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[Axiom-developer] 20070824.01.tpd.patch applied to silver
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20070824.01.tpd.patch applied to silver |
|
Date: |
Mon, 27 Aug 2007 02:47:45 -0500 |
diff --git a/changelog b/changelog
index 87548f8..eaa591a 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,68 @@
+20070824 tpd src/doc/spadhelp add DecimalExpansion
+20070824 tpd src/algebra/Makefile add DecimalExpansion.help
+20070824 tpd src/algebra/radix.spad add DecimalExpansion.help
+20070824 tpd src/algebra/radix.spad add DecimalExpansion.input
+20070824 tpd src/doc/spadhelp add DeRhamComplex
+20070824 tpd src/algebra/Makefile add DeRhamComplex.help
+20070824 tpd src/algebra/derham.spad add DeRhamComplex.help
+20070824 tpd src/algebra/derham.spad add DeRhamComplex.input
+20070824 tpd src/doc/spadhelp add CycleIndicators
+20070824 tpd src/algebra/Makefile add CycleIndicators.help
+20070824 tpd src/algebra/cycles.spad add CycleIndicators.help
+20070824 tpd src/algebra/cycles.spad add CycleIndicators.input
+20070824 tpd src/input/Makefile implement new algebra regression testing
+20070823 tpd src/doc/spadhelp add ContinuedFraction
+20070823 tpd src/algebra/Makefile add ContinuedFraction.help
+20070823 tpd src/algebra/contfrac.spad add ContinuedFraction.help
+20070823 tpd src/algebra/contfrac.spad add ContinuedFraction.input
+20070823 tpd src/doc/spadhelp add Complex
+20070823 tpd src/algebra/Makefile add Complex.help
+20070823 tpd src/algebra/gaussian.spad add Complex.help
+20070823 tpd src/algebra/gaussian.spad add Complex.input
+20070823 tpd src/doc/spadhelp add CliffordAlgebra
+20070823 tpd src/algebra/Makefile add CliffordAlgebra.help
+20070823 tpd src/algebra/clifford.spad add CliffordAlgebra.help
+20070823 tpd src/algebra/clifford.spad add CliffordAlgebra.input
+20070823 tpd src/doc/spadhelp add CharacterClass
+20070823 tpd src/algebra/Makefile add CharacterClass.help
+20070823 tpd src/algebra/string.spad add CharacterClass.help
+20070823 tpd src/algebra/string.spad add CharacterClass.input
+20070823 tpd src/doc/spadhelp add Character
+20070823 tpd src/algebra/Makefile add Character.help
+20070823 tpd src/algebra/string.spad add Character.help
+20070823 tpd src/algebra/string.spad add Character.input
+20070823 tpd src/doc/spadhelp add CartesianTensor
+20070823 tpd src/algebra/Makefile add CartesianTensor.help
+20070823 tpd src/algebra/carten.spad add CartesianTensor.help
+20070823 tpd src/algebra/carten.spad add CartesianTensor.input
+20070823 tpd src/doc/spadhelp add CardinalNumber
+20070823 tpd src/algebra/Makefile add CardinalNumber.help
+20070823 tpd src/algebra/card.spad add CardinalNumber.help
+20070823 tpd src/algebra/card.spad add CardinalNumber.input
+20070823 tpd src/doc/spadhelp add BinarySearchTree
+20070823 tpd src/algebra/Makefile add BinarySearchTree.help
+20070823 tpd src/algebra/tree.spad add BinarySearchTree.help
+20070823 tpd src/algebra/tree.spad add BinarySearchTree.input
+20070823 tpd src/doc/spadhelp add BinaryExpansion
+20070823 tpd src/algebra/Makefile add BinaryExpansion.help
+20070823 tpd src/algebra/radix.spad add BinaryExpansion.help
+20070823 tpd src/algebra/radix.spad add BinaryExpansion.input
+20070823 tpd src/doc/spadhelp add BasicOperator
+20070823 tpd src/algebra/Makefile add BasicOperator.help
+20070823 tpd src/algebra/op.spad add BasicOperator.help
+20070823 tpd src/algebra/op.spad add BasicOperator.input
+20070823 tpd src/doc/spadhelp add BalancedBinaryTree
+20070823 tpd src/algebra/Makefile add BalancedBinaryTree.help
+20070823 tpd src/algebra/tree.spad add BalancedBinaryTree.help
+20070823 tpd src/algebra/tree.spad add BalancedBinaryTree.input
+20070823 tpd src/doc/spadhelp add AssociationList
+20070823 tpd src/algebra/Makefile add AssociationList.help
+20070823 tpd src/algebra/list.spad add AssociationList.help
+20070823 tpd src/algebra/list.spad add AssociationList.input
+20070823 tpd src/algebra/Makefile create spadhelp
+20070823 tpd src/Makefile mkdir int/input
+20070823 tpd src/Makefile mkdir mnt/sys/doc/spadhelp
+20070823 tpd src/doc/book.pamphlet fix typos
20070823 tpd src/doc/spadhelp add additional spadhelp files
20070823 tpd src/doc/Makefile add additional spadhelp files
20070822 tpd src/doc/spadhelp add language help
diff --git a/src/Makefile.pamphlet b/src/Makefile.pamphlet
index 13bf3ff..5df04f4 100644
--- a/src/Makefile.pamphlet
+++ b/src/Makefile.pamphlet
@@ -307,6 +307,7 @@ Once {\bf bootsys} exists we need to build {\bf depsys}
and {\bf interpsys}. Since these two images share a lot of
files they are built in the interp subdirectory using the
same Makefile.
+
<<interpdir>>=
interpdir: ${SRC}/interp/Makefile
@echo 25 making ${SRC}/interp
@@ -372,6 +373,13 @@ stanzas that describe the steps to extract the algebra
from the
[[src/algebra/*.pamphlet]] files into the [[int/algebra/*.spad]] files.
Further details are provided in Makefile for src/algebra.
+The doc/spadhelp directory contains flat files of help text
+for the help system command. Algebra pamphlet contain examples
+that can be shown from these commands.
+
+We need to make the int/input file here because the algebra Makefile
+will extract input files for regression testing from the algebra pamphlets.
+
<<algebradir>>=
algebradir: ${SRC}/algebra/Makefile
@echo 29 making ${SRC}/algebra
@@ -380,7 +388,9 @@ algebradir: ${SRC}/algebra/Makefile
@mkdir -p ${OBJ}/${SYS}/algebra
@mkdir -p ${MNT}/${SYS}/algebra
@mkdir -p ${MNT}/${SYS}/doc/src/algebra
+ @mkdir -p ${MNT}/${SYS}/doc/spadhelp
@mkdir -p ${MNT}/${SYS}/src/algebra
+ @mkdir -p ${INT}/input
@(cd algebra ; ${ENV} ${MAKE} )
${SRC}/algebra/Makefile: ${SRC}/algebra/Makefile.pamphlet
diff --git a/src/algebra/Makefile.pamphlet b/src/algebra/Makefile.pamphlet
index 6a281bc..3fa3009 100644
--- a/src/algebra/Makefile.pamphlet
+++ b/src/algebra/Makefile.pamphlet
@@ -1148,6 +1148,9 @@ The [[OUTSRC=${MNT}/${SYS}/src/algebra]] subdirectory
contains the
algebra source files extracted from the pamphlet files. These sources
allow the end user to change the algebra if needed.
+The HELP variable points to the location of the files used for the
+[[)help]] system command. These are flat files used to provide command
+line help to Axiom. They are extracted from the algebra pamphlets.
<<environment>>=
IN=${SRC}/algebra
@@ -1156,6 +1159,7 @@ OUT=${MNT}/${SYS}/algebra
DOC=${MNT}/${SYS}/doc/src/algebra
OUTSRC=${MNT}/${SYS}/src/algebra
INPUT=${INT}/input
+HELP=${MNT}/${SYS}/doc/spadhelp
@
\subsection{The depsys variable}
@@ -2013,6 +2017,156 @@ layer23done:
@ echo ==================================
@
+<<environment>>=
+SPADHELP=\
+ ${HELP}/AssociationList.help ${HELP}/BalancedBinaryTree.help \
+ ${HELP}/BasicOperator.help ${HELP}/BinaryExpansion.help \
+ ${HELP}/BinarySearchTree.help ${HELP}/CardinalNumber.help \
+ ${HELP}/CartesianTensor.help ${HELP}/Character.help \
+ ${HELP}/CharacterClass.help ${HELP}/CliffordAlgebra.help \
+ ${HELP}/Complex.help ${HELP}/ContinuedFraction.help \
+ ${HELP}/CycleIndicators.help ${HELP}/DeRhamComplex.help \
+ ${HELP}/DecimalExpansion.help
+
+@
+The algebra files contain input chunks in regress format.
+This stanza is extracted by the src/input/Makefile after
+all of the other regression tests are complete. This stanza
+is put into a int/Makefile.algebra and then executed by make.
+<<algebra.regress>>=
+TESTSYS= ${OBJ}/${SYS}/bin/interpsys
+
+REGRESS=\
+ AssociationList.regress BalancedBinaryTree.regress \
+ BasicOperator.regress BinaryExpansion.regress \
+ BinarySearchTree.regress CardinalNumber.regress \
+ CartesianTensor.regress Character.regress \
+ CharacterClass.regress CliffordAlgebra.regress \
+ Complex.regress ContinuedFraction.regress \
+ CycleIndicators.regress DeRhamComplex.regress \
+ DecimalExpansion.regress
+
+%.regress: %.input
+ @ echo algebra regression testing $*
+ @ rm -f $*.output
+ @ echo ')read $*.input' | ${TESTSYS}
+ @ echo ')lisp (regress "$*.output")' | ${TESTSYS} \
+ | egrep -v '(Timestamp|Version)' | tee $*.regress
+
+all: ${REGRESS}
+ @echo algebra test cases complete.
+@
+<<spadhelp>>=
+${HELP}/AssociationList.help: ${IN}/list.spad.pamphlet
+ @echo 7000 create AssociationList.help from ${IN}/list.spad.pamphlet
+ @${TANGLE} -R"AssociationList.help" ${IN}/list.spad.pamphlet \
+ >${HELP}/AssociationList.help
+ @${TANGLE} -R"AssociationList.input" ${IN}/list.spad.pamphlet \
+ >${INPUT}/AssociationList.input
+
+${HELP}/BalancedBinaryTree.help: ${IN}/tree.spad.pamphlet
+ @echo 7001 create BalancedBinaryTree.help from ${IN}/tree.spad.pamphlet
+ @${TANGLE} -R"BalancedBinaryTree.help" ${IN}/tree.spad.pamphlet \
+ >${HELP}/BalancedBinaryTree.help
+ @${TANGLE} -R"BalancedBinaryTree.input" ${IN}/tree.spad.pamphlet \
+ >${INPUT}/BalancedBinaryTree.input
+
+${HELP}/BasicOperator.help: ${IN}/op.spad.pamphlet
+ @echo 7002 create BasicOperator.help from ${IN}/op.spad.pamphlet
+ @${TANGLE} -R"BasicOperator.help" ${IN}/op.spad.pamphlet \
+ >${HELP}/BasicOperator.help
+ @${TANGLE} -R"BasicOperator.input" ${IN}/op.spad.pamphlet \
+ >${INPUT}/BasicOperator.input
+
+${HELP}/BinaryExpansion.help: ${IN}/radix.spad.pamphlet
+ @echo 7003 create BinaryExpansion.help from ${IN}/radix.spad.pamphlet
+ @${TANGLE} -R"BinaryExpansion.help" ${IN}/radix.spad.pamphlet \
+ >${HELP}/BinaryExpansion.help
+ @${TANGLE} -R"BinaryExpansion.input" ${IN}/radix.spad.pamphlet \
+ >${INPUT}/BinaryExpansion.input
+
+${HELP}/BinarySearchTree.help: ${IN}/tree.spad.pamphlet
+ @echo 7004 create BinarySearchTree.help from ${IN}/tree.spad.pamphlet
+ @${TANGLE} -R"BinarySearchTree.help" ${IN}/tree.spad.pamphlet \
+ >${HELP}/BinarySearchTree.help
+ @${TANGLE} -R"BinarySearchTree.input" ${IN}/tree.spad.pamphlet \
+ >${INPUT}/BinarySearchTree.input
+
+${HELP}/CardinalNumber.help: ${IN}/card.spad.pamphlet
+ @echo 7005 create CardinalNumber.help from ${IN}/card.spad.pamphlet
+ @${TANGLE} -R"CardinalNumber.help" ${IN}/card.spad.pamphlet \
+ >${HELP}/CardinalNumber.help
+ @${TANGLE} -R"CardinalNumber.input" ${IN}/card.spad.pamphlet \
+ >${INPUT}/CardinalNumber.input
+
+${HELP}/CartesianTensor.help: ${IN}/carten.spad.pamphlet
+ @echo 7006 create CartesianTensor.help from ${IN}/carten.spad.pamphlet
+ @${TANGLE} -R"CartesianTensor.help" ${IN}/carten.spad.pamphlet \
+ >${HELP}/CartesianTensor.help
+ @${TANGLE} -R"CartesianTensor.input" ${IN}/carten.spad.pamphlet \
+ >${INPUT}/CartesianTensor.input
+
+${HELP}/Character.help: ${IN}/string.spad.pamphlet
+ @echo 7007 create Character.help from ${IN}/string.spad.pamphlet
+ @${TANGLE} -R"Character.help" ${IN}/string.spad.pamphlet \
+ >${HELP}/Character.help
+ @${TANGLE} -R"Character.input" ${IN}/string.spad.pamphlet \
+ >${INPUT}/Character.input
+
+${HELP}/CharacterClass.help: ${IN}/string.spad.pamphlet
+ @echo 7008 create CharacterClass.help from ${IN}/string.spad.pamphlet
+ @${TANGLE} -R"CharacterClass.help" ${IN}/string.spad.pamphlet \
+ >${HELP}/CharacterClass.help
+ @${TANGLE} -R"CharacterClass.input" ${IN}/string.spad.pamphlet \
+ >${INPUT}/CharacterClass.input
+
+${HELP}/CliffordAlgebra.help: ${IN}/clifford.spad.pamphlet
+ @echo 7009 create CliffordAlgebra.help from \
+ ${IN}/clifford.spad.pamphlet
+ @${TANGLE} -R"CliffordAlgebra.help" ${IN}/clifford.spad.pamphlet \
+ >${HELP}/CliffordAlgebra.help
+ @${TANGLE} -R"CliffordAlgebra.input" ${IN}/clifford.spad.pamphlet \
+ >${INPUT}/CliffordAlgebra.input
+
+${HELP}/Complex.help: ${IN}/gaussian.spad.pamphlet
+ @echo 7010 create Complex.help from ${IN}/gaussian.spad.pamphlet
+ @${TANGLE} -R"Complex.help" ${IN}/gaussian.spad.pamphlet \
+ >${HELP}/Complex.help
