diff --git a/changelog b/changelog index 297e140..eff0ef8 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20141116 tpd src/axiom-website/patches.html 20141116.03.tpd.patch +20141116 tpd src/input/Makefile remove gonshor.input 20141116 tpd src/axiom-website/patches.html 20141116.02.tpd.patch 20141116 tpd books/bookvol10.3 help for AlgebraGivenByStructuralConstants 20141116 tpd src/axiom-website/patches.html 20141116.01.tpd.patch diff --git a/patch b/patch index d30d487..150182f 100644 --- a/patch +++ b/patch @@ -1,6 +1,6 @@ -books/bookvol10.3 help file for AlgebraGivenByStructuralConstants (ALGSC) +src/input/Makefile remove gonshor.input + +gonshor.input has been merged with AlgebraGivenByStructuralConstants (ALGSC) -AlgebraGivenByStructuralConstants (ALGSC) help file -coerce from ALGSC help file diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index f301dbe..5fc0c70 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4704,6 +4704,8 @@ books/bookvol5 inline object structures using macros
books/bookvol10.3 prototype man-page style help for functions
20141116.02.tpd.patch books/bookvol10.3 help file for AlgebraGivenByStructuralConstants
+20141116.03.tpd.patch +src/input/Makefile remove gonshor.input diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 72a999e..435c5e2 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -344,7 +344,7 @@ REGRESSTESTS= ackermann.regress \ fr2.regress frac.regress fr.regress free.regress \ function.regress functioncode.regress \ galois.regress gamma.regress \ - gbf.regress genups.regress gonshor.regress graphviz.regress \ + gbf.regress genups.regress graphviz.regress \ groeb.regress grpthry.regress \ gstbl.regress guess.regress \ heap.regress heat.regress help.regress \ @@ -766,7 +766,7 @@ FILES= ${OUT}/ackermann.input \ ${OUT}/fr.input ${OUT}/frame.input \ ${OUT}/fr1.input ${OUT}/gary1.input \ ${OUT}/gbf.input ${OUT}/genups.input ${OUT}/gnarly1.input \ - ${OUT}/gonshor.input ${OUT}/graphviz.input ${OUT}/grdef.input \ + ${OUT}/graphviz.input ${OUT}/grdef.input \ ${OUT}/gstbl.input ${OUT}/guess.input \ ${OUT}/heap.input ${OUT}/heat.input ${OUT}/helix.input \ ${OUT}/herm.input ${OUT}/heugcd.input \ @@ -1183,7 +1183,7 @@ DOCFILES= \ ${DOC}/gamma.input.dvi \ ${DOC}/gary1.input.dvi ${DOC}/gbf.input.dvi \ ${DOC}/genups.input.dvi ${DOC}/gnarly1.input.dvi \ - ${DOC}/gonshor.input.dvi ${DOC}/graphics.input.dvi \ + ${DOC}/graphics.input.dvi \ ${DOC}/graphviz.input.dvi \ ${DOC}/grdef.input.dvi ${DOC}/groeb.input.dvi \ ${DOC}/grpthry.input.dvi \ diff --git a/src/input/gonshor.input.pamphlet b/src/input/gonshor.input.pamphlet deleted file mode 100644 index f851a52..0000000 --- a/src/input/gonshor.input.pamphlet +++ /dev/null @@ -1,1065 +0,0 @@ -\documentclass{article} -\usepackage{axiom} -\setlength{\textwidth}{400pt} -\begin{document} -\title{\$SPAD/src/input gonshor.input} -\author{Timothy Daly} -\maketitle -\begin{abstract} -\end{abstract} -\eject -\tableofcontents -\eject -\section{License} -\begin{chunk}{license} ---Copyright The Numerical Algorithms Group Limited 1991. -\end{chunk} -\begin{chunk}{*} -)set