path: root/src/algebra/op.spad.pamphlet

\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra op.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain BOP BasicOperator}
<<domain BOP BasicOperator>>=
)abbrev domain BOP BasicOperator
++ Basic system operators
++ Author: Manuel Bronstein
++ Date Created: 22 March 1988
++ Date Last Updated: 11 October 1993
++ Description:
++   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.
++ Keywords: operator, kernel.
BasicOperator(): Exports == Implementation where
  O   ==> OutputForm
  P   ==> AssociationList(String, None)
  L   ==> List Record(key:String, entry:None)
  SEX ==> InputForm
-- some internal properties
  LESS?   ==> "%less?"
  EQUAL?  ==> "%equal?"
  WEIGHT  ==> "%weight"
  DISPLAY ==> "%display"
  SEXPR   ==> "%input"

  Exports ==> OrderedSet with
    name      : $ -> Symbol
      ++ name(op) returns the name of op.
    properties: $ -> P
      ++ properties(op) returns the list of all the properties
      ++ currently attached to op.
    copy      : $ -> $
      ++ copy(op) returns a copy of op.
    operator  : Symbol -> $
      ++ operator(f) makes f into an operator with arbitrary arity.
    operator  : (Symbol, NonNegativeInteger) -> $
      ++ operator(f, n) makes f into an n-ary operator.
    arity     : $ -> Union(NonNegativeInteger, "failed")
      ++ arity(op) returns n if op is n-ary, and
      ++ "failed" if op has arbitrary arity.
    nullary?  : $ -> Boolean
      ++ nullary?(op) tests if op is nullary.
    unary?    : $ -> Boolean
      ++ unary?(op) tests if op is unary.
    nary?     : $ -> Boolean
      ++ nary?(op) tests if op has arbitrary arity.
    weight    : $ -> NonNegativeInteger
      ++ weight(op) returns the weight attached to op.
    weight    : ($, NonNegativeInteger) -> $
      ++ weight(op, n) attaches the weight n to op.
    equality   : ($, ($, $) -> Boolean) -> $
      ++ equality(op, foo?) attaches foo? as the "%equal?" property
      ++ to op. If op1 and op2 have the same name, and one of them
      ++ has an "%equal?" property f, then \spad{f(op1, op2)} is called to
      ++ decide whether op1 and op2 should be considered equal.
    comparison : ($, ($, $) -> Boolean) -> $
      ++ comparison(op, foo?) attaches foo? as the "%less?" property
      ++ to op. If op1 and op2 have the same name, and one of them
      ++ has a "%less?" property f, then \spad{f(op1, op2)} is called to
      ++ decide whether \spad{op1 < op2}.
    display    : $ -> Union(List O -> O, "failed")
      ++ display(op) returns the "%display" property of op if
      ++ it has one attached, and "failed" otherwise.
    display    : ($, List O -> O)      -> $
      ++ display(op, foo) attaches foo as the "%display" property
      ++ of op. If op has a "%display" property f, then \spad{op(a1,...,an)}
      ++ gets converted to OutputForm as \spad{f(a1,...,an)}.
    display    : ($, O -> O)           -> $
      ++ display(op, foo) attaches foo as the "%display" property
      ++ of op. If op has a "%display" property f, then \spad{op(a)}
      ++ gets converted to OutputForm as \spad{f(a)}.
      ++ Argument op must be unary.
    input      : ($, List SEX -> SEX)  -> $
      ++ input(op, foo) attaches foo as the "%input" property
      ++ of op. If op has a "%input" property f, then \spad{op(a1,...,an)}
      ++ gets converted to InputForm as \spad{f(a1,...,an)}.
    input      : $ -> Union(List SEX -> SEX, "failed")
      ++ input(op) returns the "%input" property of op if
      ++ it has one attached, "failed" otherwise.
    is?        : ($, Symbol) -> Boolean
      ++ is?(op, s) tests if the name of op is s.
    has?       : ($, String) -> Boolean
      ++ has?(op, s) tests if property s is attached to op.
    assert     : ($, String) -> $
      ++ assert(op, s) attaches property s to op.
      ++ Argument op is modified "in place", i.e. no copy is made.
    deleteProperty_!: ($, String) -> $
      ++ deleteProperty!(op, s) unattaches property s from op.
      ++ Argument op is modified "in place", i.e. no copy is made.
    property      : ($, String) -> Union(None, "failed")
      ++ property(op, s) returns the value of property s if
      ++ it is attached to op, and "failed" otherwise.
    setProperty   : ($, String, None) -> $
      ++ setProperty(op, s, v) attaches property s to op,
      ++ and sets its value to v.
      ++ Argument op is modified "in place", i.e. no copy is made.
    setProperties : ($, P) -> $
      ++ setProperties(op, l) sets the property list of op to l.
      ++ Argument op is modified "in place", i.e. no copy is made.

