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[Axiom-developer] 20080104.01.tpd.patch (7093)
From: |
daly |
Subject: |
[Axiom-developer] 20080104.01.tpd.patch (7093) |
Date: |
Fri, 4 Jan 2008 22:33:09 -0600 |
Martin fixed a number of misnamed functions in the input files.
These patches are redone and applied to the regression test files.
=======================================================================
diff --git a/changelog b/changelog
index 8acaf43..f6db6a4 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,11 @@
+20080104 mxr src/input/repa6.input fix function names (7093)
+20080104 mxr src/input/knot2.input fix function names (7093)
+20080104 mxr src/input/grpthry.input fix function names (7093)
+20080104 mxr src/input/exsum.input fix function names (7093)
+20080104 mxr src/input/exlap.input fix function names (7093)
+20080104 mxr src/input/easter.input fix function names (7093)
+20080104 mxr src/input/collect.input fix function names (7093)
+20080104 mxr src/input/calculus2.input fix function names (7093)
20080103 wxh src/algebra/sf.spad handle besselK (7090/355)
20080103 wxh src/algebra/op.spad handle besselK (7090/355)
20080103 wxh src/algebra/combfunc.spad handle besselK (7090/355)
diff --git a/src/input/calculus2.input.pamphlet
b/src/input/calculus2.input.pamphlet
index 3db6893..031f0ec 100644
--- a/src/input/calculus2.input.pamphlet
+++ b/src/input/calculus2.input.pamphlet
@@ -370,7 +370,7 @@ eq := differentiate(y(x), x, 3) - sin(differentiate(y(x),
x, 2)) * exp(y(x)) = c
--S 29 of 112
seriesSolve(eq, y, x = 0, [1, 0, 0])
--R
---R Compiling function %B with type List UnivariateTaylorSeries(
+--I Compiling function %B with type List UnivariateTaylorSeries(
--R Expression Integer,x,0) -> UnivariateTaylorSeries(Expression
--R Integer,x,0)
--R
@@ -388,7 +388,7 @@ seriesSolve(eq, y, x = 0, [1, 0, 0])
--S 30 of 112
x := operator 'x
--R
---R Compiled code for %B has been cleared.
+--I Compiled code for %B has been cleared.
--R
--R (4) x
--R Type:
BasicOperator
@@ -417,10 +417,10 @@ eq2 := differentiate(y(t), t) = x(t) * y(t)
--S 33 of 112
seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])
--R
---R Compiling function %D with type List UnivariateTaylorSeries(
+--I Compiling function %D with type List UnivariateTaylorSeries(
--R Expression Integer,t,0) -> UnivariateTaylorSeries(Expression
--R Integer,t,0)
---R Compiling function %E with type List UnivariateTaylorSeries(
+--I Compiling function %E with type List UnivariateTaylorSeries(
--R Expression Integer,t,0) -> UnivariateTaylorSeries(Expression
--R Integer,t,0)
--R
@@ -493,37 +493,27 @@ laplace((cos(a*t) - cos(b*t))/t, t, s)
--E 39
--S 40 of 112
-laplace(exp(a*t+b)*ei(c*t), t, s)
---R
---R There are no library operations named ei
---R Use HyperDoc Browse or issue
---R )what op ei
---R to learn if there is any operation containing " ei " in its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named ei
---R with argument type(s)
---R Polynomial Integer
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+laplace(exp(a*t+b)*Ei(c*t), t, s)
+--R
+--R b s + c - a
+--R %e log(---------)
+--R c
+--R (7) -----------------
+--R s - a
+--R Type: Expression
Integer
--E 40
--S 41 of 112
-laplace(a*ci(b*t) + c*si(d*t), t, s)
---R
---R There are no library operations named ci
---R Use HyperDoc Browse or issue
---R )what op ci
---R to learn if there is any operation containing " ci " in its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named ci
---R with argument type(s)
---R Polynomial Integer
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+laplace(a*Ci(b*t) + c*Si(d*t), t, s)
+--R
+--R 2 2
+--R s + b d
+--R a log(-------) + 2c atan(-)
+--R 2 s
+--R b
+--R (8) ---------------------------
+--R 2s
+--R Type: Expression
Integer
--E 41
--S 42 of 112
@@ -533,7 +523,7 @@ laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)
--R 2
--R 4 2 2 4 t 3
--R (s + 2a s + a )laplace(%e ,t,s) + 2a
---R (7) ----------------------------------------
+--R (9) ----------------------------------------
--R 4 2 2 4
--R s + 2a s + a
--R Type: Expression
Integer
@@ -637,9 +627,9 @@ integrate(g, x)
--R
--R
--R x +--------+
---R ++ log(\|b + %G a + 1)
---R (4) | -------------------- d%G
---R ++ %G
+--I ++ log(\|b + %G a + 1)
+--I (4) | -------------------- d%G
+--I ++ %G
--R Type: Union(Expression
Integer,...)
--E 51
diff --git a/src/input/collect.input.pamphlet b/src/input/collect.input.pamphlet
index 5e92eda..9e07896 100644
--- a/src/input/collect.input.pamphlet
+++ b/src/input/collect.input.pamphlet
@@ -91,10 +91,71 @@ e := reverse [i**3 for i in 10..0 by -2 | even? i]
--R Type: List
Integer
--E 9
+--S 10 of 55
+[x**3 - y for x in b | even? x for y in e]
+--R
+--R (10) [0,- 56,- 448]
+--R Type: List
Integer
+--E 10
+
+--S 11 of 55
+f := [i**3 for i in 0..]
+--R
+--R (11) [0,1,8,27,64,125,216,343,512,729,...]
+--R Type: Stream
NonNegativeInteger
+--E 11
+
+--S 12 of 55
+[i**3 for i in 0..10]
+--R
+--R (12) [0,1,8,27,64,125,216,343,512,729,1000]
+--R Type: List
NonNegativeInteger
+--E 12
+
+--S 13 of 55
+[i**3 for i in 0.. while i < 11]
+--R
+--R (13) [0,1,8,27,64,125,216,343,512,729,...]
+--R Type: Stream
NonNegativeInteger
+--E 13
+
+--S 14 of 55
+[i**3 for i in 0.. for x in 0..10]
+--R
+--R (14) [0,1,8,27,64,125,216,343,512,729,...]
+--R Type: Stream
NonNegativeInteger
+--E 14
+
+--S 15 of 55
+[ [i**j for j in 0..3] for i in 0..]
+--R
+--R (15)
+--R [[1,0,0,0], [1,1,1,1], [1,2,4,8], [1,3,9,27], [1,4,16,64], [1,5,25,125],
+--R [1,6,36,216], [1,7,49,343], [1,8,64,512], [1,9,81,729], ...]
+--R Type: Stream List
NonNegativeInteger
+--E 15
+
+--S 16 of 55
+[ [i**j for j in 0..] for i in 0..3]
+--R
+--R (16)
+--R [[1,0,0,0,0,0,0,0,0,0,...], [1,1,1,1,1,1,1,1,1,1,...],
+--R [1,2,4,8,16,32,64,128,256,512,...],
+--R [1,3,9,27,81,243,729,2187,6561,19683,...]]
+--R Type: List Stream Fraction
Integer
+--E 16
+
+--S 17 of 55
+brace [i**3 for i in 10..0 by -2]
+--R
+--R (17) {0,8,64,216,512,1000}
+--R Type: Set
NonNegativeInteger
+--E 17
+
-- Input generated from ContinuedFractionXmpPage
)clear all
---S 10 of 55
+--S 18 of 55
c := continuedFraction(314159/100000)
--R
--R
@@ -102,17 +163,17 @@ c := continuedFraction(314159/100000)
--R (1) 3 + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
--R | 7 | 15 | 1 | 25 | 1 | 7 | 4
--R Type: ContinuedFraction
Integer
---E 10
+--E 18
---S 11 of 55
+--S 19 of 55
partialQuotients c
--R
--R
--R (2) [3,7,15,1,25,1,7,4]
--R Type: Stream
Integer
---E 11
+--E 19
---S 12 of 55
+--S 20 of 55
convergents c
--R
--R
@@ -120,9 +181,9 @@ convergents c
--R (3) [3,--,---,---,----,----,-----,------]
--R 7 106 113 2931 3044 24239 100000
--R Type: Stream Fraction
Integer
---E 12
+--E 20
---S 13 of 55
+--S 21 of 55
approximants c
--R
--R
@@ -131,17 +192,17 @@ approximants c
--R (4) [3,--,---,---,----,----,-----,------]
--R 7 106 113 2931 3044 24239 100000
--R Type: Stream Fraction
Integer
---E 13
+--E 21
---S 14 of 55
+--S 22 of 55
pq := partialQuotients(1/c)
--R
--R
--R (5) [0,3,7,15,1,25,1,7,4]
--R Type: Stream
Integer
---E 14
+--E 22
---S 15 of 55
+--S 23 of 55
continuedFraction(first pq,repeating [1],rest pq)
--R
--R
@@ -149,9 +210,9 @@ continuedFraction(first pq,repeating [1],rest pq)
--R (6) +---+ + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
--R | 3 | 7 | 15 | 1 | 25 | 1 | 7 | 4
--R Type: ContinuedFraction
Integer
---E 15
+--E 23
---S 16 of 55
+--S 24 of 55
z:=continuedFraction(3,repeating [1],repeating [3,6])
--R
--R
@@ -164,17 +225,17 @@ z:=continuedFraction(3,repeating [1],repeating [3,6])
--R +---+ + ...
--R | 6
--R Type: ContinuedFraction
Integer
---E 16
+--E 24
---S 17 of 55
+--S 25 of 55
dens:Stream Integer := cons(1,generate((x+->x+4),6))
--R
--R
--R (8) [1,6,10,14,18,22,26,30,34,38,...]
--R Type: Stream
Integer
---E 17
+--E 25
---S 18 of 55
+--S 26 of 55
cf := continuedFraction(0,repeating [1],dens)
--R
--R
@@ -187,9 +248,9 @@ cf := continuedFraction(0,repeating [1],dens)
--R +----+ + +----+ + ...
--R | 34 | 38
--R Type: ContinuedFraction
Integer
---E 18
+--E 26
---S 19 of 55
+--S 27 of 55
ccf := convergents cf
--R
--R
@@ -197,9 +258,9 @@ ccf := convergents cf
--R (10) [0,1,-,--,----,-----,------,--------,---------,-----------,...]
--R 7 71 1001 18089 398959 10391023 312129649 10622799089
--R Type: Stream Fraction
Integer
---E 19
+--E 27
---S 20 of 55
+--S 28 of 55
eConvergents := [2*e + 1 for e in ccf]
--R
--R
@@ -207,9 +268,9 @@ eConvergents := [2*e + 1 for e in ccf]
--R (11) [1,3,--,---,----,-----,-------,--------,---------,-----------,...]
--R 7 71 1001 18089 398959 10391023 312129649 10622799089
--R Type: Stream Fraction
Integer
---E 20
+--E 28
---S 21 of 55
+--S 29 of 55
eConvergents :: Stream Float
--R
--R
@@ -219,17 +280,17 @@ eConvergents :: Stream Float
--R 2.7182818284 590458514, 2.7182818284 590452348, 2.7182818284 590452354,
--R ...]
--R Type: Stream
Float
---E 21
+--E 29
---S 22 of 55
+--S 30 of 55
exp 1.0
--R
--R
--R (13) 2.7182818284 590452354
--R Type:
Float
---E 22
+--E 30
---S 23 of 55
+--S 31 of 55
cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])
--R
--R
@@ -242,9 +303,9 @@ cf := continuedFraction(1,[(2*i+1)**2 for i in
0..],repeating [2])
--R +-----+ + +-----+ + ...
--R | 2 | 2
--R Type: ContinuedFraction
Integer
---E 23
+--E 31
---S 24 of 55
+--S 32 of 55
ccf := convergents cf
--R
--R
@@ -252,9 +313,9 @@ ccf := convergents cf
--R (15) [1,-,--,---,---,----,-----,-----,------,--------,...]
--R 2 13 76 263 2578 36979 33976 622637 11064338
--R Type: Stream Fraction
Integer
---E 24
+--E 32
---S 25 of 55
+--S 33 of 55
piConvergents := [4/p for p in ccf]
--R
--R
@@ -262,9 +323,9 @@ piConvergents := [4/p for p in ccf]
--R (16) [4,-,--,---,----,-----,------,------,-------,--------,...]
--R 3 15 105 315 3465 45045 45045 765765 14549535
--R Type: Stream Fraction
Integer
---E 25
+--E 33
---S 26 of 55
+--S 34 of 55
piConvergents :: Stream Float
--R
--R
@@ -274,9 +335,9 @@ piConvergents :: Stream Float
--R 3.2837384837 384837385, 3.0170718170 718170718, 3.2523659347 188758953,
--R 3.0418396189 294022111, ...]
--R Type: Stream
Float
---E 26
+--E 34
---S 27 of 55
+--S 35 of 55
continuedFraction((- 122 + 597*%i)/(4 - 4*%i))
--R
--R
@@ -284,15 +345,15 @@ continuedFraction((- 122 + 597*%i)/(4 - 4*%i))
--R (18) - 90 + 59%i + +---------+ + +-----------+
--R | 1 - 2%i | - 1 + 2%i
--R Type: ContinuedFraction Complex
Integer
---E 27
+--E 35
---S 28 of 55
+--S 36 of 55
r : Fraction UnivariatePolynomial(x,Fraction Integer)
--R
--R Type:
Void
---E 28
+--E 36
---S 29 of 55
+--S 37 of 55
r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))
--R
--R
@@ -302,9 +363,9 @@ r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))
--R 2
--R x - 7x + 12
--R Type: Fraction UnivariatePolynomial(x,Fraction
Integer)
---E 29
+--E 37
---S 30 of 55
+--S 38 of 55
continuedFraction r
--R
--R
@@ -314,9 +375,9 @@ continuedFraction r
--R | - x - - | -- x - --
--R | 4 8 | 3 3
--R Type: ContinuedFraction UnivariatePolynomial(x,Fraction
Integer)
---E 30
+--E 38
---S 31 of 55
+--S 39 of 55
[i*i for i in convergents(z) :: Stream Float]
--R
--R
@@ -326,77 +387,6 @@ continuedFraction r
--R 11.0000000017 53603304, 10.9999999999 12099531, 11.0000000000 04406066,
--R ...]
--R Type: Stream
Float
---E 31
-
---S 32 of 55
-[x**3 - y for x in b | even? x for y in e]
---R
---R
---RDaly Bug
---R AXIOM cannot iterate with x over your form now. Perhaps you should
---R try using a conversion to make sure your form is a list or
---R stream, for example.
---E 32
-
---S 33 of 55
-f := [i**3 for i in 0..]
---R
---R
---R (23) [0,1,8,27,64,125,216,343,512,729,...]
---R Type: Stream
NonNegativeInteger
---E 33
-
---S 34 of 55
-[i**3 for i in 0..10]
---R
---R
---R (24) [0,1,8,27,64,125,216,343,512,729,1000]
---R Type: List
NonNegativeInteger
---E 34
-
---S 35 of 55
-[i**3 for i in 0.. while i < 11]
---R
---R
---R (25) [0,1,8,27,64,125,216,343,512,729,...]
---R Type: Stream
NonNegativeInteger
---E 35
-
---S 36 of 55
-[i**3 for i in 0.. for x in 0..10]
---R
---R
---R (26) [0,1,8,27,64,125,216,343,512,729,...]
---R Type: Stream
NonNegativeInteger
---E 36
-
---S 37 of 55
-[[i**j for j in 0..3] for i in 0..]
---R
---R
---R (27)
---R [[1,0,0,0], [1,1,1,1], [1,2,4,8], [1,3,9,27], [1,4,16,64], [1,5,25,125],
---R [1,6,36,216], [1,7,49,343], [1,8,64,512], [1,9,81,729], ...]
---R Type: Stream List
NonNegativeInteger
---E 37
-
---S 38 of 55
-[[i**j for j in 0..] for i in 0..3]
---R
---R
---R (28)
---R [[1,0,0,0,0,0,0,0,0,0,...], [1,1,1,1,1,1,1,1,1,1,...],
---R [1,2,4,8,16,32,64,128,256,512,...],
---R [1,3,9,27,81,243,729,2187,6561,19683,...]]