break resume -)spool gonshor.output -)set message test on -)set message auto off -)clear all - -\end{chunk} -\section{Some examples of algebras in genetics} -Literature: -[WB] A. Woerz-Busekros: Algebras in Genetics, LNB 36, -Springer-Verlag, Berlin etc. 1980. - -\subsection{Commutative, non-associative algebras} -A Gonshor genetic algebra ([WB], p. 41-42) of dimension 4: - -The coefficient ring: -\begin{chunk}{*} ---S 1 of 98 -R := FRAC POLY INT ---R ---R ---R (1) Fraction(Polynomial(Integer)) ---R Type: Domain ---E 1 - -\end{chunk} -The following multiplication constants may be chosen arbitrarily -(notice that we write ckij for $c_(i,j)^k$): -\begin{chunk}{*} ---S 2 of 98 -(c100, c101, _ -c200, c201, c202, c211, _ -c300, c301, c302, c303, c311, c312, c322) : R ---R ---R Type: Void ---E 2 - ---S 3 of 98 -c100 := 1 ; c101 := -1 ; ---R ---R ---R Type: Fraction(Polynomial(Integer)) ---E 3 - ---S 4 of 98 -c200 := 0 ; c201 := 1 ; c202 := -1 ; - c211 := 2 ; ---R ---R ---R Type: Fraction(Polynomial(Integer)) ---E 4 - ---S 5 of 98 -c300 := 1 ; c301 := 0 ; c302 := -1 ; c303 := 1 ; - c311 := 1 ; c312 := 0 ; - c322 := 2 ; ---R ---R ---R Type: Fraction(Polynomial(Integer)) ---E 5 - -\end{chunk} -The matrices of the multiplication constants: -\begin{chunk}{*} ---S 6 of 98 -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] ] ] ; ---R ---R ---R Type: List(SquareMatrix(4,Fraction(Polynomial(Integer)))) ---E 6 - ---S 7 of 98 -basisSymbols : List Symbol := [subscript(e,[i]) for i in 0..3] ---R ---R ---R (7) [e ,e ,e ,e ] ---R 0 1 2 3 ---R Type: List(Symbol) ---E 7 - ---S 8 of 98 -GonshorGenetic := ALGSC(R, 4, basisSymbols, gonshor) ---R ---R ---R (8) ---R AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e ---R 1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---R Type: Domain ---E 8 - ---S 9 of 98 -commutative?()$GonshorGenetic ---R ---R algebra is commutative ---R ---R (9) true ---R Type: Boolean ---E 9 - ---S 10 of 98 -associative?()$GonshorGenetic ---R ---R algebra is not associative ---R ---R (10) false ---R Type: Boolean ---E 10 - -\end{chunk} -The canonical basis: -\begin{chunk}{*} ---S 11 of 98 -e0 : GonshorGenetic := [1, 0, 0, 0] :: Vector R ; ---R ---R ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 11 - ---S 12 of 98 -e1 : GonshorGenetic := [0, 1, 0, 0] :: Vector R ; ---R ---R ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 12 - ---S 13 of 98 -e2 : GonshorGenetic := [0, 0, 1, 0] :: Vector R ; ---R ---R ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 13 - ---S 14 of 98 -e3 : GonshorGenetic := [0, 0, 0, 1] :: Vector R ; ---R ---R ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 14 - -\end{chunk} -A generic element of the algebra: -\begin{chunk}{*} ---S 15 of 98 -x : GonshorGenetic := x0*e0 + x1*e1 + x2*e2 + x3*e3 ---R ---R ---R (15) x3 e + x2 e + x1 e + x0 e ---R 3 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 15 - -\end{chunk} -The matrix of the left multiplication with x : -\begin{chunk}{*} ---S 16 of 98 -Lx := leftRegularRepresentation x ---R ---R ---R +x0 - x1 + x0 - x2 + x1 x3 - x2 + x0+ ---R | | ---R |0 - x0 2x1 + x0 x1 | ---R (16) | | ---R |0 0 - x0 2x2 - x0 | ---R | | ---R +0 0 0 x0 + ---R Type: Matrix(Fraction(Polynomial(Integer))) ---E 16 - -\end{chunk} -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 : -\begin{chunk}{*} ---S 17 of 98 -p := characteristicPolynomial(Lx,Y) ---R ---R ---R 4 2 2 4 ---R (17) x0 - 2Y x0 + Y ---R Type: Polynomial(Integer) ---E 17 - -\end{chunk} -The left minimal polynomial of x divides Y * p(Y) : -\begin{chunk}{*} ---S 18 of 98 -leftMinimalPolynomial x ---R ---R ---R 5 2 3 4 ---R (18) ? - 2x0 ? + x0 ? ---R Type: SparseUnivariatePolynomial(Fraction(Polynomial(Integer))) ---E 18 - -)clear prop A a b c r s - ---S 19 of 98 -A := GonshorGenetic ---R ---R ---R (19) ---R AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e ---R 1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---R Type: Domain ---E 19 - ---S 20 of 98 -a := x ---R ---R ---R (20) x3 e + x2 e + x1 e + x0 e ---R 3 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 20 - ---S 21 of 98 -b := (1/4)*e1 + (1/5)*e2 + (3/20)*e3 + (2/5)*e0 ---R ---R ---R 3 1 1 2 ---R (21) -- e + - e + - e + - e ---R 20 3 5 2 4 1 5 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 21 - ---S 22 of 98 -c := (1/3)*e1 + (1/7)*e2 + (8/21)*e3 + (1/7)*e0 ---R ---R ---R 8 1 1 1 ---R (22) -- e + - e + - e + - e ---R 21 3 7 2 3 1 7 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 22 - ---S 23 of 98 -r : R := r ---R ---R ---R (23) r ---R Type: Fraction(Polynomial(Integer)) ---E 23 - ---S 24 of 98 -s : R := s ---R ---R ---R (24) s ---R Type: Fraction(Polynomial(Integer)) ---E 24 - ---S 25 of 98 -b*c ---R ---R ---R 2 1 47 2 ---R (25) - e + - e - --- e + -- e ---R 7 3 4 2 420 1 35 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 25 - ---S 26 of 98 -(b*c)*b ---R ---R ---R 893 277 4 4 ---R (26) ---- e - ---- e + -- e + --- e ---R 8400 3 1400 2 75 1 175 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 26 - ---S 27 of 98 -b*(c*b) ---R ---R ---R 893 277 4 4 ---R (27) ---- e - ---- e + -- e + --- e ---R 8400 3 1400 2 75 1 175 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 27 - -\end{chunk} -A: Algebra -a,b,c : A -r,s : R -\begin{chunk}{*} - -)clear prop AP ---S 28 of 98 -AP := ALGPKG(R,A) ---R ---R ---R (28) ---R AlgebraPackage(Fraction(Polynomial(Integer)),AlgebraGivenByStructuralConstant ---R s(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MA ---R TRIX,MATRIX])) ---R Type: Domain ---E 28 - ---S 29 of 98 -r*a ---R ---R ---R (29) r x3 e + r x2 e + r x1 e + r x0 e ---R 3 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 29 - ---S 30 of 98 -a*r ---R ---R ---R (30) r x3 e + r x2 e + r x1 e + r x0 e ---R 3 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 30 - ---S 31 of 98 -a*b ---R ---R ---R 8x3 + 5x1 + 7x0 - 8x2 + 18x1 + x0 - 8x1 + 3x0 2x0 ---R (31) --------------- e + ----------------- e + ----------- e + --- e ---R 20 3 20 2 20 1 5 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 31 - ---S 32 of 98 -b*c ---R ---R ---R 2 1 47 2 ---R (32) - e + - e - --- e + -- e ---R 7 3 4 2 420 1 35 