  Implementation ==> add
    -- if narg < 0 then the operator ahs variable arity.
    Rep := Record(opname:Symbol, narg:SingleInteger, props:P)

    oper: (Symbol, SingleInteger, P) -> $

    is?(op, s)           == name(op) = s
    name op              == op.opname
    properties op        == op.props
    setProperties(op, l) == (op.props := l; op)
    operator s           == oper(s, -1::SingleInteger, table())
    operator(s, n)       == oper(s, n::Integer::SingleInteger, table())
    property(op, name)   == search(name, op.props)
    assert(op, s)        == setProperty(op, s, NIL$Lisp)
    has?(op, name)       == key?(name, op.props)
    oper(se, n, prop)    == [se, n, prop]
    weight(op, n)        == setProperty(op, WEIGHT, n pretend None)
    nullary? op          == zero?(op.narg)
--    unary? op            == one?(op.narg)
    unary? op            == ((op.narg) = 1)
    nary? op             == negative?(op.narg)
    equality(op, func)   == setProperty(op, EQUAL?, func pretend None)
    comparison(op, func) == setProperty(op, LESS?, func pretend None)
    display(op:$, f:O -> O)        == display(op, f first #1)
    deleteProperty_!(op, name)     == (remove_!(name, properties op); op)
    setProperty(op, name, valu)    == (op.props.name := valu; op)
    coerce(op:$):OutputForm        == name(op)::OutputForm
    input(op:$, f:List SEX -> SEX) == setProperty(op, SEXPR, f pretend None)
    display(op:$, f:List O -> O)   == setProperty(op, DISPLAY, f pretend None)

    display op ==
      (u := property(op, DISPLAY)) case "failed" => "failed"
      (u::None) pretend (List O -> O)

    input op ==
      (u := property(op, SEXPR)) case "failed" => "failed"
      (u::None) pretend (List SEX -> SEX)

    arity op ==
      negative?(n := op.narg) => "failed"
      convert(n)@Integer :: NonNegativeInteger

    copy op ==
      oper(name op, op.narg,
          table([[r.key, r.entry] for r in entries(properties op)@L]$L))

-- property EQUAL? contains a function f: (BOP, BOP) -> Boolean
-- such that f(o1, o2) is true iff o1 = o2
    op1 = op2 ==
      name(op1) ~= name(op2) => false
      op1.narg ~= op2.narg => false
      brace(keys properties op1)~=$Set(String) brace(keys properties op2) => false
      (func := property(op1, EQUAL?)) case None =>
                   ((func::None) pretend (($, $) -> Boolean)) (op1, op2)
      true

-- property WEIGHT allows one to change the ordering around
-- by default, every operator has weigth 1
    weight op ==
      (w := property(op, WEIGHT)) case "failed" => 1
      (w::None) pretend NonNegativeInteger

-- property LESS? contains a function f: (BOP, BOP) -> Boolean
-- such that f(o1, o2) is true iff o1 < o2
    op1 < op2 ==
      (w1 := weight op1) ~= (w2 := weight op2) => w1 < w2
      op1.narg ~= op2.narg => op1.narg < op2.narg
      name(op1) ~= name(op2) => name(op1) < name(op2)
      n1 := #(k1 := brace(keys(properties op1))$Set(String))
      n2 := #(k2 := brace(keys(properties op2))$Set(String))
      n1 ~= n2 => n1 < n2
      not zero?(n1 := #(d1 := difference(k1, k2))) =>
        n1 ~= (n2 := #(d2 := difference(k2, k1))) => n1 < n2
        inspect(d1) < inspect(d2)
      (func := property(op1, LESS?)) case None =>
                   ((func::None) pretend (($, $) -> Boolean)) (op1, op2)
      (func := property(op1, EQUAL?)) case None =>
              not(((func::None) pretend (($, $) -> Boolean)) (op1, op2))
      false

@
\section{package BOP1 BasicOperatorFunctions1}
<<package BOP1 BasicOperatorFunctions1>>=
)abbrev package BOP1 BasicOperatorFunctions1
++ Tools to set/get common properties of operators
++ Author: Manuel Bronstein
++ Date Created: 28 Mar 1988
++ Date Last Updated: 15 May 1990
++ Description:
++   This package exports functions to set some commonly used properties
++   of operators, including properties which contain functions.
++ Keywords: operator.
BasicOperatorFunctions1(A:SetCategory): Exports == Implementation where
  OP   ==> BasicOperator
  EVAL    ==> "%eval"
  CONST   ==> "%constant"
  DIFF    ==> "%diff"