---R Type: List Stream Fraction
Integer
---E 38
-
---S 39 of 55
-brace [i**3 for i in 10..0 by -2]
---R
---R
---R (29) {0,8,64,216,512,1000}
---R Type: Set
NonNegativeInteger
--E 39
-- Input for page ForCollectionDetailPage
diff --git a/src/input/easter.input.pamphlet b/src/input/easter.input.pamphlet
index 5db2309..64c9c44 100644
--- a/src/input/easter.input.pamphlet
+++ b/src/input/easter.input.pamphlet
@@ -28,7 +28,7 @@
\section{Numbers}
Let's begin by playing with numbers: infinite precision integers
<<*>>=
---S 1 of 201
+--S 1 of 200
factorial(50)
--R
--R
@@ -36,7 +36,7 @@ factorial(50)
--R Type:
PositiveInteger
--E 1
---S 2 of 201
+--S 2 of 200
factor(%)
--R
--R
@@ -48,7 +48,7 @@ factor(%)
@
Infinite precision rational numbers
<<*>>=
---S 3 of 201
+--S 3 of 200
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10
--R
--R
@@ -61,7 +61,7 @@ Infinite precision rational numbers
@
Arbitrary precision floating point numbers
<<*>>=
---S 4 of 201
+--S 4 of 200
digits(50);
--R
--R
@@ -71,7 +71,7 @@ digits(50);
@
This number is nearly an integer
<<*>>=
---S 5 of 201
+--S 5 of 200
exp(sqrt(163.)*%pi)
--R
--R
@@ -79,7 +79,7 @@ exp(sqrt(163.)*%pi)
--R Type:
Float
--E 5
---S 6 of 201
+--S 6 of 200
digits(20);
--R
--R
@@ -89,7 +89,7 @@ digits(20);
@
Special functions
<<*>>=
---S 7 of 201
+--S 7 of 200
besselJ(2, 1 + %i)
--R
--R
@@ -100,7 +100,7 @@ besselJ(2, 1 + %i)
@
Complete decimal expansion of a rational number
<<*>>=
---S 8 of 201
+--S 8 of 200
decimal(1/7)
--R
--R
@@ -112,7 +112,7 @@ decimal(1/7)
@
Continued fractions
<<*>>=
---S 9 of 201
+--S 9 of 200
continuedFraction(3.1415926535)
--R
--R
@@ -125,7 +125,7 @@ continuedFraction(3.1415926535)
@
Simplify an expression with nested square roots
<<*>>=
---S 10 of 201
+--S 10 of 200
sqrt(2*sqrt(3) + 4)
--R
--R
@@ -135,7 +135,7 @@ sqrt(2*sqrt(3) + 4)
--R Type:
AlgebraicNumber
--E 10
---S 11 of 201
+--S 11 of 200
simplify(%)
--R
--R
@@ -148,7 +148,7 @@ simplify(%)
@
Try a more complicated example (from the Putnam exam)
<<*>>=
---S 12 of 201
+--S 12 of 200
sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2)))))
--R
--R
@@ -161,7 +161,7 @@ sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2)))))
--R Type:
AlgebraicNumber
--E 12
---S 13 of 201
+--S 13 of 200
simplify(%)
--R
--R
@@ -177,7 +177,7 @@ simplify(%)
@
Cardinal numbers
<<*>>=
---S 14 of 201
+--S 14 of 200
2*Aleph(0) - 3
--R
--R
@@ -190,7 +190,7 @@ Cardinal numbers
Numbers are nice, but symbols allow for variability---try some high school
algebra: rational simplification
<<*>>=
---S 15 of 201
+--S 15 of 200
(x**2 - 4)/(x**2 + 4*x + 4)
--R
--R
@@ -203,7 +203,7 @@ algebra: rational simplification
@
This example requires more sophistication
<<*>>=
---S 16 of 201
+--S 16 of 200
(%e**x - 1)/(%e**(x/2) + 1)
--R
--R
@@ -217,7 +217,7 @@ This example requires more sophistication
--R Type: Expression
Integer
--E 16
---S 17 of 201
+--S 17 of 200
normalize(%)
--R
--R
@@ -231,7 +231,7 @@ normalize(%)
@
Expand and factor polynomials
<<*>>=
---S 18 of 201
+--S 18 of 200
(x + 1)**20
--R
--R
@@ -247,7 +247,7 @@ Expand and factor polynomials
--R Type: Polynomial
Integer
--E 18
---S 19 of 201
+--S 19 of 200
D(%, x)
--R
--R
@@ -263,7 +263,7 @@ D(%, x)
--R Type: Polynomial
Integer
--E 19
---S 20 of 201
+--S 20 of 200
factor(%)
--R
--R
@@ -272,7 +272,7 @@ factor(%)
--R Type: Factored Polynomial
Integer
--E 20
---S 21 of 201
+--S 21 of 200
x**100 - 1
--R
--R
@@ -281,7 +281,7 @@ x**100 - 1
--R Type: Polynomial
Integer
--E 21
---S 22 of 201
+--S 22 of 200
factor(%)
--R
--R
@@ -300,7 +300,7 @@ factor(%)
@
Factor polynomials over finite fields and field extensions
<<*>>=
---S 23 of 201
+--S 23 of 200
p:= x**4 - 3*x**2 + 1
--R
--R
@@ -309,7 +309,7 @@ p:= x**4 - 3*x**2 + 1
--R Type: Polynomial
Integer
--E 23
---S 24 of 201
+--S 24 of 200
factor(p)
--R
--R
@@ -318,14 +318,14 @@ factor(p)
--R Type: Factored Polynomial
Integer
--E 24
---S 25 of 201
+--S 25 of 200
phi:= rootOf(phi**2 - phi - 1);
--R
--R
--R Type:
AlgebraicNumber
--E 25
---S 26 of 201
+--S 26 of 200
factor(p, [phi])
--R
--R
@@ -333,7 +333,7 @@ factor(p, [phi])
--R Type: Factored Polynomial
AlgebraicNumber
--E 26
---S 27 of 201
+--S 27 of 200
factor(p :: Polynomial(PrimeField(5)))
--R
--R
@@ -342,7 +342,7 @@ factor(p :: Polynomial(PrimeField(5)))
--R Type: Factored Polynomial PrimeField
5
--E 27
---S 28 of 201
+--S 28 of 200
expand(%)
--R
--R
@@ -354,7 +354,7 @@ expand(%)
@
Partial fraction decomposition
<<*>>=
---S 29 of 201
+--S 29 of 200
(x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2)
--R
--R
@@ -366,7 +366,7 @@ Partial fraction decomposition
--R Type: Fraction Polynomial
Integer
--E 29
---S 30 of 201
+--S 30 of 200
padicFraction(
partialFraction(numerator(%) :: UnivariatePolynomial(x, Fraction Integer),
factor(denominator(%) :: Polynomial Integer) ::
@@ -384,7 +384,7 @@ padicFraction(
\section{Trigonometry}
Trigonometric manipulations---these are typically difficult for students
<<*>>=
---S 31 of 201
+--S 31 of 200
r:= cos(3*x)/cos(x)
--R
--R
@@ -394,7 +394,7 @@ r:= cos(3*x)/cos(x)
--R Type: Expression
Integer
--E 31
---S 32 of 201
+--S 32 of 200
real(complexNormalize(%))
--R
--R
@@ -403,7 +403,7 @@ real(complexNormalize(%))
--R Type: Expression
Integer
--E 32
---S 33 of 201
+--S 33 of 200
real(normalize(simplify(complexNormalize(r))))
--R
--R
@@ -414,7 +414,7 @@ real(normalize(simplify(complexNormalize(r))))
@
Use rewrite rules
<<*>>=
---S 34 of 201
+--S 34 of 200
sincosAngles:= rule _
(cos((n | integer?(n)) * x) == _
cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x); _
@@ -428,7 +428,7 @@ sincosAngles:= rule _
--R Type: Ruleset(Integer,Integer,Expression
Integer)
--E 34
---S 35 of 201
+--S 35 of 200
sincosAngles r
--R
--R
@@ -437,7 +437,7 @@ sincosAngles r
--R Type: Expression
Integer
--E 35
---S 36 of 201
+--S 36 of 200
r:= 'r;
--R
--R
@@ -448,7 +448,7 @@ r:= 'r;
\section{Determining Zero Equivalence}
The following expressions are all equal to zero
<<*>>=
---S 37 of 201
+--S 37 of 200
sqrt(997) - (997**3)**(1/6)
--R
--R
@@ -456,7 +456,7 @@ sqrt(997) - (997**3)**(1/6)
--R Type:
AlgebraicNumber
--E 37
---S 38 of 201
+--S 38 of 200
sqrt(999983) - (999983**3)**(1/6)
--R
--R
@@ -464,7 +464,7 @@ sqrt(999983) - (999983**3)**(1/6)
--R Type:
AlgebraicNumber
--E 38
---S 39 of 201
+--S 39 of 200
(2**(1/3) + 4**(1/3))**3 - 6*(2**(1/3) + 4**(1/3)) - 6
--R
--R
@@ -473,7 +473,7 @@ sqrt(999983) - (999983**3)**(1/6)
--R Type:
AlgebraicNumber
--E 39
---S 40 of 201
+--S 40 of 200
simplify(%)
--R
--R
@@ -485,7 +485,7 @@ simplify(%)
@
This expression is zero for $x, y > 0$ and $n$ not equal to zero
<<*>>=
---S 41 of 201
+--S 41 of 200
x**(1/n)*y**(1/n) - (x*y)**(1/n)
--R
--R
@@ -496,7 +496,7 @@ x**(1/n)*y**(1/n) - (x*y)**(1/n)
--R Type: Expression
Integer
--E 41
---S 42 of 201
+--S 42 of 200
normalize(%)
--R
--R
@@ -508,7 +508,7 @@ normalize(%)
See Joel Moses, ``Algebraic Simplification: A Guide for the Perplexed'',
CACM, Volume 14, Number 8, August 1971
<<*>>=
---S 43 of 201
+--S 43 of 200
expr:= log(tan(1/2*x + %pi/4)) - asinh(tan(x))
--R
--R
@@ -518,7 +518,7 @@ expr:= log(tan(1/2*x + %pi/4)) - asinh(tan(x))
--R Type: Expression
Integer
--E 43
---S 44 of 201
+--S 44 of 200
complexNormalize(%)
--R
--R
@@ -573,7 +573,7 @@ complexNormalize(%)
@
Use a roundabout method---show that expr is a constant equal to zero
<<*>>=
---S 45 of 201
+--S 45 of 200
D(expr, x)
--R
--R
@@ -590,7 +590,7 @@ D(expr, x)
--R Type: Expression
Integer
--E 45
---S 46 of 201
+--S 46 of 200
simplify(real(complexNormalize(expand(simplify(%)))))
--R
--R
@@ -611,7 +611,7 @@ simplify(real(complexNormalize(expand(simplify(%)))))
--R Type: Expression
Integer
--E 46
---S 47 of 201
+--S 47 of 200
normalize(eval(expr, x = 0))
--R
--R
@@ -619,7 +619,7 @@ normalize(eval(expr, x = 0))
--R Type: Expression
Integer
--E 47
---S 48 of 201
+--S 48 of 200
log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1))
--R
--R
@@ -632,7 +632,7 @@ log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1))
--R Type: Expression
Integer
--E 48
---S 49 of 201
+--S 49 of 200
simplify(%)
--R
--R
@@ -645,7 +645,7 @@ simplify(%)
--R Type: Expression
Integer
--E 49
---S 50 of 201
+--S 50 of 200
(4*r + 4*sqrt(r) + 1)**(sqrt(r)/(2*sqrt(r) + 1)) _
* (2*sqrt(r) + 1)**(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1
--R
@@ -659,7 +659,7 @@ simplify(%)
--R Type: Expression
Integer
--E 50
---S 51 of 201
+--S 51 of 200
normalize(%)
--R
--R
@@ -671,13 +671,13 @@ normalize(%)
\section{The Complex Domain}
Complex functions---separate into their real and imaginary parts
<<*>>=
---S 52 of 201
+--S 52 of 200
rectform(z) == real(z) + %i*imag(z)
--R
--R Type:
Void
--E 52
---S 53 of 201
+--S 53 of 200
rectform(log(3 + 4*%i))
--R
--R Compiling function rectform with type Expression Complex Integer ->
@@ -691,7 +691,7 @@ rectform(log(3 + 4*%i))
--R Type: Expression Complex
Integer
--E 53
---S 54 of 201
+--S 54 of 200
simplify(rectform(tan(x + %i*y)))
--R
--R
@@ -710,7 +710,7 @@ September 1991. This first expression can simplify to
$\sqrt{(x y)}/\sqrt{(x)}$,
but no further in general (consider what happens when x, y = -1).
<<*>>=
---S 55 of 201
+--S 55 of 200
sqrt(x*y*abs(z)**2) / (sqrt(x)*abs(z))
--R
--R
@@ -726,7 +726,7 @@ sqrt(x*y*abs(z)**2) / (sqrt(x)*abs(z))
@
If $z = -1$, $\sqrt(1/z)$ is not equal to $1/\sqrt(z)$
<<*>>=
---S 56 of 201
+--S 56 of 200
sqrt(1/z) - 1/sqrt(z)
--R
--R
@@ -743,7 +743,7 @@ sqrt(1/z) - 1/sqrt(z)
@
If $z = 3 \pi i$, $\log(\exp(z))$ is not equal to $z$
<<*>>=
---S 57 of 201
+--S 57 of 200
log(%e**z)
--R
--R
@@ -751,7 +751,7 @@ log(%e**z)
--R Type: Expression
Integer
--E 57
---S 58 of 201
+--S 58 of 200
normalize(%)
--R
--R
@@ -762,7 +762,7 @@ normalize(%)
@
The principal value of this expression is $(10 - 4 \pi) i$
<<*>>=
---S 59 of 201
+--S 59 of 200
log(%e**(10*%i))
--R
--R
@@ -771,7 +771,7 @@ log(%e**(10*%i))
--R Type: Expression Complex
Integer
--E 59
---S 60 of 201
+--S 60 of 200
normalize(%)
--R
--R
@@ -783,7 +783,7 @@ normalize(%)
@
If $z = \pi$, $\arctan(\tan(z))$ is not equal to $z$
<<*>>=
---S 61 of 201
+--S 61 of 200
atan(tan(z))
--R
--R
@@ -794,7 +794,7 @@ atan(tan(z))
@
If $z = 2 \pi i$, $\sqrt(\exp(z))$ is not equal to $\exp(z/2)$
<<*>>=
---S 62 of 201
+--S 62 of 200
sqrt(%e**z) - %e**(z/2)
--R
--R
@@ -809,7 +809,7 @@ sqrt(%e**z) - %e**(z/2)
\section{Equations}
Manipulate an equation using a natural syntax
<<*>>=
---S 63 of 201
+--S 63 of 200
(x = 0)/2 + 1
--R
--R
@@ -822,7 +822,7 @@ Manipulate an equation using a natural syntax
@
Solve various nonlinear equations---this cubic polynomial has all real roots
<<*>>=
---S 64 of 201
+--S 64 of 200
radicalSolve(3*x**3 - 18*x**2 + 33*x - 19 = 0, x)
--R
--R
@@ -869,7 +869,7 @@ radicalSolve(3*x**3 - 18*x**2 + 33*x - 19 = 0, x)
--R Type: List Equation Expression
Integer
--E 64
---S 65 of 201
+--S 65 of 200
map(e +-> lhs(e) = rectform(rhs(e)), %)
--R
--R Compiling function rectform with type Expression Integer ->
@@ -924,7 +924,7 @@ map(e +-> lhs(e) = rectform(rhs(e)), %)
@
Some simple seeming problems can have messy answers
<<*>>=
---S 66 of 201
+--S 66 of 200
eqn:= x**4 + x**3 + x**2 + x + 1 = 0
--R
--R
@@ -933,7 +933,7 @@ eqn:= x**4 + x**3 + x**2 + x + 1 = 0
--R Type: Equation Polynomial
Integer
--E 66
---S 67 of 201
+--S 67 of 200
radicalSolve(eqn, x)
--R
--R
@@ -1284,7 +1284,7 @@ radicalSolve(eqn, x)
@
Check one of the answers
<<*>>=
---S 68 of 201
+--S 68 of 200
eval(eqn, %.1)
--R
--R
@@ -1373,7 +1373,7 @@ eval(eqn, %.1)
--R Type: Equation Expression
Integer
--E 68
---S 69 of 201
+--S 69 of 200
%e**(2*x) + 2*%e**x + 1 = z
--R
--R
@@ -1382,7 +1382,7 @@ eval(eqn, %.1)
--R Type: Equation Expression
Integer
--E 69
---S 70 of 201
+--S 70 of 200
solve(%, x)
--R
--R
@@ -1394,7 +1394,7 @@ solve(%, x)
@
This equation is already factored and so {\sl should} be easy to solve
<<*>>=
---S 71 of 201
+--S 71 of 200
(x + 1) * (sin(x)**2 + 1)**2 * cos(3*x)**3 = 0
--R
--R
@@ -1403,7 +1403,7 @@ This equation is already factored and so {\sl should} be
easy to solve
--R Type: Equation Expression
Integer
--E 71
---S 72 of 201
+--S 72 of 200
solve(%, x)
--R
--R
@@ -1417,7 +1417,7 @@ solve(%, x)
The following equations have an infinite number of solutions (let $n$ be an
arbitrary integer): $z = 0 [+ n 2 \pi i]$
<<*>>=
---S 73 of 201
+--S 73 of 200
solve(%e**z = 1, z)
--R
--R
@@ -1428,7 +1428,7 @@ solve(%e**z = 1, z)
@
$x = \pi/4 [+ n \pi]$
<<*>>=
---S 74 of 201
+--S 74 of 200
solve(sin(x) = cos(x), x)
--R
--R
@@ -1438,7 +1438,7 @@ solve(sin(x) = cos(x), x)
--R Type: List Equation Expression
Integer
--E 74
---S 75 of 201
+--S 75 of 200
solve(tan(x) = 1, x)
--R
--R
@@ -1451,7 +1451,7 @@ solve(tan(x) = 1, x)
@
$x = 0$, $0 [+ n \pi, + n 2 \pi]$
<<*>>=
---S 76 of 201
+--S 76 of 200
solve(sin(x) = tan(x), x)
--R
--R
@@ -1462,7 +1462,7 @@ solve(sin(x) = tan(x), x)
@
This equation has no solutions
<<*>>=
---S 77 of 201
+--S 77 of 200
solve(sqrt(x**2 + 1) = x - 2, x)
--R
--R
@@ -1473,7 +1473,7 @@ solve(sqrt(x**2 + 1) = x - 2, x)
@
Solve a system of linear equations
<<*>>=
---S 78 of 201
+--S 78 of 200
eq1:= x + y + z = 6
--R
--R
@@ -1481,7 +1481,7 @@ eq1:= x + y + z = 6
--R Type: Equation Polynomial
Integer
--E 78
---S 79 of 201
+--S 79 of 200
eq2:= 2*x + y + 2*z = 10
--R
--R
@@ -1489,7 +1489,7 @@ eq2:= 2*x + y + 2*z = 10
--R Type: Equation Polynomial
Integer
--E 79
---S 80 of 201
+--S 80 of 200
eq3:= x + 3*y + z = 10
--R
--R
@@ -1500,17 +1500,17 @@ eq3:= x + 3*y + z = 10
@
Note that the solution is parametric
<<*>>=
---S 81 of 201
+--S 81 of 200
solve([eq1, eq2, eq3], [x, y, z])
--R
--R
---R (81) [[x= - %BU + 4,y= 2,z= %BU]]
+--I (81) [[x= - %BU + 4,y= 2,z= %BU]]
--R Type: List List Equation Fraction Polynomial
Integer
--E 81
@
Solve a system of nonlinear equations
<<*>>=
---S 82 of 201
+--S 82 of 200
eq1:= x**2*y + 3*y*z - 4 = 0
--R
--R
@@ -1519,7 +1519,7 @@ eq1:= x**2*y + 3*y*z - 4 = 0
--R Type: Equation Polynomial
Integer
--E 82
---S 83 of 201
+--S 83 of 200
eq2:= -3*x**2*z + 2*y**2 + 1 = 0
--R
--R
@@ -1528,7 +1528,7 @@ eq2:= -3*x**2*z + 2*y**2 + 1 = 0
--R Type: Equation Polynomial
Integer
--E 83
---S 84 of 201
+--S 84 of 200
eq3:= 2*y*z**2 - z**2 - 1 = 0
--R
--R
@@ -1540,7 +1540,7 @@ eq3:= 2*y*z**2 - z**2 - 1 = 0
@
Solving this by hand would be a nightmare
<<*>>=
---S 85 of 201
+--S 85 of 200
solve([eq1, eq2, eq3], [x, y, z])
--R
--R
@@ -1562,7 +1562,7 @@ solve([eq1, eq2, eq3], [x, y, z])
@
\section{Matrix Algebra}
<<*>>=
---S 86 of 201
+--S 86 of 200
m:= matrix([[a, b], [1, a*b]])
--R
--R
@@ -1575,7 +1575,7 @@ m:= matrix([[a, b], [1, a*b]])
@
Invert the matrix
<<*>>=
---S 87 of 201
+--S 87 of 200
minv:= inverse(m)
--R
--R
@@ -1591,7 +1591,7 @@ minv:= inverse(m)
--R Type: Union(Matrix Fraction Polynomial
Integer,...)