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 32 - ---S 33 of 98 -12 * c ---R ---R ---R 32 12 12 ---R (33) -- e + -- e + 4e + -- e ---R 7 3 7 2 1 7 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 32 - ---S 34 of 98 -(-3) * a ---R ---R ---R (34) - 3x3 e - 3x2 e - 3x1 e - 3x0 e ---R 3 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 34 - ---S 35 of 98 -d := a ** 12 ---R ---R ---R (35) ---R 11 10 2 9 2 10 11 8 4 ---R 12x0 x3 + 4x0 x2 + (144x0 x1 + 144x0 x1 - 68x0 )x2 + 248x0 x1 ---R + ---R 9 3 10 2 11 12 ---R - 784x0 x1 - 86x0 x1 + 204x0 x1 - 24x0 ---R * ---R e ---R 3 ---R + ---R 11 10 2 11 11 12 12 ---R (4x0 x2 - 92x0 x1 + 28x0 x1)e + (4x0 x1 - x0 )e + x0 e ---R 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 35 - ---S 36 of 98 --d ---R ---R ---R (36) ---R 11 10 2 9 2 10 11 ---R - 12x0 x3 - 4x0 x2 + (- 144x0 x1 - 144x0 x1 + 68x0 )x2 ---R + ---R 8 4 9 3 10 2 11 12 ---R - 248x0 x1 + 784x0 x1 + 86x0 x1 - 204x0 x1 + 24x0 ---R * ---R e ---R 3 ---R + ---R 11 10 2 11 11 12 12 ---R (- 4x0 x2 + 92x0 x1 - 28x0 x1)e + (- 4x0 x1 + x0 )e - x0 e ---R 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 36 - ---S 37 of 98 -a + b ---R ---R ---R 20x3 + 3 5x2 + 1 4x1 + 1 5x0 + 2 ---R (37) -------- e + ------- e + ------- e + ------- e ---R 20 3 5 2 4 1 5 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 37 - ---S 38 of 98 -d-c ---R ---R ---R (38) ---R 11 10 2 9 2 10 11 ---R 252x0 x3 + 84x0 x2 + (3024x0 x1 + 3024x0 x1 - 1428x0 )x2 ---R + ---R 8 4 9 3 10 2 11 12 ---R 5208x0 x1 - 16464x0 x1 - 1806x0 x1 + 4284x0 x1 - 504x0 - 8 ---R / ---R 21 ---R * ---R e ---R 3 ---R + ---R 11 10 2 11 11 12 ---R 28x0 x2 - 644x0 x1 + 196x0 x1 - 1 12x0 x1 - 3x0 - 1 ---R ------------------------------------- e + -------------------- e ---R 7 2 3 1 ---R + ---R 12 ---R 7x0 - 1 ---R --------- e ---R 7 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 38 - ---S 39 of 98 -(a*(a*a) = leftPower(a,3)) :: Boolean ---R ---R ---R (39) true ---R Type: Boolean ---E 39 - ---S 40 of 98 -(a ** 11 = (a**8 * a**2) * a) :: Boolean ---R ---R ---R (40) true ---R Type: Boolean ---E 40 - ---S 41 of 98 -(a ** 11 = a**8 * (a**2 * a)) :: Boolean ---R ---R ---R (41) false ---R Type: Boolean ---E 41 - ---S 42 of 98 -zero := 0$A ---R ---R ---R (42) 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 42 - ---S 43 of 98 -zero : A := 0 ---R ---R ---R (43) 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 43 - ---S 44 of 98 -alternative?()$A ---R ---R algebra is not left alternative ---R ---R (44) false ---R Type: Boolean ---E 44 - ---S 45 of 98 -antiCommutative?()$A ---R ---R algebra is not anti-commutative ---R ---R (45) false ---R Type: Boolean ---E 45 - ---S 46 of 98 -associative?()$A ---R ---R algebra is not associative ---R ---R (46) false ---R Type: Boolean ---E 46 - ---S 47 of 98 -commutative?