtype{T}.")) (|case| (((|Boolean|) $ (|[\|\|]| |nothing|)) "\\spad{x case nothing} holds if the value for \\spad{x} is missing.") (((|Boolean|) $ (|[\|\|]| |#1|)) "\\spad{x case T} returns \\spad{true} if \\spad{x} is actually a data of type \\spad{T}.")) (|just| (($ |#1|) "\\spad{just x} injects the value `x' into \\%."))) NIL NIL -(-630 R Q) +(-633 R Q) ((|constructor| (NIL "MatrixCommonDenominator provides functions to compute the common denominator of a matrix of elements of the quotient field of an integral domain.")) (|splitDenominator| (((|Record| (|:| |num| (|Matrix| |#1|)) (|:| |den| |#1|)) (|Matrix| |#2|)) "\\spad{splitDenominator(q)} returns \\spad{[p, d]} such that \\spad{q = p/d} and \\spad{d} is a common denominator for the elements of \\spad{q}.")) (|clearDenominator| (((|Matrix| |#1|) (|Matrix| |#2|)) "\\spad{clearDenominator(q)} returns \\spad{p} such that \\spad{q = p/d} where \\spad{d} is a common denominator for the elements of \\spad{q}.")) (|commonDenominator| ((|#1| (|Matrix| |#2|)) "\\spad{commonDenominator(q)} returns a common denominator \\spad{d} for the elements of \\spad{q}."))) NIL NIL -(-631 S) +(-634 S) ((|constructor| (NIL "A multi-dictionary is a dictionary which may contain duplicates. As for any dictionary,{} its size is assumed large so that copying (non-destructive) operations are generally to be avoided.")) (|duplicates| (((|List| (|Record| (|:| |entry| |#1|) (|:| |count| (|NonNegativeInteger|)))) $) "\\spad{duplicates(d)} returns a list of values which have duplicates in \\spad{d}")) (|removeDuplicates!| (($ $) "\\spad{removeDuplicates!(d)} destructively removes any duplicate values in dictionary \\spad{d}.")) (|insert!| (($ |#1| $ (|NonNegativeInteger|)) "\\spad{insert!(x,d,n)} destructively inserts \\spad{n} copies of \\spad{x} into dictionary \\spad{d}."))) -((-3979 . T)) +((-3990 . T)) NIL -(-632 U) +(-635 U) ((|constructor| (NIL "This package supports factorization and gcds of univariate polynomials over the integers modulo different primes. The inputs are given as polynomials over the integers with the prime passed explicitly as an extra argument.")) (|exptMod| ((|#1| |#1| (|Integer|) |#1| (|Integer|)) "\\spad{exptMod(f,n,g,p)} raises the univariate polynomial \\spad{f} to the \\spad{n}th power modulo the polynomial \\spad{g} and the prime \\spad{p}.")) (|separateFactors| (((|List| |#1|) (|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|)))) (|Integer|)) "\\spad{separateFactors(ddl, p)} refines the distinct degree factorization produced by \\spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} to give a complete list of factors.")) (|ddFact| (((|List| (|Record| (|:| |factor| |#1|) (|:| |degree| (|Integer|)))) |#1| (|Integer|)) "\\spad{ddFact(f,p)} computes a distinct degree factorization of the polynomial \\spad{f} modulo the prime \\spad{p},{} \\spadignore{i.e.} such that each factor is a product of irreducibles of the same degrees. The input polynomial \\spad{f} is assumed to be square-free modulo \\spad{p}.")) (|factor| (((|List| |#1|) |#1| (|Integer|)) "\\spad{factor(f1,p)} returns the list of factors of the univariate polynomial \\spad{f1} modulo the integer prime \\spad{p}. Error: if \\spad{f1} is not square-free modulo \\spad{p}.")) (|linears| ((|#1| |#1| (|Integer|)) "\\spad{linears(f,p)} returns the product of all the linear factors of \\spad{f} modulo \\spad{p}. Potentially incorrect result if \\spad{f} is not square-free modulo \\spad{p}.")) (|gcd| ((|#1| |#1| |#1| (|Integer|)) "\\spad{gcd(f1,f2,p)} computes the gcd of the univariate polynomials \\spad{f1} and \\spad{f2} modulo the integer prime \\spad{p}."))) NIL NIL -(-633) +(-636) ((|constructor| (NIL "\\indented{1}{<description of package>} Author: Jim Wen Date Created: ?? Date Last Updated: October 1991 by Jon Steinbach Keywords: Examples: References:")) (|ptFunc| (((|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|))) "\\spad{ptFunc(a,b,c,d)} is an internal function exported in order to compile packages.")) (|meshPar1Var| (((|ThreeSpace| (|DoubleFloat|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Expression| (|Integer|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar1Var(s,t,u,f,s1,l)} \\undocumented")) (|meshFun2Var| (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Union| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) #1="undefined") (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshFun2Var(f,g,s1,s2,l)} \\undocumented")) (|meshPar2Var| (((|ThreeSpace| (|DoubleFloat|)) (|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(sp,f,s1,s2,l)} \\undocumented") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(f,s1,s2,l)} \\undocumented") (((|ThreeSpace| (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) (|Union| (|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) #1#) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{meshPar2Var(f,g,h,j,s1,s2,l)} \\undocumented"))) NIL NIL -(-634 OV E -3078 PG) +(-637 OV E -3088 PG) ((|constructor| (NIL "Package for factorization of multivariate polynomials over finite fields.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factor(p)} produces the complete factorization of the multivariate polynomial \\spad{p} over a finite field. \\spad{p} is represented as a univariate polynomial with multivariate coefficients over a finite field.") (((|Factored| |#4|) |#4|) "\\spad{factor(p)} produces the complete factorization of the multivariate polynomial \\spad{p} over a finite field."))) NIL NIL -(-635 R) +(-638 R) ((|constructor| (NIL "\\indented{1}{Modular hermitian row reduction.} Author: Manuel Bronstein Date Created: 22 February 1989 Date Last Updated: 24 November 1993 Keywords: matrix,{} reduction.")) (|normalizedDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{normalizedDivide(n,d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \\spadignore{e.g.} positive remainders")) (|rowEchelonLocal| (((|Matrix| |#1|) (|Matrix| |#1|) |#1| |#1|) "\\spad{rowEchelonLocal(m, d, p)} computes the row-echelon form of \\spad{m} concatenated with \\spad{d} times the identity matrix over a local ring where \\spad{p} is the only prime.")) (|rowEchLocal| (((|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rowEchLocal(m,p)} computes a modular row-echelon form of \\spad{m},{} finding an appropriate modulus over a local ring where \\spad{p} is the only prime.")) (|rowEchelon| (((|Matrix| |#1|) (|Matrix| |#1|) |#1|) "\\spad{rowEchelon(m, d)} computes a modular row-echelon form mod \\spad{d} of \\indented{3}{[\\spad{d}\\space{5}]} \\indented{3}{[\\space{2}\\spad{d}\\space{3}]} \\indented{3}{[\\space{4}. ]} \\indented{3}{[\\space{5}\\spad{d}]} \\indented{3}{[\\space{3}\\spad{M}\\space{2}]} where \\spad{M = m mod d}.")) (|rowEch| (((|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{rowEch(m)} computes a modular row-echelon form of \\spad{m},{} finding an appropriate modulus."))) NIL NIL -(-636 S D1 D2 I) +(-639 S D1 D2 I) ((|constructor| (NIL "transforms top-level objects into compiled functions.")) (|compiledFunction| (((|Mapping| |#4| |#2| |#3|) |#1| (|Symbol|) (|Symbol|)) "\\spad{compiledFunction(expr,x,y)} returns a function \\spad{f: (D1, D2) -> I} defined by \\spad{f(x, y) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\spad{(D1, D2)}")) (|binaryFunction| (((|Mapping| |#4| |#2| |#3|) (|Symbol|)) "\\spad{binaryFunction(s)} is a local function"))) NIL NIL -(-637 S) +(-640 S) ((|constructor| (NIL "MakeFloatCompiledFunction transforms top-level objects into compiled Lisp functions whose arguments are Lisp floats. This by-passes the \\Language{} compiler and interpreter,{} thereby gaining several orders of magnitude.")) (|makeFloatFunction| (((|Mapping| (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) |#1| (|Symbol|) (|Symbol|)) "\\spad{makeFloatFunction(expr, x, y)} returns a Lisp function \\spad{f: (\\axiomType{DoubleFloat}, \\axiomType{DoubleFloat}) -> \\axiomType{DoubleFloat}} defined by \\spad{f(x, y) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\spad{(\\axiomType{DoubleFloat}, \\axiomType{DoubleFloat})}.") (((|Mapping| (|DoubleFloat|) (|DoubleFloat|)) |#1| (|Symbol|)) "\\spad{makeFloatFunction(expr, x)} returns a Lisp function \\spad{f: \\axiomType{DoubleFloat} -> \\axiomType{DoubleFloat}} defined by \\spad{f(x) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\axiomType{DoubleFloat}."))) NIL NIL -(-638 S) +(-641 S) ((|constructor| (NIL "transforms top-level objects into interpreter functions.")) (|function| (((|Symbol|) |#1| (|Symbol|) (|List| (|Symbol|))) "\\spad{function(e, foo, [x1,...,xn])} creates a function \\spad{foo(x1,...,xn) == e}.") (((|Symbol|) |#1| (|Symbol|) (|Symbol|) (|Symbol|)) "\\spad{function(e, foo, x, y)} creates a function \\spad{foo(x, y) = e}.") (((|Symbol|) |#1| (|Symbol|) (|Symbol|)) "\\spad{function(e, foo, x)} creates a function \\spad{foo(x) == e}.") (((|Symbol|) |#1| (|Symbol|)) "\\spad{function(e, foo)} creates a function \\spad{foo() == e}."))) NIL NIL -(-639 S T$) +(-642 S T$) ((|constructor| (NIL "MakeRecord is used internally by the interpreter to create record types which are used for doing parallel iterations on streams.")) (|makeRecord| (((|Record| (|:| |part1| |#1|) (|:| |part2| |#2|)) |#1| |#2|) "\\spad{makeRecord(a,b)} creates a record object with type Record(part1:S,{} part2:R),{} where \\spad{part1} is \\spad{a} and \\spad{part2} is \\spad{b}."))) NIL NIL -(-640 S -2655 I) +(-643 S -2665 I) ((|constructor| (NIL "transforms top-level objects into compiled functions.")) (|compiledFunction| (((|Mapping| |#3| |#2|) |#1| (|Symbol|)) "\\spad{compiledFunction(expr, x)} returns a function \\spad{f: D -> I} defined by \\spad{f(x) == expr}. Function \\spad{f} is compiled and directly applicable to objects of type \\spad{D}.")) (|unaryFunction| (((|Mapping| |#3| |#2|) (|Symbol|)) "\\spad{unaryFunction(a)} is a local function"))) NIL NIL -(-641 E OV R P) +(-644 E OV R P) ((|constructor| (NIL "This package provides the functions for the multivariate \"lifting\",{} using an algorithm of Paul Wang. This package will work for every euclidean domain \\spad{R} which has property \\spad{F},{} \\spadignore{i.e.} there exists a factor operation in \\spad{R[x]}.")) (|lifting1| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|SparseUnivariatePolynomial| |#4|)) (|List| |#3|) (|List| |#4|) (|List| (|List| (|Record| (|:| |expt| (|NonNegativeInteger|)) (|:| |pcoef| |#4|)))) (|List| (|NonNegativeInteger|)) (|Vector| (|List| (|SparseUnivariatePolynomial| |#3|))) |#3|) "\\spad{lifting1(u,lv,lu,lr,lp,lt,ln,t,r)} \\undocumented")) (|lifting| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|SparseUnivariatePolynomial| |#3|)) (|List| |#3|) (|List| |#4|) (|List| (|NonNegativeInteger|)) |#3|) "\\spad{lifting(u,lv,lu,lr,lp,ln,r)} \\undocumented")) (|corrPoly| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| |#3|) (|List| (|NonNegativeInteger|)) (|List| (|SparseUnivariatePolynomial| |#4|)) (|Vector| (|List| (|SparseUnivariatePolynomial| |#3|))) |#3|) "\\spad{corrPoly(u,lv,lr,ln,lu,t,r)} \\undocumented"))) NIL NIL -(-642 R) +(-645 R) ((|constructor| (NIL "This is the category of linear operator rings with one generator. The generator is not named by the category but can always be constructed as \\spad{monomial(1,1)}. \\blankline For convenience,{} call the generator \\spad{G}. Then each value is equal to \\indented{4}{\\spad{sum(a(i)*G**i, i = 0..n)}} for some unique \\spad{n} and \\spad{a(i)} in \\spad{R}. \\blankline Note that multiplication is not necessarily commutative. In fact,{} if \\spad{a} is in \\spad{R},{} it is quite normal to have \\spad{a*G \\~= G*a}.")) (|monomial| (($ |#1| (|NonNegativeInteger|)) "\\spad{monomial(c,k)} produces \\spad{c} times the \\spad{k}-th power of the generating operator,{} \\spad{monomial(1,1)}.")) (|coefficient| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coefficient(l,k)} is \\spad{a(k)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|reductum| (($ $) "\\spad{reductum(l)} is \\spad{l - monomial(a(n),n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(l)} is \\spad{a(n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|minimumDegree| (((|NonNegativeInteger|) $) "\\spad{minimumDegree(l)} is the smallest \\spad{k} such that \\spad{a(k) \\~= 0} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(l)} is \\spad{n} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}"))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-643 R1 UP1 UPUP1 R2 UP2 UPUP2) +(-646 R1 UP1 UPUP1 R2 UP2 UPUP2) ((|constructor| (NIL "Lifting of a map through 2 levels of polynomials.")) (|map| ((|#6| (|Mapping| |#4| |#1|) |#3|) "\\spad{map(f, p)} lifts \\spad{f} to the domain of \\spad{p} then applies it to \\spad{p}."))) NIL NIL -(-644) +(-647) ((|constructor| (NIL "\\spadtype{MathMLFormat} provides a coercion from \\spadtype{OutputForm} to MathML format.")) (|display| (((|Void|) (|String|)) "prints the string returned by coerce,{} adding <math ...> tags.")) (|exprex| (((|String|) (|OutputForm|)) "coverts \\spadtype{OutputForm} to \\spadtype{String} with the structure preserved with braces. Actually this is not quite accurate. The function \\spadfun{precondition} is first applied to the \\spadtype{OutputForm} expression before \\spadfun{exprex}. The raw \\spadtype{OutputForm} and the nature of the \\spadfun{precondition} function is still obscure to me at the time of this writing (2007-02-14).")) (|coerceL| (((|String|) (|OutputForm|)) "coerceS(\\spad{o}) changes \\spad{o} in the standard output format to MathML format and displays result as one long string.")) (|coerceS| (((|String|) (|OutputForm|)) "\\spad{coerceS(o)} changes \\spad{o} in the standard output format to MathML format and displays formatted result.")) (|coerce| (((|String|) (|OutputForm|)) "coerceS(\\spad{o}) changes \\spad{o} in the standard output format to MathML format."))) NIL NIL -(-645 R |Mod| -2025 -3501 |exactQuo|) +(-648 R |Mod| -2033 -3512 |exactQuo|) ((|constructor| (NIL "\\indented{1}{These domains are used for the factorization and gcds} of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{ModularRing},{} \\spadtype{EuclideanModularRing}")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#1| |#2|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#1| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#2| $) "\\spad{modulus(x)} \\undocumented"))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-646 R P) +(-649 R P) ((|constructor| (NIL "This package \\undocumented")) (|frobenius| (($ $) "\\spad{frobenius(x)} \\undocumented")) (|computePowers| (((|PrimitiveArray| $)) "\\spad{computePowers()} \\undocumented")) (|pow| (((|PrimitiveArray| $)) "\\spad{pow()} \\undocumented")) (|An| (((|Vector| |#1|) $) "\\spad{An(x)} \\undocumented")) (|UnVectorise| (($ (|Vector| |#1|)) "\\spad{UnVectorise(v)} \\undocumented")) (|Vectorise| (((|Vector| |#1|) $) "\\spad{Vectorise(x)} \\undocumented")) (|lift| ((|#2| $) "\\spad{lift(x)} \\undocumented")) (|reduce| (($ |#2|) "\\spad{reduce(x)} \\undocumented")) (|modulus| ((|#2|) "\\spad{modulus()} \\undocumented")) (|setPoly| ((|#2| |#2|) "\\spad{setPoly(x)} \\undocumented"))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3974 |has| |#1| (-309)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| (-988) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| (-988) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-988) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-988) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| (-988) (QUOTE (-550 (-469))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-816)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-1057))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-188))) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-647 IS E |ff|) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3985 |has| |#1| (-311)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| (-993) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| (-993) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| (-993) (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-1064))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-298))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-190))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-650 IS E |ff|) ((|constructor| (NIL "This package \\undocumented")) (|construct| (($ |#1| |#2|) "\\spad{construct(i,e)} \\undocumented")) (|index| ((|#1| $) "\\spad{index(x)} \\undocumented")) (|exponent| ((|#2| $) "\\spad{exponent(x)} \\undocumented"))) NIL NIL -(-648 R M) +(-651 R M) ((|constructor| (NIL "Algebra of ADDITIVE operators on a module.")) (|makeop| (($ |#1| (|FreeGroup| (|BasicOperator|))) "\\spad{makeop should} be local but conditional")) (|opeval| ((|#2| (|BasicOperator|) |#2|) "\\spad{opeval should} be local but conditional")) (** (($ $ (|Integer|)) "\\spad{op**n} \\undocumented") (($ (|BasicOperator|) (|Integer|)) "\\spad{op**n} \\undocumented")) (|evaluateInverse| (($ $ (|Mapping| |#2| |#2|)) "\\spad{evaluateInverse(x,f)} \\undocumented")) (|evaluate| (($ $ (|Mapping| |#2| |#2|)) "\\spad{evaluate(f, u +-> g u)} attaches the map \\spad{g} to \\spad{f}. \\spad{f} must be a basic operator \\spad{g} MUST be additive,{} \\spadignore{i.e.} \\spad{g(a + b) = g(a) + g(b)} for any \\spad{a},{} \\spad{b} in \\spad{M}. This implies that \\spad{g(n a) = n g(a)} for any \\spad{a} in \\spad{M} and integer \\spad{n > 0}.")) (|conjug| ((|#1| |#1|) "\\spad{conjug(x)}should be local but conditional")) (|adjoint| (($ $ $) "\\spad{adjoint(op1, op2)} sets the adjoint of \\spad{op1} to be \\spad{op2}. \\spad{op1} must be a basic operator") (($ $) "\\spad{adjoint(op)} returns the adjoint of the operator \\spad{op}."))) -((-3973 |has| |#1| (-144)) (-3972 |has| |#1| (-144)) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118)))) -(-649 R |Mod| -2025 -3501 |exactQuo|) +((-3984 |has| |#1| (-146)) (-3983 |has| |#1| (-146)) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120)))) +(-652 R |Mod| -2033 -3512 |exactQuo|) ((|constructor| (NIL "These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \\spadtype{EuclideanModularRing} ,{}\\spadtype{ModularField}")) (|inv| (($ $) "\\spad{inv(x)} \\undocumented")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} \\undocumented")) (|exQuo| (((|Union| $ "failed") $ $) "\\spad{exQuo(x,y)} \\undocumented")) (|reduce| (($ |#1| |#2|) "\\spad{reduce(r,m)} \\undocumented")) (|coerce| ((|#1| $) "\\spad{coerce(x)} \\undocumented")) (|modulus| ((|#2| $) "\\spad{modulus(x)} \\undocumented"))) -((-3975 . T)) +((-3986 . T)) NIL -(-650 S R) +(-653 S R) ((|constructor| (NIL "The category of modules over a commutative ring. \\blankline"))) NIL NIL -(-651 R) +(-654 R) ((|constructor| (NIL "The category of modules over a commutative ring. \\blankline"))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-652 -3078) +(-655 -3088) ((|constructor| (NIL "\\indented{1}{MoebiusTransform(\\spad{F}) is the domain of fractional linear (Moebius)} transformations over \\spad{F}.")) (|eval| (((|OnePointCompletion| |#1|) $ (|OnePointCompletion| |#1|)) "\\spad{eval(m,x)} returns \\spad{(a*x + b)/(c*x + d)} where \\spad{m = moebius(a,b,c,d)} (see \\spadfunFrom{moebius}{MoebiusTransform}).") ((|#1| $ |#1|) "\\spad{eval(m,x)} returns \\spad{(a*x + b)/(c*x + d)} where \\spad{m = moebius(a,b,c,d)} (see \\spadfunFrom{moebius}{MoebiusTransform}).")) (|recip| (($ $) "\\spad{recip(m)} = recip() * \\spad{m}") (($) "\\spad{recip()} returns \\spad{matrix [[0,1],[1,0]]} representing the map \\spad{x -> 1 / x}.")) (|scale| (($ $ |#1|) "\\spad{scale(m,h)} returns \\spad{scale(h) * m} (see \\spadfunFrom{shift}{MoebiusTransform}).") (($ |#1|) "\\spad{scale(k)} returns \\spad{matrix [[k,0],[0,1]]} representing the map \\spad{x -> k * x}.")) (|shift| (($ $ |#1|) "\\spad{shift(m,h)} returns \\spad{shift(h) * m} (see \\spadfunFrom{shift}{MoebiusTransform}).") (($ |#1|) "\\spad{shift(k)} returns \\spad{matrix [[1,k],[0,1]]} representing the map \\spad{x -> x + k}.")) (|moebius| (($ |#1| |#1| |#1| |#1|) "\\spad{moebius(a,b,c,d)} returns \\spad{matrix [[a,b],[c,d]]}."))) -((-3975 . T)) +((-3986 . T)) NIL -(-653 S) +(-656 S) ((|constructor| (NIL "Monad is the class of all multiplicative monads,{} \\spadignore{i.e.} sets with a binary operation.")) (** (($ $ (|PositiveInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|PositiveInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,1) := a}.")) (|rightPower| (($ $ (|PositiveInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,1) := a}.")) (* (($ $ $) "\\spad{a*b} is the product of \\spad{a} and \\spad{b} in a set with a binary operation."))) NIL NIL -(-654) +(-657) ((|constructor| (NIL "Monad is the class of all multiplicative monads,{} \\spadignore{i.e.} sets with a binary operation.")) (** (($ $ (|PositiveInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|PositiveInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,1) := a}.")) (|rightPower| (($ $ (|PositiveInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,1) := a}.")) (* (($ $ $) "\\spad{a*b} is the product of \\spad{a} and \\spad{b} in a set with a binary operation."))) NIL NIL -(-655 S) +(-658 S) ((|constructor| (NIL "\\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \\indented{1}{\\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \\indented{3}{leftIdentity(\"*\":(\\%,{}\\%)->\\%,{}1)\\space{3}\\tab{30} 1*x=x} \\indented{3}{rightIdentity(\"*\":(\\%,{}\\%)->\\%,{}1)\\space{2}\\tab{30} x*1=x} Common Additional Axioms \\indented{3}{unitsKnown---if \"recip\" says \"failed\",{} that PROVES input wasn't a unit}")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|NonNegativeInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,0) := 1}.")) (|rightPower| (($ $ (|NonNegativeInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,0) := 1}.")) (|one?| (((|Boolean|) $) "\\spad{one?(a)} tests whether \\spad{a} is the unit 1.")) (|One| (($) "1 returns the unit element,{} denoted by 1."))) NIL NIL -(-656) +(-659) ((|constructor| (NIL "\\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \\indented{1}{\\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \\indented{3}{leftIdentity(\"*\":(\\%,{}\\%)->\\%,{}1)\\space{3}\\tab{30} 1*x=x} \\indented{3}{rightIdentity(\"*\":(\\%,{}\\%)->\\%,{}1)\\space{2}\\tab{30} x*1=x} Common Additional Axioms \\indented{3}{unitsKnown---if \"recip\" says \"failed\",{} that PROVES input wasn't a unit}")) (|rightRecip| (((|Union| $ "failed") $) "\\spad{rightRecip(a)} returns an element,{} which is a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|leftRecip| (((|Union| $ "failed") $) "\\spad{leftRecip(a)} returns an element,{} which is a left inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \\spad{a},{} or \\spad{\"failed\"} if such an element doesn't exist or cannot be determined (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{a**n} returns the \\spad{n}\\spad{-}th power of \\spad{a},{} defined by repeated squaring.")) (|leftPower| (($ $ (|NonNegativeInteger|)) "\\spad{leftPower(a,n)} returns the \\spad{n}\\spad{-}th left power of \\spad{a},{} \\spadignore{i.e.} \\spad{leftPower(a,n) := a * leftPower(a,n-1)} and \\spad{leftPower(a,0) := 1}.")) (|rightPower| (($ $ (|NonNegativeInteger|)) "\\spad{rightPower(a,n)} returns the \\spad{n}\\spad{-}th right power of \\spad{a},{} \\spadignore{i.e.} \\spad{rightPower(a,n) := rightPower(a,n-1) * a} and \\spad{rightPower(a,0) := 1}.")) (|one?| (((|Boolean|) $) "\\spad{one?(a)} tests whether \\spad{a} is the unit 1.")) (|One| (($) "1 returns the unit element,{} denoted by 1."))) NIL NIL -(-657 S R UP) +(-660 S R UP) ((|constructor| (NIL "A \\spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.")) (|derivationCoordinates| (((|Matrix| |#2|) (|Vector| $) (|Mapping| |#2| |#2|)) "\\spad{derivationCoordinates(b, ')} returns \\spad{M} such that \\spad{b' = M b}.")) (|lift| ((|#3| $) "\\spad{lift(z)} returns a minimal degree univariate polynomial up such that \\spad{z=reduce up}.")) (|convert| (($ |#3|) "\\spad{convert(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|reduce| (((|Union| $ "failed") (|Fraction| |#3|)) "\\spad{reduce(frac)} converts the fraction \\spad{frac} to an algebra element.") (($ |#3|) "\\spad{reduce(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|definingPolynomial| ((|#3|) "\\spad{definingPolynomial()} returns the minimal polynomial which \\spad{generator()} satisfies.")) (|generator| (($) "\\spad{generator()} returns the generator for this domain."))) NIL -((|HasCategory| |#2| (QUOTE (-296))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-315)))) -(-658 R UP) +((|HasCategory| |#2| (QUOTE (-298))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-317)))) +(-661 R UP) ((|constructor| (NIL "A \\spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.")) (|derivationCoordinates| (((|Matrix| |#1|) (|Vector| $) (|Mapping| |#1| |#1|)) "\\spad{derivationCoordinates(b, ')} returns \\spad{M} such that \\spad{b' = M b}.")) (|lift| ((|#2| $) "\\spad{lift(z)} returns a minimal degree univariate polynomial up such that \\spad{z=reduce up}.")) (|convert| (($ |#2|) "\\spad{convert(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|reduce| (((|Union| $ "failed") (|Fraction| |#2|)) "\\spad{reduce(frac)} converts the fraction \\spad{frac} to an algebra element.") (($ |#2|) "\\spad{reduce(up)} converts the univariate polynomial \\spad{up} to an algebra element,{} reducing by the \\spad{definingPolynomial()} if necessary.")) (|definingPolynomial| ((|#2|) "\\spad{definingPolynomial()} returns the minimal polynomial which \\spad{generator()} satisfies.")) (|generator| (($) "\\spad{generator()} returns the generator for this domain."))) -((-3971 |has| |#1| (-309)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 |has| |#1| (-311)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-659 S) +(-662 S) ((|constructor| (NIL "The class of multiplicative monoids,{} \\spadignore{i.e.} semigroups with a multiplicative identity element. \\blankline")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} tries to compute the multiplicative inverse for \\spad{x} or \"failed\" if it cannot find the inverse (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (|one?| (((|Boolean|) $) "\\spad{one?(x)} tests if \\spad{x} is equal to 1.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|One| (($) "1 is the multiplicative identity."))) NIL NIL -(-660) +(-663) ((|constructor| (NIL "The class of multiplicative monoids,{} \\spadignore{i.e.} semigroups with a multiplicative identity element. \\blankline")) (|recip| (((|Union| $ "failed") $) "\\spad{recip(x)} tries to compute the multiplicative inverse for \\spad{x} or \"failed\" if it cannot find the inverse (see unitsKnown).")) (** (($ $ (|NonNegativeInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (|one?| (((|Boolean|) $) "\\spad{one?(x)} tests if \\spad{x} is equal to 1.")) (|sample| (($) "\\spad{sample yields} a value of type \\%")) (|One| (($) "1 is the multiplicative identity."))) NIL NIL -(-661 -3078 UP) +(-664 T$) +((|constructor| (NIL "This domain implements monoid operations.")) (|monoidOperation| (($ (|Mapping| |#1| |#1| |#1|) |#1|) "\\spad{monoidOperation(f,e)} constructs a operation from the binary mapping \\spad{f} with neutral value \\spad{e}."))) +(((|%Rule| |neutrality| (|%Forall| (|%Sequence| (|:| |f| $) (|:| |x| |#1|)) (SEQ (-3052 (|f| |x| (-2408 |f|)) |x|) (|exit| 1 (-3052 (|f| (-2408 |f|) |x|) |x|))))) . T) ((|%Rule| |associativity| (|%Forall| (|%Sequence| (|:| |f| $) (|:| |x| |#1|) (|:| |y| |#1|) (|:| |z| |#1|)) (-3052 (|f| (|f| |x| |y|) |z|) (|f| |x| (|f| |y| |z|))))) . T)) +NIL +(-665 T$) +((|constructor| (NIL "This is the category of all domains that implement monoid operations")) (|neutralValue| ((|#1| $) "\\spad{neutralValue f} returns the neutral value of the monoid operation \\spad{f}."))) +(((|%Rule| |neutrality| (|%Forall| (|%Sequence| (|:| |f| $) (|:| |x| |#1|)) (SEQ (-3052 (|f| |x| (-2408 |f|)) |x|) (|exit| 1 (-3052 (|f| (-2408 |f|) |x|) |x|))))) . T) ((|%Rule| |associativity| (|%Forall| (|%Sequence| (|:| |f| $) (|:| |x| |#1|) (|:| |y| |#1|) (|:| |z| |#1|)) (-3052 (|f| (|f| |x| |y|) |z|) (|f| |x| (|f| |y| |z|))))) . T)) +NIL +(-666 -3088 UP) ((|constructor| (NIL "Tools for handling monomial extensions.")) (|decompose| (((|Record| (|:| |poly| |#2|) (|:| |normal| (|Fraction| |#2|)) (|:| |special| (|Fraction| |#2|))) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{decompose(f, D)} returns \\spad{[p,n,s]} such that \\spad{f = p+n+s},{} all the squarefree factors of \\spad{denom(n)} are normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} \\spad{denom(s)} is special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and \\spad{n} and \\spad{s} are proper fractions (no pole at infinity). \\spad{D} is the derivation to use.")) (|normalDenom| ((|#2| (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{normalDenom(f, D)} returns the product of all the normal factors of \\spad{denom(f)}. \\spad{D} is the derivation to use.")) (|splitSquarefree| (((|Record| (|:| |normal| (|Factored| |#2|)) (|:| |special| (|Factored| |#2|))) |#2| (|Mapping| |#2| |#2|)) "\\spad{splitSquarefree(p, D)} returns \\spad{[n_1 n_2\\^2 ... n_m\\^m, s_1 s_2\\^2 ... s_q\\^q]} such that \\spad{p = n_1 n_2\\^2 ... n_m\\^m s_1 s_2\\^2 ... s_q\\^q},{} each \\spad{n_i} is normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D} and each \\spad{s_i} is special \\spad{w}.\\spad{r}.\\spad{t} \\spad{D}. \\spad{D} is the derivation to use.")) (|split| (((|Record| (|:| |normal| |#2|) (|:| |special| |#2|)) |#2| (|Mapping| |#2| |#2|)) "\\spad{split(p, D)} returns \\spad{[n,s]} such that \\spad{p = n s},{} all the squarefree factors of \\spad{n} are normal \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D},{} and \\spad{s} is special \\spad{w}.\\spad{r}.\\spad{t}. \\spad{D}. \\spad{D} is the derivation to use."))) NIL NIL -(-662 |VarSet| E1 E2 R S PR PS) +(-667 |VarSet| E1 E2 R S PR PS) ((|constructor| (NIL "\\indented{1}{Utilities for MPolyCat} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 28 March 1990 (PG)")) (|reshape| ((|#7| (|List| |#5|) |#6|) "\\spad{reshape(l,p)} \\undocumented")) (|map| ((|#7| (|Mapping| |#5| |#4|) |#6|) "\\spad{map(f,p)} \\undocumented "))) NIL NIL -(-663 |Vars1| |Vars2| E1 E2 R PR1 PR2) +(-668 |Vars1| |Vars2| E1 E2 R PR1 PR2) ((|constructor| (NIL "This package \\undocumented")) (|map| ((|#7| (|Mapping| |#2| |#1|) |#6|) "\\spad{map(f,x)} \\undocumented"))) NIL NIL -(-664 E OV R PPR) +(-669 E OV R PPR) ((|constructor| (NIL "\\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are polynomials over some ring \\spad{R} over which we can factor. It is used internally by packages such as the solve package which need to work with polynomials in a specific set of variables with coefficients which are polynomials in all the other variables.")) (|factor| (((|Factored| |#4|) |#4|) "\\spad{factor(p)} factors a polynomial with polynomial coefficients.")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} makes an element from symbol \\spad{s} or fails.")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) NIL NIL -(-665 |vl| R) +(-670 |vl| R) ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are from a user specified list of symbols. The ordering is specified by the position of the variable in the list. The coefficient ring may be non commutative,{} but the variables are assumed to commute."))) -(((-3980 "*") |has| |#2| (-144)) (-3971 |has| |#2| (-491)) (-3976 |has| |#2| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-816))) (OR (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-816)))) (OR (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-816)))) (OR (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-816)))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-144))) (OR (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-491)))) (-12 (|HasCategory| |#2| (QUOTE (-791 (-325)))) (|HasCategory| (-768 |#1|) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#2| (QUOTE (-791 (-480)))) (|HasCategory| (-768 |#1|) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#2| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-768 |#1|) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#2| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-768 |#1|) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| (-768 |#1|) (QUOTE (-550 (-469))))) (|HasCategory| |#2| (QUOTE (-577 (-480)))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#2| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-309))) (|HasAttribute| |#2| (QUOTE -3976)) (|HasCategory| |#2| (QUOTE (-387))) (-12 (|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#2| (QUOTE (-116))))) -(-666 E OV R PRF) +(((-3991 "*") |has| |#2| (-146)) (-3982 |has| |#2| (-494)) (-3987 |has| |#2| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-821))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-821)))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-494)))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-327)))) (|HasCategory| (-773 |#1|) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-483)))) (|HasCategory| (-773 |#1|) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| (-773 |#1|) (QUOTE (-553 (-472))))) (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-311))) (|HasAttribute| |#2| (QUOTE -3987)) (|HasCategory| |#2| (QUOTE (-389))) (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#2| (QUOTE (-118))))) +(-671 E OV R PRF) ((|constructor| (NIL "\\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are rational functions over some ring \\spad{R} over which we can factor. It is used internally by packages such as primary decomposition which need to work with polynomials with rational function coefficients,{} \\spadignore{i.e.} themselves fractions of polynomials.")) (|factor| (((|Factored| |#4|) |#4|) "\\spad{factor(prf)} factors a polynomial with rational function coefficients.")) (|pushuconst| ((|#4| (|Fraction| (|Polynomial| |#3|)) |#2|) "\\spad{pushuconst(r,var)} takes a rational function and raises all occurances of the variable \\spad{var} to the polynomial level.")) (|pushucoef| ((|#4| (|SparseUnivariatePolynomial| (|Polynomial| |#3|)) |#2|) "\\spad{pushucoef(upoly,var)} converts the anonymous univariate polynomial \\spad{upoly} to a polynomial in \\spad{var} over rational functions.")) (|pushup| ((|#4| |#4| |#2|) "\\spad{pushup(prf,var)} raises all occurences of the variable \\spad{var} in the coefficients of the polynomial \\spad{prf} back to the polynomial level.")) (|pushdterm| ((|#4| (|SparseUnivariatePolynomial| |#4|) |#2|) "\\spad{pushdterm(monom,var)} pushes all top level occurences of the variable \\spad{var} into the coefficient domain for the monomial \\spad{monom}.")) (|pushdown| ((|#4| |#4| |#2|) "\\spad{pushdown(prf,var)} pushes all top level occurences of the variable \\spad{var} into the coefficient domain for the polynomial \\spad{prf}.")) (|totalfract| (((|Record| (|:| |sup| (|Polynomial| |#3|)) (|:| |inf| (|Polynomial| |#3|))) |#4|) "\\spad{totalfract(prf)} takes a polynomial whose coefficients are themselves fractions of polynomials and returns a record containing the numerator and denominator resulting from putting \\spad{prf} over a common denominator.")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) NIL NIL -(-667 E OV R P) +(-672 E OV R P) ((|constructor| (NIL "\\indented{1}{MRationalFactorize contains the factor function for multivariate} polynomials over the quotient field of a ring \\spad{R} such that the package MultivariateFactorize can factor multivariate polynomials over \\spad{R}.")) (|factor| (((|Factored| |#4|) |#4|) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} with coefficients which are fractions of elements of \\spad{R}."))) NIL NIL -(-668 R S M) +(-673 R S M) ((|constructor| (NIL "\\spad{MonoidRingFunctions2} implements functions between two monoid rings defined with the same monoid over different rings.")) (|map| (((|MonoidRing| |#2| |#3|) (|Mapping| |#2| |#1|) (|MonoidRing| |#1| |#3|)) "\\spad{map(f,u)} maps \\spad{f} onto the coefficients \\spad{f} the element \\spad{u} of the monoid ring to create an element of a monoid ring with the same monoid \\spad{b}."))) NIL NIL -(-669 R M) +(-674 R M) ((|constructor| (NIL "\\spadtype{MonoidRing}(\\spad{R},{}\\spad{M}),{} implements the algebra of all maps from the monoid \\spad{M} to the commutative ring \\spad{R} with finite support. Multiplication of two maps \\spad{f} and \\spad{g} is defined to map an element \\spad{c} of \\spad{M} to the (convolution) sum over {\\em f(a)g(b)} such that {\\em ab = c}. Thus \\spad{M} can be identified with a canonical basis and the maps can also be considered as formal linear combinations of the elements in \\spad{M}. Scalar multiples of a basis element are called monomials. A prominent example is the class of polynomials where the monoid is a direct product of the natural numbers with pointwise addition. When \\spad{M} is \\spadtype{FreeMonoid Symbol},{} one gets polynomials in infinitely many non-commuting variables. Another application area is representation theory of finite groups \\spad{G},{} where modules over \\spadtype{MonoidRing}(\\spad{R},{}\\spad{G}) are studied.")) (|reductum| (($ $) "\\spad{reductum(f)} is \\spad{f} minus its leading monomial.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(f)} gives the coefficient of \\spad{f},{} whose corresponding monoid element is the greatest among all those with non-zero coefficients.")) (|leadingMonomial| ((|#2| $) "\\spad{leadingMonomial(f)} gives the monomial of \\spad{f} whose corresponding monoid element is the greatest among all those with non-zero coefficients.")) (|numberOfMonomials| (((|NonNegativeInteger|) $) "\\spad{numberOfMonomials(f)} is the number of non-zero coefficients with respect to the canonical basis.")) (|monomials| (((|List| $) $) "\\spad{monomials(f)} gives the list of all monomials whose sum is \\spad{f}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(f)} lists all non-zero coefficients.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(f)} tests if \\spad{f} is a single monomial.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,u)} maps function \\spad{fn} onto the coefficients of the non-zero monomials of \\spad{u}.")) (|terms| (((|List| (|Record| (|:| |coef| |#1|) (|:| |monom| |#2|))) $) "\\spad{terms(f)} gives the list of non-zero coefficients combined with their corresponding basis element as records. This is the internal representation.")) (|coerce| (($ (|List| (|Record| (|:| |coef| |#1|) (|:| |monom| |#2|)))) "\\spad{coerce(lt)} converts a list of terms and coefficients to a member of the domain.")) (|coefficient| ((|#1| $ |#2|) "\\spad{coefficient(f,m)} extracts the coefficient of \\spad{m} in \\spad{f} with respect to the canonical basis \\spad{M}.")) (|monomial| (($ |#1| |#2|) "\\spad{monomial(r,m)} creates a scalar multiple of the basis element \\spad{m}."))) -((-3973 |has| |#1| (-144)) (-3972 |has| |#1| (-144)) (-3975 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#2| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-751)))) -(-670 S) +((-3984 |has| |#1| (-146)) (-3983 |has| |#1| (-146)) (-3986 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#2| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-756)))) +(-675 S) ((|constructor| (NIL "A multiset is a set with multiplicities.")) (|remove!| (($ (|Mapping| (|Boolean|) |#1|) $ (|Integer|)) "\\spad{remove!(p,ms,number)} removes destructively at most \\spad{number} copies of elements \\spad{x} such that \\spad{p(x)} is \\spadfun{\\spad{true}} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.") (($ |#1| $ (|Integer|)) "\\spad{remove!(x,ms,number)} removes destructively at most \\spad{number} copies of element \\spad{x} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.")) (|remove| (($ (|Mapping| (|Boolean|) |#1|) $ (|Integer|)) "\\spad{remove(p,ms,number)} removes at most \\spad{number} copies of elements \\spad{x} such that \\spad{p(x)} is \\spadfun{\\spad{true}} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.") (($ |#1| $ (|Integer|)) "\\spad{remove(x,ms,number)} removes at most \\spad{number} copies of element \\spad{x} if \\spad{number} is positive,{} all of them if \\spad{number} equals zero,{} and all but at most \\spad{-number} if \\spad{number} is negative.")) (|members| (((|List| |#1|) $) "\\spad{members(ms)} returns a list of the elements of \\spad{ms} {\\em without} their multiplicity. See also \\spadfun{parts}.")) (|multiset| (($ (|List| |#1|)) "\\spad{multiset(ls)} creates a multiset with elements from \\spad{ls}.") (($ |#1|) "\\spad{multiset(s)} creates a multiset with singleton \\spad{s}.") (($) "\\spad{multiset()}\\$\\spad{D} creates an empty multiset of domain \\spad{D}."))) -((-3978 . T) (-3968 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-671 S) +((-3989 . T) (-3979 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-676 S) ((|constructor| (NIL "A multi-set aggregate is a set which keeps track of the multiplicity of its elements."))) -((-3968 . T) (-3979 . T)) +((-3979 . T) (-3990 . T)) NIL -(-672) +(-677) ((|constructor| (NIL "\\spadtype{MoreSystemCommands} implements an interface with the system command facility. These are the commands that are issued from source files or the system interpreter and they start with a close parenthesis,{} \\spadignore{e.g.} \\spadsyscom{what} commands.")) (|systemCommand| (((|Void|) (|String|)) "\\spad{systemCommand(cmd)} takes the string \\spadvar{\\spad{cmd}} and passes it to the runtime environment for execution as a system command. Although various things may be printed,{} no usable value is returned."))) NIL NIL -(-673 S) +(-678 S) ((|constructor| (NIL "This package exports tools for merging lists")) (|mergeDifference| (((|List| |#1|) (|List| |#1|) (|List| |#1|)) "\\spad{mergeDifference(l1,l2)} returns a list of elements in \\spad{l1} not present in \\spad{l2}. Assumes lists are ordered and all \\spad{x} in \\spad{l2} are also in \\spad{l1}."))) NIL NIL -(-674 |Coef| |Var|) +(-679 |Coef| |Var|) ((|constructor| (NIL "\\spadtype{MultivariateTaylorSeriesCategory} is the most general multivariate Taylor series category.")) (|integrate| (($ $ |#2|) "\\spad{integrate(f,x)} returns the anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{x} with constant coefficient 1. We may integrate a series when we can divide coefficients by integers.")) (|polynomial| (((|Polynomial| |#1|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| (((|NonNegativeInteger|) $ |#2| (|NonNegativeInteger|)) "\\spad{order(f,x,n)} returns \\spad{min(n,order(f,x))}.") (((|NonNegativeInteger|) $ |#2|) "\\spad{order(f,x)} returns the order of \\spad{f} viewed as a series in \\spad{x} may result in an infinite loop if \\spad{f} has no non-zero terms.")) (|monomial| (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{monomial(a,[x1,x2,...,xk],[n1,n2,...,nk])} returns \\spad{a * x1^n1 * ... * xk^nk}.") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{monomial(a,x,n)} returns \\spad{a*x^n}.")) (|extend| (($ $ (|NonNegativeInteger|)) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree \\spad{<= n} to be computed.")) (|coefficient| (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{coefficient(f,[x1,x2,...,xk],[n1,n2,...,nk])} returns the coefficient of \\spad{x1^n1 * ... * xk^nk} in \\spad{f}.") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{coefficient(f,x,n)} returns the coefficient of \\spad{x^n} in \\spad{f}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3973 . T) (-3972 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-675 OV E R P) +(-680 OV E R P) ((|constructor| (NIL "\\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \\spadtype{Integer} or \\spadtype{Fraction Integer}.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} over its coefficient domain where \\spad{p} is represented as a univariate polynomial with multivariate coefficients") (((|Factored| |#4|) |#4|) "\\spad{factor(p)} factors the multivariate polynomial \\spad{p} over its coefficient domain"))) NIL NIL -(-676 E OV R P) +(-681 E OV R P) ((|constructor| (NIL "Author : \\spad{P}.Gianni This package provides the functions for the computation of the square free decomposition of a multivariate polynomial. It uses the package GenExEuclid for the resolution of the equation \\spad{Af + Bg = h} and its generalization to \\spad{n} polynomials over an integral domain and the package \\spad{MultivariateLifting} for the \"multivariate\" lifting.")) (|normDeriv2| (((|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#3|) (|Integer|)) "\\spad{normDeriv2 should} be local")) (|myDegree| (((|List| (|NonNegativeInteger|)) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|NonNegativeInteger|)) "\\spad{myDegree should} be local")) (|lift| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#3|) (|SparseUnivariatePolynomial| |#3|) |#4| (|List| |#2|) (|List| (|NonNegativeInteger|)) (|List| |#3|)) "\\spad{lift should} be local")) (|check| (((|Boolean|) (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|)))) (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|))))) "\\spad{check should} be local")) (|coefChoose| ((|#4| (|Integer|) (|Factored| |#4|)) "\\spad{coefChoose should} be local")) (|intChoose| (((|Record| (|:| |upol| (|SparseUnivariatePolynomial| |#3|)) (|:| |Lval| (|List| |#3|)) (|:| |Lfact| (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|))))) (|:| |ctpol| |#3|)) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|List| |#3|))) "\\spad{intChoose should} be local")) (|nsqfree| (((|Record| (|:| |unitPart| |#4|) (|:| |suPart| (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#4|)) (|:| |exponent| (|Integer|)))))) (|SparseUnivariatePolynomial| |#4|) (|List| |#2|) (|List| (|List| |#3|))) "\\spad{nsqfree should} be local")) (|consnewpol| (((|Record| (|:| |pol| (|SparseUnivariatePolynomial| |#4|)) (|:| |polval| (|SparseUnivariatePolynomial| |#3|))) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#3|) (|Integer|)) "\\spad{consnewpol should} be local")) (|univcase| (((|Factored| |#4|) |#4| |#2|) "\\spad{univcase should} be local")) (|compdegd| (((|Integer|) (|List| (|Record| (|:| |factor| (|SparseUnivariatePolynomial| |#3|)) (|:| |exponent| (|Integer|))))) "\\spad{compdegd should} be local")) (|squareFreePrim| (((|Factored| |#4|) |#4|) "\\spad{squareFreePrim(p)} compute the square free decomposition of a primitive multivariate polynomial \\spad{p}.")) (|squareFree| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{squareFree(p)} computes the square free decomposition of a multivariate polynomial \\spad{p} presented as a univariate polynomial with multivariate coefficients.") (((|Factored| |#4|) |#4|) "\\spad{squareFree(p)} computes the square free decomposition of a multivariate polynomial \\spad{p}."))) NIL NIL -(-677 S R) +(-682 S R) ((|constructor| (NIL "NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \\indented{3}{r*(a*b) = (r*a)*b = a*(r*b)}")) (|plenaryPower| (($ $ (|PositiveInteger|)) "\\spad{plenaryPower(a,n)} is recursively defined to be \\spad{plenaryPower(a,n-1)*plenaryPower(a,n-1)} for \\spad{n>1} and \\spad{a} for \\spad{n=1}."))) NIL NIL -(-678 R) +(-683 R) ((|constructor| (NIL "NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \\indented{3}{r*(a*b) = (r*a)*b = a*(r*b)}")) (|plenaryPower| (($ $ (|PositiveInteger|)) "\\spad{plenaryPower(a,n)} is recursively defined to be \\spad{plenaryPower(a,n-1)*plenaryPower(a,n-1)} for \\spad{n>1} and \\spad{a} for \\spad{n=1}."))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-679 S) +(-684 S) ((|constructor| (NIL "NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \\indented{2}{x*(y+z) = x*y + x*z} \\indented{2}{(x+y)*z = x*z + y*z} Common Additional Axioms \\indented{2}{noZeroDivisors\\space{2}ab = 0 => \\spad{a=0} or \\spad{b=0}}")) (|antiCommutator| (($ $ $) "\\spad{antiCommutator(a,b)} returns \\spad{a*b+b*a}.")) (|commutator| (($ $ $) "\\spad{commutator(a,b)} returns \\spad{a*b-b*a}.")) (|associator| (($ $ $ $) "\\spad{associator(a,b,c)} returns \\spad{(a*b)*c-a*(b*c)}."))) NIL NIL -(-680) +(-685) ((|constructor| (NIL "NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \\indented{2}{x*(y+z) = x*y + x*z} \\indented{2}{(x+y)*z = x*z + y*z} Common Additional Axioms \\indented{2}{noZeroDivisors\\space{2}ab = 0 => \\spad{a=0} or \\spad{b=0}}")) (|antiCommutator| (($ $ $) "\\spad{antiCommutator(a,b)} returns \\spad{a*b+b*a}.")) (|commutator| (($ $ $) "\\spad{commutator(a,b)} returns \\spad{a*b-b*a}.")) (|associator| (($ $ $ $) "\\spad{associator(a,b,c)} returns \\spad{(a*b)*c-a*(b*c)}."))) NIL NIL -(-681 S) +(-686 S) ((|constructor| (NIL "A NonAssociativeRing is a non associative rng which has a unit,{} the multiplication is not necessarily commutative or associative.")) (|coerce| (($ (|Integer|)) "\\spad{coerce(n)} coerces the integer \\spad{n} to an element of the ring.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring."))) NIL NIL -(-682) +(-687) ((|constructor| (NIL "A NonAssociativeRing is a non associative rng which has a unit,{} the multiplication is not necessarily commutative or associative.")) (|coerce| (($ (|Integer|)) "\\spad{coerce(n)} coerces the integer \\spad{n} to an element of the ring.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring."))) NIL NIL -(-683 |Par|) +(-688 |Par|) ((|constructor| (NIL "This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the complex rational numbers. The results are expressed either as complex floating numbers or as complex rational numbers depending on the type of the precision parameter.")) (|complexEigenvectors| (((|List| (|Record| (|:| |outval| (|Complex| |#1|)) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| (|Complex| |#1|)))))) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) |#1|) "\\spad{complexEigenvectors(m,eps)} returns a list of records each one containing a complex eigenvalue,{} its algebraic multiplicity,{} and a list of associated eigenvectors. All these results are computed to precision \\spad{eps} and are expressed as complex floats or complex rational numbers depending on the type of \\spad{eps} (float or rational).")) (|complexEigenvalues| (((|List| (|Complex| |#1|)) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) |#1|) "\\spad{complexEigenvalues(m,eps)} computes the eigenvalues of the matrix \\spad{m} to precision \\spad{eps}. The eigenvalues are expressed as complex floats or complex rational numbers depending on the type of \\spad{eps} (float or rational).")) (|characteristicPolynomial| (((|Polynomial| (|Complex| (|Fraction| (|Integer|)))) (|Matrix| (|Complex| (|Fraction| (|Integer|)))) (|Symbol|)) "\\spad{characteristicPolynomial(m,x)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over Complex Rationals with variable \\spad{x}.") (((|Polynomial| (|Complex| (|Fraction| (|Integer|)))) (|Matrix| (|Complex| (|Fraction| (|Integer|))))) "\\spad{characteristicPolynomial(m)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over complex rationals with a new symbol as variable."))) NIL NIL -(-684 -3078) +(-689 -3088) ((|constructor| (NIL "\\spadtype{NumericContinuedFraction} provides functions \\indented{2}{for converting floating point numbers to continued fractions.}")) (|continuedFraction| (((|ContinuedFraction| (|Integer|)) |#1|) "\\spad{continuedFraction(f)} converts the floating point number \\spad{f} to a reduced continued fraction."))) NIL NIL -(-685 P -3078) +(-690 P -3088) ((|constructor| (NIL "This package provides a division and related operations for \\spadtype{MonogenicLinearOperator}\\spad{s} over a \\spadtype{Field}. Since the multiplication is in general non-commutative,{} these operations all have left- and right-hand versions. This package provides the operations based on left-division.")) (|leftLcm| ((|#1| |#1| |#1|) "\\spad{leftLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = a*aa = b*bb} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using left-division.")) (|leftGcd| ((|#1| |#1| |#1|) "\\spad{leftGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = aa*g}} \\indented{3}{\\spad{b = bb*g}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using left-division.")) (|leftExactQuotient| (((|Union| |#1| "failed") |#1| |#1|) "\\spad{leftExactQuotient(a,b)} computes the value \\spad{q},{} if it exists,{} \\indented{1}{such that \\spad{a = b*q}.}")) (|leftRemainder| ((|#1| |#1| |#1|) "\\spad{leftRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|leftQuotient| ((|#1| |#1| |#1|) "\\spad{leftQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|leftDivide| (((|Record| (|:| |quotient| |#1|) (|:| |remainder| |#1|)) |#1| |#1|) "\\spad{leftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division''."))) NIL NIL -(-686 T$) +(-691 T$) NIL NIL NIL -(-687 UP -3078) +(-692 UP -3088) ((|constructor| (NIL "In this package \\spad{F} is a framed algebra over the integers (typically \\spad{F = Z[a]} for some algebraic integer a). The package provides functions to compute the integral closure of \\spad{Z} in the quotient quotient field of \\spad{F}.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| (|Integer|))) (|:| |basisDen| (|Integer|)) (|:| |basisInv| (|Matrix| (|Integer|)))) (|Integer|)) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{Z} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{Z}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| (|Integer|))) (|:| |basisDen| (|Integer|)) (|:| |basisInv| (|Matrix| (|Integer|))))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{Z} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{Z}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|discriminant| (((|Integer|)) "\\spad{discriminant()} returns the discriminant of the integral closure of \\spad{Z} in the quotient field of the framed algebra \\spad{F}."))) NIL NIL -(-688 R) +(-693 R) ((|constructor| (NIL "NonLinearSolvePackage is an interface to \\spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving. The solutions are given in the algebraic closure of \\spad{R} whenever possible.")) (|solve| (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{solve(lp)} finds the solution in the algebraic closure of \\spad{R} of the list \\spad{lp} of rational functions with respect to all the symbols appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{solve(lp,lv)} finds the solutions in the algebraic closure of \\spad{R} of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}.")) (|solveInField| (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{solveInField(lp)} finds the solution of the list \\spad{lp} of rational functions with respect to all the symbols appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{solveInField(lp,lv)} finds the solutions of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}."))) NIL NIL -(-689) +(-694) ((|constructor| (NIL "\\spadtype{NonNegativeInteger} provides functions for non \\indented{2}{negative integers.}")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} means multiplication is commutative : \\spad{x*y = y*x}.")) (|random| (($ $) "\\spad{random(n)} returns a random integer from 0 to \\spad{n-1}.")) (|shift| (($ $ (|Integer|)) "\\spad{shift(a,i)} shift \\spad{a} by \\spad{i} bits.")) (|exquo| (((|Union| $ "failed") $ $) "\\spad{exquo(a,b)} returns the quotient of \\spad{a} and \\spad{b},{} or \"failed\" if \\spad{b} is zero or \\spad{a} rem \\spad{b} is zero.")) (|divide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{divide(a,b)} returns a record containing both remainder and quotient.")) (|gcd| (($ $ $) "\\spad{gcd(a,b)} computes the greatest common divisor of two non negative integers \\spad{a} and \\spad{b}.")) (|rem| (($ $ $) "\\spad{a rem b} returns the remainder of \\spad{a} and \\spad{b}.")) (|quo| (($ $ $) "\\spad{a quo b} returns the quotient of \\spad{a} and \\spad{b},{} forgetting the remainder."))) -(((-3980 "*") . T)) +(((-3991 "*") . T)) NIL -(-690 R -3078) +(-695 R -3088) ((|constructor| (NIL "NonLinearFirstOrderODESolver provides a function for finding closed form first integrals of nonlinear ordinary differential equations of order 1.")) (|solve| (((|Union| |#2| "failed") |#2| |#2| (|BasicOperator|) (|Symbol|)) "\\spad{solve(M(x,y), N(x,y), y, x)} returns \\spad{F(x,y)} such that \\spad{F(x,y) = c} for a constant \\spad{c} is a first integral of the equation \\spad{M(x,y) dx + N(x,y) dy = 0},{} or \"failed\" if no first-integral can be found."))) NIL NIL -(-691) +(-696) ((|constructor| (NIL "\\spadtype{None} implements a type with no objects. It is mainly used in technical situations where such a thing is needed (\\spadignore{e.g.} the interpreter and some of the internal \\spadtype{Expression} code)."))) NIL NIL -(-692 S) +(-697 S) ((|constructor| (NIL "\\spadtype{NoneFunctions1} implements functions on \\spadtype{None}. It particular it includes a particulary dangerous coercion from any other type to \\spadtype{None}.")) (|coerce| (((|None|) |#1|) "\\spad{coerce(x)} changes \\spad{x} into an object of type \\spadtype{None}."))) NIL NIL -(-693 R |PolR| E |PolE|) +(-698 R |PolR| E |PolE|) ((|constructor| (NIL "This package implements the norm of a polynomial with coefficients in a monogenic algebra (using resultants)")) (|norm| ((|#2| |#4|) "\\spad{norm q} returns the norm of \\spad{q},{} \\spadignore{i.e.} the product of all the conjugates of \\spad{q}."))) NIL NIL -(-694 R E V P TS) +(-699 R E V P TS) ((|constructor| (NIL "A package for computing normalized assocites of univariate polynomials with coefficients in a tower of simple extensions of a field.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of gcd over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of \\spad{AAECC11}} \\indented{5}{Paris,{} 1995.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.}")) (|normInvertible?| (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{normInvertible?(\\spad{p},{}ts)} is an internal subroutine,{} exported only for developement.")) (|outputArgs| (((|Void|) (|String|) (|String|) |#4| |#5|) "\\axiom{outputArgs(\\spad{s1},{}\\spad{s2},{}\\spad{p},{}ts)} is an internal subroutine,{} exported only for developement.")) (|normalize| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{normalize(\\spad{p},{}ts)} normalizes \\axiom{\\spad{p}} \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}.")) (|normalizedAssociate| ((|#4| |#4| |#5|) "\\axiom{normalizedAssociate(\\spad{p},{}ts)} returns a normalized polynomial \\axiom{\\spad{n}} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts} such that \\axiom{\\spad{n}} and \\axiom{\\spad{p}} are associates \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts} and assuming that \\axiom{\\spad{p}} is invertible \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}.")) (|recip| (((|Record| (|:| |num| |#4|) (|:| |den| |#4|)) |#4| |#5|) "\\axiom{recip(\\spad{p},{}ts)} returns the inverse of \\axiom{\\spad{p}} \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts} assuming that \\axiom{\\spad{p}} is invertible \\spad{w}.\\spad{r}.\\spad{t} \\spad{ts}."))) NIL NIL -(-695 -3078 |ExtF| |SUEx| |ExtP| |n|) +(-700 -3088 |ExtF| |SUEx| |ExtP| |n|) ((|constructor| (NIL "This package \\undocumented")) (|Frobenius| ((|#4| |#4|) "\\spad{Frobenius(x)} \\undocumented")) (|retractIfCan| (((|Union| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|)) "failed") |#4|) "\\spad{retractIfCan(x)} \\undocumented")) (|normFactors| (((|List| |#4|) |#4|) "\\spad{normFactors(x)} \\undocumented"))) NIL NIL -(-696 BP E OV R P) +(-701 BP E OV R P) ((|constructor| (NIL "Package for the determination of the coefficients in the lifting process. Used by \\spadtype{MultivariateLifting}. This package will work for every euclidean domain \\spad{R} which has property \\spad{F},{} \\spadignore{i.e.} there exists a factor operation in \\spad{R[x]}.")) (|listexp| (((|List| (|NonNegativeInteger|)) |#1|) "\\spad{listexp }\\undocumented")) (|npcoef| (((|Record| (|:| |deter| (|List| (|SparseUnivariatePolynomial| |#5|))) (|:| |dterm| (|List| (|List| (|Record| (|:| |expt| (|NonNegativeInteger|)) (|:| |pcoef| |#5|))))) (|:| |nfacts| (|List| |#1|)) (|:| |nlead| (|List| |#5|))) (|SparseUnivariatePolynomial| |#5|) (|List| |#1|) (|List| |#5|)) "\\spad{npcoef }\\undocumented"))) NIL NIL -(-697 |Par|) +(-702 |Par|) ((|constructor| (NIL "This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the Rational Numbers. The results are expressed as floating numbers or as rational numbers depending on the type of the parameter Par.")) (|realEigenvectors| (((|List| (|Record| (|:| |outval| |#1|) (|:| |outmult| (|Integer|)) (|:| |outvect| (|List| (|Matrix| |#1|))))) (|Matrix| (|Fraction| (|Integer|))) |#1|) "\\spad{realEigenvectors(m,eps)} returns a list of records each one containing a real eigenvalue,{} its algebraic multiplicity,{} and a list of associated eigenvectors. All these results are computed to precision \\spad{eps} as floats or rational numbers depending on the type of \\spad{eps} .")) (|realEigenvalues| (((|List| |#1|) (|Matrix| (|Fraction| (|Integer|))) |#1|) "\\spad{realEigenvalues(m,eps)} computes the eigenvalues of the matrix \\spad{m} to precision \\spad{eps}. The eigenvalues are expressed as floats or rational numbers depending on the type of \\spad{eps} (float or rational).")) (|characteristicPolynomial| (((|Polynomial| (|Fraction| (|Integer|))) (|Matrix| (|Fraction| (|Integer|))) (|Symbol|)) "\\spad{characteristicPolynomial(m,x)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over RN with variable \\spad{x}. Fraction \\spad{P} RN.") (((|Polynomial| (|Fraction| (|Integer|))) (|Matrix| (|Fraction| (|Integer|)))) "\\spad{characteristicPolynomial(m)} returns the characteristic polynomial of the matrix \\spad{m} expressed as polynomial over RN with a new symbol as variable."))) NIL NIL -(-698 R |VarSet|) +(-703 R |VarSet|) ((|constructor| (NIL "A post-facto extension for \\axiomType{SMP} in order to speed up operations related to pseudo-division and gcd. This domain is based on the \\axiomType{NSUP} constructor which is itself a post-facto extension of the \\axiomType{SUP} constructor."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . 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(|halfExtendedResultant2| (((|Record| (|:| |resultant| |#1|) (|:| |coef2| $)) $ $) "\\axiom{\\spad{halfExtendedResultant2}(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca]} such that \\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{} cb]}")) (|halfExtendedResultant1| (((|Record| (|:| |resultant| |#1|) (|:| |coef1| $)) $ $) "\\axiom{\\spad{halfExtendedResultant1}(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca]} such that \\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{} cb]}")) (|extendedResultant| (((|Record| (|:| |resultant| |#1|) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedResultant(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}ca,{}cb]} such that \\axiom{\\spad{r}} is the resultant of \\axiom{a} and \\axiom{\\spad{b}} and \\axiom{\\spad{r} = ca * a + cb * \\spad{b}}")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{\\spad{halfExtendedSubResultantGcd2}(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}cb]} such that \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} cb]}")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{\\spad{halfExtendedSubResultantGcd1}(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} such that \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} cb]}")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{} cb]} such that \\axiom{\\spad{g}} is a gcd of \\axiom{a} and \\axiom{\\spad{b}} in \\axiom{R^(\\spad{-1}) \\spad{P}} and \\axiom{\\spad{g} = ca * a + cb * \\spad{b}}")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns \\axiom{resultant(a,{}\\spad{b})} if \\axiom{a} and \\axiom{\\spad{b}} has no non-trivial gcd in \\axiom{R^(\\spad{-1}) \\spad{P}} otherwise the non-zero sub-resultant with smallest index.")) (|subResultantsChain| (((|List| $) $ $) "\\axiom{subResultantsChain(a,{}\\spad{b})} returns the list of the non-zero sub-resultants of \\axiom{a} and \\axiom{\\spad{b}} sorted by increasing degree.")) (|lazyPseudoQuotient| (($ $ $) "\\axiom{lazyPseudoQuotient(a,{}\\spad{b})} returns \\axiom{\\spad{q}} if \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}")) (|lazyPseudoDivide| (((|Record| (|:| |coef| |#1|) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{c^n * a = q*b +r} and \\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]} where \\axiom{\\spad{n} + \\spad{g} = max(0,{} degree(\\spad{b}) - degree(a) + 1)}.")) (|lazyPseudoRemainder| (($ $ $) "\\axiom{lazyPseudoRemainder(a,{}\\spad{b})} returns \\axiom{\\spad{r}} if \\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]}. This lazy pseudo-remainder is computed by means of the \\axiomOpFrom{fmecg}{NewSparseUnivariatePolynomial} operation.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| |#1|) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{r},{}\\spad{c},{}\\spad{n}]} such that \\axiom{\\spad{r}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{b}} divides \\axiom{c^n * a - \\spad{r}} where \\axiom{\\spad{c}} is \\axiom{leadingCoefficient(\\spad{b})} and \\axiom{\\spad{n}} is as small as possible with the previous properties.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} returns \\axiom{\\spad{r}} such that \\axiom{\\spad{r}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{b}} divides \\axiom{a -r} where \\axiom{\\spad{b}} is monic.")) (|fmecg| (($ $ (|NonNegativeInteger|) |#1| $) "\\axiom{fmecg(\\spad{p1},{}\\spad{e},{}\\spad{r},{}\\spad{p2})} returns \\axiom{\\spad{p1} - \\spad{r} * X**e * \\spad{p2}} where \\axiom{\\spad{X}} is \\axiom{monomial(1,{}1)}"))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3974 |has| |#1| (-309)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| (-988) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| (-988) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-988) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-988) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| (-988) (QUOTE (-550 (-469))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-816)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-1057))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-188))) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-700 R S) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3985 |has| |#1| (-311)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| (-993) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| (-993) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| (-993) (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-1064))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-190))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-705 R S) ((|constructor| (NIL "This package lifts a mapping from coefficient rings \\spad{R} to \\spad{S} to a mapping from sparse univariate polynomial over \\spad{R} to a sparse univariate polynomial over \\spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.")) (|map| (((|NewSparseUnivariatePolynomial| |#2|) (|Mapping| |#2| |#1|) (|NewSparseUnivariatePolynomial| |#1|)) "\\axiom{map(func,{} poly)} creates a new polynomial by applying func to every non-zero coefficient of the polynomial poly."))) NIL NIL -(-701 R) +(-706 R) ((|constructor| (NIL "This package provides polynomials as functions on a ring.")) (|eulerE| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{eulerE(n,r)} \\undocumented")) (|bernoulliB| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{bernoulliB(n,r)} \\undocumented")) (|cyclotomic| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{cyclotomic(n,r)} \\undocumented"))) NIL -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480)))))) -(-702 R E V P) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483)))))) +(-707 R E V P) ((|constructor| (NIL "The category of normalized triangular sets. A triangular set \\spad{ts} is said normalized if for every algebraic variable \\spad{v} of \\spad{ts} the polynomial \\spad{select(ts,v)} is normalized \\spad{w}.\\spad{r}.\\spad{t}. every polynomial in \\spad{collectUnder(ts,v)}. A polynomial \\spad{p} is said normalized \\spad{w}.\\spad{r}.\\spad{t}. a non-constant polynomial \\spad{q} if \\spad{p} is constant or \\spad{degree(p,mdeg(q)) = 0} and \\spad{init(p)} is normalized \\spad{w}.\\spad{r}.\\spad{t}. \\spad{q}. One of the important features of normalized triangular sets is that they are regular sets.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)} \\indented{1}{[3] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of gcd over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of \\spad{AAECC11}} \\indented{5}{Paris,{} 1995.} \\indented{1}{[4] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.}"))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-703 S) +(-708 S) ((|constructor| (NIL "Numeric provides real and complex numerical evaluation functions for various symbolic types.")) (|numericIfCan| (((|Union| (|Float|) "failed") (|Expression| |#1|) (|PositiveInteger|)) "\\spad{numericIfCan(x, n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Expression| |#1|)) "\\spad{numericIfCan(x)} returns a real approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{numericIfCan(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Fraction| (|Polynomial| |#1|))) "\\spad{numericIfCan(x)} returns a real approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{numericIfCan(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Float|) "failed") (|Polynomial| |#1|)) "\\spad{numericIfCan(x)} returns a real approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.")) (|complexNumericIfCan| (((|Union| (|Complex| (|Float|)) "failed") (|Expression| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Expression| (|Complex| |#1|))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Expression| |#1|) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Expression| |#1|)) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| (|Complex| |#1|))) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| (|Complex| |#1|)))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Fraction| (|Polynomial| |#1|))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| |#1|)) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumericIfCan(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places,{} or \"failed\" if \\axiom{\\spad{x}} is not a constant.") (((|Union| (|Complex| (|Float|)) "failed") (|Polynomial| (|Complex| |#1|))) "\\spad{complexNumericIfCan(x)} returns a complex approximation of \\spad{x},{} or \"failed\" if \\axiom{\\spad{x}} is not constant.")) (|complexNumeric| (((|Complex| (|Float|)) (|Expression| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Expression| (|Complex| |#1|))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Expression| |#1|) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Expression| |#1|)) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| (|Complex| |#1|))) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| (|Complex| |#1|)))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x}") (((|Complex| (|Float|)) (|Fraction| (|Polynomial| |#1|))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Polynomial| |#1|)) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Polynomial| (|Complex| |#1|)) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Polynomial| (|Complex| |#1|))) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) (|Complex| |#1|) (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) (|Complex| |#1|)) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.") (((|Complex| (|Float|)) |#1| (|PositiveInteger|)) "\\spad{complexNumeric(x, n)} returns a complex approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Complex| (|Float|)) |#1|) "\\spad{complexNumeric(x)} returns a complex approximation of \\spad{x}.")) (|numeric| (((|Float|) (|Expression| |#1|) (|PositiveInteger|)) "\\spad{numeric(x, n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) (|Expression| |#1|)) "\\spad{numeric(x)} returns a real approximation of \\spad{x}.") (((|Float|) (|Fraction| (|Polynomial| |#1|)) (|PositiveInteger|)) "\\spad{numeric(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) (|Fraction| (|Polynomial| |#1|))) "\\spad{numeric(x)} returns a real approximation of \\spad{x}.") (((|Float|) (|Polynomial| |#1|) (|PositiveInteger|)) "\\spad{numeric(x,n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) (|Polynomial| |#1|)) "\\spad{numeric(x)} returns a real approximation of \\spad{x}.") (((|Float|) |#1| (|PositiveInteger|)) "\\spad{numeric(x, n)} returns a real approximation of \\spad{x} up to \\spad{n} decimal places.") (((|Float|) |#1|) "\\spad{numeric(x)} returns a real approximation of \\spad{x}."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-751)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-956))) (|HasCategory| |#1| (QUOTE (-144)))) -(-704) +((-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-756)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-961))) (|HasCategory| |#1| (QUOTE (-146)))) +(-709) ((|constructor| (NIL "NumberFormats provides function to format and read arabic and roman numbers,{} to convert numbers to strings and to read floating-point numbers.")) (|ScanFloatIgnoreSpacesIfCan| (((|Union| (|Float|) "failed") (|String|)) "\\spad{ScanFloatIgnoreSpacesIfCan(s)} tries to form a floating point number from the string \\spad{s} ignoring any spaces.")) (|ScanFloatIgnoreSpaces| (((|Float|) (|String|)) "\\spad{ScanFloatIgnoreSpaces(s)} forms a floating point number from the string \\spad{s} ignoring any spaces. Error is generated if the string is not recognised as a floating point number.")) (|ScanRoman| (((|PositiveInteger|) (|String|)) "\\spad{ScanRoman(s)} forms an integer from a Roman numeral string \\spad{s}.")) (|FormatRoman| (((|String|) (|PositiveInteger|)) "\\spad{FormatRoman(n)} forms a Roman numeral string from an integer \\spad{n}.")) (|ScanArabic| (((|PositiveInteger|) (|String|)) "\\spad{ScanArabic(s)} forms an integer from an Arabic numeral string \\spad{s}.")) (|FormatArabic| (((|String|) (|PositiveInteger|)) "\\spad{FormatArabic(n)} forms an Arabic numeral string from an integer \\spad{n}."))) NIL NIL -(-705) +(-710) ((|constructor| (NIL "This package is a suite of functions for the numerical integration of an ordinary differential equation of \\spad{n} variables: \\blankline \\indented{8}{\\center{dy/dx = \\spad{f}(\\spad{y},{}\\spad{x})\\space{5}\\spad{y} is an \\spad{n}-vector}} \\blankline \\par All the routines are based on a 4-th order Runge-Kutta kernel. These routines generally have as arguments: \\spad{n},{} the number of dependent variables; \\spad{x1},{} the initial point; \\spad{h},{} the step size; \\spad{y},{} a vector of initial conditions of length \\spad{n} which upon exit contains the solution at \\spad{x1 + h}; \\spad{derivs},{} a function which computes the right hand side of the ordinary differential equation: \\spad{derivs(dydx,y,x)} computes \\spad{dydx},{} a vector which contains the derivative information. \\blankline \\par In order of increasing complexity:\\begin{items} \\blankline \\item \\spad{rk4(y,n,x1,h,derivs)} advances the solution vector to \\spad{x1 + h} and return the values in \\spad{y}. \\blankline \\item \\spad{rk4(y,n,x1,h,derivs,t1,t2,t3,t4)} is the same as \\spad{rk4(y,n,x1,h,derivs)} except that you must provide 4 scratch arrays \\spad{t1}-\\spad{t4} of size \\spad{n}. \\blankline \\item Starting with \\spad{y} at \\spad{x1},{} \\spad{rk4f(y,n,x1,x2,ns,derivs)} uses \\spad{ns} fixed steps of a 4-th order Runge-Kutta integrator to advance the solution vector to \\spad{x2} and return the values in \\spad{y}. Argument \\spad{x2},{} is the final point,{} and \\spad{ns},{} the number of steps to take. \\blankline \\item \\spad{rk4qc(y,n,x1,step,eps,yscal,derivs)} takes a 5-th order Runge-Kutta step with monitoring of local truncation to ensure accuracy and adjust stepsize. The function takes two half steps and one full step and scales the difference in solutions at the final point. If the error is within \\spad{eps},{} the step is taken and the result is returned. If the error is not within \\spad{eps},{} the stepsize if decreased and the procedure is tried again until the desired accuracy is reached. Upon input,{} an trial step size must be given and upon return,{} an estimate of the next step size to use is returned as well as the step size which produced the desired accuracy. The scaled error is computed as \\center{\\spad{error = MAX(ABS((y2steps(i) - y1step(i))/yscal(i)))}} and this is compared against \\spad{eps}. If this is greater than \\spad{eps},{} the step size is reduced accordingly to \\center{\\spad{hnew = 0.9 * hdid * (error/eps)**(-1/4)}} If the error criterion is satisfied,{} then we check if the step size was too fine and return a more efficient one. If \\spad{error > \\spad{eps} * (6.0E-04)} then the next step size should be \\center{\\spad{hnext = 0.9 * hdid * (error/\\spad{eps})**(\\spad{-1/5})}} Otherwise \\spad{hnext = 4.0 * hdid} is returned. A more detailed discussion of this and related topics can be found in the book \"Numerical Recipies\" by \\spad{W}.Press,{} \\spad{B}.\\spad{P}. Flannery,{} \\spad{S}.A. Teukolsky,{} \\spad{W}.\\spad{T}. Vetterling published by Cambridge University Press. Argument \\spad{step} is a record of 3 floating point numbers \\spad{(try , did , next)},{} \\spad{eps} is the required accuracy,{} \\spad{yscal} is the scaling vector for the difference in solutions. On input,{} \\spad{step.try} should be the guess at a step size to achieve the accuracy. On output,{} \\spad{step.did} contains the step size which achieved the accuracy and \\spad{step.next} is the next step size to use. \\blankline \\item \\spad{rk4qc(y,n,x1,step,eps,yscal,derivs,t1,t2,t3,t4,t5,t6,t7)} is the same as \\spad{rk4qc(y,n,x1,step,eps,yscal,derivs)} except that the user must provide the 7 scratch arrays \\spad{t1-t7} of size \\spad{n}. \\blankline \\item \\spad{rk4a(y,n,x1,x2,eps,h,ns,derivs)} is a driver program which uses \\spad{rk4qc} to integrate \\spad{n} ordinary differential equations starting at \\spad{x1} to \\spad{x2},{} keeping the local truncation error to within \\spad{eps} by changing the local step size. The scaling vector is defined as \\center{\\spad{yscal(i) = abs(y(i)) + abs(h*dydx(i)) + tiny}} where \\spad{y(i)} is the solution at location \\spad{x},{} \\spad{dydx} is the ordinary differential equation's right hand side,{} \\spad{h} is the current step size and \\spad{tiny} is 10 times the smallest positive number representable. The user must supply an estimate for a trial step size and the maximum number of calls to \\spad{rk4qc} to use. Argument \\spad{x2} is the final point,{} \\spad{eps} is local truncation,{} \\spad{ns} is the maximum number of call to \\spad{rk4qc} to use. \\end{items}")) (|rk4f| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Integer|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4f(y,n,x1,x2,ns,derivs)} uses a 4-th order Runge-Kutta method to numerically integrate the ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector. Starting with \\spad{y} at \\spad{x1},{} this function uses \\spad{ns} fixed steps of a 4-th order Runge-Kutta integrator to advance the solution vector to \\spad{x2} and return the values in \\spad{y}. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.")) (|rk4qc| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Record| (|:| |tryValue| (|Float|)) (|:| |did| (|Float|)) (|:| |next| (|Float|))) (|Float|) (|Vector| (|Float|)) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|))) "\\spad{rk4qc(y,n,x1,step,eps,yscal,derivs,t1,t2,t3,t4,t5,t6,t7)} is a subfunction for the numerical integration of an ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector using a 4-th order Runge-Kutta method. This function takes a 5-th order Runge-Kutta \\spad{step} with monitoring of local truncation to ensure accuracy and adjust stepsize. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.") (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Record| (|:| |tryValue| (|Float|)) (|:| |did| (|Float|)) (|:| |next| (|Float|))) (|Float|) (|Vector| (|Float|)) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4qc(y,n,x1,step,eps,yscal,derivs)} is a subfunction for the numerical integration of an ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector using a 4-th order Runge-Kutta method. This function takes a 5-th order Runge-Kutta \\spad{step} with monitoring of local truncation to ensure accuracy and adjust stepsize. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.")) (|rk4a| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4a(y,n,x1,x2,eps,h,ns,derivs)} is a driver function for the numerical integration of an ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector using a 4-th order Runge-Kutta method. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.")) (|rk4| (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|))) "\\spad{rk4(y,n,x1,h,derivs,t1,t2,t3,t4)} is the same as \\spad{rk4(y,n,x1,h,derivs)} except that you must provide 4 scratch arrays \\spad{t1}-\\spad{t4} of size \\spad{n}. For details,{} see \\con{NumericalOrdinaryDifferentialEquations}.") (((|Void|) (|Vector| (|Float|)) (|Integer|) (|Float|) (|Float|) (|Mapping| (|Void|) (|Vector| (|Float|)) (|Vector| (|Float|)) (|Float|))) "\\spad{rk4(y,n,x1,h,derivs)} uses a 4-th order Runge-Kutta method to numerically integrate the ordinary differential equation {\\em dy/dx = f(y,x)} of \\spad{n} variables,{} where \\spad{y} is an \\spad{n}-vector. Argument \\spad{y} is a vector of initial conditions of length \\spad{n} which upon exit contains the solution at \\spad{x1 + h},{} \\spad{n} is the number of dependent variables,{} \\spad{x1} is the initial point,{} \\spad{h} is the step size,{} and \\spad{derivs} is a function which computes the right hand side of the ordinary differential equation. For details,{} see \\spadtype{NumericalOrdinaryDifferentialEquations}."))) NIL NIL -(-706) +(-711) ((|constructor| (NIL "This suite of routines performs numerical quadrature using algorithms derived from the basic trapezoidal rule. Because the error term of this rule contains only even powers of the step size (for open and closed versions),{} fast convergence can be obtained if the integrand is sufficiently smooth. \\blankline Each routine returns a Record of type TrapAns,{} which contains\\indent{3} \\newline value (\\spadtype{Float}):\\tab{20} estimate of the integral \\newline error (\\spadtype{Float}):\\tab{20} estimate of the error in the computation \\newline totalpts (\\spadtype{Integer}):\\tab{20} total number of function evaluations \\newline success (\\spadtype{Boolean}):\\tab{20} if the integral was computed within the user specified error criterion \\indent{0}\\indent{0} To produce this estimate,{} each routine generates an internal sequence of sub-estimates,{} denoted by {\\em S(i)},{} depending on the routine,{} to which the various convergence criteria are applied. The user must supply a relative accuracy,{} \\spad{eps_r},{} and an absolute accuracy,{} \\spad{eps_a}. Convergence is obtained when either \\center{\\spad{ABS(S(i) - S(i-1)) < eps_r * ABS(S(i-1))}} \\center{or \\spad{ABS(S(i) - S(i-1)) < eps_a}} are \\spad{true} statements. \\blankline The routines come in three families and three flavors: \\newline\\tab{3} closed:\\tab{20}romberg,{}\\tab{30}simpson,{}\\tab{42}trapezoidal \\newline\\tab{3} open: \\tab{20}rombergo,{}\\tab{30}simpsono,{}\\tab{42}trapezoidalo \\newline\\tab{3} adaptive closed:\\tab{20}aromberg,{}\\tab{30}asimpson,{}\\tab{42}atrapezoidal \\par The {\\em S(i)} for the trapezoidal family is the value of the integral using an equally spaced absicca trapezoidal rule for that level of refinement. \\par The {\\em S(i)} for the simpson family is the value of the integral using an equally spaced absicca simpson rule for that level of refinement. \\par The {\\em S(i)} for the romberg family is the estimate of the integral using an equally spaced absicca romberg method. For the \\spad{i}\\spad{-}th level,{} this is an appropriate combination of all the previous trapezodial estimates so that the error term starts with the \\spad{2*(i+1)} power only. \\par The three families come in a closed version,{} where the formulas include the endpoints,{} an open version where the formulas do not include the endpoints and an adaptive version,{} where the user is required to input the number of subintervals over which the appropriate closed family integrator will apply with the usual convergence parmeters for each subinterval. This is useful where a large number of points are needed only in a small fraction of the entire domain. \\par Each routine takes as arguments: \\newline \\spad{f}\\tab{10} integrand \\newline a\\tab{10} starting point \\newline \\spad{b}\\tab{10} ending point \\newline \\spad{eps_r}\\tab{10} relative error \\newline \\spad{eps_a}\\tab{10} absolute error \\newline \\spad{nmin} \\tab{10} refinement level when to start checking for convergence (> 1) \\newline \\spad{nmax} \\tab{10} maximum level of refinement \\par The adaptive routines take as an additional parameter \\newline \\spad{nint}\\tab{10} the number of independent intervals to apply a closed \\indented{1}{family integrator of the same name.} \\par Notes: \\newline Closed family level \\spad{i} uses \\spad{1 + 2**i} points. \\newline Open family level \\spad{i} uses \\spad{1 + 3**i} points.")) (|trapezoidalo| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{trapezoidalo(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the trapezoidal method to numerically integrate function \\spad{fn} over the open interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|simpsono| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{simpsono(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the simpson method to numerically integrate function \\spad{fn} over the open interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|rombergo| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{rombergo(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the romberg method to numerically integrate function \\spad{fn} over the open interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|trapezoidal| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{trapezoidal(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the trapezoidal method to numerically integrate function \\spadvar{\\spad{fn}} over the closed interval \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|simpson| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{simpson(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the simpson method to numerically integrate function \\spad{fn} over the closed interval \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|romberg| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|)) "\\spad{romberg(fn,a,b,epsrel,epsabs,nmin,nmax)} uses the romberg method to numerically integrate function \\spadvar{\\spad{fn}} over the closed interval \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|atrapezoidal| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{atrapezoidal(fn,a,b,epsrel,epsabs,nmin,nmax,nint)} uses the adaptive trapezoidal method to numerically integrate function \\spad{fn} over the closed interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax},{} and where \\spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|asimpson| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{asimpson(fn,a,b,epsrel,epsabs,nmin,nmax,nint)} uses the adaptive simpson method to numerically integrate function \\spad{fn} over the closed interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax},{} and where \\spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details.")) (|aromberg| (((|Record| (|:| |value| (|Float|)) (|:| |error| (|Float|)) (|:| |totalpts| (|Integer|)) (|:| |success| (|Boolean|))) (|Mapping| (|Float|) (|Float|)) (|Float|) (|Float|) (|Float|) (|Float|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{aromberg(fn,a,b,epsrel,epsabs,nmin,nmax,nint)} uses the adaptive romberg method to numerically integrate function \\spad{fn} over the closed interval from \\spad{a} to \\spad{b},{} with relative accuracy \\spad{epsrel} and absolute accuracy \\spad{epsabs},{} with the refinement levels for convergence checking vary from \\spad{nmin} to \\spad{nmax},{} and where \\spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \\spad{true} if the integral was computed within the user specified error criterion. See \\spadtype{NumericalQuadrature} for details."))) NIL NIL -(-707 |Curve|) +(-712 |Curve|) ((|constructor| (NIL "\\indented{1}{Author: Clifton \\spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.")) (|tube| (((|TubePlot| |#1|) |#1| (|DoubleFloat|) (|Integer|)) "\\spad{tube(c,r,n)} creates a tube of radius \\spad{r} around the curve \\spad{c}."))) NIL NIL -(-708 S) +(-713 S) ((|constructor| (NIL "Ordered sets which are also abelian groups,{} such that the addition preserves the ordering.")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x}.")) (|sign| (((|Integer|) $) "\\spad{sign(x)} is \\spad{1} if \\spad{x} is positive,{} \\spad{-1} if \\spad{x} is negative,{} and \\spad{0} otherwise.")) (|negative?| (((|Boolean|) $) "\\spad{negative?(x)} holds when \\spad{x} is less than \\spad{0}."))) NIL NIL -(-709) +(-714) ((|constructor| (NIL "Ordered sets which are also abelian groups,{} such that the addition preserves the ordering.")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x}.")) (|sign| (((|Integer|) $) "\\spad{sign(x)} is \\spad{1} if \\spad{x} is positive,{} \\spad{-1} if \\spad{x} is negative,{} and \\spad{0} otherwise.")) (|negative?| (((|Boolean|) $) "\\spad{negative?(x)} holds when \\spad{x} is less than \\spad{0}."))) NIL NIL -(-710 S) +(-715 S) ((|constructor| (NIL "Ordered sets which are also abelian monoids,{} such that the addition preserves the ordering.")) (|positive?| (((|Boolean|) $) "\\spad{positive?(x)} holds when \\spad{x} is greater than \\spad{0}."))) NIL NIL -(-711) +(-716) ((|constructor| (NIL "Ordered sets which are also abelian monoids,{} such that the addition preserves the ordering.")) (|positive?| (((|Boolean|) $) "\\spad{positive?(x)} holds when \\spad{x} is greater than \\spad{0}."))) NIL NIL -(-712) +(-717) ((|constructor| (NIL "This domain is an OrderedAbelianMonoid with a \\spadfun{sup} operation added. The purpose of the \\spadfun{sup} operator in this domain is to act as a supremum with respect to the partial order imposed by \\spadop{-},{} rather than with respect to the total \\spad{>} order (since that is \"max\"). \\blankline")) (|sup| (($ $ $) "\\spad{sup(x,y)} returns the least element from which both \\spad{x} and \\spad{y} can be subtracted."))) NIL NIL -(-713) +(-718) ((|constructor| (NIL "Ordered sets which are also abelian semigroups,{} such that the addition preserves the ordering. \\indented{2}{\\spad{ x < y => x+z < y+z}}"))) NIL NIL -(-714 S R) +(-719 S R) ((|constructor| (NIL "OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.")) (|inv| (($ $) "\\spad{inv(o)} returns the inverse of \\spad{o} if it exists.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(o)} returns the real part if all seven imaginary parts are 0,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(o)} returns the real part if all seven imaginary parts are 0. Error: if \\spad{o} is not rational.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(o)} tests if \\spad{o} is rational,{} \\spadignore{i.e.} that all seven imaginary parts are 0.")) (|abs| ((|#2| $) "\\spad{abs(o)} computes the absolute value of an octonion,{} equal to the square root of the \\spadfunFrom{norm}{Octonion}.")) (|octon| (($ |#2| |#2| |#2| |#2| |#2| |#2| |#2| |#2|) "\\spad{octon(re,ri,rj,rk,rE,rI,rJ,rK)} constructs an octonion from scalars.")) (|norm| ((|#2| $) "\\spad{norm(o)} returns the norm of an octonion,{} equal to the sum of the squares of its coefficients.")) (|imagK| ((|#2| $) "\\spad{imagK(o)} extracts the imaginary \\spad{K} part of octonion \\spad{o}.")) (|imagJ| ((|#2| $) "\\spad{imagJ(o)} extracts the imaginary \\spad{J} part of octonion \\spad{o}.")) (|imagI| ((|#2| $) "\\spad{imagI(o)} extracts the imaginary \\spad{I} part of octonion \\spad{o}.")) (|imagE| ((|#2| $) "\\spad{imagE(o)} extracts the imaginary \\spad{E} part of octonion \\spad{o}.")) (|imagk| ((|#2| $) "\\spad{imagk(o)} extracts the \\spad{k} part of octonion \\spad{o}.")) (|imagj| ((|#2| $) "\\spad{imagj(o)} extracts the \\spad{j} part of octonion \\spad{o}.")) (|imagi| ((|#2| $) "\\spad{imagi(o)} extracts the \\spad{i} part of octonion \\spad{o}.")) (|real| ((|#2| $) "\\spad{real(o)} extracts real part of octonion \\spad{o}.")) (|conjugate| (($ $) "\\spad{conjugate(o)} negates the imaginary parts \\spad{i},{}\\spad{j},{}\\spad{k},{}\\spad{E},{}\\spad{I},{}\\spad{J},{}\\spad{K} of octonian \\spad{o}."))) NIL -((|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-967))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-315)))) -(-715 R) +((|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-482))) (|HasCategory| |#2| (QUOTE (-972))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-317)))) +(-720 R) ((|constructor| (NIL "OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.")) (|inv| (($ $) "\\spad{inv(o)} returns the inverse of \\spad{o} if it exists.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(o)} returns the real part if all seven imaginary parts are 0,{} and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(o)} returns the real part if all seven imaginary parts are 0. Error: if \\spad{o} is not rational.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(o)} tests if \\spad{o} is rational,{} \\spadignore{i.e.} that all seven imaginary parts are 0.")) (|abs| ((|#1| $) "\\spad{abs(o)} computes the absolute value of an octonion,{} equal to the square root of the \\spadfunFrom{norm}{Octonion}.")) (|octon| (($ |#1| |#1| |#1| |#1| |#1| |#1| |#1| |#1|) "\\spad{octon(re,ri,rj,rk,rE,rI,rJ,rK)} constructs an octonion from scalars.")) (|norm| ((|#1| $) "\\spad{norm(o)} returns the norm of an octonion,{} equal to the sum of the squares of its coefficients.")) (|imagK| ((|#1| $) "\\spad{imagK(o)} extracts the imaginary \\spad{K} part of octonion \\spad{o}.")) (|imagJ| ((|#1| $) "\\spad{imagJ(o)} extracts the imaginary \\spad{J} part of octonion \\spad{o}.")) (|imagI| ((|#1| $) "\\spad{imagI(o)} extracts the imaginary \\spad{I} part of octonion \\spad{o}.")) (|imagE| ((|#1| $) "\\spad{imagE(o)} extracts the imaginary \\spad{E} part of octonion \\spad{o}.")) (|imagk| ((|#1| $) "\\spad{imagk(o)} extracts the \\spad{k} part of octonion \\spad{o}.")) (|imagj| ((|#1| $) "\\spad{imagj(o)} extracts the \\spad{j} part of octonion \\spad{o}.")) (|imagi| ((|#1| $) "\\spad{imagi(o)} extracts the \\spad{i} part of octonion \\spad{o}.")) (|real| ((|#1| $) "\\spad{real(o)} extracts real part of octonion \\spad{o}.")) (|conjugate| (($ $) "\\spad{conjugate(o)} negates the imaginary parts \\spad{i},{}\\spad{j},{}\\spad{k},{}\\spad{E},{}\\spad{I},{}\\spad{J},{}\\spad{K} of octonian \\spad{o}."))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-716) +(-721) ((|constructor| (NIL "Ordered sets which are also abelian cancellation monoids,{} such that the addition preserves the ordering."))) NIL NIL -(-717 R) +(-722 R) ((|constructor| (NIL "Octonion implements octonions (Cayley-Dixon algebra) over a commutative ring,{} an eight-dimensional non-associative algebra,{} doubling the quaternions in the same way as doubling the complex numbers to get the quaternions the main constructor function is {\\em octon} which takes 8 arguments: the real part,{} the \\spad{i} imaginary part,{} the \\spad{j} imaginary part,{} the \\spad{k} imaginary part,{} (as with quaternions) and in addition the imaginary parts \\spad{E},{} \\spad{I},{} \\spad{J},{} \\spad{K}.")) (|octon| (($ (|Quaternion| |#1|) (|Quaternion| |#1|)) "\\spad{octon(qe,qE)} constructs an octonion from two quaternions using the relation {\\em O = Q + QE}."))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (|%list| (QUOTE -449) (QUOTE (-1081)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -239) (|devaluate| |#1|) (|devaluate| |#1|))) (OR (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| (-904 |#1|) (QUOTE (-945 (-345 (-480)))))) (OR (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| (-904 |#1|) (QUOTE (-945 (-480))))) (|HasCategory| |#1| (QUOTE (-967))) (|HasCategory| |#1| (QUOTE (-479))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| (-904 |#1|) (QUOTE (-945 (-345 (-480))))) (|HasCategory| (-904 |#1|) (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480))))) -(-718 OR R OS S) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (|%list| (QUOTE -452) (QUOTE (-1088)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -241) (|devaluate| |#1|) (|devaluate| |#1|))) (OR (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| (-909 |#1|) (QUOTE (-950 (-347 (-483)))))) (OR (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| (-909 |#1|) (QUOTE (-950 (-483))))) (|HasCategory| |#1| (QUOTE (-972))) (|HasCategory| |#1| (QUOTE (-482))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| (-909 |#1|) (QUOTE (-950 (-347 (-483))))) (|HasCategory| (-909 |#1|) (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483))))) +(-723 OR R OS S) ((|constructor| (NIL "\\spad{OctonionCategoryFunctions2} implements functions between two octonion domains defined over different rings. The function map is used to coerce between octonion types.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the component parts of the octonion \\spad{u}."))) NIL NIL -(-719 R -3078 L) +(-724 R -3088 L) ((|constructor| (NIL "Solution of linear ordinary differential equations,{} constant coefficient case.")) (|constDsolve| (((|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#3| |#2| (|Symbol|)) "\\spad{constDsolve(op, g, x)} returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular solution of the equation \\spad{op y = g},{} and the \\spad{yi}'s form a basis for the solutions of \\spad{op y = 0}."))) NIL NIL -(-720 R -3078) +(-725 R -3088) ((|constructor| (NIL "\\spad{ElementaryFunctionODESolver} provides the top-level functions for finding closed form solutions of ordinary differential equations and initial value problems.")) (|solve| (((|Union| |#2| #1="failed") |#2| (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{solve(eq, y, x = a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{eq, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \\spad{dy/dx = f(x,y)}.") (((|Union| |#2| #1#) (|Equation| |#2|) (|BasicOperator|) (|Equation| |#2|) (|List| |#2|)) "\\spad{solve(eq, y, x = a, [y0,...,ym])} returns either the solution of the initial value problem \\spad{eq, y(a) = y0, y'(a) = y1,...} or \"failed\" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \\spad{dy/dx = f(x,y)}.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#2| #2="failed") |#2| (|BasicOperator|) (|Symbol|)) "\\spad{solve(eq, y, x)} returns either a solution of the ordinary differential equation \\spad{eq} or \"failed\" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{f(x,y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \\spad{f}(\\spad{x},{}\\spad{y})},{} a solution is of the form \\spad{h(x,y)} where \\spad{h(x,y) = c} is a first integral of the equation for any constant \\spad{c}.") (((|Union| (|Record| (|:| |particular| |#2|) (|:| |basis| (|List| |#2|))) |#2| #2#) (|Equation| |#2|) (|BasicOperator|) (|Symbol|)) "\\spad{solve(eq, y, x)} returns either a solution of the ordinary differential equation \\spad{eq} or \"failed\" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \\spad{[h, [b1,...,bm]]} where \\spad{h} is a particular solution and \\spad{[b1,...bm]} are linearly independent solutions of the associated homogenuous equation \\spad{f(x,y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \\spad{f}(\\spad{x},{}\\spad{y})},{} a solution is of the form \\spad{h(x,y)} where \\spad{h(x,y) = c} is a first integral of the equation for any constant \\spad{c}; error if the equation is not one of those 2 forms.") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|List| |#2|) (|List| (|BasicOperator|)) (|Symbol|)) "\\spad{solve([eq_1,...,eq_n], [y_1,...,y_n], x)} returns either \"failed\" or,{} if the equations form a fist order linear system,{} a solution of the form \\spad{[y_p, [b_1,...,b_n]]} where \\spad{h_p} is a particular solution and \\spad{[b_1,...b_m]} are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|List| (|Equation| |#2|)) (|List| (|BasicOperator|)) (|Symbol|)) "\\spad{solve([eq_1,...,eq_n], [y_1,...,y_n], x)} returns either \"failed\" or,{} if the equations form a fist order linear system,{} a solution of the form \\spad{[y_p, [b_1,...,b_n]]} where \\spad{h_p} is a particular solution and \\spad{[b_1,...b_m]} are linearly independent solutions of the associated homogenuous system. error if the equations do not form a first order linear system") (((|Union| (|List| (|Vector| |#2|)) "failed") (|Matrix| |#2|) (|Symbol|)) "\\spad{solve(m, x)} returns a basis for the solutions of \\spad{D y = m y}. \\spad{x} is the dependent variable.") (((|Union| (|Record| (|:| |particular| (|Vector| |#2|)) (|:| |basis| (|List| (|Vector| |#2|)))) "failed") (|Matrix| |#2|) (|Vector| |#2|) (|Symbol|)) "\\spad{solve(m, v, x)} returns \\spad{[v_p, [v_1,...,v_m]]} such that the solutions of the system \\spad{D y = m y + v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{D y = m y}. \\spad{x} is the dependent variable."))) NIL NIL -(-721 R -3078) +(-726 R -3088) ((|constructor| (NIL "\\spadtype{ODEIntegration} provides an interface to the integrator. This package is intended for use by the differential equations solver but not at top-level.")) (|diff| (((|Mapping| |#2| |#2|) (|Symbol|)) "\\spad{diff(x)} returns the derivation with respect to \\spad{x}.")) (|expint| ((|#2| |#2| (|Symbol|)) "\\spad{expint(f, x)} returns e^{the integral of \\spad{f} with respect to \\spad{x}}.")) (|int| ((|#2| |#2| (|Symbol|)) "\\spad{int(f, x)} returns the integral of \\spad{f} with respect to \\spad{x}."))) NIL NIL -(-722 -3078 UP UPUP R) +(-727 -3088 UP UPUP R) ((|constructor| (NIL "In-field solution of an linear ordinary differential equation,{} pure algebraic case.")) (|algDsolve| (((|Record| (|:| |particular| (|Union| |#4| "failed")) (|:| |basis| (|List| |#4|))) (|LinearOrdinaryDifferentialOperator1| |#4|) |#4|) "\\spad{algDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no solution in \\spad{R}. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{y_i's} form a basis for the solutions in \\spad{R} of the homogeneous equation."))) NIL NIL -(-723 -3078 UP L LQ) +(-728 -3088 UP L LQ) ((|constructor| (NIL "\\spad{PrimitiveRatDE} provides functions for in-field solutions of linear \\indented{1}{ordinary differential equations,{} in the transcendental case.} \\indented{1}{The derivation to use is given by the parameter \\spad{L}.}")) (|splitDenominator| (((|Record| (|:| |eq| |#3|) (|:| |rh| (|List| (|Fraction| |#2|)))) |#4| (|List| (|Fraction| |#2|))) "\\spad{splitDenominator(op, [g1,...,gm])} returns \\spad{op0, [h1,...,hm]} such that the equations \\spad{op y = c1 g1 + ... + cm gm} and \\spad{op0 y = c1 h1 + ... + cm hm} have the same solutions.")) (|indicialEquation| ((|#2| |#4| |#1|) "\\spad{indicialEquation(op, a)} returns the indicial equation of \\spad{op} at \\spad{a}.") ((|#2| |#3| |#1|) "\\spad{indicialEquation(op, a)} returns the indicial equation of \\spad{op} at \\spad{a}.")) (|indicialEquations| (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#4| |#2|) "\\spad{indicialEquations(op, p)} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}'s are the affine singularities of \\spad{op} above the roots of \\spad{p},{} and the \\spad{e_i}'s are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#4|) "\\spad{indicialEquations op} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}'s are the affine singularities of \\spad{op},{} and the \\spad{e_i}'s are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#3| |#2|) "\\spad{indicialEquations(op, p)} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}'s are the affine singularities of \\spad{op} above the roots of \\spad{p},{} and the \\spad{e_i}'s are the indicial equations at each \\spad{d_i}.") (((|List| (|Record| (|:| |center| |#2|) (|:| |equation| |#2|))) |#3|) "\\spad{indicialEquations op} returns \\spad{[[d1,e1],...,[dq,eq]]} where the \\spad{d_i}'s are the affine singularities of \\spad{op},{} and the \\spad{e_i}'s are the indicial equations at each \\spad{d_i}.")) (|denomLODE| ((|#2| |#3| (|List| (|Fraction| |#2|))) "\\spad{denomLODE(op, [g1,...,gm])} returns a polynomial \\spad{d} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{p/d} for some polynomial \\spad{p}.") (((|Union| |#2| "failed") |#3| (|Fraction| |#2|)) "\\spad{denomLODE(op, g)} returns a polynomial \\spad{d} such that any rational solution of \\spad{op y = g} is of the form \\spad{p/d} for some polynomial \\spad{p},{} and \"failed\",{} if the equation has no rational solution."))) NIL NIL -(-724 -3078 UP L LQ) +(-729 -3088 UP L LQ) ((|constructor| (NIL "In-field solution of Riccati equations,{} primitive case.")) (|changeVar| ((|#3| |#3| (|Fraction| |#2|)) "\\spad{changeVar(+/[ai D^i], a)} returns the operator \\spad{+/[ai (D+a)^i]}.") ((|#3| |#3| |#2|) "\\spad{changeVar(+/[ai D^i], a)} returns the operator \\spad{+/[ai (D+a)^i]}.")) (|singRicDE| (((|List| (|Record| (|:| |frac| (|Fraction| |#2|)) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#2|) |#2| (|SparseUnivariatePolynomial| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{singRicDE(op, zeros, ezfactor)} returns \\spad{[[f1, L1], [f2, L2], ... , [fk, Lk]]} such that the singular part of any rational solution of the associated Riccati equation of \\spad{op y=0} must be one of the \\spad{fi}'s (up to the constant coefficient),{} in which case the equation for \\spad{z=y e^{-int p}} is \\spad{Li z=0}. \\spad{zeros(C(x),H(x,y))} returns all the \\spad{P_i(x)}'s such that \\spad{H(x,P_i(x)) = 0 modulo C(x)}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.")) (|polyRicDE| (((|List| (|Record| (|:| |poly| |#2|) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#1|) |#2|)) "\\spad{polyRicDE(op, zeros)} returns \\spad{[[p1, L1], [p2, L2], ... , [pk, Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y=0} must be one of the \\spad{pi}'s (up to the constant coefficient),{} in which case the equation for \\spad{z=y e^{-int p}} is \\spad{Li z =0}. \\spad{zeros} is a zero finder in \\spad{UP}.")) (|constantCoefficientRicDE| (((|List| (|Record| (|:| |constant| |#1|) (|:| |eq| |#3|))) |#3| (|Mapping| (|List| |#1|) |#2|)) "\\spad{constantCoefficientRicDE(op, ric)} returns \\spad{[[a1, L1], [a2, L2], ... , [ak, Lk]]} such that any rational solution with no polynomial part of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{ai}'s in which case the equation for \\spad{z = y e^{-int ai}} is \\spad{Li z = 0}. \\spad{ric} is a Riccati equation solver over \\spad{F},{} whose input is the associated linear equation.")) (|leadingCoefficientRicDE| (((|List| (|Record| (|:| |deg| (|NonNegativeInteger|)) (|:| |eq| |#2|))) |#3|) "\\spad{leadingCoefficientRicDE(op)} returns \\spad{[[m1, p1], [m2, p2], ... , [mk, pk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must have degree mj for some \\spad{j},{} and its leading coefficient is then a zero of pj. In addition,{}\\spad{m1>m2> ... >mk}.")) (|denomRicDE| ((|#2| |#3|) "\\spad{denomRicDE(op)} returns a polynomial \\spad{d} such that any rational solution of the associated Riccati equation of \\spad{op y = 0} is of the form \\spad{p/d + q'/q + r} for some polynomials \\spad{p} and \\spad{q} and a reduced \\spad{r}. Also,{} \\spad{deg(p) < deg(d)} and {gcd(\\spad{d},{}\\spad{q}) = 1}."))) NIL NIL -(-725 -3078 UP) +(-730 -3088 UP) ((|constructor| (NIL "\\spad{RationalLODE} provides functions for in-field solutions of linear \\indented{1}{ordinary differential equations,{} in the rational case.}")) (|indicialEquationAtInfinity| ((|#2| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))) "\\spad{indicialEquationAtInfinity op} returns the indicial equation of \\spad{op} at infinity.") ((|#2| (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{indicialEquationAtInfinity op} returns the indicial equation of \\spad{op} at infinity.")) (|ratDsolve| (((|Record| (|:| |basis| (|List| (|Fraction| |#2|))) (|:| |mat| (|Matrix| |#1|))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|List| (|Fraction| |#2|))) "\\spad{ratDsolve(op, [g1,...,gm])} returns \\spad{[[h1,...,hq], M]} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{d1 h1 + ... + dq hq} where \\spad{M [d1,...,dq,c1,...,cm] = 0}.") (((|Record| (|:| |particular| (|Union| (|Fraction| |#2|) #1="failed")) (|:| |basis| (|List| (|Fraction| |#2|)))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{ratDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no rational solution. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{yi}'s form a basis for the rational solutions of the homogeneous equation.") (((|Record| (|:| |basis| (|List| (|Fraction| |#2|))) (|:| |mat| (|Matrix| |#1|))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|List| (|Fraction| |#2|))) "\\spad{ratDsolve(op, [g1,...,gm])} returns \\spad{[[h1,...,hq], M]} such that any rational solution of \\spad{op y = c1 g1 + ... + cm gm} is of the form \\spad{d1 h1 + ... + dq hq} where \\spad{M [d1,...,dq,c1,...,cm] = 0}.") (((|Record| (|:| |particular| (|Union| (|Fraction| |#2|) #1#)) (|:| |basis| (|List| (|Fraction| |#2|)))) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Fraction| |#2|)) "\\spad{ratDsolve(op, g)} returns \\spad{[\"failed\", []]} if the equation \\spad{op y = g} has no rational solution. Otherwise,{} it returns \\spad{[f, [y1,...,ym]]} where \\spad{f} is a particular rational solution and the \\spad{yi}'s form a basis for the rational solutions of the homogeneous equation."))) NIL NIL -(-726 -3078 L UP A LO) +(-731 -3088 L UP A LO) ((|constructor| (NIL "Elimination of an algebraic from the coefficentss of a linear ordinary differential equation.")) (|reduceLODE| (((|Record| (|:| |mat| (|Matrix| |#2|)) (|:| |vec| (|Vector| |#1|))) |#5| |#4|) "\\spad{reduceLODE(op, g)} returns \\spad{[m, v]} such that any solution in \\spad{A} of \\spad{op z = g} is of the form \\spad{z = (z_1,...,z_m) . (b_1,...,b_m)} where the \\spad{b_i's} are the basis of \\spad{A} over \\spad{F} returned by \\spadfun{basis}() from \\spad{A},{} and the \\spad{z_i's} satisfy the differential system \\spad{M.z = v}."))) NIL NIL -(-727 -3078 UP) +(-732 -3088 UP) ((|constructor| (NIL "In-field solution of Riccati equations,{} rational case.")) (|polyRicDE| (((|List| (|Record| (|:| |poly| |#2|) (|:| |eq| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{polyRicDE(op, zeros)} returns \\spad{[[p1, L1], [p2, L2], ... , [pk,Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{pi}'s (up to the constant coefficient),{} in which case the equation for \\spad{z = y e^{-int p}} is \\spad{Li z = 0}. \\spad{zeros} is a zero finder in \\spad{UP}.")) (|singRicDE| (((|List| (|Record| (|:| |frac| (|Fraction| |#2|)) (|:| |eq| (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))))) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{singRicDE(op, ezfactor)} returns \\spad{[[f1,L1], [f2,L2],..., [fk,Lk]]} such that the singular ++ part of any rational solution of the associated Riccati equation of \\spad{op y = 0} must be one of the \\spad{fi}'s (up to the constant coefficient),{} in which case the equation for \\spad{z = y e^{-int ai}} is \\spad{Li z = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.")) (|ricDsolve| (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|))) "\\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|))) "\\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, zeros, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator2| |#2| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{ricDsolve(op, zeros)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|) (|Mapping| (|Factored| |#2|) |#2|)) "\\spad{ricDsolve(op, zeros, ezfactor)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}. Argument \\spad{ezfactor} is a factorisation in \\spad{UP},{} not necessarily into irreducibles.") (((|List| (|Fraction| |#2|)) (|LinearOrdinaryDifferentialOperator1| (|Fraction| |#2|)) (|Mapping| (|List| |#1|) |#2|)) "\\spad{ricDsolve(op, zeros)} returns the rational solutions of the associated Riccati equation of \\spad{op y = 0}. \\spad{zeros} is a zero finder in \\spad{UP}."))) NIL ((|HasCategory| |#1| (QUOTE (-27)))) -(-728 -3078 LO) +(-733 -3088 LO) ((|constructor| (NIL "SystemODESolver provides tools for triangulating and solving some systems of linear ordinary differential equations.")) (|solveInField| (((|Record| (|:| |particular| (|Union| (|Vector| |#1|) "failed")) (|:| |basis| (|List| (|Vector| |#1|)))) (|Matrix| |#2|) (|Vector| |#1|) (|Mapping| (|Record| (|:| |particular| (|Union| |#1| "failed")) (|:| |basis| (|List| |#1|))) |#2| |#1|)) "\\spad{solveInField(m, v, solve)} returns \\spad{[[v_1,...,v_m], v_p]} such that the solutions in \\spad{F} of the system \\spad{m x = v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{m x = 0}. Argument \\spad{solve} is a function for solving a single linear ordinary differential equation in \\spad{F}.")) (|solve| (((|Union| (|Record| (|:| |particular| (|Vector| |#1|)) (|:| |basis| (|Matrix| |#1|))) "failed") (|Matrix| |#1|) (|Vector| |#1|) (|Mapping| (|Union| (|Record| (|:| |particular| |#1|) (|:| |basis| (|List| |#1|))) "failed") |#2| |#1|)) "\\spad{solve(m, v, solve)} returns \\spad{[[v_1,...,v_m], v_p]} such that the solutions in \\spad{F} of the system \\spad{D x = m x + v} are \\spad{v_p + c_1 v_1 + ... + c_m v_m} where the \\spad{c_i's} are constants,{} and the \\spad{v_i's} form a basis for the solutions of \\spad{D x = m x}. Argument \\spad{solve} is a function for solving a single linear ordinary differential equation in \\spad{F}.")) (|triangulate| (((|Record| (|:| |mat| (|Matrix| |#2|)) (|:| |vec| (|Vector| |#1|))) (|Matrix| |#2|) (|Vector| |#1|)) "\\spad{triangulate(m, v)} returns \\spad{[m_0, v_0]} such that \\spad{m_0} is upper triangular and the system \\spad{m_0 x = v_0} is equivalent to \\spad{m x = v}.") (((|Record| (|:| A (|Matrix| |#1|)) (|:| |eqs| (|List| (|Record| (|:| C (|Matrix| |#1|)) (|:| |g| (|Vector| |#1|)) (|:| |eq| |#2|) (|:| |rh| |#1|))))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{triangulate(M,v)} returns \\spad{A,[[C_1,g_1,L_1,h_1],...,[C_k,g_k,L_k,h_k]]} such that under the change of variable \\spad{y = A z},{} the first order linear system \\spad{D y = M y + v} is uncoupled as \\spad{D z_i = C_i z_i + g_i} and each \\spad{C_i} is a companion matrix corresponding to the scalar equation \\spad{L_i z_j = h_i}."))) NIL NIL -(-729 -3078 LODO) +(-734 -3088 LODO) ((|constructor| (NIL "\\spad{ODETools} provides tools for the linear ODE solver.")) (|particularSolution| (((|Union| |#1| "failed") |#2| |#1| (|List| |#1|) (|Mapping| |#1| |#1|)) "\\spad{particularSolution(op, g, [f1,...,fm], I)} returns a particular solution \\spad{h} of the equation \\spad{op y = g} where \\spad{[f1,...,fm]} are linearly independent and \\spad{op(fi)=0}. The value \"failed\" is returned if no particular solution is found. Note: the method of variations of parameters is used.")) (|variationOfParameters| (((|Union| (|Vector| |#1|) "failed") |#2| |#1| (|List| |#1|)) "\\spad{variationOfParameters(op, g, [f1,...,fm])} returns \\spad{[u1,...,um]} such that a particular solution of the equation \\spad{op y = g} is \\spad{f1 int(u1) + ... + fm int(um)} where \\spad{[f1,...,fm]} are linearly independent and \\spad{op(fi)=0}. The value \"failed\" is returned if \\spad{m < n} and no particular solution is found.")) (|wronskianMatrix| (((|Matrix| |#1|) (|List| |#1|) (|NonNegativeInteger|)) "\\spad{wronskianMatrix([f1,...,fn], q, D)} returns the \\spad{q x n} matrix \\spad{m} whose i^th row is \\spad{[f1^(i-1),...,fn^(i-1)]}.") (((|Matrix| |#1|) (|List| |#1|)) "\\spad{wronskianMatrix([f1,...,fn])} returns the \\spad{n x n} matrix \\spad{m} whose i^th row is \\spad{[f1^(i-1),...,fn^(i-1)]}."))) NIL NIL -(-730 -2607 S |f|) +(-735 -2617 S |f|) ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The ordering on the type is determined by its third argument which represents the less than function on vectors. This type is a suitable third argument for \\spadtype{GeneralDistributedMultivariatePolynomial}."))) -((-3972 |has| |#2| (-956)) (-3973 |has| |#2| (-956)) (-3975 |has| |#2| (-6 -3975)) (-3978 . 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implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is orderly. This is analogous to the domain \\spadtype{Polynomial}. \\blankline"))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-816)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| (-733 (-1081)) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| (-733 (-1081)) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-733 (-1081)) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-733 (-1081)) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| (-733 (-1081)) (QUOTE (-550 (-469))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-732 |Kernels| R |var|) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| (-738 (-1088)) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| (-738 (-1088)) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-738 (-1088)) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-738 (-1088)) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| (-738 (-1088)) (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-737 |Kernels| R |var|) ((|constructor| (NIL "This constructor produces an ordinary differential ring from a partial differential ring by specifying a variable."))) -(((-3980 "*") |has| |#2| (-309)) (-3971 |has| |#2| (-309)) (-3976 |has| |#2| (-309)) (-3970 |has| |#2| (-309)) (-3975 . T) (-3973 . T) (-3972 . T)) -((|HasCategory| |#2| (QUOTE (-309)))) -(-733 S) +(((-3991 "*") |has| |#2| (-311)) (-3982 |has| |#2| (-311)) (-3987 |has| |#2| (-311)) (-3981 |has| |#2| (-311)) (-3986 . T) (-3984 . T) (-3983 . T)) +((|HasCategory| |#2| (QUOTE (-311)))) +(-738 S) ((|constructor| (NIL "\\spadtype{OrderlyDifferentialVariable} adds a commonly used orderly ranking to the set of derivatives of an ordered list of differential indeterminates. An orderly ranking is a ranking \\spadfun{<} of the derivatives with the property that for two derivatives \\spad{u} and \\spad{v},{} \\spad{u} \\spadfun{<} \\spad{v} if the \\spadfun{order} of \\spad{u} is less than that of \\spad{v}. This domain belongs to \\spadtype{DifferentialVariableCategory}. It defines \\spadfun{weight} to be just \\spadfun{order},{} and it defines an orderly ranking \\spadfun{<} on derivatives \\spad{u} via the lexicographic order on the pair (\\spadfun{order}(\\spad{u}),{} \\spadfun{variable}(\\spad{u}))."))) NIL NIL -(-734 S) +(-739 S) ((|constructor| (NIL "\\indented{3}{The free monoid on a set \\spad{S} is the monoid of finite products of} the form \\spad{reduce(*,[si ** ni])} where the \\spad{si}'s are in \\spad{S},{} and the \\spad{ni}'s are non-negative integers. The multiplication is not commutative. For two elements \\spad{x} and \\spad{y} the relation \\spad{x < y} holds if either \\spad{length(x) < length(y)} holds or if these lengths are equal and if \\spad{x} is smaller than \\spad{y} \\spad{w}.\\spad{r}.\\spad{t}. the lexicographical ordering induced by \\spad{S}. This domain inherits implementation from \\spadtype{FreeMonoid}.")) (|varList| (((|List| |#1|) $) "\\spad{varList(x)} returns the list of variables of \\spad{x}.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length(x)} returns the length of \\spad{x}.")) (|div| (((|Union| (|Record| (|:| |lm| $) (|:| |rm| $)) "failed") $ $) "\\spad{x div y} returns the left and right exact quotients of \\spad{x} by \\spad{y},{} that is \\spad{[l, r]} such that \\spad{x = l * y * r}. \"failed\" is returned iff \\spad{x} is not of the form \\spad{l * y * r}. monomial of \\spad{x}.")) (|rquo| (((|Union| $ "failed") $ |#1|) "\\spad{rquo(x, s)} returns the exact right quotient of \\spad{x} by \\spad{s}.")) (|lquo| (((|Union| $ "failed") $ |#1|) "\\spad{lquo(x, s)} returns the exact left quotient of \\spad{x} by \\spad{s}.")) (|lexico| (((|Boolean|) $ $) "\\spad{lexico(x,y)} returns \\spad{true} iff \\spad{x} is smaller than \\spad{y} \\spad{w}.\\spad{r}.\\spad{t}. the pure lexicographical ordering induced by \\spad{S}.")) (|mirror| (($ $) "\\spad{mirror(x)} returns the reversed word of \\spad{x}.")) (|rest| (($ $) "\\spad{rest(x)} returns \\spad{x} except the first letter.")) (|first| ((|#1| $) "\\spad{first(x)} returns the first letter of \\spad{x}."))) NIL -((|HasCategory| |#1| (QUOTE (-751)))) -(-735) +((|HasCategory| |#1| (QUOTE (-756)))) +(-740) ((|constructor| (NIL "The category of ordered commutative integral domains,{} where ordering and the arithmetic operations are compatible \\blankline"))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-736 P R) +(-741 P R) ((|constructor| (NIL "This constructor creates the \\spadtype{MonogenicLinearOperator} domain which is ``opposite'' in the ring sense to \\spad{P}. That is,{} as sets \\spad{P = \\$} but \\spad{a * b} in \\spad{\\$} is equal to \\spad{b * a} in \\spad{P}.")) (|po| ((|#1| $) "\\spad{po(q)} creates a value in \\spad{P} equal to \\spad{q} in \\$.")) (|op| (($ |#1|) "\\spad{op(p)} creates a value in \\$ equal to \\spad{p} in \\spad{P}."))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-188)))) -(-737 S) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-190)))) +(-742 S) ((|constructor| (NIL "to become an in order iterator")) (|min| ((|#1| $) "\\spad{min(u)} returns the smallest entry in the multiset aggregate \\spad{u}."))) -((-3978 . T) (-3968 . T) (-3979 . T)) +((-3989 . T) (-3979 . T) (-3990 . T)) NIL -(-738 R) +(-743 R) ((|constructor| (NIL "Adjunction of a complex infinity to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a finite rational number if it is one,{} \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a finite rational number. Error: if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a finite rational number.")) (|infinite?| (((|Boolean|) $) "\\spad{infinite?(x)} tests if \\spad{x} is infinite.")) (|finite?| (((|Boolean|) $) "\\spad{finite?(x)} tests if \\spad{x} is finite.")) (|infinity| (($) "\\spad{infinity()} returns infinity."))) -((-3975 |has| |#1| (-750))) -((|HasCategory| |#1| (QUOTE (-750))) (|HasCategory| |#1| (QUOTE (-21))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-750)))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (OR (|HasCategory| |#1| (QUOTE (-750))) (|HasCategory| |#1| (QUOTE (-945 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-479)))) -(-739 R S) +((-3986 |has| |#1| (-755))) +((|HasCategory| |#1| (QUOTE (-755))) (|HasCategory| |#1| (QUOTE (-21))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-755)))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (OR (|HasCategory| |#1| (QUOTE (-755))) (|HasCategory| |#1| (QUOTE (-950 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-482)))) +(-744 R S) ((|constructor| (NIL "Lifting of maps to one-point completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989")) (|map| (((|OnePointCompletion| |#2|) (|Mapping| |#2| |#1|) (|OnePointCompletion| |#1|) (|OnePointCompletion| |#2|)) "\\spad{map(f, r, i)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(infinity) = \\spad{i}.") (((|OnePointCompletion| |#2|) (|Mapping| |#2| |#1|) (|OnePointCompletion| |#1|)) "\\spad{map(f, r)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(infinity) = infinity."))) NIL NIL -(-740 R) +(-745 R) ((|constructor| (NIL "Algebra of ADDITIVE operators over a ring."))) -((-3973 |has| |#1| (-144)) (-3972 |has| |#1| (-144)) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118)))) -(-741 A S) +((-3984 |has| |#1| (-146)) (-3983 |has| |#1| (-146)) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120)))) +(-746 A S) ((|constructor| (NIL "This category specifies the interface for operators used to build terms,{} in the sense of Universal Algebra. The domain parameter \\spad{S} provides representation for the `external name' of an operator.")) (|is?| (((|Boolean|) $ |#2|) "\\spad{is?(op,n)} holds if the name of the operator \\spad{op} is \\spad{n}.")) (|arity| (((|Arity|) $) "\\spad{arity(op)} returns the arity of the operator \\spad{op}.")) (|name| ((|#2| $) "\\spad{name(op)} returns the externam name of \\spad{op}."))) NIL NIL -(-742 S) +(-747 S) ((|constructor| (NIL "This category specifies the interface for operators used to build terms,{} in the sense of Universal Algebra. The domain parameter \\spad{S} provides representation for the `external name' of an operator.")) (|is?| (((|Boolean|) $ |#1|) "\\spad{is?(op,n)} holds if the name of the operator \\spad{op} is \\spad{n}.")) (|arity| (((|Arity|) $) "\\spad{arity(op)} returns the arity of the operator \\spad{op}.")) (|name| ((|#1| $) "\\spad{name(op)} returns the externam name of \\spad{op}."))) NIL NIL -(-743) +(-748) ((|constructor| (NIL "This package exports tools to create AXIOM Library information databases.")) (|getDatabase| (((|Database| (|IndexCard|)) (|String|)) "\\spad{getDatabase(\"char\")} returns a list of appropriate entries in the browser database. The legal values for \\spad{\"char\"} are \"o\" (operations),{} \"k\" (constructors),{} \"d\" (domains),{} \"c\" (categories) or \"p\" (packages)."))) NIL NIL -(-744) +(-749) ((|constructor| (NIL "This the datatype for an operator-signature pair.")) (|construct| (($ (|Identifier|) (|Signature|)) "\\spad{construct(op,sig)} construct a signature-operator with operator name `op',{} and signature `sig'.")) (|signature| (((|Signature|) $) "\\spad{signature(x)} returns the signature of `x'."))) NIL NIL -(-745 R) +(-750 R) ((|constructor| (NIL "Adjunction of two real infinites quantities to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(x)} returns \\spad{x} as a finite rational number if it is one and \"failed\" otherwise.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(x)} returns \\spad{x} as a finite rational number. Error: if \\spad{x} cannot be so converted.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(x)} tests if \\spad{x} is a finite rational number.")) (|whatInfinity| (((|SingleInteger|) $) "\\spad{whatInfinity(x)} returns 0 if \\spad{x} is finite,{} 1 if \\spad{x} is +infinity,{} and \\spad{-1} if \\spad{x} is -infinity.")) (|infinite?| (((|Boolean|) $) "\\spad{infinite?(x)} tests if \\spad{x} is +infinity or -infinity,{}")) (|finite?| (((|Boolean|) $) "\\spad{finite?(x)} tests if \\spad{x} is finite.")) (|minusInfinity| (($) "\\spad{minusInfinity()} returns -infinity.")) (|plusInfinity| (($) "\\spad{plusInfinity()} returns +infinity."))) -((-3975 |has| |#1| (-750))) -((|HasCategory| |#1| (QUOTE (-750))) (|HasCategory| |#1| (QUOTE (-21))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-750)))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (OR (|HasCategory| |#1| (QUOTE (-750))) (|HasCategory| |#1| (QUOTE (-945 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-479)))) -(-746 R S) +((-3986 |has| |#1| (-755))) +((|HasCategory| |#1| (QUOTE (-755))) (|HasCategory| |#1| (QUOTE (-21))) (OR (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-755)))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (OR (|HasCategory| |#1| (QUOTE (-755))) (|HasCategory| |#1| (QUOTE (-950 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-482)))) +(-751 R S) ((|constructor| (NIL "Lifting of maps to ordered completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989")) (|map| (((|OrderedCompletion| |#2|) (|Mapping| |#2| |#1|) (|OrderedCompletion| |#1|) (|OrderedCompletion| |#2|) (|OrderedCompletion| |#2|)) "\\spad{map(f, r, p, m)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(plusInfinity) = \\spad{p} and that \\spad{f}(minusInfinity) = \\spad{m}.") (((|OrderedCompletion| |#2|) (|Mapping| |#2| |#1|) (|OrderedCompletion| |#1|)) "\\spad{map(f, r)} lifts \\spad{f} and applies it to \\spad{r},{} assuming that \\spad{f}(plusInfinity) = plusInfinity and that \\spad{f}(minusInfinity) = minusInfinity."))) NIL NIL -(-747) +(-752) ((|constructor| (NIL "Ordered finite sets.")) (|max| (($) "\\spad{max} is the maximum value of \\%.")) (|min| (($) "\\spad{min} is the minimum value of \\%."))) NIL NIL -(-748 -2607 S) +(-753 -2617 S) ((|constructor| (NIL "\\indented{3}{This package provides ordering functions on vectors which} are suitable parameters for OrderedDirectProduct.")) (|reverseLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{reverseLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by the reverse lexicographic ordering.")) (|totalLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{totalLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the ordering which is total degree refined by lexicographic ordering.")) (|pureLex| (((|Boolean|) (|Vector| |#2|) (|Vector| |#2|)) "\\spad{pureLex(v1,v2)} return \\spad{true} if the vector \\spad{v1} is less than the vector \\spad{v2} in the lexicographic ordering."))) NIL NIL -(-749) +(-754) ((|constructor| (NIL "Ordered sets which are also monoids,{} such that multiplication preserves the ordering. \\blankline"))) NIL NIL -(-750) +(-755) ((|constructor| (NIL "Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \\blankline"))) -((-3975 . T)) +((-3986 . T)) NIL -(-751) +(-756) ((|constructor| (NIL "The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \\spad{(a,b)} exactly one of the following relations holds \\spad{a<b or a=b or b<a} and the relation is transitive,{} \\spadignore{i.e.} \\spad{a<b and b<c => a<c}."))) NIL NIL -(-752 T$ |f|) +(-757 T$ |f|) ((|constructor| (NIL "This domain turns any total ordering \\spad{f} on a type \\spad{T} into a model of the category \\spadtype{OrderedType}."))) NIL -((|HasCategory| |#1| (QUOTE (-549 (-767))))) -(-753 S) +((|HasCategory| |#1| (QUOTE (-552 (-772))))) +(-758 S) ((|constructor| (NIL "Category of types equipped with a total ordering.")) (|min| (($ $ $) "\\spad{min(x,y)} returns the minimum of \\spad{x} and \\spad{y} relative to the ordering.")) (|max| (($ $ $) "\\spad{max(x,y)} returns the maximum of \\spad{x} and \\spad{y} relative to the ordering.")) (>= (((|Boolean|) $ $) "\\spad{x <= y} holds if \\spad{x} is greater or equal than \\spad{y} in the current domain.")) (<= (((|Boolean|) $ $) "\\spad{x <= y} holds if \\spad{x} is less or equal than \\spad{y} in the current domain.")) (> (((|Boolean|) $ $) "\\spad{x > y} holds if \\spad{x} is greater than \\spad{y} in the current domain.")) (< (((|Boolean|) $ $) "\\spad{x < y} holds if \\spad{x} is less than \\spad{y} in the current domain."))) NIL NIL -(-754) +(-759) ((|constructor| (NIL "Category of types equipped with a total ordering.")) (|min| (($ $ $) "\\spad{min(x,y)} returns the minimum of \\spad{x} and \\spad{y} relative to the ordering.")) (|max| (($ $ $) "\\spad{max(x,y)} returns the maximum of \\spad{x} and \\spad{y} relative to the ordering.")) (>= (((|Boolean|) $ $) "\\spad{x <= y} holds if \\spad{x} is greater or equal than \\spad{y} in the current domain.")) (<= (((|Boolean|) $ $) "\\spad{x <= y} holds if \\spad{x} is less or equal than \\spad{y} in the current domain.")) (> (((|Boolean|) $ $) "\\spad{x > y} holds if \\spad{x} is greater than \\spad{y} in the current domain.")) (< (((|Boolean|) $ $) "\\spad{x < y} holds if \\spad{x} is less than \\spad{y} in the current domain."))) NIL NIL -(-755 S R) +(-760 S R) ((|constructor| (NIL "This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}. This category is an evolution of the types \\indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \\indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \\spad{M}. Watt.")) (|leftLcm| (($ $ $) "\\spad{leftLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = aa*a = bb*b} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using right-division.")) (|rightExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{rightExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = c * a + d * b = rightGcd(a, b)}.")) (|rightGcd| (($ $ $) "\\spad{rightGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = aa*g}} \\indented{3}{\\spad{b = bb*g}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using right-division.")) (|rightExactQuotient| (((|Union| $ "failed") $ $) "\\spad{rightExactQuotient(a,b)} computes the value \\spad{q},{} if it exists such that \\spad{a = q*b}.")) (|rightRemainder| (($ $ $) "\\spad{rightRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|rightQuotient| (($ $ $) "\\spad{rightQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|rightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{rightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``right division''.")) (|rightLcm| (($ $ $) "\\spad{rightLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = a*aa = b*bb} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using left-division.")) (|leftExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{leftExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = a * c + b * d = leftGcd(a, b)}.")) (|leftGcd| (($ $ $) "\\spad{leftGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = g*aa}} \\indented{3}{\\spad{b = g*bb}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using left-division.")) (|leftExactQuotient| (((|Union| $ "failed") $ $) "\\spad{leftExactQuotient(a,b)} computes the value \\spad{q},{} if it exists,{} \\indented{1}{such that \\spad{a = b*q}.}")) (|leftRemainder| (($ $ $) "\\spad{leftRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|leftQuotient| (($ $ $) "\\spad{leftQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|leftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{leftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division''.")) (|primitivePart| (($ $) "\\spad{primitivePart(l)} returns \\spad{l0} such that \\spad{l = a * l0} for some a in \\spad{R},{} and \\spad{content(l0) = 1}.")) (|content| ((|#2| $) "\\spad{content(l)} returns the gcd of all the coefficients of \\spad{l}.")) (|monicRightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicRightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``right division''.")) (|monicLeftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicLeftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``left division''.")) (|exquo| (((|Union| $ "failed") $ |#2|) "\\spad{exquo(l, a)} returns the exact quotient of \\spad{l} by a,{} returning \\axiom{\"failed\"} if this is not possible.")) (|apply| ((|#2| $ |#2| |#2|) "\\spad{apply(p, c, m)} returns \\spad{p(m)} where the action is given by \\spad{x m = c sigma(m) + delta(m)}.")) (|coefficients| (((|List| |#2|) $) "\\spad{coefficients(l)} returns the list of all the nonzero coefficients of \\spad{l}.")) (|monomial| (($ |#2| (|NonNegativeInteger|)) "\\spad{monomial(c,k)} produces \\spad{c} times the \\spad{k}-th power of the generating operator,{} \\spad{monomial(1,1)}.")) (|coefficient| ((|#2| $ (|NonNegativeInteger|)) "\\spad{coefficient(l,k)} is \\spad{a(k)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|reductum| (($ $) "\\spad{reductum(l)} is \\spad{l - monomial(a(n),n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(l)} is \\spad{a(n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|minimumDegree| (((|NonNegativeInteger|) $) "\\spad{minimumDegree(l)} is the smallest \\spad{k} such that \\spad{a(k) ~= 0} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(l)} is \\spad{n} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}"))) NIL -((|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-144)))) -(-756 R) +((|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146)))) +(-761 R) ((|constructor| (NIL "This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}. This category is an evolution of the types \\indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \\indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \\spad{M}. Watt.")) (|leftLcm| (($ $ $) "\\spad{leftLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = aa*a = bb*b} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using right-division.")) (|rightExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{rightExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = c * a + d * b = rightGcd(a, b)}.")) (|rightGcd| (($ $ $) "\\spad{rightGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = aa*g}} \\indented{3}{\\spad{b = bb*g}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using right-division.")) (|rightExactQuotient| (((|Union| $ "failed") $ $) "\\spad{rightExactQuotient(a,b)} computes the value \\spad{q},{} if it exists such that \\spad{a = q*b}.")) (|rightRemainder| (($ $ $) "\\spad{rightRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|rightQuotient| (($ $ $) "\\spad{rightQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|rightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{rightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``right division''.")) (|rightLcm| (($ $ $) "\\spad{rightLcm(a,b)} computes the value \\spad{m} of lowest degree such that \\spad{m = a*aa = b*bb} for some values \\spad{aa} and \\spad{bb}. The value \\spad{m} is computed using left-division.")) (|leftExtendedGcd| (((|Record| (|:| |coef1| $) (|:| |coef2| $) (|:| |generator| $)) $ $) "\\spad{leftExtendedGcd(a,b)} returns \\spad{[c,d]} such that \\spad{g = a * c + b * d = leftGcd(a, b)}.")) (|leftGcd| (($ $ $) "\\spad{leftGcd(a,b)} computes the value \\spad{g} of highest degree such that \\indented{3}{\\spad{a = g*aa}} \\indented{3}{\\spad{b = g*bb}} for some values \\spad{aa} and \\spad{bb}. The value \\spad{g} is computed using left-division.")) (|leftExactQuotient| (((|Union| $ "failed") $ $) "\\spad{leftExactQuotient(a,b)} computes the value \\spad{q},{} if it exists,{} \\indented{1}{such that \\spad{a = b*q}.}")) (|leftRemainder| (($ $ $) "\\spad{leftRemainder(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{r} is returned.")) (|leftQuotient| (($ $ $) "\\spad{leftQuotient(a,b)} computes the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. The value \\spad{q} is returned.")) (|leftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{leftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division''.")) (|primitivePart| (($ $) "\\spad{primitivePart(l)} returns \\spad{l0} such that \\spad{l = a * l0} for some a in \\spad{R},{} and \\spad{content(l0) = 1}.")) (|content| ((|#1| $) "\\spad{content(l)} returns the gcd of all the coefficients of \\spad{l}.")) (|monicRightDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicRightDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``right division''.")) (|monicLeftDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicLeftDivide(a,b)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``left division''.")) (|exquo| (((|Union| $ "failed") $ |#1|) "\\spad{exquo(l, a)} returns the exact quotient of \\spad{l} by a,{} returning \\axiom{\"failed\"} if this is not possible.")) (|apply| ((|#1| $ |#1| |#1|) "\\spad{apply(p, c, m)} returns \\spad{p(m)} where the action is given by \\spad{x m = c sigma(m) + delta(m)}.")) (|coefficients| (((|List| |#1|) $) "\\spad{coefficients(l)} returns the list of all the nonzero coefficients of \\spad{l}.")) (|monomial| (($ |#1| (|NonNegativeInteger|)) "\\spad{monomial(c,k)} produces \\spad{c} times the \\spad{k}-th power of the generating operator,{} \\spad{monomial(1,1)}.")) (|coefficient| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coefficient(l,k)} is \\spad{a(k)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|reductum| (($ $) "\\spad{reductum(l)} is \\spad{l - monomial(a(n),n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(l)} is \\spad{a(n)} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|minimumDegree| (((|NonNegativeInteger|) $) "\\spad{minimumDegree(l)} is the smallest \\spad{k} such that \\spad{a(k) ~= 0} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(l)} is \\spad{n} if \\indented{2}{\\spad{l = sum(monomial(a(i),i), i = 0..n)}.}"))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-757 R C) +(-762 R C) ((|constructor| (NIL "\\spad{UnivariateSkewPolynomialCategoryOps} provides products and \\indented{1}{divisions of univariate skew polynomials.}")) (|rightDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{rightDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``right division''. \\spad{\\sigma} is the morphism to use.")) (|leftDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{leftDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. This process is called ``left division''. \\spad{\\sigma} is the morphism to use.")) (|monicRightDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{monicRightDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = q*b + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``right division''. \\spad{\\sigma} is the morphism to use.")) (|monicLeftDivide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2| (|Automorphism| |#1|)) "\\spad{monicLeftDivide(a, b, sigma)} returns the pair \\spad{[q,r]} such that \\spad{a = b*q + r} and the degree of \\spad{r} is less than the degree of \\spad{b}. \\spad{b} must be monic. This process is called ``left division''. \\spad{\\sigma} is the morphism to use.")) (|apply| ((|#1| |#2| |#1| |#1| (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{apply(p, c, m, sigma, delta)} returns \\spad{p(m)} where the action is given by \\spad{x m = c sigma(m) + delta(m)}.")) (|times| ((|#2| |#2| |#2| (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{times(p, q, sigma, delta)} returns \\spad{p * q}. \\spad{\\sigma} and \\spad{\\delta} are the maps to use."))) NIL -((|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) -(-758 R |sigma| -3229) +((|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) +(-763 R |sigma| -3239) ((|constructor| (NIL "This is the domain of sparse univariate skew polynomials over an Ore coefficient field. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}.")) (|outputForm| (((|OutputForm|) $ (|OutputForm|)) "\\spad{outputForm(p, x)} returns the output form of \\spad{p} using \\spad{x} for the otherwise anonymous variable."))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-309)))) -(-759 |x| R |sigma| -3229) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-311)))) +(-764 |x| R |sigma| -3239) ((|constructor| (NIL "This is the domain of univariate skew polynomials over an Ore coefficient field in a named variable. The multiplication is given by \\spad{x a = \\sigma(a) x + \\delta a}."))) -((-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-309)))) -(-760 R) +((-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-311)))) +(-765 R) ((|constructor| (NIL "This package provides orthogonal polynomials as functions on a ring.")) (|legendreP| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{legendreP(n,x)} is the \\spad{n}-th Legendre polynomial,{} \\spad{P[n](x)}. These are defined by \\spad{1/sqrt(1-2*x*t+t**2) = sum(P[n](x)*t**n, n = 0..)}.")) (|laguerreL| ((|#1| (|NonNegativeInteger|) (|NonNegativeInteger|) |#1|) "\\spad{laguerreL(m,n,x)} is the associated Laguerre polynomial,{} \\spad{L<m>[n](x)}. This is the \\spad{m}-th derivative of \\spad{L[n](x)}.") ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{laguerreL(n,x)} is the \\spad{n}-th Laguerre polynomial,{} \\spad{L[n](x)}. These are defined by \\spad{exp(-t*x/(1-t))/(1-t) = sum(L[n](x)*t**n/n!, n = 0..)}.")) (|hermiteH| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{hermiteH(n,x)} is the \\spad{n}-th Hermite polynomial,{} \\spad{H[n](x)}. These are defined by \\spad{exp(2*t*x-t**2) = sum(H[n](x)*t**n/n!, n = 0..)}.")) (|chebyshevU| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{chebyshevU(n,x)} is the \\spad{n}-th Chebyshev polynomial of the second kind,{} \\spad{U[n](x)}. These are defined by \\spad{1/(1-2*t*x+t**2) = sum(T[n](x) *t**n, n = 0..)}.")) (|chebyshevT| ((|#1| (|NonNegativeInteger|) |#1|) "\\spad{chebyshevT(n,x)} is the \\spad{n}-th Chebyshev polynomial of the first kind,{} \\spad{T[n](x)}. These are defined by \\spad{(1-t*x)/(1-2*t*x+t**2) = sum(T[n](x) *t**n, n = 0..)}."))) NIL -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480)))))) -(-761) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483)))))) +(-766) ((|constructor| (NIL "Semigroups with compatible ordering."))) NIL NIL -(-762) +(-767) ((|constructor| (NIL "\\indented{1}{Author : Larry Lambe} Date created : 14 August 1988 Date Last Updated : 11 March 1991 Description : A domain used in order to take the free \\spad{R}-module on the Integers \\spad{I}. This is actually the forgetful functor from OrderedRings to OrderedSets applied to \\spad{I}")) (|value| (((|Integer|) $) "\\spad{value(x)} returns the integer associated with \\spad{x}")) (|coerce| (($ (|Integer|)) "\\spad{coerce(i)} returns the element corresponding to \\spad{i}"))) NIL NIL -(-763) +(-768) ((|constructor| (NIL "OutPackage allows pretty-printing from programs.")) (|outputList| (((|Void|) (|List| (|Any|))) "\\spad{outputList(l)} displays the concatenated components of the list \\spad{l} on the ``algebra output'' stream,{} as defined by \\spadsyscom{set output algebra}; quotes are stripped from strings.")) (|output| (((|Void|) (|String|) (|OutputForm|)) "\\spad{output(s,x)} displays the string \\spad{s} followed by the form \\spad{x} on the ``algebra output'' stream,{} as defined by \\spadsyscom{set output algebra}.") (((|Void|) (|OutputForm|)) "\\spad{output(x)} displays the output form \\spad{x} on the ``algebra output'' stream,{} as defined by \\spadsyscom{set output algebra}.") (((|Void|) (|String|)) "\\spad{output(s)} displays the string \\spad{s} on the ``algebra output'' stream,{} as defined by \\spadsyscom{set output algebra}."))) NIL NIL -(-764 S) +(-769 S) ((|constructor| (NIL "This category describes output byte stream conduits.")) (|writeBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{writeBytes!(c,b)} write bytes from buffer `b' onto the conduit `c'. The actual number of written bytes is returned.")) (|writeUInt8!| (((|Maybe| (|UInt8|)) $ (|UInt8|)) "\\spad{writeUInt8!(c,b)} attempts to write the unsigned 8-bit value `v' on the conduit `c'. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeInt8!| (((|Maybe| (|Int8|)) $ (|Int8|)) "\\spad{writeInt8!(c,b)} attempts to write the 8-bit value `v' on the conduit `c'. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeByte!| (((|Maybe| (|Byte|)) $ (|Byte|)) "\\spad{writeByte!(c,b)} attempts to write the byte `b' on the conduit `c'. Returns the written byte if successful,{} otherwise,{} returns \\spad{nothing}."))) NIL NIL -(-765) +(-770) ((|constructor| (NIL "This category describes output byte stream conduits.")) (|writeBytes!| (((|NonNegativeInteger|) $ (|ByteBuffer|)) "\\spad{writeBytes!(c,b)} write bytes from buffer `b' onto the conduit `c'. The actual number of written bytes is returned.")) (|writeUInt8!| (((|Maybe| (|UInt8|)) $ (|UInt8|)) "\\spad{writeUInt8!(c,b)} attempts to write the unsigned 8-bit value `v' on the conduit `c'. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeInt8!| (((|Maybe| (|Int8|)) $ (|Int8|)) "\\spad{writeInt8!(c,b)} attempts to write the 8-bit value `v' on the conduit `c'. Returns the written value if successful,{} otherwise,{} returns \\spad{nothing}.")) (|writeByte!| (((|Maybe| (|Byte|)) $ (|Byte|)) "\\spad{writeByte!(c,b)} attempts to write the byte `b' on the conduit `c'. Returns the written byte if successful,{} otherwise,{} returns \\spad{nothing}."))) NIL NIL -(-766) +(-771) ((|constructor| (NIL "This domain provides representation for binary files open for output operations. `Binary' here means that the conduits do not interpret their contents.")) (|isOpen?| (((|Boolean|) $) "open?(ifile) holds if `ifile' is in open state.")) (|outputBinaryFile| (($ (|String|)) "\\spad{outputBinaryFile(f)} returns an output conduit obtained by opening the file named by `f' as a binary file.") (($ (|FileName|)) "\\spad{outputBinaryFile(f)} returns an output conduit obtained by opening the file named by `f' as a binary file."))) NIL NIL -(-767) +(-772) ((|constructor| (NIL "This domain is used to create and manipulate mathematical expressions for output. It is intended to provide an insulating layer between the expression rendering software (\\spadignore{e.g.} TeX,{} or Script) and the output coercions in the various domains.")) (SEGMENT (($ $) "\\spad{SEGMENT(x)} creates the prefix form: \\spad{x..}.") (($ $ $) "\\spad{SEGMENT(x,y)} creates the infix form: \\spad{x..y}.")) (|not| (($ $) "\\spad{not f} creates the equivalent prefix form.")) (|or| (($ $ $) "\\spad{f or g} creates the equivalent infix form.")) (|and| (($ $ $) "\\spad{f and g} creates the equivalent infix form.")) (|exquo| (($ $ $) "\\spad{exquo(f,g)} creates the equivalent infix form.")) (|quo| (($ $ $) "\\spad{f quo g} creates the equivalent infix form.")) (|rem| (($ $ $) "\\spad{f rem g} creates the equivalent infix form.")) (|div| (($ $ $) "\\spad{f div g} creates the equivalent infix form.")) (** (($ $ $) "\\spad{f ** g} creates the equivalent infix form.")) (/ (($ $ $) "\\spad{f / g} creates the equivalent infix form.")) (* (($ $ $) "\\spad{f * g} creates the equivalent infix form.")) (- (($ $) "\\spad{- f} creates the equivalent prefix form.") (($ $ $) "\\spad{f - g} creates the equivalent infix form.")) (+ (($ $ $) "\\spad{f + g} creates the equivalent infix form.")) (>= (($ $ $) "\\spad{f >= g} creates the equivalent infix form.")) (<= (($ $ $) "\\spad{f <= g} creates the equivalent infix form.")) (> (($ $ $) "\\spad{f > g} creates the equivalent infix form.")) (< (($ $ $) "\\spad{f < g} creates the equivalent infix form.")) (~= (($ $ $) "\\spad{f ~= g} creates the equivalent infix form.")) (= (($ $ $) "\\spad{f = g} creates the equivalent infix form.")) (|blankSeparate| (($ (|List| $)) "\\spad{blankSeparate(l)} creates the form separating the elements of \\spad{l} by blanks.")) (|semicolonSeparate| (($ (|List| $)) "\\spad{semicolonSeparate(l)} creates the form separating the elements of \\spad{l} by semicolons.")) (|commaSeparate| (($ (|List| $)) "\\spad{commaSeparate(l)} creates the form separating the elements of \\spad{l} by commas.")) (|pile| (($ (|List| $)) "\\spad{pile(l)} creates the form consisting of the elements of \\spad{l} which displays as a pile,{} \\spadignore{i.e.