--E 87
---S 88 of 201
+--S 88 of 200
m * minv
--R
--R
@@ -1604,7 +1604,7 @@ m * minv
@
Define a Vandermonde matrix (useful for doing polynomial interpolations)
<<*>>=
---S 89 of 201
+--S 89 of 200
matrix([[1, 1, 1, 1 ], _
[w, x, y, z ], _
[w**2, x**2, y**2, z**2], _
@@ -1623,7 +1623,7 @@ matrix([[1, 1, 1, 1 ], _
--R Type: Matrix Polynomial
Integer
--E 89
---S 90 of 201
+--S 90 of 200
determinant(%)
--R
--R
@@ -1645,7 +1645,7 @@ determinant(%)
@
The following formula implies a general result
<<*>>=
---S 91 of 201
+--S 91 of 200
factor(%)
--R
--R
@@ -1656,7 +1656,7 @@ factor(%)
@
Compute the eigenvalues of a matrix from its characteristic polynomial
<<*>>=
---S 92 of 201
+--S 92 of 200
m:= matrix([[ 5, -3, -7], _
[-2, 1, 2], _
[ 2, -3, -4]])
@@ -1670,7 +1670,7 @@ m:= matrix([[ 5, -3, -7], _
--R Type: Matrix
Integer
--E 92
---S 93 of 201
+--S 93 of 200
characteristicPolynomial(m, lambda)
--R
--R
@@ -1679,7 +1679,7 @@ characteristicPolynomial(m, lambda)
--R Type: Polynomial
Integer
--E 93
---S 94 of 201
+--S 94 of 200
solve(% = 0, lambda)
--R
--R
@@ -1687,7 +1687,7 @@ solve(% = 0, lambda)
--R Type: List Equation Fraction Polynomial
Integer
--E 94
---S 95 of 201
+--S 95 of 200
m:= 'm;
--R
--R
@@ -1698,7 +1698,7 @@ m:= 'm;
\section{Sums and Products}
\subsection{Sums: finite and infinite}
<<*>>=
---S 96 of 201
+--S 96 of 200
summation(k**3, k = 1..n)
--R
--R
@@ -1710,7 +1710,7 @@ summation(k**3, k = 1..n)
--R Type: Expression
Integer
--E 96
---S 97 of 201
+--S 97 of 200
sum(k**3, k = 1..n)
--R
--R
@@ -1721,7 +1721,7 @@ sum(k**3, k = 1..n)
--R Type: Fraction Polynomial
Integer
--E 97
---S 98 of 201
+--S 98 of 200
limit(sum(1/k**2 + 1/k**3, k = 1..n), n = %plusInfinity)
--R
--R
@@ -1731,7 +1731,7 @@ limit(sum(1/k**2 + 1/k**3, k = 1..n), n = %plusInfinity)
@
\subsection{Products}
<<*>>=
---S 99 of 201
+--S 99 of 200
product(k, k = 1..n)
--R
--R
@@ -1747,7 +1747,7 @@ product(k, k = 1..n)
\section{Calculus}
\subsection{Limits --- start with a famous example}
<<*>>=
---S 100 of 201
+--S 100 of 200
limit((1 + 1/n)**n, n = %plusInfinity)
--R
--R
@@ -1755,7 +1755,7 @@ limit((1 + 1/n)**n, n = %plusInfinity)
--R Type: Union(OrderedCompletion Expression
Integer,...)
--E 100
---S 101 of 201
+--S 101 of 200
limit((1 - cos(x))/x**2, x = 0)
--R
--R
@@ -1769,21 +1769,21 @@ limit((1 - cos(x))/x**2, x = 0)
Apply the chain rule---this is important for PDEs and many other
applications
<<*>>=
---S 102 of 201
+--S 102 of 200
y:= operator('y);
--R
--R
--R Type:
BasicOperator
--E 102
---S 103 of 201
+--S 103 of 200
x:= operator('x);
--R
--R
--R Type:
BasicOperator
--E 103
---S 104 of 201
+--S 104 of 200
D(y(x(t)), t, 2)
--R
--R
@@ -1798,7 +1798,7 @@ D(y(x(t)), t, 2)
@
\subsection{Indefinite Integrals}
<<*>>=
---S 105 of 201
+--S 105 of 200
1/(x**3 + 2)
--R
--R
@@ -1812,7 +1812,7 @@ D(y(x(t)), t, 2)
@
This would be very difficult to do by hand
<<*>>=
---S 106 of 201
+--S 106 of 200
integrate(%, x)
--R
--R
@@ -1830,7 +1830,7 @@ integrate(%, x)
--R Type: Union(Expression
Integer,...)
--E 106
---S 107 of 201
+--S 107 of 200
D(%, x)
--R
--R
@@ -1844,7 +1844,7 @@ D(%, x)
@
This example involves several symbolic parameters
<<*>>=
---S 108 of 201
+--S 108 of 200
integrate(1/(a + b*cos(x)), x)
--R
--R
@@ -1870,7 +1870,7 @@ integrate(1/(a + b*cos(x)), x)
--R Type: Union(List Expression
Integer,...)
--E 108
---S 109 of 201
+--S 109 of 200
map(simplify, map(f +-> D(f, x), %))
--R
--R
@@ -1883,7 +1883,7 @@ map(simplify, map(f +-> D(f, x), %))
@
Calculus on a non-smooth (but well defined) function
<<*>>=
---S 110 of 201
+--S 110 of 200
D(abs(x), x)
--R
--R
@@ -1893,13 +1893,13 @@ D(abs(x), x)
--R Type: Expression
Integer
--E 110
---S 111 of 201
+--S 111 of 200
integrate(abs(x), x)
--R
--R
--R x
--R ++
---R (111) | abs(%J)d%J
+--I (111) | abs(%J)d%J
--R ++
--R Type: Union(Expression
Integer,...)
--E 111
@@ -1907,13 +1907,13 @@ integrate(abs(x), x)
@
Calculus on a piecewise defined function
<<*>>=
---S 112 of 201
+--S 112 of 200
a(x) == if x < 0 then -x else x
--R
--R Type:
Void
--E 112
---S 113 of 201
+--S 113 of 200
D(a(x), x)
--R
--R Compiling function a with type Variable x -> Polynomial Integer
@@ -1922,7 +1922,7 @@ D(a(x), x)
--R Type: Polynomial
Integer
--E 113
---S 114 of 201
+--S 114 of 200
integrate(a(x), x)
--R
--R
@@ -1938,7 +1938,7 @@ integrate(a(x), x)
The following two integrals should be equivalent. The correct solution is
$[(1 + x)^(3/2) + (1 - x)^(3/2)] / 3$
<<*>>=
---S 115 of 201
+--S 115 of 200
integrate(x/(sqrt(1 + x) + sqrt(1 - x)), x)
--R
--R
@@ -1949,7 +1949,7 @@ integrate(x/(sqrt(1 + x) + sqrt(1 - x)), x)
--R Type: Union(Expression
Integer,...)
--E 115
---S 116 of 201
+--S 116 of 200
integrate((sqrt(1 + x) - sqrt(1 - x))/2, x)
--R
--R
@@ -1964,7 +1964,7 @@ integrate((sqrt(1 + x) - sqrt(1 - x))/2, x)
\subsection{Definite Integrals}
The following two functions have a pole at zero
<<*>>=
---S 117 of 201
+--S 117 of 200
integrate(1/x, x = -1..1)
--R
--R
@@ -1976,7 +1976,7 @@ integrate(1/x, x = -1..1)
--R
--E 117
---S 118 of 201
+--S 118 of 200
integrate(1/x**2, x = -1..1)
--R
--R
@@ -1993,7 +1993,7 @@ Different branches of the square root need to be chosen
in the intervals
[0, 1] and [1, 2]. The correct results are $4/3$, $[4 - \sqrt{(8)}]/3$,
$[8 - \sqrt{(8)}]/3$, respectively.
<<*>>=
---S 119 of 201
+--S 119 of 200
integrate(sqrt(x + 1/x - 2), x = 0..1)
--R
--R
@@ -2001,7 +2001,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..1)
--R Type: Union(pole:
potentialPole,...)
--E 119
---S 120 of 201
+--S 120 of 200
integrate(sqrt(x + 1/x - 2), x = 0..1, "noPole")
--R
--R
@@ -2011,7 +2011,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..1, "noPole")
--R Type: Union(f1: OrderedCompletion Expression
Integer,...)
--E 120
---S 121 of 201
+--S 121 of 200
integrate(sqrt(x + 1/x - 2), x = 1..2)
--R
--R
@@ -2019,7 +2019,7 @@ integrate(sqrt(x + 1/x - 2), x = 1..2)
--R Type: Union(pole:
potentialPole,...)
--E 121
---S 122 of 201
+--S 122 of 200
integrate(sqrt(x + 1/x - 2), x = 1..2, "noPole")
--R
--R
@@ -2030,7 +2030,7 @@ integrate(sqrt(x + 1/x - 2), x = 1..2, "noPole")
--R Type: Union(f1: OrderedCompletion Expression
Integer,...)
--E 122
---S 123 of 201
+--S 123 of 200
integrate(sqrt(x + 1/x - 2), x = 0..2)
--R
--R
@@ -2038,7 +2038,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..2)
--R Type: Union(pole:
potentialPole,...)
--E 123
---S 124 of 201
+--S 124 of 200
integrate(sqrt(x + 1/x - 2), x = 0..2, "noPole")
--R
--R
@@ -2052,7 +2052,7 @@ integrate(sqrt(x + 1/x - 2), x = 0..2, "noPole")
@
\subsection{Contour integrals}
<<*>>=
---S 125 of 201
+--S 125 of 200
integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity)
--R
--R
@@ -2060,7 +2060,7 @@ integrate(cos(x)/(x**2 + a**2), x =
%minusInfinity..%plusInfinity)
--R Type: Union(pole:
potentialPole,...)
--E 125
---S 126 of 201
+--S 126 of 200
integrate(cos(x)/(x**2 + a**2), x = %minusInfinity..%plusInfinity, "noPole")
--R
--R
@@ -2071,7 +2071,7 @@ integrate(cos(x)/(x**2 + a**2), x =
%minusInfinity..%plusInfinity, "noPole")
@
\subsection{Integrand with a branch point}
<<*>>=
---S 127 of 201
+--S 127 of 200
integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity)
--R
--R
@@ -2079,7 +2079,7 @@ integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity)
--R Type: Union(pole:
potentialPole,...)
--E 127
---S 128 of 201
+--S 128 of 200
integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity, "noPole")
--R
--R
@@ -2090,7 +2090,7 @@ integrate(t**(a - 1)/(1 + t), t = 0..%plusInfinity,
"noPole")
@
Multiple integrals: volume of a tetrahedron
<<*>>=
---S 129 of 201
+--S 129 of 200
integrate(integrate(integrate(1, z = 0..c*(1 - x/a - y/b)), _
y = 0..b*(1 - x/a)), _
x = 0..a)
@@ -2106,7 +2106,7 @@ integrate(integrate(integrate(1, z = 0..c*(1 - x/a -
y/b)), _
\subsection{Series}
Taylor series---this first example comes from special relativity
<<*>>=
---S 130 of 201
+--S 130 of 200
1/sqrt(1 - (v/c)**2)
--R
--R
@@ -2121,7 +2121,7 @@ Taylor series---this first example comes from special
relativity
--R Type: Expression
Integer
--E 130
---S 131 of 201
+--S 131 of 200
series(%, v = 0)
--R
--R
@@ -2132,7 +2132,7 @@ series(%, v = 0)
--R Type: UnivariatePuiseuxSeries(Expression
Integer,v,0)
--E 131
---S 132 of 201
+--S 132 of 200
1/%**2
--R
--R
@@ -2143,7 +2143,7 @@ series(%, v = 0)
--R Type: UnivariatePuiseuxSeries(Expression
Integer,v,0)
--E 132
---S 133 of 201
+--S 133 of 200
tsin:= series(sin(x), x = 0)
--R
--R
@@ -2153,7 +2153,7 @@ tsin:= series(sin(x), x = 0)
--R Type: UnivariatePuiseuxSeries(Expression
Integer,x,0)
--E 133
---S 134 of 201
+--S 134 of 200
tcos:= series(cos(x), x = 0)
--R
--R
@@ -2166,7 +2166,7 @@ tcos:= series(cos(x), x = 0)
@
Note that additional terms will be computed as needed
<<*>>=
---S 135 of 201
+--S 135 of 200
tsin/tcos
--R
--R
@@ -2176,7 +2176,7 @@ tsin/tcos
--R Type: UnivariatePuiseuxSeries(Expression
Integer,x,0)
--E 135
---S 136 of 201
+--S 136 of 200
series(tan(x), x = 0)
--R
--R
@@ -2192,7 +2192,7 @@ Look at the Taylor series around $x = 1$
)set streams calculate 1
---S 137 of 201
+--S 137 of 200
log(x)**a*exp(-b*x)
--R
--R
@@ -2201,7 +2201,7 @@ log(x)**a*exp(-b*x)
--R Type: Expression
Integer
--E 137
---S 138 of 201
+--S 138 of 200
series(%, x = 1)
--R
--R
@@ -2218,7 +2218,7 @@ series(%, x = 1)
@
Compare the Taylor series of two different formulations of a function
<<*>>=
---S 139 of 201
+--S 139 of 200
taylor(log(sinh(z)) + log(cosh(z + w)), z = 0)
--R
--R
@@ -2230,7 +2230,7 @@ taylor(log(sinh(z)) + log(cosh(z + w)), z = 0)
--R
--E 139
---S 140 of 201
+--S 140 of 200
% - taylor(log(sinh(z) * cosh(z + w)), z = 0)
--R
--R
@@ -2246,7 +2246,7 @@ taylor(log(sinh(z)) + log(cosh(z + w)), z = 0)
\subsection{Power series}
Compute the general formula
<<*>>=
---S 141 of 201
+--S 141 of 200
log(sin(x)/x)
--R
--R
@@ -2256,7 +2256,7 @@ log(sin(x)/x)
--R Type: Expression
Integer
--E 141
---S 142 of 201
+--S 142 of 200
series(%, x = 0)
--R
--R
@@ -2266,7 +2266,7 @@ series(%, x = 0)
--R Type: UnivariatePuiseuxSeries(Expression
Integer,x,0)
--E 142
---S 143 of 201
+--S 143 of 200
exp(-x)*sin(x)
--R
--R
@@ -2275,7 +2275,7 @@ exp(-x)*sin(x)
--R Type: Expression
Integer
--E 143
---S 144 of 201
+--S 144 of 200
series(%, x = 0)
--R
--R
@@ -2289,14 +2289,14 @@ series(%, x = 0)
Derive an explicit Taylor series solution of y as a function of x from the
following implicit relation
<<*>>=
---S 145 of 201
+--S 145 of 200
y:= operator('y);
--R
--R
--R Type:
BasicOperator
--E 145
---S 146 of 201
+--S 146 of 200
x = sin(y(x)) + cos(y(x))
--R
--R
@@ -2304,7 +2304,7 @@ x = sin(y(x)) + cos(y(x))
--R Type: Equation Expression
Integer
--E 146
---S 147 of 201
+--S 147 of 200
seriesSolve(%, y, x = 1, 0)
--R
--R
@@ -2321,7 +2321,7 @@ seriesSolve(%, y, x = 1, 0)
@
\subsection{Pade (rational function) approximation}
<<*>>=
---S 148 of 201
+--S 148 of 200
pade(1, 1, taylor(exp(-x), x = 0))
--R
--R
@@ -2335,7 +2335,7 @@ pade(1, 1, taylor(exp(-x), x = 0))
\section{Transforms}
\subsection{Laplace and inverse Laplace transforms}
<<*>>=
---S 149 of 201
+--S 149 of 200
laplace(cos((w - 1)*t), t, s)
--R
--R
@@ -2346,7 +2346,7 @@ laplace(cos((w - 1)*t), t, s)
--R Type: Expression
Integer
--E 149
---S 150 of 201
+--S 150 of 200
inverseLaplace(%, s, t)
--R
--R
@@ -2360,14 +2360,14 @@ inverseLaplace(%, s, t)
\section{Difference and Differential Equations}
\subsection{Second order linear recurrence equation}
<<*>>=
---S 151 of 201
+--S 151 of 200
r:= operator('r);
--R
--R
--R Type:
BasicOperator
--E 151
---S 152 of 201
+--S 152 of 200
r(n + 2) - 2 * r(n + 1) + r(n) = 2
--R
--R
@@ -2375,7 +2375,7 @@ r(n + 2) - 2 * r(n + 1) + r(n) = 2
--R Type: Equation Expression
Integer
--E 152
---S 153 of 201
+--S 153 of 200
[%, r(0) = 1, r(1) = m]
--R
--R
@@ -2389,14 +2389,14 @@ r(n + 2) - 2 * r(n + 1) + r(n) = 2
\subsection{Second order ODE with initial conditions}
solve first using Laplace transforms
<<*>>=
---S 154 of 201
+--S 154 of 200
f:= operator('f);
--R
--R
--R Type:
BasicOperator
--E 154
---S 155 of 201
+--S 155 of 200
ode:= D(f(t), t, 2) + 4*f(t) = sin(2*t)
--R
--R
@@ -2406,7 +2406,7 @@ ode:= D(f(t), t, 2) + 4*f(t) = sin(2*t)
--R Type: Equation Expression
Integer
--E 155
---S 156 of 201
+--S 156 of 200
map(e +-> laplace(e, t, s), %)
--R
--R
@@ -2420,7 +2420,7 @@ map(e +-> laplace(e, t, s), %)
@
Now, solve the ODE directly
<<*>>=
---S 157 of 201
+--S 157 of 200
solve(ode, f, t = 0, [0, 0])
--R
--R
@@ -2433,14 +2433,14 @@ solve(ode, f, t = 0, [0, 0])
@
\subsection{First order linear ODE}
<<*>>=
---S 158 of 201
+--S 158 of 200
y:= operator('y);
--R
--R
--R Type:
BasicOperator
--E 158
---S 159 of 201
+--S 159 of 200
x**2 * D(y(x), x) + 3*x*y(x) = sin(x)/x
--R
--R
@@ -2450,7 +2450,7 @@ x**2 * D(y(x), x) + 3*x*y(x) = sin(x)/x
--R Type: Equation Expression
Integer
--E 159
---S 160 of 201
+--S 160 of 200
solve(%, y, x)
--R
--R
@@ -2464,7 +2464,7 @@ solve(%, y, x)
@
\subsection{Nonlinear ODE}
<<*>>=
---S 161 of 201
+--S 161 of 200
D(y(x), x, 2) + y(x)*D(y(x), x)**3 = 0
--R
--R
@@ -2474,7 +2474,7 @@ D(y(x), x, 2) + y(x)*D(y(x), x)**3 = 0
--R Type: Equation Expression
Integer
--E 161
---S 162 of 201
+--S 162 of 200
solve(%, y, x)
--R
--R
@@ -2489,7 +2489,7 @@ solve(%, y, x)
@
A simple parametric ODE
<<*>>=
---S 163 of 201
+--S 163 of 200
D(y(x, a), x) = a*y(x, a)
--R
--R
@@ -2498,7 +2498,7 @@ D(y(x, a), x) = a*y(x, a)
--R Type: Equation Expression
Integer
--E 163
---S 164 of 201
+--S 164 of 200
solve(%, y, x);
--R
--R
@@ -2515,7 +2515,7 @@ solve(%, y, x);
This problem has nontrivial solutions
$y(x) = A \sin([\pi/2 + n \pi] x)$ for $n$ an arbitrary integer.