()$A ---R ---R algebra is commutative ---R ---R (47) true ---R Type: Boolean ---E 47 - ---S 48 of 98 -commutator(a,b) ---R ---R ---R (48) 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 48 - ---S 49 of 98 -antiCommutator(a,b) ---R ---R ---R 8x3 + 5x1 + 7x0 - 8x2 + 18x1 + x0 - 8x1 + 3x0 4x0 ---R (49) --------------- e + ----------------- e + ----------- e + --- e ---R 10 3 10 2 10 1 5 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 49 - ---S 50 of 98 -associator(a,b,c) ---R ---R ---R - 21x2 + 6x1 + 7x0 12x2 - 30x1 + 58x0 12x1 - 28x0 ---R (50) ------------------ e + ------------------ e + ----------- e ---R 42 3 105 2 105 1 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 50 - ---S 51 of 98 -basis()$A ---R ---R ---R (51) [e ,e ,e ,e ] ---R 0 1 2 3 ---RType: Vector(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 51 - ---S 52 of 98 -n := rank()$A ---R ---R ---R (52) 4 ---R Type: PositiveInteger ---E 52 - ---S 53 of 98 -v : Vector R := [i for i in 1..n] ---R ---R ---R (53) [1,2,3,4] ---R Type: Vector(Fraction(Polynomial(Integer))) ---E 53 - ---S 54 of 98 -g : A := represents v ---R ---R ---R (54) 4e + 3e + 2e + e ---R 3 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 54 - ---S 55 of 98 -coordinates a ---R ---R ---R (55) [x0,x1,x2,x3] ---R Type: Vector(Fraction(Polynomial(Integer))) ---E 55 - ---S 56 of 98 -coordinates [a,b] ---R ---R ---R +x0 x1 x2 x3+ ---R | | ---R (56) |2 1 1 3| ---R |- - - --| ---R +5 4 5 20+ ---R Type: Matrix(Fraction(Polynomial(Integer))) ---E 56 - ---S 57 of 98 -a.3 ---R ---R ---R (57) x2 ---R Type: Fraction(Polynomial(Integer)) ---E 57 - ---S 58 of 98 -flexible?()$A ---R ---R algebra is flexible ---R ---R (58) true ---R Type: Boolean ---E 58 - ---S 59 of 98 -leftAlternative?()$A ---R ---R algebra is not left alternative ---R ---R (59) false ---R Type: Boolean ---E 59 - ---S 60 of 98 -rightAlternative?()$A ---R ---R algebra is not right alternative ---R ---R (60) false ---R Type: Boolean ---E 60 - ---S 61 of 98 -sB := someBasis()$A ---R ---R ---R (61) [e ,e ,e ,e ] ---R 0 1 2 3 ---RType: Vector(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 61 - ---S 62 of 98 -zero? a ---R ---R ---R (62) false ---R Type: Boolean ---E 62 - ---S 63 of 98 -associatorDependence()$A ---R ---R ---R (63) [[1,1,1,0,0,0],[0,1,0,1,0,0],[1,0,0,0,1,0],[- 1,- 1,0,0,0,1]] ---R Type: List(Vector(Fraction(Polynomial(Integer)))) ---E 63 - -\end{chunk} -ConditionsForIdempotents()\$A -\begin{chunk}{*} ---S 64 of 98 -jacobiIdentity?()$A ---R ---R Jacobi identity does not hold ---R ---R (64) false ---R Type: Boolean ---E 64 - ---S 65 of 98 -jordanAlgebra?()$A ---R ---R algebra is commutative ---R this is not a Jordan algebra ---R ---R (65) false ---R Type: Boolean ---E 65 - ---S 66 of 98 -jordanAdmissible?()$A ---R ---R algebra is not Jordan admissible ---R ---R (66) false ---R Type: Boolean ---E 66 - ---S 67 of 98 -lieAdmissible?