} the elements begin on a new line and are indented right to the same margin.")) (|paren| (($ (|List| $)) "\\spad{paren(lf)} creates the form separating the elements of \\spad{lf} by commas and encloses the result in parentheses.") (($ $) "\\spad{paren(f)} creates the form enclosing \\spad{f} in parentheses.")) (|bracket| (($ (|List| $)) "\\spad{bracket(lf)} creates the form separating the elements of \\spad{lf} by commas and encloses the result in square brackets.") (($ $) "\\spad{bracket(f)} creates the form enclosing \\spad{f} in square brackets.")) (|brace| (($ (|List| $)) "\\spad{brace(lf)} creates the form separating the elements of \\spad{lf} by commas and encloses the result in curly brackets.") (($ $) "\\spad{brace(f)} creates the form enclosing \\spad{f} in braces (curly brackets).")) (|int| (($ $ $ $) "\\spad{int(expr,lowerlimit,upperlimit)} creates the form prefixing \\spad{expr} by an integral sign with both a \\spad{lowerlimit} and \\spad{upperlimit}.") (($ $ $) "\\spad{int(expr,lowerlimit)} creates the form prefixing \\spad{expr} by an integral sign with a \\spad{lowerlimit}.") (($ $) "\\spad{int(expr)} creates the form prefixing \\spad{expr} with an integral sign.")) (|prod| (($ $ $ $) "\\spad{prod(expr,lowerlimit,upperlimit)} creates the form prefixing \\spad{expr} by a capital \\spad{pi} with both a \\spad{lowerlimit} and \\spad{upperlimit}.") (($ $ $) "\\spad{prod(expr,lowerlimit)} creates the form prefixing \\spad{expr} by a capital \\spad{pi} with a \\spad{lowerlimit}.") (($ $) "\\spad{prod(expr)} creates the form prefixing \\spad{expr} by a capital \\spad{pi}.")) (|sum| (($ $ $ $) "\\spad{sum(expr,lowerlimit,upperlimit)} creates the form prefixing \\spad{expr} by a capital sigma with both a \\spad{lowerlimit} and \\spad{upperlimit}.") (($ $ $) "\\spad{sum(expr,lowerlimit)} creates the form prefixing \\spad{expr} by a capital sigma with a \\spad{lowerlimit}.") (($ $) "\\spad{sum(expr)} creates the form prefixing \\spad{expr} by a capital sigma.")) (|overlabel| (($ $ $) "\\spad{overlabel(x,f)} creates the form \\spad{f} with \"x overbar\" over the top.")) (|overbar| (($ $) "\\spad{overbar(f)} creates the form \\spad{f} with an overbar.")) (|prime| (($ $ (|NonNegativeInteger|)) "\\spad{prime(f,n)} creates the form \\spad{f} followed by \\spad{n} primes.") (($ $) "\\spad{prime(f)} creates the form \\spad{f} followed by a suffix prime (single quote).")) (|dot| (($ $ (|NonNegativeInteger|)) "\\spad{dot(f,n)} creates the form \\spad{f} with \\spad{n} dots overhead.") (($ $) "\\spad{dot(f)} creates the form with a one dot overhead.")) (|quote| (($ $) "\\spad{quote(f)} creates the form \\spad{f} with a prefix quote.")) (|supersub| (($ $ (|List| $)) "\\spad{supersub(a,[sub1,super1,sub2,super2,...])} creates a form with each subscript aligned under each superscript.")) (|scripts| (($ $ (|List| $)) "\\spad{scripts(f, [sub, super, presuper, presub])} \\indented{1}{creates a form for \\spad{f} with scripts on all 4 corners.}")) (|presuper| (($ $ $) "\\spad{presuper(f,n)} creates a form for \\spad{f} presuperscripted by \\spad{n}.")) (|presub| (($ $ $) "\\spad{presub(f,n)} creates a form for \\spad{f} presubscripted by \\spad{n}.")) (|super| (($ $ $) "\\spad{super(f,n)} creates a form for \\spad{f} superscripted by \\spad{n}.")) (|sub| (($ $ $) "\\spad{sub(f,n)} creates a form for \\spad{f} subscripted by \\spad{n}.")) (|binomial| (($ $ $) "\\spad{binomial(n,m)} creates a form for the binomial coefficient of \\spad{n} and \\spad{m}.")) (|differentiate| (($ $ (|NonNegativeInteger|)) "\\spad{differentiate(f,n)} creates a form for the \\spad{n}th derivative of \\spad{f},{} \\spadignore{e.g.} \\spad{f'},{} \\spad{f''},{} \\spad{f'''},{} \"f super \\spad{iv}\".")) (|rarrow| (($ $ $) "\\spad{rarrow(f,g)} creates a form for the mapping \\spad{f -> g}.")) (|assign| (($ $ $) "\\spad{assign(f,g)} creates a form for the assignment \\spad{f := g}.")) (|slash| (($ $ $) "\\spad{slash(f,g)} creates a form for the horizontal fraction of \\spad{f} over \\spad{g}.")) (|over| (($ $ $) "\\spad{over(f,g)} creates a form for the vertical fraction of \\spad{f} over \\spad{g}.")) (|root| (($ $ $) "\\spad{root(f,n)} creates a form for the \\spad{n}th root of form \\spad{f}.") (($ $) "\\spad{root(f)} creates a form for the square root of form \\spad{f}.")) (|zag| (($ $ $) "\\spad{zag(f,g)} creates a form for the continued fraction form for \\spad{f} over \\spad{g}.")) (|matrix| (($ (|List| (|List| $))) "\\spad{matrix(llf)} makes \\spad{llf} (a list of lists of forms) into a form which displays as a matrix.")) (|box| (($ $) "\\spad{box(f)} encloses \\spad{f} in a box.")) (|label| (($ $ $) "\\spad{label(n,f)} gives form \\spad{f} an equation label \\spad{n}.")) (|string| (($ $) "\\spad{string(f)} creates \\spad{f} with string quotes.")) (|elt| (($ $ (|List| $)) "\\spad{elt(op,l)} creates a form for application of \\spad{op} to list of arguments \\spad{l}.")) (|infix?| (((|Boolean|) $) "\\spad{infix?(op)} returns \\spad{true} if \\spad{op} is an infix operator,{} and \\spad{false} otherwise.")) (|postfix| (($ $ $) "\\spad{postfix(op, a)} creates a form which prints as: a \\spad{op}.")) (|infix| (($ $ $ $) "\\spad{infix(op, a, b)} creates a form which prints as: a \\spad{op} \\spad{b}.") (($ $ (|List| $)) "\\spad{infix(f,l)} creates a form depicting the \\spad{n}-ary application of infix operation \\spad{f} to a tuple of arguments \\spad{l}.")) (|prefix| (($ $ (|List| $)) "\\spad{prefix(f,l)} creates a form depicting the \\spad{n}-ary prefix application of \\spad{f} to a tuple of arguments given by list \\spad{l}.")) (|vconcat| (($ (|List| $)) "\\spad{vconcat(u)} vertically concatenates all forms in list \\spad{u}.") (($ $ $) "\\spad{vconcat(f,g)} vertically concatenates forms \\spad{f} and \\spad{g}.")) (|hconcat| (($ (|List| $)) "\\spad{hconcat(u)} horizontally concatenates all forms in list \\spad{u}.") (($ $ $) "\\spad{hconcat(f,g)} horizontally concatenate forms \\spad{f} and \\spad{g}.")) (|center| (($ $) "\\spad{center(f)} centers form \\spad{f} in total space.") (($ $ (|Integer|)) "\\spad{center(f,n)} centers form \\spad{f} within space of width \\spad{n}.")) (|right| (($ $) "\\spad{right(f)} right-justifies form \\spad{f} in total space.") (($ $ (|Integer|)) "\\spad{right(f,n)} right-justifies form \\spad{f} within space of width \\spad{n}.")) (|left| (($ $) "\\spad{left(f)} left-justifies form \\spad{f} in total space.") (($ $ (|Integer|)) "\\spad{left(f,n)} left-justifies form \\spad{f} within space of width \\spad{n}.")) (|rspace| (($ (|Integer|) (|Integer|)) "\\spad{rspace(n,m)} creates rectangular white space,{} \\spad{n} wide by \\spad{m} high.")) (|vspace| (($ (|Integer|)) "\\spad{vspace(n)} creates white space of height \\spad{n}.")) (|hspace| (($ (|Integer|)) "\\spad{hspace(n)} creates white space of width \\spad{n}.")) (|superHeight| (((|Integer|) $) "\\spad{superHeight(f)} returns the height of form \\spad{f} above the base line.")) (|subHeight| (((|Integer|) $) "\\spad{subHeight(f)} returns the height of form \\spad{f} below the base line.")) (|height| (((|Integer|)) "\\spad{height()} returns the height of the display area (an integer).") (((|Integer|) $) "\\spad{height(f)} returns the height of form \\spad{f} (an integer).")) (|width| (((|Integer|)) "\\spad{width()} returns the width of the display area (an integer).") (((|Integer|) $) "\\spad{width(f)} returns the width of form \\spad{f} (an integer).")) (|doubleFloatFormat| (((|String|) (|String|)) "change the output format for doublefloats using lisp format strings")) (|empty| (($) "\\spad{empty()} creates an empty form.")) (|outputForm| (($ (|DoubleFloat|)) "\\spad{outputForm(sf)} creates an form for small float \\spad{sf}.") (($ (|String|)) "\\spad{outputForm(s)} creates an form for string \\spad{s}.") (($ (|Symbol|)) "\\spad{outputForm(s)} creates an form for symbol \\spad{s}.") (($ (|Integer|)) "\\spad{outputForm(n)} creates an form for integer \\spad{n}.")) (|messagePrint| (((|Void|) (|String|)) "\\spad{messagePrint(s)} prints \\spad{s} without string quotes. Note: \\spad{messagePrint(s)} is equivalent to \\spad{print message(s)}.")) (|message| (($ (|String|)) "\\spad{message(s)} creates an form with no string quotes from string \\spad{s}.")) (|print| (((|Void|) $) "\\spad{print(u)} prints the form \\spad{u}."))) NIL NIL -(-768 |VariableList|) +(-773 |VariableList|) ((|constructor| (NIL "This domain implements ordered variables")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} returns a member of the variable set or failed"))) NIL NIL -(-769) +(-774) ((|constructor| (NIL "This domain represents set of overloaded operators (in fact operator descriptors).")) (|members| (((|List| (|FunctionDescriptor|)) $) "\\spad{members(x)} returns the list of operator descriptors,{} \\spadignore{e.g.} signature and implementation slots,{} of the overload set \\spad{x}.")) (|name| (((|Identifier|) $) "\\spad{name(x)} returns the name of the overload set \\spad{x}."))) NIL NIL -(-770 R |vl| |wl| |wtlevel|) +(-775 R |vl| |wl| |wtlevel|) ((|constructor| (NIL "This domain represents truncated weighted polynomials over the \"Polynomial\" type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.")) (|changeWeightLevel| (((|Void|) (|NonNegativeInteger|)) "\\spad{changeWeightLevel(n)} This changes the weight level to the new value given: NB: previously calculated terms are not affected")) (/ (((|Union| $ "failed") $ $) "\\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \\spad{R} is a Field)"))) -((-3973 |has| |#1| (-144)) (-3972 |has| |#1| (-144)) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309)))) -(-771 R PS UP) +((-3984 |has| |#1| (-146)) (-3983 |has| |#1| (-146)) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311)))) +(-776 R PS UP) ((|constructor| (NIL "\\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \\indented{2}{\"Pade Approximants,{} Part I: Basic Theory\",{} Baker & Graves-Morris.}")) (|padecf| (((|Union| (|ContinuedFraction| |#3|) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) |#2| |#2|) "\\spad{padecf(nd,dd,ns,ds)} computes the approximant as a continued fraction of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function).")) (|pade| (((|Union| (|Fraction| |#3|) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) |#2| |#2|) "\\spad{pade(nd,dd,ns,ds)} computes the approximant as a quotient of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function)."))) NIL NIL -(-772 R |x| |pt|) +(-777 R |x| |pt|) ((|constructor| (NIL "\\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Trager,{}Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \\indented{2}{\"Pade Approximants,{} Part I: Basic Theory\",{} Baker & Graves-Morris.}")) (|pade| (((|Union| (|Fraction| (|UnivariatePolynomial| |#2| |#1|)) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) (|UnivariateTaylorSeries| |#1| |#2| |#3|)) "\\spad{pade(nd,dd,s)} computes the quotient of polynomials (if it exists) with numerator degree at most \\spad{nd} and denominator degree at most \\spad{dd} which matches the series \\spad{s} to order \\spad{nd + dd}.") (((|Union| (|Fraction| (|UnivariatePolynomial| |#2| |#1|)) "failed") (|NonNegativeInteger|) (|NonNegativeInteger|) (|UnivariateTaylorSeries| |#1| |#2| |#3|) (|UnivariateTaylorSeries| |#1| |#2| |#3|)) "\\spad{pade(nd,dd,ns,ds)} computes the approximant as a quotient of polynomials (if it exists) for arguments \\spad{nd} (numerator degree of approximant),{} \\spad{dd} (denominator degree of approximant),{} \\spad{ns} (numerator series of function),{} and \\spad{ds} (denominator series of function)."))) NIL NIL -(-773 |p|) +(-778 |p|) ((|constructor| (NIL "Stream-based implementation of Zp: \\spad{p}-adic numbers are represented as sum(\\spad{i} = 0..,{} a[\\spad{i}] * p^i),{} where the a[\\spad{i}] lie in 0,{}1,{}...,{}(\\spad{p} - 1)."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-774 |p|) +(-779 |p|) ((|constructor| (NIL "This is the catefory of stream-based representations of \\indented{2}{the \\spad{p}-adic integers.}")) (|root| (($ (|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{root(f,a)} returns a root of the polynomial \\spad{f}. Argument \\spad{a} must be a root of \\spad{f} \\spad{(mod p)}.")) (|sqrt| (($ $ (|Integer|)) "\\spad{sqrt(b,a)} returns a square root of \\spad{b}. Argument \\spad{a} is a square root of \\spad{b} \\spad{(mod p)}.")) (|approximate| (((|Integer|) $ (|Integer|)) "\\spad{approximate(x,n)} returns an integer \\spad{y} such that \\spad{y = x (mod p^n)} when \\spad{n} is positive,{} and 0 otherwise.")) (|quotientByP| (($ $) "\\spad{quotientByP(x)} returns \\spad{b},{} where \\spad{x = a + b p}.")) (|moduloP| (((|Integer|) $) "\\spad{modulo(x)} returns a,{} where \\spad{x = a + b p}.")) (|modulus| (((|Integer|)) "\\spad{modulus()} returns the value of \\spad{p}.")) (|complete| (($ $) "\\spad{complete(x)} forces the computation of all digits.")) (|extend| (($ $ (|Integer|)) "\\spad{extend(x,n)} forces the computation of digits up to order \\spad{n}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(x)} returns the exponent of the highest power of \\spad{p} dividing \\spad{x}.")) (|digits| (((|Stream| (|Integer|)) $) "\\spad{digits(x)} returns a stream of \\spad{p}-adic digits of \\spad{x}."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-775 |p|) +(-780 |p|) ((|constructor| (NIL "Stream-based implementation of Qp: numbers are represented as sum(\\spad{i} = \\spad{k}..,{} a[\\spad{i}] * p^i) where the a[\\spad{i}] lie in 0,{}1,{}...,{}(\\spad{p} - 1)."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-773 |#1|) (QUOTE (-816))) (|HasCategory| (-773 |#1|) (QUOTE (-945 (-1081)))) (|HasCategory| (-773 |#1|) (QUOTE (-116))) (|HasCategory| (-773 |#1|) (QUOTE (-118))) (|HasCategory| (-773 |#1|) (QUOTE (-550 (-469)))) (|HasCategory| (-773 |#1|) (QUOTE (-928))) (|HasCategory| (-773 |#1|) (QUOTE (-735))) (|HasCategory| (-773 |#1|) (QUOTE (-751))) (OR (|HasCategory| (-773 |#1|) (QUOTE (-735))) (|HasCategory| (-773 |#1|) (QUOTE (-751)))) (|HasCategory| (-773 |#1|) (QUOTE (-945 (-480)))) (|HasCategory| (-773 |#1|) (QUOTE (-1057))) (|HasCategory| (-773 |#1|) (QUOTE (-791 (-325)))) (|HasCategory| (-773 |#1|) (QUOTE (-791 (-480)))) (|HasCategory| (-773 |#1|) (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-773 |#1|) (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-773 |#1|) (QUOTE (-577 (-480)))) (|HasCategory| (-773 |#1|) (QUOTE (-187))) (|HasCategory| (-773 |#1|) (QUOTE (-806 (-1081)))) (|HasCategory| (-773 |#1|) (QUOTE (-188))) (|HasCategory| (-773 |#1|) (QUOTE (-804 (-1081)))) (|HasCategory| (-773 |#1|) (|%list| (QUOTE -449) (QUOTE (-1081)) (|%list| (QUOTE -773) (|devaluate| |#1|)))) (|HasCategory| (-773 |#1|) (|%list| (QUOTE -257) (|%list| (QUOTE -773) (|devaluate| |#1|)))) (|HasCategory| (-773 |#1|) (|%list| (QUOTE -239) (|%list| (QUOTE -773) (|devaluate| |#1|)) (|%list| (QUOTE -773) (|devaluate| |#1|)))) (|HasCategory| (-773 |#1|) (QUOTE (-255))) (|HasCategory| (-773 |#1|) (QUOTE (-479))) (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-773 |#1|) (QUOTE (-816)))) (OR (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-773 |#1|) (QUOTE (-816)))) (|HasCategory| (-773 |#1|) (QUOTE (-116))))) -(-776 |p| PADIC) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-778 |#1|) (QUOTE (-821))) (|HasCategory| (-778 |#1|) (QUOTE (-950 (-1088)))) (|HasCategory| (-778 |#1|) (QUOTE (-118))) (|HasCategory| (-778 |#1|) (QUOTE (-120))) (|HasCategory| (-778 |#1|) (QUOTE (-553 (-472)))) (|HasCategory| (-778 |#1|) (QUOTE (-933))) (|HasCategory| (-778 |#1|) (QUOTE (-740))) (|HasCategory| (-778 |#1|) (QUOTE (-756))) (OR (|HasCategory| (-778 |#1|) (QUOTE (-740))) (|HasCategory| (-778 |#1|) (QUOTE (-756)))) (|HasCategory| (-778 |#1|) (QUOTE (-950 (-483)))) (|HasCategory| (-778 |#1|) (QUOTE (-1064))) (|HasCategory| (-778 |#1|) (QUOTE (-796 (-327)))) (|HasCategory| (-778 |#1|) (QUOTE (-796 (-483)))) (|HasCategory| (-778 |#1|) (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-778 |#1|) (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-778 |#1|) (QUOTE (-580 (-483)))) (|HasCategory| (-778 |#1|) (QUOTE (-189))) (|HasCategory| (-778 |#1|) (QUOTE (-811 (-1088)))) (|HasCategory| (-778 |#1|) (QUOTE (-190))) (|HasCategory| (-778 |#1|) (QUOTE (-809 (-1088)))) (|HasCategory| (-778 |#1|) (|%list| (QUOTE -452) (QUOTE (-1088)) (|%list| (QUOTE -778) (|devaluate| |#1|)))) (|HasCategory| (-778 |#1|) (|%list| (QUOTE -259) (|%list| (QUOTE -778) (|devaluate| |#1|)))) (|HasCategory| (-778 |#1|) (|%list| (QUOTE -241) (|%list| (QUOTE -778) (|devaluate| |#1|)) (|%list| (QUOTE -778) (|devaluate| |#1|)))) (|HasCategory| (-778 |#1|) (QUOTE (-257))) (|HasCategory| (-778 |#1|) (QUOTE (-482))) (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-778 |#1|) (QUOTE (-821)))) (OR (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-778 |#1|) (QUOTE (-821)))) (|HasCategory| (-778 |#1|) (QUOTE (-118))))) +(-781 |p| PADIC) ((|constructor| (NIL "This is the category of stream-based representations of Qp.")) (|removeZeroes| (($ (|Integer|) $) "\\spad{removeZeroes(n,x)} removes up to \\spad{n} leading zeroes from the \\spad{p}-adic rational \\spad{x}.") (($ $) "\\spad{removeZeroes(x)} removes leading zeroes from the representation of the \\spad{p}-adic rational \\spad{x}. A \\spad{p}-adic rational is represented by (1) an exponent and (2) a \\spad{p}-adic integer which may have leading zero digits. When the \\spad{p}-adic integer has a leading zero digit,{} a 'leading zero' is removed from the \\spad{p}-adic rational as follows: the number is rewritten by increasing the exponent by 1 and dividing the \\spad{p}-adic integer by \\spad{p}. Note: \\spad{removeZeroes(f)} removes all leading zeroes from \\spad{f}.")) (|continuedFraction| (((|ContinuedFraction| (|Fraction| (|Integer|))) $) "\\spad{continuedFraction(x)} converts the \\spad{p}-adic rational number \\spad{x} to a continued fraction.")) (|approximate| (((|Fraction| (|Integer|)) $ (|Integer|)) "\\spad{approximate(x,n)} returns a rational number \\spad{y} such that \\spad{y = x (mod p^n)}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| |#2| (QUOTE (-945 (-1081)))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-928))) (|HasCategory| |#2| (QUOTE (-735))) (|HasCategory| |#2| (QUOTE (-751))) (OR (|HasCategory| |#2| (QUOTE (-735))) (|HasCategory| |#2| (QUOTE (-751)))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (|HasCategory| |#2| (QUOTE (-1057))) (|HasCategory| |#2| (QUOTE (-791 (-325)))) (|HasCategory| |#2| (QUOTE (-791 (-480)))) (|HasCategory| |#2| (QUOTE (-550 (-795 (-325))))) (|HasCategory| |#2| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#2| (QUOTE (-577 (-480)))) (|HasCategory| |#2| (QUOTE (-187))) (|HasCategory| |#2| (QUOTE (-806 (-1081)))) (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (QUOTE (-804 (-1081)))) (|HasCategory| |#2| (|%list| (QUOTE -449) (QUOTE (-1081)) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -239) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-255))) (|HasCategory| |#2| (QUOTE (-479))) (-12 (|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#2| (QUOTE (-116))))) -(-777 S T$) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| |#2| (QUOTE (-950 (-1088)))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-933))) (|HasCategory| |#2| (QUOTE (-740))) (|HasCategory| |#2| (QUOTE (-756))) (OR (|HasCategory| |#2| (QUOTE (-740))) (|HasCategory| |#2| (QUOTE (-756)))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (|HasCategory| |#2| (QUOTE (-1064))) (|HasCategory| |#2| (QUOTE (-796 (-327)))) (|HasCategory| |#2| (QUOTE (-796 (-483)))) (|HasCategory| |#2| (QUOTE (-553 (-800 (-327))))) (|HasCategory| |#2| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (QUOTE (-189))) (|HasCategory| |#2| (QUOTE (-811 (-1088)))) (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasCategory| |#2| (|%list| (QUOTE -452) (QUOTE (-1088)) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|))) (|HasCategory| |#2| (|%list| (QUOTE -241) (|devaluate| |#2|) (|devaluate| |#2|))) (|HasCategory| |#2| (QUOTE (-257))) (|HasCategory| |#2| (QUOTE (-482))) (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#2| (QUOTE (-118))))) +(-782 S T$) ((|constructor| (NIL "\\indented{1}{This domain provides a very simple representation} of the notion of `pair of objects'. It does not try to achieve all possible imaginable things.")) (|second| ((|#2| $) "\\spad{second(p)} extracts the second components of `p'.")) (|first| ((|#1| $) "\\spad{first(p)} extracts the first component of `p'.")) (|construct| (($ |#1| |#2|) "\\spad{construct(s,t)} is same as pair(\\spad{s},{}\\spad{t}),{} with syntactic sugar.")) (|pair| (($ |#1| |#2|) "\\spad{pair(s,t)} returns a pair object composed of `s' and `t'."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-1007))))) (-12 (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767)))))) -(-778) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-1012))))) (-12 (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772)))))) +(-783) ((|constructor| (NIL "This domain describes four groups of color shades (palettes).")) (|shade| (((|Integer|) $) "\\spad{shade(p)} returns the shade index of the indicated palette \\spad{p}.")) (|hue| (((|Color|) $) "\\spad{hue(p)} returns the hue field of the indicated palette \\spad{p}.")) (|light| (($ (|Color|)) "\\spad{light(c)} sets the shade of a hue,{} \\spad{c},{} to it's highest value.")) (|pastel| (($ (|Color|)) "\\spad{pastel(c)} sets the shade of a hue,{} \\spad{c},{} above bright,{} but below light.")) (|bright| (($ (|Color|)) "\\spad{bright(c)} sets the shade of a hue,{} \\spad{c},{} above dim,{} but below pastel.")) (|dim| (($ (|Color|)) "\\spad{dim(c)} sets the shade of a hue,{} \\spad{c},{} above dark,{} but below bright.")) (|dark| (($ (|Color|)) "\\spad{dark(c)} sets the shade of the indicated hue of \\spad{c} to it's lowest value."))) NIL NIL -(-779) +(-784) ((|constructor| (NIL "This package provides a coerce from polynomials over algebraic numbers to \\spadtype{Expression AlgebraicNumber}.")) (|coerce| (((|Expression| (|Integer|)) (|Fraction| (|Polynomial| (|AlgebraicNumber|)))) "\\spad{coerce(rf)} converts \\spad{rf},{} a fraction of polynomial \\spad{p} with algebraic number coefficients to \\spadtype{Expression Integer}.") (((|Expression| (|Integer|)) (|Polynomial| (|AlgebraicNumber|))) "\\spad{coerce(p)} converts the polynomial \\spad{p} with algebraic number coefficients to \\spadtype{Expression Integer}."))) NIL NIL -(-780) +(-785) ((|constructor| (NIL "Representation of parameters to functions or constructors. For the most part,{} they are Identifiers. However,{} in very cases,{} they are \"flags\",{} \\spadignore{e.g.} string literals.")) (|autoCoerce| (((|String|) $) "\\spad{autoCoerce(x)@String} implicitly coerce the object \\spad{x} to \\spadtype{String}. This function is left at the discretion of the compiler.") (((|Identifier|) $) "\\spad{autoCoerce(x)@Identifier} implicitly coerce the object \\spad{x} to \\spadtype{Identifier}. This function is left at the discretion of the compiler.")) (|case| (((|Boolean|) $ (|[\|\|]| (|String|))) "\\spad{x case String} if the parameter AST object \\spad{x} designates a flag.") (((|Boolean|) $ (|[\|\|]| (|Identifier|))) "\\spad{x case Identifier} if the parameter AST object \\spad{x} designates an \\spadtype{Identifier}."))) NIL NIL -(-781 CF1 CF2) +(-786 CF1 CF2) ((|constructor| (NIL "This package \\undocumented")) (|map| (((|ParametricPlaneCurve| |#2|) (|Mapping| |#2| |#1|) (|ParametricPlaneCurve| |#1|)) "\\spad{map(f,x)} \\undocumented"))) NIL NIL -(-782 |ComponentFunction|) +(-787 |ComponentFunction|) ((|constructor| (NIL "ParametricPlaneCurve is used for plotting parametric plane curves in the affine plane.")) (|coordinate| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coordinate(c,i)} returns a coordinate function for \\spad{c} using 1-based indexing according to \\spad{i}. This indicates what the function for the coordinate component \\spad{i} of the plane curve is.")) (|curve| (($ |#1| |#1|) "\\spad{curve(c1,c2)} creates a plane curve from 2 component functions \\spad{c1} and \\spad{c2}."))) NIL NIL -(-783 CF1 CF2) +(-788 CF1 CF2) ((|constructor| (NIL "This package \\undocumented")) (|map| (((|ParametricSpaceCurve| |#2|) (|Mapping| |#2| |#1|) (|ParametricSpaceCurve| |#1|)) "\\spad{map(f,x)} \\undocumented"))) NIL NIL -(-784 |ComponentFunction|) +(-789 |ComponentFunction|) ((|constructor| (NIL "ParametricSpaceCurve is used for plotting parametric space curves in affine 3-space.")) (|coordinate| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coordinate(c,i)} returns a coordinate function of \\spad{c} using 1-based indexing according to \\spad{i}. This indicates what the function for the coordinate component,{} \\spad{i},{} of the space curve is.")) (|curve| (($ |#1| |#1| |#1|) "\\spad{curve(c1,c2,c3)} creates a space curve from 3 component functions \\spad{c1},{} \\spad{c2},{} and \\spad{c3}."))) NIL NIL -(-785) +(-790) ((|constructor| (NIL "\\indented{1}{This package provides a simple Spad script parser.} Related Constructors: Syntax. See Also: Syntax.")) (|getSyntaxFormsFromFile| (((|List| (|Syntax|)) (|String|)) "\\spad{getSyntaxFormsFromFile(f)} parses the source file \\spad{f} (supposedly containing Spad scripts) and returns a List Syntax. The filename \\spad{f} is supposed to have the proper extension. Note that source location information is not part of result."))) NIL NIL -(-786 CF1 CF2) +(-791 CF1 CF2) ((|constructor| (NIL "This package \\undocumented")) (|map| (((|ParametricSurface| |#2|) (|Mapping| |#2| |#1|) (|ParametricSurface| |#1|)) "\\spad{map(f,x)} \\undocumented"))) NIL NIL -(-787 |ComponentFunction|) +(-792 |ComponentFunction|) ((|constructor| (NIL "ParametricSurface is used for plotting parametric surfaces in affine 3-space.")) (|coordinate| ((|#1| $ (|NonNegativeInteger|)) "\\spad{coordinate(s,i)} returns a coordinate function of \\spad{s} using 1-based indexing according to \\spad{i}. This indicates what the function for the coordinate component,{} \\spad{i},{} of the surface is.")) (|surface| (($ |#1| |#1| |#1|) "\\spad{surface(c1,c2,c3)} creates a surface from 3 parametric component functions \\spad{c1},{} \\spad{c2},{} and \\spad{c3}."))) NIL NIL -(-788) +(-793) ((|constructor| (NIL "PartitionsAndPermutations contains functions for generating streams of integer partitions,{} and streams of sequences of integers composed from a multi-set.")) (|permutations| (((|Stream| (|List| (|Integer|))) (|Integer|)) "\\spad{permutations(n)} is the stream of permutations \\indented{1}{formed from \\spad{1,2,3,...,n}.}")) (|sequences| (((|Stream| (|List| (|Integer|))) (|List| (|Integer|))) "\\spad{sequences([l0,l1,l2,..,ln])} is the set of \\indented{1}{all sequences formed from} \\spad{l0} 0's,{}\\spad{l1} 1's,{}\\spad{l2} 2's,{}...,{}\\spad{ln} \\spad{n}'s.") (((|Stream| (|List| (|Integer|))) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{sequences(l1,l2)} is the stream of all sequences that \\indented{1}{can be composed from the multiset defined from} \\indented{1}{two lists of integers \\spad{l1} and \\spad{l2}.} \\indented{1}{For example,{}the pair \\spad{([1,2,4],[2,3,5])} represents} \\indented{1}{multi-set with 1 \\spad{2},{} 2 \\spad{3}'s,{} and 4 \\spad{5}'s.}")) (|shufflein| (((|Stream| (|List| (|Integer|))) (|List| (|Integer|)) (|Stream| (|List| (|Integer|)))) "\\spad{shufflein(l,st)} maps shuffle(\\spad{l},{}\\spad{u}) on to all \\indented{1}{members \\spad{u} of \\spad{st},{} concatenating the results.}")) (|shuffle| (((|Stream| (|List| (|Integer|))) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{shuffle(l1,l2)} forms the stream of all shuffles of \\spad{l1} \\indented{1}{and \\spad{l2},{} \\spadignore{i.e.} all sequences that can be formed from} \\indented{1}{merging \\spad{l1} and \\spad{l2}.}")) (|conjugates| (((|Stream| (|List| (|PositiveInteger|))) (|Stream| (|List| (|PositiveInteger|)))) "\\spad{conjugates(lp)} is the stream of conjugates of a stream \\indented{1}{of partitions \\spad{lp}.}")) (|conjugate| (((|List| (|PositiveInteger|)) (|List| (|PositiveInteger|))) "\\spad{conjugate(pt)} is the conjugate of the partition \\spad{pt}."))) NIL NIL -(-789 R) +(-794 R) ((|constructor| (NIL "An object \\spad{S} is Patternable over an object \\spad{R} if \\spad{S} can lift the conversions from \\spad{R} into \\spadtype{Pattern(Integer)} and \\spadtype{Pattern(Float)} to itself."))) NIL NIL -(-790 R S L) +(-795 R S L) ((|constructor| (NIL "A PatternMatchListResult is an object internally returned by the pattern matcher when matching on lists. It is either a failed match,{} or a pair of PatternMatchResult,{} one for atoms (elements of the list),{} and one for lists.")) (|lists| (((|PatternMatchResult| |#1| |#3|) $) "\\spad{lists(r)} returns the list of matches that match lists.")) (|atoms| (((|PatternMatchResult| |#1| |#2|) $) "\\spad{atoms(r)} returns the list of matches that match atoms (elements of the lists).")) (|makeResult| (($ (|PatternMatchResult| |#1| |#2|) (|PatternMatchResult| |#1| |#3|)) "\\spad{makeResult(r1,r2)} makes the combined result [\\spad{r1},{}\\spad{r2}].")) (|new| (($) "\\spad{new()} returns a new empty match result.")) (|failed| (($) "\\spad{failed()} returns a failed match.")) (|failed?| (((|Boolean|) $) "\\spad{failed?(r)} tests if \\spad{r} is a failed match."))) NIL NIL -(-791 S) +(-796 S) ((|constructor| (NIL "A set \\spad{R} is PatternMatchable over \\spad{S} if elements of \\spad{R} can be matched to patterns over \\spad{S}.")) (|patternMatch| (((|PatternMatchResult| |#1| $) $ (|Pattern| |#1|) (|PatternMatchResult| |#1| $)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}. res contains the variables of \\spad{pat} which are already matched and their matches (necessary for recursion). Initially,{} res is just the result of \\spadfun{new} which is an empty list of matches."))) NIL NIL -(-792 |Base| |Subject| |Pat|) +(-797 |Base| |Subject| |Pat|) ((|constructor| (NIL "This package provides the top-level pattern macthing functions.")) (|Is| (((|PatternMatchResult| |#1| |#2|) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a match of the form \\spad{[v1 = e1,...,vn = en]}; returns an empty match if \\spad{expr} is exactly equal to pat. returns a \\spadfun{failed} match if pat does not match \\spad{expr}.") (((|List| (|Equation| (|Polynomial| |#2|))) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a list of matches \\spad{[v1 = e1,...,vn = en]}; returns an empty list if either \\spad{expr} is exactly equal to pat or if pat does not match \\spad{expr}.") (((|List| (|Equation| |#2|)) |#2| |#3|) "\\spad{Is(expr, pat)} matches the pattern pat on the expression \\spad{expr} and returns a list of matches \\spad{[v1 = e1,...,vn = en]}; returns an empty list if either \\spad{expr} is exactly equal to pat or if pat does not match \\spad{expr}.") (((|PatternMatchListResult| |#1| |#2| (|List| |#2|)) (|List| |#2|) |#3|) "\\spad{Is([e1,...,en], pat)} matches the pattern pat on the list of expressions \\spad{[e1,...,en]} and returns the result.")) (|is?| (((|Boolean|) (|List| |#2|) |#3|) "\\spad{is?([e1,...,en], pat)} tests if the list of expressions \\spad{[e1,...,en]} matches the pattern pat.") (((|Boolean|) |#2| |#3|) "\\spad{is?(expr, pat)} tests if the expression \\spad{expr} matches the pattern pat."))) NIL -((-12 (-2546 (|HasCategory| |#2| (QUOTE (-945 (-1081))))) (-2546 (|HasCategory| |#2| (QUOTE (-956))))) (-12 (|HasCategory| |#2| (QUOTE (-956))) (-2546 (|HasCategory| |#2| (QUOTE (-945 (-1081)))))) (|HasCategory| |#2| (QUOTE (-945 (-1081))))) -(-793 R S) +((-12 (-2556 (|HasCategory| |#2| (QUOTE (-950 (-1088))))) (-2556 (|HasCategory| |#2| (QUOTE (-961))))) (-12 (|HasCategory| |#2| (QUOTE (-961))) (-2556 (|HasCategory| |#2| (QUOTE (-950 (-1088)))))) (|HasCategory| |#2| (QUOTE (-950 (-1088))))) +(-798 R S) ((|constructor| (NIL "A PatternMatchResult is an object internally returned by the pattern matcher; It is either a failed match,{} or a list of matches of the form (var,{} expr) meaning that the variable var matches the expression expr.")) (|satisfy?| (((|Union| (|Boolean|) "failed") $ (|Pattern| |#1|)) "\\spad{satisfy?(r, p)} returns \\spad{true} if the matches satisfy the top-level predicate of \\spad{p},{} \\spad{false} if they don't,{} and \"failed\" if not enough variables of \\spad{p} are matched in \\spad{r} to decide.")) (|construct| (($ (|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| |#2|)))) "\\spad{construct([v1,e1],...,[vn,en])} returns the match result containing the matches (\\spad{v1},{}\\spad{e1}),{}...,{}(vn,{}en).")) (|destruct| (((|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| |#2|))) $) "\\spad{destruct(r)} returns the list of matches (var,{} expr) in \\spad{r}. Error: if \\spad{r} is a failed match.")) (|addMatchRestricted| (($ (|Pattern| |#1|) |#2| $ |#2|) "\\spad{addMatchRestricted(var, expr, r, val)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} provided that \\spad{expr} satisfies the predicates attached to \\spad{var},{} that \\spad{var} is not matched to another expression already,{} and that either \\spad{var} is an optional pattern variable or that \\spad{expr} is not equal to val (usually an identity).")) (|insertMatch| (($ (|Pattern| |#1|) |#2| $) "\\spad{insertMatch(var, expr, r)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} without checking predicates or previous matches for \\spad{var}.")) (|addMatch| (($ (|Pattern| |#1|) |#2| $) "\\spad{addMatch(var, expr, r)} adds the match (\\spad{var},{} \\spad{expr}) in \\spad{r},{} provided that \\spad{expr} satisfies the predicates attached to \\spad{var},{} and that \\spad{var} is not matched to another expression already.")) (|getMatch| (((|Union| |#2| "failed") (|Pattern| |#1|) $) "\\spad{getMatch(var, r)} returns the expression that \\spad{var} matches in the result \\spad{r},{} and \"failed\" if \\spad{var} is not matched in \\spad{r}.")) (|union| (($ $ $) "\\spad{union(a, b)} makes the set-union of two match results.")) (|new| (($) "\\spad{new()} returns a new empty match result.")) (|failed| (($) "\\spad{failed()} returns a failed match.")) (|failed?| (((|Boolean|) $) "\\spad{failed?(r)} tests if \\spad{r} is a failed match."))) NIL NIL -(-794 R A B) +(-799 R A B) ((|constructor| (NIL "Lifts maps to pattern matching results.")) (|map| (((|PatternMatchResult| |#1| |#3|) (|Mapping| |#3| |#2|) (|PatternMatchResult| |#1| |#2|)) "\\spad{map(f, [(v1,a1),...,(vn,an)])} returns the matching result [(\\spad{v1},{}\\spad{f}(\\spad{a1})),{}...,{}(vn,{}\\spad{f}(an))]."))) NIL NIL -(-795 R) +(-800 R) ((|constructor| (NIL "Patterns for use by the pattern matcher.")) (|optpair| (((|Union| (|List| $) "failed") (|List| $)) "\\spad{optpair(l)} returns \\spad{l} has the form \\spad{[a, b]} and a is optional,{} and \"failed\" otherwise.")) (|variables| (((|List| $) $) "\\spad{variables(p)} returns the list of matching variables appearing in \\spad{p}.")) (|getBadValues| (((|List| (|Any|)) $) "\\spad{getBadValues(p)} returns the list of \"bad values\" for \\spad{p}. Note: \\spad{p} is not allowed to match any of its \"bad values\".")) (|addBadValue| (($ $ (|Any|)) "\\spad{addBadValue(p, v)} adds \\spad{v} to the list of \"bad values\" for \\spad{p}. Note: \\spad{p} is not allowed to match any of its \"bad values\".")) (|resetBadValues| (($ $) "\\spad{resetBadValues(p)} initializes the list of \"bad values\" for \\spad{p} to \\spad{[]}. Note: \\spad{p} is not allowed to match any of its \"bad values\".")) (|hasTopPredicate?| (((|Boolean|) $) "\\spad{hasTopPredicate?(p)} tests if \\spad{p} has a top-level predicate.")) (|topPredicate| (((|Record| (|:| |var| (|List| (|Symbol|))) (|:| |pred| (|Any|))) $) "\\spad{topPredicate(x)} returns \\spad{[[a1,...,an], f]} where the top-level predicate of \\spad{x} is \\spad{f(a1,...,an)}. Note: \\spad{n} is 0 if \\spad{x} has no top-level predicate.")) (|setTopPredicate| (($ $ (|List| (|Symbol|)) (|Any|)) "\\spad{setTopPredicate(x, [a1,...,an], f)} returns \\spad{x} with the top-level predicate set to \\spad{f(a1,...,an)}.")) (|patternVariable| (($ (|Symbol|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\spad{patternVariable(x, c?, o?, m?)} creates a pattern variable \\spad{x},{} which is constant if \\spad{c? = true},{} optional if \\spad{o? = true},{} and multiple if \\spad{m? = true}.")) (|withPredicates| (($ $ (|List| (|Any|))) "\\spad{withPredicates(p, [p1,...,pn])} makes a copy of \\spad{p} and attaches the predicate \\spad{p1} and ... and pn to the copy,{} which is returned.")) (|setPredicates| (($ $ (|List| (|Any|))) "\\spad{setPredicates(p, [p1,...,pn])} attaches the predicate \\spad{p1} and ... and pn to \\spad{p}.")) (|predicates| (((|List| (|Any|)) $) "\\spad{predicates(p)} returns \\spad{[p1,...,pn]} such that the predicate attached to \\spad{p} is \\spad{p1} and ... and pn.")) (|hasPredicate?| (((|Boolean|) $) "\\spad{hasPredicate?(p)} tests if \\spad{p} has predicates attached to it.")) (|optional?| (((|Boolean|) $) "\\spad{optional?(p)} tests if \\spad{p} is a single matching variable which can match an identity.")) (|multiple?| (((|Boolean|) $) "\\spad{multiple?(p)} tests if \\spad{p} is a single matching variable allowing list matching or multiple term matching in a sum or product.")) (|generic?| (((|Boolean|) $) "\\spad{generic?(p)} tests if \\spad{p} is a single matching variable.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(p)} tests if \\spad{p} contains no matching variables.")) (|symbol?| (((|Boolean|) $) "\\spad{symbol?(p)} tests if \\spad{p} is a symbol.")) (|quoted?| (((|Boolean|) $) "\\spad{quoted?(p)} tests if \\spad{p} is of the form 's for a symbol \\spad{s}.")) (|inR?| (((|Boolean|) $) "\\spad{inR?(p)} tests if \\spad{p} is an atom (\\spadignore{i.e.} an element of \\spad{R}).")) (|copy| (($ $) "\\spad{copy(p)} returns a recursive copy of \\spad{p}.")) (|convert| (($ (|List| $)) "\\spad{convert([a1,...,an])} returns the pattern \\spad{[a1,...,an]}.")) (|depth| (((|NonNegativeInteger|) $) "\\spad{depth(p)} returns the nesting level of \\spad{p}.")) (/ (($ $ $) "\\spad{a / b} returns the pattern \\spad{a / b}.")) (** (($ $ $) "\\spad{a ** b} returns the pattern \\spad{a ** b}.") (($ $ (|NonNegativeInteger|)) "\\spad{a ** n} returns the pattern \\spad{a ** n}.")) (* (($ $ $) "\\spad{a * b} returns the pattern \\spad{a * b}.")) (+ (($ $ $) "\\spad{a + b} returns the pattern \\spad{a + b}.")) (|elt| (($ (|BasicOperator|) (|List| $)) "\\spad{elt(op, [a1,...,an])} returns \\spad{op(a1,...,an)}.")) (|isPower| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| $)) "failed") $) "\\spad{isPower(p)} returns \\spad{[a, b]} if \\spad{p = a ** b},{} and \"failed\" otherwise.")) (|isList| (((|Union| (|List| $) "failed") $) "\\spad{isList(p)} returns \\spad{[a1,...,an]} if \\spad{p = [a1,...,an]},{} \"failed\" otherwise.")) (|isQuotient| (((|Union| (|Record| (|:| |num| $) (|:| |den| $)) "failed") $) "\\spad{isQuotient(p)} returns \\spad{[a, b]} if \\spad{p = a / b},{} and \"failed\" otherwise.")) (|isExpt| (((|Union| (|Record| (|:| |val| $) (|:| |exponent| (|NonNegativeInteger|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[q, n]} if \\spad{n > 0} and \\spad{p = q ** n},{} and \"failed\" otherwise.")) (|isOp| (((|Union| (|Record| (|:| |op| (|BasicOperator|)) (|:| |arg| (|List| $))) "failed") $) "\\spad{isOp(p)} returns \\spad{[op, [a1,...,an]]} if \\spad{p = op(a1,...,an)},{} and \"failed\" otherwise.") (((|Union| (|List| $) "failed") $ (|BasicOperator|)) "\\spad{isOp(p, op)} returns \\spad{[a1,...,an]} if \\spad{p = op(a1,...,an)},{} and \"failed\" otherwise.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{n > 1} and \\spad{p = a1 * ... * an},{} and \"failed\" otherwise.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[a1,...,an]} if \\spad{n > 1} \\indented{1}{and \\spad{p = a1 + ... + an},{}} and \"failed\" otherwise.")) (|One| (($) "1")) (|Zero| (($) "0"))) NIL NIL -(-796 R -2655) +(-801 R -2665) ((|constructor| (NIL "Tools for patterns.")) (|badValues| (((|List| |#2|) (|Pattern| |#1|)) "\\spad{badValues(p)} returns the list of \"bad values\" for \\spad{p}; \\spad{p} is not allowed to match any of its \"bad values\".")) (|addBadValue| (((|Pattern| |#1|) (|Pattern| |#1|) |#2|) "\\spad{addBadValue(p, v)} adds \\spad{v} to the list of \"bad values\" for \\spad{p}; \\spad{p} is not allowed to match any of its \"bad values\".")) (|satisfy?| (((|Boolean|) (|List| |#2|) (|Pattern| |#1|)) "\\spad{satisfy?([v1,...,vn], p)} returns \\spad{f(v1,...,vn)} where \\spad{f} is the top-level predicate attached to \\spad{p}.") (((|Boolean|) |#2| (|Pattern| |#1|)) "\\spad{satisfy?(v, p)} returns \\spad{f}(\\spad{v}) where \\spad{f} is the predicate attached to \\spad{p}.")) (|predicate| (((|Mapping| (|Boolean|) |#2|) (|Pattern| |#1|)) "\\spad{predicate(p)} returns the predicate attached to \\spad{p},{} the constant function \\spad{true} if \\spad{p} has no predicates attached to it.")) (|suchThat| (((|Pattern| |#1|) (|Pattern| |#1|) (|List| (|Symbol|)) (|Mapping| (|Boolean|) (|List| |#2|))) "\\spad{suchThat(p, [a1,...,an], f)} returns a copy of \\spad{p} with the top-level predicate set to \\spad{f(a1,...,an)}.") (((|Pattern| |#1|) (|Pattern| |#1|) (|List| (|Mapping| (|Boolean|) |#2|))) "\\spad{suchThat(p, [f1,...,fn])} makes a copy of \\spad{p} and adds the predicate \\spad{f1} and ... and fn to the copy,{} which is returned.") (((|Pattern| |#1|) (|Pattern| |#1|) (|Mapping| (|Boolean|) |#2|)) "\\spad{suchThat(p, f)} makes a copy of \\spad{p} and adds the predicate \\spad{f} to the copy,{} which is returned."))) NIL NIL -(-797 R S) +(-802 R S) ((|constructor| (NIL "Lifts maps to patterns.")) (|map| (((|Pattern| |#2|) (|Mapping| |#2| |#1|) (|Pattern| |#1|)) "\\spad{map(f, p)} applies \\spad{f} to all the leaves of \\spad{p} and returns the result as a pattern over \\spad{S}."))) NIL NIL -(-798 |VarSet|) +(-803 |VarSet|) ((|constructor| (NIL "This domain provides the internal representation of polynomials in non-commutative variables written over the Poincare-Birkhoff-Witt basis. See the \\spadtype{XPBWPolynomial} domain constructor. See Free Lie Algebras by \\spad{C}. Reutenauer (Oxford science publications). \\newline Author: Michel Petitot (petitot@lifl.fr).")) (|varList| (((|List| |#1|) $) "\\spad{varList([l1]*[l2]*...[ln])} returns the list of variables in the word \\spad{l1*l2*...*ln}.")) (|retractable?| (((|Boolean|) $) "\\spad{retractable?([l1]*[l2]*...[ln])} returns \\spad{true} iff \\spad{n} equals \\spad{1}.")) (|rest| (($ $) "\\spad{rest([l1]*[l2]*...[ln])} returns the list \\spad{l2, .... ln}.")) (|ListOfTerms| (((|List| (|LyndonWord| |#1|)) $) "\\spad{ListOfTerms([l1]*[l2]*...[ln])} returns the list of words \\spad{l1, l2, .... ln}.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length([l1]*[l2]*...[ln])} returns the length of the word \\spad{l1*l2*...*ln}.")) (|first| (((|LyndonWord| |#1|) $) "\\spad{first([l1]*[l2]*...[ln])} returns the Lyndon word \\spad{l1}.")) (|coerce| (($ |#1|) "\\spad{coerce(v)} return \\spad{v}") (((|OrderedFreeMonoid| |#1|) $) "\\spad{coerce([l1]*[l2]*...[ln])} returns the word \\spad{l1*l2*...*ln},{} where \\spad{[l_i]} is the backeted form of the Lyndon word \\spad{l_i}.")) (|One| (($) "\\spad{1} returns the empty list."))) NIL NIL -(-799 UP R) +(-804 UP R) ((|constructor| (NIL "This package \\undocumented")) (|compose| ((|#1| |#1| |#1|) "\\spad{compose(p,q)} \\undocumented"))) NIL NIL -(-800 A T$ S) +(-805 A T$ S) ((|constructor| (NIL "\\indented{2}{This category captures the interface of domains with a distinguished} \\indented{2}{operation named \\spad{differentiate} for partial differentiation with} \\indented{2}{respect to some domain of variables.} See Also: \\indented{2}{DifferentialDomain,{} PartialDifferentialSpace}")) (D ((|#2| $ |#3|) "\\spad{D(x,v)} is a shorthand for \\spad{differentiate(x,v)}")) (|differentiate| ((|#2| $ |#3|) "\\spad{differentiate(x,v)} computes the partial derivative of \\spad{x} with respect to \\spad{v}."))) NIL NIL -(-801 T$ S) +(-806 T$ S) ((|constructor| (NIL "\\indented{2}{This category captures the interface of domains with a distinguished} \\indented{2}{operation named \\spad{differentiate} for partial differentiation with} \\indented{2}{respect to some domain of variables.} See Also: \\indented{2}{DifferentialDomain,{} PartialDifferentialSpace}")) (D ((|#1| $ |#2|) "\\spad{D(x,v)} is a shorthand for \\spad{differentiate(x,v)}")) (|differentiate| ((|#1| $ |#2|) "\\spad{differentiate(x,v)} computes the partial derivative of \\spad{x} with respect to \\spad{v}."))) NIL NIL -(-802 UP -3078) +(-807 UP -3088) ((|constructor| (NIL "This package \\undocumented")) (|rightFactorCandidate| ((|#1| |#1| (|NonNegativeInteger|)) "\\spad{rightFactorCandidate(p,n)} \\undocumented")) (|leftFactor| (((|Union| |#1| "failed") |#1| |#1|) "\\spad{leftFactor(p,q)} \\undocumented")) (|decompose| (((|Union| (|Record| (|:| |left| |#1|) (|:| |right| |#1|)) "failed") |#1| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{decompose(up,m,n)} \\undocumented") (((|List| |#1|) |#1|) "\\spad{decompose(up)} \\undocumented"))) NIL NIL -(-803 R S) +(-808 R S) ((|constructor| (NIL "A partial differential \\spad{R}-module with differentiations indexed by a parameter type \\spad{S}. \\blankline"))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-804 S) +(-809 S) ((|constructor| (NIL "A partial differential ring with differentiations indexed by a parameter type \\spad{S}. \\blankline"))) -((-3975 . T)) +((-3986 . T)) NIL -(-805 A S) +(-810 A S) ((|constructor| (NIL "\\indented{2}{This category captures the interface of domains stable by partial} \\indented{2}{differentiation with respect to variables from some domain.} See Also: \\indented{2}{PartialDifferentialDomain}")) (D (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{D(x,[s1,...,sn],[n1,...,nn])} is a shorthand for \\spad{differentiate(x,[s1,...,sn],[n1,...,nn])}.") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{D(x,s,n)} is a shorthand for \\spad{differentiate(x,s,n)}.") (($ $ (|List| |#2|)) "\\spad{D(x,[s1,...sn])} is a shorthand for \\spad{differentiate(x,[s1,...sn])}.")) (|differentiate| (($ $ (|List| |#2|) (|List| (|NonNegativeInteger|))) "\\spad{differentiate(x,[s1,...,sn],[n1,...,nn])} computes multiple partial derivatives,{} \\spadignore{i.e.}") (($ $ |#2| (|NonNegativeInteger|)) "\\spad{differentiate(x,s,n)} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{n}\\spad{-}th derivative of \\spad{x} with respect to \\spad{s}.") (($ $ (|List| |#2|)) "\\spad{differentiate(x,[s1,...sn])} computes successive partial derivatives,{} \\spadignore{i.e.} \\spad{differentiate(...differentiate(x, s1)..., sn)}."))) NIL NIL -(-806 S) +(-811 S) ((|constructor| (NIL "\\indented{2}{This category captures the interface of domains stable by partial} \\indented{2}{differentiation with respect to variables from some domain.} See Also: \\indented{2}{PartialDifferentialDomain}")) (D (($ $ (|List| |#1|) (|List| (|NonNegativeInteger|))) "\\spad{D(x,[s1,...,sn],[n1,...,nn])} is a shorthand for \\spad{differentiate(x,[s1,...,sn],[n1,...,nn])}.") (($ $ |#1| (|NonNegativeInteger|)) "\\spad{D(x,s,n)} is a shorthand for \\spad{differentiate(x,s,n)}.") (($ $ (|List| |#1|)) "\\spad{D(x,[s1,...sn])} is a shorthand for \\spad{differentiate(x,[s1,...sn])}.")) (|differentiate| (($ $ (|List| |#1|) (|List| (|NonNegativeInteger|))) "\\spad{differentiate(x,[s1,...,sn],[n1,...,nn])} computes multiple partial derivatives,{} \\spadignore{i.e.}") (($ $ |#1| (|NonNegativeInteger|)) "\\spad{differentiate(x,s,n)} computes multiple partial derivatives,{} \\spadignore{i.e.} \\spad{n}\\spad{-}th derivative of \\spad{x} with respect to \\spad{s}.") (($ $ (|List| |#1|)) "\\spad{differentiate(x,[s1,...sn])} computes successive partial derivatives,{} \\spadignore{i.e.} \\spad{differentiate(...differentiate(x, s1)..., sn)}."))) NIL NIL -(-807 S) +(-812 S) ((|constructor| (NIL "\\indented{1}{A PendantTree(\\spad{S})is either a leaf? and is an \\spad{S} or has} a left and a right both PendantTree(\\spad{S})'s")) (|ptree| (($ $ $) "\\spad{ptree(x,y)} \\undocumented") (($ |#1|) "\\spad{ptree(s)} is a leaf? pendant tree"))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-808 S) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-813 S) ((|constructor| (NIL "Permutation(\\spad{S}) implements the group of all bijections \\indented{2}{on a set \\spad{S},{} which move only a finite number of points.} \\indented{2}{A permutation is considered as a map from \\spad{S} into \\spad{S}. In particular} \\indented{2}{multiplication is defined as composition of maps:} \\indented{2}{{\\em pi1 * pi2 = pi1 o pi2}.} \\indented{2}{The internal representation of permuatations are two lists} \\indented{2}{of equal length representing preimages and images.}")) (|coerceImages| (($ (|List| |#1|)) "\\spad{coerceImages(ls)} coerces the list {\\em ls} to a permutation whose image is given by {\\em ls} and the preimage is fixed to be {\\em [1,...,n]}. Note: {coerceImages(\\spad{ls})=coercePreimagesImages([1,{}...,{}\\spad{n}],{}\\spad{ls})}. We assume that both preimage and image do not contain repetitions.")) (|fixedPoints| (((|Set| |#1|) $) "\\spad{fixedPoints(p)} returns the points fixed by the permutation \\spad{p}.")) (|sort| (((|List| $) (|List| $)) "\\spad{sort(lp)} sorts a list of permutations {\\em lp} according to cycle structure first according to length of cycles,{} second,{} if \\spad{S} has \\spadtype{Finite} or \\spad{S} has \\spadtype{OrderedSet} according to lexicographical order of entries in cycles of equal length.")) (|odd?| (((|Boolean|) $) "\\spad{odd?(p)} returns \\spad{true} if and only if \\spad{p} is an odd permutation \\spadignore{i.e.} {\\em sign(p)} is {\\em -1}.")) (|even?| (((|Boolean|) $) "\\spad{even?(p)} returns \\spad{true} if and only if \\spad{p} is an even permutation,{} \\spadignore{i.e.} {\\em sign(p)} is 1.")) (|sign| (((|Integer|) $) "\\spad{sign(p)} returns the signum of the permutation \\spad{p},{} \\spad{+1} or \\spad{-1}.")) (|numberOfCycles| (((|NonNegativeInteger|) $) "\\spad{numberOfCycles(p)} returns the number of non-trivial cycles of the permutation \\spad{p}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(p)} returns the order of a permutation \\spad{p} as a group element.")) (|cyclePartition| (((|Partition|) $) "\\spad{cyclePartition(p)} returns the cycle structure of a permutation \\spad{p} including cycles of length 1 only if \\spad{S} is finite.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} retuns the number of points moved by the permutation \\spad{p}.")) (|coerceListOfPairs| (($ (|List| (|List| |#1|))) "\\spad{coerceListOfPairs(lls)} coerces a list of pairs {\\em lls} to a permutation. Error: if not consistent,{} \\spadignore{i.e.} the set of the first elements coincides with the set of second elements. coerce(\\spad{p}) generates output of the permutation \\spad{p} with domain OutputForm.")) (|coerce| (($ (|List| |#1|)) "\\spad{coerce(ls)} coerces a cycle {\\em ls},{} \\spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur.") (($ (|List| (|List| |#1|))) "\\spad{coerce(lls)} coerces a list of cycles {\\em lls} to a permutation,{} each cycle being a list with no repetitions,{} is coerced to the permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.")) (|coercePreimagesImages| (($ (|List| (|List| |#1|))) "\\spad{coercePreimagesImages(lls)} coerces the representation {\\em lls} of a permutation as a list of preimages and images to a permutation. We assume that both preimage and image do not contain repetitions.")) (|listRepresentation| (((|Record| (|:| |preimage| (|List| |#1|)) (|:| |image| (|List| |#1|))) $) "\\spad{listRepresentation(p)} produces a representation {\\em rep} of the permutation \\spad{p} as a list of preimages and images,{} \\spad{i}.\\spad{e} \\spad{p} maps {\\em (rep.preimage).k} to {\\em (rep.image).k} for all indices \\spad{k}. Elements of \\spad{S} not in {\\em (rep.preimage).k} are fixed points,{} and these are the only fixed points of the permutation."))) -((-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-751)))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-751)))) -(-809 |n| R) +((-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-756)))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-756)))) +(-814 |n| R) ((|constructor| (NIL "Permanent implements the functions {\\em permanent},{} the permanent for square matrices.")) (|permanent| ((|#2| (|SquareMatrix| |#1| |#2|)) "\\spad{permanent(x)} computes the permanent of a square matrix \\spad{x}. The {\\em permanent} is equivalent to the \\spadfun{determinant} except that coefficients have no change of sign. This function is much more difficult to compute than the {\\em determinant}. The formula used is by \\spad{H}.\\spad{J}. Ryser,{} improved by [Nijenhuis and Wilf,{} Ch. 19]. Note: permanent(\\spad{x}) choose one of three algorithms,{} depending on the underlying ring \\spad{R} and on \\spad{n},{} the number of rows (and columns) of x:\\begin{items} \\item 1. if 2 has an inverse in \\spad{R} we can use the algorithm of \\indented{3}{[Nijenhuis and Wilf,{} ch.19,{}\\spad{p}.158]; if 2 has no inverse,{}} \\indented{3}{some modifications are necessary:} \\item 2. if {\\em n > 6} and \\spad{R} is an integral domain with characteristic \\indented{3}{different from 2 (the algorithm works if and only 2 is not a} \\indented{3}{zero-divisor of \\spad{R} and {\\em characteristic()\\$R ~= 2},{}} \\indented{3}{but how to check that for any given \\spad{R} ?),{}} \\indented{3}{the local function {\\em permanent2} is called;} \\item 3. else,{} the local function {\\em permanent3} is called \\indented{3}{(works for all commutative rings \\spad{R}).} \\end{items}"))) NIL NIL -(-810 S) +(-815 S) ((|constructor| (NIL "PermutationCategory provides a categorial environment \\indented{1}{for subgroups of bijections of a set (\\spadignore{i.e.} permutations)}")) (< (((|Boolean|) $ $) "\\spad{p < q} is an order relation on permutations. Note: this order is only total if and only if \\spad{S} is totally ordered or \\spad{S} is finite.")) (|orbit| (((|Set| |#1|) $ |#1|) "\\spad{orbit(p, el)} returns the orbit of {\\em el} under the permutation \\spad{p},{} \\spadignore{i.e.} the set which is given by applications of the powers of \\spad{p} to {\\em el}.")) (|support| (((|Set| |#1|) $) "\\spad{support p} returns the set of points not fixed by the permutation \\spad{p}.")) (|cycles| (($ (|List| (|List| |#1|))) "\\spad{cycles(lls)} coerces a list list of cycles {\\em lls} to a permutation,{} each cycle being a list with not repetitions,{} is coerced to the permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.")) (|cycle| (($ (|List| |#1|)) "\\spad{cycle(ls)} coerces a cycle {\\em ls},{} \\spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\\em ls.i} to {\\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur."))) -((-3975 . T)) +((-3986 . T)) NIL -(-811 S) +(-816 S) ((|constructor| (NIL "PermutationGroup implements permutation groups acting on a set \\spad{S},{} \\spadignore{i.e.} all subgroups of the symmetric group of \\spad{S},{} represented as a list of permutations (generators). Note that therefore the objects are not members of the \\Language category \\spadtype{Group}. Using the idea of base and strong generators by Sims,{} basic routines and algorithms are implemented so that the word problem for permutation groups can be solved.")) (|initializeGroupForWordProblem| (((|Void|) $ (|Integer|) (|Integer|)) "\\spad{initializeGroupForWordProblem(gp,m,n)} initializes the group {\\em gp} for the word problem. Notes: (1) with a small integer you get shorter words,{} but the routine takes longer than the standard routine for longer words. (2) be careful: invoking this routine will destroy the possibly stored information about your group (but will recompute it again). (3) users need not call this function normally for the soultion of the word problem.") (((|Void|) $) "\\spad{initializeGroupForWordProblem(gp)} initializes the group {\\em gp} for the word problem. Notes: it calls the other function of this name with parameters 0 and 1: {\\em initializeGroupForWordProblem(gp,0,1)}. Notes: (1) be careful: invoking this routine will destroy the possibly information about your group (but will recompute it again) (2) users need not call this function normally for the soultion of the word problem.")) (<= (((|Boolean|) $ $) "\\spad{gp1 <= gp2} returns \\spad{true} if and only if {\\em gp1} is a subgroup of {\\em gp2}. Note: because of a bug in the parser you have to call this function explicitly by {\\em gp1 <=\\$(PERMGRP S) gp2}.")) (< (((|Boolean|) $ $) "\\spad{gp1 < gp2} returns \\spad{true} if and only if {\\em gp1} is a proper subgroup of {\\em gp2}.")) (|support| (((|Set| |#1|) $) "\\spad{support(gp)} returns the points moved by the group {\\em gp}.")) (|wordInGenerators| (((|List| (|NonNegativeInteger|)) (|Permutation| |#1|) $) "\\spad{wordInGenerators(p,gp)} returns the word for the permutation \\spad{p} in the original generators of the group {\\em gp},{} represented by the indices of the list,{} given by {\\em generators}.")) (|wordInStrongGenerators| (((|List| (|NonNegativeInteger|)) (|Permutation| |#1|) $) "\\spad{wordInStrongGenerators(p,gp)} returns the word for the permutation \\spad{p} in the strong generators of the group {\\em gp},{} represented by the indices of the list,{} given by {\\em strongGenerators}.")) (|member?| (((|Boolean|) (|Permutation| |#1|) $) "\\spad{member?(pp,gp)} answers the question,{} whether the permutation {\\em pp} is in the group {\\em gp} or not.")) (|orbits| (((|Set| (|Set| |#1|)) $) "\\spad{orbits(gp)} returns the orbits of the group {\\em gp},{} \\spadignore{i.e.} it partitions the (finite) of all moved points.")) (|orbit| (((|Set| (|List| |#1|)) $ (|List| |#1|)) "\\spad{orbit(gp,ls)} returns the orbit of the ordered list {\\em ls} under the group {\\em gp}. Note: return type is \\spad{L} \\spad{L} \\spad{S} temporarily because FSET \\spad{L} \\spad{S} has an error.") (((|Set| (|Set| |#1|)) $ (|Set| |#1|)) "\\spad{orbit(gp,els)} returns the orbit of the unordered set {\\em els} under the group {\\em gp}.") (((|Set| |#1|) $ |#1|) "\\spad{orbit(gp,el)} returns the orbit of the element {\\em el} under the group {\\em gp},{} \\spadignore{i.e.} the set of all points gained by applying each group element to {\\em el}.")) (|permutationGroup| (($ (|List| (|Permutation| |#1|))) "\\spad{permutationGroup(ls)} coerces a list of permutations {\\em ls} to the group generated by this list.")) (|wordsForStrongGenerators| (((|List| (|List| (|NonNegativeInteger|))) $) "\\spad{wordsForStrongGenerators(gp)} returns the words for the strong generators of the group {\\em gp} in the original generators of {\\em gp},{} represented by their indices in the list,{} given by {\\em generators}.")) (|strongGenerators| (((|List| (|Permutation| |#1|)) $) "\\spad{strongGenerators(gp)} returns strong generators for the group {\\em gp}.")) (|base| (((|List| |#1|) $) "\\spad{base(gp)} returns a base for the group {\\em gp}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(gp)} returns the number of points moved by all permutations of the group {\\em gp}.")) (|order| (((|NonNegativeInteger|) $) "\\spad{order(gp)} returns the order of the group {\\em gp}.")) (|random| (((|Permutation| |#1|) $) "\\spad{random(gp)} returns a random product of maximal 20 generators of the group {\\em gp}. Note: {\\em random(gp)=random(gp,20)}.") (((|Permutation| |#1|) $ (|Integer|)) "\\spad{random(gp,i)} returns a random product of maximal \\spad{i} generators of the group {\\em gp}.")) (|elt| (((|Permutation| |#1|) $ (|NonNegativeInteger|)) "\\spad{elt(gp,i)} returns the \\spad{i}-th generator of the group {\\em gp}.")) (|generators| (((|List| (|Permutation| |#1|)) $) "\\spad{generators(gp)} returns the generators of the group {\\em gp}.")) (|coerce| (($ (|List| (|Permutation| |#1|))) "\\spad{coerce(ls)} coerces a list of permutations {\\em ls} to the group generated by this list.") (((|List| (|Permutation| |#1|)) $) "\\spad{coerce(gp)} returns the generators of the group {\\em gp}."))) NIL NIL -(-812 |p|) +(-817 |p|) ((|constructor| (NIL "PrimeField(\\spad{p}) implements the field with \\spad{p} elements if \\spad{p} is a prime number. Error: if \\spad{p} is not prime. Note: this domain does not check that argument is a prime."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| $ (QUOTE (-118))) (|HasCategory| $ (QUOTE (-116))) (|HasCategory| $ (QUOTE (-315)))) -(-813 R E |VarSet| S) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| $ (QUOTE (-120))) (|HasCategory| $ (QUOTE (-118))) (|HasCategory| $ (QUOTE (-317)))) +(-818 R E |VarSet| S) ((|constructor| (NIL "PolynomialFactorizationByRecursion(\\spad{R},{}\\spad{E},{}\\spad{VarSet},{}\\spad{S}) is used for factorization of sparse univariate polynomials over a domain \\spad{S} of multivariate polynomials over \\spad{R}.")) (|factorSFBRlcUnit| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|List| |#3|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorSFBRlcUnit(p)} returns the square free factorization of polynomial \\spad{p} (see \\spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate}) in the case where the leading coefficient of \\spad{p} is a unit.")) (|bivariateSLPEBR| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|List| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|) |#3|) "\\spad{bivariateSLPEBR(lp,p,v)} implements the bivariate case of \\spadfunFrom{solveLinearPolynomialEquationByRecursion}{PolynomialFactorizationByRecursionUnivariate}; its implementation depends on \\spad{R}")) (|randomR| ((|#1|) "\\spad{randomR produces} a random element of \\spad{R}")) (|factorSquareFreeByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorSquareFreeByRecursion(p)} returns the square free factorization of \\spad{p}. This functions performs the recursion step for factorSquareFreePolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorSquareFreePolynomial}).")) (|factorByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{factorByRecursion(p)} factors polynomial \\spad{p}. This function performs the recursion step for factorPolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorPolynomial})")) (|solveLinearPolynomialEquationByRecursion| (((|Union| (|List| (|SparseUnivariatePolynomial| |#4|)) "failed") (|List| (|SparseUnivariatePolynomial| |#4|)) (|SparseUnivariatePolynomial| |#4|)) "\\spad{solveLinearPolynomialEquationByRecursion([p1,...,pn],p)} returns the list of polynomials \\spad{[q1,...,qn]} such that \\spad{sum qi/pi = p / prod pi},{} a recursion step for solveLinearPolynomialEquation as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{solveLinearPolynomialEquation}). If no such list of \\spad{qi} exists,{} then \"failed\" is returned."))) NIL NIL -(-814 R S) +(-819 R S) ((|constructor| (NIL "\\indented{1}{PolynomialFactorizationByRecursionUnivariate} \\spad{R} is a \\spadfun{PolynomialFactorizationExplicit} domain,{} \\spad{S} is univariate polynomials over \\spad{R} We are interested in handling SparseUnivariatePolynomials over \\spad{S},{} is a variable we shall call \\spad{z}")) (|factorSFBRlcUnit| (((|Factored| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{factorSFBRlcUnit(p)} returns the square free factorization of polynomial \\spad{p} (see \\spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate}) in the case where the leading coefficient of \\spad{p} is a unit.")) (|randomR| ((|#1|) "\\spad{randomR()} produces a random element of \\spad{R}")) (|factorSquareFreeByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{factorSquareFreeByRecursion(p)} returns the square free factorization of \\spad{p}. This functions performs the recursion step for factorSquareFreePolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorSquareFreePolynomial}).")) (|factorByRecursion| (((|Factored| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{factorByRecursion(p)} factors polynomial \\spad{p}. This function performs the recursion step for factorPolynomial,{} as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{factorPolynomial})")) (|solveLinearPolynomialEquationByRecursion| (((|Union| (|List| (|SparseUnivariatePolynomial| |#2|)) "failed") (|List| (|SparseUnivariatePolynomial| |#2|)) (|SparseUnivariatePolynomial| |#2|)) "\\spad{solveLinearPolynomialEquationByRecursion([p1,...,pn],p)} returns the list of polynomials \\spad{[q1,...,qn]} such that \\spad{sum qi/pi = p / prod pi},{} a recursion step for solveLinearPolynomialEquation as defined in \\spadfun{PolynomialFactorizationExplicit} category (see \\spadfun{solveLinearPolynomialEquation}). If no such list of \\spad{qi} exists,{} then \"failed\" is returned."))) NIL NIL -(-815 S) +(-820 S) ((|constructor| (NIL "This is the category of domains that know \"enough\" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.")) (|charthRoot| (((|Maybe| $) $) "\\spad{charthRoot(r)} returns the \\spad{p}\\spad{-}th root of \\spad{r},{} or \\spad{nothing} if none exists in the domain.")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(m)} returns a vector of elements,{} not all zero,{} whose \\spad{p}\\spad{-}th powers (\\spad{p} is the characteristic of the domain) are a solution of the homogenous linear system represented by \\spad{m},{} or \"failed\" is there is no such vector.")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| $)) "failed") (|List| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}'s exists.")) (|gcdPolynomial| (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $)) "\\spad{gcdPolynomial(p,q)} returns the gcd of the univariate polynomials \\spad{p} qnd \\spad{q}.")) (|factorSquareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorSquareFreePolynomial(p)} factors the univariate polynomial \\spad{p} into irreducibles where \\spad{p} is known to be square free and primitive with respect to its main variable.")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} returns the factorization into irreducibles of the univariate polynomial \\spad{p}.")) (|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} returns the square-free factorization of the univariate polynomial \\spad{p}."))) NIL -((|HasCategory| |#1| (QUOTE (-116)))) -(-816) +((|HasCategory| |#1| (QUOTE (-118)))) +(-821) ((|constructor| (NIL "This is the category of domains that know \"enough\" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.")) (|charthRoot| (((|Maybe| $) $) "\\spad{charthRoot(r)} returns the \\spad{p}\\spad{-}th root of \\spad{r},{} or \\spad{nothing} if none exists in the domain.")) (|conditionP| (((|Union| (|Vector| $) "failed") (|Matrix| $)) "\\spad{conditionP(m)} returns a vector of elements,{} not all zero,{} whose \\spad{p}\\spad{-}th powers (\\spad{p} is the characteristic of the domain) are a solution of the homogenous linear system represented by \\spad{m},{} or \"failed\" is there is no such vector.")) (|solveLinearPolynomialEquation| (((|Union| (|List| (|SparseUnivariatePolynomial| $)) "failed") (|List| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{solveLinearPolynomialEquation([f1, ..., fn], g)} (where the \\spad{fi} are relatively prime to each other) returns a list of \\spad{ai} such that \\spad{g/prod fi = sum ai/fi} or returns \"failed\" if no such list of \\spad{ai}'s exists.")) (|gcdPolynomial| (((|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $) (|SparseUnivariatePolynomial| $)) "\\spad{gcdPolynomial(p,q)} returns the gcd of the univariate polynomials \\spad{p} qnd \\spad{q}.")) (|factorSquareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorSquareFreePolynomial(p)} factors the univariate polynomial \\spad{p} into irreducibles where \\spad{p} is known to be square free and primitive with respect to its main variable.")) (|factorPolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{factorPolynomial(p)} returns the factorization into irreducibles of the univariate polynomial \\spad{p}.")) (|squareFreePolynomial| (((|Factored| (|SparseUnivariatePolynomial| $)) (|SparseUnivariatePolynomial| $)) "\\spad{squareFreePolynomial(p)} returns the square-free factorization of the univariate polynomial \\spad{p}."))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-817 R0 -3078 UP UPUP R) +(-822 R0 -3088 UP UPUP R) ((|constructor| (NIL "This package provides function for testing whether a divisor on a curve is a torsion divisor.")) (|torsionIfCan| (((|Union| (|Record| (|:| |order| (|NonNegativeInteger|)) (|:| |function| |#5|)) "failed") (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{torsionIfCan(f)}\\\\ undocumented")) (|torsion?| (((|Boolean|) (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{torsion?(f)} \\undocumented")) (|order| (((|Union| (|NonNegativeInteger|) "failed") (|FiniteDivisor| |#2| |#3| |#4| |#5|)) "\\spad{order(f)} \\undocumented"))) NIL NIL -(-818 UP UPUP R) +(-823 UP UPUP R) ((|constructor| (NIL "This package provides function for testing whether a divisor on a curve is a torsion divisor.")) (|torsionIfCan| (((|Union| (|Record| (|:| |order| (|NonNegativeInteger|)) (|:| |function| |#3|)) "failed") (|FiniteDivisor| (|Fraction| (|Integer|)) |#1| |#2| |#3|)) "\\spad{torsionIfCan(f)} \\undocumented")) (|torsion?| (((|Boolean|) (|FiniteDivisor| (|Fraction| (|Integer|)) |#1| |#2| |#3|)) "\\spad{torsion?(f)} \\undocumented")) (|order| (((|Union| (|NonNegativeInteger|) "failed") (|FiniteDivisor| (|Fraction| (|Integer|)) |#1| |#2| |#3|)) "\\spad{order(f)} \\undocumented"))) NIL NIL -(-819 UP UPUP) +(-824 UP UPUP) ((|constructor| (NIL "\\indented{1}{Utilities for PFOQ and PFO} Author: Manuel Bronstein Date Created: 25 Aug 1988 Date Last Updated: 11 Jul 1990")) (|polyred| ((|#2| |#2|) "\\spad{polyred(u)} \\undocumented")) (|doubleDisc| (((|Integer|) |#2|) "\\spad{doubleDisc(u)} \\undocumented")) (|mix| (((|Integer|) (|List| (|Record| (|:| |den| (|Integer|)) (|:| |gcdnum| (|Integer|))))) "\\spad{mix(l)} \\undocumented")) (|badNum| (((|Integer|) |#2|) "\\spad{badNum(u)} \\undocumented") (((|Record| (|:| |den| (|Integer|)) (|:| |gcdnum| (|Integer|))) |#1|) "\\spad{badNum(p)} \\undocumented")) (|getGoodPrime| (((|PositiveInteger|) (|Integer|)) "\\spad{getGoodPrime n} returns the smallest prime not dividing \\spad{n}"))) NIL NIL -(-820 R) +(-825 R) ((|constructor| (NIL "The domain \\spadtype{PartialFraction} implements partial fractions over a euclidean domain \\spad{R}. This requirement on the argument domain allows us to normalize the fractions. Of particular interest are the 2 forms for these fractions. The ``compact'' form has only one fractional term per prime in the denominator,{} while the ``p-adic'' form expands each numerator \\spad{p}-adically via the prime \\spad{p} in the denominator. For computational efficiency,{} the compact form is used,{} though the \\spad{p}-adic form may be gotten by calling the function \\spadfunFrom{padicFraction}{PartialFraction}. For a general euclidean domain,{} it is not known how to factor the denominator. Thus the function \\spadfunFrom{partialFraction}{PartialFraction} takes as its second argument an element of \\spadtype{Factored(R)}.")) (|wholePart| ((|#1| $) "\\spad{wholePart(p)} extracts the whole part of the partial fraction \\spad{p}.")) (|partialFraction| (($ |#1| (|Factored| |#1|)) "\\spad{partialFraction(numer,denom)} is the main function for constructing partial fractions. The second argument is the denominator and should be factored.")) (|padicFraction| (($ $) "\\spad{padicFraction(q)} expands the fraction \\spad{p}-adically in the primes \\spad{p} in the denominator of \\spad{q}. For example,{} \\spad{padicFraction(3/(2**2)) = 1/2 + 1/(2**2)}. Use \\spadfunFrom{compactFraction}{PartialFraction} to return to compact form.")) (|padicallyExpand| (((|SparseUnivariatePolynomial| |#1|) |#1| |#1|) "\\spad{padicallyExpand(p,x)} is a utility function that expands the second argument \\spad{x} ``p-adically'' in the first.")) (|numberOfFractionalTerms| (((|Integer|) $) "\\spad{numberOfFractionalTerms(p)} computes the number of fractional terms in \\spad{p}. This returns 0 if there is no fractional part.")) (|nthFractionalTerm| (($ $ (|Integer|)) "\\spad{nthFractionalTerm(p,n)} extracts the \\spad{n}th fractional term from the partial fraction \\spad{p}. This returns 0 if the index \\spad{n} is out of range.")) (|firstNumer| ((|#1| $) "\\spad{firstNumer(p)} extracts the numerator of the first fractional term. This returns 0 if there is no fractional part (use \\spadfunFrom{wholePart}{PartialFraction} to get the whole part).")) (|firstDenom| (((|Factored| |#1|) $) "\\spad{firstDenom(p)} extracts the denominator of the first fractional term. This returns 1 if there is no fractional part (use \\spadfunFrom{wholePart}{PartialFraction} to get the whole part).")) (|compactFraction| (($ $) "\\spad{compactFraction(p)} normalizes the partial fraction \\spad{p} to the compact representation. In this form,{} the partial fraction has only one fractional term per prime in the denominator.")) (|coerce| (($ (|Fraction| (|Factored| |#1|))) "\\spad{coerce(f)} takes a fraction with numerator and denominator in factored form and creates a partial fraction. It is necessary for the parts to be factored because it is not known in general how to factor elements of \\spad{R} and this is needed to decompose into partial fractions.") (((|Fraction| |#1|) $) "\\spad{coerce(p)} sums up the components of the partial fraction and returns a single fraction."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-821 R) +(-826 R) ((|constructor| (NIL "The package \\spadtype{PartialFractionPackage} gives an easier to use interfact the domain \\spadtype{PartialFraction}. The user gives a fraction of polynomials,{} and a variable and the package converts it to the proper datatype for the \\spadtype{PartialFraction} domain.")) (|partialFraction| (((|Any|) (|Polynomial| |#1|) (|Factored| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{partialFraction(num, facdenom, var)} returns the partial fraction decomposition of the rational function whose numerator is \\spad{num} and whose factored denominator is \\spad{facdenom} with respect to the variable var.") (((|Any|) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{partialFraction(rf, var)} returns the partial fraction decomposition of the rational function \\spad{rf} with respect to the variable var."))) NIL NIL -(-822 E OV R P) +(-827 E OV R P) ((|gcdPrimitive| ((|#4| (|List| |#4|)) "\\spad{gcdPrimitive lp} computes the gcd of the list of primitive polynomials lp.") (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{gcdPrimitive(p,q)} computes the gcd of the primitive polynomials \\spad{p} and \\spad{q}.") ((|#4| |#4| |#4|) "\\spad{gcdPrimitive(p,q)} computes the gcd of the primitive polynomials \\spad{p} and \\spad{q}.")) (|gcd| (((|SparseUnivariatePolynomial| |#4|) (|List| (|SparseUnivariatePolynomial| |#4|))) "\\spad{gcd(lp)} computes the gcd of the list of polynomials \\spad{lp}.") (((|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|) (|SparseUnivariatePolynomial| |#4|)) "\\spad{gcd(p,q)} computes the gcd of the two polynomials \\spad{p} and \\spad{q}.") ((|#4| (|List| |#4|)) "\\spad{gcd(lp)} computes the gcd of the list of polynomials \\spad{lp}.") ((|#4| |#4| |#4|) "\\spad{gcd(p,q)} computes the gcd of the two polynomials \\spad{p} and \\spad{q}."))) NIL NIL -(-823) +(-828) ((|constructor| (NIL "PermutationGroupExamples provides permutation groups for some classes of groups: symmetric,{} alternating,{} dihedral,{} cyclic,{} direct products of cyclic,{} which are in fact the finite abelian groups of symmetric groups called Young subgroups. Furthermore,{} Rubik's group as permutation group of 48 integers and a list of sporadic simple groups derived from the atlas of finite groups.")) (|youngGroup| (((|PermutationGroup| (|Integer|)) (|Partition|)) "\\spad{youngGroup(lambda)} constructs the direct product of the symmetric groups given by the parts of the partition {\\em lambda}.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{youngGroup([n1,...,nk])} constructs the direct product of the symmetric groups {\\em Sn1},{}...,{}{\\em Snk}.")) (|rubiksGroup| (((|PermutationGroup| (|Integer|))) "\\spad{rubiksGroup constructs} the permutation group representing Rubic's Cube acting on integers {\\em 10*i+j} for {\\em 1 <= i <= 6},{} {\\em 1 <= j <= 8}. The faces of Rubik's Cube are labelled in the obvious way Front,{} Right,{} Up,{} Down,{} Left,{} Back and numbered from 1 to 6 in this given ordering,{} the pieces on each face (except the unmoveable center piece) are clockwise numbered from 1 to 8 starting with the piece in the upper left corner. The moves of the cube are represented as permutations on these pieces,{} represented as a two digit integer {\\em ij} where \\spad{i} is the numer of theface (1 to 6) and \\spad{j} is the number of the piece on this face. The remaining ambiguities are resolved by looking at the 6 generators,{} which represent a 90 degree turns of the faces,{} or from the following pictorial description. Permutation group representing Rubic's Cube acting on integers 10*i+j for 1 <= \\spad{i} <= 6,{} 1 <= \\spad{j} \\spad{<=8}. \\blankline\\begin{verbatim}Rubik's Cube: +-----+ +-- B where: marks Side # : / U /|/ / / | F(ront) <-> 1 L --> +-----+ R| R(ight) <-> 2 | | + U(p) <-> 3 | F | / D(own) <-> 4 | |/ L(eft) <-> 5 +-----+ B(ack) <-> 6 ^ | DThe Cube's surface: The pieces on each side +---+ (except the unmoveable center |567| piece) are clockwise numbered |4U8| from 1 to 8 starting with the |321| piece in the upper left +---+---+---+ corner (see figure on the |781|123|345| left). The moves of the cube |6L2|8F4|2R6| are represented as |543|765|187| permutations on these pieces. +---+---+---+ Each of the pieces is |123| represented as a two digit |8D4| integer ij where i is the |765| # of the side ( 1 to 6 for +---+ F to B (see table above )) |567| and j is the # of the piece. |4B8| |321| +---+\\end{verbatim}")) (|janko2| (((|PermutationGroup| (|Integer|))) "\\spad{janko2 constructs} the janko group acting on the integers 1,{}...,{}100.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{janko2(li)} constructs the janko group acting on the 100 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 100 different entries")) (|mathieu24| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu24 constructs} the mathieu group acting on the integers 1,{}...,{}24.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu24(li)} constructs the mathieu group acting on the 24 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 24 different entries.")) (|mathieu23| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu23 constructs} the mathieu group acting on the integers 1,{}...,{}23.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu23(li)} constructs the mathieu group acting on the 23 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 23 different entries.")) (|mathieu22| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu22 constructs} the mathieu group acting on the integers 1,{}...,{}22.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu22(li)} constructs the mathieu group acting on the 22 integers given in the list {\\em li}. Note: duplicates in the list will be removed. Error: if {\\em li} has less or more than 22 different entries.")) (|mathieu12| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu12 constructs} the mathieu group acting on the integers 1,{}...,{}12.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu12(li)} constructs the mathieu group acting on the 12 integers given in the list {\\em li}. Note: duplicates in the list will be removed Error: if {\\em li} has less or more than 12 different entries.")) (|mathieu11| (((|PermutationGroup| (|Integer|))) "\\spad{mathieu11 constructs} the mathieu group acting on the integers 1,{}...,{}11.") (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{mathieu11(li)} constructs the mathieu group acting on the 11 integers given in the list {\\em li}. Note: duplicates in the list will be removed. error,{} if {\\em li} has less or more than 11 different entries.")) (|dihedralGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{dihedralGroup([i1,...,ik])} constructs the dihedral group of order 2k acting on the integers out of {\\em i1},{}...,{}{\\em ik}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{dihedralGroup(n)} constructs the dihedral group of order 2n acting on integers 1,{}...,{}\\spad{N}.")) (|cyclicGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{cyclicGroup([i1,...,ik])} constructs the cyclic group of order \\spad{k} acting on the integers {\\em i1},{}...,{}{\\em ik}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{cyclicGroup(n)} constructs the cyclic group of order \\spad{n} acting on the integers 1,{}...,{}\\spad{n}.")) (|abelianGroup| (((|PermutationGroup| (|Integer|)) (|List| (|PositiveInteger|))) "\\spad{abelianGroup([n1,...,nk])} constructs the abelian group that is the direct product of cyclic groups with order {\\em ni}.")) (|alternatingGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{alternatingGroup(li)} constructs the alternating group acting on the integers in the list {\\em li},{} generators are in general the {\\em n-2}-cycle {\\em (li.3,...,li.n)} and the 3-cycle {\\em (li.1,li.2,li.3)},{} if \\spad{n} is odd and product of the 2-cycle {\\em (li.1,li.2)} with {\\em n-2}-cycle {\\em (li.3,...,li.n)} and the 3-cycle {\\em (li.1,li.2,li.3)},{} if \\spad{n} is even. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{alternatingGroup(n)} constructs the alternating group {\\em An} acting on the integers 1,{}...,{}\\spad{n},{} generators are in general the {\\em n-2}-cycle {\\em (3,...,n)} and the 3-cycle {\\em (1,2,3)} if \\spad{n} is odd and the product of the 2-cycle {\\em (1,2)} with {\\em n-2}-cycle {\\em (3,...,n)} and the 3-cycle {\\em (1,2,3)} if \\spad{n} is even.")) (|symmetricGroup| (((|PermutationGroup| (|Integer|)) (|List| (|Integer|))) "\\spad{symmetricGroup(li)} constructs the symmetric group acting on the integers in the list {\\em li},{} generators are the cycle given by {\\em li} and the 2-cycle {\\em (li.1,li.2)}. Note: duplicates in the list will be removed.") (((|PermutationGroup| (|Integer|)) (|PositiveInteger|)) "\\spad{symmetricGroup(n)} constructs the symmetric group {\\em Sn} acting on the integers 1,{}...,{}\\spad{n},{} generators are the {\\em n}-cycle {\\em (1,...,n)} and the 2-cycle {\\em (1,2)}."))) NIL NIL -(-824 -3078) +(-829 -3088) ((|constructor| (NIL "Groebner functions for \\spad{P} \\spad{F} \\indented{2}{This package is an interface package to the groebner basis} package which allows you to compute groebner bases for polynomials in either lexicographic ordering or total degree ordering refined by reverse lex. The input is the ordinary polynomial type which is internally converted to a type with the required ordering. The resulting grobner basis is converted back to ordinary polynomials. The ordering among the variables is controlled by an explicit list of variables which is passed as a second argument. The coefficient domain is allowed to be any gcd domain,{} but the groebner basis is computed as if the polynomials were over a field.")) (|totalGroebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{totalGroebner(lp,lv)} computes Groebner basis for the list of polynomials \\spad{lp} with the terms ordered first by total degree and then refined by reverse lexicographic ordering. The variables are ordered by their position in the list \\spad{lv}.")) (|lexGroebner| (((|List| (|Polynomial| |#1|)) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{lexGroebner(lp,lv)} computes Groebner basis for the list of polynomials \\spad{lp} in lexicographic order. The variables are ordered by their position in the list \\spad{lv}."))) NIL NIL -(-825) +(-830) ((|constructor| (NIL "\\spadtype{PositiveInteger} provides functions for \\indented{2}{positive integers.}")) (|commutative| ((|attribute| "*") "\\spad{commutative(\"*\")} means multiplication is commutative : x*y = y*x")) (|gcd| (($ $ $) "\\spad{gcd(a,b)} computes the greatest common divisor of two positive integers \\spad{a} and \\spad{b}."))) -(((-3980 "*") . T)) +(((-3991 "*") . T)) NIL -(-826 R) +(-831 R) ((|constructor| (NIL "\\indented{1}{Provides a coercion from the symbolic fractions in \\%\\spad{pi} with} integer coefficients to any Expression type. Date Created: 21 Feb 1990 Date Last Updated: 21 Feb 1990")) (|coerce| (((|Expression| |#1|) (|Pi|)) "\\spad{coerce(f)} returns \\spad{f} as an Expression(\\spad{R})."))) NIL NIL -(-827) +(-832) ((|constructor| (NIL "The category of constructive principal ideal domains,{} \\spadignore{i.e.} where a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.")) (|expressIdealMember| (((|Maybe| (|List| $)) (|List| $) $) "\\spad{expressIdealMember([f1,...,fn],h)} returns a representation of \\spad{h} as a linear combination of the \\spad{fi} or \\spad{nothing} if \\spad{h} is not in the ideal generated by the \\spad{fi}.")) (|principalIdeal| (((|Record| (|:| |coef| (|List| $)) (|:| |generator| $)) (|List| $)) "\\spad{principalIdeal([f1,...,fn])} returns a record whose generator component is a generator of the ideal generated by \\spad{[f1,...,fn]} whose coef component satisfies \\spad{generator = sum (input.i * coef.i)}"))) -((-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-828 |xx| -3078) +(-833 |xx| -3088) ((|constructor| (NIL "This package exports interpolation algorithms")) (|interpolate| (((|SparseUnivariatePolynomial| |#2|) (|List| |#2|) (|List| |#2|)) "\\spad{interpolate(lf,lg)} \\undocumented") (((|UnivariatePolynomial| |#1| |#2|) (|UnivariatePolynomial| |#1| |#2|) (|List| |#2|) (|List| |#2|)) "\\spad{interpolate(u,lf,lg)} \\undocumented"))) NIL NIL -(-829 -3078 P) +(-834 -3088 P) ((|constructor| (NIL "This package exports interpolation algorithms")) (|LagrangeInterpolation| ((|#2| (|List| |#1|) (|List| |#1|)) "\\spad{LagrangeInterpolation(l1,l2)} \\undocumented"))) NIL NIL -(-830 R |Var| |Expon| GR) +(-835 R |Var| |Expon| GR) ((|constructor| (NIL "Author: William Sit,{} spring 89")) (|inconsistent?| (((|Boolean|) (|List| (|Polynomial| |#1|))) "inconsistant?(pl) returns \\spad{true} if the system of equations \\spad{p} = 0 for \\spad{p} in pl is inconsistent. It is assumed that pl is a groebner basis.") (((|Boolean|) (|List| |#4|)) "inconsistant?(pl) returns \\spad{true} if the system of equations \\spad{p} = 0 for \\spad{p} in pl is inconsistent. It is assumed that pl is a groebner basis.")) (|sqfree| ((|#4| |#4|) "\\spad{sqfree(p)} returns the product of square free factors of \\spad{p}")) (|regime| (((|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))))) (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))) (|Matrix| |#4|) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|List| |#4|)) (|NonNegativeInteger|) (|NonNegativeInteger|) (|Integer|)) "\\spad{regime(y,c, w, p, r, rm, m)} returns a regime,{} a list of polynomials specifying the consistency conditions,{} a particular solution and basis representing the general solution of the parametric linear system \\spad{c} \\spad{z} = \\spad{w} on that regime. The regime returned depends on the subdeterminant \\spad{y}.det and the row and column indices. The solutions are simplified using the assumption that the system has rank \\spad{r} and maximum rank \\spad{rm}. The list \\spad{p} represents a list of list of factors of polynomials in a groebner basis of the ideal generated by higher order subdeterminants,{} and ius used for the simplification. The mode \\spad{m} distinguishes the cases when the system is homogeneous,{} or the right hand side is arbitrary,{} or when there is no new right hand side variables.")) (|redmat| (((|Matrix| |#4|) (|Matrix| |#4|) (|List| |#4|)) "\\spad{redmat(m,g)} returns a matrix whose entries are those of \\spad{m} modulo the ideal generated by the groebner basis \\spad{g}")) (|ParCond| (((|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|))))) (|Matrix| |#4|) (|NonNegativeInteger|)) "\\spad{ParCond(m,k)} returns the list of all \\spad{k} by \\spad{k} subdeterminants in the matrix \\spad{m}")) (|overset?| (((|Boolean|) (|List| |#4|) (|List| (|List| |#4|))) "\\spad{overset?(s,sl)} returns \\spad{true} if \\spad{s} properly a sublist of a member of \\spad{sl}; otherwise it returns \\spad{false}")) (|nextSublist| (((|List| (|List| (|Integer|))) (|Integer|) (|Integer|)) "\\spad{nextSublist(n,k)} returns a list of \\spad{k}-subsets of {1,{} ...,{} \\spad{n}}.")) (|minset| (((|List| (|List| |#4|)) (|List| (|List| |#4|))) "\\spad{minset(sl)} returns the sublist of \\spad{sl} consisting of the minimal lists (with respect to inclusion) in the list \\spad{sl} of lists")) (|minrank| (((|NonNegativeInteger|) (|List| (|Record| (|:| |rank| (|NonNegativeInteger|)) (|:| |eqns| (|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))))) (|:| |fgb| (|List| |#4|))))) "\\spad{minrank(r)} returns the minimum rank in the list \\spad{r} of regimes")) (|maxrank| (((|NonNegativeInteger|) (|List| (|Record| (|:| |rank| (|NonNegativeInteger|)) (|:| |eqns| (|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))))) (|:| |fgb| (|List| |#4|))))) "\\spad{maxrank(r)} returns the maximum rank in the list \\spad{r} of regimes")) (|factorset| (((|List| |#4|) |#4|) "\\spad{factorset(p)} returns the set of irreducible factors of \\spad{p}.")) (|B1solve| (((|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))) (|Record| (|:| |mat| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|:| |vec| (|List| (|Fraction| (|Polynomial| |#1|)))) (|:| |rank| (|NonNegativeInteger|)) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|))))) "\\spad{B1solve(s)} solves the system (\\spad{s}.mat) \\spad{z} = \\spad{s}.vec for the variables given by the column indices of \\spad{s}.cols in terms of the other variables and the right hand side \\spad{s}.vec by assuming that the rank is \\spad{s}.rank,{} that the system is consistent,{} with the linearly independent equations indexed by the given row indices \\spad{s}.rows; the coefficients in \\spad{s}.mat involving parameters are treated as polynomials. B1solve(\\spad{s}) returns a particular solution to the system and a basis of the homogeneous system (\\spad{s}.mat) \\spad{z} = 0.")) (|redpps| (((|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))) (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))) (|List| |#4|)) "\\spad{redpps(s,g)} returns the simplified form of \\spad{s} after reducing modulo a groebner basis \\spad{g}")) (|ParCondList| (((|List| (|Record| (|:| |rank| (|NonNegativeInteger|)) (|:| |eqns| (|List| (|Record| (|:| |det| |#4|) (|:| |rows| (|List| (|Integer|))) (|:| |cols| (|List| (|Integer|)))))) (|:| |fgb| (|List| |#4|)))) (|Matrix| |#4|) (|NonNegativeInteger|)) "\\spad{ParCondList(c,r)} computes a list of subdeterminants of each rank >= \\spad{r} of the matrix \\spad{c} and returns a groebner basis for the ideal they generate")) (|hasoln| (((|Record| (|:| |sysok| (|Boolean|)) (|:| |z0| (|List| |#4|)) (|:| |n0| (|List| |#4|))) (|List| |#4|) (|List| |#4|)) "\\spad{hasoln(g, l)} tests whether the quasi-algebraic set defined by \\spad{p} = 0 for \\spad{p} in \\spad{g} and \\spad{q} ~= 0 for \\spad{q} in \\spad{l} is empty or not and returns a simplified definition of the quasi-algebraic set")) (|pr2dmp| ((|#4| (|Polynomial| |#1|)) "\\spad{pr2dmp(p)} converts \\spad{p} to target domain")) (|se2rfi| (((|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{se2rfi(l)} converts \\spad{l} to target domain")) (|dmp2rfi| (((|List| (|Fraction| (|Polynomial| |#1|))) (|List| |#4|)) "\\spad{dmp2rfi(l)} converts \\spad{l} to target domain") (((|Matrix| (|Fraction| (|Polynomial| |#1|))) (|Matrix| |#4|)) "\\spad{dmp2rfi(m)} converts \\spad{m} to target domain") (((|Fraction| (|Polynomial| |#1|)) |#4|) "\\spad{dmp2rfi(p)} converts \\spad{p} to target domain")) (|bsolve| (((|Record| (|:| |rgl| (|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|)))))))))) (|:| |rgsz| (|Integer|))) (|Matrix| |#4|) (|List| (|Fraction| (|Polynomial| |#1|))) (|NonNegativeInteger|) (|String|) (|Integer|)) "\\spad{bsolve(c, w, r, s, m)} returns a list of regimes and solutions of the system \\spad{c} \\spad{z} = \\spad{w} for ranks at least \\spad{r}; depending on the mode \\spad{m} chosen,{} it writes the output to a file given by the string \\spad{s}.")) (|rdregime| (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|String|)) "\\spad{rdregime(s)} reads in a list from a file with name \\spad{s}")) (|wrregime| (((|Integer|) (|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|String|)) "\\spad{wrregime(l,s)} writes a list of regimes to a file named \\spad{s} and returns the number of regimes written")) (|psolve| (((|Integer|) (|Matrix| |#4|) (|PositiveInteger|) (|String|)) "\\spad{psolve(c,k,s)} solves \\spad{c} \\spad{z} = 0 for all possible ranks >= \\spad{k} of the matrix \\spad{c},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| (|Symbol|)) (|PositiveInteger|) (|String|)) "\\spad{psolve(c,w,k,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks >= \\spad{k} of the matrix \\spad{c} and indeterminate right hand side \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| |#4|) (|PositiveInteger|) (|String|)) "\\spad{psolve(c,w,k,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks >= \\spad{k} of the matrix \\spad{c} and given right hand side \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|String|)) "\\spad{psolve(c,s)} solves \\spad{c} \\spad{z} = 0 for all possible ranks of the matrix \\spad{c} and given right hand side vector \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| (|Symbol|)) (|String|)) "\\spad{psolve(c,w,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and indeterminate right hand side \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|Integer|) (|Matrix| |#4|) (|List| |#4|) (|String|)) "\\spad{psolve(c,w,s)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and given right hand side vector \\spad{w},{} writes the results to a file named \\spad{s},{} and returns the number of regimes") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|PositiveInteger|)) "\\spad{psolve(c)} solves the homogeneous linear system \\spad{c} \\spad{z} = 0 for all possible ranks >= \\spad{k} of the matrix \\spad{c}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| (|Symbol|)) (|PositiveInteger|)) "\\spad{psolve(c,w,k)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks >= \\spad{k} of the matrix \\spad{c} and indeterminate right hand side \\spad{w}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| |#4|) (|PositiveInteger|)) "\\spad{psolve(c,w,k)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks >= \\spad{k} of the matrix \\spad{c} and given right hand side vector \\spad{w}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|)) "\\spad{psolve(c)} solves the homogeneous linear system \\spad{c} \\spad{z} = 0 for all possible ranks of the matrix \\spad{c}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| (|Symbol|))) "\\spad{psolve(c,w)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and indeterminate right hand side \\spad{w}") (((|List| (|Record| (|:| |eqzro| (|List| |#4|)) (|:| |neqzro| (|List| |#4|)) (|:| |wcond| (|List| (|Polynomial| |#1|))) (|:| |bsoln| (|Record| (|:| |partsol| (|Vector| (|Fraction| (|Polynomial| |#1|)))) (|:| |basis| (|List| (|Vector| (|Fraction| (|Polynomial| |#1|))))))))) (|Matrix| |#4|) (|List| |#4|)) "\\spad{psolve(c,w)} solves \\spad{c} \\spad{z} = \\spad{w} for all possible ranks of the matrix \\spad{c} and given right hand side vector \\spad{w}"))) NIL NIL -(-831) +(-836) ((|constructor| (NIL "The Plot domain supports plotting of functions defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example floating point numbers and infinite continued fractions. The facilities at this point are limited to 2-dimensional plots or either a single function or a parametric function.")) (|debug| (((|Boolean|) (|Boolean|)) "\\spad{debug(true)} turns debug mode on \\spad{debug(false)} turns debug mode off")) (|numFunEvals| (((|Integer|)) "\\spad{numFunEvals()} returns the number of points computed")) (|setAdaptive| (((|Boolean|) (|Boolean|)) "\\spad{setAdaptive(true)} turns adaptive plotting on \\spad{setAdaptive(false)} turns adaptive plotting off")) (|adaptive?| (((|Boolean|)) "\\spad{adaptive?()} determines whether plotting be done adaptively")) (|setScreenResolution| (((|Integer|) (|Integer|)) "\\spad{setScreenResolution(i)} sets the screen resolution to \\spad{i}")) (|screenResolution| (((|Integer|)) "\\spad{screenResolution()} returns the screen resolution")) (|setMaxPoints| (((|Integer|) (|Integer|)) "\\spad{setMaxPoints(i)} sets the maximum number of points in a plot to \\spad{i}")) (|maxPoints| (((|Integer|)) "\\spad{maxPoints()} returns the maximum number of points in a plot")) (|setMinPoints| (((|Integer|) (|Integer|)) "\\spad{setMinPoints(i)} sets the minimum number of points in a plot to \\spad{i}")) (|minPoints| (((|Integer|)) "\\spad{minPoints()} returns the minimum number of points in a plot")) (|tRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{tRange(p)} returns the range of the parameter in a parametric plot \\spad{p}")) (|refine| (($ $) "\\spad{refine(p)} performs a refinement on the plot \\spad{p}") (($ $ (|Segment| (|DoubleFloat|))) "\\spad{refine(x,r)} \\undocumented")) (|zoom| (($ $ (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{zoom(x,r,s)} \\undocumented") (($ $ (|Segment| (|DoubleFloat|))) "\\spad{zoom(x,r)} \\undocumented")) (|parametric?| (((|Boolean|) $) "\\spad{parametric? determines} whether it is a parametric plot?")) (|plotPolar| (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) "\\spad{plotPolar(f)} plots the polar curve \\spad{r = f(theta)} as theta ranges over the interval \\spad{[0,2*\\%pi]}; this is the same as the parametric curve \\spad{x = f(t) * cos(t)},{} \\spad{y = f(t) * sin(t)}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plotPolar(f,a..b)} plots the polar curve \\spad{r = f(theta)} as theta ranges over the interval \\spad{[a,b]}; this is the same as the parametric curve \\spad{x = f(t) * cos(t)},{} \\spad{y = f(t) * sin(t)}.")) (|pointPlot| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(t +-> (f(t),g(t)),a..b,c..d,e..f)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}; \\spad{x}-range of \\spad{[c,d]} and \\spad{y}-range of \\spad{[e,f]} are noted in Plot object.") (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(t +-> (f(t),g(t)),a..b)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}.")) (|plot| (($ $ (|Segment| (|DoubleFloat|))) "\\spad{plot(x,r)} \\undocumented") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,a..b,c..d,e..f)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}; \\spad{x}-range of \\spad{[c,d]} and \\spad{y}-range of \\spad{[e,f]} are noted in Plot object.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,a..b)} plots the parametric curve \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over the interval \\spad{[a,b]}.") (($ (|List| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot([f1,...,fm],a..b,c..d)} plots the functions \\spad{y = f1(x)},{}...,{} \\spad{y = fm(x)} on the interval \\spad{a..b}; \\spad{y}-range of \\spad{[c,d]} is noted in Plot object.") (($ (|List| (|Mapping| (|DoubleFloat|) (|DoubleFloat|))) (|Segment| (|DoubleFloat|))) "\\spad{plot([f1,...,fm],a..b)} plots the functions \\spad{y = f1(x)},{}...,{} \\spad{y = fm(x)} on the interval \\spad{a..b}.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,a..b,c..d)} plots the function \\spad{f(x)} on the interval \\spad{[a,b]}; \\spad{y}-range of \\spad{[c,d]} is noted in Plot object.") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,a..b)} plots the function \\spad{f(x)} on the interval \\spad{[a,b]}."))) NIL NIL -(-832 S) +(-837 S) ((|constructor| (NIL "\\spad{PlotFunctions1} provides facilities for plotting curves where functions SF -> SF are specified by giving an expression")) (|plotPolar| (((|Plot|) |#1| (|Symbol|)) "\\spad{plotPolar(f,theta)} plots the graph of \\spad{r = f(theta)} as \\spad{theta} ranges from 0 to 2 \\spad{pi}") (((|Plot|) |#1| (|Symbol|) (|Segment| (|DoubleFloat|))) "\\spad{plotPolar(f,theta,seg)} plots the graph of \\spad{r = f(theta)} as \\spad{theta} ranges over an interval")) (|plot| (((|Plot|) |#1| |#1| (|Symbol|) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,t,seg)} plots the graph of \\spad{x = f(t)},{} \\spad{y = g(t)} as \\spad{t} ranges over an interval.") (((|Plot|) |#1| (|Symbol|) (|Segment| (|DoubleFloat|))) "\\spad{plot(fcn,x,seg)} plots the graph of \\spad{y = f(x)} on a interval"))) NIL NIL -(-833) +(-838) ((|constructor| (NIL "Plot3D supports parametric plots defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example,{} floating point numbers and infinite continued fractions are real number systems. The facilities at this point are limited to 3-dimensional parametric plots.")) (|debug3D| (((|Boolean|) (|Boolean|)) "\\spad{debug3D(true)} turns debug mode on; debug3D(\\spad{false}) turns debug mode off.")) (|numFunEvals3D| (((|Integer|)) "\\spad{numFunEvals3D()} returns the number of points computed.")) (|setAdaptive3D| (((|Boolean|) (|Boolean|)) "\\spad{setAdaptive3D(true)} turns adaptive plotting on; setAdaptive3D(\\spad{false}) turns adaptive plotting off.")) (|adaptive3D?| (((|Boolean|)) "\\spad{adaptive3D?()} determines whether plotting be done adaptively.")) (|setScreenResolution3D| (((|Integer|) (|Integer|)) "\\spad{setScreenResolution3D(i)} sets the screen resolution for a 3d graph to \\spad{i}.")) (|screenResolution3D| (((|Integer|)) "\\spad{screenResolution3D()} returns the screen resolution for a 3d graph.")) (|setMaxPoints3D| (((|Integer|) (|Integer|)) "\\spad{setMaxPoints3D(i)} sets the maximum number of points in a plot to \\spad{i}.")) (|maxPoints3D| (((|Integer|)) "\\spad{maxPoints3D()} returns the maximum number of points in a plot.")) (|setMinPoints3D| (((|Integer|) (|Integer|)) "\\spad{setMinPoints3D(i)} sets the minimum number of points in a plot to \\spad{i}.")) (|minPoints3D| (((|Integer|)) "\\spad{minPoints3D()} returns the minimum number of points in a plot.")) (|tValues| (((|List| (|List| (|DoubleFloat|))) $) "\\spad{tValues(p)} returns a list of lists of the values of the parameter for which a point is computed,{} one list for each curve in the plot \\spad{p}.")) (|tRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{tRange(p)} returns the range of the parameter in a parametric plot \\spad{p}.")) (|refine| (($ $) "\\spad{refine(x)} \\undocumented") (($ $ (|Segment| (|DoubleFloat|))) "\\spad{refine(x,r)} \\undocumented")) (|zoom| (($ $ (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{zoom(x,r,s,t)} \\undocumented")) (|plot| (($ $ (|Segment| (|DoubleFloat|))) "\\spad{plot(x,r)} \\undocumented") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f1,f2,f3,f4,x,y,z,w)} \\undocumented") (($ (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Mapping| (|DoubleFloat|) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{plot(f,g,h,a..b)} plots {/emx = \\spad{f}(\\spad{t}),{} \\spad{y} = \\spad{g}(\\spad{t}),{} \\spad{z} = \\spad{h}(\\spad{t})} as \\spad{t} ranges over {/em[a,{}\\spad{b}]}.")) (|pointPlot| (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(f,x,y,z,w)} \\undocumented") (($ (|Mapping| (|Point| (|DoubleFloat|)) (|DoubleFloat|)) (|Segment| (|DoubleFloat|))) "\\spad{pointPlot(f,g,h,a..b)} plots {/emx = \\spad{f}(\\spad{t}),{} \\spad{y} = \\spad{g}(\\spad{t}),{} \\spad{z} = \\spad{h}(\\spad{t})} as \\spad{t} ranges over {/em[a,{}\\spad{b}]}."))) NIL NIL -(-834) +(-839) ((|constructor| (NIL "This package exports plotting tools")) (|calcRanges| (((|List| (|Segment| (|DoubleFloat|))) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{calcRanges(l)} \\undocumented"))) NIL NIL -(-835) +(-840) ((|constructor| (NIL "Attaching assertions to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|multiple| (((|Expression| (|Integer|)) (|Symbol|)) "\\spad{multiple(x)} tells the pattern matcher that \\spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\\spad{x}) tells the pattern matcher that \\spad{x} should match a list instead of an element of a list.")) (|optional| (((|Expression| (|Integer|)) (|Symbol|)) "\\spad{optional(x)} tells the pattern matcher that \\spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation)..")) (|constant| (((|Expression| (|Integer|)) (|Symbol|)) "\\spad{constant(x)} tells the pattern matcher that \\spad{x} should match only the symbol 'x and no other quantity.")) (|assert| (((|Expression| (|Integer|)) (|Symbol|) (|Identifier|)) "\\spad{assert(x, s)} makes the assertion \\spad{s} about \\spad{x}."))) NIL NIL -(-836 R -3078) +(-841 R -3088) ((|constructor| (NIL "Attaching assertions to symbols for pattern matching; Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|multiple| ((|#2| |#2|) "\\spad{multiple(x)} tells the pattern matcher that \\spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\\spad{x}) tells the pattern matcher that \\spad{x} should match a list instead of an element of a list. Error: if \\spad{x} is not a symbol.")) (|optional| ((|#2| |#2|) "\\spad{optional(x)} tells the pattern matcher that \\spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation). Error: if \\spad{x} is not a symbol.")) (|constant| ((|#2| |#2|) "\\spad{constant(x)} tells the pattern matcher that \\spad{x} should match only the symbol 'x and no other quantity. Error: if \\spad{x} is not a symbol.")) (|assert| ((|#2| |#2| (|Identifier|)) "\\spad{assert(x, s)} makes the assertion \\spad{s} about \\spad{x}. Error: if \\spad{x} is not a symbol."))) NIL NIL -(-837 S A B) +(-842 S A B) ((|constructor| (NIL "This packages provides tools for matching recursively in type towers.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#2| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches. Note: this function handles type towers by changing the predicates and calling the matching function provided by \\spad{A}.")) (|fixPredicate| (((|Mapping| (|Boolean|) |#2|) (|Mapping| (|Boolean|) |#3|)) "\\spad{fixPredicate(f)} returns \\spad{g} defined by \\spad{g}(a) = \\spad{f}(a::B)."))) NIL NIL -(-838 S R -3078) +(-843 S R -3088) ((|constructor| (NIL "This package provides pattern matching functions on function spaces.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches."))) NIL NIL -(-839 I) +(-844 I) ((|constructor| (NIL "This package provides pattern matching functions on integers.")) (|patternMatch| (((|PatternMatchResult| (|Integer|) |#1|) |#1| (|Pattern| (|Integer|)) (|PatternMatchResult| (|Integer|) |#1|)) "\\spad{patternMatch(n, pat, res)} matches the pattern \\spad{pat} to the integer \\spad{n}; res contains the variables of \\spad{pat} which are already matched and their matches."))) NIL NIL -(-840 S E) +(-845 S E) ((|constructor| (NIL "This package provides pattern matching functions on kernels.")) (|patternMatch| (((|PatternMatchResult| |#1| |#2|) (|Kernel| |#2|) (|Pattern| |#1|) (|PatternMatchResult| |#1| |#2|)) "\\spad{patternMatch(f(e1,...,en), pat, res)} matches the pattern \\spad{pat} to \\spad{f(e1,...,en)}; res contains the variables of \\spad{pat} which are already matched and their matches."))) NIL NIL -(-841 S R L) +(-846 S R L) ((|constructor| (NIL "This package provides pattern matching functions on lists.")) (|patternMatch| (((|PatternMatchListResult| |#1| |#2| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchListResult| |#1| |#2| |#3|)) "\\spad{patternMatch(l, pat, res)} matches the pattern \\spad{pat} to the list \\spad{l}; res contains the variables of \\spad{pat} which are already matched and their matches."))) NIL NIL -(-842 S E V R P) +(-847 S E V R P) ((|constructor| (NIL "This package provides pattern matching functions on polynomials.")) (|patternMatch| (((|PatternMatchResult| |#1| |#5|) |#5| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|)) "\\spad{patternMatch(p, pat, res)} matches the pattern \\spad{pat} to the polynomial \\spad{p}; res contains the variables of \\spad{pat} which are already matched and their matches.") (((|PatternMatchResult| |#1| |#5|) |#5| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|) (|Mapping| (|PatternMatchResult| |#1| |#5|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#5|))) "\\spad{patternMatch(p, pat, res, vmatch)} matches the pattern \\spad{pat} to the polynomial \\spad{p}. \\spad{res} contains the variables of \\spad{pat} which are already matched and their matches; vmatch is the matching function to use on the variables."))) NIL -((|HasCategory| |#3| (|%list| (QUOTE -791) (|devaluate| |#1|)))) -(-843 -2655) +((|HasCategory| |#3| (|%list| (QUOTE -796) (|devaluate| |#1|)))) +(-848 -2665) ((|constructor| (NIL "Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|suchThat| (((|Expression| (|Integer|)) (|Symbol|) (|List| (|Mapping| (|Boolean|) |#1|))) "\\spad{suchThat(x, [f1, f2, ..., fn])} attaches the predicate \\spad{f1} and \\spad{f2} and ... and fn to \\spad{x}.") (((|Expression| (|Integer|)) (|Symbol|) (|Mapping| (|Boolean|) |#1|)) "\\spad{suchThat(x, foo)} attaches the predicate foo to \\spad{x}."))) NIL NIL -(-844 R -3078 -2655) +(-849 R -3088 -2665) ((|constructor| (NIL "Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990")) (|suchThat| ((|#2| |#2| (|List| (|Mapping| (|Boolean|) |#3|))) "\\spad{suchThat(x, [f1, f2, ..., fn])} attaches the predicate \\spad{f1} and \\spad{f2} and ... and fn to \\spad{x}. Error: if \\spad{x} is not a symbol.") ((|#2| |#2| (|Mapping| (|Boolean|) |#3|)) "\\spad{suchThat(x, foo)} attaches the predicate foo to \\spad{x}; error if \\spad{x} is not a symbol."))) NIL NIL -(-845 S R Q) +(-850 S R Q) ((|constructor| (NIL "This package provides pattern matching functions on quotients.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|)) "\\spad{patternMatch(a/b, pat, res)} matches the pattern \\spad{pat} to the quotient \\spad{a/b}; res contains the variables of \\spad{pat} which are already matched and their matches."))) NIL NIL -(-846 S) +(-851 S) ((|constructor| (NIL "This package provides pattern matching functions on symbols.")) (|patternMatch| (((|PatternMatchResult| |#1| (|Symbol|)) (|Symbol|) (|Pattern| |#1|) (|PatternMatchResult| |#1| (|Symbol|))) "\\spad{patternMatch(expr, pat, res)} matches the pattern \\spad{pat} to the expression \\spad{expr}; res contains the variables of \\spad{pat} which are already matched and their matches (necessary for recursion)."))) NIL NIL -(-847 S R P) +(-852 S R P) ((|constructor| (NIL "This package provides tools for the pattern matcher.")) (|patternMatchTimes| (((|PatternMatchResult| |#1| |#3|) (|List| |#3|) (|List| (|Pattern| |#1|)) (|PatternMatchResult| |#1| |#3|) (|Mapping| (|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|))) "\\spad{patternMatchTimes(lsubj, lpat, res, match)} matches the product of patterns \\spad{reduce(*,lpat)} to the product of subjects \\spad{reduce(*,lsubj)}; \\spad{r} contains the previous matches and match is a pattern-matching function on \\spad{P}.")) (|patternMatch| (((|PatternMatchResult| |#1| |#3|) (|List| |#3|) (|List| (|Pattern| |#1|)) (|Mapping| |#3| (|List| |#3|)) (|PatternMatchResult| |#1| |#3|) (|Mapping| (|PatternMatchResult| |#1| |#3|) |#3| (|Pattern| |#1|) (|PatternMatchResult| |#1| |#3|))) "\\spad{patternMatch(lsubj, lpat, op, res, match)} matches the list of patterns \\spad{lpat} to the list of subjects \\spad{lsubj},{} allowing for commutativity; \\spad{op} is the operator such that \\spad{op}(\\spad{lpat}) should match \\spad{op}(\\spad{lsubj}) at the end,{} \\spad{r} contains the previous matches,{} and match is a pattern-matching function on \\spad{P}."))) NIL NIL -(-848) +(-853) ((|constructor| (NIL "This package provides various polynomial number theoretic functions over the integers.")) (|legendre| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{legendre(n)} returns the \\spad{n}th Legendre polynomial \\spad{P[n](x)}. Note: Legendre polynomials,{} denoted \\spad{P[n](x)},{} are computed from the two term recurrence. The generating function is: \\spad{1/sqrt(1-2*t*x+t**2) = sum(P[n](x)*t**n, n=0..infinity)}.")) (|laguerre| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{laguerre(n)} returns the \\spad{n}th Laguerre polynomial \\spad{L[n](x)}. Note: Laguerre polynomials,{} denoted \\spad{L[n](x)},{} are computed from the two term recurrence. The generating function is: \\spad{exp(x*t/(t-1))/(1-t) = sum(L[n](x)*t**n/n!, n=0..infinity)}.")) (|hermite| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{hermite(n)} returns the \\spad{n}th Hermite polynomial \\spad{H[n](x)}. Note: Hermite polynomials,{} denoted \\spad{H[n](x)},{} are computed from the two term recurrence. The generating function is: \\spad{exp(2*t*x-t**2) = sum(H[n](x)*t**n/n!, n=0..infinity)}.")) (|fixedDivisor| (((|Integer|) (|SparseUnivariatePolynomial| (|Integer|))) "\\spad{fixedDivisor(a)} for \\spad{a(x)} in \\spad{Z[x]} is the largest integer \\spad{f} such that \\spad{f} divides \\spad{a(x=k)} for all integers \\spad{k}. Note: fixed divisor of \\spad{a} is \\spad{reduce(gcd,[a(x=k) for k in 0..degree(a)])}.")) (|euler| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{euler(n)} returns the \\spad{n}th Euler polynomial \\spad{E[n](x)}. Note: Euler polynomials denoted \\spad{E(n,x)} computed by solving the differential equation \\spad{differentiate(E(n,x),x) = n E(n-1,x)} where \\spad{E(0,x) = 1} and initial condition comes from \\spad{E(n) = 2**n E(n,1/2)}.")) (|cyclotomic| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{cyclotomic(n)} returns the \\spad{n}th cyclotomic polynomial \\spad{phi[n](x)}. Note: \\spad{phi[n](x)} is the factor of \\spad{x**n - 1} whose roots are the primitive \\spad{n}th roots of unity.")) (|chebyshevU| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{chebyshevU(n)} returns the \\spad{n}th Chebyshev polynomial \\spad{U[n](x)}. Note: Chebyshev polynomials of the second kind,{} denoted \\spad{U[n](x)},{} computed from the two term recurrence. The generating function \\spad{1/(1-2*t*x+t**2) = sum(T[n](x)*t**n, n=0..infinity)}.")) (|chebyshevT| (((|SparseUnivariatePolynomial| (|Integer|)) (|Integer|)) "\\spad{chebyshevT(n)} returns the \\spad{n}th Chebyshev polynomial \\spad{T[n](x)}. Note: Chebyshev polynomials of the first kind,{} denoted \\spad{T[n](x)},{} computed from the two term recurrence. The generating function \\spad{(1-t*x)/(1-2*t*x+t**2) = sum(T[n](x)*t**n, n=0..infinity)}.")) (|bernoulli| (((|SparseUnivariatePolynomial| (|Fraction| (|Integer|))) (|Integer|)) "\\spad{bernoulli(n)} returns the \\spad{n}th Bernoulli polynomial \\spad{B[n](x)}. Note: Bernoulli polynomials denoted \\spad{B(n,x)} computed by solving the differential equation \\spad{differentiate(B(n,x),x) = n B(n-1,x)} where \\spad{B(0,x) = 1} and initial condition comes from \\spad{B(n) = B(n,0)}."))) NIL NIL -(-849 R) +(-854 R) ((|constructor| (NIL "This domain implements points in coordinate space"))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#1| (QUOTE (-956))) (-12 (|HasCategory| |#1| (QUOTE (-910))) (|HasCategory| |#1| (QUOTE (-956)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-850 |lv| R) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#1| (QUOTE (-961))) (-12 (|HasCategory| |#1| (QUOTE (-915))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-855 |lv| R) ((|constructor| (NIL "Package with the conversion functions among different kind of polynomials")) (|pToDmp| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|Polynomial| |#2|)) "\\spad{pToDmp(p)} converts \\spad{p} from a \\spadtype{POLY} to a \\spadtype{DMP}.")) (|dmpToP| (((|Polynomial| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{dmpToP(p)} converts \\spad{p} from a \\spadtype{DMP} to a \\spadtype{POLY}.")) (|hdmpToP| (((|Polynomial| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{hdmpToP(p)} converts \\spad{p} from a \\spadtype{HDMP} to a \\spadtype{POLY}.")) (|pToHdmp| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|Polynomial| |#2|)) "\\spad{pToHdmp(p)} converts \\spad{p} from a \\spadtype{POLY} to a \\spadtype{HDMP}.")) (|hdmpToDmp| (((|DistributedMultivariatePolynomial| |#1| |#2|) (|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{hdmpToDmp(p)} converts \\spad{p} from a \\spadtype{HDMP} to a \\spadtype{DMP}.")) (|dmpToHdmp| (((|HomogeneousDistributedMultivariatePolynomial| |#1| |#2|) (|DistributedMultivariatePolynomial| |#1| |#2|)) "\\spad{dmpToHdmp(p)} converts \\spad{p} from a \\spadtype{DMP} to a \\spadtype{HDMP}."))) NIL NIL -(-851 |TheField| |ThePols|) +(-856 |TheField| |ThePols|) ((|constructor| (NIL "\\axiomType{RealPolynomialUtilitiesPackage} provides common functions used by interval coding.")) (|lazyVariations| (((|NonNegativeInteger|) (|List| |#1|) (|Integer|) (|Integer|)) "\\axiom{lazyVariations(\\spad{l},{}\\spad{s1},{}sn)} is the number of sign variations in the list of non null numbers [s1::l]@sn,{}")) (|sturmVariationsOf| (((|NonNegativeInteger|) (|List| |#1|)) "\\axiom{sturmVariationsOf(\\spad{l})} is the number of sign variations in the list of numbers \\spad{l},{} note that the first term counts as a sign")) (|boundOfCauchy| ((|#1| |#2|) "\\axiom{boundOfCauchy(\\spad{p})} bounds the roots of \\spad{p}")) (|sturmSequence| (((|List| |#2|) |#2|) "\\axiom{sturmSequence(\\spad{p}) = sylvesterSequence(\\spad{p},{}p')}")) (|sylvesterSequence| (((|List| |#2|) |#2| |#2|) "\\axiom{sylvesterSequence(\\spad{p},{}\\spad{q})} is the negated remainder sequence of \\spad{p} and \\spad{q} divided by the last computed term"))) NIL -((|HasCategory| |#1| (QUOTE (-750)))) -(-852 R) +((|HasCategory| |#1| (QUOTE (-755)))) +(-857 R) ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are arbitrary symbols. The ordering is alphabetic determined by the Symbol type. The coefficient ring may be non commutative,{} but the variables are assumed to commute.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(p,x)} computes the integral of \\spad{p*dx},{} \\spadignore{i.e.} integrates the polynomial \\spad{p} with respect to the variable \\spad{x}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-816)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| (-1081) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| (-1081) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-1081) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-1081) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| (-1081) (QUOTE (-550 (-469))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-309))) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-853 R S) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| (-1088) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| (-1088) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-1088) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-1088) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| (-1088) (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-311))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-858 R S) ((|constructor| (NIL "\\indented{2}{This package takes a mapping between coefficient rings,{} and lifts} it to a mapping between polynomials over those rings.")) (|map| (((|Polynomial| |#2|) (|Mapping| |#2| |#1|) (|Polynomial| |#1|)) "\\spad{map(f, p)} produces a new polynomial as a result of applying the function \\spad{f} to every coefficient of the polynomial \\spad{p}."))) NIL NIL -(-854 |x| R) +(-859 |x| R) ((|constructor| (NIL "This package is primarily to help the interpreter do coercions. It allows you to view a polynomial as a univariate polynomial in one of its variables with coefficients which are again a polynomial in all the other variables.")) (|univariate| (((|UnivariatePolynomial| |#1| (|Polynomial| |#2|)) (|Polynomial| |#2|) (|Variable| |#1|)) "\\spad{univariate(p, x)} converts the polynomial \\spad{p} to a one of type \\spad{UnivariatePolynomial(x,Polynomial(R))},{} ie. as a member of \\spad{R[...][x]}."))) NIL NIL -(-855 S R E |VarSet|) +(-860 S R E |VarSet|) ((|constructor| (NIL "The category for general multi-variate polynomials over a ring \\spad{R},{} in variables from VarSet,{} with exponents from the \\spadtype{OrderedAbelianMonoidSup}.")) (|canonicalUnitNormal| ((|attribute|) "we can choose a unique representative for each associate class. This normalization is chosen to be normalization of leading coefficient (by default).")) (|squareFreePart| (($ $) "\\spad{squareFreePart(p)} returns product of all the irreducible factors of polynomial \\spad{p} each taken with multiplicity one.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(p)} returns the square free factorization of the polynomial \\spad{p}.")) (|primitivePart| (($ $ |#4|) "\\spad{primitivePart(p,v)} returns the unitCanonical associate of the polynomial \\spad{p} with its content with respect to the variable \\spad{v} divided out.") (($ $) "\\spad{primitivePart(p)} returns the unitCanonical associate of the polynomial \\spad{p} with its content divided out.")) (|content| (($ $ |#4|) "\\spad{content(p,v)} is the gcd of the coefficients of the polynomial \\spad{p} when \\spad{p} is viewed as a univariate polynomial with respect to the variable \\spad{v}. Thus,{} for polynomial 7*x**2*y + 14*x*y**2,{} the gcd of the coefficients with respect to \\spad{x} is 7*y.")) (|discriminant| (($ $ |#4|) "\\spad{discriminant(p,v)} returns the disriminant of the polynomial \\spad{p} with respect to the variable \\spad{v}.")) (|resultant| (($ $ $ |#4|) "\\spad{resultant(p,q,v)} returns the resultant of the polynomials \\spad{p} and \\spad{q} with respect to the variable \\spad{v}.")) (|primitiveMonomials| (((|List| $) $) "\\spad{primitiveMonomials(p)} gives the list of monomials of the polynomial \\spad{p} with their coefficients removed. Note: \\spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),...,X^(n)]}.")) (|variables| (((|List| |#4|) $) "\\spad{variables(p)} returns the list of those variables actually appearing in the polynomial \\spad{p}.")) (|totalDegree| (((|NonNegativeInteger|) $ (|List| |#4|)) "\\spad{totalDegree(p, lv)} returns the maximum sum (over all monomials of polynomial \\spad{p}) of the variables in the list lv.") (((|NonNegativeInteger|) $) "\\spad{totalDegree(p)} returns the largest sum over all monomials of all exponents of a monomial.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#4|) (|:| |exponent| (|NonNegativeInteger|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if polynomial \\spad{p} has the form \\spad{x**n} and \\spad{n > 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if polynomial \\spad{p = a1 ... an} and \\spad{n >= 2},{} and,{} for each \\spad{i},{} \\spad{ai} is either a nontrivial constant in \\spad{R} or else of the form \\spad{x**e},{} where \\spad{e > 0} is an integer and \\spad{x} in a member of VarSet.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if polynomial \\spad{p = m1 + ... + mn} and \\spad{n >= 2} and each \\spad{mi} is a nonzero monomial.")) (|multivariate| (($ (|SparseUnivariatePolynomial| $) |#4|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.") (($ (|SparseUnivariatePolynomial| |#2|) |#4|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.")) (|monomial| (($ $ (|List| |#4|) (|List| (|NonNegativeInteger|))) "\\spad{monomial(a,[v1..vn],[e1..en])} returns \\spad{a*prod(vi**ei)}.") (($ $ |#4| (|NonNegativeInteger|)) "\\spad{monomial(a,x,n)} creates the monomial \\spad{a*x**n} where \\spad{a} is a polynomial,{} \\spad{x} is a variable and \\spad{n} is a nonnegative integer.")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $ |#4|) "\\spad{monicDivide(a,b,v)} divides the polynomial a by the polynomial \\spad{b},{} with each viewed as a univariate polynomial in \\spad{v} returning both the quotient and remainder. Error: if \\spad{b} is not monic with respect to \\spad{v}.")) (|minimumDegree| (((|List| (|NonNegativeInteger|)) $ (|List| |#4|)) "\\spad{minimumDegree(p, lv)} gives the list of minimum degrees of the polynomial \\spad{p} with respect to each of the variables in the list lv") (((|NonNegativeInteger|) $ |#4|) "\\spad{minimumDegree(p,v)} gives the minimum degree of polynomial \\spad{p} with respect to \\spad{v},{} \\spadignore{i.e.} viewed a univariate polynomial in \\spad{v}")) (|mainVariable| (((|Union| |#4| "failed") $) "\\spad{mainVariable(p)} returns the biggest variable which actually occurs in the polynomial \\spad{p},{} or \"failed\" if no variables are present. fails precisely if polynomial satisfies ground?")) (|univariate| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{univariate(p)} converts the multivariate polynomial \\spad{p},{} which should actually involve only one variable,{} into a univariate polynomial in that variable,{} whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate") (((|SparseUnivariatePolynomial| $) $ |#4|) "\\spad{univariate(p,v)} converts the multivariate polynomial \\spad{p} into a univariate polynomial in \\spad{v},{} whose coefficients are still multivariate polynomials (in all the other variables).")) (|monomials| (((|List| $) $) "\\spad{monomials(p)} returns the list of non-zero monomials of polynomial \\spad{p},{} \\spadignore{i.e.} \\spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}.")) (|coefficient| (($ $ (|List| |#4|) (|List| (|NonNegativeInteger|))) "\\spad{coefficient(p, lv, ln)} views the polynomial \\spad{p} as a polynomial in the variables of \\spad{lv} and returns the coefficient of the term \\spad{lv**ln},{} \\spadignore{i.e.} \\spad{prod(lv_i ** ln_i)}.") (($ $ |#4| (|NonNegativeInteger|)) "\\spad{coefficient(p,v,n)} views the polynomial \\spad{p} as a univariate polynomial in \\spad{v} and returns the coefficient of the \\spad{v**n} term.")) (|degree| (((|List| (|NonNegativeInteger|)) $ (|List| |#4|)) "\\spad{degree(p,lv)} gives the list of degrees of polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $ |#4|) "\\spad{degree(p,v)} gives the degree of polynomial \\spad{p} with respect to the variable \\spad{v}."))) NIL -((|HasCategory| |#2| (QUOTE (-816))) (|HasAttribute| |#2| (QUOTE -3976)) (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#4| (QUOTE (-791 (-325)))) (|HasCategory| |#2| (QUOTE (-791 (-325)))) (|HasCategory| |#4| (QUOTE (-791 (-480)))) (|HasCategory| |#2| (QUOTE (-791 (-480)))) (|HasCategory| |#4| (QUOTE (-550 (-795 (-325))))) (|HasCategory| |#2| (QUOTE (-550 (-795 (-325))))) (|HasCategory| |#4| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#2| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#4| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-550 (-469))))) -(-856 R E |VarSet|) +((|HasCategory| |#2| (QUOTE (-821))) (|HasAttribute| |#2| (QUOTE -3987)) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#4| (QUOTE (-796 (-327)))) (|HasCategory| |#2| (QUOTE (-796 (-327)))) (|HasCategory| |#4| (QUOTE (-796 (-483)))) (|HasCategory| |#2| (QUOTE (-796 (-483)))) (|HasCategory| |#4| (QUOTE (-553 (-800 (-327))))) (|HasCategory| |#2| (QUOTE (-553 (-800 (-327))))) (|HasCategory| |#4| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#2| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#4| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-553 (-472))))) +(-861 R E |VarSet|) ((|constructor| (NIL "The category for general multi-variate polynomials over a ring \\spad{R},{} in variables from VarSet,{} with exponents from the \\spadtype{OrderedAbelianMonoidSup}.")) (|canonicalUnitNormal| ((|attribute|) "we can choose a unique representative for each associate class. This normalization is chosen to be normalization of leading coefficient (by default).")) (|squareFreePart| (($ $) "\\spad{squareFreePart(p)} returns product of all the irreducible factors of polynomial \\spad{p} each taken with multiplicity one.")) (|squareFree| (((|Factored| $) $) "\\spad{squareFree(p)} returns the square free factorization of the polynomial \\spad{p}.")) (|primitivePart| (($ $ |#3|) "\\spad{primitivePart(p,v)} returns the unitCanonical associate of the polynomial \\spad{p} with its content with respect to the variable \\spad{v} divided out.") (($ $) "\\spad{primitivePart(p)} returns the unitCanonical associate of the polynomial \\spad{p} with its content divided out.")) (|content| (($ $ |#3|) "\\spad{content(p,v)} is the gcd of the coefficients of the polynomial \\spad{p} when \\spad{p} is viewed as a univariate polynomial with respect to the variable \\spad{v}. Thus,{} for polynomial 7*x**2*y + 14*x*y**2,{} the gcd of the coefficients with respect to \\spad{x} is 7*y.")) (|discriminant| (($ $ |#3|) "\\spad{discriminant(p,v)} returns the disriminant of the polynomial \\spad{p} with respect to the variable \\spad{v}.")) (|resultant| (($ $ $ |#3|) "\\spad{resultant(p,q,v)} returns the resultant of the polynomials \\spad{p} and \\spad{q} with respect to the variable \\spad{v}.")) (|primitiveMonomials| (((|List| $) $) "\\spad{primitiveMonomials(p)} gives the list of monomials of the polynomial \\spad{p} with their coefficients removed. Note: \\spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),...,X^(n)]}.")) (|variables| (((|List| |#3|) $) "\\spad{variables(p)} returns the list of those variables actually appearing in the polynomial \\spad{p}.")) (|totalDegree| (((|NonNegativeInteger|) $ (|List| |#3|)) "\\spad{totalDegree(p, lv)} returns the maximum sum (over all monomials of polynomial \\spad{p}) of the variables in the list lv.") (((|NonNegativeInteger|) $) "\\spad{totalDegree(p)} returns the largest sum over all monomials of all exponents of a monomial.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#3|) (|:| |exponent| (|NonNegativeInteger|))) "failed") $) "\\spad{isExpt(p)} returns \\spad{[x, n]} if polynomial \\spad{p} has the form \\spad{x**n} and \\spad{n > 0}.")) (|isTimes| (((|Union| (|List| $) "failed") $) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if polynomial \\spad{p = a1 ... an} and \\spad{n >= 2},{} and,{} for each \\spad{i},{} \\spad{ai} is either a nontrivial constant in \\spad{R} or else of the form \\spad{x**e},{} where \\spad{e > 0} is an integer and \\spad{x} in a member of VarSet.")) (|isPlus| (((|Union| (|List| $) "failed") $) "\\spad{isPlus(p)} returns \\spad{[m1,...,mn]} if polynomial \\spad{p = m1 + ... + mn} and \\spad{n >= 2} and each \\spad{mi} is a nonzero monomial.")) (|multivariate| (($ (|SparseUnivariatePolynomial| $) |#3|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.") (($ (|SparseUnivariatePolynomial| |#1|) |#3|) "\\spad{multivariate(sup,v)} converts an anonymous univariable polynomial \\spad{sup} to a polynomial in the variable \\spad{v}.")) (|monomial| (($ $ (|List| |#3|) (|List| (|NonNegativeInteger|))) "\\spad{monomial(a,[v1..vn],[e1..en])} returns \\spad{a*prod(vi**ei)}.") (($ $ |#3| (|NonNegativeInteger|)) "\\spad{monomial(a,x,n)} creates the monomial \\spad{a*x**n} where \\spad{a} is a polynomial,{} \\spad{x} is a variable and \\spad{n} is a nonnegative integer.")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $ |#3|) "\\spad{monicDivide(a,b,v)} divides the polynomial a by the polynomial \\spad{b},{} with each viewed as a univariate polynomial in \\spad{v} returning both the quotient and remainder. Error: if \\spad{b} is not monic with respect to \\spad{v}.")) (|minimumDegree| (((|List| (|NonNegativeInteger|)) $ (|List| |#3|)) "\\spad{minimumDegree(p, lv)} gives the list of minimum degrees of the polynomial \\spad{p} with respect to each of the variables in the list lv") (((|NonNegativeInteger|) $ |#3|) "\\spad{minimumDegree(p,v)} gives the minimum degree of polynomial \\spad{p} with respect to \\spad{v},{} \\spadignore{i.e.} viewed a univariate polynomial in \\spad{v}")) (|mainVariable| (((|Union| |#3| "failed") $) "\\spad{mainVariable(p)} returns the biggest variable which actually occurs in the polynomial \\spad{p},{} or \"failed\" if no variables are present. fails precisely if polynomial satisfies ground?")) (|univariate| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{univariate(p)} converts the multivariate polynomial \\spad{p},{} which should actually involve only one variable,{} into a univariate polynomial in that variable,{} whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate") (((|SparseUnivariatePolynomial| $) $ |#3|) "\\spad{univariate(p,v)} converts the multivariate polynomial \\spad{p} into a univariate polynomial in \\spad{v},{} whose coefficients are still multivariate polynomials (in all the other variables).")) (|monomials| (((|List| $) $) "\\spad{monomials(p)} returns the list of non-zero monomials of polynomial \\spad{p},{} \\spadignore{i.e.} \\spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),...,a_(n) X^(n)]}.")) (|coefficient| (($ $ (|List| |#3|) (|List| (|NonNegativeInteger|))) "\\spad{coefficient(p, lv, ln)} views the polynomial \\spad{p} as a polynomial in the variables of \\spad{lv} and returns the coefficient of the term \\spad{lv**ln},{} \\spadignore{i.e.} \\spad{prod(lv_i ** ln_i)}.") (($ $ |#3| (|NonNegativeInteger|)) "\\spad{coefficient(p,v,n)} views the polynomial \\spad{p} as a univariate polynomial in \\spad{v} and returns the coefficient of the \\spad{v**n} term.")) (|degree| (((|List| (|NonNegativeInteger|)) $ (|List| |#3|)) "\\spad{degree(p,lv)} gives the list of degrees of polynomial \\spad{p} with respect to each of the variables in the list \\spad{lv}.") (((|NonNegativeInteger|) $ |#3|) "\\spad{degree(p,v)} gives the degree of polynomial \\spad{p} with respect to the variable \\spad{v}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-857 E V R P -3078) +(-862 E V R P -3088) ((|constructor| (NIL "This package transforms multivariate polynomials or fractions into univariate polynomials or fractions,{} and back.")) (|isPower| (((|Union| (|Record| (|:| |val| |#5|) (|:| |exponent| (|Integer|))) "failed") |#5|) "\\spad{isPower(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0},{} \"failed\" otherwise.")) (|isExpt| (((|Union| (|Record| (|:| |var| |#2|) (|:| |exponent| (|Integer|))) "failed") |#5|) "\\spad{isExpt(p)} returns \\spad{[x, n]} if \\spad{p = x**n} and \\spad{n <> 0},{} \"failed\" otherwise.")) (|isTimes| (((|Union| (|List| |#5|) "failed") |#5|) "\\spad{isTimes(p)} returns \\spad{[a1,...,an]} if \\spad{p = a1 ... an} and \\spad{n > 1},{} \"failed\" otherwise.")) (|isPlus| (((|Union| (|List| |#5|) "failed") |#5|) "\\spad{isPlus(p)} returns [\\spad{m1},{}...,{}mn] if \\spad{p = m1 + ... + mn} and \\spad{n > 1},{} \"failed\" otherwise.")) (|multivariate| ((|#5| (|Fraction| (|SparseUnivariatePolynomial| |#5|)) |#2|) "\\spad{multivariate(f, v)} applies both the numerator and denominator of \\spad{f} to \\spad{v}.")) (|univariate| (((|SparseUnivariatePolynomial| |#5|) |#5| |#2| (|SparseUnivariatePolynomial| |#5|)) "\\spad{univariate(f, x, p)} returns \\spad{f} viewed as a univariate polynomial in \\spad{x},{} using the side-condition \\spad{p(x) = 0}.") (((|Fraction| (|SparseUnivariatePolynomial| |#5|)) |#5| |#2|) "\\spad{univariate(f, v)} returns \\spad{f} viewed as a univariate rational function in \\spad{v}.")) (|mainVariable| (((|Union| |#2| "failed") |#5|) "\\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \\spad{f},{} \"failed\" if \\spad{f} has no variables.")) (|variables| (((|List| |#2|) |#5|) "\\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \\spad{f}."))) NIL NIL -(-858 E |Vars| R P S) +(-863 E |Vars| R P S) ((|constructor| (NIL "This package provides a very general map function,{} which given a set \\spad{S} and polynomials over \\spad{R} with maps from the variables into \\spad{S} and the coefficients into \\spad{S},{} maps polynomials into \\spad{S}. \\spad{S} is assumed to support \\spad{+},{} \\spad{*} and \\spad{**}.")) (|map| ((|#5| (|Mapping| |#5| |#2|) (|Mapping| |#5| |#3|) |#4|) "\\spad{map(varmap, coefmap, p)} takes a \\spad{varmap},{} a mapping from the variables of polynomial \\spad{p} into \\spad{S},{} \\spad{coefmap},{} a mapping from coefficients of \\spad{p} into \\spad{S},{} and \\spad{p},{} and produces a member of \\spad{S} using the corresponding arithmetic. in \\spad{S}"))) NIL NIL -(-859 E V R P -3078) +(-864 E V R P -3088) ((|constructor| (NIL "computes \\spad{n}-th roots of quotients of multivariate polynomials")) (|nthr| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#4|) (|:| |radicand| (|List| |#4|))) |#4| (|NonNegativeInteger|)) "\\spad{nthr(p,n)} should be local but conditional")) (|froot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) |#5| (|NonNegativeInteger|)) "\\spad{froot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|qroot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) (|Fraction| (|Integer|)) (|NonNegativeInteger|)) "\\spad{qroot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|rroot| (((|Record| (|:| |exponent| (|NonNegativeInteger|)) (|:| |coef| |#5|) (|:| |radicand| |#5|)) |#3| (|NonNegativeInteger|)) "\\spad{rroot(f, n)} returns \\spad{[m,c,r]} such that \\spad{f**(1/n) = c * r**(1/m)}.")) (|denom| ((|#4| $) "\\spad{denom(x)} \\undocumented")) (|numer| ((|#4| $) "\\spad{numer(x)} \\undocumented"))) NIL -((|HasCategory| |#3| (QUOTE (-387)))) -(-860) +((|HasCategory| |#3| (QUOTE (-389)))) +(-865) ((|constructor| (NIL "This domain represents network port numbers (notable TCP and UDP).")) (|port| (($ (|SingleInteger|)) "\\spad{port(n)} constructs a PortNumber from the integer `n'."))) NIL NIL -(-861) +(-866) ((|constructor| (NIL "PlottablePlaneCurveCategory is the category of curves in the plane which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the \\spad{x}-coordinates and \\spad{y}-coordinates of the points on the curve.")) (|yRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{yRange(c)} returns the range of the \\spad{y}-coordinates of the points on the curve \\spad{c}.")) (|xRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{xRange(c)} returns the range of the \\spad{x}-coordinates of the points on the curve \\spad{c}.")) (|listBranches| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{listBranches(c)} returns a list of lists of points,{} representing the branches of the curve \\spad{c}."))) NIL NIL -(-862 R E) +(-867 R E) ((|constructor| (NIL "This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and terms indexed by their exponents (from an arbitrary ordered abelian monoid). This type is used,{} for example,{} by the \\spadtype{DistributedMultivariatePolynomial} domain where the exponent domain is a direct product of non negative integers.")) (|canonicalUnitNormal| ((|attribute|) "canonicalUnitNormal guarantees that the function unitCanonical returns the same representative for all associates of any particular element.")) (|fmecg| (($ $ |#2| |#1| $) "\\spad{fmecg(p1,e,r,p2)} finds \\spad{X} : \\spad{p1} - \\spad{r} * X**e * \\spad{p2}"))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3972 . T) (-3973 . T) (-3975 . 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T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-104)))) (|HasAttribute| |#1| (QUOTE -3987))) +(-868 R L) ((|constructor| (NIL "\\spadtype{PrecomputedAssociatedEquations} stores some generic precomputations which speed up the computations of the associated equations needed for factoring operators.")) (|firstUncouplingMatrix| (((|Union| (|Matrix| |#1|) "failed") |#2| (|PositiveInteger|)) "\\spad{firstUncouplingMatrix(op, m)} returns the matrix A such that \\spad{A w = (W',W'',...,W^N)} in the corresponding associated equations for right-factors of order \\spad{m} of \\spad{op}. Returns \"failed\" if the matrix A has not been precomputed for the particular combination \\spad{degree(L), m}."))) NIL NIL -(-864 S) +(-869 S) ((|constructor| (NIL "\\indented{1}{This provides a fast array type with no bound checking on elt's.} Minimum index is 0 in this type,{} cannot be changed"))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-865 A B) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-870 A B) ((|constructor| (NIL "\\indented{1}{This package provides tools for operating on primitive arrays} with unary and binary functions involving different underlying types")) (|map| (((|PrimitiveArray| |#2|) (|Mapping| |#2| |#1|) (|PrimitiveArray| |#1|)) "\\spad{map(f,a)} applies function \\spad{f} to each member of primitive array \\spad{a} resulting in a new primitive array over a possibly different underlying domain.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|PrimitiveArray| |#1|) |#2|) "\\spad{reduce(f,a,r)} applies function \\spad{f} to each successive element of the primitive array \\spad{a} and an accumulant initialized to \\spad{r}. For example,{} \\spad{reduce(_+\\$Integer,[1,2,3],0)} does \\spad{3+(2+(1+0))}. Note: third argument \\spad{r} may be regarded as the identity element for the function \\spad{f}.")) (|scan| (((|PrimitiveArray| |#2|) (|Mapping| |#2| |#1| |#2|) (|PrimitiveArray| |#1|) |#2|) "\\spad{scan(f,a,r)} successively applies \\spad{reduce(f,x,r)} to more and more leading sub-arrays \\spad{x} of primitive array \\spad{a}. More precisely,{} if \\spad{a} is \\spad{[a1,a2,...]},{} then \\spad{scan(f,a,r)} returns \\spad{[reduce(f,[a1],r),reduce(f,[a1,a2],r),...]}."))) NIL NIL -(-866) +(-871) ((|constructor| (NIL "Category for the functions defined by integrals.")) (|integral| (($ $ (|SegmentBinding| $)) "\\spad{integral(f, x = a..b)} returns the formal definite integral of \\spad{f} dx for \\spad{x} between \\spad{a} and \\spad{b}.") (($ $ (|Symbol|)) "\\spad{integral(f, x)} returns the formal integral of \\spad{f} dx."))) NIL NIL -(-867 -3078) +(-872 -3088) ((|constructor| (NIL "PrimitiveElement provides functions to compute primitive elements in algebraic extensions.")) (|primitiveElement| (((|Record| (|:| |coef| (|List| (|Integer|))) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#1|))) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|)) (|Symbol|)) "\\spad{primitiveElement([p1,...,pn], [a1,...,an], a)} returns \\spad{[[c1,...,cn], [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} where \\spad{a = a1 c1 + ... + an cn},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The \\spad{pi}'s are the defining polynomials for the \\spad{ai}'s. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}.") (((|Record| (|:| |coef| (|List| (|Integer|))) (|:| |poly| (|List| (|SparseUnivariatePolynomial| |#1|))) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|List| (|Polynomial| |#1|)) (|List| (|Symbol|))) "\\spad{primitiveElement([p1,...,pn], [a1,...,an])} returns \\spad{[[c1,...,cn], [q1,...,qn], q]} such that then \\spad{k(a1,...,an) = k(a)},{} where \\spad{a = a1 c1 + ... + an cn},{} \\spad{ai = qi(a)},{} and \\spad{q(a) = 0}. The \\spad{pi}'s are the defining polynomials for the \\spad{ai}'s. This operation uses the technique of \\spadglossSee{groebner bases}{Groebner basis}.") (((|Record| (|:| |coef1| (|Integer|)) (|:| |coef2| (|Integer|)) (|:| |prim| (|SparseUnivariatePolynomial| |#1|))) (|Polynomial| |#1|) (|Symbol|) (|Polynomial| |#1|) (|Symbol|)) "\\spad{primitiveElement(p1, a1, p2, a2)} returns \\spad{[c1, c2, q]} such that \\spad{k(a1, a2) = k(a)} where \\spad{a = c1 a1 + c2 a2, and q(a) = 0}. The \\spad{pi}'s are the defining polynomials for the \\spad{ai}'s. The \\spad{p2} may involve \\spad{a1},{} but \\spad{p1} must not involve \\spad{a2}. This operation uses \\spadfun{resultant}."))) NIL NIL -(-868 I) +(-873 I) ((|constructor| (NIL "The \\spadtype{IntegerPrimesPackage} implements a modification of Rabin's probabilistic primality test and the utility functions \\spadfun{nextPrime},{} \\spadfun{prevPrime} and \\spadfun{primes}.")) (|primes| (((|List| |#1|) |#1| |#1|) "\\spad{primes(a,b)} returns a list of all primes \\spad{p} with \\spad{a <= p <= b}")) (|prevPrime| ((|#1| |#1|) "\\spad{prevPrime(n)} returns the largest prime strictly smaller than \\spad{n}")) (|nextPrime| ((|#1| |#1|) "\\spad{nextPrime(n)} returns the smallest prime strictly larger than \\spad{n}")) (|prime?| (((|Boolean|) |#1|) "\\spad{prime?(n)} returns \\spad{true} if \\spad{n} is prime and \\spad{false} if not. The algorithm used is Rabin's probabilistic primality test (reference: Knuth Volume 2 Semi Numerical Algorithms). If \\spad{prime? n} returns \\spad{false},{} \\spad{n} is proven composite. If \\spad{prime? n} returns \\spad{true},{} prime? may be in error however,{} the probability of error is very low. and is zero below 25*10**9 (due to a result of Pomerance et al),{} below 10**12 and 10**13 due to results of Pinch,{} and below 341550071728321 due to a result of Jaeschke. Specifically,{} this implementation does at least 10 pseudo prime tests and so the probability of error is \\spad{< 4**(-10)}. The running time of this method is cubic in the length of the input \\spad{n},{} that is \\spad{O( (log n)**3 )},{} for \\spad{n<10**20}. beyond that,{} the algorithm is quartic,{} \\spad{O( (log n)**4 )}. Two improvements due to Davenport have been incorporated which catches some trivial strong pseudo-primes,{} such as [Jaeschke,{} 1991] 1377161253229053 * 413148375987157,{} which the original algorithm regards as prime"))) NIL NIL -(-869) +(-874) ((|constructor| (NIL "PrintPackage provides a print function for output forms.")) (|print| (((|Void|) (|OutputForm|)) "\\spad{print(o)} writes the output form \\spad{o} on standard output using the two-dimensional formatter."))) NIL NIL -(-870 A B) +(-875 A B) ((|constructor| (NIL "This domain implements cartesian product")) (|selectsecond| ((|#2| $) "\\spad{selectsecond(x)} \\undocumented")) (|selectfirst| ((|#1| $) "\\spad{selectfirst(x)} \\undocumented")) (|makeprod| (($ |#1| |#2|) "\\spad{makeprod(a,b)} \\undocumented"))) -((-3975 -12 (|has| |#2| (-408)) (|has| |#1| (-408)))) -((OR (-12 (|HasCategory| |#1| (QUOTE (-712))) (|HasCategory| |#2| (QUOTE (-712)))) (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-751))))) (-12 (|HasCategory| |#1| (QUOTE (-712))) (|HasCategory| |#2| (QUOTE (-712)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-102))) (|HasCategory| |#2| (QUOTE (-102)))) (-12 (|HasCategory| |#1| (QUOTE (-712))) (|HasCategory| |#2| (QUOTE (-712)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21))))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-102))) (|HasCategory| |#2| (QUOTE (-102)))) (-12 (|HasCategory| |#1| (QUOTE (-712))) (|HasCategory| |#2| (QUOTE (-712)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23))))) (-12 (|HasCategory| |#1| (QUOTE (-408))) (|HasCategory| |#2| (QUOTE (-408)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-408))) (|HasCategory| |#2| (QUOTE (-408)))) (-12 (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#2| (QUOTE (-660))))) (-12 (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#2| (QUOTE (-315)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-102))) (|HasCategory| |#2| (QUOTE (-102)))) (-12 (|HasCategory| |#1| (QUOTE (-712))) (|HasCategory| |#2| (QUOTE (-712)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-408))) (|HasCategory| |#2| (QUOTE (-408)))) (-12 (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#2| (QUOTE (-660))))) (-12 (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#2| (QUOTE (-660)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-102))) (|HasCategory| |#2| (QUOTE (-102)))) (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-751))))) -(-871) +((-3986 -12 (|has| |#2| (-410)) (|has| |#1| (-410)))) +((OR (-12 (|HasCategory| |#1| (QUOTE (-717))) (|HasCategory| |#2| (QUOTE (-717)))) (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-756))))) (-12 (|HasCategory| |#1| (QUOTE (-717))) (|HasCategory| |#2| (QUOTE (-717)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-104))) (|HasCategory| |#2| (QUOTE (-104)))) (-12 (|HasCategory| |#1| (QUOTE (-717))) (|HasCategory| |#2| (QUOTE (-717)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21))))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-104))) (|HasCategory| |#2| (QUOTE (-104)))) (-12 (|HasCategory| |#1| (QUOTE (-717))) (|HasCategory| |#2| (QUOTE (-717)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23))))) (-12 (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#2| (QUOTE (-410)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#2| (QUOTE (-410)))) (-12 (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#2| (QUOTE (-663))))) (-12 (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#2| (QUOTE (-317)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-104))) (|HasCategory| |#2| (QUOTE (-104)))) (-12 (|HasCategory| |#1| (QUOTE (-717))) (|HasCategory| |#2| (QUOTE (-717)))) (-12 (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-21)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-410))) (|HasCategory| |#2| (QUOTE (-410)))) (-12 (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#2| (QUOTE (-663))))) (-12 (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#2| (QUOTE (-663)))) (-12 (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-23)))) (-12 (|HasCategory| |#1| (QUOTE (-104))) (|HasCategory| |#2| (QUOTE (-104)))) (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-756))))) +(-876) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. An `Property' is a pair of name and value.")) (|property| (($ (|Identifier|) (|SExpression|)) "\\spad{property(n,val)} constructs a property with name `n' and value `val'.")) (|value| (((|SExpression|) $) "\\spad{value(p)} returns value of property \\spad{p}")) (|name| (((|Identifier|) $) "\\spad{name(p)} returns the name of property \\spad{p}"))) NIL NIL -(-872 T$) +(-877 T$) ((|constructor| (NIL "This domain implements propositional formula build over a term domain,{} that itself belongs to PropositionalLogic")) (|disjunction| (($ $ $) "\\spad{disjunction(p,q)} returns a formula denoting the disjunction of \\spad{p} and \\spad{q}.")) (|conjunction| (($ $ $) "\\spad{conjunction(p,q)} returns a formula denoting the conjunction of \\spad{p} and \\spad{q}.")) (|isEquiv| (((|Maybe| (|Pair| $ $)) $) "\\spad{isEquiv f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is an equivalence formula.")) (|isImplies| (((|Maybe| (|Pair| $ $)) $) "\\spad{isImplies f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is an implication formula.")) (|isOr| (((|Maybe| (|Pair| $ $)) $) "\\spad{isOr f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is a disjunction formula.")) (|isAnd| (((|Maybe| (|Pair| $ $)) $) "\\spad{isAnd f} returns a value \\spad{v} such that \\spad{v case Pair(\\%,\\%)} holds if the formula \\spad{f} is a conjunction formula.")) (|isNot| (((|Maybe| $) $) "\\spad{isNot f} returns a value \\spad{v} such that \\spad{v case \\%} holds if the formula \\spad{f} is a negation.")) (|isAtom| (((|Maybe| |#1|) $) "\\spad{isAtom f} returns a value \\spad{v} such that \\spad{v case T} holds if the formula \\spad{f} is a term."))) NIL NIL -(-873 T$) +(-878 T$) ((|constructor| (NIL "This package collects unary functions operating on propositional formulae.")) (|simplify| (((|PropositionalFormula| |#1|) (|PropositionalFormula| |#1|)) "\\spad{simplify f} returns a formula logically equivalent to \\spad{f} where obvious tautologies have been removed.")) (|atoms| (((|Set| |#1|) (|PropositionalFormula| |#1|)) "\\spad{atoms f} ++ returns the set of atoms appearing in the formula \\spad{f}.")) (|dual| (((|PropositionalFormula| |#1|) (|PropositionalFormula| |#1|)) "\\spad{dual f} returns the dual of the proposition \\spad{f}."))) NIL NIL -(-874 S T$) +(-879 S T$) ((|constructor| (NIL "This package collects binary functions operating on propositional formulae.")) (|map| (((|PropositionalFormula| |#2|) (|Mapping| |#2| |#1|) (|PropositionalFormula| |#1|)) "\\spad{map(f,x)} returns a propositional formula where all atoms in \\spad{x} have been replaced by the result of applying the function \\spad{f} to them."))) NIL NIL -(-875) +(-880) ((|constructor| (NIL "This category declares the connectives of Propositional Logic.")) (|equiv| (($ $ $) "\\spad{equiv(p,q)} returns the logical equivalence of `p',{} `q'.")) (|implies| (($ $ $) "\\spad{implies(p,q)} returns the logical implication of `q' by `p'.")) (|false| (($) "\\spad{false} is a logical constant.")) (|true| (($) "\\spad{true} is a logical constant."))) NIL NIL -(-876 S) +(-881 S) ((|constructor| (NIL "A priority queue is a bag of items from an ordered set where the item extracted is always the maximum element.")) (|merge!