<<*>>=
---S 165 of 201
+--S 165 of 200
solve(D(y(x), x, 2) + k**2*y(x) = 0, y, x)
--R
--R
@@ -2528,14 +2528,14 @@ solve(D(y(x), x, 2) + k**2*y(x) = 0, y, x)
@
\subsection{System of two linear, constant coefficient ODEs}
<<*>>=
---S 166 of 201
+--S 166 of 200
x:= operator('x);
--R
--R
--R Type:
BasicOperator
--E 166
---S 167 of 201
+--S 167 of 200
system:= [D(x(t), t) = x(t) - y(t), D(y(t), t) = x(t) + y(t)]
--R
--R
@@ -2548,7 +2548,7 @@ system:= [D(x(t), t) = x(t) - y(t), D(y(t), t) = x(t) +
y(t)]
@
Check the answer. Triangular system of two ODEs
<<*>>=
---S 168 of 201
+--S 168 of 200
system:= [D(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), _
D(y(t), t) = x(t) - y(t)]
--R
@@ -2562,8 +2562,8 @@ system:= [D(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), _
@
Try solving this system one equation at a time
<<*>>=
---S 169 of 201
-solve(system.1, x, t)
+--S 169 of 200
+s:=solve(system.1, x, t)
--R
--R
--R t t
@@ -2571,185 +2571,125 @@ solve(system.1, x, t)
--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
--E 169
---S 170 of 201
-isTimes(subst(%.basis.1, cos(t) = sqrt(1 - sin(t)**2)))
+--S 170 of 200
+eq1 := x(t) = C1 * s.basis.1
--R
--R
---R (162) "failed"
---R Type:
Union("failed",...)
+--R t t
+--R (162) x(t)= C1 %e sin(t) + 2C1 %e
+--R Type: Equation Expression
Integer
--E 170
---S 171 of 201
-reduce(*, cons(subst(
- factors(factor(subst(%.1**2, sin(t) = u) :: Polynomial Integer)).1.factor,
- u = sin(t)),
- rest(%)))
---R
---R There are 30 exposed and 3 unexposed library operations named elt
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op elt
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
+--S 171 of 200
+s1:=solve(map(e +-> subst(e, eq1), system.2), y, t)
--R
---RDaly Bug
---R Cannot find a definition or applicable library operation named elt
---R with argument type(s)
---R failed
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (163)
+--R - t t 2 - t t 2
+--R 2C1 %e (%e ) sin(t) + (- C1 cos(t) + 5C1)%e (%e )
+--R [particular= ------------------------------------------------------,
+--R 5
+--R - t
+--R basis= [%e ]]
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
--E 171
---S 172 of 201
-x(t) = C1 * %
---R
---R There are 34 exposed and 23 unexposed library operations named *
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op *
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
+--S 172 of 200
+eq2 := y(t) = simplify(s1.particular) + C2 * s1.basis.1
--R
---RDaly Bug
---R Cannot find a definition or applicable library operation named *
---R with argument type(s)
---R Variable C1
---R failed
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R t t - t
+--R 2C1 %e sin(t) + (- C1 cos(t) + 5C1)%e + 5C2 %e
+--R (164) y(t)= --------------------------------------------------
+--R 5
+--R Type: Equation Expression
Integer
--E 172
---S 173 of 201
-solve(map(e +-> subst(e, %), system.2), y, t)
---R
---R There are 3 exposed and 0 unexposed library operations named subst
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op subst
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R Cannot find a definition or applicable library operation named subst
---R with argument type(s)
---R Expression Integer
---R Union(List Expression Integer,"failed")
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
---R AXIOM will attempt to step through and interpret the code.
+--S 173 of 200
+map(e +-> rightZero eval(e, [eq1, D(eq1,t), eq2 , D(eq2,t)]), system)
--R
---RDaly Bug
---R Anonymous user functions created with +-> that are processed in
---R interpret-code mode must have result target information
---R available. This information is not present so AXIOM cannot
---R proceed any further. This may be remedied by declaring the
---R function.
+--R
+--R (165) [0= 0,0= 0]
+--R Type: List Equation Expression
Integer
--E 173
-
---S 174 of 201
-y(t) = simplify(%.particular) + C2 * %.basis.1
---R
---R There are 30 exposed and 3 unexposed library operations named elt
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op elt
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named elt
---R with argument type(s)
---R failed
---R Variable particular
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
---E 174
-
)clear properties x y
@
\section{Operators}
\subsection{Linear differential operator}
<<*>>=
---S 175 of 201
+--S 174 of 200
DD:= operator("D") :: Operator(Expression Integer)
--R
--R
---R (163) D
+--R (166) D
--R Type: Operator Expression
Integer
---E 175
+--E 174
---S 176 of 201
+--S 175 of 200
evaluate(DD, e +-> D(e, x))$Operator(Expression Integer)
--R
--R
---R (164) D
+--R (167) D
--R Type: Operator Expression
Integer
---E 176
+--E 175
---S 177 of 201
+--S 176 of 200
L:= (DD - 1) * (DD + 2)
--R
--R
--R 2
---R (165) D 2 + D - D - 2
+--R (168) D 2 + D - D - 2
--R Type: Operator Expression
Integer
---E 177
+--E 176
---S 178 of 201
+--S 177 of 200
g:= operator('g)
--R
--R
---R (166) g
+--R (169) g
--R Type:
BasicOperator
---E 178
+--E 177
---S 179 of 201
+--S 178 of 200
L(f(x))
--R
--R
--R ,, ,
---R (167) f (x) + f (x) - 2f(x)
+--R (170) f (x) + f (x) - 2f(x)
--R
--R Type: Expression
Integer
---E 179
+--E 178
---S 180 of 201
+--S 179 of 200
subst(L(subst(g(y), y = x)), x = y)
--R
--R
--R ,, ,
---R (168) g (y) + g (y) - 2g(y)
+--R (171) g (y) + g (y) - 2g(y)
--R
--R Type: Expression
Integer
---E 180
+--E 179
---S 181 of 201
+--S 180 of 200
subst(L(subst(A * sin(z**2), z = x)), x = z)
--R
--R
--R 2 2 2
---R (169) (- 4A z - 2A)sin(z ) + (2A z + 2A)cos(z )
+--R (172) (- 4A z - 2A)sin(z ) + (2A z + 2A)cos(z )
--R Type: Expression
Integer
---E 181
+--E 180
@
\subsection{Truncated Taylor series operator}
<<*>>=
---S 182 of 201
+--S 181 of 200
T:= (f, xx, a) +-> subst((DD**0)(f(x)), x = a)/factorial(0) * (xx - a)**0 + _
subst((DD**1)(f(x)), x = a)/factorial(1) * (xx - a)**1 + _
subst((DD**2)(f(x)), x = a)/factorial(2) * (xx - a)**2
--R
--R
---R (170)
+--R (173)
--R (f,xx,a)
--R +->
--R 0 1
@@ -2762,70 +2702,70 @@ T:= (f, xx, a) +-> subst((DD**0)(f(x)), x =
a)/factorial(0) * (xx - a)**0 + _
--R --------------------- (xx - a)
--R factorial(2)
--R Type:
AnonymousFunction
---E 182
+--E 181
---S 183 of 201
+--S 182 of 200
T(f, x, a)
--R
--R
--R 2 2 ,, ,
--R (x - 2a x + a )f (a) + (2x - 2a)f (a) + 2f(a)
--R
---R (171) -----------------------------------------------
+--R (174) -----------------------------------------------
--R 2
--R Type: Expression
Integer
---E 183
+--E 182
---S 184 of 201
+--S 183 of 200
T(g, y, b)
--R
--R
--R 2 2 ,, ,
--R (y - 2b y + b )g (b) + (2y - 2b)g (b) + 2g(b)
--R
---R (172) -----------------------------------------------
+--R (175) -----------------------------------------------
--R 2
--R Type: Expression
Integer
---E 184
+--E 183
---S 185 of 201
+--S 184 of 200
Sin:= operator("sin") :: Operator(Expression Integer)
--R
--R
---R (173) sin
+--R (176) sin
--R Type: Operator Expression
Integer
---E 185
+--E 184
---S 186 of 201
+--S 185 of 200
evaluate(Sin, x +-> sin(x))$Operator(Expression Integer)
--R
--R
---R (174) sin
+--R (177) sin
--R Type: Operator Expression
Integer
---E 186
+--E 185
---S 187 of 201
+--S 186 of 200
T(Sin, z, c)
--R
--R
--R 2 2
--R (- z + 2c z - c + 2)sin(c) + (2z - 2c)cos(c)
---R (175) ----------------------------------------------
+--R (178) ----------------------------------------------
--R 2
--R Type: Expression
Integer
---E 187
+--E 186
@
\section{Programming}
Write a simple program to compute Legendre polynomials
<<*>>=
---S 188 of 201
+--S 187 of 200
p(n, x) == 1/(2**n*factorial(n)) * D((x**2 - 1)**n, x, n)
--R
--R Type:
Void
---E 188
+--E 187
---S 189 of 201
+--S 188 of 200
for i in 0..4 repeat { output(""); output(concat(["p(", string(i), ", x) =
"])); output(p(i, x))}
--R
--R Compiling function p with type (NonNegativeInteger,Variable x) ->
@@ -2852,40 +2792,40 @@ for i in 0..4 repeat { output("");
output(concat(["p(", string(i), ", x) = "
--R -- x - -- x + -
--R 8 4 8
--R Type:
Void
---E 189
+--E 188
---S 190 of 201
+--S 189 of 200
eval(p(4, x), x = 1)
--R
--R Compiling function p with type (PositiveInteger,Variable x) ->
--R Polynomial Fraction Integer
--R
---R (178) 1
+--R (181) 1
--R Type: Polynomial Fraction
Integer
---E 190
+--E 189
@
Now, perform the same computation using a recursive definition
<<*>>=
---S 191 of 201
+--S 190 of 200
pp(0, x) == 1
--R
--R Type:
Void
---E 191
+--E 190
---S 192 of 201
+--S 191 of 200
pp(1, x) == x
--R
--R Type:
Void
---E 192
+--E 191
---S 193 of 201
+--S 192 of 200
pp(n, x) == ((2*n - 1)*x*pp(n - 1, x) - (n - 1)*pp(n - 2, x))/n
--R
--R Type:
Void
---E 193
+--E 192
---S 194 of 201
+--S 193 of 200
for i in 0..4 repeat { output(""); output(concat(["pp(", string(i), ", x)
= "])); output(pp(i, x))}
--R
--R Compiling function pp with type (Integer,Variable x) -> Polynomial
@@ -2912,7 +2852,7 @@ for i in 0..4 repeat { output("");
output(concat(["pp(", string(i), ", x) =
--R -- x - -- x + -
--R 8 4 8
--R Type:
Void
---E 194
+--E 193
)clear properties p pp
@@ -2921,31 +2861,31 @@ for i in 0..4 repeat { output("");
output(concat(["pp(", string(i), ", x) =
\subsection{Horner's rule}
This is important for numerical algorithms
<<*>>=
---S 195 of 201
+--S 194 of 200
a:= operator('a)
--R
--R
---R (183) a
+--R (186) a
--R Type:
BasicOperator
---E 195
+--E 194
---S 196 of 201
+--S 195 of 200
sum(a(i)*x**i, i = 1..5)
--R
--R
--R 5 4 3 2
---R (184) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x
+--R (187) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x
--R Type: Expression
Integer
---E 196
+--E 195
---S 197 of 201
+--S 196 of 200
p:= factor(%)
--R
--R
--R 5 4 3 2
---R (185) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x
+--R (188) a(5)x + a(4)x + a(3)x + a(2)x + a(1)x
--R Type: Factored Expression
Integer
---E 197
+--E 196
@
Convert the result into FORTRAN syntax
@@ -2953,40 +2893,40 @@ Convert the result into FORTRAN syntax
)set fortran ints2floats off
---S 198 of 201
+--S 197 of 200
outputAsFortran('p = p)
--R
--R p=a(5)*x**5+a(4)*x**4+a(3)*x**3+a(2)*x*x+a(1)*x
--R Type:
Void
---E 198
+--E 197
@
\section{Boolean Logic}
\subsection{Simplify logical expressions}
<<*>>=
---S 199 of 201
+--S 198 of 200
true and false
--R
--R
---R (187) false
+--R (190) false
--R Type:
Boolean
---E 199
+--E 198
---S 200 of 201
+--S 199 of 200
x or (not x)
--R
--R
--RDaly Bug
--R Argument number 1 to "or" must be a Boolean.
---E 200
+--E 199
---S 201 of 201
+--S 200 of 200
x or y or (x and y)
--R
--R
--RDaly Bug
--R Argument number 1 to "or" must be a Boolean.
---E 201
+--E 200
)spool
)lisp (bye)
diff --git a/src/input/exlap.input.pamphlet b/src/input/exlap.input.pamphlet
index 432b063..d290295 100644
--- a/src/input/exlap.input.pamphlet
+++ b/src/input/exlap.input.pamphlet
@@ -37,39 +37,31 @@ laplace((exp(a*t) - exp(b*t))/t, t, s)
--E 2
--S 3 of 6
-laplace(exp(a*t+b)*ei(c*t), t, s)
+laplace(exp(a*t+b)*Ei(c*t), t, s)
--R
---R There are no library operations named ei
---R Use HyperDoc Browse or issue
---R )what op ei
---R to learn if there is any operation containing " ei " in its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named ei
---R with argument type(s)
---R Polynomial Integer
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R b s + c - a
+--R %e log(---------)
+--R c
+--R (3) -----------------
+--R s - a
+--R Type: Expression
Integer
--E 3
)clear all
--S 4 of 6
-laplace(a*ci(b*t) + c*si(d*t), t, s)
+laplace(a*Ci(b*t) + c*Si(d*t), t, s)
--R
---R There are no library operations named ci
---R Use HyperDoc Browse or issue
---R )what op ci
---R to learn if there is any operation containing " ci " in its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named ci
---R with argument type(s)
---R Polynomial Integer
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R 2 2
+--R s + b d
+--R a log(-------) + 2c atan(-)
+--R 2 s
+--R b
+--R (1) ---------------------------
+--R 2s
+--R Type: Expression
Integer
--E 4
)clear all
diff --git a/src/input/exsum.input.pamphlet b/src/input/exsum.input.pamphlet
index 7c2f9d0..df444c0 100644
--- a/src/input/exsum.input.pamphlet
+++ b/src/input/exsum.input.pamphlet
@@ -93,32 +93,19 @@ sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))
)clear all
--S 7 of 13
-[1..15]
+[i for i in 1..15]
--R
--R
---R (1) [1..15]
---R Type: List Segment
PositiveInteger
+--R (1) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
+--R Type: List
PositiveInteger
--E 7
--S 8 of 13
-reduce(+,[1..15])
---R
---R There are 1 exposed and 3 unexposed library operations named reduce
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op reduce
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
+reduce(+,[i for i in 1..15])
--R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R reduce with argument type(s)
---R Variable +
---R List Segment PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (2) 120
+--R Type:
PositiveInteger
--E 8
)clear all
diff --git a/src/input/grpthry.input.pamphlet b/src/input/grpthry.input.pamphlet
index 544b278..b0296a4 100644
--- a/src/input/grpthry.input.pamphlet
+++ b/src/input/grpthry.input.pamphlet
@@ -19,7 +19,7 @@
)set message auto off
)clear all
---S 1 of 67
+--S 1 of 68
x : PERM INT := [[1,3,5],[7,11,9]]
--R
--R
@@ -27,7 +27,7 @@ x : PERM INT := [[1,3,5],[7,11,9]]
--R Type: Permutation
Integer
--E 1
---S 2 of 67
+--S 2 of 68
y : PERM INT := [[3,5,7,9]]
--R
--R
@@ -35,7 +35,7 @@ y : PERM INT := [[3,5,7,9]]
--R Type: Permutation
Integer
--E 2
---S 3 of 67
+--S 3 of 68
z : PERM INT := [1,3,11]
--R
--R
@@ -43,155 +43,113 @@ z : PERM INT := [1,3,11]
--R Type: Permutation
Integer
--E 3
---S 4 of 67
-g1 : PERMGRPS INT := [ x , y ]
+--S 4 of 68
+g1 : PERMGRP INT := [ x , y ]
--R
---R
---RDaly Bug
---R Category, domain or package constructor PERMGRPS is not available.
+--R
+--R (4) <(1 3 5)(7 11 9),(3 5 7 9)>
+--R Type: PermutationGroup
Integer
--E 4
---S 5 of 67
-g2 : PERMGRPS INT := [ x , z ]
+--S 5 of 68
+g2 : PERMGRP INT := [ x , z ]
--R
---R
---RDaly Bug
---R Category, domain or package constructor PERMGRPS is not available.
+--R
+--R (5) <(1 3 5)(7 11 9),(1 3 11)>
+--R Type: PermutationGroup
Integer
--E 5
---S 6 of 67
-g3 : PERMGRPS INT := [ y , z ]
+--S 6 of 68
+g3 : PERMGRP INT := [ y , z ]
--R
---R
---RDaly Bug
---R Category, domain or package constructor PERMGRPS is not available.