  Exports ==> with
    evaluate        : (OP, List A)      -> Union(A, "failed")
      ++ evaluate(op, [a1,...,an]) checks if op has an "%eval"
      ++ property f. If it has, then \spad{f(a1,...,an)} is returned, and
      ++ "failed" otherwise.
    evaluate        : (OP, List A -> A) -> OP
      ++ evaluate(op, foo) attaches foo as the "%eval" property
      ++ of op. If op has an "%eval" property f, then applying op
      ++ to \spad{(a1,...,an)} returns the result of \spad{f(a1,...,an)}.
    evaluate        : (OP, A -> A)      -> OP
      ++ evaluate(op, foo) attaches foo as the "%eval" property
      ++ of op. If op has an "%eval" property f, then applying op
      ++ to a returns the result of \spad{f(a)}. Argument op must be unary.
    evaluate        : OP                -> Union(List A -> A, "failed")
      ++ evaluate(op) returns the value of the "%eval" property of
      ++ op if it has one, and "failed" otherwise.
    derivative      : (OP, List (List A -> A)) -> OP
      ++ derivative(op, [foo1,...,foon]) attaches [foo1,...,foon] as
      ++ the "%diff" property of op. If op has an "%diff" property
      ++ \spad{[f1,...,fn]} then applying a derivation D to \spad{op(a1,...,an)}
      ++ returns \spad{f1(a1,...,an) * D(a1) + ... + fn(a1,...,an) * D(an)}.
    derivative      : (OP, A -> A) -> OP
      ++ derivative(op, foo) attaches foo as the "%diff" property
      ++ of op. If op has an "%diff" property f, then applying a
      ++ derivation D to op(a) returns \spad{f(a) * D(a)}. Argument op must be unary.
    derivative      : OP -> Union(List(List A -> A), "failed")
      ++ derivative(op) returns the value of the "%diff" property of
      ++ op if it has one, and "failed" otherwise.
    if A has OrderedSet then
      constantOperator: A -> OP
        ++ constantOperator(a) returns a nullary operator op
        ++ such that \spad{op()} always evaluate to \spad{a}.
      constantOpIfCan : OP -> Union(A, "failed")
        ++ constantOpIfCan(op) returns \spad{a} if op is the constant
        ++ nullary operator always returning \spad{a}, "failed" otherwise.

  Implementation ==> add
    evaluate(op:OP, func:A -> A) == evaluate(op, func first #1)

    evaluate op ==
      (func := property(op, EVAL)) case "failed" => "failed"
      (func::None) pretend (List A -> A)

    evaluate(op:OP, args:List A) ==
      (func := property(op, EVAL)) case "failed" => "failed"
      ((func::None) pretend (List A -> A)) args

    evaluate(op:OP, func:List A -> A) ==
      setProperty(op, EVAL, func pretend None)

    derivative op ==
      (func := property(op, DIFF)) case "failed" => "failed"
      ((func::None) pretend List(List A -> A))

    derivative(op:OP, grad:List(List A -> A)) ==
      setProperty(op, DIFF, grad pretend None)

    derivative(op:OP, f:A -> A) ==
      unary? op or nary? op =>
        derivative(op, [f first #1]$List(List A -> A))
      error "Operator is not unary"

    if A has OrderedSet then
      cdisp   : (OutputForm, List OutputForm) -> OutputForm
      csex    : (InputForm,  List InputForm) -> InputForm
      eqconst?: (OP, OP) -> Boolean
      ltconst?: (OP, OP) -> Boolean
      constOp : A -> OP

      opconst:OP :=
        comparison(equality(operator("constant"::Symbol, 0), eqconst?),
                                                               ltconst?)

      cdisp(a, l) == a
      csex(a, l)  == a

      eqconst?(a, b) ==
        (va := property(a, CONST)) case "failed" => not has?(b, CONST)
        ((vb := property(b, CONST)) case None) and
           ((va::None) pretend A) = ((vb::None) pretend A)

      ltconst?(a, b) ==
        (va := property(a, CONST)) case "failed" => has?(b, CONST)
        ((vb := property(b, CONST)) case None) and
           ((va::None) pretend A) < ((vb::None) pretend A)

      constOp a ==
        setProperty(display(copy opconst, cdisp(a::OutputForm, #1)),
                                                  CONST, a pretend None)

      constantOpIfCan op ==
        is?(op, "constant"::Symbol) and
          ((u := property(op, CONST)) case None) => (u::None) pretend A
        "failed"

      if A has ConvertibleTo InputForm then
        constantOperator a == input(constOp a, csex(convert a, #1))
      else
        constantOperator a == constOp a