+ @${TANGLE} -R"Complex.input" ${IN}/gaussian.spad.pamphlet \
+ >${INPUT}/Complex.input
+
+${HELP}/ContinuedFraction.help: ${IN}/contfrac.spad.pamphlet
+ @echo 7011 create ContinuedFraction.help from \
+ ${IN}/contfrac.spad.pamphlet
+ @${TANGLE} -R"ContinuedFraction.help" ${IN}/contfrac.spad.pamphlet \
+ >${HELP}/ContinuedFraction.help
+ @${TANGLE} -R"ContinuedFraction.input" ${IN}/contfrac.spad.pamphlet \
+ >${INPUT}/ContinuedFraction.input
+
+${HELP}/CycleIndicators.help: ${IN}/cycles.spad.pamphlet
+ @echo 7012 create CycleIndicators.help from \
+ ${IN}/cycles.spad.pamphlet
+ @${TANGLE} -R"CycleIndicators.help" ${IN}/cycles.spad.pamphlet \
+ >${HELP}/CycleIndicators.help
+ @${TANGLE} -R"CycleIndicators.input" ${IN}/cycles.spad.pamphlet \
+ >${INPUT}/CycleIndicators.input
+
+${HELP}/DeRhamComplex.help: ${IN}/derham.spad.pamphlet
+ @echo 7013 create DeRhamComplex.help from ${IN}/derham.spad.pamphlet
+ @${TANGLE} -R"DeRhamComplex.help" ${IN}/derham.spad.pamphlet \
+ >${HELP}/DeRhamComplex.help
+ @${TANGLE} -R"DeRhamComplex.input" ${IN}/derham.spad.pamphlet \
+ >${INPUT}/DeRhamComplex.input
+
+${HELP}/DecimalExpansion.help: ${IN}/radix.spad.pamphlet
+ @echo 7014 create DecimalExpansion.help from ${IN}/radix.spad.pamphlet
+ @${TANGLE} -R"DecimalExpansion.help" ${IN}/radix.spad.pamphlet \
+ >${HELP}/DecimalExpansion.help
+ @${TANGLE} -R"DecimalExpansion.input" ${IN}/radix.spad.pamphlet \
+ >${INPUT}/DecimalExpansion.input
+
+@
+
\section{The Makefile}
<<*>>=
@@ -2048,7 +2202,8 @@ layer23done:
<<USERLAYER>>
<<order>>
-all: src ${OUT}/libdb.text ${DOCFILES} ${TESTS} ${SPADBIN}/index.html
+all: src ${OUT}/libdb.text ${DOCFILES} ${TESTS} ${SPADBIN}/index.html \
+ ${SPADHELP}
@ echo 4302 finished ${IN}
${SPADBIN}/index.html:
@@ -2086,6 +2241,7 @@ document: ${DOCFILES}
<<genericRules>>
+<<spadhelp>>
<<testrules>>
<<ps (DOC from SRC)>>
<<libdb.text (OUT from IN)>>
diff --git a/src/algebra/card.spad.pamphlet b/src/algebra/card.spad.pamphlet
index a158508..4d02a4d 100644
--- a/src/algebra/card.spad.pamphlet
+++ b/src/algebra/card.spad.pamphlet
@@ -10,6 +10,324 @@
\tableofcontents
\eject
\section{domain CARD CardinalNumber}
+<<CardinalNumber.input>>=
+-- card.spad.pamphlet CardinalNumber.input
+)spool CardinalNumber.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 20
+c0 := 0 :: CardinalNumber
+--R
+--R
+--R (1) 0
+--R Type:
CardinalNumber
+--E 1
+
+--S 2 of 20
+c1 := 1 :: CardinalNumber
+--R
+--R
+--R (2) 1
+--R Type:
CardinalNumber
+--E 2
+
+--S 3 of 20
+c2 := 2 :: CardinalNumber
+--R
+--R
+--R (3) 2
+--R Type:
CardinalNumber
+--E 3
+
+--S 4 of 20
+c3 := 3 :: CardinalNumber
+--R
+--R
+--R (4) 3
+--R Type:
CardinalNumber
+--E 4
+
+--S 5 of 20
+A0 := Aleph 0
+--R
+--R
+--R (5) Aleph(0)
+--R Type:
CardinalNumber
+--E 5
+
+--S 6 of 20
+A1 := Aleph 1
+--R
+--R
+--R (6) Aleph(1)
+--R Type:
CardinalNumber
+--E 6
+
+--S 7 of 20
+finite? c2
+--R
+--R
+--R (7) true
+--R Type:
Boolean
+--E 7
+
+--S 8 of 20
+finite? A0
+--R
+--R
+--R (8) false
+--R Type:
Boolean
+--E 8
+
+--S 9 of 20
+countable? c2
+--R
+--R
+--R (9) true
+--R Type:
Boolean
+--E 9
+
+--S 10 of 20
+countable? A0
+--R
+--R
+--R (10) true
+--R Type:
Boolean
+--E 10
+
+--S 11 of 20
+countable? A1
+--R
+--R
+--R (11) false
+--R Type:
Boolean
+--E 11
+
+--S 12 of 20
+[c2 + c2, c2 + A1]
+--R
+--R
+--R (12) [4,Aleph(1)]
+--R Type: List
CardinalNumber
+--E 12
+
+--S 13 of 20
+[c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1]
+--R
+--R
+--R (13) [0,2,4,0,Aleph(1),Aleph(1),Aleph(1)]
+--R Type: List
CardinalNumber
+--E 13
+
+--S 14 of 20
+[c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2]
+--R
+--R
+--R (14) [1,2,4,1,Aleph(1),Aleph(1)]
+--R Type: List
CardinalNumber
+--E 14
+
+--S 15 of 20
+[c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1]
+--R
+--R
+--R (15) [1,0,"failed",Aleph(1),Aleph(1),"failed"]
+--R Type: List
Union(CardinalNumber,"failed")
+--E 15
+
+--S 16 of 20
+generalizedContinuumHypothesisAssumed true
+--R
+--R
+--R (16) true
+--R Type:
Boolean
+--E 16
+
+--S 17 of 20
+[c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1]
+--R
+--R
+--R (17) [0,1,Aleph(1),Aleph(1),Aleph(2),Aleph(1),Aleph(2)]
+--R Type: List
CardinalNumber
+--E 17
+
+--S 18 of 20
+a := Aleph 0
+--R
+--R
+--R (18) Aleph(0)
+--R Type:
CardinalNumber
+--E 18
+
+--S 19 of 20
+c := 2**a
+--R
+--R
+--R (19) Aleph(1)
+--R Type:
CardinalNumber
+--E 19
+
+--S 20 of 20
+f := 2**c
+--R
+--R
+--R (20) Aleph(2)
+--R Type:
CardinalNumber
+--E 20
+)spool
+)lisp (bye)
+@
+<<CardinalNumber.help>>=
+====================================================================
+CardinalNumber examples
+====================================================================
+
+The CardinalNumber domain can be used for values indicating the
+cardinality of sets, both finite and infinite. For example, the
+dimension operation in the category VectorSpace returns a cardinal
+number.
+
+The non-negative integers have a natural construction as cardinals
+
+ 0 = #{ }, 1 = {0}, 2 = {0, 1}, ..., n = {i | 0 <= i < n}.
+
+The fact that 0 acts as a zero for the multiplication of cardinals is
+equivalent to the axiom of choice.
+
+Cardinal numbers can be created by conversion from non-negative integers.
+
+ c0 := 0 :: CardinalNumber
+ 0
+ Type: CardinalNumber
+
+ c1 := 1 :: CardinalNumber
+ 1
+ Type: CardinalNumber
+
+ c2 := 2 :: CardinalNumber
+ 2
+ Type: CardinalNumber
+
+ c3 := 3 :: CardinalNumber
+ 3
+ Type: CardinalNumber
+
+They can also be obtained as the named cardinal Aleph(n).
+
+ A0 := Aleph 0
+ Aleph(0)
+ Type: CardinalNumber
+
+ A1 := Aleph 1
+ Aleph(1)
+ Type: CardinalNumber
+
+The finite? operation tests whether a value is a finite cardinal, that
+is, a non-negative integer.
+
+ finite? c2
+ true
+ Type: Boolean
+
+ finite? A0
+ false
+ Type: Boolean
+
+Similarly, the countable? operation determines whether a value is a
+countable cardinal, that is, finite or Aleph(0).
+
+ countable? c2
+ true
+ Type: Boolean
+
+ countable? A0
+ true
+ Type: Boolean
+
+ countable? A1
+ false
+ Type: Boolean
+
+Arithmetic operations are defined on cardinal numbers as follows:
+If x = #X and y = #Y then
+
+ x+y = #(X+Y) cardinality of the disjoint union
+ x-y = #(X-Y) cardinality of the relative complement
+ x*y = #(X*Y) cardinality of the Cartesian product
+ x**y = #(X**Y) cardinality of the set of maps from Y to X
+
+Here are some arithmetic examples.
+
+ [c2 + c2, c2 + A1]
+ [4, Aleph(1)]
+ Type: List CardinalNumber
+
+ [c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1]
+ [0, 2, 4, 0, Aleph(1), Aleph(1), Aleph(1)]
+ Type: List CardinalNumber
+
+ [c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2]
+ [1, 2, 4, 1, Aleph(1), Aleph(1)]
+ Type: List CardinalNumber
+
+Subtraction is a partial operation: it is not defined when subtracting
+a larger cardinal from a smaller one, nor when subtracting two equal
+infinite cardinals.
+
+ [c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1]
+ [1, 0, "failed", Aleph(1), Aleph(1), "failed"]
+ Type: List Union(CardinalNumber,"failed")
+
+The generalized continuum hypothesis asserts that
+
+ 2**Aleph i = Aleph(i+1)
+
+and is independent of the axioms of set theory.
+
+(reference: Goedel, The consistency of the continuum hypothesis,
+Ann. Math. Studies, Princeton Univ. Press, 1940.)
+
+The CardinalNumber domain provides an operation to assert whether the
+hypothesis is to be assumed.
+
+ generalizedContinuumHypothesisAssumed true
+ true
+ Type: Boolean
+
+When the generalized continuum hypothesis is assumed, exponentiation
+to a transfinite power is allowed.
+
+ [c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1]
+ [0, 1, Aleph(1), Aleph(1), Aleph(2), Aleph(1), Aleph(2)]
+ Type: List CardinalNumber
+
+Three commonly encountered cardinal numbers are
+
+ a = #Z countable infinity
+ c = #R the continuum
+ f = #{g| g: [0,1] -> R}
+
+In this domain, these values are obtained under the generalized
+continuum hypothesis in this way.
+
+ a := Aleph 0
+ Aleph(0)
+ Type: CardinalNumber
+
+ c := 2**a
+ Aleph(1)
+ Type: CardinalNumber
+
+ f := 2**c
+ Aleph(2)
+ Type: CardinalNumber
+
+See Also:
+o )show CardinalNumber
+o $AXIOM/doc/src/algebra/card.spad.dvi
+
+@
<<domain CARD CardinalNumber>>=
)abbrev domain CARD CardinalNumber
++ Author: S.M. Watt
diff --git a/src/algebra/carten.spad.pamphlet b/src/algebra/carten.spad.pamphlet
index 0f1d295..5bcaa75 100644
--- a/src/algebra/carten.spad.pamphlet
+++ b/src/algebra/carten.spad.pamphlet
@@ -102,6 +102,954 @@ GradedAlgebra(R: CommutativeRing, E: AbelianMonoid):
Category ==
@
\section{domain CARTEN CartesianTensor}
+<<CartesianTensor.input>>=
+-- carten.spad.pamphlet CartesianTensor.input
+)spool CartesianTensor.output
+)set message test on
+)set message auto off
+)clear all
+--S 1
+CT := CARTEN(i0 := 1, 2, Integer)
+--R
+--R
+--R (1) CartesianTensor(1,2,Integer)
+--R Type:
Domain
+--E 1
+
+--S 2
+t0: CT := 8
+--R
+--R
+--R (2) 8
+--R Type:
CartesianTensor(1,2,Integer)
+--E 2
+
+--S 3
+rank t0
+--R
+--R
+--R (3) 0
+--R Type:
NonNegativeInteger
+--E 3
+
+--S 4
+v: DirectProduct(2, Integer) := directProduct [3,4]
+--R
+--R
+--R (4) [3,4]
+--R Type:
DirectProduct(2,Integer)
+--E 4
+
+--S 5
+Tv: CT := v
+--R
+--R
+--R (5) [3,4]
+--R Type:
CartesianTensor(1,2,Integer)
+--E 5
+
+--S 6
+m: SquareMatrix(2, Integer) := matrix [ [1,2],[4,5] ]
+--R
+--R
+--R +1 2+
+--R (6) | |
+--R +4 5+
+--R Type:
SquareMatrix(2,Integer)
+--E 6
+
+--S 7
+Tm: CT := m
+--R
+--R
+--R +1 2+
+--R (7) | |
+--R +4 5+
+--R Type:
CartesianTensor(1,2,Integer)
+--E 7
+
+--S 8
+n: SquareMatrix(2, Integer) := matrix [ [2,3],[0,1] ]
+--R
+--R
+--R +2 3+
+--R (8) | |
+--R +0 1+
+--R Type:
SquareMatrix(2,Integer)
+--E 8
+
+--S 9
+Tn: CT := n
+--R
+--R
+--R +2 3+
+--R (9) | |
+--R +0 1+
+--R Type:
CartesianTensor(1,2,Integer)
+--E 9
+
+--S 10
+t1: CT := [2, 3]
+--R
+--R
+--R (10) [2,3]
+--R Type:
CartesianTensor(1,2,Integer)
+--E 10
+
+--S 11
+rank t1
+--R
+--R
+--R (11) 1
+--R Type:
PositiveInteger
+--E 11
+
+--S 12
+t2: CT := [t1, t1]
+--R
+--R
+--R +2 3+
+--R (12) | |
+--R +2 3+
+--R Type:
CartesianTensor(1,2,Integer)
+--E 12
+
+--S 13
+t3: CT := [t2, t2]
+--R
+--R
+--R +2 3+ +2 3+
+--R (13) [| |,| |]
+--R +2 3+ +2 3+
+--R Type:
CartesianTensor(1,2,Integer)
+--E 13
+
+--S 14
+tt: CT := [t3, t3]; tt := [tt, tt]
+--R
+--R
+--R ++2 3+ +2 3++ ++2 3+ +2 3++
+--R || | | || || | | ||
+--R |+2 3+ +2 3+| |+2 3+ +2 3+|
+--R (14) [| |,| |]
+--R |+2 3+ +2 3+| |+2 3+ +2 3+|
+--R || | | || || | | ||
+--R ++2 3+ +2 3++ ++2 3+ +2 3++
+--R Type:
CartesianTensor(1,2,Integer)
+--E 14
+
+--S 15
+rank tt
+--R
+--R
+--R (15) 5
+--R Type:
PositiveInteger
+--E 15
+
+--S 16
+Tmn := product(Tm, Tn)
+--R
+--R
+--R ++2 3+ +4 6+ +
+--R || | | | |
+--R |+0 1+ +0 2+ |
+--R (16) | |
+--R |+8 12+ +10 15+|
+--R || | | ||
+--R ++0 4 + +0 5 ++
+--R Type:
CartesianTensor(1,2,Integer)
+--E 16
+
+--S 17
+Tmv := contract(Tm,2,Tv,1)
+--R
+--R
+--R (17) [11,32]
+--R Type:
CartesianTensor(1,2,Integer)
+--E 17
+
+--S 18
+Tm*Tv
+--R
+--R
+--R (18) [11,32]
+--R Type:
CartesianTensor(1,2,Integer)
+--E 18
+
+--S 19
+Tmv = m * v
+--R
+--R
+--R (19) [11,32]= [11,32]
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 19
+
+--S 20
+t0()
+--R
+--R
+--R (20) 8
+--R Type:
PositiveInteger
+--E 20
+
+--S 21
+t1(1+1)
+--R
+--R
+--R (21) 3
+--R Type:
PositiveInteger
+--E 21
+
+--S 22
+t2(2,1)
+--R
+--R
+--R (22) 2
+--R Type:
PositiveInteger
+--E 22
+
+--S 23
+t3(2,1,2)
+--R
+--R
+--R (23) 3
+--R Type:
PositiveInteger
+--E 23
+
+--S 24
+Tmn(2,1,2,1)
+--R
+--R
+--R (24) 0
+--R Type:
NonNegativeInteger
+--E 24
+
+--S 25
+t0[]
+--R
+--R
+--R (25) 8