()$A ---R ---R algebra is Lie admissible ---R ---R (67) true ---R Type: Boolean ---E 67 - -\end{chunk} -ConditionsForIdempotents -\begin{chunk}{*} ---S 68 of 98 -b2 := [reduce(+,[sB.i for i in 1..k]) for k in 1..n] ---R ---R ---R (68) [e ,e + e ,e + e + e ,e + e + e + e ] ---R 0 1 0 2 1 0 3 2 1 0 ---RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 68 - ---S 69 of 98 -coordinates (a ,b2 :: Vector A) ---R ---R ---R (69) [- x1 + x0,- x2 + x1,- x3 + x2,x3] ---R Type: Vector(Fraction(Polynomial(Integer))) ---E 69 - ---S 70 of 98 -coordinates ([a,b] ,bb := (b2 :: Vector A)) ---R ---R ---R +- x1 + x0 - x2 + x1 - x3 + x2 x3+ ---R | | ---R (70) | 3 1 1 3| ---R | -- -- -- --| ---R + 20 20 20 20+ ---R Type: Matrix(Fraction(Polynomial(Integer))) ---E 70 - ---S 71 of 98 -leftMinimalPolynomial a ---R ---R ---R 5 2 3 4 ---R (71) ? - 2x0 ? + x0 ? ---R Type: SparseUnivariatePolynomial(Fraction(Polynomial(Integer))) ---E 71 - ---S 72 of 98 -leftPower (a,10) ---R ---R ---R (72) ---R 9 8 2 7 2 8 9 8 2 10 ---R (10x0 x3 - 6x0 x2 + (- 32x0 x1 + 8x0 x1 + 2x0 )x2 + 13x0 x1 + 5x0 )e ---R 3 ---R + ---R 9 8 2 9 10 9 10 10 ---R (- 2x0 x2 + 26x0 x1 + 6x0 x1 - 4x0 )e + (- 2x0 x1 + x0 )e + x0 e ---R 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 72 - ---S 73 of 98 -rightPower(a,10) ---R ---R ---R (73) ---R 9 8 2 7 2 8 9 8 2 10 ---R (10x0 x3 - 6x0 x2 + (- 32x0 x1 + 8x0 x1 + 2x0 )x2 + 13x0 x1 + 5x0 )e ---R 3 ---R + ---R 9 8 2 9 10 9 10 10 ---R (- 2x0 x2 + 26x0 x1 + 6x0 x1 - 4x0 )e + (- 2x0 x1 + x0 )e + x0 e ---R 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 73 - ---S 74 of 98 -leftRegularRepresentation a ---R ---R ---R +x0 - x1 + x0 - x2 + x1 x3 - x2 + x0+ ---R | | ---R |0 - x0 2x1 + x0 x1 | ---R (74) | | ---R |0 0 - x0 2x2 - x0 | ---R | | ---R +0 0 0 x0 + ---R Type: Matrix(Fraction(Polynomial(Integer))) ---E 74 - ---S 75 of 98 -leftRegularRepresentation (a,bb) ---R ---R ---R + x1 x2 - 2x1 + x0 - x3 + x1 - x0 x3 - x2 + x0 + ---R | | ---R |x1 + x0 x2 - 4x1 - x0 - x3 + 2x1 x3 - x2 + x1 + x0| ---R (75) | | ---R |x1 + x0 x2 - 4x1 - x3 - 2x2 + 2x1 x3 + x2 + x1 | ---R | | ---R +x1 + x0 x2 - 4x1 - x3 - 2x2 + 2x1 - x0 x3 + x2 + x1 + x0+ ---R Type: Matrix(Fraction(Polynomial(Integer))) ---E 75 - ---S 76 of 98 -leftUnit()$A ---R ---R this algebra has no left unit ---R ---R (76) "failed" ---R Type: Union("failed",...) ---E 76 - ---S 77 of 98 -represents (v,bb) ---R ---R ---R (77) 4e + 7e + 9e + 10e ---R 3 2 1 0 ---RType: AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX]) ---E 77 - ---S 78 of 98 -rightMinimalPolynomial a ---R ---R ---R 5 2 3 4 ---R (78) ? - 2x0 ? + x0 ? ---R Type: SparseUnivariatePolynomial(Fraction(Polynomial(Integer))) ---E 78 - ---S 79 of 98 -rightRegularRepresentation a ---R ---R ---R +x0 - x1 + x0 - x2 + x1 x3 - x2 + x0+ ---R | | ---R |0 - x0 2x1 + x0 x1 | ---R (79) | | ---R |0 0 - x0 2x2 - x0 | ---R | | ---R +0 0 0 x0 + ---R Type: Matrix(Fraction(Polynomial(Integer))) ---E 79 - ---S 80 of 98 -rightRegularRepresentation (a,bb) ---R ---R ---R + x1 x2 - 2x1 + x0 - x3 + x1 - x0 x3 - x2 + x0 + ---R | | ---R |x1 + x0 x2 - 4x1 - x0 - x3 + 2x1 x3 - x2 + x1 + x0| ---R (80) | | ---R |x1 + x0 x2 - 4x1 - x3 - 2x2 + 2x1 x3 + x2 + x1 | ---R | | ---R +x1 + x0 x2 - 4x1 - x3 - 2x2 + 2x1 - x0 x3 + x2 + x1 + x0+ ---R Type: Matrix(Fraction(Polynomial(Integer))) ---E 80 - ---S 81 of 98 -rightUnit()$A ---R ---R this algebra has no right unit ---R ---R (81) "failed" ---R Type: Union("failed",...) ---E 81 - ---S 82 of 98 -structuralConstants()$A ---R ---R ---R +1 0 0 0+ + 1 - 1 0 0+ + 0 