| (($ $ $) "\\spad{merge!(q,q1)} destructively changes priority queue \\spad{q} to include the values from priority queue \\spad{q1}.")) (|merge| (($ $ $) "\\spad{merge(q1,q2)} returns combines priority queues \\spad{q1} and \\spad{q2} to return a single priority queue \\spad{q}.")) (|max| ((|#1| $) "\\spad{max(q)} returns the maximum element of priority queue \\spad{q}."))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL -(-877 R |polR|) +(-882 R |polR|) ((|constructor| (NIL "This package contains some functions: \\axiomOpFrom{discriminant}{PseudoRemainderSequence},{} \\axiomOpFrom{resultant}{PseudoRemainderSequence},{} \\axiomOpFrom{subResultantGcd}{PseudoRemainderSequence},{} \\axiomOpFrom{chainSubResultants}{PseudoRemainderSequence},{} \\axiomOpFrom{degreeSubResultant}{PseudoRemainderSequence},{} \\axiomOpFrom{lastSubResultant}{PseudoRemainderSequence},{} \\axiomOpFrom{resultantEuclidean}{PseudoRemainderSequence},{} \\axiomOpFrom{subResultantGcdEuclidean}{PseudoRemainderSequence},{} \\axiomOpFrom{\\spad{semiSubResultantGcdEuclidean1}}{PseudoRemainderSequence},{} \\axiomOpFrom{\\spad{semiSubResultantGcdEuclidean2}}{PseudoRemainderSequence},{} etc. This procedures are coming from improvements of the subresultants algorithm. \\indented{2}{Version : 7} \\indented{2}{References : Lionel Ducos \"Optimizations of the subresultant algorithm\"} \\indented{2}{to appear in the Journal of Pure and Applied Algebra.} \\indented{2}{Author : Ducos Lionel \\axiom{Lionel.Ducos@mathlabo.univ-poitiers.fr}}")) (|semiResultantEuclideannaif| (((|Record| (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{resultantEuclidean_naif(\\spad{P},{}\\spad{Q})} returns the semi-extended resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}} computed by means of the naive algorithm.")) (|resultantEuclideannaif| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{resultantEuclidean_naif(\\spad{P},{}\\spad{Q})} returns the extended resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}} computed by means of the naive algorithm.")) (|resultantnaif| ((|#1| |#2| |#2|) "\\axiom{resultantEuclidean_naif(\\spad{P},{}\\spad{Q})} returns the resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}} computed by means of the naive algorithm.")) (|nextsousResultant2| ((|#2| |#2| |#2| |#2| |#1|) "\\axiom{\\spad{nextsousResultant2}(\\spad{P},{} \\spad{Q},{} \\spad{Z},{} \\spad{s})} returns the subresultant \\axiom{S_{\\spad{e}-1}} where \\axiom{\\spad{P} ~ S_d,{} \\spad{Q} = S_{\\spad{d}-1},{} \\spad{Z} = S_e,{} \\spad{s} = lc(S_d)}")) (|Lazard2| ((|#2| |#2| |#1| |#1| (|NonNegativeInteger|)) "\\axiom{\\spad{Lazard2}(\\spad{F},{} \\spad{x},{} \\spad{y},{} \\spad{n})} computes \\axiom{(x/y)**(\\spad{n}-1) * \\spad{F}}")) (|Lazard| ((|#1| |#1| |#1| (|NonNegativeInteger|)) "\\axiom{Lazard(\\spad{x},{} \\spad{y},{} \\spad{n})} computes \\axiom{x**n/y**(\\spad{n}-1)}")) (|divide| (((|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2|) "\\axiom{divide(\\spad{F},{}\\spad{G})} computes quotient and rest of the exact euclidean division of \\axiom{\\spad{F}} by \\axiom{\\spad{G}}.")) (|pseudoDivide| (((|Record| (|:| |coef| |#1|) (|:| |quotient| |#2|) (|:| |remainder| |#2|)) |#2| |#2|) "\\axiom{pseudoDivide(\\spad{P},{}\\spad{Q})} computes the pseudoDivide of \\axiom{\\spad{P}} by \\axiom{\\spad{Q}}.")) (|exquo| (((|Vector| |#2|) (|Vector| |#2|) |#1|) "\\axiom{\\spad{v} exquo \\spad{r}} computes the exact quotient of \\axiom{\\spad{v}} by \\axiom{\\spad{r}}")) (* (((|Vector| |#2|) |#1| (|Vector| |#2|)) "\\axiom{\\spad{r} * \\spad{v}} computes the product of \\axiom{\\spad{r}} and \\axiom{\\spad{v}}")) (|gcd| ((|#2| |#2| |#2|) "\\axiom{gcd(\\spad{P},{} \\spad{Q})} returns the gcd of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiResultantReduitEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |resultantReduit| |#1|)) |#2| |#2|) "\\axiom{semiResultantReduitEuclidean(\\spad{P},{}\\spad{Q})} returns the \"reduce resultant\" and carries out the equality \\axiom{...\\spad{P} + coef2*Q = resultantReduit(\\spad{P},{}\\spad{Q})}.")) (|resultantReduitEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |resultantReduit| |#1|)) |#2| |#2|) "\\axiom{resultantReduitEuclidean(\\spad{P},{}\\spad{Q})} returns the \"reduce resultant\" and carries out the equality \\axiom{coef1*P + coef2*Q = resultantReduit(\\spad{P},{}\\spad{Q})}.")) (|resultantReduit| ((|#1| |#2| |#2|) "\\axiom{resultantReduit(\\spad{P},{}\\spad{Q})} returns the \"reduce resultant\" of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|schema| (((|List| (|NonNegativeInteger|)) |#2| |#2|) "\\axiom{schema(\\spad{P},{}\\spad{Q})} returns the list of degrees of non zero subresultants of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|chainSubResultants| (((|List| |#2|) |#2| |#2|) "\\axiom{chainSubResultants(\\spad{P},{} \\spad{Q})} computes the list of non zero subresultants of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiDiscriminantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |discriminant| |#1|)) |#2|) "\\axiom{discriminantEuclidean(\\spad{P})} carries out the equality \\axiom{...\\spad{P} + \\spad{coef2} * \\spad{D}(\\spad{P}) = discriminant(\\spad{P})}. Warning: \\axiom{degree(\\spad{P}) >= degree(\\spad{Q})}.")) (|discriminantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |discriminant| |#1|)) |#2|) "\\axiom{discriminantEuclidean(\\spad{P})} carries out the equality \\axiom{\\spad{coef1} * \\spad{P} + \\spad{coef2} * \\spad{D}(\\spad{P}) = discriminant(\\spad{P})}.")) (|discriminant| ((|#1| |#2|) "\\axiom{discriminant(\\spad{P},{} \\spad{Q})} returns the discriminant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiSubResultantGcdEuclidean1| (((|Record| (|:| |coef1| |#2|) (|:| |gcd| |#2|)) |#2| |#2|) "\\axiom{\\spad{semiSubResultantGcdEuclidean1}(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{coef1*P + ? \\spad{Q} = +/- S_i(\\spad{P},{}\\spad{Q})} where the degree (not the indice) of the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} is the smaller as possible.")) (|semiSubResultantGcdEuclidean2| (((|Record| (|:| |coef2| |#2|) (|:| |gcd| |#2|)) |#2| |#2|) "\\axiom{\\spad{semiSubResultantGcdEuclidean2}(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{...\\spad{P} + coef2*Q = +/- S_i(\\spad{P},{}\\spad{Q})} where the degree (not the indice) of the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} is the smaller as possible. Warning: \\axiom{degree(\\spad{P}) >= degree(\\spad{Q})}.")) (|subResultantGcdEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |gcd| |#2|)) |#2| |#2|) "\\axiom{subResultantGcdEuclidean(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{coef1*P + coef2*Q = +/- S_i(\\spad{P},{}\\spad{Q})} where the degree (not the indice) of the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} is the smaller as possible.")) (|subResultantGcd| ((|#2| |#2| |#2|) "\\axiom{subResultantGcd(\\spad{P},{} \\spad{Q})} returns the gcd of two primitive polynomials \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}.")) (|semiLastSubResultantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2|) "\\axiom{semiLastSubResultantEuclidean(\\spad{P},{} \\spad{Q})} computes the last non zero subresultant \\axiom{\\spad{S}} and carries out the equality \\axiom{...\\spad{P} + coef2*Q = \\spad{S}}. Warning: \\axiom{degree(\\spad{P}) >= degree(\\spad{Q})}.")) (|lastSubResultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2|) "\\axiom{lastSubResultantEuclidean(\\spad{P},{} \\spad{Q})} computes the last non zero subresultant \\axiom{\\spad{S}} and carries out the equality \\axiom{coef1*P + coef2*Q = \\spad{S}}.")) (|lastSubResultant| ((|#2| |#2| |#2|) "\\axiom{lastSubResultant(\\spad{P},{} \\spad{Q})} computes the last non zero subresultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}")) (|semiDegreeSubResultantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns a subresultant \\axiom{\\spad{S}} of degree \\axiom{\\spad{d}} and carries out the equality \\axiom{...\\spad{P} + coef2*Q = S_i}. Warning: \\axiom{degree(\\spad{P}) >= degree(\\spad{Q})}.")) (|degreeSubResultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns a subresultant \\axiom{\\spad{S}} of degree \\axiom{\\spad{d}} and carries out the equality \\axiom{coef1*P + coef2*Q = S_i}.")) (|degreeSubResultant| ((|#2| |#2| |#2| (|NonNegativeInteger|)) "\\axiom{degreeSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{d})} computes a subresultant of degree \\axiom{\\spad{d}}.")) (|semiIndiceSubResultantEuclidean| (((|Record| (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{semiIndiceSubResultantEuclidean(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} and carries out the equality \\axiom{...\\spad{P} + coef2*Q = S_i(\\spad{P},{}\\spad{Q})} Warning: \\axiom{degree(\\spad{P}) >= degree(\\spad{Q})}.")) (|indiceSubResultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |subResultant| |#2|)) |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns the subresultant \\axiom{S_i(\\spad{P},{}\\spad{Q})} and carries out the equality \\axiom{coef1*P + coef2*Q = S_i(\\spad{P},{}\\spad{Q})}")) (|indiceSubResultant| ((|#2| |#2| |#2| (|NonNegativeInteger|)) "\\axiom{indiceSubResultant(\\spad{P},{} \\spad{Q},{} \\spad{i})} returns the subresultant of indice \\axiom{\\spad{i}}")) (|semiResultantEuclidean1| (((|Record| (|:| |coef1| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{\\spad{semiResultantEuclidean1}(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{\\spad{coef1}.\\spad{P} + ? \\spad{Q} = resultant(\\spad{P},{}\\spad{Q})}.")) (|semiResultantEuclidean2| (((|Record| (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{\\spad{semiResultantEuclidean2}(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{...\\spad{P} + coef2*Q = resultant(\\spad{P},{}\\spad{Q})}. Warning: \\axiom{degree(\\spad{P}) >= degree(\\spad{Q})}.")) (|resultantEuclidean| (((|Record| (|:| |coef1| |#2|) (|:| |coef2| |#2|) (|:| |resultant| |#1|)) |#2| |#2|) "\\axiom{resultantEuclidean(\\spad{P},{}\\spad{Q})} carries out the equality \\axiom{coef1*P + coef2*Q = resultant(\\spad{P},{}\\spad{Q})}")) (|resultant| ((|#1| |#2| |#2|) "\\axiom{resultant(\\spad{P},{} \\spad{Q})} returns the resultant of \\axiom{\\spad{P}} and \\axiom{\\spad{Q}}"))) NIL -((|HasCategory| |#1| (QUOTE (-387)))) -(-878) +((|HasCategory| |#1| (QUOTE (-389)))) +(-883) ((|constructor| (NIL "This domain represents `pretend' expressions.")) (|target| (((|TypeAst|) $) "\\spad{target(e)} returns the target type of the conversion..")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression being converted."))) NIL NIL -(-879) +(-884) ((|constructor| (NIL "Partition is an OrderedCancellationAbelianMonoid which is used as the basis for symmetric polynomial representation of the sums of powers in SymmetricPolynomial. Thus,{} \\spad{(5 2 2 1)} will represent \\spad{s5 * s2**2 * s1}.")) (|conjugate| (($ $) "\\spad{conjugate(p)} returns the conjugate partition of a partition \\spad{p}")) (|pdct| (((|PositiveInteger|) $) "\\spad{pdct(a1**n1 a2**n2 ...)} returns \\spad{n1! * a1**n1 * n2! * a2**n2 * ...}. This function is used in the package \\spadtype{CycleIndicators}.")) (|powers| (((|List| (|Pair| (|PositiveInteger|) (|PositiveInteger|))) $) "\\spad{powers(x)} returns a list of pairs. The second component of each pair is the multiplicity with which the first component occurs in \\spad{li}.")) (|partitions| (((|Stream| $) (|NonNegativeInteger|)) "\\spad{partitions n} returns the stream of all partitions of size \\spad{n}.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\#x} returns the sum of all parts of the partition \\spad{x}.")) (|parts| (((|List| (|PositiveInteger|)) $) "\\spad{parts x} returns the list of decreasing integer sequence making up the partition \\spad{x}.")) (|partition| (($ (|List| (|PositiveInteger|))) "\\spad{partition(li)} converts a list of integers \\spad{li} to a partition"))) NIL NIL -(-880 S |Coef| |Expon| |Var|) +(-885 S |Coef| |Expon| |Var|) ((|constructor| (NIL "\\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.")) (|complete| (($ $) "\\spad{complete(f)} causes all terms of \\spad{f} to be computed. Note: this results in an infinite loop if \\spad{f} has infinitely many terms.")) (|pole?| (((|Boolean|) $) "\\spad{pole?(f)} determines if the power series \\spad{f} has a pole.")) (|variables| (((|List| |#4|) $) "\\spad{variables(f)} returns a list of the variables occuring in the power series \\spad{f}.")) (|degree| ((|#3| $) "\\spad{degree(f)} returns the exponent of the lowest order term of \\spad{f}.")) (|leadingCoefficient| ((|#2| $) "\\spad{leadingCoefficient(f)} returns the coefficient of the lowest order term of \\spad{f}")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(f)} returns the monomial of \\spad{f} of lowest order.")) (|monomial| (($ $ (|List| |#4|) (|List| |#3|)) "\\spad{monomial(a,[x1,..,xk],[n1,..,nk])} computes \\spad{a * x1**n1 * .. * xk**nk}.") (($ $ |#4| |#3|) "\\spad{monomial(a,x,n)} computes \\spad{a*x**n}."))) NIL NIL -(-881 |Coef| |Expon| |Var|) +(-886 |Coef| |Expon| |Var|) ((|constructor| (NIL "\\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.")) (|complete| (($ $) "\\spad{complete(f)} causes all terms of \\spad{f} to be computed. Note: this results in an infinite loop if \\spad{f} has infinitely many terms.")) (|pole?| (((|Boolean|) $) "\\spad{pole?(f)} determines if the power series \\spad{f} has a pole.")) (|variables| (((|List| |#3|) $) "\\spad{variables(f)} returns a list of the variables occuring in the power series \\spad{f}.")) (|degree| ((|#2| $) "\\spad{degree(f)} returns the exponent of the lowest order term of \\spad{f}.")) (|leadingCoefficient| ((|#1| $) "\\spad{leadingCoefficient(f)} returns the coefficient of the lowest order term of \\spad{f}")) (|leadingMonomial| (($ $) "\\spad{leadingMonomial(f)} returns the monomial of \\spad{f} of lowest order.")) (|monomial| (($ $ (|List| |#3|) (|List| |#2|)) "\\spad{monomial(a,[x1,..,xk],[n1,..,nk])} computes \\spad{a * x1**n1 * .. * xk**nk}.") (($ $ |#3| |#2|) "\\spad{monomial(a,x,n)} computes \\spad{a*x**n}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-882) +(-887) ((|constructor| (NIL "PlottableSpaceCurveCategory is the category of curves in 3-space which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the x-,{} y-,{} and \\spad{z}-coordinates of the points on the curve.")) (|zRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{zRange(c)} returns the range of the \\spad{z}-coordinates of the points on the curve \\spad{c}.")) (|yRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{yRange(c)} returns the range of the \\spad{y}-coordinates of the points on the curve \\spad{c}.")) (|xRange| (((|Segment| (|DoubleFloat|)) $) "\\spad{xRange(c)} returns the range of the \\spad{x}-coordinates of the points on the curve \\spad{c}.")) (|listBranches| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{listBranches(c)} returns a list of lists of points,{} representing the branches of the curve \\spad{c}."))) NIL NIL -(-883 S R E |VarSet| P) +(-888 S R E |VarSet| P) ((|constructor| (NIL "A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \\spad{R} being an integral domain,{} a set of polynomials may be viewed as a representation of the ideal it generates in the polynomial ring \\spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.")) (|triangular?| (((|Boolean|) $) "\\axiom{triangular?(ps)} returns \\spad{true} iff \\axiom{ps} is a triangular set,{} \\spadignore{i.e.} two distinct polynomials have distinct main variables and no constant lies in \\axiom{ps}.")) (|rewriteIdealWithRemainder| (((|List| |#5|) (|List| |#5|) $) "\\axiom{rewriteIdealWithRemainder(lp,{}cs)} returns \\axiom{lr} such that every polynomial in \\axiom{lr} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{cs} and \\axiom{(lp,{}cs)} and \\axiom{(lr,{}cs)} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|rewriteIdealWithHeadRemainder| (((|List| |#5|) (|List| |#5|) $) "\\axiom{rewriteIdealWithHeadRemainder(lp,{}cs)} returns \\axiom{lr} such that the leading monomial of every polynomial in \\axiom{lr} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{cs} and \\axiom{(lp,{}cs)} and \\axiom{(lr,{}cs)} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|remainder| (((|Record| (|:| |rnum| |#2|) (|:| |polnum| |#5|) (|:| |den| |#2|)) |#5| $) "\\axiom{remainder(a,{}ps)} returns \\axiom{[\\spad{c},{}\\spad{b},{}\\spad{r}]} such that \\axiom{\\spad{b}} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ps},{} \\axiom{r*a - c*b} lies in the ideal generated by \\axiom{ps}. Furthermore,{} if \\axiom{\\spad{R}} is a gcd-domain,{} \\axiom{\\spad{b}} is primitive.")) (|headRemainder| (((|Record| (|:| |num| |#5|) (|:| |den| |#2|)) |#5| $) "\\axiom{headRemainder(a,{}ps)} returns \\axiom{[\\spad{b},{}\\spad{r}]} such that the leading monomial of \\axiom{\\spad{b}} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ps} and \\axiom{r*a - \\spad{b}} lies in the ideal generated by \\axiom{ps}.")) (|roughUnitIdeal?| (((|Boolean|) $) "\\axiom{roughUnitIdeal?(ps)} returns \\spad{true} iff \\axiom{ps} contains some non null element lying in the base ring \\axiom{\\spad{R}}.")) (|roughEqualIdeals?| (((|Boolean|) $ $) "\\axiom{roughEqualIdeals?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that \\axiom{\\spad{ps1}} and \\axiom{\\spad{ps2}} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}} without computing Groebner bases.")) (|roughSubIdeal?| (((|Boolean|) $ $) "\\axiom{roughSubIdeal?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that all polynomials in \\axiom{\\spad{ps1}} lie in the ideal generated by \\axiom{\\spad{ps2}} in \\axiom{\\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}} without computing Groebner bases.")) (|roughBase?| (((|Boolean|) $) "\\axiom{roughBase?(ps)} returns \\spad{true} iff for every pair \\axiom{{\\spad{p},{}\\spad{q}}} of polynomials in \\axiom{ps} their leading monomials are relatively prime.")) (|trivialIdeal?| (((|Boolean|) $) "\\axiom{trivialIdeal?(ps)} returns \\spad{true} iff \\axiom{ps} does not contain non-zero elements.")) (|sort| (((|Record| (|:| |under| $) (|:| |floor| $) (|:| |upper| $)) $ |#4|) "\\axiom{sort(\\spad{v},{}ps)} returns \\axiom{us,{}vs,{}ws} such that \\axiom{us} is \\axiom{collectUnder(ps,{}\\spad{v})},{} \\axiom{vs} is \\axiom{collect(ps,{}\\spad{v})} and \\axiom{ws} is \\axiom{collectUpper(ps,{}\\spad{v})}.")) (|collectUpper| (($ $ |#4|) "\\axiom{collectUpper(ps,{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{ps} with main variable greater than \\axiom{\\spad{v}}.")) (|collect| (($ $ |#4|) "\\axiom{collect(ps,{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{ps} with \\axiom{\\spad{v}} as main variable.")) (|collectUnder| (($ $ |#4|) "\\axiom{collectUnder(ps,{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{ps} with main variable less than \\axiom{\\spad{v}}.")) (|mainVariable?| (((|Boolean|) |#4| $) "\\axiom{mainVariable?(\\spad{v},{}ps)} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{ps}.")) (|mainVariables| (((|List| |#4|) $) "\\axiom{mainVariables(ps)} returns the decreasingly sorted list of the variables which are main variables of some polynomial in \\axiom{ps}.")) (|variables| (((|List| |#4|) $) "\\axiom{variables(ps)} returns the decreasingly sorted list of the variables which are variables of some polynomial in \\axiom{ps}.")) (|mvar| ((|#4| $) "\\axiom{mvar(ps)} returns the main variable of the non constant polynomial with the greatest main variable,{} if any,{} else an error is returned.")) (|retract| (($ (|List| |#5|)) "\\axiom{retract(lp)} returns an element of the domain whose elements are the members of \\axiom{lp} if such an element exists,{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|List| |#5|)) "\\axiom{retractIfCan(lp)} returns an element of the domain whose elements are the members of \\axiom{lp} if such an element exists,{} otherwise \\axiom{\"failed\"} is returned."))) NIL -((|HasCategory| |#2| (QUOTE (-491)))) -(-884 R E |VarSet| P) +((|HasCategory| |#2| (QUOTE (-494)))) +(-889 R E |VarSet| P) ((|constructor| (NIL "A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \\spad{R} being an integral domain,{} a set of polynomials may be viewed as a representation of the ideal it generates in the polynomial ring \\spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.")) (|triangular?| (((|Boolean|) $) "\\axiom{triangular?(ps)} returns \\spad{true} iff \\axiom{ps} is a triangular set,{} \\spadignore{i.e.} two distinct polynomials have distinct main variables and no constant lies in \\axiom{ps}.")) (|rewriteIdealWithRemainder| (((|List| |#4|) (|List| |#4|) $) "\\axiom{rewriteIdealWithRemainder(lp,{}cs)} returns \\axiom{lr} such that every polynomial in \\axiom{lr} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{cs} and \\axiom{(lp,{}cs)} and \\axiom{(lr,{}cs)} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|rewriteIdealWithHeadRemainder| (((|List| |#4|) (|List| |#4|) $) "\\axiom{rewriteIdealWithHeadRemainder(lp,{}cs)} returns \\axiom{lr} such that the leading monomial of every polynomial in \\axiom{lr} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{cs} and \\axiom{(lp,{}cs)} and \\axiom{(lr,{}cs)} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}.")) (|remainder| (((|Record| (|:| |rnum| |#1|) (|:| |polnum| |#4|) (|:| |den| |#1|)) |#4| $) "\\axiom{remainder(a,{}ps)} returns \\axiom{[\\spad{c},{}\\spad{b},{}\\spad{r}]} such that \\axiom{\\spad{b}} is fully reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ps},{} \\axiom{r*a - c*b} lies in the ideal generated by \\axiom{ps}. Furthermore,{} if \\axiom{\\spad{R}} is a gcd-domain,{} \\axiom{\\spad{b}} is primitive.")) (|headRemainder| (((|Record| (|:| |num| |#4|) (|:| |den| |#1|)) |#4| $) "\\axiom{headRemainder(a,{}ps)} returns \\axiom{[\\spad{b},{}\\spad{r}]} such that the leading monomial of \\axiom{\\spad{b}} is reduced in the sense of Groebner bases \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ps} and \\axiom{r*a - \\spad{b}} lies in the ideal generated by \\axiom{ps}.")) (|roughUnitIdeal?| (((|Boolean|) $) "\\axiom{roughUnitIdeal?(ps)} returns \\spad{true} iff \\axiom{ps} contains some non null element lying in the base ring \\axiom{\\spad{R}}.")) (|roughEqualIdeals?| (((|Boolean|) $ $) "\\axiom{roughEqualIdeals?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that \\axiom{\\spad{ps1}} and \\axiom{\\spad{ps2}} generate the same ideal in \\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}} without computing Groebner bases.")) (|roughSubIdeal?| (((|Boolean|) $ $) "\\axiom{roughSubIdeal?(\\spad{ps1},{}\\spad{ps2})} returns \\spad{true} iff it can proved that all polynomials in \\axiom{\\spad{ps1}} lie in the ideal generated by \\axiom{\\spad{ps2}} in \\axiom{\\axiom{(\\spad{R})^(\\spad{-1}) \\spad{P}}} without computing Groebner bases.")) (|roughBase?| (((|Boolean|) $) "\\axiom{roughBase?(ps)} returns \\spad{true} iff for every pair \\axiom{{\\spad{p},{}\\spad{q}}} of polynomials in \\axiom{ps} their leading monomials are relatively prime.")) (|trivialIdeal?| (((|Boolean|) $) "\\axiom{trivialIdeal?(ps)} returns \\spad{true} iff \\axiom{ps} does not contain non-zero elements.")) (|sort| (((|Record| (|:| |under| $) (|:| |floor| $) (|:| |upper| $)) $ |#3|) "\\axiom{sort(\\spad{v},{}ps)} returns \\axiom{us,{}vs,{}ws} such that \\axiom{us} is \\axiom{collectUnder(ps,{}\\spad{v})},{} \\axiom{vs} is \\axiom{collect(ps,{}\\spad{v})} and \\axiom{ws} is \\axiom{collectUpper(ps,{}\\spad{v})}.")) (|collectUpper| (($ $ |#3|) "\\axiom{collectUpper(ps,{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{ps} with main variable greater than \\axiom{\\spad{v}}.")) (|collect| (($ $ |#3|) "\\axiom{collect(ps,{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{ps} with \\axiom{\\spad{v}} as main variable.")) (|collectUnder| (($ $ |#3|) "\\axiom{collectUnder(ps,{}\\spad{v})} returns the set consisting of the polynomials of \\axiom{ps} with main variable less than \\axiom{\\spad{v}}.")) (|mainVariable?| (((|Boolean|) |#3| $) "\\axiom{mainVariable?(\\spad{v},{}ps)} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{ps}.")) (|mainVariables| (((|List| |#3|) $) "\\axiom{mainVariables(ps)} returns the decreasingly sorted list of the variables which are main variables of some polynomial in \\axiom{ps}.")) (|variables| (((|List| |#3|) $) "\\axiom{variables(ps)} returns the decreasingly sorted list of the variables which are variables of some polynomial in \\axiom{ps}.")) (|mvar| ((|#3| $) "\\axiom{mvar(ps)} returns the main variable of the non constant polynomial with the greatest main variable,{} if any,{} else an error is returned.")) (|retract| (($ (|List| |#4|)) "\\axiom{retract(lp)} returns an element of the domain whose elements are the members of \\axiom{lp} if such an element exists,{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|List| |#4|)) "\\axiom{retractIfCan(lp)} returns an element of the domain whose elements are the members of \\axiom{lp} if such an element exists,{} otherwise \\axiom{\"failed\"} is returned."))) -((-3978 . T)) +((-3989 . T)) NIL -(-885 R E V P) +(-890 R E V P) ((|constructor| (NIL "This package provides modest routines for polynomial system solving. The aim of many of the operations of this package is to remove certain factors in some polynomials in order to avoid unnecessary computations in algorithms involving splitting techniques by partial factorization.")) (|removeIrreducibleRedundantFactors| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeIrreducibleRedundantFactors(lp,{}lq)} returns the same as \\axiom{irreducibleFactors(concat(lp,{}lq))} assuming that \\axiom{irreducibleFactors(lp)} returns \\axiom{lp} up to replacing some polynomial \\axiom{pj} in \\axiom{lp} by some polynomial \\axiom{qj} associated to \\axiom{pj}.")) (|lazyIrreducibleFactors| (((|List| |#4|) (|List| |#4|)) "\\axiom{lazyIrreducibleFactors(lp)} returns \\axiom{lf} such that if \\axiom{lp = [\\spad{p1},{}...,{}pn]} and \\axiom{lf = [\\spad{f1},{}...,{}fm]} then \\axiom{p1*p2*...\\spad{*pn=0}} means \\axiom{f1*f2*...\\spad{*fm=0}},{} and the \\axiom{\\spad{fi}} are irreducible over \\axiom{\\spad{R}} and are pairwise distinct. The algorithm tries to avoid factorization into irreducible factors as far as possible and makes previously use of gcd techniques over \\axiom{\\spad{R}}.")) (|irreducibleFactors| (((|List| |#4|) (|List| |#4|)) "\\axiom{irreducibleFactors(lp)} returns \\axiom{lf} such that if \\axiom{lp = [\\spad{p1},{}...,{}pn]} and \\axiom{lf = [\\spad{f1},{}...,{}fm]} then \\axiom{p1*p2*...\\spad{*pn=0}} means \\axiom{f1*f2*...\\spad{*fm=0}},{} and the \\axiom{\\spad{fi}} are irreducible over \\axiom{\\spad{R}} and are pairwise distinct.")) (|removeRedundantFactorsInPols| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactorsInPols(lp,{}lf)} returns \\axiom{newlp} where \\axiom{newlp} is obtained from \\axiom{lp} by removing in every polynomial \\axiom{\\spad{p}} of \\axiom{lp} any non trivial factor of any polynomial \\axiom{\\spad{f}} in \\axiom{lf}. Moreover,{} squares over \\axiom{\\spad{R}} are first removed in every polynomial \\axiom{lp}.")) (|removeRedundantFactorsInContents| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactorsInContents(lp,{}lf)} returns \\axiom{newlp} where \\axiom{newlp} is obtained from \\axiom{lp} by removing in the content of every polynomial of \\axiom{lp} any non trivial factor of any polynomial \\axiom{\\spad{f}} in \\axiom{lf}. Moreover,{} squares over \\axiom{\\spad{R}} are first removed in the content of every polynomial of \\axiom{lp}.")) (|removeRoughlyRedundantFactorsInContents| (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRoughlyRedundantFactorsInContents(lp,{}lf)} returns \\axiom{newlp}where \\axiom{newlp} is obtained from \\axiom{lp} by removing in the content of every polynomial of \\axiom{lp} any occurence of a polynomial \\axiom{\\spad{f}} in \\axiom{lf}. Moreover,{} squares over \\axiom{\\spad{R}} are first removed in the content of every polynomial of \\axiom{lp}.")) (|univariatePolynomialsGcds| (((|List| |#4|) (|List| |#4|) (|Boolean|)) "\\axiom{univariatePolynomialsGcds(lp,{}opt)} returns the same as \\axiom{univariatePolynomialsGcds(lp)} if \\axiom{opt} is \\axiom{\\spad{false}} and if the previous operation does not return any non null and constant polynomial,{} else return \\axiom{[1]}.") (((|List| |#4|) (|List| |#4|)) "\\axiom{univariatePolynomialsGcds(lp)} returns \\axiom{lg} where \\axiom{lg} is a list of the gcds of every pair in \\axiom{lp} of univariate polynomials in the same main variable.")) (|squareFreeFactors| (((|List| |#4|) |#4|) "\\axiom{squareFreeFactors(\\spad{p})} returns the square-free factors of \\axiom{\\spad{p}} over \\axiom{\\spad{R}}")) (|rewriteIdealWithQuasiMonicGenerators| (((|List| |#4|) (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{rewriteIdealWithQuasiMonicGenerators(lp,{}redOp?,{}redOp)} returns \\axiom{lq} where \\axiom{lq} and \\axiom{lp} generate the same ideal in \\axiom{R^(\\spad{-1}) \\spad{P}} and \\axiom{lq} has rank not higher than the one of \\axiom{lp}. Moreover,{} \\axiom{lq} is computed by reducing \\axiom{lp} \\spad{w}.\\spad{r}.\\spad{t}. some basic set of the ideal generated by the quasi-monic polynomials in \\axiom{lp}.")) (|rewriteSetByReducingWithParticularGenerators| (((|List| |#4|) (|List| |#4|) (|Mapping| (|Boolean|) |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{rewriteSetByReducingWithParticularGenerators(lp,{}pred?,{}redOp?,{}redOp)} returns \\axiom{lq} where \\axiom{lq} is computed by the following algorithm. Chose a basic set \\spad{w}.\\spad{r}.\\spad{t}. the reduction-test \\axiom{redOp?} among the polynomials satisfying property \\axiom{pred?},{} if it is empty then leave,{} else reduce the other polynomials by this basic set \\spad{w}.\\spad{r}.\\spad{t}. the reduction-operation \\axiom{redOp}. Repeat while another basic set with smaller rank can be computed. See code. If \\axiom{pred?} is \\axiom{quasiMonic?} the ideal is unchanged.")) (|crushedSet| (((|List| |#4|) (|List| |#4|)) "\\axiom{crushedSet(lp)} returns \\axiom{lq} such that \\axiom{lp} and and \\axiom{lq} generate the same ideal and no rough basic sets reduce (in the sense of Groebner bases) the other polynomials in \\axiom{lq}.")) (|roughBasicSet| (((|Union| (|Record| (|:| |bas| (|GeneralTriangularSet| |#1| |#2| |#3| |#4|)) (|:| |top| (|List| |#4|))) "failed") (|List| |#4|)) "\\axiom{roughBasicSet(lp)} returns the smallest (with Ritt-Wu ordering) triangular set contained in \\axiom{lp}.")) (|interReduce| (((|List| |#4|) (|List| |#4|)) "\\axiom{interReduce(lp)} returns \\axiom{lq} such that \\axiom{lp} and \\axiom{lq} generate the same ideal and no polynomial in \\axiom{lq} is reducuble by the others in the sense of Groebner bases. Since no assumptions are required the result may depend on the ordering the reductions are performed.")) (|removeRoughlyRedundantFactorsInPol| ((|#4| |#4| (|List| |#4|)) "\\axiom{removeRoughlyRedundantFactorsInPol(\\spad{p},{}lf)} returns the same as removeRoughlyRedundantFactorsInPols([\\spad{p}],{}lf,{}\\spad{true})")) (|removeRoughlyRedundantFactorsInPols| (((|List| |#4|) (|List| |#4|) (|List| |#4|) (|Boolean|)) "\\axiom{removeRoughlyRedundantFactorsInPols(lp,{}lf,{}opt)} returns the same as \\axiom{removeRoughlyRedundantFactorsInPols(lp,{}lf)} if \\axiom{opt} is \\axiom{\\spad{false}} and if the previous operation does not return any non null and constant polynomial,{} else return \\axiom{[1]}.") (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRoughlyRedundantFactorsInPols(lp,{}lf)} returns \\axiom{newlp}where \\axiom{newlp} is obtained from \\axiom{lp} by removing in every polynomial \\axiom{\\spad{p}} of \\axiom{lp} any occurence of a polynomial \\axiom{\\spad{f}} in \\axiom{lf}. This may involve a lot of exact-quotients computations.")) (|bivariatePolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{bivariatePolynomials(lp)} returns \\axiom{bps,{}nbps} where \\axiom{bps} is a list of the bivariate polynomials,{} and \\axiom{nbps} are the other ones.")) (|bivariate?| (((|Boolean|) |#4|) "\\axiom{bivariate?(\\spad{p})} returns \\spad{true} iff \\axiom{\\spad{p}} involves two and only two variables.")) (|linearPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{linearPolynomials(lp)} returns \\axiom{lps,{}nlps} where \\axiom{lps} is a list of the linear polynomials in lp,{} and \\axiom{nlps} are the other ones.")) (|linear?| (((|Boolean|) |#4|) "\\axiom{linear?(\\spad{p})} returns \\spad{true} iff \\axiom{\\spad{p}} does not lie in the base ring \\axiom{\\spad{R}} and has main degree \\axiom{1}.")) (|univariatePolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{univariatePolynomials(lp)} returns \\axiom{ups,{}nups} where \\axiom{ups} is a list of the univariate polynomials,{} and \\axiom{nups} are the other ones.")) (|univariate?| (((|Boolean|) |#4|) "\\axiom{univariate?(\\spad{p})} returns \\spad{true} iff \\axiom{\\spad{p}} involves one and only one variable.")) (|quasiMonicPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| |#4|)) "\\axiom{quasiMonicPolynomials(lp)} returns \\axiom{qmps,{}nqmps} where \\axiom{qmps} is a list of the quasi-monic polynomials in \\axiom{lp} and \\axiom{nqmps} are the other ones.")) (|selectAndPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| (|Mapping| (|Boolean|) |#4|)) (|List| |#4|)) "\\axiom{selectAndPolynomials(lpred?,{}ps)} returns \\axiom{gps,{}bps} where \\axiom{gps} is a list of the polynomial \\axiom{\\spad{p}} in \\axiom{ps} such that \\axiom{pred?(\\spad{p})} holds for every \\axiom{pred?} in \\axiom{lpred?} and \\axiom{bps} are the other ones.")) (|selectOrPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|List| (|Mapping| (|Boolean|) |#4|)) (|List| |#4|)) "\\axiom{selectOrPolynomials(lpred?,{}ps)} returns \\axiom{gps,{}bps} where \\axiom{gps} is a list of the polynomial \\axiom{\\spad{p}} in \\axiom{ps} such that \\axiom{pred?(\\spad{p})} holds for some \\axiom{pred?} in \\axiom{lpred?} and \\axiom{bps} are the other ones.")) (|selectPolynomials| (((|Record| (|:| |goodPols| (|List| |#4|)) (|:| |badPols| (|List| |#4|))) (|Mapping| (|Boolean|) |#4|) (|List| |#4|)) "\\axiom{selectPolynomials(pred?,{}ps)} returns \\axiom{gps,{}bps} where \\axiom{gps} is a list of the polynomial \\axiom{\\spad{p}} in \\axiom{ps} such that \\axiom{pred?(\\spad{p})} holds and \\axiom{bps} are the other ones.")) (|probablyZeroDim?| (((|Boolean|) (|List| |#4|)) "\\axiom{probablyZeroDim?(lp)} returns \\spad{true} iff the number of polynomials in \\axiom{lp} is not smaller than the number of variables occurring in these polynomials.")) (|possiblyNewVariety?| (((|Boolean|) (|List| |#4|) (|List| (|List| |#4|))) "\\axiom{possiblyNewVariety?(newlp,{}llp)} returns \\spad{true} iff for every \\axiom{lp} in \\axiom{llp} certainlySubVariety?(newlp,{}lp) does not hold.")) (|certainlySubVariety?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{certainlySubVariety?(newlp,{}lp)} returns \\spad{true} iff for every \\axiom{\\spad{p}} in \\axiom{lp} the remainder of \\axiom{\\spad{p}} by \\axiom{newlp} using the division algorithm of Groebner techniques is zero.")) (|unprotectedRemoveRedundantFactors| (((|List| |#4|) |#4| |#4|) "\\axiom{unprotectedRemoveRedundantFactors(\\spad{p},{}\\spad{q})} returns the same as \\axiom{removeRedundantFactors(\\spad{p},{}\\spad{q})} but does assume that neither \\axiom{\\spad{p}} nor \\axiom{\\spad{q}} lie in the base ring \\axiom{\\spad{R}} and assumes that \\axiom{infRittWu?(\\spad{p},{}\\spad{q})} holds. Moreover,{} if \\axiom{\\spad{R}} is gcd-domain,{} then \\axiom{\\spad{p}} and \\axiom{\\spad{q}} are assumed to be square free.")) (|removeSquaresIfCan| (((|List| |#4|) (|List| |#4|)) "\\axiom{removeSquaresIfCan(lp)} returns \\axiom{removeDuplicates [squareFreePart(\\spad{p})\\$\\spad{P} for \\spad{p} in lp]} if \\axiom{\\spad{R}} is gcd-domain else returns \\axiom{lp}.")) (|removeRedundantFactors| (((|List| |#4|) (|List| |#4|) (|List| |#4|) (|Mapping| (|List| |#4|) (|List| |#4|))) "\\axiom{removeRedundantFactors(lp,{}lq,{}remOp)} returns the same as \\axiom{concat(remOp(removeRoughlyRedundantFactorsInPols(lp,{}lq)),{}lq)} assuming that \\axiom{remOp(lq)} returns \\axiom{lq} up to similarity.") (((|List| |#4|) (|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactors(lp,{}lq)} returns the same as \\axiom{removeRedundantFactors(concat(lp,{}lq))} assuming that \\axiom{removeRedundantFactors(lp)} returns \\axiom{lp} up to replacing some polynomial \\axiom{pj} in \\axiom{lp} by some polynomial \\axiom{qj} associated to \\axiom{pj}.") (((|List| |#4|) (|List| |#4|) |#4|) "\\axiom{removeRedundantFactors(lp,{}\\spad{q})} returns the same as \\axiom{removeRedundantFactors(cons(\\spad{q},{}lp))} assuming that \\axiom{removeRedundantFactors(lp)} returns \\axiom{lp} up to replacing some polynomial \\axiom{pj} in \\axiom{lp} by some some polynomial \\axiom{qj} associated to \\axiom{pj}.") (((|List| |#4|) |#4| |#4|) "\\axiom{removeRedundantFactors(\\spad{p},{}\\spad{q})} returns the same as \\axiom{removeRedundantFactors([\\spad{p},{}\\spad{q}])}") (((|List| |#4|) (|List| |#4|)) "\\axiom{removeRedundantFactors(lp)} returns \\axiom{lq} such that if \\axiom{lp = [\\spad{p1},{}...,{}pn]} and \\axiom{lq = [\\spad{q1},{}...,{}qm]} then the product \\axiom{p1*p2*...*pn} vanishes iff the product \\axiom{q1*q2*...*qm} vanishes,{} and the product of degrees of the \\axiom{\\spad{qi}} is not greater than the one of the \\axiom{pj},{} and no polynomial in \\axiom{lq} divides another polynomial in \\axiom{lq}. In particular,{} polynomials lying in the base ring \\axiom{\\spad{R}} are removed. Moreover,{} \\axiom{lq} is sorted \\spad{w}.\\spad{r}.\\spad{t} \\axiom{infRittWu?}. Furthermore,{} if \\spad{R} is gcd-domain,{} the polynomials in \\axiom{lq} are pairwise without common non trivial factor."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-255)))) (|HasCategory| |#1| (QUOTE (-387)))) -(-886 K) +((-12 (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-257)))) (|HasCategory| |#1| (QUOTE (-389)))) +(-891 K) ((|constructor| (NIL "PseudoLinearNormalForm provides a function for computing a block-companion form for pseudo-linear operators.")) (|companionBlocks| (((|List| (|Record| (|:| C (|Matrix| |#1|)) (|:| |g| (|Vector| |#1|)))) (|Matrix| |#1|) (|Vector| |#1|)) "\\spad{companionBlocks(m, v)} returns \\spad{[[C_1, g_1],...,[C_k, g_k]]} such that each \\spad{C_i} is a companion block and \\spad{m = diagonal(C_1,...,C_k)}.")) (|changeBase| (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|) (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{changeBase(M, A, sig, der)}: computes the new matrix of a pseudo-linear transform given by the matrix \\spad{M} under the change of base A")) (|normalForm| (((|Record| (|:| R (|Matrix| |#1|)) (|:| A (|Matrix| |#1|)) (|:| |Ainv| (|Matrix| |#1|))) (|Matrix| |#1|) (|Automorphism| |#1|) (|Mapping| |#1| |#1|)) "\\spad{normalForm(M, sig, der)} returns \\spad{[R, A, A^{-1}]} such that the pseudo-linear operator whose matrix in the basis \\spad{y} is \\spad{M} had matrix \\spad{R} in the basis \\spad{z = A y}. \\spad{der} is a \\spad{sig}-derivation."))) NIL NIL -(-887 |VarSet| E RC P) +(-892 |VarSet| E RC P) ((|constructor| (NIL "This package computes square-free decomposition of multivariate polynomials over a coefficient ring which is an arbitrary gcd domain. The requirement on the coefficient domain guarantees that the \\spadfun{content} can be removed so that factors will be primitive as well as square-free. Over an infinite ring of finite characteristic,{}it may not be possible to guarantee that the factors are square-free.")) (|squareFree| (((|Factored| |#4|) |#4|) "\\spad{squareFree(p)} returns the square-free factorization of the polynomial \\spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime."))) NIL NIL -(-888 R) +(-893 R) ((|constructor| (NIL "PointCategory is the category of points in space which may be plotted via the graphics facilities. Functions are provided for defining points and handling elements of points.")) (|extend| (($ $ (|List| |#1|)) "\\spad{extend(x,l,r)} \\undocumented")) (|cross| (($ $ $) "\\spad{cross(p,q)} computes the cross product of the two points \\spad{p} and \\spad{q}. Error if the \\spad{p} and \\spad{q} are not 3 dimensional")) (|dimension| (((|PositiveInteger|) $) "\\spad{dimension(s)} returns the dimension of the point category \\spad{s}.")) (|point| (($ (|List| |#1|)) "\\spad{point(l)} returns a point category defined by a list \\spad{l} of elements from the domain \\spad{R}."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-889 R1 R2) +(-894 R1 R2) ((|constructor| (NIL "This package \\undocumented")) (|map| (((|Point| |#2|) (|Mapping| |#2| |#1|) (|Point| |#1|)) "\\spad{map(f,p)} \\undocumented"))) NIL NIL -(-890 R) +(-895 R) ((|constructor| (NIL "This package \\undocumented")) (|shade| ((|#1| (|Point| |#1|)) "\\spad{shade(pt)} returns the fourth element of the two dimensional point,{} \\spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} shade to express a fourth dimension.")) (|hue| ((|#1| (|Point| |#1|)) "\\spad{hue(pt)} returns the third element of the two dimensional point,{} \\spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} hue to express a third dimension.")) (|color| ((|#1| (|Point| |#1|)) "\\spad{color(pt)} returns the fourth element of the point,{} \\spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} color to express a fourth dimension.")) (|phiCoord| ((|#1| (|Point| |#1|)) "\\spad{phiCoord(pt)} returns the third element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical coordinate system.")) (|thetaCoord| ((|#1| (|Point| |#1|)) "\\spad{thetaCoord(pt)} returns the second element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical or a cylindrical coordinate system.")) (|rCoord| ((|#1| (|Point| |#1|)) "\\spad{rCoord(pt)} returns the first element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical or a cylindrical coordinate system.")) (|zCoord| ((|#1| (|Point| |#1|)) "\\spad{zCoord(pt)} returns the third element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian or a cylindrical coordinate system.")) (|yCoord| ((|#1| (|Point| |#1|)) "\\spad{yCoord(pt)} returns the second element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian coordinate system.")) (|xCoord| ((|#1| (|Point| |#1|)) "\\spad{xCoord(pt)} returns the first element of the point,{} \\spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian coordinate system."))) NIL NIL -(-891 K) +(-896 K) ((|constructor| (NIL "This is the description of any package which provides partial functions on a domain belonging to TranscendentalFunctionCategory.")) (|acschIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acschIfCan(z)} returns acsch(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asechIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asechIfCan(z)} returns asech(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acothIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acothIfCan(z)} returns acoth(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|atanhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{atanhIfCan(z)} returns atanh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acoshIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acoshIfCan(z)} returns acosh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asinhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asinhIfCan(z)} returns asinh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cschIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cschIfCan(z)} returns csch(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|sechIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{sechIfCan(z)} returns sech(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cothIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cothIfCan(z)} returns coth(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|tanhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{tanhIfCan(z)} returns tanh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|coshIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{coshIfCan(z)} returns cosh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|sinhIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{sinhIfCan(z)} returns sinh(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acscIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acscIfCan(z)} returns acsc(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asecIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asecIfCan(z)} returns asec(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acotIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acotIfCan(z)} returns acot(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|atanIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{atanIfCan(z)} returns atan(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|acosIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{acosIfCan(z)} returns acos(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|asinIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{asinIfCan(z)} returns asin(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cscIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cscIfCan(z)} returns csc(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|secIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{secIfCan(z)} returns sec(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cotIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cotIfCan(z)} returns cot(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|tanIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{tanIfCan(z)} returns tan(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|cosIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{cosIfCan(z)} returns cos(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|sinIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{sinIfCan(z)} returns sin(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|logIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{logIfCan(z)} returns log(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|expIfCan| (((|Union| |#1| "failed") |#1|) "\\spad{expIfCan(z)} returns exp(\\spad{z}) if possible,{} and \"failed\" otherwise.")) (|nthRootIfCan| (((|Union| |#1| "failed") |#1| (|NonNegativeInteger|)) "\\spad{nthRootIfCan(z,n)} returns the \\spad{n}th root of \\spad{z} if possible,{} and \"failed\" otherwise."))) NIL NIL -(-892 R E OV PPR) +(-897 R E OV PPR) ((|constructor| (NIL "This package \\undocumented{}")) (|map| ((|#4| (|Mapping| |#4| (|Polynomial| |#1|)) |#4|) "\\spad{map(f,p)} \\undocumented{}")) (|pushup| ((|#4| |#4| (|List| |#3|)) "\\spad{pushup(p,lv)} \\undocumented{}") ((|#4| |#4| |#3|) "\\spad{pushup(p,v)} \\undocumented{}")) (|pushdown| ((|#4| |#4| (|List| |#3|)) "\\spad{pushdown(p,lv)} \\undocumented{}") ((|#4| |#4| |#3|) "\\spad{pushdown(p,v)} \\undocumented{}")) (|variable| (((|Union| $ "failed") (|Symbol|)) "\\spad{variable(s)} makes an element from symbol \\spad{s} or fails")) (|convert| (((|Symbol|) $) "\\spad{convert(x)} converts \\spad{x} to a symbol"))) NIL NIL -(-893 K R UP -3078) +(-898 K R UP -3088) ((|constructor| (NIL "In this package \\spad{K} is a finite field,{} \\spad{R} is a ring of univariate polynomials over \\spad{K},{} and \\spad{F} is a monogenic algebra over \\spad{R}. We require that \\spad{F} is monogenic,{} \\spadignore{i.e.} that \\spad{F = K[x,y]/(f(x,y))},{} because the integral basis algorithm used will factor the polynomial \\spad{f(x,y)}. The package provides a function to compute the integral closure of \\spad{R} in the quotient field of \\spad{F} as well as a function to compute a \"local integral basis\" at a specific prime.")) (|reducedDiscriminant| ((|#2| |#3|) "\\spad{reducedDiscriminant(up)} \\undocumented")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) |#2|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv] } containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of the framed algebra \\spad{F}. \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If 'basis' is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix 'basisInv' contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if 'basisInv' is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv] } containing information regarding the integral closure of \\spad{R} in the quotient field of the framed algebra \\spad{F}. \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If 'basis' is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of 'basis' contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix 'basisInv' contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if 'basisInv' is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}."))) NIL NIL -(-894 R |Var| |Expon| |Dpoly|) +(-899 R |Var| |Expon| |Dpoly|) ((|constructor| (NIL "\\spadtype{QuasiAlgebraicSet} constructs a domain representing quasi-algebraic sets,{} which is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the common zeros of a polynomial \\spad{f} (the defining polynomial for the inequation). This domain provides simplification of a user-given representation using groebner basis computations. There are two simplification routines: the first function \\spadfun{idealSimplify} uses groebner basis of ideals alone,{} while the second,{} \\spadfun{simplify} uses both groebner basis and factorization. The resulting defining equations \\spad{L} always form a groebner basis,{} and the resulting defining inequation \\spad{f} is always reduced. The function \\spadfun{simplify} may be applied several times if desired. A third simplification routine \\spadfun{radicalSimplify} is provided in \\spadtype{QuasiAlgebraicSet2} for comparison study only,{} as it is inefficient compared to the other two,{} as well as is restricted to only certain coefficient domains. For detail analysis and a comparison of the three methods,{} please consult the reference cited. \\blankline A polynomial function \\spad{q} defined on the quasi-algebraic set is equivalent to its reduced form with respect to \\spad{L}. While this may be obtained using the usual normal form algorithm,{} there is no canonical form for \\spad{q}. \\blankline The ordering in groebner basis computation is determined by the data type of the input polynomials. If it is possible we suggest to use refinements of total degree orderings.")) (|simplify| (($ $) "\\spad{simplify(s)} returns a different and presumably simpler representation of \\spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using a heuristic algorithm based on factoring.")) (|idealSimplify| (($ $) "\\spad{idealSimplify(s)} returns a different and presumably simpler representation of \\spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using Buchberger's algorithm.")) (|definingInequation| ((|#4| $) "\\spad{definingInequation(s)} returns a single defining polynomial for the inequation,{} that is,{} the Zariski open part of \\spad{s}.")) (|definingEquations| (((|List| |#4|) $) "\\spad{definingEquations(s)} returns a list of defining polynomials for equations,{} that is,{} for the Zariski closed part of \\spad{s}.")) (|empty?| (((|Boolean|) $) "\\spad{empty?(s)} returns \\spad{true} if the quasialgebraic set \\spad{s} has no points,{} and \\spad{false} otherwise.")) (|setStatus| (($ $ (|Union| (|Boolean|) #1="failed")) "\\spad{setStatus(s,t)} returns the same representation for \\spad{s},{} but asserts the following: if \\spad{t} is \\spad{true},{} then \\spad{s} is empty,{} if \\spad{t} is \\spad{false},{} then \\spad{s} is non-empty,{} and if \\spad{t} = \"failed\",{} then no assertion is made (that is,{} \"don't know\"). Note: for internal use only,{} with care.")) (|status| (((|Union| (|Boolean|) #1#) $) "\\spad{status(s)} returns \\spad{true} if the quasi-algebraic set is empty,{} \\spad{false} if it is not,{} and \"failed\" if not yet known")) (|quasiAlgebraicSet| (($ (|List| |#4|) |#4|) "\\spad{quasiAlgebraicSet(pl,q)} returns the quasi-algebraic set with defining equations \\spad{p} = 0 for \\spad{p} belonging to the list \\spad{pl},{} and defining inequation \\spad{q} ~= 0.")) (|empty| (($) "\\spad{empty()} returns the empty quasi-algebraic set"))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-255))))) -(-895 |vl| |nv|) +((-12 (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-257))))) +(-900 |vl| |nv|) ((|constructor| (NIL "\\spadtype{QuasiAlgebraicSet2} adds a function \\spadfun{radicalSimplify} which uses \\spadtype{IdealDecompositionPackage} to simplify the representation of a quasi-algebraic set. A quasi-algebraic set is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the common zeros of a polynomial \\spad{f} (the defining polynomial for the inequation). Quasi-algebraic sets are implemented in the domain \\spadtype{QuasiAlgebraicSet},{} where two simplification routines are provided: \\spadfun{idealSimplify} and \\spadfun{simplify}. The function \\spadfun{radicalSimplify} is added for comparison study only. Because the domain \\spadtype{IdealDecompositionPackage} provides facilities for computing with radical ideals,{} it is necessary to restrict the ground ring to the domain \\spadtype{Fraction Integer},{} and the polynomial ring to be of type \\spadtype{DistributedMultivariatePolynomial}. The routine \\spadfun{radicalSimplify} uses these to compute groebner basis of radical ideals and is inefficient and restricted when compared to the two in \\spadtype{QuasiAlgebraicSet}.")) (|radicalSimplify| (((|QuasiAlgebraicSet| (|Fraction| (|Integer|)) (|OrderedVariableList| |#1|) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|)))) (|QuasiAlgebraicSet| (|Fraction| (|Integer|)) (|OrderedVariableList| |#1|) (|DirectProduct| |#2| (|NonNegativeInteger|)) (|DistributedMultivariatePolynomial| |#1| (|Fraction| (|Integer|))))) "\\spad{radicalSimplify(s)} returns a different and presumably simpler representation of \\spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using using groebner basis of radical ideals"))) NIL NIL -(-896 R E V P TS) +(-901 R E V P TS) ((|constructor| (NIL "A package for removing redundant quasi-components and redundant branches when decomposing a variety by means of quasi-components of regular triangular sets. \\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|branchIfCan| (((|Union| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|))) "failed") (|List| |#4|) |#5| (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{branchIfCan(leq,{}ts,{}lineq,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement.")) (|prepareDecompose| (((|List| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|)))) (|List| |#4|) (|List| |#5|) (|Boolean|) (|Boolean|)) "\\axiom{prepareDecompose(lp,{}lts,{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousCases| (((|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) (|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)))) "\\axiom{removeSuperfluousCases(llpwt)} is an internal subroutine,{} exported only for developement.")) (|subCase?| (((|Boolean|) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) "\\axiom{subCase?(\\spad{lpwt1},{}\\spad{lpwt2})} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousQuasiComponents| (((|List| |#5|) (|List| |#5|)) "\\axiom{removeSuperfluousQuasiComponents(lts)} removes from \\axiom{lts} any \\spad{ts} such that \\axiom{subQuasiComponent?(ts,{}us)} holds for another \\spad{us} in \\axiom{lts}.")) (|subQuasiComponent?| (((|Boolean|) |#5| (|List| |#5|)) "\\axiom{subQuasiComponent?(ts,{}lus)} returns \\spad{true} iff \\axiom{subQuasiComponent?(ts,{}us)} holds for one \\spad{us} in \\spad{lus}.") (((|Boolean|) |#5| |#5|) "\\axiom{subQuasiComponent?(ts,{}us)} returns \\spad{true} iff \\axiomOpFrom{internalSubQuasiComponent?}{QuasiComponentPackage} returs \\spad{true}.")) (|internalSubQuasiComponent?| (((|Union| (|Boolean|) "failed") |#5| |#5|) "\\axiom{internalSubQuasiComponent?(ts,{}us)} returns a boolean \\spad{b} value if the fact that the regular zero set of \\axiom{us} contains that of \\axiom{ts} can be decided (and in that case \\axiom{\\spad{b}} gives this inclusion) otherwise returns \\axiom{\"failed\"}.")) (|infRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{infRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalInfRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalInfRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalSubPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalSubPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}} assuming that these lists are sorted increasingly \\spad{w}.\\spad{r}.\\spad{t}. \\axiomOpFrom{infRittWu?}{RecursivePolynomialCategory}.")) (|subPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{subPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}}.")) (|subTriSet?| (((|Boolean|) |#5| |#5|) "\\axiom{subTriSet?(ts,{}us)} returns \\spad{true} iff \\axiom{ts} is a sub-set of \\axiom{us}.")) (|moreAlgebraic?| (((|Boolean|) |#5| |#5|) "\\axiom{moreAlgebraic?(ts,{}us)} returns \\spad{false} iff \\axiom{ts} and \\axiom{us} are both empty,{} or \\axiom{ts} has less elements than \\axiom{us},{} or some variable is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{us} and is not \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ts}.")) (|algebraicSort| (((|List| |#5|) (|List| |#5|)) "\\axiom{algebraicSort(lts)} sorts \\axiom{lts} \\spad{w}.\\spad{r}.\\spad{t} \\axiomOpFrom{supDimElseRittWu?}{QuasiComponentPackage}.")) (|supDimElseRittWu?| (((|Boolean|) |#5| |#5|) "\\axiom{supDimElseRittWu(ts,{}us)} returns \\spad{true} iff \\axiom{ts} has less elements than \\axiom{us} otherwise if \\axiom{ts} has higher rank than \\axiom{us} \\spad{w}.\\spad{r}.\\spad{t}. Riit and Wu ordering.")) (|stopTable!| (((|Void|)) "\\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.")) (|startTable!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableGcd!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement."))) NIL NIL -(-897) +(-902) ((|constructor| (NIL "This domain implements simple database queries")) (|value| (((|String|) $) "\\spad{value(q)} returns the value (\\spadignore{i.e.} right hand side) of \\axiom{\\spad{q}}.")) (|variable| (((|Symbol|) $) "\\spad{variable(q)} returns the variable (\\spadignore{i.e.} left hand side) of \\axiom{\\spad{q}}.")) (|equation| (($ (|Symbol|) (|String|)) "\\spad{equation(s,\"a\")} creates a new equation."))) NIL NIL -(-898 A S) +(-903 A S) ((|constructor| (NIL "QuotientField(\\spad{S}) is the category of fractions of an Integral Domain \\spad{S}.")) (|floor| ((|#2| $) "\\spad{floor(x)} returns the largest integral element below \\spad{x}.")) (|ceiling| ((|#2| $) "\\spad{ceiling(x)} returns the smallest integral element above \\spad{x}.")) (|random| (($) "\\spad{random()} returns a random fraction.")) (|fractionPart| (($ $) "\\spad{fractionPart(x)} returns the fractional part of \\spad{x}. \\spad{x} = wholePart(\\spad{x}) + fractionPart(\\spad{x})")) (|wholePart| ((|#2| $) "\\spad{wholePart(x)} returns the whole part of the fraction \\spad{x} \\spadignore{i.e.} the truncated quotient of the numerator by the denominator.")) (|denominator| (($ $) "\\spad{denominator(x)} is the denominator of the fraction \\spad{x} converted to \\%.")) (|numerator| (($ $) "\\spad{numerator(x)} is the numerator of the fraction \\spad{x} converted to \\%.")) (|denom| ((|#2| $) "\\spad{denom(x)} returns the denominator of the fraction \\spad{x}.")) (|numer| ((|#2| $) "\\spad{numer(x)} returns the numerator of the fraction \\spad{x}.")) (/ (($ |#2| |#2|) "\\spad{d1 / d2} returns the fraction \\spad{d1} divided by \\spad{d2}."))) NIL -((|HasCategory| |#2| (QUOTE (-816))) (|HasCategory| |#2| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-255))) (|HasCategory| |#2| (QUOTE (-945 (-1081)))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-928))) (|HasCategory| |#2| (QUOTE (-735))) (|HasCategory| |#2| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (|HasCategory| |#2| (QUOTE (-1057)))) -(-899 S) +((|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| |#2| (QUOTE (-482))) (|HasCategory| |#2| (QUOTE (-257))) (|HasCategory| |#2| (QUOTE (-950 (-1088)))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-933))) (|HasCategory| |#2| (QUOTE (-740))) (|HasCategory| |#2| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (|HasCategory| |#2| (QUOTE (-1064)))) +(-904 S) ((|constructor| (NIL "QuotientField(\\spad{S}) is the category of fractions of an Integral Domain \\spad{S}.")) (|floor| ((|#1| $) "\\spad{floor(x)} returns the largest integral element below \\spad{x}.")) (|ceiling| ((|#1| $) "\\spad{ceiling(x)} returns the smallest integral element above \\spad{x}.")) (|random| (($) "\\spad{random()} returns a random fraction.")) (|fractionPart| (($ $) "\\spad{fractionPart(x)} returns the fractional part of \\spad{x}. \\spad{x} = wholePart(\\spad{x}) + fractionPart(\\spad{x})")) (|wholePart| ((|#1| $) "\\spad{wholePart(x)} returns the whole part of the fraction \\spad{x} \\spadignore{i.e.} the truncated quotient of the numerator by the denominator.")) (|denominator| (($ $) "\\spad{denominator(x)} is the denominator of the fraction \\spad{x} converted to \\%.")) (|numerator| (($ $) "\\spad{numerator(x)} is the numerator of the fraction \\spad{x} converted to \\%.")) (|denom| ((|#1| $) "\\spad{denom(x)} returns the denominator of the fraction \\spad{x}.")) (|numer| ((|#1| $) "\\spad{numer(x)} returns the numerator of the fraction \\spad{x}.")) (/ (($ |#1| |#1|) "\\spad{d1 / d2} returns the fraction \\spad{d1} divided by \\spad{d2}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-900 A B R S) +(-905 A B R S) ((|constructor| (NIL "This package extends a function between integral domains to a mapping between their quotient fields.")) (|map| ((|#4| (|Mapping| |#2| |#1|) |#3|) "\\spad{map(func,frac)} applies the function \\spad{func} to the numerator and denominator of \\spad{frac}."))) NIL NIL -(-901 |n| K) +(-906 |n| K) ((|constructor| (NIL "This domain provides modest support for quadratic forms.")) (|matrix| (((|SquareMatrix| |#1| |#2|) $) "\\spad{matrix(qf)} creates a square matrix from the quadratic form \\spad{qf}.")) (|quadraticForm| (($ (|SquareMatrix| |#1| |#2|)) "\\spad{quadraticForm(m)} creates a quadratic form from a symmetric,{} square matrix \\spad{m}."))) NIL NIL -(-902) +(-907) ((|constructor| (NIL "This domain represents the syntax of a quasiquote \\indented{2}{expression.}")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the syntax for the expression being quoted."))) NIL NIL -(-903 S) +(-908 S) ((|constructor| (NIL "A queue is a bag where the first item inserted is the first item extracted.")) (|back| ((|#1| $) "\\spad{back(q)} returns the element at the back of the queue. The queue \\spad{q} is unchanged by this operation. Error: if \\spad{q} is empty.")) (|front| ((|#1| $) "\\spad{front(q)} returns the element at the front of the queue. The queue \\spad{q} is unchanged by this operation. Error: if \\spad{q} is empty.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length(q)} returns the number of elements in the queue. Note: \\axiom{length(\\spad{q}) = \\#q}.")) (|rotate!| (($ $) "\\spad{rotate! q} rotates queue \\spad{q} so that the element at the front of the queue goes to the back of the queue. Note: rotate! \\spad{q} is equivalent to enqueue!(dequeue!(\\spad{q})).")) (|dequeue!| ((|#1| $) "\\spad{dequeue! s} destructively extracts the first (top) element from queue \\spad{q}. The element previously second in the queue becomes the first element. Error: if \\spad{q} is empty.")) (|enqueue!| ((|#1| |#1| $) "\\spad{enqueue!(x,q)} inserts \\spad{x} into the queue \\spad{q} at the back end."))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL -(-904 R) +(-909 R) ((|constructor| (NIL "\\spadtype{Quaternion} implements quaternions over a \\indented{2}{commutative ring. The main constructor function is \\spadfun{quatern}} \\indented{2}{which takes 4 arguments: the real part,{} the \\spad{i} imaginary part,{} the \\spad{j}} \\indented{2}{imaginary part and the \\spad{k} imaginary part.}"))) -((-3971 |has| |#1| (-243)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-309))) (OR (|HasCategory| |#1| (QUOTE (-243))) (|HasCategory| |#1| (QUOTE (-309)))) (|HasCategory| |#1| (QUOTE (-243))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (|%list| (QUOTE -449) (QUOTE (-1081)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -239) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-967))) (|HasCategory| |#1| (QUOTE (-479)))) -(-905 S R) +((-3982 |has| |#1| (-245)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-311))) (OR (|HasCategory| |#1| (QUOTE (-245))) (|HasCategory| |#1| (QUOTE (-311)))) (|HasCategory| |#1| (QUOTE (-245))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (|%list| (QUOTE -452) (QUOTE (-1088)) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))) (|HasCategory| |#1| (|%list| (QUOTE -241) (|devaluate| |#1|) (|devaluate| |#1|))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-972))) (|HasCategory| |#1| (QUOTE (-482)))) +(-910 S R) ((|constructor| (NIL "\\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(q)} returns \\spad{q} as a rational number,{} or \"failed\" if this is not possible. Note: if \\spad{rational?(q)} is \\spad{true},{} the conversion can be done and the rational number will be returned.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(q)} tries to convert \\spad{q} into a rational number. Error: if this is not possible. If \\spad{rational?(q)} is \\spad{true},{} the conversion will be done and the rational number returned.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(q)} returns {\\it \\spad{true}} if all the imaginary parts of \\spad{q} are zero and the real part can be converted into a rational number,{} and {\\it \\spad{false}} otherwise.")) (|abs| ((|#2| $) "\\spad{abs(q)} computes the absolute value of quaternion \\spad{q} (sqrt of norm).")) (|real| ((|#2| $) "\\spad{real(q)} extracts the real part of quaternion \\spad{q}.")) (|quatern| (($ |#2| |#2| |#2| |#2|) "\\spad{quatern(r,i,j,k)} constructs a quaternion from scalars.")) (|norm| ((|#2| $) "\\spad{norm(q)} computes the norm of \\spad{q} (the sum of the squares of the components).")) (|imagK| ((|#2| $) "\\spad{imagK(q)} extracts the imaginary \\spad{k} part of quaternion \\spad{q}.")) (|imagJ| ((|#2| $) "\\spad{imagJ(q)} extracts the imaginary \\spad{j} part of quaternion \\spad{q}.")) (|imagI| ((|#2| $) "\\spad{imagI(q)} extracts the imaginary \\spad{i} part of quaternion \\spad{q}.")) (|conjugate| (($ $) "\\spad{conjugate(q)} negates the imaginary parts of quaternion \\spad{q}."))) NIL -((|HasCategory| |#2| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-967))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-243)))) -(-906 R) +((|HasCategory| |#2| (QUOTE (-482))) (|HasCategory| |#2| (QUOTE (-972))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-245)))) +(-911 R) ((|constructor| (NIL "\\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") $) "\\spad{rationalIfCan(q)} returns \\spad{q} as a rational number,{} or \"failed\" if this is not possible. Note: if \\spad{rational?(q)} is \\spad{true},{} the conversion can be done and the rational number will be returned.")) (|rational| (((|Fraction| (|Integer|)) $) "\\spad{rational(q)} tries to convert \\spad{q} into a rational number. Error: if this is not possible. If \\spad{rational?(q)} is \\spad{true},{} the conversion will be done and the rational number returned.")) (|rational?| (((|Boolean|) $) "\\spad{rational?(q)} returns {\\it \\spad{true}} if all the imaginary parts of \\spad{q} are zero and the real part can be converted into a rational number,{} and {\\it \\spad{false}} otherwise.")) (|abs| ((|#1| $) "\\spad{abs(q)} computes the absolute value of quaternion \\spad{q} (sqrt of norm).")) (|real| ((|#1| $) "\\spad{real(q)} extracts the real part of quaternion \\spad{q}.")) (|quatern| (($ |#1| |#1| |#1| |#1|) "\\spad{quatern(r,i,j,k)} constructs a quaternion from scalars.")) (|norm| ((|#1| $) "\\spad{norm(q)} computes the norm of \\spad{q} (the sum of the squares of the components).")) (|imagK| ((|#1| $) "\\spad{imagK(q)} extracts the imaginary \\spad{k} part of quaternion \\spad{q}.")) (|imagJ| ((|#1| $) "\\spad{imagJ(q)} extracts the imaginary \\spad{j} part of quaternion \\spad{q}.")) (|imagI| ((|#1| $) "\\spad{imagI(q)} extracts the imaginary \\spad{i} part of quaternion \\spad{q}.")) (|conjugate| (($ $) "\\spad{conjugate(q)} negates the imaginary parts of quaternion \\spad{q}."))) -((-3971 |has| |#1| (-243)) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3982 |has| |#1| (-245)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-907 QR R QS S) +(-912 QR R QS S) ((|constructor| (NIL "\\spadtype{QuaternionCategoryFunctions2} implements functions between two quaternion domains. The function \\spadfun{map} is used by the system interpreter to coerce between quaternion types.")) (|map| ((|#3| (|Mapping| |#4| |#2|) |#1|) "\\spad{map(f,u)} maps \\spad{f} onto the component parts of the quaternion \\spad{u}."))) NIL NIL -(-908 S) +(-913 S) ((|constructor| (NIL "Linked List implementation of a Queue")) (|queue| (($ (|List| |#1|)) "\\spad{queue([x,y,...,z])} creates a queue with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last (bottom) element \\spad{z}."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-909 S) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-914 S) ((|constructor| (NIL "The \\spad{RadicalCategory} is a model for the rational numbers.")) (** (($ $ (|Fraction| (|Integer|))) "\\spad{x ** y} is the rational exponentiation of \\spad{x} by the power \\spad{y}.")) (|nthRoot| (($ $ (|Integer|)) "\\spad{nthRoot(x,n)} returns the \\spad{n}th root of \\spad{x}.")) (|sqrt| (($ $) "\\spad{sqrt(x)} returns the square root of \\spad{x}."))) NIL NIL -(-910) +(-915) ((|constructor| (NIL "The \\spad{RadicalCategory} is a model for the rational numbers.")) (** (($ $ (|Fraction| (|Integer|))) "\\spad{x ** y} is the rational exponentiation of \\spad{x} by the power \\spad{y}.")) (|nthRoot| (($ $ (|Integer|)) "\\spad{nthRoot(x,n)} returns the \\spad{n}th root of \\spad{x}.")) (|sqrt| (($ $) "\\spad{sqrt(x)} returns the square root of \\spad{x}."))) NIL NIL -(-911 -3078 UP UPUP |radicnd| |n|) +(-916 -3088 UP UPUP |radicnd| |n|) ((|constructor| (NIL "Function field defined by y**n = \\spad{f}(\\spad{x})."))) -((-3971 |has| (-345 |#2|) (-309)) (-3976 |has| (-345 |#2|) (-309)) (-3970 |has| (-345 |#2|) (-309)) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . 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T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-347 |#2|) (QUOTE (-118))) (|HasCategory| (-347 |#2|) (QUOTE (-120))) (|HasCategory| (-347 |#2|) (QUOTE (-298))) (OR (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-298)))) (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-317))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-190))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (|HasCategory| (-347 |#2|) (QUOTE (-298)))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-190))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-189))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (|HasCategory| (-347 |#2|) (QUOTE (-298)))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088))))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-298))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088)))))) (OR (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088))))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-811 (-1088)))))) (|HasCategory| (-347 |#2|) (QUOTE (-580 (-483)))) (OR (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-950 (-347 (-483)))))) (|HasCategory| (-347 |#2|) (QUOTE (-950 (-347 (-483))))) (|HasCategory| (-347 |#2|) (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-317))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-189))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-811 (-1088))))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-190))) (|HasCategory| (-347 |#2|) (QUOTE (-311)))) (-12 (|HasCategory| (-347 |#2|) (QUOTE (-311))) (|HasCategory| (-347 |#2|) (QUOTE (-809 (-1088)))))) +(-917 |bb|) ((|constructor| (NIL "This domain allows rational numbers to be presented as repeating decimal expansions or more generally as repeating expansions in any base.")) (|fractRadix| (($ (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{fractRadix(pre,cyc)} creates a fractional radix expansion from a list of prefix ragits and a list of cyclic ragits. For example,{} \\spad{fractRadix([1],[6])} will return \\spad{0.16666666...}.")) (|wholeRadix| (($ (|List| (|Integer|))) "\\spad{wholeRadix(l)} creates an integral radix expansion from a list of ragits. For example,{} \\spad{wholeRadix([1,3,4])} will return \\spad{134}.")) (|cycleRagits| (((|List| (|Integer|)) $) "\\spad{cycleRagits(rx)} returns the cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \\spad{x = 3/28 = 0.10 714285 714285 ...},{} then \\spad{cycleRagits(x) = [7,1,4,2,8,5]}.")) (|prefixRagits| (((|List| (|Integer|)) $) "\\spad{prefixRagits(rx)} returns the non-cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \\spad{x = 3/28 = 0.10 714285 714285 ...},{} then \\spad{prefixRagits(x)=[1,0]}.")) (|fractRagits| (((|Stream| (|Integer|)) $) "\\spad{fractRagits(rx)} returns the ragits of the fractional part of a radix expansion.")) (|wholeRagits| (((|List| (|Integer|)) $) "\\spad{wholeRagits(rx)} returns the ragits of the integer part of a radix expansion.")) (|fractionPart| (((|Fraction| (|Integer|)) $) "\\spad{fractionPart(rx)} returns the fractional part of a radix expansion."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-480) (QUOTE (-816))) (|HasCategory| (-480) (QUOTE (-945 (-1081)))) (|HasCategory| (-480) (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-118))) (|HasCategory| (-480) (QUOTE (-550 (-469)))) (|HasCategory| (-480) (QUOTE (-928))) (|HasCategory| (-480) (QUOTE (-735))) (|HasCategory| (-480) (QUOTE (-751))) (OR (|HasCategory| (-480) (QUOTE (-735))) (|HasCategory| (-480) (QUOTE (-751)))) (|HasCategory| (-480) (QUOTE (-945 (-480)))) (|HasCategory| (-480) (QUOTE (-1057))) (|HasCategory| (-480) (QUOTE (-791 (-325)))) (|HasCategory| (-480) (QUOTE (-791 (-480)))) (|HasCategory| (-480) (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-480) (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-480) (QUOTE (-187))) (|HasCategory| (-480) (QUOTE (-806 (-1081)))) (|HasCategory| (-480) (QUOTE (-188))) (|HasCategory| (-480) (QUOTE (-804 (-1081)))) (|HasCategory| (-480) (QUOTE (-449 (-1081) (-480)))) (|HasCategory| (-480) (QUOTE (-257 (-480)))) (|HasCategory| (-480) (QUOTE (-239 (-480) (-480)))) (|HasCategory| (-480) (QUOTE (-255))) (|HasCategory| (-480) (QUOTE (-479))) (|HasCategory| (-480) (QUOTE (-577 (-480)))) (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-816)))) (OR (-12 (|HasCategory| $ (QUOTE (-116))) (|HasCategory| (-480) (QUOTE (-816)))) (|HasCategory| (-480) (QUOTE (-116))))) -(-913) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-483) (QUOTE (-821))) (|HasCategory| (-483) (QUOTE (-950 (-1088)))) (|HasCategory| (-483) (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-120))) (|HasCategory| (-483) (QUOTE (-553 (-472)))) (|HasCategory| (-483) (QUOTE (-933))) (|HasCategory| (-483) (QUOTE (-740))) (|HasCategory| (-483) (QUOTE (-756))) (OR (|HasCategory| (-483) (QUOTE (-740))) (|HasCategory| (-483) (QUOTE (-756)))) (|HasCategory| (-483) (QUOTE (-950 (-483)))) (|HasCategory| (-483) (QUOTE (-1064))) (|HasCategory| (-483) (QUOTE (-796 (-327)))) (|HasCategory| (-483) (QUOTE (-796 (-483)))) (|HasCategory| (-483) (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-483) (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-483) (QUOTE (-189))) (|HasCategory| (-483) (QUOTE (-811 (-1088)))) (|HasCategory| (-483) (QUOTE (-190))) (|HasCategory| (-483) (QUOTE (-809 (-1088)))) (|HasCategory| (-483) (QUOTE (-452 (-1088) (-483)))) (|HasCategory| (-483) (QUOTE (-259 (-483)))) (|HasCategory| (-483) (QUOTE (-241 (-483) (-483)))) (|HasCategory| (-483) (QUOTE (-257))) (|HasCategory| (-483) (QUOTE (-482))) (|HasCategory| (-483) (QUOTE (-580 (-483)))) (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-821)))) (OR (-12 (|HasCategory| $ (QUOTE (-118))) (|HasCategory| (-483) (QUOTE (-821)))) (|HasCategory| (-483) (QUOTE (-118))))) +(-918) ((|constructor| (NIL "This package provides tools for creating radix expansions.")) (|radix| (((|Any|) (|Fraction| (|Integer|)) (|Integer|)) "\\spad{radix(x,b)} converts \\spad{x} to a radix expansion in base \\spad{b}."))) NIL NIL -(-914) +(-919) ((|constructor| (NIL "Random number generators \\indented{2}{All random numbers used in the system should originate from} \\indented{2}{the same generator.\\space{2}This package is intended to be the source.}")) (|seed| (((|Integer|)) "\\spad{seed()} returns the current seed value.")) (|reseed| (((|Void|) (|Integer|)) "\\spad{reseed(n)} restarts the random number generator at \\spad{n}.")) (|size| (((|Integer|)) "\\spad{size()} is the base of the random number generator")) (|randnum| (((|Integer|) (|Integer|)) "\\spad{randnum(n)} is a random number between 0 and \\spad{n}.") (((|Integer|)) "\\spad{randnum()} is a random number between 0 and size()."))) NIL NIL -(-915 RP) +(-920 RP) ((|factorSquareFree| (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(p)} factors an extended squareFree polynomial \\spad{p} over the rational numbers.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} factors an extended polynomial \\spad{p} over the rational numbers."))) NIL NIL -(-916 S) +(-921 S) ((|constructor| (NIL "rational number testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991")) (|rationalIfCan| (((|Union| (|Fraction| (|Integer|)) "failed") |#1|) "\\spad{rationalIfCan(x)} returns \\spad{x} as a rational number,{} \"failed\" if \\spad{x} is not a rational number.")) (|rational?| (((|Boolean|) |#1|) "\\spad{rational?(x)} returns \\spad{true} if \\spad{x} is a rational number,{} \\spad{false} otherwise.")) (|rational| (((|Fraction| (|Integer|)) |#1|) "\\spad{rational(x)} returns \\spad{x} as a rational number; error if \\spad{x} is not a rational number."))) NIL NIL -(-917 A S) +(-922 A S) ((|constructor| (NIL "A recursive aggregate over a type \\spad{S} is a model for a a directed graph containing values of type \\spad{S}. Recursively,{} a recursive aggregate is a {\\em node} consisting of a \\spadfun{value} from \\spad{S} and 0 or more \\spadfun{children} which are recursive aggregates. A node with no children is called a \\spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infinite loop.")) (|setvalue!| ((|#2| $ |#2|) "\\spad{setvalue!(u,x)} sets the value of node \\spad{u} to \\spad{x}.")) (|setelt| ((|#2| $ "value" |#2|) "\\spad{setelt(a,\"value\",x)} (also written \\axiom{a . value := \\spad{x}}) is equivalent to \\axiom{setvalue!(a,{}\\spad{x})}")) (|setchildren!| (($ $ (|List| $)) "\\spad{setchildren!(u,v)} replaces the current children of node \\spad{u} with the members of \\spad{v} in left-to-right order.")) (|node?| (((|Boolean|) $ $) "\\spad{node?(u,v)} tests if node \\spad{u} is contained in node \\spad{v} (either as a child,{} a child of a child,{} etc.).")) (|child?| (((|Boolean|) $ $) "\\spad{child?(u,v)} tests if node \\spad{u} is a child of node \\spad{v}.")) (|distance| (((|Integer|) $ $) "\\spad{distance(u,v)} returns the path length (an integer) from node \\spad{u} to \\spad{v}.")) (|leaves| (((|List| |#2|) $) "\\spad{leaves(t)} returns the list of values in obtained by visiting the nodes of tree \\axiom{\\spad{t}} in left-to-right order.")) (|cyclic?| (((|Boolean|) $) "\\spad{cyclic?(u)} tests if \\spad{u} has a cycle.")) (|elt| ((|#2| $ "value") "\\spad{elt(u,\"value\")} (also written: \\axiom{a. value}) is equivalent to \\axiom{value(a)}.")) (|value| ((|#2| $) "\\spad{value(u)} returns the value of the node \\spad{u}.")) (|leaf?| (((|Boolean|) $) "\\spad{leaf?(u)} tests if \\spad{u} is a terminal node.")) (|nodes| (((|List| $) $) "\\spad{nodes(u)} returns a list of all of the nodes of aggregate \\spad{u}.")) (|children| (((|List| $) $) "\\spad{children(u)} returns a list of the children of aggregate \\spad{u}."))) NIL -((|HasAttribute| |#1| (QUOTE -3979)) (|HasCategory| |#2| (QUOTE (-1007)))) -(-918 S) +((|HasAttribute| |#1| (QUOTE -3990)) (|HasCategory| |#2| (QUOTE (-1012)))) +(-923 S) ((|constructor| (NIL "A recursive aggregate over a type \\spad{S} is a model for a a directed graph containing values of type \\spad{S}. Recursively,{} a recursive aggregate is a {\\em node} consisting of a \\spadfun{value} from \\spad{S} and 0 or more \\spadfun{children} which are recursive aggregates. A node with no children is called a \\spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infinite loop.")) (|setvalue!| ((|#1| $ |#1|) "\\spad{setvalue!(u,x)} sets the value of node \\spad{u} to \\spad{x}.")) (|setelt| ((|#1| $ "value" |#1|) "\\spad{setelt(a,\"value\",x)} (also written \\axiom{a . value := \\spad{x}}) is equivalent to \\axiom{setvalue!(a,{}\\spad{x})}")) (|setchildren!| (($ $ (|List| $)) "\\spad{setchildren!(u,v)} replaces the current children of node \\spad{u} with the members of \\spad{v} in left-to-right order.")) (|node?| (((|Boolean|) $ $) "\\spad{node?(u,v)} tests if node \\spad{u} is contained in node \\spad{v} (either as a child,{} a child of a child,{} etc.).")) (|child?| (((|Boolean|) $ $) "\\spad{child?(u,v)} tests if node \\spad{u} is a child of node \\spad{v}.")) (|distance| (((|Integer|) $ $) "\\spad{distance(u,v)} returns the path length (an integer) from node \\spad{u} to \\spad{v}.")) (|leaves| (((|List| |#1|) $) "\\spad{leaves(t)} returns the list of values in obtained by visiting the nodes of tree \\axiom{\\spad{t}} in left-to-right order.")) (|cyclic?| (((|Boolean|) $) "\\spad{cyclic?(u)} tests if \\spad{u} has a cycle.")) (|elt| ((|#1| $ "value") "\\spad{elt(u,\"value\")} (also written: \\axiom{a. value}) is equivalent to \\axiom{value(a)}.")) (|value| ((|#1| $) "\\spad{value(u)} returns the value of the node \\spad{u}.")) (|leaf?| (((|Boolean|) $) "\\spad{leaf?(u)} tests if \\spad{u} is a terminal node.")) (|nodes| (((|List| $) $) "\\spad{nodes(u)} returns a list of all of the nodes of aggregate \\spad{u}.")) (|children| (((|List| $) $) "\\spad{children(u)} returns a list of the children of aggregate \\spad{u}."))) NIL NIL -(-919 S) +(-924 S) ((|constructor| (NIL "\\axiomType{RealClosedField} provides common acces functions for all real closed fields.")) (|approximate| (((|Fraction| (|Integer|)) $ $) "\\axiom{approximate(\\spad{n},{}\\spad{p})} gives an approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|rename| (($ $ (|OutputForm|)) "\\axiom{rename(\\spad{x},{}name)} gives a new number that prints as name")) (|rename!| (($ $ (|OutputForm|)) "\\axiom{rename!(\\spad{x},{}name)} changes the way \\axiom{\\spad{x}} is printed")) (|sqrt| (($ (|Integer|)) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} ** (1/2)}") (($ (|Fraction| (|Integer|))) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} ** (1/2)}") (($ $) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} ** (1/2)}") (($ $ (|PositiveInteger|)) "\\axiom{sqrt(\\spad{x},{}\\spad{n})} is \\axiom{\\spad{x} ** (1/n)}")) (|allRootsOf| (((|List| $) (|Polynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely")) (|rootOf| (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} creates the \\spad{n}th root for the order of \\axiom{pol} and gives it unique name") (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|) (|OutputForm|)) "\\axiom{rootOf(pol,{}\\spad{n},{}name)} creates the \\spad{n}th root for the order of \\axiom{pol} and names it \\axiom{name}")) (|mainValue| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainValue(\\spad{x})} is the expression of \\axiom{\\spad{x}} in terms of \\axiom{SparseUnivariatePolynomial(\\$)}")) (|mainDefiningPolynomial| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainDefiningPolynomial(\\spad{x})} is the defining polynomial for the main algebraic quantity of \\axiom{\\spad{x}}")) (|mainForm| (((|Union| (|OutputForm|) "failed") $) "\\axiom{mainForm(\\spad{x})} is the main algebraic quantity name of \\axiom{\\spad{x}}"))) NIL NIL -(-920) +(-925) ((|constructor| (NIL "\\axiomType{RealClosedField} provides common acces functions for all real closed fields.")) (|approximate| (((|Fraction| (|Integer|)) $ $) "\\axiom{approximate(\\spad{n},{}\\spad{p})} gives an approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|rename| (($ $ (|OutputForm|)) "\\axiom{rename(\\spad{x},{}name)} gives a new number that prints as name")) (|rename!| (($ $ (|OutputForm|)) "\\axiom{rename!(\\spad{x},{}name)} changes the way \\axiom{\\spad{x}} is printed")) (|sqrt| (($ (|Integer|)) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} ** (1/2)}") (($ (|Fraction| (|Integer|))) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} ** (1/2)}") (($ $) "\\axiom{sqrt(\\spad{x})} is \\axiom{\\spad{x} ** (1/2)}") (($ $ (|PositiveInteger|)) "\\axiom{sqrt(\\spad{x},{}\\spad{n})} is \\axiom{\\spad{x} ** (1/n)}")) (|allRootsOf| (((|List| $) (|Polynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|Polynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Integer|))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| (|Fraction| (|Integer|)))) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely") (((|List| $) (|SparseUnivariatePolynomial| $)) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} naming each uniquely")) (|rootOf| (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} creates the \\spad{n}th root for the order of \\axiom{pol} and gives it unique name") (((|Union| $ "failed") (|SparseUnivariatePolynomial| $) (|PositiveInteger|) (|OutputForm|)) "\\axiom{rootOf(pol,{}\\spad{n},{}name)} creates the \\spad{n}th root for the order of \\axiom{pol} and names it \\axiom{name}")) (|mainValue| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainValue(\\spad{x})} is the expression of \\axiom{\\spad{x}} in terms of \\axiom{SparseUnivariatePolynomial(\\$)}")) (|mainDefiningPolynomial| (((|Union| (|SparseUnivariatePolynomial| $) "failed") $) "\\axiom{mainDefiningPolynomial(\\spad{x})} is the defining polynomial for the main algebraic quantity of \\axiom{\\spad{x}}")) (|mainForm| (((|Union| (|OutputForm|) "failed") $) "\\axiom{mainForm(\\spad{x})} is the main algebraic quantity name of \\axiom{\\spad{x}}"))) -((-3971 . T) (-3976 . T) (-3970 . T) (-3973 . T) (-3972 . T) ((-3980 "*") . T) (-3975 . T)) +((-3982 . T) (-3987 . T) (-3981 . T) (-3984 . T) (-3983 . T) ((-3991 "*") . T) (-3986 . T)) NIL -(-921 R -3078) +(-926 R -3088) ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 1 February 1988 Date Last Updated: 2 November 1995 Keywords: elementary,{} function,{} integration.")) (|rischDE| (((|Record| (|:| |ans| |#2|) (|:| |right| |#2|) (|:| |sol?| (|Boolean|))) (|Integer|) |#2| |#2| (|Symbol|) (|Mapping| (|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|List| |#2|)) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| |#2|)) "\\spad{rischDE(n, f, g, x, lim, ext)} returns \\spad{[y, h, b]} such that \\spad{dy/dx + n df/dx y = h} and \\spad{b := h = g}. The equation \\spad{dy/dx + n df/dx y = g} has no solution if \\spad{h \\~~= g} (\\spad{y} is a partial solution in that case). Notes: \\spad{lim} is a limited integration function,{} and ext is an extended integration function."))) NIL NIL -(-922 R -3078) +(-927 R -3088) ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 12 August 1992 Date Last Updated: 17 August 1992 Keywords: elementary,{} function,{} integration.")) (|rischDEsys| (((|Union| (|List| |#2|) "failed") (|Integer|) |#2| |#2| |#2| (|Symbol|) (|Mapping| (|Union| (|Record| (|:| |mainpart| |#2|) (|:| |limitedlogs| (|List| (|Record| (|:| |coeff| |#2|) (|:| |logand| |#2|))))) "failed") |#2| (|List| |#2|)) (|Mapping| (|Union| (|Record| (|:| |ratpart| |#2|) (|:| |coeff| |#2|)) "failed") |#2| |#2|)) "\\spad{rischDEsys(n, f, g_1, g_2, x,lim,ext)} returns \\spad{y_1.y_2} such that \\spad{(dy1/dx,dy2/dx) + ((0, - n df/dx),(n df/dx,0)) (y1,y2) = (g1,g2)} if \\spad{y_1,y_2} exist,{} \"failed\" otherwise. \\spad{lim} is a limited integration function,{} \\spad{ext} is an extended integration function."))) NIL NIL -(-923 -3078 UP) +(-928 -3088 UP) ((|constructor| (NIL "\\indented{1}{Risch differential equation,{} transcendental case.} Author: Manuel Bronstein Date Created: Jan 1988 Date Last Updated: 2 November 1995")) (|polyRDE| (((|Union| (|:| |ans| (|Record| (|:| |ans| |#2|) (|:| |nosol| (|Boolean|)))) (|:| |eq| (|Record| (|:| |b| |#2|) (|:| |c| |#2|) (|:| |m| (|Integer|)) (|:| |alpha| |#2|) (|:| |beta| |#2|)))) |#2| |#2| |#2| (|Integer|) (|Mapping| |#2| |#2|)) "\\spad{polyRDE(a, B, C, n, D)} returns either: 1. \\spad{[Q, b]} such that \\spad{degree(Q) <= n} and \\indented{3}{\\spad{a Q'+ B Q = C} if \\spad{b = true},{} \\spad{Q} is a partial solution} \\indented{3}{otherwise.} 2. \\spad{[B1, C1, m, \\alpha, \\beta]} such that any polynomial solution \\indented{3}{of degree at most \\spad{n} of \\spad{A Q' + BQ = C} must be of the form} \\indented{3}{\\spad{Q = \\alpha H + \\beta} where \\spad{degree(H) <= m} and} \\indented{3}{\\spad{H} satisfies \\spad{H' + B1 H = C1}.} \\spad{D} is the derivation to use.")) (|baseRDE| (((|Record| (|:| |ans| (|Fraction| |#2|)) (|:| |nosol| (|Boolean|))) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{baseRDE(f, g)} returns a \\spad{[y, b]} such that \\spad{y' + fy = g} if \\spad{b = true},{} \\spad{y} is a partial solution otherwise (no solution in that case). \\spad{D} is the derivation to use.")) (|monomRDE| (((|Union| (|Record| (|:| |a| |#2|) (|:| |b| (|Fraction| |#2|)) (|:| |c| (|Fraction| |#2|)) (|:| |t| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomRDE(f,g,D)} returns \\spad{[A, B, C, T]} such that \\spad{y' + f y = g} has a solution if and only if \\spad{y = Q / T},{} where \\spad{Q} satisfies \\spad{A Q' + B Q = C} and has no normal pole. A and \\spad{T} are polynomials and \\spad{B} and \\spad{C} have no normal poles. \\spad{D} is the derivation to use."))) NIL NIL -(-924 -3078 UP) +(-929 -3088 UP) ((|constructor| (NIL "\\indented{1}{Risch differential equation system,{} transcendental case.} Author: Manuel Bronstein Date Created: 17 August 1992 Date Last Updated: 3 February 1994")) (|baseRDEsys| (((|Union| (|List| (|Fraction| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|)) "\\spad{baseRDEsys(f, g1, g2)} returns fractions \\spad{y_1.y_2} such that \\spad{(y1', y2') + ((0, -f), (f, 0)) (y1,y2) = (g1,g2)} if \\spad{y_1,y_2} exist,{} \"failed\" otherwise.")) (|monomRDEsys| (((|Union| (|Record| (|:| |a| |#2|) (|:| |b| (|Fraction| |#2|)) (|:| |h| |#2|) (|:| |c1| (|Fraction| |#2|)) (|:| |c2| (|Fraction| |#2|)) (|:| |t| |#2|)) "failed") (|Fraction| |#2|) (|Fraction| |#2|) (|Fraction| |#2|) (|Mapping| |#2| |#2|)) "\\spad{monomRDEsys(f,g1,g2,D)} returns \\spad{[A, B, H, C1, C2, T]} such that \\spad{(y1', y2') + ((0, -f), (f, 0)) (y1,y2) = (g1,g2)} has a solution if and only if \\spad{y1 = Q1 / T, y2 = Q2 / T},{} where \\spad{B,C1,C2,Q1,Q2} have no normal poles and satisfy A \\spad{(Q1', Q2') + ((H, -B), (B, H)) (Q1,Q2) = (C1,C2)} \\spad{D} is the derivation to use."))) NIL NIL -(-925 S) +(-930 S) ((|constructor| (NIL "This package exports random distributions")) (|rdHack1| (((|Mapping| |#1|) (|Vector| |#1|) (|Vector| (|Integer|)) (|Integer|)) "\\spad{rdHack1(v,u,n)} \\undocumented")) (|weighted| (((|Mapping| |#1|) (|List| (|Record| (|:| |value| |#1|) (|:| |weight| (|Integer|))))) "\\spad{weighted(l)} \\undocumented")) (|uniform| (((|Mapping| |#1|) (|Set| |#1|)) "\\spad{uniform(s)} \\undocumented"))) NIL NIL -(-926 F1 UP UPUP R F2) +(-931 F1 UP UPUP R F2) ((|constructor| (NIL "\\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 8 November 1994")) (|order| (((|NonNegativeInteger|) (|FiniteDivisor| |#1| |#2| |#3| |#4|) |#3| (|Mapping| |#5| |#1|)) "\\spad{order(f,u,g)} \\undocumented"))) NIL NIL -(-927) +(-932) ((|constructor| (NIL "This domain represents list reduction syntax.")) (|body| (((|SpadAst|) $) "\\spad{body(e)} return the list of expressions being redcued.")) (|operator| (((|SpadAst|) $) "\\spad{operator(e)} returns the magma operation being applied."))) NIL NIL -(-928) +(-933) ((|constructor| (NIL "The category of real numeric domains,{} \\spadignore{i.e.} convertible to floats."))) NIL NIL -(-929 |Pol|) +(-934 |Pol|) ((|constructor| (NIL "\\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the integers to arbitrary user-specified precision. The results are returned as a list of isolating intervals which are expressed as records with \"left\" and \"right\" rational number components.")) (|midpoints| (((|List| (|Fraction| (|Integer|))) (|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))))) "\\spad{midpoints(isolist)} returns the list of midpoints for the list of intervals \\spad{isolist}.")) (|midpoint| (((|Fraction| (|Integer|)) (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{midpoint(int)} returns the midpoint of the interval \\spad{int}.")) (|refine| (((|Union| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) "failed") |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{refine(pol, int, range)} takes a univariate polynomial \\spad{pol} and and isolating interval \\spad{int} containing exactly one real root of \\spad{pol}; the operation returns an isolating interval which is contained within range,{} or \"failed\" if no such isolating interval exists.") (((|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{refine(pol, int, eps)} refines the interval \\spad{int} containing exactly one root of the univariate polynomial \\spad{pol} to size less than the rational number eps.")) (|realZeros| (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{realZeros(pol, int, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol} which lie in the interval expressed by the record \\spad{int}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Fraction| (|Integer|))) "\\spad{realZeros(pol, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{realZeros(pol, range)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol} which lie in the interval expressed by the record range.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1|) "\\spad{realZeros(pol)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol}."))) NIL NIL -(-930 |Pol|) +(-935 |Pol|) ((|constructor| (NIL "\\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the rational numbers to arbitrary user-specified precision. The results are returned as a list of isolating intervals,{} expressed as records with \"left\" and \"right\" rational number components.")) (|refine| (((|Union| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) "failed") |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{refine(pol, int, range)} takes a univariate polynomial \\spad{pol} and and isolating interval \\spad{int} which must contain exactly one real root of \\spad{pol},{} and returns an isolating interval which is contained within range,{} or \"failed\" if no such isolating interval exists.") (((|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{refine(pol, int, eps)} refines the interval \\spad{int} containing exactly one root of the univariate polynomial \\spad{pol} to size less than the rational number eps.")) (|realZeros| (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|)))) (|Fraction| (|Integer|))) "\\spad{realZeros(pol, int, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol} which lie in the interval expressed by the record \\spad{int}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Fraction| (|Integer|))) "\\spad{realZeros(pol, eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \\spad{pol}.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) "\\spad{realZeros(pol, range)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol} which lie in the interval expressed by the record range.") (((|List| (|Record| (|:| |left| (|Fraction| (|Integer|))) (|:| |right| (|Fraction| (|Integer|))))) |#1|) "\\spad{realZeros(pol)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \\spad{pol}."))) NIL NIL -(-931) +(-936) ((|constructor| (NIL "\\indented{1}{This package provides numerical solutions of systems of polynomial} equations for use in ACPLOT.")) (|realSolve| (((|List| (|List| (|Float|))) (|List| (|Polynomial| (|Integer|))) (|List| (|Symbol|)) (|Float|)) "\\spad{realSolve(lp,lv,eps)} = compute the list of the real solutions of the list \\spad{lp} of polynomials with integer coefficients with respect to the variables in \\spad{lv},{} with precision \\spad{eps}.")) (|solve| (((|List| (|Float|)) (|Polynomial| (|Integer|)) (|Float|)) "\\spad{solve(p,eps)} finds the real zeroes of a univariate integer polynomial \\spad{p} with precision \\spad{eps}.") (((|List| (|Float|)) (|Polynomial| (|Fraction| (|Integer|))) (|Float|)) "\\spad{solve(p,eps)} finds the real zeroes of a univariate rational polynomial \\spad{p} with precision \\spad{eps}."))) NIL NIL -(-932 |TheField|) +(-937 |TheField|) ((|constructor| (NIL "This domain implements the real closure of an ordered field.")) (|relativeApprox| (((|Fraction| (|Integer|)) $ $) "\\axiom{relativeApprox(\\spad{n},{}\\spad{p})} gives a relative approximation of \\axiom{\\spad{n}} that has precision \\axiom{\\spad{p}}")) (|mainCharacterization| (((|Union| (|RightOpenIntervalRootCharacterization| $ (|SparseUnivariatePolynomial| $)) "failed") $) "\\axiom{mainCharacterization(\\spad{x})} is the main algebraic quantity of \\axiom{\\spad{x}} (\\axiom{SEG})")) (|algebraicOf| (($ (|RightOpenIntervalRootCharacterization| $ (|SparseUnivariatePolynomial| $)) (|OutputForm|)) "\\axiom{algebraicOf(char)} is the external number"))) -((-3971 . T) (-3976 . T) (-3970 . T) (-3973 . T) (-3972 . T) ((-3980 "*") . T) (-3975 . T)) -((OR (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| (-345 (-480)) (QUOTE (-945 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| (-345 (-480)) (QUOTE (-945 (-345 (-480))))) (|HasCategory| (-345 (-480)) (QUOTE (-945 (-480))))) -(-933 -3078 L) +((-3982 . T) (-3987 . T) (-3981 . T) (-3984 . T) (-3983 . T) ((-3991 "*") . T) (-3986 . T)) +((OR (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| (-347 (-483)) (QUOTE (-950 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| (-347 (-483)) (QUOTE (-950 (-347 (-483))))) (|HasCategory| (-347 (-483)) (QUOTE (-950 (-483))))) +(-938 -3088 L) ((|constructor| (NIL "\\spadtype{ReductionOfOrder} provides functions for reducing the order of linear ordinary differential equations once some solutions are known.")) (|ReduceOrder| (((|Record| (|:| |eq| |#2|) (|:| |op| (|List| |#1|))) |#2| (|List| |#1|)) "\\spad{ReduceOrder(op, [f1,...,fk])} returns \\spad{[op1,[g1,...,gk]]} such that for any solution \\spad{z} of \\spad{op1 z = 0},{} \\spad{y = gk \\int(g_{k-1} \\int(... \\int(g1 \\int z)...)} is a solution of \\spad{op y = 0}. Each \\spad{fi} must satisfy \\spad{op fi = 0}.") ((|#2| |#2| |#1|) "\\spad{ReduceOrder(op, s)} returns \\spad{op1} such that for any solution \\spad{z} of \\spad{op1 z = 0},{} \\spad{y = s \\int z} is a solution of \\spad{op y = 0}. \\spad{s} must satisfy \\spad{op s = 0}."))) NIL NIL -(-934 S) +(-939 S) ((|constructor| (NIL "\\indented{1}{\\spadtype{Reference} is for making a changeable instance} of something.")) (= (((|Boolean|) $ $) "\\spad{a=b} tests if \\spad{a} and \\spad{b} are equal.")) (|setref| ((|#1| $ |#1|) "\\spad{setref(r,s)} reset the reference \\spad{r} to refer to \\spad{s}")) (|deref| ((|#1| $) "\\spad{deref(r)} returns the object referenced by \\spad{r}")) (|ref| (($ |#1|) "\\spad{ref(s)} creates a reference to the object \\spad{s}."))) NIL NIL -(-935 R E V P) +(-940 R E V P) ((|constructor| (NIL "This domain provides an implementation of regular chains. Moreover,{} the operation \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|preprocess| (((|Record| (|:| |val| (|List| |#4|)) (|:| |towers| (|List| $))) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{pre_process(lp,{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|internalZeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalZeroSetSplit(lp,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3})} is an internal subroutine,{} exported only for developement.")) (|zeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(lp,{}\\spad{b1},{}\\spad{b2}.\\spad{b3},{}\\spad{b4})} is an internal subroutine,{} exported only for developement.") (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(lp,{}clos?,{}info?)} has the same specifications as \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory}. Moreover,{} if \\axiom{clos?} then solves in the sense of the Zariski closure else solves in the sense of the regular zeros. If \\axiom{info?} then do print messages during the computations.")) (|internalAugment| (((|List| $) |#4| $ (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalAugment(\\spad{p},{}ts,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement."))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#4| (|%list| (QUOTE -257) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-550 (-469)))) (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#3| (QUOTE (-315))) (|HasCategory| |#4| (QUOTE (-549 (-767)))) (|HasCategory| |#4| (QUOTE (-72)))) -(-936) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#4| (|%list| (QUOTE -259) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-553 (-472)))) (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#3| (QUOTE (-317))) (|HasCategory| |#4| (QUOTE (-552 (-772)))) (|HasCategory| |#4| (QUOTE (-72)))) +(-941) ((|constructor| (NIL "Package for the computation of eigenvalues and eigenvectors. This package works for matrices with coefficients which are rational functions over the integers. (see \\spadtype{Fraction Polynomial Integer}). The eigenvalues and eigenvectors are expressed in terms of radicals.")) (|orthonormalBasis| (((|List| (|Matrix| (|Expression| (|Integer|)))) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{orthonormalBasis(m)} returns the orthogonal matrix \\spad{b} such that \\spad{b*m*(inverse b)} is diagonal. Error: if \\spad{m} is not a symmetric matrix.")) (|gramschmidt| (((|List| (|Matrix| (|Expression| (|Integer|)))) (|List| (|Matrix| (|Expression| (|Integer|))))) "\\spad{gramschmidt(lv)} converts the list of column vectors \\spad{lv} into a set of orthogonal column vectors of euclidean length 1 using the Gram-Schmidt algorithm.")) (|normalise| (((|Matrix| (|Expression| (|Integer|))) (|Matrix| (|Expression| (|Integer|)))) "\\spad{normalise(v)} returns the column vector \\spad{v} divided by its euclidean norm; when possible,{} the vector \\spad{v} is expressed in terms of radicals.")) (|eigenMatrix| (((|Union| (|Matrix| (|Expression| (|Integer|))) "failed") (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{eigenMatrix(m)} returns the matrix \\spad{b} such that \\spad{b*m*(inverse b)} is diagonal,{} or \"failed\" if no such \\spad{b} exists.")) (|radicalEigenvalues| (((|List| (|Expression| (|Integer|))) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{radicalEigenvalues(m)} computes the eigenvalues of the matrix \\spad{m}; when possible,{} the eigenvalues are expressed in terms of radicals.")) (|radicalEigenvector| (((|List| (|Matrix| (|Expression| (|Integer|)))) (|Expression| (|Integer|)) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{radicalEigenvector(c,m)} computes the eigenvector(\\spad{s}) of the matrix \\spad{m} corresponding to the eigenvalue \\spad{c}; when possible,{} values are expressed in terms of radicals.")) (|radicalEigenvectors| (((|List| (|Record| (|:| |radval| (|Expression| (|Integer|))) (|:| |radmult| (|Integer|)) (|:| |radvect| (|List| (|Matrix| (|Expression| (|Integer|))))))) (|Matrix| (|Fraction| (|Polynomial| (|Integer|))))) "\\spad{radicalEigenvectors(m)} computes the eigenvalues and the corresponding eigenvectors of the matrix \\spad{m}; when possible,{} values are expressed in terms of radicals."))) NIL NIL -(-937 R) +(-942 R) ((|constructor| (NIL "\\spad{RepresentationPackage1} provides functions for representation theory for finite groups and algebras. The package creates permutation representations and uses tensor products and its symmetric and antisymmetric components to create new representations of larger degree from given ones. Note: instead of having parameters from \\spadtype{Permutation} this package allows list notation of permutations as well: \\spadignore{e.g.} \\spad{[1,4,3,2]} denotes permutes 2 and 4 and fixes 1 and 3.")) (|permutationRepresentation| (((|List| (|Matrix| (|Integer|))) (|List| (|List| (|Integer|)))) "\\spad{permutationRepresentation([pi1,...,pik],n)} returns the list of matrices {\\em [(deltai,pi1(i)),...,(deltai,pik(i))]} if the permutations {\\em pi1},{}...,{}{\\em pik} are in list notation and are permuting {\\em {1,2,...,n}}.") (((|List| (|Matrix| (|Integer|))) (|List| (|Permutation| (|Integer|))) (|Integer|)) "\\spad{permutationRepresentation([pi1,...,pik],n)} returns the list of matrices {\\em [(deltai,pi1(i)),...,(deltai,pik(i))]} (Kronecker delta) for the permutations {\\em pi1,...,pik} of {\\em {1,2,...,n}}.") (((|Matrix| (|Integer|)) (|List| (|Integer|))) "\\spad{permutationRepresentation(pi,n)} returns the matrix {\\em (deltai,pi(i))} (Kronecker delta) if the permutation {\\em pi} is in list notation and permutes {\\em {1,2,...,n}}.") (((|Matrix| (|Integer|)) (|Permutation| (|Integer|)) (|Integer|)) "\\spad{permutationRepresentation(pi,n)} returns the matrix {\\em (deltai,pi(i))} (Kronecker delta) for a permutation {\\em pi} of {\\em {1,2,...,n}}.")) (|tensorProduct| (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{tensorProduct([a1,...ak])} calculates the list of Kronecker products of each matrix {\\em ai} with itself for {1 <= \\spad{i} <= \\spad{k}}. Note: If the list of matrices corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the representation with itself.") (((|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{tensorProduct(a)} calculates the Kronecker product of the matrix {\\em a} with itself.") (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{tensorProduct([a1,...,ak],[b1,...,bk])} calculates the list of Kronecker products of the matrices {\\em ai} and {\\em bi} for {1 <= \\spad{i} <= \\spad{k}}. Note: If each list of matrices corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the two representations.") (((|Matrix| |#1|) (|Matrix| |#1|) (|Matrix| |#1|)) "\\spad{tensorProduct(a,b)} calculates the Kronecker product of the matrices {\\em a} and \\spad{b}. Note: if each matrix corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the two representations.")) (|symmetricTensors| (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|PositiveInteger|)) "\\spad{symmetricTensors(la,n)} applies to each \\spad{m}-by-\\spad{m} square matrix in the list {\\em la} the irreducible,{} polynomial representation of the general linear group {\\em GLm} which corresponds to the partition {\\em (n,0,...,0)} of \\spad{n}. Error: if the matrices in {\\em la} are not square matrices. Note: this corresponds to the symmetrization of the representation with the trivial representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the symmetric tensors of the \\spad{n}-fold tensor product.") (((|Matrix| |#1|) (|Matrix| |#1|) (|PositiveInteger|)) "\\spad{symmetricTensors(a,n)} applies to the \\spad{m}-by-\\spad{m} square matrix {\\em a} the irreducible,{} polynomial representation of the general linear group {\\em GLm} which corresponds to the partition {\\em (n,0,...,0)} of \\spad{n}. Error: if {\\em a} is not a square matrix. Note: this corresponds to the symmetrization of the representation with the trivial representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the symmetric tensors of the \\spad{n}-fold tensor product.")) (|createGenericMatrix| (((|Matrix| (|Polynomial| |#1|)) (|NonNegativeInteger|)) "\\spad{createGenericMatrix(m)} creates a square matrix of dimension \\spad{k} whose entry at the \\spad{i}-th row and \\spad{j}-th column is the indeterminate {\\em x[i,j]} (double subscripted).")) (|antisymmetricTensors| (((|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|PositiveInteger|)) "\\spad{antisymmetricTensors(la,n)} applies to each \\spad{m}-by-\\spad{m} square matrix in the list {\\em la} the irreducible,{} polynomial representation of the general linear group {\\em GLm} which corresponds to the partition {\\em (1,1,...,1,0,0,...,0)} of \\spad{n}. Error: if \\spad{n} is greater than \\spad{m}. Note: this corresponds to the symmetrization of the representation with the sign representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the antisymmetric tensors of the \\spad{n}-fold tensor product.") (((|Matrix| |#1|) (|Matrix| |#1|) (|PositiveInteger|)) "\\spad{antisymmetricTensors(a,n)} applies to the square matrix {\\em a} the irreducible,{} polynomial representation of the general linear group {\\em GLm},{} where \\spad{m} is the number of rows of {\\em a},{} which corresponds to the partition {\\em (1,1,...,1,0,0,...,0)} of \\spad{n}. Error: if \\spad{n} is greater than \\spad{m}. Note: this corresponds to the symmetrization of the representation with the sign representation of the symmetric group {\\em Sn}. The carrier spaces of the representation are the antisymmetric tensors of the \\spad{n}-fold tensor product."))) NIL -((|HasAttribute| |#1| (QUOTE (-3980 "*")))) -(-938 R) +((|HasAttribute| |#1| (QUOTE (-3991 "*")))) +(-943 R) ((|constructor| (NIL "\\spad{RepresentationPackage2} provides functions for working with modular representations of finite groups and algebra. The routines in this package are created,{} using ideas of \\spad{R}. Parker,{} (the meat-Axe) to get smaller representations from bigger ones,{} \\spadignore{i.e.} finding sub- and factormodules,{} or to show,{} that such the representations are irreducible. Note: most functions are randomized functions of Las Vegas type \\spadignore{i.e.} every answer is correct,{} but with small probability the algorithm fails to get an answer.")) (|scanOneDimSubspaces| (((|Vector| |#1|) (|List| (|Vector| |#1|)) (|Integer|)) "\\spad{scanOneDimSubspaces(basis,n)} gives a canonical representative of the {\\em n}\\spad{-}th one-dimensional subspace of the vector space generated by the elements of {\\em basis},{} all from {\\em R**n}. The coefficients of the representative are of shape {\\em (0,...,0,1,*,...,*)},{} {\\em *} in \\spad{R}. If the size of \\spad{R} is \\spad{q},{} then there are {\\em (q**n-1)/(q-1)} of them. We first reduce \\spad{n} modulo this number,{} then find the largest \\spad{i} such that {\\em +/[q**i for i in 0..i-1] <= n}. Subtracting this sum of powers from \\spad{n} results in an \\spad{i}-digit number to \\spad{basis} \\spad{q}. This fills the positions of the stars.")) (|meatAxe| (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|PositiveInteger|)) "\\spad{meatAxe(aG, numberOfTries)} calls {\\em meatAxe(aG,true,numberOfTries,7)}. Notes: 7 covers the case of three-dimensional kernels over the field with 2 elements.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Boolean|)) "\\spad{meatAxe(aG, randomElements)} calls {\\em meatAxe(aG,false,6,7)},{} only using Parker's fingerprints,{} if {\\em randomElemnts} is \\spad{false}. If it is \\spad{true},{} it calls {\\em meatAxe(aG,true,25,7)},{} only using random elements. Note: the choice of 25 was rather arbitrary. Also,{} 7 covers the case of three-dimensional kernels over the field with 2 elements.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|))) "\\spad{meatAxe(aG)} calls {\\em meatAxe(aG,false,25,7)} returns a 2-list of representations as follows. All matrices of argument \\spad{aG} are assumed to be square and of equal size. Then \\spad{aG} generates a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an A-module in the usual way. meatAxe(\\spad{aG}) creates at most 25 random elements of the algebra,{} tests them for singularity. If singular,{} it tries at most 7 elements of its kernel to generate a proper submodule. If successful a list which contains first the list of the representations of the submodule,{} then a list of the representations of the factor module is returned. Otherwise,{} if we know that all the kernel is already scanned,{} Norton's irreducibility test can be used either to prove irreducibility or to find the splitting. Notes: the first 6 tries use Parker's fingerprints. Also,{} 7 covers the case of three-dimensional kernels over the field with 2 elements.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Boolean|) (|Integer|) (|Integer|)) "\\spad{meatAxe(aG,randomElements,numberOfTries, maxTests)} returns a 2-list of representations as follows. All matrices of argument \\spad{aG} are assumed to be square and of equal size. Then \\spad{aG} generates a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an A-module in the usual way. meatAxe(\\spad{aG},{}\\spad{numberOfTries},{} maxTests) creates at most {\\em numberOfTries} random elements of the algebra,{} tests them for singularity. If singular,{} it tries at most {\\em maxTests} elements of its kernel to generate a proper submodule. If successful,{} a 2-list is returned: first,{} a list containing first the list of the representations of the submodule,{} then a list of the representations of the factor module. Otherwise,{} if we know that all the kernel is already scanned,{} Norton's irreducibility test can be used either to prove irreducibility or to find the splitting. If {\\em randomElements} is {\\em false},{} the first 6 tries use Parker's fingerprints.")) (|split| (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Vector| (|Vector| |#1|))) "\\spad{split(aG,submodule)} uses a proper \\spad{submodule} of {\\em R**n} to create the representations of the \\spad{submodule} and of the factor module.") (((|List| (|List| (|Matrix| |#1|))) (|List| (|Matrix| |#1|)) (|Vector| |#1|)) "\\spad{split(aG, vector)} returns a subalgebra \\spad{A} of all square matrix of dimension \\spad{n} as a list of list of matrices,{} generated by the list of matrices \\spad{aG},{} where \\spad{n} denotes both the size of vector as well as the dimension of each of the square matrices. {\\em V R} is an A-module in the natural way. split(\\spad{aG},{} vector) then checks whether the cyclic submodule generated by {\\em vector} is a proper submodule of {\\em V R}. If successful,{} it returns a two-element list,{} which contains first the list of the representations of the submodule,{} then the list of the representations of the factor module. If the vector generates the whole module,{} a one-element list of the old representation is given. Note: a later version this should call the other split.")) (|isAbsolutelyIrreducible?| (((|Boolean|) (|List| (|Matrix| |#1|))) "\\spad{isAbsolutelyIrreducible?(aG)} calls {\\em isAbsolutelyIrreducible?(aG,25)}. Note: the choice of 25 was rather arbitrary.") (((|Boolean|) (|List| (|Matrix| |#1|)) (|Integer|)) "\\spad{isAbsolutelyIrreducible?(aG, numberOfTries)} uses Norton's irreducibility test to check for absolute irreduciblity,{} assuming if a one-dimensional kernel is found. As no field extension changes create \"new\" elements in a one-dimensional space,{} the criterium stays \\spad{true} for every extension. The method looks for one-dimensionals only by creating random elements (no fingerprints) since a run of {\\em meatAxe} would have proved absolute irreducibility anyway.")) (|areEquivalent?| (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|Integer|)) "\\spad{areEquivalent?(aG0,aG1,numberOfTries)} calls {\\em areEquivalent?(aG0,aG1,true,25)}. Note: the choice of 25 was rather arbitrary.") (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{areEquivalent?(aG0,aG1)} calls {\\em areEquivalent?(aG0,aG1,true,25)}. Note: the choice of 25 was rather arbitrary.") (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|List| (|Matrix| |#1|)) (|Boolean|) (|Integer|)) "\\spad{areEquivalent?(aG0,aG1,randomelements,numberOfTries)} tests whether the two lists of matrices,{} all assumed of same square shape,{} can be simultaneously conjugated by a non-singular matrix. If these matrices represent the same group generators,{} the representations are equivalent. The algorithm tries {\\em numberOfTries} times to create elements in the generated algebras in the same fashion. If their ranks differ,{} they are not equivalent. If an isomorphism is assumed,{} then the kernel of an element of the first algebra is mapped to the kernel of the corresponding element in the second algebra. Now consider the one-dimensional ones. If they generate the whole space (\\spadignore{e.g.} irreducibility !) we use {\\em standardBasisOfCyclicSubmodule} to create the only possible transition matrix. The method checks whether the matrix conjugates all corresponding matrices from {\\em aGi}. The way to choose the singular matrices is as in {\\em meatAxe}. If the two representations are equivalent,{} this routine returns the transformation matrix {\\em TM} with {\\em aG0.i * TM = TM * aG1.i} for all \\spad{i}. If the representations are not equivalent,{} a small 0-matrix is returned. Note: the case with different sets of group generators cannot be handled.")) (|standardBasisOfCyclicSubmodule| (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|Vector| |#1|)) "\\spad{standardBasisOfCyclicSubmodule(lm,v)} returns a matrix as follows. It is assumed that the size \\spad{n} of the vector equals the number of rows and columns of the matrices. Then the matrices generate a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an \\spad{A}-module in the natural way. standardBasisOfCyclicSubmodule(\\spad{lm},{}\\spad{v}) calculates a matrix whose non-zero column vectors are the \\spad{R}-Basis of {\\em Av} achieved in the way as described in section 6 of \\spad{R}. A. Parker's \"The Meat-Axe\". Note: in contrast to {\\em cyclicSubmodule},{} the result is not in echelon form.")) (|cyclicSubmodule| (((|Vector| (|Vector| |#1|)) (|List| (|Matrix| |#1|)) (|Vector| |#1|)) "\\spad{cyclicSubmodule(lm,v)} generates a basis as follows. It is assumed that the size \\spad{n} of the vector equals the number of rows and columns of the matrices. Then the matrices generate a subalgebra,{} say \\spad{A},{} of the algebra of all square matrices of dimension \\spad{n}. {\\em V R} is an \\spad{A}-module in the natural way. cyclicSubmodule(\\spad{lm},{}\\spad{v}) generates the \\spad{R}-Basis of {\\em Av} as described in section 6 of \\spad{R}. A. Parker's \"The Meat-Axe\". Note: in contrast to the description in \"The Meat-Axe\" and to {\\em standardBasisOfCyclicSubmodule} the result is in echelon form.")) (|createRandomElement| (((|Matrix| |#1|) (|List| (|Matrix| |#1|)) (|Matrix| |#1|)) "\\spad{createRandomElement(aG,x)} creates a random element of the group algebra generated by {\\em aG}.")) (|completeEchelonBasis| (((|Matrix| |#1|) (|Vector| (|Vector| |#1|))) "\\spad{completeEchelonBasis(lv)} completes the basis {\\em lv} assumed to be in echelon form of a subspace of {\\em R**n} (\\spad{n} the length of all the vectors in {\\em lv}) with unit vectors to a basis of {\\em R**n}. It is assumed that the argument is not an empty vector and that it is not the basis of the 0-subspace. Note: the rows of the result correspond to the vectors of the basis."))) NIL -((-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-315)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-255)))) -(-939 S) +((-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-317)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-257)))) +(-944 S) ((|constructor| (NIL "Implements multiplication by repeated addition")) (|double| ((|#1| (|PositiveInteger|) |#1|) "\\spad{double(i, r)} multiplies \\spad{r} by \\spad{i} using repeated doubling.")) (+ (($ $ $) "\\spad{x+y} returns the sum of \\spad{x} and \\spad{y}"))) NIL NIL -(-940 S) +(-945 S) ((|constructor| (NIL "Implements exponentiation by repeated squaring")) (|expt| ((|#1| |#1| (|PositiveInteger|)) "\\spad{expt(r, i)} computes r**i by repeated squaring")) (* (($ $ $) "\\spad{x*y} returns the product of \\spad{x} and \\spad{y}"))) NIL NIL -(-941 S) +(-946 S) ((|constructor| (NIL "This package provides coercions for the special types \\spadtype{Exit} and \\spadtype{Void}.")) (|coerce| ((|#1| (|Exit|)) "\\spad{coerce(e)} is never really evaluated. This coercion is used for formal type correctness when a function will not return directly to its caller.") (((|Void|) |#1|) "\\spad{coerce(s)} throws all information about \\spad{s} away. This coercion allows values of any type to appear in contexts where they will not be used. For example,{} it allows the resolution of different types in the \\spad{then} and \\spad{else} branches when an \\spad{if} is in a context where the resulting value is not used."))) NIL NIL -(-942 -3078 |Expon| |VarSet| |FPol| |LFPol|) +(-947 -3088 |Expon| |VarSet| |FPol| |LFPol|) ((|constructor| (NIL "ResidueRing is the quotient of a polynomial ring by an ideal. The ideal is given as a list of generators. The elements of the domain are equivalence classes expressed in terms of reduced elements")) (|lift| ((|#4| $) "\\spad{lift(x)} return the canonical representative of the equivalence class \\spad{x}")) (|coerce| (($ |#4|) "\\spad{coerce(f)} produces the equivalence class of \\spad{f} in the residue ring")) (|reduce| (($ |#4|) "\\spad{reduce(f)} produces the equivalence class of \\spad{f} in the residue ring"))) -(((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-943) +(-948) ((|constructor| (NIL "This domain represents `return' expressions.")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression returned by `e'."))) NIL NIL -(-944 A S) +(-949 A S) ((|constructor| (NIL "A is retractable to \\spad{B} means that some elementsif A can be converted into elements of \\spad{B} and any element of \\spad{B} can be converted into an element of A.")) (|retract| ((|#2| $) "\\spad{retract(a)} transforms a into an element of \\spad{S} if possible. Error: if a cannot be made into an element of \\spad{S}.")) (|retractIfCan| (((|Union| |#2| "failed") $) "\\spad{retractIfCan(a)} transforms a into an element of \\spad{S} if possible. Returns \"failed\" if a cannot be made into an element of \\spad{S}."))) NIL NIL -(-945 S) +(-950 S) ((|constructor| (NIL "A is retractable to \\spad{B} means that some elementsif A can be converted into elements of \\spad{B} and any element of \\spad{B} can be converted into an element of A.")) (|retract| ((|#1| $) "\\spad{retract(a)} transforms a into an element of \\spad{S} if possible. Error: if a cannot be made into an element of \\spad{S}.")) (|retractIfCan| (((|Union| |#1| "failed") $) "\\spad{retractIfCan(a)} transforms a into an element of \\spad{S} if possible. Returns \"failed\" if a cannot be made into an element of \\spad{S}."))) NIL NIL -(-946 Q R) +(-951 Q R) ((|constructor| (NIL "RetractSolvePackage is an interface to \\spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving.")) (|solveRetract| (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#2|))))) (|List| (|Polynomial| |#2|)) (|List| (|Symbol|))) "\\spad{solveRetract(lp,lv)} finds the solutions of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}. The function tries to retract all the coefficients of the equations to \\spad{Q} before solving if possible."))) NIL NIL -(-947 R) +(-952 R) ((|constructor| (NIL "Utilities that provide the same top-level manipulations on fractions than on polynomials.")) (|coerce| (((|Fraction| (|Polynomial| |#1|)) |#1|) "\\spad{coerce(r)} returns \\spad{r} viewed as a rational function over \\spad{R}.")) (|eval| (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) "\\spad{eval(f, [v1 = g1,...,vn = gn])} returns \\spad{f} with each \\spad{vi} replaced by \\spad{gi} in parallel,{} \\spadignore{i.e.} \\spad{vi}'s appearing inside the \\spad{gi}'s are not replaced. Error: if any \\spad{vi} is not a symbol.") (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eval(f, v = g)} returns \\spad{f} with \\spad{v} replaced by \\spad{g}. Error: if \\spad{v} is not a symbol.") (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|List| (|Symbol|)) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{eval(f, [v1,...,vn], [g1,...,gn])} returns \\spad{f} with each \\spad{vi} replaced by \\spad{gi} in parallel,{} \\spadignore{i.e.} \\spad{vi}'s appearing inside the \\spad{gi}'s are not replaced.") (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|Fraction| (|Polynomial| |#1|))) "\\spad{eval(f, v, g)} returns \\spad{f} with \\spad{v} replaced by \\spad{g}.")) (|multivariate| (((|Fraction| (|Polynomial| |#1|)) (|Fraction| (|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) (|Symbol|)) "\\spad{multivariate(f, v)} applies both the numerator and denominator of \\spad{f} to \\spad{v}.")) (|univariate| (((|Fraction| (|SparseUnivariatePolynomial| (|Fraction| (|Polynomial| |#1|)))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{univariate(f, v)} returns \\spad{f} viewed as a univariate rational function in \\spad{v}.")) (|mainVariable| (((|Union| (|Symbol|) "failed") (|Fraction| (|Polynomial| |#1|))) "\\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \\spad{f},{} \"failed\" if \\spad{f} has no variables.")) (|variables| (((|List| (|Symbol|)) (|Fraction| (|Polynomial| |#1|))) "\\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \\spad{f}."))) NIL NIL -(-948) +(-953) ((|t| (((|Mapping| (|Float|)) (|NonNegativeInteger|)) "\\spad{t(n)} \\undocumented")) (F (((|Mapping| (|Float|)) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{F(n,m)} \\undocumented")) (|Beta| (((|Mapping| (|Float|)) (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{Beta(n,m)} \\undocumented")) (|chiSquare| (((|Mapping| (|Float|)) (|NonNegativeInteger|)) "\\spad{chiSquare(n)} \\undocumented")) (|exponential| (((|Mapping| (|Float|)) (|Float|)) "\\spad{exponential(f)} \\undocumented")) (|normal| (((|Mapping| (|Float|)) (|Float|) (|Float|)) "\\spad{normal(f,g)} \\undocumented")) (|uniform| (((|Mapping| (|Float|)) (|Float|) (|Float|)) "\\spad{uniform(f,g)} \\undocumented")) (|chiSquare1| (((|Float|) (|NonNegativeInteger|)) "\\spad{chiSquare1(n)} \\undocumented")) (|exponential1| (((|Float|)) "\\spad{exponential1()} \\undocumented")) (|normal01| (((|Float|)) "\\spad{normal01()} \\undocumented")) (|uniform01| (((|Float|)) "\\spad{uniform01()} \\undocumented"))) NIL NIL -(-949 UP) +(-954 UP) ((|constructor| (NIL "Factorization of univariate polynomials with coefficients which are rational functions with integer coefficients.")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p}."))) NIL NIL -(-950 R) +(-955 R) ((|constructor| (NIL "\\spadtype{RationalFunctionFactorizer} contains the factor function (called factorFraction) which factors fractions of polynomials by factoring the numerator and denominator. Since any non zero fraction is a unit the usual factor operation will just return the original fraction.")) (|factorFraction| (((|Fraction| (|Factored| (|Polynomial| |#1|))) (|Fraction| (|Polynomial| |#1|))) "\\spad{factorFraction(r)} factors the numerator and the denominator of the polynomial fraction \\spad{r}."))) NIL NIL -(-951 T$) +(-956 T$) ((|constructor| (NIL "This category defines the common interface for RGB color models.")) (|componentUpperBound| ((|#1|) "componentUpperBound is an upper bound for all component values.")) (|blue| ((|#1| $) "\\spad{blue(c)} returns the `blue' component of `c'.")) (|green| ((|#1| $) "\\spad{green(c)} returns the `green' component of `c'.")) (|red| ((|#1| $) "\\spad{red(c)} returns the `red' component of `c'."))) NIL NIL -(-952 T$) +(-957 T$) ((|constructor| (NIL "This category defines the common interface for RGB color spaces.")) (|whitePoint| (($) "whitePoint is the contant indicating the white point of this color space."))) NIL NIL -(-953 R |ls|) +(-958 R |ls|) ((|constructor| (NIL "A domain for regular chains (\\spadignore{i.e.} regular triangular sets) over a Gcd-Domain and with a fix list of variables. This is just a front-end for the \\spadtype{RegularTriangularSet} domain constructor.")) (|zeroSetSplit| (((|List| $) (|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) (|Boolean|) (|Boolean|)) "\\spad{zeroSetSplit(lp,clos?,info?)} returns a list \\spad{lts} of regular chains such that the union of the closures of their regular zero sets equals the affine variety associated with \\spad{lp}. Moreover,{} if \\spad{clos?} is \\spad{false} then the union of the regular zero set of the \\spad{ts} (for \\spad{ts} in \\spad{lts}) equals this variety. If \\spad{info?} is \\spad{true} then some information is displayed during the computations. See \\axiomOpFrom{zeroSetSplit}{RegularTriangularSet}."))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| (-698 |#1| (-768 |#2|)) (QUOTE (-1007))) (|HasCategory| (-698 |#1| (-768 |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -698) (|devaluate| |#1|) (|%list| (QUOTE -768) (|devaluate| |#2|)))))) (|HasCategory| (-698 |#1| (-768 |#2|)) (QUOTE (-550 (-469)))) (|HasCategory| (-698 |#1| (-768 |#2|)) (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| (-768 |#2|) (QUOTE (-315))) (|HasCategory| (-698 |#1| (-768 |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-698 |#1| (-768 |#2|)) (QUOTE (-72)))) -(-954) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| (-703 |#1| (-773 |#2|)) (QUOTE (-1012))) (|HasCategory| (-703 |#1| (-773 |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -703) (|devaluate| |#1|) (|%list| (QUOTE -773) (|devaluate| |#2|)))))) (|HasCategory| (-703 |#1| (-773 |#2|)) (QUOTE (-553 (-472)))) (|HasCategory| (-703 |#1| (-773 |#2|)) (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| (-773 |#2|) (QUOTE (-317))) (|HasCategory| (-703 |#1| (-773 |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-703 |#1| (-773 |#2|)) (QUOTE (-72)))) +(-959) ((|constructor| (NIL "This package exports integer distributions")) (|ridHack1| (((|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Integer|)) "\\spad{ridHack1(i,j,k,l)} \\undocumented")) (|geometric| (((|Mapping| (|Integer|)) |RationalNumber|) "\\spad{geometric(f)} \\undocumented")) (|poisson| (((|Mapping| (|Integer|)) |RationalNumber|) "\\spad{poisson(f)} \\undocumented")) (|binomial| (((|Mapping| (|Integer|)) (|Integer|) |RationalNumber|) "\\spad{binomial(n,f)} \\undocumented")) (|uniform| (((|Mapping| (|Integer|)) (|Segment| (|Integer|))) "\\spad{uniform(s)} \\undocumented"))) NIL NIL -(-955 S) +(-960 S) ((|constructor| (NIL "The category of rings with unity,{} always associative,{} but not necessarily commutative.")) (|unitsKnown| ((|attribute|) "recip truly yields reciprocal or \"failed\" if not a unit. Note: \\spad{recip(0) = \"failed\"}.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring this is the smallest positive integer \\spad{n} such that \\spad{n*x=0} for all \\spad{x} in the ring,{} or zero if no such \\spad{n} exists."))) NIL NIL -(-956) +(-961) ((|constructor| (NIL "The category of rings with unity,{} always associative,{} but not necessarily commutative.")) (|unitsKnown| ((|attribute|) "recip truly yields reciprocal or \"failed\" if not a unit. Note: \\spad{recip(0) = \"failed\"}.")) (|characteristic| (((|NonNegativeInteger|)) "\\spad{characteristic()} returns the characteristic of the ring this is the smallest positive integer \\spad{n} such that \\spad{n*x=0} for all \\spad{x} in the ring,{} or zero if no such \\spad{n} exists."))) -((-3975 . T)) +((-3986 . T)) NIL -(-957 |xx| -3078) +(-962 |xx| -3088) ((|constructor| (NIL "This package exports rational interpolation algorithms"))) NIL NIL -(-958 S) +(-963 S) ((|constructor| (NIL "\\indented{2}{A set is an \\spad{S}-right linear set if it is stable by right-dilation} \\indented{2}{by elements in the semigroup \\spad{S}.} See Also: LeftLinearSet.")) (* (($ $ |#1|) "\\spad{x*s} is the right-dilation of \\spad{x} by \\spad{s}."))) NIL NIL -(-959 S |m| |n| R |Row| |Col|) +(-964 S |m| |n| R |Row| |Col|) ((|constructor| (NIL "\\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \\spad{R}-modules and will be non-mutable.")) (|nullSpace| (((|List| |#6|) $) "\\spad{nullSpace(m)}+ returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#4|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#4|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (|map| (($ (|Mapping| |#4| |#4| |#4|) $ $) "\\spad{map(f,a,b)} returns \\spad{c},{} where \\spad{c} is such that \\spad{c(i,j) = f(a(i,j),b(i,j))} for all \\spad{i},{} \\spad{j}.") (($ (|Mapping| |#4| |#4|) $) "\\spad{map(f,a)} returns \\spad{b},{} where \\spad{b(i,j) = a(i,j)} for all \\spad{i},{} \\spad{j}.")) (|column| ((|#6| $ (|Integer|)) "\\spad{column(m,j)} returns the \\spad{j}th column of the matrix \\spad{m}. Error: if the index outside the proper range.")) (|row| ((|#5| $ (|Integer|)) "\\spad{row(m,i)} returns the \\spad{i}th row of the matrix \\spad{m}. Error: if the index is outside the proper range.")) (|qelt| ((|#4| $ (|Integer|) (|Integer|)) "\\spad{qelt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Note: there is NO error check to determine if indices are in the proper ranges.")) (|elt| ((|#4| $ (|Integer|) (|Integer|) |#4|) "\\spad{elt(m,i,j,r)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{} and returns \\spad{r} otherwise.") ((|#4| $ (|Integer|) (|Integer|)) "\\spad{elt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Error: if indices are outside the proper ranges.")) (|listOfLists| (((|List| (|List| |#4|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|ncols| (((|NonNegativeInteger|) $) "\\spad{ncols(m)} returns the number of columns in the matrix \\spad{m}.")) (|nrows| (((|NonNegativeInteger|) $) "\\spad{nrows(m)} returns the number of rows in the matrix \\spad{m}.")) (|maxColIndex| (((|Integer|) $) "\\spad{maxColIndex(m)} returns the index of the 'last' column of the matrix \\spad{m}.")) (|minColIndex| (((|Integer|) $) "\\spad{minColIndex(m)} returns the index of the 'first' column of the matrix \\spad{m}.")) (|maxRowIndex| (((|Integer|) $) "\\spad{maxRowIndex(m)} returns the index of the 'last' row of the matrix \\spad{m}.")) (|minRowIndex| (((|Integer|) $) "\\spad{minRowIndex(m)} returns the index of the 'first' row of the matrix \\spad{m}.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|matrix| (($ (|List| (|List| |#4|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|finiteAggregate| ((|attribute|) "matrices are finite"))) NIL -((|HasCategory| |#4| (QUOTE (-255))) (|HasCategory| |#4| (QUOTE (-309))) (|HasCategory| |#4| (QUOTE (-491))) (|HasCategory| |#4| (QUOTE (-144)))) -(-960 |m| |n| R |Row| |Col|) +((|HasCategory| |#4| (QUOTE (-257))) (|HasCategory| |#4| (QUOTE (-311))) (|HasCategory| |#4| (QUOTE (-494))) (|HasCategory| |#4| (QUOTE (-146)))) +(-965 |m| |n| R |Row| |Col|) ((|constructor| (NIL "\\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \\spad{R}-modules and will be non-mutable.")) (|nullSpace| (((|List| |#5|) $) "\\spad{nullSpace(m)}+ returns a basis for the null space of the matrix \\spad{m}.")) (|nullity| (((|NonNegativeInteger|) $) "\\spad{nullity(m)} returns the nullity of the matrix \\spad{m}. This is the dimension of the null space of the matrix \\spad{m}.")) (|rank| (((|NonNegativeInteger|) $) "\\spad{rank(m)} returns the rank of the matrix \\spad{m}.")) (|rowEchelon| (($ $) "\\spad{rowEchelon(m)} returns the row echelon form of the matrix \\spad{m}.")) (/ (($ $ |#3|) "\\spad{m/r} divides the elements of \\spad{m} by \\spad{r}. Error: if \\spad{r = 0}.")) (|exquo| (((|Union| $ "failed") $ |#3|) "\\spad{exquo(m,r)} computes the exact quotient of the elements of \\spad{m} by \\spad{r},{} returning \\axiom{\"failed\"} if this is not possible.")) (|map| (($ (|Mapping| |#3| |#3| |#3|) $ $) "\\spad{map(f,a,b)} returns \\spad{c},{} where \\spad{c} is such that \\spad{c(i,j) = f(a(i,j),b(i,j))} for all \\spad{i},{} \\spad{j}.") (($ (|Mapping| |#3| |#3|) $) "\\spad{map(f,a)} returns \\spad{b},{} where \\spad{b(i,j) = a(i,j)} for all \\spad{i},{} \\spad{j}.")) (|column| ((|#5| $ (|Integer|)) "\\spad{column(m,j)} returns the \\spad{j}th column of the matrix \\spad{m}. Error: if the index outside the proper range.")) (|row| ((|#4| $ (|Integer|)) "\\spad{row(m,i)} returns the \\spad{i}th row of the matrix \\spad{m}. Error: if the index is outside the proper range.")) (|qelt| ((|#3| $ (|Integer|) (|Integer|)) "\\spad{qelt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Note: there is NO error check to determine if indices are in the proper ranges.")) (|elt| ((|#3| $ (|Integer|) (|Integer|) |#3|) "\\spad{elt(m,i,j,r)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m},{} if \\spad{m} has an \\spad{i}th row and a \\spad{j}th column,{} and returns \\spad{r} otherwise.") ((|#3| $ (|Integer|) (|Integer|)) "\\spad{elt(m,i,j)} returns the element in the \\spad{i}th row and \\spad{j}th column of the matrix \\spad{m}. Error: if indices are outside the proper ranges.")) (|listOfLists| (((|List| (|List| |#3|)) $) "\\spad{listOfLists(m)} returns the rows of the matrix \\spad{m} as a list of lists.")) (|ncols| (((|NonNegativeInteger|) $) "\\spad{ncols(m)} returns the number of columns in the matrix \\spad{m}.")) (|nrows| (((|NonNegativeInteger|) $) "\\spad{nrows(m)} returns the number of rows in the matrix \\spad{m}.")) (|maxColIndex| (((|Integer|) $) "\\spad{maxColIndex(m)} returns the index of the 'last' column of the matrix \\spad{m}.")) (|minColIndex| (((|Integer|) $) "\\spad{minColIndex(m)} returns the index of the 'first' column of the matrix \\spad{m}.")) (|maxRowIndex| (((|Integer|) $) "\\spad{maxRowIndex(m)} returns the index of the 'last' row of the matrix \\spad{m}.")) (|minRowIndex| (((|Integer|) $) "\\spad{minRowIndex(m)} returns the index of the 'first' row of the matrix \\spad{m}.")) (|antisymmetric?| (((|Boolean|) $) "\\spad{antisymmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and antisymmetric (\\spadignore{i.e.} \\spad{m[i,j] = -m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|symmetric?| (((|Boolean|) $) "\\spad{symmetric?(m)} returns \\spad{true} if the matrix \\spad{m} is square and symmetric (\\spadignore{i.e.} \\spad{m[i,j] = m[j,i]} for all \\spad{i} and \\spad{j}) and \\spad{false} otherwise.")) (|diagonal?| (((|Boolean|) $) "\\spad{diagonal?(m)} returns \\spad{true} if the matrix \\spad{m} is square and diagonal (\\spadignore{i.e.} all entries of \\spad{m} not on the diagonal are zero) and \\spad{false} otherwise.")) (|square?| (((|Boolean|) $) "\\spad{square?(m)} returns \\spad{true} if \\spad{m} is a square matrix (\\spadignore{i.e.} if \\spad{m} has the same number of rows as columns) and \\spad{false} otherwise.")) (|matrix| (($ (|List| (|List| |#3|))) "\\spad{matrix(l)} converts the list of lists \\spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.")) (|finiteAggregate| ((|attribute|) "matrices are finite"))) -((-3978 . T) (-3973 . T) (-3972 . T)) +((-3989 . T) (-3984 . T) (-3983 . T)) NIL -(-961 |m| |n| R) +(-966 |m| |n| R) ((|constructor| (NIL "\\spadtype{RectangularMatrix} is a matrix domain where the number of rows and the number of columns are parameters of the domain.")) (|rectangularMatrix| (($ (|Matrix| |#3|)) "\\spad{rectangularMatrix(m)} converts a matrix of type \\spadtype{Matrix} to a matrix of type \\spad{RectangularMatrix}."))) -((-3978 . T) (-3973 . T) (-3972 . T)) -((|HasCategory| |#3| (QUOTE (-144))) (OR (-12 (|HasCategory| |#3| (QUOTE (-144))) (|HasCategory| |#3| (|%list| (QUOTE -257) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-309))) (|HasCategory| |#3| (|%list| (QUOTE -257) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1007))) (|HasCategory| |#3| (|%list| (QUOTE -257) (|devaluate| |#3|))))) (|HasCategory| |#3| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#3| (QUOTE (-144))) (|HasCategory| |#3| (QUOTE (-309)))) (|HasCategory| |#3| (QUOTE (-309))) (|HasCategory| |#3| (QUOTE (-1007))) (|HasCategory| |#3| (QUOTE (-255))) (|HasCategory| |#3| (QUOTE (-491))) (-12 (|HasCategory| |#3| (QUOTE (-1007))) (|HasCategory| |#3| (|%list| (QUOTE -257) (|devaluate| |#3|)))) (|HasCategory| |#3| (QUOTE (-72))) (|HasCategory| |#3| (QUOTE (-549 (-767))))) -(-962 |m| |n| R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) +((-3989 . T) (-3984 . T) (-3983 . T)) +((|HasCategory| |#3| (QUOTE (-146))) (OR (-12 (|HasCategory| |#3| (QUOTE (-146))) (|HasCategory| |#3| (|%list| (QUOTE -259) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-311))) (|HasCategory| |#3| (|%list| (QUOTE -259) (|devaluate| |#3|)))) (-12 (|HasCategory| |#3| (QUOTE (-1012))) (|HasCategory| |#3| (|%list| (QUOTE -259) (|devaluate| |#3|))))) (|HasCategory| |#3| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#3| (QUOTE (-146))) (|HasCategory| |#3| (QUOTE (-311)))) (|HasCategory| |#3| (QUOTE (-311))) (|HasCategory| |#3| (QUOTE (-1012))) (|HasCategory| |#3| (QUOTE (-257))) (|HasCategory| |#3| (QUOTE (-494))) (-12 (|HasCategory| |#3| (QUOTE (-1012))) (|HasCategory| |#3| (|%list| (QUOTE -259) (|devaluate| |#3|)))) (|HasCategory| |#3| (QUOTE (-72))) (|HasCategory| |#3| (QUOTE (-552 (-772))))) +(-967 |m| |n| R1 |Row1| |Col1| M1 R2 |Row2| |Col2| M2) ((|constructor| (NIL "\\spadtype{RectangularMatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \\spadfun{map} and \\spadfun{reduce}.")) (|reduce| ((|#7| (|Mapping| |#7| |#3| |#7|) |#6| |#7|) "\\spad{reduce(f,m,r)} returns a matrix \\spad{n} where \\spad{n[i,j] = f(m[i,j],r)} for all indices spad{\\spad{i}} and \\spad{j}.")) (|map| ((|#10| (|Mapping| |#7| |#3|) |#6|) "\\spad{map(f,m)} applies the function \\spad{f} to the elements of the matrix \\spad{m}."))) NIL NIL -(-963 R) +(-968 R) ((|constructor| (NIL "The category of right modules over an rng (ring not necessarily with unit). This is an abelian group which supports right multiplation by elements of the rng. \\blankline"))) NIL NIL -(-964) +(-969) ((|constructor| (NIL "The category of associative rings,{} not necessarily commutative,{} and not necessarily with a 1. This is a combination of an abelian group and a semigroup,{} with multiplication distributing over addition. \\blankline"))) NIL NIL -(-965 S T$) +(-970 S T$) ((|constructor| (NIL "This domain represents the notion of binding a variable to range over a specific segment (either bounded,{} or half bounded).")) (|segment| ((|#1| $) "\\spad{segment(x)} returns the segment from the right hand side of the \\spadtype{RangeBinding}. For example,{} if \\spad{x} is \\spad{v=s},{} then \\spad{segment(x)} returns \\spad{s}.")) (|variable| (((|Symbol|) $) "\\spad{variable(x)} returns the variable from the left hand side of the \\spadtype{RangeBinding}. For example,{} if \\spad{x} is \\spad{v=s},{} then \\spad{variable(x)} returns \\spad{v}.")) (|equation| (($ (|Symbol|) |#1|) "\\spad{equation(v,s)} creates a segment binding value with variable \\spad{v} and segment \\spad{s}. Note that the interpreter parses \\spad{v=s} to this form."))) NIL -((|HasCategory| |#1| (QUOTE (-1007)))) -(-966 S) +((|HasCategory| |#1| (QUOTE (-1012)))) +(-971 S) ((|constructor| (NIL "The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \\spadtype{DifferentialRing} or the elementary functions (see \\spadtype{TranscendentalFunctionCategory}) in the definition.")) (|round| (($ $) "\\spad{round x} computes the integer closest to \\spad{x}.")) (|truncate| (($ $) "\\spad{truncate x} returns the integer between \\spad{x} and 0 closest to \\spad{x}.")) (|fractionPart| (($ $) "\\spad{fractionPart x} returns the fractional part of \\spad{x}.")) (|wholePart| (((|Integer|) $) "\\spad{wholePart x} returns the integer part of \\spad{x}.")) (|floor| (($ $) "\\spad{floor x} returns the largest integer \\spad{<= x}.")) (|ceiling| (($ $) "\\spad{ceiling x} returns the small integer \\spad{>= x}.")) (|norm| (($ $) "\\spad{norm x} returns the same as absolute value."))) NIL NIL -(-967) +(-972) ((|constructor| (NIL "The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \\spadtype{DifferentialRing} or the elementary functions (see \\spadtype{TranscendentalFunctionCategory}) in the definition.")) (|round| (($ $) "\\spad{round x} computes the integer closest to \\spad{x}.")) (|truncate| (($ $) "\\spad{truncate x} returns the integer between \\spad{x} and 0 closest to \\spad{x}.")) (|fractionPart| (($ $) "\\spad{fractionPart x} returns the fractional part of \\spad{x}.")) (|wholePart| (((|Integer|) $) "\\spad{wholePart x} returns the integer part of \\spad{x}.")) (|floor| (($ $) "\\spad{floor x} returns the largest integer \\spad{<= x}.")) (|ceiling| (($ $) "\\spad{ceiling x} returns the small integer \\spad{>= x}.")) (|norm| (($ $) "\\spad{norm x} returns the same as absolute value."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-968 |TheField| |ThePolDom|) +(-973 |TheField| |ThePolDom|) ((|constructor| (NIL "\\axiomType{RightOpenIntervalRootCharacterization} provides work with interval root coding.")) (|relativeApprox| ((|#1| |#2| $ |#1|) "\\axiom{relativeApprox(exp,{}\\spad{c},{}\\spad{p}) = a} is relatively close to exp as a polynomial in \\spad{c} ip to precision \\spad{p}")) (|mightHaveRoots| (((|Boolean|) |#2| $) "\\axiom{mightHaveRoots(\\spad{p},{}\\spad{r})} is \\spad{false} if \\axiom{\\spad{p}.\\spad{r}} is not 0")) (|refine| (($ $) "\\axiom{refine(rootChar)} shrinks isolating interval around \\axiom{rootChar}")) (|middle| ((|#1| $) "\\axiom{middle(rootChar)} is the middle of the isolating interval")) (|size| ((|#1| $) "The size of the isolating interval")) (|right| ((|#1| $) "\\axiom{right(rootChar)} is the right bound of the isolating interval")) (|left| ((|#1| $) "\\axiom{left(rootChar)} is the left bound of the isolating interval"))) NIL NIL -(-969) +(-974) ((|constructor| (NIL "\\spadtype{RomanNumeral} provides functions for converting \\indented{1}{integers to roman numerals.}")) (|roman| (($ (|Integer|)) "\\spad{roman(n)} creates a roman numeral for \\spad{n}.") (($ (|Symbol|)) "\\spad{roman(n)} creates a roman numeral for symbol \\spad{n}.")) (|noetherian| ((|attribute|) "ascending chain condition on ideals.")) (|canonicalsClosed| ((|attribute|) "two positives multiply to give positive.")) (|canonical| ((|attribute|) "mathematical equality is data structure equality."))) -((-3966 . T) (-3970 . T) (-3965 . T) (-3976 . T) (-3977 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3977 . T) (-3981 . T) (-3976 . T) (-3987 . T) (-3988 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-970 S R E V) +(-975 S R E V) ((|constructor| (NIL "A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \\axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.")) (|mainSquareFreePart| (($ $) "\\axiom{mainSquareFreePart(\\spad{p})} returns the square free part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainPrimitivePart| (($ $) "\\axiom{mainPrimitivePart(\\spad{p})} returns the primitive part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainContent| (($ $) "\\axiom{mainContent(\\spad{p})} returns the content of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|primitivePart!| (($ $) "\\axiom{primitivePart!(\\spad{p})} replaces \\axiom{\\spad{p}} by its primitive part.")) (|gcd| ((|#2| |#2| $) "\\axiom{gcd(\\spad{r},{}\\spad{p})} returns the gcd of \\axiom{\\spad{r}} and the content of \\axiom{\\spad{p}}.")) (|nextsubResultant2| (($ $ $ $ $) "\\axiom{\\spad{nextsubResultant2}(\\spad{p},{}\\spad{q},{}\\spad{z},{}\\spad{s})} is the multivariate version of the operation \\axiomOpFrom{\\spad{next_sousResultant2}}{PseudoRemainderSequence} from the \\axiomType{PseudoRemainderSequence} constructor.")) (|LazardQuotient2| (($ $ $ $ (|NonNegativeInteger|)) "\\axiom{\\spad{LazardQuotient2}(\\spad{p},{}a,{}\\spad{b},{}\\spad{n})} returns \\axiom{(a**(\\spad{n}-1) * \\spad{p}) exquo b**(\\spad{n}-1)} assuming that this quotient does not fail.")) (|LazardQuotient| (($ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient(a,{}\\spad{b},{}\\spad{n})} returns \\axiom{a**n exquo b**(\\spad{n}-1)} assuming that this quotient does not fail.")