+--R Type:
PositiveInteger
+--E 25
+
+--S 26
+t1[2]
+--R
+--R
+--R (26) 3
+--R Type:
PositiveInteger
+--E 26
+
+--S 27
+t2[2,1]
+--R
+--R
+--R (27) 2
+--R Type:
PositiveInteger
+--E 27
+
+--S 28
+t3[2,1,2]
+--R
+--R
+--R (28) 3
+--R Type:
PositiveInteger
+--E 28
+
+--S 29
+Tmn[2,1,2,1]
+--R
+--R
+--R (29) 0
+--R Type:
NonNegativeInteger
+--E 29
+
+--S 30
+cTmn := contract(Tmn,1,2)
+--R
+--R
+--R +12 18+
+--R (30) | |
+--R +0 6 +
+--R Type:
CartesianTensor(1,2,Integer)
+--E 30
+
+--S 31
+trace(m) * n
+--R
+--R
+--R +12 18+
+--R (31) | |
+--R +0 6 +
+--R Type:
SquareMatrix(2,Integer)
+--E 31
+
+--S 32
+contract(Tmn,1,2) = trace(m) * n
+--R
+--R
+--R +12 18+ +12 18+
+--R (32) | |= | |
+--R +0 6 + +0 6 +
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 32
+
+--S 33
+contract(Tmn,1,3) = transpose(m) * n
+--R
+--R
+--R +2 7 + +2 7 +
+--R (33) | |= | |
+--R +4 11+ +4 11+
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 33
+
+--S 34
+contract(Tmn,1,4) = transpose(m) * transpose(n)
+--R
+--R
+--R +14 4+ +14 4+
+--R (34) | |= | |
+--R +19 5+ +19 5+
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 34
+
+--S 35
+contract(Tmn,2,3) = m * n
+--R
+--R
+--R +2 5 + +2 5 +
+--R (35) | |= | |
+--R +8 17+ +8 17+
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 35
+
+--S 36
+contract(Tmn,2,4) = m * transpose(n)
+--R
+--R
+--R +8 2+ +8 2+
+--R (36) | |= | |
+--R +23 5+ +23 5+
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 36
+
+--S 37
+contract(Tmn,3,4) = trace(n) * m
+--R
+--R
+--R +3 6 + +3 6 +
+--R (37) | |= | |
+--R +12 15+ +12 15+
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 37
+
+--S 38
+tTmn := transpose(Tmn,1,3)
+--R
+--R
+--R ++2 3 + +4 6 ++
+--R || | | ||
+--R |+8 12+ +10 15+|
+--R (38) | |
+--R |+0 1+ +0 2+ |
+--R || | | | |
+--R ++0 4+ +0 5+ +
+--R Type:
CartesianTensor(1,2,Integer)
+--E 38
+
+--S 39
+transpose Tmn
+--R
+--R
+--R ++2 8+ +4 10++
+--R || | | ||
+--R |+0 0+ +0 0 +|
+--R (39) | |
+--R |+3 12+ +6 15+|
+--R || | | ||
+--R ++1 4 + +2 5 ++
+--R Type:
CartesianTensor(1,2,Integer)
+--E 39
+
+--S 40
+transpose Tm = transpose m
+--R
+--R
+--R +1 4+ +1 4+
+--R (40) | |= | |
+--R +2 5+ +2 5+
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 40
+
+--S 41
+rTmn := reindex(Tmn, [1,4,2,3])
+--R
+--R
+--R ++2 0+ +3 1+ +
+--R || | | | |
+--R |+4 0+ +6 2+ |
+--R (41) | |
+--R |+8 0+ +12 4+|
+--R || | | ||
+--R ++10 0+ +15 5++
+--R Type:
CartesianTensor(1,2,Integer)
+--E 41
+
+--S 42
+tt := transpose(Tm)*Tn - Tn*transpose(Tm)
+--R
+--R
+--R +- 6 - 16+
+--R (42) | |
+--R + 2 6 +
+--R Type:
CartesianTensor(1,2,Integer)
+--E 42
+
+--S 43
+Tv*(tt+Tn)
+--R
+--R
+--R (43) [- 4,- 11]
+--R Type:
CartesianTensor(1,2,Integer)
+--E 43
+
+--S 44
+reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)
+--R
+--R
+--R ++46 84 + +57 114++
+--R || | | ||
+--R |+174 212+ +228 285+|
+--R (44) | |
+--R | +18 24+ +17 30+ |
+--R | | | | | |
+--R + +57 63+ +63 76+ +
+--R Type:
CartesianTensor(1,2,Integer)
+--E 44
+
+--S 45
+delta: CT := kroneckerDelta()
+--R
+--R
+--R +1 0+
+--R (45) | |
+--R +0 1+
+--R Type:
CartesianTensor(1,2,Integer)
+--E 45
+
+--S 46
+contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])
+--R
+--R
+--R + +2 4+ +0 0++ + +2 4+ +0 0++
+--R | | | | || | | | | ||
+--R | +3 6+ +1 2+| | +3 6+ +1 2+|
+--R (46) | |= | |
+--R |+8 10+ +0 0+| |+8 10+ +0 0+|
+--R || | | || || | | ||
+--R ++12 15+ +4 5++ ++12 15+ +4 5++
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 46
+
+--S 47
+epsilon:CT := leviCivitaSymbol()
+--R
+--R
+--R + 0 1+
+--R (47) | |
+--R +- 1 0+
+--R Type:
CartesianTensor(1,2,Integer)
+--E 47
+
+--S 48
+contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m
+--R
+--R
+--R (48) - 6= - 6
+--R Type: Equation
CartesianTensor(1,2,Integer)
+--E 48
+)spool
+)lisp (bye)
+@
+<<CartesianTensor.help>>=
+====================================================================
+CartesianTensor
+====================================================================
+
+CartesianTensor(i0,dim,R) provides Cartesian tensors with components
+belonging to a commutative ring R. Tensors can be described as a
+generalization of vectors and matrices. This gives a concise tensor
+algebra for multilinear objects supported by the CartesianTensor
+domain. You can form the inner or outer product of any two tensors
+and you can add or subtract tensors with the same number of components.
+Additionally, various forms of traces and transpositions are useful.
+
+The CartesianTensor constructor allows you to specify the minimum
+index for subscripting. In what follows we discuss in detail how to
+manipulate tensors.
+
+Here we construct the domain of Cartesian tensors of dimension 2 over the
+integers, with indices starting at 1.
+
+ CT := CARTEN(i0 := 1, 2, Integer)
+ CartesianTensor(1,2,Integer)
+ Type: Domain
+
+====================================================================
+Forming tensors
+====================================================================
+
+Scalars can be converted to tensors of rank zero.
+
+ t0: CT := 8
+ 8
+ Type: CartesianTensor(1,2,Integer)
+
+ rank t0
+ 0
+ Type: NonNegativeInteger
+
+Vectors (mathematical direct products, rather than one dimensional array
+structures) can be converted to tensors of rank one.
+
+ v: DirectProduct(2, Integer) := directProduct [3,4]
+ [3, 4]
+ Type: DirectProduct(2,Integer)
+
+ Tv: CT := v
+ [3, 4]
+ Type: CartesianTensor(1,2,Integer)
+
+Matrices can be converted to tensors of rank two.
+
+ m: SquareMatrix(2, Integer) := matrix [ [1,2],[4,5] ]
+ +1 2+
+ | |
+ +4 5+
+ Type: SquareMatrix(2,Integer)
+
+ Tm: CT := m
+ +1 2+
+ | |
+ +4 5+
+ Type: CartesianTensor(1,2,Integer)
+
+ n: SquareMatrix(2, Integer) := matrix [ [2,3],[0,1] ]
+ +2 3+
+ | |
+ +0 1+
+ Type: SquareMatrix(2,Integer)
+
+ Tn: CT := n
+ +2 3+
+ | |
+ +0 1+
+ Type: CartesianTensor(1,2,Integer)
+
+In general, a tensor of rank k can be formed by making a list of
+rank k-1 tensors or, alternatively, a k-deep nested list of lists.
+
+ t1: CT := [2, 3]
+ [2, 3]
+ Type: CartesianTensor(1,2,Integer)
+
+ rank t1
+ 1
+ Type: PositiveInteger
+
+ t2: CT := [t1, t1]
+ +2 3+
+ | |
+ +2 3+
+ Type: CartesianTensor(1,2,Integer)
+
+ t3: CT := [t2, t2]
+
+ +2 3+ +2 3+
+ [| |,| |]
+ +2 3+ +2 3+
+ Type: CartesianTensor(1,2,Integer)
+
+ tt: CT := [t3, t3]; tt := [tt, tt]
+ ++2 3+ +2 3++ ++2 3+ +2 3++
+ || | | || || | | ||
+ |+2 3+ +2 3+| |+2 3+ +2 3+|
+ [| |,| |]
+ |+2 3+ +2 3+| |+2 3+ +2 3+|
+ || | | || || | | ||
+ ++2 3+ +2 3++ ++2 3+ +2 3++
+ Type: CartesianTensor(1,2,Integer)
+
+ rank tt
+ 5
+ Type: PositiveInteger
+
+====================================================================
+Multiplication
+====================================================================
+
+Given two tensors of rank k1 and k2, the outer product forms a new
+tensor of rank k1+k2. Here
+
+ Tmn(i,j,k,l) = Tm(i,j)Tn(k,l)
+
+ Tmn := product(Tm, Tn)
+ ++2 3+ +4 6+ +
+ || | | | |
+ |+0 1+ +0 2+ |
+ | |
+ |+8 12+ +10 15+|
+ || | | ||
+ ++0 4 + +0 5 ++
+ Type: CartesianTensor(1,2,Integer)
+
+The inner product (contract) forms a tensor of rank k1+k2-2. This
+product generalizes the vector dot product and matrix-vector product
+by summing component products along two indices.
+
+Here we sum along the second index of Tm and the first index of Tv. Here
+
+ Tmv = sum {j=1..dim} Tm(i,j) Tv(j)
+
+ Tmv := contract(Tm,2,Tv,1)
+ [11,32]
+ Type: CartesianTensor(1,2,Integer)
+
+The multiplication operator * is scalar multiplication or an inner
+product depending on the ranks of the arguments.
+
+If either argument is rank zero it is treated as scalar multiplication.
+Otherwise, a*b is the inner product summing the last index of a with the
+first index of b.
+
+ Tm*Tv
+ [11,32]
+ Type: CartesianTensor(1,2,Integer)
+
+This definition is consistent with the inner product on matrices
+and vectors.
+
+ Tmv = m * v
+ [11,32] = [11,32]
+ Type: Equation CartesianTensor(1,2,Integer)
+
+====================================================================
+Selecting Components
+====================================================================
+
+For tensors of low rank (that is, four or less), components can be selected
+by applying the tensor to its indices.
+
+ t0()
+ 8
+ Type: PositiveInteger
+
+ t1(1+1)
+ 3
+ Type: PositiveInteger
+
+ t2(2,1)
+ 2
+ Type: PositiveInteger
+
+ t3(2,1,2)
+ 3
+ Type: PositiveInteger
+
+ Tmn(2,1,2,1)
+ 0
+ Type: NonNegativeInteger
+
+A general indexing mechanism is provided for a list of indices.
+
+ t0[]
+ 8
+ Type: PositiveInteger
+
+ t1[2]
+ 3
+ Type: PositiveInteger
+
+ t2[2,1]
+ 2
+ Type: PositiveInteger
+
+The general mechanism works for tensors of arbitrary rank, but is
+somewhat less efficient since the intermediate index list must be created.
+
+ t3[2,1,2]
+ 3
+ Type: PositiveInteger
+
+ Tmn[2,1,2,1]
+ 0
+ Type: NonNegativeInteger
+
+====================================================================
+Contraction
+====================================================================
+
+A "contraction" between two tensors is an inner product, as we have
+seen above. You can also contract a pair of indices of a single
+tensor. This corresponds to a "trace" in linear algebra. The
+expression contract(t,k1,k2) forms a new tensor by summing the
+diagonal given by indices in position k1 and k2.
+
+This is the tensor given by
+ xTmn = sum{k=1..dim} Tmn(k,k,i,j)
+
+ cTmn := contract(Tmn,1,2)
+ +12 18+
+ | |
+ +0 6 +
+ Type: CartesianTensor(1,2,Integer)
+
+Since Tmn is the outer product of matrix m and matrix n, the above is
+equivalent to this.
+
+ trace(m) * n
+ +12 18+
+ | |
+ +0 6 +
+ Type: SquareMatrix(2,Integer)
+
+In this and the next few examples, we show all possible contractions
+of Tmn and their matrix algebra equivalents.
+
+ contract(Tmn,1,2) = trace(m) * n
+ +12 18+ +12 18+
+ | |= | |
+ +0 6 + +0 6 +
+ Type: Equation CartesianTensor(1,2,Integer)
+
+ contract(Tmn,1,3) = transpose(m) * n
+ +2 7 + +2 7 +
+ | |= | |
+ +4 11+ +4 11+
+ Type: Equation CartesianTensor(1,2,Integer)
+
+ contract(Tmn,1,4) = transpose(m) * transpose(n)
+ +14 4+ +14 4+
+ | |= | |
+ +19 5+ +19 5+
+ Type: Equation CartesianTensor(1,2,Integer)
+
+ contract(Tmn,2,3) = m * n
+ +2 5 + +2 5 +
+ | |= | |
+ +8 17+ +8 17+
+ Type: Equation CartesianTensor(1,2,Integer)
+
+ contract(Tmn,2,4) = m * transpose(n)
+ +8 2+ +8 2+
+ | |= | |
+ +23 5+ +23 5+
+ Type: Equation CartesianTensor(1,2,Integer)
+
+ contract(Tmn,3,4) = trace(n) * m
+ +3 6 + +3 6 +
+ | |= | |
+ +12 15+ +12 15+
+ Type: Equation CartesianTensor(1,2,Integer)
+
+====================================================================
+Transpositions
+====================================================================
+
+You can exchange any desired pair of indices using the transpose
+operation.
+
+Here the indices in positions one and three are exchanged, that is,
+ tTmn(i,j,k,l) = Tmn(k,j,i,l)
+
+ tTmn := transpose(Tmn,1,3)
+ ++2 3 + +4 6 ++
+ || | | ||
+ |+8 12+ +10 15+|
+ | |
+ |+0 1+ +0 2+ |
+ || | | | |
+ ++0 4+ +0 5+ +
+ Type: CartesianTensor(1,2,Integer)
+
+If no indices are specified, the first and last index are exchanged.
+
+ transpose Tmn
+ ++2 8+ +4 10++
+ || | | ||
+ |+0 0+ +0 0 +|
+ | |
+ |+3 12+ +6 15+|
+ || | | ||
+ ++1 4 + +2 5 ++
+ Type: CartesianTensor(1,2,Integer)
+
+This is consistent with the matrix transpose.
+
+ transpose Tm = transpose m
+ +1 4+ +1 4+
+ | |= | |
+ +2 5+ +2 5+
+ Type: Equation CartesianTensor(1,2,Integer)
+
+
+If a more complicated reordering of the indices is required, then the
+reindex operation can be used. This operation allows the indices to
+be arbitrarily permuted.