@
\section{package COMMONOP CommonOperators}
<<package COMMONOP CommonOperators>>=
)abbrev package COMMONOP CommonOperators
++ Provides commonly used operators
++ Author: Manuel Bronstein
++ Date Created: 25 Mar 1988
++ Date Last Updated: 2 December 1994
++ Description:
++ 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.
++ Keywords: operator.
CommonOperators(): Exports == Implementation where
  OP  ==> BasicOperator
  O   ==> OutputForm
  POWER ==> "%power"::Symbol
  ALGOP ==> "%alg"
  EVEN  ==> "even"
  ODD   ==> "odd"
  DUMMYVAR ==> "%dummyVar"

  Exports ==> with
    operator: Symbol -> OP
        ++ operator(s) returns an operator with name s, with the
        ++ appropriate semantics if s is known. If s is not known,
        ++ the result has no semantics.

  Implementation ==> add
    dpi        : List O -> O
    dgamma     : List O -> O
    dquote     : List O -> O
    dexp       : O -> O
    dfact      : O -> O
    startUp    : Boolean -> Void
    setDummyVar: (OP, NonNegativeInteger) -> OP

    brandNew?:Reference(Boolean) := ref true

    opalg   := operator("rootOf"::Symbol, 2)$OP
    oproot  := operator("nthRoot"::Symbol, 2)
    oppi    := operator("pi"::Symbol, 0)
    oplog   := operator("log"::Symbol, 1)
    opexp   := operator("exp"::Symbol, 1)
    opabs   := operator("abs"::Symbol, 1)
    opsin   := operator("sin"::Symbol, 1)
    opcos   := operator("cos"::Symbol, 1)
    optan   := operator("tan"::Symbol, 1)
    opcot   := operator("cot"::Symbol, 1)
    opsec   := operator("sec"::Symbol, 1)
    opcsc   := operator("csc"::Symbol, 1)
    opasin  := operator("asin"::Symbol, 1)
    opacos  := operator("acos"::Symbol, 1)
    opatan  := operator("atan"::Symbol, 1)
    opacot  := operator("acot"::Symbol, 1)
    opasec  := operator("asec"::Symbol, 1)
    opacsc  := operator("acsc"::Symbol, 1)
    opsinh  := operator("sinh"::Symbol, 1)
    opcosh  := operator("cosh"::Symbol, 1)
    optanh  := operator("tanh"::Symbol, 1)
    opcoth  := operator("coth"::Symbol, 1)
    opsech  := operator("sech"::Symbol, 1)
    opcsch  := operator("csch"::Symbol, 1)
    opasinh := operator("asinh"::Symbol, 1)
    opacosh := operator("acosh"::Symbol, 1)
    opatanh := operator("atanh"::Symbol, 1)
    opacoth := operator("acoth"::Symbol, 1)
    opasech := operator("asech"::Symbol, 1)
    opacsch := operator("acsch"::Symbol, 1)
    opbox   := operator("%box"::Symbol)$OP
    oppren  := operator("%paren"::Symbol)$OP
    opquote := operator("applyQuote"::Symbol)$OP
    opdiff  := operator("%diff"::Symbol, 3)
    opsi    := operator("Si"::Symbol, 1)
    opci    := operator("Ci"::Symbol, 1)
    opei    := operator("Ei"::Symbol, 1)
    opli    := operator("li"::Symbol, 1)
    operf   := operator("erf"::Symbol, 1)
    opli2   := operator("dilog"::Symbol, 1)
    opGamma     := operator("Gamma"::Symbol, 1)
    opGamma2    := operator("Gamma2"::Symbol, 2)
    opBeta      := operator("Beta"::Symbol, 2)
    opdigamma   := operator("digamma"::Symbol, 1)
    oppolygamma := operator("polygamma"::Symbol, 2)
    opBesselJ   := operator("besselJ"::Symbol, 2)
    opBesselY   := operator("besselY"::Symbol, 2)
    opBesselI   := operator("besselI"::Symbol, 2)
    opBesselK   := operator("besselK"::Symbol, 2)
    opAiryAi    := operator("airyAi"::Symbol,  1)
    opAiryBi    := operator("airyBi"::Symbol , 1)
    opint   := operator("integral"::Symbol, 3)
    opdint  := operator("%defint"::Symbol, 5)
    opfact  := operator("factorial"::Symbol, 1)
    opperm  := operator("permutation"::Symbol, 2)
    opbinom := operator("binomial"::Symbol, 2)
    oppow   := operator(POWER, 2)
    opsum   := operator("summation"::Symbol, 3)
    opdsum  := operator("%defsum"::Symbol, 5)
    opprod  := operator("product"::Symbol, 3)
    opdprod := operator("%defprod"::Symbol, 5)