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns the last non-zero subresultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|subResultantChain| (((|List| $) $ $) "\\axiom{subResultantChain(a,{}\\spad{b})},{} where \\axiom{a} and \\axiom{\\spad{b}} are not contant polynomials with the same main variable,{} returns the subresultant chain of \\axiom{a} and \\axiom{\\spad{b}}.")) (|resultant| (($ $ $) "\\axiom{resultant(a,{}\\spad{b})} computes the resultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{\\spad{halfExtendedSubResultantGcd2}(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}cb]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}cb]} otherwise produces an error.")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{\\spad{halfExtendedSubResultantGcd1}(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}cb]} otherwise produces an error.")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[ca,{}cb,{}\\spad{r}]} such that \\axiom{\\spad{r}} is \\axiom{subResultantGcd(a,{}\\spad{b})} and we have \\axiom{ca * a + cb * cb = \\spad{r}} .")) (|subResultantGcd| (($ $ $) "\\axiom{subResultantGcd(a,{}\\spad{b})} computes a gcd of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}} with coefficients in the fraction field of the polynomial ring generated by their other variables over \\axiom{\\spad{R}}.")) (|exactQuotient!| (($ $ $) "\\axiom{exactQuotient!(a,{}\\spad{b})} replaces \\axiom{a} by \\axiom{exactQuotient(a,{}\\spad{b})}") (($ $ |#2|) "\\axiom{exactQuotient!(\\spad{p},{}\\spad{r})} replaces \\axiom{\\spad{p}} by \\axiom{exactQuotient(\\spad{p},{}\\spad{r})}.")) (|exactQuotient| (($ $ $) "\\axiom{exactQuotient(a,{}\\spad{b})} computes the exact quotient of \\axiom{a} by \\axiom{\\spad{b}},{} which is assumed to be a divisor of \\axiom{a}. No error is returned if this exact quotient fails!") (($ $ |#2|) "\\axiom{exactQuotient(\\spad{p},{}\\spad{r})} computes the exact quotient of \\axiom{\\spad{p}} by \\axiom{\\spad{r}},{} which is assumed to be a divisor of \\axiom{\\spad{p}}. No error is returned if this exact quotient fails!")) (|primPartElseUnitCanonical!| (($ $) "\\axiom{primPartElseUnitCanonical!(\\spad{p})} replaces \\axiom{\\spad{p}} by \\axiom{primPartElseUnitCanonical(\\spad{p})}.")) (|primPartElseUnitCanonical| (($ $) "\\axiom{primPartElseUnitCanonical(\\spad{p})} returns \\axiom{primitivePart(\\spad{p})} if \\axiom{\\spad{R}} is a gcd-domain,{} otherwise \\axiom{unitCanonical(\\spad{p})}.")) (|convert| (($ (|Polynomial| |#2|)) "\\axiom{convert(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}},{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.")) (|retract| (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#2|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#2|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.")) (|initiallyReduce| (($ $ $) "\\axiom{initiallyReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|headReduce| (($ $ $) "\\axiom{headReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| $) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{p},{}\\spad{q},{}\\spad{n}]} where \\axiom{\\spad{p} / q**n} represents the residue class of \\axiom{a} modulo \\axiom{\\spad{b}} and \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{q}} is \\axiom{init(\\spad{b})}.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} computes \\axiom{a mod \\spad{b}},{} if \\axiom{\\spad{b}} is monic as univariate polynomial in its main variable.")) (|pseudoDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{pseudoDivide(a,{}\\spad{b})} computes \\axiom{[pquo(a,{}\\spad{b}),{}prem(a,{}\\spad{b})]},{} both polynomials viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}},{} if \\axiom{\\spad{b}} is not a constant polynomial.")) (|lazyPseudoDivide| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $ |#4|) "\\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})},{} \\axiom{(c**g)*r = prem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}] = lazyPremWithDefault(a,{}\\spad{b})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.")) (|lazyPremWithDefault| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $ |#4|) "\\axiom{lazyPremWithDefault(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{(c**g)*r = prem(a,{}\\spad{b},{}\\spad{v})}.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $) "\\axiom{lazyPremWithDefault(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b})} and \\axiom{(c**g)*r = prem(a,{}\\spad{b})}.")) (|lazyPquo| (($ $ $ |#4|) "\\axiom{lazyPquo(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.") (($ $ $) "\\axiom{lazyPquo(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.")) (|lazyPrem| (($ $ $ |#4|) "\\axiom{lazyPrem(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} viewed as univariate polynomials in the variable \\axiom{\\spad{v}} such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.") (($ $ $) "\\axiom{lazyPrem(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.")) (|pquo| (($ $ $ |#4|) "\\axiom{pquo(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{pquo(a,{}\\spad{b})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|prem| (($ $ $ |#4|) "\\axiom{prem(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{prem(a,{}\\spad{b})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|normalized?| (((|Boolean|) $ (|List| $)) "\\axiom{normalized?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{normalized?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{normalized?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{a} and its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variable of \\axiom{\\spad{b}}")) (|initiallyReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{initiallyReduced?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{initiallyReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{initiallyReduced?(a,{}\\spad{b})} returns \\spad{false} iff there exists an iterated initial of \\axiom{a} which is not reduced \\spad{w}.\\spad{r}.\\spad{t} \\axiom{\\spad{b}}.")) (|headReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{headReduced?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{headReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{headReduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(head(a),{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|reduced?| (((|Boolean|) $ (|List| $)) "\\axiom{reduced?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{reduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{reduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(a,{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|supRittWu?| (((|Boolean|) $ $) "\\axiom{supRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is greater than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is less than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|RittWuCompare| (((|Union| (|Boolean|) "failed") $ $) "\\axiom{RittWuCompare(a,{}\\spad{b})} returns \\axiom{\"failed\"} if \\axiom{a} and \\axiom{\\spad{b}} have same rank \\spad{w}.\\spad{r}.\\spad{t}. Ritt and Wu Wen Tsun ordering using the refinement of Lazard,{} otherwise returns \\axiom{infRittWu?(a,{}\\spad{b})}.")) (|mainMonomials| (((|List| $) $) "\\axiom{mainMonomials(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [1],{} otherwise returns the list of the monomials of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainCoefficients| (((|List| $) $) "\\axiom{mainCoefficients(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [\\spad{p}],{} otherwise returns the list of the coefficients of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|leastMonomial| (($ $) "\\axiom{leastMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} the monomial of \\axiom{\\spad{p}} with lowest degree,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainMonomial| (($ $) "\\axiom{mainMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} \\axiom{mvar(\\spad{p})} raised to the power \\axiom{mdeg(\\spad{p})}.")) (|quasiMonic?| (((|Boolean|) $) "\\axiom{quasiMonic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff the initial of \\axiom{\\spad{p}} lies in the base ring \\axiom{\\spad{R}}.")) (|monic?| (((|Boolean|) $) "\\axiom{monic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff \\axiom{\\spad{p}} is monic as a univariate polynomial in its main variable.")) (|reductum| (($ $ |#4|) "\\axiom{reductum(\\spad{p},{}\\spad{v})} returns the reductum of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in \\axiom{\\spad{v}}.")) (|leadingCoefficient| (($ $ |#4|) "\\axiom{leadingCoefficient(\\spad{p},{}\\spad{v})} returns the leading coefficient of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as A univariate polynomial in \\axiom{\\spad{v}}.")) (|deepestInitial| (($ $) "\\axiom{deepestInitial(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the last term of \\axiom{iteratedInitials(\\spad{p})}.")) (|iteratedInitials| (((|List| $) $) "\\axiom{iteratedInitials(\\spad{p})} returns \\axiom{[]} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the list of the iterated initials of \\axiom{\\spad{p}}.")) (|deepestTail| (($ $) "\\axiom{deepestTail(\\spad{p})} returns \\axiom{0} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns tail(\\spad{p}),{} if \\axiom{tail(\\spad{p})} belongs to \\axiom{\\spad{R}} or \\axiom{mvar(tail(\\spad{p})) < mvar(\\spad{p})},{} otherwise returns \\axiom{deepestTail(tail(\\spad{p}))}.")) (|tail| (($ $) "\\axiom{tail(\\spad{p})} returns its reductum,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|head| (($ $) "\\axiom{head(\\spad{p})} returns \\axiom{\\spad{p}} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading term (monomial in the AXIOM sense),{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|init| (($ $) "\\axiom{init(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading coefficient,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mdeg| (((|NonNegativeInteger|) $) "\\axiom{mdeg(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{0},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{0},{} otherwise,{} returns the degree of \\axiom{\\spad{p}} in its main variable.")) (|mvar| ((|#4| $) "\\axiom{mvar(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in \\axiom{\\spad{V}}."))) NIL -((|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (|HasCategory| |#2| (QUOTE (-479))) (|HasCategory| |#2| (QUOTE (-38 (-480)))) (|HasCategory| |#2| (QUOTE (-899 (-480)))) (|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#4| (QUOTE (-550 (-1081))))) -(-971 R E V) +((|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (|HasCategory| |#2| (QUOTE (-482))) (|HasCategory| |#2| (QUOTE (-38 (-483)))) (|HasCategory| |#2| (QUOTE (-904 (-483)))) (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#4| (QUOTE (-553 (-1088))))) +(-976 R E V) ((|constructor| (NIL "A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \\axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.")) (|mainSquareFreePart| (($ $) "\\axiom{mainSquareFreePart(\\spad{p})} returns the square free part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainPrimitivePart| (($ $) "\\axiom{mainPrimitivePart(\\spad{p})} returns the primitive part of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|mainContent| (($ $) "\\axiom{mainContent(\\spad{p})} returns the content of \\axiom{\\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \\axiom{\\spad{R}}.")) (|primitivePart!| (($ $) "\\axiom{primitivePart!(\\spad{p})} replaces \\axiom{\\spad{p}} by its primitive part.")) (|gcd| ((|#1| |#1| $) "\\axiom{gcd(\\spad{r},{}\\spad{p})} returns the gcd of \\axiom{\\spad{r}} and the content of \\axiom{\\spad{p}}.")) (|nextsubResultant2| (($ $ $ $ $) "\\axiom{\\spad{nextsubResultant2}(\\spad{p},{}\\spad{q},{}\\spad{z},{}\\spad{s})} is the multivariate version of the operation \\axiomOpFrom{\\spad{next_sousResultant2}}{PseudoRemainderSequence} from the \\axiomType{PseudoRemainderSequence} constructor.")) (|LazardQuotient2| (($ $ $ $ (|NonNegativeInteger|)) "\\axiom{\\spad{LazardQuotient2}(\\spad{p},{}a,{}\\spad{b},{}\\spad{n})} returns \\axiom{(a**(\\spad{n}-1) * \\spad{p}) exquo b**(\\spad{n}-1)} assuming that this quotient does not fail.")) (|LazardQuotient| (($ $ $ (|NonNegativeInteger|)) "\\axiom{LazardQuotient(a,{}\\spad{b},{}\\spad{n})} returns \\axiom{a**n exquo b**(\\spad{n}-1)} assuming that this quotient does not fail.")) (|lastSubResultant| (($ $ $) "\\axiom{lastSubResultant(a,{}\\spad{b})} returns the last non-zero subresultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|subResultantChain| (((|List| $) $ $) "\\axiom{subResultantChain(a,{}\\spad{b})},{} where \\axiom{a} and \\axiom{\\spad{b}} are not contant polynomials with the same main variable,{} returns the subresultant chain of \\axiom{a} and \\axiom{\\spad{b}}.")) (|resultant| (($ $ $) "\\axiom{resultant(a,{}\\spad{b})} computes the resultant of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}}.")) (|halfExtendedSubResultantGcd2| (((|Record| (|:| |gcd| $) (|:| |coef2| $)) $ $) "\\axiom{\\spad{halfExtendedSubResultantGcd2}(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}cb]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}cb]} otherwise produces an error.")) (|halfExtendedSubResultantGcd1| (((|Record| (|:| |gcd| $) (|:| |coef1| $)) $ $) "\\axiom{\\spad{halfExtendedSubResultantGcd1}(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca]} if \\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[\\spad{g},{}ca,{}cb]} otherwise produces an error.")) (|extendedSubResultantGcd| (((|Record| (|:| |gcd| $) (|:| |coef1| $) (|:| |coef2| $)) $ $) "\\axiom{extendedSubResultantGcd(a,{}\\spad{b})} returns \\axiom{[ca,{}cb,{}\\spad{r}]} such that \\axiom{\\spad{r}} is \\axiom{subResultantGcd(a,{}\\spad{b})} and we have \\axiom{ca * a + cb * cb = \\spad{r}} .")) (|subResultantGcd| (($ $ $) "\\axiom{subResultantGcd(a,{}\\spad{b})} computes a gcd of \\axiom{a} and \\axiom{\\spad{b}} where \\axiom{a} and \\axiom{\\spad{b}} are assumed to have the same main variable \\axiom{\\spad{v}} and are viewed as univariate polynomials in \\axiom{\\spad{v}} with coefficients in the fraction field of the polynomial ring generated by their other variables over \\axiom{\\spad{R}}.")) (|exactQuotient!| (($ $ $) "\\axiom{exactQuotient!(a,{}\\spad{b})} replaces \\axiom{a} by \\axiom{exactQuotient(a,{}\\spad{b})}") (($ $ |#1|) "\\axiom{exactQuotient!(\\spad{p},{}\\spad{r})} replaces \\axiom{\\spad{p}} by \\axiom{exactQuotient(\\spad{p},{}\\spad{r})}.")) (|exactQuotient| (($ $ $) "\\axiom{exactQuotient(a,{}\\spad{b})} computes the exact quotient of \\axiom{a} by \\axiom{\\spad{b}},{} which is assumed to be a divisor of \\axiom{a}. No error is returned if this exact quotient fails!") (($ $ |#1|) "\\axiom{exactQuotient(\\spad{p},{}\\spad{r})} computes the exact quotient of \\axiom{\\spad{p}} by \\axiom{\\spad{r}},{} which is assumed to be a divisor of \\axiom{\\spad{p}}. No error is returned if this exact quotient fails!")) (|primPartElseUnitCanonical!| (($ $) "\\axiom{primPartElseUnitCanonical!(\\spad{p})} replaces \\axiom{\\spad{p}} by \\axiom{primPartElseUnitCanonical(\\spad{p})}.")) (|primPartElseUnitCanonical| (($ $) "\\axiom{primPartElseUnitCanonical(\\spad{p})} returns \\axiom{primitivePart(\\spad{p})} if \\axiom{\\spad{R}} is a gcd-domain,{} otherwise \\axiom{unitCanonical(\\spad{p})}.")) (|convert| (($ (|Polynomial| |#1|)) "\\axiom{convert(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}},{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.") (($ (|Polynomial| (|Integer|))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{convert(\\spad{p})} returns the same as \\axiom{retract(\\spad{p})}.")) (|retract| (($ (|Polynomial| |#1|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#1|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| |#1|)) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Integer|))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.") (($ (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retract(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if \\axiom{retractIfCan(\\spad{p})} does not return \"failed\",{} otherwise an error is produced.")) (|retractIfCan| (((|Union| $ "failed") (|Polynomial| |#1|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#1|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| |#1|)) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Integer|))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.") (((|Union| $ "failed") (|Polynomial| (|Fraction| (|Integer|)))) "\\axiom{retractIfCan(\\spad{p})} returns \\axiom{\\spad{p}} as an element of the current domain if all its variables belong to \\axiom{\\spad{V}}.")) (|initiallyReduce| (($ $ $) "\\axiom{initiallyReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|headReduce| (($ $ $) "\\axiom{headReduce(a,{}\\spad{b})} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduced?(\\spad{r},{}\\spad{b})} holds and there exists an integer \\axiom{\\spad{e}} such that \\axiom{init(\\spad{b})^e a - \\spad{r}} is zero modulo \\axiom{\\spad{b}}.")) (|lazyResidueClass| (((|Record| (|:| |polnum| $) (|:| |polden| $) (|:| |power| (|NonNegativeInteger|))) $ $) "\\axiom{lazyResidueClass(a,{}\\spad{b})} returns \\axiom{[\\spad{p},{}\\spad{q},{}\\spad{n}]} where \\axiom{\\spad{p} / q**n} represents the residue class of \\axiom{a} modulo \\axiom{\\spad{b}} and \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and \\axiom{\\spad{q}} is \\axiom{init(\\spad{b})}.")) (|monicModulo| (($ $ $) "\\axiom{monicModulo(a,{}\\spad{b})} computes \\axiom{a mod \\spad{b}},{} if \\axiom{\\spad{b}} is monic as univariate polynomial in its main variable.")) (|pseudoDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{pseudoDivide(a,{}\\spad{b})} computes \\axiom{[pquo(a,{}\\spad{b}),{}prem(a,{}\\spad{b})]},{} both polynomials viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}},{} if \\axiom{\\spad{b}} is not a constant polynomial.")) (|lazyPseudoDivide| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $ |#3|) "\\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})},{} \\axiom{(c**g)*r = prem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]} such that \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}] = lazyPremWithDefault(a,{}\\spad{b})} and \\axiom{\\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.")) (|lazyPremWithDefault| (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $ |#3|) "\\axiom{lazyPremWithDefault(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b},{}\\spad{v})} and \\axiom{(c**g)*r = prem(a,{}\\spad{b},{}\\spad{v})}.") (((|Record| (|:| |coef| $) (|:| |gap| (|NonNegativeInteger|)) (|:| |remainder| $)) $ $) "\\axiom{lazyPremWithDefault(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{r}]} such that \\axiom{\\spad{r} = lazyPrem(a,{}\\spad{b})} and \\axiom{(c**g)*r = prem(a,{}\\spad{b})}.")) (|lazyPquo| (($ $ $ |#3|) "\\axiom{lazyPquo(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b},{}\\spad{v})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.") (($ $ $) "\\axiom{lazyPquo(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{q}} such that \\axiom{lazyPseudoDivide(a,{}\\spad{b})} returns \\axiom{[\\spad{c},{}\\spad{g},{}\\spad{q},{}\\spad{r}]}.")) (|lazyPrem| (($ $ $ |#3|) "\\axiom{lazyPrem(a,{}\\spad{b},{}\\spad{v})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} viewed as univariate polynomials in the variable \\axiom{\\spad{v}} such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.") (($ $ $) "\\axiom{lazyPrem(a,{}\\spad{b})} returns the polynomial \\axiom{\\spad{r}} reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{\\spad{b}} and such that \\axiom{\\spad{b}} divides \\axiom{init(\\spad{b})^e a - \\spad{r}} where \\axiom{\\spad{e}} is the number of steps of this pseudo-division.")) (|pquo| (($ $ $ |#3|) "\\axiom{pquo(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{pquo(a,{}\\spad{b})} computes the pseudo-quotient of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|prem| (($ $ $ |#3|) "\\axiom{prem(a,{}\\spad{b},{}\\spad{v})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in \\axiom{\\spad{v}}.") (($ $ $) "\\axiom{prem(a,{}\\spad{b})} computes the pseudo-remainder of \\axiom{a} by \\axiom{\\spad{b}},{} both viewed as univariate polynomials in the main variable of \\axiom{\\spad{b}}.")) (|normalized?| (((|Boolean|) $ (|List| $)) "\\axiom{normalized?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{normalized?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{normalized?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{a} and its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variable of \\axiom{\\spad{b}}")) (|initiallyReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{initiallyReduced?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{initiallyReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{initiallyReduced?(a,{}\\spad{b})} returns \\spad{false} iff there exists an iterated initial of \\axiom{a} which is not reduced \\spad{w}.\\spad{r}.\\spad{t} \\axiom{\\spad{b}}.")) (|headReduced?| (((|Boolean|) $ (|List| $)) "\\axiom{headReduced?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{headReduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{headReduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(head(a),{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|reduced?| (((|Boolean|) $ (|List| $)) "\\axiom{reduced?(\\spad{q},{}lp)} returns \\spad{true} iff \\axiom{reduced?(\\spad{q},{}\\spad{p})} holds for every \\axiom{\\spad{p}} in \\axiom{lp}.") (((|Boolean|) $ $) "\\axiom{reduced?(a,{}\\spad{b})} returns \\spad{true} iff \\axiom{degree(a,{}mvar(\\spad{b})) < mdeg(\\spad{b})}.")) (|supRittWu?| (((|Boolean|) $ $) "\\axiom{supRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is greater than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(a,{}\\spad{b})} returns \\spad{true} if \\axiom{a} is less than \\axiom{\\spad{b}} \\spad{w}.\\spad{r}.\\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.")) (|RittWuCompare| (((|Union| (|Boolean|) "failed") $ $) "\\axiom{RittWuCompare(a,{}\\spad{b})} returns \\axiom{\"failed\"} if \\axiom{a} and \\axiom{\\spad{b}} have same rank \\spad{w}.\\spad{r}.\\spad{t}. Ritt and Wu Wen Tsun ordering using the refinement of Lazard,{} otherwise returns \\axiom{infRittWu?(a,{}\\spad{b})}.")) (|mainMonomials| (((|List| $) $) "\\axiom{mainMonomials(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [1],{} otherwise returns the list of the monomials of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainCoefficients| (((|List| $) $) "\\axiom{mainCoefficients(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns [\\spad{p}],{} otherwise returns the list of the coefficients of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|leastMonomial| (($ $) "\\axiom{leastMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} the monomial of \\axiom{\\spad{p}} with lowest degree,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mainMonomial| (($ $) "\\axiom{mainMonomial(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{\\spad{O}},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{1},{} otherwise,{} \\axiom{mvar(\\spad{p})} raised to the power \\axiom{mdeg(\\spad{p})}.")) (|quasiMonic?| (((|Boolean|) $) "\\axiom{quasiMonic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff the initial of \\axiom{\\spad{p}} lies in the base ring \\axiom{\\spad{R}}.")) (|monic?| (((|Boolean|) $) "\\axiom{monic?(\\spad{p})} returns \\spad{false} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns \\spad{true} iff \\axiom{\\spad{p}} is monic as a univariate polynomial in its main variable.")) (|reductum| (($ $ |#3|) "\\axiom{reductum(\\spad{p},{}\\spad{v})} returns the reductum of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in \\axiom{\\spad{v}}.")) (|leadingCoefficient| (($ $ |#3|) "\\axiom{leadingCoefficient(\\spad{p},{}\\spad{v})} returns the leading coefficient of \\axiom{\\spad{p}},{} where \\axiom{\\spad{p}} is viewed as A univariate polynomial in \\axiom{\\spad{v}}.")) (|deepestInitial| (($ $) "\\axiom{deepestInitial(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the last term of \\axiom{iteratedInitials(\\spad{p})}.")) (|iteratedInitials| (((|List| $) $) "\\axiom{iteratedInitials(\\spad{p})} returns \\axiom{[]} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns the list of the iterated initials of \\axiom{\\spad{p}}.")) (|deepestTail| (($ $) "\\axiom{deepestTail(\\spad{p})} returns \\axiom{0} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns tail(\\spad{p}),{} if \\axiom{tail(\\spad{p})} belongs to \\axiom{\\spad{R}} or \\axiom{mvar(tail(\\spad{p})) < mvar(\\spad{p})},{} otherwise returns \\axiom{deepestTail(tail(\\spad{p}))}.")) (|tail| (($ $) "\\axiom{tail(\\spad{p})} returns its reductum,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|head| (($ $) "\\axiom{head(\\spad{p})} returns \\axiom{\\spad{p}} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading term (monomial in the AXIOM sense),{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|init| (($ $) "\\axiom{init(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its leading coefficient,{} where \\axiom{\\spad{p}} is viewed as a univariate polynomial in its main variable.")) (|mdeg| (((|NonNegativeInteger|) $) "\\axiom{mdeg(\\spad{p})} returns an error if \\axiom{\\spad{p}} is \\axiom{0},{} otherwise,{} if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}} returns \\axiom{0},{} otherwise,{} returns the degree of \\axiom{\\spad{p}} in its main variable.")) (|mvar| ((|#3| $) "\\axiom{mvar(\\spad{p})} returns an error if \\axiom{\\spad{p}} belongs to \\axiom{\\spad{R}},{} otherwise returns its main variable \\spad{w}. \\spad{r}. \\spad{t}. to the total ordering on the elements in \\axiom{\\spad{V}}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-972) +(-977) ((|constructor| (NIL "This domain represents the `repeat' iterator syntax.")) (|body| (((|SpadAst|) $) "\\spad{body(e)} returns the body of the loop `e'.")) (|iterators| (((|List| (|SpadAst|)) $) "\\spad{iterators(e)} returns the list of iterators controlling the loop `e'."))) NIL NIL -(-973 S |TheField| |ThePols|) +(-978 S |TheField| |ThePols|) ((|constructor| (NIL "\\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.")) (|relativeApprox| ((|#2| |#3| $ |#2|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|approximate| ((|#2| |#3| $ |#2|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|rootOf| (((|Union| $ "failed") |#3| (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} gives the \\spad{n}th root for the order of the Real Closure")) (|allRootsOf| (((|List| $) |#3|) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} in the Real Closure,{} assumed in order.")) (|definingPolynomial| ((|#3| $) "\\axiom{definingPolynomial(aRoot)} gives a polynomial such that \\axiom{definingPolynomial(aRoot).aRoot = 0}")) (|recip| (((|Union| |#3| "failed") |#3| $) "\\axiom{recip(pol,{}aRoot)} tries to inverse \\axiom{pol} interpreted as \\axiom{aRoot}")) (|positive?| (((|Boolean|) |#3| $) "\\axiom{positive?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is positive")) (|negative?| (((|Boolean|) |#3| $) "\\axiom{negative?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is negative")) (|zero?| (((|Boolean|) |#3| $) "\\axiom{zero?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is \\axiom{0}")) (|sign| (((|Integer|) |#3| $) "\\axiom{sign(pol,{}aRoot)} gives the sign of \\axiom{pol} interpreted as \\axiom{aRoot}"))) NIL NIL -(-974 |TheField| |ThePols|) +(-979 |TheField| |ThePols|) ((|constructor| (NIL "\\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.")) (|relativeApprox| ((|#1| |#2| $ |#1|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|approximate| ((|#1| |#2| $ |#1|) "\\axiom{approximate(term,{}root,{}prec)} gives an approximation of \\axiom{term} over \\axiom{root} with precision \\axiom{prec}")) (|rootOf| (((|Union| $ "failed") |#2| (|PositiveInteger|)) "\\axiom{rootOf(pol,{}\\spad{n})} gives the \\spad{n}th root for the order of the Real Closure")) (|allRootsOf| (((|List| $) |#2|) "\\axiom{allRootsOf(pol)} creates all the roots of \\axiom{pol} in the Real Closure,{} assumed in order.")) (|definingPolynomial| ((|#2| $) "\\axiom{definingPolynomial(aRoot)} gives a polynomial such that \\axiom{definingPolynomial(aRoot).aRoot = 0}")) (|recip| (((|Union| |#2| "failed") |#2| $) "\\axiom{recip(pol,{}aRoot)} tries to inverse \\axiom{pol} interpreted as \\axiom{aRoot}")) (|positive?| (((|Boolean|) |#2| $) "\\axiom{positive?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is positive")) (|negative?| (((|Boolean|) |#2| $) "\\axiom{negative?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is negative")) (|zero?| (((|Boolean|) |#2| $) "\\axiom{zero?(pol,{}aRoot)} answers if \\axiom{pol} interpreted as \\axiom{aRoot} is \\axiom{0}")) (|sign| (((|Integer|) |#2| $) "\\axiom{sign(pol,{}aRoot)} gives the sign of \\axiom{pol} interpreted as \\axiom{aRoot}"))) NIL NIL -(-975 R E V P TS) +(-980 R E V P TS) ((|constructor| (NIL "A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are proposed: in the sense of Zariski closure (like in Kalkbrener's algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard methods). This algorithm is valid for nay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \\spad{TS} and the packages \\axiomType{QCMPACK}(\\spad{R},{}\\spad{E},{}\\spad{V},{}\\spad{P},{}TS) and \\axiomType{RSETGCD}(\\spad{R},{}\\spad{E},{}\\spad{V},{}\\spad{P},{}TS). The same way it does not care about the way univariate polynomial gcd (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these gcd need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \\axiom{TS}. Thus,{} the operations of this package are not documented.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) NIL NIL -(-976 S R E V P) +(-981 S R E V P) ((|constructor| (NIL "The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polynomial ring \\spad{k[x1,...,xn]} where \\spad{k} is the fraction field of \\spad{R}. The triangular set \\spad{[t1,...,tm]} is regular iff for every \\spad{i} the initial of \\spad{ti+1} is invertible in the tower of simple extensions associated with \\spad{[t1,...,ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \\spad{I} iff the radical of \\spad{I} is equal to the intersection of the radical ideals generated by the saturated ideals of the \\spad{[T1,...,Ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \\spad{T} iff it is a split of Kalkbrener of the saturated ideal of \\spad{T}. Let \\spad{K} be an algebraic closure of \\spad{k}. Assume that \\spad{V} is finite with cardinality \\spad{n} and let \\spad{A} be the affine space \\spad{K^n}. For a regular triangular set \\spad{T} let denote by \\spad{W(T)} the set of regular zeros of \\spad{T}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given subset \\spad{S} of \\spad{A} iff the union of the \\spad{W(Ti)} contains \\spad{S} and is contained in the closure of \\spad{S} (\\spad{w}.\\spad{r}.\\spad{t}. Zariski topology). A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given triangular set \\spad{T} if it is a split of Lazard of \\spad{W(T)}. Note that if \\spad{[T1,...,Ts]} is a split of Lazard of \\spad{T} then it is also a split of Kalkbrener of \\spad{T}. The converse is \\spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \\newline References : \\indented{1}{[1] \\spad{M}. KALKBRENER \"Three contributions to elimination theory\"} \\indented{5}{Phd Thesis,{} University of Linz,{} Austria,{} 1991.} \\indented{1}{[2] \\spad{M}. KALKBRENER \"Algorithmic properties of polynomial rings\"} \\indented{5}{Journal of Symbol. Comp. 1998} \\indented{1}{[3] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)} \\indented{1}{[4] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|zeroSetSplit| (((|List| $) (|List| |#5|) (|Boolean|)) "\\spad{zeroSetSplit(lp,clos?)} returns \\spad{lts} a split of Kalkbrener of the radical ideal associated with \\spad{lp}. If \\spad{clos?} is \\spad{false},{} it is also a decomposition of the variety associated with \\spad{lp} into the regular zero set of the \\spad{ts} in \\spad{lts} (or,{} in other words,{} a split of Lazard of this variety). See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets.")) (|extend| (((|List| $) (|List| |#5|) (|List| $)) "\\spad{extend(lp,lts)} returns the same as \\spad{concat([extend(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#5|) $) "\\spad{extend(lp,ts)} returns \\spad{ts} if \\spad{empty? lp} \\spad{extend(p,ts)} if \\spad{lp = [p]} else \\spad{extend(first lp, extend(rest lp, ts))}") (((|List| $) |#5| (|List| $)) "\\spad{extend(p,lts)} returns the same as \\spad{concat([extend(p,ts) for ts in lts])|}") (((|List| $) |#5| $) "\\spad{extend(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is not a regular triangular set.")) (|internalAugment| (($ (|List| |#5|) $) "\\spad{internalAugment(lp,ts)} returns \\spad{ts} if \\spad{lp} is empty otherwise returns \\spad{internalAugment(rest lp, internalAugment(first lp, ts))}") (($ |#5| $) "\\spad{internalAugment(p,ts)} assumes that \\spad{augment(p,ts)} returns a singleton and returns it.")) (|augment| (((|List| $) (|List| |#5|) (|List| $)) "\\spad{augment(lp,lts)} returns the same as \\spad{concat([augment(lp,ts) for ts in lts])}") (((|List| $) (|List| |#5|) $) "\\spad{augment(lp,ts)} returns \\spad{ts} if \\spad{empty? lp},{} \\spad{augment(p,ts)} if \\spad{lp = [p]},{} otherwise \\spad{augment(first lp, augment(rest lp, ts))}") (((|List| $) |#5| (|List| $)) "\\spad{augment(p,lts)} returns the same as \\spad{concat([augment(p,ts) for ts in lts])}") (((|List| $) |#5| $) "\\spad{augment(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. This operation assumes also that if \\spad{p} is added to \\spad{ts} the resulting set,{} say \\spad{ts+p},{} is a regular triangular set. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is required to be square-free.")) (|intersect| (((|List| $) |#5| (|List| $)) "\\spad{intersect(p,lts)} returns the same as \\spad{intersect([p],lts)}") (((|List| $) (|List| |#5|) (|List| $)) "\\spad{intersect(lp,lts)} returns the same as \\spad{concat([intersect(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#5|) $) "\\spad{intersect(lp,ts)} returns \\spad{lts} a split of Lazard of the intersection of the affine variety associated with \\spad{lp} and the regular zero set of \\spad{ts}.") (((|List| $) |#5| $) "\\spad{intersect(p,ts)} returns the same as \\spad{intersect([p],ts)}")) (|squareFreePart| (((|List| (|Record| (|:| |val| |#5|) (|:| |tower| $))) |#5| $) "\\spad{squareFreePart(p,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a square-free polynomial \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} this polynomial being associated with \\spad{p} modulo \\spad{lpwt.i.tower},{} for every \\spad{i}. Moreover,{} the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. WARNING: This assumes that \\spad{p} is a non-constant polynomial such that if \\spad{p} is added to \\spad{ts},{} then the resulting set is a regular triangular set.")) (|lastSubResultant| (((|List| (|Record| (|:| |val| |#5|) (|:| |tower| $))) |#5| |#5| $) "\\spad{lastSubResultant(p1,p2,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a quasi-monic gcd of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} for every \\spad{i},{} and such that the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. Moreover,{} if \\spad{p1} and \\spad{p2} do not have a non-trivial gcd \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower} then \\spad{lpwt.i.val} is the resultant of these polynomials \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|lastSubResultantElseSplit| (((|Union| |#5| (|List| $)) |#5| |#5| $) "\\spad{lastSubResultantElseSplit(p1,p2,ts)} returns either \\spad{g} a quasi-monic gcd of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. the \\spad{ts} or a split of Kalkbrener of \\spad{ts}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|invertibleSet| (((|List| $) |#5| $) "\\spad{invertibleSet(p,ts)} returns a split of Kalkbrener of the quotient ideal of the ideal \\axiom{\\spad{I}} by \\spad{p} where \\spad{I} is the radical of saturated of \\spad{ts}.")) (|invertible?| (((|Boolean|) |#5| $) "\\spad{invertible?(p,ts)} returns \\spad{true} iff \\spad{p} is invertible in the tower associated with \\spad{ts}.") (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| $))) |#5| $) "\\spad{invertible?(p,ts)} returns \\spad{lbwt} where \\spad{lbwt.i} is the result of \\spad{invertibleElseSplit?(p,lbwt.i.tower)} and the list of the \\spad{(lqrwt.i).tower} is a split of Kalkbrener of \\spad{ts}.")) (|invertibleElseSplit?| (((|Union| (|Boolean|) (|List| $)) |#5| $) "\\spad{invertibleElseSplit?(p,ts)} returns \\spad{true} (resp. \\spad{false}) if \\spad{p} is invertible in the tower associated with \\spad{ts} or returns a split of Kalkbrener of \\spad{ts}.")) (|purelyAlgebraicLeadingMonomial?| (((|Boolean|) |#5| $) "\\spad{purelyAlgebraicLeadingMonomial?(p,ts)} returns \\spad{true} iff the main variable of any non-constant iterarted initial of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|algebraicCoefficients?| (((|Boolean|) |#5| $) "\\spad{algebraicCoefficients?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} which is not the main one of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|purelyTranscendental?| (((|Boolean|) |#5| $) "\\spad{purelyTranscendental?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is not algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}")) (|purelyAlgebraic?| (((|Boolean|) $) "\\spad{purelyAlgebraic?(ts)} returns \\spad{true} iff for every algebraic variable \\spad{v} of \\spad{ts} we have \\spad{algebraicCoefficients?(t_v,ts_v_-)} where \\spad{ts_v} is \\axiomOpFrom{select}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}) and \\spad{ts_v_-} is \\axiomOpFrom{collectUnder}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}).") (((|Boolean|) |#5| $) "\\spad{purelyAlgebraic?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}."))) NIL NIL -(-977 R E V P) +(-982 R E V P) ((|constructor| (NIL "The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polynomial ring \\spad{k[x1,...,xn]} where \\spad{k} is the fraction field of \\spad{R}. The triangular set \\spad{[t1,...,tm]} is regular iff for every \\spad{i} the initial of \\spad{ti+1} is invertible in the tower of simple extensions associated with \\spad{[t1,...,ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \\spad{I} iff the radical of \\spad{I} is equal to the intersection of the radical ideals generated by the saturated ideals of the \\spad{[T1,...,Ti]}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \\spad{T} iff it is a split of Kalkbrener of the saturated ideal of \\spad{T}. Let \\spad{K} be an algebraic closure of \\spad{k}. Assume that \\spad{V} is finite with cardinality \\spad{n} and let \\spad{A} be the affine space \\spad{K^n}. For a regular triangular set \\spad{T} let denote by \\spad{W(T)} the set of regular zeros of \\spad{T}. A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given subset \\spad{S} of \\spad{A} iff the union of the \\spad{W(Ti)} contains \\spad{S} and is contained in the closure of \\spad{S} (\\spad{w}.\\spad{r}.\\spad{t}. Zariski topology). A family \\spad{[T1,...,Ts]} of regular triangular sets is a split of Lazard of a given triangular set \\spad{T} if it is a split of Lazard of \\spad{W(T)}. Note that if \\spad{[T1,...,Ts]} is a split of Lazard of \\spad{T} then it is also a split of Kalkbrener of \\spad{T}. The converse is \\spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \\newline References : \\indented{1}{[1] \\spad{M}. KALKBRENER \"Three contributions to elimination theory\"} \\indented{5}{Phd Thesis,{} University of Linz,{} Austria,{} 1991.} \\indented{1}{[2] \\spad{M}. KALKBRENER \"Algorithmic properties of polynomial rings\"} \\indented{5}{Journal of Symbol. Comp. 1998} \\indented{1}{[3] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)} \\indented{1}{[4] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|zeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|)) "\\spad{zeroSetSplit(lp,clos?)} returns \\spad{lts} a split of Kalkbrener of the radical ideal associated with \\spad{lp}. If \\spad{clos?} is \\spad{false},{} it is also a decomposition of the variety associated with \\spad{lp} into the regular zero set of the \\spad{ts} in \\spad{lts} (or,{} in other words,{} a split of Lazard of this variety). See the example illustrating the \\spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets.")) (|extend| (((|List| $) (|List| |#4|) (|List| $)) "\\spad{extend(lp,lts)} returns the same as \\spad{concat([extend(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#4|) $) "\\spad{extend(lp,ts)} returns \\spad{ts} if \\spad{empty? lp} \\spad{extend(p,ts)} if \\spad{lp = [p]} else \\spad{extend(first lp, extend(rest lp, ts))}") (((|List| $) |#4| (|List| $)) "\\spad{extend(p,lts)} returns the same as \\spad{concat([extend(p,ts) for ts in lts])|}") (((|List| $) |#4| $) "\\spad{extend(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is not a regular triangular set.")) (|internalAugment| (($ (|List| |#4|) $) "\\spad{internalAugment(lp,ts)} returns \\spad{ts} if \\spad{lp} is empty otherwise returns \\spad{internalAugment(rest lp, internalAugment(first lp, ts))}") (($ |#4| $) "\\spad{internalAugment(p,ts)} assumes that \\spad{augment(p,ts)} returns a singleton and returns it.")) (|augment| (((|List| $) (|List| |#4|) (|List| $)) "\\spad{augment(lp,lts)} returns the same as \\spad{concat([augment(lp,ts) for ts in lts])}") (((|List| $) (|List| |#4|) $) "\\spad{augment(lp,ts)} returns \\spad{ts} if \\spad{empty? lp},{} \\spad{augment(p,ts)} if \\spad{lp = [p]},{} otherwise \\spad{augment(first lp, augment(rest lp, ts))}") (((|List| $) |#4| (|List| $)) "\\spad{augment(p,lts)} returns the same as \\spad{concat([augment(p,ts) for ts in lts])}") (((|List| $) |#4| $) "\\spad{augment(p,ts)} assumes that \\spad{p} is a non-constant polynomial whose main variable is greater than any variable of \\spad{ts}. This operation assumes also that if \\spad{p} is added to \\spad{ts} the resulting set,{} say \\spad{ts+p},{} is a regular triangular set. Then it returns a split of Kalkbrener of \\spad{ts+p}. This may not be \\spad{ts+p} itself,{} if for instance \\spad{ts+p} is required to be square-free.")) (|intersect| (((|List| $) |#4| (|List| $)) "\\spad{intersect(p,lts)} returns the same as \\spad{intersect([p],lts)}") (((|List| $) (|List| |#4|) (|List| $)) "\\spad{intersect(lp,lts)} returns the same as \\spad{concat([intersect(lp,ts) for ts in lts])|}") (((|List| $) (|List| |#4|) $) "\\spad{intersect(lp,ts)} returns \\spad{lts} a split of Lazard of the intersection of the affine variety associated with \\spad{lp} and the regular zero set of \\spad{ts}.") (((|List| $) |#4| $) "\\spad{intersect(p,ts)} returns the same as \\spad{intersect([p],ts)}")) (|squareFreePart| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| $))) |#4| $) "\\spad{squareFreePart(p,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a square-free polynomial \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} this polynomial being associated with \\spad{p} modulo \\spad{lpwt.i.tower},{} for every \\spad{i}. Moreover,{} the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. WARNING: This assumes that \\spad{p} is a non-constant polynomial such that if \\spad{p} is added to \\spad{ts},{} then the resulting set is a regular triangular set.")) (|lastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| $))) |#4| |#4| $) "\\spad{lastSubResultant(p1,p2,ts)} returns \\spad{lpwt} such that \\spad{lpwt.i.val} is a quasi-monic gcd of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower},{} for every \\spad{i},{} and such that the list of the \\spad{lpwt.i.tower} is a split of Kalkbrener of \\spad{ts}. Moreover,{} if \\spad{p1} and \\spad{p2} do not have a non-trivial gcd \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower} then \\spad{lpwt.i.val} is the resultant of these polynomials \\spad{w}.\\spad{r}.\\spad{t}. \\spad{lpwt.i.tower}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|lastSubResultantElseSplit| (((|Union| |#4| (|List| $)) |#4| |#4| $) "\\spad{lastSubResultantElseSplit(p1,p2,ts)} returns either \\spad{g} a quasi-monic gcd of \\spad{p1} and \\spad{p2} \\spad{w}.\\spad{r}.\\spad{t}. the \\spad{ts} or a split of Kalkbrener of \\spad{ts}. This assumes that \\spad{p1} and \\spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \\spad{ts}.")) (|invertibleSet| (((|List| $) |#4| $) "\\spad{invertibleSet(p,ts)} returns a split of Kalkbrener of the quotient ideal of the ideal \\axiom{\\spad{I}} by \\spad{p} where \\spad{I} is the radical of saturated of \\spad{ts}.")) (|invertible?| (((|Boolean|) |#4| $) "\\spad{invertible?(p,ts)} returns \\spad{true} iff \\spad{p} is invertible in the tower associated with \\spad{ts}.") (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| $))) |#4| $) "\\spad{invertible?(p,ts)} returns \\spad{lbwt} where \\spad{lbwt.i} is the result of \\spad{invertibleElseSplit?(p,lbwt.i.tower)} and the list of the \\spad{(lqrwt.i).tower} is a split of Kalkbrener of \\spad{ts}.")) (|invertibleElseSplit?| (((|Union| (|Boolean|) (|List| $)) |#4| $) "\\spad{invertibleElseSplit?(p,ts)} returns \\spad{true} (resp. \\spad{false}) if \\spad{p} is invertible in the tower associated with \\spad{ts} or returns a split of Kalkbrener of \\spad{ts}.")) (|purelyAlgebraicLeadingMonomial?| (((|Boolean|) |#4| $) "\\spad{purelyAlgebraicLeadingMonomial?(p,ts)} returns \\spad{true} iff the main variable of any non-constant iterarted initial of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|algebraicCoefficients?| (((|Boolean|) |#4| $) "\\spad{algebraicCoefficients?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} which is not the main one of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}.")) (|purelyTranscendental?| (((|Boolean|) |#4| $) "\\spad{purelyTranscendental?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is not algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}")) (|purelyAlgebraic?| (((|Boolean|) $) "\\spad{purelyAlgebraic?(ts)} returns \\spad{true} iff for every algebraic variable \\spad{v} of \\spad{ts} we have \\spad{algebraicCoefficients?(t_v,ts_v_-)} where \\spad{ts_v} is \\axiomOpFrom{select}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}) and \\spad{ts_v_-} is \\axiomOpFrom{collectUnder}{TriangularSetCategory}(\\spad{ts},{}\\spad{v}).") (((|Boolean|) |#4| $) "\\spad{purelyAlgebraic?(p,ts)} returns \\spad{true} iff every variable of \\spad{p} is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ts}."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-978 R E V P TS) +(-983 R E V P TS) ((|constructor| (NIL "An internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of gcd over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of \\spad{AAECC11}} \\indented{5}{Paris,{} 1995.} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|toseSquareFreePart| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{toseSquareFreePart(\\spad{p},{}ts)} has the same specifications as \\axiomOpFrom{squareFreePart}{RegularTriangularSetCategory}.")) (|toseInvertibleSet| (((|List| |#5|) |#4| |#5|) "\\axiom{toseInvertibleSet(\\spad{p1},{}\\spad{p2},{}ts)} has the same specifications as \\axiomOpFrom{invertibleSet}{RegularTriangularSetCategory}.")) (|toseInvertible?| (((|List| (|Record| (|:| |val| (|Boolean|)) (|:| |tower| |#5|))) |#4| |#5|) "\\axiom{toseInvertible?(\\spad{p1},{}\\spad{p2},{}ts)} has the same specifications as \\axiomOpFrom{invertible?}{RegularTriangularSetCategory}.") (((|Boolean|) |#4| |#5|) "\\axiom{toseInvertible?(\\spad{p1},{}\\spad{p2},{}ts)} has the same specifications as \\axiomOpFrom{invertible?}{RegularTriangularSetCategory}.")) (|toseLastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#4| |#5|) "\\axiom{toseLastSubResultant(\\spad{p1},{}\\spad{p2},{}ts)} has the same specifications as \\axiomOpFrom{lastSubResultant}{RegularTriangularSetCategory}.")) (|integralLastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#4| |#5|) "\\axiom{integralLastSubResultant(\\spad{p1},{}\\spad{p2},{}ts)} is an internal subroutine,{} exported only for developement.")) (|internalLastSubResultant| (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) (|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) |#3| (|Boolean|)) "\\axiom{internalLastSubResultant(lpwt,{}\\spad{v},{}flag)} is an internal subroutine,{} exported only for developement.") (((|List| (|Record| (|:| |val| |#4|) (|:| |tower| |#5|))) |#4| |#4| |#5| (|Boolean|) (|Boolean|)) "\\axiom{internalLastSubResultant(\\spad{p1},{}\\spad{p2},{}ts,{}inv?,{}break?)} is an internal subroutine,{} exported only for developement.")) (|prepareSubResAlgo| (((|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) |#4| |#4| |#5|) "\\axiom{prepareSubResAlgo(\\spad{p1},{}\\spad{p2},{}ts)} is an internal subroutine,{} exported only for developement.")) (|stopTableInvSet!| (((|Void|)) "\\axiom{stopTableInvSet!()} is an internal subroutine,{} exported only for developement.")) (|startTableInvSet!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableInvSet!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement.")) (|stopTableGcd!| (((|Void|)) "\\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.")) (|startTableGcd!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableGcd!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement."))) NIL NIL -(-979) +(-984) ((|constructor| (NIL "This domain represents `restrict' expressions.")) (|target| (((|TypeAst|) $) "\\spad{target(e)} returns the target type of the conversion..")) (|expression| (((|SpadAst|) $) "\\spad{expression(e)} returns the expression being converted."))) NIL NIL -(-980) +(-985) ((|constructor| (NIL "This is the datatype of OpenAxiom runtime values. It exists solely for internal purposes.")) (|eq| (((|Boolean|) $ $) "\\spad{eq(x,y)} holds if both values \\spad{x} and \\spad{y} resides at the same address in memory."))) NIL NIL -(-981 |Base| R -3078) +(-986 |Base| R -3088) ((|constructor| (NIL "\\indented{1}{Rules for the pattern matcher} Author: Manuel Bronstein Date Created: 24 Oct 1988 Date Last Updated: 26 October 1993 Keywords: pattern,{} matching,{} rule.")) (|quotedOperators| (((|List| (|Symbol|)) $) "\\spad{quotedOperators(r)} returns the list of operators on the right hand side of \\spad{r} that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.")) (|elt| ((|#3| $ |#3| (|PositiveInteger|)) "\\spad{elt(r,f,n)} or \\spad{r}(\\spad{f},{} \\spad{n}) applies the rule \\spad{r} to \\spad{f} at most \\spad{n} times.")) (|rhs| ((|#3| $) "\\spad{rhs(r)} returns the right hand side of the rule \\spad{r}.")) (|lhs| ((|#3| $) "\\spad{lhs(r)} returns the left hand side of the rule \\spad{r}.")) (|pattern| (((|Pattern| |#1|) $) "\\spad{pattern(r)} returns the pattern corresponding to the left hand side of the rule \\spad{r}.")) (|suchThat| (($ $ (|List| (|Symbol|)) (|Mapping| (|Boolean|) (|List| |#3|))) "\\spad{suchThat(r, [a1,...,an], f)} returns the rewrite rule \\spad{r} with the predicate \\spad{f(a1,...,an)} attached to it.")) (|rule| (($ |#3| |#3| (|List| (|Symbol|))) "\\spad{rule(f, g, [f1,...,fn])} creates the rewrite rule \\spad{f == eval(eval(g, g is f), [f1,...,fn])},{} that is a rule with left-hand side \\spad{f} and right-hand side \\spad{g}; The symbols \\spad{f1},{}...,{}fn are the operators that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.") (($ |#3| |#3|) "\\spad{rule(f, g)} creates the rewrite rule: \\spad{f == eval(g, g is f)},{} with left-hand side \\spad{f} and right-hand side \\spad{g}."))) NIL NIL -(-982 |f|) +(-987 |f|) ((|constructor| (NIL "This domain implements named rules")) (|name| (((|Symbol|) $) "\\spad{name(x)} returns the symbol"))) NIL NIL -(-983 |Base| R -3078) +(-988 |Base| R -3088) ((|constructor| (NIL "A ruleset is a set of pattern matching rules grouped together.")) (|elt| ((|#3| $ |#3| (|PositiveInteger|)) "\\spad{elt(r,f,n)} or \\spad{r}(\\spad{f},{} \\spad{n}) applies all the rules of \\spad{r} to \\spad{f} at most \\spad{n} times.")) (|rules| (((|List| (|RewriteRule| |#1| |#2| |#3|)) $) "\\spad{rules(r)} returns the rules contained in \\spad{r}.")) (|ruleset| (($ (|List| (|RewriteRule| |#1| |#2| |#3|))) "\\spad{ruleset([r1,...,rn])} creates the rule set \\spad{{r1,...,rn}}."))) NIL NIL -(-984 R |ls|) +(-989 R |ls|) ((|constructor| (NIL "\\indented{1}{A package for computing the rational univariate representation} \\indented{1}{of a zero-dimensional algebraic variety given by a regular} \\indented{1}{triangular set. This package is essentially an interface for the} \\spadtype{InternalRationalUnivariateRepresentationPackage} constructor. It is used in the \\spadtype{ZeroDimensionalSolvePackage} for solving polynomial systems with finitely many solutions.")) (|rur| (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{rur(lp,univ?,check?)} returns the same as \\spad{rur(lp,true)}. Moreover,{} if \\spad{check?} is \\spad{true} then the result is checked.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{rur(lp)} returns the same as \\spad{rur(lp,true)}") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{rur(lp,univ?)} returns a rational univariate representation of \\spad{lp}. This assumes that \\spad{lp} defines a regular triangular \\spad{ts} whose associated variety is zero-dimensional over \\spad{R}. \\spad{rur(lp,univ?)} returns a list of items \\spad{[u,lc]} where \\spad{u} is an irreducible univariate polynomial and each \\spad{c} in \\spad{lc} involves two variables: one from \\spad{ls},{} called the coordinate of \\spad{c},{} and an extra variable which represents any root of \\spad{u}. Every root of \\spad{u} leads to a tuple of values for the coordinates of \\spad{lc}. Moreover,{} a point \\spad{x} belongs to the variety associated with \\spad{lp} iff there exists an item \\spad{[u,lc]} in \\spad{rur(lp,univ?)} and a root \\spad{r} of \\spad{u} such that \\spad{x} is given by the tuple of values for the coordinates of \\spad{lc} evaluated at \\spad{r}. If \\spad{univ?} is \\spad{true} then each polynomial \\spad{c} will have a constant leading coefficient \\spad{w}.\\spad{r}.\\spad{t}. its coordinate. See the example which illustrates the \\spadtype{ZeroDimensionalSolvePackage} package constructor."))) NIL NIL -(-985 R UP M) +(-990 R UP M) ((|constructor| (NIL "Domain which represents simple algebraic extensions of arbitrary rings. The first argument to the domain,{} \\spad{R},{} is the underlying ring,{} the second argument is a domain of univariate polynomials over \\spad{K},{} while the last argument specifies the defining minimal polynomial. The elements of the domain are canonically represented as polynomials of degree less than that of the minimal polynomial with coefficients in \\spad{R}. The second argument is both the type of the third argument and the underlying representation used by \\spadtype{SAE} itself."))) -((-3971 |has| |#1| (-309)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-296))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-296)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-315))) (OR (-12 (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-309)))) (|HasCategory| |#1| (QUOTE (-296)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-309)))) (-12 (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-309)))) (|HasCategory| |#1| (QUOTE (-296)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-804 (-1081))))) (-12 (|HasCategory| |#1| (QUOTE (-296))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-804 (-1081))))) (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-309)))) (|HasCategory| |#1| (QUOTE (-296)))) (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-806 (-1081))))) (-12 (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-309)))) (-12 (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-309)))) (-12 (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))))) -(-986 UP SAE UPA) +((-3982 |has| |#1| (-311)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-298))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-298)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-317))) (OR (-12 (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-311)))) (|HasCategory| |#1| (QUOTE (-298)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-311)))) (-12 (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-311)))) (|HasCategory| |#1| (QUOTE (-298)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-809 (-1088))))) (-12 (|HasCategory| |#1| (QUOTE (-298))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-809 (-1088))))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-311)))) (|HasCategory| |#1| (QUOTE (-298)))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-811 (-1088))))) (-12 (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-311)))) (-12 (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-311)))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))))) +(-991 UP SAE UPA) ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of the rational numbers (\\spadtype{Fraction Integer}).")) (|factor| (((|Factored| |#3|) |#3|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p}."))) NIL NIL -(-987 UP SAE UPA) +(-992 UP SAE UPA) ((|constructor| (NIL "Factorization of univariate polynomials with coefficients in an algebraic extension of \\spadtype{Fraction Polynomial Integer}.")) (|factor| (((|Factored| |#3|) |#3|) "\\spad{factor(p)} returns a prime factorisation of \\spad{p}."))) NIL NIL -(-988) +(-993) ((|constructor| (NIL "This trivial domain lets us build Univariate Polynomials in an anonymous variable"))) NIL NIL -(-989) +(-994) ((|constructor| (NIL "This is the category of Spad syntax objects."))) NIL NIL -(-990 S) +(-995 S) ((|constructor| (NIL "\\indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \\indented{2}{A sorted cache of a cachable set \\spad{S} is a dynamic structure that} \\indented{2}{keeps the elements of \\spad{S} sorted and assigns an integer to each} \\indented{2}{element of \\spad{S} once it is in the cache. This way,{} equality and ordering} \\indented{2}{on \\spad{S} are tested directly on the integers associated with the elements} \\indented{2}{of \\spad{S},{} once they have been entered in the cache.}")) (|enterInCache| ((|#1| |#1| (|Mapping| (|Integer|) |#1| |#1|)) "\\spad{enterInCache(x, f)} enters \\spad{x} in the cache,{} calling \\spad{f(x, y)} to determine whether \\spad{x < y (f(x,y) < 0), x = y (f(x,y) = 0)},{} or \\spad{x > y (f(x,y) > 0)}. It returns \\spad{x} with an integer associated with it.") ((|#1| |#1| (|Mapping| (|Boolean|) |#1|)) "\\spad{enterInCache(x, f)} enters \\spad{x} in the cache,{} calling \\spad{f(y)} to determine whether \\spad{x} is equal to \\spad{y}. It returns \\spad{x} with an integer associated with it.")) (|cache| (((|List| |#1|)) "\\spad{cache()} returns the current cache as a list.")) (|clearCache| (((|Void|)) "\\spad{clearCache()} empties the cache."))) NIL NIL -(-991) +(-996) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: October 24,{} 2007 Date Last Modified: January 18,{} 2008. A `Scope' is a sequence of contours.")) (|currentCategoryFrame| (($) "\\spad{currentCategoryFrame()} returns the category frame currently in effect.")) (|currentScope| (($) "\\spad{currentScope()} returns the scope currently in effect")) (|pushNewContour| (($ (|Binding|) $) "\\spad{pushNewContour(b,s)} pushs a new contour with sole binding `b'.")) (|findBinding| (((|Maybe| (|Binding|)) (|Identifier|) $) "\\spad{findBinding(n,s)} returns the first binding of `n' in `s'; otherwise `nothing'.")) (|contours| (((|List| (|Contour|)) $) "\\spad{contours(s)} returns the list of contours in scope \\spad{s}.")) (|empty| (($) "\\spad{empty()} returns an empty scope."))) NIL NIL -(-992 R) +(-997 R) ((|constructor| (NIL "StructuralConstantsPackage provides functions creating structural constants from a multiplication tables or a basis of a matrix algebra and other useful functions in this context.")) (|coordinates| (((|Vector| |#1|) (|Matrix| |#1|) (|List| (|Matrix| |#1|))) "\\spad{coordinates(a,[v1,...,vn])} returns the coordinates of \\spad{a} with respect to the \\spad{R}-module basis \\spad{v1},{}...,{}\\spad{vn}.")) (|structuralConstants| (((|Vector| (|Matrix| |#1|)) (|List| (|Matrix| |#1|))) "\\spad{structuralConstants(basis)} takes the \\spad{basis} of a matrix algebra,{} \\spadignore{e.g.} the result of \\spadfun{basisOfCentroid} and calculates the structural constants. Note,{} that the it is not checked,{} whether \\spad{basis} really is a \\spad{basis} of a matrix algebra.") (((|Vector| (|Matrix| (|Polynomial| |#1|))) (|List| (|Symbol|)) (|Matrix| (|Polynomial| |#1|))) "\\spad{structuralConstants(ls,mt)} determines the structural constants of an algebra with generators \\spad{ls} and multiplication table \\spad{mt},{} the entries of which must be given as linear polynomials in the indeterminates given by \\spad{ls}. The result is in particular useful \\indented{1}{as fourth argument for \\spadtype{AlgebraGivenByStructuralConstants}} \\indented{1}{and \\spadtype{GenericNonAssociativeAlgebra}.}") (((|Vector| (|Matrix| (|Fraction| (|Polynomial| |#1|)))) (|List| (|Symbol|)) (|Matrix| (|Fraction| (|Polynomial| |#1|)))) "\\spad{structuralConstants(ls,mt)} determines the structural constants of an algebra with generators \\spad{ls} and multiplication table \\spad{mt},{} the entries of which must be given as linear polynomials in the indeterminates given by \\spad{ls}. The result is in particular useful \\indented{1}{as fourth argument for \\spadtype{AlgebraGivenByStructuralConstants}} \\indented{1}{and \\spadtype{GenericNonAssociativeAlgebra}.}"))) NIL NIL -(-993 R) +(-998 R) ((|constructor| (NIL "\\spadtype{SequentialDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is sequential. \\blankline"))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-816)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| (-994 (-1081)) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| (-994 (-1081)) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-994 (-1081)) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-994 (-1081)) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| (-994 (-1081)) (QUOTE (-550 (-469))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-188))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-994 S) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| (-999 (-1088)) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| (-999 (-1088)) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-999 (-1088)) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-999 (-1088)) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| (-999 (-1088)) (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-190))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-999 S) ((|constructor| (NIL "\\spadtype{OrderlyDifferentialVariable} adds a commonly used sequential ranking to the set of derivatives of an ordered list of differential indeterminates. A sequential ranking is a ranking \\spadfun{<} of the derivatives with the property that for any derivative \\spad{v},{} there are only a finite number of derivatives \\spad{u} with \\spad{u} \\spadfun{<} \\spad{v}. This domain belongs to \\spadtype{DifferentialVariableCategory}. It defines \\spadfun{weight} to be just \\spadfun{order},{} and it defines a sequential ranking \\spadfun{<} on derivatives \\spad{u} by the lexicographic order on the pair (\\spadfun{variable}(\\spad{u}),{} \\spadfun{order}(\\spad{u}))."))) NIL NIL -(-995 S) +(-1000 S) ((|constructor| (NIL "This type is used to specify a range of values from type \\spad{S}."))) NIL -((|HasCategory| |#1| (QUOTE (-750))) (|HasCategory| |#1| (QUOTE (-1007)))) -(-996 R S) +((|HasCategory| |#1| (QUOTE (-755))) (|HasCategory| |#1| (QUOTE (-1012)))) +(-1001 R S) ((|constructor| (NIL "This package provides operations for mapping functions onto segments.")) (|map| (((|List| |#2|) (|Mapping| |#2| |#1|) (|Segment| |#1|)) "\\spad{map(f,s)} expands the segment \\spad{s},{} applying \\spad{f} to each value. For example,{} if \\spad{s = l..h by k},{} then the list \\spad{[f(l), f(l+k),..., f(lN)]} is computed,{} where \\spad{lN <= h < lN+k}.") (((|Segment| |#2|) (|Mapping| |#2| |#1|) (|Segment| |#1|)) "\\spad{map(f,l..h)} returns a new segment \\spad{f(l)..f(h)}."))) NIL -((|HasCategory| |#1| (QUOTE (-750)))) -(-997) +((|HasCategory| |#1| (QUOTE (-755)))) +(-1002) ((|constructor| (NIL "This domain represents segement expressions.")) (|bounds| (((|List| (|SpadAst|)) $) "\\spad{bounds(s)} returns the bounds of the segment `s'. If `s' designates an infinite interval,{} then the returns list a singleton list."))) NIL NIL -(-998 S) +(-1003 S) ((|constructor| (NIL "This domain is used to provide the function argument syntax \\spad{v=a..b}. This is used,{} for example,{} by the top-level \\spadfun{draw} functions."))) NIL -((|HasCategory| (-995 |#1|) (QUOTE (-1007)))) -(-999 R S) +((|HasCategory| (-1000 |#1|) (QUOTE (-1012)))) +(-1004 R S) ((|constructor| (NIL "This package provides operations for mapping functions onto \\spadtype{SegmentBinding}\\spad{s}.")) (|map| (((|SegmentBinding| |#2|) (|Mapping| |#2| |#1|) (|SegmentBinding| |#1|)) "\\spad{map(f,v=a..b)} returns the value given by \\spad{v=f(a)..f(b)}."))) NIL NIL -(-1000 S) +(-1005 S) ((|constructor| (NIL "This category provides operations on ranges,{} or {\\em segments} as they are called.")) (|segment| (($ |#1| |#1|) "\\spad{segment(i,j)} is an alternate way to create the segment \\spad{i..j}.")) (|incr| (((|Integer|) $) "\\spad{incr(s)} returns \\spad{n},{} where \\spad{s} is a segment in which every \\spad{n}\\spad{-}th element is used. Note: \\spad{incr(l..h by n) = n}.")) (|high| ((|#1| $) "\\spad{high(s)} returns the second endpoint of \\spad{s}. Note: \\spad{high(l..h) = h}.")) (|low| ((|#1| $) "\\spad{low(s)} returns the first endpoint of \\spad{s}. Note: \\spad{low(l..h) = l}.")) (|hi| ((|#1| $) "\\spad{hi(s)} returns the second endpoint of \\spad{s}. Note: \\spad{hi(l..h) = h}.")) (|lo| ((|#1| $) "\\spad{lo(s)} returns the first endpoint of \\spad{s}. Note: \\spad{lo(l..h) = l}.")) (BY (($ $ (|Integer|)) "\\spad{s by n} creates a new segment in which only every \\spad{n}\\spad{-}th element is used.")) (SEGMENT (($ |#1| |#1|) "\\spad{l..h} creates a segment with \\spad{l} and \\spad{h} as the endpoints."))) NIL NIL -(-1001 S L) +(-1006 S L) ((|constructor| (NIL "This category provides an interface for expanding segments to a stream of elements.")) (|map| ((|#2| (|Mapping| |#1| |#1|) $) "\\spad{map(f,l..h by k)} produces a value of type \\spad{L} by applying \\spad{f} to each of the succesive elements of the segment,{} that is,{} \\spad{[f(l), f(l+k), ..., f(lN)]},{} where \\spad{lN <= h < lN+k}.")) (|expand| ((|#2| $) "\\spad{expand(l..h by k)} creates value of type \\spad{L} with elements \\spad{l, l+k, ... lN} where \\spad{lN <= h < lN+k}. For example,{} \\spad{expand(1..5 by 2) = [1,3,5]}.") ((|#2| (|List| $)) "\\spad{expand(l)} creates a new value of type \\spad{L} in which each segment \\spad{l..h by k} is replaced with \\spad{l, l+k, ... lN},{} where \\spad{lN <= h < lN+k}. For example,{} \\spad{expand [1..4, 7..9] = [1,2,3,4,7,8,9]}."))) NIL NIL -(-1002) +(-1007) ((|constructor| (NIL "This domain represents a block of expressions.")) (|last| (((|SpadAst|) $) "\\spad{last(e)} returns the last instruction in `e'.")) (|body| (((|List| (|SpadAst|)) $) "\\spad{body(e)} returns the list of expressions in the sequence of instruction `e'."))) NIL NIL -(-1003 S) +(-1008 S) ((|constructor| (NIL "A set over a domain \\spad{D} models the usual mathematical notion of a finite set of elements from \\spad{D}. Sets are unordered collections of distinct elements (that is,{} order and duplication does not matter). The notation \\spad{set [a,b,c]} can be used to create a set and the usual operations such as union and intersection are available to form new sets. In our implementation,{} \\Language{} maintains the entries in sorted order. Specifically,{} the parts function returns the entries as a list in ascending order and the extract operation returns the maximum entry. Given two sets \\spad{s} and \\spad{t} where \\spad{\\#s = m} and \\spad{\\#t = n},{} the complexity of \\indented{2}{\\spad{s = t} is \\spad{O(min(n,m))}} \\indented{2}{\\spad{s < t} is \\spad{O(max(n,m))}} \\indented{2}{\\spad{union(s,t)},{} \\spad{intersect(s,t)},{} \\spad{minus(s,t)},{} \\spad{symmetricDifference(s,t)} is \\spad{O(max(n,m))}} \\indented{2}{\\spad{member(x,t)} is \\spad{O(n log n)}} \\indented{2}{\\spad{insert(x,t)} and \\spad{remove(x,t)} is \\spad{O(n)}}"))) -((-3978 . T) (-3968 . T) (-3979 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#1| (QUOTE (-315))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-1004 A S) +((-3989 . T) (-3979 . T) (-3990 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#1| (QUOTE (-317))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-1009 A S) ((|constructor| (NIL "A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.")) (|union| (($ |#2| $) "\\spad{union(x,u)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{x},{}\\spad{u})} returns a copy of \\spad{u}.") (($ $ |#2|) "\\spad{union(u,x)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{u},{}\\spad{x})} returns a copy of \\spad{u}.") (($ $ $) "\\spad{union(u,v)} returns the set aggregate of elements which are members of either set aggregate \\spad{u} or \\spad{v}.")) (|subset?| (((|Boolean|) $ $) "\\spad{subset?(u,v)} tests if \\spad{u} is a subset of \\spad{v}. Note: equivalent to \\axiom{reduce(and,{}{member?(\\spad{x},{}\\spad{v}) for \\spad{x} in \\spad{u}},{}\\spad{true},{}\\spad{false})}.")) (|symmetricDifference| (($ $ $) "\\spad{symmetricDifference(u,v)} returns the set aggregate of elements \\spad{x} which are members of set aggregate \\spad{u} or set aggregate \\spad{v} but not both. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{symmetricDifference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: \\axiom{symmetricDifference(\\spad{u},{}\\spad{v}) = union(difference(\\spad{u},{}\\spad{v}),{}difference(\\spad{v},{}\\spad{u}))}")) (|difference| (($ $ |#2|) "\\spad{difference(u,x)} returns the set aggregate \\spad{u} with element \\spad{x} removed. If \\spad{u} does not contain \\spad{x},{} a copy of \\spad{u} is returned. Note: \\axiom{difference(\\spad{s},{} \\spad{x}) = difference(\\spad{s},{} {\\spad{x}})}.") (($ $ $) "\\spad{difference(u,v)} returns the set aggregate \\spad{w} consisting of elements in set aggregate \\spad{u} but not in set aggregate \\spad{v}. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{difference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: equivalent to the notation (not currently supported) \\axiom{{\\spad{x} for \\spad{x} in \\spad{u} | not member?(\\spad{x},{}\\spad{v})}}.")) (|intersect| (($ $ $) "\\spad{intersect(u,v)} returns the set aggregate \\spad{w} consisting of elements common to both set aggregates \\spad{u} and \\spad{v}. Note: equivalent to the notation (not currently supported) {\\spad{x} for \\spad{x} in \\spad{u} | member?(\\spad{x},{}\\spad{v})}.")) (|set| (($ (|List| |#2|)) "\\spad{set([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.") (($) "\\spad{set()}\\$\\spad{D} creates an empty set aggregate of type \\spad{D}.")) (|brace| (($ (|List| |#2|)) "\\spad{brace([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}. This form is considered obsolete. Use \\axiomFun{set} instead.") (($) "\\spad{brace()}\\$\\spad{D} (otherwise written {}\\$\\spad{D}) creates an empty set aggregate of type \\spad{D}. This form is considered obsolete. Use \\axiomFun{set} instead.")) (|part?| (((|Boolean|) $ $) "\\spad{s} < \\spad{t} returns \\spad{true} if all elements of set aggregate \\spad{s} are also elements of set aggregate \\spad{t}."))) NIL NIL -(-1005 S) +(-1010 S) ((|constructor| (NIL "A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.")) (|union| (($ |#1| $) "\\spad{union(x,u)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{x},{}\\spad{u})} returns a copy of \\spad{u}.") (($ $ |#1|) "\\spad{union(u,x)} returns the set aggregate \\spad{u} with the element \\spad{x} added. If \\spad{u} already contains \\spad{x},{} \\axiom{union(\\spad{u},{}\\spad{x})} returns a copy of \\spad{u}.") (($ $ $) "\\spad{union(u,v)} returns the set aggregate of elements which are members of either set aggregate \\spad{u} or \\spad{v}.")) (|subset?| (((|Boolean|) $ $) "\\spad{subset?(u,v)} tests if \\spad{u} is a subset of \\spad{v}. Note: equivalent to \\axiom{reduce(and,{}{member?(\\spad{x},{}\\spad{v}) for \\spad{x} in \\spad{u}},{}\\spad{true},{}\\spad{false})}.")) (|symmetricDifference| (($ $ $) "\\spad{symmetricDifference(u,v)} returns the set aggregate of elements \\spad{x} which are members of set aggregate \\spad{u} or set aggregate \\spad{v} but not both. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{symmetricDifference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: \\axiom{symmetricDifference(\\spad{u},{}\\spad{v}) = union(difference(\\spad{u},{}\\spad{v}),{}difference(\\spad{v},{}\\spad{u}))}")) (|difference| (($ $ |#1|) "\\spad{difference(u,x)} returns the set aggregate \\spad{u} with element \\spad{x} removed. If \\spad{u} does not contain \\spad{x},{} a copy of \\spad{u} is returned. Note: \\axiom{difference(\\spad{s},{} \\spad{x}) = difference(\\spad{s},{} {\\spad{x}})}.") (($ $ $) "\\spad{difference(u,v)} returns the set aggregate \\spad{w} consisting of elements in set aggregate \\spad{u} but not in set aggregate \\spad{v}. If \\spad{u} and \\spad{v} have no elements in common,{} \\axiom{difference(\\spad{u},{}\\spad{v})} returns a copy of \\spad{u}. Note: equivalent to the notation (not currently supported) \\axiom{{\\spad{x} for \\spad{x} in \\spad{u} | not member?(\\spad{x},{}\\spad{v})}}.")) (|intersect| (($ $ $) "\\spad{intersect(u,v)} returns the set aggregate \\spad{w} consisting of elements common to both set aggregates \\spad{u} and \\spad{v}. Note: equivalent to the notation (not currently supported) {\\spad{x} for \\spad{x} in \\spad{u} | member?(\\spad{x},{}\\spad{v})}.")) (|set| (($ (|List| |#1|)) "\\spad{set([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}.") (($) "\\spad{set()}\\$\\spad{D} creates an empty set aggregate of type \\spad{D}.")) (|brace| (($ (|List| |#1|)) "\\spad{brace([x,y,...,z])} creates a set aggregate containing items \\spad{x},{}\\spad{y},{}...,{}\\spad{z}. This form is considered obsolete. Use \\axiomFun{set} instead.") (($) "\\spad{brace()}\\$\\spad{D} (otherwise written {}\\$\\spad{D}) creates an empty set aggregate of type \\spad{D}. This form is considered obsolete. Use \\axiomFun{set} instead.")) (|part?| (((|Boolean|) $ $) "\\spad{s} < \\spad{t} returns \\spad{true} if all elements of set aggregate \\spad{s} are also elements of set aggregate \\spad{t}."))) -((-3968 . T)) +((-3979 . T)) NIL -(-1006 S) +(-1011 S) ((|constructor| (NIL "\\spadtype{SetCategory} is the basic category for describing a collection of elements with \\spadop{=} (equality) and \\spadfun{coerce} to output form. \\blankline Conditional Attributes: \\indented{3}{canonical\\tab{15}data structure equality is the same as \\spadop{=}}")) (|latex| (((|String|) $) "\\spad{latex(s)} returns a LaTeX-printable output representation of \\spad{s}.")) (|hash| (((|SingleInteger|) $) "\\spad{hash(s)} calculates a hash code for \\spad{s}."))) NIL NIL -(-1007) +(-1012) ((|constructor| (NIL "\\spadtype{SetCategory} is the basic category for describing a collection of elements with \\spadop{=} (equality) and \\spadfun{coerce} to output form. \\blankline Conditional Attributes: \\indented{3}{canonical\\tab{15}data structure equality is the same as \\spadop{=}}")) (|latex| (((|String|) $) "\\spad{latex(s)} returns a LaTeX-printable output representation of \\spad{s}.")) (|hash| (((|SingleInteger|) $) "\\spad{hash(s)} calculates a hash code for \\spad{s}."))) NIL NIL -(-1008 |m| |n|) +(-1013 |m| |n|) ((|constructor| (NIL "\\spadtype{SetOfMIntegersInOneToN} implements the subsets of \\spad{M} integers in the interval \\spad{[1..n]}")) (|delta| (((|NonNegativeInteger|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{delta(S,k,p)} returns the number of elements of \\spad{S} which are strictly between \\spad{p} and the k^{th} element of \\spad{S}.")) (|member?| (((|Boolean|) (|PositiveInteger|) $) "\\spad{member?(p, s)} returns \\spad{true} is \\spad{p} is in \\spad{s},{} \\spad{false} otherwise.")) (|enumerate| (((|Vector| $)) "\\spad{enumerate()} returns a vector of all the sets of \\spad{M} integers in \\spad{1..n}.")) (|setOfMinN| (($ (|List| (|PositiveInteger|))) "\\spad{setOfMinN([a_1,...,a_m])} returns the set {\\spad{a_1},{}...,{}a_m}. Error if {\\spad{a_1},{}...,{}a_m} is not a set of \\spad{M} integers in \\spad{1..n}.")) (|elements| (((|List| (|PositiveInteger|)) $) "\\spad{elements(S)} returns the list of the elements of \\spad{S} in increasing order.")) (|replaceKthElement| (((|Union| $ #1="failed") $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{replaceKthElement(S,k,p)} replaces the k^{th} element of \\spad{S} by \\spad{p},{} and returns \"failed\" if the result is not a set of \\spad{M} integers in \\spad{1..n} any more.")) (|incrementKthElement| (((|Union| $ #1#) $ (|PositiveInteger|)) "\\spad{incrementKthElement(S,k)} increments the k^{th} element of \\spad{S},{} and returns \"failed\" if the result is not a set of \\spad{M} integers in \\spad{1..n} any more."))) NIL NIL -(-1009) +(-1014) ((|constructor| (NIL "This domain allows the manipulation of the usual Lisp values."))) NIL NIL -(-1010 |Str| |Sym| |Int| |Flt| |Expr|) +(-1015 |Str| |Sym| |Int| |Flt| |Expr|) ((|constructor| (NIL "This category allows the manipulation of Lisp values while keeping the grunge fairly localized.")) (|#| (((|Integer|) $) "\\spad{\\#((a1,...,an))} returns \\spad{n}.")) (|cdr| (($ $) "\\spad{cdr((a1,...,an))} returns \\spad{(a2,...,an)}.")) (|car| (($ $) "\\spad{car((a1,...,an))} returns \\spad{a1}.")) (|expr| ((|#5| $) "\\spad{expr(s)} returns \\spad{s} as an element of Expr; Error: if \\spad{s} is not an atom that also belongs to Expr.")) (|float| ((|#4| $) "\\spad{float(s)} returns \\spad{s} as an element of Flt; Error: if \\spad{s} is not an atom that also belongs to Flt.")) (|integer| ((|#3| $) "\\spad{integer(s)} returns \\spad{s} as an element of Int. Error: if \\spad{s} is not an atom that also belongs to Int.")) (|symbol| ((|#2| $) "\\spad{symbol(s)} returns \\spad{s} as an element of Sym. Error: if \\spad{s} is not an atom that also belongs to Sym.")) (|string| ((|#1| $) "\\spad{string(s)} returns \\spad{s} as an element of Str. Error: if \\spad{s} is not an atom that also belongs to Str.")) (|destruct| (((|List| $) $) "\\spad{destruct((a1,...,an))} returns the list [\\spad{a1},{}...,{}an].")) (|float?| (((|Boolean|) $) "\\spad{float?(s)} is \\spad{true} if \\spad{s} is an atom and belong to Flt.")) (|integer?| (((|Boolean|) $) "\\spad{integer?(s)} is \\spad{true} if \\spad{s} is an atom and belong to Int.")) (|symbol?| (((|Boolean|) $) "\\spad{symbol?(s)} is \\spad{true} if \\spad{s} is an atom and belong to Sym.")) (|string?| (((|Boolean|) $) "\\spad{string?(s)} is \\spad{true} if \\spad{s} is an atom and belong to Str.")) (|list?| (((|Boolean|) $) "\\spad{list?(s)} is \\spad{true} if \\spad{s} is a Lisp list,{} possibly ().")) (|pair?| (((|Boolean|) $) "\\spad{pair?(s)} is \\spad{true} if \\spad{s} has is a non-null Lisp list.")) (|atom?| (((|Boolean|) $) "\\spad{atom?(s)} is \\spad{true} if \\spad{s} is a Lisp atom.")) (|null?| (((|Boolean|) $) "\\spad{null?(s)} is \\spad{true} if \\spad{s} is the \\spad{S}-expression ().")) (|eq| (((|Boolean|) $ $) "\\spad{eq(s, t)} is \\spad{true} if \\%peq(\\spad{s},{}\\spad{t}) is \\spad{true} for pointers."))) NIL NIL -(-1011 |Str| |Sym| |Int| |Flt| |Expr|) +(-1016 |Str| |Sym| |Int| |Flt| |Expr|) ((|constructor| (NIL "This domain allows the manipulation of Lisp values over arbitrary atomic types."))) NIL NIL -(-1012 R E V P TS) +(-1017 R E V P TS) ((|constructor| (NIL "\\indented{2}{A internal package for removing redundant quasi-components and redundant} \\indented{2}{branches when decomposing a variety by means of quasi-components} \\indented{2}{of regular triangular sets. \\newline} References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{5}{Tech. Report (PoSSo project)} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}")) (|branchIfCan| (((|Union| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|))) "failed") (|List| |#4|) |#5| (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{branchIfCan(leq,{}ts,{}lineq,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement.")) (|prepareDecompose| (((|List| (|Record| (|:| |eq| (|List| |#4|)) (|:| |tower| |#5|) (|:| |ineq| (|List| |#4|)))) (|List| |#4|) (|List| |#5|) (|Boolean|) (|Boolean|)) "\\axiom{prepareDecompose(lp,{}lts,{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousCases| (((|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) (|List| (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)))) "\\axiom{removeSuperfluousCases(llpwt)} is an internal subroutine,{} exported only for developement.")) (|subCase?| (((|Boolean|) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|)) (|Record| (|:| |val| (|List| |#4|)) (|:| |tower| |#5|))) "\\axiom{subCase?(\\spad{lpwt1},{}\\spad{lpwt2})} is an internal subroutine,{} exported only for developement.")) (|removeSuperfluousQuasiComponents| (((|List| |#5|) (|List| |#5|)) "\\axiom{removeSuperfluousQuasiComponents(lts)} removes from \\axiom{lts} any \\spad{ts} such that \\axiom{subQuasiComponent?(ts,{}us)} holds for another \\spad{us} in \\axiom{lts}.")) (|subQuasiComponent?| (((|Boolean|) |#5| (|List| |#5|)) "\\axiom{subQuasiComponent?(ts,{}lus)} returns \\spad{true} iff \\axiom{subQuasiComponent?(ts,{}us)} holds for one \\spad{us} in \\spad{lus}.") (((|Boolean|) |#5| |#5|) "\\axiom{subQuasiComponent?(ts,{}us)} returns \\spad{true} iff \\axiomOpFrom{internalSubQuasiComponent?(ts,{}us)}{QuasiComponentPackage} returs \\spad{true}.")) (|internalSubQuasiComponent?| (((|Union| (|Boolean|) "failed") |#5| |#5|) "\\axiom{internalSubQuasiComponent?(ts,{}us)} returns a boolean \\spad{b} value if the fact the regular zero set of \\axiom{us} contains that of \\axiom{ts} can be decided (and in that case \\axiom{\\spad{b}} gives this inclusion) otherwise returns \\axiom{\"failed\"}.")) (|infRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{infRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalInfRittWu?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalInfRittWu?(\\spad{lp1},{}\\spad{lp2})} is an internal subroutine,{} exported only for developement.")) (|internalSubPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{internalSubPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}} assuming that these lists are sorted increasingly \\spad{w}.\\spad{r}.\\spad{t}. \\axiomOpFrom{infRittWu?}{RecursivePolynomialCategory}.")) (|subPolSet?| (((|Boolean|) (|List| |#4|) (|List| |#4|)) "\\axiom{subPolSet?(\\spad{lp1},{}\\spad{lp2})} returns \\spad{true} iff \\axiom{\\spad{lp1}} is a sub-set of \\axiom{\\spad{lp2}}.")) (|subTriSet?| (((|Boolean|) |#5| |#5|) "\\axiom{subTriSet?(ts,{}us)} returns \\spad{true} iff \\axiom{ts} is a sub-set of \\axiom{us}.")) (|moreAlgebraic?| (((|Boolean|) |#5| |#5|) "\\axiom{moreAlgebraic?(ts,{}us)} returns \\spad{false} iff \\axiom{ts} and \\axiom{us} are both empty,{} or \\axiom{ts} has less elements than \\axiom{us},{} or some variable is algebraic \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{us} and is not \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ts}.")) (|algebraicSort| (((|List| |#5|) (|List| |#5|)) "\\axiom{algebraicSort(lts)} sorts \\axiom{lts} \\spad{w}.\\spad{r}.\\spad{t} \\axiomOpFrom{supDimElseRittWu}{QuasiComponentPackage}.")) (|supDimElseRittWu?| (((|Boolean|) |#5| |#5|) "\\axiom{supDimElseRittWu(ts,{}us)} returns \\spad{true} iff \\axiom{ts} has less elements than \\axiom{us} otherwise if \\axiom{ts} has higher rank than \\axiom{us} \\spad{w}.\\spad{r}.\\spad{t}. Riit and Wu ordering.")) (|stopTable!| (((|Void|)) "\\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.")) (|startTable!| (((|Void|) (|String|) (|String|) (|String|)) "\\axiom{startTableGcd!(\\spad{s1},{}\\spad{s2},{}\\spad{s3})} is an internal subroutine,{} exported only for developement."))) NIL NIL -(-1013 R E V P TS) +(-1018 R E V P TS) ((|constructor| (NIL "A internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field. There is no need to use directly this package since its main operations are available from \\spad{TS}. \\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA and \\spad{R}. RIOBOO \"Computations of gcd over} \\indented{5}{algebraic towers of simple extensions\" In proceedings of \\spad{AAECC11}} \\indented{5}{Paris,{} 1995.} \\indented{1}{[2] \\spad{M}. MORENO MAZA \"Calculs de pgcd au-dessus des tours} \\indented{5}{d'extensions simples et resolution des systemes d'equations} \\indented{5}{algebriques\" These,{} Universite \\spad{P}.etM. Curie,{} Paris,{} 1997.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) NIL NIL -(-1014 R E V P) +(-1019 R E V P) ((|constructor| (NIL "The category of square-free regular triangular sets. A regular triangular set \\spad{ts} is square-free if the gcd of any polynomial \\spad{p} in \\spad{ts} and \\spad{differentiate(p,mvar(p))} \\spad{w}.\\spad{r}.\\spad{t}. \\axiomOpFrom{collectUnder}{TriangularSetCategory}(ts,{}\\axiomOpFrom{mvar}{RecursivePolynomialCategory}(\\spad{p})) has degree zero \\spad{w}.\\spad{r}.\\spad{t}. \\spad{mvar(p)}. Thus any square-free regular set defines a tower of square-free simple extensions.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991} \\indented{1}{[2] \\spad{M}. KALKBRENER \"Algorithmic properties of polynomial rings\"} \\indented{5}{Habilitation Thesis,{} ETZH,{} Zurich,{} 1995.} \\indented{1}{[3] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-1015) +(-1020) ((|constructor| (NIL "SymmetricGroupCombinatoricFunctions contains combinatoric functions concerning symmetric groups and representation theory: list young tableaus,{} improper partitions,{} subsets bijection of Coleman.")) (|unrankImproperPartitions1| (((|List| (|Integer|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{unrankImproperPartitions1(n,m,k)} computes the {\\em k}\\spad{-}th improper partition of nonnegative \\spad{n} in at most \\spad{m} nonnegative parts ordered as follows: first,{} in reverse lexicographically according to their non-zero parts,{} then according to their positions (\\spadignore{i.e.} lexicographical order using {\\em subSet}: {\\em [3,0,0] < [0,3,0] < [0,0,3] < [2,1,0] < [2,0,1] < [0,2,1] < [1,2,0] < [1,0,2] < [0,1,2] < [1,1,1]}). Note: counting of subtrees is done by {\\em numberOfImproperPartitionsInternal}.")) (|unrankImproperPartitions0| (((|List| (|Integer|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{unrankImproperPartitions0(n,m,k)} computes the {\\em k}\\spad{-}th improper partition of nonnegative \\spad{n} in \\spad{m} nonnegative parts in reverse lexicographical order. Example: {\\em [0,0,3] < [0,1,2] < [0,2,1] < [0,3,0] < [1,0,2] < [1,1,1] < [1,2,0] < [2,0,1] < [2,1,0] < [3,0,0]}. Error: if \\spad{k} is negative or too big. Note: counting of subtrees is done by \\spadfunFrom{numberOfImproperPartitions}{SymmetricGroupCombinatoricFunctions}.")) (|subSet| (((|List| (|Integer|)) (|Integer|) (|Integer|) (|Integer|)) "\\spad{subSet(n,m,k)} calculates the {\\em k}\\spad{-}th {\\em m}-subset of the set {\\em 0,1,...,(n-1)} in the lexicographic order considered as a decreasing map from {\\em 0,...,(m-1)} into {\\em 0,...,(n-1)}. See \\spad{S}.\\spad{G}. Williamson: Theorem 1.60. Error: if not {\\em (0 <= m <= n and 0 < = k < (n choose m))}.")) (|numberOfImproperPartitions| (((|Integer|) (|Integer|) (|Integer|)) "\\spad{numberOfImproperPartitions(n,m)} computes the number of partitions of the nonnegative integer \\spad{n} in \\spad{m} nonnegative parts with regarding the order (improper partitions). Example: {\\em numberOfImproperPartitions (3,3)} is 10,{} since {\\em [0,0,3], [0,1,2], [0,2,1], [0,3,0], [1,0,2], [1,1,1], [1,2,0], [2,0,1], [2,1,0], [3,0,0]} are the possibilities. Note: this operation has a recursive implementation.")) (|nextPartition| (((|Vector| (|Integer|)) (|List| (|Integer|)) (|Vector| (|Integer|)) (|Integer|)) "\\spad{nextPartition(gamma,part,number)} generates the partition of {\\em number} which follows {\\em part} according to the right-to-left lexicographical order. The partition has the property that its components do not exceed the corresponding components of {\\em gamma}. the first partition is achieved by {\\em part=[]}. Also,{} {\\em []} indicates that {\\em part} is the last partition.") (((|Vector| (|Integer|)) (|Vector| (|Integer|)) (|Vector| (|Integer|)) (|Integer|)) "\\spad{nextPartition(gamma,part,number)} generates the partition of {\\em number} which follows {\\em part} according to the right-to-left lexicographical order. The partition has the property that its components do not exceed the corresponding components of {\\em gamma}. The first partition is achieved by {\\em part=[]}. Also,{} {\\em []} indicates that {\\em part} is the last partition.")) (|nextLatticePermutation| (((|List| (|Integer|)) (|List| (|PositiveInteger|)) (|List| (|Integer|)) (|Boolean|)) "\\spad{nextLatticePermutation(lambda,lattP,constructNotFirst)} generates the lattice permutation according to the proper partition {\\em lambda} succeeding the lattice permutation {\\em lattP} in lexicographical order as long as {\\em constructNotFirst} is \\spad{true}. If {\\em constructNotFirst} is \\spad{false},{} the first lattice permutation is returned. The result {\\em nil} indicates that {\\em lattP} has no successor.")) (|nextColeman| (((|Matrix| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|)) (|Matrix| (|Integer|))) "\\spad{nextColeman(alpha,beta,C)} generates the next Coleman matrix of column sums {\\em alpha} and row sums {\\em beta} according to the lexicographical order from bottom-to-top. The first Coleman matrix is achieved by {\\em C=new(1,1,0)}. Also,{} {\\em new(1,1,0)} indicates that \\spad{C} is the last Coleman matrix.")) (|makeYoungTableau| (((|Matrix| (|Integer|)) (|List| (|PositiveInteger|)) (|List| (|Integer|))) "\\spad{makeYoungTableau(lambda,gitter)} computes for a given lattice permutation {\\em gitter} and for an improper partition {\\em lambda} the corresponding standard tableau of shape {\\em lambda}. Notes: see {\\em listYoungTableaus}. The entries are from {\\em 0,...,n-1}.")) (|listYoungTableaus| (((|List| (|Matrix| (|Integer|))) (|List| (|PositiveInteger|))) "\\spad{listYoungTableaus(lambda)} where {\\em lambda} is a proper partition generates the list of all standard tableaus of shape {\\em lambda} by means of lattice permutations. The numbers of the lattice permutation are interpreted as column labels. Hence the contents of these lattice permutations are the conjugate of {\\em lambda}. Notes: the functions {\\em nextLatticePermutation} and {\\em makeYoungTableau} are used. The entries are from {\\em 0,...,n-1}.")) (|inverseColeman| (((|List| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|)) (|Matrix| (|Integer|))) "\\spad{inverseColeman(alpha,beta,C)}: there is a bijection from the set of matrices having nonnegative entries and row sums {\\em alpha},{} column sums {\\em beta} to the set of {\\em Salpha - Sbeta} double cosets of the symmetric group {\\em Sn}. ({\\em Salpha} is the Young subgroup corresponding to the improper partition {\\em alpha}). For such a matrix \\spad{C},{} inverseColeman(\\spad{alpha},{}\\spad{beta},{}\\spad{C}) calculates the lexicographical smallest {\\em pi} in the corresponding double coset. Note: the resulting permutation {\\em pi} of {\\em {1,2,...,n}} is given in list form. Notes: the inverse of this map is {\\em coleman}. For details,{} see James/Kerber.")) (|coleman| (((|Matrix| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|)) (|List| (|Integer|))) "\\spad{coleman(alpha,beta,pi)}: there is a bijection from the set of matrices having nonnegative entries and row sums {\\em alpha},{} column sums {\\em beta} to the set of {\\em Salpha - Sbeta} double cosets of the symmetric group {\\em Sn}. ({\\em Salpha} is the Young subgroup corresponding to the improper partition {\\em alpha}). For a representing element {\\em pi} of such a double coset,{} coleman(\\spad{alpha},{}\\spad{beta},{}\\spad{pi}) generates the Coleman-matrix corresponding to {\\em alpha, beta, pi}. Note: The permutation {\\em pi} of {\\em {1,2,...,n}} has to be given in list form. Note: the inverse of this map is {\\em inverseColeman} (if {\\em pi} is the lexicographical smallest permutation in the coset). For details see James/Kerber."))) NIL NIL -(-1016 S) +(-1021 T$) +((|constructor| (NIL "This domain implements semigroup operations.")) (|semiGroupOperation| (($ (|Mapping| |#1| |#1| |#1|)) "\\spad{semiGroupOperation f} constructs a semigroup operation out of a binary homogeneous mapping known to be associative."))) +(((|%Rule| |associativity| (|%Forall| (|%Sequence| (|:| |f| $) (|:| |x| |#1|) (|:| |y| |#1|) (|:| |z| |#1|)) (-3052 (|f| (|f| |x| |y|) |z|) (|f| |x| (|f| |y| |z|))))) . T)) +NIL +(-1022 T$) +((|constructor| (NIL "This is the category of all domains that implement semigroup operations"))) +(((|%Rule| |associativity| (|%Forall| (|%Sequence| (|:| |f| $) (|:| |x| |#1|) (|:| |y| |#1|) (|:| |z| |#1|)) (-3052 (|f| (|f| |x| |y|) |z|) (|f| |x| (|f| |y| |z|))))) . T)) +NIL +(-1023 S) ((|constructor| (NIL "the class of all multiplicative semigroups,{} \\spadignore{i.e.} a set with an associative operation \\spadop{*}. \\blankline")) (** (($ $ (|PositiveInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (* (($ $ $) "\\spad{x*y} returns the product of \\spad{x} and \\spad{y}."))) NIL NIL -(-1017) +(-1024) ((|constructor| (NIL "the class of all multiplicative semigroups,{} \\spadignore{i.e.} a set with an associative operation \\spadop{*}. \\blankline")) (** (($ $ (|PositiveInteger|)) "\\spad{x**n} returns the repeated product of \\spad{x} \\spad{n} times,{} \\spadignore{i.e.} exponentiation.")) (* (($ $ $) "\\spad{x*y} returns the product of \\spad{x} and \\spad{y}."))) NIL NIL -(-1018 |dimtot| |dim1| S) +(-1025 |dimtot| |dim1| S) ((|constructor| (NIL "\\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The vectors are ordered as if they were split into two blocks. The \\spad{dim1} parameter specifies the length of the first block. The ordering is lexicographic between the blocks but acts like \\spadtype{HomogeneousDirectProduct} within each block. This type is a suitable third argument for \\spadtype{GeneralDistributedMultivariatePolynomial}."))) -((-3972 |has| |#3| (-956)) (-3973 |has| |#3| (-956)) (-3975 |has| |#3| (-6 -3975)) (-3978 . 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(|SturmHabichtCoefficients| (((|List| |#1|) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtCoefficients(p1,p2)} computes the principal Sturm-Habicht coefficients of \\spad{p1} and \\spad{p2}")) (|SturmHabichtSequence| (((|List| (|UnivariatePolynomial| |#2| |#1|)) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{SturmHabichtSequence(p1,p2)} computes the Sturm-Habicht sequence of \\spad{p1} and \\spad{p2}")) (|subresultantSequence| (((|List| (|UnivariatePolynomial| |#2| |#1|)) (|UnivariatePolynomial| |#2| |#1|) (|UnivariatePolynomial| |#2| |#1|)) "\\spad{subresultantSequence(p1,p2)} computes the (standard) subresultant sequence of \\spad{p1} and \\spad{p2}"))) NIL -((|HasCategory| |#1| (QUOTE (-387)))) -(-1020) +((|HasCategory| |#1| (QUOTE (-389)))) +(-1027) ((|constructor| (NIL "This is the datatype for operation signatures as \\indented{2}{used by the compiler and the interpreter.\\space{2}Note that this domain} \\indented{2}{differs from SignatureAst.} See also: ConstructorCall,{} Domain.")) (|source| (((|List| (|Syntax|)) $) "\\spad{source(s)} returns the list of parameter types of `s'.")) (|target| (((|Syntax|) $) "\\spad{target(s)} returns the target type of the signature `s'.")) (|signature| (($ (|List| (|Syntax|)) (|Syntax|)) "\\spad{signature(s,t)} constructs a Signature object with parameter types indicaded by `s',{} and return type indicated by `t'."))) NIL NIL -(-1021) +(-1028) ((|constructor| (NIL "This domain represents a signature AST. A signature AST \\indented{2}{is a description of an exported operation,{} \\spadignore{e.g.} its name,{} result} \\indented{2}{type,{} and the list of its argument types.}")) (|signature| (((|Signature|) $) "\\spad{signature(s)} returns AST of the declared signature for `s'.")) (|name| (((|Identifier|) $) "\\spad{name(s)} returns the name of the signature `s'.")) (|signatureAst| (($ (|Identifier|) (|Signature|)) "\\spad{signatureAst(n,s,t)} builds the signature AST n: \\spad{s} -> \\spad{t}"))) NIL NIL -(-1022 R -3078) +(-1029 R -3088) ((|constructor| (NIL "This package provides functions to determine the sign of an elementary function around a point or infinity.")) (|sign| (((|Union| (|Integer|) #1="failed") |#2| (|Symbol|) |#2| (|String|)) "\\spad{sign(f, x, a, s)} returns the sign of \\spad{f} as \\spad{x} nears \\spad{a} from below if \\spad{s} is \"left\",{} or above if \\spad{s} is \"right\".") (((|Union| (|Integer|) #1#) |#2| (|Symbol|) (|OrderedCompletion| |#2|)) "\\spad{sign(f, x, a)} returns the sign of \\spad{f} as \\spad{x} nears \\spad{a},{} from both sides if \\spad{a} is finite.") (((|Union| (|Integer|) #1#) |#2|) "\\spad{sign(f)} returns the sign of \\spad{f} if it is constant everywhere."))) NIL NIL -(-1023 R) +(-1030 R) ((|constructor| (NIL "Find the sign of a rational function around a point or infinity.")) (|sign| (((|Union| (|Integer|) #1="failed") (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|Fraction| (|Polynomial| |#1|)) (|String|)) "\\spad{sign(f, x, a, s)} returns the sign of \\spad{f} as \\spad{x} nears \\spad{a} from the left (below) if \\spad{s} is the string \\spad{\"left\"},{} or from the right (above) if \\spad{s} is the string \\spad{\"right\"}.") (((|Union| (|Integer|) #1#) (|Fraction| (|Polynomial| |#1|)) (|Symbol|) (|OrderedCompletion| (|Fraction| (|Polynomial| |#1|)))) "\\spad{sign(f, x, a)} returns the sign of \\spad{f} as \\spad{x} approaches \\spad{a},{} from both sides if \\spad{a} is finite.") (((|Union| (|Integer|) #1#) (|Fraction| (|Polynomial| |#1|))) "\\spad{sign f} returns the sign of \\spad{f} if it is constant everywhere."))) NIL NIL -(-1024) +(-1031) ((|constructor| (NIL "\\indented{1}{Package to allow simplify to be called on AlgebraicNumbers} by converting to EXPR(INT)")) (|simplify| (((|Expression| (|Integer|)) (|AlgebraicNumber|)) "\\spad{simplify(an)} applies simplifications to \\spad{an}"))) NIL NIL -(-1025) +(-1032) ((|constructor| (NIL "SingleInteger is intended to support machine integer arithmetic.")) (|xor| (($ $ $) "\\spad{xor(n,m)} returns the bit-by-bit logical {\\em xor} of the single integers \\spad{n} and \\spad{m}.")) (|noetherian| ((|attribute|) "\\spad{noetherian} all ideals are finitely generated (in fact principal).")) (|canonicalsClosed| ((|attribute|) "\\spad{canonicalClosed} means two positives multiply to give positive.")) (|canonical| ((|attribute|) "\\spad{canonical} means that mathematical equality is implied by data structure equality."))) -((-3966 . T) (-3970 . T) (-3965 . T) (-3976 . T) (-3977 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3977 . T) (-3981 . T) (-3976 . T) (-3987 . T) (-3988 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1026 S) +(-1033 S) ((|constructor| (NIL "A stack is a bag where the last item inserted is the first item extracted.")) (|depth| (((|NonNegativeInteger|) $) "\\spad{depth(s)} returns the number of elements of stack \\spad{s}. Note: \\axiom{depth(\\spad{s}) = \\#s}.")) (|top| ((|#1| $) "\\spad{top(s)} returns the top element \\spad{x} from \\spad{s}; \\spad{s} remains unchanged. Note: Use \\axiom{pop!(\\spad{s})} to obtain \\spad{x} and remove it from \\spad{s}.")) (|pop!| ((|#1| $) "\\spad{pop!(s)} returns the top element \\spad{x},{} destructively removing \\spad{x} from \\spad{s}. Note: Use \\axiom{top(\\spad{s})} to obtain \\spad{x} without removing it from \\spad{s}. Error: if \\spad{s} is empty.")) (|push!| ((|#1| |#1| $) "\\spad{push!(x,s)} pushes \\spad{x} onto stack \\spad{s},{} \\spadignore{i.e.} destructively changing \\spad{s} so as to have a new first (top) element \\spad{x}. Afterwards,{} pop!(\\spad{s}) produces \\spad{x} and pop!(\\spad{s}) produces the original \\spad{s}."))) -((-3978 . T) (-3979 . T)) +((-3989 . T) (-3990 . T)) NIL -(-1027 S |ndim| R |Row| |Col|) +(-1034 S |ndim| R |Row| |Col|) ((|constructor| (NIL "\\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if the matrix is not invertible.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m},{} if that matrix is invertible and returns \"failed\" otherwise.")) (|minordet| ((|#3| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors.")) (|determinant| ((|#3| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}.")) (* ((|#4| |#4| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#5| $ |#5|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.")) (|diagonalProduct| ((|#3| $) "\\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \\spad{m}.")) (|trace| ((|#3| $) "\\spad{trace(m)} returns the trace of the matrix \\spad{m}. this is the sum of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonal| ((|#4| $) "\\spad{diagonal(m)} returns a row consisting of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonalMatrix| (($ (|List| |#3|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ |#3|) "\\spad{scalarMatrix(r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}'s on the diagonal and zeroes elsewhere."))) NIL -((|HasCategory| |#3| (QUOTE (-309))) (|HasAttribute| |#3| (QUOTE (-3980 "*"))) (|HasCategory| |#3| (QUOTE (-144)))) -(-1028 |ndim| R |Row| |Col|) +((|HasCategory| |#3| (QUOTE (-311))) (|HasAttribute| |#3| (QUOTE (-3991 "*"))) (|HasCategory| |#3| (QUOTE (-146)))) +(-1035 |ndim| R |Row| |Col|) ((|constructor| (NIL "\\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.")) (** (($ $ (|Integer|)) "\\spad{m**n} computes an integral power of the matrix \\spad{m}. Error: if the matrix is not invertible.")) (|inverse| (((|Union| $ "failed") $) "\\spad{inverse(m)} returns the inverse of the matrix \\spad{m},{} if that matrix is invertible and returns \"failed\" otherwise.")) (|minordet| ((|#2| $) "\\spad{minordet(m)} computes the determinant of the matrix \\spad{m} using minors.")) (|determinant| ((|#2| $) "\\spad{determinant(m)} returns the determinant of the matrix \\spad{m}.")) (* ((|#3| |#3| $) "\\spad{r * x} is the product of the row vector \\spad{r} and the matrix \\spad{x}. Error: if the dimensions are incompatible.") ((|#4| $ |#4|) "\\spad{x * c} is the product of the matrix \\spad{x} and the column vector \\spad{c}. Error: if the dimensions are incompatible.")) (|diagonalProduct| ((|#2| $) "\\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \\spad{m}.")) (|trace| ((|#2| $) "\\spad{trace(m)} returns the trace of the matrix \\spad{m}. this is the sum of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonal| ((|#3| $) "\\spad{diagonal(m)} returns a row consisting of the elements on the diagonal of the matrix \\spad{m}.")) (|diagonalMatrix| (($ (|List| |#2|)) "\\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \\spad{l} on the diagonal.")) (|scalarMatrix| (($ |#2|) "\\spad{scalarMatrix(r)} returns an \\spad{n}-by-\\spad{n} matrix with \\spad{r}'s on the diagonal and zeroes elsewhere."))) -((-3978 . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3989 . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1029 R |Row| |Col| M) +(-1036 R |Row| |Col| M) ((|constructor| (NIL "\\spadtype{SmithNormalForm} is a package which provides some standard canonical forms for matrices.")) (|diophantineSystem| (((|Record| (|:| |particular| (|Union| |#3| "failed")) (|:| |basis| (|List| |#3|))) |#4| |#3|) "\\spad{diophantineSystem(A,B)} returns a particular integer solution and an integer basis of the equation \\spad{AX = B}.")) (|completeSmith| (((|Record| (|:| |Smith| |#4|) (|:| |leftEqMat| |#4|) (|:| |rightEqMat| |#4|)) |#4|) "\\spad{completeSmith} returns a record that contains the Smith normal form \\spad{H} of the matrix and the left and right equivalence matrices \\spad{U} and \\spad{V} such that U*m*v = \\spad{H}")) (|smith| ((|#4| |#4|) "\\spad{smith(m)} returns the Smith Normal form of the matrix \\spad{m}.")) (|completeHermite| (((|Record| (|:| |Hermite| |#4|) (|:| |eqMat| |#4|)) |#4|) "\\spad{completeHermite} returns a record that contains the Hermite normal form \\spad{H} of the matrix and the equivalence matrix \\spad{U} such that U*m = \\spad{H}")) (|hermite| ((|#4| |#4|) "\\spad{hermite(m)} returns the Hermite normal form of the matrix \\spad{m}."))) NIL NIL -(-1030 R |VarSet|) +(-1037 R |VarSet|) ((|constructor| (NIL "\\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials. It is parameterized by the coefficient ring and the variable set which may be infinite. The variable ordering is determined by the variable set parameter. The coefficient ring may be non-commutative,{} but the variables are assumed to commute."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-816)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| |#2| (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| |#2| (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| |#2| (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| |#2| (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-550 (-469))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-309))) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-1031 |Coef| |Var| SMP) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| |#2| (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| |#2| (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| |#2| (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| |#2| (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-311))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-1038 |Coef| |Var| SMP) ((|constructor| (NIL "This domain provides multivariate Taylor series with variables from an arbitrary ordered set. A Taylor series is represented by a stream of polynomials from the polynomial domain SMP. The \\spad{n}th element of the stream is a form of degree \\spad{n}. SMTS is an internal domain.")) (|fintegrate| (($ (|Mapping| $) |#2| |#1|) "\\spad{fintegrate(f,v,c)} is the integral of \\spad{f()} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.} \\indented{1}{The evaluation of \\spad{f()} is delayed.}")) (|integrate| (($ $ |#2| |#1|) "\\spad{integrate(s,v,c)} is the integral of \\spad{s} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.}")) (|csubst| (((|Mapping| (|Stream| |#3|) |#3|) (|List| |#2|) (|List| (|Stream| |#3|))) "\\spad{csubst(a,b)} is for internal use only")) (* (($ |#3| $) "\\spad{smp*ts} multiplies a TaylorSeries by a monomial SMP.")) (|coerce| (($ |#3|) "\\spad{coerce(poly)} regroups the terms by total degree and forms a series.") (($ |#2|) "\\spad{coerce(var)} converts a variable to a Taylor series")) (|coefficient| ((|#3| $ (|NonNegativeInteger|)) "\\spad{coefficient(s, n)} gives the terms of total degree \\spad{n}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-309)))) -(-1032 R E V P) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-311)))) +(-1039 R E V P) ((|constructor| (NIL "The category of square-free and normalized triangular sets. Thus,{} up to the primitivity axiom of [1],{} these sets are Lazard triangular sets.\\newline References : \\indented{1}{[1] \\spad{D}. LAZARD \"A new method for solving algebraic systems of} \\indented{5}{positive dimension\" Discr. App. Math. 33:147-160,{}1991}"))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-1033 UP -3078) +(-1040 UP -3088) ((|constructor| (NIL "This package factors the formulas out of the general solve code,{} allowing their recursive use over different domains. Care is taken to introduce few radicals so that radical extension domains can more easily simplify the results.")) (|aQuartic| ((|#2| |#2| |#2| |#2| |#2| |#2|) "\\spad{aQuartic(f,g,h,i,k)} \\undocumented")) (|aCubic| ((|#2| |#2| |#2| |#2| |#2|) "\\spad{aCubic(f,g,h,j)} \\undocumented")) (|aQuadratic| ((|#2| |#2| |#2| |#2|) "\\spad{aQuadratic(f,g,h)} \\undocumented")) (|aLinear| ((|#2| |#2| |#2|) "\\spad{aLinear(f,g)} \\undocumented")) (|quartic| (((|List| |#2|) |#2| |#2| |#2| |#2| |#2|) "\\spad{quartic(f,g,h,i,j)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{quartic(u)} \\undocumented")) (|cubic| (((|List| |#2|) |#2| |#2| |#2| |#2|) "\\spad{cubic(f,g,h,i)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{cubic(u)} \\undocumented")) (|quadratic| (((|List| |#2|) |#2| |#2| |#2|) "\\spad{quadratic(f,g,h)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{quadratic(u)} \\undocumented")) (|linear| (((|List| |#2|) |#2| |#2|) "\\spad{linear(f,g)} \\undocumented") (((|List| |#2|) |#1|) "\\spad{linear(u)} \\undocumented")) (|mapSolve| (((|Record| (|:| |solns| (|List| |#2|)) (|:| |maps| (|List| (|Record| (|:| |arg| |#2|) (|:| |res| |#2|))))) |#1| (|Mapping| |#2| |#2|)) "\\spad{mapSolve(u,f)} \\undocumented")) (|particularSolution| ((|#2| |#1|) "\\spad{particularSolution(u)} \\undocumented")) (|solve| (((|List| |#2|) |#1|) "\\spad{solve(u)} \\undocumented"))) NIL NIL -(-1034 R) +(-1041 R) ((|constructor| (NIL "This package tries to find solutions expressed in terms of radicals for systems of equations of rational functions with coefficients in an integral domain \\spad{R}.")) (|contractSolve| (((|SuchThat| (|List| (|Expression| |#1|)) (|List| (|Equation| (|Expression| |#1|)))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{contractSolve(rf,x)} finds the solutions expressed in terms of radicals of the equation \\spad{rf} = 0 with respect to the symbol \\spad{x},{} where \\spad{rf} is a rational function. The result contains new symbols for common subexpressions in order to reduce the size of the output.") (((|SuchThat| (|List| (|Expression| |#1|)) (|List| (|Equation| (|Expression| |#1|)))) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{contractSolve(eq,x)} finds the solutions expressed in terms of radicals of the equation of rational functions \\spad{eq} with respect to the symbol \\spad{x}. The result contains new symbols for common subexpressions in order to reduce the size of the output.")) (|radicalRoots| (((|List| (|List| (|Expression| |#1|))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{radicalRoots(lrf,lvar)} finds the roots expressed in terms of radicals of the list of rational functions \\spad{lrf} with respect to the list of symbols \\spad{lvar}.") (((|List| (|Expression| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{radicalRoots(rf,x)} finds the roots expressed in terms of radicals of the rational function \\spad{rf} with respect to the symbol \\spad{x}.")) (|radicalSolve| (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) "\\spad{radicalSolve(leq)} finds the solutions expressed in terms of radicals of the system of equations of rational functions \\spad{leq} with respect to the unique symbol \\spad{x} appearing in \\spad{leq}.") (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|List| (|Symbol|))) "\\spad{radicalSolve(leq,lvar)} finds the solutions expressed in terms of radicals of the system of equations of rational functions \\spad{leq} with respect to the list of symbols \\spad{lvar}.") (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{radicalSolve(lrf)} finds the solutions expressed in terms of radicals of the system of equations \\spad{lrf} = 0,{} where \\spad{lrf} is a system of univariate rational functions.") (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{radicalSolve(lrf,lvar)} finds the solutions expressed in terms of radicals of the system of equations \\spad{lrf} = 0 with respect to the list of symbols \\spad{lvar},{} where \\spad{lrf} is a list of rational functions.") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{radicalSolve(eq)} finds the solutions expressed in terms of radicals of the equation of rational functions \\spad{eq} with respect to the unique symbol \\spad{x} appearing in \\spad{eq}.") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{radicalSolve(eq,x)} finds the solutions expressed in terms of radicals of the equation of rational functions \\spad{eq} with respect to the symbol \\spad{x}.") (((|List| (|Equation| (|Expression| |#1|))) (|Fraction| (|Polynomial| |#1|))) "\\spad{radicalSolve(rf)} finds the solutions expressed in terms of radicals of the equation \\spad{rf} = 0,{} where \\spad{rf} is a univariate rational function.") (((|List| (|Equation| (|Expression| |#1|))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{radicalSolve(rf,x)} finds the solutions expressed in terms of radicals of the equation \\spad{rf} = 0 with respect to the symbol \\spad{x},{} where \\spad{rf} is a rational function."))) NIL NIL -(-1035 R) +(-1042 R) ((|constructor| (NIL "This package finds the function \\spad{func3} where \\spad{func1} and \\spad{func2} \\indented{1}{are given and\\space{2}\\spad{func1} = \\spad{func3}(\\spad{func2}) .\\space{2}If there is no solution then} \\indented{1}{function \\spad{func1} will be returned.} \\indented{1}{An example would be\\space{2}\\spad{func1:= 8*X**3+32*X**2-14*X ::EXPR INT} and} \\indented{1}{\\spad{func2:=2*X ::EXPR INT} convert them via univariate} \\indented{1}{to FRAC SUP EXPR INT and then the solution is \\spad{func3:=X**3+X**2-X}} \\indented{1}{of type FRAC SUP EXPR INT}")) (|unvectorise| (((|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Vector| (|Expression| |#1|)) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Integer|)) "\\spad{unvectorise(vect, var, n)} returns \\spad{vect(1) + vect(2)*var + ... + vect(n+1)*var**(n)} where \\spad{vect} is the vector of the coefficients of the polynomail ,{} \\spad{var} the new variable and \\spad{n} the degree.")) (|decomposeFunc| (((|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|))) (|Fraction| (|SparseUnivariatePolynomial| (|Expression| |#1|)))) "\\spad{decomposeFunc(func1, func2, newvar)} returns a function \\spad{func3} where \\spad{func1} = \\spad{func3}(\\spad{func2}) and expresses it in the new variable newvar. If there is no solution then \\spad{func1} will be returned."))) NIL NIL -(-1036 R) +(-1043 R) ((|constructor| (NIL "This package tries to find solutions of equations of type Expression(\\spad{R}). This means expressions involving transcendental,{} exponential,{} logarithmic and nthRoot functions. After trying to transform different kernels to one kernel by applying several rules,{} it calls zerosOf for the SparseUnivariatePolynomial in the remaining kernel. For example the expression \\spad{sin(x)*cos(x)-2} will be transformed to \\indented{3}{\\spad{-2 tan(x/2)**4 -2 tan(x/2)**3 -4 tan(x/2)**2 +2 tan(x/2) -2}} by using the function normalize and then to \\indented{3}{\\spad{-2 tan(x)**2 + tan(x) -2}} with help of subsTan. This function tries to express the given function in terms of \\spad{tan(x/2)} to express in terms of \\spad{tan(x)} . Other examples are the expressions \\spad{sqrt(x+1)+sqrt(x+7)+1} or \\indented{1}{\\spad{sqrt(sin(x))+1} .}")) (|solve| (((|List| (|List| (|Equation| (|Expression| |#1|)))) (|List| (|Equation| (|Expression| |#1|))) (|List| (|Symbol|))) "\\spad{solve(leqs, lvar)} returns a list of solutions to the list of equations \\spad{leqs} with respect to the list of symbols lvar.") (((|List| (|Equation| (|Expression| |#1|))) (|Expression| |#1|) (|Symbol|)) "\\spad{solve(expr,x)} finds the solutions of the equation \\spad{expr} = 0 with respect to the symbol \\spad{x} where \\spad{expr} is a function of type Expression(\\spad{R}).") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Expression| |#1|)) (|Symbol|)) "\\spad{solve(eq,x)} finds the solutions of the equation \\spad{eq} where \\spad{eq} is an equation of functions of type Expression(\\spad{R}) with respect to the symbol \\spad{x}.") (((|List| (|Equation| (|Expression| |#1|))) (|Equation| (|Expression| |#1|))) "\\spad{solve(eq)} finds the solutions of the equation \\spad{eq} where \\spad{eq} is an equation of functions of type Expression(\\spad{R}) with respect to the unique symbol \\spad{x} appearing in \\spad{eq}.") (((|List| (|Equation| (|Expression| |#1|))) (|Expression| |#1|)) "\\spad{solve(expr)} finds the solutions of the equation \\spad{expr} = 0 where \\spad{expr} is a function of type Expression(\\spad{R}) with respect to the unique symbol \\spad{x} appearing in eq."))) NIL NIL -(-1037 S A) +(-1044 S A) ((|constructor| (NIL "This package exports sorting algorithnms")) (|insertionSort!| ((|#2| |#2|) "\\spad{insertionSort! }\\undocumented") ((|#2| |#2| (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{insertionSort!(a,f)} \\undocumented")) (|bubbleSort!| ((|#2| |#2|) "\\spad{bubbleSort!(a)} \\undocumented") ((|#2| |#2| (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{bubbleSort!(a,f)} \\undocumented"))) NIL -((|HasCategory| |#1| (QUOTE (-751)))) -(-1038 R) +((|HasCategory| |#1| (QUOTE (-756)))) +(-1045 R) ((|constructor| (NIL "The domain ThreeSpace is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them."))) NIL NIL -(-1039 R) +(-1046 R) ((|constructor| (NIL "The category ThreeSpaceCategory is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(s)} returns the \\spadtype{ThreeSpace} \\spad{s} to Output format.")) (|subspace| (((|SubSpace| 3 |#1|) $) "\\spad{subspace(s)} returns the \\spadtype{SubSpace} which holds all the point information in the \\spadtype{ThreeSpace},{} \\spad{s}.")) (|check| (($ $) "\\spad{check(s)} returns lllpt,{} list of lists of lists of point information about the \\spadtype{ThreeSpace} \\spad{s}.")) (|objects| (((|Record| (|:| |points| (|NonNegativeInteger|)) (|:| |curves| (|NonNegativeInteger|)) (|:| |polygons| (|NonNegativeInteger|)) (|:| |constructs| (|NonNegativeInteger|))) $) "\\spad{objects(s)} returns the \\spadtype{ThreeSpace},{} \\spad{s},{} in the form of a 3D object record containing information on the number of points,{} curves,{} polygons and constructs comprising the \\spadtype{ThreeSpace}..")) (|lprop| (((|List| (|SubSpaceComponentProperty|)) $) "\\spad{lprop(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of subspace component properties,{} and if so,{} returns the list; An error is signaled otherwise.")) (|llprop| (((|List| (|List| (|SubSpaceComponentProperty|))) $) "\\spad{llprop(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of curves which are lists of the subspace component properties of the curves,{} and if so,{} returns the list of lists; An error is signaled otherwise.")) (|lllp| (((|List| (|List| (|List| (|Point| |#1|)))) $) "\\spad{lllp(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of components,{} which are lists of curves,{} which are lists of points,{} and if so,{} returns the list of lists of lists; An error is signaled otherwise.")) (|lllip| (((|List| (|List| (|List| (|NonNegativeInteger|)))) $) "\\spad{lllip(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a list of components,{} which are lists of curves,{} which are lists of indices to points,{} and if so,{} returns the list of lists of lists; An error is signaled otherwise.")) (|lp| (((|List| (|Point| |#1|)) $) "\\spad{lp(s)} returns the list of points component which the \\spadtype{ThreeSpace},{} \\spad{s},{} contains; these points are used by reference,{} \\spadignore{i.e.} the component holds indices referring to the points rather than the points themselves. This allows for sharing of the points.")) (|mesh?| (((|Boolean|) $) "\\spad{mesh?(s)} returns \\spad{true} if the \\spadtype{ThreeSpace} \\spad{s} is composed of one component,{} a mesh comprising a list of curves which are lists of points,{} or returns \\spad{false} if otherwise")) (|mesh| (((|List| (|List| (|Point| |#1|))) $) "\\spad{mesh(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single surface component defined by a list curves which contain lists of points,{} and if so,{} returns the list of lists of points; An error is signaled otherwise.") (($ (|List| (|List| (|Point| |#1|))) (|Boolean|) (|Boolean|)) "\\spad{mesh([[p0],[p1],...,[pn]], close1, close2)} creates a surface defined over a list of curves,{} \\spad{p0} through pn,{} which are lists of points; the booleans \\spad{close1} and \\spad{close2} indicate how the surface is to be closed: \\spad{close1} set to \\spad{true} means that each individual list (a curve) is to be closed (that is,{} the last point of the list is to be connected to the first point); \\spad{close2} set to \\spad{true} means that the boundary at one end of the surface is to be connected to the boundary at the other end (the boundaries are defined as the first list of points (curve) and the last list of points (curve)); the \\spadtype{ThreeSpace} containing this surface is returned.") (($ (|List| (|List| (|Point| |#1|)))) "\\spad{mesh([[p0],[p1],...,[pn]])} creates a surface defined by a list of curves which are lists,{} \\spad{p0} through pn,{} of points,{} and returns a \\spadtype{ThreeSpace} whose component is the surface.") (($ $ (|List| (|List| (|List| |#1|))) (|Boolean|) (|Boolean|)) "\\spad{mesh(s,[ [[r10]...,[r1m]], [[r20]...,[r2m]],..., [[rn0]...,[rnm]] ], close1, close2)} adds a surface component to the \\spadtype{ThreeSpace} \\spad{s},{} which is defined over a rectangular domain of size WxH where \\spad{W} is the number of lists of points from the domain \\spad{PointDomain(R)} and \\spad{H} is the number of elements in each of those lists; the booleans \\spad{close1} and \\spad{close2} indicate how the surface is to be closed: if \\spad{close1} is \\spad{true} this means that each individual list (a curve) is to be closed (\\spadignore{i.e.} the last point of the list is to be connected to the first point); if \\spad{close2} is \\spad{true},{} this means that the boundary at one end of the surface is to be connected to the boundary at the other end (the boundaries are defined as the first list of points (curve) and the last list of points (curve)).") (($ $ (|List| (|List| (|Point| |#1|))) (|Boolean|) (|Boolean|)) "\\spad{mesh(s,[[p0],[p1],...,[pn]], close1, close2)} adds a surface component to the \\spadtype{ThreeSpace},{} which is defined over a list of curves,{} in which each of these curves is a list of points. The boolean arguments \\spad{close1} and \\spad{close2} indicate how the surface is to be closed. Argument \\spad{close1} equal \\spad{true} means that each individual list (a curve) is to be closed,{} \\spadignore{i.e.} the last point of the list is to be connected to the first point. Argument \\spad{close2} equal \\spad{true} means that the boundary at one end of the surface is to be connected to the boundary at the other end,{} \\spadignore{i.e.} the boundaries are defined as the first list of points (curve) and the last list of points (curve).") (($ $ (|List| (|List| (|List| |#1|))) (|List| (|SubSpaceComponentProperty|)) (|SubSpaceComponentProperty|)) "\\spad{mesh(s,[ [[r10]...,[r1m]], [[r20]...,[r2m]],..., [[rn0]...,[rnm]] ], [props], prop)} adds a surface component to the \\spadtype{ThreeSpace} \\spad{s},{} which is defined over a rectangular domain of size WxH where \\spad{W} is the number of lists of points from the domain \\spad{PointDomain(R)} and \\spad{H} is the number of elements in each of those lists; lprops is the list of the subspace component properties for each curve list,{} and prop is the subspace component property by which the points are defined.") (($ $ (|List| (|List| (|Point| |#1|))) (|List| (|SubSpaceComponentProperty|)) (|SubSpaceComponentProperty|)) "\\spad{mesh(s,[[p0],[p1],...,[pn]],[props],prop)} adds a surface component,{} defined over a list curves which contains lists of points,{} to the \\spadtype{ThreeSpace} \\spad{s}; props is a list which contains the subspace component properties for each surface parameter,{} and \\spad{prop} is the subspace component property by which the points are defined.")) (|polygon?| (((|Boolean|) $) "\\spad{polygon?(s)} returns \\spad{true} if the \\spadtype{ThreeSpace} \\spad{s} contains a single polygon component,{} or \\spad{false} otherwise.")) (|polygon| (((|List| (|Point| |#1|)) $) "\\spad{polygon(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single polygon component defined by a list of points,{} and if so,{} returns the list of points; An error is signaled otherwise.") (($ (|List| (|Point| |#1|))) "\\spad{polygon([p0,p1,...,pn])} creates a polygon defined by a list of points,{} \\spad{p0} through pn,{} and returns a \\spadtype{ThreeSpace} whose component is the polygon.") (($ $ (|List| (|List| |#1|))) "\\spad{polygon(s,[[r0],[r1],...,[rn]])} adds a polygon component defined by a list of points \\spad{r0} through \\spad{rn},{} which are lists of elements from the domain \\spad{PointDomain(m,R)} to the \\spadtype{ThreeSpace} \\spad{s},{} where \\spad{m} is the dimension of the points and \\spad{R} is the \\spadtype{Ring} over which the points are defined.") (($ $ (|List| (|Point| |#1|))) "\\spad{polygon(s,[p0,p1,...,pn])} adds a polygon component defined by a list of points,{} \\spad{p0} throught pn,{} to the \\spadtype{ThreeSpace} \\spad{s}.")) (|closedCurve?| (((|Boolean|) $) "\\spad{closedCurve?(s)} returns \\spad{true} if the \\spadtype{ThreeSpace} \\spad{s} contains a single closed curve component,{} \\spadignore{i.e.} the first element of the curve is also the last element,{} or \\spad{false} otherwise.")) (|closedCurve| (((|List| (|Point| |#1|)) $) "\\spad{closedCurve(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single closed curve component defined by a list of points in which the first point is also the last point,{} all of which are from the domain \\spad{PointDomain(m,R)} and if so,{} returns the list of points. An error is signaled otherwise.") (($ (|List| (|Point| |#1|))) "\\spad{closedCurve(lp)} sets a list of points defined by the first element of \\spad{lp} through the last element of \\spad{lp} and back to the first elelment again and returns a \\spadtype{ThreeSpace} whose component is the closed curve defined by \\spad{lp}.") (($ $ (|List| (|List| |#1|))) "\\spad{closedCurve(s,[[lr0],[lr1],...,[lrn],[lr0]])} adds a closed curve component defined by a list of points \\spad{lr0} through \\spad{lrn},{} which are lists of elements from the domain \\spad{PointDomain(m,R)},{} where \\spad{R} is the \\spadtype{Ring} over which the point elements are defined and \\spad{m} is the dimension of the points,{} in which the last element of the list of points contains a copy of the first element list,{} \\spad{lr0}. The closed curve is added to the \\spadtype{ThreeSpace},{} \\spad{s}.") (($ $ (|List| (|Point| |#1|))) "\\spad{closedCurve(s,[p0,p1,...,pn,p0])} adds a closed curve component which is a list of points defined by the first element \\spad{p0} through the last element \\spad{pn} and back to the first element \\spad{p0} again,{} to the \\spadtype{ThreeSpace} \\spad{s}.")) (|curve?| (((|Boolean|) $) "\\spad{curve?(s)} queries whether the \\spadtype{ThreeSpace},{} \\spad{s},{} is a curve,{} \\spadignore{i.e.} has one component,{} a list of list of points,{} and returns \\spad{true} if it is,{} or \\spad{false} otherwise.")) (|curve| (((|List| (|Point| |#1|)) $) "\\spad{curve(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single curve defined by a list of points and if so,{} returns the curve,{} \\spadignore{i.e.} list of points. An error is signaled otherwise.") (($ (|List| (|Point| |#1|))) "\\spad{curve([p0,p1,p2,...,pn])} creates a space curve defined by the list of points \\spad{p0} through \\spad{pn},{} and returns the \\spadtype{ThreeSpace} whose component is the curve.") (($ $ (|List| (|List| |#1|))) "\\spad{curve(s,[[p0],[p1],...,[pn]])} adds a space curve which is a list of points \\spad{p0} through pn defined by lists of elements from the domain \\spad{PointDomain(m,R)},{} where \\spad{R} is the \\spadtype{Ring} over which the point elements are defined and \\spad{m} is the dimension of the points,{} to the \\spadtype{ThreeSpace} \\spad{s}.") (($ $ (|List| (|Point| |#1|))) "\\spad{curve(s,[p0,p1,...,pn])} adds a space curve component defined by a list of points \\spad{p0} through \\spad{pn},{} to the \\spadtype{ThreeSpace} \\spad{s}.")) (|point?| (((|Boolean|) $) "\\spad{point?(s)} queries whether the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of a single component which is a point and returns the boolean result.")) (|point| (((|Point| |#1|) $) "\\spad{point(s)} checks to see if the \\spadtype{ThreeSpace},{} \\spad{s},{} is composed of only a single point and if so,{} returns the point. An error is signaled otherwise.") (($ (|Point| |#1|)) "\\spad{point(p)} returns a \\spadtype{ThreeSpace} object which is composed of one component,{} the point \\spad{p}.") (($ $ (|NonNegativeInteger|)) "\\spad{point(s,i)} adds a point component which is placed into a component list of the \\spadtype{ThreeSpace},{} \\spad{s},{} at the index given by \\spad{i}.") (($ $ (|List| |#1|)) "\\spad{point(s,[x,y,z])} adds a point component defined by a list of elements which are from the \\spad{PointDomain(R)} to the \\spadtype{ThreeSpace},{} \\spad{s},{} where \\spad{R} is the \\spadtype{Ring} over which the point elements are defined.") (($ $ (|Point| |#1|)) "\\spad{point(s,p)} adds a point component defined by the point,{} \\spad{p},{} specified as a list from \\spad{List(R)},{} to the \\spadtype{ThreeSpace},{} \\spad{s},{} where \\spad{R} is the \\spadtype{Ring} over which the point is defined.")) (|modifyPointData| (($ $ (|NonNegativeInteger|) (|Point| |#1|)) "\\spad{modifyPointData(s,i,p)} changes the point at the indexed location \\spad{i} in the \\spadtype{ThreeSpace},{} \\spad{s},{} to that of point \\spad{p}. This is useful for making changes to a point which has been transformed.")) (|enterPointData| (((|NonNegativeInteger|) $ (|List| (|Point| |#1|))) "\\spad{enterPointData(s,[p0,p1,...,pn])} adds a list of points from \\spad{p0} through pn to the \\spadtype{ThreeSpace},{} \\spad{s},{} and returns the index,{} to the starting point of the list.")) (|copy| (($ $) "\\spad{copy(s)} returns a new \\spadtype{ThreeSpace} that is an exact copy of \\spad{s}.")) (|composites| (((|List| $) $) "\\spad{composites(s)} takes the \\spadtype{ThreeSpace} \\spad{s},{} and creates a list containing a unique \\spadtype{ThreeSpace} for each single composite of \\spad{s}. If \\spad{s} has no composites defined (composites need to be explicitly created),{} the list returned is empty. Note that not all the components need to be part of a composite.")) (|components| (((|List| $) $) "\\spad{components(s)} takes the \\spadtype{ThreeSpace} \\spad{s},{} and creates a list containing a unique \\spadtype{ThreeSpace} for each single component of \\spad{s}. If \\spad{s} has no components defined,{} the list returned is empty.")) (|composite| (($ (|List| $)) "\\spad{composite([s1,s2,...,sn])} will create a new \\spadtype{ThreeSpace} that is a union of all the components from each \\spadtype{ThreeSpace} in the parameter list,{} grouped as a composite.")) (|merge| (($ $ $) "\\spad{merge(s1,s2)} will create a new \\spadtype{ThreeSpace} that has the components of \\spad{s1} and \\spad{s2}; Groupings of components into composites are maintained.") (($ (|List| $)) "\\spad{merge([s1,s2,...,sn])} will create a new \\spadtype{ThreeSpace} that has the components of all the ones in the list; Groupings of components into composites are maintained.")) (|numberOfComposites| (((|NonNegativeInteger|) $) "\\spad{numberOfComposites(s)} returns the number of supercomponents,{} or composites,{} in the \\spadtype{ThreeSpace},{} \\spad{s}; Composites are arbitrary groupings of otherwise distinct and unrelated components; A \\spadtype{ThreeSpace} need not have any composites defined at all and,{} outside of the requirement that no component can belong to more than one composite at a time,{} the definition and interpretation of composites are unrestricted.")) (|numberOfComponents| (((|NonNegativeInteger|) $) "\\spad{numberOfComponents(s)} returns the number of distinct object components in the indicated \\spadtype{ThreeSpace},{} \\spad{s},{} such as points,{} curves,{} polygons,{} and constructs.")) (|create3Space| (($ (|SubSpace| 3 |#1|)) "\\spad{create3Space(s)} creates a \\spadtype{ThreeSpace} object containing objects pre-defined within some \\spadtype{SubSpace} \\spad{s}.") (($) "\\spad{create3Space()} creates a \\spadtype{ThreeSpace} object capable of holding point,{} curve,{} mesh components and any combination."))) NIL NIL -(-1040) +(-1047) ((|constructor| (NIL "This domain represents a kind of base domain \\indented{2}{for Spad syntax domain.\\space{2}It merely exists as a kind of} \\indented{2}{of abstract base in object-oriented programming language.} \\indented{2}{However,{} this is not an abstract class.}"))) NIL NIL -(-1041) +(-1048) ((|constructor| (NIL "\\indented{1}{This package provides a simple Spad algebra parser.} Related Constructors: Syntax. See Also: Syntax.")) (|parse| (((|List| (|Syntax|)) (|String|)) "\\spad{parse(f)} parses the source file \\spad{f} (supposedly containing Spad algebras) and returns a List Syntax. The filename \\spad{f} is supposed to have the proper extension. Note that this function has the side effect of executing any system command contained in the file \\spad{f},{} even if it might not be meaningful."))) NIL NIL -(-1042) +(-1049) ((|constructor| (NIL "This category describes the exported \\indented{2}{signatures of the SpadAst domain.}")) (|autoCoerce| (((|Integer|) $) "\\spad{autoCoerce(s)} returns the Integer view of `s'. Left at the discretion of the compiler.") (((|String|) $) "\\spad{autoCoerce(s)} returns the String view of `s'. Left at the discretion of the compiler.") (((|Identifier|) $) "\\spad{autoCoerce(s)} returns the Identifier view of `s'. Left at the discretion of the compiler.") (((|IsAst|) $) "\\spad{autoCoerce(s)} returns the IsAst view of `s'. Left at the discretion of the compiler.") (((|HasAst|) $) "\\spad{autoCoerce(s)} returns the HasAst view of `s'. Left at the discretion of the compiler.") (((|CaseAst|) $) "\\spad{autoCoerce(s)} returns the CaseAst view of `s'. Left at the discretion of the compiler.") (((|ColonAst|) $) "\\spad{autoCoerce(s)} returns the ColoonAst view of `s'. Left at the discretion of the compiler.") (((|SuchThatAst|) $) "\\spad{autoCoerce(s)} returns the SuchThatAst view of `s'. Left at the discretion of the compiler.") (((|LetAst|) $) "\\spad{autoCoerce(s)} returns the LetAst view of `s'. Left at the discretion of the compiler.") (((|SequenceAst|) $) "\\spad{autoCoerce(s)} returns the SequenceAst view of `s'. Left at the discretion of the compiler.") (((|SegmentAst|) $) "\\spad{autoCoerce(s)} returns the SegmentAst view of `s'. Left at the discretion of the compiler.") (((|RestrictAst|) $) "\\spad{autoCoerce(s)} returns the RestrictAst view of `s'. Left at the discretion of the compiler.") (((|PretendAst|) $) "\\spad{autoCoerce(s)} returns the PretendAst view of `s'. Left at the discretion of the compiler.") (((|CoerceAst|) $) "\\spad{autoCoerce(s)} returns the CoerceAst view of `s'. Left at the discretion of the compiler.") (((|ReturnAst|) $) "\\spad{autoCoerce(s)} returns the ReturnAst view of `s'. Left at the discretion of the compiler.") (((|ExitAst|) $) "\\spad{autoCoerce(s)} returns the ExitAst view of `s'. Left at the discretion of the compiler.") (((|ConstructAst|) $) "\\spad{autoCoerce(s)} returns the ConstructAst view of `s'. Left at the discretion of the compiler.") (((|CollectAst|) $) "\\spad{autoCoerce(s)} returns the CollectAst view of `s'. Left at the discretion of the compiler.") (((|StepAst|) $) "\\spad{autoCoerce(s)} returns the InAst view of \\spad{s}. Left at the discretion of the compiler.") (((|InAst|) $) "\\spad{autoCoerce(s)} returns the InAst view of `s'. Left at the discretion of the compiler.") (((|WhileAst|) $) "\\spad{autoCoerce(s)} returns the WhileAst view of `s'. Left at the discretion of the compiler.") (((|RepeatAst|) $) "\\spad{autoCoerce(s)} returns the RepeatAst view of `s'. Left at the discretion of the compiler.") (((|IfAst|) $) "\\spad{autoCoerce(s)} returns the IfAst view of `s'. Left at the discretion of the compiler.") (((|MappingAst|) $) "\\spad{autoCoerce(s)} returns the MappingAst view of `s'. Left at the discretion of the compiler.") (((|AttributeAst|) $) "\\spad{autoCoerce(s)} returns the AttributeAst view of `s'. Left at the discretion of the compiler.") (((|SignatureAst|) $) "\\spad{autoCoerce(s)} returns the SignatureAst view of `s'. Left at the discretion of the compiler.") (((|CapsuleAst|) $) "\\spad{autoCoerce(s)} returns the CapsuleAst view of `s'. Left at the discretion of the compiler.") (((|JoinAst|) $) "\\spad{autoCoerce(s)} returns the \\spadype{JoinAst} view of of the AST object \\spad{s}. Left at the discretion of the compiler.") (((|CategoryAst|) $) "\\spad{autoCoerce(s)} returns the CategoryAst view of `s'. Left at the discretion of the compiler.") (((|WhereAst|) $) "\\spad{autoCoerce(s)} returns the WhereAst view of `s'. Left at the discretion of the compiler.") (((|MacroAst|) $) "\\spad{autoCoerce(s)} returns the MacroAst view of `s'. Left at the discretion of the compiler.") (((|DefinitionAst|) $) "\\spad{autoCoerce(s)} returns the DefinitionAst view of `s'. Left at the discretion of the compiler.") (((|ImportAst|) $) "\\spad{autoCoerce(s)} returns the ImportAst view of `s'. Left at the discretion of the compiler.")) (|case| (((|Boolean|) $ (|[\|\|]| (|Integer|))) "\\spad{s case Integer} holds if `s' represents an integer literal.") (((|Boolean|) $ (|[\|\|]| (|String|))) "\\spad{s case String} holds if `s' represents a string literal.") (((|Boolean|) $ (|[\|\|]| (|Identifier|))) "\\spad{s case Identifier} holds if `s' represents an identifier.") (((|Boolean|) $ (|[\|\|]| (|IsAst|))) "\\spad{s case IsAst} holds if `s' represents an is-expression.") (((|Boolean|) $ (|[\|\|]| (|HasAst|))) "\\spad{s case HasAst} holds if `s' represents a has-expression.") (((|Boolean|) $ (|[\|\|]| (|CaseAst|))) "\\spad{s case CaseAst} holds if `s' represents a case-expression.") (((|Boolean|) $ (|[\|\|]| (|ColonAst|))) "\\spad{s case ColonAst} holds if `s' represents a colon-expression.") (((|Boolean|) $ (|[\|\|]| (|SuchThatAst|))) "\\spad{s case SuchThatAst} holds if `s' represents a qualified-expression.") (((|Boolean|) $ (|[\|\|]| (|LetAst|))) "\\spad{s case LetAst} holds if `s' represents an assignment-expression.") (((|Boolean|) $ (|[\|\|]| (|SequenceAst|))) "\\spad{s case SequenceAst} holds if `s' represents a sequence-of-statements.") (((|Boolean|) $ (|[\|\|]| (|SegmentAst|))) "\\spad{s case SegmentAst} holds if `s' represents a segment-expression.") (((|Boolean|) $ (|[\|\|]| (|RestrictAst|))) "\\spad{s case RestrictAst} holds if `s' represents a restrict-expression.") (((|Boolean|) $ (|[\|\|]| (|PretendAst|))) "\\spad{s case PretendAst} holds if `s' represents a pretend-expression.") (((|Boolean|) $ (|[\|\|]| (|CoerceAst|))) "\\spad{s case ReturnAst} holds if `s' represents a coerce-expression.") (((|Boolean|) $ (|[\|\|]| (|ReturnAst|))) "\\spad{s case ReturnAst} holds if `s' represents a return-statement.") (((|Boolean|) $ (|[\|\|]| (|ExitAst|))) "\\spad{s case ExitAst} holds if `s' represents an exit-expression.") (((|Boolean|) $ (|[\|\|]| (|ConstructAst|))) "\\spad{s case ConstructAst} holds if `s' represents a list-expression.") (((|Boolean|) $ (|[\|\|]| (|CollectAst|))) "\\spad{s case CollectAst} holds if `s' represents a list-comprehension.") (((|Boolean|) $ (|[\|\|]| (|StepAst|))) "\\spad{s case StepAst} holds if \\spad{s} represents an arithmetic progression iterator.") (((|Boolean|) $ (|[\|\|]| (|InAst|))) "\\spad{s case InAst} holds if `s' represents a in-iterator") (((|Boolean|) $ (|[\|\|]| (|WhileAst|))) "\\spad{s case WhileAst} holds if `s' represents a while-iterator") (((|Boolean|) $ (|[\|\|]| (|RepeatAst|))) "\\spad{s case RepeatAst} holds if `s' represents an repeat-loop.") (((|Boolean|) $ (|[\|\|]| (|IfAst|))) "\\spad{s case IfAst} holds if `s' represents an if-statement.") (((|Boolean|) $ (|[\|\|]| (|MappingAst|))) "\\spad{s case MappingAst} holds if `s' represents a mapping type.") (((|Boolean|) $ (|[\|\|]| (|AttributeAst|))) "\\spad{s case AttributeAst} holds if `s' represents an attribute.") (((|Boolean|) $ (|[\|\|]| (|SignatureAst|))) "\\spad{s case SignatureAst} holds if `s' represents a signature export.") (((|Boolean|) $ (|[\|\|]| (|CapsuleAst|))) "\\spad{s case CapsuleAst} holds if `s' represents a domain capsule.") (((|Boolean|) $ (|[\|\|]| (|JoinAst|))) "\\spad{s case JoinAst} holds is the syntax object \\spad{s} denotes the join of several categories.") (((|Boolean|) $ (|[\|\|]| (|CategoryAst|))) "\\spad{s case CategoryAst} holds if `s' represents an unnamed category.") (((|Boolean|) $ (|[\|\|]| (|WhereAst|))) "\\spad{s case WhereAst} holds if `s' represents an expression with local definitions.") (((|Boolean|) $ (|[\|\|]| (|MacroAst|))) "\\spad{s case MacroAst} holds if `s' represents a macro definition.") (((|Boolean|) $ (|[\|\|]| (|DefinitionAst|))) "\\spad{s case DefinitionAst} holds if `s' represents a definition.") (((|Boolean|) $ (|[\|\|]| (|ImportAst|))) "\\spad{s case ImportAst} holds if `s' represents an `import' statement."))) NIL NIL -(-1043) +(-1050) ((|constructor| (NIL "SpecialOutputPackage allows FORTRAN,{} Tex and \\indented{2}{Script Formula Formatter output from programs.}")) (|outputAsTex| (((|Void|) (|List| (|OutputForm|))) "\\spad{outputAsTex(l)} sends (for each expression in the list \\spad{l}) output in Tex format to the destination as defined by \\spadsyscom{set output tex}.") (((|Void|) (|OutputForm|)) "\\spad{outputAsTex(o)} sends output \\spad{o} in Tex format to the destination defined by \\spadsyscom{set output tex}.")) (|outputAsScript| (((|Void|) (|List| (|OutputForm|))) "\\spad{outputAsScript(l)} sends (for each expression in the list \\spad{l}) output in Script Formula Formatter format to the destination defined. by \\spadsyscom{set output forumula}.") (((|Void|) (|OutputForm|)) "\\spad{outputAsScript(o)} sends output \\spad{o} in Script Formula Formatter format to the destination defined by \\spadsyscom{set output formula}.")) (|outputAsFortran| (((|Void|) (|List| (|OutputForm|))) "\\spad{outputAsFortran(l)} sends (for each expression in the list \\spad{l}) output in FORTRAN format to the destination defined by \\spadsyscom{set output fortran}.") (((|Void|) (|OutputForm|)) "\\spad{outputAsFortran(o)} sends output \\spad{o} in FORTRAN format.") (((|Void|) (|String|) (|OutputForm|)) "\\spad{outputAsFortran(v,o)} sends output \\spad{v} = \\spad{o} in FORTRAN format to the destination defined by \\spadsyscom{set output fortran}."))) NIL NIL -(-1044) +(-1051) ((|constructor| (NIL "Category for the other special functions.")) (|airyBi| (($ $) "\\spad{airyBi(x)} is the Airy function \\spad{Bi(x)}.")) (|airyAi| (($ $) "\\spad{airyAi(x)} is the Airy function \\spad{Ai(x)}.")) (|besselK| (($ $ $) "\\spad{besselK(v,z)} is the modified Bessel function of the second kind.")) (|besselI| (($ $ $) "\\spad{besselI(v,z)} is the modified Bessel function of the first kind.")) (|besselY| (($ $ $) "\\spad{besselY(v,z)} is the Bessel function of the second kind.")) (|besselJ| (($ $ $) "\\spad{besselJ(v,z)} is the Bessel function of the first kind.")) (|polygamma| (($ $ $) "\\spad{polygamma(k,x)} is the \\spad{k-th} derivative of \\spad{digamma(x)},{} (often written \\spad{psi(k,x)} in the literature).")) (|digamma| (($ $) "\\spad{digamma(x)} is the logarithmic derivative of \\spad{Gamma(x)} (often written \\spad{psi(x)} in the literature).")) (|Beta| (($ $ $) "\\spad{Beta(x,y)} is \\spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.")) (|Gamma| (($ $ $) "\\spad{Gamma(a,x)} is the incomplete Gamma function.") (($ $) "\\spad{Gamma(x)} is the Euler Gamma function.")) (|abs| (($ $) "\\spad{abs(x)} returns the absolute value of \\spad{x}."))) NIL NIL -(-1045 V C) +(-1052 V C) ((|constructor| (NIL "This domain exports a modest implementation for the vertices of splitting trees. These vertices are called here splitting nodes. Every of these nodes store 3 informations. The first one is its value,{} that is the current expression to evaluate. The second one is its condition,{} that is the hypothesis under which the value has to be evaluated. The last one is its status,{} that is a boolean flag which is \\spad{true} iff the value is the result of its evaluation under its condition. Two splitting vertices are equal iff they have the sane values and the same conditions (so their status do not matter).")) (|subNode?| (((|Boolean|) $ $ (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{subNode?(\\spad{n1},{}\\spad{n2},{}\\spad{o2})} returns \\spad{true} iff \\axiom{value(\\spad{n1}) = value(\\spad{n2})} and \\axiom{\\spad{o2}(condition(\\spad{n1}),{}condition(\\spad{n2}))}")) (|infLex?| (((|Boolean|) $ $ (|Mapping| (|Boolean|) |#1| |#1|) (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{infLex?(\\spad{n1},{}\\spad{n2},{}\\spad{o1},{}\\spad{o2})} returns \\spad{true} iff \\axiom{\\spad{o1}(value(\\spad{n1}),{}value(\\spad{n2}))} or \\axiom{value(\\spad{n1}) = value(\\spad{n2})} and \\axiom{\\spad{o2}(condition(\\spad{n1}),{}condition(\\spad{n2}))}.")) (|setEmpty!| (($ $) "\\axiom{setEmpty!(\\spad{n})} replaces \\spad{n} by \\axiom{empty()\\$\\%}.")) (|setStatus!| (($ $ (|Boolean|)) "\\axiom{setStatus!(\\spad{n},{}\\spad{b})} returns \\spad{n} whose status has been replaced by \\spad{b} if it is not empty,{} else an error is produced.")) (|setCondition!| (($ $ |#2|) "\\axiom{setCondition!(\\spad{n},{}\\spad{t})} returns \\spad{n} whose condition has been replaced by \\spad{t} if it is not empty,{} else an error is produced.")) (|setValue!| (($ $ |#1|) "\\axiom{setValue!(\\spad{n},{}\\spad{v})} returns \\spad{n} whose value has been replaced by \\spad{v} if it is not empty,{} else an error is produced.")) (|copy| (($ $) "\\axiom{copy(\\spad{n})} returns a copy of \\spad{n}.")) (|construct| (((|List| $) |#1| (|List| |#2|)) "\\axiom{construct(\\spad{v},{}lt)} returns the same as \\axiom{[construct(\\spad{v},{}\\spad{t}) for \\spad{t} in lt]}") (((|List| $) (|List| (|Record| (|:| |val| |#1|) (|:| |tower| |#2|)))) "\\axiom{construct(lvt)} returns the same as \\axiom{[construct(vt.val,{}vt.tower) for vt in lvt]}") (($ (|Record| (|:| |val| |#1|) (|:| |tower| |#2|))) "\\axiom{construct(vt)} returns the same as \\axiom{construct(vt.val,{}vt.tower)}") (($ |#1| |#2|) "\\axiom{construct(\\spad{v},{}\\spad{t})} returns the same as \\axiom{construct(\\spad{v},{}\\spad{t},{}\\spad{false})}") (($ |#1| |#2| (|Boolean|)) "\\axiom{construct(\\spad{v},{}\\spad{t},{}\\spad{b})} returns the non-empty node with value \\spad{v},{} condition \\spad{t} and flag \\spad{b}")) (|status| (((|Boolean|) $) "\\axiom{status(\\spad{n})} returns the status of the node \\spad{n}.")) (|condition| ((|#2| $) "\\axiom{condition(\\spad{n})} returns the condition of the node \\spad{n}.")) (|value| ((|#1| $) "\\axiom{value(\\spad{n})} returns the value of the node \\spad{n}.")) (|empty?| (((|Boolean|) $) "\\axiom{empty?(\\spad{n})} returns \\spad{true} iff the node \\spad{n} is \\axiom{empty()\\$\\%}.")) (|empty| (($) "\\axiom{empty()} returns the same as \\axiom{[empty()\\$\\spad{V},{}empty()\\$\\spad{C},{}\\spad{false}]\\$\\%}"))) NIL NIL -(-1046 V C) +(-1053 V C) ((|constructor| (NIL "This domain exports a modest implementation of splitting trees. Spliiting trees are needed when the evaluation of some quantity under some hypothesis requires to split the hypothesis into sub-cases. For instance by adding some new hypothesis on one hand and its negation on another hand. The computations are terminated is a splitting tree \\axiom{a} when \\axiom{status(value(a))} is \\axiom{\\spad{true}}. Thus,{} if for the splitting tree \\axiom{a} the flag \\axiom{status(value(a))} is \\axiom{\\spad{true}},{} then \\axiom{status(value(\\spad{d}))} is \\axiom{\\spad{true}} for any subtree \\axiom{\\spad{d}} of \\axiom{a}. This property of splitting trees is called the termination condition. If no vertex in a splitting tree \\axiom{a} is equal to another,{} \\axiom{a} is said to satisfy the no-duplicates condition. The splitting tree \\axiom{a} will satisfy this condition if nodes are added to \\axiom{a} by mean of \\axiom{splitNodeOf!} and if \\axiom{construct} is only used to create the root of \\axiom{a} with no children.")) (|splitNodeOf!| (($ $ $ (|List| (|SplittingNode| |#1| |#2|)) (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{splitNodeOf!(\\spad{l},{}a,{}ls,{}sub?)} returns \\axiom{a} where the children list of \\axiom{\\spad{l}} has been set to \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in ls | not subNodeOf?(\\spad{s},{}a,{}sub?)]}. Thus,{} if \\axiom{\\spad{l}} is not a node of \\axiom{a},{} this latter splitting tree is unchanged.") (($ $ $ (|List| (|SplittingNode| |#1| |#2|))) "\\axiom{splitNodeOf!(\\spad{l},{}a,{}ls)} returns \\axiom{a} where the children list of \\axiom{\\spad{l}} has been set to \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in ls | not nodeOf?(\\spad{s},{}a)]}. Thus,{} if \\axiom{\\spad{l}} is not a node of \\axiom{a},{} this latter splitting tree is unchanged.")) (|remove!| (($ (|SplittingNode| |#1| |#2|) $) "\\axiom{remove!(\\spad{s},{}a)} replaces a by remove(\\spad{s},{}a)")) (|remove| (($ (|SplittingNode| |#1| |#2|) $) "\\axiom{remove(\\spad{s},{}a)} returns the splitting tree obtained from a by removing every sub-tree \\axiom{\\spad{b}} such that \\axiom{value(\\spad{b})} and \\axiom{\\spad{s}} have the same value,{} condition and status.")) (|subNodeOf?| (((|Boolean|) (|SplittingNode| |#1| |#2|) $ (|Mapping| (|Boolean|) |#2| |#2|)) "\\axiom{subNodeOf?(\\spad{s},{}a,{}sub?)