+--R
+--R (6) <(3 5 7 9),(1 3 11)>
+--R Type: PermutationGroup
Integer
--E 6
---S 7 of 67
+--S 7 of 68
order g1
--R
---R There are 9 exposed and 5 unexposed library operations named order
---R having 1 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op order
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named order
---R with argument type(s)
---R Variable g1
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (7) 720
+--R Type:
PositiveInteger
--E 7
---S 8 of 67
+--S 8 of 68
degree g3
--R
--R
---R (4) g3
---R Type: IndexedExponents
Symbol
+--R (8) 6
+--R Type:
PositiveInteger
--E 8
---S 9 of 67
+--S 9 of 68
movedPoints g2
--R
---R There are 2 exposed and 0 unexposed library operations named
---R movedPoints having 1 argument(s) but none was determined to be
---R applicable. Use HyperDoc Browse, or issue
---R )display op movedPoints
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R movedPoints with argument type(s)
---R Variable g2
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (9) {1,3,5,7,9,11}
+--R Type: Set
Integer
--E 9
---S 10 of 67
+--S 10 of 68
orbit (g1, 3)
--R
---R There are 4 exposed and 0 unexposed library operations named orbit
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op orbit
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named orbit
---R with argument type(s)
---R Variable g1
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (10) {1,3,5,7,9,11}
+--R Type: Set
Integer
--E 10
---S 11 of 67
+--S 11 of 68
orbits g3
--R
---R There are 1 exposed and 0 unexposed library operations named orbits
---R having 1 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op orbits
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R orbits with argument type(s)
---R Variable g3
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (11) {{1,3,5,7,9,11}}
+--R Type: Set Set
Integer
--E 11
---S 12 of 67
+--S 12 of 68
member? ( y , g2 )
--R
---R There are 2 exposed and 1 unexposed library operations named member?
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op member?
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R member? with argument type(s)
---R Permutation Integer
---R Variable g2
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (12) false
+--R Type:
Boolean
--E 12
---S 13 of 67
-)sh PERMGRPS
+--S 13 of 68
+)sh PERMGRP
--R
---R The )show system command is used to display information about types
---R or partial types. For example, )show Integer will show
---R information about Integer .
---R
---R PERMGRPS is not the name of a known type constructor. If you want
---R to see information about any operations named PERMGRPS , issue
---R )display operations PERMGRPS
+--R PermutationGroup S: SetCategory is a domain constructor
+--R Abbreviation for PermutationGroup is PERMGRP
+--R This constructor is exposed in this frame.
+--R Issue )edit permgrps.spad.pamphlet to see algebra source code for PERMGRP
+--R
+--R------------------------------- Operations --------------------------------
+--R ?<? : (%,%) -> Boolean ?<=? : (%,%) -> Boolean
+--R ?=? : (%,%) -> Boolean base : % -> List S
+--R coerce : List Permutation S -> % coerce : % -> List Permutation S
+--R coerce : % -> OutputForm degree : % -> NonNegativeInteger
+--R hash : % -> SingleInteger latex : % -> String
+--R movedPoints : % -> Set S orbit : (%,List S) -> Set List S
+--R orbit : (%,Set S) -> Set Set S orbit : (%,S) -> Set S
+--R orbits : % -> Set Set S order : % -> NonNegativeInteger
+--R random : % -> Permutation S ?~=? : (%,%) -> Boolean
+--R ?.? : (%,NonNegativeInteger) -> Permutation S
+--R generators : % -> List Permutation S
+--R initializeGroupForWordProblem : (%,Integer,Integer) -> Void
+--R initializeGroupForWordProblem : % -> Void
+--R member? : (Permutation S,%) -> Boolean
+--R permutationGroup : List Permutation S -> %
+--R random : (%,Integer) -> Permutation S
+--R strongGenerators : % -> List Permutation S
+--R wordInGenerators : (Permutation S,%) -> List NonNegativeInteger
+--R wordInStrongGenerators : (Permutation S,%) -> List NonNegativeInteger
+--R wordsForStrongGenerators : % -> List List NonNegativeInteger
+--R
--E 13
)clear all
---S 14 of 67
+--S 14 of 68
ptn9 := partitions 9
--R
--R
@@ -200,112 +158,1133 @@ ptn9 := partitions 9
--R Type: Stream List
Integer
--E 14
---S 15 of 67
-map(dimIrrRepSym, ptn9)
---R
---R There are 68 exposed and 8 unexposed library operations named map
---R having 2 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op map
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named map
---R with argument type(s)
---R Variable dimIrrRepSym
---R Stream List Integer
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--S 15 of 68
+map(dimensionOfIrreducibleRepresentation, ptn9)
+--R
+--R
+--R (2) [1,8,27,28,48,105,56,42,162,120,...]
+--R Type: Stream
NonNegativeInteger
--E 15
---S 16 of 67
-yt := listYoungTableaus [4,2]; yt :: (LIST TABLEAU I)
---R
+--S 16 of 68
+yt := listYoungTableaus [4,2]
--R
---RDaly Bug
---R I is not a valid type.
+--R
+--R (3)
+--R +0 2 4 5+ +0 2 3 5+ +0 2 3 4+ +0 1 4 5+ +0 1 3 5+
+--R [| |, | |, | |, | |, | |,
+--R +1 3 0 0+ +1 4 0 0+ +1 5 0 0+ +2 3 0 0+ +2 4 0 0+
+--R +0 1 3 4+ +0 1 2 5+ +0 1 2 4+ +0 1 2 3+
+--R | |, | |, | |, | |]
+--R +2 5 0 0+ +3 4 0 0+ +3 5 0 0+ +4 5 0 0+
+--R Type: List Matrix
Integer
--E 16
---S 17 of 67
-r1 := irrRepSymNat([4,2],[1,2,4,5,3,6])
---R
---R There are no library operations named irrRepSymNat
---R Use HyperDoc Browse or issue
---R )what op irrRepSymNat
---R to learn if there is any operation containing " irrRepSymNat " in
---R its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R irrRepSymNat with argument type(s)
---R List PositiveInteger
---R List PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--S 17 of 68
+r1 := irreducibleRepresentation([4,2],[1,2,4,5,3,6])
+--R
+--R
+--R + 0 - 1 - 1 0 0 0 0 0 1 +
+--R | |
+--R |- 1 0 0 0 0 0 0 0 0 |
+--R | |
+--R | 1 1 1 0 0 0 0 0 0 |
+--R | |
+--R | 0 1 0 0 0 0 0 0 - 1|
+--R | |
+--R (4) | 0 0 0 0 0 0 1 0 0 |
+--R | |
+--R | 0 0 0 0 1 0 0 0 0 |
+--R | |
+--R | 1 0 0 0 0 0 - 1 - 1 0 |
+--R | |
+--R |- 1 - 1 - 1 - 1 - 1 - 1 0 0 0 |
+--R | |
+--R + 0 0 0 1 0 0 0 0 0 +
+--R Type: Matrix
Integer
--E 17
---S 18 of 67
-r2 := irrRepSymNat([4,2],[3,2,1,5,6,4])
---R
---R There are no library operations named irrRepSymNat
---R Use HyperDoc Browse or issue
---R )what op irrRepSymNat
---R to learn if there is any operation containing " irrRepSymNat " in
---R its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R irrRepSymNat with argument type(s)
---R List PositiveInteger
---R List PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--S 18 of 68
+r2 := irreducibleRepresentation([4,2],[3,2,1,5,6,4])
+--R
+--R
+--R + 0 0 - 1 0 0 0 0 - 1 0 +
+--R | |
+--R | 1 0 1 0 - 1 0 - 1 0 0 |
+--R | |
+--R | 0 0 0 0 1 0 0 0 0 |
+--R | |
+--R | 0 0 0 0 0 0 0 1 0 |
+--R | |
+--R (5) |- 1 0 0 - 1 0 0 0 0 0 |
+--R | |
+--R | 0 0 0 0 0 0 1 0 0 |
+--R | |
+--R | 0 0 - 1 0 0 - 1 0 - 1 - 1|
+--R | |
+--R | 0 0 0 0 0 0 0 0 1 |
+--R | |
+--R + 0 - 1 0 0 - 1 0 - 1 0 0 +
+--R Type: Matrix
Integer
--E 18
---S 19 of 67
-r3 := irrRepSymNat([4,2],[4,2,1,3,6,5])
---R
---R There are no library operations named irrRepSymNat
---R Use HyperDoc Browse or issue
---R )what op irrRepSymNat
---R to learn if there is any operation containing " irrRepSymNat " in
---R its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R irrRepSymNat with argument type(s)
---R List PositiveInteger
---R List PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--S 19 of 68
+r3 := irreducibleRepresentation([4,2],[4,2,1,3,6,5])
+--R
+--R
+--R +0 0 0 0 1 0 1 0 1 +
+--R | |
+--R |0 0 0 0 0 1 0 1 0 |
+--R | |
+--R |0 0 0 1 0 0 0 0 0 |
+--R | |
+--R |0 - 1 0 0 - 1 0 - 1 0 0 |
+--R | |
+--R (6) |0 0 - 1 0 0 - 1 0 - 1 - 1|
+--R | |
+--R |1 1 1 0 0 0 0 0 0 |
+--R | |
+--R |0 0 0 0 0 0 0 0 1 |
+--R | |
+--R |0 0 0 0 1 0 0 0 0 |
+--R | |
+--R +0 0 0 0 0 1 0 0 0 +
+--R Type: Matrix
Integer
--E 19
---S 20 of 67
+--S 20 of 68
(r3 = r1*r2) :: Boolean
--R
--R
---R (2) false
+--R (7) false
--R Type:
Boolean
--E 20
---S 21 of 67
-irrRepSymNat [4,4,1]
+--S 21 of 68
+irreducibleRepresentation [4,4,1]
--R
--R
---R (3) irrRepSymNat
---R 4,4,1
---R Type:
Symbol
+--R (8)
+--R [
+--R [
+--R [- 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, 0, 0, 1, 0, -
1,
+--R 0, - 1, - 1, - 1, 0, 1, 1, 0, 0, 0, 0, - 1, 0, - 1, 1, - 1, - 1, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, 0, 0, 1,
1, 0,
+--R 1, 1, 0, - 1, - 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, - 1, 0,
1,
+--R 0, 0, 1, 1, 0, 0, - 1, 0, - 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, - 1, 0, - 1,
+--R - 1, 0, 0, 0, 0, 1, 0, 0, - 1, 0, - 1, - 1, 0, 0, 0, - 1, 1, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
- 1,
+--R - 1, - 1, 0, 0, 1, 0, 0, 0, 0, 0, - 1, - 1, - 1, 1, - 1, - 1, - 1, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 1, 1, 0,
0, 0,
+--R 0, - 1, 0, - 1, 1, - 1, - 1, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, 0, 1,
0,
+--R 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
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1,
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0,
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0,
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1,
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0,
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0,
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0,
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0,
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1,
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+--R ,
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0, 0,
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0,
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0,
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0,
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1,
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0, 0,
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0,
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0, 1,
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1,
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0,
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0,
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0,
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- 1,
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0,
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1,
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0,
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1,
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0,
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0,
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0,
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0,
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1,
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0,
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1,
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0,
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0,
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0, 0,
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+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, - 1, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1, - 1, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - 1, - 1,
0,
+--R 0, 0, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -
1, 0,
+--R 0, - 1, 0, 0, 0, - 1, 0, 0, 0, - 1, 0, 0]
+--R ,
+--R
+--R [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0,
+--R 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
- 1,
+--R 0, 0, - 1, 0, 0, 0, - 1, 0, 0, 0, - 1, 0]
+--R ]
+--R ]
+--R Type: List Matrix
Integer
--E 21
)clear all
---S 22 of 67
+--S 22 of 68
permutationRepresentation [2,3,1,4,6,5,11,10,7,8,9]
--R
--R
@@ -333,7 +1312,7 @@ permutationRepresentation [2,3,1,4,6,5,11,10,7,8,9]
--R Type: Matrix
Integer
--E 22
---S 23 of 67
+--S 23 of 68
gm2 := createGenericMatrix 2
--R
--R
@@ -345,7 +1324,7 @@ gm2 := createGenericMatrix 2
--R Type: Matrix Polynomial
Integer
--E 23
---S 24 of 67
+--S 24 of 68
symmetricTensors (gm2,2)
--R
--R
@@ -362,7 +1341,7 @@ symmetricTensors (gm2,2)
--R Type: Matrix Polynomial
Integer
--E 24
---S 25 of 67
+--S 25 of 68
gm3 := createGenericMatrix 3
--R
--R
@@ -377,7 +1356,7 @@ gm3 := createGenericMatrix 3
--R Type: Matrix Polynomial
Integer
--E 25
---S 26 of 67
+--S 26 of 68
antisymmetricTensors (gm3,2)
--R
--R
@@ -392,7 +1371,7 @@ antisymmetricTensors (gm3,2)
--R Type: Matrix Polynomial
Integer
--E 26
---S 27 of 67
+--S 27 of 68
tensorProduct(gm2,gm2)
--R
--R
@@ -412,7 +1391,7 @@ tensorProduct(gm2,gm2)
--R Type: Matrix Polynomial
Integer
--E 27
---S 28 of 67
+--S 28 of 68
)sh REP1
--R
--R RepresentationPackage1 R: Ring is a package constructor
@@ -439,207 +1418,402 @@ tensorProduct(gm2,gm2)
)clear all
---S 29 of 67
-r0 := irrRepSymNat [2,2,2,1,1]; r28 := meatAxe (r0::(LIST MATRIX PF 2))
---R
+--S 29 of 68
+r0 := irreducibleRepresentation [2,2,2,1,1];
--R
---RDaly Bug
---R Cannot convert from type Symbol to List Matrix PrimeField 2 for
---R value
---R irrRepSymNat
---R 2,2,2,1,1
--R
+--R Type: List Matrix
Integer
--E 29
---S 30 of 67
-areEquivalent? (r28.1, r28.2)
+--S 30 of 68
+r28 := meatAxe (r0::(LIST MATRIX PF 2))
--R
---R There are no library operations named r28
---R Use HyperDoc Browse or issue
---R )what op r28
---R to learn if there is any operation containing " r28 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named r28
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Fingerprint element in generated algebra is singular
+--R A proper cyclic submodule is found.
+--R Transition matrix computed
+--R The inverse of the transition matrix computed
+--R Now transform the matrices
+--R
+--R (2)
+--R [
+--R +0 1 1 1 1 1 1 0 0 1 1 1 0 0+
+--R | |
+--R |1 0 1 1 1 0 0 1 1 1 0 0 1 1|
+--R | |
+--R |1 1 0 1 0 1 0 0 1 0 1 0 0 1|
+--R | |
+--R |1 1 1 0 0 0 1 1 0 0 0 1 1 0|
+--R | |
+--R |1 1 0 0 0 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |1 0 1 0 1 0 1 0 1 1 1 1 0 1|
+--R | |
+--R |1 0 0 1 1 1 0 1 0 1 1 1 1 0|
+--R [| |,
+--R |0 1 1 0 1 1 0 1 0 1 1 0 0 0|
+--R | |
+--R |0 1 0 1 1 0 1 0 1 1 0 1 0 0|
+--R | |
+--R |1 1 0 0 1 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 0 1 0 0 1 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 0 0 1 0 0 1 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 1 0 0 0 0 1 0 0 0 0 0 0|
+--R | |
+--R +0 1 0 1 0 0 0 0 1 0 0 0 0 0+
+--R +1 1 1 1 0 0 0 0 0 0 0 0 0 0+
+--R | |
+--R |1 1 1 0 0 0 1 0 0 1 1 0 0 0|
+--R | |
+--R |1 1 1 0 0 0 0 0 1 1 0 0 1 0|
+--R | |
+--R |1 1 1 0 0 0 0 1 1 0 1 0 1 0|
+--R | |
+--R |1 1 1 0 0 0 0 1 0 1 1 1 1 0|
+--R | |
+--R |1 1 1 0 0 1 0 0 0 1 1 0 1 1|
+--R | |
+--R |1 1 1 0 1 0 0 0 0 1 1 0 0 1|
+--R | |]
+--R |1 1 0 0 0 0 0 0 0 0 1 1 1 1|
+--R | |
+--R |1 0 1 0 0 0 0 0 0 1 0 1 0 1|
+--R | |
+--R |0 0 0 1 0 0 1 0 0 1 1 1 1 0|
+--R | |
+--R |0 0 0 1 0 0 0 0 1 1 1 0 1 1|
+--R | |
+--R |0 0 0 1 0 0 0 1 1 1 1 0 0 1|
+--R | |
+--R |0 0 0 0 0 0 1 0 1 0 1 1 1 1|
+--R | |
+--R +0 0 0 0 0 0 1 1 1 1 0 1 0 1+
+--R ,
+--R
+--R +1 0 0 0 0 0 0 0 1 1 1 1 1 1+
+--R | |
+--R |0 1 0 0 0 0 0 0 1 1 1 0 0 0|
+--R | |
+--R |0 0 1 0 0 1 1 0 1 0 0 1 0 0|
+--R | |
+--R |0 0 0 1 0 1 0 1 0 1 0 0 1 0|
+--R | |
+--R |0 0 0 0 1 0 1 1 1 1 0 0 0 1|
+--R | |
+--R |0 0 0 0 0 1 1 1 1 1 0 1 1 0|
+--R | |
+--R |0 0 0 0 0 1 1 1 1 0 1 1 0 1|
+--R [| |,
+--R |0 0 0 0 0 1 1 1 0 1 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 1 1 0 1 1 1 1 0 0|
+--R | |
+--R |0 0 0 0 0 1 0 1 1 1 1 0 1 0|
+--R | |
+--R |0 0 0 0 0 0 1 1 1 1 1 1 1 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 1 0 1|
+--R | |
+--R +0 0 0 0 0 0 0 0 0 0 0 0 0 1+
+--R +0 0 1 1 1 1 1 0 1 0 0 0 0 0+
+--R | |
+--R |0 0 1 0 0 0 0 0 0 0 0 0 1 1|
+--R | |
+--R |0 0 0 0 0 0 0 1 0 1 0 0 1 0|
+--R | |
+--R |0 0 0 0 0 0 0 1 0 0 1 0 0 1|
+--R | |
+--R |0 0 1 0 0 0 0 0 0 1 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 0 0 1 1 0 0 1 0 0|
+--R | |
+--R |0 0 0 0 0 0 1 0 0 1 0 1 0 0|
+--R | |]
+--R |1 1 0 0 0 1 0 0 0 0 1 1 0 0|
+--R | |
+--R |0 0 1 1 0 0 1 0 1 0 0 0 1 0|
+--R | |
+--R |1 0 1 0 1 1 0 0 1 0 0 0 0 1|
+--R | |
+--R |1 0 1 1 1 0 0 0 1 0 0 0 1 1|
+--R | |
+--R |0 0 1 1 0 0 1 1 1 1 0 1 1 0|
+--R | |
+--R |0 1 1 0 1 1 0 1 1 0 1 1 0 1|
+--R | |
+--R +0 1 1 1 1 0 0 0 1 1 1 1 1 1+
+--R ]
+--R Type: List List Matrix PrimeField
2
--E 30
---S 31 of 67
-meatAxe r28.2
+--S 31 of 68
+areEquivalent? (r28.1, r28.2)
--R
---R There are no library operations named r28
---R Use HyperDoc Browse or issue
---R )what op r28
---R to learn if there is any operation containing " r28 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named r28
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Dimensions of kernels differ
+--R
+--R Representations are not equivalent.