1 - 1 0+ + 1 0 - 1 1+ ---R | | | | | | | | ---R |0 0 0 0| |- 1 0 0 0| | 1 2 0 0| | 0 1 0 0| ---R (82) [| |,| |,| |,| |] ---R |0 0 0 0| | 0 0 0 0| |- 1 0 0 0| |- 1 0 2 0| ---R | | | | | | | | ---R +0 0 0 0+ + 0 0 0 0+ + 0 0 0 0+ + 1 0 0 0+ ---R Type: Vector(Matrix(Fraction(Polynomial(Integer)))) ---E 82 - ---S 83 of 98 -structuralConstants(bb) ---R ---R ---R +0 1 1 1+ + 1 - 1 0 0 + +- 1 0 0 - 1+ +1 1 0 1+ ---R | | | | | | | | ---R |1 2 2 2| |- 1 - 5 - 4 - 4| | 0 2 2 1 | |1 2 1 2| ---R (83) [| |,| |,| |,| |] ---R |1 2 2 2| | 0 - 4 - 3 - 3| | 0 2 0 - 1| |0 1 2 3| ---R | | | | | | | | ---R +1 2 2 2+ + 0 - 4 - 3 - 3+ +- 1 1 - 1 - 2+ +1 2 3 4+ ---R Type: Vector(Matrix(Fraction(Polynomial(Integer)))) ---E 83 - ---S 84 of 98 -unit()$A ---R ---R this algebra has no unit ---R ---R (84) "failed" ---R Type: Union("failed",...) ---E 84 - ---S 85 of 98 -biRank a ---R ---R ---R (85) 4 ---R Type: PositiveInteger ---E 85 - ---S 86 of 98 -leftRank a ---R ---R ---R (86) 4 ---R Type: PositiveInteger ---E 86 - ---S 87 of 98 -doubleRank a ---R ---R ---R (87) 4 ---R Type: PositiveInteger ---E 87 - ---S 88 of 98 -rightRank a ---R ---R ---R (88) 4 ---R Type: PositiveInteger ---E 88 - ---S 89 of 98 -weakBiRank a ---R ---R ---R (89) 4 ---R Type: PositiveInteger ---E 89 - ---S 90 of 98 -basisOfCenter()$AP ---R ---R ---R (90) [e ] ---R 3 ---RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 90 - ---S 91 of 98 -basisOfLeftNucleus()$AP ---R ---R ---R (91) [e ] ---R 3 ---RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 91 - ---S 92 of 98 -basisOfNucleus()$AP ---R ---R ---R (92) [e ] ---R 3 ---RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 92 - ---S 93 of 98 -basisOfRightNucleus()$AP ---R ---R ---R (93) [e ] ---R 3 ---RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 93 - ---S 94 of 98 -basisOfCentroid()$AP ---R ---R ---R +0 0 0 0+ +1 0 0 0+ ---R | | | | ---R |0 0 0 0| |0 1 0 0| ---R (94) [| |,| |] ---R |0 0 0 0| |0 0 1 0| ---R | | | | ---R +1 0 0 0+ +0 0 0 1+ ---R Type: List(Matrix(Fraction(Polynomial(Integer)))) ---E 94 - ---S 95 of 98 -basisOfCommutingElements()$AP ---R ---R ---R (95) [e ,e ,e ,e ] ---R 3 2 1 0 ---RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 95 - ---S 96 of 98 -basisOfLeftNucloid()$AP ---R ---R ---R +0 0 0 0+ +1 0 0 0+ ---R | | | | ---R |0 0 0 0| |0 1 0 0| ---R (96) [| |,| |] ---R |0 0 0 0| |0 0 1 0| ---R | | | | ---R +1 0 0 0+ +0 0 0 1+ ---R Type: List(Matrix(Fraction(Polynomial(Integer)))) ---E 96 - ---S 97 of 98 -basisOfMiddleNucleus()$AP ---R ---R ---R (97) [e ] ---R 3 ---RType: List(AlgebraGivenByStructuralConstants(Fraction(Polynomial(Integer)),4,[*01e0,*01e1,*01e2,*01e3],[MATRIX,MATRIX,MATRIX,MATRIX])) ---E 97 - ---S 98 of 98 -basisOfRightNucloid()$AP ---R ---R ---R +0 0 0 0+ +1 0 0 0+ ---R | | | | ---R |0 0 0 0| |0 1 0 0| ---R (98) [| |,| |] ---R |0 0 0 0| |0 0 1 0| ---R | | | | ---R +1 0 0 0+ +0 0 0 1+ ---R Type: List(Matrix(Fraction(Polynomial(Integer)))) ---E 98 -)spool -)lisp (bye) - -\end{chunk} -\eject -\begin{thebibliography}{99} -\bibitem{1} nothing -\end{thebibliography} -\end{document} - -