} returns \\spad{true} iff for some node \\axiom{\\spad{n}} in \\axiom{a} we have \\axiom{\\spad{s} = \\spad{n}} or \\axiom{status(\\spad{n})} and \\axiom{subNode?(\\spad{s},{}\\spad{n},{}sub?)}.")) (|nodeOf?| (((|Boolean|) (|SplittingNode| |#1| |#2|) $) "\\axiom{nodeOf?(\\spad{s},{}a)} returns \\spad{true} iff some node of \\axiom{a} is equal to \\axiom{\\spad{s}}")) (|result| (((|List| (|Record| (|:| |val| |#1|) (|:| |tower| |#2|))) $) "\\axiom{result(a)} where \\axiom{ls} is the leaves list of \\axiom{a} returns \\axiom{[[value(\\spad{s}),{}condition(\\spad{s})]\\$VT for \\spad{s} in ls]} if the computations are terminated in \\axiom{a} else an error is produced.")) (|conditions| (((|List| |#2|) $) "\\axiom{conditions(a)} returns the list of the conditions of the leaves of a")) (|construct| (($ |#1| |#2| |#1| (|List| |#2|)) "\\axiom{construct(\\spad{v1},{}\\spad{t},{}\\spad{v2},{}lt)} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with children list given by \\axiom{[[[\\spad{v},{}\\spad{t}]\\$\\spad{S}]\\$\\% for \\spad{s} in ls]}.") (($ |#1| |#2| (|List| (|SplittingNode| |#1| |#2|))) "\\axiom{construct(\\spad{v},{}\\spad{t},{}ls)} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with children list given by \\axiom{[[\\spad{s}]\\$\\% for \\spad{s} in ls]}.") (($ |#1| |#2| (|List| $)) "\\axiom{construct(\\spad{v},{}\\spad{t},{}la)} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{[\\spad{v},{}\\spad{t}]\\$\\spad{S}} and with \\axiom{la} as children list.") (($ (|SplittingNode| |#1| |#2|)) "\\axiom{construct(\\spad{s})} creates a splitting tree with value (\\spadignore{i.e.} root vertex) given by \\axiom{\\spad{s}} and no children. Thus,{} if the status of \\axiom{\\spad{s}} is \\spad{false},{} \\axiom{[\\spad{s}]} represents the starting point of the evaluation \\axiom{value(\\spad{s})} under the hypothesis \\axiom{condition(\\spad{s})}.")) (|updateStatus!| (($ $) "\\axiom{updateStatus!(a)} returns a where the status of the vertices are updated to satisfy the \"termination condition\".")) (|extractSplittingLeaf| (((|Union| $ "failed") $) "\\axiom{extractSplittingLeaf(a)} returns the left most leaf (as a tree) whose status is \\spad{false} if any,{} else \"failed\" is returned."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| (-1045 |#1| |#2|) (|%list| (QUOTE -257) (|%list| (QUOTE -1045) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1045 |#1| |#2|) (QUOTE (-1007)))) (|HasCategory| (-1045 |#1| |#2|) (QUOTE (-1007))) (OR (|HasCategory| (-1045 |#1| |#2|) (QUOTE (-72))) (|HasCategory| (-1045 |#1| |#2|) (QUOTE (-1007)))) (|HasCategory| (-1045 |#1| |#2|) (QUOTE (-549 (-767)))) (|HasCategory| (-1045 |#1| |#2|) (QUOTE (-72)))) -(-1047 |ndim| R) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| (-1052 |#1| |#2|) (|%list| (QUOTE -259) (|%list| (QUOTE -1052) (|devaluate| |#1|) (|devaluate| |#2|)))) (|HasCategory| (-1052 |#1| |#2|) (QUOTE (-1012)))) (|HasCategory| (-1052 |#1| |#2|) (QUOTE (-1012))) (OR (|HasCategory| (-1052 |#1| |#2|) (QUOTE (-72))) (|HasCategory| (-1052 |#1| |#2|) (QUOTE (-1012)))) (|HasCategory| (-1052 |#1| |#2|) (QUOTE (-552 (-772)))) (|HasCategory| (-1052 |#1| |#2|) (QUOTE (-72)))) +(-1054 |ndim| R) ((|constructor| (NIL "\\spadtype{SquareMatrix} is a matrix domain of square matrices,{} where the number of rows (= number of columns) is a parameter of the type.")) (|unitsKnown| ((|attribute|) "the invertible matrices are simply the matrices whose determinants are units in the Ring \\spad{R}.")) (|central| ((|attribute|) "the elements of the Ring \\spad{R},{} viewed as diagonal matrices,{} commute with all matrices and,{} indeed,{} are the only matrices which commute with all matrices.")) (|squareMatrix| (($ (|Matrix| |#2|)) "\\spad{squareMatrix(m)} converts a matrix of type \\spadtype{Matrix} to a matrix of type \\spadtype{SquareMatrix}.")) (|transpose| (($ $) "\\spad{transpose(m)} returns the transpose of the matrix \\spad{m}.")) (|new| (($ |#2|) "\\spad{new(c)} constructs a new \\spadtype{SquareMatrix} object of dimension \\spad{ndim} with initial entries equal to \\spad{c}."))) -((-3975 . T) (-3967 |has| |#2| (-6 (-3980 "*"))) (-3978 . T) (-3972 . T) (-3973 . T)) -((|HasCategory| |#2| (QUOTE (-804 (-1081)))) (|HasCategory| |#2| (QUOTE (-806 (-1081)))) (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (QUOTE (-187))) (|HasAttribute| |#2| (QUOTE (-3980 #1="*"))) (|HasCategory| |#2| (QUOTE (-577 (-480)))) (|HasCategory| |#2| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-945 (-480)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-577 (-480)))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-804 (-1081)))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|))))) (|HasCategory| |#2| (QUOTE (-550 (-469)))) (|HasCategory| |#2| (QUOTE (-255))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-309))) (OR (|HasAttribute| |#2| (QUOTE (-3980 #1#))) (|HasCategory| |#2| (QUOTE (-188))) (|HasCategory| |#2| (QUOTE (-804 (-1081))))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-72))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-144)))) -(-1048 S) +((-3986 . T) (-3978 |has| |#2| (-6 (-3991 "*"))) (-3989 . T) (-3983 . T) (-3984 . T)) +((|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasCategory| |#2| (QUOTE (-811 (-1088)))) (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (QUOTE (-189))) (|HasAttribute| |#2| (QUOTE (-3991 #1="*"))) (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|))))) (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| |#2| (QUOTE (-257))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-311))) (OR (|HasAttribute| |#2| (QUOTE (-3991 #1#))) (|HasCategory| |#2| (QUOTE (-190))) (|HasCategory| |#2| (QUOTE (-809 (-1088))))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-72))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| |#2| (QUOTE (-146)))) +(-1055 S) ((|constructor| (NIL "A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.")) (|elt| (($ $ $) "\\spad{elt(s,t)} returns the concatenation of \\spad{s} and \\spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \\axiom{\"smoo\" \"shed\"} returns \\axiom{\"smooshed\"}.")) (|rightTrim| (($ $ (|CharacterClass|)) "\\spad{rightTrim(s,cc)} returns \\spad{s} with all trailing occurences of characters in \\spad{cc} deleted. For example,{} \\axiom{rightTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"(abc\"}.") (($ $ (|Character|)) "\\spad{rightTrim(s,c)} returns \\spad{s} with all trailing occurrences of \\spad{c} deleted. For example,{} \\axiom{rightTrim(\" abc \",{} char \" \")} returns \\axiom{\" abc\"}.")) (|leftTrim| (($ $ (|CharacterClass|)) "\\spad{leftTrim(s,cc)} returns \\spad{s} with all leading characters in \\spad{cc} deleted. For example,{} \\axiom{leftTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc)\"}.") (($ $ (|Character|)) "\\spad{leftTrim(s,c)} returns \\spad{s} with all leading characters \\spad{c} deleted. For example,{} \\axiom{leftTrim(\" abc \",{} char \" \")} returns \\axiom{\"abc \"}.")) (|trim| (($ $ (|CharacterClass|)) "\\spad{trim(s,cc)} returns \\spad{s} with all characters in \\spad{cc} deleted from right and left ends. For example,{} \\axiom{trim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc\"}.") (($ $ (|Character|)) "\\spad{trim(s,c)} returns \\spad{s} with all characters \\spad{c} deleted from right and left ends. For example,{} \\axiom{trim(\" abc \",{} char \" \")} returns \\axiom{\"abc\"}.")) (|split| (((|List| $) $ (|CharacterClass|)) "\\spad{split(s,cc)} returns a list of substrings delimited by characters in \\spad{cc}.") (((|List| $) $ (|Character|)) "\\spad{split(s,c)} returns a list of substrings delimited by character \\spad{c}.")) (|coerce| (($ (|Character|)) "\\spad{coerce(c)} returns \\spad{c} as a string \\spad{s} with the character \\spad{c}.")) (|position| (((|Integer|) (|CharacterClass|) $ (|Integer|)) "\\spad{position(cc,t,i)} returns the position \\axiom{\\spad{j} >= \\spad{i}} in \\spad{t} of the first character belonging to \\spad{cc}.") (((|Integer|) $ $ (|Integer|)) "\\spad{position(s,t,i)} returns the position \\spad{j} of the substring \\spad{s} in string \\spad{t},{} where \\axiom{\\spad{j} >= \\spad{i}} is required.")) (|replace| (($ $ (|UniversalSegment| (|Integer|)) $) "\\spad{replace(s,i..j,t)} replaces the substring \\axiom{\\spad{s}(\\spad{i}..\\spad{j})} of \\spad{s} by string \\spad{t}.")) (|match?| (((|Boolean|) $ $ (|Character|)) "\\spad{match?(s,t,c)} tests if \\spad{s} matches \\spad{t} except perhaps for multiple and consecutive occurrences of character \\spad{c}. Typically \\spad{c} is the blank character.")) (|match| (((|NonNegativeInteger|) $ $ (|Character|)) "\\spad{match(p,s,wc)} tests if pattern \\axiom{\\spad{p}} matches subject \\axiom{\\spad{s}} where \\axiom{\\spad{wc}} is a wild card character. If no match occurs,{} the index \\axiom{0} is returned; otheriwse,{} the value returned is the first index of the first character in the subject matching the subject (excluding that matched by an initial wild-card). For example,{} \\axiom{match(\"*to*\",{}\"yorktown\",{}\"*\")} returns \\axiom{5} indicating a successful match starting at index \\axiom{5} of \\axiom{\"yorktown\"}.")) (|substring?| (((|Boolean|) $ $ (|Integer|)) "\\spad{substring?(s,t,i)} tests if \\spad{s} is a substring of \\spad{t} beginning at index \\spad{i}. Note: \\axiom{substring?(\\spad{s},{}\\spad{t},{}0) = prefix?(\\spad{s},{}\\spad{t})}.")) (|suffix?| (((|Boolean|) $ $) "\\spad{suffix?(s,t)} tests if the string \\spad{s} is the final substring of \\spad{t}. Note: \\axiom{suffix?(\\spad{s},{}\\spad{t}) == reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.(\\spad{n} - \\spad{m} + \\spad{i}) for \\spad{i} in 0..maxIndex \\spad{s}])} where \\spad{m} and \\spad{n} denote the maxIndex of \\spad{s} and \\spad{t} respectively.")) (|prefix?| (((|Boolean|) $ $) "\\spad{prefix?(s,t)} tests if the string \\spad{s} is the initial substring of \\spad{t}. Note: \\axiom{prefix?(\\spad{s},{}\\spad{t}) == reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.\\spad{i} for \\spad{i} in 0..maxIndex \\spad{s}])}.")) (|upperCase!| (($ $) "\\spad{upperCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by upper case characters.")) (|upperCase| (($ $) "\\spad{upperCase(s)} returns the string with all characters in upper case.")) (|lowerCase!| (($ $) "\\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by lower case.")) (|lowerCase| (($ $) "\\spad{lowerCase(s)} returns the string with all characters in lower case."))) NIL NIL -(-1049) +(-1056) ((|constructor| (NIL "A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.")) (|elt| (($ $ $) "\\spad{elt(s,t)} returns the concatenation of \\spad{s} and \\spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \\axiom{\"smoo\" \"shed\"} returns \\axiom{\"smooshed\"}.")) (|rightTrim| (($ $ (|CharacterClass|)) "\\spad{rightTrim(s,cc)} returns \\spad{s} with all trailing occurences of characters in \\spad{cc} deleted. For example,{} \\axiom{rightTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"(abc\"}.") (($ $ (|Character|)) "\\spad{rightTrim(s,c)} returns \\spad{s} with all trailing occurrences of \\spad{c} deleted. For example,{} \\axiom{rightTrim(\" abc \",{} char \" \")} returns \\axiom{\" abc\"}.")) (|leftTrim| (($ $ (|CharacterClass|)) "\\spad{leftTrim(s,cc)} returns \\spad{s} with all leading characters in \\spad{cc} deleted. For example,{} \\axiom{leftTrim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc)\"}.") (($ $ (|Character|)) "\\spad{leftTrim(s,c)} returns \\spad{s} with all leading characters \\spad{c} deleted. For example,{} \\axiom{leftTrim(\" abc \",{} char \" \")} returns \\axiom{\"abc \"}.")) (|trim| (($ $ (|CharacterClass|)) "\\spad{trim(s,cc)} returns \\spad{s} with all characters in \\spad{cc} deleted from right and left ends. For example,{} \\axiom{trim(\"(abc)\",{} charClass \"()\")} returns \\axiom{\"abc\"}.") (($ $ (|Character|)) "\\spad{trim(s,c)} returns \\spad{s} with all characters \\spad{c} deleted from right and left ends. For example,{} \\axiom{trim(\" abc \",{} char \" \")} returns \\axiom{\"abc\"}.")) (|split| (((|List| $) $ (|CharacterClass|)) "\\spad{split(s,cc)} returns a list of substrings delimited by characters in \\spad{cc}.") (((|List| $) $ (|Character|)) "\\spad{split(s,c)} returns a list of substrings delimited by character \\spad{c}.")) (|coerce| (($ (|Character|)) "\\spad{coerce(c)} returns \\spad{c} as a string \\spad{s} with the character \\spad{c}.")) (|position| (((|Integer|) (|CharacterClass|) $ (|Integer|)) "\\spad{position(cc,t,i)} returns the position \\axiom{\\spad{j} >= \\spad{i}} in \\spad{t} of the first character belonging to \\spad{cc}.") (((|Integer|) $ $ (|Integer|)) "\\spad{position(s,t,i)} returns the position \\spad{j} of the substring \\spad{s} in string \\spad{t},{} where \\axiom{\\spad{j} >= \\spad{i}} is required.")) (|replace| (($ $ (|UniversalSegment| (|Integer|)) $) "\\spad{replace(s,i..j,t)} replaces the substring \\axiom{\\spad{s}(\\spad{i}..\\spad{j})} of \\spad{s} by string \\spad{t}.")) (|match?| (((|Boolean|) $ $ (|Character|)) "\\spad{match?(s,t,c)} tests if \\spad{s} matches \\spad{t} except perhaps for multiple and consecutive occurrences of character \\spad{c}. Typically \\spad{c} is the blank character.")) (|match| (((|NonNegativeInteger|) $ $ (|Character|)) "\\spad{match(p,s,wc)} tests if pattern \\axiom{\\spad{p}} matches subject \\axiom{\\spad{s}} where \\axiom{\\spad{wc}} is a wild card character. If no match occurs,{} the index \\axiom{0} is returned; otheriwse,{} the value returned is the first index of the first character in the subject matching the subject (excluding that matched by an initial wild-card). For example,{} \\axiom{match(\"*to*\",{}\"yorktown\",{}\"*\")} returns \\axiom{5} indicating a successful match starting at index \\axiom{5} of \\axiom{\"yorktown\"}.")) (|substring?| (((|Boolean|) $ $ (|Integer|)) "\\spad{substring?(s,t,i)} tests if \\spad{s} is a substring of \\spad{t} beginning at index \\spad{i}. Note: \\axiom{substring?(\\spad{s},{}\\spad{t},{}0) = prefix?(\\spad{s},{}\\spad{t})}.")) (|suffix?| (((|Boolean|) $ $) "\\spad{suffix?(s,t)} tests if the string \\spad{s} is the final substring of \\spad{t}. Note: \\axiom{suffix?(\\spad{s},{}\\spad{t}) == reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.(\\spad{n} - \\spad{m} + \\spad{i}) for \\spad{i} in 0..maxIndex \\spad{s}])} where \\spad{m} and \\spad{n} denote the maxIndex of \\spad{s} and \\spad{t} respectively.")) (|prefix?| (((|Boolean|) $ $) "\\spad{prefix?(s,t)} tests if the string \\spad{s} is the initial substring of \\spad{t}. Note: \\axiom{prefix?(\\spad{s},{}\\spad{t}) == reduce(and,{}[\\spad{s}.\\spad{i} = \\spad{t}.\\spad{i} for \\spad{i} in 0..maxIndex \\spad{s}])}.")) (|upperCase!| (($ $) "\\spad{upperCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by upper case characters.")) (|upperCase| (($ $) "\\spad{upperCase(s)} returns the string with all characters in upper case.")) (|lowerCase!| (($ $) "\\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \\spad{s} by lower case.")) (|lowerCase| (($ $) "\\spad{lowerCase(s)} returns the string with all characters in lower case."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-1050 R E V P TS) +(-1057 R E V P TS) ((|constructor| (NIL "A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are provided: in the sense of Zariski closure (like in Kalkbrener's algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard- Moreno methods). This algorithm is valid for nay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \\spad{TS} and the packages \\spad{QCMPPK(R,E,V,P,TS)} and \\spad{RSETGCD(R,E,V,P,TS)}. The same way it does not care about the way univariate polynomial gcds (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these gcds need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \\axiomType{TS}. Thus,{} the operations of this package are not documented.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.}"))) NIL NIL -(-1051 R E V P) +(-1058 R E V P) ((|constructor| (NIL "This domain provides an implementation of square-free regular chains. Moreover,{} the operation \\axiomOpFrom{zeroSetSplit}{SquareFreeRegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\\newline References : \\indented{1}{[1] \\spad{M}. MORENO MAZA \"A new algorithm for computing triangular} \\indented{5}{decomposition of algebraic varieties\" NAG Tech. Rep. 4/98.} \\indented{2}{Version: 2}")) (|preprocess| (((|Record| (|:| |val| (|List| |#4|)) (|:| |towers| (|List| $))) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{pre_process(lp,{}\\spad{b1},{}\\spad{b2})} is an internal subroutine,{} exported only for developement.")) (|internalZeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalZeroSetSplit(lp,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3})} is an internal subroutine,{} exported only for developement.")) (|zeroSetSplit| (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(lp,{}\\spad{b1},{}\\spad{b2}.\\spad{b3},{}\\spad{b4})} is an internal subroutine,{} exported only for developement.") (((|List| $) (|List| |#4|) (|Boolean|) (|Boolean|)) "\\axiom{zeroSetSplit(lp,{}clos?,{}info?)} has the same specifications as \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory} from \\spadtype{RegularTriangularSetCategory} Moreover,{} if \\axiom{clos?} then solves in the sense of the Zariski closure else solves in the sense of the regular zeros. If \\axiom{info?} then do print messages during the computations.")) (|internalAugment| (((|List| $) |#4| $ (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|) (|Boolean|)) "\\axiom{internalAugment(\\spad{p},{}ts,{}\\spad{b1},{}\\spad{b2},{}\\spad{b3},{}\\spad{b4},{}\\spad{b5})} is an internal subroutine,{} exported only for developement."))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#4| (|%list| (QUOTE -257) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-550 (-469)))) (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#3| (QUOTE (-315))) (|HasCategory| |#4| (QUOTE (-549 (-767)))) (|HasCategory| |#4| (QUOTE (-72)))) -(-1052) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#4| (|%list| (QUOTE -259) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-553 (-472)))) (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#3| (QUOTE (-317))) (|HasCategory| |#4| (QUOTE (-552 (-772)))) (|HasCategory| |#4| (QUOTE (-72)))) +(-1059) ((|constructor| (NIL "The category of all semiring structures,{} \\spadignore{e.g.} triples (\\spad{D},{}+,{}*) such that (\\spad{D},{}+) is an Abelian monoid and (\\spad{D},{}*) is a monoid with the following laws:"))) NIL NIL -(-1053 S) +(-1060 S) ((|constructor| (NIL "Linked List implementation of a Stack")) (|stack| (($ (|List| |#1|)) "\\spad{stack([x,y,...,z])} creates a stack with first (top) element \\spad{x},{} second element \\spad{y},{}...,{}and last element \\spad{z}."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-1054 A S) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-1061 A S) ((|constructor| (NIL "A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \\spadtype{Stream}. From streams,{} a number of infinite structures such power series can be built. A stream aggregate may also be infinite since it may be cyclic. For example,{} see \\spadtype{DecimalExpansion}.")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{size?(u,n)} tests if \\spad{u} has exactly \\spad{n} elements.")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{more?(u,n)} tests if \\spad{u} has greater than \\spad{n} elements.")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{less?(u,n)} tests if \\spad{u} has less than \\spad{n} elements.")) (|possiblyInfinite?| (((|Boolean|) $) "\\spad{possiblyInfinite?(s)} tests if the stream \\spad{s} could possibly have an infinite number of elements. Note: for many datatypes,{} \\axiom{possiblyInfinite?(\\spad{s}) = not explictlyFinite?(\\spad{s})}.")) (|explicitlyFinite?| (((|Boolean|) $) "\\spad{explicitlyFinite?(s)} tests if the stream has a finite number of elements,{} and \\spad{false} otherwise. Note: for many datatypes,{} \\axiom{explicitlyFinite?(\\spad{s}) = not possiblyInfinite?(\\spad{s})}."))) NIL NIL -(-1055 S) +(-1062 S) ((|constructor| (NIL "A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \\spadtype{Stream}. From streams,{} a number of infinite structures such power series can be built. A stream aggregate may also be infinite since it may be cyclic. For example,{} see \\spadtype{DecimalExpansion}.")) (|size?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{size?(u,n)} tests if \\spad{u} has exactly \\spad{n} elements.")) (|more?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{more?(u,n)} tests if \\spad{u} has greater than \\spad{n} elements.")) (|less?| (((|Boolean|) $ (|NonNegativeInteger|)) "\\spad{less?(u,n)} tests if \\spad{u} has less than \\spad{n} elements.")) (|possiblyInfinite?| (((|Boolean|) $) "\\spad{possiblyInfinite?(s)} tests if the stream \\spad{s} could possibly have an infinite number of elements. Note: for many datatypes,{} \\axiom{possiblyInfinite?(\\spad{s}) = not explictlyFinite?(\\spad{s})}.")) (|explicitlyFinite?| (((|Boolean|) $) "\\spad{explicitlyFinite?(s)} tests if the stream has a finite number of elements,{} and \\spad{false} otherwise. Note: for many datatypes,{} \\axiom{explicitlyFinite?(\\spad{s}) = not possiblyInfinite?(\\spad{s})}."))) NIL NIL -(-1056 |Key| |Ent| |dent|) +(-1063 |Key| |Ent| |dent|) ((|constructor| (NIL "A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key."))) -((-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) -(-1057) +((-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) +(-1064) ((|constructor| (NIL "A class of objects which can be 'stepped through'. Repeated applications of \\spadfun{nextItem} is guaranteed never to return duplicate items and only return \"failed\" after exhausting all elements of the domain. This assumes that the sequence starts with \\spad{init()}. For non-fiinite domains,{} repeated application of \\spadfun{nextItem} is not required to reach all possible domain elements starting from any initial element. \\blankline")) (|nextItem| (((|Maybe| $) $) "\\spad{nextItem(x)} returns the next item,{} or \\spad{failed} if domain is exhausted.")) (|init| (($) "\\spad{init()} chooses an initial object for stepping."))) NIL NIL -(-1058) +(-1065) ((|constructor| (NIL "This domain represents an arithmetic progression iterator syntax.")) (|step| (((|SpadAst|) $) "\\spad{step(i)} returns the Spad AST denoting the step of the arithmetic progression represented by the iterator \\spad{i}.")) (|upperBound| (((|Maybe| (|SpadAst|)) $) "If the set of values assumed by the iteration variable is bounded from above,{} \\spad{upperBound(i)} returns the upper bound. Otherwise,{} its returns \\spad{nothing}.")) (|lowerBound| (((|SpadAst|) $) "\\spad{lowerBound(i)} returns the lower bound on the values assumed by the iteration variable.")) (|iterationVar| (((|Identifier|) $) "\\spad{iterationVar(i)} returns the name of the iterating variable of the arithmetic progression iterator \\spad{i}."))) NIL NIL -(-1059 |Coef|) +(-1066 |Coef|) ((|constructor| (NIL "This package computes infinite products of Taylor series over an integral domain of characteristic 0. Here Taylor series are represented by streams of Taylor coefficients.")) (|generalInfiniteProduct| (((|Stream| |#1|) (|Stream| |#1|) (|Integer|) (|Integer|)) "\\spad{generalInfiniteProduct(f(x),a,d)} computes \\spad{product(n=a,a+d,a+2*d,...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|oddInfiniteProduct| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{oddInfiniteProduct(f(x))} computes \\spad{product(n=1,3,5...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|evenInfiniteProduct| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{evenInfiniteProduct(f(x))} computes \\spad{product(n=2,4,6...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1.")) (|infiniteProduct| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{infiniteProduct(f(x))} computes \\spad{product(n=1,2,3...,f(x**n))}. The series \\spad{f(x)} should have constant coefficient 1."))) NIL NIL -(-1060 S) +(-1067 S) ((|constructor| (NIL "A stream is an implementation of an infinite sequence using a list of terms that have been computed and a function closure to compute additional terms when needed.")) (|filterUntil| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterUntil(p,s)} returns \\spad{[x0,x1,...,x(n)]} where \\spad{s = [x0,x1,x2,..]} and \\spad{n} is the smallest index such that \\spad{p(xn) = true}.")) (|filterWhile| (($ (|Mapping| (|Boolean|) |#1|) $) "\\spad{filterWhile(p,s)} returns \\spad{[x0,x1,...,x(n-1)]} where \\spad{s = [x0,x1,x2,..]} and \\spad{n} is the smallest index such that \\spad{p(xn) = false}.")) (|generate| (($ (|Mapping| |#1| |#1|) |#1|) "\\spad{generate(f,x)} creates an infinite stream whose first element is \\spad{x} and whose \\spad{n}th element (\\spad{n > 1}) is \\spad{f} applied to the previous element. Note: \\spad{generate(f,x) = [x,f(x),f(f(x)),...]}.") (($ (|Mapping| |#1|)) "\\spad{generate(f)} creates an infinite stream all of whose elements are equal to \\spad{f()}. Note: \\spad{generate(f) = [f(),f(),f(),...]}.")) (|setrest!| (($ $ (|Integer|) $) "\\spad{setrest!(x,n,y)} sets rest(\\spad{x},{}\\spad{n}) to \\spad{y}. The function will expand cycles if necessary.")) (|showAll?| (((|Boolean|)) "\\spad{showAll?()} returns \\spad{true} if all computed entries of streams will be displayed.")) (|showAllElements| (((|OutputForm|) $) "\\spad{showAllElements(s)} creates an output form which displays all computed elements.")) (|output| (((|Void|) (|Integer|) $) "\\spad{output(n,st)} computes and displays the first \\spad{n} entries of \\spad{st}.")) (|cons| (($ |#1| $) "\\spad{cons(a,s)} returns a stream whose \\spad{first} is \\spad{a} and whose \\spad{rest} is \\spad{s}. Note: \\spad{cons(a,s) = concat(a,s)}.")) (|delay| (($ (|Mapping| $)) "\\spad{delay(f)} creates a stream with a lazy evaluation defined by function \\spad{f}. Caution: This function can only be called in compiled code.")) (|findCycle| (((|Record| (|:| |cycle?| (|Boolean|)) (|:| |prefix| (|NonNegativeInteger|)) (|:| |period| (|NonNegativeInteger|))) (|NonNegativeInteger|) $) "\\spad{findCycle(n,st)} determines if \\spad{st} is periodic within \\spad{n}.")) (|repeating?| (((|Boolean|) (|List| |#1|) $) "\\spad{repeating?(l,s)} returns \\spad{true} if a stream \\spad{s} is periodic with period \\spad{l},{} and \\spad{false} otherwise.")) (|repeating| (($ (|List| |#1|)) "\\spad{repeating(l)} is a repeating stream whose period is the list \\spad{l}.")) (|shallowlyMutable| ((|attribute|) "one may destructively alter a stream by assigning new values to its entries."))) -((-3979 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-72)))) -(-1061 S) +((-3990 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-72)))) +(-1068 S) ((|constructor| (NIL "Functions defined on streams with entries in one set.")) (|concat| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{concat(u)} returns the left-to-right concatentation of the streams in \\spad{u}. Note: \\spad{concat(u) = reduce(concat,u)}."))) NIL NIL -(-1062 A B) +(-1069 A B) ((|constructor| (NIL "Functions defined on streams with entries in two sets.")) (|reduce| ((|#2| |#2| (|Mapping| |#2| |#1| |#2|) (|Stream| |#1|)) "\\spad{reduce(b,f,u)},{} where \\spad{u} is a finite stream \\spad{[x0,x1,...,xn]},{} returns the value \\spad{r(n)} computed as follows: \\spad{r0 = f(x0,b), r1 = f(x1,r0),..., r(n) = f(xn,r(n-1))}.")) (|scan| (((|Stream| |#2|) |#2| (|Mapping| |#2| |#1| |#2|) (|Stream| |#1|)) "\\spad{scan(b,h,[x0,x1,x2,...])} returns \\spad{[y0,y1,y2,...]},{} where \\spad{y0 = h(x0,b)},{} \\spad{y1 = h(x1,y0)},{}\\spad{...} \\spad{yn = h(xn,y(n-1))}.")) (|map| (((|Stream| |#2|) (|Mapping| |#2| |#1|) (|Stream| |#1|)) "\\spad{map(f,s)} returns a stream whose elements are the function \\spad{f} applied to the corresponding elements of \\spad{s}. Note: \\spad{map(f,[x0,x1,x2,...]) = [f(x0),f(x1),f(x2),..]}."))) NIL NIL -(-1063 A B C) +(-1070 A B C) ((|constructor| (NIL "Functions defined on streams with entries in three sets.")) (|map| (((|Stream| |#3|) (|Mapping| |#3| |#1| |#2|) (|Stream| |#1|) (|Stream| |#2|)) "\\spad{map(f,st1,st2)} returns the stream whose elements are the function \\spad{f} applied to the corresponding elements of \\spad{st1} and \\spad{st2}. Note: \\spad{map(f,[x0,x1,x2,..],[y0,y1,y2,..]) = [f(x0,y0),f(x1,y1),..]}."))) NIL NIL -(-1064) +(-1071) ((|constructor| (NIL "This is the domain of character strings.")) (|string| (($ (|Identifier|)) "\\spad{string id} is the string representation of the identifier \\spad{id}") (($ (|DoubleFloat|)) "\\spad{string f} returns the decimal representation of \\spad{f} in a string") (($ (|Integer|)) "\\spad{string i} returns the decimal representation of \\spad{i} in a string"))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| (-115) (QUOTE (-257 (-115)))) (|HasCategory| (-115) (QUOTE (-751)))) (-12 (|HasCategory| (-115) (QUOTE (-257 (-115)))) (|HasCategory| (-115) (QUOTE (-1007))))) (|HasCategory| (-115) (QUOTE (-549 (-767)))) (|HasCategory| (-115) (QUOTE (-550 (-469)))) (OR (|HasCategory| (-115) (QUOTE (-751))) (|HasCategory| (-115) (QUOTE (-1007)))) (|HasCategory| (-115) (QUOTE (-751))) (OR (|HasCategory| (-115) (QUOTE (-72))) (|HasCategory| (-115) (QUOTE (-751))) (|HasCategory| (-115) (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| (-115) (QUOTE (-1007))) (|HasCategory| (-115) (QUOTE (-72))) (-12 (|HasCategory| (-115) (QUOTE (-257 (-115)))) (|HasCategory| (-115) (QUOTE (-1007))))) -(-1065 |Entry|) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| (-117) (QUOTE (-259 (-117)))) (|HasCategory| (-117) (QUOTE (-756)))) (-12 (|HasCategory| (-117) (QUOTE (-259 (-117)))) (|HasCategory| (-117) (QUOTE (-1012))))) (|HasCategory| (-117) (QUOTE (-552 (-772)))) (|HasCategory| (-117) (QUOTE (-553 (-472)))) (OR (|HasCategory| (-117) (QUOTE (-756))) (|HasCategory| (-117) (QUOTE (-1012)))) (|HasCategory| (-117) (QUOTE (-756))) (OR (|HasCategory| (-117) (QUOTE (-72))) (|HasCategory| (-117) (QUOTE (-756))) (|HasCategory| (-117) (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| (-117) (QUOTE (-1012))) (|HasCategory| (-117) (QUOTE (-72))) (-12 (|HasCategory| (-117) (QUOTE (-259 (-117)))) (|HasCategory| (-117) (QUOTE (-1012))))) +(-1072 |Entry|) ((|constructor| (NIL "This domain provides tables where the keys are strings. A specialized hash function for strings is used."))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (QUOTE (|:| -3843 (-1064))) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-1007)))) (OR (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-1007)))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-1007))) (|HasCategory| (-1064) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-72)))) (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 (-1064)) (|:| |entry| |#1|)) (QUOTE (-72)))) -(-1066 A) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (QUOTE (|:| -3854 (-1071))) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#1|))))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-1012)))) (OR (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-1012)))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-1012))) (|HasCategory| (-1071) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-72)))) (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 (-1071)) (|:| |entry| |#1|)) (QUOTE (-72)))) +(-1073 A) ((|constructor| (NIL "StreamTaylorSeriesOperations implements Taylor series arithmetic,{} where a Taylor series is represented by a stream of its coefficients.")) (|power| (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{power(a,f)} returns the power series \\spad{f} raised to the power \\spad{a}.")) (|lazyGintegrate| (((|Stream| |#1|) (|Mapping| |#1| (|Integer|)) |#1| (|Mapping| (|Stream| |#1|))) "\\spad{lazyGintegrate(f,r,g)} is used for fixed point computations.")) (|mapdiv| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{mapdiv([a0,a1,..],[b0,b1,..])} returns \\spad{[a0/b0,a1/b1,..]}.")) (|powern| (((|Stream| |#1|) (|Fraction| (|Integer|)) (|Stream| |#1|)) "\\spad{powern(r,f)} raises power series \\spad{f} to the power \\spad{r}.")) (|nlde| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{nlde(u)} solves a first order non-linear differential equation described by \\spad{u} of the form \\spad{[[b<0,0>,b<0,1>,...],[b<1,0>,b<1,1>,.],...]}. the differential equation has the form \\spad{y' = sum(i=0 to infinity,j=0 to infinity,b<i,j>*(x**i)*(y**j))}.")) (|lazyIntegrate| (((|Stream| |#1|) |#1| (|Mapping| (|Stream| |#1|))) "\\spad{lazyIntegrate(r,f)} is a local function used for fixed point computations.")) (|integrate| (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{integrate(r,a)} returns the integral of the power series \\spad{a} with respect to the power series variableintegration where \\spad{r} denotes the constant of integration. Thus \\spad{integrate(a,[a0,a1,a2,...]) = [a,a0,a1/2,a2/3,...]}.")) (|invmultisect| (((|Stream| |#1|) (|Integer|) (|Integer|) (|Stream| |#1|)) "\\spad{invmultisect(a,b,st)} substitutes \\spad{x**((a+b)*n)} for \\spad{x**n} and multiplies by \\spad{x**b}.")) (|multisect| (((|Stream| |#1|) (|Integer|) (|Integer|) (|Stream| |#1|)) "\\spad{multisect(a,b,st)} selects the coefficients of \\spad{x**((a+b)*n+a)},{} and changes them to \\spad{x**n}.")) (|generalLambert| (((|Stream| |#1|) (|Stream| |#1|) (|Integer|) (|Integer|)) "\\spad{generalLambert(f(x),a,d)} returns \\spad{f(x**a) + f(x**(a + d)) + f(x**(a + 2 d)) + ...}. \\spad{f(x)} should have zero constant coefficient and \\spad{a} and \\spad{d} should be positive.")) (|evenlambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{evenlambert(st)} computes \\spad{f(x**2) + f(x**4) + f(x**6) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f(x)} is a power series with constant coefficient 1,{} then \\spad{prod(f(x**(2*n)),n=1..infinity) = exp(evenlambert(log(f(x))))}.")) (|oddlambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{oddlambert(st)} computes \\spad{f(x) + f(x**3) + f(x**5) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f}(\\spad{x}) is a power series with constant coefficient 1 then \\spad{prod(f(x**(2*n-1)),n=1..infinity) = exp(oddlambert(log(f(x))))}.")) (|lambert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{lambert(st)} computes \\spad{f(x) + f(x**2) + f(x**3) + ...} if \\spad{st} is a stream representing \\spad{f(x)}. This function is used for computing infinite products. If \\spad{f(x)} is a power series with constant coefficient 1 then \\spad{prod(f(x**n),n = 1..infinity) = exp(lambert(log(f(x))))}.")) (|addiag| (((|Stream| |#1|) (|Stream| (|Stream| |#1|))) "\\spad{addiag(x)} performs diagonal addition of a stream of streams. if \\spad{x} = \\spad{[[a<0,0>,a<0,1>,..],[a<1,0>,a<1,1>,..],[a<2,0>,a<2,1>,..],..]} and \\spad{addiag(x) = [b<0,b<1>,...], then b<k> = sum(i+j=k,a<i,j>)}.")) (|revert| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{revert(a)} computes the inverse of a power series \\spad{a} with respect to composition. the series should have constant coefficient 0 and first order coefficient should be invertible.")) (|lagrange| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{lagrange(g)} produces the power series for \\spad{f} where \\spad{f} is implicitly defined as \\spad{f(z) = z*g(f(z))}.")) (|compose| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{compose(a,b)} composes the power series \\spad{a} with the power series \\spad{b}.")) (|eval| (((|Stream| |#1|) (|Stream| |#1|) |#1|) "\\spad{eval(a,r)} returns a stream of partial sums of the power series \\spad{a} evaluated at the power series variable equal to \\spad{r}.")) (|coerce| (((|Stream| |#1|) |#1|) "\\spad{coerce(r)} converts a ring element \\spad{r} to a stream with one element.")) (|gderiv| (((|Stream| |#1|) (|Mapping| |#1| (|Integer|)) (|Stream| |#1|)) "\\spad{gderiv(f,[a0,a1,a2,..])} returns \\spad{[f(0)*a0,f(1)*a1,f(2)*a2,..]}.")) (|deriv| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{deriv(a)} returns the derivative of the power series with respect to the power series variable. Thus \\spad{deriv([a0,a1,a2,...])} returns \\spad{[a1,2 a2,3 a3,...]}.")) (|mapmult| (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{mapmult([a0,a1,..],[b0,b1,..])} returns \\spad{[a0*b0,a1*b1,..]}.")) (|int| (((|Stream| |#1|) |#1|) "\\spad{int(r)} returns [\\spad{r},{}\\spad{r+1},{}\\spad{r+2},{}...],{} where \\spad{r} is a ring element.")) (|oddintegers| (((|Stream| (|Integer|)) (|Integer|)) "\\spad{oddintegers(n)} returns \\spad{[n,n+2,n+4,...]}.")) (|integers| (((|Stream| (|Integer|)) (|Integer|)) "\\spad{integers(n)} returns \\spad{[n,n+1,n+2,...]}.")) (|monom| (((|Stream| |#1|) |#1| (|Integer|)) "\\spad{monom(deg,coef)} is a monomial of degree \\spad{deg} with coefficient \\spad{coef}.")) (|recip| (((|Union| (|Stream| |#1|) "failed") (|Stream| |#1|)) "\\spad{recip(a)} returns the power series reciprocal of \\spad{a},{} or \"failed\" if not possible.")) (/ (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a / b} returns the power series quotient of \\spad{a} by \\spad{b}. An error message is returned if \\spad{b} is not invertible. This function is used in fixed point computations.")) (|exquo| (((|Union| (|Stream| |#1|) "failed") (|Stream| |#1|) (|Stream| |#1|)) "\\spad{exquo(a,b)} returns the power series quotient of \\spad{a} by \\spad{b},{} if the quotient exists,{} and \"failed\" otherwise")) (* (((|Stream| |#1|) (|Stream| |#1|) |#1|) "\\spad{a * r} returns the power series scalar multiplication of \\spad{a} by r: \\spad{[a0,a1,...] * r = [a0 * r,a1 * r,...]}") (((|Stream| |#1|) |#1| (|Stream| |#1|)) "\\spad{r * a} returns the power series scalar multiplication of \\spad{r} by \\spad{a}: \\spad{r * [a0,a1,...] = [r * a0,r * a1,...]}") (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a * b} returns the power series (Cauchy) product of \\spad{a} and b: \\spad{[a0,a1,...] * [b0,b1,...] = [c0,c1,...]} where \\spad{ck = sum(i + j = k,ai * bk)}.")) (- (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{- a} returns the power series negative of \\spad{a}: \\spad{- [a0,a1,...] = [- a0,- a1,...]}") (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a - b} returns the power series difference of \\spad{a} and \\spad{b}: \\spad{[a0,a1,..] - [b0,b1,..] = [a0 - b0,a1 - b1,..]}")) (+ (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{a + b} returns the power series sum of \\spad{a} and \\spad{b}: \\spad{[a0,a1,..] + [b0,b1,..] = [a0 + b0,a1 + b1,..]}"))) NIL -((|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480)))))) -(-1067 |Coef|) +((|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483)))))) +(-1074 |Coef|) ((|constructor| (NIL "StreamTranscendentalFunctions implements transcendental functions on Taylor series,{} where a Taylor series is represented by a stream of its coefficients.")) (|acsch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsch(st)} computes the inverse hyperbolic cosecant of a power series \\spad{st}.")) (|asech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asech(st)} computes the inverse hyperbolic secant of a power series \\spad{st}.")) (|acoth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acoth(st)} computes the inverse hyperbolic cotangent of a power series \\spad{st}.")) (|atanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atanh(st)} computes the inverse hyperbolic tangent of a power series \\spad{st}.")) (|acosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acosh(st)} computes the inverse hyperbolic cosine of a power series \\spad{st}.")) (|asinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asinh(st)} computes the inverse hyperbolic sine of a power series \\spad{st}.")) (|csch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csch(st)} computes the hyperbolic cosecant of a power series \\spad{st}.")) (|sech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sech(st)} computes the hyperbolic secant of a power series \\spad{st}.")) (|coth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{coth(st)} computes the hyperbolic cotangent of a power series \\spad{st}.")) (|tanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tanh(st)} computes the hyperbolic tangent of a power series \\spad{st}.")) (|cosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cosh(st)} computes the hyperbolic cosine of a power series \\spad{st}.")) (|sinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sinh(st)} computes the hyperbolic sine of a power series \\spad{st}.")) (|sinhcosh| (((|Record| (|:| |sinh| (|Stream| |#1|)) (|:| |cosh| (|Stream| |#1|))) (|Stream| |#1|)) "\\spad{sinhcosh(st)} returns a record containing the hyperbolic sine and cosine of a power series \\spad{st}.")) (|acsc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsc(st)} computes arccosecant of a power series \\spad{st}.")) (|asec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asec(st)} computes arcsecant of a power series \\spad{st}.")) (|acot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acot(st)} computes arccotangent of a power series \\spad{st}.")) (|atan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atan(st)} computes arctangent of a power series \\spad{st}.")) (|acos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acos(st)} computes arccosine of a power series \\spad{st}.")) (|asin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asin(st)} computes arcsine of a power series \\spad{st}.")) (|csc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csc(st)} computes cosecant of a power series \\spad{st}.")) (|sec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sec(st)} computes secant of a power series \\spad{st}.")) (|cot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cot(st)} computes cotangent of a power series \\spad{st}.")) (|tan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tan(st)} computes tangent of a power series \\spad{st}.")) (|cos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cos(st)} computes cosine of a power series \\spad{st}.")) (|sin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sin(st)} computes sine of a power series \\spad{st}.")) (|sincos| (((|Record| (|:| |sin| (|Stream| |#1|)) (|:| |cos| (|Stream| |#1|))) (|Stream| |#1|)) "\\spad{sincos(st)} returns a record containing the sine and cosine of a power series \\spad{st}.")) (** (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{st1 ** st2} computes the power of a power series \\spad{st1} by another power series \\spad{st2}.")) (|log| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{log(st)} computes the log of a power series.")) (|exp| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{exp(st)} computes the exponential of a power series \\spad{st}."))) NIL NIL -(-1068 |Coef|) +(-1075 |Coef|) ((|constructor| (NIL "StreamTranscendentalFunctionsNonCommutative implements transcendental functions on Taylor series over a non-commutative ring,{} where a Taylor series is represented by a stream of its coefficients.")) (|acsch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsch(st)} computes the inverse hyperbolic cosecant of a power series \\spad{st}.")) (|asech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asech(st)} computes the inverse hyperbolic secant of a power series \\spad{st}.")) (|acoth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acoth(st)} computes the inverse hyperbolic cotangent of a power series \\spad{st}.")) (|atanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atanh(st)} computes the inverse hyperbolic tangent of a power series \\spad{st}.")) (|acosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acosh(st)} computes the inverse hyperbolic cosine of a power series \\spad{st}.")) (|asinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asinh(st)} computes the inverse hyperbolic sine of a power series \\spad{st}.")) (|csch| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csch(st)} computes the hyperbolic cosecant of a power series \\spad{st}.")) (|sech| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sech(st)} computes the hyperbolic secant of a power series \\spad{st}.")) (|coth| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{coth(st)} computes the hyperbolic cotangent of a power series \\spad{st}.")) (|tanh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tanh(st)} computes the hyperbolic tangent of a power series \\spad{st}.")) (|cosh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cosh(st)} computes the hyperbolic cosine of a power series \\spad{st}.")) (|sinh| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sinh(st)} computes the hyperbolic sine of a power series \\spad{st}.")) (|acsc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acsc(st)} computes arccosecant of a power series \\spad{st}.")) (|asec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asec(st)} computes arcsecant of a power series \\spad{st}.")) (|acot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acot(st)} computes arccotangent of a power series \\spad{st}.")) (|atan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{atan(st)} computes arctangent of a power series \\spad{st}.")) (|acos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{acos(st)} computes arccosine of a power series \\spad{st}.")) (|asin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{asin(st)} computes arcsine of a power series \\spad{st}.")) (|csc| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{csc(st)} computes cosecant of a power series \\spad{st}.")) (|sec| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sec(st)} computes secant of a power series \\spad{st}.")) (|cot| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cot(st)} computes cotangent of a power series \\spad{st}.")) (|tan| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{tan(st)} computes tangent of a power series \\spad{st}.")) (|cos| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{cos(st)} computes cosine of a power series \\spad{st}.")) (|sin| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{sin(st)} computes sine of a power series \\spad{st}.")) (** (((|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) "\\spad{st1 ** st2} computes the power of a power series \\spad{st1} by another power series \\spad{st2}.")) (|log| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{log(st)} computes the log of a power series.")) (|exp| (((|Stream| |#1|) (|Stream| |#1|)) "\\spad{exp(st)} computes the exponential of a power series \\spad{st}."))) NIL NIL -(-1069 R UP) +(-1076 R UP) ((|constructor| (NIL "This package computes the subresultants of two polynomials which is needed for the `Lazard Rioboo' enhancement to Tragers integrations formula For efficiency reasons this has been rewritten to call Lionel Ducos package which is currently the best one. \\blankline")) (|primitivePart| ((|#2| |#2| |#1|) "\\spad{primitivePart(p, q)} reduces the coefficient of \\spad{p} modulo \\spad{q},{} takes the primitive part of the result,{} and ensures that the leading coefficient of that result is monic.")) (|subresultantVector| (((|PrimitiveArray| |#2|) |#2| |#2|) "\\spad{subresultantVector(p, q)} returns \\spad{[p0,...,pn]} where \\spad{pi} is the \\spad{i}-th subresultant of \\spad{p} and \\spad{q}. In particular,{} \\spad{p0 = resultant(p, q)}."))) NIL -((|HasCategory| |#1| (QUOTE (-255)))) -(-1070 |n| R) +((|HasCategory| |#1| (QUOTE (-257)))) +(-1077 |n| R) ((|constructor| (NIL "This domain \\undocumented")) (|pointData| (((|List| (|Point| |#2|)) $) "\\spad{pointData(s)} returns the list of points from the point data field of the 3 dimensional subspace \\spad{s}.")) (|parent| (($ $) "\\spad{parent(s)} returns the subspace which is the parent of the indicated 3 dimensional subspace \\spad{s}. If \\spad{s} is the top level subspace an error message is returned.")) (|level| (((|NonNegativeInteger|) $) "\\spad{level(s)} returns a non negative integer which is the current level field of the indicated 3 dimensional subspace \\spad{s}.")) (|extractProperty| (((|SubSpaceComponentProperty|) $) "\\spad{extractProperty(s)} returns the property of domain \\spadtype{SubSpaceComponentProperty} of the indicated 3 dimensional subspace \\spad{s}.")) (|extractClosed| (((|Boolean|) $) "\\spad{extractClosed(s)} returns the \\spadtype{Boolean} value of the closed property for the indicated 3 dimensional subspace \\spad{s}. If the property is closed,{} \\spad{True} is returned,{} otherwise \\spad{False} is returned.")) (|extractIndex| (((|NonNegativeInteger|) $) "\\spad{extractIndex(s)} returns a non negative integer which is the current index of the 3 dimensional subspace \\spad{s}.")) (|extractPoint| (((|Point| |#2|) $) "\\spad{extractPoint(s)} returns the point which is given by the current index location into the point data field of the 3 dimensional subspace \\spad{s}.")) (|traverse| (($ $ (|List| (|NonNegativeInteger|))) "\\spad{traverse(s,li)} follows the branch list of the 3 dimensional subspace,{} \\spad{s},{} along the path dictated by the list of non negative integers,{} \\spad{li},{} which points to the component which has been traversed to. The subspace,{} \\spad{s},{} is returned,{} where \\spad{s} is now the subspace pointed to by \\spad{li}.")) (|defineProperty| (($ $ (|List| (|NonNegativeInteger|)) (|SubSpaceComponentProperty|)) "\\spad{defineProperty(s,li,p)} defines the component property in the 3 dimensional subspace,{} \\spad{s},{} to be that of \\spad{p},{} where \\spad{p} is of the domain \\spadtype{SubSpaceComponentProperty}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component whose property is being defined. The subspace,{} \\spad{s},{} is returned with the component property definition.")) (|closeComponent| (($ $ (|List| (|NonNegativeInteger|)) (|Boolean|)) "\\spad{closeComponent(s,li,b)} sets the property of the component in the 3 dimensional subspace,{} \\spad{s},{} to be closed if \\spad{b} is \\spad{true},{} or open if \\spad{b} is \\spad{false}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component whose closed property is to be set. The subspace,{} \\spad{s},{} is returned with the component property modification.")) (|modifyPoint| (($ $ (|NonNegativeInteger|) (|Point| |#2|)) "\\spad{modifyPoint(s,ind,p)} modifies the point referenced by the index location,{} \\spad{ind},{} by replacing it with the point,{} \\spad{p} in the 3 dimensional subspace,{} \\spad{s}. An error message occurs if \\spad{s} is empty,{} otherwise the subspace \\spad{s} is returned with the point modification.") (($ $ (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{modifyPoint(s,li,i)} replaces an existing point in the 3 dimensional subspace,{} \\spad{s},{} with the 4 dimensional point indicated by the index location,{} \\spad{i}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the existing point is to be modified. An error message occurs if \\spad{s} is empty,{} otherwise the subspace \\spad{s} is returned with the point modification.") (($ $ (|List| (|NonNegativeInteger|)) (|Point| |#2|)) "\\spad{modifyPoint(s,li,p)} replaces an existing point in the 3 dimensional subspace,{} \\spad{s},{} with the 4 dimensional point,{} \\spad{p}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the existing point is to be modified. An error message occurs if \\spad{s} is empty,{} otherwise the subspace \\spad{s} is returned with the point modification.")) (|addPointLast| (($ $ $ (|Point| |#2|) (|NonNegativeInteger|)) "\\spad{addPointLast(s,s2,li,p)} adds the 4 dimensional point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s}. \\spad{s2} point to the end of the subspace \\spad{s}. \\spad{n} is the path in the \\spad{s2} component. The subspace \\spad{s} is returned with the additional point.")) (|addPoint2| (($ $ (|Point| |#2|)) "\\spad{addPoint2(s,p)} adds the 4 dimensional point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s}. The subspace \\spad{s} is returned with the additional point.")) (|addPoint| (((|NonNegativeInteger|) $ (|Point| |#2|)) "\\spad{addPoint(s,p)} adds the point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s},{} and returns the new total number of points in \\spad{s}.") (($ $ (|List| (|NonNegativeInteger|)) (|NonNegativeInteger|)) "\\spad{addPoint(s,li,i)} adds the 4 dimensional point indicated by the index location,{} \\spad{i},{} to the 3 dimensional subspace,{} \\spad{s}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the point is to be added. It's length should range from 0 to \\spad{n - 1} where \\spad{n} is the dimension of the subspace. If the length is \\spad{n - 1},{} then a specific lowest level component is being referenced. If it is less than \\spad{n - 1},{} then some higher level component (0 indicates top level component) is being referenced and a component of that level with the desired point is created. The subspace \\spad{s} is returned with the additional point.") (($ $ (|List| (|NonNegativeInteger|)) (|Point| |#2|)) "\\spad{addPoint(s,li,p)} adds the 4 dimensional point,{} \\spad{p},{} to the 3 dimensional subspace,{} \\spad{s}. The list of non negative integers,{} \\spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the point is to be added. It's length should range from 0 to \\spad{n - 1} where \\spad{n} is the dimension of the subspace. If the length is \\spad{n - 1},{} then a specific lowest level component is being referenced. If it is less than \\spad{n - 1},{} then some higher level component (0 indicates top level component) is being referenced and a component of that level with the desired point is created. The subspace \\spad{s} is returned with the additional point.")) (|separate| (((|List| $) $) "\\spad{separate(s)} makes each of the components of the \\spadtype{SubSpace},{} \\spad{s},{} into a list of separate and distinct subspaces and returns the list.")) (|merge| (($ (|List| $)) "\\spad{merge(ls)} a list of subspaces,{} \\spad{ls},{} into one subspace.") (($ $ $) "\\spad{merge(s1,s2)} the subspaces \\spad{s1} and \\spad{s2} into a single subspace.")) 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(|sum| ((|#2| |#2| (|SegmentBinding| |#2|)) "\\spad{sum(f(n), n = a..b)} returns \\spad{f}(a) + \\spad{f}(\\spad{a+1}) + ... + \\spad{f}(\\spad{b}).") ((|#2| |#2| (|Symbol|)) "\\spad{sum(a(n), n)} returns A(\\spad{n}) such that A(\\spad{n+1}) - A(\\spad{n}) = a(\\spad{n})."))) NIL NIL -(-1075 R) +(-1082 R) ((|constructor| (NIL "Computes sums of rational functions.")) (|sum| (((|Union| (|Fraction| (|Polynomial| |#1|)) (|Expression| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|SegmentBinding| (|Fraction| (|Polynomial| |#1|)))) "\\spad{sum(f(n), n = a..b)} returns \\spad{f(a) + f(a+1) + ... f(b)}.") (((|Fraction| (|Polynomial| |#1|)) (|Polynomial| |#1|) (|SegmentBinding| (|Polynomial| |#1|))) "\\spad{sum(f(n), n = a..b)} returns \\spad{f(a) + f(a+1) + ... f(b)}.") (((|Union| (|Fraction| (|Polynomial| |#1|)) (|Expression| |#1|)) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{sum(a(n), n)} returns \\spad{A} which is the indefinite sum of \\spad{a} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{A(n+1) - A(n) = a(n)}.") (((|Fraction| (|Polynomial| |#1|)) (|Polynomial| |#1|) (|Symbol|)) "\\spad{sum(a(n), n)} returns \\spad{A} which is the indefinite sum of \\spad{a} with respect to upward difference on \\spad{n},{} \\spadignore{i.e.} \\spad{A(n+1) - A(n) = a(n)}."))) NIL NIL -(-1076 R) +(-1083 R) ((|constructor| (NIL "This domain represents univariate polynomials over arbitrary (not necessarily commutative) coefficient rings. The variable is unspecified so that the variable displays as \\spad{?} on output. If it is necessary to specify the variable name,{} use type \\spadtype{UnivariatePolynomial}. The representation is sparse in the sense that only non-zero terms are represented.")) (|fmecg| (($ $ (|NonNegativeInteger|) |#1| $) "\\spad{fmecg(p1,e,r,p2)} finds \\spad{X} : \\spad{p1} - \\spad{r} * X**e * \\spad{p2}")) (|outputForm| (((|OutputForm|) $ (|OutputForm|)) "\\spad{outputForm(p,var)} converts the SparseUnivariatePolynomial \\spad{p} to an output form (see \\spadtype{OutputForm}) printed as a polynomial in the output form variable."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3974 |has| |#1| (-309)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-325)))) (|HasCategory| (-988) (QUOTE (-791 (-325))))) (-12 (|HasCategory| |#1| (QUOTE (-791 (-480)))) (|HasCategory| (-988) (QUOTE (-791 (-480))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-325))))) (|HasCategory| (-988) (QUOTE (-550 (-795 (-325)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-795 (-480))))) (|HasCategory| (-988) (QUOTE (-550 (-795 (-480)))))) (-12 (|HasCategory| |#1| (QUOTE (-550 (-469)))) (|HasCategory| (-988) (QUOTE (-550 (-469))))) (|HasCategory| |#1| (QUOTE (-577 (-480)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-816)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (|HasCategory| |#1| (QUOTE (-816)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-1057))) (|HasCategory| |#1| (QUOTE (-806 (-1081)))) (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasCategory| |#1| (QUOTE (-187))) (|HasCategory| |#1| (QUOTE (-188))) (|HasAttribute| |#1| (QUOTE -3976)) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-816))) (|HasCategory| $ (QUOTE (-116)))) (|HasCategory| |#1| (QUOTE (-116))))) -(-1077 R S) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3985 |has| |#1| (-311)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-327)))) (|HasCategory| (-993) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#1| (QUOTE (-796 (-483)))) (|HasCategory| (-993) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#1| (QUOTE (-553 (-472)))) (|HasCategory| (-993) (QUOTE (-553 (-472))))) (|HasCategory| |#1| (QUOTE (-580 (-483)))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-821)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (|HasCategory| |#1| (QUOTE (-821)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-1064))) (|HasCategory| |#1| (QUOTE (-811 (-1088)))) (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasCategory| |#1| (QUOTE (-189))) (|HasCategory| |#1| (QUOTE (-190))) (|HasAttribute| |#1| (QUOTE -3987)) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))))) +(-1084 R S) ((|constructor| (NIL "This package lifts a mapping from coefficient rings \\spad{R} to \\spad{S} to a mapping from sparse univariate polynomial over \\spad{R} to a sparse univariate polynomial over \\spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.")) (|map| (((|SparseUnivariatePolynomial| |#2|) (|Mapping| |#2| |#1|) (|SparseUnivariatePolynomial| |#1|)) "\\spad{map(func, poly)} creates a new polynomial by applying \\spad{func} to every non-zero coefficient of the polynomial poly."))) NIL NIL -(-1078 E OV R P) +(-1085 E OV R P) ((|constructor| (NIL "\\indented{1}{SupFractionFactorize} contains the factor function for univariate polynomials over the quotient field of a ring \\spad{S} such that the package MultivariateFactorize works for \\spad{S}")) (|squareFree| (((|Factored| (|SparseUnivariatePolynomial| (|Fraction| |#4|))) (|SparseUnivariatePolynomial| (|Fraction| |#4|))) "\\spad{squareFree(p)} returns the square-free factorization of the univariate polynomial \\spad{p} with coefficients which are fractions of polynomials over \\spad{R}. Each factor has no repeated roots and the factors are pairwise relatively prime.")) (|factor| (((|Factored| (|SparseUnivariatePolynomial| (|Fraction| |#4|))) (|SparseUnivariatePolynomial| (|Fraction| |#4|))) "\\spad{factor(p)} factors the univariate polynomial \\spad{p} with coefficients which are fractions of polynomials over \\spad{R}."))) NIL NIL -(-1079 |Coef| |var| |cen|) +(-1086 |Coef| |var| |cen|) ((|constructor| (NIL "Sparse Puiseux series in one variable \\indented{2}{\\spadtype{SparseUnivariatePuiseuxSeries} is a domain representing Puiseux} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{SparseUnivariatePuiseuxSeries(Integer,x,3)} represents Puiseux} \\indented{2}{series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|)))) (|HasCategory| (-345 (-480)) (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-309))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-29 (-480)))) (|HasCategory| |#1| (QUOTE (-866))) (|HasCategory| |#1| (QUOTE (-1106)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1081))))) (|HasSignature| |#1| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#1|))))))) -(-1080 |Coef| |var| |cen|) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|)))) (|HasCategory| (-347 (-483)) (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-311))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-29 (-483)))) (|HasCategory| |#1| (QUOTE (-871))) (|HasCategory| |#1| (QUOTE (-1113)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1088))))) (|HasSignature| |#1| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#1|))))))) +(-1087 |Coef| |var| |cen|) ((|constructor| (NIL "Sparse Taylor series in one variable \\indented{2}{\\spadtype{SparseUnivariateTaylorSeries} is a domain representing Taylor} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spadtype{SparseUnivariateTaylorSeries}(Integer,{}\\spad{x},{}3) represents Taylor} \\indented{2}{series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x),x)} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-689)) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-689)) (|devaluate| |#1|)))) (|HasCategory| (-689) (QUOTE (-1017))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-689))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-689))))) (|HasCategory| |#1| (QUOTE (-309))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-29 (-480)))) (|HasCategory| |#1| (QUOTE (-866))) (|HasCategory| |#1| (QUOTE (-1106)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1081))))) (|HasSignature| |#1| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#1|))))))) -(-1081) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-694)) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-694)) (|devaluate| |#1|)))) (|HasCategory| (-694) (QUOTE (-1024))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-694))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-694))))) (|HasCategory| |#1| (QUOTE (-311))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-29 (-483)))) (|HasCategory| |#1| (QUOTE (-871))) (|HasCategory| |#1| (QUOTE (-1113)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1088))))) (|HasSignature| |#1| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#1|))))))) +(-1088) ((|constructor| (NIL "Basic and scripted symbols.")) (|sample| (($) "\\spad{sample()} returns a sample of \\%")) (|list| (((|List| $) $) "\\spad{list(sy)} takes a scripted symbol and produces a list of the name followed by the scripts.")) (|string| (((|String|) $) "\\spad{string(s)} converts the symbol \\spad{s} to a string. Error: if the symbol is subscripted.")) (|elt| (($ $ (|List| (|OutputForm|))) "\\spad{elt(s,[a1,...,an])} or \\spad{s}([\\spad{a1},{}...,{}an]) returns \\spad{s} subscripted by \\spad{[a1,...,an]}.")) (|argscript| (($ $ (|List| (|OutputForm|))) "\\spad{argscript(s, [a1,...,an])} returns \\spad{s} arg-scripted by \\spad{[a1,...,an]}.")) (|superscript| (($ $ (|List| (|OutputForm|))) "\\spad{superscript(s, [a1,...,an])} returns \\spad{s} superscripted by \\spad{[a1,...,an]}.")) (|subscript| (($ $ (|List| (|OutputForm|))) "\\spad{subscript(s, [a1,...,an])} returns \\spad{s} subscripted by \\spad{[a1,...,an]}.")) (|script| (($ $ (|Record| (|:| |sub| (|List| (|OutputForm|))) (|:| |sup| (|List| (|OutputForm|))) (|:| |presup| (|List| (|OutputForm|))) (|:| |presub| (|List| (|OutputForm|))) (|:| |args| (|List| (|OutputForm|))))) "\\spad{script(s, [a,b,c,d,e])} returns \\spad{s} with subscripts a,{} superscripts \\spad{b},{} pre-superscripts \\spad{c},{} pre-subscripts \\spad{d},{} and argument-scripts \\spad{e}.") (($ $ (|List| (|List| (|OutputForm|)))) "\\spad{script(s, [a,b,c,d,e])} returns \\spad{s} with subscripts a,{} superscripts \\spad{b},{} pre-superscripts \\spad{c},{} pre-subscripts \\spad{d},{} and argument-scripts \\spad{e}. Omitted components are taken to be empty. For example,{} \\spad{script(s, [a,b,c])} is equivalent to \\spad{script(s,[a,b,c,[],[]])}.")) (|scripts| (((|Record| (|:| |sub| (|List| (|OutputForm|))) (|:| |sup| (|List| (|OutputForm|))) (|:| |presup| (|List| (|OutputForm|))) (|:| |presub| (|List| (|OutputForm|))) (|:| |args| (|List| (|OutputForm|)))) $) "\\spad{scripts(s)} returns all the scripts of \\spad{s}.")) (|scripted?| (((|Boolean|) $) "\\spad{scripted?(s)} is \\spad{true} if \\spad{s} has been given any scripts.")) (|name| (($ $) "\\spad{name(s)} returns \\spad{s} without its scripts.")) (|resetNew| (((|Void|)) "\\spad{resetNew()} resets the internals counters that new() and new(\\spad{s}) use to return distinct symbols every time.")) (|new| (($ $) "\\spad{new(s)} returns a new symbol whose name starts with \\%\\spad{s}.") (($) "\\spad{new()} returns a new symbol whose name starts with \\%."))) NIL NIL -(-1082 R) +(-1089 R) ((|constructor| (NIL "Computes all the symmetric functions in \\spad{n} variables.")) (|symFunc| (((|Vector| |#1|) |#1| (|PositiveInteger|)) "\\spad{symFunc(r, n)} returns the vector of the elementary symmetric functions in \\spad{[r,r,...,r]} \\spad{n} times.") (((|Vector| |#1|) (|List| |#1|)) "\\spad{symFunc([r1,...,rn])} returns the vector of the elementary symmetric functions in the \\spad{ri's}: \\spad{[r1 + ... + rn, r1 r2 + ... + r(n-1) rn, ..., r1 r2 ... rn]}."))) NIL NIL -(-1083 R) +(-1090 R) ((|constructor| (NIL "This domain implements symmetric polynomial"))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-6 -3976)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (OR (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480)))))) (|HasCategory| |#1| (QUOTE (-945 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-945 (-480)))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-387))) (-12 (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| (-879) (QUOTE (-102)))) (|HasAttribute| |#1| (QUOTE -3976))) -(-1084) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-6 -3987)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (OR (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#1| (QUOTE (-950 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-950 (-483)))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-389))) (-12 (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| (-884) (QUOTE (-104)))) (|HasAttribute| |#1| (QUOTE -3987))) +(-1091) ((|constructor| (NIL "Creates and manipulates one global symbol table for FORTRAN code generation,{} containing details of types,{} dimensions,{} and argument lists.")) (|symbolTableOf| (((|SymbolTable|) (|Symbol|) $) "\\spad{symbolTableOf(f,tab)} returns the symbol table of \\spad{f}")) (|argumentListOf| (((|List| (|Symbol|)) (|Symbol|) $) "\\spad{argumentListOf(f,tab)} returns the argument list of \\spad{f}")) (|returnTypeOf| (((|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1="void")) (|Symbol|) $) "\\spad{returnTypeOf(f,tab)} returns the type of the object returned by \\spad{f}")) (|empty| (($) "\\spad{empty()} creates a new,{} empty symbol table.")) (|printTypes| (((|Void|) (|Symbol|)) "\\spad{printTypes(tab)} produces FORTRAN type declarations from \\spad{tab},{} on the current FORTRAN output stream")) (|printHeader| (((|Void|)) "\\spad{printHeader()} produces the FORTRAN header for the current subprogram in the global symbol table on the current FORTRAN output stream.") (((|Void|) (|Symbol|)) "\\spad{printHeader(f)} produces the FORTRAN header for subprogram \\spad{f} in the global symbol table on the current FORTRAN output stream.") (((|Void|) (|Symbol|) $) "\\spad{printHeader(f,tab)} produces the FORTRAN header for subprogram \\spad{f} in symbol table \\spad{tab} on the current FORTRAN output stream.")) (|returnType!| (((|Void|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#))) "\\spad{returnType!(t)} declares that the return type of he current subprogram in the global symbol table is \\spad{t}.") (((|Void|) (|Symbol|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#))) "\\spad{returnType!(f,t)} declares that the return type of subprogram \\spad{f} in the global symbol table is \\spad{t}.") (((|Void|) (|Symbol|) (|Union| (|:| |fst| (|FortranScalarType|)) (|:| |void| #1#)) $) "\\spad{returnType!(f,t,tab)} declares that the return type of subprogram \\spad{f} in symbol table \\spad{tab} is \\spad{t}.")) (|argumentList!| (((|Void|) (|List| (|Symbol|))) "\\spad{argumentList!(l)} declares that the argument list for the current subprogram in the global symbol table is \\spad{l}.") (((|Void|) (|Symbol|) (|List| (|Symbol|))) "\\spad{argumentList!(f,l)} declares that the argument list for subprogram \\spad{f} in the global symbol table is \\spad{l}.") (((|Void|) (|Symbol|) (|List| (|Symbol|)) $) "\\spad{argumentList!(f,l,tab)} declares that the argument list for subprogram \\spad{f} in symbol table \\spad{tab} is \\spad{l}.")) (|endSubProgram| (((|Symbol|)) "\\spad{endSubProgram()} asserts that we are no longer processing the current subprogram.")) (|currentSubProgram| (((|Symbol|)) "\\spad{currentSubProgram()} returns the name of the current subprogram being processed")) (|newSubProgram| (((|Void|) (|Symbol|)) "\\spad{newSubProgram(f)} asserts that from now on type declarations are part of subprogram \\spad{f}.")) (|declare!| (((|FortranType|) (|Symbol|) (|FortranType|) (|Symbol|)) "\\spad{declare!(u,t,asp)} declares the parameter \\spad{u} to have type \\spad{t} in \\spad{asp}.") (((|FortranType|) (|Symbol|) (|FortranType|)) "\\spad{declare!(u,t)} declares the parameter \\spad{u} to have type \\spad{t} in the current level of the symbol table.") (((|FortranType|) (|List| (|Symbol|)) (|FortranType|) (|Symbol|) $) "\\spad{declare!(u,t,asp,tab)} declares the parameters \\spad{u} of subprogram \\spad{asp} to have type \\spad{t} in symbol table \\spad{tab}.") (((|FortranType|) (|Symbol|) (|FortranType|) (|Symbol|) $) "\\spad{declare!(u,t,asp,tab)} declares the parameter \\spad{u} of subprogram \\spad{asp} to have type \\spad{t} in symbol table \\spad{tab}.")) (|clearTheSymbolTable| (((|Void|) (|Symbol|)) "\\spad{clearTheSymbolTable(x)} removes the symbol \\spad{x} from the table") (((|Void|)) "\\spad{clearTheSymbolTable()} clears the current symbol table.")) (|showTheSymbolTable| (($) "\\spad{showTheSymbolTable()} returns the current symbol table."))) NIL NIL -(-1085) +(-1092) ((|constructor| (NIL "Create and manipulate a symbol table for generated FORTRAN code")) (|symbolTable| (($ (|List| (|Record| (|:| |key| (|Symbol|)) (|:| |entry| (|FortranType|))))) "\\spad{symbolTable(l)} creates a symbol table from the elements of \\spad{l}.")) (|printTypes| (((|Void|) $) "\\spad{printTypes(tab)} produces FORTRAN type declarations from \\spad{tab},{} on the current FORTRAN output stream")) (|newTypeLists| (((|SExpression|) $) "\\spad{newTypeLists(x)} \\undocumented")) (|typeLists| (((|List| (|List| (|Union| (|:| |name| (|Symbol|)) (|:| |bounds| (|List| (|Union| (|:| S (|Symbol|)) (|:| P (|Polynomial| (|Integer|))))))))) $) "\\spad{typeLists(tab)} returns a list of lists of types of objects in \\spad{tab}")) (|externalList| (((|List| (|Symbol|)) $) "\\spad{externalList(tab)} returns a list of all the external symbols in \\spad{tab}")) (|typeList| (((|List| (|Union| (|:| |name| (|Symbol|)) (|:| |bounds| (|List| (|Union| (|:| S (|Symbol|)) (|:| P (|Polynomial| (|Integer|)))))))) (|FortranScalarType|) $) "\\spad{typeList(t,tab)} returns a list of all the objects of type \\spad{t} in \\spad{tab}")) (|parametersOf| (((|List| (|Symbol|)) $) "\\spad{parametersOf(tab)} returns a list of all the symbols declared in \\spad{tab}")) (|fortranTypeOf| (((|FortranType|) (|Symbol|) $) "\\spad{fortranTypeOf(u,tab)} returns the type of \\spad{u} in \\spad{tab}")) (|declare!| (((|FortranType|) (|Symbol|) (|FortranType|) $) "\\spad{declare!(u,t,tab)} creates a new entry in \\spad{tab},{} declaring \\spad{u} to be of type \\spad{t}") (((|FortranType|) (|List| (|Symbol|)) (|FortranType|) $) "\\spad{declare!(l,t,tab)} creates new entrys in \\spad{tab},{} declaring each of \\spad{l} to be of type \\spad{t}")) (|empty| (($) "\\spad{empty()} returns a new,{} empty symbol table")) (|coerce| (((|Table| (|Symbol|) (|FortranType|)) $) "\\spad{coerce(x)} returns a table view of \\spad{x}"))) NIL NIL -(-1086) +(-1093) ((|constructor| (NIL "\\indented{1}{This domain provides a simple domain,{} general enough for} \\indented{2}{building complete representation of Spad programs as objects} \\indented{2}{of a term algebra built from ground terms of type integers,{} foats,{}} \\indented{2}{identifiers,{} and strings.} \\indented{2}{This domain differs from InputForm in that it represents} \\indented{2}{any entity in a Spad program,{} not just expressions.\\space{2}Furthermore,{}} \\indented{2}{while InputForm may contain atoms like vectors and other Lisp} \\indented{2}{objects,{} the Syntax domain is supposed to contain only that} \\indented{2}{initial algebra build from the primitives listed above.} Related Constructors: \\indented{2}{Integer,{} DoubleFloat,{} Identifier,{} String,{} SExpression.} See Also: SExpression,{} InputForm. The equality supported by this domain is structural.")) (|case| (((|Boolean|) $ (|[\|\|]| (|String|))) "\\spad{x case String} is \\spad{true} if `x' really is a String") (((|Boolean|) $ (|[\|\|]| (|Identifier|))) "\\spad{x case Identifier} is \\spad{true} if `x' really is an Identifier") (((|Boolean|) $ (|[\|\|]| (|DoubleFloat|))) "\\spad{x case DoubleFloat} is \\spad{true} if `x' really is a DoubleFloat") (((|Boolean|) $ (|[\|\|]| (|Integer|))) "\\spad{x case Integer} is \\spad{true} if `x' really is an Integer")) (|compound?| (((|Boolean|) $) "\\spad{compound? x} is \\spad{true} when `x' is not an atomic syntax.")) (|getOperands| (((|List| $) $) "\\spad{getOperands(x)} returns the list of operands to the operator in `x'.")) (|getOperator| (((|Union| (|Integer|) (|DoubleFloat|) (|Identifier|) (|String|) $) $) "\\spad{getOperator(x)} returns the operator,{} or tag,{} of the syntax `x'. The value returned is itself a syntax if `x' really is an application of a function symbol as opposed to being an atomic ground term.")) (|nil?| (((|Boolean|) $) "\\spad{nil?(s)} is \\spad{true} when `s' is a syntax for the constant nil.")) (|buildSyntax| (($ $ (|List| $)) "\\spad{buildSyntax(op, [a1, ..., an])} builds a syntax object for \\spad{op}(\\spad{a1},{}...,{}an).") (($ (|Identifier|) (|List| $)) "\\spad{buildSyntax(op, [a1, ..., an])} builds a syntax object for \\spad{op}(\\spad{a1},{}...,{}an).")) (|autoCoerce| (((|String|) $) "\\spad{autoCoerce(s)} forcibly extracts a string value from the syntax `s'; no check performed. To be called only at the discretion of the compiler.") (((|Identifier|) $) "\\spad{autoCoerce(s)} forcibly extracts an identifier from the Syntax domain `s'; no check performed. To be called only at at the discretion of the compiler.") (((|DoubleFloat|) $) "\\spad{autoCoerce(s)} forcibly extracts a float value from the syntax `s'; no check performed. To be called only at the discretion of the compiler") (((|Integer|) $) "\\spad{autoCoerce(s)} forcibly extracts an integer value from the syntax `s'; no check performed. To be called only at the discretion of the compiler.")) (|coerce| (((|String|) $) "\\spad{coerce(s)} extracts a string value from the syntax `s'.") (((|Identifier|) $) "\\spad{coerce(s)} extracts an identifier from the syntax `s'.") (((|DoubleFloat|) $) "\\spad{coerce(s)} extracts a float value from the syntax `s'.") (((|Integer|) $) "\\spad{coerce(s)} extracts and integer value from the syntax `s'")) (|convert| (($ (|SExpression|)) "\\spad{convert(s)} converts an \\spad{s}-expression to Syntax. Note,{} when `s' is not an atom,{} it is expected that it designates a proper list,{} \\spadignore{e.g.} a sequence of cons cells ending with nil.") (((|SExpression|) $) "\\spad{convert(s)} returns the \\spad{s}-expression representation of a syntax."))) NIL NIL -(-1087 N) +(-1094 N) ((|constructor| (NIL "This domain implements sized (signed) integer datatypes parameterized by the precision (or width) of the underlying representation. The intent is that they map directly to the hosting hardware natural integer datatypes. Consequently,{} natural values for \\spad{N} are: 8,{} 16,{} 32,{} 64,{} etc. These datatypes are mostly useful for system programming tasks,{} \\spadignore{i.e.} interfacting with the hosting operating system,{} reading/writing external binary format files.")) (|sample| (($) "\\spad{sample} gives a sample datum of this type."))) NIL NIL -(-1088 N) +(-1095 N) ((|constructor| (NIL "This domain implements sized (unsigned) integer datatypes parameterized by the precision (or width) of the underlying representation. The intent is that they map directly to the hosting hardware natural integer datatypes. Consequently,{} natural values for \\spad{N} are: 8,{} 16,{} 32,{} 64,{} etc. These datatypes are mostly useful for system programming tasks,{} \\spadignore{i.e.} interfacting with the hosting operating system,{} reading/writing external binary format files.")) (|sample| (($) "\\spad{sample} gives a sample datum of type Byte.")) (|bitior| (($ $ $) "\\spad{bitior(x,y)} returns the bitwise `inclusive or' of `x' and `y'.")) (|bitand| (($ $ $) "\\spad{bitand(x,y)} returns the bitwise `and' of `x' and `y'."))) NIL NIL -(-1089) +(-1096) ((|constructor| (NIL "This domain is a datatype system-level pointer values."))) NIL NIL -(-1090 R) +(-1097 R) ((|triangularSystems| (((|List| (|List| (|Polynomial| |#1|))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{triangularSystems(lf,lv)} solves the system of equations defined by \\spad{lf} with respect to the list of symbols \\spad{lv}; the system of equations is obtaining by equating to zero the list of rational functions \\spad{lf}. The output is a list of solutions where each solution is expressed as a \"reduced\" triangular system of polynomials.")) (|solve| (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Equation| (|Fraction| (|Polynomial| |#1|)))) "\\spad{solve(eq)} finds the solutions of the equation \\spad{eq} with respect to the unique variable appearing in \\spad{eq}.") (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Fraction| (|Polynomial| |#1|))) "\\spad{solve(p)} finds the solution of a rational function \\spad{p} = 0 with respect to the unique variable appearing in \\spad{p}.") (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Equation| (|Fraction| (|Polynomial| |#1|))) (|Symbol|)) "\\spad{solve(eq,v)} finds the solutions of the equation \\spad{eq} with respect to the variable \\spad{v}.") (((|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|Fraction| (|Polynomial| |#1|)) (|Symbol|)) "\\spad{solve(p,v)} solves the equation \\spad{p=0},{} where \\spad{p} is a rational function with respect to the variable \\spad{v}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) "\\spad{solve(le)} finds the solutions of the list \\spad{le} of equations of rational functions with respect to all symbols appearing in \\spad{le}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Fraction| (|Polynomial| |#1|)))) "\\spad{solve(lp)} finds the solutions of the list \\spad{lp} of rational functions with respect to all symbols appearing in \\spad{lp}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Equation| (|Fraction| (|Polynomial| |#1|)))) (|List| (|Symbol|))) "\\spad{solve(le,lv)} finds the solutions of the list \\spad{le} of equations of rational functions with respect to the list of symbols \\spad{lv}.") (((|List| (|List| (|Equation| (|Fraction| (|Polynomial| |#1|))))) (|List| (|Fraction| (|Polynomial| |#1|))) (|List| (|Symbol|))) "\\spad{solve(lp,lv)} finds the solutions of the list \\spad{lp} of rational functions with respect to the list of symbols \\spad{lv}."))) NIL NIL -(-1091) +(-1098) ((|constructor| (NIL "The package \\spadtype{System} provides information about the runtime system and its characteristics.")) (|loadNativeModule| (((|Void|) (|String|)) "\\spad{loadNativeModule(path)} loads the native modile designated by \\spadvar{\\spad{path}}.")) (|nativeModuleExtension| (((|String|)) "\\spad{nativeModuleExtension} is a string representation of a filename extension for native modules.")) (|hostByteOrder| (((|ByteOrder|)) "\\sapd{hostByteOrder}")) (|hostPlatform| (((|String|)) "\\spad{hostPlatform} is a string `triplet' description of the platform hosting the running OpenAxiom system.")) (|rootDirectory| (((|String|)) "\\spad{rootDirectory()} returns the pathname of the root directory for the running OpenAxiom system."))) NIL NIL -(-1092 S) +(-1099 S) ((|constructor| (NIL "TableauBumpers implements the Schenstead-Knuth correspondence between sequences and pairs of Young tableaux. The 2 Young tableaux are represented as a single tableau with pairs as components.")) (|mr| (((|Record| (|:| |f1| (|List| |#1|)) (|:| |f2| (|List| (|List| (|List| |#1|)))) (|:| |f3| (|List| (|List| |#1|))) (|:| |f4| (|List| (|List| (|List| |#1|))))) (|List| (|List| (|List| |#1|)))) "\\spad{mr(t)} is an auxiliary function which finds the position of the maximum element of a tableau \\spad{t} which is in the lowest row,{} producing a record of results")) (|maxrow| (((|Record| (|:| |f1| (|List| |#1|)) (|:| |f2| (|List| (|List| (|List| |#1|)))) (|:| |f3| (|List| (|List| |#1|))) (|:| |f4| (|List| (|List| (|List| |#1|))))) (|List| |#1|) (|List| (|List| (|List| |#1|))) (|List| (|List| |#1|)) (|List| (|List| (|List| |#1|))) (|List| (|List| (|List| |#1|))) (|List| (|List| (|List| |#1|)))) "\\spad{maxrow(a,b,c,d,e)} is an auxiliary function for mr")) (|inverse| (((|List| |#1|) (|List| |#1|)) "\\spad{inverse(ls)} forms the inverse of a sequence \\spad{ls}")) (|slex| (((|List| (|List| |#1|)) (|List| |#1|)) "\\spad{slex(ls)} sorts the argument sequence \\spad{ls},{} then zips (see \\spadfunFrom{map}{\\spad{ListFunctions3}}) the original argument sequence with the sorted result to a list of pairs")) (|lex| (((|List| (|List| |#1|)) (|List| (|List| |#1|))) "\\spad{lex(ls)} sorts a list of pairs to lexicographic order")) (|tab| (((|Tableau| (|List| |#1|)) (|List| |#1|)) "\\spad{tab(ls)} creates a tableau from \\spad{ls} by first creating a list of pairs using \\spadfunFrom{slex}{TableauBumpers},{} then creating a tableau using \\spadfunFrom{\\spad{tab1}}{TableauBumpers}.")) (|tab1| (((|List| (|List| (|List| |#1|))) (|List| (|List| |#1|))) "\\spad{tab1(lp)} creates a tableau from a list of pairs \\spad{lp}")) (|bat| (((|List| (|List| |#1|)) (|Tableau| (|List| |#1|))) "\\spad{bat(ls)} unbumps a tableau \\spad{ls}")) (|bat1| (((|List| (|List| |#1|)) (|List| (|List| (|List| |#1|)))) "\\spad{bat1(llp)} unbumps a tableau \\spad{llp}. Operation \\spad{bat1} is the inverse of \\spad{tab1}.")) (|untab| (((|List| (|List| |#1|)) (|List| (|List| |#1|)) (|List| (|List| (|List| |#1|)))) "\\spad{untab(lp,llp)} is an auxiliary function which unbumps a tableau \\spad{llp},{} using \\spad{lp} to accumulate pairs")) (|bumptab1| (((|List| (|List| (|List| |#1|))) (|List| |#1|) (|List| (|List| (|List| |#1|)))) "\\spad{bumptab1(pr,t)} bumps a tableau \\spad{t} with a pair \\spad{pr} using comparison function \\spadfun{<},{} returning a new tableau")) (|bumptab| (((|List| (|List| (|List| |#1|))) (|Mapping| (|Boolean|) |#1| |#1|) (|List| |#1|) (|List| (|List| (|List| |#1|)))) "\\spad{bumptab(cf,pr,t)} bumps a tableau \\spad{t} with a pair \\spad{pr} using comparison function \\spad{cf},{} returning a new tableau")) (|bumprow| (((|Record| (|:| |fs| (|Boolean|)) (|:| |sd| (|List| |#1|)) (|:| |td| (|List| (|List| |#1|)))) (|Mapping| (|Boolean|) |#1| |#1|) (|List| |#1|) (|List| (|List| |#1|))) "\\spad{bumprow(cf,pr,r)} is an auxiliary function which bumps a row \\spad{r} with a pair \\spad{pr} using comparison function \\spad{cf},{} and returns a record"))) NIL NIL -(-1093 |Key| |Entry|) +(-1100 |Key| |Entry|) ((|constructor| (NIL "This is the general purpose table type. The keys are hashed to look up the entries. This creates a \\spadtype{HashTable} if equal for the Key domain is consistent with Lisp EQUAL otherwise an \\spadtype{AssociationList}"))) -((-3978 . T) (-3979 . T)) -((-12 (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -257) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3843) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007)))) (OR (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| |#2| (QUOTE (-549 (-767))))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-550 (-469)))) (-12 (|HasCategory| |#2| (QUOTE (-1007))) (|HasCategory| |#2| (|%list| (QUOTE -257) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#2| (QUOTE (-1007))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-549 (-767)))) (|HasCategory| (-2 (|:| -3843 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) -(-1094 S) +((-3989 . T) (-3990 . T)) +((-12 (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (|%list| (QUOTE -259) (|%list| (QUOTE -2) (|%list| (QUOTE |:|) (QUOTE -3854) (|devaluate| |#1|)) (|%list| (QUOTE |:|) (QUOTE |entry|) (|devaluate| |#2|))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012)))) (OR (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| |#2| (QUOTE (-552 (-772))))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-553 (-472)))) (-12 (|HasCategory| |#2| (QUOTE (-1012))) (|HasCategory| |#2| (|%list| (QUOTE -259) (|devaluate| |#2|)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#2| (QUOTE (-1012))) (OR (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) (|HasCategory| |#2| (QUOTE (-72))) (|HasCategory| |#2| (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-552 (-772)))) (|HasCategory| (-2 (|:| -3854 |#1|) (|:| |entry| |#2|)) (QUOTE (-72)))) +(-1101 S) ((|constructor| (NIL "\\indented{1}{The tableau domain is for printing Young tableaux,{} and} coercions to and from List List \\spad{S} where \\spad{S} is a set.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(t)} converts a tableau \\spad{t} to an output form.")) (|listOfLists| (((|List| (|List| |#1|)) $) "\\spad{listOfLists t} converts a tableau \\spad{t} to a list of lists.")) (|tableau| (($ (|List| (|List| |#1|))) "\\spad{tableau(ll)} converts a list of lists \\spad{ll} to a tableau."))) NIL NIL -(-1095 S) +(-1102 S) ((|constructor| (NIL "\\indented{1}{Author: Gabriel Dos Reis} Date Created: April 17,{} 2010 Date Last Modified: April 17,{} 2010")) (|operator| (($ |#1| (|Arity|)) "\\spad{operator(n,a)} returns an operator named \\spad{n} and with arity \\spad{a}."))) NIL NIL -(-1096 R) +(-1103 R) ((|constructor| (NIL "Expands tangents of sums and scalar products.")) (|tanNa| ((|#1| |#1| (|Integer|)) "\\spad{tanNa(a, n)} returns \\spad{f(a)} such that if \\spad{a = tan(u)} then \\spad{f(a) = tan(n * u)}.")) (|tanAn| (((|SparseUnivariatePolynomial| |#1|) |#1| (|PositiveInteger|)) "\\spad{tanAn(a, n)} returns \\spad{P(x)} such that if \\spad{a = tan(u)} then \\spad{P(tan(u/n)) = 0}.")) (|tanSum| ((|#1| (|List| |#1|)) "\\spad{tanSum([a1,...,an])} returns \\spad{f(a1,...,an)} such that if \\spad{ai = tan(ui)} then \\spad{f(a1,...,an) = tan(u1 + ... + un)}."))) NIL NIL -(-1097 S |Key| |Entry|) +(-1104 S |Key| |Entry|) ((|constructor| (NIL "A table aggregate is a model of a table,{} \\spadignore{i.e.} a discrete many-to-one mapping from keys to entries.")) (|map| (($ (|Mapping| |#3| |#3| |#3|) $ $) "\\spad{map(fn,t1,t2)} creates a new table \\spad{t} from given tables \\spad{t1} and \\spad{t2} with elements \\spad{fn}(\\spad{x},{}\\spad{y}) where \\spad{x} and \\spad{y} are corresponding elements from \\spad{t1} and \\spad{t2} respectively.")) (|table| (($ (|List| (|Record| (|:| |key| |#2|) (|:| |entry| |#3|)))) "\\spad{table([x,y,...,z])} creates a table consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{table()}\\$\\spad{T} creates an empty table of type \\spad{T}.")) (|setelt| ((|#3| $ |#2| |#3|) "\\spad{setelt(t,k,e)} (also written \\axiom{\\spad{t}.\\spad{k} := \\spad{e}}) is equivalent to \\axiom{(insert([\\spad{k},{}\\spad{e}],{}\\spad{t}); \\spad{e})}."))) NIL NIL -(-1098 |Key| |Entry|) +(-1105 |Key| |Entry|) ((|constructor| (NIL "A table aggregate is a model of a table,{} \\spadignore{i.e.} a discrete many-to-one mapping from keys to entries.")) (|map| (($ (|Mapping| |#2| |#2| |#2|) $ $) "\\spad{map(fn,t1,t2)} creates a new table \\spad{t} from given tables \\spad{t1} and \\spad{t2} with elements \\spad{fn}(\\spad{x},{}\\spad{y}) where \\spad{x} and \\spad{y} are corresponding elements from \\spad{t1} and \\spad{t2} respectively.")) (|table| (($ (|List| (|Record| (|:| |key| |#1|) (|:| |entry| |#2|)))) "\\spad{table([x,y,...,z])} creates a table consisting of entries \\axiom{\\spad{x},{}\\spad{y},{}...,{}\\spad{z}}.") (($) "\\spad{table()}\\$\\spad{T} creates an empty table of type \\spad{T}.")) (|setelt| ((|#2| $ |#1| |#2|) "\\spad{setelt(t,k,e)} (also written \\axiom{\\spad{t}.\\spad{k} := \\spad{e}}) is equivalent to \\axiom{(insert([\\spad{k},{}\\spad{e}],{}\\spad{t}); \\spad{e})}."))) -((-3979 . T)) +((-3990 . T)) NIL -(-1099 |Key| |Entry|) +(-1106 |Key| |Entry|) ((|constructor| (NIL "\\axiom{TabulatedComputationPackage(Key ,{}Entry)} provides some modest support for dealing with operations with type \\axiom{Key -> Entry}. The result of such operations can be stored and retrieved with this package by using a hash-table. The user does not need to worry about the management of this hash-table. However,{} onnly one hash-table is built by calling \\axiom{TabulatedComputationPackage(Key ,{}Entry)}.")) (|insert!| (((|Void|) |#1| |#2|) "\\axiom{insert!(\\spad{x},{}\\spad{y})} stores the item whose key is \\axiom{\\spad{x}} and whose entry is \\axiom{\\spad{y}}.")) (|extractIfCan| (((|Union| |#2| "failed") |#1|) "\\axiom{extractIfCan(\\spad{x})} searches the item whose key is \\axiom{\\spad{x}}.")) (|makingStats?| (((|Boolean|)) "\\axiom{makingStats?()} returns \\spad{true} iff the statisitics process is running.")) (|printingInfo?| (((|Boolean|)) "\\axiom{printingInfo?()} returns \\spad{true} iff messages are printed when manipulating items from the hash-table.")) (|usingTable?| (((|Boolean|)) "\\axiom{usingTable?()} returns \\spad{true} iff the hash-table is used")) (|clearTable!| (((|Void|)) "\\axiom{clearTable!()} clears the hash-table and assumes that it will no longer be used.")) (|printStats!| (((|Void|)) "\\axiom{printStats!()} prints the statistics.")) (|startStats!| (((|Void|) (|String|)) "\\axiom{startStats!(\\spad{x})} initializes the statisitics process and sets the comments to display when statistics are printed")) (|printInfo!| (((|Void|) (|String|) (|String|)) "\\axiom{printInfo!(\\spad{x},{}\\spad{y})} initializes the mesages to be printed when manipulating items from the hash-table. If a key is retrieved then \\axiom{\\spad{x}} is displayed. If an item is stored then \\axiom{\\spad{y}} is displayed.")) (|initTable!| (((|Void|)) "\\axiom{initTable!()} initializes the hash-table."))) NIL NIL -(-1100) +(-1107) ((|constructor| (NIL "\\spadtype{TexFormat} provides a coercion from \\spadtype{OutputForm} to \\TeX{} format. The particular dialect of \\TeX{} used is \\LaTeX{}. The basic object consists of three parts: a prologue,{} a tex part and an epilogue. The functions \\spadfun{prologue},{} \\spadfun{tex} and \\spadfun{epilogue} extract these parts,{} respectively. The main guts of the expression go into the tex part. The other parts can be set (\\spadfun{setPrologue!},{} \\spadfun{setEpilogue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain ``\\verb+\\[+'' and ``\\verb+\\]+'',{} respectively,{} so that the TeX section will be printed in LaTeX display math mode.")) (|setPrologue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setPrologue!(t,strings)} sets the prologue section of a TeX form \\spad{t} to \\spad{strings}.")) (|setTex!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setTex!(t,strings)} sets the TeX section of a TeX form \\spad{t} to \\spad{strings}.")) (|setEpilogue!| (((|List| (|String|)) $ (|List| (|String|))) "\\spad{setEpilogue!(t,strings)} sets the epilogue section of a TeX form \\spad{t} to \\spad{strings}.")) (|prologue| (((|List| (|String|)) $) "\\spad{prologue(t)} extracts the prologue section of a TeX form \\spad{t}.")) (|new| (($) "\\spad{new()} create a new,{} empty object. Use \\spadfun{setPrologue!},{} \\spadfun{setTex!} and \\spadfun{setEpilogue!} to set the various components of this object.")) (|tex| (((|List| (|String|)) $) "\\spad{tex(t)} extracts the TeX section of a TeX form \\spad{t}.")) (|epilogue| (((|List| (|String|)) $) "\\spad{epilogue(t)} extracts the epilogue section of a TeX form \\spad{t}.")) (|display| (((|Void|) $) "\\spad{display(t)} outputs the TeX formatted code \\spad{t} so that each line has length less than or equal to the value set by the system command \\spadsyscom{set output length}.") (((|Void|) $ (|Integer|)) "\\spad{display(t,width)} outputs the TeX formatted code \\spad{t} so that each line has length less than or equal to \\spadvar{\\spad{width}}.")) (|convert| (($ (|OutputForm|) (|Integer|) (|OutputForm|)) "\\spad{convert(o,step,type)} changes \\spad{o} in standard output format to TeX format and also adds the given \\spad{step} number and \\spad{type}. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \\spad{step} numbers.") (($ (|OutputForm|) (|Integer|)) "\\spad{convert(o,step)} changes \\spad{o} in standard output format to TeX format and also adds the given \\spad{step} number. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \\spad{step} numbers."))) NIL NIL -(-1101 S) +(-1108 S) ((|constructor| (NIL "\\spadtype{TexFormat1} provides a utility coercion for changing to TeX format anything that has a coercion to the standard output format.")) (|coerce| (((|TexFormat|) |#1|) "\\spad{coerce(s)} provides a direct coercion from a domain \\spad{S} to TeX format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to TeX format."))) NIL NIL -(-1102) +(-1109) ((|constructor| (NIL "This domain provides an implementation of text files. Text is stored in these files using the native character set of the computer.")) (|endOfFile?| (((|Boolean|) $) "\\spad{endOfFile?(f)} tests whether the file \\spad{f} is positioned after the end of all text. If the file is open for output,{} then this test is always \\spad{true}.")) (|readIfCan!| (((|Union| (|String|) "failed") $) "\\spad{readIfCan!(f)} returns a string of the contents of a line from file \\spad{f},{} if possible. If \\spad{f} is not readable or if it is positioned at the end of file,{} then \\spad{\"failed\"} is returned.")) (|readLineIfCan!| (((|Union| (|String|) "failed") $) "\\spad{readLineIfCan!(f)} returns a string of the contents of a line from file \\spad{f},{} if possible. If \\spad{f} is not readable or if it is positioned at the end of file,{} then \\spad{\"failed\"} is returned.")) (|readLine!| (((|String|) $) "\\spad{readLine!(f)} returns a string of the contents of a line from the file \\spad{f}.")) (|writeLine!| (((|String|) $) "\\spad{writeLine!(f)} finishes the current line in the file \\spad{f}. An empty string is returned. The call \\spad{writeLine!(f)} is equivalent to \\spad{writeLine!(f,\"\")}.") (((|String|) $ (|String|)) "\\spad{writeLine!(f,s)} writes the contents of the string \\spad{s} and finishes the current line in the file \\spad{f}. The value of \\spad{s} is returned."))) NIL NIL -(-1103 R) +(-1110 R) ((|constructor| (NIL "Tools for the sign finding utilities.")) (|direction| (((|Integer|) (|String|)) "\\spad{direction(s)} \\undocumented")) (|nonQsign| (((|Union| (|Integer|) "failed") |#1|) "\\spad{nonQsign(r)} \\undocumented")) (|sign| (((|Union| (|Integer|) "failed") |#1|) "\\spad{sign(r)} \\undocumented"))) NIL NIL -(-1104) +(-1111) ((|constructor| (NIL "This package exports a function for making a \\spadtype{ThreeSpace}")) (|createThreeSpace| (((|ThreeSpace| (|DoubleFloat|))) "\\spad{createThreeSpace()} creates a \\spadtype{ThreeSpace(DoubleFloat)} object capable of holding point,{} curve,{} mesh components and any combination."))) NIL NIL -(-1105 S) +(-1112 S) ((|constructor| (NIL "Category for the transcendental elementary functions.")) (|pi| (($) "\\spad{pi()} returns the constant \\spad{pi}."))) NIL NIL -(-1106) +(-1113) ((|constructor| (NIL "Category for the transcendental elementary functions.")) (|pi| (($) "\\spad{pi()} returns the constant \\spad{pi}."))) NIL NIL -(-1107 S) +(-1114 S) ((|constructor| (NIL "\\spadtype{Tree(S)} is a basic domains of tree structures. Each tree is either empty or else is a {\\it node} consisting of a value and a list of (sub)trees.")) (|cyclicParents| (((|List| $) $) "\\spad{cyclicParents(t)} returns a list of cycles that are parents of \\spad{t}.")) (|cyclicEqual?| (((|Boolean|) $ $) "\\spad{cyclicEqual?(t1, t2)} tests of two cyclic trees have the same structure.")) (|cyclicEntries| (((|List| $) $) "\\spad{cyclicEntries(t)} returns a list of top-level cycles in tree \\spad{t}.")) (|cyclicCopy| (($ $) "\\spad{cyclicCopy(l)} makes a copy of a (possibly) cyclic tree \\spad{l}.")) (|cyclic?| (((|Boolean|) $) "\\spad{cyclic?(t)} tests if \\spad{t} is a cyclic tree.")) (|tree| (($ |#1|) "\\spad{tree(nd)} creates a tree with value \\spad{nd},{} and no children") (($ (|List| |#1|)) "\\spad{tree(ls)} creates a tree from a list of elements of \\spad{s}.") (($ |#1| (|List| $)) "\\spad{tree(nd,ls)} creates a tree with value \\spad{nd},{} and children \\spad{ls}."))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1007))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-72)))) -(-1108 S) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (|HasCategory| |#1| (QUOTE (-1012))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-72)))) +(-1115 S) ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) NIL NIL -(-1109) +(-1116) ((|constructor| (NIL "Category for the trigonometric functions.")) (|tan| (($ $) "\\spad{tan(x)} returns the tangent of \\spad{x}.")) (|sin| (($ $) "\\spad{sin(x)} returns the sine of \\spad{x}.")) (|sec| (($ $) "\\spad{sec(x)} returns the secant of \\spad{x}.")) (|csc| (($ $) "\\spad{csc(x)} returns the cosecant of \\spad{x}.")) (|cot| (($ $) "\\spad{cot(x)} returns the cotangent of \\spad{x}.")) (|cos| (($ $) "\\spad{cos(x)} returns the cosine of \\spad{x}."))) NIL NIL -(-1110 R -3078) +(-1117 R -3088) ((|constructor| (NIL "\\spadtype{TrigonometricManipulations} provides transformations from trigonometric functions to complex exponentials and logarithms,{} and back.")) (|complexForm| (((|Complex| |#2|) |#2|) "\\spad{complexForm(f)} returns \\spad{[real f, imag f]}.")) (|real?| (((|Boolean|) |#2|) "\\spad{real?(f)} returns \\spad{true} if \\spad{f = real f}.")) (|imag| ((|#2| |#2|) "\\spad{imag(f)} returns the imaginary part of \\spad{f} where \\spad{f} is a complex function.")) (|real| ((|#2| |#2|) "\\spad{real(f)} returns the real part of \\spad{f} where \\spad{f} is a complex function.")) (|trigs| ((|#2| |#2|) "\\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \\spad{f} in terms of trigonometric functions.")) (|complexElementary| ((|#2| |#2| (|Symbol|)) "\\spad{complexElementary(f, x)} rewrites the kernels of \\spad{f} involving \\spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.") ((|#2| |#2|) "\\spad{complexElementary(f)} rewrites \\spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \\spad{log, exp}.")) (|complexNormalize| ((|#2| |#2| (|Symbol|)) "\\spad{complexNormalize(f, x)} rewrites \\spad{f} using the least possible number of complex independent kernels involving \\spad{x}.") ((|#2| |#2|) "\\spad{complexNormalize(f)} rewrites \\spad{f} using the least possible number of complex independent kernels."))) NIL NIL -(-1111 R |Row| |Col| M) +(-1118 R |Row| |Col| M) ((|constructor| (NIL "This package provides functions that compute \"fraction-free\" inverses of upper and lower triangular matrices over a integral domain. By \"fraction-free inverses\" we mean the following: given a matrix \\spad{B} with entries in \\spad{R} and an element \\spad{d} of \\spad{R} such that \\spad{d} * inv(\\spad{B}) also has entries in \\spad{R},{} we return \\spad{d} * inv(\\spad{B}). Thus,{} it is not necessary to pass to the quotient field in any of our computations.")) (|LowTriBddDenomInv| ((|#4| |#4| |#1|) "\\spad{LowTriBddDenomInv(B,d)} returns \\spad{M},{} where \\spad{B} is a non-singular lower triangular matrix and \\spad{d} is an element of \\spad{R} such that \\spad{M = d * inv(B)} has entries in \\spad{R}.")) (|UpTriBddDenomInv| ((|#4| |#4| |#1|) "\\spad{UpTriBddDenomInv(B,d)} returns \\spad{M},{} where \\spad{B} is a non-singular upper triangular matrix and \\spad{d} is an element of \\spad{R} such that \\spad{M = d * inv(B)} has entries in \\spad{R}."))) NIL NIL -(-1112 R -3078) +(-1119 R -3088) ((|constructor| (NIL "TranscendentalManipulations provides functions to simplify and expand expressions involving transcendental operators.")) (|expandTrigProducts| ((|#2| |#2|) "\\spad{expandTrigProducts(e)} replaces \\axiom{sin(\\spad{x})*sin(\\spad{y})} by \\spad{(cos(x-y)-cos(x+y))/2},{} \\axiom{cos(\\spad{x})*cos(\\spad{y})} by \\spad{(cos(x-y)+cos(x+y))/2},{} and \\axiom{sin(\\spad{x})*cos(\\spad{y})} by \\spad{(sin(x-y)+sin(x+y))/2}. Note that this operation uses the pattern matcher and so is relatively expensive. To avoid getting into an infinite loop the transformations are applied at most ten times.")) (|removeSinhSq| ((|#2| |#2|) "\\spad{removeSinhSq(f)} converts every \\spad{sinh(u)**2} appearing in \\spad{f} into \\spad{1 - cosh(x)**2},{} and also reduces higher powers of \\spad{sinh(u)} with that formula.")) (|removeCoshSq| ((|#2| |#2|) "\\spad{removeCoshSq(f)} converts every \\spad{cosh(u)**2} appearing in \\spad{f} into \\spad{1 - sinh(x)**2},{} and also reduces higher powers of \\spad{cosh(u)} with that formula.")) (|removeSinSq| ((|#2| |#2|) "\\spad{removeSinSq(f)} converts every \\spad{sin(u)**2} appearing in \\spad{f} into \\spad{1 - cos(x)**2},{} and also reduces higher powers of \\spad{sin(u)} with that formula.")) (|removeCosSq| ((|#2| |#2|) "\\spad{removeCosSq(f)} converts every \\spad{cos(u)**2} appearing in \\spad{f} into \\spad{1 - sin(x)**2},{} and also reduces higher powers of \\spad{cos(u)} with that formula.")) (|coth2tanh| ((|#2| |#2|) "\\spad{coth2tanh(f)} converts every \\spad{coth(u)} appearing in \\spad{f} into \\spad{1/tanh(u)}.")) (|cot2tan| ((|#2| |#2|) "\\spad{cot2tan(f)} converts every \\spad{cot(u)} appearing in \\spad{f} into \\spad{1/tan(u)}.")) (|tanh2coth| ((|#2| |#2|) "\\spad{tanh2coth(f)} converts every \\spad{tanh(u)} appearing in \\spad{f} into \\spad{1/coth(u)}.")) (|tan2cot| ((|#2| |#2|) "\\spad{tan2cot(f)} converts every \\spad{tan(u)} appearing in \\spad{f} into \\spad{1/cot(u)}.")) (|tanh2trigh| ((|#2| |#2|) "\\spad{tanh2trigh(f)} converts every \\spad{tanh(u)} appearing in \\spad{f} into \\spad{sinh(u)/cosh(u)}.")) (|tan2trig| ((|#2| |#2|) "\\spad{tan2trig(f)} converts every \\spad{tan(u)} appearing in \\spad{f} into \\spad{sin(u)/cos(u)}.")) (|sinh2csch| ((|#2| |#2|) "\\spad{sinh2csch(f)} converts every \\spad{sinh(u)} appearing in \\spad{f} into \\spad{1/csch(u)}.")) (|sin2csc| ((|#2| |#2|) "\\spad{sin2csc(f)} converts every \\spad{sin(u)} appearing in \\spad{f} into \\spad{1/csc(u)}.")) (|sech2cosh| ((|#2| |#2|) "\\spad{sech2cosh(f)} converts every \\spad{sech(u)} appearing in \\spad{f} into \\spad{1/cosh(u)}.")) (|sec2cos| ((|#2| |#2|) "\\spad{sec2cos(f)} converts every \\spad{sec(u)} appearing in \\spad{f} into \\spad{1/cos(u)}.")) (|csch2sinh| ((|#2| |#2|) "\\spad{csch2sinh(f)} converts every \\spad{csch(u)} appearing in \\spad{f} into \\spad{1/sinh(u)}.")) (|csc2sin| ((|#2| |#2|) "\\spad{csc2sin(f)} converts every \\spad{csc(u)} appearing in \\spad{f} into \\spad{1/sin(u)}.")) (|coth2trigh| ((|#2| |#2|) "\\spad{coth2trigh(f)} converts every \\spad{coth(u)} appearing in \\spad{f} into \\spad{cosh(u)/sinh(u)}.")) (|cot2trig| ((|#2| |#2|) "\\spad{cot2trig(f)} converts every \\spad{cot(u)} appearing in \\spad{f} into \\spad{cos(u)/sin(u)}.")) (|cosh2sech| ((|#2| |#2|) "\\spad{cosh2sech(f)} converts every \\spad{cosh(u)} appearing in \\spad{f} into \\spad{1/sech(u)}.")) (|cos2sec| ((|#2| |#2|) "\\spad{cos2sec(f)} converts every \\spad{cos(u)} appearing in \\spad{f} into \\spad{1/sec(u)}.")) (|expandLog| ((|#2| |#2|) "\\spad{expandLog(f)} converts every \\spad{log(a/b)} appearing in \\spad{f} into \\spad{log(a) - log(b)},{} and every \\spad{log(a*b)} into \\spad{log(a) + log(b)}..")) (|expandPower| ((|#2| |#2|) "\\spad{expandPower(f)} converts every power \\spad{(a/b)**c} appearing in \\spad{f} into \\spad{a**c * b**(-c)}.")) (|simplifyLog| ((|#2| |#2|) "\\spad{simplifyLog(f)} converts every \\spad{log(a) - log(b)} appearing in \\spad{f} into \\spad{log(a/b)},{} every \\spad{log(a) + log(b)} into \\spad{log(a*b)} and every \\spad{n*log(a)} into \\spad{log(a^n)}.")) (|simplifyExp| ((|#2| |#2|) "\\spad{simplifyExp(f)} converts every product \\spad{exp(a)*exp(b)} appearing in \\spad{f} into \\spad{exp(a+b)}.")) (|htrigs| ((|#2| |#2|) "\\spad{htrigs(f)} converts all the exponentials in \\spad{f} into hyperbolic sines and cosines.")) (|simplify| ((|#2| |#2|) "\\spad{simplify(f)} performs the following simplifications on f:\\begin{items} \\item 1. rewrites trigs and hyperbolic trigs in terms of \\spad{sin} ,{}\\spad{cos},{} \\spad{sinh},{} \\spad{cosh}. \\item 2. rewrites \\spad{sin**2} and \\spad{sinh**2} in terms of \\spad{cos} and \\spad{cosh},{} \\item 3. rewrites \\spad{exp(a)*exp(b)} as \\spad{exp(a+b)}. \\item 4. rewrites \\spad{(a**(1/n))**m * (a**(1/s))**t} as a single power of a single radical of \\spad{a}. \\end{items}")) (|expand| ((|#2| |#2|) "\\spad{expand(f)} performs the following expansions on f:\\begin{items} \\item 1. logs of products are expanded into sums of logs,{} \\item 2. trigonometric and hyperbolic trigonometric functions of sums are expanded into sums of products of trigonometric and hyperbolic trigonometric functions. \\item 3. formal powers of the form \\spad{(a/b)**c} are expanded into \\spad{a**c * b**(-c)}. \\end{items}"))) NIL -((-12 (|HasCategory| |#1| (|%list| (QUOTE -550) (|%list| (QUOTE -795) (|devaluate| |#1|)))) (|HasCategory| |#1| (|%list| (QUOTE -791) (|devaluate| |#1|))) (|HasCategory| |#2| (|%list| (QUOTE -550) (|%list| (QUOTE -795) (|devaluate| |#1|)))) (|HasCategory| |#2| (|%list| (QUOTE -791) (|devaluate| |#1|))))) -(-1113 |Coef|) +((-12 (|HasCategory| |#1| (|%list| (QUOTE -553) (|%list| (QUOTE -800) (|devaluate| |#1|)))) (|HasCategory| |#1| (|%list| (QUOTE -796) (|devaluate| |#1|))) (|HasCategory| |#2| (|%list| (QUOTE -553) (|%list| (QUOTE -800) (|devaluate| |#1|)))) (|HasCategory| |#2| (|%list| (QUOTE -796) (|devaluate| |#1|))))) +(-1120 |Coef|) ((|constructor| (NIL "\\spadtype{TaylorSeries} is a general multivariate Taylor series domain over the ring Coef and with variables of type Symbol.")) (|fintegrate| (($ (|Mapping| $) (|Symbol|) |#1|) "\\spad{fintegrate(f,v,c)} is the integral of \\spad{f()} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.} \\indented{1}{The evaluation of \\spad{f()} is delayed.}")) (|integrate| (($ $ (|Symbol|) |#1|) "\\spad{integrate(s,v,c)} is the integral of \\spad{s} with respect \\indented{1}{to \\spad{v} and having \\spad{c} as the constant of integration.}")) (|coerce| (($ (|Polynomial| |#1|)) "\\spad{coerce(s)} regroups terms of \\spad{s} by total degree \\indented{1}{and forms a series.}") (($ (|Symbol|)) "\\spad{coerce(s)} converts a variable to a Taylor series")) (|coefficient| (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{coefficient(s, n)} gives the terms of total degree \\spad{n}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-116))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-309)))) -(-1114 S R E V P) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-120))) (|HasCategory| |#1| (QUOTE (-118))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-311)))) +(-1121 S R E V P) ((|constructor| (NIL "The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \\axiom{\\spad{R}} be an integral domain and \\axiom{\\spad{V}} a finite ordered set of variables,{} say \\axiom{\\spad{X1} < \\spad{X2} < ... < Xn}. A set \\axiom{\\spad{S}} of polynomials in \\axiom{\\spad{R}[\\spad{X1},{}\\spad{X2},{}...,{}Xn]} is triangular if no elements of \\axiom{\\spad{S}} lies in \\axiom{\\spad{R}},{} and if two distinct elements of \\axiom{\\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a non-constant polynomial \\axiom{\\spad{Q}} if the degree of \\axiom{\\spad{P}} in the main variable of \\axiom{\\spad{Q}} is less than the main degree of \\axiom{\\spad{Q}}. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a triangular set \\axiom{\\spad{T}} if it is reduced \\spad{w}.\\spad{r}.\\spad{t}. every polynomial of \\axiom{\\spad{T}}. \\newline References : \\indented{1}{[1] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)}")) (|coHeight| (((|NonNegativeInteger|) $) "\\axiom{coHeight(ts)} returns \\axiom{size()\\$\\spad{V}} minus \\axiom{\\#ts}.")) (|extend| (($ $ |#5|) "\\axiom{extend(ts,{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{ts},{} according to the properties of triangular sets of the current category If the required properties do not hold an error is returned.")) (|extendIfCan| (((|Union| $ "failed") $ |#5|) "\\axiom{extendIfCan(ts,{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{ts},{} according to the properties of triangular sets of the current domain. If the required properties do not hold then \"failed\" is returned. This operation encodes in some sense the properties of the triangular sets of the current category. Is is used to implement the \\axiom{construct} operation to guarantee that every triangular set build from a list of polynomials has the required properties.")) (|select| (((|Union| |#5| "failed") $ |#4|) "\\axiom{select(ts,{}\\spad{v})} returns the polynomial of \\axiom{ts} with \\axiom{\\spad{v}} as main variable,{} if any.")) (|algebraic?| (((|Boolean|) |#4| $) "\\axiom{algebraic?(\\spad{v},{}ts)} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{ts}.")) (|algebraicVariables| (((|List| |#4|) $) "\\axiom{algebraicVariables(ts)} returns the decreasingly sorted list of the main variables of the polynomials of \\axiom{ts}.")) (|rest| (((|Union| $ "failed") $) "\\axiom{rest(ts)} returns the polynomials of \\axiom{ts} with smaller main variable than \\axiom{mvar(ts)} if \\axiom{ts} is not empty,{} otherwise returns \"failed\"")) (|last| (((|Union| |#5| "failed") $) "\\axiom{last(ts)} returns the polynomial of \\axiom{ts} with smallest main variable if \\axiom{ts} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|first| (((|Union| |#5| "failed") $) "\\axiom{first(ts)} returns the polynomial of \\axiom{ts} with greatest main variable if \\axiom{ts} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|zeroSetSplitIntoTriangularSystems| (((|List| (|Record| (|:| |close| $) (|:| |open| (|List| |#5|)))) (|List| |#5|)) "\\axiom{zeroSetSplitIntoTriangularSystems(lp)} returns a list of triangular systems \\axiom{[[\\spad{ts1},{}\\spad{qs1}],{}...,{}[tsn,{}qsn]]} such that the zero set of \\axiom{lp} is the union of the closures of the \\axiom{W_i} where \\axiom{W_i} consists of the zeros of \\axiom{ts} which do not cancel any polynomial in \\axiom{qsi}.")) (|zeroSetSplit| (((|List| $) (|List| |#5|)) "\\axiom{zeroSetSplit(lp)} returns a list \\axiom{lts} of triangular sets such that the zero set of \\axiom{lp} is the union of the closures of the regular zero sets of the members of \\axiom{lts}.")) (|reduceByQuasiMonic| ((|#5| |#5| $) "\\axiom{reduceByQuasiMonic(\\spad{p},{}ts)} returns the same as \\axiom{remainder(\\spad{p},{}collectQuasiMonic(ts)).polnum}.")) (|collectQuasiMonic| (($ $) "\\axiom{collectQuasiMonic(ts)} returns the subset of \\axiom{ts} consisting of the polynomials with initial in \\axiom{\\spad{R}}.")) (|removeZero| ((|#5| |#5| $) "\\axiom{removeZero(\\spad{p},{}ts)} returns \\axiom{0} if \\axiom{\\spad{p}} reduces to \\axiom{0} by pseudo-division \\spad{w}.\\spad{r}.\\spad{t} \\axiom{ts} otherwise returns a polynomial \\axiom{\\spad{q}} computed from \\axiom{\\spad{p}} by removing any coefficient in \\axiom{\\spad{p}} reducing to \\axiom{0}.")) (|initiallyReduce| ((|#5| |#5| $) "\\axiom{initiallyReduce(\\spad{p},{}ts)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}ts)} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}.")) (|headReduce| ((|#5| |#5| $) "\\axiom{headReduce(\\spad{p},{}ts)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduce?(\\spad{r},{}ts)} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}.")) (|stronglyReduce| ((|#5| |#5| $) "\\axiom{stronglyReduce(\\spad{p},{}ts)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{stronglyReduced?(\\spad{r},{}ts)} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}.")) (|rewriteSetWithReduction| (((|List| |#5|) (|List| |#5|) $ (|Mapping| |#5| |#5| |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{rewriteSetWithReduction(lp,{}ts,{}redOp,{}redOp?)} returns a list \\axiom{lq} of polynomials such that \\axiom{[reduce(\\spad{p},{}ts,{}redOp,{}redOp?) for \\spad{p} in lp]} and \\axiom{lp} have the same zeros inside the regular zero set of \\axiom{ts}. Moreover,{} for every polynomial \\axiom{\\spad{q}} in \\axiom{lq} and every polynomial \\axiom{\\spad{t}} in \\axiom{ts} \\axiom{redOp?(\\spad{q},{}\\spad{t})} holds and there exists a polynomial \\axiom{\\spad{p}} in the ideal generated by \\axiom{lp} and a product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = f*q + redOp(\\spad{p},{}\\spad{q})}.")) (|reduce| ((|#5| |#5| $ (|Mapping| |#5| |#5| |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{reduce(\\spad{p},{}ts,{}redOp,{}redOp?)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{redOp?(\\spad{r},{}\\spad{p})} holds for every \\axiom{\\spad{p}} of \\axiom{ts} and there exists some product \\axiom{\\spad{h}} of the initials of the members of \\axiom{ts} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = f*q + redOp(\\spad{p},{}\\spad{q})}.")) (|autoReduced?| (((|Boolean|) $ (|Mapping| (|Boolean|) |#5| (|List| |#5|))) "\\axiom{autoReduced?(ts,{}redOp?)} returns \\spad{true} iff every element of \\axiom{ts} is reduced \\spad{w}.\\spad{r}.\\spad{t} to every other in the sense of \\axiom{redOp?}")) (|initiallyReduced?| (((|Boolean|) $) "\\spad{initiallyReduced?(ts)} returns \\spad{true} iff for every element \\axiom{\\spad{p}} of \\axiom{\\spad{ts}} \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the other elements of \\axiom{\\spad{ts}} with the same main variable.") (((|Boolean|) |#5| $) "\\axiom{initiallyReduced?(\\spad{p},{}ts)} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the elements of \\axiom{ts} with the same main variable.")) (|headReduced?| (((|Boolean|) $) "\\spad{headReduced?(ts)} returns \\spad{true} iff the head of every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{\\spad{ts}}.") (((|Boolean|) |#5| $) "\\axiom{headReduced?(\\spad{p},{}ts)} returns \\spad{true} iff the head of \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ts}.")) (|stronglyReduced?| (((|Boolean|) $) "\\axiom{stronglyReduced?(ts)} returns \\spad{true} iff every element of \\axiom{ts} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{ts}.") (((|Boolean|) |#5| $) "\\axiom{stronglyReduced?(\\spad{p},{}ts)} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ts}.")) (|reduced?| (((|Boolean|) |#5| $ (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{reduced?(\\spad{p},{}ts,{}redOp?)} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. in the sense of the operation \\axiom{redOp?},{} that is if for every \\axiom{\\spad{t}} in \\axiom{ts} \\axiom{redOp?(\\spad{p},{}\\spad{t})} holds.")) (|normalized?| (((|Boolean|) $) "\\axiom{normalized?(ts)} returns \\spad{true} iff for every axiom{\\spad{p}} in axiom{ts} we have \\axiom{normalized?(\\spad{p},{}us)} where \\axiom{us} is \\axiom{collectUnder(ts,{}mvar(\\spad{p}))}.") (((|Boolean|) |#5| $) "\\axiom{normalized?(\\spad{p},{}ts)} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variables of the polynomials of \\axiom{ts}")) (|quasiComponent| (((|Record| (|:| |close| (|List| |#5|)) (|:| |open| (|List| |#5|))) $) "\\axiom{quasiComponent(ts)} returns \\axiom{[lp,{}lq]} where \\axiom{lp} is the list of the members of \\axiom{ts} and \\axiom{lq}is \\axiom{initials(ts)}.")) (|degree| (((|NonNegativeInteger|) $) "\\axiom{degree(ts)} returns the product of main degrees of the members of \\axiom{ts}.")) (|initials| (((|List| |#5|) $) "\\axiom{initials(ts)} returns the list of the non-constant initials of the members of \\axiom{ts}.")) (|basicSet| (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#5|))) "failed") (|List| |#5|) (|Mapping| (|Boolean|) |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{basicSet(ps,{}pred?,{}redOp?)} returns the same as \\axiom{basicSet(qs,{}redOp?)} where \\axiom{qs} consists of the polynomials of \\axiom{ps} satisfying property \\axiom{pred?}.") (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#5|))) "failed") (|List| |#5|) (|Mapping| (|Boolean|) |#5| |#5|)) "\\axiom{basicSet(ps,{}redOp?)} returns \\axiom{[bs,{}ts]} where \\axiom{concat(bs,{}ts)} is \\axiom{ps} and \\axiom{bs} is a basic set in Wu Wen Tsun sense of \\axiom{ps} \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?},{} if no non-zero constant polynomial lie in \\axiom{ps},{} otherwise \\axiom{\"failed\"} is returned.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(\\spad{ts1},{}\\spad{ts2})} returns \\spad{true} iff \\axiom{\\spad{ts2}} has higher rank than \\axiom{\\spad{ts1}} in Wu Wen Tsun sense."))) NIL -((|HasCategory| |#4| (QUOTE (-315)))) -(-1115 R E V P) +((|HasCategory| |#4| (QUOTE (-317)))) +(-1122 R E V P) ((|constructor| (NIL "The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \\axiom{\\spad{R}} be an integral domain and \\axiom{\\spad{V}} a finite ordered set of variables,{} say \\axiom{\\spad{X1} < \\spad{X2} < ... < Xn}. A set \\axiom{\\spad{S}} of polynomials in \\axiom{\\spad{R}[\\spad{X1},{}\\spad{X2},{}...,{}Xn]} is triangular if no elements of \\axiom{\\spad{S}} lies in \\axiom{\\spad{R}},{} and if two distinct elements of \\axiom{\\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a non-constant polynomial \\axiom{\\spad{Q}} if the degree of \\axiom{\\spad{P}} in the main variable of \\axiom{\\spad{Q}} is less than the main degree of \\axiom{\\spad{Q}}. A polynomial \\axiom{\\spad{P}} is reduced \\spad{w}.\\spad{r}.\\spad{t} a triangular set \\axiom{\\spad{T}} if it is reduced \\spad{w}.\\spad{r}.\\spad{t}. every polynomial of \\axiom{\\spad{T}}. \\newline References : \\indented{1}{[1] \\spad{P}. AUBRY,{} \\spad{D}. LAZARD and \\spad{M}. MORENO MAZA \"On the Theories} \\indented{5}{of Triangular Sets\" Journal of Symbol. Comp. (to appear)}")) (|coHeight| (((|NonNegativeInteger|) $) "\\axiom{coHeight(ts)} returns \\axiom{size()\\$\\spad{V}} minus \\axiom{\\#ts}.")) (|extend| (($ $ |#4|) "\\axiom{extend(ts,{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{ts},{} according to the properties of triangular sets of the current category If the required properties do not hold an error is returned.")) (|extendIfCan| (((|Union| $ "failed") $ |#4|) "\\axiom{extendIfCan(ts,{}\\spad{p})} returns a triangular set which encodes the simple extension by \\axiom{\\spad{p}} of the extension of the base field defined by \\axiom{ts},{} according to the properties of triangular sets of the current domain. If the required properties do not hold then \"failed\" is returned. This operation encodes in some sense the properties of the triangular sets of the current category. Is is used to implement the \\axiom{construct} operation to guarantee that every triangular set build from a list of polynomials has the required properties.")) (|select| (((|Union| |#4| "failed") $ |#3|) "\\axiom{select(ts,{}\\spad{v})} returns the polynomial of \\axiom{ts} with \\axiom{\\spad{v}} as main variable,{} if any.")) (|algebraic?| (((|Boolean|) |#3| $) "\\axiom{algebraic?(\\spad{v},{}ts)} returns \\spad{true} iff \\axiom{\\spad{v}} is the main variable of some polynomial in \\axiom{ts}.")) (|algebraicVariables| (((|List| |#3|) $) "\\axiom{algebraicVariables(ts)} returns the decreasingly sorted list of the main variables of the polynomials of \\axiom{ts}.")) (|rest| (((|Union| $ "failed") $) "\\axiom{rest(ts)} returns the polynomials of \\axiom{ts} with smaller main variable than \\axiom{mvar(ts)} if \\axiom{ts} is not empty,{} otherwise returns \"failed\"")) (|last| (((|Union| |#4| "failed") $) "\\axiom{last(ts)} returns the polynomial of \\axiom{ts} with smallest main variable if \\axiom{ts} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|first| (((|Union| |#4| "failed") $) "\\axiom{first(ts)} returns the polynomial of \\axiom{ts} with greatest main variable if \\axiom{ts} is not empty,{} otherwise returns \\axiom{\"failed\"}.")) (|zeroSetSplitIntoTriangularSystems| (((|List| (|Record| (|:| |close| $) (|:| |open| (|List| |#4|)))) (|List| |#4|)) "\\axiom{zeroSetSplitIntoTriangularSystems(lp)} returns a list of triangular systems \\axiom{[[\\spad{ts1},{}\\spad{qs1}],{}...,{}[tsn,{}qsn]]} such that the zero set of \\axiom{lp} is the union of the closures of the \\axiom{W_i} where \\axiom{W_i} consists of the zeros of \\axiom{ts} which do not cancel any polynomial in \\axiom{qsi}.")) (|zeroSetSplit| (((|List| $) (|List| |#4|)) "\\axiom{zeroSetSplit(lp)} returns a list \\axiom{lts} of triangular sets such that the zero set of \\axiom{lp} is the union of the closures of the regular zero sets of the members of \\axiom{lts}.")) (|reduceByQuasiMonic| ((|#4| |#4| $) "\\axiom{reduceByQuasiMonic(\\spad{p},{}ts)} returns the same as \\axiom{remainder(\\spad{p},{}collectQuasiMonic(ts)).polnum}.")) (|collectQuasiMonic| (($ $) "\\axiom{collectQuasiMonic(ts)} returns the subset of \\axiom{ts} consisting of the polynomials with initial in \\axiom{\\spad{R}}.")) (|removeZero| ((|#4| |#4| $) "\\axiom{removeZero(\\spad{p},{}ts)} returns \\axiom{0} if \\axiom{\\spad{p}} reduces to \\axiom{0} by pseudo-division \\spad{w}.\\spad{r}.\\spad{t} \\axiom{ts} otherwise returns a polynomial \\axiom{\\spad{q}} computed from \\axiom{\\spad{p}} by removing any coefficient in \\axiom{\\spad{p}} reducing to \\axiom{0}.")) (|initiallyReduce| ((|#4| |#4| $) "\\axiom{initiallyReduce(\\spad{p},{}ts)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{initiallyReduced?(\\spad{r},{}ts)} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}.")) (|headReduce| ((|#4| |#4| $) "\\axiom{headReduce(\\spad{p},{}ts)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{headReduce?(\\spad{r},{}ts)} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}.")) (|stronglyReduce| ((|#4| |#4| $) "\\axiom{stronglyReduce(\\spad{p},{}ts)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{stronglyReduced?(\\spad{r},{}ts)} holds and there exists some product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}.")) (|rewriteSetWithReduction| (((|List| |#4|) (|List| |#4|) $ (|Mapping| |#4| |#4| |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{rewriteSetWithReduction(lp,{}ts,{}redOp,{}redOp?)} returns a list \\axiom{lq} of polynomials such that \\axiom{[reduce(\\spad{p},{}ts,{}redOp,{}redOp?) for \\spad{p} in lp]} and \\axiom{lp} have the same zeros inside the regular zero set of \\axiom{ts}. Moreover,{} for every polynomial \\axiom{\\spad{q}} in \\axiom{lq} and every polynomial \\axiom{\\spad{t}} in \\axiom{ts} \\axiom{redOp?(\\spad{q},{}\\spad{t})} holds and there exists a polynomial \\axiom{\\spad{p}} in the ideal generated by \\axiom{lp} and a product \\axiom{\\spad{h}} of \\axiom{initials(ts)} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = f*q + redOp(\\spad{p},{}\\spad{q})}.")) (|reduce| ((|#4| |#4| $ (|Mapping| |#4| |#4| |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{reduce(\\spad{p},{}ts,{}redOp,{}redOp?)} returns a polynomial \\axiom{\\spad{r}} such that \\axiom{redOp?(\\spad{r},{}\\spad{p})} holds for every \\axiom{\\spad{p}} of \\axiom{ts} and there exists some product \\axiom{\\spad{h}} of the initials of the members of \\axiom{ts} such that \\axiom{h*p - \\spad{r}} lies in the ideal generated by \\axiom{ts}. The operation \\axiom{redOp} must satisfy the following conditions. For every \\axiom{\\spad{p}} and \\axiom{\\spad{q}} we have \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that \\axiom{init(\\spad{q})^e*p = f*q + redOp(\\spad{p},{}\\spad{q})}.")) (|autoReduced?| (((|Boolean|) $ (|Mapping| (|Boolean|) |#4| (|List| |#4|))) "\\axiom{autoReduced?(ts,{}redOp?)} returns \\spad{true} iff every element of \\axiom{ts} is reduced \\spad{w}.\\spad{r}.\\spad{t} to every other in the sense of \\axiom{redOp?}")) (|initiallyReduced?| (((|Boolean|) $) "\\spad{initiallyReduced?(ts)} returns \\spad{true} iff for every element \\axiom{\\spad{p}} of \\axiom{\\spad{ts}} \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the other elements of \\axiom{\\spad{ts}} with the same main variable.") (((|Boolean|) |#4| $) "\\axiom{initiallyReduced?(\\spad{p},{}ts)} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials are reduced \\spad{w}.\\spad{r}.\\spad{t}. to the elements of \\axiom{ts} with the same main variable.")) (|headReduced?| (((|Boolean|) $) "\\spad{headReduced?(ts)} returns \\spad{true} iff the head of every element of \\axiom{\\spad{ts}} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{\\spad{ts}}.") (((|Boolean|) |#4| $) "\\axiom{headReduced?(\\spad{p},{}ts)} returns \\spad{true} iff the head of \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ts}.")) (|stronglyReduced?| (((|Boolean|) $) "\\axiom{stronglyReduced?(ts)} returns \\spad{true} iff every element of \\axiom{ts} is reduced \\spad{w}.\\spad{r}.\\spad{t} to any other element of \\axiom{ts}.") (((|Boolean|) |#4| $) "\\axiom{stronglyReduced?(\\spad{p},{}ts)} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. \\axiom{ts}.")) (|reduced?| (((|Boolean|) |#4| $ (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{reduced?(\\spad{p},{}ts,{}redOp?)} returns \\spad{true} iff \\axiom{\\spad{p}} is reduced \\spad{w}.\\spad{r}.\\spad{t}. in the sense of the operation \\axiom{redOp?},{} that is if for every \\axiom{\\spad{t}} in \\axiom{ts} \\axiom{redOp?(\\spad{p},{}\\spad{t})} holds.")) (|normalized?| (((|Boolean|) $) "\\axiom{normalized?(ts)} returns \\spad{true} iff for every axiom{\\spad{p}} in axiom{ts} we have \\axiom{normalized?(\\spad{p},{}us)} where \\axiom{us} is \\axiom{collectUnder(ts,{}mvar(\\spad{p}))}.") (((|Boolean|) |#4| $) "\\axiom{normalized?(\\spad{p},{}ts)} returns \\spad{true} iff \\axiom{\\spad{p}} and all its iterated initials have degree zero \\spad{w}.\\spad{r}.\\spad{t}. the main variables of the polynomials of \\axiom{ts}")) (|quasiComponent| (((|Record| (|:| |close| (|List| |#4|)) (|:| |open| (|List| |#4|))) $) "\\axiom{quasiComponent(ts)} returns \\axiom{[lp,{}lq]} where \\axiom{lp} is the list of the members of \\axiom{ts} and \\axiom{lq}is \\axiom{initials(ts)}.")) (|degree| (((|NonNegativeInteger|) $) "\\axiom{degree(ts)} returns the product of main degrees of the members of \\axiom{ts}.")) (|initials| (((|List| |#4|) $) "\\axiom{initials(ts)} returns the list of the non-constant initials of the members of \\axiom{ts}.")) (|basicSet| (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#4|))) "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{basicSet(ps,{}pred?,{}redOp?)} returns the same as \\axiom{basicSet(qs,{}redOp?)} where \\axiom{qs} consists of the polynomials of \\axiom{ps} satisfying property \\axiom{pred?}.") (((|Union| (|Record| (|:| |bas| $) (|:| |top| (|List| |#4|))) "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|)) "\\axiom{basicSet(ps,{}redOp?)} returns \\axiom{[bs,{}ts]} where \\axiom{concat(bs,{}ts)} is \\axiom{ps} and \\axiom{bs} is a basic set in Wu Wen Tsun sense of \\axiom{ps} \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?},{} if no non-zero constant polynomial lie in \\axiom{ps},{} otherwise \\axiom{\"failed\"} is returned.")) (|infRittWu?| (((|Boolean|) $ $) "\\axiom{infRittWu?(\\spad{ts1},{}\\spad{ts2})} returns \\spad{true} iff \\axiom{\\spad{ts2}} has higher rank than \\axiom{\\spad{ts1}} in Wu Wen Tsun sense."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-1116 |Curve|) +(-1123 |Curve|) ((|constructor| (NIL "\\indented{2}{Package for constructing tubes around 3-dimensional parametric curves.} Domain of tubes around 3-dimensional parametric curves.")) (|tube| (($ |#1| (|List| (|List| (|Point| (|DoubleFloat|)))) (|Boolean|)) "\\spad{tube(c,ll,b)} creates a tube of the domain \\spadtype{TubePlot} from a space curve \\spad{c} of the category \\spadtype{PlottableSpaceCurveCategory},{} a list of lists of points (loops) \\spad{ll} and a boolean \\spad{b} which if \\spad{true} indicates a closed tube,{} or if \\spad{false} an open tube.")) (|setClosed| (((|Boolean|) $ (|Boolean|)) "\\spad{setClosed(t,b)} declares the given tube plot \\spad{t} to be closed if \\spad{b} is \\spad{true},{} or if \\spad{b} is \\spad{false},{} \\spad{t} is set to be open.")) (|open?| (((|Boolean|) $) "\\spad{open?(t)} tests whether the given tube plot \\spad{t} is open.")) (|closed?| (((|Boolean|) $) "\\spad{closed?(t)} tests whether the given tube plot \\spad{t} is closed.")) (|listLoops| (((|List| (|List| (|Point| (|DoubleFloat|)))) $) "\\spad{listLoops(t)} returns the list of lists of points,{} or the 'loops',{} of the given tube plot \\spad{t}.")) (|getCurve| ((|#1| $) "\\spad{getCurve(t)} returns the \\spadtype{PlottableSpaceCurveCategory} representing the parametric curve of the given tube plot \\spad{t}."))) NIL NIL -(-1117) +(-1124) ((|constructor| (NIL "Tools for constructing tubes around 3-dimensional parametric curves.")) (|loopPoints| (((|List| (|Point| (|DoubleFloat|))) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|DoubleFloat|) (|List| (|List| (|DoubleFloat|)))) "\\spad{loopPoints(p,n,b,r,lls)} creates and returns a list of points which form the loop with radius \\spad{r},{} around the center point indicated by the point \\spad{p},{} with the principal normal vector of the space curve at point \\spad{p} given by the point(vector) \\spad{n},{} and the binormal vector given by the point(vector) \\spad{b},{} and a list of lists,{} \\spad{lls},{} which is the \\spadfun{cosSinInfo} of the number of points defining the loop.")) (|cosSinInfo| (((|List| (|List| (|DoubleFloat|))) (|Integer|)) "\\spad{cosSinInfo(n)} returns the list of lists of values for \\spad{n},{} in the form: \\spad{[[cos(n - 1) a,sin(n - 1) a],...,[cos 2 a,sin 2 a],[cos a,sin a]]} where \\spad{a = 2 pi/n}. Note: \\spad{n} should be greater than 2.")) (|unitVector| (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{unitVector(p)} creates the unit vector of the point \\spad{p} and returns the result as a point. Note: \\spad{unitVector(p) = p/|p|}.")) (|cross| (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{cross(p,q)} computes the cross product of the two points \\spad{p} and \\spad{q} using only the first three coordinates,{} and keeping the color of the first point \\spad{p}. The result is returned as a point.")) (|dot| (((|DoubleFloat|) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{dot(p,q)} computes the dot product of the two points \\spad{p} and \\spad{q} using only the first three coordinates,{} and returns the resulting \\spadtype{DoubleFloat}.")) (- (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{p - q} computes and returns a point whose coordinates are the differences of the coordinates of two points \\spad{p} and \\spad{q},{} using the color,{} or fourth coordinate,{} of the first point \\spad{p} as the color also of the point \\spad{q}.")) (+ (((|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|)) (|Point| (|DoubleFloat|))) "\\spad{p + q} computes and returns a point whose coordinates are the sums of the coordinates of the two points \\spad{p} and \\spad{q},{} using the color,{} or fourth coordinate,{} of the first point \\spad{p} as the color also of the point \\spad{q}.")) (* (((|Point| (|DoubleFloat|)) (|DoubleFloat|) (|Point| (|DoubleFloat|))) "\\spad{s * p} returns a point whose coordinates are the scalar multiple of the point \\spad{p} by the scalar \\spad{s},{} preserving the color,{} or fourth coordinate,{} of \\spad{p}.")) (|point| (((|Point| (|DoubleFloat|)) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|)) "\\spad{point(x1,x2,x3,c)} creates and returns a point from the three specified coordinates \\spad{x1},{} \\spad{x2},{} \\spad{x3},{} and also a fourth coordinate,{} \\spad{c},{} which is generally used to specify the color of the point."))) NIL NIL -(-1118 S) +(-1125 S) ((|constructor| (NIL "\\indented{1}{This domain is used to interface with the interpreter's notion} of comma-delimited sequences of values.")) (|length| (((|NonNegativeInteger|) $) "\\spad{length(x)} returns the number of elements in tuple \\spad{x}")) (|select| ((|#1| $ (|NonNegativeInteger|)) "\\spad{select(x,n)} returns the \\spad{n}-th element of tuple \\spad{x}. tuples are 0-based"))) NIL -((|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-549 (-767))))) -(-1119 -3078) +((|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-552 (-772))))) +(-1126 -3088) ((|constructor| (NIL "A basic package for the factorization of bivariate polynomials over a finite field. The functions here represent the base step for the multivariate factorizer.")) (|twoFactor| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|)) (|Integer|)) "\\spad{twoFactor(p,n)} returns the factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}. Also,{} \\spad{p} is assumed primitive and square-free and \\spad{n} is the degree of the inner variable of \\spad{p} (maximum of the degrees of the coefficients of \\spad{p}).")) (|generalSqFr| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) "\\spad{generalSqFr(p)} returns the square-free factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}.")) (|generalTwoFactor| (((|Factored| (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) (|SparseUnivariatePolynomial| (|SparseUnivariatePolynomial| |#1|))) "\\spad{generalTwoFactor(p)} returns the factorisation of polynomial \\spad{p},{} a sparse univariate polynomial (sup) over a sup over \\spad{F}."))) NIL NIL -(-1120) +(-1127) ((|constructor| (NIL "The fundamental Type."))) NIL NIL -(-1121) +(-1128) ((|constructor| (NIL "This domain represents a type AST."))) NIL NIL -(-1122 S) +(-1129 S) ((|constructor| (NIL "Provides functions to force a partial ordering on any set.")) (|more?| (((|Boolean|) |#1| |#1|) "\\spad{more?(a, b)} compares \\spad{a} and \\spad{b} in the partial ordering induced by setOrder,{} and uses the ordering on \\spad{S} if \\spad{a} and \\spad{b} are not comparable in the partial ordering.")) (|userOrdered?| (((|Boolean|)) "\\spad{userOrdered?()} tests if the partial ordering induced by \\spadfunFrom{setOrder}{UserDefinedPartialOrdering} is not empty.")) (|largest| ((|#1| (|List| |#1|)) "\\spad{largest l} returns the largest element of \\spad{l} where the partial ordering induced by setOrder is completed into a total one by the ordering on \\spad{S}.") ((|#1| (|List| |#1|) (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{largest(l, fn)} returns the largest element of \\spad{l} where the partial ordering induced by setOrder is completed into a total one by fn.")) (|less?| (((|Boolean|) |#1| |#1| (|Mapping| (|Boolean|) |#1| |#1|)) "\\spad{less?(a, b, fn)} compares \\spad{a} and \\spad{b} in the partial ordering induced by setOrder,{} and returns \\spad{fn(a, b)} if \\spad{a} and \\spad{b} are not comparable in that ordering.") (((|Union| (|Boolean|) "failed") |#1| |#1|) "\\spad{less?(a, b)} compares \\spad{a} and \\spad{b} in the partial ordering induced by setOrder.")) (|getOrder| (((|Record| (|:| |low| (|List| |#1|)) (|:| |high| (|List| |#1|)))) "\\spad{getOrder()} returns \\spad{[[b1,...,bm], [a1,...,an]]} such that the partial ordering on \\spad{S} was given by \\spad{setOrder([b1,...,bm],[a1,...,an])}.")) 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The ordering affects the display only and not the computations.")) (|resetVariableOrder| (((|Void|)) "\\spad{resetVariableOrder()} cancels any previous use of setVariableOrder and returns to the default system ordering.")) (|getVariableOrder| (((|Record| (|:| |high| (|List| (|Symbol|))) (|:| |low| (|List| (|Symbol|))))) "\\spad{getVariableOrder()} returns \\spad{[[b1,...,bm], [a1,...,an]]} such that the ordering on the variables was given by \\spad{setVariableOrder([b1,...,bm], [a1,...,an])}.")) (|setVariableOrder| (((|Void|) (|List| (|Symbol|)) (|List| (|Symbol|))) "\\spad{setVariableOrder([b1,...,bm], [a1,...,an])} defines an ordering on the variables given by \\spad{b1 > b2 > ... > bm >} other variables \\spad{> a1 > a2 > ... > an}.") (((|Void|) (|List| (|Symbol|))) "\\spad{setVariableOrder([a1,...,an])} defines an ordering on the variables given by \\spad{a1 > a2 > ... > an > other variables}."))) 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A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable.") (($ $) "\\spad{removeZeroes(f(x))} removes leading zeroes from the representation of the Laurent series \\spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing the Taylor series by its variable. Note: \\spad{removeZeroes(f)} removes all leading zeroes from \\spad{f}")) (|taylorRep| ((|#3| $) "\\spad{taylorRep(f(x))} returns \\spad{g(x)},{} where \\spad{f = x**n * g(x)} is represented by \\spad{[n,g(x)]}.")) (|degree| (((|Integer|) $) "\\spad{degree(f(x))} returns the degree of the lowest order term of \\spad{f(x)},{} which may have zero as a coefficient.")) (|laurent| (($ (|Integer|) |#3|) "\\spad{laurent(n,f(x))} returns \\spad{x**n * f(x)}."))) NIL -((|HasCategory| |#2| (QUOTE (-309)))) -(-1134 |Coef| UTS) +((|HasCategory| |#2| (QUOTE (-311)))) +(-1141 |Coef| UTS) ((|constructor| (NIL "This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair \\spad{[n,f(x)]},{} where \\spad{n} is an arbitrary integer and \\spad{f(x)} is a Taylor series. This pair represents the Laurent series \\spad{x**n * f(x)}.")) 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(|HasCategory| |#2| (QUOTE (-811 (-1088))))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-189)))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-118)))))) +(-1143 ZP) ((|constructor| (NIL "Package for the factorization of univariate polynomials with integer coefficients. The factorization is done by \"lifting\" (HENSEL) the factorization over a finite field.")) (|henselFact| (((|Record| (|:| |contp| (|Integer|)) (|:| |factors| (|List| (|Record| (|:| |irr| |#1|) (|:| |pow| (|Integer|)))))) |#1| (|Boolean|)) "\\spad{henselFact(m,flag)} returns the factorization of \\spad{m},{} FinalFact is a Record \\spad{s}.\\spad{t}. FinalFact.contp=content \\spad{m},{} FinalFact.factors=List of irreducible factors of \\spad{m} with exponent ,{} if \\spad{flag} =true the polynomial is assumed square free.")) (|factorSquareFree| (((|Factored| |#1|) |#1|) "\\spad{factorSquareFree(m)} returns the factorization of \\spad{m} square free polynomial")) (|factor| (((|Factored| |#1|) |#1|) "\\spad{factor(m)} returns the factorization of \\spad{m}"))) NIL NIL -(-1137 S) +(-1144 S) ((|constructor| (NIL "This domain provides segments which may be half open. That is,{} ranges of the form \\spad{a..} or \\spad{a..b}.")) (|hasHi| (((|Boolean|) $) "\\spad{hasHi(s)} tests whether the segment \\spad{s} has an upper bound.")) (|coerce| (($ (|Segment| |#1|)) "\\spad{coerce(x)} allows \\spadtype{Segment} values to be used as \\%.")) (|segment| (($ |#1|) "\\spad{segment(l)} is an alternate way to construct the segment \\spad{l..}.")) (SEGMENT (($ |#1|) "\\spad{l..} produces a half open segment,{} that is,{} one with no upper bound."))) NIL -((|HasCategory| |#1| (QUOTE (-750))) (|HasCategory| |#1| (QUOTE (-1007)))) -(-1138 R S) +((|HasCategory| |#1| (QUOTE (-755))) (|HasCategory| |#1| (QUOTE (-1012)))) +(-1145 R S) ((|constructor| (NIL "This package provides operations for mapping functions onto segments.")) (|map| (((|Stream| |#2|) (|Mapping| |#2| |#1|) (|UniversalSegment| |#1|)) "\\spad{map(f,s)} expands the segment \\spad{s},{} applying \\spad{f} to each value.") 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T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-494)))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-327)))) (|HasCategory| (-993) (QUOTE (-796 (-327))))) (-12 (|HasCategory| |#2| (QUOTE (-796 (-483)))) (|HasCategory| (-993) (QUOTE (-796 (-483))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-327))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-327)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-800 (-483))))) (|HasCategory| (-993) (QUOTE (-553 (-800 (-483)))))) (-12 (|HasCategory| |#2| (QUOTE (-553 (-472)))) (|HasCategory| (-993) (QUOTE (-553 (-472))))) (|HasCategory| |#2| (QUOTE (-580 (-483)))) (|HasCategory| |#2| (QUOTE (-120))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-483)))) (OR (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483)))))) (|HasCategory| |#2| (QUOTE (-950 (-347 (-483))))) (OR (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-821)))) (OR (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-821)))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-1064))) (|HasCategory| |#2| (QUOTE (-811 (-1088)))) (|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasCategory| |#2| (QUOTE (-189))) (|HasCategory| |#2| (QUOTE (-190))) (|HasAttribute| |#2| (QUOTE -3987)) (|HasCategory| |#2| (QUOTE (-389))) (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (OR (-12 (|HasCategory| |#2| (QUOTE (-821))) (|HasCategory| $ (QUOTE (-118)))) (|HasCategory| |#2| (QUOTE (-118))))) +(-1147 |x| R |y| S) ((|constructor| (NIL "This package lifts a mapping from coefficient rings \\spad{R} to \\spad{S} to a mapping from \\spadtype{UnivariatePolynomial}(\\spad{x},{}\\spad{R}) to \\spadtype{UnivariatePolynomial}(\\spad{y},{}\\spad{S}). Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.")) (|map| (((|UnivariatePolynomial| |#3| |#4|) (|Mapping| |#4| |#2|) (|UnivariatePolynomial| |#1| |#2|)) "\\spad{map(func, poly)} creates a new polynomial by applying \\spad{func} to every non-zero coefficient of the polynomial poly."))) NIL NIL -(-1141 R Q UP) +(-1148 R Q UP) ((|constructor| (NIL "UnivariatePolynomialCommonDenominator provides functions to compute the common denominator of the coefficients of univariate polynomials over the quotient field of a gcd domain.")) (|splitDenominator| (((|Record| (|:| |num| |#3|) (|:| |den| |#1|)) |#3|) "\\spad{splitDenominator(q)} returns \\spad{[p, d]} such that \\spad{q = p/d} and \\spad{d} is a common denominator for the coefficients of \\spad{q}.")) (|clearDenominator| ((|#3| |#3|) "\\spad{clearDenominator(q)} returns \\spad{p} such that \\spad{q = p/d} where \\spad{d} is a common denominator for the coefficients of \\spad{q}.")) (|commonDenominator| ((|#1| |#3|) "\\spad{commonDenominator(q)} returns a common denominator \\spad{d} for the coefficients of \\spad{q}."))) NIL NIL -(-1142 R UP) +(-1149 R UP) ((|constructor| (NIL "UnivariatePolynomialDecompositionPackage implements functional decomposition of univariate polynomial with coefficients in an \\spad{IntegralDomain} of \\spad{CharacteristicZero}.")) (|monicCompleteDecompose| (((|List| |#2|) |#2|) "\\spad{monicCompleteDecompose(f)} returns a list of factors of \\spad{f} for the functional decomposition ([ \\spad{f1},{} ...,{} fn ] means \\spad{f} = \\spad{f1} \\spad{o} ... \\spad{o} fn).")) (|monicDecomposeIfCan| (((|Union| (|Record| (|:| |left| |#2|) (|:| |right| |#2|)) "failed") |#2|) "\\spad{monicDecomposeIfCan(f)} returns a functional decomposition of the monic polynomial \\spad{f} of \"failed\" if it has not found any.")) (|leftFactorIfCan| (((|Union| |#2| "failed") |#2| |#2|) "\\spad{leftFactorIfCan(f,h)} returns the left factor (\\spad{g} in \\spad{f} = \\spad{g} \\spad{o} \\spad{h}) of the functional decomposition of the polynomial \\spad{f} with given \\spad{h} or \\spad{\"failed\"} if \\spad{g} does not exist.")) (|rightFactorIfCan| (((|Union| |#2| "failed") |#2| (|NonNegativeInteger|) |#1|) "\\spad{rightFactorIfCan(f,d,c)} returns a candidate to be the right factor (\\spad{h} in \\spad{f} = \\spad{g} \\spad{o} \\spad{h}) of degree \\spad{d} with leading coefficient \\spad{c} of a functional decomposition of the polynomial \\spad{f} or \\spad{\"failed\"} if no such candidate.")) (|monicRightFactorIfCan| (((|Union| |#2| "failed") |#2| (|NonNegativeInteger|)) "\\spad{monicRightFactorIfCan(f,d)} returns a candidate to be the monic right factor (\\spad{h} in \\spad{f} = \\spad{g} \\spad{o} \\spad{h}) of degree \\spad{d} of a functional decomposition of the polynomial \\spad{f} or \\spad{\"failed\"} if no such candidate."))) NIL NIL -(-1143 R UP) +(-1150 R UP) ((|constructor| (NIL "UnivariatePolynomialDivisionPackage provides a division for non monic univarite polynomials with coefficients in an \\spad{IntegralDomain}.")) (|divideIfCan| (((|Union| (|Record| (|:| |quotient| |#2|) (|:| |remainder| |#2|)) "failed") |#2| |#2|) "\\spad{divideIfCan(f,g)} returns quotient and remainder of the division of \\spad{f} by \\spad{g} or \"failed\" if it has not succeeded."))) NIL NIL -(-1144 R U) +(-1151 R U) ((|constructor| (NIL "This package implements Karatsuba's trick for multiplying (large) univariate polynomials. It could be improved with a version doing the work on place and also with a special case for squares. We've done this in Basicmath,{} but we believe that this out of the scope of AXIOM.")) (|karatsuba| ((|#2| |#2| |#2| (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{karatsuba(a,b,l,k)} returns \\spad{a*b} by applying Karatsuba's trick provided that both \\spad{a} and \\spad{b} have at least \\spad{l} terms and \\spad{k > 0} holds and by calling \\spad{noKaratsuba} otherwise. The other multiplications are performed by recursive calls with the same third argument and \\spad{k-1} as fourth argument.")) (|karatsubaOnce| ((|#2| |#2| |#2|) "\\spad{karatsuba(a,b)} returns \\spad{a*b} by applying Karatsuba's trick once. The other multiplications are performed by calling \\spad{*} from \\spad{U}.")) (|noKaratsuba| ((|#2| |#2| |#2|) "\\spad{noKaratsuba(a,b)} returns \\spad{a*b} without using Karatsuba's trick at all."))) NIL NIL -(-1145 S R) +(-1152 S R) ((|constructor| (NIL "The category of univariate polynomials over a ring \\spad{R}. No particular model is assumed - implementations can be either sparse or dense.")) (|integrate| (($ $) "\\spad{integrate(p)} integrates the univariate polynomial \\spad{p} with respect to its distinguished variable.")) (|additiveValuation| ((|attribute|) "euclideanSize(a*b) = euclideanSize(a) + euclideanSize(\\spad{b})")) (|separate| (((|Record| (|:| |primePart| $) (|:| |commonPart| $)) $ $) "\\spad{separate(p, q)} returns \\spad{[a, b]} such that polynomial \\spad{p = a b} and \\spad{a} is relatively prime to \\spad{q}.")) (|pseudoDivide| (((|Record| (|:| |coef| |#2|) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{pseudoDivide(p,q)} returns \\spad{[c, q, r]},{} when \\spad{p' := p*lc(q)**(deg p - deg q + 1) = c * p} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|pseudoQuotient| (($ $ $) "\\spad{pseudoQuotient(p,q)} returns \\spad{r},{} the quotient when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|composite| (((|Union| (|Fraction| $) "failed") (|Fraction| $) $) "\\spad{composite(f, q)} returns \\spad{h} if \\spad{f} = \\spad{h}(\\spad{q}),{} and \"failed\" is no such \\spad{h} exists.") (((|Union| $ "failed") $ $) "\\spad{composite(p, q)} returns \\spad{h} if \\spad{p = h(q)},{} and \"failed\" no such \\spad{h} exists.")) (|subResultantGcd| (($ $ $) "\\spad{subResultantGcd(p,q)} computes the gcd of the polynomials \\spad{p} and \\spad{q} using the SubResultant GCD algorithm.")) (|order| (((|NonNegativeInteger|) $ $) "\\spad{order(p, q)} returns the largest \\spad{n} such that \\spad{q**n} divides polynomial \\spad{p} \\spadignore{i.e.