+--R
+--R (3) [0]
+--R Type: Matrix PrimeField
2
--E 31
---S 32 of 67
-isAbsolutelyIrreducible? r28.2
+--S 32 of 68
+meatAxe r28.2
--R
---R There are no library operations named r28
---R Use HyperDoc Browse or issue
---R )what op r28
---R to learn if there is any operation containing " r28 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named r28
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is irreducible, but we don't know
+--R whether it is absolutely irreducible
+--R
+--R (4)
+--R [
+--R +1 0 0 0 0 0 0 0 0 0 0 0 0 0+
+--R | |
+--R |0 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 1 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 1 1 0 1 1 1 1 1 0 0 0 0|
+--R | |
+--R |0 0 1 0 1 1 1 1 1 0 1 0 0 0|
+--R [| |,
+--R |0 0 0 1 1 1 1 1 0 1 1 0 0 0|
+--R | |
+--R |1 1 1 0 1 1 1 0 1 1 1 0 0 0|
+--R | |
+--R |1 1 0 1 1 1 0 1 1 1 1 0 0 0|
+--R | |
+--R |1 1 0 0 0 0 1 1 1 1 1 0 0 0|
+--R | |
+--R |1 0 1 0 0 1 1 0 1 0 1 0 1 0|
+--R | |
+--R |1 0 0 1 0 1 0 1 0 1 1 1 0 0|
+--R | |
+--R +1 0 0 0 1 0 1 1 0 0 0 1 1 1+
+--R +0 0 0 0 0 0 0 1 0 1 1 0 0 0+
+--R | |
+--R |0 0 0 0 0 0 0 1 0 0 0 0 1 1|
+--R | |
+--R |1 1 0 0 1 0 0 0 1 1 1 1 1 1|
+--R | |
+--R |1 0 0 0 0 0 0 0 1 0 1 1 0 1|
+--R | |
+--R |1 0 0 0 0 0 0 0 0 1 1 0 1 1|
+--R | |
+--R |1 0 0 0 0 0 0 1 0 1 0 0 1 0|
+--R | |
+--R |1 0 0 0 0 0 1 0 1 0 0 1 0 0|
+--R | |]
+--R |0 0 1 1 0 1 0 0 0 0 0 1 1 0|
+--R | |
+--R |1 0 0 0 0 1 0 0 1 1 1 1 1 1|
+--R | |
+--R |0 0 1 0 1 0 1 0 0 0 0 1 0 1|
+--R | |
+--R |0 0 0 1 1 0 0 1 0 0 0 0 1 1|
+--R | |
+--R |0 0 0 0 0 1 1 1 0 0 0 1 1 1|
+--R | |
+--R |0 1 1 0 1 0 0 0 1 0 1 1 0 1|
+--R | |
+--R +0 1 0 1 1 0 0 0 0 1 1 0 1 1+
+--R ]
+--R Type: List List Matrix PrimeField
2
--E 32
---S 33 of 67
-ma := meatAxe r28.1
+--S 33 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? r28.2
--R
---R There are no library operations named r28
---R Use HyperDoc Browse or issue
---R )what op r28
---R to learn if there is any operation containing " r28 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named r28
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (5) true
+--R Type:
Boolean
--E 33
---S 34 of 67
-isAbsolutelyIrreducible? ma.1
+--S 34 of 68
+ma := meatAxe r28.1
--R
---R There are no library operations named ma
---R Use HyperDoc Browse or issue
---R )what op ma
---R to learn if there is any operation containing " ma " in its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named ma
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R A proper cyclic submodule is found.
+--R Transition matrix computed
+--R The inverse of the transition matrix computed
+--R Now transform the matrices
+--R
+--R (6)
+--R +0 0 0 0 1 0 1 1+ +1 1 1 1 1 1 0 0+
+--R | | | |
+--R |0 0 0 0 0 1 0 1| |1 0 0 1 1 0 1 0|
+--R | | | |
+--R |0 0 0 0 0 0 1 1| |0 0 1 1 0 0 1 0|
+--R | | | |
+--R |0 0 0 0 0 0 0 1| |1 1 0 1 1 1 1 1|
+--R [[| |,| |],
+--R |1 0 1 0 0 0 0 0| |1 1 1 1 0 0 1 0|
+--R | | | |
+--R |0 1 0 1 0 0 0 0| |1 0 0 1 1 1 1 1|
+--R | | | |
+--R |0 0 1 1 0 0 0 0| |0 1 1 0 1 0 1 1|
+--R | | | |
+--R +0 0 0 1 0 0 0 0+ +1 0 0 1 0 1 0 1+
+--R +0 1 1 0 0 1+ +1 1 0 0 0 0+
+--R | | | |
+--R |1 0 1 0 0 1| |1 0 1 1 0 0|
+--R | | | |
+--R |1 1 0 0 0 1| |1 0 0 1 0 1|
+--R [| |,| |]]
+--R |0 0 0 1 0 0| |1 0 1 1 1 0|
+--R | | | |
+--R |0 0 0 0 1 0| |1 0 0 0 1 1|
+--R | | | |
+--R +1 1 1 0 0 0+ +0 1 1 1 0 1+
+--R Type: List List Matrix PrimeField
2
--E 34
---S 35 of 67
-isAbsolutelyIrreducible? ma.2
+--S 35 of 68
+isAbsolutelyIrreducible? ma.1
--R
---R There are no library operations named ma
---R Use HyperDoc Browse or issue
---R )what op ma
---R to learn if there is any operation containing " ma " in its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named ma
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (7) true
+--R Type:
Boolean
--E 35
+--S 36 of 68
+isAbsolutelyIrreducible? ma.2
+--R
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (8) true
+--R Type:
Boolean
+--E 36
+
)clear all
---S 36 of 67
+--S 37 of 68
px : PERM PF 29 := cycles [[1,3,5],[7,11,9]]
--R
--R
--R (1) (1 3 5)(7 11 9)
--R Type: Permutation PrimeField
29
---E 36
+--E 37
---S 37 of 67
+--S 38 of 68
py : PERM PF 29 := cycles [[3,5,7,9]]
--R
--R
--R (2) (3 5 7 9)
--R Type: Permutation PrimeField
29
---E 37
+--E 38
---S 38 of 67
+--S 39 of 68
pz : PERM PF 29 := cycle [1,3,11]
--R
--R
--R (3) (1 3 11)
--R Type: Permutation PrimeField
29
---E 38
+--E 39
---S 39 of 67
+--S 40 of 68
px * pz
--R
--R
--R (4) (1 5)(3 9 7 11)
--R Type: Permutation PrimeField
29
---E 39
+--E 40
---S 40 of 67
+--S 41 of 68
py ** 3
--R
--R
--R (5) (3 9 7 5)
--R Type: Permutation PrimeField
29
---E 40
+--E 41
---S 41 of 67
+--S 42 of 68
inv px
--R
--R
--R (6) (1 5 3)(7 9 11)
--R Type: Permutation PrimeField
29
---E 41
+--E 42
---S 42 of 67
+--S 43 of 68
order px
--R
--R
--R (7) 3
--R Type:
PositiveInteger
---E 42
+--E 43
---S 43 of 67
+--S 44 of 68
movedPoints py
--R
--R
--R (8) {3,5,7,9}
--R Type: Set PrimeField
29
---E 43
+--E 44
---S 44 of 67
+--S 45 of 68
orbit ( pz , 3 )
--R
--R
--R (9) {3,11,1}
--R Type: Set PrimeField
29
---E 44
+--E 45
---S 45 of 67
+--S 46 of 68
eval ( py , 7 )
--R
--R
--R (10) 9
--R Type: PrimeField
29
---E 45
+--E 46
---S 46 of 67
+--S 47 of 68
)sh PERM
--R
--R Permutation S: SetCategory is a domain constructor
@@ -679,19 +1853,19 @@ eval ( py , 7 )
--R min : (%,%) -> % if S has FINITE or S has ORDSET
--R numberOfCycles : % -> NonNegativeInteger
--R
---E
+--E 47
)clear all
---S 47 of 67
+--S 48 of 68
genA6 : List PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]]
--R
--R
--R (1) [(1 2 3),(2 3 4 5 6)]
--R Type: List Permutation
Integer
---E 47
+--E 48
---S 48 of 67
+--S 49 of 68
pRA6 := permutationRepresentation (genA6,6)
--R
--R
@@ -707,9 +1881,9 @@ pRA6 := permutationRepresentation (genA6,6)
--R | | | |
--R +0 0 0 0 0 1+ +0 0 0 0 1 0+
--R Type: List Matrix
Integer
---E 48
+--E 49
---S 49 of 67
+--S 50 of 68
sp0 := meatAxe (pRA6::(List Matrix PF 2))
--R
--R Fingerprint element in generated algebra is singular
@@ -728,9 +1902,9 @@ sp0 := meatAxe (pRA6::(List Matrix PF 2))
--R | | | |
--R +0 0 0 0 1+ +0 0 0 1 0+
--R Type: List List Matrix PrimeField
2
---E 49
+--E 50
---S 50 of 67
+--S 51 of 68
sp1 := meatAxe sp0.1
--R
--R Fingerprint element in generated algebra is singular
@@ -754,9 +1928,9 @@ sp1 := meatAxe sp0.1
--R | | | |
--R +0 0 0 1+ +1 1 1 1+
--R Type: List List Matrix PrimeField
2
---E 50
+--E 51
---S 51 of 67 random generation, failure ok.
+--S 52 of 68 random generation, FAILURE OK.
isAbsolutelyIrreducible? sp1.2
--R
--R Random element in generated algebra has
@@ -769,260 +1943,600 @@ isAbsolutelyIrreducible? sp1.2
--R
--I (5) true
--R Type:
Boolean
---E 51
-
---S 52 of 67
-d2211 := irrRepSymNat ([2,2,1,1],genA6)
---R
---R There are no library operations named irrRepSymNat
---R Use HyperDoc Browse or issue
---R )what op irrRepSymNat
---R to learn if there is any operation containing " irrRepSymNat " in
---R its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R irrRepSymNat with argument type(s)
---R List PositiveInteger
---R List Permutation Integer
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
--E 52
---S 53 of 67
-d2211m2 := (d2211::(List Matrix PF 2)); sp2 := meatAxe d2211m2
+--S 53 of 68
+d2211 := irreducibleRepresentation ([2,2,1,1],genA6)
--R
---R
---RDaly Bug
---R Cannot convert from type Variable d2211 to List Matrix PrimeField 2
---R for value
---R d2211
--R
+--R (6)
+--R +1 0 0 - 1 1 0 0 0 0 + + 0 0 1 0 0 0 1 0
0+
+--R | | |
|
+--R |0 1 0 1 0 1 0 0 0 | | 0 0 0 0 1 0 - 1 0
0|
+--R | | |
|
+--R |0 0 1 0 1 - 1 0 0 0 | | 0 0 0 0 0 1 1 0
0|
+--R | | |
|
+--R |0 0 0 - 1 0 0 - 1 0 0 | | 0 0 0 0 0 0 1 1
0|
+--R | | |
|
+--R [|0 0 0 0 - 1 0 0 - 1 0 |,| 0 0 0 0 0 0 - 1 0
1|]
+--R | | |
|
+--R |0 0 0 0 0 - 1 0 0 - 1| | 0 0 0 0 0 0 1 0
0|
+--R | | |
|
+--R |0 0 0 1 0 0 0 0 0 | |- 1 0 0 0 0 0 - 1 0
0|
+--R | | |
|
+--R |0 0 0 0 1 0 0 0 0 | | 0 - 1 0 0 0 0 1 0
0|
+--R | | |
|
+--R +0 0 0 0 0 1 0 0 0 + + 0 0 0 - 1 0 0 - 1 0
0+
+--R Type: List Matrix
Integer
--E 53
---S 54 of 67
-isAbsolutelyIrreducible? sp2.1
+--S 54 of 68
+d2211m2 := (d2211::(List Matrix PF 2)); sp2 := meatAxe d2211m2
--R
---R There are no library operations named sp2
---R Use HyperDoc Browse or issue
---R )what op sp2
---R to learn if there is any operation containing " sp2 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp2
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Fingerprint element in generated algebra is singular
+--R A proper cyclic submodule is found.
+--R Transition matrix computed
+--R The inverse of the transition matrix computed
+--R Now transform the matrices
+--R
+--R +1 0 0 0 0+ +1 1 1 0 0+
+--R +1 0 1 1+ +0 0 1 0+ | | | |
+--R | | | | |0 1 1 1 1| |0 0 1 1 1|
+--R |0 1 0 1| |1 1 1 1| | | | |
+--R (7) [[| |,| |],[|0 1 1 0 0|,|1 0 0 1 0|]]
+--R |1 1 0 0| |1 0 1 1| | | | |
+--R | | | | |0 1 0 1 0| |0 0 1 0 1|
+--R +0 1 0 0+ +0 1 0 1+ | | | |
+--R +0 1 1 1 0+ +1 0 0 1 1+
+--R Type: List List Matrix PrimeField
2
--E 54
---S 55 of 67
-areEquivalent? (sp2.1, sp1.2)
+--S 55 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? sp2.1
--R
---R There are no library operations named sp2
---R Use HyperDoc Browse or issue
---R )what op sp2
---R to learn if there is any operation containing " sp2 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp2
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (8) true
+--R Type:
Boolean
--E 55
---S 56 of 67
-dA6d16 := tensorProduct(sp2.1,sp1.2); meatAxe dA6d16
+--S 56 of 68 random generation, FAILURE OK.
+areEquivalent? (sp2.1, sp1.2)
--R
---R There are no library operations named sp2
---R Use HyperDoc Browse or issue
---R )what op sp2
---R to learn if there is any operation containing " sp2 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp2
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Dimensions of kernels differ
+--R
+--R Representations are not equivalent.
+--R
+--R (9) [0]
+--R Type: Matrix PrimeField
2
--E 56
---S 57 of 67
-isAbsolutelyIrreducible? dA6d16
+--S 57 of 68
+dA6d16 := tensorProduct(sp2.1,sp1.2); meatAxe dA6d16
--R
---R There are 1 exposed and 0 unexposed library operations named
---R isAbsolutelyIrreducible? having 1 argument(s) but none was
---R determined to be applicable. Use HyperDoc Browse, or issue
---R )display op isAbsolutelyIrreducible?
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R isAbsolutelyIrreducible? with argument type(s)
---R Variable dA6d16
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is irreducible, but we don't know
+--R whether it is absolutely irreducible
+--R
+--R (10)
+--R [
+--R +0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0+
+--R | |
+--R |1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0|
+--R | |
+--R |0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1|
+--R [| |,
+--R |0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R +0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0+
+--R +0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0+
+--R | |
+--R |0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1|
+--R | |
+--R |0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1|
+--R | |
+--R |0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1|
+--R | |
+--R |0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1|
+--R | |]
+--R |0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0|
+--R | |
+--R |1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0|
+--R | |
+--R |1 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1|
+--R | |
+--R |0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1|
+--R | |
+--R +0 0 0 0 1 1 0 1 1 1 0 1 1 1 0 1+
+--R ]
+--R Type: List List Matrix PrimeField
2
--E 57
---S 58 of 67
-sp3 := meatAxe (dA6d16 :: (List Matrix FF(2,2)))
---R
+--S 58 of 68
+isAbsolutelyIrreducible? dA6d16
--R
---RDaly Bug
---R Cannot convert from type Variable dA6d16 to List Matrix FiniteField(
---R 2,2) for value
---R dA6d16
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R We have not found a one-dimensional kernel so far,
+--R as we do a random search you could try again
--R
+--R (11) false
+--R Type:
Boolean
--E 58
---S 59 of 67
-isAbsolutelyIrreducible? sp3.1
+--S 59 of 68
+sp3 := meatAxe (dA6d16 :: (List Matrix FF(2,2)))
--R
---R There are no library operations named sp3
---R Use HyperDoc Browse or issue
---R )what op sp3
---R to learn if there is any operation containing " sp3 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp3
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R A proper cyclic submodule is found.
+--R Transition matrix computed
+--R The inverse of the transition matrix computed
+--R Now transform the matrices
+--R
+--R (12)
+--R [
+--R +%A + 1 %A + 1 0 %A 1 %A %A %A + 1+
+--R | |
+--R | %A 0 %A + 1 %A + 1 1 %A + 1 %A + 1 %A |
+--R | |
+--R | %A %A + 1 %A 0 1 %A %A + 1 0 |
+--R | |
+--R | 0 %A %A + 1 %A 0 1 1 0 |
+--R [| |,
+--R | %A %A %A + 1 1 %A + 1 %A 0 %A |
+--R | |
+--R |%A + 1 %A %A + 1 1 %A 0 %A %A + 1|
+--R | |
+--R | 1 1 1 0 %A + 1 %A + 1 %A 0 |
+--R | |
+--R + 0 %A + 1 %A 0 0 %A %A + 1 %A + 1+
+--R + 0 %A + 1 %A + 1 %A 1 1 0 %A +
+--R | |
+--R |%A + 1 %A + 1 1 0 1 %A + 1 1 %A + 1|
+--R | |
+--R | %A 0 1 1 %A %A + 1 %A + 1 0 |
+--R | |
+--R | %A 1 0 %A %A 0 1 %A |
+--R | |]
+--R | 1 1 0 %A + 1 0 1 1 0 |
+--R | |
+--R | 1 %A 1 0 1 0 0 %A |
+--R | |
+--R |%A + 1 0 1 1 0 %A %A + 1 1 |
+--R | |
+--R +%A + 1 %A %A %A + 1 0 %A + 1 %A 0 +
+--R ,
+--R
+--R +%A + 1 %A %A 0 %A 1 %A + 1 0 +
+--R | |
+--R |%A + 1 1 0 %A + 1 1 %A + 1 1 %A |
+--R | |
+--R | 1 1 %A %A 1 %A 1 %A + 1|
+--R | |
+--R | 1 0 1 %A + 1 %A + 1 0 %A 1 |
+--R [| |,
+--R | 1 1 1 0 %A + 1 %A + 1 %A 0 |
+--R | |
+--R |%A + 1 %A %A 1 %A + 1 1 1 %A + 1|
+--R | |
+--R |%A + 1 %A + 1 %A 1 0 1 %A %A |
+--R | |
+--R + %A 0 %A + 1 0 1 0 1 %A +
+--R + 1 1 %A %A + 1 0 %A %A + 1 %A + 1+
+--R | |
+--R |%A + 1 0 0 1 %A + 1 1 1 %A + 1|
+--R | |
+--R | %A 0 1 0 %A + 1 0 %A + 1 1 |
+--R | |
+--R | 1 1 %A + 1 %A %A %A 1 0 |
+--R | |]
+--R | 1 %A 0 1 1 %A 1 0 |
+--R | |
+--R | 1 0 1 %A + 1 0 %A + 1 1 %A + 1|
+--R | |
+--R | 0 1 %A + 1 1 1 %A + 1 %A + 1 1 |
+--R | |
+--R + %A %A %A + 1 %A + 1 %A %A 0 1 +
+--R ]
+--R Type: List List Matrix
FiniteField(2,2)
--E 59
---S 60 of 67
-isAbsolutelyIrreducible? sp3.2
+--S 60 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? sp3.1
--R
---R There are no library operations named sp3
---R Use HyperDoc Browse or issue
---R )what op sp3
---R to learn if there is any operation containing " sp3 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp3
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (13) true
+--R Type:
Boolean
--E 60
---S 61 of 67
-areEquivalent? (sp3.1,sp3.2)
+--S 61 of 68 random generation, FAILURE OK.