+
+ rTmn(i,j,k,l) = Tmn(i,l,j,k)
+
+ rTmn := reindex(Tmn, [1,4,2,3])
+ ++2 0+ +3 1+ +
+ || | | | |
+ |+4 0+ +6 2+ |
+ | |
+ |+8 0+ +12 4+|
+ || | | ||
+ ++10 0+ +15 5++
+ Type: CartesianTensor(1,2,Integer)
+
+====================================================================
+Arithmetic
+====================================================================
+
+Tensors of equal rank can be added or subtracted so arithmetic
+expressions can be used to produce new tensors.
+
+ tt := transpose(Tm)*Tn - Tn*transpose(Tm)
+ +- 6 - 16+
+ | |
+ + 2 6 +
+ Type: CartesianTensor(1,2,Integer)
+
+
+ Tv*(tt+Tn)
+ [- 4,- 11]
+ Type: CartesianTensor(1,2,Integer)
+
+ reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)
+ ++46 84 + +57 114++
+ || | | ||
+ |+174 212+ +228 285+|
+ | |
+ | +18 24+ +17 30+ |
+ | | | | | |
+ + +57 63+ +63 76+ +
+ Type: CartesianTensor(1,2,Integer)
+
+====================================================================
+Specific Tensors
+====================================================================
+
+Two specific tensors have properties which depend only on the dimension.
+
+The Kronecker delta satisfies
+
+ +- -+
+ | 1 if i = j |
+delta(i,j) = | |
+ | 0 if i ^= j |
+ +- -+
+
+
+ delta: CT := kroneckerDelta()
+ +1 0+
+ | |
+ +0 1+
+ Type: CartesianTensor(1,2,Integer)
+
+This can be used to reindex via contraction.
+
+ contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])
+ + +2 4+ +0 0++ + +2 4+ +0 0++
+ | | | | || | | | | ||
+ | +3 6+ +1 2+| | +3 6+ +1 2+|
+ | |= | |
+ |+8 10+ +0 0+| |+8 10+ +0 0+|
+ || | | || || | | ||
+ ++12 15+ +4 5++ ++12 15+ +4 5++
+ Type: Equation CartesianTensor(1,2,Integer)
+
+The Levi Civita symbol determines the sign of a permutation of indices.
+
+ epsilon:CT := leviCivitaSymbol()
+ + 0 1+
+ | |
+ +- 1 0+
+ Type: CartesianTensor(1,2,Integer)
+
+Here we have:
+
+ epsilon(i1,...,idim)
+ = +1 if i1,...,idim is an even permutation of i0,...,i0+dim-1
+ = -1 if i1,...,idim is an odd permutation of i0,...,i0+dim-1
+ = 0 if i1,...,idim is not a permutation of i0,...,i0+dim-1
+
+This property can be used to form determinants.
+
+ contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m
+ - 6= - 6
+ Type: Equation CartesianTensor(1,2,Integer)
+
+
+====================================================================
+Properties of the CartesianTensor domain
+====================================================================
+
+GradedModule(R,E) denotes "E-graded R-module", that is, a collection
+of R-modules indexed by an abelian monoid E. An element g of G[s] for
+some specific s in E is said to be an element of G with degree s.
+Sums are defined in each module G[s] so two elements of G can be added
+if they have the same degree. Morphisms can be defined and composed
+by degree to give the mathematical category of graded modules.
+
+GradedAlgebra(R,E) denotes "E-graded R-algebra". A graded algebra is
+a graded module together with a degree preserving R-bilinear map,
+called the product.
+
+ degree(product(a,b)) = degree(a) + degree(b)
+
+ product(r*a,b) = product(a,r*b) = r*product(a,b)
+ product(a1+a2,b) = product(a1,b) + product(a2,b)
+ product(a,b1+b2) = product(a,b1) + product(a,b2)
+ product(a,product(b,c)) = product(product(a,b),c)
+
+The domain CartesianTensor(i0, dim, R) belongs to the category
+GradedAlgebra(R, NonNegativeInteger). The non-negative integer degree
+is the tensor rank and the graded algebra product is the tensor outer
+product. The graded module addition captures the notion that only
+tensors of equal rank can be added.
+
+If V is a vector space of dimension dim over R, then the tensor module
+T[k](V) is defined as
+
+ T[0](V) = R
+ T[k](V) = T[k-1](V) * V
+
+where * denotes the R-module tensor product. CartesianTensor(i0,dim,R)
+is the graded algebra in which the degree k module is T[k](V).
+
+====================================================================
+Tensor Calculus
+====================================================================
+
+It should be noted here that often tensors are used in the context of
+tensor-valued manifold maps. This leads to the notion of covariant
+and contravariant bases with tensor component functions transforming
+in specific ways under a change of coordinates on the manifold. This
+is no more directly supported by the CartesianTensor domain than it is
+by the Vector domain. However, it is possible to have the components
+implicitly represent component maps by choosing a polynomial or
+expression type for the components. In this case, it is up to the
+user to satisfy any constraints which arise on the basis of this
+interpretation.
+
+See Also
+o )show CartesianTensor
+o $AXIOM/doc/src/algebra/carten.spad.dvi
+
+@
<<domain CARTEN CartesianTensor>>=
)abbrev domain CARTEN CartesianTensor
++ Author: Stephen M. Watt
diff --git a/src/algebra/clifford.spad.pamphlet
b/src/algebra/clifford.spad.pamphlet
index 88b7b3d..cd94dd5 100644
--- a/src/algebra/clifford.spad.pamphlet
+++ b/src/algebra/clifford.spad.pamphlet
@@ -236,6 +236,624 @@ $\mathbb{R}_{m,m}$ & --------$>$ &
$\mathbb{R}^{4^m}$ \\
$\mathbb{R}(2^m)$ & --------$>$ & $\mathbb{R}^{4^m}$\\
& reshape & \\
\end{tabular}
+<<CliffordAlgebra.input>>=
+-- clifford.spad.pamphlet CliffordAlgebra.input
+)spool CliffordAlgebra.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 36
+K := Fraction Polynomial Integer
+--R
+--R
+--R (1) Fraction Polynomial Integer
+--R Type:
Domain
+--E 1
+
+--S 2 of 36
+m := matrix [ [-1] ]
+--R
+--R
+--R (2) [- 1]
+--R Type: Matrix
Integer
+--E 2
+
+--S 3 of 36
+C := CliffordAlgebra(1, K, quadraticForm m)
+--R
+--R
+--R (3) CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+--R Type:
Domain
+--E 3
+
+--S 4 of 36
+i: C := e(1)
+--R
+--R
+--R (4) e
+--R 1
+--R Type: CliffordAlgebra(1,Fraction Polynomial
Integer,MATRIX)
+--E 4
+
+--S 5 of 36
+x := a + b * i
+--R
+--R
+--R (5) a + b e
+--R 1
+--R Type: CliffordAlgebra(1,Fraction Polynomial
Integer,MATRIX)
+--E 5
+
+--S 6 of 36
+y := c + d * i
+--R
+--R
+--R (6) c + d e
+--R 1
+--R Type: CliffordAlgebra(1,Fraction Polynomial
Integer,MATRIX)
+--E 6
+
+--S 7 of 36
+x * y
+--R
+--R
+--R (7) - b d + a c + (a d + b c)e
+--R 1
+--R Type: CliffordAlgebra(1,Fraction Polynomial
Integer,MATRIX)
+--E 7
+)clear all
+
+--S 8 of 36
+K := Fraction Polynomial Integer
+--R
+--R
+--R (1) Fraction Polynomial Integer
+--R Type:
Domain
+--E 8
+
+--S 9 of 36
+m := matrix [ [-1,0],[0,-1] ]
+--R
+--R
+--R +- 1 0 +
+--R (2) | |
+--R + 0 - 1+
+--R Type: Matrix
Integer
+--E 9
+
+--S 10 of 36
+H := CliffordAlgebra(2, K, quadraticForm m)
+--R
+--R
+--R (3) CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+--R Type:
Domain
+--E 10
+
+--S 11 of 36
+i: H := e(1)
+--R
+--R
+--R (4) e
+--R 1
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 11
+
+--S 12 of 36
+j: H := e(2)
+--R
+--R
+--R (5) e
+--R 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 12
+
+--S 13 of 36
+k: H := i * j
+--R
+--R
+--R (6) e e
+--R 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 13
+
+--S 14 of 36
+x := a + b * i + c * j + d * k
+--R
+--R
+--R (7) a + b e + c e + d e e
+--R 1 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 14
+
+--S 15 of 36
+y := e + f * i + g * j + h * k
+--R
+--R
+--R (8) e + f e + g e + h e e
+--R 1 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 15
+
+--S 16 of 36
+x + y
+--R
+--R
+--R (9) e + a + (f + b)e + (g + c)e + (h + d)e e
+--R 1 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 16
+
+--S 17 of 36
+x * y
+--R
+--R
+--R (10)
+--R - d h - c g - b f + a e + (c h - d g + a f + b e)e
+--R 1
+--R +
+--R (- b h + a g + d f + c e)e + (a h + b g - c f + d e)e e
+--R 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 17
+
+--S 18 of 36
+y * x
+--R
+--R
+--R (11)
+--R - d h - c g - b f + a e + (- c h + d g + a f + b e)e
+--R 1
+--R +
+--R (b h + a g - d f + c e)e + (a h - b g + c f + d e)e e
+--R 2 1 2
+--R Type: CliffordAlgebra(2,Fraction Polynomial
Integer,MATRIX)
+--E 18
+)clear all
+
+--S 19 of 36
+K := Fraction Polynomial Integer
+--R
+--R
+--R (1) Fraction Polynomial Integer
+--R Type:
Domain
+--E 19
+
+--S 20 of 36
+Ext := CliffordAlgebra(3, K, quadraticForm 0)
+--R
+--R
+--R (2) CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+--R Type:
Domain
+--E 20
+
+--S 21 of 36
+i: Ext := e(1)
+--R
+--R
+--R (3) e
+--R 1
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 21
+
+--S 22 of 36
+j: Ext := e(2)
+--R
+--R
+--R (4) e
+--R 2
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 22
+
+--S 23 of 36
+k: Ext := e(3)
+--R
+--R
+--R (5) e
+--R 3
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 23
+
+--S 24 of 36
+x := x1*i + x2*j + x3*k
+--R
+--R
+--R (6) x1 e + x2 e + x3 e
+--R 1 2 3
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 24
+
+--S 25 of 36
+y := y1*i + y2*j + y3*k
+--R
+--R
+--R (7) y1 e + y2 e + y3 e
+--R 1 2 3
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 25
+
+--S 26 of 36
+x + y
+--R
+--R
+--R (8) (y1 + x1)e + (y2 + x2)e + (y3 + x3)e
+--R 1 2 3
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 26
+
+--S 27 of 36
+x * y + y * x
+--R
+--R
+--R (9) 0
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 27
+
+--S 28 of 36
+dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j +
coefficient(a,[1,2]) * k
+--R
+--R Type:
Void
+--E 28
+
+--S 29 of 36
+dual2(x*y)
+--R
+--R Compiling function dual2 with type CliffordAlgebra(3,Fraction
+--R Polynomial Integer,MATRIX) -> CliffordAlgebra(3,Fraction
+--R Polynomial Integer,MATRIX)
+--R
+--R (11) (x2 y3 - x3 y2)e + (- x1 y3 + x3 y1)e + (x1 y2 - x2 y1)e
+--R 1 2 3
+--R Type: CliffordAlgebra(3,Fraction Polynomial
Integer,MATRIX)
+--E 29
+)clear all
+
+--S 30 of 36
+K := Fraction Integer
+--R
+--R
+--R (1) Fraction Integer
+--R Type:
Domain
+--E 30
+
+--S 31 of 36
+g := matrix [ [1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1] ]
+--R
+--R
+--R +1 0 0 0 +
+--R | |
+--R |0 - 1 0 0 |
+--R (2) | |
+--R |0 0 - 1 0 |
+--R | |
+--R +0 0 0 - 1+
+--R Type: Matrix
Integer
+--E 31
+
+--S 32 of 36
+D := CliffordAlgebra(4,K, quadraticForm g)
+--R
+--R
+--R (3) CliffordAlgebra(4,Fraction Integer,MATRIX)
+--R Type:
Domain
+--E 32
+
+--S 33 of 36
+gam := [e(i)$D for i in 1..4]
+--R
+--R
+--R (4) [e ,e ,e ,e ]
+--R 1 2 3 4
+--R Type: List CliffordAlgebra(4,Fraction
Integer,MATRIX)
+--E 33
+
+--S 34 of 36
+m := 1; n:= 2; r := 3; s := 4;
+--R
+--R
+--R Type:
PositiveInteger
+--E 34
+
+--S 35 of 36
+lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t)
for l in 1..4]) for t in 1..4])
+--R
+--R
+--R (6) - 4e e e e
+--R 1 2 3 4
+--R Type: CliffordAlgebra(4,Fraction
Integer,MATRIX)
+--E 35
+
+--S 36 of 36
+rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s)
+--R
+--R
+--R (7) - 4e e e e
+--R 1 2 3 4
+--R Type: CliffordAlgebra(4,Fraction
Integer,MATRIX)
+--E 36
+)spool
+)lisp (bye)
+@
+<<CliffordAlgebra.help>>=
+====================================================================
+CliffordAlgebra examples
+====================================================================
+
+CliffordAlgebra(n,K,Q) defines a vector space of dimension 2^n over
+the field K with a given quadratic form Q. If {e1..en} is a basis for
+K^n then
+
+
+{ 1,
+ e(i) 1 <= i <= n,
+ e(i1)*e(i2) 1 <= i1 < i2 <=n,
+ ...,
+ e(1)*e(2)*...*e(n) }
+
+is a basis for the Clifford algebra. The algebra is defined by the relations
+
+ e(i)*e(i) = Q(e(i))
+ e(i)*e(j) = -e(j)*e(i), for i ^= j
+
+Examples of Clifford Algebras are gaussians (complex numbers),
+quaternions, exterior algebras and spin algebras.
+
+====================================================================
+The Complex Numbers as a Clifford Algebra
+====================================================================
+
+This is the field over which we will work, rational functions with
+integer coefficients.
+
+ K := Fraction Polynomial Integer
+ Fraction Polynomial Integer
+ Type: Domain
+
+We use this matrix for the quadratic form.
+
+ m := matrix [ [-1] ]
+ [- 1]
+ Type: Matrix Integer
+
+We get complex arithmetic by using this domain.
+
+ C := CliffordAlgebra(1, K, quadraticForm m)
+ CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+ Type: Domain
+
+Here is i, the usual square root of -1.
+
+ i: C := e(1)
+ e
+ 1
+ Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+
+Here are some examples of the arithmetic.
+
+ x := a + b * i
+ a + b e
+ 1
+ Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+
+ y := c + d * i
+ c + d e
+ 1
+ Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+
+ x * y
+ - b d + a c + (a d + b c)e
+ 1
+ Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+
+====================================================================
+The Quaternion Numbers as a Clifford Algebra
+====================================================================
+
+This is the field over which we will work, rational functions with
+integer coefficients.
+
+ K := Fraction Polynomial Integer
+ Fraction Polynomial Integer
+ Type: Domain
+
+We use this matrix for the quadratic form.
+
+ m := matrix [ [-1,0],[0,-1] ]
+ +- 1 0 +
+ | |
+ + 0 - 1+
+ Type: Matrix Integer
+
+The resulting domain is the quaternions.
+
+ H := CliffordAlgebra(2, K, quadraticForm m)
+ CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+ Type: Domain
+
+We use Hamilton's notation for i, j, k.