} the order of \\spad{p(x)} at \\spad{q(x)=0}.")) (|elt| ((|#2| (|Fraction| $) |#2|) "\\spad{elt(a,r)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by the constant \\spad{r}.") (((|Fraction| $) (|Fraction| $) (|Fraction| $)) "\\spad{elt(a,b)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by \\spad{b}.")) (|resultant| ((|#2| $ $) "\\spad{resultant(p,q)} returns the resultant of the polynomials \\spad{p} and \\spad{q}.")) (|discriminant| ((|#2| $) "\\spad{discriminant(p)} returns the discriminant of the polynomial \\spad{p}.")) (|differentiate| (($ $ (|Mapping| |#2| |#2|) $) "\\spad{differentiate(p, d, x')} extends the \\spad{R}-derivation \\spad{d} to an extension \\spad{D} in \\spad{R[x]} where Dx is given by x',{} and returns \\spad{Dp}.")) (|pseudoRemainder| (($ $ $) "\\spad{pseudoRemainder(p,q)} = \\spad{r},{} for polynomials \\spad{p} and \\spad{q},{} returns the remainder when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|shiftLeft| (($ $ (|NonNegativeInteger|)) "\\spad{shiftLeft(p,n)} returns \\spad{p * monomial(1,n)}")) (|shiftRight| (($ $ (|NonNegativeInteger|)) "\\spad{shiftRight(p,n)} returns \\spad{monicDivide(p,monomial(1,n)).quotient}")) (|karatsubaDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ (|NonNegativeInteger|)) "\\spad{karatsubaDivide(p,n)} returns the same as \\spad{monicDivide(p,monomial(1,n))}")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicDivide(p,q)} divide the polynomial \\spad{p} by the monic polynomial \\spad{q},{} returning the pair \\spad{[quotient, remainder]}. Error: if \\spad{q} isn't monic.")) (|divideExponents| (((|Union| $ "failed") $ (|NonNegativeInteger|)) "\\spad{divideExponents(p,n)} returns a new polynomial resulting from dividing all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n},{} or \"failed\" if some exponent is not exactly divisible by \\spad{n}.")) (|multiplyExponents| (($ $ (|NonNegativeInteger|)) "\\spad{multiplyExponents(p,n)} returns a new polynomial resulting from multiplying all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n}.")) (|unmakeSUP| (($ (|SparseUnivariatePolynomial| |#2|)) "\\spad{unmakeSUP(sup)} converts \\spad{sup} of type \\spadtype{SparseUnivariatePolynomial(R)} to be a member of the given type. Note: converse of makeSUP.")) (|makeSUP| (((|SparseUnivariatePolynomial| |#2|) $) "\\spad{makeSUP(p)} converts the polynomial \\spad{p} to be of type SparseUnivariatePolynomial over the same coefficients.")) (|vectorise| (((|Vector| |#2|) $ (|NonNegativeInteger|)) "\\spad{vectorise(p, n)} returns \\spad{[a0,...,a(n-1)]} where \\spad{p = a0 + a1*x + ... + a(n-1)*x**(n-1)} + higher order terms. The degree of polynomial \\spad{p} can be different from \\spad{n-1}."))) NIL -((|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-309))) (|HasCategory| |#2| (QUOTE (-387))) (|HasCategory| |#2| (QUOTE (-491))) (|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-1057)))) -(-1146 R) +((|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-311))) (|HasCategory| |#2| (QUOTE (-389))) (|HasCategory| |#2| (QUOTE (-494))) (|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-1064)))) +(-1153 R) ((|constructor| (NIL "The category of univariate polynomials over a ring \\spad{R}. No particular model is assumed - implementations can be either sparse or dense.")) (|integrate| (($ $) "\\spad{integrate(p)} integrates the univariate polynomial \\spad{p} with respect to its distinguished variable.")) (|additiveValuation| ((|attribute|) "euclideanSize(a*b) = euclideanSize(a) + euclideanSize(\\spad{b})")) (|separate| (((|Record| (|:| |primePart| $) (|:| |commonPart| $)) $ $) "\\spad{separate(p, q)} returns \\spad{[a, b]} such that polynomial \\spad{p = a b} and \\spad{a} is relatively prime to \\spad{q}.")) (|pseudoDivide| (((|Record| (|:| |coef| |#1|) (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{pseudoDivide(p,q)} returns \\spad{[c, q, r]},{} when \\spad{p' := p*lc(q)**(deg p - deg q + 1) = c * p} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|pseudoQuotient| (($ $ $) "\\spad{pseudoQuotient(p,q)} returns \\spad{r},{} the quotient when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|composite| (((|Union| (|Fraction| $) "failed") (|Fraction| $) $) "\\spad{composite(f, q)} returns \\spad{h} if \\spad{f} = \\spad{h}(\\spad{q}),{} and \"failed\" is no such \\spad{h} exists.") (((|Union| $ "failed") $ $) "\\spad{composite(p, q)} returns \\spad{h} if \\spad{p = h(q)},{} and \"failed\" no such \\spad{h} exists.")) (|subResultantGcd| (($ $ $) "\\spad{subResultantGcd(p,q)} computes the gcd of the polynomials \\spad{p} and \\spad{q} using the SubResultant GCD algorithm.")) (|order| (((|NonNegativeInteger|) $ $) "\\spad{order(p, q)} returns the largest \\spad{n} such that \\spad{q**n} divides polynomial \\spad{p} \\spadignore{i.e.} the order of \\spad{p(x)} at \\spad{q(x)=0}.")) (|elt| ((|#1| (|Fraction| $) |#1|) "\\spad{elt(a,r)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by the constant \\spad{r}.") (((|Fraction| $) (|Fraction| $) (|Fraction| $)) "\\spad{elt(a,b)} evaluates the fraction of univariate polynomials \\spad{a} with the distinguished variable replaced by \\spad{b}.")) (|resultant| ((|#1| $ $) "\\spad{resultant(p,q)} returns the resultant of the polynomials \\spad{p} and \\spad{q}.")) (|discriminant| ((|#1| $) "\\spad{discriminant(p)} returns the discriminant of the polynomial \\spad{p}.")) (|differentiate| (($ $ (|Mapping| |#1| |#1|) $) "\\spad{differentiate(p, d, x')} extends the \\spad{R}-derivation \\spad{d} to an extension \\spad{D} in \\spad{R[x]} where Dx is given by x',{} and returns \\spad{Dp}.")) (|pseudoRemainder| (($ $ $) "\\spad{pseudoRemainder(p,q)} = \\spad{r},{} for polynomials \\spad{p} and \\spad{q},{} returns the remainder when \\spad{p' := p*lc(q)**(deg p - deg q + 1)} is pseudo right-divided by \\spad{q},{} \\spadignore{i.e.} \\spad{p' = s q + r}.")) (|shiftLeft| (($ $ (|NonNegativeInteger|)) "\\spad{shiftLeft(p,n)} returns \\spad{p * monomial(1,n)}")) (|shiftRight| (($ $ (|NonNegativeInteger|)) "\\spad{shiftRight(p,n)} returns \\spad{monicDivide(p,monomial(1,n)).quotient}")) (|karatsubaDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ (|NonNegativeInteger|)) "\\spad{karatsubaDivide(p,n)} returns the same as \\spad{monicDivide(p,monomial(1,n))}")) (|monicDivide| (((|Record| (|:| |quotient| $) (|:| |remainder| $)) $ $) "\\spad{monicDivide(p,q)} divide the polynomial \\spad{p} by the monic polynomial \\spad{q},{} returning the pair \\spad{[quotient, remainder]}. Error: if \\spad{q} isn't monic.")) (|divideExponents| (((|Union| $ "failed") $ (|NonNegativeInteger|)) "\\spad{divideExponents(p,n)} returns a new polynomial resulting from dividing all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n},{} or \"failed\" if some exponent is not exactly divisible by \\spad{n}.")) (|multiplyExponents| (($ $ (|NonNegativeInteger|)) "\\spad{multiplyExponents(p,n)} returns a new polynomial resulting from multiplying all exponents of the polynomial \\spad{p} by the non negative integer \\spad{n}.")) (|unmakeSUP| (($ (|SparseUnivariatePolynomial| |#1|)) "\\spad{unmakeSUP(sup)} converts \\spad{sup} of type \\spadtype{SparseUnivariatePolynomial(R)} to be a member of the given type. Note: converse of makeSUP.")) (|makeSUP| (((|SparseUnivariatePolynomial| |#1|) $) "\\spad{makeSUP(p)} converts the polynomial \\spad{p} to be of type SparseUnivariatePolynomial over the same coefficients.")) (|vectorise| (((|Vector| |#1|) $ (|NonNegativeInteger|)) "\\spad{vectorise(p, n)} returns \\spad{[a0,...,a(n-1)]} where \\spad{p = a0 + a1*x + ... + a(n-1)*x**(n-1)} + higher order terms. The degree of polynomial \\spad{p} can be different from \\spad{n-1}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3974 |has| |#1| (-309)) (-3976 |has| |#1| (-6 -3976)) (-3973 . T) (-3972 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3985 |has| |#1| (-311)) (-3987 |has| |#1| (-6 -3987)) (-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-1147 R PR S PS) +(-1154 R PR S PS) ((|constructor| (NIL "Mapping from polynomials over \\spad{R} to polynomials over \\spad{S} given a map from \\spad{R} to \\spad{S} assumed to send zero to zero.")) (|map| ((|#4| (|Mapping| |#3| |#1|) |#2|) "\\spad{map(f, p)} takes a function \\spad{f} from \\spad{R} to \\spad{S},{} and applies it to each (non-zero) coefficient of a polynomial \\spad{p} over \\spad{R},{} getting a new polynomial over \\spad{S}. Note: since the map is not applied to zero elements,{} it may map zero to zero."))) NIL NIL -(-1148 S |Coef| |Expon|) +(-1155 S |Coef| |Expon|) ((|constructor| (NIL "\\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.")) (|eval| (((|Stream| |#2|) $ |#2|) "\\spad{eval(f,a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.")) (|extend| (($ $ |#3|) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree <= \\spad{n} to be computed.")) (|approximate| ((|#2| $ |#3|) "\\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.")) (|truncate| (($ $ |#3| |#3|) "\\spad{truncate(f,k1,k2)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (($ $ |#3|) "\\spad{truncate(f,k)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| ((|#3| $ |#3|) "\\spad{order(f,n) = min(m,n)},{} where \\spad{m} is the degree of the lowest order non-zero term in \\spad{f}.") ((|#3| $) "\\spad{order(f)} is the degree of the lowest order non-zero term in \\spad{f}. This will result in an infinite loop if \\spad{f} has no non-zero terms.")) (|multiplyExponents| (($ $ (|PositiveInteger|)) "\\spad{multiplyExponents(f,n)} multiplies all exponents of the power series \\spad{f} by the positive integer \\spad{n}.")) (|center| ((|#2| $) "\\spad{center(f)} returns the point about which the series \\spad{f} is expanded.")) (|variable| (((|Symbol|) $) "\\spad{variable(f)} returns the (unique) power series variable of the power series \\spad{f}.")) (|terms| (((|Stream| (|Record| (|:| |k| |#3|) (|:| |c| |#2|))) $) "\\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents."))) NIL -((|HasCategory| |#2| (QUOTE (-804 (-1081)))) (|HasSignature| |#2| (|%list| (QUOTE *) (|%list| (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#2|)))) (|HasCategory| |#3| (QUOTE (-1017))) (|HasSignature| |#2| (|%list| (QUOTE **) (|%list| (|devaluate| |#2|) (|devaluate| |#2|) (|devaluate| |#3|)))) (|HasSignature| |#2| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#2|) (QUOTE (-1081)))))) -(-1149 |Coef| |Expon|) +((|HasCategory| |#2| (QUOTE (-809 (-1088)))) (|HasSignature| |#2| (|%list| (QUOTE *) (|%list| (|devaluate| |#2|) (|devaluate| |#3|) (|devaluate| |#2|)))) (|HasCategory| |#3| (QUOTE (-1024))) (|HasSignature| |#2| (|%list| (QUOTE **) (|%list| (|devaluate| |#2|) (|devaluate| |#2|) (|devaluate| |#3|)))) (|HasSignature| |#2| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#2|) (QUOTE (-1088)))))) +(-1156 |Coef| |Expon|) ((|constructor| (NIL "\\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.")) (|eval| (((|Stream| |#1|) $ |#1|) "\\spad{eval(f,a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.")) (|extend| (($ $ |#2|) "\\spad{extend(f,n)} causes all terms of \\spad{f} of degree <= \\spad{n} to be computed.")) (|approximate| ((|#1| $ |#2|) "\\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.")) (|truncate| (($ $ |#2| |#2|) "\\spad{truncate(f,k1,k2)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (($ $ |#2|) "\\spad{truncate(f,k)} returns a (finite) power series consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|order| ((|#2| $ |#2|) "\\spad{order(f,n) = min(m,n)},{} where \\spad{m} is the degree of the lowest order non-zero term in \\spad{f}.") ((|#2| $) "\\spad{order(f)} is the degree of the lowest order non-zero term in \\spad{f}. This will result in an infinite loop if \\spad{f} has no non-zero terms.")) (|multiplyExponents| (($ $ (|PositiveInteger|)) "\\spad{multiplyExponents(f,n)} multiplies all exponents of the power series \\spad{f} by the positive integer \\spad{n}.")) (|center| ((|#1| $) "\\spad{center(f)} returns the point about which the series \\spad{f} is expanded.")) (|variable| (((|Symbol|) $) "\\spad{variable(f)} returns the (unique) power series variable of the power series \\spad{f}.")) (|terms| (((|Stream| (|Record| (|:| |k| |#2|) (|:| |c| |#1|))) $) "\\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1150 RC P) +(-1157 RC P) ((|constructor| (NIL "This package provides for square-free decomposition of univariate polynomials over arbitrary rings,{} \\spadignore{i.e.} a partial factorization such that each factor is a product of irreducibles with multiplicity one and the factors are pairwise relatively prime. If the ring has characteristic zero,{} the result is guaranteed to satisfy this condition. If the ring is an infinite ring of finite characteristic,{} then it may not be possible to decide when polynomials contain factors which are \\spad{p}th powers. In this case,{} the flag associated with that polynomial is set to \"nil\" (meaning that that polynomials are not guaranteed to be square-free).")) (|BumInSepFFE| (((|Record| (|:| |flg| (|Union| #1="nil" #2="sqfr" #3="irred" #4="prime")) (|:| |fctr| |#2|) (|:| |xpnt| (|Integer|))) (|Record| (|:| |flg| (|Union| #1# #2# #3# #4#)) (|:| |fctr| |#2|) (|:| |xpnt| (|Integer|)))) "\\spad{BumInSepFFE(f)} is a local function,{} exported only because it has multiple conditional definitions.")) (|squareFreePart| ((|#2| |#2|) "\\spad{squareFreePart(p)} returns a polynomial which has the same irreducible factors as the univariate polynomial \\spad{p},{} but each factor has multiplicity one.")) (|squareFree| (((|Factored| |#2|) |#2|) "\\spad{squareFree(p)} computes the square-free factorization of the univariate polynomial \\spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime.")) (|gcd| (($ $ $) "\\spad{gcd(p,q)} computes the greatest-common-divisor of \\spad{p} and \\spad{q}."))) NIL NIL -(-1151 |Coef| |var| |cen|) +(-1158 |Coef| |var| |cen|) ((|constructor| (NIL "Dense Puiseux series in one variable \\indented{2}{\\spadtype{UnivariatePuiseuxSeries} is a domain representing Puiseux} \\indented{2}{series in one variable with coefficients in an arbitrary ring.\\space{2}The} \\indented{2}{parameters of the type specify the coefficient ring,{} the power series} \\indented{2}{variable,{} and the center of the power series expansion.\\space{2}For example,{}} \\indented{2}{\\spad{UnivariatePuiseuxSeries(Integer,x,3)} represents Puiseux series in} \\indented{2}{\\spad{(x - 3)} with \\spadtype{Integer} coefficients.}")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|)))) (|HasCategory| (-345 (-480)) (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-309))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-29 (-480)))) (|HasCategory| |#1| (QUOTE (-866))) (|HasCategory| |#1| (QUOTE (-1106)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1081))))) (|HasSignature| |#1| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#1|))))))) -(-1152 |Coef1| |Coef2| |var1| |var2| |cen1| |cen2|) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|)))) (|HasCategory| (-347 (-483)) (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-311))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-29 (-483)))) (|HasCategory| |#1| (QUOTE (-871))) (|HasCategory| |#1| (QUOTE (-1113)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1088))))) (|HasSignature| |#1| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#1|))))))) +(-1159 |Coef1| |Coef2| |var1| |var2| |cen1| |cen2|) ((|constructor| (NIL "Mapping package for univariate Puiseux series. This package allows one to apply a function to the coefficients of a univariate Puiseux series.")) (|map| (((|UnivariatePuiseuxSeries| |#2| |#4| |#6|) (|Mapping| |#2| |#1|) (|UnivariatePuiseuxSeries| |#1| |#3| |#5|)) "\\spad{map(f,g(x))} applies the map \\spad{f} to the coefficients of the Puiseux series \\spad{g(x)}."))) NIL NIL -(-1153 |Coef|) +(-1160 |Coef|) ((|constructor| (NIL "\\spadtype{UnivariatePuiseuxSeriesCategory} is the category of Puiseux series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),var)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{var}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 1. We may integrate a series when we can divide coefficients by rational numbers.")) (|multiplyExponents| (($ $ (|Fraction| (|Integer|))) "\\spad{multiplyExponents(f,r)} multiplies all exponents of the power series \\spad{f} by the positive rational number \\spad{r}.")) (|series| (($ (|NonNegativeInteger|) (|Stream| (|Record| (|:| |k| (|Fraction| (|Integer|))) (|:| |c| |#1|)))) "\\spad{series(n,st)} creates a series from a common denomiator and a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents and \\spad{n} should be a common denominator for the exponents in the stream of terms."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1154 S |Coef| ULS) +(-1161 S |Coef| ULS) ((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x^r)}.")) (|laurentIfCan| (((|Union| |#3| "failed") $) "\\spad{laurentIfCan(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. If this is not possible,{} \"failed\" is returned.")) (|laurent| ((|#3| $) "\\spad{laurent(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. Error: if this is not possible.")) (|degree| (((|Fraction| (|Integer|)) $) "\\spad{degree(f(x))} returns the degree of the leading term of the Puiseux series \\spad{f(x)},{} which may have zero as a coefficient.")) (|laurentRep| ((|#3| $) "\\spad{laurentRep(f(x))} returns \\spad{g(x)} where the Puiseux series \\spad{f(x) = g(x^r)} is represented by \\spad{[r,g(x)]}.")) (|rationalPower| (((|Fraction| (|Integer|)) $) "\\spad{rationalPower(f(x))} returns \\spad{r} where the Puiseux series \\spad{f(x) = g(x^r)}.")) (|puiseux| (($ (|Fraction| (|Integer|)) |#3|) "\\spad{puiseux(r,f(x))} returns \\spad{f(x^r)}."))) NIL NIL -(-1155 |Coef| ULS) +(-1162 |Coef| ULS) ((|constructor| (NIL "This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x^r)}.")) (|laurentIfCan| (((|Union| |#2| "failed") $) "\\spad{laurentIfCan(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. If this is not possible,{} \"failed\" is returned.")) (|laurent| ((|#2| $) "\\spad{laurent(f(x))} converts the Puiseux series \\spad{f(x)} to a Laurent series if possible. Error: if this is not possible.")) (|degree| (((|Fraction| (|Integer|)) $) "\\spad{degree(f(x))} returns the degree of the leading term of the Puiseux series \\spad{f(x)},{} which may have zero as a coefficient.")) (|laurentRep| ((|#2| $) "\\spad{laurentRep(f(x))} returns \\spad{g(x)} where the Puiseux series \\spad{f(x) = g(x^r)} is represented by \\spad{[r,g(x)]}.")) (|rationalPower| (((|Fraction| (|Integer|)) $) "\\spad{rationalPower(f(x))} returns \\spad{r} where the Puiseux series \\spad{f(x) = g(x^r)}.")) (|puiseux| (($ (|Fraction| (|Integer|)) |#2|) "\\spad{puiseux(r,f(x))} returns \\spad{f(x^r)}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1156 |Coef| ULS) +(-1163 |Coef| ULS) ((|constructor| (NIL "This package enables one to construct a univariate Puiseux series domain from a univariate Laurent series domain. Univariate Puiseux series are represented by a pair \\spad{[r,f(x)]},{} where \\spad{r} is a positive rational number and \\spad{f(x)} is a Laurent series. This pair represents the Puiseux series \\spad{f(x^r)}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3976 |has| |#1| (-309)) (-3970 |has| |#1| (-309)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#1| (QUOTE (-144))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480))) (|devaluate| |#1|)))) (|HasCategory| (-345 (-480)) (QUOTE (-1017))) (|HasCategory| |#1| (QUOTE (-309))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (OR (|HasCategory| |#1| (QUOTE (-309))) (|HasCategory| |#1| (QUOTE (-491)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -345) (QUOTE (-480)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-29 (-480)))) (|HasCategory| |#1| (QUOTE (-866))) (|HasCategory| |#1| (QUOTE (-1106)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1081))))) (|HasSignature| |#1| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#1|)))))) (|HasCategory| |#1| (QUOTE (-38 (-345 (-480)))))) -(-1157 R FE |var| |cen|) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3987 |has| |#1| (-311)) (-3981 |has| |#1| (-311)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#1| (QUOTE (-146))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483))) (|devaluate| |#1|)))) (|HasCategory| (-347 (-483)) (QUOTE (-1024))) (|HasCategory| |#1| (QUOTE (-311))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (OR (|HasCategory| |#1| (QUOTE (-311))) (|HasCategory| |#1| (QUOTE (-494)))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (|%list| (QUOTE -347) (QUOTE (-483)))))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-29 (-483)))) (|HasCategory| |#1| (QUOTE (-871))) (|HasCategory| |#1| (QUOTE (-1113)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1088))))) (|HasSignature| |#1| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#1|)))))) (|HasCategory| |#1| (QUOTE (-38 (-347 (-483)))))) +(-1164 R FE |var| |cen|) ((|constructor| (NIL "UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent functions with essential singularities. Objects in this domain are sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux series. Thus,{} the elements of this domain are sums of expressions of the form \\spad{g(x) * exp(f(x))},{} where \\spad{g}(\\spad{x}) is a univariate Puiseux series and \\spad{f}(\\spad{x}) is a univariate Puiseux series with no terms of non-negative degree.")) (|dominantTerm| (((|Union| (|Record| (|:| |%term| (|Record| (|:| |%coef| (|UnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expon| (|ExponentialOfUnivariatePuiseuxSeries| |#2| |#3| |#4|)) (|:| |%expTerms| (|List| (|Record| (|:| |k| (|Fraction| (|Integer|))) (|:| |c| |#2|)))))) (|:| |%type| (|String|))) "failed") $) "\\spad{dominantTerm(f(var))} returns the term that dominates the limiting behavior of \\spad{f(var)} as \\spad{var -> cen+} together with a \\spadtype{String} which briefly describes that behavior. The value of the \\spadtype{String} will be \\spad{\"zero\"} (resp. \\spad{\"infinity\"}) if the term tends to zero (resp. infinity) exponentially and will \\spad{\"series\"} if the term is a Puiseux series.")) (|limitPlus| (((|Union| (|OrderedCompletion| |#2|) "failed") $) "\\spad{limitPlus(f(var))} returns \\spad{limit(var -> cen+,f(var))}."))) -(((-3980 "*") |has| (-1151 |#2| |#3| |#4|) (-144)) (-3971 |has| (-1151 |#2| |#3| |#4|) (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) -((|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-38 (-345 (-480))))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-116))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-118))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-144))) (OR (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-38 (-345 (-480))))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-945 (-345 (-480)))))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-945 (-345 (-480))))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-945 (-480)))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-309))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-387))) (|HasCategory| (-1151 |#2| |#3| |#4|) (QUOTE (-491)))) -(-1158 A S) +(((-3991 "*") |has| (-1158 |#2| |#3| |#4|) (-146)) (-3982 |has| (-1158 |#2| |#3| |#4|) (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-38 (-347 (-483))))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-118))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-120))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-146))) (OR (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-38 (-347 (-483))))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-950 (-347 (-483)))))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-950 (-347 (-483))))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-950 (-483)))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-311))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-389))) (|HasCategory| (-1158 |#2| |#3| |#4|) (QUOTE (-494)))) +(-1165 A S) ((|constructor| (NIL "A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \\spadfun{value} of the list designating the head,{} or \\spadfun{first},{} of the list,{} and the child designating the tail,{} or \\spadfun{rest},{} of the list. A node with no child then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.")) (|split!| (($ $ (|Integer|)) "\\spad{split!(u,n)} splits \\spad{u} into two aggregates: \\axiom{\\spad{v} = rest(\\spad{u},{}\\spad{n})} and \\axiom{\\spad{w} = first(\\spad{u},{}\\spad{n})},{} returning \\axiom{\\spad{v}}. Note: afterwards \\axiom{rest(\\spad{u},{}\\spad{n})} returns \\axiom{empty()}.")) (|setlast!| ((|#2| $ |#2|) "\\spad{setlast!(u,x)} destructively changes the last element of \\spad{u} to \\spad{x}.")) (|setrest!| (($ $ $) "\\spad{setrest!(u,v)} destructively changes the rest of \\spad{u} to \\spad{v}.")) (|setelt| ((|#2| $ "last" |#2|) "\\spad{setelt(u,\"last\",x)} (also written: \\axiom{\\spad{u}.last := \\spad{b}}) is equivalent to \\axiom{setlast!(\\spad{u},{}\\spad{v})}.") (($ $ "rest" $) "\\spad{setelt(u,\"rest\",v)} (also written: \\axiom{\\spad{u}.rest := \\spad{v}}) is equivalent to \\axiom{setrest!(\\spad{u},{}\\spad{v})}.") ((|#2| $ "first" |#2|) "\\spad{setelt(u,\"first\",x)} (also written: \\axiom{\\spad{u}.first := \\spad{x}}) is equivalent to \\axiom{setfirst!(\\spad{u},{}\\spad{x})}.")) (|setfirst!| ((|#2| $ |#2|) "\\spad{setfirst!(u,x)} destructively changes the first element of a to \\spad{x}.")) (|cycleSplit!| (($ $) "\\spad{cycleSplit!(u)} splits the aggregate by dropping off the cycle. The value returned is the cycle entry,{} or nil if none exists. For example,{} if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} is the cyclic list where \\spad{v} is the head of the cycle,{} \\axiom{cycleSplit!(\\spad{w})} will drop \\spad{v} off \\spad{w} thus destructively changing \\spad{w} to \\spad{u},{} and returning \\spad{v}.")) (|concat!| (($ $ |#2|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}. Note: \\axiom{concat!(a,{}\\spad{x}) = setlast!(a,{}[\\spad{x}])}.") (($ $ $) "\\spad{concat!(u,v)} destructively concatenates \\spad{v} to the end of \\spad{u}. Note: \\axiom{concat!(\\spad{u},{}\\spad{v}) = setlast!(\\spad{u},{}\\spad{v})}.")) (|cycleTail| (($ $) "\\spad{cycleTail(u)} returns the last node in the cycle,{} or empty if none exists.")) (|cycleLength| (((|NonNegativeInteger|) $) "\\spad{cycleLength(u)} returns the length of a top-level cycle contained in aggregate \\spad{u},{} or 0 is \\spad{u} has no such cycle.")) (|cycleEntry| (($ $) "\\spad{cycleEntry(u)} returns the head of a top-level cycle contained in aggregate \\spad{u},{} or \\axiom{empty()} if none exists.")) (|third| ((|#2| $) "\\spad{third(u)} returns the third element of \\spad{u}. Note: \\axiom{third(\\spad{u}) = first(rest(rest(\\spad{u})))}.")) (|second| ((|#2| $) "\\spad{second(u)} returns the second element of \\spad{u}. Note: \\axiom{second(\\spad{u}) = first(rest(\\spad{u}))}.")) (|tail| (($ $) "\\spad{tail(u)} returns the last node of \\spad{u}. Note: if \\spad{u} is \\axiom{shallowlyMutable},{} \\axiom{setrest(tail(\\spad{u}),{}\\spad{v}) = concat(\\spad{u},{}\\spad{v})}.")) (|last| (($ $ (|NonNegativeInteger|)) "\\spad{last(u,n)} returns a copy of the last \\spad{n} (\\axiom{\\spad{n} >= 0}) nodes of \\spad{u}. Note: \\axiom{last(\\spad{u},{}\\spad{n})} is a list of \\spad{n} elements.") ((|#2| $) "\\spad{last(u)} resturn the last element of \\spad{u}. Note: for lists,{} \\axiom{last(\\spad{u}) = \\spad{u} . (maxIndex \\spad{u}) = \\spad{u} . (\\# \\spad{u} - 1)}.")) (|rest| (($ $ (|NonNegativeInteger|)) "\\spad{rest(u,n)} returns the \\axiom{\\spad{n}}th (\\spad{n} >= 0) node of \\spad{u}. Note: \\axiom{rest(\\spad{u},{}0) = \\spad{u}}.") (($ $) "\\spad{rest(u)} returns an aggregate consisting of all but the first element of \\spad{u} (equivalently,{} the next node of \\spad{u}).")) (|elt| ((|#2| $ "last") "\\spad{elt(u,\"last\")} (also written: \\axiom{\\spad{u} . last}) is equivalent to last \\spad{u}.") (($ $ "rest") "\\spad{elt(\\%,\"rest\")} (also written: \\axiom{\\spad{u}.rest}) is equivalent to \\axiom{rest \\spad{u}}.") ((|#2| $ "first") "\\spad{elt(u,\"first\")} (also written: \\axiom{\\spad{u} . first}) is equivalent to first \\spad{u}.")) (|first| (($ $ (|NonNegativeInteger|)) "\\spad{first(u,n)} returns a copy of the first \\spad{n} (\\axiom{\\spad{n} >= 0}) elements of \\spad{u}.") ((|#2| $) "\\spad{first(u)} returns the first element of \\spad{u} (equivalently,{} the value at the current node).")) (|concat| (($ |#2| $) "\\spad{concat(x,u)} returns aggregate consisting of \\spad{x} followed by the elements of \\spad{u}. Note: if \\axiom{\\spad{v} = concat(\\spad{x},{}\\spad{u})} then \\axiom{\\spad{x} = first \\spad{v}} and \\axiom{\\spad{u} = rest \\spad{v}}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate \\spad{w} consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: \\axiom{\\spad{v} = rest(\\spad{w},{}\\#a)}."))) NIL -((|HasAttribute| |#1| (QUOTE -3979))) -(-1159 S) +((|HasAttribute| |#1| (QUOTE -3990))) +(-1166 S) ((|constructor| (NIL "A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \\spadfun{value} of the list designating the head,{} or \\spadfun{first},{} of the list,{} and the child designating the tail,{} or \\spadfun{rest},{} of the list. A node with no child then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.")) (|split!| (($ $ (|Integer|)) "\\spad{split!(u,n)} splits \\spad{u} into two aggregates: \\axiom{\\spad{v} = rest(\\spad{u},{}\\spad{n})} and \\axiom{\\spad{w} = first(\\spad{u},{}\\spad{n})},{} returning \\axiom{\\spad{v}}. Note: afterwards \\axiom{rest(\\spad{u},{}\\spad{n})} returns \\axiom{empty()}.")) (|setlast!| ((|#1| $ |#1|) "\\spad{setlast!(u,x)} destructively changes the last element of \\spad{u} to \\spad{x}.")) (|setrest!| (($ $ $) "\\spad{setrest!(u,v)} destructively changes the rest of \\spad{u} to \\spad{v}.")) (|setelt| ((|#1| $ "last" |#1|) "\\spad{setelt(u,\"last\",x)} (also written: \\axiom{\\spad{u}.last := \\spad{b}}) is equivalent to \\axiom{setlast!(\\spad{u},{}\\spad{v})}.") (($ $ "rest" $) "\\spad{setelt(u,\"rest\",v)} (also written: \\axiom{\\spad{u}.rest := \\spad{v}}) is equivalent to \\axiom{setrest!(\\spad{u},{}\\spad{v})}.") ((|#1| $ "first" |#1|) "\\spad{setelt(u,\"first\",x)} (also written: \\axiom{\\spad{u}.first := \\spad{x}}) is equivalent to \\axiom{setfirst!(\\spad{u},{}\\spad{x})}.")) (|setfirst!| ((|#1| $ |#1|) "\\spad{setfirst!(u,x)} destructively changes the first element of a to \\spad{x}.")) (|cycleSplit!| (($ $) "\\spad{cycleSplit!(u)} splits the aggregate by dropping off the cycle. The value returned is the cycle entry,{} or nil if none exists. For example,{} if \\axiom{\\spad{w} = concat(\\spad{u},{}\\spad{v})} is the cyclic list where \\spad{v} is the head of the cycle,{} \\axiom{cycleSplit!(\\spad{w})} will drop \\spad{v} off \\spad{w} thus destructively changing \\spad{w} to \\spad{u},{} and returning \\spad{v}.")) (|concat!| (($ $ |#1|) "\\spad{concat!(u,x)} destructively adds element \\spad{x} to the end of \\spad{u}. Note: \\axiom{concat!(a,{}\\spad{x}) = setlast!(a,{}[\\spad{x}])}.") (($ $ $) "\\spad{concat!(u,v)} destructively concatenates \\spad{v} to the end of \\spad{u}. Note: \\axiom{concat!(\\spad{u},{}\\spad{v}) = setlast!(\\spad{u},{}\\spad{v})}.")) (|cycleTail| (($ $) "\\spad{cycleTail(u)} returns the last node in the cycle,{} or empty if none exists.")) (|cycleLength| (((|NonNegativeInteger|) $) "\\spad{cycleLength(u)} returns the length of a top-level cycle contained in aggregate \\spad{u},{} or 0 is \\spad{u} has no such cycle.")) (|cycleEntry| (($ $) "\\spad{cycleEntry(u)} returns the head of a top-level cycle contained in aggregate \\spad{u},{} or \\axiom{empty()} if none exists.")) (|third| ((|#1| $) "\\spad{third(u)} returns the third element of \\spad{u}. Note: \\axiom{third(\\spad{u}) = first(rest(rest(\\spad{u})))}.")) (|second| ((|#1| $) "\\spad{second(u)} returns the second element of \\spad{u}. Note: \\axiom{second(\\spad{u}) = first(rest(\\spad{u}))}.")) (|tail| (($ $) "\\spad{tail(u)} returns the last node of \\spad{u}. Note: if \\spad{u} is \\axiom{shallowlyMutable},{} \\axiom{setrest(tail(\\spad{u}),{}\\spad{v}) = concat(\\spad{u},{}\\spad{v})}.")) (|last| (($ $ (|NonNegativeInteger|)) "\\spad{last(u,n)} returns a copy of the last \\spad{n} (\\axiom{\\spad{n} >= 0}) nodes of \\spad{u}. Note: \\axiom{last(\\spad{u},{}\\spad{n})} is a list of \\spad{n} elements.") ((|#1| $) "\\spad{last(u)} resturn the last element of \\spad{u}. Note: for lists,{} \\axiom{last(\\spad{u}) = \\spad{u} . (maxIndex \\spad{u}) = \\spad{u} . (\\# \\spad{u} - 1)}.")) (|rest| (($ $ (|NonNegativeInteger|)) "\\spad{rest(u,n)} returns the \\axiom{\\spad{n}}th (\\spad{n} >= 0) node of \\spad{u}. Note: \\axiom{rest(\\spad{u},{}0) = \\spad{u}}.") (($ $) "\\spad{rest(u)} returns an aggregate consisting of all but the first element of \\spad{u} (equivalently,{} the next node of \\spad{u}).")) (|elt| ((|#1| $ "last") "\\spad{elt(u,\"last\")} (also written: \\axiom{\\spad{u} . last}) is equivalent to last \\spad{u}.") (($ $ "rest") "\\spad{elt(\\%,\"rest\")} (also written: \\axiom{\\spad{u}.rest}) is equivalent to \\axiom{rest \\spad{u}}.") ((|#1| $ "first") "\\spad{elt(u,\"first\")} (also written: \\axiom{\\spad{u} . first}) is equivalent to first \\spad{u}.")) (|first| (($ $ (|NonNegativeInteger|)) "\\spad{first(u,n)} returns a copy of the first \\spad{n} (\\axiom{\\spad{n} >= 0}) elements of \\spad{u}.") ((|#1| $) "\\spad{first(u)} returns the first element of \\spad{u} (equivalently,{} the value at the current node).")) (|concat| (($ |#1| $) "\\spad{concat(x,u)} returns aggregate consisting of \\spad{x} followed by the elements of \\spad{u}. Note: if \\axiom{\\spad{v} = concat(\\spad{x},{}\\spad{u})} then \\axiom{\\spad{x} = first \\spad{v}} and \\axiom{\\spad{u} = rest \\spad{v}}.") (($ $ $) "\\spad{concat(u,v)} returns an aggregate \\spad{w} consisting of the elements of \\spad{u} followed by the elements of \\spad{v}. Note: \\axiom{\\spad{v} = rest(\\spad{w},{}\\#a)}."))) NIL NIL -(-1160 |Coef| |var| |cen|) +(-1167 |Coef| |var| |cen|) ((|constructor| (NIL "Dense Taylor series in one variable \\spadtype{UnivariateTaylorSeries} is a domain representing Taylor series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring,{} the power series variable,{} and the center of the power series expansion. For example,{} \\spadtype{UnivariateTaylorSeries}(Integer,{}\\spad{x},{}3) represents Taylor series in \\spad{(x - 3)} with \\spadtype{Integer} coefficients.")) (|integrate| (($ $ (|Variable| |#2|)) "\\spad{integrate(f(x),x)} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (|invmultisect| (($ (|Integer|) (|Integer|) $) "\\spad{invmultisect(a,b,f(x))} substitutes \\spad{x^((a+b)*n)} \\indented{1}{for \\spad{x^n} and multiples by \\spad{x^b}.}")) (|multisect| (($ (|Integer|) (|Integer|) $) "\\spad{multisect(a,b,f(x))} selects the coefficients of \\indented{1}{\\spad{x^((a+b)*n+a)},{} and changes this monomial to \\spad{x^n}.}")) (|revert| (($ $) "\\spad{revert(f(x))} returns a Taylor series \\spad{g(x)} such that \\spad{f(g(x)) = g(f(x)) = x}. Series \\spad{f(x)} should have constant coefficient 0 and invertible 1st order coefficient.")) (|generalLambert| (($ $ (|Integer|) (|Integer|)) "\\spad{generalLambert(f(x),a,d)} returns \\spad{f(x^a) + f(x^(a + d)) + \\indented{1}{f(x^(a + 2 d)) + ... }. \\spad{f(x)} should have zero constant} \\indented{1}{coefficient and \\spad{a} and \\spad{d} should be positive.}")) (|evenlambert| (($ $) "\\spad{evenlambert(f(x))} returns \\spad{f(x^2) + f(x^4) + f(x^6) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n))) = exp(log(evenlambert(f(x))))}.}")) (|oddlambert| (($ $) "\\spad{oddlambert(f(x))} returns \\spad{f(x) + f(x^3) + f(x^5) + ...}. \\indented{1}{\\spad{f(x)} should have a zero constant coefficient.} \\indented{1}{This function is used for computing infinite products.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n=1..infinity,f(x^(2*n-1)))=exp(log(oddlambert(f(x))))}.}")) (|lambert| (($ $) "\\spad{lambert(f(x))} returns \\spad{f(x) + f(x^2) + f(x^3) + ...}. \\indented{1}{This function is used for computing infinite products.} \\indented{1}{\\spad{f(x)} should have zero constant coefficient.} \\indented{1}{If \\spad{f(x)} is a Taylor series with constant term 1,{} then} \\indented{1}{\\spad{product(n = 1..infinity,f(x^n)) = exp(log(lambert(f(x))))}.}")) (|lagrange| (($ $) "\\spad{lagrange(g(x))} produces the Taylor series for \\spad{f(x)} \\indented{1}{where \\spad{f(x)} is implicitly defined as \\spad{f(x) = x*g(f(x))}.}")) (|univariatePolynomial| (((|UnivariatePolynomial| |#2| |#1|) $ (|NonNegativeInteger|)) "\\spad{univariatePolynomial(f,k)} returns a univariate polynomial \\indented{1}{consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.}")) (|coerce| (($ (|Variable| |#2|)) "\\spad{coerce(var)} converts the series variable \\spad{var} into a \\indented{1}{Taylor series.}") (($ (|UnivariatePolynomial| |#2| |#1|)) "\\spad{coerce(p)} converts a univariate polynomial \\spad{p} in the variable \\spad{var} to a univariate Taylor series in \\spad{var}."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . 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T)) -((|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-491))) (OR (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-491)))) (|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-116))) (|HasCategory| |#1| (QUOTE (-118))) (-12 (|HasCategory| |#1| (QUOTE (-804 (-1081)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-689)) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-689)) (|devaluate| |#1|)))) (|HasCategory| (-689) (QUOTE (-1017))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-689))))) (|HasSignature| |#1| (|%list| (QUOTE -3929) (|%list| (|devaluate| |#1|) (QUOTE (-1081)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-689))))) (|HasCategory| |#1| (QUOTE (-309))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#1| (QUOTE (-29 (-480)))) (|HasCategory| |#1| (QUOTE (-866))) (|HasCategory| |#1| (QUOTE (-1106)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-345 (-480))))) (|HasSignature| |#1| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1081))))) (|HasSignature| |#1| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#1|))))))) -(-1161 |Coef1| |Coef2| UTS1 UTS2) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-494))) (OR (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-494)))) (|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-118))) (|HasCategory| |#1| (QUOTE (-120))) (-12 (|HasCategory| |#1| (QUOTE (-809 (-1088)))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-694)) (|devaluate| |#1|))))) (|HasSignature| |#1| (|%list| (QUOTE *) (|%list| (|devaluate| |#1|) (QUOTE (-694)) (|devaluate| |#1|)))) (|HasCategory| (-694) (QUOTE (-1024))) (-12 (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-694))))) (|HasSignature| |#1| (|%list| (QUOTE -3940) (|%list| (|devaluate| |#1|) (QUOTE (-1088)))))) (|HasSignature| |#1| (|%list| (QUOTE **) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-694))))) (|HasCategory| |#1| (QUOTE (-311))) (OR (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#1| (QUOTE (-29 (-483)))) (|HasCategory| |#1| (QUOTE (-871))) (|HasCategory| |#1| (QUOTE (-1113)))) (-12 (|HasCategory| |#1| (QUOTE (-38 (-347 (-483))))) (|HasSignature| |#1| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#1|) (|devaluate| |#1|) (QUOTE (-1088))))) (|HasSignature| |#1| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#1|))))))) +(-1168 |Coef1| |Coef2| UTS1 UTS2) ((|constructor| (NIL "Mapping package for univariate Taylor series. \\indented{2}{This package allows one to apply a function to the coefficients of} \\indented{2}{a univariate Taylor series.}")) (|map| ((|#4| (|Mapping| |#2| |#1|) |#3|) "\\spad{map(f,g(x))} applies the map \\spad{f} to the coefficients of \\indented{1}{the Taylor series \\spad{g(x)}.}"))) NIL NIL -(-1162 S |Coef|) +(-1169 S |Coef|) ((|constructor| (NIL "\\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (** (($ $ |#2|) "\\spad{f(x) ** a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.")) (|polynomial| (((|Polynomial| |#2|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#2|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|multiplyCoefficients| (($ (|Mapping| |#2| (|Integer|)) $) "\\spad{multiplyCoefficients(f,sum(n = 0..infinity,a[n] * x**n))} returns \\spad{sum(n = 0..infinity,f(n) * a[n] * x**n)}. This function is used when Laurent series are represented by a Taylor series and an order.")) (|quoByVar| (($ $) "\\spad{quoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.")) (|coefficients| (((|Stream| |#2|) $) "\\spad{coefficients(a0 + a1 x + a2 x**2 + ...)} returns a stream of coefficients: \\spad{[a0,a1,a2,...]}. The entries of the stream may be zero.")) (|series| (($ (|Stream| |#2|)) "\\spad{series([a0,a1,a2,...])} is the Taylor series \\spad{a0 + a1 x + a2 x**2 + ...}.") (($ (|Stream| (|Record| (|:| |k| (|NonNegativeInteger|)) (|:| |c| |#2|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents."))) NIL -((|HasCategory| |#2| (QUOTE (-29 (-480)))) (|HasCategory| |#2| (QUOTE (-866))) (|HasCategory| |#2| (QUOTE (-1106))) (|HasSignature| |#2| (|%list| (QUOTE -3067) (|%list| (|%list| (QUOTE -580) (QUOTE (-1081))) (|devaluate| |#2|)))) (|HasSignature| |#2| (|%list| (QUOTE -3795) (|%list| (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1081))))) (|HasCategory| |#2| (QUOTE (-38 (-345 (-480))))) (|HasCategory| |#2| (QUOTE (-309)))) -(-1163 |Coef|) +((|HasCategory| |#2| (QUOTE (-29 (-483)))) (|HasCategory| |#2| (QUOTE (-871))) (|HasCategory| |#2| (QUOTE (-1113))) (|HasSignature| |#2| (|%list| (QUOTE -3077) (|%list| (|%list| (QUOTE -583) (QUOTE (-1088))) (|devaluate| |#2|)))) (|HasSignature| |#2| (|%list| (QUOTE -3806) (|%list| (|devaluate| |#2|) (|devaluate| |#2|) (QUOTE (-1088))))) (|HasCategory| |#2| (QUOTE (-38 (-347 (-483))))) (|HasCategory| |#2| (QUOTE (-311)))) +(-1170 |Coef|) ((|constructor| (NIL "\\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.")) (|integrate| (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $ (|Symbol|)) "\\spad{integrate(f(x),y)} returns an anti-derivative of the power series \\spad{f(x)} with respect to the variable \\spad{y}.") (($ $) "\\spad{integrate(f(x))} returns an anti-derivative of the power series \\spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.")) (** (($ $ |#1|) "\\spad{f(x) ** a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.")) (|polynomial| (((|Polynomial| |#1|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{polynomial(f,k1,k2)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{d} with \\spad{k1 <= d <= k2}.") (((|Polynomial| |#1|) $ (|NonNegativeInteger|)) "\\spad{polynomial(f,k)} returns a polynomial consisting of the sum of all terms of \\spad{f} of degree \\spad{<= k}.")) (|multiplyCoefficients| (($ (|Mapping| |#1| (|Integer|)) $) "\\spad{multiplyCoefficients(f,sum(n = 0..infinity,a[n] * x**n))} returns \\spad{sum(n = 0..infinity,f(n) * a[n] * x**n)}. This function is used when Laurent series are represented by a Taylor series and an order.")) (|quoByVar| (($ $) "\\spad{quoByVar(a0 + a1 x + a2 x**2 + ...)} returns \\spad{a1 + a2 x + a3 x**2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.")) (|coefficients| (((|Stream| |#1|) $) "\\spad{coefficients(a0 + a1 x + a2 x**2 + ...)} returns a stream of coefficients: \\spad{[a0,a1,a2,...]}. The entries of the stream may be zero.")) (|series| (($ (|Stream| |#1|)) "\\spad{series([a0,a1,a2,...])} is the Taylor series \\spad{a0 + a1 x + a2 x**2 + ...}.") (($ (|Stream| (|Record| (|:| |k| (|NonNegativeInteger|)) (|:| |c| |#1|)))) "\\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents."))) -(((-3980 "*") |has| |#1| (-144)) (-3971 |has| |#1| (-491)) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") |has| |#1| (-146)) (-3982 |has| |#1| (-494)) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1164 |Coef| UTS) +(-1171 |Coef| UTS) ((|constructor| (NIL "\\indented{1}{This package provides Taylor series solutions to regular} linear or non-linear ordinary differential equations of arbitrary order.")) (|mpsode| (((|List| |#2|) (|List| |#1|) (|List| (|Mapping| |#2| (|List| |#2|)))) "\\spad{mpsode(r,f)} solves the system of differential equations \\spad{dy[i]/dx =f[i] [x,y[1],y[2],...,y[n]]},{} \\spad{y[i](a) = r[i]} for \\spad{i} in 1..\\spad{n}.")) (|ode| ((|#2| (|Mapping| |#2| (|List| |#2|)) (|List| |#1|)) "\\spad{ode(f,cl)} is the solution to \\spad{y<n>=f(y,y',..,y<n-1>)} such that \\spad{y<i>(a) = cl.i} for \\spad{i} in 1..\\spad{n}.")) (|ode2| ((|#2| (|Mapping| |#2| |#2| |#2|) |#1| |#1|) "\\spad{ode2(f,c0,c1)} is the solution to \\spad{y'' = f(y,y')} such that \\spad{y(a) = c0} and \\spad{y'(a) = c1}.")) (|ode1| ((|#2| (|Mapping| |#2| |#2|) |#1|) "\\spad{ode1(f,c)} is the solution to \\spad{y' = f(y)} such that \\spad{y(a) = c}.")) (|fixedPointExquo| ((|#2| |#2| |#2|) "\\spad{fixedPointExquo(f,g)} computes the exact quotient of \\spad{f} and \\spad{g} using a fixed point computation.")) (|stFuncN| (((|Mapping| (|Stream| |#1|) (|List| (|Stream| |#1|))) (|Mapping| |#2| (|List| |#2|))) "\\spad{stFuncN(f)} is a local function xported due to compiler problem. This function is of no interest to the top-level user.")) (|stFunc2| (((|Mapping| (|Stream| |#1|) (|Stream| |#1|) (|Stream| |#1|)) (|Mapping| |#2| |#2| |#2|)) "\\spad{stFunc2(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user.")) (|stFunc1| (((|Mapping| (|Stream| |#1|) (|Stream| |#1|)) (|Mapping| |#2| |#2|)) "\\spad{stFunc1(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user."))) NIL NIL -(-1165 -3078 UP L UTS) +(-1172 -3088 UP L UTS) ((|constructor| (NIL "\\spad{RUTSodetools} provides tools to interface with the series \\indented{1}{ODE solver when presented with linear ODEs.}")) (RF2UTS ((|#4| (|Fraction| |#2|)) "\\spad{RF2UTS(f)} converts \\spad{f} to a Taylor series.")) (LODO2FUN (((|Mapping| |#4| (|List| |#4|)) |#3|) "\\spad{LODO2FUN(op)} returns the function to pass to the series ODE solver in order to solve \\spad{op y = 0}.")) (UTS2UP ((|#2| |#4| (|NonNegativeInteger|)) "\\spad{UTS2UP(s, n)} converts the first \\spad{n} terms of \\spad{s} to a univariate polynomial.")) (UP2UTS ((|#4| |#2|) "\\spad{UP2UTS(p)} converts \\spad{p} to a Taylor series."))) NIL -((|HasCategory| |#1| (QUOTE (-491)))) -(-1166) +((|HasCategory| |#1| (QUOTE (-494)))) +(-1173) ((|constructor| (NIL "The category of domains that act like unions. UnionType,{} like Type or Category,{} acts mostly as a take that communicates `union-like' intended semantics to the compiler. A domain \\spad{D} that satifies UnionType should provide definitions for `case' operators,{} with corresponding `autoCoerce' operators."))) NIL NIL -(-1167 |sym|) +(-1174 |sym|) ((|constructor| (NIL "This domain implements variables")) (|variable| (((|Symbol|)) "\\spad{variable()} returns the symbol")) (|coerce| (((|Symbol|) $) "\\spad{coerce(x)} returns the symbol"))) NIL NIL -(-1168 S R) +(-1175 S R) ((|constructor| (NIL "\\spadtype{VectorCategory} represents the type of vector like objects,{} \\spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.")) (|magnitude| ((|#2| $) "\\spad{magnitude(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the length")) (|length| ((|#2| $) "\\spad{length(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the magnitude")) (|cross| (($ $ $) "vectorProduct(\\spad{u},{}\\spad{v}) constructs the cross product of \\spad{u} and \\spad{v}. Error: if \\spad{u} and \\spad{v} are not of length 3.")) (|outerProduct| (((|Matrix| |#2|) $ $) "\\spad{outerProduct(u,v)} constructs the matrix whose (\\spad{i},{}\\spad{j})\\spad{'}th element is \\spad{u}(\\spad{i})*v(\\spad{j}).")) (|dot| ((|#2| $ $) "\\spad{dot(x,y)} computes the inner product of the two vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.")) (* (($ $ |#2|) "\\spad{y * r} multiplies each component of the vector \\spad{y} by the element \\spad{r}.") (($ |#2| $) "\\spad{r * y} multiplies the element \\spad{r} times each component of the vector \\spad{y}.") (($ (|Integer|) $) "\\spad{n * y} multiplies each component of the vector \\spad{y} by the integer \\spad{n}.")) (- (($ $ $) "\\spad{x - y} returns the component-wise difference of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.") (($ $) "\\spad{-x} negates all components of the vector \\spad{x}.")) (|zero| (($ (|NonNegativeInteger|)) "\\spad{zero(n)} creates a zero vector of length \\spad{n}.")) (+ (($ $ $) "\\spad{x + y} returns the component-wise sum of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length."))) NIL -((|HasCategory| |#2| (QUOTE (-910))) (|HasCategory| |#2| (QUOTE (-956))) (|HasCategory| |#2| (QUOTE (-660))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-25)))) -(-1169 R) +((|HasCategory| |#2| (QUOTE (-915))) (|HasCategory| |#2| (QUOTE (-961))) (|HasCategory| |#2| (QUOTE (-663))) (|HasCategory| |#2| (QUOTE (-21))) (|HasCategory| |#2| (QUOTE (-23))) (|HasCategory| |#2| (QUOTE (-25)))) +(-1176 R) ((|constructor| (NIL "\\spadtype{VectorCategory} represents the type of vector like objects,{} \\spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.")) (|magnitude| ((|#1| $) "\\spad{magnitude(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the length")) (|length| ((|#1| $) "\\spad{length(v)} computes the sqrt(dot(\\spad{v},{}\\spad{v})),{} \\spadignore{i.e.} the magnitude")) (|cross| (($ $ $) "vectorProduct(\\spad{u},{}\\spad{v}) constructs the cross product of \\spad{u} and \\spad{v}. Error: if \\spad{u} and \\spad{v} are not of length 3.")) (|outerProduct| (((|Matrix| |#1|) $ $) "\\spad{outerProduct(u,v)} constructs the matrix whose (\\spad{i},{}\\spad{j})\\spad{'}th element is \\spad{u}(\\spad{i})*v(\\spad{j}).")) (|dot| ((|#1| $ $) "\\spad{dot(x,y)} computes the inner product of the two vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.")) (* (($ $ |#1|) "\\spad{y * r} multiplies each component of the vector \\spad{y} by the element \\spad{r}.") (($ |#1| $) "\\spad{r * y} multiplies the element \\spad{r} times each component of the vector \\spad{y}.") (($ (|Integer|) $) "\\spad{n * y} multiplies each component of the vector \\spad{y} by the integer \\spad{n}.")) (- (($ $ $) "\\spad{x - y} returns the component-wise difference of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length.") (($ $) "\\spad{-x} negates all components of the vector \\spad{x}.")) (|zero| (($ (|NonNegativeInteger|)) "\\spad{zero(n)} creates a zero vector of length \\spad{n}.")) (+ (($ $ $) "\\spad{x + y} returns the component-wise sum of the vectors \\spad{x} and \\spad{y}. Error: if \\spad{x} and \\spad{y} are not of the same length."))) -((-3979 . T) (-3978 . T)) +((-3990 . T) (-3989 . T)) NIL -(-1170 R) +(-1177 R) ((|constructor| (NIL "This type represents vector like objects with varying lengths and indexed by a finite segment of integers starting at 1.")) (|vector| (($ (|List| |#1|)) "\\spad{vector(l)} converts the list \\spad{l} to a vector."))) -((-3979 . T) (-3978 . T)) -((OR (-12 (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-549 (-767)))) (|HasCategory| |#1| (QUOTE (-550 (-469)))) (OR (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| |#1| (QUOTE (-751))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007)))) (|HasCategory| (-480) (QUOTE (-751))) (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-660))) (|HasCategory| |#1| (QUOTE (-956))) (-12 (|HasCategory| |#1| (QUOTE (-910))) (|HasCategory| |#1| (QUOTE (-956)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1007))) (|HasCategory| |#1| (|%list| (QUOTE -257) (|devaluate| |#1|))))) -(-1171 A B) +((-3990 . T) (-3989 . T)) +((OR (-12 (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|)))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) (|HasCategory| |#1| (QUOTE (-552 (-772)))) (|HasCategory| |#1| (QUOTE (-553 (-472)))) (OR (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| |#1| (QUOTE (-756))) (OR (|HasCategory| |#1| (QUOTE (-72))) (|HasCategory| |#1| (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012)))) (|HasCategory| (-483) (QUOTE (-756))) (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-25))) (|HasCategory| |#1| (QUOTE (-23))) (|HasCategory| |#1| (QUOTE (-21))) (|HasCategory| |#1| (QUOTE (-663))) (|HasCategory| |#1| (QUOTE (-961))) (-12 (|HasCategory| |#1| (QUOTE (-915))) (|HasCategory| |#1| (QUOTE (-961)))) (|HasCategory| |#1| (QUOTE (-72))) (-12 (|HasCategory| |#1| (QUOTE (-1012))) (|HasCategory| |#1| (|%list| (QUOTE -259) (|devaluate| |#1|))))) +(-1178 A B) ((|constructor| (NIL "\\indented{2}{This package provides operations which all take as arguments} vectors of elements of some type \\spad{A} and functions from \\spad{A} to another of type \\spad{B}. The operations all iterate over their vector argument and either return a value of type \\spad{B} or a vector over \\spad{B}.")) (|map| (((|Union| (|Vector| |#2|) "failed") (|Mapping| (|Union| |#2| "failed") |#1|) (|Vector| |#1|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values or \\spad{\"failed\"}.") (((|Vector| |#2|) (|Mapping| |#2| |#1|) (|Vector| |#1|)) "\\spad{map(f, v)} applies the function \\spad{f} to every element of the vector \\spad{v} producing a new vector containing the values.")) (|reduce| ((|#2| (|Mapping| |#2| |#1| |#2|) (|Vector| |#1|) |#2|) "\\spad{reduce(func,vec,ident)} combines the elements in \\spad{vec} using the binary function \\spad{func}. Argument \\spad{ident} is returned if \\spad{vec} is empty.")) (|scan| (((|Vector| |#2|) (|Mapping| |#2| |#1| |#2|) (|Vector| |#1|) |#2|) "\\spad{scan(func,vec,ident)} creates a new vector whose elements are the result of applying reduce to the binary function \\spad{func},{} increasing initial subsequences of the vector \\spad{vec},{} and the element \\spad{ident}."))) NIL NIL -(-1172) +(-1179) ((|constructor| (NIL "ViewportPackage provides functions for creating GraphImages and TwoDimensionalViewports from lists of lists of points.")) (|coerce| (((|TwoDimensionalViewport|) (|GraphImage|)) "\\spad{coerce(gi)} converts the indicated \\spadtype{GraphImage},{} \\spad{gi},{} into the \\spadtype{TwoDimensionalViewport} form.")) (|drawCurves| (((|TwoDimensionalViewport|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|DrawOption|))) "\\spad{drawCurves([[p0],[p1],...,[pn]],[options])} creates a \\spadtype{TwoDimensionalViewport} from the list of lists of points,{} \\spad{p0} throught pn,{} using the options specified in the list \\spad{options}.") (((|TwoDimensionalViewport|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|Palette|) (|Palette|) (|PositiveInteger|) (|List| (|DrawOption|))) "\\spad{drawCurves([[p0],[p1],...,[pn]],ptColor,lineColor,ptSize,[options])} creates a \\spadtype{TwoDimensionalViewport} from the list of lists of points,{} \\spad{p0} throught pn,{} using the options specified in the list \\spad{options}. The point color is specified by \\spad{ptColor},{} the line color is specified by \\spad{lineColor},{} and the point size is specified by \\spad{ptSize}.")) (|graphCurves| (((|GraphImage|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|List| (|DrawOption|))) "\\spad{graphCurves([[p0],[p1],...,[pn]],[options])} creates a \\spadtype{GraphImage} from the list of lists of points,{} \\spad{p0} throught pn,{} using the options specified in the list \\spad{options}.") (((|GraphImage|) (|List| (|List| (|Point| (|DoubleFloat|))))) "\\spad{graphCurves([[p0],[p1],...,[pn]])} creates a \\spadtype{GraphImage} from the list of lists of points indicated by \\spad{p0} through pn.") (((|GraphImage|) (|List| (|List| (|Point| (|DoubleFloat|)))) (|Palette|) (|Palette|) (|PositiveInteger|) (|List| (|DrawOption|))) "\\spad{graphCurves([[p0],[p1],...,[pn]],ptColor,lineColor,ptSize,[options])} creates a \\spadtype{GraphImage} from the list of lists of points,{} \\spad{p0} throught pn,{} using the options specified in the list \\spad{options}. The graph point color is specified by \\spad{ptColor},{} the graph line color is specified by \\spad{lineColor},{} and the size of the points is specified by \\spad{ptSize}."))) NIL NIL -(-1173) +(-1180) ((|constructor| (NIL "TwoDimensionalViewport creates viewports to display graphs.")) (|coerce| (((|OutputForm|) $) "\\spad{coerce(v)} returns the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport} as output of the domain \\spadtype{OutputForm}.")) (|key| (((|Integer|) $) "\\spad{key(v)} returns the process ID number of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport}.")) (|reset| (((|Void|) $) "\\spad{reset(v)} sets the current state of the graph characteristics of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} back to their initial settings.")) (|write| (((|String|) $ (|String|) (|List| (|String|))) "\\spad{write(v,s,lf)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v} and the optional file types indicated by the list \\spad{lf}.") (((|String|) $ (|String|) (|String|)) "\\spad{write(v,s,f)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v} and an optional file type \\spad{f}.") (((|String|) $ (|String|)) "\\spad{write(v,s)} takes the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data files for \\spad{v}.")) (|resize| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{resize(v,w,h)} displays the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with a width of \\spad{w} and a height of \\spad{h},{} keeping the upper left-hand corner position unchanged.")) (|update| (((|Void|) $ (|GraphImage|) (|PositiveInteger|)) "\\spad{update(v,gr,n)} drops the graph \\spad{gr} in slot \\spad{n} of viewport \\spad{v}. The graph \\spad{gr} must have been transmitted already and acquired an integer key.")) (|move| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{move(v,x,y)} displays the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the upper left-hand corner of the viewport window at the screen coordinate position \\spad{x},{} \\spad{y}.")) (|show| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{show(v,n,s)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the graph if \\spad{s} is \"off\".")) (|translate| (((|Void|) $ (|PositiveInteger|) (|Float|) (|Float|)) "\\spad{translate(v,n,dx,dy)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} translated by \\spad{dx} in the \\spad{x}-coordinate direction from the center of the viewport,{} and by \\spad{dy} in the \\spad{y}-coordinate direction from the center. Setting \\spad{dx} and \\spad{dy} to \\spad{0} places the center of the graph at the center of the viewport.")) (|scale| (((|Void|) $ (|PositiveInteger|) (|Float|) (|Float|)) "\\spad{scale(v,n,sx,sy)} displays the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} scaled by the factor \\spad{sx} in the \\spad{x}-coordinate direction and by the factor \\spad{sy} in the \\spad{y}-coordinate direction.")) (|dimensions| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{dimensions(v,x,y,width,height)} sets the position of the upper left-hand corner of the two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} to the window coordinate \\spad{x},{} \\spad{y},{} and sets the dimensions of the window to that of \\spad{width},{} \\spad{height}. The new dimensions are not displayed until the function \\spadfun{makeViewport2D} is executed again for \\spad{v}.")) (|close| (((|Void|) $) "\\spad{close(v)} closes the viewport window of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} and terminates the corresponding process ID.")) (|controlPanel| (((|Void|) $ (|String|)) "\\spad{controlPanel(v,s)} displays the control panel of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or hides the control panel if \\spad{s} is \"off\".")) (|connect| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{connect(v,n,s)} displays the lines connecting the graph points in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the lines if \\spad{s} is \"off\".")) (|region| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{region(v,n,s)} displays the bounding box of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the bounding box if \\spad{s} is \"off\".")) (|points| (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{points(v,n,s)} displays the points of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the points if \\spad{s} is \"off\".")) (|units| (((|Void|) $ (|PositiveInteger|) (|Palette|)) "\\spad{units(v,n,c)} displays the units of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the units color set to the given palette color \\spad{c}.") (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{units(v,n,s)} displays the units of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the units if \\spad{s} is \"off\".")) (|axes| (((|Void|) $ (|PositiveInteger|) (|Palette|)) "\\spad{axes(v,n,c)} displays the axes of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} with the axes color set to the given palette color \\spad{c}.") (((|Void|) $ (|PositiveInteger|) (|String|)) "\\spad{axes(v,n,s)} displays the axes of the graph in field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the axes if \\spad{s} is \"off\".")) (|getGraph| (((|GraphImage|) $ (|PositiveInteger|)) "\\spad{getGraph(v,n)} returns the graph which is of the domain \\spadtype{GraphImage} which is located in graph field \\spad{n} of the given two-dimensional viewport,{} \\spad{v},{} which is of the domain \\spadtype{TwoDimensionalViewport}.")) (|putGraph| (((|Void|) $ (|GraphImage|) (|PositiveInteger|)) "\\spad{putGraph(v,gi,n)} sets the graph field indicated by \\spad{n},{} of the indicated two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport},{} to be the graph,{} \\spad{gi} of domain \\spadtype{GraphImage}. The contents of viewport,{} \\spad{v},{} will contain \\spad{gi} when the function \\spadfun{makeViewport2D} is called to create the an updated viewport \\spad{v}.")) (|title| (((|Void|) $ (|String|)) "\\spad{title(v,s)} changes the title which is shown in the two-dimensional viewport window,{} \\spad{v} of domain \\spadtype{TwoDimensionalViewport}.")) (|graphs| (((|Vector| (|Union| (|GraphImage|) "undefined")) $) "\\spad{graphs(v)} returns a vector,{} or list,{} which is a union of all the graphs,{} of the domain \\spadtype{GraphImage},{} which are allocated for the two-dimensional viewport,{} \\spad{v},{} of domain \\spadtype{TwoDimensionalViewport}. Those graphs which have no data are labeled \"undefined\",{} otherwise their contents are shown.")) (|graphStates| (((|Vector| (|Record| (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)) (|:| |points| (|Integer|)) (|:| |connect| (|Integer|)) (|:| |spline| (|Integer|)) (|:| |axes| (|Integer|)) (|:| |axesColor| (|Palette|)) (|:| |units| (|Integer|)) (|:| |unitsColor| (|Palette|)) (|:| |showing| (|Integer|)))) $) "\\spad{graphStates(v)} returns and shows a listing of a record containing the current state of the characteristics of each of the ten graph records in the given two-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{TwoDimensionalViewport}.")) (|graphState| (((|Void|) $ (|PositiveInteger|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|DoubleFloat|) (|Integer|) (|Integer|) (|Integer|) (|Integer|) (|Palette|) (|Integer|) (|Palette|) (|Integer|)) "\\spad{graphState(v,num,sX,sY,dX,dY,pts,lns,box,axes,axesC,un,unC,cP)} sets the state of the characteristics for the graph indicated by \\spad{num} in the given two-dimensional viewport \\spad{v},{} of domain \\spadtype{TwoDimensionalViewport},{} to the values given as parameters. The scaling of the graph in the \\spad{x} and \\spad{y} component directions is set to be \\spad{sX} and \\spad{sY}; the window translation in the \\spad{x} and \\spad{y} component directions is set to be \\spad{dX} and \\spad{dY}; The graph points,{} lines,{} bounding \\spad{box},{} \\spad{axes},{} or units will be shown in the viewport if their given parameters \\spad{pts},{} \\spad{lns},{} \\spad{box},{} \\spad{axes} or \\spad{un} are set to be \\spad{1},{} but will not be shown if they are set to \\spad{0}. The color of the \\spad{axes} and the color of the units are indicated by the palette colors \\spad{axesC} and \\spad{unC} respectively. To display the control panel when the viewport window is displayed,{} set \\spad{cP} to \\spad{1},{} otherwise set it to \\spad{0}.")) (|options| (($ $ (|List| (|DrawOption|))) "\\spad{options(v,lopt)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and returns \\spad{v} with it's draw options modified to be those which are indicated in the given list,{} \\spad{lopt} of domain \\spadtype{DrawOption}.") (((|List| (|DrawOption|)) $) "\\spad{options(v)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and returns a list containing the draw options from the domain \\spadtype{DrawOption} for \\spad{v}.")) (|makeViewport2D| (($ (|GraphImage|) (|List| (|DrawOption|))) "\\spad{makeViewport2D(gi,lopt)} creates and displays a viewport window of the domain \\spadtype{TwoDimensionalViewport} whose graph field is assigned to be the given graph,{} \\spad{gi},{} of domain \\spadtype{GraphImage},{} and whose options field is set to be the list of options,{} \\spad{lopt} of domain \\spadtype{DrawOption}.") (($ $) "\\spad{makeViewport2D(v)} takes the given two-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{TwoDimensionalViewport} and displays a viewport window on the screen which contains the contents of \\spad{v}.")) (|viewport2D| (($) "\\spad{viewport2D()} returns an undefined two-dimensional viewport of the domain \\spadtype{TwoDimensionalViewport} whose contents are empty.")) (|getPickedPoints| (((|List| (|Point| (|DoubleFloat|))) $) "\\spad{getPickedPoints(x)} returns a list of small floats for the points the user interactively picked on the viewport for full integration into the system,{} some design issues need to be addressed: \\spadignore{e.g.} how to go through the GraphImage interface,{} how to default to graphs,{} etc."))) NIL NIL -(-1174) +(-1181) ((|key| (((|Integer|) $) "\\spad{key(v)} returns the process ID number of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|close| (((|Void|) $) "\\spad{close(v)} closes the viewport window of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and terminates the corresponding process ID.")) (|write| (((|String|) $ (|String|) (|List| (|String|))) "\\spad{write(v,s,lf)} takes the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data file for \\spad{v} and the optional file types indicated by the list \\spad{lf}.") (((|String|) $ (|String|) (|String|)) "\\spad{write(v,s,f)} takes the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data file for \\spad{v} and an optional file type \\spad{f}.") (((|String|) $ (|String|)) "\\spad{write(v,s)} takes the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \\spad{s},{} which contains the graph data file for \\spad{v}.")) (|colorDef| (((|Void|) $ (|Color|) (|Color|)) "\\spad{colorDef(v,c1,c2)} sets the range of colors along the colormap so that the lower end of the colormap is defined by \\spad{c1} and the top end of the colormap is defined by \\spad{c2},{} for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|reset| (((|Void|) $) "\\spad{reset(v)} sets the current state of the graph characteristics of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} back to their initial settings.")) (|intensity| (((|Void|) $ (|Float|)) "\\spad{intensity(v,i)} sets the intensity of the light source to \\spad{i},{} for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|lighting| (((|Void|) $ (|Float|) (|Float|) (|Float|)) "\\spad{lighting(v,x,y,z)} sets the position of the light source to the coordinates \\spad{x},{} \\spad{y},{} and \\spad{z} and displays the graph for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|clipSurface| (((|Void|) $ (|String|)) "\\spad{clipSurface(v,s)} displays the graph with the specified clipping region removed if \\spad{s} is \"on\",{} or displays the graph without clipping implemented if \\spad{s} is \"off\",{} for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|showClipRegion| (((|Void|) $ (|String|)) "\\spad{showClipRegion(v,s)} displays the clipping region of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the region if \\spad{s} is \"off\".")) (|showRegion| (((|Void|) $ (|String|)) "\\spad{showRegion(v,s)} displays the bounding box of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the box if \\spad{s} is \"off\".")) (|hitherPlane| (((|Void|) $ (|Float|)) "\\spad{hitherPlane(v,h)} sets the hither clipping plane of the graph to \\spad{h},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|eyeDistance| (((|Void|) $ (|Float|)) "\\spad{eyeDistance(v,d)} sets the distance of the observer from the center of the graph to \\spad{d},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|perspective| (((|Void|) $ (|String|)) "\\spad{perspective(v,s)} displays the graph in perspective if \\spad{s} is \"on\",{} or does not display perspective if \\spad{s} is \"off\" for the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport}.")) (|translate| (((|Void|) $ (|Float|) (|Float|)) "\\spad{translate(v,dx,dy)} sets the horizontal viewport offset to \\spad{dx} and the vertical viewport offset to \\spad{dy},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|zoom| (((|Void|) $ (|Float|) (|Float|) (|Float|)) "\\spad{zoom(v,sx,sy,sz)} sets the graph scaling factors for the \\spad{x}-coordinate axis to \\spad{sx},{} the \\spad{y}-coordinate axis to \\spad{sy} and the \\spad{z}-coordinate axis to \\spad{sz} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.") (((|Void|) $ (|Float|)) "\\spad{zoom(v,s)} sets the graph scaling factor to \\spad{s},{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|rotate| (((|Void|) $ (|Integer|) (|Integer|)) "\\spad{rotate(v,th,phi)} rotates the graph to the longitudinal view angle \\spad{th} degrees and the latitudinal view angle \\spad{phi} degrees for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new rotation position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.") (((|Void|) $ (|Float|) (|Float|)) "\\spad{rotate(v,th,phi)} rotates the graph to the longitudinal view angle \\spad{th} radians and the latitudinal view angle \\spad{phi} radians for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}.")) (|drawStyle| (((|Void|) $ (|String|)) "\\spad{drawStyle(v,s)} displays the surface for the given three-dimensional viewport \\spad{v} which is of domain \\spadtype{ThreeDimensionalViewport} in the style of drawing indicated by \\spad{s}. If \\spad{s} is not a valid drawing style the style is wireframe by default. Possible styles are \\spad{\"shade\"},{} \\spad{\"solid\"} or \\spad{\"opaque\"},{} \\spad{\"smooth\"},{} and \\spad{\"wireMesh\"}.")) (|outlineRender| (((|Void|) $ (|String|)) "\\spad{outlineRender(v,s)} displays the polygon outline showing either triangularized surface or a quadrilateral surface outline depending on the whether the \\spadfun{diagonals} function has been set,{} for the given three-dimensional viewport \\spad{v} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the polygon outline if \\spad{s} is \"off\".")) (|diagonals| (((|Void|) $ (|String|)) "\\spad{diagonals(v,s)} displays the diagonals of the polygon outline showing a triangularized surface instead of a quadrilateral surface outline,{} for the given three-dimensional viewport \\spad{v} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the diagonals if \\spad{s} is \"off\".")) (|axes| (((|Void|) $ (|String|)) "\\spad{axes(v,s)} displays the axes of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or does not display the axes if \\spad{s} is \"off\".")) (|controlPanel| (((|Void|) $ (|String|)) "\\spad{controlPanel(v,s)} displays the control panel of the given three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} if \\spad{s} is \"on\",{} or hides the control panel if \\spad{s} is \"off\".")) (|viewpoint| (((|Void|) $ (|Float|) (|Float|) (|Float|)) "\\spad{viewpoint(v,rotx,roty,rotz)} sets the rotation about the \\spad{x}-axis to be \\spad{rotx} radians,{} sets the rotation about the \\spad{y}-axis to be \\spad{roty} radians,{} and sets the rotation about the \\spad{z}-axis to be \\spad{rotz} radians,{} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport} and displays \\spad{v} with the new view position.") (((|Void|) $ (|Float|) (|Float|)) "\\spad{viewpoint(v,th,phi)} sets the longitudinal view angle to \\spad{th} radians and the latitudinal view angle to \\spad{phi} radians for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.") (((|Void|) $ (|Integer|) (|Integer|) (|Float|) (|Float|) (|Float|)) "\\spad{viewpoint(v,th,phi,s,dx,dy)} sets the longitudinal view angle to \\spad{th} degrees,{} the latitudinal view angle to \\spad{phi} degrees,{} the scale factor to \\spad{s},{} the horizontal viewport offset to \\spad{dx},{} and the vertical viewport offset to \\spad{dy} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.") (((|Void|) $ (|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|)))) "\\spad{viewpoint(v,viewpt)} sets the viewpoint for the viewport. The viewport record consists of the latitudal and longitudal angles,{} the zoom factor,{} the \\spad{X},{} \\spad{Y},{} and \\spad{Z} scales,{} and the \\spad{X} and \\spad{Y} displacements.") (((|Record| (|:| |theta| (|DoubleFloat|)) (|:| |phi| (|DoubleFloat|)) (|:| |scale| (|DoubleFloat|)) (|:| |scaleX| (|DoubleFloat|)) (|:| |scaleY| (|DoubleFloat|)) (|:| |scaleZ| (|DoubleFloat|)) (|:| |deltaX| (|DoubleFloat|)) (|:| |deltaY| (|DoubleFloat|))) $) "\\spad{viewpoint(v)} returns the current viewpoint setting of the given viewport,{} \\spad{v}. This function is useful in the situation where the user has created a viewport,{} proceeded to interact with it via the control panel and desires to save the values of the viewpoint as the default settings for another viewport to be created using the system.") (((|Void|) $ (|Float|) (|Float|) (|Float|) (|Float|) (|Float|)) "\\spad{viewpoint(v,th,phi,s,dx,dy)} sets the longitudinal view angle to \\spad{th} radians,{} the latitudinal view angle to \\spad{phi} radians,{} the scale factor to \\spad{s},{} the horizontal viewport offset to \\spad{dx},{} and the vertical viewport offset to \\spad{dy} for the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.")) (|dimensions| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|) (|PositiveInteger|) (|PositiveInteger|)) "\\spad{dimensions(v,x,y,width,height)} sets the position of the upper left-hand corner of the three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} to the window coordinate \\spad{x},{} \\spad{y},{} and sets the dimensions of the window to that of \\spad{width},{} \\spad{height}. The new dimensions are not displayed until the function \\spadfun{makeViewport3D} is executed again for \\spad{v}.")) (|title| (((|Void|) $ (|String|)) "\\spad{title(v,s)} changes the title which is shown in the three-dimensional viewport window,{} \\spad{v} of domain \\spadtype{ThreeDimensionalViewport}.")) (|resize| (((|Void|) $ (|PositiveInteger|) (|PositiveInteger|)) "\\spad{resize(v,w,h)} displays the three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} with a width of \\spad{w} and a height of \\spad{h},{} keeping the upper left-hand corner position unchanged.")) (|move| (((|Void|) $ (|NonNegativeInteger|) (|NonNegativeInteger|)) "\\spad{move(v,x,y)} displays the three-dimensional viewport,{} \\spad{v},{} which is of domain \\spadtype{ThreeDimensionalViewport},{} with the upper left-hand corner of the viewport window at the screen coordinate position \\spad{x},{} \\spad{y}.")) (|options| (($ $ (|List| (|DrawOption|))) "\\spad{options(v,lopt)} takes the viewport,{} \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport} and sets the draw options being used by \\spad{v} to those indicated in the list,{} \\spad{lopt},{} which is a list of options from the domain \\spad{DrawOption}.") (((|List| (|DrawOption|)) $) "\\spad{options(v)} takes the viewport,{} \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport} and returns a list of all the draw options from the domain \\spad{DrawOption} which are being used by \\spad{v}.")) (|modifyPointData| (((|Void|) $ (|NonNegativeInteger|) (|Point| (|DoubleFloat|))) "\\spad{modifyPointData(v,ind,pt)} takes the viewport,{} \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport},{} and places the data point,{} \\spad{pt} into the list of points database of \\spad{v} at the index location given by \\spad{ind}.")) (|subspace| (($ $ (|ThreeSpace| (|DoubleFloat|))) "\\spad{subspace(v,sp)} places the contents of the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport},{} in the subspace \\spad{sp},{} which is of the domain \\spad{ThreeSpace}.") (((|ThreeSpace| (|DoubleFloat|)) $) "\\spad{subspace(v)} returns the contents of the viewport \\spad{v},{} which is of the domain \\spadtype{ThreeDimensionalViewport},{} as a subspace of the domain \\spad{ThreeSpace}.")) (|makeViewport3D| (($ (|ThreeSpace| (|DoubleFloat|)) (|List| (|DrawOption|))) "\\spad{makeViewport3D(sp,lopt)} takes the given space,{} \\spad{sp} which is of the domain \\spadtype{ThreeSpace} and displays a viewport window on the screen which contains the contents of \\spad{sp},{} and whose draw options are indicated by the list \\spad{lopt},{} which is a list of options from the domain \\spad{DrawOption}.") (($ (|ThreeSpace| (|DoubleFloat|)) (|String|)) "\\spad{makeViewport3D(sp,s)} takes the given space,{} \\spad{sp} which is of the domain \\spadtype{ThreeSpace} and displays a viewport window on the screen which contains the contents of \\spad{sp},{} and whose title is given by \\spad{s}.") (($ $) "\\spad{makeViewport3D(v)} takes the given three-dimensional viewport,{} \\spad{v},{} of the domain \\spadtype{ThreeDimensionalViewport} and displays a viewport window on the screen which contains the contents of \\spad{v}.")) (|viewport3D| (($) "\\spad{viewport3D()} returns an undefined three-dimensional viewport of the domain \\spadtype{ThreeDimensionalViewport} whose contents are empty.")) (|viewDeltaYDefault| (((|Float|) (|Float|)) "\\spad{viewDeltaYDefault(dy)} sets the current default vertical offset from the center of the viewport window to be \\spad{dy} and returns \\spad{dy}.") (((|Float|)) "\\spad{viewDeltaYDefault()} returns the current default vertical offset from the center of the viewport window.")) (|viewDeltaXDefault| (((|Float|) (|Float|)) "\\spad{viewDeltaXDefault(dx)} sets the current default horizontal offset from the center of the viewport window to be \\spad{dx} and returns \\spad{dx}.") (((|Float|)) "\\spad{viewDeltaXDefault()} returns the current default horizontal offset from the center of the viewport window.")) (|viewZoomDefault| (((|Float|) (|Float|)) "\\spad{viewZoomDefault(s)} sets the current default graph scaling value to \\spad{s} and returns \\spad{s}.") (((|Float|)) "\\spad{viewZoomDefault()} returns the current default graph scaling value.")) (|viewPhiDefault| (((|Float|) (|Float|)) "\\spad{viewPhiDefault(p)} sets the current default latitudinal view angle in radians to the value \\spad{p} and returns \\spad{p}.") (((|Float|)) "\\spad{viewPhiDefault()} returns the current default latitudinal view angle in radians.")) (|viewThetaDefault| (((|Float|) (|Float|)) "\\spad{viewThetaDefault(t)} sets the current default longitudinal view angle in radians to the value \\spad{t} and returns \\spad{t}.") (((|Float|)) "\\spad{viewThetaDefault()} returns the current default longitudinal view angle in radians."))) NIL NIL -(-1175) +(-1182) ((|constructor| (NIL "ViewportDefaultsPackage describes default and user definable values for graphics")) (|tubeRadiusDefault| (((|DoubleFloat|)) "\\spad{tubeRadiusDefault()} returns the radius used for a 3D tube plot.") (((|DoubleFloat|) (|Float|)) "\\spad{tubeRadiusDefault(r)} sets the default radius for a 3D tube plot to \\spad{r}.")) (|tubePointsDefault| (((|PositiveInteger|)) "\\spad{tubePointsDefault()} returns the number of points to be used when creating the circle to be used in creating a 3D tube plot.") (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{tubePointsDefault(i)} sets the number of points to use when creating the circle to be used in creating a 3D tube plot to \\spad{i}.")) (|var2StepsDefault| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{var2StepsDefault(i)} sets the number of steps to take when creating a 3D mesh in the direction of the first defined free variable to \\spad{i} (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).") (((|PositiveInteger|)) "\\spad{var2StepsDefault()} is the current setting for the number of steps to take when creating a 3D mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).")) (|var1StepsDefault| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{var1StepsDefault(i)} sets the number of steps to take when creating a 3D mesh in the direction of the first defined free variable to \\spad{i} (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).") (((|PositiveInteger|)) "\\spad{var1StepsDefault()} is the current setting for the number of steps to take when creating a 3D mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (\\spadignore{e.g.} \\spad{x=0}..10)).")) (|viewWriteAvailable| (((|List| (|String|))) "\\spad{viewWriteAvailable()} returns a list of available methods for writing,{} such as BITMAP,{} POSTSCRIPT,{} etc.")) (|viewWriteDefault| (((|List| (|String|)) (|List| (|String|))) "\\spad{viewWriteDefault(l)} sets the default list of things to write in a viewport data file to the strings in \\spad{l}; a viewAlone file is always genereated.") (((|List| (|String|))) "\\spad{viewWriteDefault()} returns the list of things to write in a viewport data file; a viewAlone file is always generated.")) (|viewDefaults| (((|Void|)) "\\spad{viewDefaults()} resets all the default graphics settings.")) (|viewSizeDefault| (((|List| (|PositiveInteger|)) (|List| (|PositiveInteger|))) "\\spad{viewSizeDefault([w,h])} sets the default viewport width to \\spad{w} and height to \\spad{h}.") (((|List| (|PositiveInteger|))) "\\spad{viewSizeDefault()} returns the default viewport width and height.")) (|viewPosDefault| (((|List| (|NonNegativeInteger|)) (|List| (|NonNegativeInteger|))) "\\spad{viewPosDefault([x,y])} sets the default \\spad{X} and \\spad{Y} position of a viewport window unless overriden explicityly,{} newly created viewports will have th \\spad{X} and \\spad{Y} coordinates \\spad{x},{} \\spad{y}.") (((|List| (|NonNegativeInteger|))) "\\spad{viewPosDefault()} returns the default \\spad{X} and \\spad{Y} position of a viewport window unless overriden explicityly,{} newly created viewports will have this \\spad{X} and \\spad{Y} coordinate.")) (|pointSizeDefault| (((|PositiveInteger|) (|PositiveInteger|)) "\\spad{pointSizeDefault(i)} sets the default size of the points in a 2D viewport to \\spad{i}.") (((|PositiveInteger|)) "\\spad{pointSizeDefault()} returns the default size of the points in a 2D viewport.")) (|unitsColorDefault| (((|Palette|) (|Palette|)) "\\spad{unitsColorDefault(p)} sets the default color of the unit ticks in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{unitsColorDefault()} returns the default color of the unit ticks in a 2D viewport.")) (|axesColorDefault| (((|Palette|) (|Palette|)) "\\spad{axesColorDefault(p)} sets the default color of the axes in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{axesColorDefault()} returns the default color of the axes in a 2D viewport.")) (|lineColorDefault| (((|Palette|) (|Palette|)) "\\spad{lineColorDefault(p)} sets the default color of lines connecting points in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{lineColorDefault()} returns the default color of lines connecting points in a 2D viewport.")) (|pointColorDefault| (((|Palette|) (|Palette|)) "\\spad{pointColorDefault(p)} sets the default color of points in a 2D viewport to the palette \\spad{p}.") (((|Palette|)) "\\spad{pointColorDefault()} returns the default color of points in a 2D viewport."))) NIL NIL -(-1176) +(-1183) ((|constructor| (NIL "This type is used when no value is needed,{} \\spadignore{e.g.} in the \\spad{then} part of a one armed \\spad{if}. All values can be coerced to type Void. Once a value has been coerced to Void,{} it cannot be recovered.")) (|void| (($) "\\spad{void()} produces a void object."))) NIL NIL -(-1177 A S) +(-1184 A S) ((|constructor| (NIL "Vector Spaces (not necessarily finite dimensional) over a field.")) (|dimension| (((|CardinalNumber|)) "\\spad{dimension()} returns the dimensionality of the vector space.")) (/ (($ $ |#2|) "\\spad{x/y} divides the vector \\spad{x} by the scalar \\spad{y}."))) NIL NIL -(-1178 S) +(-1185 S) ((|constructor| (NIL "Vector Spaces (not necessarily finite dimensional) over a field.")) (|dimension| (((|CardinalNumber|)) "\\spad{dimension()} returns the dimensionality of the vector space.")) (/ (($ $ |#1|) "\\spad{x/y} divides the vector \\spad{x} by the scalar \\spad{y}."))) -((-3973 . T) (-3972 . T)) +((-3984 . T) (-3983 . T)) NIL -(-1179 R) +(-1186 R) ((|constructor| (NIL "This package implements the Weierstrass preparation theorem \\spad{f} or multivariate power series. weierstrass(\\spad{v},{}\\spad{p}) where \\spad{v} is a variable,{} and \\spad{p} is a TaylorSeries(\\spad{R}) in which the terms of lowest degree \\spad{s} must include c*v**s where \\spad{c} is a constant,{}\\spad{s>0},{} is a list of TaylorSeries coefficients A[\\spad{i}] of the equivalent polynomial A = A[0] + A[1]*v + A[2]\\spad{*v**2} + ... + A[\\spad{s}-1]*v**(\\spad{s}-1) + v**s such that p=A*B ,{} \\spad{B} being a TaylorSeries of minimum degree 0")) (|qqq| (((|Mapping| (|Stream| (|TaylorSeries| |#1|)) (|Stream| (|TaylorSeries| |#1|))) (|NonNegativeInteger|) (|TaylorSeries| |#1|) (|Stream| (|TaylorSeries| |#1|))) "\\spad{qqq(n,s,st)} is used internally.")) (|weierstrass| (((|List| (|TaylorSeries| |#1|)) (|Symbol|) (|TaylorSeries| |#1|)) "\\spad{weierstrass(v,ts)} where \\spad{v} is a variable and \\spad{ts} is \\indented{1}{a TaylorSeries,{} impements the Weierstrass Preparation} \\indented{1}{Theorem. The result is a list of TaylorSeries that} \\indented{1}{are the coefficients of the equivalent series.}")) (|clikeUniv| (((|Mapping| (|SparseUnivariatePolynomial| (|Polynomial| |#1|)) (|Polynomial| |#1|)) (|Symbol|)) "\\spad{clikeUniv(v)} is used internally.")) (|sts2stst| (((|Stream| (|Stream| (|Polynomial| |#1|))) (|Symbol|) (|Stream| (|Polynomial| |#1|))) "\\spad{sts2stst(v,s)} is used internally.")) (|cfirst| (((|Mapping| (|Stream| (|Polynomial| |#1|)) (|Stream| (|Polynomial| |#1|))) (|NonNegativeInteger|)) "\\spad{cfirst n} is used internally.")) (|crest| (((|Mapping| (|Stream| (|Polynomial| |#1|)) (|Stream| (|Polynomial| |#1|))) (|NonNegativeInteger|)) "\\spad{crest n} is used internally."))) NIL NIL -(-1180 K R UP -3078) +(-1187 K R UP -3088) ((|constructor| (NIL "In this package \\spad{K} is a finite field,{} \\spad{R} is a ring of univariate polynomials over \\spad{K},{} and \\spad{F} is a framed algebra over \\spad{R}. The package provides a function to compute the integral closure of \\spad{R} in the quotient field of \\spad{F} as well as a function to compute a \"local integral basis\" at a specific prime.")) (|localIntegralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|))) |#2|) "\\spad{integralBasis(p)} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the local integral closure of \\spad{R} at the prime \\spad{p} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the local integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}.")) (|integralBasis| (((|Record| (|:| |basis| (|Matrix| |#2|)) (|:| |basisDen| |#2|) (|:| |basisInv| (|Matrix| |#2|)))) "\\spad{integralBasis()} returns a record \\spad{[basis,basisDen,basisInv]} containing information regarding the integral closure of \\spad{R} in the quotient field of \\spad{F},{} where \\spad{F} is a framed algebra with \\spad{R}-module basis \\spad{w1,w2,...,wn}. If \\spad{basis} is the matrix \\spad{(aij, i = 1..n, j = 1..n)},{} then the \\spad{i}th element of the integral basis is \\spad{vi = (1/basisDen) * sum(aij * wj, j = 1..n)},{} \\spadignore{i.e.} the \\spad{i}th row of \\spad{basis} contains the coordinates of the \\spad{i}th basis vector. Similarly,{} the \\spad{i}th row of the matrix \\spad{basisInv} contains the coordinates of \\spad{wi} with respect to the basis \\spad{v1,...,vn}: if \\spad{basisInv} is the matrix \\spad{(bij, i = 1..n, j = 1..n)},{} then \\spad{wi = sum(bij * vj, j = 1..n)}."))) NIL NIL -(-1181) +(-1188) ((|constructor| (NIL "This domain represents the syntax of a `where' expression.")) (|qualifier| (((|SpadAst|) $) "\\spad{qualifier(e)} returns the qualifier of the expression `e'.")) (|mainExpression| (((|SpadAst|) $) "\\spad{mainExpression(e)} returns the main expression of the `where' expression `e'."))) NIL NIL -(-1182) +(-1189) ((|constructor| (NIL "This domain represents the `while' iterator syntax.")) (|condition| (((|SpadAst|) $) "\\spad{condition(i)} returns the condition of the while iterator `i'."))) NIL NIL -(-1183 R |VarSet| E P |vl| |wl| |wtlevel|) +(-1190 R |VarSet| E P |vl| |wl| |wtlevel|) ((|constructor| (NIL "This domain represents truncated weighted polynomials over a general (not necessarily commutative) polynomial type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.")) (|changeWeightLevel| (((|Void|) (|NonNegativeInteger|)) "\\spad{changeWeightLevel(n)} changes the weight level to the new value given: NB: previously calculated terms are not affected")) (/ (((|Union| $ "failed") $ $) "\\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \\spad{R} is a Field)"))) -((-3973 |has| |#1| (-144)) (-3972 |has| |#1| (-144)) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309)))) -(-1184 R E V P) +((-3984 |has| |#1| (-146)) (-3983 |has| |#1| (-146)) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311)))) +(-1191 R E V P) ((|constructor| (NIL "A domain constructor of the category \\axiomType{GeneralTriangularSet}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. The \\axiomOpFrom{construct}{WuWenTsunTriangularSet} operation does not check the previous requirement. Triangular sets are stored as sorted lists \\spad{w}.\\spad{r}.\\spad{t}. the main variables of their members. Furthermore,{} this domain exports operations dealing with the characteristic set method of Wu Wen Tsun and some optimizations mainly proposed by Dong Ming Wang.\\newline References : \\indented{1}{[1] \\spad{W}. \\spad{T}. WU \"A Zero Structure Theorem for polynomial equations solving\"} \\indented{6}{MM Research Preprints,{} 1987.} \\indented{1}{[2] \\spad{D}. \\spad{M}. WANG \"An implementation of the characteristic set method in Maple\"} \\indented{6}{Proc. \\spad{DISCO'92}. Bath,{} England.}")) (|characteristicSerie| (((|List| $) (|List| |#4|)) "\\axiom{characteristicSerie(ps)} returns the same as \\axiom{characteristicSerie(ps,{}initiallyReduced?,{}initiallyReduce)}.") (((|List| $) (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{characteristicSerie(ps,{}redOp?,{}redOp)} returns a list \\axiom{lts} of triangular sets such that the zero set of \\axiom{ps} is the union of the regular zero sets of the members of \\axiom{lts}. This is made by the Ritt and Wu Wen Tsun process applying the operation \\axiom{characteristicSet(ps,{}redOp?,{}redOp)} to compute characteristic sets in Wu Wen Tsun sense.")) (|characteristicSet| (((|Union| $ "failed") (|List| |#4|)) "\\axiom{characteristicSet(ps)} returns the same as \\axiom{characteristicSet(ps,{}initiallyReduced?,{}initiallyReduce)}.") (((|Union| $ "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{characteristicSet(ps,{}redOp?,{}redOp)} returns a non-contradictory characteristic set of \\axiom{ps} in Wu Wen Tsun sense \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?} (using \\axiom{redOp} to reduce polynomials \\spad{w}.\\spad{r}.\\spad{t} a \\axiom{redOp?} basic set),{} if no non-zero constant polynomial appear during those reductions,{} else \\axiom{\"failed\"} is returned. The operations \\axiom{redOp} and \\axiom{redOp?} must satisfy the following conditions: \\axiom{redOp?(redOp(\\spad{p},{}\\spad{q}),{}\\spad{q})} holds for every polynomials \\axiom{\\spad{p},{}\\spad{q}} and there exists an integer \\axiom{\\spad{e}} and a polynomial \\axiom{\\spad{f}} such that we have \\axiom{init(\\spad{q})^e*p = f*q + redOp(\\spad{p},{}\\spad{q})}.")) (|medialSet| (((|Union| $ "failed") (|List| |#4|)) "\\axiom{medial(ps)} returns the same as \\axiom{medialSet(ps,{}initiallyReduced?,{}initiallyReduce)}.") (((|Union| $ "failed") (|List| |#4|) (|Mapping| (|Boolean|) |#4| |#4|) (|Mapping| |#4| |#4| |#4|)) "\\axiom{medialSet(ps,{}redOp?,{}redOp)} returns \\axiom{bs} a basic set (in Wu Wen Tsun sense \\spad{w}.\\spad{r}.\\spad{t} the reduction-test \\axiom{redOp?}) of some set generating the same ideal as \\axiom{ps} (with rank not higher than any basic set of \\axiom{ps}),{} if no non-zero constant polynomials appear during the computatioms,{} else \\axiom{\"failed\"} is returned. In the former case,{} \\axiom{bs} has to be understood as a candidate for being a characteristic set of \\axiom{ps}. In the original algorithm,{} \\axiom{bs} is simply a basic set of \\axiom{ps}."))) -((-3979 . T) (-3978 . T)) -((-12 (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#4| (|%list| (QUOTE -257) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-550 (-469)))) (|HasCategory| |#4| (QUOTE (-1007))) (|HasCategory| |#1| (QUOTE (-491))) (|HasCategory| |#3| (QUOTE (-315))) (|HasCategory| |#4| (QUOTE (-549 (-767)))) (|HasCategory| |#4| (QUOTE (-72)))) -(-1185 R) +((-3990 . T) (-3989 . T)) +((-12 (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#4| (|%list| (QUOTE -259) (|devaluate| |#4|)))) (|HasCategory| |#4| (QUOTE (-553 (-472)))) (|HasCategory| |#4| (QUOTE (-1012))) (|HasCategory| |#1| (QUOTE (-494))) (|HasCategory| |#3| (QUOTE (-317))) (|HasCategory| |#4| (QUOTE (-552 (-772)))) (|HasCategory| |#4| (QUOTE (-72)))) +(-1192 R) ((|constructor| (NIL "This is the category of algebras over non-commutative rings. It is used by constructors of non-commutative algebras such as: \\indented{4}{\\spadtype{XPolynomialRing}.} \\indented{4}{\\spadtype{XFreeAlgebra}} Author: Michel Petitot (petitot@lifl.fr)"))) -((-3972 . T) (-3973 . T) (-3975 . T)) +((-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1186 |vl| R) +(-1193 |vl| R) ((|constructor| (NIL "\\indented{2}{This type supports distributed multivariate polynomials} whose variables do not commute. The coefficient ring may be non-commutative too. However,{} coefficients and variables commute."))) -((-3975 . T) (-3971 |has| |#2| (-6 -3971)) (-3973 . T) (-3972 . T)) -((|HasCategory| |#2| (QUOTE (-144))) (|HasAttribute| |#2| (QUOTE -3971))) -(-1187 R |VarSet| XPOLY) +((-3986 . T) (-3982 |has| |#2| (-6 -3982)) (-3984 . T) (-3983 . T)) +((|HasCategory| |#2| (QUOTE (-146))) (|HasAttribute| |#2| (QUOTE -3982))) +(-1194 R |VarSet| XPOLY) ((|constructor| (NIL "This package provides computations of logarithms and exponentials for polynomials in non-commutative variables. \\newline Author: Michel Petitot (petitot@lifl.fr).")) (|Hausdorff| ((|#3| |#3| |#3| (|NonNegativeInteger|)) "\\axiom{Hausdorff(a,{}\\spad{b},{}\\spad{n})} returns log(exp(a)*exp(\\spad{b})) truncated at order \\axiom{\\spad{n}}.")) (|log| ((|#3| |#3| (|NonNegativeInteger|)) "\\axiom{log(\\spad{p},{} \\spad{n})} returns the logarithm of \\axiom{\\spad{p}} truncated at order \\axiom{\\spad{n}}.")) (|exp| ((|#3| |#3| (|NonNegativeInteger|)) "\\axiom{exp(\\spad{p},{} \\spad{n})} returns the exponential of \\axiom{\\spad{p}} truncated at order \\axiom{\\spad{n}}."))) NIL NIL -(-1188 S -3078) +(-1195 S -3088) ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|NonNegativeInteger|)) "\\spad{Frobenius(a,s)} returns \\spad{a**(q**s)} where \\spad{q} is the size()\\$\\spad{F}.") (($ $) "\\spad{Frobenius(a)} returns \\spad{a ** q} where \\spad{q} is the \\spad{size()\\$F}.")) (|transcendenceDegree| (((|NonNegativeInteger|)) "\\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.")) (|extensionDegree| (((|OnePointCompletion| (|PositiveInteger|))) "\\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \\spad{infinity} if it is not.")) (|degree| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{degree(a)} returns the degree of minimal polynomial of an element \\spad{a} if \\spad{a} is algebraic with respect to the ground field \\spad{F},{} and \\spad{infinity} otherwise.")) (|inGroundField?| (((|Boolean|) $) "\\spad{inGroundField?(a)} tests whether an element \\spad{a} is already in the ground field \\spad{F}.")) (|transcendent?| (((|Boolean|) $) "\\spad{transcendent?(a)} tests whether an element \\spad{a} is transcendent with respect to the ground field \\spad{F}.")) (|algebraic?| (((|Boolean|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) NIL -((|HasCategory| |#2| (QUOTE (-315))) (|HasCategory| |#2| (QUOTE (-116))) (|HasCategory| |#2| (QUOTE (-118)))) -(-1189 -3078) +((|HasCategory| |#2| (QUOTE (-317))) (|HasCategory| |#2| (QUOTE (-118))) (|HasCategory| |#2| (QUOTE (-120)))) +(-1196 -3088) ((|constructor| (NIL "ExtensionField {\\em F} is the category of fields which extend the field \\spad{F}")) (|Frobenius| (($ $ (|NonNegativeInteger|)) "\\spad{Frobenius(a,s)} returns \\spad{a**(q**s)} where \\spad{q} is the size()\\$\\spad{F}.") (($ $) "\\spad{Frobenius(a)} returns \\spad{a ** q} where \\spad{q} is the \\spad{size()\\$F}.")) (|transcendenceDegree| (((|NonNegativeInteger|)) "\\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.")) (|extensionDegree| (((|OnePointCompletion| (|PositiveInteger|))) "\\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \\spad{infinity} if it is not.")) (|degree| (((|OnePointCompletion| (|PositiveInteger|)) $) "\\spad{degree(a)} returns the degree of minimal polynomial of an element \\spad{a} if \\spad{a} is algebraic with respect to the ground field \\spad{F},{} and \\spad{infinity} otherwise.")) (|inGroundField?| (((|Boolean|) $) "\\spad{inGroundField?(a)} tests whether an element \\spad{a} is already in the ground field \\spad{F}.")) (|transcendent?| (((|Boolean|) $) "\\spad{transcendent?(a)} tests whether an element \\spad{a} is transcendent with respect to the ground field \\spad{F}.")) (|algebraic?| (((|Boolean|) $) "\\spad{algebraic?(a)} tests whether an element \\spad{a} is algebraic with respect to the ground field \\spad{F}."))) -((-3970 . T) (-3976 . T) (-3971 . T) ((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +((-3981 . T) (-3987 . T) (-3982 . T) ((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL -(-1190 |vl| R) +(-1197 |vl| R) ((|constructor| (NIL "This category specifies opeations for polynomials and formal series with non-commutative variables.")) (|varList| (((|List| |#1|) $) "\\spad{varList(x)} returns the list of variables which appear in \\spad{x}.")) (|map| (($ (|Mapping| |#2| |#2|) $) "\\spad{map(fn,x)} returns \\spad{Sum(fn(r_i) w_i)} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|sh| (($ $ (|NonNegativeInteger|)) "\\spad{sh(x,n)} returns the shuffle power of \\spad{x} to the \\spad{n}.") (($ $ $) "\\spad{sh(x,y)} returns the shuffle-product of \\spad{x} by \\spad{y}. This multiplication is associative and commutative.")) (|quasiRegular| (($ $) "\\spad{quasiRegular(x)} return \\spad{x} minus its constant term.")) (|quasiRegular?| (((|Boolean|) $) "\\spad{quasiRegular?(x)} return \\spad{true} if \\spad{constant(x)} is zero.")) (|constant| ((|#2| $) "\\spad{constant(x)} returns the constant term of \\spad{x}.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(x)} returns \\spad{true} if \\spad{x} is constant.")) (|coerce| (($ |#1|) "\\spad{coerce(v)} returns \\spad{v}.")) (|mirror| (($ $) "\\spad{mirror(x)} returns \\spad{Sum(r_i mirror(w_i))} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|monomial?| (((|Boolean|) $) "\\spad{monomial?(x)} returns \\spad{true} if \\spad{x} is a monomial")) (|monom| (($ (|OrderedFreeMonoid| |#1|) |#2|) "\\spad{monom(w,r)} returns the product of the word \\spad{w} by the coefficient \\spad{r}.")) (|rquo| (($ $ $) "\\spad{rquo(x,y)} returns the right simplification of \\spad{x} by \\spad{y}.") (($ $ (|OrderedFreeMonoid| |#1|)) "\\spad{rquo(x,w)} returns the right simplification of \\spad{x} by \\spad{w}.") (($ $ |#1|) "\\spad{rquo(x,v)} returns the right simplification of \\spad{x} by the variable \\spad{v}.")) (|lquo| (($ $ $) "\\spad{lquo(x,y)} returns the left simplification of \\spad{x} by \\spad{y}.") (($ $ (|OrderedFreeMonoid| |#1|)) "\\spad{lquo(x,w)} returns the left simplification of \\spad{x} by the word \\spad{w}.") (($ $ |#1|) "\\spad{lquo(x,v)} returns the left simplification of \\spad{x} by the variable \\spad{v}.")) (|coef| ((|#2| $ $) "\\spad{coef(x,y)} returns scalar product of \\spad{x} by \\spad{y},{} the set of words being regarded as an orthogonal basis.") ((|#2| $ (|OrderedFreeMonoid| |#1|)) "\\spad{coef(x,w)} returns the coefficient of the word \\spad{w} in \\spad{x}.")) (|mindegTerm| (((|Record| (|:| |k| (|OrderedFreeMonoid| |#1|)) (|:| |c| |#2|)) $) "\\spad{mindegTerm(x)} returns the term whose word is \\spad{mindeg(x)}.")) (|mindeg| (((|OrderedFreeMonoid| |#1|) $) "\\spad{mindeg(x)} returns the little word which appears in \\spad{x}. Error if \\spad{x=0}.")) (* (($ $ |#2|) "\\spad{x * r} returns the product of \\spad{x} by \\spad{r}. Usefull if \\spad{R} is a non-commutative Ring.") (($ |#1| $) "\\spad{v * x} returns the product of a variable \\spad{x} by \\spad{x}."))) -((-3971 |has| |#2| (-6 -3971)) (-3973 . T) (-3972 . T) (-3975 . T)) +((-3982 |has| |#2| (-6 -3982)) (-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-1191 |VarSet| R) +(-1198 |VarSet| R) ((|constructor| (NIL "This domain constructor implements polynomials in non-commutative variables written in the Poincare-Birkhoff-Witt basis from the Lyndon basis. These polynomials can be used to compute Baker-Campbell-Hausdorff relations. \\newline Author: Michel Petitot (petitot@lifl.fr).")) (|log| (($ $ (|NonNegativeInteger|)) "\\axiom{log(\\spad{p},{}\\spad{n})} returns the logarithm of \\axiom{\\spad{p}} (truncated up to order \\axiom{\\spad{n}}).")) (|exp| (($ $ (|NonNegativeInteger|)) "\\axiom{exp(\\spad{p},{}\\spad{n})} returns the exponential of \\axiom{\\spad{p}} (truncated up to order \\axiom{\\spad{n}}).")) (|product| (($ $ $ (|NonNegativeInteger|)) "\\axiom{product(a,{}\\spad{b},{}\\spad{n})} returns \\axiom{a*b} (truncated up to order \\axiom{\\spad{n}}).")) (|LiePolyIfCan| (((|Union| (|LiePolynomial| |#1| |#2|) "failed") $) "\\axiom{LiePolyIfCan(\\spad{p})} return \\axiom{\\spad{p}} if \\axiom{\\spad{p}} is a Lie polynomial.")) (|coerce| (((|XRecursivePolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{p})} returns \\axiom{\\spad{p}} as a recursive polynomial.") (((|XDistributedPolynomial| |#1| |#2|) $) "\\axiom{coerce(\\spad{p})} returns \\axiom{\\spad{p}} as a distributed polynomial.") (($ (|LiePolynomial| |#1| |#2|)) "\\axiom{coerce(\\spad{p})} returns \\axiom{\\spad{p}}."))) -((-3971 |has| |#2| (-6 -3971)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-144))) (|HasCategory| |#2| (QUOTE (-651 (-345 (-480))))) (|HasAttribute| |#2| (QUOTE -3971))) -(-1192 R) +((-3982 |has| |#2| (-6 -3982)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-146))) (|HasCategory| |#2| (QUOTE (-654 (-347 (-483))))) (|HasAttribute| |#2| (QUOTE -3982))) +(-1199 R) ((|constructor| (NIL "\\indented{2}{This type supports multivariate polynomials} whose set of variables is \\spadtype{Symbol}. The representation is recursive. The coefficient ring may be non-commutative and the variables do not commute. However,{} coefficients and variables commute."))) -((-3971 |has| |#1| (-6 -3971)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasAttribute| |#1| (QUOTE -3971))) -(-1193 |vl| R) +((-3982 |has| |#1| (-6 -3982)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasAttribute| |#1| (QUOTE -3982))) +(-1200 |vl| R) ((|constructor| (NIL "The Category of polynomial rings with non-commutative variables. The coefficient ring may be non-commutative too. However coefficients commute with vaiables.")) (|trunc| (($ $ (|NonNegativeInteger|)) "\\spad{trunc(p,n)} returns the polynomial \\spad{p} truncated at order \\spad{n}.")) (|degree| (((|NonNegativeInteger|) $) "\\spad{degree(p)} returns the degree of \\spad{p}. \\indented{1}{Note that the degree of a word is its length.}")) (|maxdeg| (((|OrderedFreeMonoid| |#1|) $) "\\spad{maxdeg(p)} returns the greatest leading word in the support of \\spad{p}."))) -((-3971 |has| |#2| (-6 -3971)) (-3973 . T) (-3972 . T) (-3975 . T)) +((-3982 |has| |#2| (-6 -3982)) (-3984 . T) (-3983 . T) (-3986 . T)) NIL -(-1194 R E) +(-1201 R E) ((|constructor| (NIL "This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and words belonging to an arbitrary \\spadtype{OrderedMonoid}. This type is used,{} for instance,{} by the \\spadtype{XDistributedPolynomial} domain constructor where the Monoid is free.")) (|canonicalUnitNormal| ((|attribute|) "canonicalUnitNormal guarantees that the function unitCanonical returns the same representative for all associates of any particular element.")) (/ (($ $ |#1|) "\\spad{p/r} returns \\spad{p*(1/r)}.")) (|map| (($ (|Mapping| |#1| |#1|) $) "\\spad{map(fn,x)} returns \\spad{Sum(fn(r_i) w_i)} if \\spad{x} writes \\spad{Sum(r_i w_i)}.")) (|quasiRegular| (($ $) "\\spad{quasiRegular(x)} return \\spad{x} minus its constant term.")) (|quasiRegular?| (((|Boolean|) $) "\\spad{quasiRegular?(x)} return \\spad{true} if \\spad{constant(p)} is zero.")) (|constant| ((|#1| $) "\\spad{constant(p)} return the constant term of \\spad{p}.")) (|constant?| (((|Boolean|) $) "\\spad{constant?(p)} tests whether the polynomial \\spad{p} belongs to the coefficient ring.")) (|coef| ((|#1| $ |#2|) "\\spad{coef(p,e)} extracts the coefficient of the monomial \\spad{e}. Returns zero if \\spad{e} is not present.")) (|reductum| (($ $) "\\spad{reductum(p)} returns \\spad{p} minus its leading term. An error is produced if \\spad{p} is zero.")) (|mindeg| ((|#2| $) "\\spad{mindeg(p)} returns the smallest word occurring in the polynomial \\spad{p} with a non-zero coefficient. An error is produced if \\spad{p} is zero.")) (|maxdeg| ((|#2| $) "\\spad{maxdeg(p)} returns the greatest word occurring in the polynomial \\spad{p} with a non-zero coefficient. An error is produced if \\spad{p} is zero.")) (|#| (((|NonNegativeInteger|) $) "\\spad{\\# p} returns the number of terms in \\spad{p}.")) (* (($ $ |#1|) "\\spad{p*r} returns the product of \\spad{p} by \\spad{r}."))) -((-3975 . T) (-3976 |has| |#1| (-6 -3976)) (-3971 |has| |#1| (-6 -3971)) (-3973 . T) (-3972 . T)) -((|HasCategory| |#1| (QUOTE (-144))) (|HasCategory| |#1| (QUOTE (-309))) (|HasAttribute| |#1| (QUOTE -3975)) (|HasAttribute| |#1| (QUOTE -3976)) (|HasAttribute| |#1| (QUOTE -3971))) -(-1195 |VarSet| R) +((-3986 . T) (-3987 |has| |#1| (-6 -3987)) (-3982 |has| |#1| (-6 -3982)) (-3984 . T) (-3983 . T)) +((|HasCategory| |#1| (QUOTE (-146))) (|HasCategory| |#1| (QUOTE (-311))) (|HasAttribute| |#1| (QUOTE -3986)) (|HasAttribute| |#1| (QUOTE -3987)) (|HasAttribute| |#1| (QUOTE -3982))) +(-1202 |VarSet| R) ((|constructor| (NIL "\\indented{2}{This type supports multivariate polynomials} whose variables do not commute. The representation is recursive. The coefficient ring may be non-commutative. Coefficients and variables commute.")) (|RemainderList| (((|List| (|Record| (|:| |k| |#1|) (|:| |c| $))) $) "\\spad{RemainderList(p)} returns the regular part of \\spad{p} as a list of terms.")) (|unexpand| (($ (|XDistributedPolynomial| |#1| |#2|)) "\\spad{unexpand(p)} returns \\spad{p} in recursive form.")) (|expand| (((|XDistributedPolynomial| |#1| |#2|) $) "\\spad{expand(p)} returns \\spad{p} in distributed form."))) -((-3971 |has| |#2| (-6 -3971)) (-3973 . T) (-3972 . T) (-3975 . T)) -((|HasCategory| |#2| (QUOTE (-144))) (|HasAttribute| |#2| (QUOTE -3971))) -(-1196) +((-3982 |has| |#2| (-6 -3982)) (-3984 . T) (-3983 . T) (-3986 . T)) +((|HasCategory| |#2| (QUOTE (-146))) (|HasAttribute| |#2| (QUOTE -3982))) +(-1203) ((|constructor| (NIL "This domain provides representations of Young diagrams.")) (|shape| (((|Partition|) $) "\\spad{shape x} returns the partition shaping \\spad{x}.")) (|youngDiagram| (($ (|List| (|PositiveInteger|))) "\\spad{youngDiagram l} returns an object representing a Young diagram with shape given by the list of integers \\spad{l}"))) NIL NIL -(-1197 A) +(-1204 A) ((|constructor| (NIL "This package implements fixed-point computations on streams.")) (Y (((|List| (|Stream| |#1|)) (|Mapping| (|List| (|Stream| |#1|)) (|List| (|Stream| |#1|))) (|Integer|)) "\\spad{Y(g,n)} computes a fixed point of the function \\spad{g},{} where \\spad{g} takes a list of \\spad{n} streams and returns a list of \\spad{n} streams.") (((|Stream| |#1|) (|Mapping| (|Stream| |#1|) (|Stream| |#1|))) "\\spad{Y(f)} computes a fixed point of the function \\spad{f}."))) NIL NIL -(-1198 R |ls| |ls2|) +(-1205 R |ls| |ls2|) ((|constructor| (NIL "A package for computing symbolically the complex and real roots of zero-dimensional algebraic systems over the integer or rational numbers. Complex roots are given by means of univariate representations of irreducible regular chains. Real roots are given by means of tuples of coordinates lying in the \\spadtype{RealClosure} of the coefficient ring. This constructor takes three arguments. The first one \\spad{R} is the coefficient ring. The second one \\spad{ls} is the list of variables involved in the systems to solve. The third one must be \\spad{concat(ls,s)} where \\spad{s} is an additional symbol used for the univariate representations. WARNING: The third argument is not checked. All operations are based on triangular decompositions. The default is to compute these decompositions directly from the input system by using the \\spadtype{RegularChain} domain constructor. The lexTriangular algorithm can also be used for computing these decompositions (see the \\spadtype{LexTriangularPackage} package constructor). For that purpose,{} the operations \\axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage},{} \\axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage} and \\axiomOpFrom{positiveSolve}{ZeroDimensionalSolvePackage} admit an optional argument. \\newline Author: Marc Moreno Maza.")) (|convert| (((|List| (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|))) (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#3|)) (|OrderedVariableList| |#3|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|)))) "\\spad{convert(st)} returns the members of \\spad{st}. ") (((|SparseUnivariatePolynomial| (|RealClosure| (|Fraction| |#1|))) (|SparseUnivariatePolynomial| |#1|)) "\\spad{convert(u)} converts \\spad{u}.") (((|Polynomial| (|RealClosure| (|Fraction| |#1|))) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|))) "\\spad{convert(q)} converts \\spad{q}.") (((|Polynomial| (|RealClosure| (|Fraction| |#1|))) (|Polynomial| |#1|)) "\\spad{convert(p)} converts \\spad{p}.") (((|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|)) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#2|))) "\\spad{convert(q)} converts \\spad{q}.")) (|squareFree| (((|List| (|SquareFreeRegularTriangularSet| |#1| (|IndexedExponents| (|OrderedVariableList| |#3|)) (|OrderedVariableList| |#3|) (|NewSparseMultivariatePolynomial| |#1| (|OrderedVariableList| |#3|)))) (|RegularChain| |#1| |#2|)) "\\spad{squareFree(ts)} returns the square-free factorization of \\spad{ts}. Moreover,{} each factor is a Lazard triangular set and the decomposition is a Kalkbrener split of \\spad{ts},{} which is enough here for the matter of solving zero-dimensional algebraic systems. WARNING: \\spad{ts} is not checked to be zero-dimensional.")) (|positiveSolve| (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|))) "\\spad{positiveSolve(lp)} returns the same as \\spad{positiveSolve(lp,false,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{positiveSolve(lp)} returns the same as \\spad{positiveSolve(lp,info?,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{positiveSolve(lp,info?,lextri?)} returns the set of the points in the variety associated with \\spad{lp} whose coordinates are (real) strictly positive. Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during decomposition into regular chains. If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\\spad{lp},{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}. WARNING: For each set of coordinates given by \\spad{positiveSolve(lp,info?,lextri?)} the ordering of the indeterminates is reversed \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ls}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|RegularChain| |#1| |#2|)) "\\spad{positiveSolve(ts)} returns the points of the regular set of \\spad{ts} with (real) strictly positive coordinates.")) (|realSolve| (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|))) "\\spad{realSolve(lp)} returns the same as \\spad{realSolve(ts,false,false,false)}") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{realSolve(ts,info?)} returns the same as \\spad{realSolve(ts,info?,false,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{realSolve(ts,info?,check?)} returns the same as \\spad{realSolve(ts,info?,check?,false)}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|) (|Boolean|)) "\\spad{realSolve(ts,info?,check?,lextri?)} returns the set of the points in the variety associated with \\spad{lp} whose coordinates are all real. Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during decomposition into regular chains. If \\spad{check?} is \\spad{true} then the result is checked. If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(lp,{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}. WARNING: For each set of coordinates given by \\spad{realSolve(ts,info?,check?,lextri?)} the ordering of the indeterminates is reversed \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ls}.") (((|List| (|List| (|RealClosure| (|Fraction| |#1|)))) (|RegularChain| |#1| |#2|)) "\\spad{realSolve(ts)} returns the set of the points in the regular zero set of \\spad{ts} whose coordinates are all real. WARNING: For each set of coordinates given by \\spad{realSolve(ts)} the ordering of the indeterminates is reversed \\spad{w}.\\spad{r}.\\spad{t}. \\spad{ls}.")) (|univariateSolve| (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|))) "\\spad{univariateSolve(lp)} returns the same as \\spad{univariateSolve(lp,false,false,false)}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{univariateSolve(lp,info?)} returns the same as \\spad{univariateSolve(lp,info?,false,false)}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{univariateSolve(lp,info?,check?)} returns the same as \\spad{univariateSolve(lp,info?,check?,false)}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|) (|Boolean|)) "\\spad{univariateSolve(lp,info?,check?,lextri?)} returns a univariate representation of the variety associated with \\spad{lp}. Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during the decomposition into regular chains. If \\spad{check?} is \\spad{true} then the result is checked. See \\axiomOpFrom{rur}{RationalUnivariateRepresentationPackage}(\\spad{lp},{}\\spad{true}). If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\\spad{lp},{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}.") (((|List| (|Record| (|:| |complexRoots| (|SparseUnivariatePolynomial| |#1|)) (|:| |coordinates| (|List| (|Polynomial| |#1|))))) (|RegularChain| |#1| |#2|)) "\\spad{univariateSolve(ts)} returns a univariate representation of \\spad{ts}. See \\axiomOpFrom{rur}{RationalUnivariateRepresentationPackage}(lp,{}\\spad{true}).")) (|triangSolve| (((|List| (|RegularChain| |#1| |#2|)) (|List| (|Polynomial| |#1|))) "\\spad{triangSolve(lp)} returns the same as \\spad{triangSolve(lp,false,false)}") (((|List| (|RegularChain| |#1| |#2|)) (|List| (|Polynomial| |#1|)) (|Boolean|)) "\\spad{triangSolve(lp,info?)} returns the same as \\spad{triangSolve(lp,false)}") (((|List| (|RegularChain| |#1| |#2|)) (|List| (|Polynomial| |#1|)) (|Boolean|) (|Boolean|)) "\\spad{triangSolve(lp,info?,lextri?)} decomposes the variety associated with \\axiom{\\spad{lp}} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \\axiom{\\spad{lp}} needs to generate a zero-dimensional ideal. If \\axiom{\\spad{lp}} is not zero-dimensional then the result is only a decomposition of its zero-set in the sense of the closure (\\spad{w}.\\spad{r}.\\spad{t}. Zarisky topology). Moreover,{} if \\spad{info?} is \\spad{true} then some information is displayed during the computations. See \\axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory}(\\spad{lp},{}\\spad{true},{}\\spad{info?}). If \\spad{lextri?} is \\spad{true} then the lexTriangular algorithm is called from the \\spadtype{LexTriangularPackage} constructor (see \\axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\\spad{lp},{}\\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \\axiomOpFrom{zeroSetSplit}{RegularChain} from \\spadtype{RegularChain}."))) NIL NIL -(-1199 R) +(-1206 R) ((|constructor| (NIL "Test for linear dependence over the integers.")) (|solveLinearlyOverQ| (((|Union| (|Vector| (|Fraction| (|Integer|))) "failed") (|Vector| |#1|) |#1|) "\\spad{solveLinearlyOverQ([v1,...,vn], u)} returns \\spad{[c1,...,cn]} such that \\spad{c1*v1 + ... + cn*vn = u},{} \"failed\" if no such rational numbers \\spad{ci}'s exist.")) (|linearDependenceOverZ| (((|Union| (|Vector| (|Integer|)) "failed") (|Vector| |#1|)) "\\spad{linearlyDependenceOverZ([v1,...,vn])} returns \\spad{[c1,...,cn]} if \\spad{c1*v1 + ... + cn*vn = 0} and not all the \\spad{ci}'s are 0,{} \"failed\" if the \\spad{vi}'s are linearly independent over the integers.")) (|linearlyDependentOverZ?| (((|Boolean|) (|Vector| |#1|)) "\\spad{linearlyDependentOverZ?([v1,...,vn])} returns \\spad{true} if the \\spad{vi}'s are linearly dependent over the integers,{} \\spad{false} otherwise."))) NIL NIL -(-1200 |p|) +(-1207 |p|) ((|constructor| (NIL "IntegerMod(\\spad{n}) creates the ring of integers reduced modulo the integer \\spad{n}."))) -(((-3980 "*") . T) (-3972 . T) (-3973 . T) (-3975 . T)) +(((-3991 "*") . T) (-3983 . T) (-3984 . T) (-3986 . T)) NIL NIL NIL @@ -4748,4 +4776,4 @@ NIL NIL NIL NIL -((-3 NIL 1962874 1962879 1962884 1962889) (-2 NIL 1962854 1962859 1962864 1962869) (-1 NIL 1962834 1962839 1962844 1962849) (0 NIL 1962814 1962819 1962824 1962829) (-1200 "ZMOD.spad" 1962623 1962636 1962752 1962809) (-1199 "ZLINDEP.spad" 1961721 1961732 1962613 1962618) (-1198 "ZDSOLVE.spad" 1951681 1951703 1961711 1961716) (-1197 "YSTREAM.spad" 1951176 1951187 1951671 1951676) (-1196 "YDIAGRAM.spad" 1950810 1950819 1951166 1951171) (-1195 "XRPOLY.spad" 1950030 1950050 1950666 1950735) (-1194 "XPR.spad" 1947825 1947838 1949748 1949847) (-1193 "XPOLYC.spad" 1947144 1947160 1947751 1947820) (-1192 "XPOLY.spad" 1946699 1946710 1947000 1947069) (-1191 "XPBWPOLY.spad" 1945170 1945190 1946505 1946574) (-1190 "XFALG.spad" 1942218 1942234 1945096 1945165) (-1189 "XF.spad" 1940681 1940696 1942120 1942213) (-1188 "XF.spad" 1939124 1939141 1940565 1940570) (-1187 "XEXPPKG.spad" 1938383 1938409 1939114 1939119) (-1186 "XDPOLY.spad" 1937997 1938013 1938239 1938308) (-1185 "XALG.spad" 1937665 1937676 1937953 1937992) (-1184 "WUTSET.spad" 1933668 1933685 1937299 1937326) (-1183 "WP.spad" 1932875 1932919 1933526 1933593) (-1182 "WHILEAST.spad" 1932673 1932682 1932865 1932870) (-1181 "WHEREAST.spad" 1932344 1932353 1932663 1932668) (-1180 "WFFINTBS.spad" 1930007 1930029 1932334 1932339) (-1179 "WEIER.spad" 1928229 1928240 1929997 1930002) (-1178 "VSPACE.spad" 1927902 1927913 1928197 1928224) (-1177 "VSPACE.spad" 1927595 1927608 1927892 1927897) (-1176 "VOID.spad" 1927272 1927281 1927585 1927590) (-1175 "VIEWDEF.spad" 1922473 1922482 1927262 1927267) (-1174 "VIEW3D.spad" 1906434 1906443 1922463 1922468) (-1173 "VIEW2D.spad" 1894333 1894342 1906424 1906429) (-1172 "VIEW.spad" 1892053 1892062 1894323 1894328) (-1171 "VECTOR2.spad" 1890692 1890705 1892043 1892048) (-1170 "VECTOR.spad" 1889411 1889422 1889662 1889689) (-1169 "VECTCAT.spad" 1887323 1887334 1889379 1889406) (-1168 "VECTCAT.spad" 1885044 1885057 1887102 1887107) (-1167 "VARIABLE.spad" 1884824 1884839 1885034 1885039) (-1166 "UTYPE.spad" 1884468 1884477 1884814 1884819) (-1165 "UTSODETL.spad" 1883763 1883787 1884424 1884429) (-1164 "UTSODE.spad" 1881979 1881999 1883753 1883758) (-1163 "UTSCAT.spad" 1879458 1879474 1881877 1881974) (-1162 "UTSCAT.spad" 1876605 1876623 1879026 1879031) (-1161 "UTS2.spad" 1876200 1876235 1876595 1876600) (-1160 "UTS.spad" 1871212 1871240 1874732 1874829) (-1159 "URAGG.spad" 1865933 1865944 1871202 1871207) (-1158 "URAGG.spad" 1860618 1860631 1865889 1865894) (-1157 "UPXSSING.spad" 1858386 1858412 1859822 1859955) (-1156 "UPXSCONS.spad" 1856204 1856224 1856577 1856726) (-1155 "UPXSCCA.spad" 1854775 1854795 1856050 1856199) (-1154 "UPXSCCA.spad" 1853488 1853510 1854765 1854770) (-1153 "UPXSCAT.spad" 1852077 1852093 1853334 1853483) (-1152 "UPXS2.spad" 1851620 1851673 1852067 1852072) (-1151 "UPXS.spad" 1848975 1849003 1849811 1849960) (-1150 "UPSQFREE.spad" 1847390 1847404 1848965 1848970) (-1149 "UPSCAT.spad" 1845185 1845209 1847288 1847385) (-1148 "UPSCAT.spad" 1842681 1842707 1844786 1844791) (-1147 "UPOLYC2.spad" 1842152 1842171 1842671 1842676) (-1146 "UPOLYC.spad" 1837232 1837243 1841994 1842147) (-1145 "UPOLYC.spad" 1832230 1832243 1836994 1836999) (-1144 "UPMP.spad" 1831162 1831175 1832220 1832225) (-1143 "UPDIVP.spad" 1830727 1830741 1831152 1831157) (-1142 "UPDECOMP.spad" 1828988 1829002 1830717 1830722) (-1141 "UPCDEN.spad" 1828205 1828221 1828978 1828983) (-1140 "UP2.spad" 1827569 1827590 1828195 1828200) (-1139 "UP.spad" 1825039 1825054 1825426 1825579) (-1138 "UNISEG2.spad" 1824536 1824549 1824995 1825000) (-1137 "UNISEG.spad" 1823889 1823900 1824455 1824460) (-1136 "UNIFACT.spad" 1822992 1823004 1823879 1823884) (-1135 "ULSCONS.spad" 1817035 1817055 1817405 1817554) (-1134 "ULSCCAT.spad" 1814772 1814792 1816881 1817030) (-1133 "ULSCCAT.spad" 1812617 1812639 1814728 1814733) (-1132 "ULSCAT.spad" 1810857 1810873 1812463 1812612) (-1131 "ULS2.spad" 1810371 1810424 1810847 1810852) (-1130 "ULS.spad" 1802637 1802665 1803582 1804005) (-1129 "UINT8.spad" 1802514 1802523 1802627 1802632) (-1128 "UINT64.spad" 1802390 1802399 1802504 1802509) (-1127 "UINT32.spad" 1802266 1802275 1802380 1802385) (-1126 "UINT16.spad" 1802142 1802151 1802256 1802261) (-1125 "UFD.spad" 1801207 1801216 1802068 1802137) (-1124 "UFD.spad" 1800334 1800345 1801197 1801202) (-1123 "UDVO.spad" 1799215 1799224 1800324 1800329) (-1122 "UDPO.spad" 1796796 1796807 1799171 1799176) (-1121 "TYPEAST.spad" 1796715 1796724 1796786 1796791) (-1120 "TYPE.spad" 1796647 1796656 1796705 1796710) (-1119 "TWOFACT.spad" 1795299 1795314 1796637 1796642) (-1118 "TUPLE.spad" 1794806 1794817 1795211 1795216) (-1117 "TUBETOOL.spad" 1791673 1791682 1794796 1794801) (-1116 "TUBE.spad" 1790320 1790337 1791663 1791668) (-1115 "TSETCAT.spad" 1778391 1778408 1790288 1790315) (-1114 "TSETCAT.spad" 1766448 1766467 1778347 1778352) (-1113 "TS.spad" 1765076 1765092 1766042 1766139) (-1112 "TRMANIP.spad" 1759440 1759457 1764764 1764769) (-1111 "TRIMAT.spad" 1758403 1758428 1759430 1759435) (-1110 "TRIGMNIP.spad" 1756930 1756947 1758393 1758398) (-1109 "TRIGCAT.spad" 1756442 1756451 1756920 1756925) (-1108 "TRIGCAT.spad" 1755952 1755963 1756432 1756437) (-1107 "TREE.spad" 1754592 1754603 1755624 1755651) (-1106 "TRANFUN.spad" 1754431 1754440 1754582 1754587) (-1105 "TRANFUN.spad" 1754268 1754279 1754421 1754426) (-1104 "TOPSP.spad" 1753942 1753951 1754258 1754263) (-1103 "TOOLSIGN.spad" 1753605 1753616 1753932 1753937) (-1102 "TEXTFILE.spad" 1752166 1752175 1753595 1753600) (-1101 "TEX1.spad" 1751722 1751733 1752156 1752161) (-1100 "TEX.spad" 1748916 1748925 1751712 1751717) (-1099 "TBCMPPK.spad" 1747017 1747040 1748906 1748911) (-1098 "TBAGG.spad" 1746075 1746098 1746997 1747012) (-1097 "TBAGG.spad" 1745141 1745166 1746065 1746070) (-1096 "TANEXP.spad" 1744549 1744560 1745131 1745136) (-1095 "TALGOP.spad" 1744273 1744284 1744539 1744544) (-1094 "TABLEAU.spad" 1743754 1743765 1744263 1744268) (-1093 "TABLE.spad" 1742029 1742052 1742299 1742326) (-1092 "TABLBUMP.spad" 1738808 1738819 1742019 1742024) (-1091 "SYSTEM.spad" 1738036 1738045 1738798 1738803) (-1090 "SYSSOLP.spad" 1735519 1735530 1738026 1738031) (-1089 "SYSPTR.spad" 1735418 1735427 1735509 1735514) (-1088 "SYSNNI.spad" 1734641 1734652 1735408 1735413) (-1087 "SYSINT.spad" 1734045 1734056 1734631 1734636) (-1086 "SYNTAX.spad" 1730379 1730388 1734035 1734040) (-1085 "SYMTAB.spad" 1728447 1728456 1730369 1730374) (-1084 "SYMS.spad" 1724476 1724485 1728437 1728442) (-1083 "SYMPOLY.spad" 1723609 1723620 1723691 1723818) (-1082 "SYMFUNC.spad" 1723110 1723121 1723599 1723604) (-1081 "SYMBOL.spad" 1720605 1720614 1723100 1723105) (-1080 "SUTS.spad" 1717718 1717746 1719137 1719234) (-1079 "SUPXS.spad" 1715060 1715088 1715909 1716058) (-1078 "SUPFRACF.spad" 1714165 1714183 1715050 1715055) (-1077 "SUP2.spad" 1713557 1713570 1714155 1714160) (-1076 "SUP.spad" 1710641 1710652 1711414 1711567) (-1075 "SUMRF.spad" 1709615 1709626 1710631 1710636) (-1074 "SUMFS.spad" 1709244 1709261 1709605 1709610) (-1073 "SULS.spad" 1701497 1701525 1702455 1702878) (-1072 "syntax.spad" 1701266 1701275 1701487 1701492) (-1071 "SUCH.spad" 1700956 1700971 1701256 1701261) (-1070 "SUBSPACE.spad" 1693087 1693102 1700946 1700951) (-1069 "SUBRESP.spad" 1692257 1692271 1693043 1693048) (-1068 "STTFNC.spad" 1688725 1688741 1692247 1692252) (-1067 "STTF.spad" 1684824 1684840 1688715 1688720) (-1066 "STTAYLOR.spad" 1677501 1677512 1684731 1684736) (-1065 "STRTBL.spad" 1675888 1675905 1676037 1676064) (-1064 "STRING.spad" 1674756 1674765 1675141 1675168) (-1063 "STREAM3.spad" 1674329 1674344 1674746 1674751) (-1062 "STREAM2.spad" 1673457 1673470 1674319 1674324) (-1061 "STREAM1.spad" 1673163 1673174 1673447 1673452) (-1060 "STREAM.spad" 1670159 1670170 1672766 1672781) (-1059 "STINPROD.spad" 1669095 1669111 1670149 1670154) (-1058 "STEPAST.spad" 1668329 1668338 1669085 1669090) (-1057 "STEP.spad" 1667646 1667655 1668319 1668324) (-1056 "STBL.spad" 1666036 1666064 1666203 1666218) (-1055 "STAGG.spad" 1664735 1664746 1666026 1666031) (-1054 "STAGG.spad" 1663432 1663445 1664725 1664730) (-1053 "STACK.spad" 1662854 1662865 1663104 1663131) (-1052 "SRING.spad" 1662614 1662623 1662844 1662849) (-1051 "SREGSET.spad" 1660346 1660363 1662248 1662275) (-1050 "SRDCMPK.spad" 1658923 1658943 1660336 1660341) (-1049 "SRAGG.spad" 1654106 1654115 1658891 1658918) (-1048 "SRAGG.spad" 1649309 1649320 1654096 1654101) (-1047 "SQMATRIX.spad" 1646986 1647004 1647902 1647989) (-1046 "SPLTREE.spad" 1641728 1641741 1646524 1646551) (-1045 "SPLNODE.spad" 1638348 1638361 1641718 1641723) (-1044 "SPFCAT.spad" 1637157 1637166 1638338 1638343) (-1043 "SPECOUT.spad" 1635709 1635718 1637147 1637152) (-1042 "SPADXPT.spad" 1627800 1627809 1635699 1635704) (-1041 "spad-parser.spad" 1627265 1627274 1627790 1627795) (-1040 "SPADAST.spad" 1626966 1626975 1627255 1627260) (-1039 "SPACEC.spad" 1611181 1611192 1626956 1626961) (-1038 "SPACE3.spad" 1610957 1610968 1611171 1611176) (-1037 "SORTPAK.spad" 1610506 1610519 1610913 1610918) (-1036 "SOLVETRA.spad" 1608269 1608280 1610496 1610501) (-1035 "SOLVESER.spad" 1606725 1606736 1608259 1608264) (-1034 "SOLVERAD.spad" 1602751 1602762 1606715 1606720) (-1033 "SOLVEFOR.spad" 1601213 1601231 1602741 1602746) (-1032 "SNTSCAT.spad" 1600813 1600830 1601181 1601208) (-1031 "SMTS.spad" 1599130 1599156 1600407 1600504) (-1030 "SMP.spad" 1596938 1596958 1597328 1597455) (-1029 "SMITH.spad" 1595783 1595808 1596928 1596933) (-1028 "SMATCAT.spad" 1593901 1593931 1595727 1595778) (-1027 "SMATCAT.spad" 1591951 1591983 1593779 1593784) (-1026 "SKAGG.spad" 1590920 1590931 1591919 1591946) (-1025 "SINT.spad" 1590219 1590228 1590786 1590915) (-1024 "SIMPAN.spad" 1589947 1589956 1590209 1590214) (-1023 "SIGNRF.spad" 1589072 1589083 1589937 1589942) (-1022 "SIGNEF.spad" 1588358 1588375 1589062 1589067) (-1021 "syntax.spad" 1587775 1587784 1588348 1588353) (-1020 "SIG.spad" 1587137 1587146 1587765 1587770) (-1019 "SHP.spad" 1585081 1585096 1587093 1587098) (-1018 "SHDP.spad" 1574574 1574601 1575091 1575188) (-1017 "SGROUP.spad" 1574182 1574191 1574564 1574569) (-1016 "SGROUP.spad" 1573788 1573799 1574172 1574177) (-1015 "SGCF.spad" 1566927 1566936 1573778 1573783) (-1014 "SFRTCAT.spad" 1565873 1565890 1566895 1566922) (-1013 "SFRGCD.spad" 1564936 1564956 1565863 1565868) (-1012 "SFQCMPK.spad" 1559749 1559769 1564926 1564931) (-1011 "SEXOF.spad" 1559592 1559632 1559739 1559744) (-1010 "SEXCAT.spad" 1557420 1557460 1559582 1559587) (-1009 "SEX.spad" 1557312 1557321 1557410 1557415) (-1008 "SETMN.spad" 1555772 1555789 1557302 1557307) (-1007 "SETCAT.spad" 1555257 1555266 1555762 1555767) (-1006 "SETCAT.spad" 1554740 1554751 1555247 1555252) (-1005 "SETAGG.spad" 1551289 1551300 1554720 1554735) (-1004 "SETAGG.spad" 1547846 1547859 1551279 1551284) (-1003 "SET.spad" 1546155 1546166 1547252 1547291) (-1002 "syntax.spad" 1545858 1545867 1546145 1546150) (-1001 "SEGXCAT.spad" 1545014 1545027 1545848 1545853) (-1000 "SEGCAT.spad" 1543939 1543950 1545004 1545009) (-999 "SEGBIND2.spad" 1543638 1543650 1543929 1543934) (-998 "SEGBIND.spad" 1543398 1543408 1543586 1543591) (-997 "SEGAST.spad" 1543129 1543137 1543388 1543393) (-996 "SEG2.spad" 1542565 1542577 1543085 1543090) (-995 "SEG.spad" 1542379 1542389 1542484 1542489) (-994 "SDVAR.spad" 1541656 1541666 1542369 1542374) (-993 "SDPOL.spad" 1539354 1539364 1539644 1539771) (-992 "SCPKG.spad" 1537444 1537454 1539344 1539349) (-991 "SCOPE.spad" 1536622 1536630 1537434 1537439) (-990 "SCACHE.spad" 1535319 1535329 1536612 1536617) (-989 "SASTCAT.spad" 1535229 1535237 1535309 1535314) (-988 "SAOS.spad" 1535102 1535110 1535219 1535224) (-987 "SAERFFC.spad" 1534816 1534835 1535092 1535097) (-986 "SAEFACT.spad" 1534518 1534537 1534806 1534811) (-985 "SAE.spad" 1532169 1532184 1532779 1532914) (-984 "RURPK.spad" 1529829 1529844 1532159 1532164) (-983 "RULESET.spad" 1529283 1529306 1529819 1529824) (-982 "RULECOLD.spad" 1529136 1529148 1529273 1529278) (-981 "RULE.spad" 1527385 1527408 1529126 1529131) (-980 "RTVALUE.spad" 1527121 1527129 1527375 1527380) (-979 "syntax.spad" 1526839 1526847 1527111 1527116) (-978 "RSETGCD.spad" 1523282 1523301 1526829 1526834) (-977 "RSETCAT.spad" 1513251 1513267 1523250 1523277) (-976 "RSETCAT.spad" 1503240 1503258 1513241 1513246) (-975 "RSDCMPK.spad" 1501741 1501760 1503230 1503235) (-974 "RRCC.spad" 1500126 1500155 1501731 1501736) (-973 "RRCC.spad" 1498509 1498540 1500116 1500121) (-972 "RPTAST.spad" 1498212 1498220 1498499 1498504) (-971 "RPOLCAT.spad" 1477717 1477731 1498080 1498207) (-970 "RPOLCAT.spad" 1457015 1457031 1477380 1477385) (-969 "ROMAN.spad" 1456344 1456352 1456881 1457010) (-968 "ROIRC.spad" 1455425 1455456 1456334 1456339) (-967 "RNS.spad" 1454402 1454410 1455327 1455420) (-966 "RNS.spad" 1453465 1453475 1454392 1454397) (-965 "RNGBIND.spad" 1452626 1452639 1453420 1453425) (-964 "RNG.spad" 1452362 1452370 1452616 1452621) (-963 "RMODULE.spad" 1452144 1452154 1452352 1452357) (-962 "RMCAT2.spad" 1451565 1451621 1452134 1452139) (-961 "RMATRIX.spad" 1450375 1450393 1450717 1450756) (-960 "RMATCAT.spad" 1445955 1445985 1450331 1450370) (-959 "RMATCAT.spad" 1441425 1441457 1445803 1445808) (-958 "RLINSET.spad" 1441130 1441140 1441415 1441420) (-957 "RINTERP.spad" 1441019 1441038 1441120 1441125) (-956 "RING.spad" 1440490 1440498 1440999 1441014) (-955 "RING.spad" 1439969 1439979 1440480 1440485) (-954 "RIDIST.spad" 1439362 1439370 1439959 1439964) (-953 "RGCHAIN.spad" 1437917 1437932 1438810 1438837) (-952 "RGBCSPC.spad" 1437707 1437718 1437907 1437912) (-951 "RGBCMDL.spad" 1437270 1437281 1437697 1437702) (-950 "RFFACTOR.spad" 1436733 1436743 1437260 1437265) (-949 "RFFACT.spad" 1436469 1436480 1436723 1436728) (-948 "RFDIST.spad" 1435466 1435474 1436459 1436464) (-947 "RF.spad" 1433141 1433151 1435456 1435461) (-946 "RETSOL.spad" 1432561 1432573 1433131 1433136) (-945 "RETRACT.spad" 1431990 1432000 1432551 1432556) (-944 "RETRACT.spad" 1431417 1431429 1431980 1431985) (-943 "RETAST.spad" 1431230 1431238 1431407 1431412) (-942 "RESRING.spad" 1430578 1430624 1431168 1431225) (-941 "RESLATC.spad" 1429903 1429913 1430568 1430573) (-940 "REPSQ.spad" 1429635 1429645 1429893 1429898) (-939 "REPDB.spad" 1429343 1429353 1429625 1429630) (-938 "REP2.spad" 1419058 1419068 1429185 1429190) (-937 "REP1.spad" 1413279 1413289 1419008 1419013) (-936 "REP.spad" 1410834 1410842 1413269 1413274) (-935 "REGSET.spad" 1408660 1408676 1410468 1410495) (-934 "REF.spad" 1408179 1408189 1408650 1408655) (-933 "REDORDER.spad" 1407386 1407402 1408169 1408174) (-932 "RECLOS.spad" 1406283 1406302 1406986 1407079) (-931 "REALSOLV.spad" 1405424 1405432 1406273 1406278) (-930 "REAL0Q.spad" 1402723 1402737 1405414 1405419) (-929 "REAL0.spad" 1399568 1399582 1402713 1402718) (-928 "REAL.spad" 1399441 1399449 1399558 1399563) (-927 "RDUCEAST.spad" 1399163 1399171 1399431 1399436) (-926 "RDIV.spad" 1398819 1398843 1399153 1399158) (-925 "RDIST.spad" 1398387 1398397 1398809 1398814) (-924 "RDETRS.spad" 1397252 1397269 1398377 1398382) (-923 "RDETR.spad" 1395392 1395409 1397242 1397247) (-922 "RDEEFS.spad" 1394492 1394508 1395382 1395387) (-921 "RDEEF.spad" 1393503 1393519 1394482 1394487) (-920 "RCFIELD.spad" 1390722 1390730 1393405 1393498) (-919 "RCFIELD.spad" 1388027 1388037 1390712 1390717) (-918 "RCAGG.spad" 1385964 1385974 1388017 1388022) (-917 "RCAGG.spad" 1383828 1383840 1385883 1385888) (-916 "RATRET.spad" 1383189 1383199 1383818 1383823) (-915 "RATFACT.spad" 1382882 1382893 1383179 1383184) (-914 "RANDSRC.spad" 1382202 1382210 1382872 1382877) (-913 "RADUTIL.spad" 1381959 1381967 1382192 1382197) (-912 "RADIX.spad" 1379004 1379017 1380549 1380642) (-911 "RADFF.spad" 1376921 1376957 1377039 1377195) (-910 "RADCAT.spad" 1376517 1376525 1376911 1376916) (-909 "RADCAT.spad" 1376111 1376121 1376507 1376512) (-908 "QUEUE.spad" 1375525 1375535 1375783 1375810) (-907 "QUATCT2.spad" 1375146 1375164 1375515 1375520) (-906 "QUATCAT.spad" 1373317 1373327 1375076 1375141) (-905 "QUATCAT.spad" 1371253 1371265 1373014 1373019) (-904 "QUAT.spad" 1369860 1369870 1370202 1370267) (-903 "QUAGG.spad" 1368694 1368704 1369828 1369855) (-902 "QQUTAST.spad" 1368463 1368471 1368684 1368689) (-901 "QFORM.spad" 1368082 1368096 1368453 1368458) (-900 "QFCAT2.spad" 1367775 1367791 1368072 1368077) (-899 "QFCAT.spad" 1366478 1366488 1367677 1367770) (-898 "QFCAT.spad" 1364814 1364826 1366015 1366020) (-897 "QEQUAT.spad" 1364373 1364381 1364804 1364809) (-896 "QCMPACK.spad" 1359288 1359307 1364363 1364368) (-895 "QALGSET2.spad" 1357284 1357302 1359278 1359283) (-894 "QALGSET.spad" 1353389 1353421 1357198 1357203) (-893 "PWFFINTB.spad" 1350805 1350826 1353379 1353384) (-892 "PUSHVAR.spad" 1350144 1350163 1350795 1350800) (-891 "PTRANFN.spad" 1346280 1346290 1350134 1350139) (-890 "PTPACK.spad" 1343368 1343378 1346270 1346275) (-889 "PTFUNC2.spad" 1343191 1343205 1343358 1343363) (-888 "PTCAT.spad" 1342446 1342456 1343159 1343186) (-887 "PSQFR.spad" 1341761 1341785 1342436 1342441) (-886 "PSEUDLIN.spad" 1340647 1340657 1341751 1341756) (-885 "PSETPK.spad" 1327352 1327368 1340525 1340530) (-884 "PSETCAT.spad" 1321752 1321775 1327332 1327347) (-883 "PSETCAT.spad" 1316126 1316151 1321708 1321713) (-882 "PSCURVE.spad" 1315125 1315133 1316116 1316121) (-881 "PSCAT.spad" 1313908 1313937 1315023 1315120) (-880 "PSCAT.spad" 1312781 1312812 1313898 1313903) (-879 "PRTITION.spad" 1311479 1311487 1312771 1312776) (-878 "PRTDAST.spad" 1311198 1311206 1311469 1311474) (-877 "PRS.spad" 1300816 1300833 1311154 1311159) (-876 "PRQAGG.spad" 1300251 1300261 1300784 1300811) (-875 "PROPLOG.spad" 1299855 1299863 1300241 1300246) (-874 "PROPFUN2.spad" 1299478 1299491 1299845 1299850) (-873 "PROPFUN1.spad" 1298884 1298895 1299468 1299473) (-872 "PROPFRML.spad" 1297452 1297463 1298874 1298879) (-871 "PROPERTY.spad" 1296948 1296956 1297442 1297447) (-870 "PRODUCT.spad" 1294645 1294657 1294929 1294984) (-869 "PRINT.spad" 1294397 1294405 1294635 1294640) (-868 "PRIMES.spad" 1292658 1292668 1294387 1294392) (-867 "PRIMELT.spad" 1290779 1290793 1292648 1292653) (-866 "PRIMCAT.spad" 1290422 1290430 1290769 1290774) (-865 "PRIMARR2.spad" 1289189 1289201 1290412 1290417) (-864 "PRIMARR.spad" 1288244 1288254 1288414 1288441) (-863 "PREASSOC.spad" 1287626 1287638 1288234 1288239) (-862 "PR.spad" 1286144 1286156 1286843 1286970) (-861 "PPCURVE.spad" 1285281 1285289 1286134 1286139) (-860 "PORTNUM.spad" 1285072 1285080 1285271 1285276) (-859 "POLYROOT.spad" 1283921 1283943 1285028 1285033) (-858 "POLYLIFT.spad" 1283186 1283209 1283911 1283916) (-857 "POLYCATQ.spad" 1281312 1281334 1283176 1283181) (-856 "POLYCAT.spad" 1274814 1274835 1281180 1281307) (-855 "POLYCAT.spad" 1267836 1267859 1274204 1274209) (-854 "POLY2UP.spad" 1267288 1267302 1267826 1267831) (-853 "POLY2.spad" 1266885 1266897 1267278 1267283) (-852 "POLY.spad" 1264553 1264563 1265068 1265195) (-851 "POLUTIL.spad" 1263518 1263547 1264509 1264514) (-850 "POLTOPOL.spad" 1262266 1262281 1263508 1263513) (-849 "POINT.spad" 1261149 1261159 1261236 1261263) (-848 "PNTHEORY.spad" 1257851 1257859 1261139 1261144) (-847 "PMTOOLS.spad" 1256626 1256640 1257841 1257846) (-846 "PMSYM.spad" 1256175 1256185 1256616 1256621) (-845 "PMQFCAT.spad" 1255766 1255780 1256165 1256170) (-844 "PMPREDFS.spad" 1255228 1255250 1255756 1255761) (-843 "PMPRED.spad" 1254715 1254729 1255218 1255223) (-842 "PMPLCAT.spad" 1253792 1253810 1254644 1254649) (-841 "PMLSAGG.spad" 1253377 1253391 1253782 1253787) (-840 "PMKERNEL.spad" 1252956 1252968 1253367 1253372) (-839 "PMINS.spad" 1252536 1252546 1252946 1252951) (-838 "PMFS.spad" 1252113 1252131 1252526 1252531) (-837 "PMDOWN.spad" 1251403 1251417 1252103 1252108) (-836 "PMASSFS.spad" 1250378 1250394 1251393 1251398) (-835 "PMASS.spad" 1249396 1249404 1250368 1250373) (-834 "PLOTTOOL.spad" 1249176 1249184 1249386 1249391) (-833 "PLOT3D.spad" 1245640 1245648 1249166 1249171) (-832 "PLOT1.spad" 1244813 1244823 1245630 1245635) (-831 "PLOT.spad" 1239736 1239744 1244803 1244808) (-830 "PLEQN.spad" 1227138 1227165 1239726 1239731) (-829 "PINTERPA.spad" 1226922 1226938 1227128 1227133) (-828 "PINTERP.spad" 1226544 1226563 1226912 1226917) (-827 "PID.spad" 1225518 1225526 1226470 1226539) (-826 "PICOERCE.spad" 1225175 1225185 1225508 1225513) (-825 "PI.spad" 1224792 1224800 1225149 1225170) (-824 "PGROEB.spad" 1223401 1223415 1224782 1224787) (-823 "PGE.spad" 1215074 1215082 1223391 1223396) (-822 "PGCD.spad" 1214028 1214045 1215064 1215069) (-821 "PFRPAC.spad" 1213177 1213187 1214018 1214023) (-820 "PFR.spad" 1209880 1209890 1213079 1213172) (-819 "PFOTOOLS.spad" 1209138 1209154 1209870 1209875) (-818 "PFOQ.spad" 1208508 1208526 1209128 1209133) (-817 "PFO.spad" 1207927 1207954 1208498 1208503) (-816 "PFECAT.spad" 1205637 1205645 1207853 1207922) (-815 "PFECAT.spad" 1203375 1203385 1205593 1205598) (-814 "PFBRU.spad" 1201263 1201275 1203365 1203370) (-813 "PFBR.spad" 1198823 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(-171 "DEFINTEF.spad" 184398 184414 185850 185855) (-170 "DEFAST.spad" 183782 183790 184388 184393) (-169 "DECIMAL.spad" 182011 182019 182372 182465) (-168 "DDFACT.spad" 179832 179849 182001 182006) (-167 "DBLRESP.spad" 179432 179456 179822 179827) (-166 "DBASIS.spad" 179058 179073 179422 179427) (-165 "DBASE.spad" 177722 177732 179048 179053) (-164 "DATAARY.spad" 177208 177221 177712 177717) (-163 "CYCLOTOM.spad" 176714 176722 177198 177203) (-162 "CYCLES.spad" 173506 173514 176704 176709) (-161 "CVMP.spad" 172923 172933 173496 173501) (-160 "CTRIGMNP.spad" 171423 171439 172913 172918) (-159 "CTORKIND.spad" 171026 171034 171413 171418) (-158 "CTORCAT.spad" 170267 170275 171016 171021) (-157 "CTORCAT.spad" 169506 169516 170257 170262) (-156 "CTORCALL.spad" 169095 169105 169496 169501) (-155 "CTOR.spad" 168786 168794 169085 169090) (-154 "CSTTOOLS.spad" 168031 168044 168776 168781) (-153 "CRFP.spad" 161803 161816 168021 168026) (-152 "CRCEAST.spad" 161523 161531 161793 161798) (-151 "CRAPACK.spad" 160590 160600 161513 161518) (-150 "CPMATCH.spad" 160091 160106 160512 160517) (-149 "CPIMA.spad" 159796 159815 160081 160086) (-148 "COORDSYS.spad" 154805 154815 159786 159791) (-147 "CONTOUR.spad" 154232 154240 154795 154800) (-146 "CONTFRAC.spad" 149982 149992 154134 154227) (-145 "CONDUIT.spad" 149740 149748 149972 149977) (-144 "COMRING.spad" 149414 149422 149678 149735) (-143 "COMPPROP.spad" 148932 148940 149404 149409) (-142 "COMPLPAT.spad" 148699 148714 148922 148927) (-141 "COMPLEX2.spad" 148414 148426 148689 148694) (-140 "COMPLEX.spad" 144120 144130 144364 144622) (-139 "COMPILER.spad" 143669 143677 144110 144115) (-138 "COMPFACT.spad" 143271 143285 143659 143664) (-137 "COMPCAT.spad" 141346 141356 143008 143266) (-136 "COMPCAT.spad" 139162 139174 140826 140831) (-135 "COMMUPC.spad" 138910 138928 139152 139157) (-134 "COMMONOP.spad" 138443 138451 138900 138905) (-133 "COMMAAST.spad" 138206 138214 138433 138438) (-132 "COMM.spad" 138017 138025 138196 138201) (-131 "COMBOPC.spad" 136940 136948 138007 138012) (-130 "COMBINAT.spad" 135707 135717 136930 136935) (-129 "COMBF.spad" 133129 133145 135697 135702) (-128 "COLOR.spad" 131966 131974 133119 133124) (-127 "COLONAST.spad" 131632 131640 131956 131961) (-126 "CMPLXRT.spad" 131343 131360 131622 131627) (-125 "CLLCTAST.spad" 131005 131013 131333 131338) (-124 "CLIP.spad" 127113 127121 130995 131000) (-123 "CLIF.spad" 125768 125784 127069 127108) (-122 "CLAGG.spad" 122305 122315 125758 125763) (-121 "CLAGG.spad" 118726 118738 122181 122186) (-120 "CINTSLPE.spad" 118081 118094 118716 118721) (-119 "CHVAR.spad" 116219 116241 118071 118076) (-118 "CHARZ.spad" 116134 116142 116199 116214) (-117 "CHARPOL.spad" 115660 115670 116124 116129) (-116 "CHARNZ.spad" 115422 115430 115640 115655) (-115 "CHAR.spad" 112790 112798 115412 115417) (-114 "CFCAT.spad" 112118 112126 112780 112785) (-113 "CDEN.spad" 111338 111352 112108 112113) (-112 "CCLASS.spad" 109518 109526 110780 110819) (-111 "CATEGORY.spad" 108592 108600 109508 109513) (-110 "CATCTOR.spad" 108483 108491 108582 108587) (-109 "CATAST.spad" 108109 108117 108473 108478) (-108 "CASEAST.spad" 107823 107831 108099 108104) (-107 "CARTEN2.spad" 107213 107240 107813 107818) (-106 "CARTEN.spad" 102965 102989 107203 107208) (-105 "CARD.spad" 100260 100268 102939 102960) (-104 "CAPSLAST.spad" 100042 100050 100250 100255) (-103 "CACHSET.spad" 99666 99674 100032 100037) (-102 "CABMON.spad" 99221 99229 99656 99661) (-101 "BYTEORD.spad" 98896 98904 99211 99216) (-100 "BYTEBUF.spad" 96882 96890 98168 98195) (-99 "BYTE.spad" 96358 96365 96872 96877) (-98 "BTREE.spad" 95497 95506 96030 96057) (-97 "BTOURN.spad" 94568 94577 95169 95196) (-96 "BTCAT.spad" 93961 93970 94536 94563) (-95 "BTCAT.spad" 93374 93385 93951 93956) (-94 "BTAGG.spad" 92841 92848 93342 93369) (-93 "BTAGG.spad" 92328 92337 92831 92836) (-92 "BSTREE.spad" 91135 91144 92000 92027) (-91 "BRILL.spad" 89341 89351 91125 91130) (-90 "BRAGG.spad" 88298 88307 89331 89336) (-89 "BRAGG.spad" 87219 87230 88254 88259) (-88 "BPADICRT.spad" 85279 85290 85525 85618) (-87 "BPADIC.spad" 84952 84963 85205 85274) (-86 "BOUNDZRO.spad" 84609 84625 84942 84947) (-85 "BOP1.spad" 82068 82077 84599 84604) (-84 "BOP.spad" 77211 77218 82058 82063) (-83 "BOOLEAN.spad" 76760 76767 77201 77206) (-82 "BOOLE.spad" 76411 76418 76750 76755) (-81 "BOOLE.spad" 76060 76069 76401 76406) (-80 "BMODULE.spad" 75773 75784 76028 76055) (-79 "BITS.spad" 75205 75212 75419 75446) (-78 "BINDING.spad" 74627 74634 75195 75200) (-77 "BINARY.spad" 72862 72869 73217 73310) (-76 "BGAGG.spad" 72068 72077 72842 72857) (-75 "BGAGG.spad" 71282 71293 72058 72063) (-74 "BEZOUT.spad" 70423 70449 71232 71237) (-73 "BBTREE.spad" 67366 67375 70095 70122) (-72 "BASTYPE.spad" 66866 66873 67356 67361) (-71 "BASTYPE.spad" 66364 66373 66856 66861) (-70 "BALFACT.spad" 65824 65836 66354 66359) (-69 "AUTOMOR.spad" 65275 65284 65804 65819) (-68 "ATTREG.spad" 61998 62005 65027 65270) (-67 "ATTRAST.spad" 61715 61722 61988 61993) (-66 "ATRIG.spad" 61185 61192 61705 61710) (-65 "ATRIG.spad" 60653 60662 61175 61180) (-64 "ASTCAT.spad" 60557 60564 60643 60648) (-63 "ASTCAT.spad" 60459 60468 60547 60552) (-62 "ASTACK.spad" 59863 59872 60131 60158) (-61 "ASSOCEQ.spad" 58697 58708 59819 59824) (-60 "ARRAY2.spad" 58130 58139 58369 58396) (-59 "ARRAY12.spad" 56843 56854 58120 58125) (-58 "ARRAY1.spad" 55722 55731 56068 56095) (-57 "ARR2CAT.spad" 51504 51525 55690 55717) (-56 "ARR2CAT.spad" 47306 47329 51494 51499) (-55 "ARITY.spad" 46678 46685 47296 47301) (-54 "APPRULE.spad" 45962 45984 46668 46673) (-53 "APPLYORE.spad" 45581 45594 45952 45957) (-52 "ANY1.spad" 44652 44661 45571 45576) (-51 "ANY.spad" 43503 43510 44642 44647) (-50 "ANTISYM.spad" 41948 41964 43483 43498) (-49 "ANON.spad" 41657 41664 41938 41943) (-48 "AN.spad" 40125 40132 41488 41581) (-47 "AMR.spad" 38310 38321 40023 40120) (-46 "AMR.spad" 36358 36371 38073 38078) (-45 "ALIST.spad" 33596 33617 33946 33973) (-44 "ALGSC.spad" 32731 32757 33468 33521) (-43 "ALGPKG.spad" 28514 28525 32687 32692) (-42 "ALGMFACT.spad" 27707 27721 28504 28509) (-41 "ALGMANIP.spad" 25208 25223 27551 27556) (-40 "ALGFF.spad" 23026 23053 23243 23399) (-39 "ALGFACT.spad" 22145 22155 23016 23021) (-38 "ALGEBRA.spad" 21978 21987 22101 22140) (-37 "ALGEBRA.spad" 21843 21854 21968 21973) (-36 "ALAGG.spad" 21355 21376 21811 21838) (-35 "AHYP.spad" 20736 20743 21345 21350) (-34 "AGG.spad" 19445 19452 20726 20731) (-33 "AGG.spad" 18118 18127 19401 19406) (-32 "AF.spad" 16563 16578 18067 18072) (-31 "ADDAST.spad" 16249 16256 16553 16558) (-30 "ACPLOT.spad" 14840 14847 16239 16244) (-29 "ACFS.spad" 12697 12706 14742 14835) (-28 "ACFS.spad" 10640 10651 12687 12692) (-27 "ACF.spad" 7394 7401 10542 10635) (-26 "ACF.spad" 4234 4243 7384 7389) (-25 "ABELSG.spad" 3775 3782 4224 4229) (-24 "ABELSG.spad" 3314 3323 3765 3770) (-23 "ABELMON.spad" 2859 2866 3304 3309) (-22 "ABELMON.spad" 2402 2411 2849 2854) (-21 "ABELGRP.spad" 2067 2074 2392 2397) (-20 "ABELGRP.spad" 1730 1739 2057 2062) (-19 "A1AGG.spad" 870 879 1698 1725) (-18 "A1AGG.spad" 30 41 860 865))