+isAbsolutelyIrreducible? sp3.2
--R
---R There are no library operations named sp3
---R Use HyperDoc Browse or issue
---R )what op sp3
---R to learn if there is any operation containing " sp3 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp3
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (14) true
+--R Type:
Boolean
--E 61
---S 62 of 67
+--S 62 of 68 random generation, FAILURE OK.
+areEquivalent? (sp3.1,sp3.2)
+--R
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R There is no isomorphism, as the only possible one
+--R fails to do the necessary base change
+--R
+--R Representations are not equivalent.
+--R
+--R (15) [0]
+--R Type: Matrix
FiniteField(2,2)
+--E 62
+
+--S 63 of 68
sp0.2
--R
--R
---R (6) [[1],[1]]
+--R (16) [[1],[1]]
--R Type: List Matrix PrimeField
2
---E 62
+--E 63
---S 63 of 67
+--S 64 of 68
sp1.2
--R
--R
---R +0 1 0 0+ +0 1 1 1+
---R | | | |
---R |0 0 1 0| |1 1 0 1|
---R (7) [| |,| |]
---R |1 0 0 0| |1 1 1 0|
---R | | | |
---R +0 0 0 1+ +1 1 1 1+
+--R +0 1 0 0+ +0 1 1 1+
+--R | | | |
+--R |0 0 1 0| |1 1 0 1|
+--R (17) [| |,| |]
+--R |1 0 0 0| |1 1 1 0|
+--R | | | |
+--R +0 0 0 1+ +1 1 1 1+
--R Type: List Matrix PrimeField
2
---E 63
+--E 64
---S 64 of 67
+--S 65 of 68
sp2.1
--R
---R There are no library operations named sp2
---R Use HyperDoc Browse or issue
---R )what op sp2
---R to learn if there is any operation containing " sp2 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp2
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
---E 64
+--R
+--R +1 0 1 1+ +0 0 1 0+
+--R | | | |
+--R |0 1 0 1| |1 1 1 1|
+--R (18) [| |,| |]
+--R |1 1 0 0| |1 0 1 1|
+--R | | | |
+--R +0 1 0 0+ +0 1 0 1+
+--R Type: List Matrix PrimeField
2
+--E 65
---S 65 of 67
+--S 66 of 68
sp3.1
--R
---R There are no library operations named sp3
---R Use HyperDoc Browse or issue
---R )what op sp3
---R to learn if there is any operation containing " sp3 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp3
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
---E 65
+--R
+--R (19)
+--R +%A + 1 %A + 1 0 %A 1 %A %A %A + 1+
+--R | |
+--R | %A 0 %A + 1 %A + 1 1 %A + 1 %A + 1 %A |
+--R | |
+--R | %A %A + 1 %A 0 1 %A %A + 1 0 |
+--R | |
+--R | 0 %A %A + 1 %A 0 1 1 0 |
+--R [| |,
+--R | %A %A %A + 1 1 %A + 1 %A 0 %A |
+--R | |
+--R |%A + 1 %A %A + 1 1 %A 0 %A %A + 1|
+--R | |
+--R | 1 1 1 0 %A + 1 %A + 1 %A 0 |
+--R | |
+--R + 0 %A + 1 %A 0 0 %A %A + 1 %A + 1+
+--R + 0 %A + 1 %A + 1 %A 1 1 0 %A +
+--R | |
+--R |%A + 1 %A + 1 1 0 1 %A + 1 1 %A + 1|
+--R | |
+--R | %A 0 1 1 %A %A + 1 %A + 1 0 |
+--R | |
+--R | %A 1 0 %A %A 0 1 %A |
+--R | |]
+--R | 1 1 0 %A + 1 0 1 1 0 |
+--R | |
+--R | 1 %A 1 0 1 0 0 %A |
+--R | |
+--R |%A + 1 0 1 1 0 %A %A + 1 1 |
+--R | |
+--R +%A + 1 %A %A %A + 1 0 %A + 1 %A 0 +
+--R Type: List Matrix
FiniteField(2,2)
+--E 66
---S 66 of 67
+--S 67 of 68
sp3.2
--R
---R There are no library operations named sp3
---R Use HyperDoc Browse or issue
---R )what op sp3
---R to learn if there is any operation containing " sp3 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp3
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
---E 66
+--R
+--R (20)
+--R +%A + 1 %A %A 0 %A 1 %A + 1 0 +
+--R | |
+--R |%A + 1 1 0 %A + 1 1 %A + 1 1 %A |
+--R | |
+--R | 1 1 %A %A 1 %A 1 %A + 1|
+--R | |
+--R | 1 0 1 %A + 1 %A + 1 0 %A 1 |
+--R [| |,
+--R | 1 1 1 0 %A + 1 %A + 1 %A 0 |
+--R | |
+--R |%A + 1 %A %A 1 %A + 1 1 1 %A + 1|
+--R | |
+--R |%A + 1 %A + 1 %A 1 0 1 %A %A |
+--R | |
+--R + %A 0 %A + 1 0 1 0 1 %A +
+--R + 1 1 %A %A + 1 0 %A %A + 1 %A + 1+
+--R | |
+--R |%A + 1 0 0 1 %A + 1 1 1 %A + 1|
+--R | |
+--R | %A 0 1 0 %A + 1 0 %A + 1 1 |
+--R | |
+--R | 1 1 %A + 1 %A %A %A 1 0 |
+--R | |]
+--R | 1 %A 0 1 1 %A 1 0 |
+--R | |
+--R | 1 0 1 %A + 1 0 %A + 1 1 %A + 1|
+--R | |
+--R | 0 1 %A + 1 1 1 %A + 1 %A + 1 1 |
+--R | |
+--R + %A %A %A + 1 %A + 1 %A %A 0 1 +
+--R Type: List Matrix
FiniteField(2,2)
+--E 67
---S 67 of 67
+--S 68 of 68
dA6d16
--R
--R
---R (8) dA6d16
---R Type: Variable
dA6d16
---E 67
+--R (21)
+--R +0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0+
+--R | |
+--R |0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0|
+--R | |
+--R |1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0|
+--R | |
+--R |0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1|
+--R | |
+--R |0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0|
+--R | |
+--R |0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0|
+--R | |
+--R |0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1|
+--R [| |,
+--R |0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R +0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0+
+--R +0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0+
+--R | |
+--R |0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0|
+--R | |
+--R |0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1|
+--R | |
+--R |1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1|
+--R | |
+--R |1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |]
+--R |0 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1|
+--R | |
+--R |1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 1|
+--R | |
+--R |1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0|
+--R | |
+--R |1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1|
+--R | |
+--R |0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1|
+--R | |
+--R |0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1|
+--R | |
+--R |0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0|
+--R | |
+--R +0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1+
+--R Type: List Matrix PrimeField
2
+--E 68
)spool
)lisp (bye)
diff --git a/src/input/knot2.input.pamphlet b/src/input/knot2.input.pamphlet
index a793b46..7973989 100644
--- a/src/input/knot2.input.pamphlet
+++ b/src/input/knot2.input.pamphlet
@@ -63,57 +63,49 @@ l := lcm(p, q) quo p
--E 4
--S 5 of 8
-maxRange := (odd? l => l * %pi::SF; 2 * l * %pi::SF)
+maxRange := (odd? l => l * %pi; 2 * l * %pi)
--R
--R
---R (5) 15.707963267948966
---R Type:
DoubleFloat
+--R (5) 5%pi
+--R Type:
Pi
--E 5
--S 6 of 8
theRange := 0..maxRange
--R
--R
---R (6) 0...15.707963267948966
---R Type: Segment
DoubleFloat
+--R (6) 0..(5%pi)
+--R Type: Segment
Pi
--E 6
@
Create the knot
-<<*>>=
---S 7 of 8
+\begin{verbatim}
knot:TUBE := tubePlot(sin t * cos(PQ*t),cos t * cos(PQ*t),cos t * sin(PQ*t),
f, theRange, 0.1::SF, 6, "open" )
+\end{verbatim}
+<<*>>=
+--S 7 of 8
+v:=draw(curve(sin t * cos(PQ*t),cos t * cos(PQ*t),cos t * sin(PQ*t)), _
+ t=theRange, tubeRadius==0.1)
--R
---R
---RDaly Bug
---R Although TubePlot is the name of a constructor, a full type must be
---R specified in the context you have used it. Issue )show TubePlot
---R for more information.
+--I Compiling function %B with type DoubleFloat -> DoubleFloat
+--I Compiling function %D with type DoubleFloat -> DoubleFloat
+--I Compiling function %F with type DoubleFloat -> DoubleFloat
+--R Transmitting data...
+--R
+--R (7) ThreeDimensionalViewport: "DCOS((3*t)/5)*DSIN(t)"
+--R Type:
ThreeDimensionalViewport
--E 7
@
-Make a viewport out of it
+close the viewport
+\begin{verbatim}
+makeViewport3D(knot, concat ["knot",p::String,q::String])$VIEW3D
+\end{verbatim}
<<*>>=
--S 8 of 8
-makeViewport3D(knot, concat ["knot",p::String,q::String])$VIEW3D
---R
---R There are 2 exposed and 0 unexposed library operations named
---R makeViewport3D having 2 argument(s) but none was determined to be
---R applicable. Use HyperDoc Browse, or issue
---R )display op makeViewport3D
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R makeViewport3D with argument type(s)
---R Symbol
---R String
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+close(v)
--E 8
)spool
)lisp (bye)
diff --git a/src/input/repa6.input.pamphlet b/src/input/repa6.input.pamphlet
index 352a022..099c75d 100644
--- a/src/input/repa6.input.pamphlet
+++ b/src/input/repa6.input.pamphlet
@@ -39,7 +39,7 @@ representations of the alternating group A6.
We generate A6 by the permutations threecycle x=(1,2,3)
and the 5-cycle y=(2,3,4,5,6)
<<*>>=
---S 1 of 33
+--S 1 of 33
genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]
--R
--R
@@ -50,7 +50,7 @@ genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]
@
pRA6 is the permutation representation over the Integers...
<<*>>=
---S 2 of 33
+--S 2 of 33
pRA6 := permutationRepresentation (genA6, 6)
--R
--R
@@ -71,7 +71,7 @@ pRA6 := permutationRepresentation (genA6, 6)
@
And pRA6m2 is the permutation representation over PrimeField 2:
<<*>>=
---S 3 of 33
+--S 3 of 33
pRA6m2 : List Matrix PrimeField 2 := pRA6
--R
--R
@@ -117,7 +117,7 @@ sp0 := meatAxe pRA6m2
We have found the trivial module as a factormodule
and a 5-dimensional submodule.
<<*>>=
---S 5 of 33
+--S 5 of 33
dA6d1 := sp0.2
--R
--R
@@ -128,7 +128,7 @@ dA6d1 := sp0.2
@
Try to split again...
<<*>>=
---S 6 of 33
+--S 6 of 33
sp1 := meatAxe sp0.1
--R
--R Fingerprint element in generated algebra is singular
@@ -158,7 +158,7 @@ sp1 := meatAxe sp0.1
And find a 4-dimensional submodule, say dA6d4a, and the
trivial one again.
<<*>>=
---S 7 of 33
+--S 7 of 33
dA6d4a := sp1.2
--R
--R
@@ -175,13 +175,13 @@ dA6d4a := sp1.2
@
Now we want to test, whether dA6d4a is absolutely irreducible...
<<*>>=
---S 8 of 33 random input, ok to fail
+--S 8 of 33 random input, FAILURE OK
isAbsolutelyIrreducible? dA6d4a
--R
--R Random element in generated algebra does
--R not have a one-dimensional kernel
---R Random element in generated algebra has
---R one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
--R Random element in generated algebra has
--R one-dimensional kernel
--R We know that all the cyclic submodules generated by all
@@ -204,7 +204,7 @@ What is the degree of the representation belonging to
partition
[2,2,1,1]?
<<*>>=
-- lambda : PRTITION := partition [2,2,1,1]
---S 9 of 33
+--S 9 of 33
lambda := [2,2,1,1]
--R
--R
@@ -213,21 +213,11 @@ lambda := [2,2,1,1]
--E 9
--S 10 of 33
-dimIrrRepSym lambda
---R
---R There are no library operations named dimIrrRepSym
---R Use HyperDoc Browse or issue
---R )what op dimIrrRepSym
---R to learn if there is any operation containing " dimIrrRepSym " in
---R its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R dimIrrRepSym with argument type(s)
---R List PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+dimensionOfIrreducibleRepresentation lambda
+--R
+--R
+--R (10) 9
+--R Type:
PositiveInteger
--E 10
@
@@ -235,22 +225,28 @@ Now create the restriction to A6:
<<*>>=
--S 11 of 33
-d2211 := irrRepSymNat(lambda, genA6)
---R
---R There are no library operations named irrRepSymNat
---R Use HyperDoc Browse or issue
---R )what op irrRepSymNat
---R to learn if there is any operation containing " irrRepSymNat " in
---R its name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R irrRepSymNat with argument type(s)
---R List PositiveInteger
---R List Permutation Integer
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+d2211 := irreducibleRepresentation(lambda, genA6)
+--R
+--R
+--R (11)
+--R +1 0 0 - 1 1 0 0 0 0 + + 0 0 1 0 0 0 1 0
0+
+--R | | |
|
+--R |0 1 0 1 0 1 0 0 0 | | 0 0 0 0 1 0 - 1 0
0|
+--R | | |
|
+--R |0 0 1 0 1 - 1 0 0 0 | | 0 0 0 0 0 1 1 0
0|
+--R | | |
|
+--R |0 0 0 - 1 0 0 - 1 0 0 | | 0 0 0 0 0 0 1 1
0|
+--R | | |
|
+--R [|0 0 0 0 - 1 0 0 - 1 0 |,| 0 0 0 0 0 0 - 1 0
1|]
+--R | | |
|
+--R |0 0 0 0 0 - 1 0 0 - 1| | 0 0 0 0 0 0 1 0
0|
+--R | | |
|
+--R |0 0 0 1 0 0 0 0 0 | |- 1 0 0 0 0 0 - 1 0
0|
+--R | | |
|
+--R |0 0 0 0 1 0 0 0 0 | | 0 - 1 0 0 0 0 1 0
0|
+--R | | |
|
+--R +0 0 0 0 0 1 0 0 0 + + 0 0 0 - 1 0 0 - 1 0
0+
+--R Type: List Matrix
Integer
--E 11
@
@@ -259,13 +255,25 @@ And d2211m2 is the representation over PrimeField 2:
--S 12 of 33
d2211m2 : List Matrix PrimeField 2 := d2211
--R
---R
---RDaly Bug
---R Cannot convert right-hand side of assignment
---R d2211
--R
---R to an object of the type List Matrix PrimeField 2 of the
---R left-hand side.
+--R +1 0 0 1 1 0 0 0 0+ +0 0 1 0 0 0 1 0 0+
+--R | | | |
+--R |0 1 0 1 0 1 0 0 0| |0 0 0 0 1 0 1 0 0|
+--R | | | |
+--R |0 0 1 0 1 1 0 0 0| |0 0 0 0 0 1 1 0 0|
+--R | | | |
+--R |0 0 0 1 0 0 1 0 0| |0 0 0 0 0 0 1 1 0|
+--R | | | |
+--R (12) [|0 0 0 0 1 0 0 1 0|,|0 0 0 0 0 0 1 0 1|]
+--R | | | |
+--R |0 0 0 0 0 1 0 0 1| |0 0 0 0 0 0 1 0 0|
+--R | | | |
+--R |0 0 0 1 0 0 0 0 0| |1 0 0 0 0 0 1 0 0|
+--R | | | |
+--R |0 0 0 0 1 0 0 0 0| |0 1 0 0 0 0 1 0 0|
+--R | | | |
+--R +0 0 0 0 0 1 0 0 0+ +0 0 0 1 0 0 1 0 0+
+--R Type: List Matrix PrimeField
2
--E 12
@
@@ -274,10 +282,22 @@ And split it:
--S 13 of 33
sp2 := meatAxe d2211m2
--R
---R
---RDaly Bug
---R d2211m2 is declared as being in List Matrix PrimeField 2 but has not
---R been given a value.
+--R Fingerprint element in generated algebra is singular
+--R A proper cyclic submodule is found.
+--R Transition matrix computed
+--R The inverse of the transition matrix computed
+--R Now transform the matrices
+--R
+--R +1 0 0 0 0+ +1 1 1 0 0+
+--R +1 0 1 1+ +0 0 1 0+ | | | |
+--R | | | | |0 1 1 1 1| |0 0 1 1 1|
+--R |0 1 0 1| |1 1 1 1| | | | |
+--R (13) [[| |,| |],[|0 1 1 0 0|,|1 0 0 1 0|]]
+--R |1 1 0 0| |1 0 1 1| | | | |
+--R | | | | |0 1 0 1 0| |0 0 1 0 1|
+--R +0 1 0 0+ +0 1 0 1+ | | | |
+--R +0 1 1 1 0+ +1 0 0 1 1+
+--R Type: List List Matrix PrimeField
2
--E 13
@
@@ -285,69 +305,52 @@ A 5 and a 4-dimensional one.
we take the 4-dimensional one, say dA6d4b:
<<*>>=
---S 14 of 33
+--S 14 of 33
dA6d4b := sp2.1
--R
---R There are no library operations named sp2
---R Use HyperDoc Browse or issue
---R )what op sp2
---R to learn if there is any operation containing " sp2 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp2
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R +1 0 1 1+ +0 0 1 0+
+--R | | | |
+--R |0 1 0 1| |1 1 1 1|
+--R (14) [| |,| |]
+--R |1 1 0 0| |1 0 1 1|
+--R | | | |
+--R +0 1 0 0+ +0 1 0 1+
+--R Type: List Matrix PrimeField
2
--E 14
@
This is absolutely irreducible, too ...
<<*>>=
---S 15 of 33
+--S 15 of 33 random generation, FAILURE OK.
isAbsolutelyIrreducible? dA6d4b
--R
---R There are 1 exposed and 0 unexposed library operations named
---R isAbsolutelyIrreducible? having 1 argument(s) but none was
---R determined to be applicable. Use HyperDoc Browse, or issue
---R )display op isAbsolutelyIrreducible?