+
+ i: H := e(1)
+ e
+ 1
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+ j: H := e(2)
+ e
+ 2
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+ k: H := i * j
+ e e
+ 1 2
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+ x := a + b * i + c * j + d * k
+ a + b e + c e + d e e
+ 1 2 1 2
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+ y := e + f * i + g * j + h * k
+ e + f e + g e + h e e
+ 1 2 1 2
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+ x + y
+ e + a + (f + b)e + (g + c)e + (h + d)e e
+ 1 2 1 2
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+ x * y
+ - d h - c g - b f + a e + (c h - d g + a f + b e)e
+ 1
+ +
+ (- b h + a g + d f + c e)e + (a h + b g - c f + d e)e e
+ 2 1 2
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+ y * x
+ - d h - c g - b f + a e + (- c h + d g + a f + b e)e
+ 1
+ +
+ (b h + a g - d f + c e)e + (a h - b g + c f + d e)e e
+ 2 1 2
+ Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+
+====================================================================
+The Exterior Algebra on a Three Space
+====================================================================
+
+This is the field over which we will work, rational functions with
+integer coefficients.
+
+ K := Fraction Polynomial Integer
+ Fraction Polynomial Integer
+ Type: Domain
+
+If we chose the three by three zero quadratic form, we obtain
+the exterior algebra on e(1),e(2),e(3).
+
+ Ext := CliffordAlgebra(3, K, quadraticForm 0)
+ CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+ Type: Domain
+
+This is a three dimensional vector algebra. We define i, j, k as the
+unit vectors.
+
+ i: Ext := e(1)
+ e
+ 1
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+ j: Ext := e(2)
+ e
+ 2
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+ k: Ext := e(3)
+ e
+ 3
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+Now it is possible to do arithmetic.
+
+ x := x1*i + x2*j + x3*k
+ x1 e + x2 e + x3 e
+ 1 2 3
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+ y := y1*i + y2*j + y3*k
+ y1 e + y2 e + y3 e
+ 1 2 3
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+ x + y
+ (y1 + x1)e + (y2 + x2)e + (y3 + x3)e
+ 1 2 3
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+ x * y + y * x
+ 0
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+On an n space, a grade p form has a dual n-p form. In particular, in
+three space the dual of a grade two element identifies
+
+ e1*e2 -> e3, e2*e3 -> e1, e3*e1 -> e2.
+
+ dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j +
coefficient(a,[1,2]) * k
+ Type: Void
+
+The vector cross product is then given by this.
+
+ dual2(x*y)
+ (x2 y3 - x3 y2)e + (- x1 y3 + x3 y1)e + (x1 y2 - x2 y1)e
+ 1 2 3
+ Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+
+====================================================================
+The Dirac Spin Algebra
+====================================================================
+
+In this section we will work over the field of rational numbers.
+
+ K := Fraction Integer
+ Fraction Integer
+ Type: Domain
+
+We define the quadratic form to be the Minkowski space-time metric.
+
+ g := matrix [ [1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1] ]
+ +1 0 0 0 +
+ | |
+ |0 - 1 0 0 |
+ | |
+ |0 0 - 1 0 |
+ | |
+ +0 0 0 - 1+
+ Type: Matrix Integer
+
+We obtain the Dirac spin algebra used in Relativistic Quantum Field Theory.
+
+ D := CliffordAlgebra(4,K, quadraticForm g)
+ CliffordAlgebra(4,Fraction Integer,MATRIX)
+ Type: Domain
+
+The usual notation for the basis is gamma with a superscript. For
+Axiom input we will use gam(i):
+
+ gam := [e(i)$D for i in 1..4]
+ [e ,e ,e ,e ]
+ 1 2 3 4
+ Type: List CliffordAlgebra(4,Fraction Integer,MATRIX)
+
+There are various contraction identities of the form
+
+ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) =
+ 2*(gam(s)gam(m)gam(n)gam(r) + gam(r)*gam(n)*gam(m)*gam(s))
+
+where a sum over l and t is implied.
+
+Verify this identity for particular values of m,n,r,s.
+
+ m := 1; n:= 2; r := 3; s := 4;
+ Type: PositiveInteger
+
+ lhs := reduce(+, [reduce(+, [
g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4])
+ - 4e e e e
+ 1 2 3 4
+ Type: CliffordAlgebra(4,Fraction Integer,MATRIX)
+
+ rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s)
+ - 4e e e e
+ 1 2 3 4
+ Type: CliffordAlgebra(4,Fraction Integer,MATRIX)
+
+See Also:
+o )help Complex
+o )help Quaternion
+o )show CliffordAlgebra
+o $AXIOM/doc/src/algebra/clifford.spad
+
+@
<<domain CLIF CliffordAlgebra>>=
)abbrev domain CLIF CliffordAlgebra
++ Author: Stephen M. Watt
diff --git a/src/algebra/contfrac.spad.pamphlet
b/src/algebra/contfrac.spad.pamphlet
index 2261d76..d6c24af 100644
--- a/src/algebra/contfrac.spad.pamphlet
+++ b/src/algebra/contfrac.spad.pamphlet
@@ -2,7 +2,7 @@
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra contfrac.spad}
-\author{Stephen M. Watt, Clifton J. Williamson}
+\author{Stephen M. Watt, Clifton J. Williamson, Timothy Daly}
\maketitle
\begin{abstract}
\end{abstract}
@@ -10,6 +10,550 @@
\tableofcontents
\eject
\section{domain CONTFRAC ContinuedFraction}
+<<ContinuedFraction.input>>=
+-- contfrac.spad.pamphlet ContinuedFraction.input
+)spool ContinuedFraction.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 22
+c := continuedFraction(314159/100000)
+--R
+--R
+--R 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+--R (1) 3 + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
+--R | 7 | 15 | 1 | 25 | 1 | 7 | 4
+--R Type: ContinuedFraction
Integer
+--E 1
+
+--S 2 of 22
+partialQuotients c
+--R
+--R
+--R (2) [3,7,15,1,25,1,7,4]
+--R Type: Stream
Integer
+--E 2
+
+--S 3 of 22
+convergents c
+--R
+--R
+--R 22 333 355 9208 9563 76149 314159
+--R (3) [3,--,---,---,----,----,-----,------]
+--R 7 106 113 2931 3044 24239 100000
+--R Type: Stream Fraction
Integer
+--E 3
+
+--S 4 of 22
+approximants c
+--R
+--R
+--R ______
+--R 22 333 355 9208 9563 76149 314159
+--R (4) [3,--,---,---,----,----,-----,------]
+--R 7 106 113 2931 3044 24239 100000
+--R Type: Stream Fraction
Integer
+--E 4
+
+--S 5 of 22
+pq := partialQuotients(1/c)
+--R
+--R
+--R (5) [0,3,7,15,1,25,1,7,4]
+--R Type: Stream
Integer
+--E 5
+
+--S 6 of 22
+continuedFraction(first pq,repeating [1],rest pq)
+--R
+--R
+--R 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+--R (6) +---+ + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
+--R | 3 | 7 | 15 | 1 | 25 | 1 | 7 | 4
+--R Type: ContinuedFraction
Integer
+--E 6
+
+--S 7 of 22
+z:=continuedFraction(3,repeating [1],repeating [3,6])
+--R
+--R
+--R (7)
+--R 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 |
+--R 3 + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ +
+---+
+--R | 3 | 6 | 3 | 6 | 3 | 6 | 3 | 6 | 3
+--R +
+--R 1 |
+--R +---+ + ...
+--R | 6
+--R Type: ContinuedFraction
Integer
+--E 7
+
+--S 8 of 22
+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 8
+
+--S 9 of 22
+cf := continuedFraction(0,repeating [1],dens)
+--R
+--R
+--R (9)
+--R 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+--R +---+ + +---+ + +----+ + +----+ + +----+ + +----+ + +----+ + +----+
+--R | 1 | 6 | 10 | 14 | 18 | 22 | 26 | 30
+--R +
+--R 1 | 1 |
+--R +----+ + +----+ + ...
+--R | 34 | 38
+--R Type: ContinuedFraction
Integer
+--E 9
+
+--S 10 of 22
+ccf := convergents cf
+--R
+--R
+--R 6 61 860 15541 342762 8927353 268163352 9126481321
+--R (10) [0,1,-,--,----,-----,------,--------,---------,-----------,...]
+--R 7 71 1001 18089 398959 10391023 312129649 10622799089
+--R Type: Stream Fraction
Integer
+--E 10
+
+--S 11 of 22
+eConvergents := [2*e + 1 for e in ccf]
+--R
+--R
+--R 19 193 2721 49171 1084483 28245729 848456353 28875761731
+--R (11) [1,3,--,---,----,-----,-------,--------,---------,-----------,...]
+--R 7 71 1001 18089 398959 10391023 312129649 10622799089
+--R Type: Stream Fraction
Integer
+--E 11
+
+--S 12 of 22
+eConvergents :: Stream Float
+--R
+--R
+--R (12)
+--R [1.0, 3.0, 2.7142857142 857142857, 2.7183098591 549295775,
+--R 2.7182817182 817182817, 2.7182818287 356957267, 2.7182818284 585634113,
+--R 2.7182818284 590458514, 2.7182818284 590452348, 2.7182818284 590452354,
+--R ...]
+--R Type: Stream
Float
+--E 12
+
+--S 13 of 22
+exp 1.0
+--R
+--R
+--R (13) 2.7182818284 590452354
+--R Type:
Float
+--E 13
+
+--S 14 of 22
+cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])
+--R
+--R
+--R (14)
+--R 1 | 9 | 25 | 49 | 81 | 121 | 169 |
225 |
+--R 1 + +---+ + +---+ + +----+ + +----+ + +----+ + +-----+ + +-----+ +
+-----+
+--R | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
+--R +
+--R 289 | 361 |
+--R +-----+ + +-----+ + ...
+--R | 2 | 2
+--R Type: ContinuedFraction
Integer
+--E 14
+
+--S 15 of 22
+ccf := convergents cf
+--R
+--R
+--R 3 15 105 315 3465 45045 45045 765765 14549535
+--R (15) [1,-,--,---,---,----,-----,-----,------,--------,...]
+--R 2 13 76 263 2578 36979 33976 622637 11064338
+--R Type: Stream Fraction
Integer
+--E 15
+
+--S 16 of 22
+piConvergents := [4/p for p in ccf]
+--R
+--R
+--R 8 52 304 1052 10312 147916 135904 2490548 44257352
+--R (16) [4,-,--,---,----,-----,------,------,-------,--------,...]
+--R 3 15 105 315 3465 45045 45045 765765 14549535
+--R Type: Stream Fraction
Integer
+--E 16
+
+--S 17 of 22
+piConvergents :: Stream Float
+--R
+--R
+--R (17)
+--R [4.0, 2.6666666666 666666667, 3.4666666666 666666667,
+--R 2.8952380952 380952381, 3.3396825396 825396825, 2.9760461760 461760462,
+--R 3.2837384837 384837385, 3.0170718170 718170718, 3.2523659347 188758953,
+--R 3.0418396189 294022111, ...]
+--R Type: Stream
Float
+--E 17
+
+--S 18 of 22
+continuedFraction((- 122 + 597*%i)/(4 - 4*%i))
+--R
+--R
+--R 1 | 1 |
+--R (18) - 90 + 59%i + +---------+ + +-----------+
+--R | 1 - 2%i | - 1 + 2%i
+--R Type: ContinuedFraction Complex
Integer
+--E 18
+
+--S 19 of 22
+r : Fraction UnivariatePolynomial(x,Fraction Integer)
+--R
+--R Type:
Void
+--E 19
+
+--S 20 of 22
+r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))
+--R
+--R
+--R 2
+--R x - 3x + 2
+--R (20) ------------
+--R 2
+--R x - 7x + 12
+--R Type: Fraction UnivariatePolynomial(x,Fraction
Integer)
+--E 20
+
+--S 21 of 22
+continuedFraction r
+--R
+--R
+--R 1 | 1 |
+--R (21) 1 + +---------+ + +-----------+
+--R | 1 9 | 16 40
+--R | - x - - | -- x - --
+--R | 4 8 | 3 3
+--R Type: ContinuedFraction UnivariatePolynomial(x,Fraction
Integer)
+--E 21
+
+--S 22 of 22
+[i*i for i in convergents(z) :: Stream Float]
+--R
+--R
+--R (22)
+--R [9.0, 11.1111111111 11111111, 10.9944598337 9501385, 11.0002777777
77777778,
+--R 10.9999860763 98799786, 11.0000006979 29731039, 10.9999999650 15834446,
+--R 11.0000000017 53603304, 10.9999999999 12099531, 11.0000000000 04406066,
+--R ...]
+--R Type: Stream
Float
+--E 22
+)spool
+)lisp (bye)
+@
+<<ContinuedFraction.help>>=
+====================================================================
+ContinuedFraction examples
+====================================================================
+
+Continued fractions have been a fascinating and useful tool in
+mathematics for well over three hundred years. Axiom implements
+continued fractions for fractions of any Euclidean domain. In
+practice, this usually means rational numbers. In this section we
+demonstrate some of the operations available for manipulating both
+finite and infinite continued fractions.
+
+The ContinuedFraction domain is a field and therefore you can add,
+subtract, multiply and divide the fractions.
+
+The continuedFraction operation converts its fractional argument to a
+continued fraction.
+
+ c := continuedFraction(314159/100000)
+ 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+ 3 + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
+ | 7 | 15 | 1 | 25 | 1 | 7 | 4
+ Type: ContinuedFraction Integer
+
+This display is a compact form of the bulkier
+
+ 3 + 1
+ -------------------------------
+ 7 + 1
+ ---------------------------
+ 15 + 1
+ ----------------------
+ 1 + 1
+ ------------------
+ 25 + 1
+ -------------
+ 1 + 1
+ ---------
+ 7 + 1
+ -----
+ 4
+
+You can write any rational number in a similar form. The fraction
+will be finite and you can always take the "numerators" to be 1.
+That is, any rational number can be written as a simple, finite
+continued fraction of the form
+
+ a(1) + 1
+ -------------------------
+ a(2) + 1
+ --------------------
+ a(3) +
+ .
+ .
+ .
+ 1
+ -------------
+ a(n-1) + 1
+ ----
+ a(n)
+
+
+The a(i) are called partial quotients and the operation partialQuotients
+creates a stream of them.
+
+ partialQuotients c
+ [3,7,15,1,25,1,7,4]
+ Type: Stream Integer
+
+By considering more and more of the fraction, you get the convergents.
+For example, the first convergent is a(1), the second is a(1) + 1/a(2)
+and so on.
+
+ convergents c
+ 22 333 355 9208 9563 76149 314159
+ [3,--,---,---,----,----,-----,------]
+ 7 106 113 2931 3044 24239 100000
+ Type: Stream Fraction Integer
+
+Since this is a finite continued fraction, the last convergent is the
+original rational number, in reduced form. The result of approximants
+is always an infinite stream, though it may just repeat the "last" value.
+
+ approximants c
+ ______
+ 22 333 355 9208 9563 76149 314159
+ [3,--,---,---,----,----,-----,------]
+ 7 106 113 2931 3044 24239 100000
+ Type: Stream Fraction Integer
+
+Inverting c only changes the partial quotients of its fraction by
+inserting a 0 at the beginning of the list.
+
+ pq := partialQuotients(1/c)
+ [0,3,7,15,1,25,1,7,4]
+ Type: Stream Integer
+
+Do this to recover the original continued fraction from this list of
+partial quotients. The three-argument form of the continuedFraction
+operation takes an element which is the whole part of the fraction, a
+stream of elements which are the numerators of the fraction, and a
+stream of elements which are the denominators of the fraction.