\ No newline at end of file +((-3 NIL 1965399 1965404 1965409 1965414) (-2 NIL 1965379 1965384 1965389 1965394) (-1 NIL 1965359 1965364 1965369 1965374) (0 NIL 1965339 1965344 1965349 1965354) (-1207 "ZMOD.spad" 1965148 1965161 1965277 1965334) (-1206 "ZLINDEP.spad" 1964246 1964257 1965138 1965143) (-1205 "ZDSOLVE.spad" 1954206 1954228 1964236 1964241) (-1204 "YSTREAM.spad" 1953701 1953712 1954196 1954201) (-1203 "YDIAGRAM.spad" 1953335 1953344 1953691 1953696) (-1202 "XRPOLY.spad" 1952555 1952575 1953191 1953260) (-1201 "XPR.spad" 1950350 1950363 1952273 1952372) (-1200 "XPOLYC.spad" 1949669 1949685 1950276 1950345) (-1199 "XPOLY.spad" 1949224 1949235 1949525 1949594) (-1198 "XPBWPOLY.spad" 1947695 1947715 1949030 1949099) (-1197 "XFALG.spad" 1944743 1944759 1947621 1947690) (-1196 "XF.spad" 1943206 1943221 1944645 1944738) (-1195 "XF.spad" 1941649 1941666 1943090 1943095) (-1194 "XEXPPKG.spad" 1940908 1940934 1941639 1941644) (-1193 "XDPOLY.spad" 1940522 1940538 1940764 1940833) (-1192 "XALG.spad" 1940190 1940201 1940478 1940517) (-1191 "WUTSET.spad" 1936193 1936210 1939824 1939851) (-1190 "WP.spad" 1935400 1935444 1936051 1936118) (-1189 "WHILEAST.spad" 1935198 1935207 1935390 1935395) (-1188 "WHEREAST.spad" 1934869 1934878 1935188 1935193) (-1187 "WFFINTBS.spad" 1932532 1932554 1934859 1934864) (-1186 "WEIER.spad" 1930754 1930765 1932522 1932527) (-1185 "VSPACE.spad" 1930427 1930438 1930722 1930749) (-1184 "VSPACE.spad" 1930120 1930133 1930417 1930422) (-1183 "VOID.spad" 1929797 1929806 1930110 1930115) (-1182 "VIEWDEF.spad" 1924998 1925007 1929787 1929792) (-1181 "VIEW3D.spad" 1908959 1908968 1924988 1924993) (-1180 "VIEW2D.spad" 1896858 1896867 1908949 1908954) (-1179 "VIEW.spad" 1894578 1894587 1896848 1896853) (-1178 "VECTOR2.spad" 1893217 1893230 1894568 1894573) (-1177 "VECTOR.spad" 1891936 1891947 1892187 1892214) (-1176 "VECTCAT.spad" 1889848 1889859 1891904 1891931) (-1175 "VECTCAT.spad" 1887569 1887582 1889627 1889632) (-1174 "VARIABLE.spad" 1887349 1887364 1887559 1887564) (-1173 "UTYPE.spad" 1886993 1887002 1887339 1887344) (-1172 "UTSODETL.spad" 1886288 1886312 1886949 1886954) (-1171 "UTSODE.spad" 1884504 1884524 1886278 1886283) (-1170 "UTSCAT.spad" 1881983 1881999 1884402 1884499) (-1169 "UTSCAT.spad" 1879130 1879148 1881551 1881556) (-1168 "UTS2.spad" 1878725 1878760 1879120 1879125) (-1167 "UTS.spad" 1873737 1873765 1877257 1877354) (-1166 "URAGG.spad" 1868458 1868469 1873727 1873732) (-1165 "URAGG.spad" 1863143 1863156 1868414 1868419) (-1164 "UPXSSING.spad" 1860911 1860937 1862347 1862480) (-1163 "UPXSCONS.spad" 1858729 1858749 1859102 1859251) (-1162 "UPXSCCA.spad" 1857300 1857320 1858575 1858724) (-1161 "UPXSCCA.spad" 1856013 1856035 1857290 1857295) (-1160 "UPXSCAT.spad" 1854602 1854618 1855859 1856008) (-1159 "UPXS2.spad" 1854145 1854198 1854592 1854597) (-1158 "UPXS.spad" 1851500 1851528 1852336 1852485) (-1157 "UPSQFREE.spad" 1849915 1849929 1851490 1851495) (-1156 "UPSCAT.spad" 1847710 1847734 1849813 1849910) (-1155 "UPSCAT.spad" 1845206 1845232 1847311 1847316) (-1154 "UPOLYC2.spad" 1844677 1844696 1845196 1845201) (-1153 "UPOLYC.spad" 1839757 1839768 1844519 1844672) (-1152 "UPOLYC.spad" 1834755 1834768 1839519 1839524) (-1151 "UPMP.spad" 1833687 1833700 1834745 1834750) (-1150 "UPDIVP.spad" 1833252 1833266 1833677 1833682) (-1149 "UPDECOMP.spad" 1831513 1831527 1833242 1833247) (-1148 "UPCDEN.spad" 1830730 1830746 1831503 1831508) (-1147 "UP2.spad" 1830094 1830115 1830720 1830725) (-1146 "UP.spad" 1827564 1827579 1827951 1828104) (-1145 "UNISEG2.spad" 1827061 1827074 1827520 1827525) (-1144 "UNISEG.spad" 1826414 1826425 1826980 1826985) (-1143 "UNIFACT.spad" 1825517 1825529 1826404 1826409) (-1142 "ULSCONS.spad" 1819560 1819580 1819930 1820079) (-1141 "ULSCCAT.spad" 1817297 1817317 1819406 1819555) (-1140 "ULSCCAT.spad" 1815142 1815164 1817253 1817258) (-1139 "ULSCAT.spad" 1813382 1813398 1814988 1815137) (-1138 "ULS2.spad" 1812896 1812949 1813372 1813377) (-1137 "ULS.spad" 1805162 1805190 1806107 1806530) (-1136 "UINT8.spad" 1805039 1805048 1805152 1805157) (-1135 "UINT64.spad" 1804915 1804924 1805029 1805034) (-1134 "UINT32.spad" 1804791 1804800 1804905 1804910) (-1133 "UINT16.spad" 1804667 1804676 1804781 1804786) (-1132 "UFD.spad" 1803732 1803741 1804593 1804662) (-1131 "UFD.spad" 1802859 1802870 1803722 1803727) (-1130 "UDVO.spad" 1801740 1801749 1802849 1802854) (-1129 "UDPO.spad" 1799321 1799332 1801696 1801701) (-1128 "TYPEAST.spad" 1799240 1799249 1799311 1799316) (-1127 "TYPE.spad" 1799172 1799181 1799230 1799235) (-1126 "TWOFACT.spad" 1797824 1797839 1799162 1799167) (-1125 "TUPLE.spad" 1797331 1797342 1797736 1797741) (-1124 "TUBETOOL.spad" 1794198 1794207 1797321 1797326) (-1123 "TUBE.spad" 1792845 1792862 1794188 1794193) (-1122 "TSETCAT.spad" 1780916 1780933 1792813 1792840) (-1121 "TSETCAT.spad" 1768973 1768992 1780872 1780877) (-1120 "TS.spad" 1767601 1767617 1768567 1768664) (-1119 "TRMANIP.spad" 1761965 1761982 1767289 1767294) (-1118 "TRIMAT.spad" 1760928 1760953 1761955 1761960) (-1117 "TRIGMNIP.spad" 1759455 1759472 1760918 1760923) (-1116 "TRIGCAT.spad" 1758967 1758976 1759445 1759450) (-1115 "TRIGCAT.spad" 1758477 1758488 1758957 1758962) (-1114 "TREE.spad" 1757117 1757128 1758149 1758176) (-1113 "TRANFUN.spad" 1756956 1756965 1757107 1757112) (-1112 "TRANFUN.spad" 1756793 1756804 1756946 1756951) (-1111 "TOPSP.spad" 1756467 1756476 1756783 1756788) (-1110 "TOOLSIGN.spad" 1756130 1756141 1756457 1756462) (-1109 "TEXTFILE.spad" 1754691 1754700 1756120 1756125) (-1108 "TEX1.spad" 1754247 1754258 1754681 1754686) (-1107 "TEX.spad" 1751441 1751450 1754237 1754242) (-1106 "TBCMPPK.spad" 1749542 1749565 1751431 1751436) (-1105 "TBAGG.spad" 1748600 1748623 1749522 1749537) (-1104 "TBAGG.spad" 1747666 1747691 1748590 1748595) (-1103 "TANEXP.spad" 1747074 1747085 1747656 1747661) (-1102 "TALGOP.spad" 1746798 1746809 1747064 1747069) (-1101 "TABLEAU.spad" 1746279 1746290 1746788 1746793) (-1100 "TABLE.spad" 1744554 1744577 1744824 1744851) (-1099 "TABLBUMP.spad" 1741333 1741344 1744544 1744549) (-1098 "SYSTEM.spad" 1740561 1740570 1741323 1741328) (-1097 "SYSSOLP.spad" 1738044 1738055 1740551 1740556) (-1096 "SYSPTR.spad" 1737943 1737952 1738034 1738039) (-1095 "SYSNNI.spad" 1737166 1737177 1737933 1737938) (-1094 "SYSINT.spad" 1736570 1736581 1737156 1737161) (-1093 "SYNTAX.spad" 1732904 1732913 1736560 1736565) (-1092 "SYMTAB.spad" 1730972 1730981 1732894 1732899) (-1091 "SYMS.spad" 1727001 1727010 1730962 1730967) (-1090 "SYMPOLY.spad" 1726134 1726145 1726216 1726343) (-1089 "SYMFUNC.spad" 1725635 1725646 1726124 1726129) (-1088 "SYMBOL.spad" 1723130 1723139 1725625 1725630) (-1087 "SUTS.spad" 1720243 1720271 1721662 1721759) (-1086 "SUPXS.spad" 1717585 1717613 1718434 1718583) (-1085 "SUPFRACF.spad" 1716690 1716708 1717575 1717580) (-1084 "SUP2.spad" 1716082 1716095 1716680 1716685) (-1083 "SUP.spad" 1713166 1713177 1713939 1714092) (-1082 "SUMRF.spad" 1712140 1712151 1713156 1713161) (-1081 "SUMFS.spad" 1711769 1711786 1712130 1712135) (-1080 "SULS.spad" 1704022 1704050 1704980 1705403) (-1079 "syntax.spad" 1703791 1703800 1704012 1704017) (-1078 "SUCH.spad" 1703481 1703496 1703781 1703786) (-1077 "SUBSPACE.spad" 1695612 1695627 1703471 1703476) (-1076 "SUBRESP.spad" 1694782 1694796 1695568 1695573) (-1075 "STTFNC.spad" 1691250 1691266 1694772 1694777) (-1074 "STTF.spad" 1687349 1687365 1691240 1691245) (-1073 "STTAYLOR.spad" 1680026 1680037 1687256 1687261) (-1072 "STRTBL.spad" 1678413 1678430 1678562 1678589) (-1071 "STRING.spad" 1677281 1677290 1677666 1677693) (-1070 "STREAM3.spad" 1676854 1676869 1677271 1677276) (-1069 "STREAM2.spad" 1675982 1675995 1676844 1676849) (-1068 "STREAM1.spad" 1675688 1675699 1675972 1675977) (-1067 "STREAM.spad" 1672684 1672695 1675291 1675306) (-1066 "STINPROD.spad" 1671620 1671636 1672674 1672679) (-1065 "STEPAST.spad" 1670854 1670863 1671610 1671615) (-1064 "STEP.spad" 1670171 1670180 1670844 1670849) (-1063 "STBL.spad" 1668561 1668589 1668728 1668743) (-1062 "STAGG.spad" 1667260 1667271 1668551 1668556) (-1061 "STAGG.spad" 1665957 1665970 1667250 1667255) (-1060 "STACK.spad" 1665379 1665390 1665629 1665656) (-1059 "SRING.spad" 1665139 1665148 1665369 1665374) (-1058 "SREGSET.spad" 1662871 1662888 1664773 1664800) (-1057 "SRDCMPK.spad" 1661448 1661468 1662861 1662866) (-1056 "SRAGG.spad" 1656631 1656640 1661416 1661443) (-1055 "SRAGG.spad" 1651834 1651845 1656621 1656626) (-1054 "SQMATRIX.spad" 1649511 1649529 1650427 1650514) (-1053 "SPLTREE.spad" 1644253 1644266 1649049 1649076) (-1052 "SPLNODE.spad" 1640873 1640886 1644243 1644248) (-1051 "SPFCAT.spad" 1639682 1639691 1640863 1640868) (-1050 "SPECOUT.spad" 1638234 1638243 1639672 1639677) (-1049 "SPADXPT.spad" 1630325 1630334 1638224 1638229) (-1048 "spad-parser.spad" 1629790 1629799 1630315 1630320) (-1047 "SPADAST.spad" 1629491 1629500 1629780 1629785) (-1046 "SPACEC.spad" 1613706 1613717 1629481 1629486) (-1045 "SPACE3.spad" 1613482 1613493 1613696 1613701) (-1044 "SORTPAK.spad" 1613031 1613044 1613438 1613443) (-1043 "SOLVETRA.spad" 1610794 1610805 1613021 1613026) (-1042 "SOLVESER.spad" 1609250 1609261 1610784 1610789) (-1041 "SOLVERAD.spad" 1605276 1605287 1609240 1609245) (-1040 "SOLVEFOR.spad" 1603738 1603756 1605266 1605271) (-1039 "SNTSCAT.spad" 1603338 1603355 1603706 1603733) (-1038 "SMTS.spad" 1601655 1601681 1602932 1603029) (-1037 "SMP.spad" 1599463 1599483 1599853 1599980) (-1036 "SMITH.spad" 1598308 1598333 1599453 1599458) (-1035 "SMATCAT.spad" 1596426 1596456 1598252 1598303) (-1034 "SMATCAT.spad" 1594476 1594508 1596304 1596309) (-1033 "SKAGG.spad" 1593445 1593456 1594444 1594471) (-1032 "SINT.spad" 1592744 1592753 1593311 1593440) (-1031 "SIMPAN.spad" 1592472 1592481 1592734 1592739) (-1030 "SIGNRF.spad" 1591597 1591608 1592462 1592467) (-1029 "SIGNEF.spad" 1590883 1590900 1591587 1591592) (-1028 "syntax.spad" 1590300 1590309 1590873 1590878) (-1027 "SIG.spad" 1589662 1589671 1590290 1590295) (-1026 "SHP.spad" 1587606 1587621 1589618 1589623) (-1025 "SHDP.spad" 1577099 1577126 1577616 1577713) (-1024 "SGROUP.spad" 1576707 1576716 1577089 1577094) (-1023 "SGROUP.spad" 1576313 1576324 1576697 1576702) (-1022 "catdef.spad" 1576023 1576035 1576134 1576308) (-1021 "catdef.spad" 1575579 1575591 1575844 1576018) (-1020 "SGCF.spad" 1568718 1568727 1575569 1575574) (-1019 "SFRTCAT.spad" 1567664 1567681 1568686 1568713) (-1018 "SFRGCD.spad" 1566727 1566747 1567654 1567659) (-1017 "SFQCMPK.spad" 1561540 1561560 1566717 1566722) (-1016 "SEXOF.spad" 1561383 1561423 1561530 1561535) (-1015 "SEXCAT.spad" 1559211 1559251 1561373 1561378) (-1014 "SEX.spad" 1559103 1559112 1559201 1559206) (-1013 "SETMN.spad" 1557563 1557580 1559093 1559098) (-1012 "SETCAT.spad" 1557048 1557057 1557553 1557558) (-1011 "SETCAT.spad" 1556531 1556542 1557038 1557043) (-1010 "SETAGG.spad" 1553080 1553091 1556511 1556526) (-1009 "SETAGG.spad" 1549637 1549650 1553070 1553075) (-1008 "SET.spad" 1547946 1547957 1549043 1549082) (-1007 "syntax.spad" 1547649 1547658 1547936 1547941) (-1006 "SEGXCAT.spad" 1546805 1546818 1547639 1547644) (-1005 "SEGCAT.spad" 1545730 1545741 1546795 1546800) (-1004 "SEGBIND2.spad" 1545428 1545441 1545720 1545725) (-1003 "SEGBIND.spad" 1545186 1545197 1545375 1545380) (-1002 "SEGAST.spad" 1544916 1544925 1545176 1545181) (-1001 "SEG2.spad" 1544351 1544364 1544872 1544877) (-1000 "SEG.spad" 1544164 1544175 1544270 1544275) (-999 "SDVAR.spad" 1543441 1543451 1544154 1544159) (-998 "SDPOL.spad" 1541139 1541149 1541429 1541556) (-997 "SCPKG.spad" 1539229 1539239 1541129 1541134) (-996 "SCOPE.spad" 1538407 1538415 1539219 1539224) (-995 "SCACHE.spad" 1537104 1537114 1538397 1538402) (-994 "SASTCAT.spad" 1537014 1537022 1537094 1537099) (-993 "SAOS.spad" 1536887 1536895 1537004 1537009) (-992 "SAERFFC.spad" 1536601 1536620 1536877 1536882) (-991 "SAEFACT.spad" 1536303 1536322 1536591 1536596) (-990 "SAE.spad" 1533954 1533969 1534564 1534699) (-989 "RURPK.spad" 1531614 1531629 1533944 1533949) (-988 "RULESET.spad" 1531068 1531091 1531604 1531609) (-987 "RULECOLD.spad" 1530921 1530933 1531058 1531063) (-986 "RULE.spad" 1529170 1529193 1530911 1530916) (-985 "RTVALUE.spad" 1528906 1528914 1529160 1529165) (-984 "syntax.spad" 1528624 1528632 1528896 1528901) (-983 "RSETGCD.spad" 1525067 1525086 1528614 1528619) (-982 "RSETCAT.spad" 1515036 1515052 1525035 1525062) (-981 "RSETCAT.spad" 1505025 1505043 1515026 1515031) (-980 "RSDCMPK.spad" 1503526 1503545 1505015 1505020) (-979 "RRCC.spad" 1501911 1501940 1503516 1503521) (-978 "RRCC.spad" 1500294 1500325 1501901 1501906) (-977 "RPTAST.spad" 1499997 1500005 1500284 1500289) (-976 "RPOLCAT.spad" 1479502 1479516 1499865 1499992) (-975 "RPOLCAT.spad" 1458800 1458816 1479165 1479170) (-974 "ROMAN.spad" 1458129 1458137 1458666 1458795) (-973 "ROIRC.spad" 1457210 1457241 1458119 1458124) (-972 "RNS.spad" 1456187 1456195 1457112 1457205) (-971 "RNS.spad" 1455250 1455260 1456177 1456182) (-970 "RNGBIND.spad" 1454411 1454424 1455205 1455210) (-969 "RNG.spad" 1454147 1454155 1454401 1454406) (-968 "RMODULE.spad" 1453929 1453939 1454137 1454142) (-967 "RMCAT2.spad" 1453350 1453406 1453919 1453924) (-966 "RMATRIX.spad" 1452160 1452178 1452502 1452541) (-965 "RMATCAT.spad" 1447740 1447770 1452116 1452155) (-964 "RMATCAT.spad" 1443210 1443242 1447588 1447593) (-963 "RLINSET.spad" 1442915 1442925 1443200 1443205) (-962 "RINTERP.spad" 1442804 1442823 1442905 1442910) (-961 "RING.spad" 1442275 1442283 1442784 1442799) (-960 "RING.spad" 1441754 1441764 1442265 1442270) (-959 "RIDIST.spad" 1441147 1441155 1441744 1441749) (-958 "RGCHAIN.spad" 1439702 1439717 1440595 1440622) (-957 "RGBCSPC.spad" 1439492 1439503 1439692 1439697) (-956 "RGBCMDL.spad" 1439055 1439066 1439482 1439487) (-955 "RFFACTOR.spad" 1438518 1438528 1439045 1439050) (-954 "RFFACT.spad" 1438254 1438265 1438508 1438513) (-953 "RFDIST.spad" 1437251 1437259 1438244 1438249) (-952 "RF.spad" 1434926 1434936 1437241 1437246) (-951 "RETSOL.spad" 1434346 1434358 1434916 1434921) (-950 "RETRACT.spad" 1433775 1433785 1434336 1434341) (-949 "RETRACT.spad" 1433202 1433214 1433765 1433770) (-948 "RETAST.spad" 1433015 1433023 1433192 1433197) (-947 "RESRING.spad" 1432363 1432409 1432953 1433010) (-946 "RESLATC.spad" 1431688 1431698 1432353 1432358) (-945 "REPSQ.spad" 1431420 1431430 1431678 1431683) (-944 "REPDB.spad" 1431128 1431138 1431410 1431415) (-943 "REP2.spad" 1420843 1420853 1430970 1430975) (-942 "REP1.spad" 1415064 1415074 1420793 1420798) (-941 "REP.spad" 1412619 1412627 1415054 1415059) (-940 "REGSET.spad" 1410445 1410461 1412253 1412280) (-939 "REF.spad" 1409964 1409974 1410435 1410440) (-938 "REDORDER.spad" 1409171 1409187 1409954 1409959) (-937 "RECLOS.spad" 1408068 1408087 1408771 1408864) (-936 "REALSOLV.spad" 1407209 1407217 1408058 1408063) (-935 "REAL0Q.spad" 1404508 1404522 1407199 1407204) (-934 "REAL0.spad" 1401353 1401367 1404498 1404503) (-933 "REAL.spad" 1401226 1401234 1401343 1401348) (-932 "RDUCEAST.spad" 1400948 1400956 1401216 1401221) (-931 "RDIV.spad" 1400604 1400628 1400938 1400943) (-930 "RDIST.spad" 1400172 1400182 1400594 1400599) (-929 "RDETRS.spad" 1399037 1399054 1400162 1400167) (-928 "RDETR.spad" 1397177 1397194 1399027 1399032) (-927 "RDEEFS.spad" 1396277 1396293 1397167 1397172) (-926 "RDEEF.spad" 1395288 1395304 1396267 1396272) (-925 "RCFIELD.spad" 1392507 1392515 1395190 1395283) (-924 "RCFIELD.spad" 1389812 1389822 1392497 1392502) (-923 "RCAGG.spad" 1387749 1387759 1389802 1389807) (-922 "RCAGG.spad" 1385613 1385625 1387668 1387673) (-921 "RATRET.spad" 1384974 1384984 1385603 1385608) (-920 "RATFACT.spad" 1384667 1384678 1384964 1384969) (-919 "RANDSRC.spad" 1383987 1383995 1384657 1384662) (-918 "RADUTIL.spad" 1383744 1383752 1383977 1383982) (-917 "RADIX.spad" 1380789 1380802 1382334 1382427) (-916 "RADFF.spad" 1378706 1378742 1378824 1378980) (-915 "RADCAT.spad" 1378302 1378310 1378696 1378701) (-914 "RADCAT.spad" 1377896 1377906 1378292 1378297) (-913 "QUEUE.spad" 1377310 1377320 1377568 1377595) (-912 "QUATCT2.spad" 1376931 1376949 1377300 1377305) (-911 "QUATCAT.spad" 1375102 1375112 1376861 1376926) (-910 "QUATCAT.spad" 1373038 1373050 1374799 1374804) (-909 "QUAT.spad" 1371645 1371655 1371987 1372052) (-908 "QUAGG.spad" 1370479 1370489 1371613 1371640) (-907 "QQUTAST.spad" 1370248 1370256 1370469 1370474) (-906 "QFORM.spad" 1369867 1369881 1370238 1370243) (-905 "QFCAT2.spad" 1369560 1369576 1369857 1369862) (-904 "QFCAT.spad" 1368263 1368273 1369462 1369555) (-903 "QFCAT.spad" 1366599 1366611 1367800 1367805) (-902 "QEQUAT.spad" 1366158 1366166 1366589 1366594) (-901 "QCMPACK.spad" 1361073 1361092 1366148 1366153) (-900 "QALGSET2.spad" 1359069 1359087 1361063 1361068) (-899 "QALGSET.spad" 1355174 1355206 1358983 1358988) (-898 "PWFFINTB.spad" 1352590 1352611 1355164 1355169) (-897 "PUSHVAR.spad" 1351929 1351948 1352580 1352585) (-896 "PTRANFN.spad" 1348065 1348075 1351919 1351924) (-895 "PTPACK.spad" 1345153 1345163 1348055 1348060) (-894 "PTFUNC2.spad" 1344976 1344990 1345143 1345148) (-893 "PTCAT.spad" 1344231 1344241 1344944 1344971) (-892 "PSQFR.spad" 1343546 1343570 1344221 1344226) (-891 "PSEUDLIN.spad" 1342432 1342442 1343536 1343541) (-890 "PSETPK.spad" 1329137 1329153 1342310 1342315) (-889 "PSETCAT.spad" 1323537 1323560 1329117 1329132) (-888 "PSETCAT.spad" 1317911 1317936 1323493 1323498) (-887 "PSCURVE.spad" 1316910 1316918 1317901 1317906) (-886 "PSCAT.spad" 1315693 1315722 1316808 1316905) (-885 "PSCAT.spad" 1314566 1314597 1315683 1315688) (-884 "PRTITION.spad" 1313264 1313272 1314556 1314561) (-883 "PRTDAST.spad" 1312983 1312991 1313254 1313259) (-882 "PRS.spad" 1302601 1302618 1312939 1312944) (-881 "PRQAGG.spad" 1302036 1302046 1302569 1302596) (-880 "PROPLOG.spad" 1301640 1301648 1302026 1302031) (-879 "PROPFUN2.spad" 1301263 1301276 1301630 1301635) (-878 "PROPFUN1.spad" 1300669 1300680 1301253 1301258) (-877 "PROPFRML.spad" 1299237 1299248 1300659 1300664) (-876 "PROPERTY.spad" 1298733 1298741 1299227 1299232) (-875 "PRODUCT.spad" 1296430 1296442 1296714 1296769) (-874 "PRINT.spad" 1296182 1296190 1296420 1296425) (-873 "PRIMES.spad" 1294443 1294453 1296172 1296177) (-872 "PRIMELT.spad" 1292564 1292578 1294433 1294438) (-871 "PRIMCAT.spad" 1292207 1292215 1292554 1292559) (-870 "PRIMARR2.spad" 1290974 1290986 1292197 1292202) (-869 "PRIMARR.spad" 1290029 1290039 1290199 1290226) (-868 "PREASSOC.spad" 1289411 1289423 1290019 1290024) (-867 "PR.spad" 1287929 1287941 1288628 1288755) (-866 "PPCURVE.spad" 1287066 1287074 1287919 1287924) (-865 "PORTNUM.spad" 1286857 1286865 1287056 1287061) (-864 "POLYROOT.spad" 1285706 1285728 1286813 1286818) (-863 "POLYLIFT.spad" 1284971 1284994 1285696 1285701) (-862 "POLYCATQ.spad" 1283097 1283119 1284961 1284966) (-861 "POLYCAT.spad" 1276599 1276620 1282965 1283092) (-860 "POLYCAT.spad" 1269621 1269644 1275989 1275994) (-859 "POLY2UP.spad" 1269073 1269087 1269611 1269616) (-858 "POLY2.spad" 1268670 1268682 1269063 1269068) (-857 "POLY.spad" 1266338 1266348 1266853 1266980) (-856 "POLUTIL.spad" 1265303 1265332 1266294 1266299) (-855 "POLTOPOL.spad" 1264051 1264066 1265293 1265298) (-854 "POINT.spad" 1262934 1262944 1263021 1263048) (-853 "PNTHEORY.spad" 1259636 1259644 1262924 1262929) (-852 "PMTOOLS.spad" 1258411 1258425 1259626 1259631) (-851 "PMSYM.spad" 1257960 1257970 1258401 1258406) (-850 "PMQFCAT.spad" 1257551 1257565 1257950 1257955) (-849 "PMPREDFS.spad" 1257013 1257035 1257541 1257546) (-848 "PMPRED.spad" 1256500 1256514 1257003 1257008) (-847 "PMPLCAT.spad" 1255577 1255595 1256429 1256434) (-846 "PMLSAGG.spad" 1255162 1255176 1255567 1255572) (-845 "PMKERNEL.spad" 1254741 1254753 1255152 1255157) (-844 "PMINS.spad" 1254321 1254331 1254731 1254736) (-843 "PMFS.spad" 1253898 1253916 1254311 1254316) (-842 "PMDOWN.spad" 1253188 1253202 1253888 1253893) (-841 "PMASSFS.spad" 1252163 1252179 1253178 1253183) (-840 "PMASS.spad" 1251181 1251189 1252153 1252158) (-839 "PLOTTOOL.spad" 1250961 1250969 1251171 1251176) (-838 "PLOT3D.spad" 1247425 1247433 1250951 1250956) (-837 "PLOT1.spad" 1246598 1246608 1247415 1247420) (-836 "PLOT.spad" 1241521 1241529 1246588 1246593) (-835 "PLEQN.spad" 1228923 1228950 1241511 1241516) (-834 "PINTERPA.spad" 1228707 1228723 1228913 1228918) (-833 "PINTERP.spad" 1228329 1228348 1228697 1228702) (-832 "PID.spad" 1227303 1227311 1228255 1228324) (-831 "PICOERCE.spad" 1226960 1226970 1227293 1227298) (-830 "PI.spad" 1226577 1226585 1226934 1226955) (-829 "PGROEB.spad" 1225186 1225200 1226567 1226572) (-828 "PGE.spad" 1216859 1216867 1225176 1225181) (-827 "PGCD.spad" 1215813 1215830 1216849 1216854) (-826 "PFRPAC.spad" 1214962 1214972 1215803 1215808) (-825 "PFR.spad" 1211665 1211675 1214864 1214957) (-824 "PFOTOOLS.spad" 1210923 1210939 1211655 1211660) (-823 "PFOQ.spad" 1210293 1210311 1210913 1210918) (-822 "PFO.spad" 1209712 1209739 1210283 1210288) (-821 "PFECAT.spad" 1207422 1207430 1209638 1209707) (-820 "PFECAT.spad" 1205160 1205170 1207378 1207383) (-819 "PFBRU.spad" 1203048 1203060 1205150 1205155) (-818 "PFBR.spad" 1200608 1200631 1203038 1203043) (-817 "PF.spad" 1200182 1200194 1200413 1200506) (-816 "PERMGRP.spad" 1194952 1194962 1200172 1200177) (-815 "PERMCAT.spad" 1193613 1193623 1194932 1194947) (-814 "PERMAN.spad" 1192169 1192183 1193603 1193608) (-813 "PERM.spad" 1187979 1187989 1192002 1192017) (-812 "PENDTREE.spad" 1187393 1187403 1187673 1187678) (-811 "PDSPC.spad" 1186206 1186216 1187383 1187388) (-810 "PDSPC.spad" 1185017 1185029 1186196 1186201) (-809 "PDRING.spad" 1184859 1184869 1184997 1185012) (-808 "PDMOD.spad" 1184675 1184687 1184827 1184854) (-807 "PDECOMP.spad" 1184145 1184162 1184665 1184670) (-806 "PDDOM.spad" 1183583 1183596 1184135 1184140) (-805 "PDDOM.spad" 1183019 1183034 1183573 1183578) (-804 "PCOMP.spad" 1182872 1182885 1183009 1183014) (-803 "PBWLB.spad" 1181470 1181487 1182862 1182867) (-802 "PATTERN2.spad" 1181208 1181220 1181460 1181465) (-801 "PATTERN1.spad" 1179552 1179568 1181198 1181203) (-800 "PATTERN.spad" 1174127 1174137 1179542 1179547) (-799 "PATRES2.spad" 1173799 1173813 1174117 1174122) (-798 "PATRES.spad" 1171382 1171394 1173789 1173794) (-797 "PATMATCH.spad" 1169623 1169654 1171134 1171139) (-796 "PATMAB.spad" 1169052 1169062 1169613 1169618) (-795 "PATLRES.spad" 1168138 1168152 1169042 1169047) (-794 "PATAB.spad" 1167902 1167912 1168128 1168133) (-793 "PARTPERM.spad" 1165958 1165966 1167892 1167897) (-792 "PARSURF.spad" 1165392 1165420 1165948 1165953) (-791 "PARSU2.spad" 1165189 1165205 1165382 1165387) (-790 "script-parser.spad" 1164709 1164717 1165179 1165184) (-789 "PARSCURV.spad" 1164143 1164171 1164699 1164704) (-788 "PARSC2.spad" 1163934 1163950 1164133 1164138) (-787 "PARPCURV.spad" 1163396 1163424 1163924 1163929) (-786 "PARPC2.spad" 1163187 1163203 1163386 1163391) (-785 "PARAMAST.spad" 1162315 1162323 1163177 1163182) (-784 "PAN2EXPR.spad" 1161727 1161735 1162305 1162310) (-783 "PALETTE.spad" 1160841 1160849 1161717 1161722) (-782 "PAIR.spad" 1159915 1159928 1160484 1160489) (-781 "PADICRC.spad" 1157320 1157338 1158483 1158576) (-780 "PADICRAT.spad" 1155380 1155392 1155593 1155686) (-779 "PADICCT.spad" 1153929 1153941 1155306 1155375) (-778 "PADIC.spad" 1153632 1153644 1153855 1153924) (-777 "PADEPAC.spad" 1152321 1152340 1153622 1153627) (-776 "PADE.spad" 1151073 1151089 1152311 1152316) (-775 "OWP.spad" 1150321 1150351 1150931 1150998) (-774 "OVERSET.spad" 1149894 1149902 1150311 1150316) (-773 "OVAR.spad" 1149675 1149698 1149884 1149889) (-772 "OUTFORM.spad" 1139083 1139091 1149665 1149670) (-771 "OUTBFILE.spad" 1138517 1138525 1139073 1139078) (-770 "OUTBCON.spad" 1137587 1137595 1138507 1138512) (-769 "OUTBCON.spad" 1136655 1136665 1137577 1137582) (-768 "OUT.spad" 1135773 1135781 1136645 1136650) (-767 "OSI.spad" 1135248 1135256 1135763 1135768) (-766 "OSGROUP.spad" 1135166 1135174 1135238 1135243) (-765 "ORTHPOL.spad" 1133677 1133687 1135109 1135114) (-764 "OREUP.spad" 1133171 1133199 1133398 1133437) (-763 "ORESUP.spad" 1132513 1132537 1132892 1132931) (-762 "OREPCTO.spad" 1130402 1130414 1132433 1132438) (-761 "OREPCAT.spad" 1124589 1124599 1130358 1130397) (-760 "OREPCAT.spad" 1118666 1118678 1124437 1124442) (-759 "ORDTYPE.spad" 1117903 1117911 1118656 1118661) (-758 "ORDTYPE.spad" 1117138 1117148 1117893 1117898) (-757 "ORDSTRCT.spad" 1116924 1116939 1117087 1117092) (-756 "ORDSET.spad" 1116624 1116632 1116914 1116919) (-755 "ORDRING.spad" 1116441 1116449 1116604 1116619) (-754 "ORDMON.spad" 1116296 1116304 1116431 1116436) (-753 "ORDFUNS.spad" 1115428 1115444 1116286 1116291) (-752 "ORDFIN.spad" 1115248 1115256 1115418 1115423) (-751 "ORDCOMP2.spad" 1114541 1114553 1115238 1115243) (-750 "ORDCOMP.spad" 1113067 1113077 1114149 1114178) (-749 "OPSIG.spad" 1112729 1112737 1113057 1113062) (-748 "OPQUERY.spad" 1112310 1112318 1112719 1112724) (-747 "OPERCAT.spad" 1111776 1111786 1112300 1112305) (-746 "OPERCAT.spad" 1111240 1111252 1111766 1111771) (-745 "OP.spad" 1110982 1110992 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"MPC3.spad" 983395 983435 983568 983573) (-667 "MPC2.spad" 983048 983081 983385 983390) (-666 "MONOTOOL.spad" 981399 981416 983038 983043) (-665 "catdef.spad" 980832 980843 981053 981394) (-664 "catdef.spad" 980230 980241 980486 980827) (-663 "MONOID.spad" 979551 979559 980220 980225) (-662 "MONOID.spad" 978870 978880 979541 979546) (-661 "MONOGEN.spad" 977618 977631 978730 978865) (-660 "MONOGEN.spad" 976388 976403 977502 977507) (-659 "MONADWU.spad" 974468 974476 976378 976383) (-658 "MONADWU.spad" 972546 972556 974458 974463) (-657 "MONAD.spad" 971706 971714 972536 972541) (-656 "MONAD.spad" 970864 970874 971696 971701) (-655 "MOEBIUS.spad" 969600 969614 970844 970859) (-654 "MODULE.spad" 969470 969480 969568 969595) (-653 "MODULE.spad" 969360 969372 969460 969465) (-652 "MODRING.spad" 968695 968734 969340 969355) (-651 "MODOP.spad" 967352 967364 968517 968584) (-650 "MODMONOM.spad" 967083 967101 967342 967347) (-649 "MODMON.spad" 964153 964165 964868 965021) (-648 "MODFIELD.spad" 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(-607 "LSAGG.spad" 901600 901612 901901 901906) (-606 "LPOLY.spad" 900562 900581 901456 901525) (-605 "LPEFRAC.spad" 899833 899843 900552 900557) (-604 "LOGIC.spad" 899435 899443 899823 899828) (-603 "LOGIC.spad" 899035 899045 899425 899430) (-602 "LODOOPS.spad" 897965 897977 899025 899030) (-601 "LODOF.spad" 897011 897028 897922 897927) (-600 "LODOCAT.spad" 895677 895687 896967 897006) (-599 "LODOCAT.spad" 894341 894353 895633 895638) (-598 "LODO2.spad" 893655 893667 894062 894101) (-597 "LODO1.spad" 893096 893106 893376 893415) (-596 "LODO.spad" 892521 892537 892817 892856) (-595 "LODEEF.spad" 891323 891341 892511 892516) (-594 "LO.spad" 890724 890738 891257 891284) (-593 "LNAGG.spad" 886911 886921 890714 890719) (-592 "LNAGG.spad" 883062 883074 886867 886872) (-591 "LMOPS.spad" 879830 879847 883052 883057) (-590 "LMODULE.spad" 879614 879624 879820 879825) (-589 "LMDICT.spad" 878995 879005 879243 879270) (-588 "LLINSET.spad" 878702 878712 878985 878990) (-587 "LITERAL.spad" 878608 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(-566 "LEXP.spad" 848308 848335 850269 850284) (-565 "LETAST.spad" 848007 848015 848298 848303) (-564 "LEADCDET.spad" 846413 846430 847997 848002) (-563 "LAZM3PK.spad" 845157 845179 846403 846408) (-562 "LAUPOL.spad" 843824 843837 844724 844793) (-561 "LAPLACE.spad" 843407 843423 843814 843819) (-560 "LALG.spad" 843183 843193 843387 843402) (-559 "LALG.spad" 842967 842979 843173 843178) (-558 "LA.spad" 842407 842421 842889 842928) (-557 "KVTFROM.spad" 842150 842160 842397 842402) (-556 "KTVLOGIC.spad" 841694 841702 842140 842145) (-555 "KRCFROM.spad" 841440 841450 841684 841689) (-554 "KOVACIC.spad" 840171 840188 841430 841435) (-553 "KONVERT.spad" 839893 839903 840161 840166) (-552 "KOERCE.spad" 839630 839640 839883 839888) (-551 "KERNEL2.spad" 839333 839345 839620 839625) (-550 "KERNEL.spad" 838053 838063 839182 839187) (-549 "KDAGG.spad" 837162 837184 838033 838048) (-548 "KDAGG.spad" 836279 836303 837152 837157) (-547 "KAFILE.spad" 835169 835185 835404 835431) (-546 "JVMOP.spad" 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197977) ((-1192 . -23) T) ((-1192 . -1007) T) ((-1192 . -549) 197959) ((-1192 . -1120) T) ((-1192 . -13) T) ((-1192 . -72) T) ((-1192 . -25) T) ((-1192 . -102) T) ((-1192 . -587) 197933) ((-1192 . -1185) 197917) ((-1192 . -651) 197887) ((-1192 . -579) 197857) ((-1192 . -963) 197841) ((-1192 . -958) 197825) ((-1192 . -80) 197804) ((-1192 . -38) 197774) ((-1192 . -1190) 197750) ((-1191 . -1193) 197729) ((-1191 . -945) 197686) ((-1191 . -552) 197615) ((-1191 . -956) T) ((-1191 . -660) T) ((-1191 . -1052) T) ((-1191 . -1017) T) ((-1191 . -964) T) ((-1191 . -21) T) ((-1191 . -585) 197574) ((-1191 . -23) T) ((-1191 . -1007) T) ((-1191 . -549) 197556) ((-1191 . -1120) T) ((-1191 . -13) T) ((-1191 . -72) T) ((-1191 . -25) T) ((-1191 . -102) T) ((-1191 . -587) 197530) ((-1191 . -1185) 197514) ((-1191 . -651) 197484) ((-1191 . -579) 197454) ((-1191 . -963) 197438) ((-1191 . -958) 197422) ((-1191 . -80) 197401) ((-1191 . -38) 197371) ((-1191 . -1190) 197350) ((-1191 . -330) 197322) ((-1186 . 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-1169) 196035) ((-1170 . -319) 196019) ((-1170 . -754) 195998) ((-1170 . -751) 195977) ((-1170 . -122) 195961) ((-1170 . -34) T) ((-1170 . -13) T) ((-1170 . -1120) T) ((-1170 . -72) 195895) ((-1170 . -549) 195810) ((-1170 . -257) 195748) ((-1170 . -449) 195681) ((-1170 . -1007) 195634) ((-1170 . -424) 195618) ((-1170 . -550) 195579) ((-1170 . -239) 195531) ((-1170 . -535) 195508) ((-1170 . -241) 195485) ((-1170 . -590) 195469) ((-1170 . -19) 195453) ((-1167 . -1007) T) ((-1167 . -549) 195419) ((-1167 . -1120) T) ((-1167 . -13) T) ((-1167 . -72) T) ((-1160 . -1163) 195403) ((-1160 . -188) 195362) ((-1160 . -552) 195244) ((-1160 . -587) 195169) ((-1160 . -585) 195079) ((-1160 . -102) T) ((-1160 . -25) T) ((-1160 . -72) T) ((-1160 . -549) 195061) ((-1160 . -1007) T) ((-1160 . -23) T) ((-1160 . -21) T) ((-1160 . -964) T) ((-1160 . -1017) T) ((-1160 . -1052) T) ((-1160 . -660) T) ((-1160 . -956) T) ((-1160 . -184) 195014) ((-1160 . -13) T) ((-1160 . -1120) T) ((-1160 . -187) 194973) ((-1160 . -239) 194938) ((-1160 . -804) 194851) ((-1160 . -801) 194739) ((-1160 . -806) 194652) ((-1160 . -881) 194622) ((-1160 . -38) 194519) ((-1160 . -80) 194384) ((-1160 . -958) 194270) ((-1160 . -963) 194156) ((-1160 . -579) 194053) ((-1160 . -651) 193950) ((-1160 . -116) 193929) ((-1160 . -118) 193908) ((-1160 . -144) 193862) ((-1160 . -491) 193841) ((-1160 . -243) 193820) ((-1160 . -47) 193797) ((-1160 . -1149) 193774) ((-1160 . -35) 193740) ((-1160 . -66) 193706) ((-1160 . -237) 193672) ((-1160 . -428) 193638) ((-1160 . -1109) 193604) ((-1160 . -1106) 193570) ((-1160 . -910) 193536) ((-1157 . -274) 193480) ((-1157 . -945) 193446) ((-1157 . -350) 193412) ((-1157 . -38) 193269) ((-1157 . -552) 193143) ((-1157 . -587) 193032) ((-1157 . -585) 192906) ((-1157 . -964) T) ((-1157 . -1017) T) ((-1157 . -1052) T) ((-1157 . -660) T) ((-1157 . -956) T) ((-1157 . -80) 192756) ((-1157 . -958) 192645) ((-1157 . -963) 192534) ((-1157 . -21) T) ((-1157 . -23) T) ((-1157 . -1007) T) ((-1157 . -549) 192516) ((-1157 . -1120) T) ((-1157 . -13) T) ((-1157 . -72) T) ((-1157 . -25) T) ((-1157 . -102) T) ((-1157 . -579) 192373) ((-1157 . -651) 192230) ((-1157 . -116) 192191) ((-1157 . -118) 192152) ((-1157 . -144) T) ((-1157 . -491) T) ((-1157 . -243) T) ((-1157 . -47) 192096) ((-1156 . -1155) 192075) ((-1156 . -309) 192054) ((-1156 . -1125) 192033) ((-1156 . -827) 192012) ((-1156 . -491) 191966) ((-1156 . -144) 191900) ((-1156 . -552) 191719) ((-1156 . -651) 191566) ((-1156 . -579) 191413) ((-1156 . -38) 191260) ((-1156 . -387) 191239) ((-1156 . -255) 191218) ((-1156 . -587) 191118) ((-1156 . -585) 191003) ((-1156 . -964) T) ((-1156 . -1017) T) ((-1156 . -1052) T) ((-1156 . -660) T) ((-1156 . -956) T) ((-1156 . -80) 190823) ((-1156 . -958) 190664) ((-1156 . -963) 190505) ((-1156 . -21) T) ((-1156 . -23) T) ((-1156 . -1007) T) ((-1156 . -549) 190487) ((-1156 . -1120) T) ((-1156 . -13) T) ((-1156 . -72) T) ((-1156 . -25) T) ((-1156 . -102) T) ((-1156 . -243) 190441) ((-1156 . -199) 190420) ((-1156 . -910) 190386) ((-1156 . -1106) 190352) ((-1156 . -1109) 190318) ((-1156 . -428) 190284) ((-1156 . -237) 190250) ((-1156 . -66) 190216) ((-1156 . -35) 190182) ((-1156 . -1149) 190152) ((-1156 . -47) 190122) ((-1156 . -118) 190101) ((-1156 . -116) 190080) ((-1156 . -881) 190043) ((-1156 . -806) 189949) ((-1156 . -801) 189853) ((-1156 . -804) 189759) ((-1156 . -239) 189717) ((-1156 . -187) 189669) ((-1156 . -184) 189615) ((-1156 . -188) 189567) ((-1156 . -1153) 189551) ((-1156 . -945) 189535) ((-1151 . -1155) 189496) ((-1151 . -309) 189475) ((-1151 . -1125) 189454) ((-1151 . -827) 189433) ((-1151 . -491) 189387) ((-1151 . -144) 189321) ((-1151 . -552) 189070) ((-1151 . -651) 188917) ((-1151 . -579) 188764) ((-1151 . -38) 188611) ((-1151 . -387) 188590) ((-1151 . -255) 188569) ((-1151 . -587) 188469) ((-1151 . -585) 188354) ((-1151 . -964) T) ((-1151 . -1017) T) ((-1151 . -1052) T) ((-1151 . -660) T) ((-1151 . -956) T) ((-1151 . -80) 188174) ((-1151 . -958) 188015) ((-1151 . -963) 187856) ((-1151 . -21) T) ((-1151 . -23) T) ((-1151 . -1007) T) ((-1151 . -549) 187838) ((-1151 . -1120) T) ((-1151 . -13) T) ((-1151 . -72) T) ((-1151 . -25) T) ((-1151 . -102) T) ((-1151 . -243) 187792) ((-1151 . -199) 187771) ((-1151 . -910) 187737) ((-1151 . -1106) 187703) ((-1151 . -1109) 187669) ((-1151 . -428) 187635) ((-1151 . -237) 187601) ((-1151 . -66) 187567) ((-1151 . -35) 187533) ((-1151 . -1149) 187503) ((-1151 . -47) 187473) ((-1151 . -118) 187452) ((-1151 . -116) 187431) ((-1151 . -881) 187394) ((-1151 . -806) 187300) ((-1151 . -801) 187181) ((-1151 . -804) 187087) ((-1151 . -239) 187045) ((-1151 . -187) 186997) ((-1151 . -184) 186943) ((-1151 . -188) 186895) ((-1151 . -1153) 186879) ((-1151 . -945) 186814) ((-1139 . -1146) 186798) ((-1139 . -1057) 186776) ((-1139 . -550) NIL) ((-1139 . -257) 186763) ((-1139 . -449) 186711) ((-1139 . -274) 186688) ((-1139 . -945) 186571) ((-1139 . -350) 186555) ((-1139 . -38) 186387) ((-1139 . -80) 186192) ((-1139 . -958) 186018) ((-1139 . -963) 185844) ((-1139 . -585) 185754) ((-1139 . -587) 185643) ((-1139 . -579) 185475) ((-1139 . -651) 185307) ((-1139 . -552) 185063) ((-1139 . -116) 185042) ((-1139 . -118) 185021) ((-1139 . -47) 184998) ((-1139 . -324) 184982) ((-1139 . -577) 184930) ((-1139 . -804) 184874) ((-1139 . -801) 184781) ((-1139 . -806) 184692) ((-1139 . -791) NIL) ((-1139 . -816) 184671) ((-1139 . -1125) 184650) ((-1139 . -856) 184620) ((-1139 . -827) 184599) ((-1139 . -491) 184513) ((-1139 . -243) 184427) ((-1139 . -144) 184321) ((-1139 . -387) 184255) ((-1139 . -255) 184234) ((-1139 . -239) 184161) ((-1139 . -188) T) ((-1139 . -102) T) ((-1139 . -25) T) ((-1139 . -72) T) ((-1139 . -549) 184143) ((-1139 . -1007) T) ((-1139 . -23) T) ((-1139 . -21) T) ((-1139 . -964) T) ((-1139 . -1017) T) ((-1139 . -1052) T) ((-1139 . -660) T) ((-1139 . -956) T) ((-1139 . -184) 184130) ((-1139 . -13) T) ((-1139 . -1120) T) ((-1139 . -187) T) ((-1139 . -223) 184114) ((-1139 . -182) 184098) ((-1137 . -1000) 184082) ((-1137 . -554) 184066) ((-1137 . -1007) 184044) ((-1137 . -549) 184011) ((-1137 . -1120) 183989) ((-1137 . -13) 183967) ((-1137 . -72) 183945) ((-1137 . -1001) 183902) ((-1135 . -1134) 183881) ((-1135 . -910) 183847) ((-1135 . -1106) 183813) ((-1135 . -1109) 183779) ((-1135 . -428) 183745) ((-1135 . -237) 183711) ((-1135 . -66) 183677) ((-1135 . -35) 183643) ((-1135 . -1149) 183620) ((-1135 . -47) 183597) ((-1135 . -552) 183352) ((-1135 . -651) 183172) ((-1135 . -579) 182992) ((-1135 . -587) 182803) ((-1135 . -585) 182661) ((-1135 . -963) 182475) ((-1135 . -958) 182289) ((-1135 . -80) 182077) ((-1135 . -38) 181897) ((-1135 . -881) 181867) ((-1135 . -239) 181767) ((-1135 . -1132) 181751) ((-1135 . -964) T) ((-1135 . -1017) T) ((-1135 . -1052) T) ((-1135 . -660) T) ((-1135 . -956) T) ((-1135 . -21) T) ((-1135 . -23) T) ((-1135 . -1007) T) ((-1135 . -549) 181733) ((-1135 . -1120) T) ((-1135 . -13) T) ((-1135 . -72) T) ((-1135 . -25) T) ((-1135 . -102) T) ((-1135 . -116) 181661) ((-1135 . -118) 181589) ((-1135 . -550) 181262) ((-1135 . -182) 181232) ((-1135 . -804) 181086) ((-1135 . -806) 180886) ((-1135 . -801) 180684) ((-1135 . -223) 180654) ((-1135 . -187) 180516) ((-1135 . -184) 180372) ((-1135 . -188) 180280) ((-1135 . -309) 180259) ((-1135 . -1125) 180238) ((-1135 . -827) 180217) ((-1135 . -491) 180171) ((-1135 . -144) 180105) ((-1135 . -387) 180084) ((-1135 . -255) 180063) ((-1135 . -243) 180017) ((-1135 . -199) 179996) ((-1135 . -285) 179966) ((-1135 . -449) 179826) ((-1135 . -257) 179765) ((-1135 . -324) 179735) ((-1135 . -577) 179643) ((-1135 . -338) 179613) ((-1135 . -791) 179486) ((-1135 . -735) 179439) ((-1135 . -709) 179392) ((-1135 . -711) 179345) ((-1135 . -751) 179247) ((-1135 . -754) 179149) ((-1135 . -713) 179102) ((-1135 . -716) 179055) ((-1135 . -750) 179008) ((-1135 . -789) 178978) ((-1135 . -816) 178931) ((-1135 . -928) 178884) ((-1135 . -945) 178673) ((-1135 . -1057) 178625) ((-1135 . -899) 178595) ((-1130 . -1134) 178556) ((-1130 . -910) 178522) ((-1130 . -1106) 178488) ((-1130 . -1109) 178454) ((-1130 . -428) 178420) ((-1130 . -237) 178386) ((-1130 . -66) 178352) ((-1130 . -35) 178318) ((-1130 . -1149) 178295) ((-1130 . -47) 178272) ((-1130 . -552) 178073) ((-1130 . -651) 177875) ((-1130 . -579) 177677) ((-1130 . -587) 177532) ((-1130 . -585) 177372) ((-1130 . -963) 177168) ((-1130 . -958) 176964) ((-1130 . -80) 176716) ((-1130 . -38) 176518) ((-1130 . -881) 176488) ((-1130 . -239) 176316) ((-1130 . -1132) 176300) ((-1130 . -964) T) ((-1130 . -1017) T) ((-1130 . -1052) T) ((-1130 . -660) T) ((-1130 . -956) T) ((-1130 . -21) T) ((-1130 . -23) T) ((-1130 . -1007) T) ((-1130 . -549) 176282) ((-1130 . -1120) T) ((-1130 . -13) T) ((-1130 . -72) T) ((-1130 . -25) T) ((-1130 . -102) T) ((-1130 . -116) 176192) ((-1130 . -118) 176102) ((-1130 . -550) NIL) ((-1130 . -182) 176054) ((-1130 . -804) 175890) ((-1130 . -806) 175654) ((-1130 . -801) 175393) ((-1130 . -223) 175345) ((-1130 . -187) 175171) ((-1130 . -184) 174991) ((-1130 . -188) 174881) ((-1130 . -309) 174860) ((-1130 . -1125) 174839) ((-1130 . -827) 174818) ((-1130 . -491) 174772) ((-1130 . -144) 174706) ((-1130 . -387) 174685) ((-1130 . -255) 174664) ((-1130 . -243) 174618) ((-1130 . -199) 174597) ((-1130 . -285) 174549) ((-1130 . -449) 174283) ((-1130 . -257) 174168) ((-1130 . -324) 174120) ((-1130 . -577) 174072) ((-1130 . -338) 174024) ((-1130 . -791) NIL) ((-1130 . -735) NIL) ((-1130 . -709) NIL) ((-1130 . -711) NIL) ((-1130 . -751) NIL) ((-1130 . -754) NIL) ((-1130 . -713) NIL) ((-1130 . -716) NIL) ((-1130 . -750) NIL) ((-1130 . -789) 173976) ((-1130 . -816) NIL) ((-1130 . -928) NIL) ((-1130 . -945) 173942) ((-1130 . -1057) NIL) ((-1130 . -899) 173894) ((-1129 . -747) T) ((-1129 . -754) T) ((-1129 . -751) T) ((-1129 . -1007) T) ((-1129 . -549) 173876) ((-1129 . -1120) T) ((-1129 . -13) T) ((-1129 . -72) T) ((-1129 . -315) T) ((-1129 . -601) T) ((-1128 . -747) T) ((-1128 . -754) T) ((-1128 . -751) T) ((-1128 . -1007) T) ((-1128 . -549) 173858) ((-1128 . -1120) T) ((-1128 . -13) T) ((-1128 . -72) T) ((-1128 . -315) T) ((-1128 . -601) T) ((-1127 . -747) T) ((-1127 . -754) T) ((-1127 . -751) T) ((-1127 . -1007) T) ((-1127 . -549) 173840) ((-1127 . -1120) T) ((-1127 . -13) T) ((-1127 . -72) T) ((-1127 . -315) T) ((-1127 . -601) T) ((-1126 . -747) T) ((-1126 . -754) T) ((-1126 . -751) T) ((-1126 . -1007) T) ((-1126 . -549) 173822) ((-1126 . -1120) T) ((-1126 . -13) T) ((-1126 . -72) T) ((-1126 . -315) T) ((-1126 . -601) T) ((-1121 . -989) T) ((-1121 . -425) 173803) ((-1121 . -549) 173769) ((-1121 . -552) 173750) ((-1121 . -1007) T) ((-1121 . -1120) T) ((-1121 . -13) T) ((-1121 . -72) T) ((-1121 . -64) T) ((-1118 . -425) 173727) ((-1118 . -549) 173668) ((-1118 . -552) 173645) ((-1118 . -1007) 173623) ((-1118 . -1120) 173601) ((-1118 . -13) 173579) ((-1118 . -72) 173557) ((-1113 . -674) 173533) ((-1113 . -35) 173499) ((-1113 . -66) 173465) ((-1113 . -237) 173431) ((-1113 . -428) 173397) ((-1113 . -1109) 173363) ((-1113 . -1106) 173329) ((-1113 . -910) 173295) ((-1113 . -47) 173264) ((-1113 . -38) 173161) ((-1113 . -579) 173058) ((-1113 . -651) 172955) ((-1113 . -552) 172837) ((-1113 . -243) 172816) ((-1113 . -491) 172795) ((-1113 . -80) 172660) ((-1113 . -958) 172546) ((-1113 . -963) 172432) ((-1113 . -144) 172386) ((-1113 . -118) 172365) ((-1113 . -116) 172344) ((-1113 . -587) 172269) ((-1113 . -585) 172179) ((-1113 . -881) 172140) ((-1113 . -806) 172121) ((-1113 . -1120) T) ((-1113 . -13) T) ((-1113 . -801) 172100) ((-1113 . -956) T) ((-1113 . -660) T) ((-1113 . -1052) T) ((-1113 . -1017) T) ((-1113 . -964) T) ((-1113 . -21) T) ((-1113 . -23) T) ((-1113 . -1007) T) ((-1113 . -549) 172082) ((-1113 . -72) T) ((-1113 . -25) T) ((-1113 . -102) T) ((-1113 . -804) 172063) ((-1113 . -449) 172030) ((-1113 . -257) 172017) ((-1107 . -918) 172001) ((-1107 . -34) T) ((-1107 . -13) T) ((-1107 . -1120) T) ((-1107 . -72) 171955) ((-1107 . -549) 171890) ((-1107 . -257) 171828) ((-1107 . -449) 171761) ((-1107 . -1007) 171739) ((-1107 . -424) 171723) ((-1102 . -311) 171697) ((-1102 . -72) T) ((-1102 . -13) T) ((-1102 . -1120) T) ((-1102 . -549) 171679) ((-1102 . -1007) T) ((-1100 . -1007) T) ((-1100 . -549) 171661) ((-1100 . -1120) T) ((-1100 . -13) T) ((-1100 . -72) T) ((-1100 . -552) 171643) ((-1095 . -742) 171627) ((-1095 . -72) T) ((-1095 . -13) T) ((-1095 . -1120) T) ((-1095 . -549) 171609) ((-1095 . -1007) T) ((-1093 . -1098) 171588) ((-1093 . -181) 171536) ((-1093 . -76) 171484) ((-1093 . -257) 171282) ((-1093 . -449) 171034) ((-1093 . -424) 170969) ((-1093 . -122) 170917) ((-1093 . -550) NIL) ((-1093 . -191) 170865) ((-1093 . -546) 170844) ((-1093 . -241) 170823) ((-1093 . -1120) T) ((-1093 . -13) T) ((-1093 . -239) 170802) ((-1093 . -1007) T) ((-1093 . -549) 170784) ((-1093 . -72) T) ((-1093 . -34) T) ((-1093 . -535) 170763) ((-1089 . -1007) T) ((-1089 . -549) 170745) ((-1089 . -1120) T) ((-1089 . -13) T) ((-1089 . -72) T) ((-1088 . -747) T) ((-1088 . -754) T) ((-1088 . -751) T) ((-1088 . -1007) T) ((-1088 . -549) 170727) ((-1088 . -1120) T) ((-1088 . -13) T) ((-1088 . -72) T) ((-1088 . -315) T) ((-1088 . -601) T) ((-1087 . -747) T) ((-1087 . -754) T) ((-1087 . -751) T) ((-1087 . -1007) T) ((-1087 . -549) 170709) ((-1087 . -1120) T) ((-1087 . -13) T) ((-1087 . -72) T) ((-1087 . -315) T) ((-1086 . -1166) T) ((-1086 . -1007) T) ((-1086 . -549) 170676) ((-1086 . -1120) T) ((-1086 . -13) T) ((-1086 . -72) T) ((-1086 . -945) 170612) ((-1086 . -552) 170548) ((-1085 . -549) 170530) ((-1084 . -549) 170512) ((-1083 . -274) 170489) ((-1083 . -945) 170387) ((-1083 . -350) 170371) ((-1083 . -38) 170268) ((-1083 . -552) 170125) ((-1083 . -587) 170050) ((-1083 . -585) 169960) ((-1083 . -964) T) ((-1083 . -1017) T) ((-1083 . -1052) T) ((-1083 . -660) T) ((-1083 . -956) T) ((-1083 . -80) 169825) ((-1083 . -958) 169711) ((-1083 . -963) 169597) ((-1083 . -21) T) ((-1083 . -23) T) ((-1083 . -1007) T) ((-1083 . -549) 169579) ((-1083 . -1120) T) ((-1083 . -13) T) ((-1083 . -72) T) ((-1083 . -25) T) ((-1083 . -102) T) ((-1083 . -579) 169476) ((-1083 . -651) 169373) ((-1083 . -116) 169352) ((-1083 . -118) 169331) ((-1083 . -144) 169285) ((-1083 . -491) 169264) ((-1083 . -243) 169243) ((-1083 . -47) 169220) ((-1081 . -751) T) ((-1081 . -549) 169202) ((-1081 . -1007) T) ((-1081 . -72) T) ((-1081 . -13) T) ((-1081 . -1120) T) ((-1081 . -754) T) ((-1081 . -550) 169124) ((-1081 . -552) 169090) ((-1081 . -945) 169072) ((-1081 . -791) 169039) ((-1080 . -1163) 169023) ((-1080 . -188) 168982) ((-1080 . -552) 168864) ((-1080 . -587) 168789) ((-1080 . -585) 168699) ((-1080 . -102) T) ((-1080 . -25) T) ((-1080 . -72) T) ((-1080 . -549) 168681) ((-1080 . -1007) T) ((-1080 . -23) T) ((-1080 . -21) T) ((-1080 . -964) T) ((-1080 . -1017) T) ((-1080 . -1052) T) ((-1080 . -660) T) ((-1080 . -956) T) ((-1080 . -184) 168634) ((-1080 . -13) T) ((-1080 . -1120) T) ((-1080 . -187) 168593) ((-1080 . -239) 168558) ((-1080 . -804) 168471) ((-1080 . -801) 168359) ((-1080 . -806) 168272) ((-1080 . -881) 168242) ((-1080 . -38) 168139) ((-1080 . -80) 168004) ((-1080 . -958) 167890) ((-1080 . -963) 167776) ((-1080 . -579) 167673) ((-1080 . -651) 167570) ((-1080 . -116) 167549) ((-1080 . -118) 167528) ((-1080 . -144) 167482) ((-1080 . -491) 167461) ((-1080 . -243) 167440) ((-1080 . -47) 167417) ((-1080 . -1149) 167394) ((-1080 . -35) 167360) ((-1080 . -66) 167326) ((-1080 . -237) 167292) ((-1080 . -428) 167258) ((-1080 . -1109) 167224) ((-1080 . -1106) 167190) ((-1080 . -910) 167156) ((-1079 . -1155) 167117) ((-1079 . -309) 167096) ((-1079 . -1125) 167075) ((-1079 . -827) 167054) ((-1079 . -491) 167008) ((-1079 . -144) 166942) ((-1079 . -552) 166691) ((-1079 . -651) 166538) ((-1079 . -579) 166385) ((-1079 . -38) 166232) ((-1079 . -387) 166211) ((-1079 . -255) 166190) ((-1079 . -587) 166090) ((-1079 . -585) 165975) ((-1079 . -964) T) ((-1079 . -1017) T) ((-1079 . -1052) T) ((-1079 . -660) T) ((-1079 . -956) T) ((-1079 . -80) 165795) ((-1079 . -958) 165636) ((-1079 . -963) 165477) ((-1079 . -21) T) ((-1079 . -23) T) ((-1079 . -1007) T) ((-1079 . -549) 165459) ((-1079 . -1120) T) ((-1079 . -13) T) ((-1079 . -72) T) ((-1079 . -25) T) ((-1079 . -102) T) ((-1079 . -243) 165413) ((-1079 . -199) 165392) ((-1079 . -910) 165358) ((-1079 . -1106) 165324) ((-1079 . -1109) 165290) ((-1079 . -428) 165256) ((-1079 . -237) 165222) ((-1079 . -66) 165188) ((-1079 . -35) 165154) ((-1079 . -1149) 165124) ((-1079 . -47) 165094) ((-1079 . -118) 165073) ((-1079 . -116) 165052) ((-1079 . -881) 165015) ((-1079 . -806) 164921) ((-1079 . -801) 164802) ((-1079 . -804) 164708) ((-1079 . -239) 164666) ((-1079 . -187) 164618) ((-1079 . -184) 164564) ((-1079 . -188) 164516) ((-1079 . -1153) 164500) ((-1079 . -945) 164435) ((-1076 . -1146) 164419) ((-1076 . -1057) 164397) ((-1076 . -550) NIL) ((-1076 . -257) 164384) ((-1076 . -449) 164332) ((-1076 . -274) 164309) ((-1076 . -945) 164192) ((-1076 . -350) 164176) ((-1076 . -38) 164008) ((-1076 . -80) 163813) ((-1076 . -958) 163639) ((-1076 . -963) 163465) ((-1076 . -585) 163375) ((-1076 . -587) 163264) ((-1076 . -579) 163096) ((-1076 . -651) 162928) ((-1076 . -552) 162705) ((-1076 . -116) 162684) ((-1076 . -118) 162663) ((-1076 . -47) 162640) ((-1076 . -324) 162624) ((-1076 . -577) 162572) ((-1076 . -804) 162516) ((-1076 . -801) 162423) ((-1076 . -806) 162334) ((-1076 . -791) NIL) ((-1076 . -816) 162313) ((-1076 . -1125) 162292) ((-1076 . -856) 162262) ((-1076 . -827) 162241) ((-1076 . -491) 162155) ((-1076 . -243) 162069) ((-1076 . -144) 161963) ((-1076 . -387) 161897) ((-1076 . -255) 161876) ((-1076 . -239) 161803) ((-1076 . -188) T) ((-1076 . -102) T) ((-1076 . -25) T) ((-1076 . -72) T) ((-1076 . -549) 161785) ((-1076 . -1007) T) ((-1076 . -23) T) ((-1076 . -21) T) ((-1076 . -964) T) ((-1076 . -1017) T) ((-1076 . -1052) T) ((-1076 . -660) T) ((-1076 . -956) T) ((-1076 . -184) 161772) ((-1076 . -13) T) ((-1076 . -1120) T) ((-1076 . -187) T) ((-1076 . -223) 161756) ((-1076 . -182) 161740) ((-1073 . -1134) 161701) ((-1073 . -910) 161667) ((-1073 . -1106) 161633) ((-1073 . -1109) 161599) ((-1073 . -428) 161565) ((-1073 . -237) 161531) ((-1073 . -66) 161497) ((-1073 . -35) 161463) ((-1073 . -1149) 161440) ((-1073 . -47) 161417) ((-1073 . -552) 161218) ((-1073 . -651) 161020) ((-1073 . -579) 160822) ((-1073 . -587) 160677) ((-1073 . -585) 160517) ((-1073 . -963) 160313) ((-1073 . -958) 160109) ((-1073 . -80) 159861) ((-1073 . -38) 159663) ((-1073 . -881) 159633) ((-1073 . -239) 159461) ((-1073 . -1132) 159445) ((-1073 . -964) T) ((-1073 . -1017) T) ((-1073 . -1052) T) ((-1073 . -660) T) ((-1073 . -956) T) ((-1073 . -21) T) ((-1073 . -23) T) ((-1073 . -1007) T) ((-1073 . -549) 159427) ((-1073 . -1120) T) ((-1073 . -13) T) ((-1073 . -72) T) ((-1073 . -25) T) ((-1073 . -102) T) ((-1073 . -116) 159337) ((-1073 . -118) 159247) ((-1073 . -550) NIL) ((-1073 . -182) 159199) ((-1073 . -804) 159035) ((-1073 . -806) 158799) ((-1073 . -801) 158538) ((-1073 . -223) 158490) ((-1073 . -187) 158316) ((-1073 . -184) 158136) ((-1073 . -188) 158026) ((-1073 . -309) 158005) ((-1073 . -1125) 157984) ((-1073 . -827) 157963) ((-1073 . -491) 157917) ((-1073 . -144) 157851) ((-1073 . -387) 157830) ((-1073 . -255) 157809) ((-1073 . -243) 157763) ((-1073 . -199) 157742) ((-1073 . -285) 157694) ((-1073 . -449) 157428) ((-1073 . -257) 157313) ((-1073 . -324) 157265) ((-1073 . -577) 157217) ((-1073 . -338) 157169) ((-1073 . -791) NIL) ((-1073 . -735) NIL) ((-1073 . -709) NIL) ((-1073 . -711) NIL) ((-1073 . -751) NIL) ((-1073 . -754) NIL) ((-1073 . -713) NIL) ((-1073 . -716) NIL) ((-1073 . -750) NIL) ((-1073 . -789) 157121) ((-1073 . -816) NIL) ((-1073 . -928) NIL) ((-1073 . -945) 157087) ((-1073 . -1057) NIL) ((-1073 . -899) 157039) ((-1072 . -989) T) ((-1072 . -425) 157020) ((-1072 . -549) 156986) ((-1072 . -552) 156967) ((-1072 . -1007) T) ((-1072 . -1120) T) ((-1072 . -13) T) ((-1072 . -72) T) ((-1072 . -64) T) ((-1071 . -1007) T) ((-1071 . -549) 156949) ((-1071 . -1120) T) ((-1071 . -13) T) ((-1071 . -72) T) ((-1070 . -1007) T) ((-1070 . -549) 156931) ((-1070 . -1120) T) ((-1070 . -13) T) ((-1070 . -72) T) ((-1065 . -1098) 156907) ((-1065 . -181) 156852) ((-1065 . -76) 156797) ((-1065 . -257) 156586) ((-1065 . -449) 156326) ((-1065 . -424) 156258) ((-1065 . -122) 156203) ((-1065 . -550) NIL) ((-1065 . -191) 156148) ((-1065 . -546) 156124) ((-1065 . -241) 156100) ((-1065 . -1120) T) ((-1065 . -13) T) ((-1065 . -239) 156076) ((-1065 . -1007) T) ((-1065 . -549) 156058) ((-1065 . -72) T) ((-1065 . -34) T) ((-1065 . -535) 156034) ((-1064 . -1049) T) ((-1064 . -319) 156016) ((-1064 . -754) T) ((-1064 . -751) T) ((-1064 . -122) 155998) ((-1064 . -34) T) ((-1064 . -13) T) ((-1064 . -1120) T) ((-1064 . -72) T) ((-1064 . -549) 155980) ((-1064 . -257) NIL) ((-1064 . -449) NIL) ((-1064 . -1007) T) ((-1064 . -424) 155962) ((-1064 . -550) NIL) ((-1064 . -239) 155912) ((-1064 . -535) 155887) ((-1064 . -241) 155862) ((-1064 . -590) 155844) ((-1064 . -19) 155826) ((-1060 . -613) 155810) ((-1060 . -590) 155794) ((-1060 . -241) 155771) ((-1060 . -239) 155723) ((-1060 . -535) 155700) ((-1060 . -550) 155661) ((-1060 . -424) 155645) ((-1060 . -1007) 155623) ((-1060 . -449) 155556) ((-1060 . -257) 155494) ((-1060 . -549) 155429) ((-1060 . -72) 155383) ((-1060 . -1120) T) ((-1060 . -13) T) ((-1060 . -34) T) ((-1060 . -122) 155367) ((-1060 . -1159) 155351) ((-1060 . -918) 155335) ((-1060 . -1055) 155319) ((-1060 . -552) 155296) ((-1058 . -989) T) ((-1058 . -425) 155277) ((-1058 . -549) 155243) ((-1058 . -552) 155224) ((-1058 . -1007) T) ((-1058 . -1120) T) ((-1058 . -13) T) ((-1058 . -72) T) ((-1058 . -64) T) ((-1056 . -1098) 155203) ((-1056 . -181) 155151) ((-1056 . -76) 155099) ((-1056 . -257) 154897) ((-1056 . -449) 154649) ((-1056 . -424) 154584) ((-1056 . -122) 154532) ((-1056 . -550) NIL) ((-1056 . -191) 154480) ((-1056 . -546) 154459) ((-1056 . -241) 154438) ((-1056 . -1120) T) ((-1056 . -13) T) ((-1056 . -239) 154417) ((-1056 . -1007) T) ((-1056 . -549) 154399) ((-1056 . -72) T) ((-1056 . -34) T) ((-1056 . -535) 154378) ((-1053 . -1026) 154362) ((-1053 . -424) 154346) ((-1053 . -1007) 154324) ((-1053 . -449) 154257) ((-1053 . -257) 154195) ((-1053 . -549) 154130) ((-1053 . -72) 154084) ((-1053 . -1120) T) ((-1053 . -13) T) ((-1053 . -34) T) ((-1053 . -76) 154068) ((-1051 . -1014) 154037) ((-1051 . -1115) 154006) ((-1051 . -549) 153968) ((-1051 . -122) 153952) ((-1051 . -34) T) ((-1051 . -13) T) ((-1051 . -1120) T) ((-1051 . -72) T) ((-1051 . -257) 153890) ((-1051 . -449) 153823) ((-1051 . -1007) T) ((-1051 . -424) 153807) ((-1051 . -550) 153768) ((-1051 . -884) 153737) ((-1051 . -977) 153706) ((-1047 . -1028) 153651) ((-1047 . -424) 153635) ((-1047 . -449) 153568) ((-1047 . -257) 153506) ((-1047 . -34) T) ((-1047 . -960) 153446) ((-1047 . -945) 153344) ((-1047 . -552) 153263) ((-1047 . -350) 153247) ((-1047 . -577) 153195) ((-1047 . -587) 153133) ((-1047 . -324) 153117) ((-1047 . -188) 153096) ((-1047 . -184) 153044) ((-1047 . -187) 152998) ((-1047 . -223) 152982) ((-1047 . -801) 152906) ((-1047 . -806) 152832) ((-1047 . -804) 152791) ((-1047 . -182) 152775) ((-1047 . -651) 152710) ((-1047 . -579) 152645) ((-1047 . -585) 152604) ((-1047 . -102) T) ((-1047 . -25) T) ((-1047 . -72) T) ((-1047 . -13) T) ((-1047 . -1120) T) ((-1047 . -549) 152566) ((-1047 . -1007) T) ((-1047 . -23) T) ((-1047 . -21) T) ((-1047 . -963) 152550) ((-1047 . -958) 152534) ((-1047 . -80) 152513) ((-1047 . -956) T) ((-1047 . -660) T) ((-1047 . -1052) T) ((-1047 . -1017) T) ((-1047 . -964) T) ((-1047 . -38) 152473) ((-1047 . -550) 152434) ((-1046 . -918) 152405) ((-1046 . -34) T) ((-1046 . -13) T) ((-1046 . -1120) T) ((-1046 . -72) T) ((-1046 . -549) 152387) ((-1046 . -257) 152313) ((-1046 . -449) 152221) ((-1046 . -1007) T) ((-1046 . -424) 152192) ((-1045 . -1007) T) ((-1045 . -549) 152174) ((-1045 . -1120) T) ((-1045 . -13) T) ((-1045 . -72) T) ((-1040 . -1042) T) ((-1040 . -1166) T) ((-1040 . -64) T) ((-1040 . -72) T) ((-1040 . -13) T) ((-1040 . -1120) T) ((-1040 . -549) 152140) ((-1040 . -1007) T) ((-1040 . -552) 152121) ((-1040 . -425) 152102) ((-1040 . -989) T) ((-1038 . -1039) 152086) ((-1038 . -72) T) ((-1038 . -13) T) ((-1038 . -1120) T) ((-1038 . -549) 152068) ((-1038 . -1007) T) ((-1031 . -674) 152047) ((-1031 . -35) 152013) ((-1031 . -66) 151979) ((-1031 . -237) 151945) ((-1031 . -428) 151911) ((-1031 . -1109) 151877) ((-1031 . -1106) 151843) ((-1031 . -910) 151809) ((-1031 . -47) 151781) ((-1031 . -38) 151678) ((-1031 . -579) 151575) ((-1031 . -651) 151472) ((-1031 . -552) 151354) ((-1031 . -243) 151333) ((-1031 . -491) 151312) ((-1031 . -80) 151177) ((-1031 . -958) 151063) ((-1031 . -963) 150949) ((-1031 . -144) 150903) ((-1031 . -118) 150882) ((-1031 . -116) 150861) ((-1031 . -587) 150786) ((-1031 . -585) 150696) ((-1031 . -881) 150663) ((-1031 . -806) 150647) ((-1031 . -1120) T) ((-1031 . -13) T) ((-1031 . -801) 150629) ((-1031 . -956) T) ((-1031 . -660) T) ((-1031 . -1052) T) ((-1031 . -1017) T) ((-1031 . -964) T) ((-1031 . -21) T) ((-1031 . -23) T) ((-1031 . -1007) T) ((-1031 . -549) 150611) ((-1031 . -72) T) ((-1031 . -25) T) ((-1031 . -102) T) ((-1031 . -804) 150595) ((-1031 . -449) 150565) ((-1031 . -257) 150552) ((-1030 . -856) 150519) ((-1030 . -552) 150318) ((-1030 . -945) 150203) ((-1030 . -1125) 150182) ((-1030 . -816) 150161) ((-1030 . -791) 150020) ((-1030 . -806) 150004) ((-1030 . -801) 149986) ((-1030 . -804) 149970) ((-1030 . -449) 149922) ((-1030 . -387) 149876) ((-1030 . -577) 149824) ((-1030 . -587) 149713) ((-1030 . -324) 149697) ((-1030 . -47) 149669) ((-1030 . -38) 149521) ((-1030 . -579) 149373) ((-1030 . -651) 149225) ((-1030 . -243) 149159) ((-1030 . -491) 149093) ((-1030 . -80) 148918) ((-1030 . -958) 148764) ((-1030 . -963) 148610) ((-1030 . -144) 148524) ((-1030 . -118) 148503) ((-1030 . -116) 148482) ((-1030 . -585) 148392) ((-1030 . -102) T) ((-1030 . -25) T) ((-1030 . -72) T) ((-1030 . -13) T) ((-1030 . -1120) T) ((-1030 . -549) 148374) ((-1030 . -1007) T) ((-1030 . -23) T) ((-1030 . -21) T) ((-1030 . -956) T) ((-1030 . -660) T) ((-1030 . -1052) T) ((-1030 . -1017) T) ((-1030 . -964) T) ((-1030 . -350) 148358) ((-1030 . -274) 148330) ((-1030 . -257) 148317) ((-1030 . -550) 148065) ((-1025 . -479) T) ((-1025 . -1125) T) ((-1025 . -1057) T) ((-1025 . -945) 148047) ((-1025 . -550) 147962) ((-1025 . -928) T) ((-1025 . -791) 147944) ((-1025 . -750) T) ((-1025 . -716) T) ((-1025 . -713) T) ((-1025 . -754) T) ((-1025 . -751) T) ((-1025 . -711) T) ((-1025 . -709) T) ((-1025 . -735) T) ((-1025 . -587) 147916) ((-1025 . -577) 147898) ((-1025 . -827) T) ((-1025 . -491) T) ((-1025 . -243) T) ((-1025 . -144) T) ((-1025 . -552) 147870) ((-1025 . -651) 147857) ((-1025 . -579) 147844) ((-1025 . -963) 147831) ((-1025 . -958) 147818) ((-1025 . -80) 147803) ((-1025 . -38) 147790) ((-1025 . -387) T) ((-1025 . -255) T) ((-1025 . -187) T) ((-1025 . -184) 147777) ((-1025 . -188) T) ((-1025 . -114) T) ((-1025 . -956) T) ((-1025 . -660) T) ((-1025 . -1052) T) ((-1025 . -1017) T) ((-1025 . -964) T) ((-1025 . -21) T) ((-1025 . -585) 147749) ((-1025 . -23) T) ((-1025 . -1007) T) ((-1025 . -549) 147731) ((-1025 . -1120) T) ((-1025 . -13) T) ((-1025 . -72) T) ((-1025 . -25) T) ((-1025 . -102) T) ((-1025 . -118) T) ((-1025 . -747) T) ((-1025 . -315) T) ((-1025 . -82) T) ((-1025 . -601) T) ((-1021 . -989) T) ((-1021 . -425) 147712) ((-1021 . -549) 147678) ((-1021 . -552) 147659) ((-1021 . -1007) T) ((-1021 . -1120) T) ((-1021 . -13) T) ((-1021 . -72) T) ((-1021 . -64) T) ((-1020 . -1007) T) ((-1020 . -549) 147641) ((-1020 . -1120) T) ((-1020 . -13) T) ((-1020 . -72) T) ((-1018 . -194) 147620) ((-1018 . -1178) 147590) ((-1018 . -716) 147569) ((-1018 . -713) 147548) ((-1018 . -754) 147502) ((-1018 . -751) 147456) ((-1018 . -711) 147435) ((-1018 . -712) 147414) ((-1018 . -651) 147359) ((-1018 . -579) 147284) ((-1018 . -241) 147261) ((-1018 . -239) 147238) ((-1018 . -424) 147222) ((-1018 . -449) 147155) ((-1018 . -257) 147093) ((-1018 . -34) T) ((-1018 . -535) 147070) ((-1018 . -945) 146899) ((-1018 . -552) 146703) ((-1018 . -350) 146672) ((-1018 . -577) 146580) ((-1018 . -587) 146419) ((-1018 . -324) 146389) ((-1018 . -315) 146368) ((-1018 . -188) 146321) ((-1018 . -585) 146109) ((-1018 . -964) 146088) ((-1018 . -1017) 146067) ((-1018 . -1052) 146046) ((-1018 . -660) 146025) ((-1018 . -956) 146004) ((-1018 . -184) 145900) ((-1018 . -187) 145802) ((-1018 . -223) 145772) ((-1018 . -801) 145644) ((-1018 . -806) 145518) ((-1018 . -804) 145451) ((-1018 . -182) 145421) ((-1018 . -549) 145118) ((-1018 . -963) 145043) ((-1018 . -958) 144948) ((-1018 . -80) 144868) ((-1018 . -102) 144743) ((-1018 . -25) 144580) ((-1018 . -72) 144317) ((-1018 . -13) T) ((-1018 . -1120) T) ((-1018 . -1007) 144073) ((-1018 . -23) 143929) ((-1018 . -21) 143844) ((-1011 . -1010) 143808) ((-1011 . -72) T) ((-1011 . -549) 143790) ((-1011 . -1007) T) ((-1011 . -239) 143746) ((-1011 . -1120) T) ((-1011 . -13) T) ((-1011 . -554) 143661) ((-1009 . -1010) 143613) ((-1009 . -72) T) ((-1009 . -549) 143595) ((-1009 . -1007) T) ((-1009 . -239) 143551) ((-1009 . -1120) T) ((-1009 . -13) T) ((-1009 . -554) 143454) ((-1008 . -315) T) ((-1008 . -72) T) ((-1008 . -13) T) ((-1008 . -1120) T) ((-1008 . -549) 143436) ((-1008 . -1007) T) ((-1003 . -364) 143420) ((-1003 . -1005) 143404) ((-1003 . -315) 143383) ((-1003 . -191) 143367) ((-1003 . -550) 143328) ((-1003 . -122) 143312) ((-1003 . -424) 143296) ((-1003 . -1007) T) ((-1003 . -449) 143229) ((-1003 . -257) 143167) ((-1003 . -549) 143149) ((-1003 . -72) T) ((-1003 . -1120) T) ((-1003 . -13) T) ((-1003 . -34) T) ((-1003 . -76) 143133) ((-1003 . -181) 143117) ((-1002 . -989) T) ((-1002 . -425) 143098) ((-1002 . -549) 143064) ((-1002 . -552) 143045) ((-1002 . -1007) T) ((-1002 . -1120) T) ((-1002 . -13) T) ((-1002 . -72) T) ((-1002 . -64) T) ((-998 . -1120) T) ((-998 . -13) T) ((-998 . -1007) 143016) ((-998 . -549) 142976) ((-998 . -72) 142947) ((-997 . -989) T) ((-997 . -425) 142928) ((-997 . -549) 142894) ((-997 . -552) 142875) ((-997 . -1007) T) ((-997 . -1120) T) ((-997 . -13) T) ((-997 . -72) T) ((-997 . -64) T) ((-995 . -1000) 142859) ((-995 . -554) 142843) ((-995 . -1007) 142821) ((-995 . -549) 142788) ((-995 . -1120) 142766) ((-995 . -13) 142744) ((-995 . -72) 142722) ((-995 . -1001) 142680) ((-994 . -226) 142664) ((-994 . -552) 142648) ((-994 . -945) 142632) ((-994 . -754) T) ((-994 . -72) T) ((-994 . -1007) T) ((-994 . -549) 142614) ((-994 . -751) T) ((-994 . -184) 142601) ((-994 . -13) T) ((-994 . -1120) T) ((-994 . -187) T) ((-993 . -211) 142540) ((-993 . -552) 142284) ((-993 . -945) 142114) ((-993 . -550) NIL) ((-993 . -274) 142076) ((-993 . -350) 142060) ((-993 . -38) 141912) ((-993 . -80) 141737) ((-993 . -958) 141583) ((-993 . -963) 141429) ((-993 . -585) 141339) ((-993 . -587) 141228) ((-993 . -579) 141080) ((-993 . -651) 140932) ((-993 . -116) 140911) ((-993 . -118) 140890) ((-993 . -144) 140804) ((-993 . -491) 140738) ((-993 . -243) 140672) ((-993 . -47) 140634) ((-993 . -324) 140618) ((-993 . -577) 140566) ((-993 . -387) 140520) ((-993 . -449) 140385) ((-993 . -804) 140321) ((-993 . -801) 140220) ((-993 . -806) 140123) ((-993 . -791) NIL) ((-993 . -816) 140102) ((-993 . -1125) 140081) ((-993 . -856) 140028) ((-993 . -257) 140015) ((-993 . -188) 139994) ((-993 . -102) T) ((-993 . -25) T) ((-993 . -72) T) ((-993 . -549) 139976) ((-993 . -1007) T) ((-993 . -23) T) ((-993 . -21) T) ((-993 . -964) T) ((-993 . -1017) T) ((-993 . -1052) T) ((-993 . -660) T) ((-993 . -956) T) ((-993 . -184) 139924) ((-993 . -13) T) ((-993 . -1120) T) ((-993 . -187) 139878) ((-993 . -223) 139862) ((-993 . -182) 139846) ((-991 . -549) 139828) ((-988 . -751) T) ((-988 . -549) 139810) ((-988 . -1007) T) ((-988 . -72) T) ((-988 . -13) T) ((-988 . -1120) T) ((-988 . -754) T) ((-988 . -550) 139791) ((-985 . -658) 139770) ((-985 . -945) 139668) ((-985 . -350) 139652) ((-985 . -577) 139600) ((-985 . -587) 139477) ((-985 . -324) 139461) ((-985 . -317) 139440) ((-985 . 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121646) ((-807 . -549) 121560) ((-807 . -72) 121514) ((-807 . -1120) T) ((-807 . -13) T) ((-807 . -34) T) ((-807 . -918) 121498) ((-798 . -751) T) ((-798 . -549) 121480) ((-798 . -1007) T) ((-798 . -72) T) ((-798 . -13) T) ((-798 . -1120) T) ((-798 . -754) T) ((-798 . -945) 121457) ((-798 . -552) 121434) ((-795 . -1007) T) ((-795 . -549) 121416) ((-795 . -1120) T) ((-795 . -13) T) ((-795 . -72) T) ((-795 . -945) 121384) ((-795 . -552) 121352) ((-793 . -1007) T) ((-793 . -549) 121334) ((-793 . -1120) T) ((-793 . -13) T) ((-793 . -72) T) ((-790 . -1007) T) ((-790 . -549) 121316) ((-790 . -1120) T) ((-790 . -13) T) ((-790 . -72) T) ((-780 . -989) T) ((-780 . -425) 121297) ((-780 . -549) 121263) ((-780 . -552) 121244) ((-780 . -1007) T) ((-780 . -1120) T) ((-780 . -13) T) ((-780 . -72) T) ((-780 . -64) T) ((-780 . -1166) T) ((-778 . -1007) T) ((-778 . -549) 121226) ((-778 . -1120) T) ((-778 . -13) T) ((-778 . -72) T) ((-778 . -552) 121208) ((-777 . -1120) T) ((-777 . -13) T) ((-777 . -549) 121083) ((-777 . -1007) 121034) ((-777 . -72) 120985) ((-776 . -899) 120969) ((-776 . -1057) 120947) ((-776 . -945) 120814) ((-776 . -552) 120713) ((-776 . -550) 120516) ((-776 . -928) 120495) ((-776 . -816) 120474) ((-776 . -789) 120458) ((-776 . -750) 120437) ((-776 . -716) 120416) ((-776 . -713) 120395) ((-776 . -754) 120349) ((-776 . -751) 120303) ((-776 . -711) 120282) ((-776 . -709) 120261) ((-776 . -735) 120240) ((-776 . -791) 120165) ((-776 . -338) 120149) ((-776 . -577) 120097) ((-776 . -587) 120013) ((-776 . -324) 119997) ((-776 . -239) 119955) ((-776 . -257) 119920) ((-776 . -449) 119832) ((-776 . -285) 119816) ((-776 . -199) T) ((-776 . -80) 119747) ((-776 . -958) 119699) ((-776 . -963) 119651) ((-776 . -243) T) ((-776 . -651) 119603) ((-776 . -579) 119555) ((-776 . -585) 119492) ((-776 . -38) 119444) ((-776 . -255) T) ((-776 . -387) T) ((-776 . -144) T) ((-776 . -491) T) ((-776 . -827) T) ((-776 . -1125) T) ((-776 . -309) T) ((-776 . -188) 119423) ((-776 . -184) 119371) ((-776 . -187) 119325) ((-776 . -223) 119309) ((-776 . -801) 119233) ((-776 . -806) 119159) ((-776 . -804) 119118) ((-776 . -182) 119102) ((-776 . -118) 119081) ((-776 . -116) 119060) ((-776 . -102) T) ((-776 . -25) T) ((-776 . -72) T) ((-776 . -13) T) ((-776 . -1120) T) ((-776 . -549) 119042) ((-776 . -1007) T) ((-776 . -23) T) ((-776 . -21) T) ((-776 . -956) T) ((-776 . -660) T) ((-776 . -1052) T) ((-776 . -1017) T) ((-776 . -964) T) ((-775 . -899) 119019) ((-775 . -1057) NIL) ((-775 . -945) 118996) ((-775 . -552) 118926) ((-775 . -550) NIL) ((-775 . -928) NIL) ((-775 . -816) NIL) ((-775 . -789) 118903) ((-775 . -750) NIL) ((-775 . -716) NIL) ((-775 . -713) NIL) ((-775 . -754) NIL) ((-775 . -751) NIL) ((-775 . -711) NIL) ((-775 . -709) NIL) ((-775 . -735) NIL) ((-775 . -791) NIL) ((-775 . -338) 118880) ((-775 . -577) 118857) ((-775 . -587) 118802) ((-775 . -324) 118779) ((-775 . -239) 118709) ((-775 . -257) 118653) ((-775 . -449) 118516) ((-775 . -285) 118493) ((-775 . -199) T) ((-775 . -80) 118410) ((-775 . -958) 118355) ((-775 . -963) 118300) ((-775 . -243) T) ((-775 . -651) 118245) ((-775 . -579) 118190) ((-775 . -585) 118120) ((-775 . -38) 118065) ((-775 . -255) T) ((-775 . -387) T) ((-775 . -144) T) ((-775 . -491) T) ((-775 . -827) T) ((-775 . -1125) T) ((-775 . -309) T) ((-775 . -188) NIL) ((-775 . -184) NIL) ((-775 . -187) NIL) ((-775 . -223) 118042) ((-775 . -801) NIL) ((-775 . -806) NIL) ((-775 . -804) NIL) ((-775 . -182) 118019) ((-775 . -118) T) ((-775 . -116) NIL) ((-775 . -102) T) ((-775 . -25) T) ((-775 . -72) T) ((-775 . -13) T) ((-775 . -1120) T) ((-775 . -549) 118001) ((-775 . -1007) T) ((-775 . -23) T) ((-775 . -21) T) ((-775 . -956) T) ((-775 . -660) T) ((-775 . -1052) T) ((-775 . -1017) T) ((-775 . -964) T) ((-773 . -774) 117985) ((-773 . -827) T) ((-773 . -491) T) ((-773 . -243) T) ((-773 . -144) T) ((-773 . -552) 117957) ((-773 . -651) 117944) ((-773 . -579) 117931) ((-773 . -963) 117918) ((-773 . -958) 117905) ((-773 . -80) 117890) ((-773 . -38) 117877) ((-773 . -387) T) ((-773 . -255) T) ((-773 . -956) T) ((-773 . -660) T) ((-773 . -1052) T) ((-773 . -1017) T) ((-773 . -964) T) ((-773 . -21) T) ((-773 . -585) 117849) ((-773 . -23) T) ((-773 . -1007) T) ((-773 . -549) 117831) ((-773 . -1120) T) ((-773 . -13) T) ((-773 . -72) T) ((-773 . -25) T) ((-773 . -102) T) ((-773 . -587) 117818) ((-773 . -118) T) ((-770 . -956) T) ((-770 . -660) T) ((-770 . -1052) T) ((-770 . -1017) T) ((-770 . -964) T) ((-770 . -21) T) ((-770 . -585) 117763) ((-770 . -23) T) ((-770 . -1007) T) ((-770 . -549) 117725) ((-770 . -1120) T) ((-770 . -13) T) ((-770 . -72) T) ((-770 . -25) T) ((-770 . -102) T) ((-770 . -587) 117685) ((-770 . -552) 117620) ((-770 . -425) 117597) ((-770 . -38) 117567) ((-770 . -80) 117532) ((-770 . -958) 117502) ((-770 . -963) 117472) ((-770 . -579) 117442) ((-770 . -651) 117412) ((-769 . -1007) T) ((-769 . -549) 117394) ((-769 . -1120) T) ((-769 . -13) T) ((-769 . -72) T) ((-768 . -747) T) ((-768 . -754) T) ((-768 . -751) T) ((-768 . -1007) T) ((-768 . -549) 117376) ((-768 . -1120) T) ((-768 . -13) T) ((-768 . -72) T) ((-768 . -315) T) ((-768 . -550) 117298) ((-767 . -1007) T) ((-767 . -549) 117280) ((-767 . -1120) T) ((-767 . -13) T) ((-767 . -72) T) ((-766 . -765) T) ((-766 . -145) T) ((-766 . -549) 117262) ((-762 . -751) T) ((-762 . -549) 117244) ((-762 . -1007) T) ((-762 . -72) T) ((-762 . -13) T) ((-762 . -1120) T) ((-762 . -754) T) ((-759 . -756) 117228) ((-759 . -945) 117126) ((-759 . -552) 117024) ((-759 . -350) 117008) ((-759 . -651) 116978) ((-759 . -579) 116948) ((-759 . -587) 116922) ((-759 . -585) 116881) ((-759 . -102) T) ((-759 . -25) T) ((-759 . -72) T) ((-759 . -13) T) ((-759 . -1120) T) ((-759 . -549) 116863) ((-759 . -1007) T) ((-759 . -23) T) ((-759 . -21) T) ((-759 . -963) 116847) ((-759 . -958) 116831) ((-759 . -80) 116810) ((-759 . -956) T) ((-759 . -660) T) ((-759 . -1052) T) ((-759 . -1017) T) ((-759 . -964) T) ((-759 . -38) 116780) ((-758 . -756) 116764) ((-758 . -945) 116662) ((-758 . -552) 116581) ((-758 . -350) 116565) ((-758 . -651) 116535) ((-758 . -579) 116505) ((-758 . -587) 116479) ((-758 . -585) 116438) ((-758 . -102) T) ((-758 . -25) T) ((-758 . -72) T) ((-758 . -13) T) ((-758 . -1120) T) ((-758 . -549) 116420) ((-758 . -1007) T) ((-758 . -23) T) ((-758 . -21) T) ((-758 . -963) 116404) ((-758 . -958) 116388) ((-758 . -80) 116367) ((-758 . -956) T) ((-758 . -660) T) ((-758 . -1052) T) ((-758 . -1017) T) ((-758 . -964) T) ((-758 . -38) 116337) ((-752 . -754) T) ((-752 . -1120) T) ((-752 . -13) T) ((-752 . -72) T) ((-752 . -425) 116321) ((-752 . -549) 116269) ((-752 . -552) 116253) ((-745 . -1007) T) ((-745 . -549) 116235) ((-745 . -1120) T) ((-745 . -13) T) ((-745 . -72) T) ((-745 . -350) 116219) ((-745 . -552) 116092) ((-745 . -945) 115990) ((-745 . -21) 115945) ((-745 . -585) 115865) ((-745 . -23) 115820) ((-745 . -25) 115775) ((-745 . -102) 115730) ((-745 . -750) 115709) ((-745 . -587) 115682) ((-745 . -964) 115661) ((-745 . -1052) 115640) ((-745 . -956) 115619) ((-745 . -716) 115598) ((-745 . -713) 115577) ((-745 . -754) 115556) ((-745 . -751) 115535) ((-745 . -711) 115514) ((-745 . -709) 115493) ((-745 . -1017) 115472) ((-745 . -660) 115451) ((-744 . -742) 115433) ((-744 . -72) T) ((-744 . -13) T) ((-744 . -1120) T) ((-744 . -549) 115415) ((-744 . -1007) T) ((-740 . -956) T) ((-740 . -660) T) ((-740 . -1052) T) ((-740 . -1017) T) ((-740 . -964) T) ((-740 . -21) T) ((-740 . -585) 115360) ((-740 . -23) T) ((-740 . -1007) T) ((-740 . -549) 115342) ((-740 . -1120) T) ((-740 . -13) T) ((-740 . -72) T) ((-740 . -25) T) ((-740 . -102) T) ((-740 . -587) 115302) ((-740 . -552) 115257) ((-740 . -945) 115227) ((-740 . -239) 115206) ((-740 . -118) 115185) ((-740 . -116) 115164) ((-740 . -38) 115134) ((-740 . -80) 115099) ((-740 . -958) 115069) ((-740 . -963) 115039) ((-740 . -579) 115009) ((-740 . -651) 114979) ((-738 . -1007) T) ((-738 . -549) 114961) ((-738 . -1120) T) ((-738 . -13) T) ((-738 . -72) T) ((-738 . -350) 114945) ((-738 . -552) 114818) ((-738 . -945) 114716) ((-738 . -21) 114671) ((-738 . -585) 114591) ((-738 . -23) 114546) ((-738 . -25) 114501) ((-738 . -102) 114456) ((-738 . -750) 114435) ((-738 . -587) 114408) ((-738 . -964) 114387) ((-738 . -1052) 114366) ((-738 . -956) 114345) ((-738 . -716) 114324) ((-738 . -713) 114303) ((-738 . -754) 114282) ((-738 . -751) 114261) ((-738 . -711) 114240) ((-738 . -709) 114219) ((-738 . -1017) 114198) ((-738 . -660) 114177) ((-736 . -642) 114161) ((-736 . -552) 114116) ((-736 . -651) 114086) ((-736 . -579) 114056) ((-736 . -587) 114030) ((-736 . -585) 113989) ((-736 . -102) T) ((-736 . -25) T) ((-736 . -72) T) ((-736 . -13) T) ((-736 . -1120) T) ((-736 . -549) 113971) ((-736 . -1007) T) ((-736 . -23) T) ((-736 . -21) T) ((-736 . -963) 113955) ((-736 . -958) 113939) ((-736 . -80) 113918) ((-736 . -956) T) ((-736 . -660) T) ((-736 . -1052) T) ((-736 . -1017) T) ((-736 . -964) T) ((-736 . -38) 113888) ((-736 . -188) 113867) ((-736 . -184) 113840) ((-736 . -187) 113819) ((-734 . -331) 113803) ((-734 . -552) 113787) ((-734 . -945) 113771) ((-734 . -754) T) ((-734 . -751) T) ((-734 . -1017) T) ((-734 . -72) T) ((-734 . -13) T) ((-734 . -1120) T) ((-734 . -549) 113753) ((-734 . -1007) T) ((-734 . -660) T) ((-734 . -749) T) ((-734 . -761) T) ((-733 . -226) 113737) ((-733 . -552) 113721) ((-733 . -945) 113705) ((-733 . -754) T) ((-733 . -72) T) ((-733 . -1007) T) ((-733 . -549) 113687) ((-733 . -751) T) ((-733 . -184) 113674) ((-733 . -13) T) ((-733 . -1120) T) ((-733 . -187) T) ((-732 . -80) 113609) ((-732 . -958) 113560) ((-732 . -963) 113511) ((-732 . -21) T) ((-732 . -585) 113447) ((-732 . -23) T) ((-732 . -1007) T) ((-732 . -549) 113416) ((-732 . -1120) T) ((-732 . -13) T) ((-732 . -72) T) ((-732 . -25) T) ((-732 . -102) T) ((-732 . -587) 113367) ((-732 . -188) T) ((-732 . -552) 113276) ((-732 . -964) T) ((-732 . -1017) T) ((-732 . -1052) T) ((-732 . -660) T) ((-732 . -956) T) ((-732 . -184) 113263) ((-732 . -187) T) ((-732 . -425) 113247) ((-732 . -309) 113226) ((-732 . -1125) 113205) ((-732 . -827) 113184) ((-732 . -491) 113163) ((-732 . -144) 113142) ((-732 . -651) 113079) ((-732 . -579) 113016) ((-732 . -38) 112953) ((-732 . -387) 112932) ((-732 . -255) 112911) ((-732 . -243) 112890) ((-732 . -199) 112869) ((-731 . -211) 112808) ((-731 . -552) 112552) ((-731 . -945) 112382) ((-731 . -550) NIL) ((-731 . -274) 112344) ((-731 . -350) 112328) ((-731 . -38) 112180) ((-731 . -80) 112005) ((-731 . -958) 111851) ((-731 . -963) 111697) ((-731 . -585) 111607) ((-731 . -587) 111496) ((-731 . -579) 111348) ((-731 . -651) 111200) ((-731 . -116) 111179) ((-731 . -118) 111158) ((-731 . -144) 111072) ((-731 . -491) 111006) ((-731 . -243) 110940) ((-731 . -47) 110902) ((-731 . -324) 110886) ((-731 . -577) 110834) ((-731 . -387) 110788) ((-731 . -449) 110653) ((-731 . -804) 110589) ((-731 . -801) 110488) ((-731 . -806) 110391) ((-731 . -791) NIL) ((-731 . -816) 110370) ((-731 . -1125) 110349) ((-731 . -856) 110296) ((-731 . -257) 110283) ((-731 . -188) 110262) ((-731 . -102) T) ((-731 . -25) T) ((-731 . -72) T) ((-731 . -549) 110244) ((-731 . -1007) T) ((-731 . -23) T) ((-731 . -21) T) ((-731 . -964) T) ((-731 . -1017) T) ((-731 . -1052) T) ((-731 . -660) T) ((-731 . -956) T) ((-731 . -184) 110192) ((-731 . -13) T) ((-731 . -1120) T) ((-731 . -187) 110146) ((-731 . -223) 110130) ((-731 . -182) 110114) ((-730 . -194) 110093) ((-730 . -1178) 110063) ((-730 . -716) 110042) ((-730 . -713) 110021) ((-730 . -754) 109975) ((-730 . -751) 109929) ((-730 . -711) 109908) ((-730 . -712) 109887) ((-730 . -651) 109832) ((-730 . -579) 109757) ((-730 . -241) 109734) ((-730 . -239) 109711) ((-730 . -424) 109695) ((-730 . -449) 109628) ((-730 . -257) 109566) ((-730 . -34) T) ((-730 . -535) 109543) ((-730 . -945) 109372) ((-730 . -552) 109176) ((-730 . -350) 109145) ((-730 . -577) 109053) ((-730 . -587) 108892) ((-730 . -324) 108862) ((-730 . -315) 108841) ((-730 . -188) 108794) ((-730 . -585) 108582) ((-730 . -964) 108561) ((-730 . -1017) 108540) ((-730 . -1052) 108519) ((-730 . -660) 108498) ((-730 . -956) 108477) ((-730 . -184) 108373) ((-730 . -187) 108275) ((-730 . -223) 108245) ((-730 . -801) 108117) ((-730 . -806) 107991) ((-730 . -804) 107924) ((-730 . -182) 107894) ((-730 . -549) 107591) ((-730 . -963) 107516) ((-730 . -958) 107421) ((-730 . -80) 107341) ((-730 . -102) 107216) ((-730 . -25) 107053) ((-730 . -72) 106790) ((-730 . -13) T) ((-730 . -1120) T) ((-730 . -1007) 106546) ((-730 . -23) 106402) ((-730 . -21) 106317) ((-717 . -715) 106301) ((-717 . -754) 106280) ((-717 . -751) 106259) ((-717 . -945) 106052) ((-717 . -552) 105905) ((-717 . -350) 105869) ((-717 . -239) 105827) ((-717 . -257) 105792) ((-717 . -449) 105704) ((-717 . -285) 105688) ((-717 . -315) 105667) ((-717 . -550) 105628) ((-717 . -118) 105607) ((-717 . -116) 105586) ((-717 . -651) 105570) ((-717 . -579) 105554) ((-717 . -587) 105528) ((-717 . -585) 105487) ((-717 . -102) T) ((-717 . -25) T) ((-717 . -72) T) ((-717 . -13) T) ((-717 . -1120) T) ((-717 . -549) 105469) ((-717 . -1007) T) ((-717 . -23) T) ((-717 . -21) T) ((-717 . -963) 105453) ((-717 . -958) 105437) ((-717 . -80) 105416) ((-717 . -956) T) ((-717 . -660) T) ((-717 . -1052) T) ((-717 . -1017) T) ((-717 . -964) T) ((-717 . -38) 105400) ((-699 . -1146) 105384) ((-699 . -1057) 105362) ((-699 . -550) NIL) ((-699 . -257) 105349) ((-699 . -449) 105297) ((-699 . -274) 105274) ((-699 . -945) 105136) ((-699 . -350) 105120) ((-699 . -38) 104952) ((-699 . -80) 104757) ((-699 . -958) 104583) ((-699 . -963) 104409) ((-699 . -585) 104319) ((-699 . -587) 104208) ((-699 . -579) 104040) ((-699 . -651) 103872) ((-699 . -552) 103628) ((-699 . -116) 103607) ((-699 . -118) 103586) ((-699 . -47) 103563) ((-699 . -324) 103547) ((-699 . -577) 103495) ((-699 . -804) 103439) ((-699 . -801) 103346) ((-699 . -806) 103257) ((-699 . -791) NIL) ((-699 . -816) 103236) ((-699 . -1125) 103215) ((-699 . -856) 103185) ((-699 . -827) 103164) ((-699 . -491) 103078) ((-699 . -243) 102992) ((-699 . -144) 102886) ((-699 . -387) 102820) ((-699 . -255) 102799) ((-699 . -239) 102726) ((-699 . -188) T) ((-699 . -102) T) ((-699 . -25) T) ((-699 . -72) T) ((-699 . -549) 102687) ((-699 . -1007) T) ((-699 . -23) T) ((-699 . -21) T) ((-699 . -964) T) ((-699 . -1017) T) ((-699 . -1052) T) ((-699 . -660) T) ((-699 . -956) T) ((-699 . -184) 102674) ((-699 . -13) T) ((-699 . -1120) T) ((-699 . -187) T) ((-699 . -223) 102658) ((-699 . -182) 102642) ((-698 . -971) 102609) ((-698 . -550) 102244) ((-698 . -257) 102231) ((-698 . -449) 102183) ((-698 . -274) 102155) ((-698 . -945) 102014) ((-698 . -350) 101998) ((-698 . -38) 101850) ((-698 . -552) 101623) ((-698 . -587) 101512) ((-698 . -585) 101422) ((-698 . -964) T) ((-698 . -1017) T) ((-698 . -1052) T) ((-698 . -660) T) ((-698 . -956) T) ((-698 . -80) 101247) ((-698 . -958) 101093) ((-698 . -963) 100939) ((-698 . -21) T) ((-698 . -23) T) ((-698 . -1007) T) ((-698 . -549) 100853) ((-698 . -1120) T) ((-698 . -13) T) ((-698 . -72) T) ((-698 . -25) T) ((-698 . -102) T) ((-698 . -579) 100705) ((-698 . -651) 100557) ((-698 . -116) 100536) ((-698 . -118) 100515) ((-698 . -144) 100429) ((-698 . -491) 100363) ((-698 . -243) 100297) ((-698 . -47) 100269) ((-698 . -324) 100253) ((-698 . -577) 100201) ((-698 . -387) 100155) ((-698 . -804) 100139) ((-698 . -801) 100121) ((-698 . -806) 100105) ((-698 . -791) 99964) ((-698 . -816) 99943) ((-698 . -1125) 99922) ((-698 . -856) 99889) ((-691 . -1007) T) ((-691 . -549) 99871) ((-691 . -1120) T) ((-691 . -13) T) ((-691 . -72) T) ((-689 . -712) T) ((-689 . -102) T) ((-689 . -25) T) ((-689 . -72) T) ((-689 . -13) T) ((-689 . -1120) T) ((-689 . -549) 99853) ((-689 . -1007) T) ((-689 . -23) T) ((-689 . -711) T) ((-689 . -751) T) ((-689 . -754) T) ((-689 . -713) T) ((-689 . -716) T) ((-689 . -660) T) ((-689 . -1017) T) ((-670 . -671) 99837) ((-670 . -1005) 99821) ((-670 . -191) 99805) ((-670 . -550) 99766) ((-670 . -122) 99750) ((-670 . -424) 99734) ((-670 . -1007) T) ((-670 . -449) 99667) ((-670 . -257) 99605) ((-670 . -549) 99587) ((-670 . -72) T) ((-670 . -1120) T) ((-670 . -13) T) ((-670 . -34) T) ((-670 . -76) 99571) ((-670 . -631) 99555) ((-669 . -956) T) ((-669 . -660) T) ((-669 . -1052) T) ((-669 . -1017) T) ((-669 . -964) T) ((-669 . -21) T) ((-669 . -585) 99500) ((-669 . -23) T) ((-669 . -1007) T) ((-669 . -549) 99482) ((-669 . -1120) T) ((-669 . -13) T) ((-669 . -72) T) ((-669 . -25) T) ((-669 . -102) T) ((-669 . -587) 99442) ((-669 . -552) 99398) ((-669 . -945) 99369) ((-669 . -118) 99348) ((-669 . -116) 99327) ((-669 . -38) 99297) ((-669 . -80) 99262) ((-669 . -958) 99232) ((-669 . -963) 99202) ((-669 . -579) 99172) ((-669 . -651) 99142) ((-669 . -315) 99095) ((-665 . -856) 99048) ((-665 . -552) 98840) ((-665 . -945) 98718) ((-665 . -1125) 98697) ((-665 . -816) 98676) ((-665 . -791) NIL) ((-665 . -806) 98653) ((-665 . -801) 98628) ((-665 . -804) 98605) ((-665 . -449) 98543) ((-665 . -387) 98497) ((-665 . -577) 98445) ((-665 . -587) 98334) ((-665 . 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T) ((-584 . -13) T) ((-584 . -72) T) ((-584 . -64) T) ((-580 . -605) 88447) ((-580 . -1159) 88431) ((-580 . -918) 88415) ((-580 . -1055) 88399) ((-580 . -751) 88378) ((-580 . -754) 88357) ((-580 . -319) 88341) ((-580 . -590) 88325) ((-580 . -241) 88302) ((-580 . -239) 88254) ((-580 . -535) 88231) ((-580 . -550) 88192) ((-580 . -424) 88176) ((-580 . -1007) 88129) ((-580 . -449) 88062) ((-580 . -257) 88000) ((-580 . -549) 87915) ((-580 . -72) 87849) ((-580 . -1120) T) ((-580 . -13) T) ((-580 . -34) T) ((-580 . -122) 87833) ((-580 . -235) 87817) ((-578 . -1178) 87801) ((-578 . -80) 87780) ((-578 . -958) 87764) ((-578 . -963) 87748) ((-578 . -21) T) ((-578 . -585) 87717) ((-578 . -23) T) ((-578 . -1007) T) ((-578 . -549) 87699) ((-578 . -1120) T) ((-578 . -13) T) ((-578 . -72) T) ((-578 . -25) T) ((-578 . -102) T) ((-578 . -587) 87683) ((-578 . -579) 87667) ((-578 . -651) 87651) ((-578 . -239) 87618) ((-576 . -1178) 87602) ((-576 . -80) 87581) ((-576 . -958) 87565) ((-576 . -963) 87549) 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((-559 . -549) 85768) ((-559 . -72) T) ((-559 . -25) T) ((-559 . -102) T) ((-559 . -587) 85755) ((-559 . -552) 85651) ((-559 . -188) 85630) ((-559 . -491) T) ((-559 . -243) T) ((-559 . -144) T) ((-559 . -651) 85617) ((-559 . -579) 85604) ((-559 . -963) 85591) ((-559 . -958) 85578) ((-559 . -80) 85563) ((-559 . -38) 85550) ((-559 . -550) 85527) ((-559 . -350) 85511) ((-559 . -945) 85396) ((-559 . -118) 85375) ((-559 . -116) 85354) ((-559 . -255) 85333) ((-559 . -387) 85312) ((-559 . -827) 85291) ((-555 . -38) 85275) ((-555 . -552) 85244) ((-555 . -587) 85218) ((-555 . -585) 85177) ((-555 . -964) T) ((-555 . -1017) T) ((-555 . -1052) T) ((-555 . -660) T) ((-555 . -956) T) ((-555 . -80) 85156) ((-555 . -958) 85140) ((-555 . -963) 85124) ((-555 . -21) T) ((-555 . -23) T) ((-555 . -1007) T) ((-555 . -549) 85106) ((-555 . -1120) T) ((-555 . -13) T) ((-555 . -72) T) ((-555 . -25) T) ((-555 . -102) T) ((-555 . -579) 85090) ((-555 . -651) 85074) ((-555 . -750) 85053) ((-555 . -716) 85032) ((-555 . -713) 85011) ((-555 . -754) 84990) ((-555 . -751) 84969) ((-555 . -711) 84948) ((-555 . -709) 84927) ((-553 . -875) T) ((-553 . -72) T) ((-553 . -549) 84909) ((-553 . -1007) T) ((-553 . -601) T) ((-553 . -13) T) ((-553 . -1120) T) ((-553 . -82) T) ((-553 . -315) T) ((-547 . -103) T) ((-547 . -72) T) ((-547 . -13) T) ((-547 . -1120) T) ((-547 . -549) 84891) ((-547 . -1007) T) ((-547 . -751) T) ((-547 . -754) T) ((-547 . -789) 84875) ((-547 . -550) 84736) ((-544 . -311) 84674) ((-544 . -72) T) ((-544 . -13) T) ((-544 . -1120) T) ((-544 . -549) 84656) ((-544 . -1007) T) ((-544 . -1098) 84632) ((-544 . -181) 84577) ((-544 . -76) 84522) ((-544 . -257) 84311) ((-544 . -449) 84051) ((-544 . -424) 83983) ((-544 . -122) 83928) ((-544 . -550) NIL) ((-544 . -191) 83873) ((-544 . -546) 83849) ((-544 . -241) 83825) ((-544 . -239) 83801) ((-544 . -34) T) ((-544 . -535) 83777) ((-543 . -1007) T) ((-543 . -549) 83730) ((-543 . -1120) T) ((-543 . -13) T) ((-543 . -72) T) ((-543 . -425) 83698) 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83235) ((-537 . -239) 83193) ((-536 . -989) T) ((-536 . -425) 83174) ((-536 . -549) 83124) ((-536 . -552) 83105) ((-536 . -1007) T) ((-536 . -1120) T) ((-536 . -13) T) ((-536 . -72) T) ((-536 . -64) T) ((-533 . -1169) 83089) ((-533 . -319) 83073) ((-533 . -754) 83052) ((-533 . -751) 83031) ((-533 . -122) 83015) ((-533 . -34) T) ((-533 . -13) T) ((-533 . -1120) T) ((-533 . -72) 82949) ((-533 . -549) 82864) ((-533 . -257) 82802) ((-533 . -449) 82735) ((-533 . -1007) 82688) ((-533 . -424) 82672) ((-533 . -550) 82633) ((-533 . -239) 82585) ((-533 . -535) 82562) ((-533 . -241) 82539) ((-533 . -590) 82523) ((-533 . -19) 82507) ((-532 . -549) 82489) ((-528 . -1007) T) ((-528 . -549) 82455) ((-528 . -1120) T) ((-528 . -13) T) ((-528 . -72) T) ((-528 . -425) 82436) ((-528 . -552) 82417) ((-527 . -956) T) ((-527 . -660) T) ((-527 . -1052) T) ((-527 . -1017) T) ((-527 . -964) T) ((-527 . -21) T) ((-527 . -585) 82376) ((-527 . -23) T) ((-527 . -1007) T) ((-527 . -549) 82358) ((-527 . -1120) T) 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. -199) T) ((-430 . -80) 73506) ((-430 . -958) 73471) ((-430 . -963) 73436) ((-430 . -243) T) ((-430 . -651) 73401) ((-430 . -579) 73366) ((-430 . -587) 73316) ((-430 . -585) 73266) ((-430 . -102) T) ((-430 . -25) T) ((-430 . -23) T) ((-430 . -21) T) ((-430 . -956) T) ((-430 . -660) T) ((-430 . -1052) T) ((-430 . -1017) T) ((-430 . -964) T) ((-430 . -38) 73231) ((-430 . -255) T) ((-430 . -387) T) ((-430 . -144) T) ((-430 . -491) T) ((-430 . -827) T) ((-430 . -1125) T) ((-430 . -309) T) ((-430 . -577) 73191) ((-430 . -928) T) ((-430 . -550) 73136) ((-430 . -118) T) ((-430 . -188) T) ((-430 . -184) 73123) ((-430 . -187) T) ((-426 . -1007) T) ((-426 . -549) 73089) ((-426 . -1120) T) ((-426 . -13) T) ((-426 . -72) T) ((-422 . -899) 73071) ((-422 . -1057) T) ((-422 . -552) 73021) ((-422 . -945) 72981) ((-422 . -550) 72911) ((-422 . -928) T) ((-422 . -816) NIL) ((-422 . -789) 72893) ((-422 . -750) T) ((-422 . -716) T) ((-422 . -713) T) ((-422 . -754) T) ((-422 . -751) T) ((-422 . -711) T) 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190667) ((-1163 . -915) 190633) ((-1163 . -1113) 190599) ((-1163 . -1116) 190565) ((-1163 . -430) 190531) ((-1163 . -239) 190497) ((-1163 . -66) 190463) ((-1163 . -35) 190429) ((-1163 . -1156) 190399) ((-1163 . -47) 190369) ((-1163 . -120) 190348) ((-1163 . -118) 190327) ((-1163 . -886) 190290) ((-1163 . -811) 190196) ((-1163 . -806) 190100) ((-1163 . -809) 190006) ((-1163 . -241) 189964) ((-1163 . -189) 189916) ((-1163 . -186) 189862) ((-1163 . -190) 189814) ((-1163 . -1160) 189798) ((-1163 . -950) 189782) ((-1158 . -1162) 189743) ((-1158 . -311) 189722) ((-1158 . -1132) 189701) ((-1158 . -832) 189680) ((-1158 . -494) 189634) ((-1158 . -146) 189568) ((-1158 . -555) 189317) ((-1158 . -654) 189164) ((-1158 . -582) 189011) ((-1158 . -38) 188858) ((-1158 . -389) 188837) ((-1158 . -257) 188816) ((-1158 . -590) 188716) ((-1158 . -588) 188601) ((-1158 . -969) T) ((-1158 . -1024) T) ((-1158 . -1059) T) ((-1158 . -663) T) ((-1158 . -961) T) ((-1158 . -82) 188421) ((-1158 . -963) 188262) ((-1158 . -968) 188103) ((-1158 . -21) T) ((-1158 . -23) T) ((-1158 . -1012) T) ((-1158 . -552) 188085) ((-1158 . -1127) T) ((-1158 . -13) T) ((-1158 . -72) T) ((-1158 . -25) T) ((-1158 . -104) T) ((-1158 . -245) 188039) ((-1158 . -201) 188018) ((-1158 . -915) 187984) ((-1158 . -1113) 187950) ((-1158 . -1116) 187916) ((-1158 . -430) 187882) ((-1158 . -239) 187848) ((-1158 . -66) 187814) ((-1158 . -35) 187780) ((-1158 . -1156) 187750) ((-1158 . -47) 187720) ((-1158 . -120) 187699) ((-1158 . -118) 187678) ((-1158 . -886) 187641) ((-1158 . -811) 187547) ((-1158 . -806) 187428) ((-1158 . -809) 187334) ((-1158 . -241) 187292) ((-1158 . -189) 187244) ((-1158 . -186) 187190) ((-1158 . -190) 187142) ((-1158 . -1160) 187126) ((-1158 . -950) 187061) ((-1146 . -1153) 187045) ((-1146 . -1064) 187023) ((-1146 . -553) NIL) ((-1146 . -259) 187010) ((-1146 . -452) 186958) ((-1146 . -276) 186935) ((-1146 . -950) 186818) ((-1146 . -352) 186802) ((-1146 . -38) 186634) ((-1146 . -82) 186439) ((-1146 . -963) 186265) ((-1146 . -968) 186091) ((-1146 . -588) 186001) ((-1146 . -590) 185890) ((-1146 . -582) 185722) ((-1146 . -654) 185554) ((-1146 . -555) 185310) ((-1146 . -118) 185289) ((-1146 . -120) 185268) ((-1146 . -47) 185245) ((-1146 . -326) 185229) ((-1146 . -580) 185177) ((-1146 . -809) 185121) ((-1146 . -806) 185028) ((-1146 . -811) 184939) ((-1146 . -796) NIL) ((-1146 . -821) 184918) ((-1146 . -1132) 184897) ((-1146 . -861) 184867) ((-1146 . -832) 184846) ((-1146 . -494) 184760) ((-1146 . -245) 184674) ((-1146 . -146) 184568) ((-1146 . -389) 184502) ((-1146 . -257) 184481) ((-1146 . -241) 184408) ((-1146 . -190) T) ((-1146 . -104) T) ((-1146 . -25) T) ((-1146 . -72) T) ((-1146 . -552) 184390) ((-1146 . -1012) T) ((-1146 . -23) T) ((-1146 . -21) T) ((-1146 . -969) T) ((-1146 . -1024) T) ((-1146 . -1059) T) ((-1146 . -663) T) ((-1146 . -961) T) ((-1146 . -186) 184377) ((-1146 . -13) T) ((-1146 . -1127) T) ((-1146 . -189) T) ((-1146 . -225) 184361) ((-1146 . -184) 184345) ((-1144 . -1005) 184329) ((-1144 . -557) 184313) ((-1144 . -1012) 184291) ((-1144 . -552) 184258) ((-1144 . -1127) 184236) ((-1144 . -13) 184214) ((-1144 . -72) 184192) ((-1144 . -1006) 184149) ((-1142 . -1141) 184128) ((-1142 . -915) 184094) ((-1142 . -1113) 184060) ((-1142 . -1116) 184026) ((-1142 . -430) 183992) ((-1142 . -239) 183958) ((-1142 . -66) 183924) ((-1142 . -35) 183890) ((-1142 . -1156) 183867) ((-1142 . -47) 183844) ((-1142 . -555) 183599) ((-1142 . -654) 183419) ((-1142 . -582) 183239) ((-1142 . -590) 183050) ((-1142 . -588) 182908) ((-1142 . -968) 182722) ((-1142 . -963) 182536) ((-1142 . -82) 182324) ((-1142 . -38) 182144) ((-1142 . -886) 182114) ((-1142 . -241) 182014) ((-1142 . -1139) 181998) ((-1142 . -969) T) ((-1142 . -1024) T) ((-1142 . -1059) T) ((-1142 . -663) T) ((-1142 . -961) T) ((-1142 . -21) T) ((-1142 . -23) T) ((-1142 . -1012) T) ((-1142 . -552) 181980) ((-1142 . -1127) T) ((-1142 . -13) T) ((-1142 . -72) T) ((-1142 . -25) T) ((-1142 . -104) T) ((-1142 . -118) 181908) ((-1142 . -120) 181836) ((-1142 . -553) 181509) ((-1142 . -184) 181479) ((-1142 . -809) 181333) ((-1142 . -811) 181133) ((-1142 . -806) 180931) ((-1142 . -225) 180901) ((-1142 . -189) 180763) ((-1142 . -186) 180619) ((-1142 . -190) 180527) ((-1142 . -311) 180506) ((-1142 . -1132) 180485) ((-1142 . -832) 180464) ((-1142 . -494) 180418) ((-1142 . -146) 180352) ((-1142 . -389) 180331) ((-1142 . -257) 180310) ((-1142 . -245) 180264) ((-1142 . -201) 180243) ((-1142 . -287) 180213) ((-1142 . -452) 180073) ((-1142 . -259) 180012) ((-1142 . -326) 179982) ((-1142 . -580) 179890) ((-1142 . -340) 179860) ((-1142 . -796) 179733) ((-1142 . -740) 179686) ((-1142 . -714) 179639) ((-1142 . -716) 179592) ((-1142 . -756) 179494) ((-1142 . -759) 179396) ((-1142 . -718) 179349) ((-1142 . -721) 179302) ((-1142 . -755) 179255) ((-1142 . -794) 179225) ((-1142 . -821) 179178) ((-1142 . -933) 179131) ((-1142 . -950) 178920) ((-1142 . -1064) 178872) ((-1142 . -904) 178842) ((-1137 . -1141) 178803) ((-1137 . -915) 178769) ((-1137 . -1113) 178735) ((-1137 . -1116) 178701) ((-1137 . -430) 178667) ((-1137 . -239) 178633) ((-1137 . -66) 178599) ((-1137 . -35) 178565) ((-1137 . -1156) 178542) ((-1137 . -47) 178519) ((-1137 . -555) 178320) ((-1137 . -654) 178122) ((-1137 . -582) 177924) ((-1137 . -590) 177779) ((-1137 . -588) 177619) ((-1137 . -968) 177415) ((-1137 . -963) 177211) ((-1137 . -82) 176963) ((-1137 . -38) 176765) ((-1137 . -886) 176735) ((-1137 . -241) 176563) ((-1137 . -1139) 176547) ((-1137 . -969) T) ((-1137 . -1024) T) ((-1137 . -1059) T) ((-1137 . -663) T) ((-1137 . -961) T) ((-1137 . -21) T) ((-1137 . -23) T) ((-1137 . -1012) T) ((-1137 . -552) 176529) ((-1137 . -1127) T) ((-1137 . -13) T) ((-1137 . -72) T) ((-1137 . -25) T) ((-1137 . -104) T) ((-1137 . -118) 176439) ((-1137 . -120) 176349) ((-1137 . -553) NIL) ((-1137 . -184) 176301) ((-1137 . -809) 176137) ((-1137 . -811) 175901) ((-1137 . -806) 175640) ((-1137 . -225) 175592) ((-1137 . -189) 175418) ((-1137 . -186) 175238) ((-1137 . -190) 175128) ((-1137 . -311) 175107) ((-1137 . -1132) 175086) ((-1137 . -832) 175065) ((-1137 . -494) 175019) ((-1137 . -146) 174953) ((-1137 . -389) 174932) ((-1137 . -257) 174911) ((-1137 . -245) 174865) ((-1137 . -201) 174844) ((-1137 . -287) 174796) ((-1137 . -452) 174530) ((-1137 . -259) 174415) ((-1137 . -326) 174367) ((-1137 . -580) 174319) ((-1137 . -340) 174271) ((-1137 . -796) NIL) ((-1137 . -740) NIL) ((-1137 . -714) NIL) ((-1137 . -716) NIL) ((-1137 . -756) NIL) ((-1137 . -759) NIL) ((-1137 . -718) NIL) ((-1137 . -721) NIL) ((-1137 . -755) NIL) ((-1137 . -794) 174223) ((-1137 . -821) NIL) ((-1137 . -933) NIL) ((-1137 . -950) 174189) ((-1137 . -1064) NIL) ((-1137 . -904) 174141) ((-1136 . -752) T) ((-1136 . -759) T) ((-1136 . -756) T) ((-1136 . -1012) T) ((-1136 . -552) 174123) ((-1136 . -1127) T) ((-1136 . -13) T) ((-1136 . -72) T) ((-1136 . -317) T) ((-1136 . -604) T) ((-1135 . -752) T) ((-1135 . -759) T) ((-1135 . -756) T) ((-1135 . -1012) T) ((-1135 . -552) 174105) ((-1135 . -1127) T) ((-1135 . -13) T) ((-1135 . -72) T) ((-1135 . -317) T) ((-1135 . -604) T) ((-1134 . -752) T) ((-1134 . -759) T) ((-1134 . -756) T) ((-1134 . -1012) T) ((-1134 . -552) 174087) ((-1134 . -1127) T) ((-1134 . -13) T) ((-1134 . -72) T) ((-1134 . -317) T) ((-1134 . -604) T) ((-1133 . -752) T) ((-1133 . -759) T) ((-1133 . -756) T) ((-1133 . -1012) T) ((-1133 . -552) 174069) ((-1133 . -1127) T) ((-1133 . -13) T) ((-1133 . -72) T) ((-1133 . -317) T) ((-1133 . -604) T) ((-1128 . -994) T) ((-1128 . -427) 174050) ((-1128 . -552) 174016) ((-1128 . -555) 173997) ((-1128 . -1012) T) ((-1128 . -1127) T) ((-1128 . -13) T) ((-1128 . -72) T) ((-1128 . -64) T) ((-1125 . -427) 173974) ((-1125 . -552) 173915) ((-1125 . -555) 173892) ((-1125 . -1012) 173870) ((-1125 . -1127) 173848) ((-1125 . -13) 173826) ((-1125 . -72) 173804) ((-1120 . -679) 173780) ((-1120 . -35) 173746) ((-1120 . -66) 173712) ((-1120 . -239) 173678) ((-1120 . -430) 173644) ((-1120 . -1116) 173610) ((-1120 . -1113) 173576) ((-1120 . -915) 173542) ((-1120 . -47) 173511) ((-1120 . -38) 173408) ((-1120 . -582) 173305) ((-1120 . -654) 173202) ((-1120 . -555) 173084) ((-1120 . -245) 173063) ((-1120 . -494) 173042) ((-1120 . -82) 172907) ((-1120 . -963) 172793) ((-1120 . -968) 172679) ((-1120 . -146) 172633) ((-1120 . -120) 172612) ((-1120 . -118) 172591) ((-1120 . -590) 172516) ((-1120 . -588) 172426) ((-1120 . -886) 172387) ((-1120 . -811) 172368) ((-1120 . -1127) T) ((-1120 . -13) T) ((-1120 . -806) 172347) ((-1120 . -961) T) ((-1120 . -663) T) ((-1120 . -1059) T) ((-1120 . -1024) T) ((-1120 . -969) T) ((-1120 . -21) T) ((-1120 . -23) T) ((-1120 . -1012) T) ((-1120 . -552) 172329) ((-1120 . -72) T) ((-1120 . -25) T) ((-1120 . -104) T) ((-1120 . -809) 172310) ((-1120 . -452) 172277) ((-1120 . -259) 172264) ((-1114 . -923) 172248) ((-1114 . -34) T) ((-1114 . -13) T) ((-1114 . -1127) T) ((-1114 . -72) 172202) ((-1114 . -552) 172137) ((-1114 . -259) 172075) ((-1114 . -452) 172008) ((-1114 . -1012) 171986) ((-1114 . -426) 171970) ((-1109 . -313) 171944) ((-1109 . -72) T) ((-1109 . -13) T) ((-1109 . -1127) T) ((-1109 . -552) 171926) ((-1109 . -1012) T) ((-1107 . -1012) T) ((-1107 . -552) 171908) ((-1107 . -1127) T) ((-1107 . -13) T) ((-1107 . -72) T) ((-1107 . -555) 171890) ((-1102 . -747) 171874) ((-1102 . -72) T) ((-1102 . -13) T) ((-1102 . -1127) T) ((-1102 . -552) 171856) ((-1102 . -1012) T) ((-1100 . -1105) 171835) ((-1100 . -183) 171783) ((-1100 . -76) 171731) ((-1100 . -259) 171529) ((-1100 . -452) 171281) ((-1100 . -426) 171216) ((-1100 . -124) 171164) ((-1100 . -553) NIL) ((-1100 . -193) 171112) ((-1100 . -549) 171091) ((-1100 . -243) 171070) ((-1100 . -1127) T) ((-1100 . -13) T) ((-1100 . -241) 171049) ((-1100 . -1012) T) ((-1100 . -552) 171031) ((-1100 . -72) T) ((-1100 . -34) T) ((-1100 . -538) 171010) ((-1096 . -1012) T) ((-1096 . -552) 170992) ((-1096 . -1127) T) ((-1096 . -13) T) ((-1096 . -72) T) ((-1095 . -752) T) ((-1095 . -759) T) ((-1095 . -756) T) ((-1095 . -1012) T) ((-1095 . -552) 170974) ((-1095 . -1127) T) ((-1095 . -13) T) ((-1095 . -72) T) ((-1095 . -317) T) ((-1095 . -604) T) ((-1094 . -752) T) ((-1094 . -759) T) ((-1094 . -756) T) ((-1094 . -1012) T) ((-1094 . -552) 170956) ((-1094 . -1127) T) ((-1094 . -13) T) ((-1094 . -72) T) ((-1094 . -317) T) ((-1093 . -1173) T) ((-1093 . -1012) T) ((-1093 . -552) 170923) ((-1093 . -1127) T) ((-1093 . -13) T) ((-1093 . -72) T) ((-1093 . -950) 170859) ((-1093 . -555) 170795) ((-1092 . -552) 170777) ((-1091 . -552) 170759) ((-1090 . -276) 170736) ((-1090 . -950) 170634) ((-1090 . -352) 170618) ((-1090 . -38) 170515) ((-1090 . -555) 170372) ((-1090 . -590) 170297) ((-1090 . -588) 170207) ((-1090 . -969) T) ((-1090 . -1024) T) ((-1090 . -1059) T) ((-1090 . -663) T) ((-1090 . -961) T) ((-1090 . -82) 170072) ((-1090 . -963) 169958) ((-1090 . -968) 169844) ((-1090 . -21) T) ((-1090 . -23) T) ((-1090 . -1012) T) ((-1090 . -552) 169826) ((-1090 . -1127) T) ((-1090 . -13) T) ((-1090 . -72) T) ((-1090 . -25) T) ((-1090 . -104) T) ((-1090 . -582) 169723) ((-1090 . -654) 169620) ((-1090 . -118) 169599) ((-1090 . -120) 169578) ((-1090 . -146) 169532) ((-1090 . -494) 169511) ((-1090 . -245) 169490) ((-1090 . -47) 169467) ((-1088 . -756) T) ((-1088 . -552) 169449) ((-1088 . -1012) T) ((-1088 . -72) T) ((-1088 . -13) T) ((-1088 . -1127) T) ((-1088 . -759) T) ((-1088 . -553) 169371) ((-1088 . -555) 169337) ((-1088 . -950) 169319) ((-1088 . -796) 169286) ((-1087 . -1170) 169270) ((-1087 . -190) 169229) ((-1087 . -555) 169111) ((-1087 . -590) 169036) ((-1087 . -588) 168946) ((-1087 . -104) T) ((-1087 . -25) T) ((-1087 . -72) T) ((-1087 . -552) 168928) ((-1087 . -1012) T) ((-1087 . -23) T) ((-1087 . -21) T) ((-1087 . -969) T) ((-1087 . -1024) T) ((-1087 . -1059) T) ((-1087 . -663) T) ((-1087 . -961) T) ((-1087 . -186) 168881) ((-1087 . -13) T) ((-1087 . -1127) T) ((-1087 . -189) 168840) ((-1087 . -241) 168805) ((-1087 . -809) 168718) ((-1087 . -806) 168606) ((-1087 . -811) 168519) ((-1087 . -886) 168489) ((-1087 . -38) 168386) ((-1087 . -82) 168251) ((-1087 . -963) 168137) ((-1087 . -968) 168023) ((-1087 . -582) 167920) ((-1087 . -654) 167817) ((-1087 . -118) 167796) ((-1087 . -120) 167775) ((-1087 . -146) 167729) ((-1087 . -494) 167708) ((-1087 . -245) 167687) ((-1087 . -47) 167664) ((-1087 . -1156) 167641) ((-1087 . -35) 167607) ((-1087 . -66) 167573) ((-1087 . -239) 167539) ((-1087 . -430) 167505) ((-1087 . -1116) 167471) ((-1087 . -1113) 167437) ((-1087 . -915) 167403) ((-1086 . -1162) 167364) ((-1086 . -311) 167343) ((-1086 . -1132) 167322) ((-1086 . -832) 167301) ((-1086 . -494) 167255) ((-1086 . -146) 167189) ((-1086 . -555) 166938) ((-1086 . -654) 166785) ((-1086 . -582) 166632) ((-1086 . -38) 166479) ((-1086 . -389) 166458) ((-1086 . -257) 166437) ((-1086 . -590) 166337) ((-1086 . -588) 166222) ((-1086 . -969) T) ((-1086 . -1024) T) ((-1086 . -1059) T) ((-1086 . -663) T) ((-1086 . -961) T) ((-1086 . -82) 166042) ((-1086 . -963) 165883) ((-1086 . -968) 165724) ((-1086 . -21) T) ((-1086 . -23) T) ((-1086 . -1012) T) ((-1086 . -552) 165706) ((-1086 . -1127) T) ((-1086 . -13) T) ((-1086 . -72) T) ((-1086 . -25) T) ((-1086 . -104) T) ((-1086 . -245) 165660) ((-1086 . -201) 165639) ((-1086 . -915) 165605) ((-1086 . -1113) 165571) ((-1086 . -1116) 165537) ((-1086 . -430) 165503) ((-1086 . -239) 165469) ((-1086 . -66) 165435) ((-1086 . -35) 165401) ((-1086 . -1156) 165371) ((-1086 . -47) 165341) ((-1086 . -120) 165320) ((-1086 . -118) 165299) ((-1086 . -886) 165262) ((-1086 . -811) 165168) ((-1086 . -806) 165049) ((-1086 . -809) 164955) ((-1086 . -241) 164913) ((-1086 . -189) 164865) ((-1086 . -186) 164811) ((-1086 . -190) 164763) ((-1086 . -1160) 164747) ((-1086 . -950) 164682) ((-1083 . -1153) 164666) ((-1083 . -1064) 164644) ((-1083 . -553) NIL) ((-1083 . -259) 164631) ((-1083 . -452) 164579) ((-1083 . -276) 164556) ((-1083 . -950) 164439) ((-1083 . -352) 164423) ((-1083 . -38) 164255) ((-1083 . -82) 164060) ((-1083 . -963) 163886) ((-1083 . -968) 163712) ((-1083 . -588) 163622) ((-1083 . -590) 163511) ((-1083 . -582) 163343) ((-1083 . -654) 163175) ((-1083 . -555) 162952) ((-1083 . -118) 162931) ((-1083 . -120) 162910) ((-1083 . -47) 162887) ((-1083 . -326) 162871) ((-1083 . -580) 162819) ((-1083 . -809) 162763) ((-1083 . -806) 162670) ((-1083 . -811) 162581) ((-1083 . -796) NIL) ((-1083 . -821) 162560) ((-1083 . -1132) 162539) ((-1083 . -861) 162509) ((-1083 . -832) 162488) ((-1083 . -494) 162402) ((-1083 . -245) 162316) ((-1083 . -146) 162210) ((-1083 . -389) 162144) ((-1083 . -257) 162123) ((-1083 . -241) 162050) ((-1083 . -190) T) ((-1083 . -104) T) ((-1083 . -25) T) ((-1083 . -72) T) ((-1083 . -552) 162032) ((-1083 . -1012) T) ((-1083 . -23) T) ((-1083 . -21) T) ((-1083 . -969) T) ((-1083 . -1024) T) ((-1083 . -1059) T) ((-1083 . -663) T) ((-1083 . -961) T) ((-1083 . -186) 162019) ((-1083 . -13) T) ((-1083 . -1127) T) ((-1083 . -189) T) ((-1083 . -225) 162003) ((-1083 . -184) 161987) ((-1080 . -1141) 161948) ((-1080 . -915) 161914) ((-1080 . -1113) 161880) ((-1080 . -1116) 161846) ((-1080 . -430) 161812) ((-1080 . -239) 161778) ((-1080 . -66) 161744) ((-1080 . -35) 161710) ((-1080 . -1156) 161687) ((-1080 . -47) 161664) ((-1080 . -555) 161465) ((-1080 . -654) 161267) ((-1080 . -582) 161069) ((-1080 . -590) 160924) ((-1080 . -588) 160764) ((-1080 . -968) 160560) ((-1080 . -963) 160356) ((-1080 . -82) 160108) ((-1080 . -38) 159910) ((-1080 . -886) 159880) ((-1080 . -241) 159708) ((-1080 . -1139) 159692) ((-1080 . -969) T) ((-1080 . -1024) T) ((-1080 . -1059) T) ((-1080 . -663) T) ((-1080 . -961) T) ((-1080 . -21) T) ((-1080 . -23) T) ((-1080 . -1012) T) ((-1080 . -552) 159674) ((-1080 . -1127) T) ((-1080 . -13) T) ((-1080 . -72) T) ((-1080 . -25) T) ((-1080 . -104) T) ((-1080 . -118) 159584) ((-1080 . -120) 159494) ((-1080 . -553) NIL) ((-1080 . -184) 159446) ((-1080 . -809) 159282) ((-1080 . -811) 159046) ((-1080 . -806) 158785) ((-1080 . -225) 158737) ((-1080 . -189) 158563) ((-1080 . -186) 158383) ((-1080 . -190) 158273) ((-1080 . -311) 158252) ((-1080 . -1132) 158231) ((-1080 . -832) 158210) ((-1080 . -494) 158164) ((-1080 . -146) 158098) ((-1080 . -389) 158077) ((-1080 . -257) 158056) ((-1080 . -245) 158010) ((-1080 . -201) 157989) ((-1080 . -287) 157941) ((-1080 . -452) 157675) ((-1080 . -259) 157560) ((-1080 . -326) 157512) ((-1080 . -580) 157464) ((-1080 . -340) 157416) ((-1080 . -796) NIL) ((-1080 . -740) NIL) ((-1080 . -714) NIL) ((-1080 . -716) NIL) ((-1080 . -756) NIL) ((-1080 . -759) NIL) ((-1080 . -718) NIL) ((-1080 . -721) NIL) ((-1080 . -755) NIL) ((-1080 . -794) 157368) ((-1080 . -821) NIL) ((-1080 . -933) NIL) ((-1080 . -950) 157334) ((-1080 . -1064) NIL) ((-1080 . -904) 157286) ((-1079 . -994) T) ((-1079 . -427) 157267) ((-1079 . -552) 157233) ((-1079 . -555) 157214) ((-1079 . -1012) T) ((-1079 . -1127) T) ((-1079 . -13) T) ((-1079 . -72) T) ((-1079 . -64) T) ((-1078 . -1012) T) ((-1078 . -552) 157196) ((-1078 . -1127) T) 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. -616) 156057) ((-1067 . -593) 156041) ((-1067 . -243) 156018) ((-1067 . -241) 155970) ((-1067 . -538) 155947) ((-1067 . -553) 155908) ((-1067 . -426) 155892) ((-1067 . -1012) 155870) ((-1067 . -452) 155803) ((-1067 . -259) 155741) ((-1067 . -552) 155676) ((-1067 . -72) 155630) ((-1067 . -1127) T) ((-1067 . -13) T) ((-1067 . -34) T) ((-1067 . -124) 155614) ((-1067 . -1166) 155598) ((-1067 . -923) 155582) ((-1067 . -1062) 155566) ((-1067 . -555) 155543) ((-1065 . -994) T) ((-1065 . -427) 155524) ((-1065 . -552) 155490) ((-1065 . -555) 155471) ((-1065 . -1012) T) ((-1065 . -1127) T) ((-1065 . -13) T) ((-1065 . -72) T) ((-1065 . -64) T) ((-1063 . -1105) 155450) ((-1063 . -183) 155398) ((-1063 . -76) 155346) ((-1063 . -259) 155144) ((-1063 . -452) 154896) ((-1063 . -426) 154831) ((-1063 . -124) 154779) ((-1063 . -553) NIL) ((-1063 . -193) 154727) ((-1063 . -549) 154706) ((-1063 . -243) 154685) ((-1063 . -1127) T) ((-1063 . -13) T) ((-1063 . -241) 154664) ((-1063 . -1012) T) ((-1063 . 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. -1015) 143979) ((-1016 . -72) T) ((-1016 . -552) 143961) ((-1016 . -1012) T) ((-1016 . -241) 143917) ((-1016 . -1127) T) ((-1016 . -13) T) ((-1016 . -557) 143832) ((-1014 . -1015) 143784) ((-1014 . -72) T) ((-1014 . -552) 143766) ((-1014 . -1012) T) ((-1014 . -241) 143722) ((-1014 . -1127) T) ((-1014 . -13) T) ((-1014 . -557) 143625) ((-1013 . -317) T) ((-1013 . -72) T) ((-1013 . -13) T) ((-1013 . -1127) T) ((-1013 . -552) 143607) ((-1013 . -1012) T) ((-1008 . -366) 143591) ((-1008 . -1010) 143575) ((-1008 . -317) 143554) ((-1008 . -193) 143538) ((-1008 . -553) 143499) ((-1008 . -124) 143483) ((-1008 . -426) 143467) ((-1008 . -1012) T) ((-1008 . -452) 143400) ((-1008 . -259) 143338) ((-1008 . -552) 143320) ((-1008 . -72) T) ((-1008 . -1127) T) ((-1008 . -13) T) ((-1008 . -34) T) ((-1008 . -76) 143304) ((-1008 . -183) 143288) ((-1007 . -994) T) ((-1007 . -427) 143269) ((-1007 . -552) 143235) ((-1007 . -555) 143216) ((-1007 . -1012) T) ((-1007 . -1127) T) ((-1007 . -13) T) ((-1007 . -72) T) ((-1007 . -64) T) ((-1003 . -1127) T) ((-1003 . -13) T) ((-1003 . -1012) 143186) ((-1003 . -552) 143145) ((-1003 . -72) 143115) ((-1002 . -994) T) ((-1002 . -427) 143096) ((-1002 . -552) 143062) ((-1002 . -555) 143043) ((-1002 . -1012) T) ((-1002 . -1127) T) ((-1002 . -13) T) ((-1002 . -72) T) ((-1002 . -64) T) ((-1000 . -1005) 143027) ((-1000 . -557) 143011) ((-1000 . -1012) 142989) ((-1000 . -552) 142956) ((-1000 . -1127) 142934) ((-1000 . -13) 142912) ((-1000 . -72) 142890) ((-1000 . -1006) 142848) ((-999 . -228) 142832) ((-999 . -555) 142816) ((-999 . -950) 142800) ((-999 . -759) T) ((-999 . -72) T) ((-999 . -1012) T) ((-999 . -552) 142782) ((-999 . -756) T) ((-999 . -186) 142769) ((-999 . -13) T) ((-999 . -1127) T) ((-999 . -189) T) ((-998 . -213) 142708) ((-998 . -555) 142452) ((-998 . -950) 142282) ((-998 . -553) NIL) ((-998 . -276) 142244) ((-998 . -352) 142228) ((-998 . -38) 142080) ((-998 . -82) 141905) ((-998 . -963) 141751) ((-998 . -968) 141597) ((-998 . -588) 141507) ((-998 . -590) 141396) ((-998 . -582) 141248) ((-998 . -654) 141100) ((-998 . -118) 141079) ((-998 . -120) 141058) ((-998 . -146) 140972) ((-998 . -494) 140906) ((-998 . -245) 140840) ((-998 . -47) 140802) ((-998 . -326) 140786) ((-998 . -580) 140734) ((-998 . -389) 140688) ((-998 . -452) 140553) ((-998 . -809) 140489) ((-998 . -806) 140388) ((-998 . -811) 140291) ((-998 . -796) NIL) ((-998 . -821) 140270) ((-998 . -1132) 140249) ((-998 . -861) 140196) ((-998 . -259) 140183) ((-998 . -190) 140162) ((-998 . -104) T) ((-998 . -25) T) ((-998 . -72) T) ((-998 . -552) 140144) ((-998 . -1012) T) ((-998 . -23) T) ((-998 . -21) T) ((-998 . -969) T) ((-998 . -1024) T) ((-998 . -1059) T) ((-998 . -663) T) ((-998 . -961) T) ((-998 . -186) 140092) ((-998 . -13) T) ((-998 . -1127) T) ((-998 . -189) 140046) ((-998 . -225) 140030) ((-998 . -184) 140014) ((-996 . -552) 139996) ((-993 . -756) T) ((-993 . -552) 139978) ((-993 . -1012) T) ((-993 . -72) T) ((-993 . -13) T) ((-993 . -1127) T) 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-552) 137377) ((-990 . -1127) T) ((-990 . -13) T) ((-990 . -72) T) ((-990 . -25) T) ((-990 . -104) T) ((-990 . -961) T) ((-990 . -663) T) ((-990 . -1059) T) ((-990 . -1024) T) ((-990 . -969) T) ((-990 . -146) T) ((-988 . -1012) T) ((-988 . -552) 137359) ((-988 . -1127) T) ((-988 . -13) T) ((-988 . -72) T) ((-988 . -241) 137338) ((-987 . -1012) T) ((-987 . -552) 137320) ((-987 . -1127) T) ((-987 . -13) T) ((-987 . -72) T) ((-986 . -1012) T) ((-986 . -552) 137302) ((-986 . -1127) T) ((-986 . -13) T) ((-986 . -72) T) ((-986 . -241) 137281) ((-986 . -950) 137258) ((-986 . -555) 137235) ((-985 . -1127) T) ((-985 . -13) T) ((-984 . -994) T) ((-984 . -427) 137216) ((-984 . -552) 137182) ((-984 . -555) 137163) ((-984 . -1012) T) ((-984 . -1127) T) ((-984 . -13) T) ((-984 . -72) T) ((-984 . -64) T) ((-977 . -994) T) ((-977 . -427) 137144) ((-977 . -552) 137110) ((-977 . -555) 137091) ((-977 . -1012) T) ((-977 . -1127) T) ((-977 . -13) T) ((-977 . -72) T) ((-977 . -64) T) ((-974 . -482) T) 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136717) ((-973 . -72) T) ((-973 . -13) T) ((-973 . -1127) T) ((-973 . -552) 136699) ((-973 . -1012) T) ((-970 . -1127) T) ((-970 . -13) T) ((-970 . -1012) 136677) ((-970 . -552) 136644) ((-970 . -72) 136622) ((-966 . -965) 136562) ((-966 . -582) 136507) ((-966 . -654) 136452) ((-966 . -34) T) ((-966 . -259) 136390) ((-966 . -452) 136323) ((-966 . -426) 136307) ((-966 . -590) 136291) ((-966 . -588) 136260) ((-966 . -104) T) ((-966 . -25) T) ((-966 . -72) T) ((-966 . -13) T) ((-966 . -1127) T) ((-966 . -552) 136222) ((-966 . -1012) T) ((-966 . -23) T) ((-966 . -21) T) ((-966 . -968) 136206) ((-966 . -963) 136190) ((-966 . -82) 136169) ((-966 . -1185) 136139) ((-966 . -553) 136100) ((-958 . -982) 136029) ((-958 . -889) 135958) ((-958 . -553) 135900) ((-958 . -426) 135865) ((-958 . -1012) T) ((-958 . -452) 135749) ((-958 . -259) 135657) ((-958 . -552) 135600) ((-958 . -72) T) ((-958 . -1127) T) ((-958 . -13) T) ((-958 . -34) T) ((-958 . -124) 135565) ((-958 . -1122) 135494) ((-948 . -994) 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. -963) 114107) ((-741 . -82) 114086) ((-741 . -961) T) ((-741 . -663) T) ((-741 . -1059) T) ((-741 . -1024) T) ((-741 . -969) T) ((-741 . -38) 114056) ((-741 . -190) 114035) ((-741 . -186) 114008) ((-741 . -189) 113987) ((-739 . -333) 113971) ((-739 . -555) 113955) ((-739 . -950) 113939) ((-739 . -759) T) ((-739 . -756) T) ((-739 . -1024) T) ((-739 . -72) T) ((-739 . -13) T) ((-739 . -1127) T) ((-739 . -552) 113921) ((-739 . -1012) T) ((-739 . -663) T) ((-739 . -754) T) ((-739 . -766) T) ((-738 . -228) 113905) ((-738 . -555) 113889) ((-738 . -950) 113873) ((-738 . -759) T) ((-738 . -72) T) ((-738 . -1012) T) ((-738 . -552) 113855) ((-738 . -756) T) ((-738 . -186) 113842) ((-738 . -13) T) ((-738 . -1127) T) ((-738 . -189) T) ((-737 . -82) 113777) ((-737 . -963) 113728) ((-737 . -968) 113679) ((-737 . -21) T) ((-737 . -588) 113615) ((-737 . -23) T) ((-737 . -1012) T) ((-737 . -552) 113584) ((-737 . -1127) T) ((-737 . -13) T) ((-737 . -72) T) ((-737 . -25) T) ((-737 . -104) T) ((-737 . 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103607) ((-704 . -806) 103514) ((-704 . -811) 103425) ((-704 . -796) NIL) ((-704 . -821) 103404) ((-704 . -1132) 103383) ((-704 . -861) 103353) ((-704 . -832) 103332) ((-704 . -494) 103246) ((-704 . -245) 103160) ((-704 . -146) 103054) ((-704 . -389) 102988) ((-704 . -257) 102967) ((-704 . -241) 102894) ((-704 . -190) T) ((-704 . -104) T) ((-704 . -25) T) ((-704 . -72) T) ((-704 . -552) 102855) ((-704 . -1012) T) ((-704 . -23) T) ((-704 . -21) T) ((-704 . -969) T) ((-704 . -1024) T) ((-704 . -1059) T) ((-704 . -663) T) ((-704 . -961) T) ((-704 . -186) 102842) ((-704 . -13) T) ((-704 . -1127) T) ((-704 . -189) T) ((-704 . -225) 102826) ((-704 . -184) 102810) ((-703 . -976) 102777) ((-703 . -553) 102412) ((-703 . -259) 102399) ((-703 . -452) 102351) ((-703 . -276) 102323) ((-703 . -950) 102182) ((-703 . -352) 102166) ((-703 . -38) 102018) ((-703 . -555) 101791) ((-703 . -590) 101680) ((-703 . -588) 101590) ((-703 . -969) T) ((-703 . -1024) T) ((-703 . -1059) T) ((-703 . -663) T) ((-703 . 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. -13) T) ((-419 . -1127) T) ((-419 . -1012) 68325) ((-419 . -23) 68181) ((-419 . -21) 68096) ((-418 . -861) 68041) ((-418 . -555) 67833) ((-418 . -950) 67711) ((-418 . -1132) 67690) ((-418 . -821) 67669) ((-418 . -796) NIL) ((-418 . -811) 67646) ((-418 . -806) 67621) ((-418 . -809) 67598) ((-418 . -452) 67536) ((-418 . -389) 67490) ((-418 . -580) 67438) ((-418 . -590) 67327) ((-418 . -326) 67311) ((-418 . -47) 67268) ((-418 . -38) 67120) ((-418 . -582) 66972) ((-418 . -654) 66824) ((-418 . -245) 66758) ((-418 . -494) 66692) ((-418 . -82) 66517) ((-418 . -963) 66363) ((-418 . -968) 66209) ((-418 . -146) 66123) ((-418 . -120) 66102) ((-418 . -118) 66081) ((-418 . -588) 65991) ((-418 . -104) T) ((-418 . -25) T) ((-418 . -72) T) ((-418 . -13) T) ((-418 . -1127) T) ((-418 . -552) 65973) ((-418 . -1012) T) ((-418 . -23) T) ((-418 . -21) T) ((-418 . -961) T) ((-418 . -663) T) ((-418 . -1059) T) ((-418 . -1024) T) ((-418 . -969) T) ((-418 . -352) 65957) ((-418 . -276) 65914) ((-418 . -259) 65901) ((-418 . -553) 65762) ((-416 . -1105) 65741) ((-416 . -183) 65689) ((-416 . -76) 65637) ((-416 . -259) 65435) ((-416 . -452) 65187) ((-416 . -426) 65122) ((-416 . -124) 65070) ((-416 . -553) NIL) ((-416 . -193) 65018) ((-416 . -549) 64997) ((-416 . -243) 64976) ((-416 . -1127) T) ((-416 . -13) T) ((-416 . -241) 64955) ((-416 . -1012) T) ((-416 . -552) 64937) ((-416 . -72) T) ((-416 . -34) T) ((-416 . -538) 64916) ((-415 . -994) T) ((-415 . -427) 64897) ((-415 . -552) 64863) ((-415 . -555) 64844) ((-415 . -1012) T) ((-415 . -1127) T) ((-415 . -13) T) ((-415 . -72) T) ((-415 . -64) T) ((-414 . -311) T) ((-414 . -1132) T) ((-414 . -832) T) ((-414 . -494) T) ((-414 . -146) T) ((-414 . -555) 64794) ((-414 . -654) 64759) ((-414 . -582) 64724) ((-414 . -38) 64689) ((-414 . -389) T) ((-414 . -257) T) ((-414 . -590) 64654) ((-414 . -588) 64604) ((-414 . -969) T) ((-414 . -1024) T) ((-414 . -1059) T) ((-414 . -663) T) ((-414 . -961) T) ((-414 . -82) 64553) ((-414 . -963) 64518) ((-414 . 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. -315) 57404) ((-390 . -1127) T) ((-390 . -13) T) ((-390 . -241) 57367) ((-378 . -1012) T) ((-378 . -552) 57349) ((-378 . -1127) T) ((-378 . -13) T) ((-378 . -72) T) ((-377 . -1012) T) ((-377 . -552) 57331) ((-377 . -1127) T) ((-377 . -13) T) ((-377 . -72) T) ((-376 . -1012) T) ((-376 . -552) 57313) ((-376 . -1127) T) ((-376 . -13) T) ((-376 . -72) T) ((-374 . -552) 57295) ((-369 . -38) 57279) ((-369 . -555) 57248) ((-369 . -590) 57222) ((-369 . -588) 57181) ((-369 . -969) T) ((-369 . -1024) T) ((-369 . -1059) T) ((-369 . -663) T) ((-369 . -961) T) ((-369 . -82) 57160) ((-369 . -963) 57144) ((-369 . -968) 57128) ((-369 . -21) T) ((-369 . -23) T) ((-369 . -1012) T) ((-369 . -552) 57110) ((-369 . -1127) T) ((-369 . -13) T) ((-369 . -72) T) ((-369 . -25) T) ((-369 . -104) T) ((-369 . -582) 57094) ((-369 . -654) 57078) ((-355 . -663) T) ((-355 . -1012) T) ((-355 . -552) 57060) ((-355 . -1127) T) ((-355 . -13) T) ((-355 . -72) T) ((-355 . -1024) T) ((-353 . -410) T) ((-353 . -1024) T) 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54703) ((-345 . -811) 54629) ((-345 . -806) 54553) ((-345 . -225) 54537) ((-345 . -189) 54491) ((-345 . -186) 54439) ((-345 . -190) 54418) ((-345 . -287) 54402) ((-345 . -452) 54244) ((-345 . -259) 54183) ((-345 . -241) 54111) ((-345 . -352) 54095) ((-345 . -950) 53993) ((-345 . -389) 53946) ((-345 . -933) 53925) ((-345 . -553) 53828) ((-345 . -1132) 53806) ((-339 . -1012) T) ((-339 . -552) 53788) ((-339 . -1127) T) ((-339 . -13) T) ((-339 . -72) T) ((-339 . -189) T) ((-339 . -186) 53775) ((-339 . -553) 53752) ((-337 . -683) 53736) ((-337 . -657) T) ((-337 . -685) T) ((-337 . -82) 53715) ((-337 . -963) 53699) ((-337 . -968) 53683) ((-337 . -21) T) ((-337 . -588) 53652) ((-337 . -23) T) ((-337 . -1012) T) ((-337 . -552) 53634) ((-337 . -1127) T) ((-337 . -13) T) ((-337 . -72) T) ((-337 . -25) T) ((-337 . -104) T) ((-337 . -590) 53618) ((-337 . -582) 53602) ((-337 . -654) 53586) ((-335 . -336) T) ((-335 . -72) T) ((-335 . -13) T) ((-335 . -1127) T) ((-335 . -552) 53552) ((-335 . -1012) T) ((-335 . -555) 53533) ((-335 . -427) 53514) ((-334 . -333) 53498) ((-334 . -555) 53482) ((-334 . -950) 53466) ((-334 . -759) 53445) ((-334 . -756) 53424) ((-334 . -1024) T) ((-334 . -72) T) ((-334 . -13) T) ((-334 . -1127) T) ((-334 . -552) 53406) ((-334 . -1012) T) ((-334 . -663) T) ((-331 . -332) 53385) ((-331 . -555) 53369) ((-331 . -950) 53353) ((-331 . -582) 53323) ((-331 . -654) 53293) ((-331 . -590) 53277) ((-331 . -588) 53246) ((-331 . -104) T) ((-331 . -25) T) ((-331 . -72) T) ((-331 . -13) T) ((-331 . -1127) T) ((-331 . -552) 53228) ((-331 . -1012) T) ((-331 . -23) T) ((-331 . -21) T) ((-331 . -968) 53212) ((-331 . -963) 53196) ((-331 . -82) 53175) ((-330 . -82) 53154) ((-330 . -963) 53138) ((-330 . -968) 53122) ((-330 . -21) T) ((-330 . -588) 53091) ((-330 . -23) T) ((-330 . -1012) T) ((-330 . -552) 53073) ((-330 . -1127) T) ((-330 . -13) T) ((-330 . -72) T) ((-330 . -25) T) ((-330 . -104) T) ((-330 . -590) 53057) ((-330 . -447) 53036) ((-330 . -654) 53006) ((-330 . -582) 52976) ((-327 . -344) T) ((-327 . -120) T) ((-327 . -555) 52926) ((-327 . -590) 52891) ((-327 . -588) 52841) ((-327 . -104) T) ((-327 . -25) T) ((-327 . -72) T) ((-327 . -13) T) ((-327 . -1127) T) ((-327 . -552) 52808) ((-327 . -1012) T) ((-327 . -23) T) ((-327 . -21) T) ((-327 . -969) T) ((-327 . -1024) T) ((-327 . -1059) T) ((-327 . -663) T) ((-327 . -961) T) ((-327 . -553) 52722) ((-327 . -311) T) ((-327 . -1132) T) ((-327 . -832) T) ((-327 . -494) T) ((-327 . -146) T) ((-327 . -654) 52687) ((-327 . -582) 52652) ((-327 . -38) 52617) ((-327 . -389) T) ((-327 . -257) T) ((-327 . -82) 52566) ((-327 . -963) 52531) ((-327 . -968) 52496) ((-327 . -245) T) ((-327 . -201) T) ((-327 . -755) T) ((-327 . -721) T) ((-327 . -718) T) ((-327 . -759) T) ((-327 . -756) T) ((-327 . -716) T) ((-327 . -714) T) ((-327 . -796) 52478) ((-327 . -915) T) ((-327 . -933) T) ((-327 . -950) 52438) ((-327 . -972) T) ((-327 . -190) T) ((-327 . -186) 52425) ((-327 . -189) T) ((-327 . -1113) T) ((-327 . -1116) T) 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-654) 51907) ((-307 . -582) 51859) ((-307 . -38) 51824) ((-307 . -389) T) ((-307 . -257) T) ((-307 . -82) 51755) ((-307 . -963) 51707) ((-307 . -968) 51659) ((-307 . -245) T) ((-307 . -201) T) ((-307 . -342) 51613) ((-307 . -118) 51567) ((-307 . -950) 51551) ((-307 . -1185) 51535) ((-307 . -1196) 51519) ((-303 . -279) 51503) ((-303 . -190) 51482) ((-303 . -186) 51455) ((-303 . -189) 51434) ((-303 . -317) 51413) ((-303 . -1064) 51392) ((-303 . -298) 51371) ((-303 . -120) 51350) ((-303 . -555) 51287) ((-303 . -590) 51239) ((-303 . -588) 51176) ((-303 . -104) T) ((-303 . -25) T) ((-303 . -72) T) ((-303 . -13) T) ((-303 . -1127) T) ((-303 . -552) 51158) ((-303 . -1012) T) ((-303 . -23) T) ((-303 . -21) T) ((-303 . -969) T) ((-303 . -1024) T) ((-303 . -1059) T) ((-303 . -663) T) ((-303 . -961) T) ((-303 . -311) T) ((-303 . -1132) T) ((-303 . -832) T) ((-303 . -494) T) ((-303 . -146) T) ((-303 . -654) 51110) ((-303 . -582) 51062) ((-303 . -38) 51027) ((-303 . -389) T) ((-303 . -257) T) 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. -98) 9221) ((-100 . -923) 9205) ((-100 . -34) T) ((-100 . -13) T) ((-100 . -1127) T) ((-100 . -72) 9159) ((-100 . -552) 9094) ((-100 . -259) 9032) ((-100 . -452) 8965) ((-100 . -1012) 8943) ((-100 . -426) 8927) ((-100 . -92) 8911) ((-99 . -98) 8895) ((-99 . -923) 8879) ((-99 . -34) T) ((-99 . -13) T) ((-99 . -1127) T) ((-99 . -72) 8833) ((-99 . -552) 8768) ((-99 . -259) 8706) ((-99 . -452) 8639) ((-99 . -1012) 8617) ((-99 . -426) 8601) ((-99 . -92) 8585) ((-94 . -98) 8569) ((-94 . -923) 8553) ((-94 . -34) T) ((-94 . -13) T) ((-94 . -1127) T) ((-94 . -72) 8507) ((-94 . -552) 8442) ((-94 . -259) 8380) ((-94 . -452) 8313) ((-94 . -1012) 8291) ((-94 . -426) 8275) ((-94 . -92) 8259) ((-90 . -904) 8237) ((-90 . -1064) NIL) ((-90 . -950) 8215) ((-90 . -555) 8146) ((-90 . -553) NIL) ((-90 . -933) NIL) ((-90 . -821) NIL) ((-90 . -794) 8124) ((-90 . -755) NIL) ((-90 . -721) NIL) ((-90 . -718) NIL) ((-90 . -759) NIL) ((-90 . -756) NIL) ((-90 . -716) NIL) ((-90 . -714) NIL) ((-90 . -740) NIL) 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. -588) 6040) ((-77 . -38) 5990) ((-77 . -257) T) ((-77 . -389) T) ((-77 . -146) T) ((-77 . -494) T) ((-77 . -832) T) ((-77 . -1132) T) ((-77 . -311) T) ((-77 . -190) T) ((-77 . -186) 5977) ((-77 . -189) T) ((-77 . -225) 5959) ((-77 . -806) NIL) ((-77 . -811) NIL) ((-77 . -809) NIL) ((-77 . -184) 5941) ((-77 . -120) T) ((-77 . -118) NIL) ((-77 . -104) T) ((-77 . -25) T) ((-77 . -72) T) ((-77 . -13) T) ((-77 . -1127) T) ((-77 . -552) 5884) ((-77 . -1012) T) ((-77 . -23) T) ((-77 . -21) T) ((-77 . -961) T) ((-77 . -663) T) ((-77 . -1059) T) ((-77 . -1024) T) ((-77 . -969) T) ((-73 . -98) 5868) ((-73 . -923) 5852) ((-73 . -34) T) ((-73 . -13) T) ((-73 . -1127) T) ((-73 . -72) 5806) ((-73 . -552) 5741) ((-73 . -259) 5679) ((-73 . -452) 5612) ((-73 . -1012) 5590) ((-73 . -426) 5574) ((-73 . -92) 5558) ((-69 . -410) T) ((-69 . -1024) T) ((-69 . -72) T) ((-69 . -13) T) ((-69 . -1127) T) ((-69 . -552) 5540) ((-69 . -1012) T) ((-69 . -663) T) ((-69 . -241) 5519) ((-67 . -994) T) ((-67 . -427) 5500) ((-67 . -552) 5466) ((-67 . -555) 5447) ((-67 . -1012) T) ((-67 . -1127) T) ((-67 . -13) T) ((-67 . -72) T) ((-67 . -64) T) ((-62 . -1033) 5431) ((-62 . -426) 5415) ((-62 . -1012) 5393) ((-62 . -452) 5326) ((-62 . -259) 5264) ((-62 . -552) 5199) ((-62 . -72) 5153) ((-62 . -1127) T) ((-62 . -13) T) ((-62 . -34) T) ((-62 . -76) 5137) ((-60 . -57) 5099) ((-60 . -34) T) ((-60 . -13) T) ((-60 . -1127) T) ((-60 . -72) 5053) ((-60 . -552) 4988) ((-60 . -259) 4926) ((-60 . -452) 4859) ((-60 . -1012) 4837) ((-60 . -426) 4821) ((-58 . -19) 4805) ((-58 . -593) 4789) ((-58 . -243) 4766) ((-58 . -241) 4718) ((-58 . -538) 4695) ((-58 . -553) 4656) ((-58 . -426) 4640) ((-58 . -1012) 4593) ((-58 . -452) 4526) ((-58 . -259) 4464) ((-58 . -552) 4379) ((-58 . -72) 4313) ((-58 . -1127) T) ((-58 . -13) T) ((-58 . -34) T) ((-58 . -124) 4297) ((-58 . -756) 4276) ((-58 . -759) 4255) ((-58 . -321) 4239) ((-55 . -1012) T) ((-55 . -552) 4221) ((-55 . -1127) T) ((-55 . -13) T) ((-55 . -72) T) ((-55 . -950) 4203) ((-55 . -555) 4185) ((-51 . -1012) T) ((-51 . -552) 4167) ((-51 . -1127) T) ((-51 . -13) T) ((-51 . -72) T) ((-50 . -560) 4151) ((-50 . -555) 4120) ((-50 . -590) 4094) ((-50 . -588) 4053) ((-50 . -969) T) ((-50 . -1024) T) ((-50 . -1059) T) ((-50 . -663) T) ((-50 . -961) T) ((-50 . -21) T) ((-50 . -23) T) ((-50 . -1012) T) ((-50 . -552) 4035) ((-50 . -1127) T) ((-50 . -13) T) ((-50 . -72) T) ((-50 . -25) T) ((-50 . -104) T) ((-50 . -950) 4019) ((-49 . -1012) T) ((-49 . -552) 4001) ((-49 . -1127) T) ((-49 . -13) T) ((-49 . -72) T) ((-48 . -253) T) ((-48 . -72) T) ((-48 . -13) T) ((-48 . -1127) T) ((-48 . -552) 3983) ((-48 . -1012) T) ((-48 . -555) 3884) ((-48 . -950) 3827) ((-48 . -452) 3793) ((-48 . -259) 3780) ((-48 . -27) T) ((-48 . -915) T) ((-48 . -201) T) ((-48 . -82) 3729) ((-48 . -963) 3694) ((-48 . -968) 3659) ((-48 . -245) T) ((-48 . -654) 3624) ((-48 . -582) 3589) ((-48 . -590) 3539) ((-48 . -588) 3489) ((-48 . -104) T) ((-48 . -25) T) ((-48 . -23) T) ((-48 . -21) T) ((-48 . -961) T) ((-48 . -663) T) ((-48 . -1059) T) ((-48 . -1024) T) ((-48 . -969) T) ((-48 . -38) 3454) ((-48 . -257) T) ((-48 . -389) T) ((-48 . -146) T) ((-48 . -494) T) ((-48 . -832) T) ((-48 . -1132) T) ((-48 . -311) T) ((-48 . -580) 3414) ((-48 . -933) T) ((-48 . -553) 3359) ((-48 . -120) T) ((-48 . -190) T) ((-48 . -186) 3346) ((-48 . -189) T) ((-45 . -36) 3325) ((-45 . -538) 3248) ((-45 . -259) 3046) ((-45 . -452) 2798) ((-45 . -426) 2733) ((-45 . -241) 2631) ((-45 . -243) 2554) ((-45 . -549) 2533) ((-45 . -193) 2481) ((-45 . -76) 2429) ((-45 . -183) 2377) ((-45 . -1105) 2356) ((-45 . -237) 2304) ((-45 . -124) 2252) ((-45 . -34) T) ((-45 . -13) T) ((-45 . -1127) T) ((-45 . -72) T) ((-45 . -552) 2234) ((-45 . -1012) T) ((-45 . -553) NIL) ((-45 . -593) 2182) ((-45 . -321) 2130) ((-45 . -759) NIL) ((-45 . -756) NIL) ((-45 . -1062) 2078) ((-45 . -923) 2026) ((-45 . -1166) 1974) ((-45 . -608) 1922) ((-44 . -358) 1906) ((-44 . -683) 1890) ((-44 . -657) T) ((-44 . -685) T) ((-44 . -82) 1869) ((-44 . -963) 1853) ((-44 . -968) 1837) ((-44 . -21) T) ((-44 . -588) 1780) ((-44 . -23) T) ((-44 . -1012) T) ((-44 . -552) 1762) ((-44 . -72) T) ((-44 . -25) T) ((-44 . -104) T) ((-44 . -590) 1720) ((-44 . -582) 1704) ((-44 . -654) 1688) ((-44 . -315) 1672) ((-44 . -1127) T) ((-44 . -13) T) ((-44 . -241) 1649) ((-40 . -290) 1623) ((-40 . -146) T) ((-40 . -555) 1553) ((-40 . -969) T) ((-40 . -1024) T) ((-40 . -1059) T) ((-40 . -663) T) ((-40 . -961) T) ((-40 . -590) 1455) ((-40 . -588) 1385) ((-40 . -104) T) ((-40 . -25) T) ((-40 . -72) T) ((-40 . -13) T) ((-40 . -1127) T) ((-40 . -552) 1367) ((-40 . -1012) T) ((-40 . -23) T) ((-40 . -21) T) ((-40 . -968) 1312) ((-40 . -963) 1257) ((-40 . -82) 1174) ((-40 . -553) 1158) ((-40 . -184) 1135) ((-40 . -809) 1087) ((-40 . -811) 999) ((-40 . -806) 909) ((-40 . -225) 886) ((-40 . -189) 826) ((-40 . -186) 760) ((-40 . -190) 732) ((-40 . -311) T) ((-40 . -1132) T) ((-40 . -832) T) ((-40 . -494) T) ((-40 . -654) 677) ((-40 . -582) 622) ((-40 . -38) 567) ((-40 . -389) T) ((-40 . -257) T) ((-40 . -245) T) ((-40 . -201) T) ((-40 . -317) NIL) ((-40 . -298) NIL) ((-40 . -1064) NIL) ((-40 . -118) 539) ((-40 . -342) NIL) ((-40 . -350) 511) ((-40 . -120) 483) ((-40 . -319) 455) ((-40 . -326) 432) ((-40 . -580) 366) ((-40 . -352) 343) ((-40 . -950) 220) ((-40 . -661) 192) ((-31 . -994) T) ((-31 . -427) 173) ((-31 . -552) 139) ((-31 . -555) 120) ((-31 . -1012) T) ((-31 . -1127) T) ((-31 . -13) T) ((-31 . -72) T) ((-31 . -64) T) ((-30 . -866) T) ((-30 . -552) 102) ((0 . |EnumerationCategory|) T) ((0 . -552) 84) ((0 . -1012) T) ((0 . -72) T) ((0 . -1127) T) ((-2 . |RecordCategory|) T) ((-2 . -552) 66) ((-2 . -1012) T) ((-2 . -72) T) ((-2 . -1127) T) ((-3 . |UnionCategory|) T) ((-3 . -552) 48) ((-3 . -1012) T) ((-3 . -72) T) ((-3 . -1127) T) ((-1 . -1012) T) ((-1 . -552) 30) ((-1 . -1127) T) ((-1 . -13) T) ((-1 . -72) T))
\ No newline at end of file diff --git a/src/share/algebra/compress.daase b/src/share/algebra/compress.daase index cdaf1982..45e17117 100644 --- a/src/share/algebra/compress.daase +++ b/src/share/algebra/compress.daase @@ -1,6 +1,6 @@ -(30 . 3538630436) -(3981 |Enumeration| |Mapping| |Record| |Union| |ofCategory| |isDomain| +(30 . 3539125282) +(3992 |Enumeration| |Mapping| |Record| |Union| |ofCategory| |isDomain| ATTRIBUTE |package| |domain| |category| CATEGORY |nobranch| AND |Join| |ofType| SIGNATURE "failed" "algebra" |OneDimensionalArrayAggregate&| |OneDimensionalArrayAggregate| |AbelianGroup&| |AbelianGroup| |AbelianMonoid&| @@ -21,19 +21,19 @@ |ArcTrigonometricFunctionCategory| |AttributeAst| |AttributeRegistry| |Automorphism| |BalancedFactorisation| |BasicType&| |BasicType| |BalancedBinaryTree| |BezoutMatrix| |BagAggregate&| |BagAggregate| - |BinaryExpansion| |Binding| |Bits| |BiModule| |BooleanLogic&| |BooleanLogic| - |Boolean| |BasicOperator| |BasicOperatorFunctions1| |BoundIntegerRoots| - |BalancedPAdicInteger| |BalancedPAdicRational| |BinaryRecursiveAggregate&| - |BinaryRecursiveAggregate| |BrillhartTests| |BinarySearchTree| |BitAggregate&| - |BitAggregate| |BinaryTreeCategory&| |BinaryTreeCategory| |BinaryTournament| - |BinaryTree| |Byte| |ByteBuffer| |ByteOrder| |CancellationAbelianMonoid| - |CachableSet| |CapsuleAst| |CardinalNumber| |CartesianTensor| - |CartesianTensorFunctions2| |CaseAst| |CategoryAst| |CategoryConstructor| - |Category| |CharacterClass| |CommonDenominator| - |CombinatorialFunctionCategory| |Character| |CharacteristicNonZero| - |CharacteristicPolynomialPackage| |CharacteristicZero| |ChangeOfVariable| - |ComplexIntegerSolveLinearPolynomialEquation| |Collection&| |Collection| - |CliffordAlgebra| |TwoDimensionalPlotClipping| |CollectAst| + |BinaryExpansion| |Binding| |BinaryOperation| |BinaryOperatorCategory| |Bits| + |BiModule| |BooleanLogic&| |BooleanLogic| |Boolean| |BasicOperator| + |BasicOperatorFunctions1| |BoundIntegerRoots| |BalancedPAdicInteger| + |BalancedPAdicRational| |BinaryRecursiveAggregate&| |BinaryRecursiveAggregate| + |BrillhartTests| |BinarySearchTree| |BitAggregate&| |BitAggregate| + |BinaryTreeCategory&| |BinaryTreeCategory| |BinaryTournament| |BinaryTree| + |Byte| |ByteBuffer| |ByteOrder| |CancellationAbelianMonoid| |CachableSet| + |CapsuleAst| |CardinalNumber| |CartesianTensor| |CartesianTensorFunctions2| + |CaseAst| |CategoryAst| |CategoryConstructor| |Category| |CharacterClass| + |CommonDenominator| |CombinatorialFunctionCategory| |Character| + |CharacteristicNonZero| |CharacteristicPolynomialPackage| |CharacteristicZero| + |ChangeOfVariable| |ComplexIntegerSolveLinearPolynomialEquation| |Collection&| + |Collection| |CliffordAlgebra| |TwoDimensionalPlotClipping| |CollectAst| |ComplexRootPackage| |ColonAst| |Color| |CombinatorialFunction| |IntegerCombinatoricFunctions| |CombinatorialOpsCategory| |Commutator| |CommaAst| |CommonOperators| |CommuteUnivariatePolynomialCategory| @@ -141,7 +141,7 @@ |IndexedTwoDimensionalArray| |ChineseRemainderToolsForIntegralBases| |IntegralBasisTools| |IndexedBits| |IntegralBasisPolynomialTools| |IndexCard| |InnerCommonDenominator| |PolynomialIdeals| |IdealDecompositionPackage| - |Identifier| |IndexedDirectProductAbelianGroup| + |IdempotentOperatorCategory| |Identifier| |IndexedDirectProductAbelianGroup| |IndexedDirectProductAbelianMonoid| |IndexedDirectProductCategory| |IndexedDirectProductObject| |IndexedDirectProductOrderedAbelianMonoid| |IndexedDirectProductOrderedAbelianMonoidSup| |InnerEvalable&| |InnerEvalable| @@ -211,18 +211,18 @@ |MultipleMap| |MathMLFormat| |ModularField| |ModMonic| |ModuleMonomial| |ModuleOperator| |ModularRing| |Module&| |Module| |MoebiusTransform| |Monad&| |Monad| |MonadWithUnit&| |MonadWithUnit| |MonogenicAlgebra&| - |MonogenicAlgebra| |Monoid&| |Monoid| |MonomialExtensionTools| - |MPolyCatFunctions2| |MPolyCatFunctions3| |MPolyCatPolyFactorizer| - |MultivariatePolynomial| |MPolyCatRationalFunctionFactorizer| - |MRationalFactorize| |MonoidRingFunctions2| |MonoidRing| |Multiset| - |MultisetAggregate| |MoreSystemCommands| |MergeThing| - |MultivariateTaylorSeriesCategory| |MultivariateFactorize| - |MultivariateSquareFree| |NonAssociativeAlgebra&| |NonAssociativeAlgebra| - |NonAssociativeRng&| |NonAssociativeRng| |NonAssociativeRing&| - |NonAssociativeRing| |NumericComplexEigenPackage| |NumericContinuedFraction| - |NonCommutativeOperatorDivision| |NetworkClientSocket| - |NumberFieldIntegralBasis| |NonLinearSolvePackage| |NonNegativeInteger| - |NonLinearFirstOrderODESolver| |None| |NoneFunctions1| + |MonogenicAlgebra| |Monoid&| |Monoid| |MonoidOperation| + |MonoidOperatorCategory| |MonomialExtensionTools| |MPolyCatFunctions2| + |MPolyCatFunctions3| |MPolyCatPolyFactorizer| |MultivariatePolynomial| + |MPolyCatRationalFunctionFactorizer| |MRationalFactorize| + |MonoidRingFunctions2| |MonoidRing| |Multiset| |MultisetAggregate| + |MoreSystemCommands| |MergeThing| |MultivariateTaylorSeriesCategory| + |MultivariateFactorize| |MultivariateSquareFree| |NonAssociativeAlgebra&| + |NonAssociativeAlgebra| |NonAssociativeRng&| |NonAssociativeRng| + |NonAssociativeRing&| |NonAssociativeRing| |NumericComplexEigenPackage| + |NumericContinuedFraction| |NonCommutativeOperatorDivision| + |NetworkClientSocket| |NumberFieldIntegralBasis| |NonLinearSolvePackage| + |NonNegativeInteger| |NonLinearFirstOrderODESolver| |None| |NoneFunctions1| |NormInMonogenicAlgebra| |NormalizationPackage| |NormRetractPackage| |NPCoef| |NumericRealEigenPackage| |NewSparseMultivariatePolynomial| |NewSparseUnivariatePolynomial| |NewSparseUnivariatePolynomialFunctions2| @@ -330,8 +330,9 @@ |SExpressionCategory| |SExpressionOf| |SquareFreeQuasiComponentPackage| |SquareFreeRegularTriangularSetGcdPackage| |SquareFreeRegularTriangularSetCategory| |SymmetricGroupCombinatoricFunctions| - |SemiGroup&| |SemiGroup| |SplitHomogeneousDirectProduct| |SturmHabichtPackage| - |Signature| |SignatureAst| |ElementaryFunctionSign| |RationalFunctionSign| + |SemiGroupOperation| |SemiGroupOperatorCategory| |SemiGroup&| |SemiGroup| + |SplitHomogeneousDirectProduct| |SturmHabichtPackage| |Signature| + |SignatureAst| |ElementaryFunctionSign| |RationalFunctionSign| |SimplifyAlgebraicNumberConvertPackage| |SingleInteger| |StackAggregate| |SquareMatrixCategory&| |SquareMatrixCategory| |SmithNormalForm| |SparseMultivariatePolynomial| |SparseMultivariateTaylorSeries| @@ -406,9 +407,9 @@ |uncouplingMatrices| |associatedEquations| |arrayStack| |morphism| |balancedFactorisation| |before?| |mapDown!| |mapUp!| |setleaves!| |balancedBinaryTree| |sylvesterMatrix| |bezoutMatrix| |bezoutResultant| - |bezoutDiscriminant| |inspect| |extract!| |bag| |binding| |setProperties| - |setProperty| |deleteProperty!| |has?| |comparison| |equality| |nary?| - |unary?| |nullary?| |properties| |derivative| |constantOperator| + |bezoutDiscriminant| |inspect| |extract!| |bag| |binding| |binaryOperation| + |setProperties| |setProperty| |deleteProperty!| |has?| |comparison| |equality| + |nary?| |unary?| |nullary?| |properties| |derivative| |constantOperator| |constantOpIfCan| |integerBound| |setright!| |setleft!| |brillhartIrreducible?| |brillhartTrials| |noLinearFactor?| |insertRoot!| |binarySearchTree| |nor| |nand| |node| |binaryTournament| |binaryTree| |byte| @@ -625,9 +626,9 @@ |corrPoly| |lifting| |lifting1| |exprex| |coerceL| |coerceS| |frobenius| |computePowers| |pow| |An| |UnVectorise| |Vectorise| |setPoly| |index| |exponent| |exQuo| |moebius| |rightRecip| |leftRecip| |leftPower| |rightPower| - |derivationCoordinates| |generator| |one?| |splitSquarefree| |normalDenom| - |reshape| |totalfract| |pushdterm| |pushucoef| |pushuconst| - |numberOfMonomials| |multiset| |systemCommand| |mergeDifference| + |derivationCoordinates| |generator| |one?| |monoidOperation| |neutralValue| + |splitSquarefree| |normalDenom| |reshape| |totalfract| |pushdterm| |pushucoef| + |pushuconst| |numberOfMonomials| |multiset| |systemCommand| |mergeDifference| |squareFreePrim| |compdegd| |univcase| |consnewpol| |nsqfree| |intChoose| |coefChoose| |myDegree| |normDeriv2| |plenaryPower| |antiCommutator| |commutator| |associator| |complexEigenvalues| |complexEigenvectors| @@ -807,39 +808,39 @@ |stoseSquareFreePart| |coleman| |inverseColeman| |listYoungTableaus| |makeYoungTableau| |nextColeman| |nextLatticePermutation| |nextPartition| |numberOfImproperPartitions| |subSet| |unrankImproperPartitions0| - |unrankImproperPartitions1| |subresultantSequence| |SturmHabichtSequence| - |SturmHabichtCoefficients| |SturmHabicht| |countRealRoots| - |SturmHabichtMultiple| |countRealRootsMultiple| |source| |target| |signature| - |signatureAst| |xor| |depth| |top| |pop!| |push!| |minordet| |determinant| - |diagonalProduct| |trace| |diagonal| |diagonalMatrix| |scalarMatrix| |hermite| - |completeHermite| |smith| |completeSmith| |diophantineSystem| |csubst| - |particularSolution| |mapSolve| |linear| |quadratic| |cubic| |quartic| - |aLinear| |aQuadratic| |aCubic| |aQuartic| |radicalSolve| |radicalRoots| - |contractSolve| |decomposeFunc| |unvectorise| |bubbleSort!| |insertionSort!| - |check| |objects| |lprop| |llprop| |lllp| |lllip| |lp| |mesh?| |mesh| - |polygon?| |polygon| |closedCurve?| |closedCurve| |curve?| |curve| |point?| - |enterPointData| |composites| |components| |numberOfComposites| - |numberOfComponents| |create3Space| |parse| |outputAsFortran| |outputAsScript| - |outputAsTex| |abs| |Beta| |digamma| |polygamma| |Gamma| |besselJ| |besselY| - |besselI| |besselK| |airyAi| |airyBi| |subNode?| |infLex?| |setEmpty!| - |setStatus!| |setCondition!| |setValue!| |copy| |status| |value| |empty?| - |splitNodeOf!| |remove!| |remove| |subNodeOf?| |nodeOf?| |result| |conditions| - |updateStatus!| |extractSplittingLeaf| |squareMatrix| |transpose| |rightTrim| - |leftTrim| |trim| |split| |position| |replace| |match?| |match| |substring?| - |suffix?| |prefix?| |upperCase!| |upperCase| |lowerCase!| |lowerCase| - |KrullNumber| |numberOfVariables| |algebraicDecompose| - |transcendentalDecompose| |internalDecompose| |decompose| |upDateBranches| - |printInfo| |preprocess| |internalZeroSetSplit| |internalAugment| |stack| - |size?| |possiblyInfinite?| |explicitlyFinite?| |nextItem| |init| |step| - |upperBound| |lowerBound| |iterationVar| |infiniteProduct| - |evenInfiniteProduct| |oddInfiniteProduct| |generalInfiniteProduct| - |filterUntil| |filterWhile| |generate| |showAll?| |showAllElements| |output| - |cons| |delay| |findCycle| |repeating?| |repeating| |exquo| |recip| |integers| - |oddintegers| |int| |mapmult| |deriv| |gderiv| |compose| |addiag| - |lazyIntegrate| |nlde| |powern| |mapdiv| |lazyGintegrate| |power| |sincos| - |sinhcosh| |asin| |acos| |atan| |acot| |asec| |acsc| |sinh| |cosh| |tanh| - |coth| |sech| |csch| |asinh| |acosh| |atanh| |acoth| |asech| |acsch| - |subresultantVector| |primitivePart| |pointData| |parent| |level| + |unrankImproperPartitions1| |semiGroupOperation| |subresultantSequence| + |SturmHabichtSequence| |SturmHabichtCoefficients| |SturmHabicht| + |countRealRoots| |SturmHabichtMultiple| |countRealRootsMultiple| |source| + |target| |signature| |signatureAst| |xor| |depth| |top| |pop!| |push!| + |minordet| |determinant| |diagonalProduct| |trace| |diagonal| |diagonalMatrix| + |scalarMatrix| |hermite| |completeHermite| |smith| |completeSmith| + |diophantineSystem| |csubst| |particularSolution| |mapSolve| |linear| + |quadratic| |cubic| |quartic| |aLinear| |aQuadratic| |aCubic| |aQuartic| + |radicalSolve| |radicalRoots| |contractSolve| |decomposeFunc| |unvectorise| + |bubbleSort!| |insertionSort!| |check| |objects| |lprop| |llprop| |lllp| + |lllip| |lp| |mesh?| |mesh| |polygon?| |polygon| |closedCurve?| |closedCurve| + |curve?| |curve| |point?| |enterPointData| |composites| |components| + |numberOfComposites| |numberOfComponents| |create3Space| |parse| + |outputAsFortran| |outputAsScript| |outputAsTex| |abs| |Beta| |digamma| + |polygamma| |Gamma| |besselJ| |besselY| |besselI| |besselK| |airyAi| |airyBi| + |subNode?| |infLex?| |setEmpty!| |setStatus!| |setCondition!| |setValue!| + |copy| |status| |value| |empty?| |splitNodeOf!| |remove!| |remove| + |subNodeOf?| |nodeOf?| |result| |conditions| |updateStatus!| + |extractSplittingLeaf| |squareMatrix| |transpose| |rightTrim| |leftTrim| + |trim| |split| |position| |replace| |match?| |match| |substring?| |suffix?| + |prefix?| |upperCase!| |upperCase| |lowerCase!| |lowerCase| |KrullNumber| + |numberOfVariables| |algebraicDecompose| |transcendentalDecompose| + |internalDecompose| |decompose| |upDateBranches| |printInfo| |preprocess| + |internalZeroSetSplit| |internalAugment| |stack| |size?| |possiblyInfinite?| + |explicitlyFinite?| |nextItem| |init| |step| |upperBound| |lowerBound| + |iterationVar| |infiniteProduct| |evenInfiniteProduct| |oddInfiniteProduct| + |generalInfiniteProduct| |filterUntil| |filterWhile| |generate| |showAll?| + |showAllElements| |output| |cons| |delay| |findCycle| |repeating?| |repeating| + |exquo| |recip| |integers| |oddintegers| |int| |mapmult| |deriv| |gderiv| + |compose| |addiag| |lazyIntegrate| |nlde| |powern| |mapdiv| |lazyGintegrate| + |power| |sincos| |sinhcosh| |asin| |acos| |atan| |acot| |asec| |acsc| |sinh| + |cosh| |tanh| |coth| |sech| |csch| |asinh| |acosh| |atanh| |acoth| |asech| + |acsch| |subresultantVector| |primitivePart| |pointData| |parent| |level| |extractProperty| |extractClosed| |extractIndex| |extractPoint| |traverse| |defineProperty| |closeComponent| |modifyPoint| |addPointLast| |addPoint2| |addPoint| |merge| |deepCopy| |shallowCopy| |numberOfChildren| |children| diff --git a/src/share/algebra/interp.daase b/src/share/algebra/interp.daase index f2d1886a..02d7945e 100644 --- a/src/share/algebra/interp.daase +++ b/src/share/algebra/interp.daase @@ -1,4010 +1,4039 @@ -(2801385 . 3538630445) -((-1721 (((-83) (-1 (-83) |#2| |#2|) $) 86 T ELT) 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T) ((-72) OR (|has| |#2| (-1007)) (|has| |#2| (-956)) (|has| |#2| (-751)) (|has| |#2| (-712)) (|has| |#2| (-660)) (|has| |#2| (-315)) (|has| |#2| (-309)) (|has| |#2| (-144)) (|has| |#2| (-102)) (|has| |#2| (-72)) (|has| |#2| (-25)) (|has| |#2| (-23)) (|has| |#2| (-21))) ((-80 |#2| |#2|) OR (|has| |#2| (-956)) (|has| |#2| (-309)) (|has| |#2| (-144))) ((-102) OR (|has| |#2| (-956)) (|has| |#2| (-712)) (|has| |#2| (-309)) (|has| |#2| (-144)) (|has| |#2| (-102)) (|has| |#2| (-21))) ((-552 (-345 (-480))) -12 (|has| |#2| (-945 (-345 (-480)))) (|has| |#2| (-1007))) ((-552 (-480)) OR (|has| |#2| (-956)) (-12 (|has| |#2| (-945 (-480))) (|has| |#2| (-1007)))) ((-552 |#2|) |has| |#2| (-1007)) ((-549 (-767)) OR (|has| |#2| (-1007)) (|has| |#2| (-956)) (|has| |#2| (-751)) (|has| |#2| (-712)) (|has| |#2| (-660)) (|has| |#2| (-315)) (|has| |#2| (-309)) (|has| |#2| (-144)) (|has| |#2| (-549 (-767))) (|has| |#2| (-102)) (|has| |#2| (-25)) (|has| |#2| (-23)) (|has| |#2| (-21))) ((-549 (-1170 |#2|)) . T) ((-184 $) OR (-12 (|has| |#2| (-187)) (|has| |#2| (-956))) (-12 (|has| |#2| (-188)) (|has| |#2| (-956)))) ((-182 |#2|) |has| |#2| (-956)) ((-188) -12 (|has| |#2| (-188)) (|has| |#2| (-956))) ((-187) OR (-12 (|has| |#2| (-187)) (|has| |#2| (-956))) (-12 (|has| |#2| (-188)) (|has| |#2| (-956)))) ((-223 |#2|) |has| |#2| (-956)) ((-239 (-480) |#2|) . T) ((-241 (-480) |#2|) . T) ((-257 |#2|) -12 (|has| |#2| (-257 |#2|)) (|has| |#2| (-1007))) ((-315) |has| |#2| (-315)) ((-324 |#2|) |has| |#2| (-956)) ((-350 |#2|) |has| |#2| (-1007)) ((-424 |#2|) . T) ((-535 (-480) |#2|) . T) ((-449 |#2| |#2|) -12 (|has| |#2| (-257 |#2|)) (|has| |#2| (-1007))) ((-13) . T) ((-585 (-480)) OR (|has| |#2| (-956)) (|has| |#2| (-309)) (|has| |#2| (-144)) (|has| |#2| (-21))) ((-585 |#2|) OR (|has| |#2| (-956)) (|has| |#2| (-660)) (|has| |#2| (-309)) (|has| |#2| (-144))) ((-585 $) |has| |#2| (-956)) ((-587 (-480)) -12 (|has| |#2| (-577 (-480))) (|has| |#2| (-956))) ((-587 |#2|) OR (|has| |#2| (-956)) (|has| |#2| (-309)) (|has| |#2| (-144))) ((-587 $) |has| |#2| (-956)) ((-579 |#2|) OR (|has| |#2| (-660)) (|has| |#2| (-309)) (|has| |#2| (-144))) ((-577 (-480)) -12 (|has| |#2| (-577 (-480))) (|has| |#2| (-956))) ((-577 |#2|) |has| |#2| (-956)) ((-651 |#2|) OR (|has| |#2| (-309)) (|has| |#2| (-144))) ((-660) |has| |#2| (-956)) ((-711) |has| |#2| (-712)) ((-712) |has| |#2| (-712)) ((-713) |has| |#2| (-712)) ((-716) |has| |#2| (-712)) ((-751) OR (|has| |#2| (-751)) (|has| |#2| (-712))) ((-754) OR (|has| |#2| (-751)) (|has| |#2| (-712))) ((-801 $ (-1081)) OR (-12 (|has| |#2| (-806 (-1081))) (|has| |#2| (-956))) (-12 (|has| |#2| (-804 (-1081))) (|has| |#2| (-956)))) ((-804 (-1081)) -12 (|has| |#2| (-804 (-1081))) (|has| |#2| (-956))) ((-806 (-1081)) OR (-12 (|has| |#2| (-806 (-1081))) (|has| |#2| (-956))) (-12 (|has| |#2| (-804 (-1081))) (|has| |#2| (-956)))) ((-945 (-345 (-480))) -12 (|has| |#2| (-945 (-345 (-480)))) (|has| |#2| (-1007))) ((-945 (-480)) -12 (|has| |#2| (-945 (-480))) (|has| |#2| (-1007))) ((-945 |#2|) |has| |#2| (-1007)) ((-958 |#2|) OR (|has| |#2| (-956)) (|has| |#2| (-660)) (|has| |#2| (-309)) (|has| |#2| (-144))) ((-963 |#2|) OR (|has| |#2| (-956)) (|has| |#2| (-309)) (|has| |#2| (-144))) ((-956) |has| |#2| (-956)) ((-964) |has| |#2| (-956)) ((-1017) |has| |#2| (-956)) ((-1052) |has| |#2| (-956)) ((-1007) OR (|has| |#2| (-1007)) (|has| |#2| (-956)) (|has| |#2| (-751)) (|has| |#2| (-712)) (|has| |#2| (-660)) (|has| |#2| (-315)) (|has| |#2| (-309)) (|has| |#2| (-144)) (|has| |#2| (-102)) (|has| |#2| (-25)) (|has| |#2| (-23)) (|has| |#2| (-21))) ((-1120) . T) ((-1178 |#2|) |has| |#2| (-309))) -((-2554 (((-83) $ $) NIL (|has| |#2| (-72)) ELT)) (-3173 (((-83) $) NIL (|has| |#2| (-23)) ELT)) (-3690 (($ (-825)) 63 (|has| |#2| (-956)) ELT)) (-2186 (((-1176) $ (-480) (-480)) NIL (|has| $ (-6 -3979)) ELT)) (-2469 (($ $ $) 69 (|has| |#2| (-712)) ELT)) (-1301 (((-3 $ #1="failed") $ $) 54 (|has| |#2| (-102)) ELT)) (-3121 (((-689)) NIL (|has| |#2| (-315)) ELT)) (-3771 ((|#2| $ (-480) |#2|) NIL (|has| $ (-6 -3979)) ELT)) (-3707 (($) NIL T CONST)) (-3142 (((-3 (-480) #1#) $) NIL (-12 (|has| |#2| (-945 (-480))) (|has| |#2| (-1007))) ELT) (((-3 (-345 (-480)) #1#) $) NIL (-12 (|has| |#2| (-945 (-345 (-480)))) (|has| |#2| (-1007))) ELT) (((-3 |#2| #1#) $) 31 (|has| |#2| (-1007)) ELT)) (-3141 (((-480) $) NIL (-12 (|has| |#2| (-945 (-480))) (|has| |#2| (-1007))) ELT) (((-345 (-480)) $) NIL (-12 (|has| |#2| (-945 (-345 (-480)))) (|has| |#2| (-1007))) ELT) ((|#2| $) 29 (|has| |#2| (-1007)) ELT)) (-2267 (((-627 (-480)) (-627 $)) NIL (-12 (|has| |#2| (-577 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T) ((-72) OR (|has| |#2| (-1012)) (|has| |#2| (-961)) (|has| |#2| (-756)) (|has| |#2| (-717)) (|has| |#2| (-663)) (|has| |#2| (-317)) (|has| |#2| (-311)) (|has| |#2| (-146)) (|has| |#2| (-104)) (|has| |#2| (-72)) (|has| |#2| (-25)) (|has| |#2| (-23)) (|has| |#2| (-21))) ((-82 |#2| |#2|) OR (|has| |#2| (-961)) (|has| |#2| (-311)) (|has| |#2| (-146))) ((-104) OR (|has| |#2| (-961)) (|has| |#2| (-717)) (|has| |#2| (-311)) (|has| |#2| (-146)) (|has| |#2| (-104)) (|has| |#2| (-21))) ((-555 (-347 (-483))) -12 (|has| |#2| (-950 (-347 (-483)))) (|has| |#2| (-1012))) ((-555 (-483)) OR (|has| |#2| (-961)) (-12 (|has| |#2| (-950 (-483))) (|has| |#2| (-1012)))) ((-555 |#2|) |has| |#2| (-1012)) ((-552 (-772)) OR (|has| |#2| (-1012)) (|has| |#2| (-961)) (|has| |#2| (-756)) (|has| |#2| (-717)) (|has| |#2| (-663)) (|has| |#2| (-317)) (|has| |#2| (-311)) (|has| |#2| (-146)) (|has| |#2| (-552 (-772))) (|has| |#2| (-104)) (|has| |#2| (-25)) (|has| |#2| (-23)) (|has| |#2| (-21))) ((-552 (-1177 |#2|)) . T) ((-186 $) OR (-12 (|has| |#2| (-189)) (|has| |#2| (-961))) (-12 (|has| |#2| (-190)) (|has| |#2| (-961)))) ((-184 |#2|) |has| |#2| (-961)) ((-190) -12 (|has| |#2| (-190)) (|has| |#2| (-961))) ((-189) OR (-12 (|has| |#2| (-189)) (|has| |#2| (-961))) (-12 (|has| |#2| (-190)) (|has| |#2| (-961)))) ((-225 |#2|) |has| |#2| (-961)) ((-241 (-483) |#2|) . T) ((-243 (-483) |#2|) . T) ((-259 |#2|) -12 (|has| |#2| (-259 |#2|)) (|has| |#2| (-1012))) ((-317) |has| |#2| (-317)) ((-326 |#2|) |has| |#2| (-961)) ((-352 |#2|) |has| |#2| (-1012)) ((-426 |#2|) . T) ((-538 (-483) |#2|) . T) ((-452 |#2| |#2|) -12 (|has| |#2| (-259 |#2|)) (|has| |#2| (-1012))) ((-13) . T) ((-588 (-483)) OR (|has| |#2| (-961)) (|has| |#2| (-311)) (|has| |#2| (-146)) (|has| |#2| (-21))) ((-588 |#2|) OR (|has| |#2| (-961)) (|has| |#2| (-663)) (|has| |#2| (-311)) (|has| |#2| (-146))) ((-588 $) |has| |#2| (-961)) ((-590 (-483)) -12 (|has| |#2| (-580 (-483))) (|has| |#2| (-961))) ((-590 |#2|) OR (|has| |#2| (-961)) (|has| |#2| (-311)) (|has| |#2| (-146))) ((-590 $) |has| |#2| (-961)) ((-582 |#2|) OR (|has| |#2| (-663)) (|has| |#2| (-311)) (|has| |#2| (-146))) ((-580 (-483)) -12 (|has| |#2| (-580 (-483))) (|has| |#2| (-961))) ((-580 |#2|) |has| |#2| (-961)) ((-654 |#2|) OR (|has| |#2| (-311)) (|has| |#2| (-146))) ((-663) |has| |#2| (-961)) ((-716) |has| |#2| (-717)) ((-717) |has| |#2| (-717)) ((-718) |has| |#2| (-717)) ((-721) |has| |#2| (-717)) ((-756) OR (|has| |#2| (-756)) (|has| |#2| (-717))) ((-759) OR (|has| |#2| (-756)) (|has| |#2| (-717))) ((-806 $ (-1088)) OR (-12 (|has| |#2| (-811 (-1088))) (|has| |#2| (-961))) (-12 (|has| |#2| (-809 (-1088))) (|has| |#2| (-961)))) ((-809 (-1088)) -12 (|has| |#2| (-809 (-1088))) (|has| |#2| (-961))) ((-811 (-1088)) OR (-12 (|has| |#2| (-811 (-1088))) (|has| |#2| (-961))) (-12 (|has| |#2| (-809 (-1088))) (|has| |#2| (-961)))) ((-950 (-347 (-483))) -12 (|has| |#2| (-950 (-347 (-483)))) (|has| |#2| (-1012))) ((-950 (-483)) -12 (|has| |#2| (-950 (-483))) (|has| |#2| (-1012))) ((-950 |#2|) |has| |#2| (-1012)) ((-963 |#2|) OR (|has| |#2| (-961)) (|has| |#2| (-663)) (|has| |#2| (-311)) (|has| |#2| (-146))) ((-968 |#2|) OR (|has| |#2| (-961)) (|has| |#2| (-311)) (|has| |#2| (-146))) ((-961) |has| |#2| (-961)) ((-969) |has| |#2| (-961)) ((-1024) |has| |#2| (-961)) ((-1059) |has| |#2| (-961)) ((-1012) OR (|has| |#2| (-1012)) (|has| |#2| (-961)) (|has| |#2| (-756)) (|has| |#2| (-717)) (|has| |#2| (-663)) (|has| |#2| (-317)) (|has| |#2| (-311)) (|has| |#2| (-146)) (|has| |#2| (-104)) (|has| |#2| (-25)) (|has| |#2| (-23)) (|has| |#2| (-21))) ((-1127) . 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T) ((-587 (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-587 (-480)) |has| |#1| (-577 (-480))) ((-587 |#1|) . T) ((-587 $) . T) ((-579 (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-579 |#1|) |has| |#1| (-144)) ((-579 $) OR (|has| |#1| (-816)) (|has| |#1| (-491)) (|has| |#1| (-387))) ((-577 (-480)) |has| |#1| (-577 (-480))) ((-577 |#1|) . T) ((-651 (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-651 |#1|) |has| |#1| (-144)) ((-651 $) OR (|has| |#1| (-816)) (|has| |#1| (-491)) (|has| |#1| (-387))) ((-660) . T) ((-801 $ |#3|) . T) ((-804 |#3|) . T) ((-806 |#3|) . T) ((-791 (-325)) -12 (|has| |#1| (-791 (-325))) (|has| |#3| (-791 (-325)))) ((-791 (-480)) -12 (|has| |#1| (-791 (-480))) (|has| |#3| (-791 (-480)))) ((-856 |#1| |#2| |#3|) . T) ((-816) |has| |#1| (-816)) ((-945 (-345 (-480))) |has| |#1| (-945 (-345 (-480)))) ((-945 (-480)) |has| |#1| (-945 (-480))) ((-945 |#1|) . T) ((-945 |#3|) . T) ((-958 (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-958 |#1|) . T) ((-958 $) OR (|has| |#1| (-816)) (|has| |#1| (-491)) (|has| |#1| (-387)) (|has| |#1| (-144))) ((-963 (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-963 |#1|) . T) ((-963 $) OR (|has| |#1| (-816)) (|has| |#1| (-491)) (|has| |#1| (-387)) (|has| |#1| (-144))) ((-956) . T) ((-964) . T) ((-1017) . T) ((-1052) . T) ((-1007) . T) ((-1120) . T) ((-1125) |has| |#1| (-816))) -((-2554 (((-83) $ $) NIL T ELT)) (-3227 (((-1064) $) NIL T ELT)) (-3166 (((-580 (-1040)) $) 18 T ELT)) (-3228 (((-1025) $) NIL T ELT)) (-3929 (((-767) $) 27 T ELT) (($ (-1086)) NIL T ELT) (((-1086) $) NIL T ELT)) (-3218 (((-1040) $) 20 T ELT)) (-1255 (((-83) $ $) NIL T ELT)) (-3042 (((-83) $ $) NIL T ELT))) -(((-972) (-13 (-989) (-10 -8 (-15 -3166 ((-580 (-1040)) $)) (-15 -3218 ((-1040) $))))) (T -972)) -((-3166 (*1 *2 *1) (-12 (-5 *2 (-580 (-1040))) (-5 *1 (-972)))) (-3218 (*1 *2 *1) (-12 (-5 *2 (-1040)) (-5 *1 (-972))))) -((-3173 (((-83) |#3| $) 15 T ELT)) (-3168 (((-3 $ #1="failed") |#3| (-825)) 29 T ELT)) (-3450 (((-3 |#3| #1#) |#3| $) 45 T ELT)) (-3171 (((-83) |#3| $) 19 T ELT)) (-3172 (((-83) |#3| $) 17 T ELT))) -(((-973 |#1| |#2| |#3|) (-10 -7 (-15 -3168 ((-3 |#1| #1="failed") |#3| (-825))) (-15 -3450 ((-3 |#3| #1#) |#3| |#1|)) (-15 -3171 ((-83) |#3| |#1|)) (-15 -3172 ((-83) |#3| |#1|)) (-15 -3173 ((-83) |#3| |#1|))) (-974 |#2| |#3|) (-13 (-750) (-309)) (-1146 |#2|)) (T -973)) -NIL -((-2554 (((-83) $ $) 7 T ELT)) (-3173 (((-83) |#2| $) 25 T ELT)) (-3606 (((-480) |#2| $) 26 T ELT)) (-3168 (((-3 $ "failed") |#2| (-825)) 19 T ELT)) (-3167 ((|#1| |#2| $ |#1|) 17 T ELT)) (-3450 (((-3 |#2| "failed") |#2| $) 22 T ELT)) (-3171 (((-83) |#2| $) 23 T ELT)) (-3172 (((-83) |#2| $) 24 T ELT)) (-3227 (((-1064) $) 11 T ELT)) (-3228 (((-1025) $) 12 T ELT)) (-3170 ((|#2| $) 21 T ELT)) (-3929 (((-767) $) 13 T ELT)) (-1255 (((-83) $ $) 6 T ELT)) (-3753 ((|#1| |#2| $ |#1|) 18 T ELT)) (-3169 (((-580 $) |#2|) 20 T ELT)) (-3042 (((-83) $ $) 8 T ELT))) -(((-974 |#1| |#2|) (-111) (-13 (-750) (-309)) (-1146 |t#1|)) (T -974)) -((-3606 (*1 *2 *3 *1) (-12 (-4 *1 (-974 *4 *3)) (-4 *4 (-13 (-750) (-309))) (-4 *3 (-1146 *4)) (-5 *2 (-480)))) (-3173 (*1 *2 *3 *1) (-12 (-4 *1 (-974 *4 *3)) (-4 *4 (-13 (-750) (-309))) (-4 *3 (-1146 *4)) (-5 *2 (-83)))) (-3172 (*1 *2 *3 *1) (-12 (-4 *1 (-974 *4 *3)) (-4 *4 (-13 (-750) (-309))) (-4 *3 (-1146 *4)) (-5 *2 (-83)))) (-3171 (*1 *2 *3 *1) (-12 (-4 *1 (-974 *4 *3)) (-4 *4 (-13 (-750) (-309))) (-4 *3 (-1146 *4)) (-5 *2 (-83)))) (-3450 (*1 *2 *2 *1) (|partial| -12 (-4 *1 (-974 *3 *2)) (-4 *3 (-13 (-750) (-309))) (-4 *2 (-1146 *3)))) (-3170 (*1 *2 *1) (-12 (-4 *1 (-974 *3 *2)) (-4 *3 (-13 (-750) (-309))) (-4 *2 (-1146 *3)))) (-3169 (*1 *2 *3) (-12 (-4 *4 (-13 (-750) (-309))) (-4 *3 (-1146 *4)) (-5 *2 (-580 *1)) (-4 *1 (-974 *4 *3)))) (-3168 (*1 *1 *2 *3) (|partial| -12 (-5 *3 (-825)) (-4 *4 (-13 (-750) (-309))) (-4 *1 (-974 *4 *2)) (-4 *2 (-1146 *4)))) (-3753 (*1 *2 *3 *1 *2) (-12 (-4 *1 (-974 *2 *3)) (-4 *2 (-13 (-750) (-309))) (-4 *3 (-1146 *2)))) (-3167 (*1 *2 *3 *1 *2) (-12 (-4 *1 (-974 *2 *3)) (-4 *2 (-13 (-750) (-309))) (-4 *3 (-1146 *2))))) -(-13 (-1007) (-10 -8 (-15 -3606 ((-480) |t#2| $)) (-15 -3173 ((-83) |t#2| $)) (-15 -3172 ((-83) |t#2| $)) (-15 -3171 ((-83) |t#2| $)) (-15 -3450 ((-3 |t#2| "failed") |t#2| $)) (-15 -3170 (|t#2| $)) (-15 -3169 ((-580 $) |t#2|)) (-15 -3168 ((-3 $ "failed") |t#2| (-825))) (-15 -3753 (|t#1| |t#2| $ |t#1|)) (-15 -3167 (|t#1| |t#2| $ |t#1|)))) -(((-72) . 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T) ((-23) . T) ((-47 |#1| (-689)) . T) ((-25) . T) ((-38 (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-38 |#1|) |has| |#1| (-144)) ((-38 $) |has| |#1| (-491)) ((-35) |has| |#1| (-38 (-345 (-480)))) ((-66) |has| |#1| (-38 (-345 (-480)))) ((-72) . T) ((-80 (-345 (-480)) (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-80 |#1| |#1|) . T) ((-80 $ $) OR (|has| |#1| (-491)) (|has| |#1| (-144))) ((-102) . T) ((-116) |has| |#1| (-116)) ((-118) |has| |#1| (-118)) ((-552 (-345 (-480))) |has| |#1| (-38 (-345 (-480)))) ((-552 (-480)) . T) ((-552 |#1|) |has| |#1| (-144)) ((-552 $) |has| |#1| (-491)) ((-549 (-767)) . T) ((-144) OR (|has| |#1| (-491)) (|has| |#1| (-144))) ((-184 $) |has| |#1| (-15 * (|#1| (-689) |#1|))) ((-188) |has| |#1| (-15 * (|#1| (-689) |#1|))) ((-187) |has| |#1| (-15 * (|#1| (-689) |#1|))) ((-237) |has| |#1| (-38 (-345 (-480)))) ((-239 (-689) |#1|) . 