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R isAbsolutelyIrreducible? with argument type(s)
---R Variable dA6d4b
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (15) true
+--R Type:
Boolean
--E 15
@
And dA6d4a and dA6d4b are not equivalent:
<<*>>=
---S 16 of 33
+--S 16 of 33 random generation, FAILURE OK.
areEquivalent? ( dA6d4a , dA6d4b )
--R
---R There are 1 exposed and 0 unexposed library operations named
---R areEquivalent? having 2 argument(s) but none was determined to be
---R applicable. Use HyperDoc Browse, or issue
---R )display op areEquivalent?
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R areEquivalent? with argument type(s)
---R List Matrix PrimeField 2
---R Variable dA6d4b
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Dimensions of kernels differ
+--R
+--R Representations are not equivalent.
+--R
+--R (16) [0]
+--R Type: Matrix PrimeField
2
--E 16
@
@@ -355,25 +358,74 @@ So the third irreducible representation is found.
Now construct a new representation with the help of the tensorproduct
<<*>>=
---S 17 of 33
+--S 17 of 33
dA6d16 := tensorProduct ( dA6d4a , dA6d4b )
--R
---R There are 2 exposed and 0 unexposed library operations named
---R tensorProduct having 2 argument(s) but none was determined to be
---R applicable. Use HyperDoc Browse, or issue
---R )display op tensorProduct
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R tensorProduct with argument type(s)
---R List Matrix PrimeField 2
---R Variable dA6d4b
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (17)
+--R +0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0+
+--R | |
+--R |0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
+--R [| |,
+--R |1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0|
+--R | |
+--R +0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0+
+--R +0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0+
+--R | |
+--R |0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1|
+--R | |
+--R |1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1|
+--R | |]
+--R |0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0|
+--R | |
+--R |1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1|
+--R | |
+--R +0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1+
+--R Type: List Matrix PrimeField
2
--E 17
@
@@ -382,149 +434,508 @@ And try to split it...
--S 18 of 33
sp3 := meatAxe dA6d16
--R
---R There are 1 exposed and 0 unexposed library operations named meatAxe
---R having 1 argument(s) but none was determined to be applicable.
---R Use HyperDoc Browse, or issue
---R )display op meatAxe
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R meatAxe with argument type(s)
---R Variable dA6d16
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is irreducible, but we don't know
+--R whether it is absolutely irreducible
+--R
+--R (18)
+--R [
+--R +0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0+
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0|
+--R | |
+--R |1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R [| |,
+--R |0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
+--R | |
+--R +0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0+
+--R +0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0+
+--R | |
+--R |0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1|
+--R | |
+--R |0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0|
+--R | |
+--R |0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1|
+--R | |
+--R |0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1|
+--R | |
+--R |1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0|
+--R | |
+--R |0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1|
+--R | |]
+--R |0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0|
+--R | |
+--R |0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1|
+--R | |
+--R |1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0|
+--R | |
+--R |0 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1|
+--R | |
+--R |0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1|
+--R | |
+--R |1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0|
+--R | |
+--R +0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1+
+--R ]
+--R Type: List List Matrix PrimeField
2
--E 18
@
The representation is irreducible, but may be not
absolutely irreducible.
<<*>>=
---S 19 of 33
+--S 19 of 33
isAbsolutelyIrreducible? dA6d16
--R
---R There are 1 exposed and 0 unexposed library operations named
---R isAbsolutelyIrreducible? having 1 argument(s) but none was
---R determined to be applicable. Use HyperDoc Browse, or issue
---R )display op isAbsolutelyIrreducible?
---R to learn more about the available operations. Perhaps
---R package-calling the operation or using coercions on the arguments
---R will allow you to apply the operation.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named
---R isAbsolutelyIrreducible? with argument type(s)
---R Variable dA6d16
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R We have not found a one-dimensional kernel so far,
+--R as we do a random search you could try again
+--R
+--R (19) false
+--R Type:
Boolean
--E 19
@
So let's try the same over the field with 4 elements:
<<*>>=
---S 20 of 33
+--S 20 of 33
gf4 := FiniteField(2,2)
--R
--R
---R (10) FiniteField(2,2)
+--R (20) FiniteField(2,2)
--R Type:
Domain
--E 20
---S 21 of 33
+--S 21 of 33
dA6d16gf4 : List Matrix gf4 := dA6d16
--R
---R
---RDaly Bug
---R Cannot convert right-hand side of assignment
---R dA6d16
--R
---R to an object of the type List Matrix FiniteField(2,2) of the
---R left-hand side.
+--R (21)
+--R +0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0+
+--R | |
+--R |0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
+--R [| |,
+--R |1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0|
+--R | |
+--R +0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0+
+--R +0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0+
+--R | |
+--R |0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1|
+--R | |
+--R |1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1|
+--R | |]
+--R |0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0|
+--R | |
+--R |1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1|
+--R | |
+--R +0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1+
+--R Type: List Matrix
FiniteField(2,2)
--E 21
--S 22 of 33
sp4 := meatAxe dA6d16gf4
--R
---R
---RDaly Bug
---R dA6d16gf4 is declared as being in List Matrix FiniteField(2,2) but
---R has not been given a value.
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is non-singular
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R Fingerprint element in generated algebra is singular
+--R The generated cyclic submodule was not proper
+--R The generated cyclic submodule was not proper
+--R A proper cyclic submodule is found.
+--R Transition matrix computed
+--R The inverse of the transition matrix computed
+--R Now transform the matrices
+--R
+--R (22)
+--R [
+--R + %A %A + 1 0 %A 1 %A + 1 0 0 +
+--R | |
+--R | 0 0 %A %A + 1 %A %A 0 0 |
+--R | |
+--R | %A %A + 1 %A 1 %A + 1 0 0 0 |
+--R | |
+--R | %A %A + 1 %A 1 %A 0 0 0 |
+--R [| |,
+--R |%A + 1 1 1 1 0 0 %A + 1 %A|
+--R | |
+--R | 0 0 %A + 1 1 0 0 %A 0 |
+--R | |
+--R | 1 0 1 1 0 0 0 0 |
+--R | |
+--R + 1 1 0 0 0 0 0 0 +
+--R + 1 0 %A 0 1 1 %A %A + 1+
+--R | |
+--R | 1 %A + 1 0 0 0 %A + 1 1 %A + 1|
+--R | |
+--R | %A 1 %A + 1 %A + 1 %A + 1 1 %A 0 |
+--R | |
+--R |%A + 1 %A + 1 0 0 1 %A + 1 1 1 |
+--R | |]
+--R | 1 0 %A + 1 0 1 1 %A %A |
+--R | |
+--R | 0 0 %A + 1 %A + 1 %A + 1 1 1 %A |
+--R | |
+--R | 0 0 1 0 0 1 0 1 |
+--R | |
+--R + 0 %A 0 %A 1 %A + 1 %A + 1 %A +
+--R ,
+--R
+--R +0 1 1 %A + 1 0 0 0 0+
+--R | |
+--R |1 1 %A + 1 0 0 0 0 0|
+--R | |
+--R |%A 0 0 0 0 0 0 0|
+--R | |
+--R |1 %A 0 0 0 0 0 0|
+--R [| |,
+--R |%A %A + 1 1 1 1 0 1 1|
+--R | |
+--R |0 0 %A 1 0 1 0 1|
+--R | |
+--R |%A 1 0 1 1 1 0 0|
+--R | |
+--R +1 %A %A + 1 %A 0 1 0 0+
+--R +%A + 1 1 %A 0 0 %A + 1 0 1 +
+--R | |
+--R | 0 %A 1 1 1 0 %A + 1 %A |
+--R | |
+--R | 0 %A + 1 0 %A + 1 %A + 1 1 %A + 1 %A |
+--R | |
+--R | 1 %A + 1 1 %A + 1 0 0 %A + 1 1 |
+--R | |]
+--R | 0 %A 0 %A + 1 %A + 1 0 0 %A + 1|
+--R | |
+--R |%A + 1 0 %A + 1 %A 0 %A + 1 0 %A + 1|
+--R | |
+--R | 0 1 0 1 %A + 1 0 %A + 1 %A + 1|
+--R | |
+--R + %A %A %A 1 %A %A 1 %A + 1+
+--R ]
+--R Type: List List Matrix
FiniteField(2,2)
--E 22
@
Now we find two 8-dimensional ones, dA6d8a and dA6d8b.
<<*>>=
---S 23 of 33
+--S 23 of 33
dA6d8a : List Matrix gf4 := sp4.1
--R
---R There are no library operations named sp4
---R Use HyperDoc Browse or issue
---R )what op sp4
---R to learn if there is any operation containing " sp4 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp4
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (23)
+--R + %A %A + 1 0 %A 1 %A + 1 0 0 +
+--R | |
+--R | 0 0 %A %A + 1 %A %A 0 0 |
+--R | |
+--R | %A %A + 1 %A 1 %A + 1 0 0 0 |
+--R | |
+--R | %A %A + 1 %A 1 %A 0 0 0 |
+--R [| |,
+--R |%A + 1 1 1 1 0 0 %A + 1 %A|
+--R | |
+--R | 0 0 %A + 1 1 0 0 %A 0 |
+--R | |
+--R | 1 0 1 1 0 0 0 0 |
+--R | |
+--R + 1 1 0 0 0 0 0 0 +
+--R + 1 0 %A 0 1 1 %A %A + 1+
+--R | |
+--R | 1 %A + 1 0 0 0 %A + 1 1 %A + 1|
+--R | |
+--R | %A 1 %A + 1 %A + 1 %A + 1 1 %A 0 |
+--R | |
+--R |%A + 1 %A + 1 0 0 1 %A + 1 1 1 |
+--R | |]
+--R | 1 0 %A + 1 0 1 1 %A %A |
+--R | |
+--R | 0 0 %A + 1 %A + 1 %A + 1 1 1 %A |
+--R | |
+--R | 0 0 1 0 0 1 0 1 |
+--R | |
+--R + 0 %A 0 %A 1 %A + 1 %A + 1 %A +
+--R Type: List Matrix
FiniteField(2,2)
--E 23
--S 24 of 33
dA6d8b : List Matrix gf4 := sp4.2
--R
---R There are no library operations named sp4
---R Use HyperDoc Browse or issue
---R )what op sp4
---R to learn if there is any operation containing " sp4 " in its
---R name.
---R
---RDaly Bug
---R Cannot find a definition or applicable library operation named sp4
---R with argument type(s)
---R PositiveInteger
---R
---R Perhaps you should use "@" to indicate the required return type,
---R or "$" to specify which version of the function you need.
+--R
+--R (24)
+--R +0 1 1 %A + 1 0 0 0 0+
+--R | |
+--R |1 1 %A + 1 0 0 0 0 0|
+--R | |
+--R |%A 0 0 0 0 0 0 0|
+--R | |
+--R |1 %A 0 0 0 0 0 0|
+--R [| |,
+--R |%A %A + 1 1 1 1 0 1 1|
+--R | |
+--R |0 0 %A 1 0 1 0 1|
+--R | |
+--R |%A 1 0 1 1 1 0 0|
+--R | |
+--R +1 %A %A + 1 %A 0 1 0 0+
+--R +%A + 1 1 %A 0 0 %A + 1 0 1 +
+--R | |
+--R | 0 %A 1 1 1 0 %A + 1 %A |
+--R | |
+--R | 0 %A + 1 0 %A + 1 %A + 1 1 %A + 1 %A |
+--R | |
+--R | 1 %A + 1 1 %A + 1 0 0 %A + 1 1 |
+--R | |]
+--R | 0 %A 0 %A + 1 %A + 1 0 0 %A + 1|
+--R | |
+--R |%A + 1 0 %A + 1 %A 0 %A + 1 0 %A + 1|
+--R | |
+--R | 0 1 0 1 %A + 1 0 %A + 1 %A + 1|
+--R | |
+--R + %A %A %A 1 %A %A 1 %A + 1+
+--R Type: List Matrix
FiniteField(2,2)
--E 24
@
Both are absolutely irreducible...
<<*>>=
---S 25 of 33
+--S 25 of 33 random generation, FAILURE OK.
isAbsolutelyIrreducible? dA6d8a
--R
---R
---RDaly Bug
---R dA6d8a is declared as being in List Matrix FiniteField(2,2) but has
---R not been given a value.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (25) true
+--R Type:
Boolean
--E 25
---S 26 of 33
+--S 26 of 33 random generation, FAILURE OK.
isAbsolutelyIrreducible? dA6d8b
--R
---R
---RDaly Bug
---R dA6d8b is declared as being in List Matrix FiniteField(2,2) but has
---R not been given a value.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R We know that all the cyclic submodules generated by all
+--R non-trivial element of the singular matrix under view are
+--R not proper, hence Norton's irreducibility test can be done:
+--R The generated cyclic submodule was not proper
+--R Representation is absolutely irreducible
+--R
+--R (26) true
+--R Type:
Boolean
--E 26
@
And they are not equivalent...
<<*>>=
---S 27 of 33
+--S 27 of 33 random generation, FAILURE OK.
areEquivalent? ( dA6d8a, dA6d8b )
--R
---R
---RDaly Bug
---R dA6d8a is declared as being in List Matrix FiniteField(2,2) but has
---R not been given a value.
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra does
+--R not have a one-dimensional kernel
+--R Random element in generated algebra has
+--R one-dimensional kernel
+--R There is no isomorphism, as the only possible one
+--R fails to do the necessary base change
+--R
+--R Representations are not equivalent.
+--R
+--R (27) [0]
+--R Type: Matrix
FiniteField(2,2)
--E 27
@
@@ -535,11 +946,11 @@ The theory tells us that there are no more irreducible
ones.
Here again are all absolutely irreducible 2-modular
representations of A6
<<*>>=
---S 28 of 33
+--S 28 of 33
dA6d1
--R
--R
---R (11) [[1],[1]]
+--R (28) [[1],[1]]
--R Type: List Matrix PrimeField
2
--E 28
@@ -550,7 +961,7 @@ dA6d4a
--R +0 1 0 0+ +0 1 1 1+
--R | | | |
--R |0 0 1 0| |1 1 0 1|
---R (12) [| |,| |]
+--R (29) [| |,| |]
--R |1 0 0 0| |1 1 1 0|
--R | | | |
--R +0 0 0 1+ +1 1 1 1+
@@ -561,26 +972,90 @@ dA6d4a
dA6d4b
--R
--R
---R (13) dA6d4b
---R Type: Variable
dA6d4b
+--R +1 0 1 1+ +0 0 1 0+
+--R | | | |
+--R |0 1 0 1| |1 1 1 1|
+--R (30) [| |,| |]
+--R |1 1 0 0| |1 0 1 1|
+--R | | | |
+--R +0 1 0 0+ +0 1 0 1+
+--R Type: List Matrix PrimeField
2
--E 30
--S 31 of 33
dA6d8a
--R
---R
---RDaly Bug
---R dA6d8a is declared as being in List Matrix FiniteField(2,2) but has
---R not been given a value.
+--R
+--R (31)
+--R + %A %A + 1 0 %A 1 %A + 1 0 0 +
+--R | |
+--R | 0 0 %A %A + 1 %A %A 0 0 |
+--R | |
+--R | %A %A + 1 %A 1 %A + 1 0 0 0 |
+--R | |
+--R | %A %A + 1 %A 1 %A 0 0 0 |
+--R [| |,
+--R |%A + 1 1 1 1 0 0 %A + 1 %A|
+--R | |
+--R | 0 0 %A + 1 1 0 0 %A 0 |
+--R | |
+--R | 1 0 1 1 0 0 0 0 |
+--R | |
+--R + 1 1 0 0 0 0 0 0 +
+--R + 1 0 %A 0 1 1 %A %A + 1+
+--R | |
+--R | 1 %A + 1 0 0 0 %A + 1 1 %A + 1|
+--R | |
+--R | %A 1 %A + 1 %A + 1 %A + 1 1 %A 0 |
+--R | |
+--R |%A + 1 %A + 1 0 0 1 %A + 1 1 1 |
+--R | |]
+--R | 1 0 %A + 1 0 1 1 %A %A |
+--R | |
+--R | 0 0 %A + 1 %A + 1 %A + 1 1 1 %A |
+--R | |
+--R | 0 0 1 0 0 1 0 1 |
+--R | |
+--R + 0 %A 0 %A 1 %A + 1 %A + 1 %A +
+--R Type: List Matrix
FiniteField(2,2)
--E 31
--S 32 of 33
dA6d8b
--R
---R
---RDaly Bug
---R dA6d8b is declared as being in List Matrix FiniteField(2,2) but has
---R not been given a value.
+--R
+--R (32)
+--R +0 1 1 %A + 1 0 0 0 0+
+--R | |
+--R |1 1 %A + 1 0 0 0 0 0|
+--R | |
+--R |%A 0 0 0 0 0 0 0|
+--R | |
+--R |1 %A 0 0 0 0 0 0|
+--R [| |,
+--R |%A %A + 1 1 1 1 0 1 1|
+--R | |
+--R |0 0 %A 1 0 1 0 1|
+--R | |
+--R |%A 1 0 1 1 1 0 0|
+--R | |
+--R +1 %A %A + 1 %A 0 1 0 0+
+--R +%A + 1 1 %A 0 0 %A + 1 0 1 +
+--R | |
+--R | 0 %A 1 1 1 0 %A + 1 %A |
+--R | |
+--R | 0 %A + 1 0 %A + 1 %A + 1 1 %A + 1 %A |
+--R | |
+--R | 1 %A + 1 1 %A + 1 0 0 %A + 1 1 |
+--R | |]
+--R | 0 %A 0 %A + 1 %A + 1 0 0 %A + 1|
+--R | |
+--R |%A + 1 0 %A + 1 %A 0 %A + 1 0 %A + 1|
+--R | |
+--R | 0 1 0 1 %A + 1 0 %A + 1 %A + 1|
+--R | |
+--R + %A %A %A 1 %A %A 1 %A + 1+
+--R Type: List Matrix
FiniteField(2,2)
--E 32
@
@@ -591,8 +1066,70 @@ representations of A6 over PrimeField 2
dA6d16
--R
--R
---R (14) dA6d16
---R Type: Variable
dA6d16
+--R (33)
+--R +0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0+
+--R | |
+--R |0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
+--R [| |,
+--R |1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0|
+--R | |
+--R +0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0+
+--R +0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0+
+--R | |
+--R |0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1|
+--R | |
+--R |0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1|
+--R | |
+--R |1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1|
+--R | |]
+--R |0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0|
+--R | |
+--R |1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0|
+--R | |
+--R |0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0|
+--R | |
+--R |0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0|
+--R | |
+--R |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1|
+--R | |
+--R |1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1|
+--R | |
+--R +0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1+
+--R Type: List Matrix PrimeField
2
--E 33
)spool
)lisp (bye)
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