+
+ continuedFraction(first pq,repeating [1],rest pq)
+ 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+ +---+ + +---+ + +----+ + +---+ + +----+ + +---+ + +---+ + +---+
+ | 3 | 7 | 15 | 1 | 25 | 1 | 7 | 4
+ Type: ContinuedFraction Integer
+
+The streams need not be finite for continuedFraction. Can you guess
+which irrational number has the following continued fraction? See the
+end of this section for the answer.
+
+ z:=continuedFraction(3,repeating [1],repeating [3,6])
+ 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+ 3 + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+
+ | 3 | 6 | 3 | 6 | 3 | 6 | 3 | 6 | 3
+ +
+ 1 |
+ +---+ + ...
+ | 6
+ Type: ContinuedFraction Integer
+
+In 1737 Euler discovered the infinite continued fraction expansion
+
+ e - 1 1
+ ----- = ---------------------
+ 2 1 + 1
+ -----------------
+ 6 + 1
+ -------------
+ 10 + 1
+ --------
+ 14 + ...
+
+
+We use this expansion to compute rational and floating point
+approximations of e. For this and other interesting expansions, see
+C. D. Olds, Continued Fractions, New Mathematical Library, (New York:
+Random House, 1963), pp. 134--139.}
+
+By looking at the above expansion, we see that the whole part is 0 and
+the numerators are all equal to 1. This constructs the stream of
+denominators.
+
+ dens:Stream Integer := cons(1,generate((x+->x+4),6))
+ [1,6,10,14,18,22,26,30,34,38,...]
+ Type: Stream Integer
+
+Therefore this is the continued fraction expansion for (e - 1) / 2.
+
+ cf := continuedFraction(0,repeating [1],dens)
+ 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
+ +---+ + +---+ + +----+ + +----+ + +----+ + +----+ + +----+ + +----+
+ | 1 | 6 | 10 | 14 | 18 | 22 | 26 | 30
+ +
+ 1 | 1 |
+ +----+ + +----+ + ...
+ | 34 | 38
+ Type: ContinuedFraction Integer
+
+These are the rational number convergents.
+
+ ccf := convergents cf
+ 6 61 860 15541 342762 8927353 268163352 9126481321
+ [0,1,-,--,----,-----,------,--------,---------,-----------,...]
+ 7 71 1001 18089 398959 10391023 312129649 10622799089
+ Type: Stream Fraction Integer
+
+You can get rational convergents for e by multiplying by 2 and adding 1.
+
+ eConvergents := [2*e + 1 for e in ccf]
+ 19 193 2721 49171 1084483 28245729 848456353 28875761731
+ [1,3,--,---,----,-----,-------,--------,---------,-----------,...]
+ 7 71 1001 18089 398959 10391023 312129649 10622799089
+ Type: Stream Fraction Integer
+
+You can also compute the floating point approximations to these convergents.
+
+ eConvergents :: Stream Float
+ [1.0, 3.0, 2.7142857142 857142857, 2.7183098591 549295775,
+ 2.7182817182 817182817, 2.7182818287 356957267, 2.7182818284 585634113,
+ 2.7182818284 590458514, 2.7182818284 590452348, 2.7182818284 590452354,
+ ...]
+ Type: Stream Float
+
+Compare this to the value of e computed by the exp operation in Float.
+
+ exp 1.0
+ 2.7182818284 590452354
+ Type: Float
+
+In about 1658, Lord Brouncker established the following expansion
+for 4 / pi,
+
+ 1 + 1
+ -----------------------
+ 2 + 9
+ -------------------
+ 2 + 25
+ ---------------
+ 2 + 49
+ -----------
+ 2 + 81
+ -------
+ 2 + ...
+
+Let's use this expansion to compute rational and floating point
+approximations for pi.
+
+ cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])
+ 1 | 9 | 25 | 49 | 81 | 121 | 169 | 225 |
+ 1 + +---+ + +---+ + +----+ + +----+ + +----+ + +-----+ + +-----+ + +-----+
+ | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
+ +
+ 289 | 361 |
+ +-----+ + +-----+ + ...
+ | 2 | 2
+ Type: ContinuedFraction Integer
+
+ ccf := convergents cf
+ 3 15 105 315 3465 45045 45045 765765 14549535
+ [1,-,--,---,---,----,-----,-----,------,--------,...]
+ 2 13 76 263 2578 36979 33976 622637 11064338
+ Type: Stream Fraction Integer
+
+ piConvergents := [4/p for p in ccf]
+ 8 52 304 1052 10312 147916 135904 2490548 44257352
+ [4,-,--,---,----,-----,------,------,-------,--------,...]
+ 3 15 105 315 3465 45045 45045 765765 14549535
+ Type: Stream Fraction Integer
+
+As you can see, the values are converging to
+pi = 3.14159265358979323846..., but not very quickly.
+
+ piConvergents :: Stream Float
+ [4.0, 2.6666666666 666666667, 3.4666666666 666666667,
+ 2.8952380952 380952381, 3.3396825396 825396825, 2.9760461760 461760462,
+ 3.2837384837 384837385, 3.0170718170 718170718, 3.2523659347 188758953,
+ 3.0418396189 294022111, ...]
+ Type: Stream Float
+
+You need not restrict yourself to continued fractions of integers.
+Here is an expansion for a quotient of Gaussian integers.
+
+ continuedFraction((- 122 + 597*%i)/(4 - 4*%i))
+ 1 | 1 |
+ - 90 + 59%i + +---------+ + +-----------+
+ | 1 - 2%i | - 1 + 2%i
+ Type: ContinuedFraction Complex Integer
+
+This is an expansion for a quotient of polynomials in one variable
+with rational number coefficients.
+
+ r : Fraction UnivariatePolynomial(x,Fraction Integer)
+ Type: Void
+
+ r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))
+ 2
+ x - 3x + 2
+ ------------
+ 2
+ x - 7x + 12
+ Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+
+ continuedFraction r
+ 1 | 1 |
+ 1 + +---------+ + +-----------+
+ | 1 9 | 16 40
+ | - x - - | -- x - --
+ | 4 8 | 3 3
+ Type: ContinuedFraction UnivariatePolynomial(x,Fraction Integer)
+
+To conclude this section, we give you evidence that
+
+ z = 3 + 1
+ -----------------------
+ 3 + 1
+ -------------------
+ 6 + 1
+ ---------------
+ 3 + 1
+ -----------
+ 6 + 1
+ -------
+ 3 + ...
+
+is the expansion of sqrt(11).
+
+ [i*i for i in convergents(z) :: Stream Float]
+ [9.0, 11.1111111111 11111111, 10.9944598337 9501385, 11.0002777777 77777778,
+ 10.9999860763 98799786, 11.0000006979 29731039, 10.9999999650 15834446,
+ 11.0000000017 53603304, 10.9999999999 12099531, 11.0000000000 04406066,
+ ...]
+ Type: Stream Float
+
+See Also:
+o )help Stream
+o )show ContinuedFraction
+o $AXIOM/doc/src/algebra/contrac.spad.dvi
+
+@
<<domain CONTFRAC ContinuedFraction>>=
)abbrev domain CONTFRAC ContinuedFraction
++ Author: Stephen M. Watt
diff --git a/src/algebra/cycles.spad.pamphlet b/src/algebra/cycles.spad.pamphlet
index d5747f5..c831683 100644
--- a/src/algebra/cycles.spad.pamphlet
+++ b/src/algebra/cycles.spad.pamphlet
@@ -10,6 +10,913 @@
\tableofcontents
\eject
\section{package CYCLES CycleIndicators}
+<<CycleIndicators.input>>=
+-- cycles.spad.pamphlet CycleIndicators.input
+)spool CycleIndicators.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 47
+complete 1
+--R
+--R
+--R (1) (1)
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 1
+
+--S 2 of 47
+complete 2
+--R
+--R
+--R 1 1 2
+--R (2) - (2) + - (1 )
+--R 2 2
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 2
+
+--S 3 of 47
+complete 3
+--R
+--R
+--R 1 1 1 3
+--R (3) - (3) + - (2 1) + - (1 )
+--R 3 2 6
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 3
+
+--S 4 of 47
+complete 7
+--R
+--R
+--R (4)
+--R 1 1 1 1 2 1 1 1
3
+--R - (7) + - (6 1) + -- (5 2) + -- (5 1 ) + -- (4 3) + - (4 2 1) + -- (4
1 )
+--R 7 6 10 10 12 8 24
+--R +
+--R 1 2 1 2 1 2 1 4 1 3 1 2 3
+--R -- (3 1) + -- (3 2 ) + -- (3 2 1 ) + -- (3 1 ) + -- (2 1) + -- (2 1 )
+--R 18 24 12 72 48 48
+--R +
+--R 1 5 1 7
+--R --- (2 1 ) + ---- (1 )
+--R 240 5040
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 4
+
+--S 5 of 47
+elementary 7
+--R
+--R
+--R (5)
+--R 1 1 1 1 2 1 1 1
3
+--R - (7) - - (6 1) - -- (5 2) + -- (5 1 ) - -- (4 3) + - (4 2 1) - -- (4
1 )
+--R 7 6 10 10 12 8 24
+--R +
+--R 1 2 1 2 1 2 1 4 1 3 1 2 3
+--R -- (3 1) + -- (3 2 ) - -- (3 2 1 ) + -- (3 1 ) - -- (2 1) + -- (2 1 )
+--R 18 24 12 72 48 48
+--R +
+--R 1 5 1 7
+--R - --- (2 1 ) + ---- (1 )
+--R 240 5040
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 5
+
+--S 6 of 47
+alternating 7
+--R
+--R
+--R (6)
+--R 2 1 2 1 1 2 1 2 1 4 1
2 3
+--R - (7) + - (5 1 ) + - (4 2 1) + - (3 1) + -- (3 2 ) + -- (3 1 ) + -- (2
1 )
+--R 7 5 4 9 12 36 24
+--R +
+--R 1 7
+--R ---- (1 )
+--R 2520
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 6
+
+--S 7 of 47
+cyclic 7
+--R
+--R
+--R 6 1 7
+--R (7) - (7) + - (1 )
+--R 7 7
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 7
+
+--S 8 of 47
+dihedral 7
+--R
+--R
+--R 3 1 3 1 7
+--R (8) - (7) + - (2 1) + -- (1 )
+--R 7 2 14
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 8
+
+--S 9 of 47
+graphs 5
+--R
+--R
+--R (9)
+--R 1 1 2 1 2 1 3 1 4 2 1 3 4 1 10
+--R - (6 3 1) + - (5 ) + - (4 2) + - (3 1) + - (2 1 ) + -- (2 1 ) + --- (1 )
+--R 6 5 4 6 8 12 120
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 9
+
+--S 10 of 47
+cap(complete 2**2, complete 2*complete 1**2)
+--R
+--R
+--R (10) 4
+--R Type: Fraction
Integer
+--E 10
+
+--S 11 of 47
+cap(elementary 2**2, complete 2*complete 1**2)
+--R
+--R
+--R (11) 2
+--R Type: Fraction
Integer
+--E 11
+
+--S 12 of 47
+cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)
+--R
+--R
+--R (12) 24
+--R Type: Fraction
Integer
+--E 12
+
+--S 13 of 47
+cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)
+--R
+--R
+--R (13) 8
+--R Type: Fraction
Integer
+--E 13
+
+--S 14 of 47
+cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)
+--R
+--R
+--R (14) 8
+--R Type: Fraction
Integer
+--E 14
+
+--S 15 of 47
+eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete
1**2,complete 2**3)))
+--R
+--R
+--R (15) 1500
+--R Type: Fraction
Integer
+--E 15
+
+--S 16 of 47
+square:=dihedral 4
+--R
+--R
+--R 1 3 2 1 2 1 4
+--R (16) - (4) + - (2 ) + - (2 1 ) + - (1 )
+--R 4 8 4 8
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 16
+
+--S 17 of 47
+cap(complete 2**2,square)
+--R
+--R
+--R (17) 2
+--R Type: Fraction
Integer
+--E 17
+
+--S 18 of 47
+cap(complete 3*complete 2**2,dihedral 7)
+--R
+--R
+--R (18) 18
+--R Type: Fraction
Integer
+--E 18
+
+--S 19 of 47
+cap(graphs 5,complete 7*complete 3)
+--R
+--R
+--R (19) 4
+--R Type: Fraction
Integer
+--E 19
+
+--S 20 of 47
+s(x) == powerSum(x)
+--R
+--R Type:
Void
+--E 20
+
+--S 21 of 47
+cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)
+--R
+--R Compiling function s with type PositiveInteger ->
+--R SymmetricPolynomial Fraction Integer
+--R
+--R 1 2 1 2 2 3 4 1 8
+--R (21) - (4 ) + - (3 1 ) + - (2 ) + -- (1 )
+--R 4 3 8 24
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 21
+
+--S 22 of 47
+cap(complete 4**2,cube)
+--R
+--R
+--R (22) 7
+--R Type: Fraction
Integer
+--E 22
+
+--S 23 of 47
+cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))
+--R
+--R
+--R (23) 7
+--R Type: Fraction
Integer
+--E 23
+
+--S 24 of 47
+cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))
+--R
+--R
+--R (24) 17
+--R Type: Fraction
Integer
+--E 24
+
+--S 25 of 47
+cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))
+--R
+--R
+--R (25) 10
+--R Type: Fraction
Integer
+--E 25
+
+--S 26 of 47
+cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))
+--R
+--R
+--R (26) 23
+--R Type: Fraction
Integer
+--E 26
+
+--S 27 of 47
+x: ULS(FRAC INT,'x,0) := 'x
+--R
+--R
+--R (27) x
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 27
+
+--S 28 of 47
+ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0)
+--R
+--R Type:
Void
+--E 28
+
+--S 29 of 47
+Integers: INT -> ULS(FRAC INT, 'x, 0)
+--R
+--R Type:
Void
+--E 29
+
+--S 30 of 47
+ZeroOrOne n == 1+x**n
+--R
+--R Type:
Void
+--E 30
+
+--S 31 of 47
+ZeroOrOne 5
+--R
+--R Compiling function ZeroOrOne with type Integer ->
+--R UnivariateLaurentSeries(Fraction Integer,x,0)
+--R
+--R 5
+--R (31) 1 + x
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 31
+
+--S 32 of 47
+Integers n == 1/(1-x**n)
+--R
+--R Type:
Void
+--E 32
+
+--S 33 of 47
+Integers 5
+--R
+--R Compiling function Integers with type Integer ->
+--R UnivariateLaurentSeries(Fraction Integer,x,0)
+--R
+--R 5 10 11
+--R (33) 1 + x + x + O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 33
+
+--S 34 of 47
+)expose EVALCYC
+--R
+--I EvaluateCycleIndicators is now explicitly exposed in frame frame0
+--E 34
+
+--S 35 of 47
+eval(ZeroOrOne, graphs 5)
+--R
+--R
+--R 2 3 4 5 6 7 8 9 10 11
+--R (34) 1 + x + 2x + 4x + 6x + 6x + 6x + 4x + 2x + x + x + O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 35
+
+--S 36 of 47
+eval(ZeroOrOne,dihedral 8)
+--R
+--R
+--R 2 3 4 5 6 7 8
+--R (35) 1 + x + 4x + 5x + 8x + 5x + 4x + x + x
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 36
+
+--S 37 of 47
+eval(Integers,complete 4)
+--R
+--R
+--R (36)
+--R 2 3 4 5 6 7 8 9 10 11
+--R 1 + x + 2x + 3x + 5x + 6x + 9x + 11x + 15x + 18x + 23x + O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 37
+
+--S 38 of 47
+eval(Integers,elementary 4)
+--R
+--R
+--R (37)
+--R 6 7 8 9 10 11 12 13 14 15
16
+--R x + x + 2x + 3x + 5x + 6x + 9x + 11x + 15x + 18x + 23x
+--R +
+--R 17
+--R O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 38
+
+--S 39 of 47
+eval(ZeroOrOne,cube)
+--R
+--R
+--R 2 3 4 5 6 7 8
+--R (38) 1 + x + 3x + 3x + 7x + 3x + 3x + x + x
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 39
+
+--S 40 of 47
+eval(Integers,cube)
+--R
+--R
+--R (39)
+--R 2 3 4 5 6 7 8 9 10
+--R 1 + x + 4x + 7x + 21x + 37x + 85x + 151x + 292x + 490x + 848x
+--R +
+--R 11
+--R O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 40
+
+--S 41 of 47
+eval(Integers,graphs 5)
+--R
+--R
+--R (40)
+--R 2 3 4 5 6 7 8 9 10
+--R 1 + x + 3x + 7x + 17x + 35x + 76x + 149x + 291x + 539x + 974x
+--R +
+--R 11
+--R O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 41
+
+--S 42 of 47
+eval(ZeroOrOne ,graphs 15)
+--R
+--R
+--R (41)
+--R 2 3 4 5 6 7 8 9
10
+--R 1 + x + 2x + 5x + 11x + 26x + 68x + 177x + 496x + 1471x + 4583x
+--R +
+--R 11
+--R O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 42
+
+--S 43 of 47
+cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)
+--R
+--R
+--R (42) 49958972383320
+--R Type: Fraction
Integer
+--E 43
+
+--S 44 of 47
+sf3221:= SFunction [3,2,2,1]
+--R
+--R
+--R (43)
+--R 1 1 2 1 2 1 1 4 1 2
+--R -- (6 2) - -- (6 1 ) - -- (4 ) + -- (4 3 1) + -- (4 1 ) - -- (3 2)
+--R 12 12 16 12 24 36
+--R +
+--R 1 2 2 1 2 1 3 1 5 1 4 1 3 2
+--R -- (3 1 ) - -- (3 2 1) - -- (3 2 1 ) - -- (3 1 ) - --- (2 ) + -- (2 1 )
+--R 36 24 36 72 192 48
+--R +
+--R 1 2 4 1 6 1 8
+--R -- (2 1 ) - --- (2 1 ) + --- (1 )
+--R 96 144 576
+--R Type: SymmetricPolynomial Fraction
Integer
+--E 44
+
+--S 45 of 47
+cap(sf3221,complete 2**4)
+--R
+--R
+--R (44) 3
+--R Type: Fraction
Integer
+--E 45
+
+--S 46 of 47
+cap(sf3221, powerSum 1**8)
+--R
+--R
+--R (45) 70
+--R Type: Fraction
Integer
+--E 46
+
+--S 47 of 47
+eval(Integers, sf3221)
+--R
+--R
+--R (46)
+--R 9 10 11 12 13 14 15 16 17
18
+--R x + 3x + 7x + 14x + 27x + 47x + 79x + 126x + 196x +
294x
+--R +
+--R 19 20
+--R 432x + O(x )
+--R Type: UnivariateLaurentSeries(Fraction
Integer,x,0)
+--E 47
+)spool
+)lisp (bye)
+@
+<<CycleIndicators.help>>=
+====================================================================
+\section{CycleIndicators}
+\label{CycleIndicatorsXmpPage}
+
+This section is based upon the paper J. H. Redfield, ``The Theory of
+Group-Reduced Distributions'', American J. Math.,49 (1927) 433-455,
+and is an application of group theory to enumeration problems. It is
+a development of the work by P. A. MacMahon on the application of
+symmetric functions and Hammond operators to combinatorial theory.