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-8 NIL NIL NIL) (-1169 2714921 2719529 2719572 "VECTCAT" 2720560 VECTCAT (NIL T) -9 NIL 2721144 NIL) (-1168 2714200 2714526 2714916 "VECTCAT-" NIL VECTCAT- (NIL T T) -7 NIL NIL NIL) (-1167 2713694 2713936 2714056 "VARIABLE" NIL VARIABLE (NIL NIL) -8 NIL NIL NIL) (-1166 2713627 2713632 2713662 "UTYPE" 2713667 UTYPE (NIL) -9 NIL NIL NIL) (-1165 2712614 2712790 2713051 "UTSODETL" NIL UTSODETL (NIL T T T T) -7 NIL NIL NIL) (-1164 2710465 2710973 2711497 "UTSODE" NIL UTSODE (NIL T T) -7 NIL NIL NIL) (-1163 2700409 2706317 2706359 "UTSCAT" 2707457 UTSCAT (NIL T) -9 NIL 2708214 NIL) (-1162 2698474 2699417 2700404 "UTSCAT-" NIL UTSCAT- (NIL T T) -7 NIL NIL NIL) (-1161 2698148 2698197 2698328 "UTS2" NIL UTS2 (NIL T T T T) -7 NIL NIL NIL) (-1160 2689923 2696344 2696823 "UTS" NIL UTS (NIL T NIL NIL) -8 NIL NIL NIL) (-1159 2683918 2686731 2686774 "URAGG" 2688844 URAGG (NIL T) -9 NIL 2689566 NIL) (-1158 2681933 2682895 2683913 "URAGG-" NIL URAGG- (NIL T T) -7 NIL NIL NIL) (-1157 2677704 2680909 2681371 "UPXSSING" NIL UPXSSING (NIL T T NIL NIL) -8 NIL NIL NIL) (-1156 2670197 2677628 2677699 "UPXSCONS" NIL UPXSCONS (NIL T T) -8 NIL NIL NIL) (-1155 2658910 2666335 2666396 "UPXSCCA" 2666964 UPXSCCA (NIL T T) -9 NIL 2667196 NIL) (-1154 2658631 2658733 2658905 "UPXSCCA-" NIL UPXSCCA- (NIL T T T) -7 NIL NIL NIL) (-1153 2647245 2654395 2654437 "UPXSCAT" 2655077 UPXSCAT (NIL T) -9 NIL 2655685 NIL) (-1152 2646758 2646843 2647020 "UPXS2" NIL UPXS2 (NIL T T NIL NIL NIL NIL) -7 NIL NIL NIL) (-1151 2638508 2646349 2646611 "UPXS" NIL UPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-1150 2637403 2637673 2638023 "UPSQFREE" NIL UPSQFREE (NIL T T) -7 NIL NIL NIL) (-1149 2630168 2633591 2633645 "UPSCAT" 2634714 UPSCAT (NIL T T) -9 NIL 2635478 NIL) (-1148 2629588 2629840 2630163 "UPSCAT-" NIL UPSCAT- (NIL T T T) -7 NIL NIL NIL) (-1147 2629262 2629311 2629442 "UPOLYC2" NIL UPOLYC2 (NIL T T T T) -7 NIL NIL NIL) (-1146 2613456 2622346 2622388 "UPOLYC" 2624466 UPOLYC (NIL T) -9 NIL 2625686 NIL) (-1145 2607511 2610359 2613451 "UPOLYC-" NIL UPOLYC- (NIL T T) -7 NIL NIL NIL) (-1144 2606947 2607072 2607235 "UPMP" NIL UPMP (NIL T T) -7 NIL NIL NIL) (-1143 2606581 2606668 2606807 "UPDIVP" NIL UPDIVP (NIL T T) -7 NIL NIL NIL) (-1142 2605394 2605661 2605965 "UPDECOMP" NIL UPDECOMP (NIL T T) -7 NIL NIL NIL) (-1141 2604727 2604857 2605042 "UPCDEN" NIL UPCDEN (NIL T T T) -7 NIL NIL NIL) (-1140 2604319 2604394 2604541 "UP2" NIL UP2 (NIL NIL T NIL T) -7 NIL NIL NIL) (-1139 2595147 2604085 2604213 "UP" NIL UP (NIL NIL T) -8 NIL NIL NIL) (-1138 2594509 2594646 2594851 "UNISEG2" NIL UNISEG2 (NIL T T) -7 NIL NIL NIL) (-1137 2593113 2593959 2594234 "UNISEG" NIL UNISEG (NIL T) -8 NIL NIL NIL) (-1136 2592342 2592539 2592764 "UNIFACT" NIL UNIFACT (NIL T) -7 NIL NIL NIL) (-1135 2579216 2592266 2592337 "ULSCONS" NIL ULSCONS (NIL T T) -8 NIL NIL NIL) (-1134 2559130 2572303 2572364 "ULSCCAT" 2572995 ULSCCAT (NIL T T) -9 NIL 2573282 NIL) (-1133 2558465 2558751 2559125 "ULSCCAT-" NIL ULSCCAT- (NIL T T T) -7 NIL NIL NIL) (-1132 2546899 2553971 2554013 "ULSCAT" 2554866 ULSCAT (NIL T) -9 NIL 2555596 NIL) (-1131 2546412 2546497 2546674 "ULS2" NIL ULS2 (NIL T T NIL NIL NIL NIL) -7 NIL NIL NIL) (-1130 2528593 2545911 2546152 "ULS" NIL ULS (NIL T NIL NIL) -8 NIL NIL NIL) (-1129 2527627 2528320 2528434 "UINT8" NIL UINT8 (NIL) -8 NIL NIL 2528545) (-1128 2526660 2527353 2527467 "UINT64" NIL UINT64 (NIL) -8 NIL NIL 2527578) (-1127 2525693 2526386 2526500 "UINT32" NIL UINT32 (NIL) -8 NIL NIL 2526611) (-1126 2524726 2525419 2525533 "UINT16" NIL UINT16 (NIL) -8 NIL NIL 2525644) (-1125 2522795 2523954 2523984 "UFD" 2524195 UFD (NIL) -9 NIL 2524308 NIL) (-1124 2522639 2522696 2522790 "UFD-" NIL UFD- (NIL T) -7 NIL NIL NIL) (-1123 2521891 2522098 2522314 "UDVO" NIL UDVO (NIL) -7 NIL NIL NIL) (-1122 2520111 2520564 2521029 "UDPO" NIL UDPO (NIL T) -7 NIL NIL NIL) (-1121 2519836 2520076 2520106 "TYPEAST" NIL TYPEAST (NIL) -8 NIL NIL NIL) (-1120 2519774 2519779 2519809 "TYPE" 2519814 TYPE (NIL) -9 NIL 2519821 NIL) (-1119 2518933 2519153 2519393 "TWOFACT" NIL TWOFACT (NIL T) -7 NIL NIL NIL) (-1118 2518111 2518542 2518777 "TUPLE" NIL TUPLE (NIL T) -8 NIL NIL NIL) (-1117 2516265 2516838 2517377 "TUBETOOL" NIL TUBETOOL (NIL) -7 NIL NIL NIL) (-1116 2515299 2515535 2515771 "TUBE" NIL TUBE (NIL T) -8 NIL NIL NIL) (-1115 2503653 2508121 2508217 "TSETCAT" 2513432 TSETCAT (NIL T T T T) -9 NIL 2514944 NIL) (-1114 2499990 2501806 2503648 "TSETCAT-" NIL TSETCAT- (NIL T T T T T) -7 NIL NIL NIL) (-1113 2494446 2499216 2499498 "TS" NIL TS (NIL T) -8 NIL NIL NIL) (-1112 2489783 2490796 2491725 "TRMANIP" NIL TRMANIP (NIL T T) -7 NIL NIL NIL) (-1111 2489280 2489355 2489518 "TRIMAT" NIL TRIMAT (NIL T T T T) -7 NIL NIL NIL) (-1110 2487356 2487646 2488001 "TRIGMNIP" NIL TRIGMNIP (NIL T T) -7 NIL NIL NIL) (-1109 2486840 2486989 2487019 "TRIGCAT" 2487232 TRIGCAT (NIL) -9 NIL NIL NIL) (-1108 2486591 2486694 2486835 "TRIGCAT-" NIL TRIGCAT- (NIL T) -7 NIL NIL NIL) (-1107 2483587 2485700 2485978 "TREE" NIL TREE (NIL T) -8 NIL NIL NIL) (-1106 2482693 2483389 2483419 "TRANFUN" 2483454 TRANFUN (NIL) -9 NIL 2483520 NIL) (-1105 2482157 2482408 2482688 "TRANFUN-" NIL TRANFUN- (NIL T) -7 NIL NIL NIL) (-1104 2481994 2482032 2482093 "TOPSP" NIL TOPSP (NIL) -7 NIL NIL NIL) (-1103 2481451 2481582 2481733 "TOOLSIGN" NIL TOOLSIGN (NIL T) -7 NIL NIL NIL) (-1102 2480192 2480849 2481085 "TEXTFILE" NIL TEXTFILE (NIL) -8 NIL NIL NIL) (-1101 2480004 2480041 2480113 "TEX1" NIL TEX1 (NIL T) -7 NIL NIL NIL) (-1100 2478218 2478864 2479293 "TEX" NIL TEX (NIL) -8 NIL NIL NIL) (-1099 2476598 2476935 2477257 "TBCMPPK" NIL TBCMPPK (NIL T T) -7 NIL NIL NIL) (-1098 2467656 2474399 2474455 "TBAGG" 2474857 TBAGG (NIL T T) -9 NIL 2475070 NIL) (-1097 2464187 2465879 2467651 "TBAGG-" NIL TBAGG- (NIL T T T) -7 NIL NIL NIL) (-1096 2463664 2463789 2463934 "TANEXP" NIL TANEXP (NIL T) -7 NIL NIL NIL) (-1095 2463174 2463494 2463584 "TALGOP" NIL TALGOP (NIL T) -8 NIL NIL NIL) (-1094 2462671 2462788 2462926 "TABLEAU" NIL TABLEAU (NIL T) -8 NIL NIL NIL) (-1093 2455758 2462573 2462666 "TABLE" NIL TABLE (NIL T T) -8 NIL NIL NIL) (-1092 2451511 2452806 2454051 "TABLBUMP" NIL TABLBUMP (NIL T) -7 NIL NIL NIL) (-1091 2450880 2451039 2451220 "SYSTEM" NIL SYSTEM (NIL) -7 NIL NIL NIL) (-1090 2448034 2448787 2449570 "SYSSOLP" NIL SYSSOLP (NIL T) -7 NIL NIL NIL) (-1089 2447808 2447998 2448029 "SYSPTR" NIL SYSPTR (NIL) -8 NIL NIL NIL) (-1088 2446762 2447447 2447573 "SYSNNI" NIL SYSNNI (NIL NIL) -8 NIL NIL 2447759) (-1087 2446026 2446574 2446653 "SYSINT" NIL SYSINT (NIL NIL) -8 NIL NIL 2446713) (-1086 2442849 2444008 2444708 "SYNTAX" NIL SYNTAX (NIL) -8 NIL NIL NIL) (-1085 2440532 2441215 2441849 "SYMTAB" NIL SYMTAB (NIL) -8 NIL NIL NIL) (-1084 2436610 2437656 2438633 "SYMS" NIL SYMS (NIL) -8 NIL NIL NIL) (-1083 2433773 2436265 2436494 "SYMPOLY" NIL SYMPOLY (NIL T) -8 NIL NIL NIL) (-1082 2433369 2433456 2433578 "SYMFUNC" NIL SYMFUNC (NIL T) -7 NIL NIL NIL) (-1081 2429993 2431467 2432286 "SYMBOL" NIL SYMBOL (NIL) -8 NIL NIL NIL) (-1080 2423017 2429190 2429483 "SUTS" NIL SUTS (NIL T NIL NIL) -8 NIL NIL NIL) (-1079 2414767 2422608 2422870 "SUPXS" NIL SUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-1078 2414046 2414185 2414402 "SUPFRACF" NIL SUPFRACF (NIL T T T T) -7 NIL NIL NIL) (-1077 2413730 2413795 2413906 "SUP2" NIL SUP2 (NIL T T) -7 NIL NIL NIL) (-1076 2404517 2413442 2413567 "SUP" NIL SUP (NIL T) -8 NIL NIL NIL) (-1075 2403253 2403549 2403902 "SUMRF" NIL SUMRF (NIL T) -7 NIL NIL NIL) (-1074 2402661 2402738 2402928 "SUMFS" NIL SUMFS (NIL T T) -7 NIL NIL NIL) (-1073 2384877 2402160 2402401 "SULS" NIL SULS (NIL T NIL NIL) -8 NIL NIL NIL) (-1072 2384476 2384748 2384817 "SUCHTAST" NIL SUCHTAST (NIL) -8 NIL NIL NIL) (-1071 2383812 2384093 2384233 "SUCH" NIL SUCH (NIL T T) -8 NIL NIL NIL) (-1070 2378414 2379673 2380626 "SUBSPACE" NIL SUBSPACE (NIL NIL T) -8 NIL NIL NIL) (-1069 2377946 2378046 2378210 "SUBRESP" NIL SUBRESP (NIL T T) -7 NIL NIL NIL) (-1068 2373057 2374339 2375486 "STTFNC" NIL STTFNC (NIL T) -7 NIL NIL NIL) (-1067 2367515 2368986 2370297 "STTF" NIL STTF (NIL T) -7 NIL NIL NIL) (-1066 2360430 2362494 2364285 "STTAYLOR" NIL STTAYLOR (NIL T) -7 NIL NIL NIL) (-1065 2353260 2360342 2360425 "STRTBL" NIL STRTBL (NIL T) -8 NIL NIL NIL) (-1064 2347954 2352974 2353089 "STRING" NIL STRING (NIL) -8 NIL NIL NIL) (-1063 2347541 2347624 2347768 "STREAM3" NIL STREAM3 (NIL T T T) -7 NIL NIL NIL) (-1062 2346692 2346893 2347128 "STREAM2" NIL STREAM2 (NIL T T) -7 NIL NIL NIL) (-1061 2346432 2346490 2346583 "STREAM1" NIL STREAM1 (NIL T) -7 NIL NIL NIL) (-1060 2339170 2344637 2345243 "STREAM" NIL STREAM (NIL T) -8 NIL NIL NIL) (-1059 2338346 2338551 2338782 "STINPROD" NIL STINPROD (NIL T) -7 NIL NIL NIL) (-1058 2337591 2337962 2338109 "STEPAST" NIL STEPAST (NIL) -8 NIL NIL NIL) (-1057 2337079 2337321 2337351 "STEP" 2337445 STEP (NIL) -9 NIL 2337516 NIL) (-1056 2330182 2336997 2337074 "STBL" NIL STBL (NIL T T NIL) -8 NIL NIL NIL) (-1055 2324397 2328980 2329023 "STAGG" 2329450 STAGG (NIL T) -9 NIL 2329624 NIL) (-1054 2322776 2323524 2324392 "STAGG-" NIL STAGG- (NIL T T) -7 NIL NIL NIL) (-1053 2320933 2322603 2322695 "STACK" NIL STACK (NIL T) -8 NIL NIL NIL) (-1052 2320244 2320752 2320782 "SRING" 2320787 SRING (NIL) -9 NIL 2320807 NIL) (-1051 2312866 2318782 2319221 "SREGSET" NIL SREGSET (NIL T T T T) -8 NIL NIL NIL) (-1050 2306640 2308079 2309583 "SRDCMPK" NIL SRDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1049 2299065 2303976 2304006 "SRAGG" 2305305 SRAGG (NIL) -9 NIL 2305909 NIL) (-1048 2298362 2298682 2299060 "SRAGG-" NIL SRAGG- (NIL T) -7 NIL NIL NIL) (-1047 2292481 2297684 2298107 "SQMATRIX" NIL SQMATRIX (NIL NIL T) -8 NIL NIL NIL) (-1046 2286694 2289863 2290585 "SPLTREE" NIL SPLTREE (NIL T T) -8 NIL NIL NIL) (-1045 2283123 2283942 2284579 "SPLNODE" NIL SPLNODE (NIL T T) -8 NIL NIL NIL) (-1044 2282098 2282403 2282433 "SPFCAT" 2282877 SPFCAT (NIL) -9 NIL NIL NIL) (-1043 2281035 2281287 2281551 "SPECOUT" NIL SPECOUT (NIL) -7 NIL NIL NIL) (-1042 2271799 2274071 2274101 "SPADXPT" 2278736 SPADXPT (NIL) -9 NIL 2280858 NIL) (-1041 2271601 2271647 2271716 "SPADPRSR" NIL SPADPRSR (NIL) -7 NIL NIL NIL) (-1040 2269259 2271565 2271596 "SPADAST" NIL SPADAST (NIL) -8 NIL NIL NIL) (-1039 2260933 2263022 2263064 "SPACEC" 2267379 SPACEC (NIL T) -9 NIL 2269184 NIL) (-1038 2258762 2260880 2260928 "SPACE3" NIL SPACE3 (NIL T) -8 NIL NIL NIL) (-1037 2257695 2257884 2258173 "SORTPAK" NIL SORTPAK (NIL T T) -7 NIL NIL NIL) (-1036 2256099 2256432 2256843 "SOLVETRA" NIL SOLVETRA (NIL T) -7 NIL NIL NIL) (-1035 2255364 2255598 2255859 "SOLVESER" NIL SOLVESER (NIL T) -7 NIL NIL NIL) (-1034 2251544 2252504 2253499 "SOLVERAD" NIL SOLVERAD (NIL T) -7 NIL NIL NIL) (-1033 2247902 2248601 2249330 "SOLVEFOR" NIL SOLVEFOR (NIL T T) -7 NIL NIL NIL) (-1032 2241688 2247242 2247338 "SNTSCAT" 2247343 SNTSCAT (NIL T T T T) -9 NIL 2247413 NIL) (-1031 2235573 2240329 2240719 "SMTS" NIL SMTS (NIL T T T) -8 NIL NIL NIL) (-1030 2229409 2235492 2235568 "SMP" NIL SMP (NIL T T) -8 NIL NIL NIL) (-1029 2227841 2228172 2228570 "SMITH" NIL SMITH (NIL T T T T) -7 NIL NIL NIL) (-1028 2219510 2224425 2224527 "SMATCAT" 2225870 SMATCAT (NIL NIL T T T) -9 NIL 2226418 NIL) (-1027 2217351 2218335 2219505 "SMATCAT-" NIL SMATCAT- (NIL T NIL T T T) -7 NIL NIL NIL) (-1026 2214943 2216557 2216600 "SKAGG" 2216861 SKAGG (NIL T) -9 NIL 2216995 NIL) (-1025 2211053 2214763 2214874 "SINT" NIL SINT (NIL) -8 NIL NIL 2214915) (-1024 2210863 2210907 2210973 "SIMPAN" NIL SIMPAN (NIL) -7 NIL NIL NIL) (-1023 2209938 2210170 2210438 "SIGNRF" NIL SIGNRF (NIL T) -7 NIL NIL NIL) (-1022 2208942 2209104 2209380 "SIGNEF" NIL SIGNEF (NIL T T) -7 NIL NIL NIL) (-1021 2208288 2208628 2208751 "SIGAST" NIL SIGAST (NIL) -8 NIL NIL NIL) (-1020 2207634 2207941 2208081 "SIG" NIL SIG (NIL) -8 NIL NIL NIL) (-1019 2205745 2206237 2206743 "SHP" NIL SHP (NIL T NIL) -7 NIL NIL NIL) (-1018 2199284 2205664 2205740 "SHDP" NIL SHDP (NIL NIL NIL T) -8 NIL NIL NIL) (-1017 2198787 2199024 2199054 "SGROUP" 2199147 SGROUP (NIL) -9 NIL 2199209 NIL) (-1016 2198677 2198709 2198782 "SGROUP-" NIL SGROUP- (NIL T) -7 NIL NIL NIL) (-1015 2196100 2196869 2197591 "SGCF" NIL SGCF (NIL) -7 NIL NIL NIL) (-1014 2189985 2195539 2195635 "SFRTCAT" 2195640 SFRTCAT (NIL T T T T) -9 NIL 2195678 NIL) (-1013 2184377 2185490 2186617 "SFRGCD" NIL SFRGCD (NIL T T T T T) -7 NIL NIL NIL) (-1012 2178553 2179714 2180878 "SFQCMPK" NIL SFQCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1011 2177525 2178427 2178548 "SEXOF" NIL SEXOF (NIL T T T T T) -8 NIL NIL NIL) (-1010 2173133 2174028 2174123 "SEXCAT" 2176736 SEXCAT (NIL T T T T T) -9 NIL 2177287 NIL) (-1009 2172106 2173060 2173128 "SEX" NIL SEX (NIL) -8 NIL NIL NIL) (-1008 2170497 2171082 2171384 "SETMN" NIL SETMN (NIL NIL NIL) -8 NIL NIL NIL) (-1007 2170020 2170205 2170235 "SETCAT" 2170352 SETCAT (NIL) -9 NIL 2170436 NIL) (-1006 2169852 2169916 2170015 "SETCAT-" NIL SETCAT- (NIL T) -7 NIL NIL NIL) (-1005 2166075 2168306 2168349 "SETAGG" 2169217 SETAGG (NIL T) -9 NIL 2169555 NIL) (-1004 2165681 2165833 2166070 "SETAGG-" NIL SETAGG- (NIL T T) -7 NIL NIL NIL) (-1003 2162635 2165628 2165676 "SET" NIL SET (NIL T) -8 NIL NIL NIL) (-1002 2162101 2162411 2162511 "SEQAST" NIL SEQAST (NIL) -8 NIL NIL NIL) (-1001 2161228 2161594 2161655 "SEGXCAT" 2161941 SEGXCAT (NIL T T) -9 NIL 2162061 NIL) (-1000 2160153 2160421 2160464 "SEGCAT" 2160986 SEGCAT (NIL T) -9 NIL 2161207 NIL) (-999 2159842 2159905 2160014 "SEGBIND2" NIL SEGBIND2 (NIL T T) -7 NIL NIL NIL) (-998 2158926 2159388 2159591 "SEGBIND" NIL SEGBIND (NIL T) -8 NIL NIL NIL) (-997 2158507 2158786 2158860 "SEGAST" NIL SEGAST (NIL) -8 NIL NIL NIL) (-996 2157885 2158018 2158217 "SEG2" NIL SEG2 (NIL T T) -7 NIL NIL NIL) (-995 2156953 2157700 2157880 "SEG" NIL SEG (NIL T) -8 NIL NIL NIL) (-994 2156208 2156903 2156948 "SDVAR" NIL SDVAR (NIL T) -8 NIL NIL NIL) (-993 2147809 2156079 2156203 "SDPOL" NIL SDPOL (NIL T) -8 NIL NIL NIL) (-992 2146669 2146959 2147276 "SCPKG" NIL SCPKG (NIL T) -7 NIL NIL NIL) (-991 2145975 2146187 2146375 "SCOPE" NIL SCOPE (NIL) -8 NIL NIL NIL) (-990 2145325 2145482 2145658 "SCACHE" NIL SCACHE (NIL T) -7 NIL NIL NIL) (-989 2144898 2145129 2145157 "SASTCAT" 2145162 SASTCAT (NIL) -9 NIL 2145175 NIL) (-988 2144365 2144790 2144864 "SAOS" NIL SAOS (NIL) -8 NIL NIL NIL) (-987 2143968 2144009 2144180 "SAERFFC" NIL SAERFFC (NIL T T T) -7 NIL NIL NIL) (-986 2143599 2143640 2143797 "SAEFACT" NIL SAEFACT (NIL T T T) -7 NIL NIL NIL) (-985 2136744 2143516 2143594 "SAE" NIL SAE (NIL T T NIL) -8 NIL NIL NIL) (-984 2135394 2135723 2136119 "RURPK" NIL RURPK (NIL T NIL) -7 NIL NIL NIL) (-983 2134155 2134516 2134816 "RULESET" NIL RULESET (NIL T T T) -8 NIL NIL NIL) (-982 2133779 2134000 2134081 "RULECOLD" NIL RULECOLD (NIL NIL) -8 NIL NIL NIL) (-981 2131239 2131873 2132326 "RULE" NIL RULE (NIL T T T) -8 NIL NIL NIL) (-980 2131078 2131111 2131179 "RTVALUE" NIL RTVALUE (NIL) -8 NIL NIL NIL) (-979 2130569 2130872 2130963 "RSTRCAST" NIL RSTRCAST (NIL) -8 NIL NIL NIL) (-978 2126197 2127065 2127976 "RSETGCD" NIL RSETGCD (NIL T T T T T) -7 NIL NIL NIL) (-977 2115016 2120570 2120664 "RSETCAT" 2124720 RSETCAT (NIL T T T T) -9 NIL 2125808 NIL) (-976 2113554 2114196 2115011 "RSETCAT-" NIL RSETCAT- (NIL T T T T T) -7 NIL NIL NIL) (-975 2107328 2108773 2110280 "RSDCMPK" NIL RSDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-974 2105210 2105767 2105839 "RRCC" 2106912 RRCC (NIL T T) -9 NIL 2107253 NIL) (-973 2104735 2104934 2105205 "RRCC-" NIL RRCC- (NIL T T T) -7 NIL NIL NIL) (-972 2104205 2104515 2104613 "RPTAST" NIL RPTAST (NIL) -8 NIL NIL NIL) (-971 2076821 2087470 2087534 "RPOLCAT" 2098008 RPOLCAT (NIL T T T) -9 NIL 2101153 NIL) (-970 2070920 2073743 2076816 "RPOLCAT-" NIL RPOLCAT- (NIL T T T T) -7 NIL NIL NIL) (-969 2067151 2070668 2070806 "ROMAN" NIL ROMAN (NIL) -8 NIL NIL NIL) (-968 2065479 2066218 2066474 "ROIRC" NIL ROIRC (NIL T T) -8 NIL NIL NIL) (-967 2061184 2063934 2063962 "RNS" 2064224 RNS (NIL) -9 NIL 2064476 NIL) (-966 2060087 2060574 2061111 "RNS-" NIL RNS- (NIL T) -7 NIL NIL NIL) (-965 2059205 2059606 2059806 "RNGBIND" NIL RNGBIND (NIL T T) -8 NIL NIL NIL) (-964 2058493 2058993 2059021 "RNG" 2059026 RNG (NIL) -9 NIL 2059047 NIL) (-963 2057786 2058260 2058300 "RMODULE" 2058305 RMODULE (NIL T) -9 NIL 2058331 NIL) (-962 2056725 2056831 2057161 "RMCAT2" NIL RMCAT2 (NIL NIL NIL T T T T T T T T) -7 NIL NIL NIL) (-961 2053603 2056315 2056608 "RMATRIX" NIL RMATRIX (NIL NIL NIL T) -8 NIL NIL NIL) (-960 2046283 2048744 2048856 "RMATCAT" 2052161 RMATCAT (NIL NIL NIL T T T) -9 NIL 2053138 NIL) (-959 2045800 2045979 2046278 "RMATCAT-" NIL RMATCAT- (NIL T NIL NIL T T T) -7 NIL NIL NIL) (-958 2045368 2045579 2045620 "RLINSET" 2045681 RLINSET (NIL T) -9 NIL 2045725 NIL) (-957 2045013 2045094 2045220 "RINTERP" NIL RINTERP (NIL NIL T) -7 NIL NIL NIL) (-956 2043921 2044590 2044618 "RING" 2044673 RING (NIL) -9 NIL 2044765 NIL) (-955 2043766 2043822 2043916 "RING-" NIL RING- (NIL T) -7 NIL NIL NIL) (-954 2042823 2043089 2043344 "RIDIST" NIL RIDIST (NIL) -7 NIL NIL NIL) (-953 2033810 2042451 2042652 "RGCHAIN" NIL RGCHAIN (NIL T NIL) -8 NIL NIL NIL) (-952 2033066 2033546 2033585 "RGBCSPC" 2033642 RGBCSPC (NIL T) -9 NIL 2033693 NIL) (-951 2032131 2032586 2032625 "RGBCMDL" 2032853 RGBCMDL (NIL T) -9 NIL 2032967 NIL) (-950 2031843 2031912 2032013 "RFFACTOR" NIL RFFACTOR (NIL T) -7 NIL NIL NIL) (-949 2031606 2031647 2031742 "RFFACT" NIL RFFACT (NIL T) -7 NIL NIL NIL) (-948 2030030 2030460 2030840 "RFDIST" NIL RFDIST (NIL) -7 NIL NIL NIL) (-947 2027617 2028285 2028953 "RF" NIL RF (NIL T) -7 NIL NIL NIL) (-946 2027167 2027265 2027425 "RETSOL" NIL RETSOL (NIL T T) -7 NIL NIL NIL) (-945 2026789 2026887 2026928 "RETRACT" 2027059 RETRACT (NIL T) -9 NIL 2027146 NIL) (-944 2026669 2026700 2026784 "RETRACT-" NIL RETRACT- (NIL T T) -7 NIL NIL NIL) (-943 2026271 2026543 2026610 "RETAST" NIL RETAST (NIL) -8 NIL NIL NIL) (-942 2024815 2025642 2025839 "RESRING" NIL RESRING (NIL T T T T NIL) -8 NIL NIL NIL) (-941 2024506 2024567 2024663 "RESLATC" NIL RESLATC (NIL T) -7 NIL NIL NIL) (-940 2024249 2024290 2024395 "REPSQ" NIL REPSQ (NIL T) -7 NIL NIL NIL) (-939 2023984 2024025 2024134 "REPDB" NIL REPDB (NIL T) -7 NIL NIL NIL) (-938 2019055 2020506 2021721 "REP2" NIL REP2 (NIL T) -7 NIL NIL NIL) (-937 2016154 2016912 2017720 "REP1" NIL REP1 (NIL T) -7 NIL NIL NIL) (-936 2014123 2014745 2015345 "REP" NIL REP (NIL) -7 NIL NIL NIL) (-935 2006758 2012674 2013110 "REGSET" NIL REGSET (NIL T T T T) -8 NIL NIL NIL) (-934 2006070 2006350 2006499 "REF" NIL REF (NIL T) -8 NIL NIL NIL) (-933 2005555 2005670 2005835 "REDORDER" NIL REDORDER (NIL T T) -7 NIL NIL NIL) (-932 2001212 2004958 2005179 "RECLOS" NIL RECLOS (NIL T) -8 NIL NIL NIL) (-931 2000444 2000643 2000856 "REALSOLV" NIL REALSOLV (NIL) -7 NIL NIL NIL) (-930 1997734 1998572 1999454 "REAL0Q" NIL REAL0Q (NIL T) -7 NIL NIL NIL) (-929 1994316 1995352 1996411 "REAL0" NIL REAL0 (NIL T) -7 NIL NIL NIL) (-928 1994152 1994205 1994233 "REAL" 1994238 REAL (NIL) -9 NIL 1994273 NIL) (-927 1993642 1993946 1994037 "RDUCEAST" NIL RDUCEAST (NIL) -8 NIL NIL NIL) (-926 1993122 1993200 1993405 "RDIV" NIL RDIV (NIL T T T T T) -7 NIL NIL NIL) (-925 1992355 1992547 1992758 "RDIST" NIL RDIST (NIL T) -7 NIL NIL NIL) (-924 1991243 1991540 1991907 "RDETRS" NIL RDETRS (NIL T T) -7 NIL NIL NIL) (-923 1989510 1989980 1990513 "RDETR" NIL RDETR (NIL T T) -7 NIL NIL NIL) (-922 1988432 1988709 1989096 "RDEEFS" NIL RDEEFS (NIL T T) -7 NIL NIL NIL) (-921 1987259 1987568 1987987 "RDEEF" NIL RDEEF (NIL T T) -7 NIL NIL NIL) (-920 1980671 1984119 1984147 "RCFIELD" 1985424 RCFIELD (NIL) -9 NIL 1986154 NIL) (-919 1979289 1979901 1980598 "RCFIELD-" NIL RCFIELD- (NIL T) -7 NIL NIL NIL) (-918 1975489 1977381 1977422 "RCAGG" 1978489 RCAGG (NIL T) -9 NIL 1978950 NIL) (-917 1975216 1975326 1975484 "RCAGG-" NIL RCAGG- (NIL T T) -7 NIL NIL NIL) (-916 1974661 1974790 1974951 "RATRET" NIL RATRET (NIL T) -7 NIL NIL NIL) (-915 1974278 1974357 1974476 "RATFACT" NIL RATFACT (NIL T) -7 NIL NIL NIL) (-914 1973693 1973843 1973993 "RANDSRC" NIL RANDSRC (NIL) -7 NIL NIL NIL) (-913 1973475 1973525 1973596 "RADUTIL" NIL RADUTIL (NIL) -7 NIL NIL NIL) (-912 1965981 1972593 1972901 "RADIX" NIL RADIX (NIL NIL) -8 NIL NIL NIL) (-911 1955747 1965848 1965976 "RADFF" NIL RADFF (NIL T T T NIL NIL) -8 NIL NIL NIL) (-910 1955381 1955474 1955502 "RADCAT" 1955659 RADCAT (NIL) -9 NIL NIL NIL) (-909 1955219 1955279 1955376 "RADCAT-" NIL RADCAT- (NIL T) -7 NIL NIL NIL) (-908 1953319 1955050 1955139 "QUEUE" NIL QUEUE (NIL T) -8 NIL NIL NIL) (-907 1953000 1953049 1953176 "QUATCT2" NIL QUATCT2 (NIL T T T T) -7 NIL NIL NIL) (-906 1945351 1949371 1949411 "QUATCAT" 1950189 QUATCAT (NIL T) -9 NIL 1950953 NIL) (-905 1942601 1943881 1945257 "QUATCAT-" NIL QUATCAT- (NIL T T) -7 NIL NIL NIL) (-904 1938505 1942551 1942596 "QUAT" NIL QUAT (NIL T) -8 NIL NIL NIL) (-903 1935892 1937559 1937600 "QUAGG" 1937975 QUAGG (NIL T) -9 NIL 1938149 NIL) (-902 1935494 1935766 1935833 "QQUTAST" NIL QQUTAST (NIL) -8 NIL NIL NIL) (-901 1934532 1935130 1935293 "QFORM" NIL QFORM (NIL NIL T) -8 NIL NIL NIL) (-900 1934213 1934262 1934389 "QFCAT2" NIL QFCAT2 (NIL T T T T) -7 NIL NIL NIL) (-899 1923900 1930007 1930047 "QFCAT" 1930705 QFCAT (NIL T) -9 NIL 1931698 NIL) (-898 1920784 1922223 1923806 "QFCAT-" NIL QFCAT- (NIL T T) -7 NIL NIL NIL) (-897 1920330 1920464 1920594 "QEQUAT" NIL QEQUAT (NIL) -8 NIL NIL NIL) (-896 1914526 1915687 1916849 "QCMPACK" NIL QCMPACK (NIL T T T T T) -7 NIL NIL NIL) (-895 1913945 1914125 1914357 "QALGSET2" NIL QALGSET2 (NIL NIL NIL) -7 NIL NIL NIL) (-894 1911767 1912295 1912718 "QALGSET" NIL QALGSET (NIL T T T T) -8 NIL NIL NIL) (-893 1910666 1910908 1911225 "PWFFINTB" NIL PWFFINTB (NIL T T T T) -7 NIL NIL NIL) (-892 1909027 1909225 1909578 "PUSHVAR" NIL PUSHVAR (NIL T T T T) -7 NIL NIL NIL) (-891 1904783 1905999 1906040 "PTRANFN" 1907924 PTRANFN (NIL T) -9 NIL NIL NIL) (-890 1903430 1903775 1904096 "PTPACK" NIL PTPACK (NIL T) -7 NIL NIL NIL) (-889 1903123 1903186 1903293 "PTFUNC2" NIL PTFUNC2 (NIL T T) -7 NIL NIL NIL) (-888 1897196 1901919 1901959 "PTCAT" 1902251 PTCAT (NIL T) -9 NIL 1902404 NIL) (-887 1896889 1896930 1897054 "PSQFR" NIL PSQFR (NIL T T T T) -7 NIL NIL NIL) (-886 1895768 1896084 1896418 "PSEUDLIN" NIL PSEUDLIN (NIL T) -7 NIL NIL NIL) (-885 1884647 1887208 1889517 "PSETPK" NIL PSETPK (NIL T T T T) -7 NIL NIL NIL) (-884 1877554 1880450 1880544 "PSETCAT" 1883518 PSETCAT (NIL T T T T) -9 NIL 1884325 NIL) (-883 1876004 1876738 1877549 "PSETCAT-" NIL PSETCAT- (NIL T T T T T) -7 NIL NIL NIL) (-882 1875332 1875524 1875552 "PSCURVE" 1875817 PSCURVE (NIL) -9 NIL 1875981 NIL) (-881 1870996 1872754 1872818 "PSCAT" 1873653 PSCAT (NIL T T T) -9 NIL 1873892 NIL) (-880 1870310 1870592 1870991 "PSCAT-" NIL PSCAT- (NIL T T T T) -7 NIL NIL NIL) (-879 1868739 1869622 1869885 "PRTITION" NIL PRTITION (NIL) -8 NIL NIL NIL) (-878 1868230 1868533 1868624 "PRTDAST" NIL PRTDAST (NIL) -8 NIL NIL NIL) (-877 1859250 1861672 1863860 "PRS" NIL PRS (NIL T T) -7 NIL NIL NIL) (-876 1856993 1858570 1858610 "PRQAGG" 1858793 PRQAGG (NIL T) -9 NIL 1858894 NIL) (-875 1856166 1856612 1856640 "PROPLOG" 1856779 PROPLOG (NIL) -9 NIL 1856893 NIL) (-874 1855841 1855904 1856027 "PROPFUN2" NIL PROPFUN2 (NIL T T) -7 NIL NIL NIL) (-873 1855277 1855416 1855588 "PROPFUN1" NIL PROPFUN1 (NIL T) -7 NIL NIL NIL) (-872 1853525 1854288 1854585 "PROPFRML" NIL PROPFRML (NIL T) -8 NIL NIL NIL) (-871 1853077 1853209 1853337 "PROPERTY" NIL PROPERTY (NIL) -8 NIL NIL NIL) (-870 1847733 1852017 1852837 "PRODUCT" NIL PRODUCT (NIL T T) -8 NIL NIL NIL) (-869 1847562 1847600 1847659 "PRINT" NIL PRINT (NIL) -7 NIL NIL NIL) (-868 1847001 1847141 1847292 "PRIMES" NIL PRIMES (NIL T) -7 NIL NIL NIL) (-867 1845469 1845888 1846354 "PRIMELT" NIL PRIMELT (NIL T) -7 NIL NIL NIL) (-866 1845189 1845249 1845277 "PRIMCAT" 1845400 PRIMCAT (NIL) -9 NIL NIL NIL) (-865 1844360 1844556 1844784 "PRIMARR2" NIL PRIMARR2 (NIL T T) -7 NIL NIL NIL) (-864 1840238 1844310 1844355 "PRIMARR" NIL PRIMARR (NIL T) -8 NIL NIL NIL) (-863 1839937 1839999 1840110 "PREASSOC" NIL PREASSOC (NIL T T) -7 NIL NIL NIL) (-862 1837137 1839586 1839819 "PR" NIL PR (NIL T T) -8 NIL NIL NIL) (-861 1836594 1836749 1836777 "PPCURVE" 1836980 PPCURVE (NIL) -9 NIL 1837114 NIL) (-860 1836207 1836452 1836535 "PORTNUM" NIL PORTNUM (NIL) -8 NIL NIL NIL) (-859 1833963 1834384 1834976 "POLYROOT" NIL POLYROOT (NIL T T T T T) -7 NIL NIL NIL) (-858 1833406 1833470 1833703 "POLYLIFT" NIL POLYLIFT (NIL T T T T T) -7 NIL NIL NIL) (-857 1830126 1830612 1831223 "POLYCATQ" NIL POLYCATQ (NIL T T T T T) -7 NIL NIL NIL) (-856 1815781 1821846 1821910 "POLYCAT" 1825395 POLYCAT (NIL T T T) -9 NIL 1827272 NIL) (-855 1811291 1813438 1815776 "POLYCAT-" NIL POLYCAT- (NIL T T T T) -7 NIL NIL NIL) (-854 1810948 1811022 1811141 "POLY2UP" NIL POLY2UP (NIL NIL T) -7 NIL NIL NIL) (-853 1810641 1810704 1810811 "POLY2" NIL POLY2 (NIL T T) -7 NIL NIL NIL) (-852 1804068 1810374 1810533 "POLY" NIL POLY (NIL T) -8 NIL NIL NIL) (-851 1802955 1803218 1803494 "POLUTIL" NIL POLUTIL (NIL T T) -7 NIL NIL NIL) (-850 1801559 1801872 1802202 "POLTOPOL" NIL POLTOPOL (NIL NIL T) -7 NIL NIL NIL) (-849 1796721 1801509 1801554 "POINT" NIL POINT (NIL T) -8 NIL NIL NIL) (-848 1795209 1795620 1795995 "PNTHEORY" NIL PNTHEORY (NIL) -7 NIL NIL NIL) (-847 1793966 1794275 1794671 "PMTOOLS" NIL PMTOOLS (NIL T T T) -7 NIL NIL NIL) (-846 1793637 1793721 1793838 "PMSYM" NIL PMSYM (NIL T) -7 NIL NIL NIL) (-845 1793216 1793291 1793465 "PMQFCAT" NIL PMQFCAT (NIL T T T) -7 NIL NIL NIL) (-844 1792702 1792798 1792958 "PMPREDFS" NIL PMPREDFS (NIL T T T) -7 NIL NIL NIL) (-843 1792174 1792294 1792448 "PMPRED" NIL PMPRED (NIL T) -7 NIL NIL NIL) (-842 1791069 1791287 1791664 "PMPLCAT" NIL PMPLCAT (NIL T T T T T) -7 NIL NIL NIL) (-841 1790680 1790765 1790917 "PMLSAGG" NIL PMLSAGG (NIL T T T) -7 NIL NIL NIL) (-840 1790231 1790313 1790494 "PMKERNEL" NIL PMKERNEL (NIL T T) -7 NIL NIL NIL) (-839 1789923 1790004 1790117 "PMINS" NIL PMINS (NIL T) -7 NIL NIL NIL) (-838 1789436 1789511 1789719 "PMFS" NIL PMFS (NIL T T T) -7 NIL NIL NIL) (-837 1788784 1788912 1789114 "PMDOWN" NIL PMDOWN (NIL T T T) -7 NIL NIL NIL) (-836 1788146 1788280 1788443 "PMASSFS" NIL PMASSFS (NIL T T) -7 NIL NIL NIL) (-835 1787450 1787632 1787813 "PMASS" NIL PMASS (NIL) -7 NIL NIL NIL) (-834 1787176 1787249 1787342 "PLOTTOOL" NIL PLOTTOOL (NIL) -7 NIL NIL NIL) (-833 1783787 1784957 1785857 "PLOT3D" NIL PLOT3D (NIL) -8 NIL NIL NIL) (-832 1782880 1783078 1783310 "PLOT1" NIL PLOT1 (NIL T) -7 NIL NIL NIL) (-831 1778503 1779864 1780985 "PLOT" NIL PLOT (NIL) -8 NIL NIL NIL) (-830 1758424 1763311 1768158 "PLEQN" NIL PLEQN (NIL T T T T) -7 NIL NIL NIL) (-829 1758164 1758217 1758320 "PINTERPA" NIL PINTERPA (NIL T T) -7 NIL NIL NIL) (-828 1757605 1757739 1757919 "PINTERP" NIL PINTERP (NIL NIL T) -7 NIL NIL NIL) (-827 1755676 1756835 1756863 "PID" 1757060 PID (NIL) -9 NIL 1757187 NIL) (-826 1755464 1755507 1755582 "PICOERCE" NIL PICOERCE (NIL T) -7 NIL NIL NIL) (-825 1754651 1755311 1755398 "PI" NIL PI (NIL) -8 NIL NIL 1755438) (-824 1754103 1754254 1754430 "PGROEB" NIL PGROEB (NIL T) -7 NIL NIL NIL) (-823 1750431 1751389 1752294 "PGE" NIL PGE (NIL) -7 NIL NIL NIL) (-822 1748795 1749084 1749450 "PGCD" NIL PGCD (NIL T T T T) -7 NIL NIL NIL) (-821 1748237 1748352 1748513 "PFRPAC" NIL PFRPAC (NIL T) -7 NIL NIL NIL) (-820 1744842 1747106 1747459 "PFR" NIL PFR (NIL T) -8 NIL NIL NIL) (-819 1743448 1743728 1744053 "PFOTOOLS" NIL PFOTOOLS (NIL T T) -7 NIL NIL NIL) (-818 1742213 1742467 1742815 "PFOQ" NIL PFOQ (NIL T T T) -7 NIL NIL NIL) (-817 1740923 1741150 1741502 "PFO" NIL PFO (NIL T T T T T) -7 NIL NIL NIL) (-816 1737995 1739493 1739521 "PFECAT" 1740114 PFECAT (NIL) -9 NIL 1740491 NIL) (-815 1737618 1737783 1737990 "PFECAT-" NIL PFECAT- (NIL T) -7 NIL NIL NIL) (-814 1736442 1736724 1737025 "PFBRU" NIL PFBRU (NIL T T) -7 NIL NIL NIL) (-813 1734624 1735011 1735441 "PFBR" NIL PFBR (NIL T T T T) -7 NIL NIL NIL) (-812 1730658 1734550 1734619 "PF" NIL PF (NIL NIL) -8 NIL NIL NIL) (-811 1726561 1727708 1728575 "PERMGRP" NIL PERMGRP (NIL T) -8 NIL NIL NIL) (-810 1724493 1725582 1725623 "PERMCAT" 1726022 PERMCAT (NIL T) -9 NIL 1726319 NIL) (-809 1724189 1724236 1724359 "PERMAN" NIL PERMAN (NIL NIL T) -7 NIL NIL NIL) (-808 1720638 1722319 1722964 "PERM" NIL PERM (NIL T) -8 NIL NIL NIL) (-807 1718103 1720393 1720514 "PENDTREE" NIL PENDTREE (NIL T) -8 NIL NIL NIL) (-806 1716972 1717235 1717276 "PDSPC" 1717809 PDSPC (NIL T) -9 NIL 1718054 NIL) (-805 1716339 1716605 1716967 "PDSPC-" NIL PDSPC- (NIL T T) -7 NIL NIL NIL) (-804 1715036 1715967 1716008 "PDRING" 1716013 PDRING (NIL T) -9 NIL 1716040 NIL) (-803 1713777 1714535 1714588 "PDMOD" 1714593 PDMOD (NIL T T) -9 NIL 1714696 NIL) (-802 1712870 1713082 1713331 "PDECOMP" NIL PDECOMP (NIL T T) -7 NIL NIL NIL) (-801 1712475 1712542 1712596 "PDDOM" 1712761 PDDOM (NIL T T) -9 NIL 1712841 NIL) (-800 1712327 1712363 1712470 "PDDOM-" NIL PDDOM- (NIL T T T) -7 NIL NIL NIL) (-799 1712113 1712152 1712241 "PCOMP" NIL PCOMP (NIL T T) -7 NIL NIL NIL) (-798 1710430 1711184 1711483 "PBWLB" NIL PBWLB (NIL T) -8 NIL NIL NIL) (-797 1710119 1710182 1710291 "PATTERN2" NIL PATTERN2 (NIL T T) -7 NIL NIL NIL) (-796 1708257 1708687 1709138 "PATTERN1" NIL PATTERN1 (NIL T T) -7 NIL NIL NIL) (-795 1701877 1703706 1704998 "PATTERN" NIL PATTERN (NIL T) -8 NIL NIL NIL) (-794 1701508 1701581 1701713 "PATRES2" NIL PATRES2 (NIL T T T) -7 NIL NIL NIL) (-793 1699210 1699890 1700371 "PATRES" NIL PATRES (NIL T T) -8 NIL NIL NIL) (-792 1697414 1697842 1698245 "PATMATCH" NIL PATMATCH (NIL T T T) -7 NIL NIL NIL) (-791 1696860 1697108 1697149 "PATMAB" 1697256 PATMAB (NIL T) -9 NIL 1697339 NIL) (-790 1695507 1695911 1696168 "PATLRES" NIL PATLRES (NIL T T T) -8 NIL NIL NIL) (-789 1695045 1695176 1695217 "PATAB" 1695222 PATAB (NIL T) -9 NIL 1695394 NIL) (-788 1693588 1694025 1694448 "PARTPERM" NIL PARTPERM (NIL) -7 NIL NIL NIL) (-787 1693266 1693341 1693443 "PARSURF" NIL PARSURF (NIL T) -8 NIL NIL NIL) (-786 1692955 1693018 1693127 "PARSU2" NIL PARSU2 (NIL T T) -7 NIL NIL NIL) (-785 1692760 1692806 1692873 "PARSER" NIL PARSER (NIL) -7 NIL NIL NIL) (-784 1692438 1692513 1692615 "PARSCURV" NIL PARSCURV (NIL T) -8 NIL NIL NIL) (-783 1692127 1692190 1692299 "PARSC2" NIL PARSC2 (NIL T T) -7 NIL NIL NIL) (-782 1691818 1691888 1691985 "PARPCURV" NIL PARPCURV (NIL T) -8 NIL NIL NIL) (-781 1691507 1691570 1691679 "PARPC2" NIL PARPC2 (NIL T T) -7 NIL NIL NIL) (-780 1690668 1691047 1691226 "PARAMAST" NIL PARAMAST (NIL) -8 NIL NIL NIL) (-779 1690275 1690373 1690492 "PAN2EXPR" NIL PAN2EXPR (NIL) -7 NIL NIL NIL) (-778 1689243 1689668 1689887 "PALETTE" NIL PALETTE (NIL) -8 NIL NIL NIL) (-777 1687908 1688562 1688922 "PAIR" NIL PAIR (NIL T T) -8 NIL NIL NIL) (-776 1681062 1687312 1687506 "PADICRC" NIL PADICRC (NIL NIL T) -8 NIL NIL NIL) (-775 1673547 1680560 1680744 "PADICRAT" NIL PADICRAT (NIL NIL) -8 NIL NIL NIL) (-774 1670334 1672187 1672227 "PADICCT" 1672808 PADICCT (NIL NIL) -9 NIL 1673090 NIL) (-773 1668388 1670284 1670329 "PADIC" NIL PADIC (NIL NIL) -8 NIL NIL NIL) (-772 1667550 1667760 1668026 "PADEPAC" NIL PADEPAC (NIL T NIL NIL) -7 NIL NIL NIL) (-771 1666892 1667035 1667239 "PADE" NIL PADE (NIL T T T) -7 NIL NIL NIL) (-770 1665337 1666300 1666578 "OWP" NIL OWP (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-769 1664861 1665120 1665217 "OVERSET" NIL OVERSET (NIL) -8 NIL NIL NIL) (-768 1663920 1664598 1664770 "OVAR" NIL OVAR (NIL NIL) -8 NIL NIL NIL) (-767 1654342 1657211 1659410 "OUTFORM" NIL OUTFORM (NIL) -8 NIL NIL NIL) (-766 1653736 1654048 1654174 "OUTBFILE" NIL OUTBFILE (NIL) -8 NIL NIL NIL) (-765 1653019 1653212 1653240 "OUTBCON" 1653556 OUTBCON (NIL) -9 NIL 1653720 NIL) (-764 1652727 1652857 1653014 "OUTBCON-" NIL OUTBCON- (NIL T) -7 NIL NIL NIL) (-763 1652108 1652253 1652414 "OUT" NIL OUT (NIL) -7 NIL NIL NIL) (-762 1651479 1651906 1651995 "OSI" NIL OSI (NIL) -8 NIL NIL NIL) (-761 1650894 1651309 1651337 "OSGROUP" 1651342 OSGROUP (NIL) -9 NIL 1651364 NIL) (-760 1649858 1650119 1650404 "ORTHPOL" NIL ORTHPOL (NIL T) -7 NIL NIL NIL) (-759 1647191 1649733 1649853 "OREUP" NIL OREUP (NIL NIL T NIL NIL) -8 NIL NIL NIL) (-758 1644396 1646942 1647068 "ORESUP" NIL ORESUP (NIL T NIL NIL) -8 NIL NIL NIL) (-757 1642414 1642942 1643502 "OREPCTO" NIL OREPCTO (NIL T T) -7 NIL NIL NIL) (-756 1635818 1638296 1638336 "OREPCAT" 1640657 OREPCAT (NIL T) -9 NIL 1641759 NIL) (-755 1633844 1634778 1635813 "OREPCAT-" NIL OREPCAT- (NIL T T) -7 NIL NIL NIL) (-754 1633041 1633312 1633340 "ORDTYPE" 1633645 ORDTYPE (NIL) -9 NIL 1633803 NIL) (-753 1632575 1632786 1633036 "ORDTYPE-" NIL ORDTYPE- (NIL T) -7 NIL NIL NIL) (-752 1632037 1632413 1632570 "ORDSTRCT" NIL ORDSTRCT (NIL T NIL) -8 NIL NIL NIL) (-751 1631531 1631894 1631922 "ORDSET" 1631927 ORDSET (NIL) -9 NIL 1631949 NIL) (-750 1630171 1631131 1631159 "ORDRING" 1631164 ORDRING (NIL) -9 NIL 1631192 NIL) (-749 1629419 1629976 1630004 "ORDMON" 1630009 ORDMON (NIL) -9 NIL 1630030 NIL) (-748 1628723 1628885 1629077 "ORDFUNS" NIL ORDFUNS (NIL NIL T) -7 NIL NIL NIL) (-747 1627934 1628442 1628470 "ORDFIN" 1628535 ORDFIN (NIL) -9 NIL 1628609 NIL) (-746 1627328 1627467 1627653 "ORDCOMP2" NIL ORDCOMP2 (NIL T T) -7 NIL NIL NIL) (-745 1624102 1626296 1626702 "ORDCOMP" NIL ORDCOMP (NIL T) -8 NIL NIL NIL) (-744 1623509 1623864 1623969 "OPSIG" NIL OPSIG (NIL) -8 NIL NIL NIL) (-743 1623317 1623362 1623428 "OPQUERY" NIL OPQUERY (NIL) -7 NIL NIL NIL) (-742 1622618 1622894 1622935 "OPERCAT" 1623146 OPERCAT (NIL T) -9 NIL 1623242 NIL) (-741 1622430 1622497 1622613 "OPERCAT-" NIL OPERCAT- (NIL T T) -7 NIL NIL NIL) (-740 1619860 1621232 1621728 "OP" NIL OP (NIL T) -8 NIL NIL NIL) (-739 1619281 1619408 1619582 "ONECOMP2" NIL ONECOMP2 (NIL T T) -7 NIL NIL NIL) (-738 1616281 1618420 1618786 "ONECOMP" NIL ONECOMP (NIL T) -8 NIL NIL NIL) (-737 1612912 1615711 1615751 "OMSAGG" 1615812 OMSAGG (NIL T) -9 NIL 1615876 NIL) (-736 1611388 1612583 1612751 "OMLO" NIL OMLO (NIL T T) -8 NIL NIL NIL) (-735 1609659 1610838 1610866 "OINTDOM" 1610871 OINTDOM (NIL) -9 NIL 1610892 NIL) (-734 1607089 1608661 1608990 "OFMONOID" NIL OFMONOID (NIL T) -8 NIL NIL NIL) (-733 1606343 1607039 1607084 "ODVAR" NIL ODVAR (NIL T) -8 NIL NIL NIL) (-732 1603609 1606184 1606338 "ODR" NIL ODR (NIL T T NIL) -8 NIL NIL NIL) (-731 1595210 1603480 1603604 "ODPOL" NIL ODPOL (NIL T) -8 NIL NIL NIL) (-730 1588720 1595101 1595205 "ODP" NIL ODP (NIL NIL T NIL) -8 NIL NIL NIL) (-729 1587692 1587929 1588202 "ODETOOLS" NIL ODETOOLS (NIL T T) -7 NIL NIL NIL) (-728 1585326 1585996 1586700 "ODESYS" NIL ODESYS (NIL T T) -7 NIL NIL NIL) (-727 1581103 1582063 1583086 "ODERTRIC" NIL ODERTRIC (NIL T T) -7 NIL NIL NIL) (-726 1580611 1580699 1580893 "ODERED" NIL ODERED (NIL T T T T T) -7 NIL NIL NIL) (-725 1578060 1578642 1579315 "ODERAT" NIL ODERAT (NIL T T) -7 NIL NIL NIL) (-724 1575455 1575963 1576559 "ODEPRRIC" NIL ODEPRRIC (NIL T T T T) -7 NIL NIL NIL) (-723 1572452 1572991 1573637 "ODEPRIM" NIL ODEPRIM (NIL T T T T) -7 NIL NIL NIL) (-722 1571807 1571915 1572173 "ODEPAL" NIL ODEPAL (NIL T T T T) -7 NIL NIL NIL) (-721 1570965 1571090 1571311 "ODEINT" NIL ODEINT (NIL T T) -7 NIL NIL NIL) (-720 1567249 1568045 1568958 "ODEEF" NIL ODEEF (NIL T T) -7 NIL NIL NIL) (-719 1566689 1566784 1567006 "ODECONST" NIL ODECONST (NIL T T T) -7 NIL NIL NIL) (-718 1566370 1566419 1566546 "OCTCT2" NIL OCTCT2 (NIL T T T T) -7 NIL NIL NIL) (-717 1563037 1566169 1566288 "OCT" NIL OCT (NIL T) -8 NIL NIL NIL) (-716 1562228 1562819 1562847 "OCAMON" 1562852 OCAMON (NIL) -9 NIL 1562873 NIL) (-715 1556504 1559254 1559294 "OC" 1560389 OC (NIL T) -9 NIL 1561245 NIL) (-714 1554504 1555430 1556410 "OC-" NIL OC- (NIL T T) -7 NIL NIL NIL) (-713 1553920 1554338 1554366 "OASGP" 1554371 OASGP (NIL) -9 NIL 1554391 NIL) (-712 1553014 1553632 1553660 "OAMONS" 1553700 OAMONS (NIL) -9 NIL 1553743 NIL) (-711 1552190 1552740 1552768 "OAMON" 1552825 OAMON (NIL) -9 NIL 1552876 NIL) (-710 1552086 1552118 1552185 "OAMON-" NIL OAMON- (NIL T) -7 NIL NIL NIL) (-709 1550868 1551611 1551639 "OAGROUP" 1551785 OAGROUP (NIL) -9 NIL 1551877 NIL) (-708 1550659 1550746 1550863 "OAGROUP-" NIL OAGROUP- (NIL T) -7 NIL NIL NIL) (-707 1550399 1550455 1550543 "NUMTUBE" NIL NUMTUBE (NIL T) -7 NIL NIL NIL) (-706 1545461 1547024 1548551 "NUMQUAD" NIL NUMQUAD (NIL) -7 NIL NIL NIL) (-705 1542156 1543190 1544225 "NUMODE" NIL NUMODE (NIL) -7 NIL NIL NIL) (-704 1541266 1541499 1541717 "NUMFMT" NIL NUMFMT (NIL) -7 NIL NIL NIL) (-703 1530127 1533155 1535603 "NUMERIC" NIL NUMERIC (NIL T) -7 NIL NIL NIL) (-702 1524014 1529568 1529662 "NTSCAT" 1529667 NTSCAT (NIL T T T T) -9 NIL 1529705 NIL) (-701 1523355 1523534 1523727 "NTPOLFN" NIL NTPOLFN (NIL T) -7 NIL NIL NIL) (-700 1523048 1523111 1523218 "NSUP2" NIL NSUP2 (NIL T T) -7 NIL NIL NIL) (-699 1510779 1520668 1521478 "NSUP" NIL NSUP (NIL T) -8 NIL NIL NIL) (-698 1499852 1510644 1510774 "NSMP" NIL NSMP (NIL T T) -8 NIL NIL NIL) (-697 1498572 1498897 1499254 "NREP" NIL NREP (NIL T) -7 NIL NIL NIL) (-696 1497408 1497672 1498030 "NPCOEF" NIL NPCOEF (NIL T T T T T) -7 NIL NIL NIL) (-695 1496575 1496708 1496924 "NORMRETR" NIL NORMRETR (NIL T T T T NIL) -7 NIL NIL NIL) (-694 1494893 1495212 1495618 "NORMPK" NIL NORMPK (NIL T T T T T) -7 NIL NIL NIL) (-693 1494606 1494640 1494764 "NORMMA" NIL NORMMA (NIL T T T T) -7 NIL NIL NIL) (-692 1494425 1494460 1494529 "NONE1" NIL NONE1 (NIL T) -7 NIL NIL NIL) (-691 1494201 1494391 1494420 "NONE" NIL NONE (NIL) -8 NIL NIL NIL) (-690 1493765 1493832 1494009 "NODE1" NIL NODE1 (NIL T T) -7 NIL NIL NIL) (-689 1492083 1493128 1493383 "NNI" NIL NNI (NIL) -8 NIL NIL 1493730) (-688 1490811 1491148 1491512 "NLINSOL" NIL NLINSOL (NIL T) -7 NIL NIL NIL) (-687 1489788 1490040 1490342 "NFINTBAS" NIL NFINTBAS (NIL T T) -7 NIL NIL NIL) (-686 1488878 1489440 1489481 "NETCLT" 1489652 NETCLT (NIL T) -9 NIL 1489733 NIL) (-685 1487782 1488049 1488330 "NCODIV" NIL NCODIV (NIL T T) -7 NIL NIL NIL) (-684 1487581 1487624 1487699 "NCNTFRAC" NIL NCNTFRAC (NIL T) -7 NIL NIL NIL) (-683 1486112 1486500 1486920 "NCEP" NIL NCEP (NIL T) -7 NIL NIL NIL) (-682 1484776 1485711 1485739 "NASRING" 1485849 NASRING (NIL) -9 NIL 1485929 NIL) (-681 1484621 1484677 1484771 "NASRING-" NIL NASRING- (NIL T) -7 NIL NIL NIL) (-680 1483581 1484228 1484256 "NARNG" 1484373 NARNG (NIL) -9 NIL 1484464 NIL) (-679 1483357 1483442 1483576 "NARNG-" NIL NARNG- (NIL T) -7 NIL NIL NIL) (-678 1482154 1482877 1482917 "NAALG" 1482996 NAALG (NIL T) -9 NIL 1483057 NIL) (-677 1482024 1482059 1482149 "NAALG-" NIL NAALG- (NIL T T) -7 NIL NIL NIL) (-676 1477003 1478188 1479374 "MULTSQFR" NIL MULTSQFR (NIL T T T T) -7 NIL NIL NIL) (-675 1476398 1476485 1476669 "MULTFACT" NIL MULTFACT (NIL T T T T) -7 NIL NIL NIL) (-674 1468472 1472902 1472954 "MTSCAT" 1474014 MTSCAT (NIL T T) -9 NIL 1474528 NIL) (-673 1468238 1468298 1468390 "MTHING" NIL MTHING (NIL T) -7 NIL NIL NIL) (-672 1468064 1468103 1468163 "MSYSCMD" NIL MSYSCMD (NIL) -7 NIL NIL NIL) (-671 1464926 1467615 1467656 "MSETAGG" 1467661 MSETAGG (NIL T) -9 NIL 1467695 NIL) (-670 1461063 1463972 1464290 "MSET" NIL MSET (NIL T) -8 NIL NIL NIL) (-669 1457401 1459160 1459900 "MRING" NIL MRING (NIL T T) -8 NIL NIL NIL) (-668 1457038 1457111 1457240 "MRF2" NIL MRF2 (NIL T T T) -7 NIL NIL NIL) (-667 1456691 1456732 1456876 "MRATFAC" NIL MRATFAC (NIL T T T T) -7 NIL NIL NIL) (-666 1454556 1454893 1455324 "MPRFF" NIL MPRFF (NIL T T T T) -7 NIL NIL NIL) (-665 1448018 1454455 1454551 "MPOLY" NIL MPOLY (NIL NIL T) -8 NIL NIL NIL) (-664 1447543 1447584 1447792 "MPCPF" NIL MPCPF (NIL T T T T) -7 NIL NIL NIL) (-663 1447102 1447151 1447334 "MPC3" NIL MPC3 (NIL T T T T T T T) -7 NIL NIL NIL) (-662 1446376 1446469 1446688 "MPC2" NIL MPC2 (NIL T T T T T T T) -7 NIL NIL NIL) (-661 1444993 1445354 1445744 "MONOTOOL" NIL MONOTOOL (NIL T T) -7 NIL NIL NIL) (-660 1444135 1444514 1444542 "MONOID" 1444760 MONOID (NIL) -9 NIL 1444904 NIL) (-659 1443794 1443944 1444130 "MONOID-" NIL MONOID- (NIL T) -7 NIL NIL NIL) (-658 1432794 1439602 1439661 "MONOGEN" 1440335 MONOGEN (NIL T T) -9 NIL 1440791 NIL) (-657 1430806 1431692 1432675 "MONOGEN-" NIL MONOGEN- (NIL T T T) -7 NIL NIL NIL) (-656 1429520 1430064 1430092 "MONADWU" 1430483 MONADWU (NIL) -9 NIL 1430718 NIL) (-655 1429068 1429268 1429515 "MONADWU-" NIL MONADWU- (NIL T) -7 NIL NIL NIL) (-654 1428345 1428646 1428674 "MONAD" 1428881 MONAD (NIL) -9 NIL 1428993 NIL) (-653 1428112 1428208 1428340 "MONAD-" NIL MONAD- (NIL T) -7 NIL NIL NIL) (-652 1426502 1427272 1427551 "MOEBIUS" NIL MOEBIUS (NIL T) -8 NIL NIL NIL) (-651 1425667 1426163 1426203 "MODULE" 1426208 MODULE (NIL T) -9 NIL 1426246 NIL) (-650 1425346 1425472 1425662 "MODULE-" NIL MODULE- (NIL T T) -7 NIL NIL NIL) (-649 1423121 1423943 1424257 "MODRING" NIL MODRING (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-648 1420364 1421717 1422230 "MODOP" NIL MODOP (NIL T T) -8 NIL NIL NIL) (-647 1418998 1419572 1419848 "MODMONOM" NIL MODMONOM (NIL T T NIL) -8 NIL NIL NIL) (-646 1408281 1417663 1418076 "MODMON" NIL MODMON (NIL T T) -8 NIL NIL NIL) (-645 1405301 1407281 1407550 "MODFIELD" NIL MODFIELD (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-644 1404385 1404752 1404942 "MMLFORM" NIL MMLFORM (NIL) -8 NIL NIL NIL) (-643 1403954 1404003 1404182 "MMAP" NIL MMAP (NIL T T T T T T) -7 NIL NIL NIL) (-642 1401841 1402775 1402815 "MLO" 1403232 MLO (NIL T) -9 NIL 1403472 NIL) (-641 1399722 1400249 1400844 "MLIFT" NIL MLIFT (NIL T T T T) -7 NIL NIL NIL) (-640 1399190 1399286 1399440 "MKUCFUNC" NIL MKUCFUNC (NIL T T T) -7 NIL NIL NIL) (-639 1398860 1398936 1399059 "MKRECORD" NIL MKRECORD (NIL T T) -7 NIL NIL NIL) (-638 1398072 1398258 1398486 "MKFUNC" NIL MKFUNC (NIL T) -7 NIL NIL NIL) (-637 1397565 1397681 1397837 "MKFLCFN" NIL MKFLCFN (NIL T) -7 NIL NIL NIL) (-636 1396937 1397051 1397236 "MKBCFUNC" NIL MKBCFUNC (NIL T T T T) -7 NIL NIL NIL) (-635 1395964 1396237 1396514 "MHROWRED" NIL MHROWRED (NIL T) -7 NIL NIL NIL) (-634 1395397 1395485 1395656 "MFINFACT" NIL MFINFACT (NIL T T T T) -7 NIL NIL NIL) (-633 1392578 1393448 1394318 "MESH" NIL MESH (NIL) -7 NIL NIL NIL) (-632 1391245 1391593 1391946 "MDDFACT" NIL MDDFACT (NIL T) -7 NIL NIL NIL) (-631 1387902 1390369 1390410 "MDAGG" 1390667 MDAGG (NIL T) -9 NIL 1390812 NIL) (-630 1387176 1387340 1387540 "MCDEN" NIL MCDEN (NIL T T) -7 NIL NIL NIL) (-629 1386254 1386540 1386770 "MAYBE" NIL MAYBE (NIL T) -8 NIL NIL NIL) (-628 1384351 1384928 1385489 "MATSTOR" NIL MATSTOR (NIL T) -7 NIL NIL NIL) (-627 1380122 1383941 1384188 "MATRIX" NIL MATRIX (NIL T) -8 NIL NIL NIL) (-626 1376471 1377240 1377974 "MATLIN" NIL MATLIN (NIL T T T T) -7 NIL NIL NIL) (-625 1375224 1375393 1375722 "MATCAT2" NIL MATCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-624 1364737 1368326 1368402 "MATCAT" 1373390 MATCAT (NIL T T T) -9 NIL 1374858 NIL) (-623 1362018 1363324 1364732 "MATCAT-" NIL MATCAT- (NIL T T T T) -7 NIL NIL NIL) (-622 1360419 1360779 1361163 "MAPPKG3" NIL MAPPKG3 (NIL T T T) -7 NIL NIL NIL) (-621 1359552 1359749 1359971 "MAPPKG2" NIL MAPPKG2 (NIL T T) -7 NIL NIL NIL) (-620 1358303 1358629 1358956 "MAPPKG1" NIL MAPPKG1 (NIL T) -7 NIL NIL NIL) (-619 1357465 1357867 1358043 "MAPPAST" NIL MAPPAST (NIL) -8 NIL NIL NIL) (-618 1357134 1357198 1357321 "MAPHACK3" NIL MAPHACK3 (NIL T T T) -7 NIL NIL NIL) (-617 1356782 1356855 1356969 "MAPHACK2" NIL MAPHACK2 (NIL T T) -7 NIL NIL NIL) (-616 1356317 1356432 1356574 "MAPHACK1" NIL MAPHACK1 (NIL T) -7 NIL NIL NIL) (-615 1354526 1355294 1355595 "MAGMA" NIL MAGMA (NIL T) -8 NIL NIL NIL) (-614 1354020 1354322 1354412 "MACROAST" NIL MACROAST (NIL) -8 NIL NIL NIL) (-613 1347529 1352335 1352376 "LZSTAGG" 1353153 LZSTAGG (NIL T) -9 NIL 1353443 NIL) (-612 1344648 1346082 1347524 "LZSTAGG-" NIL LZSTAGG- (NIL T T) -7 NIL NIL NIL) (-611 1342035 1343001 1343484 "LWORD" NIL LWORD (NIL T) -8 NIL NIL NIL) (-610 1341616 1341895 1341969 "LSTAST" NIL LSTAST (NIL) -8 NIL NIL NIL) (-609 1333844 1341477 1341611 "LSQM" NIL LSQM (NIL NIL T) -8 NIL NIL NIL) (-608 1333207 1333352 1333580 "LSPP" NIL LSPP (NIL T T T T) -7 NIL NIL NIL) (-607 1330691 1331389 1332101 "LSMP1" NIL LSMP1 (NIL T) -7 NIL NIL NIL) (-606 1328803 1329126 1329574 "LSMP" NIL LSMP (NIL T T T T) -7 NIL NIL NIL) (-605 1321972 1327890 1327931 "LSAGG" 1327993 LSAGG (NIL T) -9 NIL 1328071 NIL) (-604 1319666 1320765 1321967 "LSAGG-" NIL LSAGG- (NIL T T) -7 NIL NIL NIL) (-603 1317178 1319015 1319264 "LPOLY" NIL LPOLY (NIL T T) -8 NIL NIL NIL) (-602 1316845 1316936 1317059 "LPEFRAC" NIL LPEFRAC (NIL T) -7 NIL NIL NIL) (-601 1316516 1316595 1316623 "LOGIC" 1316734 LOGIC (NIL) -9 NIL 1316816 NIL) (-600 1316411 1316440 1316511 "LOGIC-" NIL LOGIC- (NIL T) -7 NIL NIL NIL) (-599 1315730 1315888 1316081 "LODOOPS" NIL LODOOPS (NIL T T) -7 NIL NIL NIL) (-598 1314515 1314764 1315115 "LODOF" NIL LODOF (NIL T T) -7 NIL NIL NIL) (-597 1310401 1313136 1313176 "LODOCAT" 1313608 LODOCAT (NIL T) -9 NIL 1313819 NIL) (-596 1310194 1310270 1310396 "LODOCAT-" NIL LODOCAT- (NIL T T) -7 NIL NIL NIL) (-595 1307258 1310071 1310189 "LODO2" NIL LODO2 (NIL T T) -8 NIL NIL NIL) (-594 1304420 1307208 1307253 "LODO1" NIL LODO1 (NIL T) -8 NIL NIL NIL) (-593 1301571 1304350 1304415 "LODO" NIL LODO (NIL T NIL) -8 NIL NIL NIL) (-592 1300624 1300799 1301101 "LODEEF" NIL LODEEF (NIL T T T) -7 NIL NIL NIL) (-591 1298788 1299886 1300139 "LO" NIL LO (NIL T T T) -8 NIL NIL NIL) (-590 1293883 1296947 1296988 "LNAGG" 1297850 LNAGG (NIL T) -9 NIL 1298285 NIL) (-589 1293270 1293537 1293878 "LNAGG-" NIL LNAGG- (NIL T T) -7 NIL NIL NIL) (-588 1289842 1290783 1291420 "LMOPS" NIL LMOPS (NIL T T NIL) -8 NIL NIL NIL) (-587 1289135 1289609 1289649 "LMODULE" 1289654 LMODULE (NIL T) -9 NIL 1289680 NIL) (-586 1286314 1288872 1288994 "LMDICT" NIL LMDICT (NIL T) -8 NIL NIL NIL) (-585 1285882 1286093 1286134 "LLINSET" 1286195 LLINSET (NIL T) -9 NIL 1286239 NIL) (-584 1285558 1285818 1285877 "LITERAL" NIL LITERAL (NIL T) -8 NIL NIL NIL) (-583 1285157 1285237 1285376 "LIST3" NIL LIST3 (NIL T T T) -7 NIL NIL NIL) (-582 1283608 1283956 1284355 "LIST2MAP" NIL LIST2MAP (NIL T T) -7 NIL NIL NIL) (-581 1282779 1282975 1283203 "LIST2" NIL LIST2 (NIL T T) -7 NIL NIL NIL) (-580 1275826 1282035 1282289 "LIST" NIL LIST (NIL T) -8 NIL NIL NIL) (-579 1275403 1275636 1275677 "LINSET" 1275682 LINSET (NIL T) -9 NIL 1275715 NIL) (-578 1274336 1275026 1275193 "LINFORM" NIL LINFORM (NIL T NIL) -8 NIL NIL NIL) (-577 1272633 1273357 1273397 "LINEXP" 1273883 LINEXP (NIL T) -9 NIL 1274156 NIL) (-576 1271342 1272242 1272423 "LINELT" NIL LINELT (NIL T NIL) -8 NIL NIL NIL) (-575 1270169 1270441 1270743 "LINDEP" NIL LINDEP (NIL T T) -7 NIL NIL NIL) (-574 1269382 1269971 1270081 "LINBASIS" NIL LINBASIS (NIL NIL) -8 NIL NIL NIL) (-573 1266932 1267654 1268404 "LIMITRF" NIL LIMITRF (NIL T) -7 NIL NIL NIL) (-572 1265562 1265859 1266250 "LIMITPS" NIL LIMITPS (NIL T T) -7 NIL NIL NIL) (-571 1264386 1264957 1264997 "LIECAT" 1265137 LIECAT (NIL T) -9 NIL 1265288 NIL) (-570 1264260 1264293 1264381 "LIECAT-" NIL LIECAT- (NIL T T) -7 NIL NIL NIL) (-569 1258548 1263950 1264178 "LIE" NIL LIE (NIL T T) -8 NIL NIL NIL) (-568 1250897 1258224 1258380 "LIB" NIL LIB (NIL) -8 NIL NIL NIL) (-567 1247349 1248298 1249233 "LGROBP" NIL LGROBP (NIL NIL T) -7 NIL NIL NIL) (-566 1245974 1246881 1246909 "LFCAT" 1247116 LFCAT (NIL) -9 NIL 1247255 NIL) (-565 1244216 1244545 1244889 "LF" NIL LF (NIL T T) -7 NIL NIL NIL) (-564 1241733 1242398 1243079 "LEXTRIPK" NIL LEXTRIPK (NIL T NIL) -7 NIL NIL NIL) (-563 1238745 1239723 1240226 "LEXP" NIL LEXP (NIL T T NIL) -8 NIL NIL NIL) (-562 1238236 1238539 1238630 "LETAST" NIL LETAST (NIL) -8 NIL NIL NIL) (-561 1236943 1237267 1237667 "LEADCDET" NIL LEADCDET (NIL T T T T) -7 NIL NIL NIL) (-560 1236209 1236294 1236520 "LAZM3PK" NIL LAZM3PK (NIL T T T T T T) -7 NIL NIL NIL) (-559 1231276 1234777 1235313 "LAUPOL" NIL LAUPOL (NIL T T) -8 NIL NIL NIL) (-558 1230901 1230951 1231111 "LAPLACE" NIL LAPLACE (NIL T T) -7 NIL NIL NIL) (-557 1229734 1230445 1230485 "LALG" 1230546 LALG (NIL T) -9 NIL 1230604 NIL) (-556 1229517 1229594 1229729 "LALG-" NIL LALG- (NIL T T) -7 NIL NIL NIL) (-555 1227434 1228785 1229036 "LA" NIL LA (NIL T T T) -8 NIL NIL NIL) (-554 1227263 1227293 1227334 "KVTFROM" 1227396 KVTFROM (NIL T) -9 NIL NIL NIL) (-553 1226079 1226794 1226983 "KTVLOGIC" NIL KTVLOGIC (NIL) -8 NIL NIL NIL) (-552 1225908 1225938 1225979 "KRCFROM" 1226041 KRCFROM (NIL T) -9 NIL NIL NIL) (-551 1225010 1225207 1225502 "KOVACIC" NIL KOVACIC (NIL T T) -7 NIL NIL NIL) (-550 1224839 1224869 1224910 "KONVERT" 1224972 KONVERT (NIL T) -9 NIL NIL NIL) (-549 1224668 1224698 1224739 "KOERCE" 1224801 KOERCE (NIL T) -9 NIL NIL NIL) (-548 1224238 1224331 1224463 "KERNEL2" NIL KERNEL2 (NIL T T) -7 NIL NIL NIL) (-547 1222291 1223185 1223557 "KERNEL" NIL KERNEL (NIL T) -8 NIL NIL NIL) (-546 1215468 1220483 1220537 "KDAGG" 1220913 KDAGG (NIL T T) -9 NIL 1221120 NIL) (-545 1215116 1215258 1215463 "KDAGG-" NIL KDAGG- (NIL T T T) -7 NIL NIL NIL) (-544 1207946 1214897 1215054 "KAFILE" NIL KAFILE (NIL T) -8 NIL NIL NIL) (-543 1207599 1207879 1207941 "JVMOP" NIL JVMOP (NIL) -8 NIL NIL NIL) (-542 1206569 1207068 1207317 "JVMMDACC" NIL JVMMDACC (NIL) -8 NIL NIL NIL) (-541 1205695 1206144 1206349 "JVMFDACC" NIL JVMFDACC (NIL) -8 NIL NIL NIL) (-540 1204561 1205052 1205351 "JVMCSTTG" NIL JVMCSTTG (NIL) -8 NIL NIL NIL) (-539 1203843 1204242 1204403 "JVMCFACC" NIL JVMCFACC (NIL) -8 NIL NIL NIL) (-538 1203556 1203790 1203838 "JVMBCODE" NIL JVMBCODE (NIL) -8 NIL NIL NIL) (-537 1197843 1203246 1203474 "JORDAN" NIL JORDAN (NIL T T) -8 NIL NIL NIL) (-536 1197261 1197594 1197714 "JOINAST" NIL JOINAST (NIL) -8 NIL NIL NIL) (-535 1193423 1195438 1195492 "IXAGG" 1196419 IXAGG (NIL T T) -9 NIL 1196876 NIL) (-534 1192629 1193000 1193418 "IXAGG-" NIL IXAGG- (NIL T T T) -7 NIL NIL NIL) (-533 1187883 1192565 1192624 "IVECTOR" NIL IVECTOR (NIL T NIL) -8 NIL NIL NIL) (-532 1186850 1187125 1187388 "ITUPLE" NIL ITUPLE (NIL T) -8 NIL NIL NIL) (-531 1185512 1185719 1186012 "ITRIGMNP" NIL ITRIGMNP (NIL T T T) -7 NIL NIL NIL) (-530 1184463 1184685 1184968 "ITFUN3" NIL ITFUN3 (NIL T T T) -7 NIL NIL NIL) (-529 1184138 1184201 1184324 "ITFUN2" NIL ITFUN2 (NIL T T) -7 NIL NIL NIL) (-528 1183400 1183772 1183946 "ITFORM" NIL ITFORM (NIL) -8 NIL NIL NIL) (-527 1181440 1182676 1182950 "ITAYLOR" NIL ITAYLOR (NIL T) -8 NIL NIL NIL) (-526 1171052 1176757 1177914 "ISUPS" NIL ISUPS (NIL T) -8 NIL NIL NIL) (-525 1170300 1170451 1170686 "ISUMP" NIL ISUMP (NIL T T T T) -7 NIL NIL NIL) (-524 1169791 1170094 1170185 "ISAST" NIL ISAST (NIL) -8 NIL NIL NIL) (-523 1169084 1169175 1169388 "IRURPK" NIL IRURPK (NIL T T T T T) -7 NIL NIL NIL) (-522 1168216 1168441 1168681 "IRSN" NIL IRSN (NIL) -7 NIL NIL NIL) (-521 1166629 1167010 1167438 "IRRF2F" NIL IRRF2F (NIL T) -7 NIL NIL NIL) (-520 1166414 1166458 1166534 "IRREDFFX" NIL IRREDFFX (NIL T) -7 NIL NIL NIL) (-519 1165264 1165561 1165856 "IROOT" NIL IROOT (NIL T) -7 NIL NIL NIL) (-518 1164537 1164888 1165039 "IRFORM" NIL IRFORM (NIL) -8 NIL NIL NIL) (-517 1163740 1163871 1164084 "IR2F" NIL IR2F (NIL T T) -7 NIL NIL NIL) (-516 1161895 1162392 1162936 "IR2" NIL IR2 (NIL T T) -7 NIL NIL NIL) (-515 1159008 1160244 1160933 "IR" NIL IR (NIL T) -8 NIL NIL NIL) (-514 1158833 1158873 1158933 "IPRNTPK" NIL IPRNTPK (NIL) -7 NIL NIL NIL) (-513 1154895 1158759 1158828 "IPF" NIL IPF (NIL NIL) -8 NIL NIL NIL) (-512 1152962 1154834 1154890 "IPADIC" NIL IPADIC (NIL NIL NIL) -8 NIL NIL NIL) (-511 1152336 1152634 1152763 "IP4ADDR" NIL IP4ADDR (NIL) -8 NIL NIL NIL) (-510 1151789 1152077 1152209 "IOMODE" NIL IOMODE (NIL) -8 NIL NIL NIL) (-509 1150873 1151495 1151621 "IOBFILE" NIL IOBFILE (NIL) -8 NIL NIL NIL) (-508 1150286 1150777 1150805 "IOBCON" 1150810 IOBCON (NIL) -9 NIL 1150831 NIL) (-507 1149857 1149921 1150103 "INVLAPLA" NIL INVLAPLA (NIL T T) -7 NIL NIL NIL) (-506 1141901 1144272 1146597 "INTTR" NIL INTTR (NIL T T) -7 NIL NIL NIL) (-505 1139012 1139795 1140659 "INTTOOLS" NIL INTTOOLS (NIL T T) -7 NIL NIL NIL) (-504 1138689 1138786 1138903 "INTSLPE" NIL INTSLPE (NIL) -7 NIL NIL NIL) (-503 1136195 1138625 1138684 "INTRVL" NIL INTRVL (NIL T) -8 NIL NIL NIL) (-502 1134307 1134836 1135403 "INTRF" NIL INTRF (NIL T) -7 NIL NIL NIL) (-501 1133809 1133923 1134063 "INTRET" NIL INTRET (NIL T) -7 NIL NIL NIL) (-500 1132193 1132599 1133061 "INTRAT" NIL INTRAT (NIL T T) -7 NIL NIL NIL) (-499 1129972 1130566 1131177 "INTPM" NIL INTPM (NIL T T) -7 NIL NIL NIL) (-498 1127345 1127955 1128675 "INTPAF" NIL INTPAF (NIL T T T) -7 NIL NIL NIL) (-497 1126749 1126907 1127115 "INTHERTR" NIL INTHERTR (NIL T T) -7 NIL NIL NIL) (-496 1126268 1126354 1126542 "INTHERAL" NIL INTHERAL (NIL T T T T) -7 NIL NIL NIL) (-495 1124473 1124994 1125451 "INTHEORY" NIL INTHEORY (NIL) -7 NIL NIL NIL) (-494 1117555 1119208 1120937 "INTG0" NIL INTG0 (NIL T T T) -7 NIL NIL NIL) (-493 1116921 1117083 1117256 "INTFACT" NIL INTFACT (NIL T) -7 NIL NIL NIL) (-492 1114794 1115258 1115802 "INTEF" NIL INTEF (NIL T T) -7 NIL NIL NIL) (-491 1112982 1113870 1113898 "INTDOM" 1114197 INTDOM (NIL) -9 NIL 1114402 NIL) (-490 1112535 1112737 1112977 "INTDOM-" NIL INTDOM- (NIL T) -7 NIL NIL NIL) (-489 1108406 1110814 1110868 "INTCAT" 1111664 INTCAT (NIL T) -9 NIL 1111980 NIL) (-488 1107971 1108091 1108218 "INTBIT" NIL INTBIT (NIL) -7 NIL NIL NIL) (-487 1106811 1106983 1107289 "INTALG" NIL INTALG (NIL T T T T T) -7 NIL NIL NIL) (-486 1106384 1106480 1106637 "INTAF" NIL INTAF (NIL T T) -7 NIL NIL NIL) (-485 1099424 1106239 1106379 "INTABL" NIL INTABL (NIL T T T) -8 NIL NIL NIL) (-484 1098722 1099277 1099342 "INT8" NIL INT8 (NIL) -8 NIL NIL 1099376) (-483 1098019 1098574 1098639 "INT64" NIL INT64 (NIL) -8 NIL NIL 1098673) (-482 1097316 1097871 1097936 "INT32" NIL INT32 (NIL) -8 NIL NIL 1097970) (-481 1096613 1097168 1097233 "INT16" NIL INT16 (NIL) -8 NIL NIL 1097267) (-480 1093140 1096532 1096608 "INT" NIL INT (NIL) -8 NIL NIL NIL) (-479 1087261 1090680 1090708 "INS" 1091638 INS (NIL) -9 NIL 1092297 NIL) (-478 1085323 1086241 1087188 "INS-" NIL INS- (NIL T) -7 NIL NIL NIL) (-477 1084382 1084605 1084880 "INPSIGN" NIL INPSIGN (NIL T T) -7 NIL NIL NIL) (-476 1083596 1083737 1083934 "INPRODPF" NIL INPRODPF (NIL T T) -7 NIL NIL NIL) (-475 1082586 1082727 1082964 "INPRODFF" NIL INPRODFF (NIL T T T T) -7 NIL NIL NIL) (-474 1081738 1081902 1082162 "INNMFACT" NIL INNMFACT (NIL T T T T) -7 NIL NIL NIL) (-473 1081018 1081133 1081321 "INMODGCD" NIL INMODGCD (NIL T T NIL NIL) -7 NIL NIL NIL) (-472 1079757 1080026 1080350 "INFSP" NIL INFSP (NIL T T T) -7 NIL NIL NIL) (-471 1079037 1079178 1079361 "INFPROD0" NIL INFPROD0 (NIL T T) -7 NIL NIL NIL) (-470 1078700 1078772 1078870 "INFORM1" NIL INFORM1 (NIL T) -7 NIL NIL NIL) (-469 1075778 1077264 1077787 "INFORM" NIL INFORM (NIL) -8 NIL NIL NIL) (-468 1075377 1075484 1075598 "INFINITY" NIL INFINITY (NIL) -7 NIL NIL NIL) (-467 1074536 1075178 1075279 "INETCLTS" NIL INETCLTS (NIL) -8 NIL NIL NIL) (-466 1073386 1073654 1073975 "INEP" NIL INEP (NIL T T T) -7 NIL NIL NIL) (-465 1072458 1073316 1073381 "INDE" NIL INDE (NIL T) -8 NIL NIL NIL) (-464 1072083 1072163 1072280 "INCRMAPS" NIL INCRMAPS (NIL T) -7 NIL NIL NIL) (-463 1070998 1071542 1071746 "INBFILE" NIL INBFILE (NIL) -8 NIL NIL NIL) (-462 1067093 1068148 1069091 "INBFF" NIL INBFF (NIL T) -7 NIL NIL NIL) (-461 1065950 1066272 1066300 "INBCON" 1066812 INBCON (NIL) -9 NIL 1067077 NIL) (-460 1065404 1065669 1065945 "INBCON-" NIL INBCON- (NIL T) -7 NIL NIL NIL) (-459 1064898 1065200 1065290 "INAST" NIL INAST (NIL) -8 NIL NIL NIL) (-458 1064355 1064664 1064769 "IMPTAST" NIL IMPTAST (NIL) -8 NIL NIL NIL) (-457 1060455 1064247 1064350 "IMATRIX" NIL IMATRIX (NIL T NIL NIL) -8 NIL NIL NIL) (-456 1059295 1059434 1059749 "IMATQF" NIL IMATQF (NIL T T T T T T T T) -7 NIL NIL NIL) (-455 1057719 1057986 1058323 "IMATLIN" NIL IMATLIN (NIL T T T T) -7 NIL NIL NIL) (-454 1055535 1057601 1057714 "IIARRAY2" NIL IIARRAY2 (NIL T NIL NIL T T) -8 NIL NIL NIL) (-453 1050442 1055466 1055530 "IFF" NIL IFF (NIL NIL NIL) -8 NIL NIL NIL) (-452 1049822 1050156 1050271 "IFAST" NIL IFAST (NIL) -8 NIL NIL NIL) (-451 1044629 1049260 1049446 "IFARRAY" NIL IFARRAY (NIL T NIL) -8 NIL NIL NIL) (-450 1043691 1044551 1044624 "IFAMON" NIL IFAMON (NIL T T NIL) -8 NIL NIL NIL) (-449 1043263 1043340 1043394 "IEVALAB" 1043601 IEVALAB (NIL T T) -9 NIL NIL NIL) (-448 1043018 1043098 1043258 "IEVALAB-" NIL IEVALAB- (NIL T T T) -7 NIL NIL NIL) (-447 1042091 1042938 1043013 "IDPOAMS" NIL IDPOAMS (NIL T T) -8 NIL NIL NIL) (-446 1041233 1042011 1042086 "IDPOAM" NIL IDPOAM (NIL T T) -8 NIL NIL NIL) (-445 1040636 1041167 1041228 "IDPO" NIL IDPO (NIL T T) -8 NIL NIL NIL) (-444 1039116 1039640 1039691 "IDPC" 1040197 IDPC (NIL T T) -9 NIL 1040477 NIL) (-443 1038482 1039038 1039111 "IDPAM" NIL IDPAM (NIL T T) -8 NIL NIL NIL) (-442 1037731 1038404 1038477 "IDPAG" NIL IDPAG (NIL T T) -8 NIL NIL NIL) (-441 1037424 1037637 1037697 "IDENT" NIL IDENT (NIL) -8 NIL NIL NIL) (-440 1034495 1035376 1036268 "IDECOMP" NIL IDECOMP (NIL NIL NIL) -7 NIL NIL NIL) (-439 1028121 1029398 1030437 "IDEAL" NIL IDEAL (NIL T T T T) -8 NIL NIL NIL) (-438 1027383 1027513 1027712 "ICDEN" NIL ICDEN (NIL T T T T) -7 NIL NIL NIL) (-437 1026556 1027055 1027193 "ICARD" NIL ICARD (NIL) -8 NIL NIL NIL) (-436 1024945 1025276 1025667 "IBPTOOLS" NIL IBPTOOLS (NIL T T T T) -7 NIL NIL NIL) (-435 1020714 1024901 1024940 "IBITS" NIL IBITS (NIL NIL) -8 NIL NIL NIL) (-434 1017972 1018596 1019291 "IBATOOL" NIL IBATOOL (NIL T T T) -7 NIL NIL NIL) (-433 1016198 1016678 1017211 "IBACHIN" NIL IBACHIN (NIL T T T) -7 NIL NIL NIL) (-432 1013962 1016090 1016193 "IARRAY2" NIL IARRAY2 (NIL T NIL NIL) -8 NIL NIL NIL) (-431 1009831 1013900 1013957 "IARRAY1" NIL IARRAY1 (NIL T NIL) -8 NIL NIL NIL) (-430 1003474 1008795 1009263 "IAN" NIL IAN (NIL) -8 NIL NIL NIL) (-429 1003042 1003105 1003278 "IALGFACT" NIL IALGFACT (NIL T T T T) -7 NIL NIL NIL) (-428 1002534 1002683 1002711 "HYPCAT" 1002918 HYPCAT (NIL) -9 NIL NIL NIL) (-427 1002190 1002343 1002529 "HYPCAT-" NIL HYPCAT- (NIL T) -7 NIL NIL NIL) (-426 1001803 1002048 1002131 "HOSTNAME" NIL HOSTNAME (NIL) -8 NIL NIL NIL) (-425 1001636 1001685 1001726 "HOMOTOP" 1001731 HOMOTOP (NIL T) -9 NIL 1001764 NIL) (-424 998204 999578 999619 "HOAGG" 1000594 HOAGG (NIL T) -9 NIL 1001315 NIL) (-423 997210 997680 998199 "HOAGG-" NIL HOAGG- (NIL T T) -7 NIL NIL NIL) (-422 990474 996935 997083 "HEXADEC" NIL HEXADEC (NIL) -8 NIL NIL NIL) (-421 989409 989667 989930 "HEUGCD" NIL HEUGCD (NIL T) -7 NIL NIL NIL) (-420 988376 989274 989404 "HELLFDIV" NIL HELLFDIV (NIL T T T T) -8 NIL NIL NIL) (-419 986570 988209 988297 "HEAP" NIL HEAP (NIL T) -8 NIL NIL NIL) (-418 985885 986237 986370 "HEADAST" NIL HEADAST (NIL) -8 NIL NIL NIL) (-417 979438 985818 985880 "HDP" NIL HDP (NIL NIL T) -8 NIL NIL NIL) (-416 972641 979174 979325 "HDMP" NIL HDMP (NIL NIL T) -8 NIL NIL NIL) (-415 972094 972251 972414 "HB" NIL HB (NIL) -7 NIL NIL NIL) (-414 965177 971985 972089 "HASHTBL" NIL HASHTBL (NIL T T NIL) -8 NIL NIL NIL) (-413 964668 964971 965062 "HASAST" NIL HASAST (NIL) -8 NIL NIL NIL) (-412 962282 964455 964634 "HACKPI" NIL HACKPI (NIL) -8 NIL NIL NIL) (-411 957675 962165 962277 "GTSET" NIL GTSET (NIL T T T T) -8 NIL NIL NIL) (-410 950761 957572 957670 "GSTBL" NIL GSTBL (NIL T T T NIL) -8 NIL NIL NIL) (-409 942762 950130 950385 "GSERIES" NIL GSERIES (NIL T NIL NIL) -8 NIL NIL NIL) (-408 941786 942295 942323 "GROUP" 942526 GROUP (NIL) -9 NIL 942660 NIL) (-407 941329 941530 941781 "GROUP-" NIL GROUP- (NIL T) -7 NIL NIL NIL) (-406 940001 940340 940727 "GROEBSOL" NIL GROEBSOL (NIL NIL T T) -7 NIL NIL NIL) (-405 938823 939180 939231 "GRMOD" 939760 GRMOD (NIL T T) -9 NIL 939926 NIL) (-404 938642 938690 938818 "GRMOD-" NIL GRMOD- (NIL T T T) -7 NIL NIL NIL) (-403 934773 935981 936978 "GRIMAGE" NIL GRIMAGE (NIL) -8 NIL NIL NIL) (-402 933495 933819 934134 "GRDEF" NIL GRDEF (NIL) -7 NIL NIL NIL) (-401 933048 933176 933317 "GRAY" NIL GRAY (NIL) -7 NIL NIL NIL) (-400 932121 932620 932671 "GRALG" 932824 GRALG (NIL T T) -9 NIL 932914 NIL) (-399 931840 931941 932116 "GRALG-" NIL GRALG- (NIL T T T) -7 NIL NIL NIL) (-398 928557 931522 931698 "GPOLSET" NIL GPOLSET (NIL T T T T) -8 NIL NIL NIL) (-397 927970 928033 928290 "GOSPER" NIL GOSPER (NIL T T T T T) -7 NIL NIL NIL) (-396 923856 924720 925245 "GMODPOL" NIL GMODPOL (NIL NIL T T T NIL T) -8 NIL NIL NIL) (-395 923031 923233 923471 "GHENSEL" NIL GHENSEL (NIL T T) -7 NIL NIL NIL) (-394 918034 918961 919980 "GENUPS" NIL GENUPS (NIL T T) -7 NIL NIL NIL) (-393 917782 917839 917928 "GENUFACT" NIL GENUFACT (NIL T) -7 NIL NIL NIL) (-392 917264 917353 917518 "GENPGCD" NIL GENPGCD (NIL T T T T) -7 NIL NIL NIL) (-391 916773 916814 917027 "GENMFACT" NIL GENMFACT (NIL T T T T T) -7 NIL NIL NIL) (-390 915574 915857 916161 "GENEEZ" NIL GENEEZ (NIL T T) -7 NIL NIL NIL) (-389 908913 915264 915425 "GDMP" NIL GDMP (NIL NIL T T) -8 NIL NIL NIL) (-388 898728 903703 904807 "GCNAALG" NIL GCNAALG (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-387 896842 897883 897911 "GCDDOM" 898166 GCDDOM (NIL) -9 NIL 898323 NIL) (-386 896465 896622 896837 "GCDDOM-" NIL GCDDOM- (NIL T) -7 NIL NIL NIL) (-385 887258 889728 892116 "GBINTERN" NIL GBINTERN (NIL T T T T) -7 NIL NIL NIL) (-384 885393 885718 886136 "GBF" NIL GBF (NIL T T T T) -7 NIL NIL NIL) (-383 884334 884523 884790 "GBEUCLID" NIL GBEUCLID (NIL T T T T) -7 NIL NIL NIL) (-382 883205 883412 883716 "GB" NIL GB (NIL T T T T) -7 NIL NIL NIL) (-381 882668 882810 882958 "GAUSSFAC" NIL GAUSSFAC (NIL) -7 NIL NIL NIL) (-380 881280 881628 881941 "GALUTIL" NIL GALUTIL (NIL T) -7 NIL NIL NIL) (-379 879825 880146 880468 "GALPOLYU" NIL GALPOLYU (NIL T T) -7 NIL NIL NIL) (-378 877451 877807 878212 "GALFACTU" NIL GALFACTU (NIL T T T) -7 NIL NIL NIL) (-377 870703 872364 873942 "GALFACT" NIL GALFACT (NIL T) -7 NIL NIL NIL) (-376 870355 870576 870644 "FUNDESC" NIL FUNDESC (NIL) -8 NIL NIL NIL) (-375 869979 870200 870281 "FUNCTION" NIL FUNCTION (NIL NIL) -8 NIL NIL NIL) (-374 868076 868759 869219 "FT" NIL FT (NIL) -8 NIL NIL NIL) (-373 866669 866976 867368 "FSUPFACT" NIL FSUPFACT (NIL T T T) -7 NIL NIL NIL) (-372 865324 865683 866007 "FST" NIL FST (NIL) -8 NIL NIL NIL) (-371 864627 864751 864938 "FSRED" NIL FSRED (NIL T T) -7 NIL NIL NIL) (-370 863601 863867 864214 "FSPRMELT" NIL FSPRMELT (NIL T T) -7 NIL NIL NIL) (-369 861259 861789 862271 "FSPECF" NIL FSPECF (NIL T T) -7 NIL NIL NIL) (-368 860842 860902 861071 "FSINT" NIL FSINT (NIL T T) -7 NIL NIL NIL) (-367 859206 860056 860359 "FSERIES" NIL FSERIES (NIL T T) -8 NIL NIL NIL) (-366 858354 858488 858711 "FSCINT" NIL FSCINT (NIL T T) -7 NIL NIL NIL) (-365 857525 857686 857913 "FSAGG2" NIL FSAGG2 (NIL T T T T) -7 NIL NIL NIL) (-364 853508 856459 856500 "FSAGG" 856870 FSAGG (NIL T) -9 NIL 857129 NIL) (-363 851862 852621 853413 "FSAGG-" NIL FSAGG- (NIL T T) -7 NIL NIL NIL) (-362 849818 850114 850658 "FS2UPS" NIL FS2UPS (NIL T T T T T NIL) -7 NIL NIL NIL) (-361 848865 849047 849347 "FS2EXPXP" NIL FS2EXPXP (NIL T T NIL NIL) -7 NIL NIL NIL) (-360 848546 848595 848722 "FS2" NIL FS2 (NIL T T T T) -7 NIL NIL NIL) (-359 828801 838203 838244 "FS" 842114 FS (NIL T) -9 NIL 844392 NIL) (-358 821032 824525 828504 "FS-" NIL FS- (NIL T T) -7 NIL NIL NIL) (-357 820566 820693 820845 "FRUTIL" NIL FRUTIL (NIL T) -7 NIL NIL NIL) (-356 815120 818247 818287 "FRNAALG" 819607 FRNAALG (NIL T) -9 NIL 820205 NIL) (-355 811861 813112 814370 "FRNAALG-" NIL FRNAALG- (NIL T T) -7 NIL NIL NIL) (-354 811542 811591 811718 "FRNAAF2" NIL FRNAAF2 (NIL T T T T) -7 NIL NIL NIL) (-353 810029 810586 810880 "FRMOD" NIL FRMOD (NIL T T T T NIL) -8 NIL NIL NIL) (-352 809315 809408 809695 "FRIDEAL2" NIL FRIDEAL2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-351 807149 807915 808231 "FRIDEAL" NIL FRIDEAL (NIL T T T T) -8 NIL NIL NIL) (-350 806258 806701 806742 "FRETRCT" 806747 FRETRCT (NIL T) -9 NIL 806918 NIL) (-349 805631 805909 806253 "FRETRCT-" NIL FRETRCT- (NIL T T) -7 NIL NIL NIL) (-348 802437 803895 803954 "FRAMALG" 804836 FRAMALG (NIL T T) -9 NIL 805128 NIL) (-347 801033 801584 802214 "FRAMALG-" NIL FRAMALG- (NIL T T T) -7 NIL NIL NIL) (-346 800726 800789 800896 "FRAC2" NIL FRAC2 (NIL T T) -7 NIL NIL NIL) (-345 794431 800531 800721 "FRAC" NIL FRAC (NIL T) -8 NIL NIL NIL) (-344 794124 794187 794294 "FR2" NIL FR2 (NIL T T) -7 NIL NIL NIL) (-343 786496 791003 792331 "FR" NIL FR (NIL T) -8 NIL NIL NIL) (-342 780336 783777 783805 "FPS" 784924 FPS (NIL) -9 NIL 785480 NIL) (-341 779893 780026 780190 "FPS-" NIL FPS- (NIL T) -7 NIL NIL NIL) (-340 776766 778746 778774 "FPC" 778999 FPC (NIL) -9 NIL 779141 NIL) (-339 776612 776664 776761 "FPC-" NIL FPC- (NIL T) -7 NIL NIL NIL) (-338 775389 776098 776139 "FPATMAB" 776144 FPATMAB (NIL T) -9 NIL 776296 NIL) (-337 773819 774415 774762 "FPARFRAC" NIL FPARFRAC (NIL T T) -8 NIL NIL NIL) (-336 773394 773452 773625 "FORDER" NIL FORDER (NIL T T T T) -7 NIL NIL NIL) (-335 771929 772792 772966 "FNLA" NIL FNLA (NIL NIL NIL T) -8 NIL NIL NIL) (-334 770544 771049 771077 "FNCAT" 771534 FNCAT (NIL) -9 NIL 771791 NIL) (-333 770001 770511 770539 "FNAME" NIL FNAME (NIL) -8 NIL NIL NIL) (-332 768588 769950 769996 "FMONOID" NIL FMONOID (NIL T) -8 NIL NIL NIL) (-331 765176 766534 766575 "FMONCAT" 767792 FMONCAT (NIL T) -9 NIL 768396 NIL) (-330 762065 763112 763165 "FMCAT" 764346 FMCAT (NIL T T) -9 NIL 764838 NIL) (-329 760797 761888 761987 "FM1" NIL FM1 (NIL T T) -8 NIL NIL NIL) (-328 759925 760645 760792 "FM" NIL FM (NIL T T) -8 NIL NIL NIL) (-327 758112 758564 759058 "FLOATRP" NIL FLOATRP (NIL T) -7 NIL NIL NIL) (-326 756047 756583 757161 "FLOATCP" NIL FLOATCP (NIL T) -7 NIL NIL NIL) (-325 749497 754384 754998 "FLOAT" NIL FLOAT (NIL) -8 NIL NIL NIL) (-324 748009 749079 749119 "FLINEXP" 749124 FLINEXP (NIL T) -9 NIL 749217 NIL) (-323 747418 747677 748004 "FLINEXP-" NIL FLINEXP- (NIL T T) -7 NIL NIL NIL) (-322 746633 746792 747013 "FLASORT" NIL FLASORT (NIL T T) -7 NIL NIL NIL) (-321 743547 744595 744647 "FLALG" 745874 FLALG (NIL T T) -9 NIL 746341 NIL) (-320 742718 742879 743106 "FLAGG2" NIL FLAGG2 (NIL T T T T) -7 NIL NIL NIL) (-319 736127 740137 740178 "FLAGG" 741433 FLAGG (NIL T) -9 NIL 742078 NIL) (-318 735235 735639 736122 "FLAGG-" NIL FLAGG- (NIL T T) -7 NIL NIL NIL) (-317 731858 733060 733119 "FINRALG" 734247 FINRALG (NIL T T) -9 NIL 734755 NIL) (-316 731249 731514 731853 "FINRALG-" NIL FINRALG- (NIL T T T) -7 NIL NIL NIL) (-315 730547 730843 730871 "FINITE" 731067 FINITE (NIL) -9 NIL 731174 NIL) (-314 730455 730481 730542 "FINITE-" NIL FINITE- (NIL T) -7 NIL NIL NIL) (-313 722447 725007 725047 "FINAALG" 728699 FINAALG (NIL T) -9 NIL 730137 NIL) (-312 718714 719959 721082 "FINAALG-" NIL FINAALG- (NIL T T) -7 NIL NIL NIL) (-311 717266 717685 717739 "FILECAT" 718423 FILECAT (NIL T T) -9 NIL 718639 NIL) (-310 716617 717091 717194 "FILE" NIL FILE (NIL T) -8 NIL NIL NIL) (-309 713927 715743 715771 "FIELD" 715811 FIELD (NIL) -9 NIL 715891 NIL) (-308 712952 713413 713922 "FIELD-" NIL FIELD- (NIL T) -7 NIL NIL NIL) (-307 710956 711902 712248 "FGROUP" NIL FGROUP (NIL T) -8 NIL NIL NIL) (-306 710199 710380 710599 "FGLMICPK" NIL FGLMICPK (NIL T NIL) -7 NIL NIL NIL) (-305 705533 710137 710194 "FFX" NIL FFX (NIL T NIL) -8 NIL NIL NIL) (-304 705195 705262 705397 "FFSLPE" NIL FFSLPE (NIL T T T) -7 NIL NIL NIL) (-303 704735 704777 704986 "FFPOLY2" NIL FFPOLY2 (NIL T T) -7 NIL NIL NIL) (-302 701415 702292 703069 "FFPOLY" NIL FFPOLY (NIL T) -7 NIL NIL NIL) (-301 696763 701347 701410 "FFP" NIL FFP (NIL T NIL) -8 NIL NIL NIL) (-300 691506 696252 696442 "FFNBX" NIL FFNBX (NIL T NIL) -8 NIL NIL NIL) (-299 686051 690787 691045 "FFNBP" NIL FFNBP (NIL T NIL) -8 NIL NIL NIL) (-298 680322 685502 685713 "FFNB" NIL FFNB (NIL NIL NIL) -8 NIL NIL NIL) (-297 679345 679555 679870 "FFINTBAS" NIL FFINTBAS (NIL T T T) -7 NIL NIL NIL) (-296 674848 677490 677518 "FFIELDC" 678137 FFIELDC (NIL) -9 NIL 678512 NIL) (-295 673917 674357 674843 "FFIELDC-" NIL FFIELDC- (NIL T) -7 NIL NIL NIL) (-294 673532 673590 673714 "FFHOM" NIL FFHOM (NIL T T T) -7 NIL NIL NIL) (-293 671676 672199 672716 "FFF" NIL FFF (NIL T) -7 NIL NIL NIL) (-292 666834 671475 671576 "FFCGX" NIL FFCGX (NIL T NIL) -8 NIL NIL NIL) (-291 661998 666623 666730 "FFCGP" NIL FFCGP (NIL T NIL) -8 NIL NIL NIL) (-290 656728 661789 661897 "FFCG" NIL FFCG (NIL NIL NIL) -8 NIL NIL NIL) (-289 656182 656231 656466 "FFCAT2" NIL FFCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-288 634819 645791 645877 "FFCAT" 651027 FFCAT (NIL T T T) -9 NIL 652463 NIL) (-287 631059 632285 633591 "FFCAT-" NIL FFCAT- (NIL T T T T) -7 NIL NIL NIL) (-286 625966 630990 631054 "FF" NIL FF (NIL NIL NIL) -8 NIL NIL NIL) (-285 624858 625327 625368 "FEVALAB" 625452 FEVALAB (NIL T) -9 NIL 625713 NIL) (-284 624263 624515 624853 "FEVALAB-" NIL FEVALAB- (NIL T T) -7 NIL NIL NIL) (-283 621121 622001 622116 "FDIVCAT" 623683 FDIVCAT (NIL T T T T) -9 NIL 624119 NIL) (-282 620915 620947 621116 "FDIVCAT-" NIL FDIVCAT- (NIL T T T T T) -7 NIL NIL NIL) (-281 620222 620315 620592 "FDIV2" NIL FDIV2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-280 618740 619706 619909 "FDIV" NIL FDIV (NIL T T T T) -8 NIL NIL NIL) (-279 617833 618217 618419 "FCTRDATA" NIL FCTRDATA (NIL) -8 NIL NIL NIL) (-278 616955 617444 617584 "FCOMP" NIL FCOMP (NIL T) -8 NIL NIL NIL) (-277 608604 613185 613225 "FAXF" 615026 FAXF (NIL T) -9 NIL 615716 NIL) (-276 606520 607324 608139 "FAXF-" NIL FAXF- (NIL T T) -7 NIL NIL NIL) (-275 601384 606042 606216 "FARRAY" NIL FARRAY (NIL T) -8 NIL NIL NIL) (-274 595906 598265 598317 "FAMR" 599328 FAMR (NIL T T) -9 NIL 599787 NIL) (-273 595105 595470 595901 "FAMR-" NIL FAMR- (NIL T T T) -7 NIL NIL NIL) (-272 594158 595047 595100 "FAMONOID" NIL FAMONOID (NIL T) -8 NIL NIL NIL) (-271 591783 592631 592684 "FAMONC" 593625 FAMONC (NIL T T) -9 NIL 594010 NIL) (-270 590371 591641 591778 "FAGROUP" NIL FAGROUP (NIL T) -8 NIL NIL NIL) (-269 588451 588812 589214 "FACUTIL" NIL FACUTIL (NIL T T T T) -7 NIL NIL NIL) (-268 587728 587925 588147 "FACTFUNC" NIL FACTFUNC (NIL T) -7 NIL NIL NIL) (-267 579652 587175 587374 "EXPUPXS" NIL EXPUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-266 577683 578249 578831 "EXPRTUBE" NIL EXPRTUBE (NIL) -7 NIL NIL NIL) (-265 574585 575227 575947 "EXPRODE" NIL EXPRODE (NIL T T) -7 NIL NIL NIL) (-264 569742 570449 571254 "EXPR2UPS" NIL EXPR2UPS (NIL T T) -7 NIL NIL NIL) (-263 569431 569494 569603 "EXPR2" NIL EXPR2 (NIL T T) -7 NIL NIL NIL) (-262 554384 568480 568906 "EXPR" NIL EXPR (NIL T) -8 NIL NIL NIL) (-261 544975 553704 553992 "EXPEXPAN" NIL EXPEXPAN (NIL T T NIL NIL) -8 NIL NIL NIL) (-260 544469 544771 544861 "EXITAST" NIL EXITAST (NIL) -8 NIL NIL NIL) (-259 544245 544435 544464 "EXIT" NIL EXIT (NIL) -8 NIL NIL NIL) (-258 543934 544002 544115 "EVALCYC" NIL EVALCYC (NIL T) -7 NIL NIL NIL) (-257 543451 543593 543634 "EVALAB" 543804 EVALAB (NIL T) -9 NIL 543908 NIL) (-256 543079 543225 543446 "EVALAB-" NIL EVALAB- (NIL T T) -7 NIL NIL NIL) (-255 540184 541717 541745 "EUCDOM" 542299 EUCDOM (NIL) -9 NIL 542648 NIL) (-254 539111 539604 540179 "EUCDOM-" NIL EUCDOM- (NIL T) -7 NIL NIL NIL) (-253 538836 538892 538992 "ES2" NIL ES2 (NIL T T) -7 NIL NIL NIL) (-252 538524 538588 538697 "ES1" NIL ES1 (NIL T T) -7 NIL NIL NIL) (-251 532295 534195 534223 "ES" 536965 ES (NIL) -9 NIL 538349 NIL) (-250 528810 530342 532134 "ES-" NIL ES- (NIL T) -7 NIL NIL NIL) (-249 528158 528311 528487 "ERROR" NIL ERROR (NIL) -7 NIL NIL NIL) (-248 521247 528062 528153 "EQTBL" NIL EQTBL (NIL T T) -8 NIL NIL NIL) (-247 520936 520999 521108 "EQ2" NIL EQ2 (NIL T T) -7 NIL NIL NIL) (-246 514662 517688 519121 "EQ" NIL EQ (NIL T) -8 NIL NIL NIL) (-245 510965 512061 513154 "EP" NIL EP (NIL T) -7 NIL NIL NIL) (-244 509794 510144 510449 "ENV" NIL ENV (NIL) -8 NIL NIL NIL) (-243 508741 509410 509438 "ENTIRER" 509443 ENTIRER (NIL) -9 NIL 509487 NIL) (-242 505438 507171 507520 "EMR" NIL EMR (NIL T T T NIL NIL NIL) -8 NIL NIL NIL) (-241 504530 504741 504795 "ELTAGG" 505175 ELTAGG (NIL T T) -9 NIL 505386 NIL) (-240 504310 504384 504525 "ELTAGG-" NIL ELTAGG- (NIL T T T) -7 NIL NIL NIL) (-239 504056 504091 504145 "ELTAB" 504229 ELTAB (NIL T T) -9 NIL 504281 NIL) (-238 503307 503477 503676 "ELFUTS" NIL ELFUTS (NIL T T) -7 NIL NIL NIL) (-237 503031 503105 503133 "ELEMFUN" 503238 ELEMFUN (NIL) -9 NIL NIL NIL) (-236 502931 502958 503026 "ELEMFUN-" NIL ELEMFUN- (NIL T) -7 NIL NIL NIL) (-235 497477 500972 501013 "ELAGG" 501950 ELAGG (NIL T) -9 NIL 502410 NIL) (-234 496275 496813 497472 "ELAGG-" NIL ELAGG- (NIL T T) -7 NIL NIL NIL) (-233 495693 495860 496016 "ELABOR" NIL ELABOR (NIL) -8 NIL NIL NIL) (-232 494606 494925 495204 "ELABEXPR" NIL ELABEXPR (NIL) -8 NIL NIL NIL) (-231 487999 489997 490824 "EFUPXS" NIL EFUPXS (NIL T T T T) -7 NIL NIL NIL) (-230 481978 483974 484784 "EFULS" NIL EFULS (NIL T T T) -7 NIL NIL NIL) (-229 479792 480198 480669 "EFSTRUC" NIL EFSTRUC (NIL T T) -7 NIL NIL NIL) (-228 470792 472705 474246 "EF" NIL EF (NIL T T) -7 NIL NIL NIL) (-227 469905 470406 470555 "EAB" NIL EAB (NIL) -8 NIL NIL NIL) (-226 468603 469277 469317 "DVARCAT" 469600 DVARCAT (NIL T) -9 NIL 469740 NIL) (-225 468022 468286 468598 "DVARCAT-" NIL DVARCAT- (NIL T T) -7 NIL NIL NIL) (-224 460153 467890 468017 "DSMP" NIL DSMP (NIL T T T) -8 NIL NIL NIL) (-223 458491 459282 459323 "DSEXT" 459686 DSEXT (NIL T) -9 NIL 459980 NIL) (-222 457296 457820 458486 "DSEXT-" NIL DSEXT- (NIL T T) -7 NIL NIL NIL) (-221 457020 457085 457183 "DROPT1" NIL DROPT1 (NIL T) -7 NIL NIL NIL) (-220 453176 454390 455519 "DROPT0" NIL DROPT0 (NIL) -7 NIL NIL NIL) (-219 448834 450185 451245 "DROPT" NIL DROPT (NIL) -8 NIL NIL NIL) (-218 447509 447870 448256 "DRAWPT" NIL DRAWPT (NIL) -7 NIL NIL NIL) (-217 447201 447258 447374 "DRAWHACK" NIL DRAWHACK (NIL T) -7 NIL NIL NIL) (-216 446186 446480 446766 "DRAWCX" NIL DRAWCX (NIL) -7 NIL NIL NIL) (-215 445771 445846 445996 "DRAWCURV" NIL DRAWCURV (NIL T T) -7 NIL NIL NIL) (-214 438280 440356 442435 "DRAWCFUN" NIL DRAWCFUN (NIL) -7 NIL NIL NIL) (-213 433861 434856 435911 "DRAW" NIL DRAW (NIL T) -7 NIL NIL NIL) (-212 430456 432525 432566 "DQAGG" 433195 DQAGG (NIL T) -9 NIL 433468 NIL) (-211 417063 424639 424721 "DPOLCAT" 426558 DPOLCAT (NIL T T T T) -9 NIL 427101 NIL) (-210 413471 415119 417058 "DPOLCAT-" NIL DPOLCAT- (NIL T T T T T) -7 NIL NIL NIL) (-209 406558 413369 413466 "DPMO" NIL DPMO (NIL NIL T T) -8 NIL NIL NIL) (-208 399554 406387 406553 "DPMM" NIL DPMM (NIL NIL T T T) -8 NIL NIL NIL) (-207 399147 399407 399496 "DOMTMPLT" NIL DOMTMPLT (NIL) -8 NIL NIL NIL) (-206 398561 399009 399089 "DOMCTOR" NIL DOMCTOR (NIL) -8 NIL NIL NIL) (-205 397847 398172 398323 "DOMAIN" NIL DOMAIN (NIL) -8 NIL NIL NIL) (-204 391050 397583 397734 "DMP" NIL DMP (NIL NIL T) -8 NIL NIL NIL) (-203 388830 390116 390156 "DMEXT" 390161 DMEXT (NIL T) -9 NIL 390336 NIL) (-202 388486 388548 388692 "DLP" NIL DLP (NIL T) -7 NIL NIL NIL) (-201 381811 387971 388161 "DLIST" NIL DLIST (NIL T) -8 NIL NIL NIL) (-200 378477 380634 380675 "DLAGG" 381225 DLAGG (NIL T) -9 NIL 381454 NIL) (-199 376890 377699 377727 "DIVRING" 377819 DIVRING (NIL) -9 NIL 377902 NIL) (-198 376341 376585 376885 "DIVRING-" NIL DIVRING- (NIL T) -7 NIL NIL NIL) (-197 374769 375186 375592 "DISPLAY" NIL DISPLAY (NIL) -7 NIL NIL NIL) (-196 373806 374027 374292 "DIRPROD2" NIL DIRPROD2 (NIL NIL T T) -7 NIL NIL NIL) (-195 367379 373738 373801 "DIRPROD" NIL DIRPROD (NIL NIL T) -8 NIL NIL NIL) (-194 355798 362159 362212 "DIRPCAT" 362468 DIRPCAT (NIL NIL T) -9 NIL 363341 NIL) (-193 353804 354574 355461 "DIRPCAT-" NIL DIRPCAT- (NIL 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T) ((-241 |#2| $) -12 (|has| |#1| (-311)) (|has| |#2| (-241 |#2| |#2|))) ((-241 $ $) |has| (-483) (-1024)) ((-245) OR (|has| |#1| (-494)) (|has| |#1| (-311))) ((-257) |has| |#1| (-311)) ((-259 |#2|) -12 (|has| |#1| (-311)) (|has| |#2| (-259 |#2|))) ((-311) |has| |#1| (-311)) ((-287 |#2|) |has| |#1| (-311)) ((-326 |#2|) |has| |#1| (-311)) ((-340 |#2|) |has| |#1| (-311)) ((-389) |has| |#1| (-311)) ((-430) |has| |#1| (-38 (-347 (-483)))) ((-452 (-1088) |#2|) -12 (|has| |#1| (-311)) (|has| |#2| (-452 (-1088) |#2|))) ((-452 |#2| |#2|) -12 (|has| |#1| (-311)) (|has| |#2| (-259 |#2|))) ((-494) OR (|has| |#1| (-494)) (|has| |#1| (-311))) ((-13) . T) ((-588 (-347 (-483))) OR (|has| |#1| (-311)) (|has| |#1| (-38 (-347 (-483))))) ((-588 (-483)) . T) ((-588 |#1|) . T) ((-588 |#2|) |has| |#1| (-311)) ((-588 $) . T) ((-590 (-347 (-483))) OR (|has| |#1| (-311)) (|has| |#1| (-38 (-347 (-483))))) ((-590 (-483)) -12 (|has| |#1| (-311)) (|has| |#2| (-580 (-483)))) ((-590 |#1|) . T) ((-590 |#2|) |has| |#1| (-311)) ((-590 $) . T) ((-582 (-347 (-483))) OR (|has| |#1| (-311)) (|has| |#1| (-38 (-347 (-483))))) ((-582 |#1|) |has| |#1| (-146)) ((-582 |#2|) |has| |#1| (-311)) ((-582 $) OR (|has| |#1| (-494)) (|has| |#1| (-311))) ((-580 (-483)) -12 (|has| |#1| (-311)) (|has| |#2| (-580 (-483)))) ((-580 |#2|) |has| |#1| (-311)) ((-654 (-347 (-483))) OR (|has| |#1| (-311)) (|has| |#1| (-38 (-347 (-483))))) ((-654 |#1|) |has| |#1| (-146)) ((-654 |#2|) |has| |#1| (-311)) ((-654 $) OR (|has| |#1| (-494)) (|has| |#1| (-311))) ((-663) . T) ((-714) -12 (|has| |#1| (-311)) (|has| |#2| (-740))) ((-716) -12 (|has| |#1| (-311)) (|has| |#2| (-740))) ((-718) -12 (|has| |#1| (-311)) (|has| |#2| (-740))) ((-721) -12 (|has| |#1| (-311)) (|has| |#2| (-740))) ((-740) -12 (|has| |#1| (-311)) (|has| |#2| (-740))) ((-755) -12 (|has| |#1| (-311)) (|has| |#2| (-740))) ((-756) OR (-12 (|has| |#1| (-311)) (|has| |#2| (-756))) (-12 (|has| |#1| (-311)) (|has| |#2| (-740)))) ((-759) OR (-12 (|has| |#1| (-311)) (|has| |#2| (-756))) (-12 (|has| |#1| (-311)) (|has| |#2| (-740)))) ((-806 $ (-1088)) OR (-12 (|has| |#1| (-809 (-1088))) (|has| |#1| (-15 * (|#1| (-483) |#1|)))) (-12 (|has| |#1| (-311)) (|has| |#2| (-811 (-1088)))) (-12 (|has| |#1| (-311)) (|has| |#2| (-809 (-1088))))) ((-809 (-1088)) OR (-12 (|has| |#1| (-809 (-1088))) (|has| |#1| (-15 * (|#1| (-483) |#1|)))) (-12 (|has| |#1| (-311)) (|has| |#2| (-809 (-1088))))) ((-811 (-1088)) OR (-12 (|has| |#1| (-809 (-1088))) (|has| |#1| (-15 * (|#1| (-483) |#1|)))) (-12 (|has| |#1| (-311)) (|has| |#2| (-811 (-1088)))) (-12 (|has| |#1| (-311)) (|has| |#2| (-809 (-1088))))) ((-796 (-327)) -12 (|has| |#1| (-311)) (|has| |#2| (-796 (-327)))) ((-796 (-483)) -12 (|has| |#1| (-311)) (|has| |#2| (-796 (-483)))) ((-794 |#2|) |has| |#1| (-311)) ((-821) -12 (|has| |#1| (-311)) (|has| |#2| (-821))) ((-886 |#1| (-483) (-993)) . T) ((-832) |has| |#1| (-311)) ((-904 |#2|) |has| |#1| (-311)) ((-915) |has| |#1| (-38 (-347 (-483)))) ((-933) -12 (|has| |#1| (-311)) (|has| |#2| (-933))) ((-950 (-347 (-483))) -12 (|has| |#1| (-311)) (|has| |#2| (-950 (-483)))) ((-950 (-483)) -12 (|has| |#1| (-311)) (|has| |#2| (-950 (-483)))) ((-950 (-1088)) -12 (|has| |#1| (-311)) (|has| |#2| (-950 (-1088)))) ((-950 |#2|) . T) ((-963 (-347 (-483))) OR (|has| |#1| (-311)) (|has| |#1| (-38 (-347 (-483))))) ((-963 |#1|) . T) ((-963 |#2|) |has| |#1| (-311)) ((-963 $) OR (|has| |#1| (-494)) (|has| |#1| (-311)) (|has| |#1| (-146))) ((-968 (-347 (-483))) OR (|has| |#1| (-311)) (|has| |#1| (-38 (-347 (-483))))) ((-968 |#1|) . T) ((-968 |#2|) |has| |#1| (-311)) ((-968 $) OR (|has| |#1| (-494)) (|has| |#1| (-311)) (|has| |#1| (-146))) ((-961) . T) ((-969) . T) ((-1024) . T) ((-1059) . T) ((-1012) . T) ((-1064) -12 (|has| |#1| (-311)) (|has| |#2| (-1064))) ((-1113) |has| |#1| (-38 (-347 (-483)))) ((-1116) |has| |#1| (-38 (-347 (-483)))) ((-1127) . T) ((-1132) |has| |#1| (-311)) ((-1139 |#1|) . T) ((-1156 |#1| (-483)) . 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T) ((-23) . T) ((-47 |#1| (-694)) . T) ((-25) . T) ((-38 (-347 (-483))) |has| |#1| (-38 (-347 (-483)))) ((-38 |#1|) |has| |#1| (-146)) ((-38 $) OR (|has| |#1| (-821)) (|has| |#1| (-494)) (|has| |#1| (-389)) (|has| |#1| (-311))) ((-72) . T) ((-82 (-347 (-483)) (-347 (-483))) |has| |#1| (-38 (-347 (-483)))) ((-82 |#1| |#1|) . T) ((-82 $ $) OR (|has| |#1| (-821)) (|has| |#1| (-494)) (|has| |#1| (-389)) (|has| |#1| (-311)) (|has| |#1| (-146))) ((-104) . T) ((-118) |has| |#1| (-118)) ((-120) |has| |#1| (-120)) ((-555 (-347 (-483))) OR (|has| |#1| (-950 (-347 (-483)))) (|has| |#1| (-38 (-347 (-483))))) ((-555 (-483)) . T) ((-555 (-993)) . T) ((-555 |#1|) . T) ((-555 $) OR (|has| |#1| (-821)) (|has| |#1| (-494)) (|has| |#1| (-389)) (|has| |#1| (-311))) ((-552 (-772)) . 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NIL NIL NIL) (-1113 2486117 2486813 2486843 "TRANFUN" 2486878 TRANFUN (NIL) -9 NIL 2486944 NIL) (-1112 2485581 2485832 2486112 "TRANFUN-" NIL TRANFUN- (NIL T) -7 NIL NIL NIL) (-1111 2485418 2485456 2485517 "TOPSP" NIL TOPSP (NIL) -7 NIL NIL NIL) (-1110 2484875 2485006 2485157 "TOOLSIGN" NIL TOOLSIGN (NIL T) -7 NIL NIL NIL) (-1109 2483616 2484273 2484509 "TEXTFILE" NIL TEXTFILE (NIL) -8 NIL NIL NIL) (-1108 2483428 2483465 2483537 "TEX1" NIL TEX1 (NIL T) -7 NIL NIL NIL) (-1107 2481642 2482288 2482717 "TEX" NIL TEX (NIL) -8 NIL NIL NIL) (-1106 2480022 2480359 2480681 "TBCMPPK" NIL TBCMPPK (NIL T T) -7 NIL NIL NIL) (-1105 2471080 2477823 2477879 "TBAGG" 2478281 TBAGG (NIL T T) -9 NIL 2478494 NIL) (-1104 2467611 2469303 2471075 "TBAGG-" NIL TBAGG- (NIL T T T) -7 NIL NIL NIL) (-1103 2467088 2467213 2467358 "TANEXP" NIL TANEXP (NIL T) -7 NIL NIL NIL) (-1102 2466598 2466918 2467008 "TALGOP" NIL TALGOP (NIL T) -8 NIL NIL NIL) (-1101 2466095 2466212 2466350 "TABLEAU" NIL TABLEAU (NIL T) -8 NIL NIL NIL) (-1100 2459182 2465997 2466090 "TABLE" NIL TABLE (NIL T T) -8 NIL NIL NIL) (-1099 2454935 2456230 2457475 "TABLBUMP" NIL TABLBUMP (NIL T) -7 NIL NIL NIL) (-1098 2454304 2454463 2454644 "SYSTEM" NIL SYSTEM (NIL) -7 NIL NIL NIL) (-1097 2451458 2452211 2452994 "SYSSOLP" NIL SYSSOLP (NIL T) -7 NIL NIL NIL) (-1096 2451232 2451422 2451453 "SYSPTR" NIL SYSPTR (NIL) -8 NIL NIL NIL) (-1095 2450186 2450871 2450997 "SYSNNI" NIL SYSNNI (NIL NIL) -8 NIL NIL 2451183) (-1094 2449450 2449998 2450077 "SYSINT" NIL SYSINT (NIL NIL) -8 NIL NIL 2450137) (-1093 2446273 2447432 2448132 "SYNTAX" NIL SYNTAX (NIL) -8 NIL NIL NIL) (-1092 2443956 2444639 2445273 "SYMTAB" NIL SYMTAB (NIL) -8 NIL NIL NIL) (-1091 2440034 2441080 2442057 "SYMS" NIL SYMS (NIL) -8 NIL NIL NIL) (-1090 2437197 2439689 2439918 "SYMPOLY" NIL SYMPOLY (NIL T) -8 NIL NIL NIL) (-1089 2436793 2436880 2437002 "SYMFUNC" NIL SYMFUNC (NIL T) -7 NIL NIL NIL) (-1088 2433417 2434891 2435710 "SYMBOL" NIL SYMBOL (NIL) -8 NIL NIL NIL) (-1087 2426441 2432614 2432907 "SUTS" NIL SUTS (NIL T NIL NIL) -8 NIL NIL NIL) (-1086 2418191 2426032 2426294 "SUPXS" NIL SUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-1085 2417470 2417609 2417826 "SUPFRACF" NIL SUPFRACF (NIL T T T T) -7 NIL NIL NIL) (-1084 2417154 2417219 2417330 "SUP2" NIL SUP2 (NIL T T) -7 NIL NIL NIL) (-1083 2407941 2416866 2416991 "SUP" NIL SUP (NIL T) -8 NIL NIL NIL) (-1082 2406671 2406969 2407324 "SUMRF" NIL SUMRF (NIL T) -7 NIL NIL NIL) (-1081 2406076 2406154 2406345 "SUMFS" NIL SUMFS (NIL T T) -7 NIL NIL NIL) (-1080 2388292 2405575 2405816 "SULS" NIL SULS (NIL T NIL NIL) -8 NIL NIL NIL) (-1079 2387891 2388163 2388232 "SUCHTAST" NIL SUCHTAST (NIL) -8 NIL NIL NIL) (-1078 2387227 2387508 2387648 "SUCH" NIL SUCH (NIL T T) -8 NIL NIL NIL) (-1077 2381829 2383088 2384041 "SUBSPACE" NIL SUBSPACE (NIL NIL T) -8 NIL NIL NIL) (-1076 2381361 2381461 2381625 "SUBRESP" NIL SUBRESP (NIL T T) -7 NIL NIL NIL) (-1075 2376472 2377754 2378901 "STTFNC" NIL STTFNC (NIL T) -7 NIL NIL NIL) (-1074 2370930 2372401 2373712 "STTF" NIL STTF (NIL T) -7 NIL NIL NIL) (-1073 2363845 2365909 2367700 "STTAYLOR" NIL STTAYLOR (NIL T) -7 NIL NIL NIL) (-1072 2356675 2363757 2363840 "STRTBL" NIL STRTBL (NIL T) -8 NIL NIL NIL) (-1071 2351369 2356389 2356504 "STRING" NIL STRING (NIL) -8 NIL NIL NIL) (-1070 2350956 2351039 2351183 "STREAM3" NIL STREAM3 (NIL T T T) -7 NIL NIL NIL) (-1069 2350107 2350308 2350543 "STREAM2" NIL STREAM2 (NIL T T) -7 NIL NIL NIL) (-1068 2349847 2349905 2349998 "STREAM1" NIL STREAM1 (NIL T) -7 NIL NIL NIL) (-1067 2342585 2348052 2348658 "STREAM" NIL STREAM (NIL T) -8 NIL NIL NIL) (-1066 2341761 2341966 2342197 "STINPROD" NIL STINPROD (NIL T) -7 NIL NIL NIL) (-1065 2341006 2341377 2341524 "STEPAST" NIL STEPAST (NIL) -8 NIL NIL NIL) (-1064 2340494 2340736 2340766 "STEP" 2340860 STEP (NIL) -9 NIL 2340931 NIL) (-1063 2333597 2340412 2340489 "STBL" NIL STBL (NIL T T NIL) -8 NIL NIL NIL) (-1062 2327812 2332395 2332438 "STAGG" 2332865 STAGG (NIL T) -9 NIL 2333039 NIL) (-1061 2326191 2326939 2327807 "STAGG-" NIL STAGG- (NIL T T) -7 NIL NIL NIL) (-1060 2324348 2326018 2326110 "STACK" NIL STACK (NIL T) -8 NIL NIL NIL) (-1059 2323659 2324167 2324197 "SRING" 2324202 SRING (NIL) -9 NIL 2324222 NIL) (-1058 2316281 2322197 2322636 "SREGSET" NIL SREGSET (NIL T T T T) -8 NIL NIL NIL) (-1057 2310055 2311494 2312998 "SRDCMPK" NIL SRDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1056 2302480 2307391 2307421 "SRAGG" 2308720 SRAGG (NIL) -9 NIL 2309324 NIL) (-1055 2301777 2302097 2302475 "SRAGG-" NIL SRAGG- (NIL T) -7 NIL NIL NIL) (-1054 2295896 2301099 2301522 "SQMATRIX" NIL SQMATRIX (NIL NIL T) -8 NIL NIL NIL) (-1053 2290109 2293278 2294000 "SPLTREE" NIL SPLTREE (NIL T T) -8 NIL NIL NIL) (-1052 2286538 2287357 2287994 "SPLNODE" NIL SPLNODE (NIL T T) -8 NIL NIL NIL) (-1051 2285513 2285818 2285848 "SPFCAT" 2286292 SPFCAT (NIL) -9 NIL NIL NIL) (-1050 2284450 2284702 2284966 "SPECOUT" NIL SPECOUT (NIL) -7 NIL NIL NIL) (-1049 2275208 2277482 2277512 "SPADXPT" 2282149 SPADXPT (NIL) -9 NIL 2284273 NIL) (-1048 2275010 2275056 2275125 "SPADPRSR" NIL SPADPRSR (NIL) -7 NIL NIL NIL) (-1047 2272666 2274974 2275005 "SPADAST" NIL SPADAST (NIL) -8 NIL NIL NIL) (-1046 2264340 2266429 2266471 "SPACEC" 2270786 SPACEC (NIL T) -9 NIL 2272591 NIL) (-1045 2262169 2264287 2264335 "SPACE3" NIL SPACE3 (NIL T) -8 NIL NIL NIL) (-1044 2261102 2261291 2261580 "SORTPAK" NIL SORTPAK (NIL T T) -7 NIL NIL NIL) (-1043 2259506 2259839 2260250 "SOLVETRA" NIL SOLVETRA (NIL T) -7 NIL NIL NIL) (-1042 2258771 2259005 2259266 "SOLVESER" NIL SOLVESER (NIL T) -7 NIL NIL NIL) (-1041 2254951 2255911 2256906 "SOLVERAD" NIL SOLVERAD (NIL T) -7 NIL NIL NIL) (-1040 2251309 2252008 2252737 "SOLVEFOR" NIL SOLVEFOR (NIL T T) -7 NIL NIL NIL) (-1039 2245095 2250649 2250745 "SNTSCAT" 2250750 SNTSCAT (NIL T T T T) -9 NIL 2250820 NIL) (-1038 2238980 2243736 2244126 "SMTS" NIL SMTS (NIL T T T) -8 NIL NIL NIL) (-1037 2232816 2238899 2238975 "SMP" NIL SMP (NIL T T) -8 NIL NIL NIL) (-1036 2231248 2231579 2231977 "SMITH" NIL SMITH (NIL T T T T) -7 NIL NIL NIL) (-1035 2222917 2227832 2227934 "SMATCAT" 2229277 SMATCAT (NIL NIL T T T) -9 NIL 2229825 NIL) (-1034 2220758 2221742 2222912 "SMATCAT-" NIL SMATCAT- (NIL T NIL T T T) -7 NIL NIL NIL) (-1033 2218350 2219964 2220007 "SKAGG" 2220268 SKAGG (NIL T) -9 NIL 2220402 NIL) (-1032 2214460 2218170 2218281 "SINT" NIL SINT (NIL) -8 NIL NIL 2218322) (-1031 2214270 2214314 2214380 "SIMPAN" NIL SIMPAN (NIL) -7 NIL NIL NIL) (-1030 2213345 2213577 2213845 "SIGNRF" NIL SIGNRF (NIL T) -7 NIL NIL NIL) (-1029 2212349 2212511 2212787 "SIGNEF" NIL SIGNEF (NIL T T) -7 NIL NIL NIL) (-1028 2211695 2212035 2212158 "SIGAST" NIL SIGAST (NIL) -8 NIL NIL NIL) (-1027 2211041 2211348 2211488 "SIG" NIL SIG (NIL) -8 NIL NIL NIL) (-1026 2209152 2209644 2210150 "SHP" NIL SHP (NIL T NIL) -7 NIL NIL NIL) (-1025 2202691 2209071 2209147 "SHDP" NIL SHDP (NIL NIL NIL T) -8 NIL NIL NIL) (-1024 2202194 2202431 2202461 "SGROUP" 2202554 SGROUP (NIL) -9 NIL 2202616 NIL) (-1023 2202084 2202116 2202189 "SGROUP-" NIL SGROUP- (NIL T) -7 NIL NIL NIL) (-1022 2201722 2201762 2201803 "SGPOPC" 2201808 SGPOPC (NIL T) -9 NIL 2202009 NIL) (-1021 2201256 2201533 2201639 "SGPOP" NIL SGPOP (NIL T) -8 NIL NIL NIL) (-1020 2198679 2199448 2200170 "SGCF" NIL SGCF (NIL) -7 NIL NIL NIL) (-1019 2192564 2198118 2198214 "SFRTCAT" 2198219 SFRTCAT (NIL T T T T) -9 NIL 2198257 NIL) (-1018 2186956 2188069 2189196 "SFRGCD" NIL SFRGCD (NIL T T T T T) -7 NIL NIL NIL) (-1017 2181132 2182293 2183457 "SFQCMPK" NIL SFQCMPK (NIL T T T T T) -7 NIL NIL NIL) (-1016 2180104 2181006 2181127 "SEXOF" NIL SEXOF (NIL T T T T T) -8 NIL NIL NIL) (-1015 2175712 2176607 2176702 "SEXCAT" 2179315 SEXCAT (NIL T T T T T) -9 NIL 2179866 NIL) (-1014 2174685 2175639 2175707 "SEX" NIL SEX (NIL) -8 NIL NIL NIL) (-1013 2173076 2173661 2173963 "SETMN" NIL SETMN (NIL NIL NIL) -8 NIL NIL NIL) (-1012 2172599 2172784 2172814 "SETCAT" 2172931 SETCAT (NIL) -9 NIL 2173015 NIL) (-1011 2172431 2172495 2172594 "SETCAT-" NIL SETCAT- (NIL T) -7 NIL NIL NIL) (-1010 2168654 2170885 2170928 "SETAGG" 2171796 SETAGG (NIL T) -9 NIL 2172134 NIL) (-1009 2168260 2168412 2168649 "SETAGG-" NIL SETAGG- (NIL T T) -7 NIL NIL NIL) (-1008 2165214 2168207 2168255 "SET" NIL SET (NIL T) -8 NIL NIL NIL) (-1007 2164680 2164990 2165090 "SEQAST" NIL SEQAST (NIL) -8 NIL NIL NIL) (-1006 2163807 2164173 2164234 "SEGXCAT" 2164520 SEGXCAT (NIL T T) -9 NIL 2164640 NIL) (-1005 2162732 2163000 2163043 "SEGCAT" 2163565 SEGCAT (NIL T) -9 NIL 2163786 NIL) (-1004 2162412 2162477 2162590 "SEGBIND2" NIL SEGBIND2 (NIL T T) -7 NIL NIL NIL) (-1003 2161478 2161948 2162156 "SEGBIND" NIL SEGBIND (NIL T) -8 NIL NIL NIL) (-1002 2161056 2161335 2161411 "SEGAST" NIL SEGAST (NIL) -8 NIL NIL NIL) (-1001 2160421 2160557 2160761 "SEG2" NIL SEG2 (NIL T T) -7 NIL NIL NIL) (-1000 2159487 2160234 2160416 "SEG" NIL SEG (NIL T) -8 NIL NIL NIL) (-999 2158742 2159437 2159482 "SDVAR" NIL SDVAR (NIL T) -8 NIL NIL NIL) (-998 2150343 2158613 2158737 "SDPOL" NIL SDPOL (NIL T) -8 NIL NIL NIL) (-997 2149203 2149493 2149810 "SCPKG" NIL SCPKG (NIL T) -7 NIL NIL NIL) (-996 2148509 2148721 2148909 "SCOPE" NIL SCOPE (NIL) -8 NIL NIL NIL) (-995 2147859 2148016 2148192 "SCACHE" NIL SCACHE (NIL T) -7 NIL NIL NIL) (-994 2147432 2147663 2147691 "SASTCAT" 2147696 SASTCAT (NIL) -9 NIL 2147709 NIL) (-993 2146899 2147324 2147398 "SAOS" NIL SAOS (NIL) -8 NIL NIL NIL) (-992 2146502 2146543 2146714 "SAERFFC" NIL SAERFFC (NIL T T T) -7 NIL NIL NIL) (-991 2146133 2146174 2146331 "SAEFACT" NIL SAEFACT (NIL T T T) -7 NIL NIL NIL) (-990 2139278 2146050 2146128 "SAE" NIL SAE (NIL T T NIL) -8 NIL NIL NIL) (-989 2137928 2138257 2138653 "RURPK" NIL RURPK (NIL T NIL) -7 NIL NIL NIL) (-988 2136689 2137050 2137350 "RULESET" NIL RULESET (NIL T T T) -8 NIL NIL NIL) (-987 2136313 2136534 2136615 "RULECOLD" NIL RULECOLD (NIL NIL) -8 NIL NIL NIL) (-986 2133773 2134407 2134860 "RULE" NIL RULE (NIL T T T) -8 NIL NIL NIL) (-985 2133612 2133645 2133713 "RTVALUE" NIL RTVALUE (NIL) -8 NIL NIL NIL) (-984 2133103 2133406 2133497 "RSTRCAST" NIL RSTRCAST (NIL) -8 NIL NIL NIL) (-983 2128731 2129599 2130510 "RSETGCD" NIL RSETGCD (NIL T T T T T) -7 NIL NIL NIL) (-982 2117550 2123104 2123198 "RSETCAT" 2127254 RSETCAT (NIL T T T T) -9 NIL 2128342 NIL) (-981 2116088 2116730 2117545 "RSETCAT-" NIL RSETCAT- (NIL T T T T T) -7 NIL NIL NIL) (-980 2109862 2111307 2112814 "RSDCMPK" NIL RSDCMPK (NIL T T T T T) -7 NIL NIL NIL) (-979 2107744 2108301 2108373 "RRCC" 2109446 RRCC (NIL T T) -9 NIL 2109787 NIL) (-978 2107269 2107468 2107739 "RRCC-" NIL RRCC- (NIL T T T) -7 NIL NIL NIL) (-977 2106739 2107049 2107147 "RPTAST" NIL RPTAST (NIL) -8 NIL NIL NIL) (-976 2079355 2090004 2090068 "RPOLCAT" 2100542 RPOLCAT (NIL T T T) -9 NIL 2103687 NIL) (-975 2073454 2076277 2079350 "RPOLCAT-" NIL RPOLCAT- (NIL T T T T) -7 NIL NIL NIL) (-974 2069685 2073202 2073340 "ROMAN" NIL ROMAN (NIL) -8 NIL NIL NIL) (-973 2068013 2068752 2069008 "ROIRC" NIL ROIRC (NIL T T) -8 NIL NIL NIL) (-972 2063718 2066468 2066496 "RNS" 2066758 RNS (NIL) -9 NIL 2067010 NIL) (-971 2062621 2063108 2063645 "RNS-" NIL RNS- (NIL T) -7 NIL NIL NIL) (-970 2061739 2062140 2062340 "RNGBIND" NIL RNGBIND (NIL T T) -8 NIL NIL NIL) (-969 2061027 2061527 2061555 "RNG" 2061560 RNG (NIL) -9 NIL 2061581 NIL) (-968 2060320 2060794 2060834 "RMODULE" 2060839 RMODULE (NIL T) -9 NIL 2060865 NIL) (-967 2059259 2059365 2059695 "RMCAT2" NIL RMCAT2 (NIL NIL NIL T T T T T T T T) -7 NIL NIL NIL) (-966 2056137 2058849 2059142 "RMATRIX" NIL RMATRIX (NIL NIL NIL T) -8 NIL NIL NIL) (-965 2048817 2051278 2051390 "RMATCAT" 2054695 RMATCAT (NIL NIL NIL T T T) -9 NIL 2055672 NIL) (-964 2048334 2048513 2048812 "RMATCAT-" NIL RMATCAT- (NIL T NIL NIL T T T) -7 NIL NIL NIL) (-963 2047902 2048113 2048154 "RLINSET" 2048215 RLINSET (NIL T) -9 NIL 2048259 NIL) (-962 2047547 2047628 2047754 "RINTERP" NIL RINTERP (NIL NIL T) -7 NIL NIL NIL) (-961 2046455 2047124 2047152 "RING" 2047207 RING (NIL) -9 NIL 2047299 NIL) (-960 2046300 2046356 2046450 "RING-" NIL RING- (NIL T) -7 NIL NIL NIL) (-959 2045354 2045621 2045877 "RIDIST" NIL RIDIST (NIL) -7 NIL NIL NIL) (-958 2036341 2044982 2045183 "RGCHAIN" NIL RGCHAIN (NIL T NIL) -8 NIL NIL NIL) (-957 2035597 2036077 2036116 "RGBCSPC" 2036173 RGBCSPC (NIL T) -9 NIL 2036224 NIL) (-956 2034662 2035117 2035156 "RGBCMDL" 2035384 RGBCMDL (NIL T) -9 NIL 2035498 NIL) (-955 2034374 2034443 2034544 "RFFACTOR" NIL RFFACTOR (NIL T) -7 NIL NIL NIL) (-954 2034137 2034178 2034273 "RFFACT" NIL RFFACT (NIL T) -7 NIL NIL NIL) (-953 2032561 2032991 2033371 "RFDIST" NIL RFDIST (NIL) -7 NIL NIL NIL) (-952 2030148 2030816 2031484 "RF" NIL RF (NIL T) -7 NIL NIL NIL) (-951 2029698 2029796 2029956 "RETSOL" NIL RETSOL (NIL T T) -7 NIL NIL NIL) (-950 2029320 2029418 2029459 "RETRACT" 2029590 RETRACT (NIL T) -9 NIL 2029677 NIL) (-949 2029200 2029231 2029315 "RETRACT-" NIL RETRACT- (NIL T T) -7 NIL NIL NIL) (-948 2028802 2029074 2029141 "RETAST" NIL RETAST (NIL) -8 NIL NIL NIL) (-947 2027346 2028173 2028370 "RESRING" NIL RESRING (NIL T T T T NIL) -8 NIL NIL NIL) (-946 2027037 2027098 2027194 "RESLATC" NIL RESLATC (NIL T) -7 NIL NIL NIL) (-945 2026780 2026821 2026926 "REPSQ" NIL REPSQ (NIL T) -7 NIL NIL NIL) (-944 2026515 2026556 2026665 "REPDB" NIL REPDB (NIL T) -7 NIL NIL NIL) (-943 2021586 2023037 2024252 "REP2" NIL REP2 (NIL T) -7 NIL NIL NIL) (-942 2018685 2019443 2020251 "REP1" NIL REP1 (NIL T) -7 NIL NIL NIL) (-941 2016654 2017276 2017876 "REP" NIL REP (NIL) -7 NIL NIL NIL) (-940 2009289 2015205 2015641 "REGSET" NIL REGSET (NIL T T T T) -8 NIL NIL NIL) (-939 2008601 2008881 2009030 "REF" NIL REF (NIL T) -8 NIL NIL NIL) (-938 2008086 2008201 2008366 "REDORDER" NIL REDORDER (NIL T T) -7 NIL NIL NIL) (-937 2003743 2007489 2007710 "RECLOS" NIL RECLOS (NIL T) -8 NIL NIL NIL) (-936 2002975 2003174 2003387 "REALSOLV" NIL REALSOLV (NIL) -7 NIL NIL NIL) (-935 2000265 2001103 2001985 "REAL0Q" NIL REAL0Q (NIL T) -7 NIL NIL NIL) (-934 1996847 1997883 1998942 "REAL0" NIL REAL0 (NIL T) -7 NIL NIL NIL) (-933 1996683 1996736 1996764 "REAL" 1996769 REAL (NIL) -9 NIL 1996804 NIL) (-932 1996173 1996477 1996568 "RDUCEAST" NIL RDUCEAST (NIL) -8 NIL NIL NIL) (-931 1995653 1995731 1995936 "RDIV" NIL RDIV (NIL T T T T T) -7 NIL NIL NIL) (-930 1994886 1995078 1995289 "RDIST" NIL RDIST (NIL T) -7 NIL NIL NIL) (-929 1993774 1994071 1994438 "RDETRS" NIL RDETRS (NIL T T) -7 NIL NIL NIL) (-928 1992041 1992511 1993044 "RDETR" NIL RDETR (NIL T T) -7 NIL NIL NIL) (-927 1990963 1991240 1991627 "RDEEFS" NIL RDEEFS (NIL T T) -7 NIL NIL NIL) (-926 1989790 1990099 1990518 "RDEEF" NIL RDEEF (NIL T T) -7 NIL NIL NIL) (-925 1983202 1986650 1986678 "RCFIELD" 1987955 RCFIELD (NIL) -9 NIL 1988685 NIL) (-924 1981820 1982432 1983129 "RCFIELD-" NIL RCFIELD- (NIL T) -7 NIL NIL NIL) (-923 1978020 1979912 1979953 "RCAGG" 1981020 RCAGG (NIL T) -9 NIL 1981481 NIL) (-922 1977747 1977857 1978015 "RCAGG-" NIL RCAGG- (NIL T T) -7 NIL NIL NIL) (-921 1977192 1977321 1977482 "RATRET" NIL RATRET (NIL T) -7 NIL NIL NIL) (-920 1976809 1976888 1977007 "RATFACT" NIL RATFACT (NIL T) -7 NIL NIL NIL) (-919 1976224 1976374 1976524 "RANDSRC" NIL RANDSRC (NIL) -7 NIL NIL NIL) (-918 1976006 1976056 1976127 "RADUTIL" NIL RADUTIL (NIL) -7 NIL NIL NIL) (-917 1968512 1975124 1975432 "RADIX" NIL RADIX (NIL NIL) -8 NIL NIL NIL) (-916 1958278 1968379 1968507 "RADFF" NIL RADFF (NIL T T T NIL NIL) -8 NIL NIL NIL) (-915 1957912 1958005 1958033 "RADCAT" 1958190 RADCAT (NIL) -9 NIL NIL NIL) (-914 1957750 1957810 1957907 "RADCAT-" NIL RADCAT- (NIL T) -7 NIL NIL NIL) (-913 1955850 1957581 1957670 "QUEUE" NIL QUEUE (NIL T) -8 NIL NIL NIL) (-912 1955531 1955580 1955707 "QUATCT2" NIL QUATCT2 (NIL T T T T) -7 NIL NIL NIL) (-911 1947882 1951902 1951942 "QUATCAT" 1952720 QUATCAT (NIL T) -9 NIL 1953484 NIL) (-910 1945132 1946412 1947788 "QUATCAT-" NIL QUATCAT- (NIL T T) -7 NIL NIL NIL) (-909 1941036 1945082 1945127 "QUAT" NIL QUAT (NIL T) -8 NIL NIL NIL) (-908 1938423 1940090 1940131 "QUAGG" 1940506 QUAGG (NIL T) -9 NIL 1940680 NIL) (-907 1938025 1938297 1938364 "QQUTAST" NIL QQUTAST (NIL) -8 NIL NIL NIL) (-906 1937063 1937661 1937824 "QFORM" NIL QFORM (NIL NIL T) -8 NIL NIL NIL) (-905 1936744 1936793 1936920 "QFCAT2" NIL QFCAT2 (NIL T T T T) -7 NIL NIL NIL) (-904 1926431 1932538 1932578 "QFCAT" 1933236 QFCAT (NIL T) -9 NIL 1934229 NIL) (-903 1923315 1924754 1926337 "QFCAT-" NIL QFCAT- (NIL T T) -7 NIL NIL NIL) (-902 1922861 1922995 1923125 "QEQUAT" NIL QEQUAT (NIL) -8 NIL NIL NIL) (-901 1917057 1918218 1919380 "QCMPACK" NIL QCMPACK (NIL T T T T T) -7 NIL NIL NIL) (-900 1916476 1916656 1916888 "QALGSET2" NIL QALGSET2 (NIL NIL NIL) -7 NIL NIL NIL) (-899 1914298 1914826 1915249 "QALGSET" NIL QALGSET (NIL T T T T) -8 NIL NIL NIL) (-898 1913197 1913439 1913756 "PWFFINTB" NIL PWFFINTB (NIL T T T T) -7 NIL NIL NIL) (-897 1911558 1911756 1912109 "PUSHVAR" NIL PUSHVAR (NIL T T T T) -7 NIL NIL NIL) (-896 1907314 1908530 1908571 "PTRANFN" 1910455 PTRANFN (NIL T) -9 NIL NIL NIL) (-895 1905961 1906306 1906627 "PTPACK" NIL PTPACK (NIL T) -7 NIL NIL NIL) (-894 1905654 1905717 1905824 "PTFUNC2" NIL PTFUNC2 (NIL T T) -7 NIL NIL NIL) (-893 1899727 1904450 1904490 "PTCAT" 1904782 PTCAT (NIL T) -9 NIL 1904935 NIL) (-892 1899420 1899461 1899585 "PSQFR" NIL PSQFR (NIL T T T T) -7 NIL NIL NIL) (-891 1898299 1898615 1898949 "PSEUDLIN" NIL PSEUDLIN (NIL T) -7 NIL NIL NIL) (-890 1887178 1889739 1892048 "PSETPK" NIL PSETPK (NIL T T T T) -7 NIL NIL NIL) (-889 1880085 1882981 1883075 "PSETCAT" 1886049 PSETCAT (NIL T T T T) -9 NIL 1886856 NIL) (-888 1878535 1879269 1880080 "PSETCAT-" NIL PSETCAT- (NIL T T T T T) -7 NIL NIL NIL) (-887 1877854 1878049 1878077 "PSCURVE" 1878345 PSCURVE (NIL) -9 NIL 1878512 NIL) (-886 1873518 1875276 1875340 "PSCAT" 1876175 PSCAT (NIL T T T) -9 NIL 1876414 NIL) (-885 1872832 1873114 1873513 "PSCAT-" NIL PSCAT- (NIL T T T T) -7 NIL NIL NIL) (-884 1871261 1872144 1872407 "PRTITION" NIL PRTITION (NIL) -8 NIL NIL NIL) (-883 1870752 1871055 1871146 "PRTDAST" NIL PRTDAST (NIL) -8 NIL NIL NIL) (-882 1861772 1864194 1866382 "PRS" NIL PRS (NIL T T) -7 NIL NIL NIL) (-881 1859515 1861092 1861132 "PRQAGG" 1861315 PRQAGG (NIL T) -9 NIL 1861416 NIL) (-880 1858688 1859134 1859162 "PROPLOG" 1859301 PROPLOG (NIL) -9 NIL 1859415 NIL) (-879 1858363 1858426 1858549 "PROPFUN2" NIL PROPFUN2 (NIL T T) -7 NIL NIL NIL) (-878 1857799 1857938 1858110 "PROPFUN1" NIL PROPFUN1 (NIL T) -7 NIL NIL NIL) (-877 1856047 1856810 1857107 "PROPFRML" NIL PROPFRML (NIL T) -8 NIL NIL NIL) (-876 1855599 1855731 1855859 "PROPERTY" NIL PROPERTY (NIL) -8 NIL NIL NIL) (-875 1850255 1854539 1855359 "PRODUCT" NIL PRODUCT (NIL T T) -8 NIL NIL NIL) (-874 1850084 1850122 1850181 "PRINT" NIL PRINT (NIL) -7 NIL NIL NIL) (-873 1849523 1849663 1849814 "PRIMES" NIL PRIMES (NIL T) -7 NIL NIL NIL) (-872 1847991 1848410 1848876 "PRIMELT" NIL PRIMELT (NIL T) -7 NIL NIL NIL) (-871 1847708 1847769 1847797 "PRIMCAT" 1847921 PRIMCAT (NIL) -9 NIL NIL NIL) (-870 1846879 1847075 1847303 "PRIMARR2" NIL PRIMARR2 (NIL T T) -7 NIL NIL NIL) (-869 1842757 1846829 1846874 "PRIMARR" NIL PRIMARR (NIL T) -8 NIL NIL NIL) (-868 1842456 1842518 1842629 "PREASSOC" NIL PREASSOC (NIL T T) -7 NIL NIL NIL) (-867 1839656 1842105 1842338 "PR" NIL PR (NIL T T) -8 NIL NIL NIL) (-866 1839107 1839264 1839292 "PPCURVE" 1839497 PPCURVE (NIL) -9 NIL 1839633 NIL) (-865 1838720 1838965 1839048 "PORTNUM" NIL PORTNUM (NIL) -8 NIL NIL NIL) (-864 1836476 1836897 1837489 "POLYROOT" NIL POLYROOT (NIL T T T T T) -7 NIL NIL NIL) (-863 1835919 1835983 1836216 "POLYLIFT" NIL POLYLIFT (NIL T T T T T) -7 NIL NIL NIL) (-862 1832639 1833125 1833736 "POLYCATQ" NIL POLYCATQ (NIL T T T T T) -7 NIL NIL NIL) (-861 1818294 1824359 1824423 "POLYCAT" 1827908 POLYCAT (NIL T T T) -9 NIL 1829785 NIL) (-860 1813804 1815951 1818289 "POLYCAT-" NIL POLYCAT- (NIL T T T T) -7 NIL NIL NIL) (-859 1813461 1813535 1813654 "POLY2UP" NIL POLY2UP (NIL NIL T) -7 NIL NIL NIL) (-858 1813154 1813217 1813324 "POLY2" NIL POLY2 (NIL T T) -7 NIL NIL NIL) (-857 1806581 1812887 1813046 "POLY" NIL POLY (NIL T) -8 NIL NIL NIL) (-856 1805468 1805731 1806007 "POLUTIL" NIL POLUTIL (NIL T T) -7 NIL NIL NIL) (-855 1804072 1804385 1804715 "POLTOPOL" NIL POLTOPOL (NIL NIL T) -7 NIL NIL NIL) (-854 1799234 1804022 1804067 "POINT" NIL POINT (NIL T) -8 NIL NIL NIL) (-853 1797722 1798133 1798508 "PNTHEORY" NIL PNTHEORY (NIL) -7 NIL NIL NIL) (-852 1796479 1796788 1797184 "PMTOOLS" NIL PMTOOLS (NIL T T T) -7 NIL NIL NIL) (-851 1796150 1796234 1796351 "PMSYM" NIL PMSYM (NIL T) -7 NIL NIL NIL) (-850 1795729 1795804 1795978 "PMQFCAT" NIL PMQFCAT (NIL T T T) -7 NIL NIL NIL) (-849 1795215 1795311 1795471 "PMPREDFS" NIL PMPREDFS (NIL T T T) -7 NIL NIL NIL) (-848 1794687 1794807 1794961 "PMPRED" NIL PMPRED (NIL T) -7 NIL NIL NIL) (-847 1793582 1793800 1794177 "PMPLCAT" NIL PMPLCAT (NIL T T T T T) -7 NIL NIL NIL) (-846 1793193 1793278 1793430 "PMLSAGG" NIL PMLSAGG (NIL T T T) -7 NIL NIL NIL) (-845 1792744 1792826 1793007 "PMKERNEL" NIL PMKERNEL (NIL T T) -7 NIL NIL NIL) (-844 1792436 1792517 1792630 "PMINS" NIL PMINS (NIL T) -7 NIL NIL NIL) (-843 1791949 1792024 1792232 "PMFS" NIL PMFS (NIL T T T) -7 NIL NIL NIL) (-842 1791297 1791425 1791627 "PMDOWN" NIL PMDOWN (NIL T T T) -7 NIL NIL NIL) (-841 1790659 1790793 1790956 "PMASSFS" NIL PMASSFS (NIL T T) -7 NIL NIL NIL) (-840 1789963 1790145 1790326 "PMASS" NIL PMASS (NIL) -7 NIL NIL NIL) (-839 1789686 1789760 1789854 "PLOTTOOL" NIL PLOTTOOL (NIL) -7 NIL NIL NIL) (-838 1786254 1787443 1788359 "PLOT3D" NIL PLOT3D (NIL) -8 NIL NIL NIL) (-837 1785338 1785539 1785774 "PLOT1" NIL PLOT1 (NIL T) -7 NIL NIL NIL) (-836 1780903 1782287 1783429 "PLOT" NIL PLOT (NIL) -8 NIL NIL NIL) (-835 1760824 1765711 1770558 "PLEQN" NIL PLEQN (NIL T T T T) -7 NIL NIL NIL) (-834 1760564 1760617 1760720 "PINTERPA" NIL PINTERPA (NIL T T) -7 NIL NIL NIL) (-833 1760005 1760139 1760319 "PINTERP" NIL PINTERP (NIL NIL T) -7 NIL NIL NIL) (-832 1758076 1759235 1759263 "PID" 1759460 PID (NIL) -9 NIL 1759587 NIL) (-831 1757864 1757907 1757982 "PICOERCE" NIL PICOERCE (NIL T) -7 NIL NIL NIL) (-830 1757051 1757711 1757798 "PI" NIL PI (NIL) -8 NIL NIL 1757838) (-829 1756503 1756654 1756830 "PGROEB" NIL PGROEB (NIL T) -7 NIL NIL NIL) (-828 1752831 1753789 1754694 "PGE" NIL PGE (NIL) -7 NIL NIL NIL) (-827 1751195 1751484 1751850 "PGCD" NIL PGCD (NIL T T T T) -7 NIL NIL NIL) (-826 1750637 1750752 1750913 "PFRPAC" NIL PFRPAC (NIL T) -7 NIL NIL NIL) (-825 1747242 1749506 1749859 "PFR" NIL PFR (NIL T) -8 NIL NIL NIL) (-824 1745848 1746128 1746453 "PFOTOOLS" NIL PFOTOOLS (NIL T T) -7 NIL NIL NIL) (-823 1744613 1744867 1745215 "PFOQ" NIL PFOQ (NIL T T T) -7 NIL NIL NIL) (-822 1743323 1743550 1743902 "PFO" NIL PFO (NIL T T T T T) -7 NIL NIL NIL) (-821 1740395 1741893 1741921 "PFECAT" 1742514 PFECAT (NIL) -9 NIL 1742891 NIL) (-820 1740018 1740183 1740390 "PFECAT-" NIL PFECAT- (NIL T) -7 NIL NIL NIL) (-819 1738842 1739124 1739425 "PFBRU" NIL PFBRU (NIL T T) -7 NIL NIL NIL) (-818 1737024 1737411 1737841 "PFBR" NIL PFBR (NIL T T T T) -7 NIL NIL NIL) (-817 1733058 1736950 1737019 "PF" NIL PF (NIL NIL) -8 NIL NIL NIL) (-816 1728961 1730108 1730975 "PERMGRP" NIL PERMGRP (NIL T) -8 NIL NIL NIL) (-815 1726893 1727982 1728023 "PERMCAT" 1728422 PERMCAT (NIL T) -9 NIL 1728719 NIL) (-814 1726589 1726636 1726759 "PERMAN" NIL PERMAN (NIL NIL T) -7 NIL NIL NIL) (-813 1723038 1724719 1725364 "PERM" NIL PERM (NIL T) -8 NIL NIL NIL) (-812 1720503 1722793 1722914 "PENDTREE" NIL PENDTREE (NIL T) -8 NIL NIL NIL) (-811 1719372 1719635 1719676 "PDSPC" 1720209 PDSPC (NIL T) -9 NIL 1720454 NIL) (-810 1718739 1719005 1719367 "PDSPC-" NIL PDSPC- (NIL T T) -7 NIL NIL NIL) (-809 1717436 1718367 1718408 "PDRING" 1718413 PDRING (NIL T) -9 NIL 1718440 NIL) (-808 1716177 1716935 1716988 "PDMOD" 1716993 PDMOD (NIL T T) -9 NIL 1717096 NIL) (-807 1715270 1715482 1715731 "PDECOMP" NIL PDECOMP (NIL T T) -7 NIL NIL NIL) (-806 1714875 1714942 1714996 "PDDOM" 1715161 PDDOM (NIL T T) -9 NIL 1715241 NIL) (-805 1714727 1714763 1714870 "PDDOM-" NIL PDDOM- (NIL T T T) -7 NIL NIL NIL) (-804 1714513 1714552 1714641 "PCOMP" NIL PCOMP (NIL T T) -7 NIL NIL NIL) (-803 1712830 1713584 1713883 "PBWLB" NIL PBWLB (NIL T) -8 NIL NIL NIL) (-802 1712519 1712582 1712691 "PATTERN2" NIL PATTERN2 (NIL T T) -7 NIL NIL NIL) (-801 1710657 1711087 1711538 "PATTERN1" NIL PATTERN1 (NIL T T) -7 NIL NIL NIL) (-800 1704277 1706106 1707398 "PATTERN" NIL PATTERN (NIL T) -8 NIL NIL NIL) (-799 1703908 1703981 1704113 "PATRES2" NIL PATRES2 (NIL T T T) -7 NIL NIL NIL) (-798 1701610 1702290 1702771 "PATRES" NIL PATRES (NIL T T) -8 NIL NIL NIL) (-797 1699814 1700242 1700645 "PATMATCH" NIL PATMATCH (NIL T T T) -7 NIL NIL NIL) (-796 1699260 1699508 1699549 "PATMAB" 1699656 PATMAB (NIL T) -9 NIL 1699739 NIL) (-795 1697907 1698311 1698568 "PATLRES" NIL PATLRES (NIL T T T) -8 NIL NIL NIL) (-794 1697445 1697576 1697617 "PATAB" 1697622 PATAB (NIL T) -9 NIL 1697794 NIL) (-793 1695988 1696425 1696848 "PARTPERM" NIL PARTPERM (NIL) -7 NIL NIL NIL) (-792 1695666 1695741 1695843 "PARSURF" NIL PARSURF (NIL T) -8 NIL NIL NIL) 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(-778 1670788 1672684 1672729 "PADIC" NIL PADIC (NIL NIL) -8 NIL NIL NIL) (-777 1669950 1670160 1670426 "PADEPAC" NIL PADEPAC (NIL T NIL NIL) -7 NIL NIL NIL) (-776 1669292 1669435 1669639 "PADE" NIL PADE (NIL T T T) -7 NIL NIL NIL) (-775 1667737 1668700 1668978 "OWP" NIL OWP (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-774 1667261 1667520 1667617 "OVERSET" NIL OVERSET (NIL) -8 NIL NIL NIL) (-773 1666320 1666998 1667170 "OVAR" NIL OVAR (NIL NIL) -8 NIL NIL NIL) (-772 1656742 1659611 1661810 "OUTFORM" NIL OUTFORM (NIL) -8 NIL NIL NIL) (-771 1656134 1656448 1656574 "OUTBFILE" NIL OUTBFILE (NIL) -8 NIL NIL NIL) (-770 1655411 1655606 1655634 "OUTBCON" 1655952 OUTBCON (NIL) -9 NIL 1656118 NIL) (-769 1655119 1655249 1655406 "OUTBCON-" NIL OUTBCON- (NIL T) -7 NIL NIL NIL) (-768 1654500 1654645 1654806 "OUT" NIL OUT (NIL) -7 NIL NIL NIL) (-767 1653871 1654298 1654387 "OSI" NIL OSI (NIL) -8 NIL NIL NIL) (-766 1653286 1653701 1653729 "OSGROUP" 1653734 OSGROUP (NIL) -9 NIL 1653756 NIL) (-765 1652250 1652511 1652796 "ORTHPOL" NIL ORTHPOL (NIL T) -7 NIL NIL NIL) (-764 1649583 1652125 1652245 "OREUP" NIL OREUP (NIL NIL T NIL NIL) -8 NIL NIL NIL) (-763 1646788 1649334 1649460 "ORESUP" NIL ORESUP (NIL T NIL NIL) -8 NIL NIL NIL) (-762 1644806 1645334 1645894 "OREPCTO" NIL OREPCTO (NIL T T) -7 NIL NIL NIL) (-761 1638210 1640688 1640728 "OREPCAT" 1643049 OREPCAT (NIL T) -9 NIL 1644151 NIL) (-760 1636236 1637170 1638205 "OREPCAT-" NIL OREPCAT- (NIL T T) -7 NIL NIL NIL) (-759 1635433 1635704 1635732 "ORDTYPE" 1636037 ORDTYPE (NIL) -9 NIL 1636195 NIL) (-758 1634967 1635178 1635428 "ORDTYPE-" NIL ORDTYPE- (NIL T) -7 NIL NIL NIL) (-757 1634429 1634805 1634962 "ORDSTRCT" NIL ORDSTRCT (NIL T NIL) -8 NIL NIL NIL) (-756 1633923 1634286 1634314 "ORDSET" 1634319 ORDSET (NIL) -9 NIL 1634341 NIL) (-755 1632563 1633523 1633551 "ORDRING" 1633556 ORDRING (NIL) -9 NIL 1633584 NIL) (-754 1631811 1632368 1632396 "ORDMON" 1632401 ORDMON (NIL) -9 NIL 1632422 NIL) (-753 1631115 1631277 1631469 "ORDFUNS" NIL ORDFUNS (NIL NIL T) -7 NIL NIL NIL) (-752 1630326 1630834 1630862 "ORDFIN" 1630927 ORDFIN (NIL) -9 NIL 1631001 NIL) (-751 1629720 1629859 1630045 "ORDCOMP2" NIL ORDCOMP2 (NIL T T) -7 NIL NIL NIL) (-750 1626494 1628688 1629094 "ORDCOMP" NIL ORDCOMP (NIL T) -8 NIL NIL NIL) (-749 1625901 1626256 1626361 "OPSIG" NIL OPSIG (NIL) -8 NIL NIL NIL) (-748 1625709 1625754 1625820 "OPQUERY" NIL OPQUERY (NIL) -7 NIL NIL NIL) (-747 1625010 1625286 1625327 "OPERCAT" 1625538 OPERCAT (NIL T) -9 NIL 1625634 NIL) (-746 1624822 1624889 1625005 "OPERCAT-" NIL OPERCAT- (NIL T T) -7 NIL NIL NIL) (-745 1622252 1623624 1624120 "OP" NIL OP (NIL T) -8 NIL NIL NIL) (-744 1621673 1621800 1621974 "ONECOMP2" NIL ONECOMP2 (NIL T T) -7 NIL NIL NIL) (-743 1618673 1620812 1621178 "ONECOMP" NIL ONECOMP (NIL T) -8 NIL NIL NIL) (-742 1615304 1618103 1618143 "OMSAGG" 1618204 OMSAGG (NIL T) -9 NIL 1618268 NIL) (-741 1613780 1614975 1615143 "OMLO" NIL OMLO (NIL T T) -8 NIL NIL NIL) (-740 1612051 1613230 1613258 "OINTDOM" 1613263 OINTDOM (NIL) -9 NIL 1613284 NIL) (-739 1609481 1611053 1611382 "OFMONOID" NIL OFMONOID (NIL T) -8 NIL NIL NIL) (-738 1608735 1609431 1609476 "ODVAR" NIL ODVAR (NIL T) -8 NIL NIL NIL) (-737 1606001 1608576 1608730 "ODR" NIL ODR (NIL T T NIL) -8 NIL NIL NIL) (-736 1597602 1605872 1605996 "ODPOL" NIL ODPOL (NIL T) -8 NIL NIL NIL) (-735 1591112 1597493 1597597 "ODP" NIL ODP (NIL NIL T NIL) -8 NIL NIL NIL) (-734 1590084 1590321 1590594 "ODETOOLS" NIL ODETOOLS (NIL T T) -7 NIL NIL NIL) (-733 1587718 1588388 1589092 "ODESYS" NIL ODESYS (NIL T T) -7 NIL NIL NIL) (-732 1583495 1584455 1585478 "ODERTRIC" NIL ODERTRIC (NIL T T) -7 NIL NIL NIL) (-731 1583003 1583091 1583285 "ODERED" NIL ODERED (NIL T T T T T) -7 NIL NIL NIL) (-730 1580452 1581034 1581707 "ODERAT" NIL ODERAT (NIL T T) -7 NIL NIL NIL) (-729 1577847 1578355 1578951 "ODEPRRIC" NIL ODEPRRIC (NIL T T T T) -7 NIL NIL NIL) (-728 1574844 1575383 1576029 "ODEPRIM" NIL ODEPRIM (NIL T T T T) -7 NIL NIL NIL) (-727 1574199 1574307 1574565 "ODEPAL" NIL ODEPAL (NIL T T T T) -7 NIL NIL NIL) (-726 1573357 1573482 1573703 "ODEINT" NIL ODEINT (NIL T T) -7 NIL NIL NIL) (-725 1569641 1570437 1571350 "ODEEF" NIL ODEEF (NIL T T) -7 NIL NIL NIL) (-724 1569081 1569176 1569398 "ODECONST" NIL ODECONST (NIL T T T) -7 NIL NIL NIL) (-723 1568762 1568811 1568938 "OCTCT2" NIL OCTCT2 (NIL T T T T) -7 NIL NIL NIL) (-722 1565429 1568561 1568680 "OCT" NIL OCT (NIL T) -8 NIL NIL NIL) (-721 1564620 1565211 1565239 "OCAMON" 1565244 OCAMON (NIL) -9 NIL 1565265 NIL) (-720 1558896 1561646 1561686 "OC" 1562781 OC (NIL T) -9 NIL 1563637 NIL) (-719 1556896 1557822 1558802 "OC-" NIL OC- (NIL T T) -7 NIL NIL NIL) (-718 1556312 1556730 1556758 "OASGP" 1556763 OASGP (NIL) -9 NIL 1556783 NIL) (-717 1555406 1556024 1556052 "OAMONS" 1556092 OAMONS (NIL) -9 NIL 1556135 NIL) (-716 1554582 1555132 1555160 "OAMON" 1555217 OAMON (NIL) -9 NIL 1555268 NIL) (-715 1554478 1554510 1554577 "OAMON-" NIL OAMON- (NIL T) -7 NIL NIL NIL) (-714 1553260 1554003 1554031 "OAGROUP" 1554177 OAGROUP (NIL) -9 NIL 1554269 NIL) (-713 1553051 1553138 1553255 "OAGROUP-" NIL OAGROUP- (NIL T) -7 NIL NIL NIL) (-712 1552791 1552847 1552935 "NUMTUBE" NIL NUMTUBE (NIL T) -7 NIL NIL NIL) (-711 1547853 1549416 1550943 "NUMQUAD" NIL NUMQUAD (NIL) -7 NIL NIL NIL) (-710 1544548 1545582 1546617 "NUMODE" NIL NUMODE (NIL) -7 NIL NIL NIL) (-709 1543658 1543891 1544109 "NUMFMT" NIL NUMFMT (NIL) -7 NIL NIL NIL) (-708 1532519 1535547 1537995 "NUMERIC" NIL NUMERIC (NIL T) -7 NIL NIL NIL) (-707 1526406 1531960 1532054 "NTSCAT" 1532059 NTSCAT (NIL T T T T) -9 NIL 1532097 NIL) (-706 1525747 1525926 1526119 "NTPOLFN" NIL NTPOLFN (NIL T) -7 NIL NIL NIL) (-705 1525440 1525503 1525610 "NSUP2" NIL NSUP2 (NIL T T) -7 NIL NIL NIL) (-704 1513171 1523060 1523870 "NSUP" NIL NSUP (NIL T) -8 NIL NIL NIL) (-703 1502244 1513036 1513166 "NSMP" NIL NSMP (NIL T T) -8 NIL NIL NIL) (-702 1500964 1501289 1501646 "NREP" NIL NREP (NIL T) -7 NIL NIL NIL) (-701 1499800 1500064 1500422 "NPCOEF" NIL NPCOEF (NIL T T T T T) -7 NIL NIL NIL) (-700 1498967 1499100 1499316 "NORMRETR" NIL NORMRETR (NIL T T T T NIL) -7 NIL NIL NIL) (-699 1497285 1497604 1498010 "NORMPK" NIL NORMPK (NIL T T T T T) -7 NIL NIL NIL) (-698 1496998 1497032 1497156 "NORMMA" NIL NORMMA (NIL T T T T) -7 NIL NIL NIL) (-697 1496817 1496852 1496921 "NONE1" NIL NONE1 (NIL T) -7 NIL NIL NIL) (-696 1496593 1496783 1496812 "NONE" NIL NONE (NIL) -8 NIL NIL NIL) (-695 1496157 1496224 1496401 "NODE1" NIL NODE1 (NIL T T) -7 NIL NIL NIL) (-694 1494475 1495520 1495775 "NNI" NIL NNI (NIL) -8 NIL NIL 1496122) (-693 1493203 1493540 1493904 "NLINSOL" NIL NLINSOL (NIL T) -7 NIL NIL NIL) (-692 1492180 1492432 1492734 "NFINTBAS" NIL NFINTBAS (NIL T T) -7 NIL NIL NIL) (-691 1491267 1491832 1491873 "NETCLT" 1492044 NETCLT (NIL T) -9 NIL 1492125 NIL) (-690 1490171 1490438 1490719 "NCODIV" NIL NCODIV (NIL T T) -7 NIL NIL NIL) (-689 1489970 1490013 1490088 "NCNTFRAC" NIL NCNTFRAC (NIL T) -7 NIL NIL NIL) (-688 1488501 1488889 1489309 "NCEP" NIL NCEP (NIL T) -7 NIL NIL NIL) (-687 1487165 1488100 1488128 "NASRING" 1488238 NASRING (NIL) -9 NIL 1488318 NIL) (-686 1487010 1487066 1487160 "NASRING-" NIL NASRING- (NIL T) -7 NIL NIL NIL) (-685 1485970 1486617 1486645 "NARNG" 1486762 NARNG (NIL) -9 NIL 1486853 NIL) (-684 1485746 1485831 1485965 "NARNG-" NIL NARNG- (NIL T) -7 NIL NIL NIL) (-683 1484543 1485266 1485306 "NAALG" 1485385 NAALG (NIL T) -9 NIL 1485446 NIL) (-682 1484413 1484448 1484538 "NAALG-" NIL NAALG- (NIL T T) -7 NIL NIL NIL) (-681 1479392 1480577 1481763 "MULTSQFR" NIL MULTSQFR (NIL T T T T) -7 NIL NIL NIL) (-680 1478787 1478874 1479058 "MULTFACT" NIL MULTFACT (NIL T T T T) -7 NIL NIL NIL) (-679 1470861 1475291 1475343 "MTSCAT" 1476403 MTSCAT (NIL T T) -9 NIL 1476917 NIL) (-678 1470627 1470687 1470779 "MTHING" NIL MTHING (NIL T) -7 NIL NIL NIL) (-677 1470453 1470492 1470552 "MSYSCMD" NIL MSYSCMD (NIL) -7 NIL NIL NIL) (-676 1467315 1470004 1470045 "MSETAGG" 1470050 MSETAGG (NIL T) -9 NIL 1470084 NIL) (-675 1463452 1466361 1466679 "MSET" NIL MSET (NIL T) -8 NIL NIL NIL) (-674 1459790 1461549 1462289 "MRING" NIL MRING (NIL T T) -8 NIL NIL NIL) (-673 1459427 1459500 1459629 "MRF2" NIL MRF2 (NIL T T T) -7 NIL NIL NIL) (-672 1459080 1459121 1459265 "MRATFAC" NIL MRATFAC (NIL T T T T) -7 NIL NIL NIL) (-671 1456945 1457282 1457713 "MPRFF" NIL MPRFF (NIL T T T T) -7 NIL NIL NIL) (-670 1450407 1456844 1456940 "MPOLY" NIL MPOLY (NIL NIL T) -8 NIL NIL NIL) (-669 1449932 1449973 1450181 "MPCPF" NIL MPCPF (NIL T T T T) -7 NIL NIL NIL) (-668 1449491 1449540 1449723 "MPC3" NIL MPC3 (NIL T T T T T T T) -7 NIL NIL NIL) (-667 1448765 1448858 1449077 "MPC2" NIL MPC2 (NIL T T T T T T T) -7 NIL NIL NIL) (-666 1447382 1447743 1448133 "MONOTOOL" NIL MONOTOOL (NIL T T) -7 NIL NIL NIL) (-665 1446903 1446970 1447009 "MONOPC" 1447069 MONOPC (NIL T) -9 NIL 1447288 NIL) (-664 1446403 1446710 1446818 "MONOP" NIL MONOP (NIL T) -8 NIL NIL NIL) (-663 1445545 1445924 1445952 "MONOID" 1446170 MONOID (NIL) -9 NIL 1446314 NIL) (-662 1445204 1445354 1445540 "MONOID-" NIL MONOID- (NIL T) -7 NIL NIL NIL) (-661 1434204 1441012 1441071 "MONOGEN" 1441745 MONOGEN (NIL T T) -9 NIL 1442201 NIL) (-660 1432216 1433102 1434085 "MONOGEN-" NIL MONOGEN- (NIL T T T) -7 NIL NIL NIL) (-659 1430930 1431474 1431502 "MONADWU" 1431893 MONADWU (NIL) -9 NIL 1432128 NIL) (-658 1430478 1430678 1430925 "MONADWU-" NIL MONADWU- (NIL T) -7 NIL NIL NIL) (-657 1429755 1430056 1430084 "MONAD" 1430291 MONAD (NIL) -9 NIL 1430403 NIL) (-656 1429522 1429618 1429750 "MONAD-" NIL MONAD- (NIL T) -7 NIL NIL NIL) (-655 1427912 1428682 1428961 "MOEBIUS" NIL MOEBIUS (NIL T) -8 NIL NIL NIL) (-654 1427077 1427573 1427613 "MODULE" 1427618 MODULE (NIL T) -9 NIL 1427656 NIL) (-653 1426756 1426882 1427072 "MODULE-" NIL MODULE- (NIL T T) -7 NIL NIL NIL) (-652 1424531 1425353 1425667 "MODRING" NIL MODRING (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-651 1421774 1423127 1423640 "MODOP" NIL MODOP (NIL T T) -8 NIL NIL NIL) (-650 1420408 1420982 1421258 "MODMONOM" NIL MODMONOM (NIL T T NIL) -8 NIL NIL NIL) (-649 1409691 1419073 1419486 "MODMON" NIL MODMON (NIL T T) -8 NIL NIL NIL) (-648 1406711 1408691 1408960 "MODFIELD" NIL MODFIELD (NIL T T NIL NIL NIL) -8 NIL NIL NIL) (-647 1405795 1406162 1406352 "MMLFORM" NIL MMLFORM (NIL) -8 NIL NIL NIL) (-646 1405364 1405413 1405592 "MMAP" NIL MMAP (NIL T T T T T T) -7 NIL NIL NIL) (-645 1403251 1404185 1404225 "MLO" 1404642 MLO (NIL T) -9 NIL 1404882 NIL) (-644 1401132 1401659 1402254 "MLIFT" NIL MLIFT (NIL T T T T) -7 NIL NIL NIL) (-643 1400600 1400696 1400850 "MKUCFUNC" NIL MKUCFUNC (NIL T T T) -7 NIL NIL NIL) (-642 1400270 1400346 1400469 "MKRECORD" NIL MKRECORD (NIL T T) -7 NIL NIL NIL) (-641 1399482 1399668 1399896 "MKFUNC" NIL MKFUNC (NIL T) -7 NIL NIL NIL) (-640 1398975 1399091 1399247 "MKFLCFN" NIL MKFLCFN (NIL T) -7 NIL NIL NIL) (-639 1398347 1398461 1398646 "MKBCFUNC" NIL MKBCFUNC (NIL T T T T) -7 NIL NIL NIL) (-638 1397374 1397647 1397924 "MHROWRED" NIL MHROWRED (NIL T) -7 NIL NIL NIL) (-637 1396807 1396895 1397066 "MFINFACT" NIL MFINFACT (NIL T T T T) -7 NIL NIL NIL) (-636 1393965 1394844 1395723 "MESH" NIL MESH (NIL) -7 NIL NIL NIL) (-635 1392632 1392980 1393333 "MDDFACT" NIL MDDFACT (NIL T) -7 NIL NIL NIL) (-634 1389289 1391756 1391797 "MDAGG" 1392054 MDAGG (NIL T) -9 NIL 1392199 NIL) (-633 1388563 1388727 1388927 "MCDEN" NIL MCDEN (NIL T T) -7 NIL NIL NIL) (-632 1387641 1387927 1388157 "MAYBE" NIL MAYBE (NIL T) -8 NIL NIL NIL) (-631 1385738 1386315 1386876 "MATSTOR" NIL MATSTOR (NIL T) -7 NIL NIL NIL) (-630 1381509 1385328 1385575 "MATRIX" NIL MATRIX (NIL T) -8 NIL NIL NIL) (-629 1377858 1378627 1379361 "MATLIN" NIL MATLIN (NIL T T T T) -7 NIL NIL NIL) (-628 1376611 1376780 1377109 "MATCAT2" NIL MATCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-627 1366124 1369713 1369789 "MATCAT" 1374777 MATCAT (NIL T T T) -9 NIL 1376245 NIL) (-626 1363405 1364711 1366119 "MATCAT-" NIL MATCAT- (NIL T T T T) -7 NIL NIL NIL) (-625 1361806 1362166 1362550 "MAPPKG3" NIL MAPPKG3 (NIL T T T) -7 NIL NIL NIL) (-624 1360939 1361136 1361358 "MAPPKG2" NIL MAPPKG2 (NIL T T) -7 NIL NIL NIL) (-623 1359690 1360016 1360343 "MAPPKG1" NIL MAPPKG1 (NIL T) -7 NIL NIL NIL) (-622 1358852 1359254 1359430 "MAPPAST" NIL MAPPAST (NIL) -8 NIL NIL NIL) (-621 1358521 1358585 1358708 "MAPHACK3" NIL MAPHACK3 (NIL T T T) -7 NIL NIL NIL) (-620 1358169 1358242 1358356 "MAPHACK2" NIL MAPHACK2 (NIL T T) -7 NIL NIL NIL) (-619 1357704 1357819 1357961 "MAPHACK1" NIL MAPHACK1 (NIL T) -7 NIL NIL NIL) (-618 1355913 1356681 1356982 "MAGMA" NIL MAGMA (NIL T) -8 NIL NIL NIL) (-617 1355407 1355709 1355799 "MACROAST" NIL MACROAST (NIL) -8 NIL NIL NIL) (-616 1348916 1353722 1353763 "LZSTAGG" 1354540 LZSTAGG (NIL T) -9 NIL 1354830 NIL) (-615 1346035 1347469 1348911 "LZSTAGG-" NIL LZSTAGG- (NIL T T) -7 NIL NIL NIL) (-614 1343422 1344388 1344871 "LWORD" NIL LWORD (NIL T) -8 NIL NIL NIL) (-613 1343003 1343282 1343356 "LSTAST" NIL LSTAST (NIL) -8 NIL NIL NIL) (-612 1335231 1342864 1342998 "LSQM" NIL LSQM (NIL NIL T) -8 NIL NIL NIL) (-611 1334594 1334739 1334967 "LSPP" NIL LSPP (NIL T T T T) -7 NIL NIL NIL) (-610 1332078 1332776 1333488 "LSMP1" NIL LSMP1 (NIL T) -7 NIL NIL NIL) (-609 1330190 1330513 1330961 "LSMP" NIL LSMP (NIL T T T T) -7 NIL NIL NIL) (-608 1323359 1329277 1329318 "LSAGG" 1329380 LSAGG (NIL T) -9 NIL 1329458 NIL) (-607 1321053 1322152 1323354 "LSAGG-" NIL LSAGG- (NIL T T) -7 NIL NIL NIL) (-606 1318565 1320402 1320651 "LPOLY" NIL LPOLY (NIL T T) -8 NIL NIL NIL) (-605 1318232 1318323 1318446 "LPEFRAC" NIL LPEFRAC (NIL T) -7 NIL NIL NIL) (-604 1317903 1317982 1318010 "LOGIC" 1318121 LOGIC (NIL) -9 NIL 1318203 NIL) (-603 1317798 1317827 1317898 "LOGIC-" NIL LOGIC- (NIL T) -7 NIL NIL NIL) (-602 1317117 1317275 1317468 "LODOOPS" NIL LODOOPS (NIL T T) -7 NIL NIL NIL) (-601 1315902 1316151 1316502 "LODOF" NIL LODOF (NIL T T) -7 NIL NIL NIL) (-600 1311788 1314523 1314563 "LODOCAT" 1314995 LODOCAT (NIL T) -9 NIL 1315206 NIL) (-599 1311581 1311657 1311783 "LODOCAT-" NIL LODOCAT- (NIL T T) -7 NIL NIL NIL) (-598 1308645 1311458 1311576 "LODO2" NIL LODO2 (NIL T T) -8 NIL NIL NIL) (-597 1305807 1308595 1308640 "LODO1" NIL LODO1 (NIL T) -8 NIL NIL NIL) (-596 1302958 1305737 1305802 "LODO" NIL LODO (NIL T NIL) -8 NIL NIL NIL) (-595 1302011 1302186 1302488 "LODEEF" NIL LODEEF (NIL T T T) -7 NIL NIL NIL) (-594 1300175 1301273 1301526 "LO" NIL LO (NIL T T T) -8 NIL NIL NIL) (-593 1295270 1298334 1298375 "LNAGG" 1299237 LNAGG (NIL T) -9 NIL 1299672 NIL) (-592 1294657 1294924 1295265 "LNAGG-" NIL LNAGG- (NIL T T) -7 NIL NIL NIL) (-591 1291229 1292170 1292807 "LMOPS" NIL LMOPS (NIL T T NIL) -8 NIL NIL NIL) (-590 1290522 1290996 1291036 "LMODULE" 1291041 LMODULE (NIL T) -9 NIL 1291067 NIL) (-589 1287701 1290259 1290381 "LMDICT" NIL LMDICT (NIL T) -8 NIL NIL NIL) (-588 1287269 1287480 1287521 "LLINSET" 1287582 LLINSET (NIL T) -9 NIL 1287626 NIL) (-587 1286945 1287205 1287264 "LITERAL" NIL LITERAL (NIL T) -8 NIL NIL NIL) (-586 1286544 1286624 1286763 "LIST3" NIL LIST3 (NIL T T T) -7 NIL NIL NIL) (-585 1284995 1285343 1285742 "LIST2MAP" NIL LIST2MAP (NIL T T) -7 NIL NIL NIL) (-584 1284166 1284362 1284590 "LIST2" NIL LIST2 (NIL T T) -7 NIL NIL NIL) (-583 1277213 1283422 1283676 "LIST" NIL LIST (NIL T) -8 NIL NIL NIL) (-582 1276790 1277023 1277064 "LINSET" 1277069 LINSET (NIL T) -9 NIL 1277102 NIL) (-581 1275723 1276413 1276580 "LINFORM" NIL LINFORM (NIL T NIL) -8 NIL NIL NIL) (-580 1274020 1274744 1274784 "LINEXP" 1275270 LINEXP (NIL T) -9 NIL 1275543 NIL) (-579 1272729 1273629 1273810 "LINELT" NIL LINELT (NIL T NIL) -8 NIL NIL NIL) (-578 1271556 1271828 1272130 "LINDEP" NIL LINDEP (NIL T T) -7 NIL NIL NIL) (-577 1270769 1271358 1271468 "LINBASIS" NIL LINBASIS (NIL NIL) -8 NIL NIL NIL) (-576 1268319 1269041 1269791 "LIMITRF" NIL LIMITRF (NIL T) -7 NIL NIL NIL) (-575 1266949 1267246 1267637 "LIMITPS" NIL LIMITPS (NIL T T) -7 NIL NIL NIL) (-574 1265773 1266344 1266384 "LIECAT" 1266524 LIECAT (NIL T) -9 NIL 1266675 NIL) (-573 1265647 1265680 1265768 "LIECAT-" NIL LIECAT- (NIL T T) -7 NIL NIL NIL) (-572 1259935 1265337 1265565 "LIE" NIL LIE (NIL T T) -8 NIL NIL NIL) (-571 1252284 1259611 1259767 "LIB" NIL LIB (NIL) -8 NIL NIL NIL) (-570 1248736 1249685 1250620 "LGROBP" NIL LGROBP (NIL NIL T) -7 NIL NIL NIL) (-569 1247360 1248268 1248296 "LFCAT" 1248503 LFCAT (NIL) -9 NIL 1248642 NIL) (-568 1245599 1245929 1246274 "LF" NIL LF (NIL T T) -7 NIL NIL NIL) (-567 1243116 1243781 1244462 "LEXTRIPK" NIL LEXTRIPK (NIL T NIL) -7 NIL NIL NIL) (-566 1240128 1241106 1241609 "LEXP" NIL LEXP (NIL T T NIL) -8 NIL NIL NIL) (-565 1239619 1239922 1240013 "LETAST" NIL LETAST (NIL) -8 NIL NIL NIL) (-564 1238326 1238650 1239050 "LEADCDET" NIL LEADCDET (NIL T T T T) -7 NIL NIL NIL) (-563 1237592 1237677 1237903 "LAZM3PK" NIL LAZM3PK (NIL T T T T T T) -7 NIL NIL NIL) (-562 1232659 1236160 1236696 "LAUPOL" NIL LAUPOL (NIL T T) -8 NIL NIL NIL) (-561 1232284 1232334 1232494 "LAPLACE" NIL LAPLACE (NIL T T) -7 NIL NIL NIL) (-560 1231117 1231828 1231868 "LALG" 1231929 LALG (NIL T) -9 NIL 1231987 NIL) (-559 1230900 1230977 1231112 "LALG-" NIL LALG- (NIL T T) -7 NIL NIL NIL) (-558 1228817 1230168 1230419 "LA" NIL LA (NIL T T T) -8 NIL NIL NIL) (-557 1228646 1228676 1228717 "KVTFROM" 1228779 KVTFROM (NIL T) -9 NIL NIL NIL) (-556 1227462 1228177 1228366 "KTVLOGIC" NIL KTVLOGIC (NIL) -8 NIL NIL NIL) (-555 1227291 1227321 1227362 "KRCFROM" 1227424 KRCFROM (NIL T) -9 NIL NIL NIL) (-554 1226393 1226590 1226885 "KOVACIC" NIL KOVACIC (NIL T T) -7 NIL NIL NIL) (-553 1226222 1226252 1226293 "KONVERT" 1226355 KONVERT (NIL T) -9 NIL NIL NIL) (-552 1226051 1226081 1226122 "KOERCE" 1226184 KOERCE (NIL T) -9 NIL NIL NIL) (-551 1225621 1225714 1225846 "KERNEL2" NIL KERNEL2 (NIL T T) -7 NIL NIL NIL) (-550 1223674 1224568 1224940 "KERNEL" NIL KERNEL (NIL T) -8 NIL NIL NIL) (-549 1216851 1221866 1221920 "KDAGG" 1222296 KDAGG (NIL T T) -9 NIL 1222503 NIL) (-548 1216499 1216641 1216846 "KDAGG-" NIL KDAGG- (NIL T T T) -7 NIL NIL NIL) (-547 1209329 1216280 1216437 "KAFILE" NIL KAFILE (NIL T) -8 NIL NIL NIL) (-546 1208979 1209261 1209324 "JVMOP" NIL JVMOP (NIL) -8 NIL NIL NIL) (-545 1207949 1208448 1208697 "JVMMDACC" NIL JVMMDACC (NIL) -8 NIL NIL NIL) (-544 1207075 1207524 1207729 "JVMFDACC" NIL JVMFDACC (NIL) -8 NIL NIL NIL) (-543 1205939 1206431 1206731 "JVMCSTTG" NIL JVMCSTTG (NIL) -8 NIL NIL NIL) (-542 1205221 1205620 1205781 "JVMCFACC" NIL JVMCFACC (NIL) -8 NIL NIL NIL) (-541 1204931 1205167 1205216 "JVMBCODE" NIL JVMBCODE (NIL) -8 NIL NIL NIL) (-540 1199218 1204621 1204849 "JORDAN" NIL JORDAN (NIL T T) -8 NIL NIL NIL) (-539 1198636 1198969 1199089 "JOINAST" NIL JOINAST (NIL) -8 NIL NIL NIL) (-538 1194798 1196813 1196867 "IXAGG" 1197794 IXAGG (NIL T T) -9 NIL 1198251 NIL) (-537 1194004 1194375 1194793 "IXAGG-" NIL IXAGG- (NIL T T T) -7 NIL NIL NIL) (-536 1189258 1193940 1193999 "IVECTOR" NIL IVECTOR (NIL T NIL) -8 NIL NIL NIL) (-535 1188225 1188500 1188763 "ITUPLE" NIL ITUPLE (NIL T) -8 NIL NIL NIL) (-534 1186887 1187094 1187387 "ITRIGMNP" NIL ITRIGMNP (NIL T T T) -7 NIL NIL NIL) (-533 1185838 1186060 1186343 "ITFUN3" NIL ITFUN3 (NIL T T T) -7 NIL NIL NIL) (-532 1185513 1185576 1185699 "ITFUN2" NIL ITFUN2 (NIL T T) -7 NIL NIL NIL) (-531 1184775 1185147 1185321 "ITFORM" NIL ITFORM (NIL) -8 NIL NIL NIL) (-530 1182815 1184051 1184325 "ITAYLOR" NIL ITAYLOR (NIL T) -8 NIL NIL NIL) (-529 1172427 1178132 1179289 "ISUPS" NIL ISUPS (NIL T) -8 NIL NIL NIL) (-528 1171672 1171824 1172060 "ISUMP" NIL ISUMP (NIL T T T T) -7 NIL NIL NIL) (-527 1171163 1171466 1171557 "ISAST" NIL ISAST (NIL) -8 NIL NIL NIL) (-526 1170456 1170547 1170760 "IRURPK" NIL IRURPK (NIL T T T T T) -7 NIL NIL NIL) (-525 1169588 1169813 1170053 "IRSN" NIL IRSN (NIL) -7 NIL NIL NIL) (-524 1168001 1168382 1168810 "IRRF2F" NIL IRRF2F (NIL T) -7 NIL NIL NIL) (-523 1167786 1167830 1167906 "IRREDFFX" NIL IRREDFFX (NIL T) -7 NIL NIL NIL) (-522 1166636 1166933 1167228 "IROOT" NIL IROOT (NIL T) -7 NIL NIL NIL) (-521 1165909 1166260 1166411 "IRFORM" NIL IRFORM (NIL) -8 NIL NIL NIL) (-520 1165112 1165243 1165456 "IR2F" NIL IR2F (NIL T T) -7 NIL NIL NIL) (-519 1163267 1163764 1164308 "IR2" NIL IR2 (NIL T T) -7 NIL NIL NIL) (-518 1160380 1161616 1162305 "IR" NIL IR (NIL T) -8 NIL NIL NIL) (-517 1160205 1160245 1160305 "IPRNTPK" NIL IPRNTPK (NIL) -7 NIL NIL NIL) (-516 1156267 1160131 1160200 "IPF" NIL IPF (NIL NIL) -8 NIL NIL NIL) (-515 1154334 1156206 1156262 "IPADIC" NIL IPADIC (NIL NIL NIL) -8 NIL NIL NIL) (-514 1153705 1154004 1154134 "IP4ADDR" NIL IP4ADDR (NIL) -8 NIL NIL NIL) (-513 1153158 1153446 1153578 "IOMODE" NIL IOMODE (NIL) -8 NIL NIL NIL) (-512 1152239 1152864 1152990 "IOBFILE" NIL IOBFILE (NIL) -8 NIL NIL NIL) (-511 1151649 1152143 1152171 "IOBCON" 1152176 IOBCON (NIL) -9 NIL 1152197 NIL) (-510 1151220 1151284 1151466 "INVLAPLA" NIL INVLAPLA (NIL T T) -7 NIL NIL NIL) (-509 1143264 1145635 1147960 "INTTR" NIL INTTR (NIL T T) -7 NIL NIL NIL) (-508 1140375 1141158 1142022 "INTTOOLS" NIL INTTOOLS (NIL T T) -7 NIL NIL NIL) (-507 1140052 1140149 1140266 "INTSLPE" NIL INTSLPE (NIL) -7 NIL NIL NIL) (-506 1137558 1139988 1140047 "INTRVL" NIL INTRVL (NIL T) -8 NIL NIL NIL) (-505 1135670 1136199 1136766 "INTRF" NIL INTRF (NIL T) -7 NIL NIL NIL) (-504 1135172 1135286 1135426 "INTRET" NIL INTRET (NIL T) -7 NIL NIL NIL) (-503 1133556 1133962 1134424 "INTRAT" NIL INTRAT (NIL T T) -7 NIL NIL NIL) (-502 1131335 1131929 1132540 "INTPM" NIL INTPM (NIL T T) -7 NIL NIL NIL) (-501 1128708 1129318 1130038 "INTPAF" NIL INTPAF (NIL T T T) -7 NIL NIL NIL) (-500 1128112 1128270 1128478 "INTHERTR" NIL INTHERTR (NIL T T) -7 NIL NIL NIL) (-499 1127631 1127717 1127905 "INTHERAL" NIL INTHERAL (NIL T T T T) -7 NIL NIL NIL) (-498 1125836 1126357 1126814 "INTHEORY" NIL INTHEORY (NIL) -7 NIL NIL NIL) (-497 1118918 1120571 1122300 "INTG0" NIL INTG0 (NIL T T T) -7 NIL NIL NIL) (-496 1118284 1118446 1118619 "INTFACT" NIL INTFACT (NIL T) -7 NIL NIL NIL) (-495 1116157 1116621 1117165 "INTEF" NIL INTEF (NIL T T) -7 NIL NIL NIL) (-494 1114345 1115233 1115261 "INTDOM" 1115560 INTDOM (NIL) -9 NIL 1115765 NIL) (-493 1113898 1114100 1114340 "INTDOM-" NIL INTDOM- (NIL T) -7 NIL NIL NIL) (-492 1109769 1112177 1112231 "INTCAT" 1113027 INTCAT (NIL T) -9 NIL 1113343 NIL) (-491 1109334 1109454 1109581 "INTBIT" NIL INTBIT (NIL) -7 NIL NIL NIL) (-490 1108174 1108346 1108652 "INTALG" NIL INTALG (NIL T T T T T) -7 NIL NIL NIL) (-489 1107747 1107843 1108000 "INTAF" NIL INTAF (NIL T T) -7 NIL NIL NIL) (-488 1100787 1107602 1107742 "INTABL" NIL INTABL (NIL T T T) -8 NIL NIL NIL) (-487 1100085 1100640 1100705 "INT8" NIL INT8 (NIL) -8 NIL NIL 1100739) (-486 1099382 1099937 1100002 "INT64" NIL INT64 (NIL) -8 NIL NIL 1100036) (-485 1098679 1099234 1099299 "INT32" NIL INT32 (NIL) -8 NIL NIL 1099333) (-484 1097976 1098531 1098596 "INT16" NIL INT16 (NIL) -8 NIL NIL 1098630) (-483 1094503 1097895 1097971 "INT" NIL INT (NIL) -8 NIL NIL NIL) (-482 1088624 1092043 1092071 "INS" 1093001 INS (NIL) -9 NIL 1093660 NIL) (-481 1086686 1087604 1088551 "INS-" NIL INS- (NIL T) -7 NIL NIL NIL) (-480 1085745 1085968 1086243 "INPSIGN" NIL INPSIGN (NIL T T) -7 NIL NIL NIL) (-479 1084959 1085100 1085297 "INPRODPF" NIL INPRODPF (NIL T T) -7 NIL NIL NIL) (-478 1083949 1084090 1084327 "INPRODFF" NIL INPRODFF (NIL T T T T) -7 NIL NIL NIL) (-477 1083101 1083265 1083525 "INNMFACT" NIL INNMFACT (NIL T T T T) -7 NIL NIL NIL) (-476 1082381 1082496 1082684 "INMODGCD" NIL INMODGCD (NIL T T NIL NIL) -7 NIL NIL NIL) (-475 1081120 1081389 1081713 "INFSP" NIL INFSP (NIL T T T) -7 NIL NIL NIL) (-474 1080400 1080541 1080724 "INFPROD0" NIL INFPROD0 (NIL T T) -7 NIL NIL NIL) (-473 1080063 1080135 1080233 "INFORM1" NIL INFORM1 (NIL T) -7 NIL NIL NIL) (-472 1077141 1078627 1079150 "INFORM" NIL INFORM (NIL) -8 NIL NIL NIL) (-471 1076740 1076847 1076961 "INFINITY" NIL INFINITY (NIL) -7 NIL NIL NIL) (-470 1075896 1076541 1076642 "INETCLTS" NIL INETCLTS (NIL) -8 NIL NIL NIL) (-469 1074746 1075014 1075335 "INEP" NIL INEP (NIL T T T) -7 NIL NIL NIL) (-468 1073818 1074676 1074741 "INDE" NIL INDE (NIL T) -8 NIL NIL NIL) (-467 1073443 1073523 1073640 "INCRMAPS" NIL INCRMAPS (NIL T) -7 NIL NIL NIL) (-466 1072357 1072902 1073106 "INBFILE" NIL INBFILE (NIL) -8 NIL NIL NIL) (-465 1068452 1069507 1070450 "INBFF" NIL INBFF (NIL T) -7 NIL NIL NIL) (-464 1067306 1067629 1067657 "INBCON" 1068170 INBCON (NIL) -9 NIL 1068436 NIL) (-463 1066760 1067025 1067301 "INBCON-" NIL INBCON- (NIL T) -7 NIL NIL NIL) (-462 1066254 1066556 1066646 "INAST" NIL INAST (NIL) -8 NIL NIL NIL) (-461 1065711 1066020 1066125 "IMPTAST" NIL IMPTAST (NIL) -8 NIL NIL NIL) (-460 1061811 1065603 1065706 "IMATRIX" NIL IMATRIX (NIL T NIL NIL) -8 NIL NIL NIL) (-459 1060651 1060790 1061105 "IMATQF" NIL IMATQF (NIL T T T T T T T T) -7 NIL NIL NIL) (-458 1059075 1059342 1059679 "IMATLIN" NIL IMATLIN (NIL T T T T) -7 NIL NIL NIL) (-457 1056891 1058957 1059070 "IIARRAY2" NIL IIARRAY2 (NIL T NIL NIL T T) -8 NIL NIL NIL) (-456 1051798 1056822 1056886 "IFF" NIL IFF (NIL NIL NIL) -8 NIL NIL NIL) (-455 1051178 1051512 1051627 "IFAST" NIL IFAST (NIL) -8 NIL NIL NIL) (-454 1045985 1050616 1050802 "IFARRAY" NIL IFARRAY (NIL T NIL) -8 NIL NIL NIL) (-453 1045047 1045907 1045980 "IFAMON" NIL IFAMON (NIL T T NIL) -8 NIL NIL NIL) (-452 1044619 1044696 1044750 "IEVALAB" 1044957 IEVALAB (NIL T T) -9 NIL NIL NIL) (-451 1044374 1044454 1044614 "IEVALAB-" NIL IEVALAB- (NIL T T T) -7 NIL NIL NIL) (-450 1043447 1044294 1044369 "IDPOAMS" NIL IDPOAMS (NIL T T) -8 NIL NIL NIL) (-449 1042589 1043367 1043442 "IDPOAM" NIL IDPOAM (NIL T T) -8 NIL NIL NIL) (-448 1041992 1042523 1042584 "IDPO" NIL IDPO (NIL T T) -8 NIL NIL NIL) (-447 1040472 1040996 1041047 "IDPC" 1041553 IDPC (NIL T T) -9 NIL 1041833 NIL) (-446 1039838 1040394 1040467 "IDPAM" NIL IDPAM (NIL T T) -8 NIL NIL NIL) (-445 1039087 1039760 1039833 "IDPAG" NIL IDPAG (NIL T T) -8 NIL NIL NIL) (-444 1038780 1038993 1039053 "IDENT" NIL IDENT (NIL) -8 NIL NIL NIL) (-443 1038484 1038524 1038563 "IDEMOPC" 1038568 IDEMOPC (NIL T) -9 NIL 1038705 NIL) (-442 1035555 1036436 1037328 "IDECOMP" NIL IDECOMP (NIL NIL NIL) -7 NIL NIL NIL) (-441 1029181 1030458 1031497 "IDEAL" NIL IDEAL (NIL T T T T) -8 NIL NIL NIL) (-440 1028443 1028573 1028772 "ICDEN" NIL ICDEN (NIL T T T T) -7 NIL NIL NIL) (-439 1027616 1028115 1028253 "ICARD" NIL ICARD (NIL) -8 NIL NIL NIL) (-438 1026005 1026336 1026727 "IBPTOOLS" NIL IBPTOOLS (NIL T T T T) -7 NIL NIL NIL) (-437 1021774 1025961 1026000 "IBITS" NIL IBITS (NIL NIL) -8 NIL NIL NIL) (-436 1019032 1019656 1020351 "IBATOOL" NIL IBATOOL (NIL T T T) -7 NIL NIL NIL) (-435 1017258 1017738 1018271 "IBACHIN" NIL IBACHIN (NIL T T T) -7 NIL NIL NIL) (-434 1015022 1017150 1017253 "IARRAY2" NIL IARRAY2 (NIL T NIL NIL) -8 NIL NIL NIL) (-433 1010891 1014960 1015017 "IARRAY1" NIL IARRAY1 (NIL T NIL) -8 NIL NIL NIL) (-432 1004534 1009855 1010323 "IAN" NIL IAN (NIL) -8 NIL NIL NIL) (-431 1004102 1004165 1004338 "IALGFACT" NIL IALGFACT (NIL T T T T) -7 NIL NIL NIL) (-430 1003594 1003743 1003771 "HYPCAT" 1003978 HYPCAT (NIL) -9 NIL NIL NIL) (-429 1003250 1003403 1003589 "HYPCAT-" NIL HYPCAT- (NIL T) -7 NIL NIL NIL) (-428 1002863 1003108 1003191 "HOSTNAME" NIL HOSTNAME (NIL) -8 NIL NIL NIL) (-427 1002696 1002745 1002786 "HOMOTOP" 1002791 HOMOTOP (NIL T) -9 NIL 1002824 NIL) (-426 999264 1000638 1000679 "HOAGG" 1001654 HOAGG (NIL T) -9 NIL 1002375 NIL) (-425 998270 998740 999259 "HOAGG-" NIL HOAGG- (NIL T T) -7 NIL NIL NIL) (-424 991534 997995 998143 "HEXADEC" NIL HEXADEC (NIL) -8 NIL NIL NIL) (-423 990469 990727 990990 "HEUGCD" NIL HEUGCD (NIL T) -7 NIL NIL NIL) (-422 989436 990334 990464 "HELLFDIV" NIL HELLFDIV (NIL T T T T) -8 NIL NIL NIL) (-421 987630 989269 989357 "HEAP" NIL HEAP (NIL T) -8 NIL NIL NIL) (-420 986945 987297 987430 "HEADAST" NIL HEADAST (NIL) -8 NIL NIL NIL) (-419 980498 986878 986940 "HDP" NIL HDP (NIL NIL T) -8 NIL NIL NIL) (-418 973701 980234 980385 "HDMP" NIL HDMP (NIL NIL T) -8 NIL NIL NIL) (-417 973154 973311 973474 "HB" NIL HB (NIL) -7 NIL NIL NIL) (-416 966237 973045 973149 "HASHTBL" NIL HASHTBL (NIL T T NIL) -8 NIL NIL NIL) (-415 965728 966031 966122 "HASAST" NIL HASAST (NIL) -8 NIL NIL NIL) (-414 963342 965515 965694 "HACKPI" NIL HACKPI (NIL) -8 NIL NIL NIL) (-413 958735 963225 963337 "GTSET" NIL GTSET (NIL T T T T) -8 NIL NIL NIL) (-412 951821 958632 958730 "GSTBL" NIL GSTBL (NIL T T T NIL) -8 NIL NIL NIL) (-411 943822 951190 951445 "GSERIES" NIL GSERIES (NIL T NIL NIL) -8 NIL NIL NIL) (-410 942846 943355 943383 "GROUP" 943586 GROUP (NIL) -9 NIL 943720 NIL) (-409 942389 942590 942841 "GROUP-" NIL GROUP- (NIL T) -7 NIL NIL NIL) (-408 941061 941400 941787 "GROEBSOL" NIL GROEBSOL (NIL NIL T T) -7 NIL NIL NIL) (-407 939883 940240 940291 "GRMOD" 940820 GRMOD (NIL T T) -9 NIL 940986 NIL) (-406 939702 939750 939878 "GRMOD-" NIL GRMOD- (NIL T T T) -7 NIL NIL NIL) (-405 935825 937036 938036 "GRIMAGE" NIL GRIMAGE (NIL) -8 NIL NIL NIL) (-404 934547 934871 935186 "GRDEF" NIL GRDEF (NIL) -7 NIL NIL NIL) (-403 934100 934228 934369 "GRAY" NIL GRAY (NIL) -7 NIL NIL NIL) (-402 933173 933672 933723 "GRALG" 933876 GRALG (NIL T T) -9 NIL 933966 NIL) (-401 932892 932993 933168 "GRALG-" NIL GRALG- (NIL T T T) -7 NIL NIL NIL) (-400 929609 932574 932750 "GPOLSET" NIL GPOLSET (NIL T T T T) -8 NIL NIL NIL) (-399 929022 929085 929342 "GOSPER" NIL GOSPER (NIL T T T T T) -7 NIL NIL NIL) (-398 924908 925772 926297 "GMODPOL" NIL GMODPOL (NIL NIL T T T NIL T) -8 NIL NIL NIL) (-397 924083 924285 924523 "GHENSEL" NIL GHENSEL (NIL T T) -7 NIL NIL NIL) (-396 919086 920013 921032 "GENUPS" NIL GENUPS (NIL T T) -7 NIL NIL NIL) (-395 918834 918891 918980 "GENUFACT" NIL GENUFACT (NIL T) -7 NIL NIL NIL) (-394 918316 918405 918570 "GENPGCD" NIL GENPGCD (NIL T T T T) -7 NIL NIL NIL) (-393 917825 917866 918079 "GENMFACT" NIL GENMFACT (NIL T T T T T) -7 NIL NIL NIL) (-392 916626 916909 917213 "GENEEZ" NIL GENEEZ (NIL T T) -7 NIL NIL NIL) (-391 909965 916316 916477 "GDMP" NIL GDMP (NIL NIL T T) -8 NIL NIL NIL) (-390 899780 904755 905859 "GCNAALG" NIL GCNAALG (NIL T NIL NIL NIL) -8 NIL NIL NIL) (-389 897894 898935 898963 "GCDDOM" 899218 GCDDOM (NIL) -9 NIL 899375 NIL) (-388 897517 897674 897889 "GCDDOM-" NIL GCDDOM- (NIL T) -7 NIL NIL NIL) (-387 888310 890780 893168 "GBINTERN" NIL GBINTERN (NIL T T T T) -7 NIL NIL NIL) (-386 886445 886770 887188 "GBF" NIL GBF (NIL T T T T) -7 NIL NIL NIL) (-385 885386 885575 885842 "GBEUCLID" NIL GBEUCLID (NIL T T T T) -7 NIL NIL NIL) (-384 884257 884464 884768 "GB" NIL GB (NIL T T T T) -7 NIL NIL NIL) (-383 883720 883862 884010 "GAUSSFAC" NIL GAUSSFAC (NIL) -7 NIL NIL NIL) (-382 882332 882680 882993 "GALUTIL" NIL GALUTIL (NIL T) -7 NIL NIL NIL) (-381 880877 881198 881520 "GALPOLYU" NIL GALPOLYU (NIL T T) -7 NIL NIL NIL) (-380 878503 878859 879264 "GALFACTU" NIL GALFACTU (NIL T T T) -7 NIL NIL NIL) (-379 871755 873416 874994 "GALFACT" NIL GALFACT (NIL T) -7 NIL NIL NIL) (-378 871407 871628 871696 "FUNDESC" NIL FUNDESC (NIL) -8 NIL NIL NIL) (-377 871031 871252 871333 "FUNCTION" NIL FUNCTION (NIL NIL) -8 NIL NIL NIL) (-376 869128 869811 870271 "FT" NIL FT (NIL) -8 NIL NIL NIL) (-375 867721 868028 868420 "FSUPFACT" NIL FSUPFACT (NIL T T T) -7 NIL NIL NIL) (-374 866376 866735 867059 "FST" NIL FST (NIL) -8 NIL NIL NIL) (-373 865679 865803 865990 "FSRED" NIL FSRED (NIL T T) -7 NIL NIL NIL) (-372 864653 864919 865266 "FSPRMELT" NIL FSPRMELT (NIL T T) -7 NIL NIL NIL) (-371 862311 862841 863323 "FSPECF" NIL FSPECF (NIL T T) -7 NIL NIL NIL) (-370 861894 861954 862123 "FSINT" NIL FSINT (NIL T T) -7 NIL NIL NIL) (-369 860258 861108 861411 "FSERIES" NIL FSERIES (NIL T T) -8 NIL NIL NIL) (-368 859406 859540 859763 "FSCINT" NIL FSCINT (NIL T T) -7 NIL NIL NIL) (-367 858577 858738 858965 "FSAGG2" NIL FSAGG2 (NIL T T T T) -7 NIL NIL NIL) (-366 854560 857511 857552 "FSAGG" 857922 FSAGG (NIL T) -9 NIL 858181 NIL) (-365 852914 853673 854465 "FSAGG-" NIL FSAGG- (NIL T T) -7 NIL NIL NIL) (-364 850870 851166 851710 "FS2UPS" NIL FS2UPS (NIL T T T T T NIL) -7 NIL NIL NIL) (-363 849917 850099 850399 "FS2EXPXP" NIL FS2EXPXP (NIL T T NIL NIL) -7 NIL NIL NIL) (-362 849598 849647 849774 "FS2" NIL FS2 (NIL T T T T) -7 NIL NIL NIL) (-361 829853 839255 839296 "FS" 843166 FS (NIL T) -9 NIL 845444 NIL) (-360 822084 825577 829556 "FS-" NIL FS- (NIL T T) -7 NIL NIL NIL) (-359 821618 821745 821897 "FRUTIL" NIL FRUTIL (NIL T) -7 NIL NIL NIL) (-358 816172 819299 819339 "FRNAALG" 820659 FRNAALG (NIL T) -9 NIL 821257 NIL) (-357 812913 814164 815422 "FRNAALG-" NIL FRNAALG- (NIL T T) -7 NIL NIL NIL) (-356 812594 812643 812770 "FRNAAF2" NIL FRNAAF2 (NIL T T T T) -7 NIL NIL NIL) (-355 811081 811638 811932 "FRMOD" NIL FRMOD (NIL T T T T NIL) -8 NIL NIL NIL) (-354 810367 810460 810747 "FRIDEAL2" NIL FRIDEAL2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-353 808201 808967 809283 "FRIDEAL" NIL FRIDEAL (NIL T T T T) -8 NIL NIL NIL) (-352 807310 807753 807794 "FRETRCT" 807799 FRETRCT (NIL T) -9 NIL 807970 NIL) (-351 806683 806961 807305 "FRETRCT-" NIL FRETRCT- (NIL T T) -7 NIL NIL NIL) (-350 803489 804947 805006 "FRAMALG" 805888 FRAMALG (NIL T T) -9 NIL 806180 NIL) (-349 802085 802636 803266 "FRAMALG-" NIL FRAMALG- (NIL T T T) -7 NIL NIL NIL) (-348 801778 801841 801948 "FRAC2" NIL FRAC2 (NIL T T) -7 NIL NIL NIL) (-347 795483 801583 801773 "FRAC" NIL FRAC (NIL T) -8 NIL NIL NIL) (-346 795176 795239 795346 "FR2" NIL FR2 (NIL T T) -7 NIL NIL NIL) (-345 787548 792055 793383 "FR" NIL FR (NIL T) -8 NIL NIL NIL) (-344 781388 784829 784857 "FPS" 785976 FPS (NIL) -9 NIL 786532 NIL) (-343 780945 781078 781242 "FPS-" NIL FPS- (NIL T) -7 NIL NIL NIL) (-342 777818 779798 779826 "FPC" 780051 FPC (NIL) -9 NIL 780193 NIL) (-341 777664 777716 777813 "FPC-" NIL FPC- (NIL T) -7 NIL NIL NIL) (-340 776441 777150 777191 "FPATMAB" 777196 FPATMAB (NIL T) -9 NIL 777348 NIL) (-339 774871 775467 775814 "FPARFRAC" NIL FPARFRAC (NIL T T) -8 NIL NIL NIL) (-338 774446 774504 774677 "FORDER" NIL FORDER (NIL T T T T) -7 NIL NIL NIL) (-337 772981 773844 774018 "FNLA" NIL FNLA (NIL NIL NIL T) -8 NIL NIL NIL) (-336 771596 772101 772129 "FNCAT" 772586 FNCAT (NIL) -9 NIL 772843 NIL) (-335 771053 771563 771591 "FNAME" NIL FNAME (NIL) -8 NIL NIL NIL) (-334 769640 771002 771048 "FMONOID" NIL FMONOID (NIL T) -8 NIL NIL NIL) (-333 766228 767586 767627 "FMONCAT" 768844 FMONCAT (NIL T) -9 NIL 769448 NIL) (-332 763117 764164 764217 "FMCAT" 765398 FMCAT (NIL T T) -9 NIL 765890 NIL) (-331 761849 762940 763039 "FM1" NIL FM1 (NIL T T) -8 NIL NIL NIL) (-330 760977 761697 761844 "FM" NIL FM (NIL T T) -8 NIL NIL NIL) (-329 759164 759616 760110 "FLOATRP" NIL FLOATRP (NIL T) -7 NIL NIL NIL) (-328 757099 757635 758213 "FLOATCP" NIL FLOATCP (NIL T) -7 NIL NIL NIL) (-327 750549 755436 756050 "FLOAT" NIL FLOAT (NIL) -8 NIL NIL NIL) (-326 749061 750131 750171 "FLINEXP" 750176 FLINEXP (NIL T) -9 NIL 750269 NIL) (-325 748470 748729 749056 "FLINEXP-" NIL FLINEXP- (NIL T T) -7 NIL NIL NIL) (-324 747685 747844 748065 "FLASORT" NIL FLASORT (NIL T T) -7 NIL NIL NIL) (-323 744599 745647 745699 "FLALG" 746926 FLALG (NIL T T) -9 NIL 747393 NIL) (-322 743770 743931 744158 "FLAGG2" NIL FLAGG2 (NIL T T T T) -7 NIL NIL NIL) (-321 737179 741189 741230 "FLAGG" 742485 FLAGG (NIL T) -9 NIL 743130 NIL) (-320 736287 736691 737174 "FLAGG-" NIL FLAGG- (NIL T T) -7 NIL NIL NIL) (-319 732910 734112 734171 "FINRALG" 735299 FINRALG (NIL T T) -9 NIL 735807 NIL) (-318 732301 732566 732905 "FINRALG-" NIL FINRALG- (NIL T T T) -7 NIL NIL NIL) (-317 731599 731895 731923 "FINITE" 732119 FINITE (NIL) -9 NIL 732226 NIL) (-316 731507 731533 731594 "FINITE-" NIL FINITE- (NIL T) -7 NIL NIL NIL) (-315 723499 726059 726099 "FINAALG" 729751 FINAALG (NIL T) -9 NIL 731189 NIL) (-314 719766 721011 722134 "FINAALG-" NIL FINAALG- (NIL T T) -7 NIL NIL NIL) (-313 718318 718737 718791 "FILECAT" 719475 FILECAT (NIL T T) -9 NIL 719691 NIL) (-312 717669 718143 718246 "FILE" NIL FILE (NIL T) -8 NIL NIL NIL) (-311 714979 716795 716823 "FIELD" 716863 FIELD (NIL) -9 NIL 716943 NIL) (-310 714004 714465 714974 "FIELD-" NIL FIELD- (NIL T) -7 NIL NIL NIL) (-309 712008 712954 713300 "FGROUP" NIL FGROUP (NIL T) -8 NIL NIL NIL) (-308 711251 711432 711651 "FGLMICPK" NIL FGLMICPK (NIL T NIL) -7 NIL NIL NIL) (-307 706585 711189 711246 "FFX" NIL FFX (NIL T NIL) -8 NIL NIL NIL) (-306 706247 706314 706449 "FFSLPE" NIL FFSLPE (NIL T T T) -7 NIL NIL NIL) (-305 705787 705829 706038 "FFPOLY2" NIL FFPOLY2 (NIL T T) -7 NIL NIL NIL) (-304 702467 703344 704121 "FFPOLY" NIL FFPOLY (NIL T) -7 NIL NIL NIL) (-303 697815 702399 702462 "FFP" NIL FFP (NIL T NIL) -8 NIL NIL NIL) (-302 692558 697304 697494 "FFNBX" NIL FFNBX (NIL T NIL) -8 NIL NIL NIL) (-301 687103 691839 692097 "FFNBP" NIL FFNBP (NIL T NIL) -8 NIL NIL NIL) (-300 681374 686554 686765 "FFNB" NIL FFNB (NIL NIL NIL) -8 NIL NIL NIL) (-299 680397 680607 680922 "FFINTBAS" NIL FFINTBAS (NIL T T T) -7 NIL NIL NIL) (-298 675900 678542 678570 "FFIELDC" 679189 FFIELDC (NIL) -9 NIL 679564 NIL) (-297 674969 675409 675895 "FFIELDC-" NIL FFIELDC- (NIL T) -7 NIL NIL NIL) (-296 674584 674642 674766 "FFHOM" NIL FFHOM (NIL T T T) -7 NIL NIL NIL) (-295 672728 673251 673768 "FFF" NIL FFF (NIL T) -7 NIL NIL NIL) (-294 667886 672527 672628 "FFCGX" NIL FFCGX (NIL T NIL) -8 NIL NIL NIL) (-293 663050 667675 667782 "FFCGP" NIL FFCGP (NIL T NIL) -8 NIL NIL NIL) (-292 657780 662841 662949 "FFCG" NIL FFCG (NIL NIL NIL) -8 NIL NIL NIL) (-291 657234 657283 657518 "FFCAT2" NIL FFCAT2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-290 635871 646843 646929 "FFCAT" 652079 FFCAT (NIL T T T) -9 NIL 653515 NIL) (-289 632111 633337 634643 "FFCAT-" NIL FFCAT- (NIL T T T T) -7 NIL NIL NIL) (-288 627018 632042 632106 "FF" NIL FF (NIL NIL NIL) -8 NIL NIL NIL) (-287 625910 626379 626420 "FEVALAB" 626504 FEVALAB (NIL T) -9 NIL 626765 NIL) (-286 625315 625567 625905 "FEVALAB-" NIL FEVALAB- (NIL T T) -7 NIL NIL NIL) (-285 622173 623053 623168 "FDIVCAT" 624735 FDIVCAT (NIL T T T T) -9 NIL 625171 NIL) (-284 621967 621999 622168 "FDIVCAT-" NIL FDIVCAT- (NIL T T T T T) -7 NIL NIL NIL) (-283 621274 621367 621644 "FDIV2" NIL FDIV2 (NIL T T T T T T T T) -7 NIL NIL NIL) (-282 619792 620758 620961 "FDIV" NIL FDIV (NIL T T T T) -8 NIL NIL NIL) (-281 618885 619269 619471 "FCTRDATA" NIL FCTRDATA (NIL) -8 NIL NIL NIL) (-280 618007 618496 618636 "FCOMP" NIL FCOMP (NIL T) -8 NIL NIL NIL) (-279 609656 614237 614277 "FAXF" 616078 FAXF (NIL T) -9 NIL 616768 NIL) (-278 607572 608376 609191 "FAXF-" NIL FAXF- (NIL T T) -7 NIL NIL NIL) (-277 602436 607094 607268 "FARRAY" NIL FARRAY (NIL T) -8 NIL NIL NIL) (-276 596958 599317 599369 "FAMR" 600380 FAMR (NIL T T) -9 NIL 600839 NIL) (-275 596157 596522 596953 "FAMR-" NIL FAMR- (NIL T T T) -7 NIL NIL NIL) (-274 595210 596099 596152 "FAMONOID" NIL FAMONOID (NIL T) -8 NIL NIL NIL) (-273 592835 593683 593736 "FAMONC" 594677 FAMONC (NIL T T) -9 NIL 595062 NIL) (-272 591423 592693 592830 "FAGROUP" NIL FAGROUP (NIL T) -8 NIL NIL NIL) (-271 589503 589864 590266 "FACUTIL" NIL FACUTIL (NIL T T T T) -7 NIL NIL NIL) (-270 588780 588977 589199 "FACTFUNC" NIL FACTFUNC (NIL T) -7 NIL NIL NIL) (-269 580704 588227 588426 "EXPUPXS" NIL EXPUPXS (NIL T NIL NIL) -8 NIL NIL NIL) (-268 578723 579293 579879 "EXPRTUBE" NIL EXPRTUBE (NIL) -7 NIL NIL NIL) (-267 575625 576267 576987 "EXPRODE" NIL EXPRODE (NIL T T) -7 NIL NIL NIL) (-266 570782 571489 572294 "EXPR2UPS" NIL EXPR2UPS (NIL T T) -7 NIL NIL NIL) (-265 570471 570534 570643 "EXPR2" NIL EXPR2 (NIL T T) -7 NIL NIL NIL) (-264 555421 569520 569946 "EXPR" NIL EXPR (NIL T) -8 NIL NIL NIL) (-263 546012 554741 555029 "EXPEXPAN" NIL EXPEXPAN (NIL T T NIL NIL) -8 NIL NIL NIL) (-262 545506 545808 545898 "EXITAST" NIL EXITAST (NIL) -8 NIL NIL NIL) (-261 545282 545472 545501 "EXIT" NIL EXIT (NIL) -8 NIL NIL NIL) (-260 544971 545039 545152 "EVALCYC" NIL EVALCYC (NIL T) -7 NIL NIL NIL) (-259 544488 544630 544671 "EVALAB" 544841 EVALAB (NIL T) -9 NIL 544945 NIL) (-258 544116 544262 544483 "EVALAB-" NIL EVALAB- (NIL T T) -7 NIL NIL NIL) (-257 541221 542754 542782 "EUCDOM" 543336 EUCDOM (NIL) -9 NIL 543685 NIL) (-256 540148 540641 541216 "EUCDOM-" NIL EUCDOM- (NIL T) -7 NIL NIL NIL) (-255 539873 539929 540029 "ES2" NIL ES2 (NIL T T) -7 NIL NIL NIL) (-254 539561 539625 539734 "ES1" NIL ES1 (NIL T T) -7 NIL NIL NIL) (-253 533332 535232 535260 "ES" 538002 ES (NIL) -9 NIL 539386 NIL) (-252 529847 531379 533171 "ES-" NIL ES- (NIL T) -7 NIL NIL NIL) (-251 529195 529348 529524 "ERROR" NIL ERROR (NIL) -7 NIL NIL NIL) (-250 522284 529099 529190 "EQTBL" NIL EQTBL (NIL T T) -8 NIL NIL NIL) (-249 521973 522036 522145 "EQ2" NIL EQ2 (NIL T T) -7 NIL NIL NIL) (-248 515699 518725 520158 "EQ" NIL EQ (NIL T) -8 NIL NIL NIL) (-247 512002 513098 514191 "EP" NIL EP (NIL T) -7 NIL NIL NIL) (-246 510831 511181 511486 "ENV" NIL ENV (NIL) -8 NIL NIL NIL) (-245 509778 510447 510475 "ENTIRER" 510480 ENTIRER (NIL) -9 NIL 510524 NIL) (-244 506475 508208 508557 "EMR" NIL EMR (NIL T T T NIL NIL NIL) -8 NIL NIL NIL) (-243 505567 505778 505832 "ELTAGG" 506212 ELTAGG (NIL T T) -9 NIL 506423 NIL) (-242 505347 505421 505562 "ELTAGG-" NIL ELTAGG- (NIL T T T) -7 NIL NIL NIL) (-241 505093 505128 505182 "ELTAB" 505266 ELTAB (NIL T T) -9 NIL 505318 NIL) (-240 504344 504514 504713 "ELFUTS" NIL ELFUTS (NIL T T) -7 NIL NIL NIL) (-239 504068 504142 504170 "ELEMFUN" 504275 ELEMFUN (NIL) -9 NIL NIL NIL) (-238 503968 503995 504063 "ELEMFUN-" NIL ELEMFUN- (NIL T) -7 NIL NIL NIL) (-237 498514 502009 502050 "ELAGG" 502987 ELAGG (NIL T) -9 NIL 503447 NIL) (-236 497312 497850 498509 "ELAGG-" NIL ELAGG- (NIL T T) -7 NIL NIL NIL) (-235 496730 496897 497053 "ELABOR" NIL ELABOR (NIL) -8 NIL NIL NIL) (-234 495643 495962 496241 "ELABEXPR" NIL ELABEXPR (NIL) -8 NIL NIL NIL) (-233 489036 491034 491861 "EFUPXS" NIL EFUPXS (NIL T T T T) -7 NIL NIL NIL) (-232 483015 485011 485821 "EFULS" NIL EFULS (NIL T T T) -7 NIL NIL NIL) (-231 480829 481235 481706 "EFSTRUC" NIL EFSTRUC (NIL T T) -7 NIL NIL NIL) (-230 471829 473742 475283 "EF" NIL EF (NIL T T) -7 NIL NIL NIL) (-229 470942 471443 471592 "EAB" NIL EAB (NIL) -8 NIL NIL NIL) (-228 469640 470314 470354 "DVARCAT" 470637 DVARCAT (NIL T) -9 NIL 470777 NIL) (-227 469059 469323 469635 "DVARCAT-" NIL DVARCAT- (NIL T T) -7 NIL NIL NIL) (-226 461190 468927 469054 "DSMP" NIL DSMP (NIL T T T) -8 NIL NIL NIL) (-225 459528 460319 460360 "DSEXT" 460723 DSEXT (NIL T) -9 NIL 461017 NIL) (-224 458333 458857 459523 "DSEXT-" NIL DSEXT- (NIL T T) -7 NIL NIL NIL) (-223 458057 458122 458220 "DROPT1" NIL DROPT1 (NIL T) -7 NIL NIL NIL) (-222 454208 455424 456555 "DROPT0" NIL DROPT0 (NIL) -7 NIL NIL NIL) (-221 449854 451209 452273 "DROPT" NIL DROPT (NIL) -8 NIL NIL NIL) (-220 448529 448890 449276 "DRAWPT" NIL DRAWPT (NIL) -7 NIL NIL NIL) (-219 448215 448274 448392 "DRAWHACK" NIL DRAWHACK (NIL T) -7 NIL NIL NIL) (-218 447190 447488 447778 "DRAWCX" NIL DRAWCX (NIL) -7 NIL NIL NIL) (-217 446775 446850 447000 "DRAWCURV" NIL DRAWCURV (NIL T T) -7 NIL NIL NIL) (-216 439188 441300 443415 "DRAWCFUN" NIL DRAWCFUN (NIL) -7 NIL NIL NIL) (-215 434705 435724 436803 "DRAW" NIL DRAW (NIL T) -7 NIL NIL NIL) (-214 431300 433369 433410 "DQAGG" 434039 DQAGG (NIL T) -9 NIL 434312 NIL) (-213 417907 425483 425565 "DPOLCAT" 427402 DPOLCAT (NIL T T T T) -9 NIL 427945 NIL) (-212 414315 415963 417902 "DPOLCAT-" NIL DPOLCAT- (NIL T T T T T) -7 NIL NIL NIL) (-211 407402 414213 414310 "DPMO" NIL DPMO (NIL NIL T T) -8 NIL NIL NIL) (-210 400398 407231 407397 "DPMM" NIL DPMM (NIL NIL T T T) -8 NIL NIL NIL) (-209 399991 400251 400340 "DOMTMPLT" NIL DOMTMPLT (NIL) -8 NIL NIL NIL) (-208 399405 399853 399933 "DOMCTOR" NIL DOMCTOR (NIL) -8 NIL NIL NIL) (-207 398691 399016 399167 "DOMAIN" NIL DOMAIN (NIL) -8 NIL NIL NIL) (-206 391894 398427 398578 "DMP" NIL DMP (NIL NIL T) -8 NIL NIL NIL) (-205 389674 390960 391000 "DMEXT" 391005 DMEXT (NIL T) -9 NIL 391180 NIL) (-204 389330 389392 389536 "DLP" NIL DLP (NIL T) -7 NIL NIL NIL) (-203 382655 388815 389005 "DLIST" NIL DLIST (NIL T) -8 NIL NIL NIL) (-202 379321 381478 381519 "DLAGG" 382069 DLAGG (NIL T) -9 NIL 382298 NIL) (-201 377734 378543 378571 "DIVRING" 378663 DIVRING (NIL) -9 NIL 378746 NIL) (-200 377185 377429 377729 "DIVRING-" NIL DIVRING- (NIL T) -7 NIL NIL NIL) (-199 375613 376030 376436 "DISPLAY" NIL DISPLAY (NIL) -7 NIL NIL NIL) (-198 374650 374871 375136 "DIRPROD2" NIL DIRPROD2 (NIL NIL T T) -7 NIL NIL NIL) (-197 368223 374582 374645 "DIRPROD" NIL DIRPROD (NIL NIL T) -8 NIL NIL NIL) (-196 356642 363003 363056 "DIRPCAT" 363312 DIRPCAT (NIL NIL T) -9 NIL 364185 NIL) (-195 354648 355418 356305 "DIRPCAT-" NIL DIRPCAT- (NIL T NIL T) -7 NIL NIL NIL) (-194 354095 354261 354447 "DIOSP" NIL DIOSP (NIL) -7 NIL NIL NIL) (-193 350641 352981 353022 "DIOPS" 353454 DIOPS (NIL T) -9 NIL 353680 NIL) (-192 350301 350445 350636 "DIOPS-" NIL DIOPS- (NIL T T) -7 NIL NIL NIL) (-191 349339 350054 350082 "DIOID" 350087 DIOID (NIL) -9 NIL 350109 NIL) (-190 348229 348996 349024 "DIFRING" 349029 DIFRING (NIL) -9 NIL 349050 NIL) (-189 347865 347963 347991 "DIFFSPC" 348110 DIFFSPC (NIL) -9 NIL 348185 NIL) (-188 347606 347708 347860 "DIFFSPC-" NIL DIFFSPC- (NIL T) -7 NIL NIL NIL) (-187 346540 347134 347174 "DIFFMOD" 347179 DIFFMOD (NIL T) -9 NIL 347276 NIL) (-186 346224 346281 346322 "DIFFDOM" 346443 DIFFDOM (NIL T) -9 NIL 346511 NIL) (-185 346105 346135 346219 "DIFFDOM-" NIL DIFFDOM- (NIL T T) -7 NIL NIL NIL) (-184 343840 345299 345339 "DIFEXT" 345344 DIFEXT (NIL T) -9 NIL 345496 NIL) (-183 341001 343341 343382 "DIAGG" 343387 DIAGG (NIL T) -9 NIL 343407 NIL) (-182 340557 340747 340996 "DIAGG-" NIL DIAGG- (NIL T T) -7 NIL NIL NIL) (-181 335769 339747 340024 "DHMATRIX" NIL DHMATRIX (NIL T) -8 NIL NIL NIL) (-180 332227 333280 334290 "DFSFUN" NIL DFSFUN (NIL) -7 NIL NIL NIL) (-179 326841 331381 331708 "DFLOAT" NIL DFLOAT (NIL) -8 NIL NIL NIL) (-178 325407 325699 326074 "DFINTTLS" NIL DFINTTLS (NIL T T) -7 NIL NIL NIL) (-177 322591 323779 324175 "DERHAM" NIL DERHAM (NIL T NIL) -8 NIL NIL NIL) (-176 320311 322422 322511 "DEQUEUE" NIL DEQUEUE (NIL T) -8 NIL NIL NIL) (-175 319694 319839 320021 "DEGRED" NIL DEGRED (NIL T T) -7 NIL NIL NIL) (-174 317012 317736 318536 "DEFINTRF" NIL DEFINTRF (NIL T) -7 NIL NIL NIL) (-173 315121 315579 316141 "DEFINTEF" NIL DEFINTEF (NIL T T) -7 NIL NIL NIL) (-172 314504 314837 314951 "DEFAST" NIL DEFAST (NIL) -8 NIL NIL NIL) (-171 307768 314229 314377 "DECIMAL" NIL DECIMAL (NIL) -8 NIL NIL NIL) (-170 305688 306198 306702 "DDFACT" NIL DDFACT (NIL T T) -7 NIL NIL NIL) (-169 305327 305376 305527 "DBLRESP" NIL DBLRESP (NIL T T T T) -7 NIL NIL NIL) (-168 304586 305148 305239 "DBASIS" NIL DBASIS (NIL NIL) -8 NIL NIL NIL) (-167 302610 303052 303412 "DBASE" NIL DBASE (NIL T) -8 NIL NIL NIL) (-166 301902 302191 302337 "DATAARY" NIL DATAARY (NIL NIL T) -8 NIL NIL NIL) (-165 301353 301499 301651 "CYCLOTOM" NIL CYCLOTOM (NIL) -7 NIL NIL NIL) (-164 298715 299508 300235 "CYCLES" NIL CYCLES (NIL) -7 NIL NIL NIL) (-163 298154 298300 298471 "CVMP" NIL CVMP (NIL T) -7 NIL NIL NIL) (-162 296226 296537 296904 "CTRIGMNP" NIL CTRIGMNP (NIL T T) -7 NIL NIL NIL) (-161 295783 296038 296139 "CTORKIND" NIL CTORKIND (NIL) -8 NIL NIL NIL) (-160 294984 295367 295395 "CTORCAT" 295576 CTORCAT (NIL) -9 NIL 295688 NIL) (-159 294687 294821 294979 "CTORCAT-" NIL CTORCAT- (NIL T) -7 NIL NIL NIL) (-158 294180 294437 294545 "CTORCALL" NIL CTORCALL (NIL T) -8 NIL NIL NIL) (-157 293596 294027 294100 "CTOR" NIL CTOR (NIL) -8 NIL NIL NIL) (-156 293055 293172 293325 "CSTTOOLS" NIL CSTTOOLS (NIL T T) -7 NIL NIL NIL) (-155 289449 290205 290960 "CRFP" NIL CRFP (NIL T T) -7 NIL NIL NIL) (-154 288940 289243 289334 "CRCEAST" NIL CRCEAST (NIL) -8 NIL NIL NIL) (-153 288159 288368 288596 "CRAPACK" NIL CRAPACK (NIL T) -7 NIL NIL NIL) (-152 287663 287768 287972 "CPMATCH" NIL CPMATCH (NIL T T T) -7 NIL NIL NIL) (-151 287416 287450 287556 "CPIMA" NIL CPIMA (NIL T T T) -7 NIL NIL NIL) (-150 284355 285117 285835 "COORDSYS" NIL COORDSYS (NIL T) -7 NIL NIL NIL) (-149 283874 284016 284155 "CONTOUR" NIL CONTOUR (NIL) -8 NIL NIL NIL) (-148 279831 282337 282829 "CONTFRAC" NIL CONTFRAC (NIL T) -8 NIL NIL NIL) (-147 279705 279732 279760 "CONDUIT" 279797 CONDUIT (NIL) -9 NIL NIL NIL) (-146 278646 279315 279343 "COMRING" 279348 COMRING (NIL) -9 NIL 279398 NIL) (-145 277811 278178 278356 "COMPPROP" NIL COMPPROP (NIL) -8 NIL NIL NIL) (-144 277507 277548 277676 "COMPLPAT" NIL COMPLPAT (NIL T T T) -7 NIL NIL NIL) (-143 277200 277263 277370 "COMPLEX2" NIL COMPLEX2 (NIL T T) -7 NIL NIL NIL) (-142 266106 277150 277195 "COMPLEX" NIL COMPLEX (NIL T) -8 NIL NIL NIL) (-141 265567 265706 265866 "COMPILER" NIL COMPILER (NIL) -7 NIL NIL NIL) (-140 265320 265361 265459 "COMPFACT" NIL COMPFACT (NIL T T) -7 NIL NIL NIL) (-139 246813 259001 259041 "COMPCAT" 260042 COMPCAT (NIL T) -9 NIL 261384 NIL) (-138 239351 242864 246457 "COMPCAT-" NIL COMPCAT- (NIL T T) -7 NIL NIL NIL) (-137 239110 239144 239246 "COMMUPC" NIL COMMUPC (NIL T T T) -7 NIL NIL NIL) (-136 238940 238979 239037 "COMMONOP" NIL COMMONOP (NIL) -7 NIL NIL NIL) (-135 238521 238800 238874 "COMMAAST" NIL COMMAAST (NIL) -8 NIL NIL NIL) (-134 238098 238339 238426 "COMM" NIL COMM (NIL) -8 NIL NIL NIL) (-133 237293 237541 237569 "COMBOPC" 237907 COMBOPC (NIL) -9 NIL 238082 NIL) (-132 236357 236609 236851 "COMBINAT" NIL COMBINAT (NIL T) -7 NIL NIL NIL) (-131 233289 233973 234596 "COMBF" NIL COMBF (NIL T T) -7 NIL NIL NIL) (-130 232169 232620 232855 "COLOR" NIL COLOR (NIL) -8 NIL NIL NIL) (-129 231660 231963 232054 "COLONAST" NIL COLONAST (NIL) -8 NIL NIL NIL) (-128 231347 231400 231525 "CMPLXRT" NIL CMPLXRT (NIL T T) -7 NIL NIL NIL) (-127 230817 231127 231225 "CLLCTAST" NIL CLLCTAST (NIL) -8 NIL NIL NIL) (-126 227337 228407 229487 "CLIP" NIL CLIP (NIL) -7 NIL NIL NIL) (-125 225696 226617 226855 "CLIF" NIL CLIF (NIL NIL T NIL) -8 NIL NIL NIL) (-124 221808 223816 223857 "CLAGG" 224783 CLAGG (NIL T) -9 NIL 225316 NIL) (-123 220701 221228 221803 "CLAGG-" NIL CLAGG- (NIL T T) -7 NIL NIL NIL) (-122 220330 220421 220561 "CINTSLPE" NIL CINTSLPE (NIL T T) -7 NIL NIL NIL) (-121 218267 218774 219322 "CHVAR" NIL CHVAR (NIL T T T) -7 NIL NIL NIL) (-120 217290 217959 217987 "CHARZ" 217992 CHARZ (NIL) -9 NIL 218006 NIL) (-119 217084 217130 217208 "CHARPOL" NIL CHARPOL (NIL T) -7 NIL NIL NIL) (-118 215985 216686 216714 "CHARNZ" 216775 CHARNZ (NIL) -9 NIL 216823 NIL) (-117 213463 214560 215083 "CHAR" NIL CHAR (NIL) -8 NIL NIL NIL) (-116 213171 213250 213278 "CFCAT" 213389 CFCAT (NIL) -9 NIL NIL NIL) (-115 212514 212643 212825 "CDEN" NIL CDEN (NIL T T T) -7 NIL NIL NIL) (-114 208503 211927 212207 "CCLASS" NIL CCLASS (NIL) -8 NIL NIL NIL) (-113 207881 208068 208245 "CATEGORY" NIL -10 (NIL) -8 NIL NIL NIL) (-112 207409 207828 207876 "CATCTOR" NIL CATCTOR (NIL) -8 NIL NIL NIL) (-111 206882 207191 207288 "CATAST" NIL CATAST (NIL) -8 NIL NIL NIL) (-110 206373 206676 206767 "CASEAST" NIL CASEAST (NIL) -8 NIL NIL NIL) (-109 205622 205782 206003 "CARTEN2" NIL CARTEN2 (NIL NIL NIL T T) -7 NIL NIL NIL) (-108 201722 202979 203687 "CARTEN" NIL CARTEN (NIL NIL NIL T) -8 NIL NIL NIL) (-107 200120 201119 201370 "CARD" NIL CARD (NIL) -8 NIL NIL NIL) (-106 199701 199980 200054 "CAPSLAST" NIL CAPSLAST (NIL) -8 NIL NIL NIL) (-105 199135 199388 199416 "CACHSET" 199548 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\ No newline at end of file diff --git a/src/share/algebra/operation.daase b/src/share/algebra/operation.daase index 4b39a9aa..e1142baa 100644 --- a/src/share/algebra/operation.daase +++ b/src/share/algebra/operation.daase @@ -1,14148 +1,14159 @@ -(630148 . 3538630438) +(630603 . 3539125284) (((*1 *2 *3 *4) - (|partial| -12 (-5 *3 (-1170 *4)) (-4 *4 (-13 (-956) (-577 (-480)))) - (-5 *2 (-1170 (-345 (-480)))) (-5 *1 (-1199 *4))))) + (|partial| -12 (-5 *3 (-1177 *4)) (-4 *4 (-13 (-961) (-580 (-483)))) + (-5 *2 (-1177 (-347 (-483)))) (-5 *1 (-1206 *4))))) (((*1 *2 *3) - (|partial| -12 (-5 *3 (-1170 *4)) (-4 *4 (-13 (-956) (-577 (-480)))) - (-5 *2 (-1170 (-480))) (-5 *1 (-1199 *4))))) + (|partial| -12 (-5 *3 (-1177 *4)) (-4 *4 (-13 (-961) (-580 (-483)))) + (-5 *2 (-1177 (-483))) (-5 *1 (-1206 *4))))) (((*1 *2 *3) - (-12 (-5 *3 (-1170 *4)) (-4 *4 (-13 (-956) (-577 (-480)))) (-5 *2 (-83)) - (-5 *1 (-1199 *4))))) + (-12 (-5 *3 (-1177 *4)) (-4 *4 (-13 (-961) (-580 (-483)))) (-5 *2 (-85)) + (-5 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