From ab8cc85adde879fb963c94d15675783f2cf4b183 Mon Sep 17 00:00:00 2001 From: dos-reis Date: Tue, 14 Aug 2007 05:14:52 +0000 Subject: Initial population. --- src/input/gonshor.input.pamphlet | 218 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 218 insertions(+) create mode 100644 src/input/gonshor.input.pamphlet (limited to 'src/input/gonshor.input.pamphlet') diff --git a/src/input/gonshor.input.pamphlet b/src/input/gonshor.input.pamphlet new file mode 100644 index 00000000..51f08777 --- /dev/null +++ b/src/input/gonshor.input.pamphlet @@ -0,0 +1,218 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input gonshor.input} +\author{The Axiom Team} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{License} +<>= +--Copyright The Numerical Algorithms Group Limited 1991. +@ +<<*>>= +<> + +------------- Some examples of algebras in genetics ------------- + +-- Literature: +-- [WB] A. Woerz-Busekros: Algebras in Genetics, LNB 36, +-- Springer-Verlag, Berlin etc. 1980. + + + +--------------- Commutative, non-associative algebras -- + + + +-- A Gonshor genetic algebra ([WB], p. 41-42) of dimension 4: +-- ========================================================= + +)clear all + +-- The coefficient ring: +R := FRAC POLY INT + +-- The following multiplication constants may be chosen arbitrarily +-- (notice that we write ckij for c_(i,j)^k): + +(c100, c101, _ +c200, c201, c202, c211, _ +c300, c301, c302, c303, c311, c312, c322) : R + +---------------------------------------------------------------- +c100 := 1 ; c101 := -1 ; +---------------------------------------------------------------- +c200 := 0 ; c201 := 1 ; c202 := -1 ; + c211 := 2 ; +---------------------------------------------------------------- +c300 := 1 ; c301 := 0 ; c302 := -1 ; c303 := 1 ; + c311 := 1 ; c312 := 0 ; + c322 := 2 ; +---------------------------------------------------------------- + +-- The matrices of the multiplication constants: + +gonshor : List SquareMatrix(4,R) := + [matrix [ [1, 0, 0, 0], [0, 0, 0, 0],_ + [0, 0, 0, 0], [0, 0, 0, 0] ],_ + matrix [ [c100, c101, 0, 0], [c101, 0, 0, 0],_ + [0, 0, 0, 0], [0, 0, 0, 0] ],_ + matrix [ [c200, c201, c202, 0], [c201, c211, 0, 0],_ + [c202, 0, 0, 0], [0, 0, 0, 0] ],_ + matrix [ [c300, c301, c302, c303], [c301, c311, c312, 0],_ + [c302, c312, c322, 0], [c303, 0, 0, 0] ] ] ; + + +basisSymbols : List Symbol := [subscript(e,[i]) for i in 0..3] + +GonshorGenetic := ALGSC(R, 4, basisSymbols, gonshor) + +commutative?()$GonshorGenetic +associative?()$GonshorGenetic + +-- The canonical basis: +e0 : GonshorGenetic := [1, 0, 0, 0] :: Vector R ; +e1 : GonshorGenetic := [0, 1, 0, 0] :: Vector R ; +e2 : GonshorGenetic := [0, 0, 1, 0] :: Vector R ; +e3 : GonshorGenetic := [0, 0, 0, 1] :: Vector R ; + + +-- A generic element of the algebra: +x : GonshorGenetic := x0*e0 + x1*e1 + x2*e2 + x3*e3 + +-- The matrix of the left multiplication with x : +Lx := leftRegularRepresentation x + +-- leftRegularRepresentationt 8 : GonshorGenetic -> R be the weight homomorphism +-- defined by 8(e0) := 1 and 8(ei) := 0 for i = 1,2,3 . +-- The coefficients of the characteristic polynomial +-- of Lx depend only on 8(x) = x0 : +p := characteristicPolynomial(Lx,Y) + +-- The left minimal polynomial of x divides Y * p(Y) : +leftMinimalPolynomial x + + +)clear prop A a b c r s +A := GonshorGenetic +a := x +b := (1/4)*e1 + (1/5)*e2 + (3/20)*e3 + (2/5)*e0 +c := (1/3)*e1 + (1/7)*e2 + (8/21)*e3 + (1/7)*e0 +r : R := r +s : R := s + +b*c +(b*c)*b +b*(c*b) + +-- A: Algebra +-- a,b,c : A +-- r,s : R + +)clear prop AP +AP := ALGPKG(R,A) + +r*a +a*r + +a*b +b*c + +12 * c +(-3) * a + +d := a ** 12 +-d + +a + b +d-c + +(a*(a*a) = leftPower(a,3)) :: Boolean +(a ** 11 = (a**8 * a**2) * a) :: Boolean +(a ** 11 = a**8 * (a**2 * a)) :: Boolean + + +zero := 0$A +zero : A := 0 + +alternative?()$A +antiCommutative?()$A +associative?()$A +commutative?()$A + +commutator(a,b) +antiCommutator(a,b) +associator(a,b,c) +basis()$A + +n := rank()$A +v : Vector R := [i for i in 1..n] +g : A := represents v + +coordinates a +coordinates [a,b] + +a.3 + +flexible?()$A +leftAlternative?()$A + +rightAlternative?()$A +sB := someBasis()$A +zero? a +associatorDependence()$A +--conditionsForIdempotents()$A +jacobiIdentity?()$A +jordanAlgebra?()$A +jordanAdmissible?()$A +lieAdmissible?()$A +--conditionsForIdempotents sB +b2 := [reduce(+,[sB.i for i in 1..k]) for k in 1..n] +coordinates (a ,b2 :: Vector A) +coordinates ([a,b] ,bb := (b2 :: Vector A)) +leftMinimalPolynomial a +leftPower (a,10) +rightPower(a,10) +leftRegularRepresentation a +leftRegularRepresentation (a,bb) +leftUnit()$A +represents (v,bb) +rightMinimalPolynomial a +rightRegularRepresentation a +rightRegularRepresentation (a,bb) +rightUnit()$A +structuralConstants()$A +structuralConstants(bb) +unit()$A + + +-- functions from ALGPKG + +biRank a +leftRank a +doubleRank a +rightRank a +weakBiRank a + + +basisOfCenter()$AP +basisOfLeftNucleus()$AP +basisOfNucleus()$AP +basisOfRightNucleus()$AP +basisOfCentroid()$AP +basisOfCommutingElements()$AP +basisOfLeftNucloid()$AP +basisOfMiddleNucleus()$AP +basisOfRightNucloid()$AP + + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} -- cgit v1.2.3