+
+The theory is based upon the power sum symmetric functions s(i) which
+are the sum of the i-th powers of the variables. The cycle index of a
+permutation is an expression that specifies the sizes of the cycles of
+a permutation, and may be represented as a partition. A partition of
+a non-negative integer n is a collection of positive integers called
+its parts whose sum is n. For example, the partition (3^2 2 1^2) will
+be used to represent s^2_3 s_2 s^2_1 and will indicate that the
+permutation has two cycles of length 3, one of length 2 and two of
+length 1. The cycle index of a permutation group is the sum of the
+cycle indices of its permutations divided by the number of
+permutations. The cycle indices of certain groups are provided.
+
+The operation complete returns the cycle index of the symmetric group
+of order n for argument n. Alternatively, it is the n-th complete
+homogeneous symmetric function expressed in terms of power sum
+symmetric functions.
+
+ complete 1
+ (1)
+ Type: SymmetricPolynomial Fraction Integer
+
+ complete 2
+ 1 1 2
+ - (2) + - (1 )
+ 2 2
+ Type: SymmetricPolynomial Fraction Integer
+
+ complete 3
+ 1 1 1 3
+ - (3) + - (2 1) + - (1 )
+ 3 2 6
+ Type: SymmetricPolynomial Fraction Integer
+
+ complete 7
+ 1 1 1 1 2 1 1 1 3
+ - (7) + - (6 1) + -- (5 2) + -- (5 1 ) + -- (4 3) + - (4 2 1) + -- (4 1 )
+ 7 6 10 10 12 8 24
+ +
+ 1 2 1 2 1 2 1 4 1 3 1 2 3
+ -- (3 1) + -- (3 2 ) + -- (3 2 1 ) + -- (3 1 ) + -- (2 1) + -- (2 1 )
+ 18 24 12 72 48 48
+ +
+ 1 5 1 7
+ --- (2 1 ) + ---- (1 )
+ 240 5040
+ Type: SymmetricPolynomial Fraction Integer
+
+The operation elementary computes the n-th elementary symmetric
+function for argument n.
+
+ elementary 7
+ 1 1 1 1 2 1 1 1 3
+ - (7) - - (6 1) - -- (5 2) + -- (5 1 ) - -- (4 3) + - (4 2 1) - -- (4 1 )
+ 7 6 10 10 12 8 24
+ +
+ 1 2 1 2 1 2 1 4 1 3 1 2 3
+ -- (3 1) + -- (3 2 ) - -- (3 2 1 ) + -- (3 1 ) - -- (2 1) + -- (2 1 )
+ 18 24 12 72 48 48
+ +
+ 1 5 1 7
+ - --- (2 1 ) + ---- (1 )
+ 240 5040
+ Type: SymmetricPolynomial Fraction Integer
+
+The operation alternating returns the cycle index of the alternating
+group having an even number of even parts in each cycle partition.
+
+ alternating 7
+ 2 1 2 1 1 2 1 2 1 4 1 2 3
+ - (7) + - (5 1 ) + - (4 2 1) + - (3 1) + -- (3 2 ) + -- (3 1 ) + -- (2 1 )
+ 7 5 4 9 12 36 24
+ +
+ 1 7
+ ---- (1 )
+ 2520
+ Type: SymmetricPolynomial Fraction Integer
+
+The operation cyclic returns the cycle index of the cyclic group.
+
+ cyclic 7
+ 6 1 7
+ - (7) + - (1 )
+ 7 7
+ Type: SymmetricPolynomial Fraction Integer
+
+The operation dihedral is the cycle index of the dihedral group.
+
+ dihedral 7
+ 3 1 3 1 7
+ - (7) + - (2 1) + -- (1 )
+ 7 2 14
+ Type: SymmetricPolynomial Fraction Integer
+
+The operation graphs for argument n returns the cycle index of the
+group of permutations on the edges of the complete graph with n nodes
+induced by applying the symmetric group to the nodes.
+
+ graphs 5
+ 1 1 2 1 2 1 3 1 4 2 1 3 4 1 10
+ - (6 3 1) + - (5 ) + - (4 2) + - (3 1) + - (2 1 ) + -- (2 1 ) + --- (1 )
+ 6 5 4 6 8 12 120
+ Type: SymmetricPolynomial Fraction Integer
+
+The cycle index of a direct product of two groups is the product of
+the cycle indices of the groups. Redfield provided two operations on
+two cycle indices which will be called "cup" and "cap" here. The cup
+of two cycle indices is a kind of scalar product that combines
+monomials for permutations with the same cycles. The cap operation
+provides the sum of the coefficients of the result of the cup
+operation which will be an integer that enumerates what Redfield
+called group-reduced distributions.
+
+We can, for example, represent complete 2 * complete 2 as the set of
+objects a a b b and complete 2 * complete 1 * complete 1 as c c d e.
+
+This integer is the number of different sets of four pairs.
+
+ cap(complete 2**2, complete 2*complete 1**2)
+ 4
+ Type: Fraction Integer
+
+For example,
+ a a b b a a b b a a b b a a b b
+ c c d e c d c e c e c d d e c c
+
+This integer is the number of different sets of four pairs no two
+pairs being equal.
+
+ cap(elementary 2**2, complete 2*complete 1**2)
+ 2
+ Type: Fraction Integer
+
+For example,
+
+ a a b b a a b b
+ c d c e c e c d
+
+In this case the configurations enumerated are easily constructed,
+however the theory merely enumerates them providing little help in
+actually constructing them.
+
+Here are the number of 6-pairs, first from a a a b b c, second
+from d d e e f g.
+
+ cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)
+ 24
+ Type: Fraction Integer
+
+Here it is again, but with no equal pairs.
+
+ cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)
+ 8
+ Type: Fraction Integer
+
+ cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)
+ 8
+ Type: Fraction Integer
+
+The number of 6-triples, first from a a a b b c, second from
+d d e e f g, third from h h i i j j.
+
+ eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete
1**2,complete 2**3)))
+ 1500
+ Type: Fraction Integer
+
+The cycle index of vertices of a square is dihedral 4.
+
+ square:=dihedral 4
+ 1 3 2 1 2 1 4
+ - (4) + - (2 ) + - (2 1 ) + - (1 )
+ 4 8 4 8
+ Type: SymmetricPolynomial Fraction Integer
+
+The number of different squares with 2 red vertices and 2 blue vertices.
+
+ cap(complete 2**2,square)
+ 2
+ Type: Fraction Integer
+
+The number of necklaces with 3 red beads, 2 blue beads and 2 green beads.
+
+ cap(complete 3*complete 2**2,dihedral 7)
+ 18
+ Type: Fraction Integer
+
+The number of graphs with 5 nodes and 7 edges.
+
+ cap(graphs 5,complete 7*complete 3)
+ 4
+ Type: Fraction Integer
+
+The cycle index of rotations of vertices of a cube.
+
+ s(x) == powerSum(x)
+ Type: Void
+
+ cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)
+ 1 2 1 2 2 3 4 1 8
+ - (4 ) + - (3 1 ) + - (2 ) + -- (1 )
+ 4 3 8 24
+ Type: SymmetricPolynomial Fraction Integer
+
+The number of cubes with 4 red vertices and 4 blue vertices.
+
+ cap(complete 4**2,cube)
+ 7
+ Type: Fraction Integer
+
+The number of labeled graphs with degree sequence 2 2 2 1 1 with no
+loops or multiple edges.
+
+ cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))
+ 7
+ Type: Fraction Integer
+
+Again, but with loops allowed but not multiple edges.
+
+ cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))
+ 17
+ Type: Fraction Integer
+
+Again, but with multiple edges allowed, but not loops
+
+ cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))
+ 10
+ Type: Fraction Integer
+
+Again, but with both multiple edges and loops allowed
+
+ cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))
+ 23
+ Type: Fraction Integer
+
+Having constructed a cycle index for a configuration we are at liberty
+to evaluate the s_i components any way we please. For example we can
+produce enumerating generating functions. This is done by providing a
+function f on an integer i to the value required of s_i, and then
+evaluating eval(f, cycleindex).
+
+ x: ULS(FRAC INT,'x,0) := 'x
+ x
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+ ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0)
+ Type: Void
+
+ Integers: INT -> ULS(FRAC INT, 'x, 0)
+ Type: Void
+
+For the integers 0 and 1, or two colors.
+
+ ZeroOrOne n == 1+x**n
+ Type: Void
+
+ ZeroOrOne 5
+ 5
+ 1 + x
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+For the integers 0, 1, 2, ... we have this.
+
+ Integers n == 1/(1-x**n)
+ Type: Void
+
+ Integers 5
+ 5 10 11
+ 1 + x + x + O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The coefficient of x^n is the number of graphs with 5 nodes and n edges.
+
+Note that there is an eval function that takes two arguments. It has the
+signature:
+
+ ((Integer -> D1),SymmetricPolynomial Fraction Integer) -> D1
+ from EvaluateCycleIndicators D1 if D1 has ALGEBRA FRAC INT
+
+This function is not normally exposed (it will not normally be considered
+in the list of eval functions) as it is only useful for this particular
+domain. To use it we ask that it be considered thus:
+
+ )expose EVALCYC
+
+and now we can use it:
+
+ eval(ZeroOrOne, graphs 5)
+ 2 3 4 5 6 7 8 9 10 11
+ 1 + x + 2x + 4x + 6x + 6x + 6x + 4x + 2x + x + x + O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The coefficient of x^n is the number of necklaces with n red beads
+and n-8 green beads.
+
+ eval(ZeroOrOne,dihedral 8)
+ 2 3 4 5 6 7 8
+ 1 + x + 4x + 5x + 8x + 5x + 4x + x + x
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The coefficient of x^n is the number of partitions of n into 4 or fewer parts.
+
+ eval(Integers,complete 4)
+ 2 3 4 5 6 7 8 9 10 11
+ 1 + x + 2x + 3x + 5x + 6x + 9x + 11x + 15x + 18x + 23x + O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The coefficient of x^n is the number of partitions of n into 4 boxes
+containing ordered distinct parts.
+
+ eval(Integers,elementary 4)
+ 6 7 8 9 10 11 12 13 14 15 16
+ x + x + 2x + 3x + 5x + 6x + 9x + 11x + 15x + 18x + 23x
+ +
+ 17
+ O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The coefficient of x^n is the number of different cubes with n red
+vertices and 8-n green ones.
+
+ eval(ZeroOrOne,cube)
+ 2 3 4 5 6 7 8
+ 1 + x + 3x + 3x + 7x + 3x + 3x + x + x
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The coefficient of x^n is the number of different cubes with integers
+on the vertices whose sum is n.
+
+ eval(Integers,cube)
+ 2 3 4 5 6 7 8 9 10
+ 1 + x + 4x + 7x + 21x + 37x + 85x + 151x + 292x + 490x + 848x
+ +
+ 11
+ O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The coefficient of x^n is the number of graphs with 5 nodes and with
+integers on the edges whose sum is n. In other words, the enumeration
+is of multigraphs with 5 nodes and n edges.
+
+ eval(Integers,graphs 5)
+ 2 3 4 5 6 7 8 9 10
+ 1 + x + 3x + 7x + 17x + 35x + 76x + 149x + 291x + 539x + 974x
+ +
+ 11
+ O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+Graphs with 15 nodes enumerated with respect to number of edges.
+
+ eval(ZeroOrOne ,graphs 15)
+ 2 3 4 5 6 7 8 9 10
+ 1 + x + 2x + 5x + 11x + 26x + 68x + 177x + 496x + 1471x + 4583x
+ +
+ 11
+ O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+Necklaces with 7 green beads, 8 white beads, 5 yellow beads and 10
+red beads.
+
+ cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)
+ 49958972383320
+ Type: Fraction Integer
+
+The operation SFunction is the S-function or Schur function of a
+partition written as a descending list of integers expressed in terms
+of power sum symmetric functions.