    algop   := [oproot, opalg]$List(OP)
    rtrigop := [opsin, opcos, optan, opcot, opsec, opcsc,
                         opasin, opacos, opatan, opacot, opasec, opacsc]
    htrigop := [opsinh, opcosh, optanh, opcoth, opsech, opcsch,
                   opasinh, opacosh, opatanh, opacoth, opasech, opacsch]
    trigop  := concat(rtrigop, htrigop)
    elemop  := concat(trigop, [oppi, oplog, opexp])
    primop  := [opei, opli, opsi, opci, operf, opli2, opint, opdint]
    combop  := [opfact, opperm, opbinom, oppow,
                                         opsum, opdsum, opprod, opdprod]
    specop  := [opGamma, opGamma2, opBeta, opdigamma, oppolygamma, opabs,
                opBesselJ, opBesselY, opBesselI, opBesselK]
    anyop   := [oppren, opdiff, opbox, opquote]
    allop   := concat(concat(concat(concat(concat(
                            algop,elemop),primop),combop),specop),anyop)

-- odd and even operators, must be maintained current!
    evenop := [opcos, opsec, opcosh, opsech, opabs]
    oddop  := [opsin, opcsc, optan, opcot, opasin, opacsc, opatan,
               opsinh, opcsch, optanh, opcoth, opasinh, opacsch,opatanh,opacoth,
                opsi, operf]

-- operators whose second argument is a dummy variable
    dummyvarop1 := [opdiff,opalg, opint, opsum, opprod]
-- operators whose second and third arguments are dummy variables
    dummyvarop2 := [opdint, opdsum, opdprod]

    operator s ==
      if (deref brandNew?) then startUp false
      for op in allop repeat
        is?(op, s) => return copy op
      operator(s)$OP

    dpi l    == "%pi"::Symbol::O
    dfact x  == postfix("!"::Symbol::O, (ATOM(x)$Lisp => x; paren x))
    dquote l == prefix(quote(first(l)::O), rest l)
    dgamma l == prefix(hconcat("|"::Symbol::O, overbar(" "::Symbol::O)), l)
    setDummyVar(op, n) == setProperty(op, DUMMYVAR, n pretend None)

    dexp x ==
      e := "%e"::Symbol::O
      x = 1::O => e
      e ** x

    startUp b ==
      brandNew?() := b
      display(oppren, paren)
      display(opbox, commaSeparate)
      display(oppi, dpi)
      display(opexp, dexp)
      display(opGamma, dgamma)
      display(opGamma2, dgamma)
      display(opfact, dfact)
      display(opquote, dquote)
      display(opperm, supersub("A"::Symbol::O, #1))
      display(opbinom, binomial(first #1, second #1))
      display(oppow, first(#1) ** second(#1))
      display(opsum,   sum(first #1, second #1, third #1))
      display(opprod, prod(first #1, second #1, third #1))
      display(opint, int(first #1 * hconcat("d"::Symbol::O, second #1),
                                                   empty(), third #1))
      input(oppren, convert concat(convert("("::Symbol)@InputForm,
                            concat(#1, convert(")"::Symbol)@InputForm)))
      input(oppow, convert concat(convert("**"::Symbol)@InputForm, #1))
      input(oproot,
            convert [convert("**"::Symbol)@InputForm, first #1, 1 / second #1])
      for op in algop   repeat assert(op, ALGOP)
      for op in rtrigop repeat assert(op, "rtrig")
      for op in htrigop repeat assert(op, "htrig")
      for op in trigop  repeat assert(op, "trig")
      for op in elemop  repeat assert(op, "elem")
      for op in primop  repeat assert(op, "prim")
      for op in combop  repeat assert(op, "comb")
      for op in specop  repeat assert(op, "special")
      for op in anyop   repeat assert(op, "any")
      for op in evenop  repeat assert(op, EVEN)
      for op in oddop   repeat assert(op, ODD)
      for op in dummyvarop1 repeat setDummyVar(op, 1)
      for op in dummyvarop2 repeat setDummyVar(op, 2)
      assert(oppren, "linear")
      void

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

-- SPAD files for the functional world should be compiled in the
-- following order:
--
--   OP  kl  expr function

<<domain BOP BasicOperator>>
<<package BOP1 BasicOperatorFunctions1>>
<<package COMMONOP CommonOperators>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}