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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 | --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
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