+
+In this case the argument partition represents a tableau shape. For
+example 3,2,2,1 represents a tableau with three boxes in the first
+row, two boxes in the second and third rows, and one box in the fourth
+row. SFunction [3,2,2,1] counts the number of different tableaux of
+shape 3, 2, 2, 1 filled with objects with an ascending order in the
+columns and a non-descending order in the rows.
+
+ sf3221:= SFunction [3,2,2,1]
+ 1 1 2 1 2 1 1 4 1 2
+ -- (6 2) - -- (6 1 ) - -- (4 ) + -- (4 3 1) + -- (4 1 ) - -- (3 2)
+ 12 12 16 12 24 36
+ +
+ 1 2 2 1 2 1 3 1 5 1 4 1 3 2
+ -- (3 1 ) - -- (3 2 1) - -- (3 2 1 ) - -- (3 1 ) - --- (2 ) + -- (2 1 )
+ 36 24 36 72 192 48
+ +
+ 1 2 4 1 6 1 8
+ -- (2 1 ) - --- (2 1 ) + --- (1 )
+ 96 144 576
+ Type: SymmetricPolynomial Fraction Integer
+
+This is the number filled with a a b b c c d d.
+
+ cap(sf3221,complete 2**4)
+ 3
+ Type: Fraction Integer
+
+The configurations enumerated above are:
+
+ a a b a a c a a d
+ b c b b b b
+ c d c d c c
+ d d d
+
+This is the number of tableaux filled with 1..8.
+
+ cap(sf3221, powerSum 1**8)
+ 70
+ Type: Fraction Integer
+
+The coefficient of x^n is the number of column strict reverse plane
+partitions of n of shape 3 2 2 1.
+
+ eval(Integers, sf3221)
+ 9 10 11 12 13 14 15 16 17
+ x + 3x + 7x + 14x + 27x + 47x + 79x + 126x + 196x
+ +
+ 18 19 20
+ 294x + 432x + O(x )
+ Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+
+The smallest is
+
+ 0 0 0
+ 1 1
+ 2 2
+ 3
+
+See Also:
+o )show CycleIndicators
+o $AXIOM/doc/src/algebra/cycles.spad.dvi
+
+@
<<package CYCLES CycleIndicators>>=
)abbrev package CYCLES CycleIndicators
++ Polya-Redfield enumeration by cycle indices.
diff --git a/src/algebra/derham.spad.pamphlet b/src/algebra/derham.spad.pamphlet
index eb06ae8..47d2db1 100644
--- a/src/algebra/derham.spad.pamphlet
+++ b/src/algebra/derham.spad.pamphlet
@@ -307,6 +307,584 @@ AntiSymm(R:Ring, lVar:List Symbol): Export == Implement
where
@
\section{domain DERHAM DeRhamComplex}
+<<DeRhamComplex.input>>=
+-- derham.spad.pamphlet DeRhamComplex.input
+)spool DeRhamComplex.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 34
+coefRing := Integer
+--R
+--R
+--R (1) Integer
+--R Type:
Domain
+--E 1
+
+--S 2 of 34
+lv : List Symbol := [x,y,z]
+--R
+--R
+--R (2) [x,y,z]
+--R Type: List
Symbol
+--E 2
+
+--S 3 of 34
+der := DERHAM(coefRing,lv)
+--R
+--R
+--R (3) DeRhamComplex(Integer,[x,y,z])
+--R Type:
Domain
+--E 3
+
+--S 4 of 34
+R := Expression coefRing
+--R
+--R
+--R (4) Expression Integer
+--R Type:
Domain
+--E 4
+
+--S 5 of 34
+f : R := x**2*y*z-5*x**3*y**2*z**5
+--R
+--R
+--R 3 2 5 2
+--R (5) - 5x y z + x y z
+--R Type: Expression
Integer
+--E 5
+
+--S 6 of 34
+g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2
+--R
+--R
+--R 2 3 2 2
+--R (6) - 7z sin(x y ) + y z cos(z)
+--R Type: Expression
Integer
+--E 6
+
+--S 7 of 34
+h : R :=x*y*z-2*x**3*y*z**2
+--R
+--R
+--R 3 2
+--R (7) - 2x y z + x y z
+--R Type: Expression
Integer
+--E 7
+
+--S 8 of 34
+dx : der := generator(1)
+--R
+--R
+--R (8) dx
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 8
+
+--S 9 of 34
+dy : der := generator(2)
+--R
+--R
+--R (9) dy
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 9
+
+--S 10 of 34
+dz : der := generator(3)
+--R
+--R
+--R (10) dz
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 10
+
+--S 11 of 34
+[dx,dy,dz] := [generator(i)$der for i in 1..3]
+--R
+--R
+--R (11) [dx,dy,dz]
+--R Type: List
DeRhamComplex(Integer,[x,y,z])
+--E 11
+
+--S 12 of 34
+alpha : der := f*dx + g*dy + h*dz
+--R
+--R
+--R (12)
+--R 3 2 2 3 2 2
+--R (- 2x y z + x y z)dz + (- 7z sin(x y ) + y z cos(z))dy
+--R +
+--R 3 2 5 2
+--R (- 5x y z + x y z)dx
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 12
+
+--S 13 of 34
+beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy
+--R
+--R
+--R (13) x dy + cos(tan(x y z) + x y z)dx
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 13
+
+--S 14 of 34
+exteriorDifferential alpha
+--R
+--R
+--R (14)
+--R 2 3 2 3 2
+--R (y z sin(z) + 14z sin(x y ) - 2y z cos(z) - 2x z + x z)dy dz
+--R +
+--R 3 2 4 2 2 2
+--R (25x y z - 6x y z + y z - x y)dx dz
+--R +
+--R 2 2 2 3 2 3 5 2
+--R (- 21x y z cos(x y ) + 10x y z - x z)dx dy
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 14
+
+--S 15 of 34
+exteriorDifferential %
+--R
+--R
+--R (15) 0
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 15
+
+--S 16 of 34
+gamma := alpha * beta
+--R
+--R
+--R (16)
+--R 4 2 2 3 2
+--R (2x y z - x y z)dy dz + (2x y z - x y z)cos(tan(x y z) + x y z)dx dz
+--R +
+--R 2 3 2 2 4 2 5 3
+--R ((7z sin(x y ) - y z cos(z))cos(tan(x y z) + x y z) - 5x y z + x y z)dx
dy
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 16
+
+--S 17 of 34
+exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha *
exteriorDifferential(beta))
+--R
+--R
+--R (17) 0
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 17
+
+--S 18 of 34
+a : BOP := operator('a)
+--R
+--R
+--R (18) a
+--R Type:
BasicOperator
+--E 18
+
+--S 19 of 34
+b : BOP := operator('b)
+--R
+--R
+--R (19) b
+--R Type:
BasicOperator
+--E 19
+
+--S 20 of 34
+c : BOP := operator('c)
+--R
+--R
+--R (20) c
+--R Type:
BasicOperator
+--E 20
+
+--S 21 of 34
+sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz
+--R
+--R
+--R (21) c(x,y,z)dz + b(x,y,z)dy + a(x,y,z)dx
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 21
+
+--S 22 of 34
+theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz
+--R
+--R
+--R (22) c(x,y,z)dy dz + b(x,y,z)dx dz + a(x,y,z)dx dy
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 22
+
+--S 23 of 34
+totalDifferential(a(x,y,z))$der
+--R
+--R
+--R (23) a (x,y,z)dz + a (x,y,z)dy + a (x,y,z)dx
+--R ,3 ,2 ,1
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 23
+
+--S 24 of 34
+exteriorDifferential sigma
+--R
+--R
+--R (24)
+--R (c (x,y,z) - b (x,y,z))dy dz + (c (x,y,z) - a (x,y,z))dx dz
+--R ,2 ,3 ,1 ,3
+--R +
+--R (b (x,y,z) - a (x,y,z))dx dy
+--R ,1 ,2
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 24
+
+--S 25 of 34
+exteriorDifferential theta
+--R
+--R
+--R (25) (c (x,y,z) - b (x,y,z) + a (x,y,z))dx dy dz
+--R ,1 ,2 ,3
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 25
+
+--S 26 of 34
+one : der := 1
+--R
+--R
+--R (26) 1
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 26
+
+--S 27 of 34
+g1 : der := a([x,t,y,u,v,z,e]) * one
+--R
+--R
+--R (27) a(x,t,y,u,v,z,e)
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 27
+
+--S 28 of 34
+h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one
+--R
+--R
+--R (28) a(x,y,x,t,x,z,y,r,u,x)
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 28
+
+--S 29 of 34
+exteriorDifferential g1
+--R
+--R
+--R (29) a (x,t,y,u,v,z,e)dz + a (x,t,y,u,v,z,e)dy + a (x,t,y,u,v,z,e)dx
+--R ,6 ,3 ,1
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 29
+
+--S 30 of 34
+exteriorDifferential h1
+--R
+--R
+--R (30)
+--R a (x,y,x,t,x,z,y,r,u,x)dz
+--R ,6
+--R +
+--R (a (x,y,x,t,x,z,y,r,u,x) + a (x,y,x,t,x,z,y,r,u,x))dy
+--R ,7 ,2
+--R +
+--R a (x,y,x,t,x,z,y,r,u,x) + a (x,y,x,t,x,z,y,r,u,x)
+--R ,10 ,5
+--R +
+--R a (x,y,x,t,x,z,y,r,u,x) + a (x,y,x,t,x,z,y,r,u,x)
+--R ,3 ,1
+--R *
+--R dx
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 30
+
+--S 31 of 34
+coefficient(gamma, dx*dy)
+--R
+--R
+--R 2 3 2 2 4 2 5 3
+--R (31) (7z sin(x y ) - y z cos(z))cos(tan(x y z) + x y z) - 5x y z + x y
z
+--R Type: Expression
Integer
+--E 31
+
+--S 32 of 34
+coefficient(gamma, one)
+--R
+--R
+--R (32) 0
+--R Type: Expression
Integer
+--E 32
+
+--S 33 of 34
+coefficient(g1,one)
+--R
+--R
+--R (33) a(x,t,y,u,v,z,e)
+--R Type: Expression
Integer
+--E 33
+
+--S 34 of 34
+gamma := alpha * beta
+--R
+--R
+--R (34)
+--R 4 2 2 3 2
+--R (2x y z - x y z)dy dz + (2x y z - x y z)cos(tan(x y z) + x y z)dx dz
+--R +
+--R 2 3 2 2 4 2 5 3
+--R ((7z sin(x y ) - y z cos(z))cos(tan(x y z) + x y z) - 5x y z + x y z)dx
dy
+--R Type:
DeRhamComplex(Integer,[x,y,z])
+--E 34
+)spool
+)lisp (bye)
+@
+<<DeRhamComplex.help>>=
+====================================================================
+DeRhamComplex
+====================================================================
+
+The domain constructor DeRhamComplex creates the class of differential
+forms of arbitrary degree over a coefficient ring. The De Rham
+complex constructor takes two arguments: a ring, coefRing, and a list
+of coordinate variables.
+
+This is the ring of coefficients.
+
+ coefRing := Integer
+ Integer
+ Type: Domain
+
+These are the coordinate variables.
+
+ lv : List Symbol := [x,y,z]
+ [x,y,z]
+ Type: List Symbol
+
+
+This is the De Rham complex of Euclidean three-space using coordinates
+x, y and z.
+
+ der := DERHAM(coefRing,lv)
+ DeRhamComplex(Integer,[x,y,z])
+ Type: Domain
+
+This complex allows us to describe differential forms having
+expressions of integers as coefficients. These coefficients can
+involve any number of variables, for example, f(x,t,r,y,u,z). As
+we've chosen to work with ordinary Euclidean three-space, expressions
+involving these forms are treated as functions of x, y and z with the
+additional arguments t, r and u regarded as symbolic constants.
+
+Here are some examples of coefficients.
+
+ R := Expression coefRing
+ Expression Integer
+ Type: Domain
+
+ f : R := x**2*y*z-5*x**3*y**2*z**5
+ 3 2 5 2
+ - 5x y z + x y z
+ Type: Expression Integer
+
+ g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2
+ 2 3 2 2
+ - 7z sin(x y ) + y z cos(z)
+ Type: Expression Integer
+
+ h : R :=x*y*z-2*x**3*y*z**2
+ 3 2
+ - 2x y z + x y z
+ Type: Expression Integer
+
+We now define the multiplicative basis elements for the exterior
+algebra over R.
+
+ dx : der := generator(1)
+ dx
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+ dy : der := generator(2)
+ dy
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+ dz : der := generator(3)
+ dz
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+This is an alternative way to give the above assignments.
+
+ [dx,dy,dz] := [generator(i)$der for i in 1..3]
+ [dx,dy,dz]
+ Type: List DeRhamComplex(Integer,[x,y,z])
+
+Now we define some one-forms.
+
+ alpha : der := f*dx + g*dy + h*dz
+ 3 2 2 3 2 2
+ (- 2x y z + x y z)dz + (- 7z sin(x y ) + y z cos(z))dy
+ +
+ 3 2 5 2
+ (- 5x y z + x y z)dx
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+ beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy
+ x dy + cos(tan(x y z) + x y z)dx
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+A well-known theorem states that the composition of exteriorDifferential
+with itself is the zero map for continuous forms. Let's verify this
+theorem for alpha.
+
+ exteriorDifferential alpha
+ 2 3 2 3 2
+ (y z sin(z) + 14z sin(x y ) - 2y z cos(z) - 2x z + x z)dy dz
+ +
+ 3 2 4 2 2 2
+ (25x y z - 6x y z + y z - x y)dx dz
+ +
+ 2 2 2 3 2 3 5 2
+ (- 21x y z cos(x y ) + 10x y z - x z)dx dy
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+We see a lengthy output of the last expression, but nevertheless, the
+composition is zero.
+
+ exteriorDifferential %
+ 0
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+Now we check that exteriorDifferential is a "graded derivation" D,
+that is, D satisfies:
+
+ D(a*b) = D(a)*b + (-1)**degree(a)*a*D(b)
+
+ gamma := alpha * beta
+ 4 2 2 3 2
+ (2x y z - x y z)dy dz + (2x y z - x y z)cos(tan(x y z) + x y z)dx dz
+ +
+ 2 3 2 2 4 2 5 3
+ ((7z sin(x y ) - y z cos(z))cos(tan(x y z) + x y z) - 5x y z + x y z)dx dy
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+We try this for the one-forms alpha and beta.
+
+ exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha *
exteriorDifferential(beta))
+ 0
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+Now we define some "basic operators"
+
+ a : BOP := operator('a)
+ a
+ Type: BasicOperator
+
+ b : BOP := operator('b)
+ b
+ Type: BasicOperator
+
+ c : BOP := operator('c)
+ c
+ Type: BasicOperator
+
+We also define some indeterminate one- and two-forms using these
+operators.
+
+ sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz
+ c(x,y,z)dz + b(x,y,z)dy + a(x,y,z)dx
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+ theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz
+ c(x,y,z)dy dz + b(x,y,z)dx dz + a(x,y,z)dx dy
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+This allows us to get formal definitions for the "gradient" ...
+
+ totalDifferential(a(x,y,z))$der
+ (23) a (x,y,z)dz + a (x,y,z)dy + a (x,y,z)dx
+ ,3 ,2 ,1
+ Type: DeRhamComplex(Integer,[x,y,z])
+the "curl" ...
+
+ exteriorDifferential sigma
+ (c (x,y,z) - b (x,y,z))dy dz + (c (x,y,z) - a (x,y,z))dx dz
+ ,2 ,3 ,1 ,3
+ +
+ (b (x,y,z) - a (x,y,z))dx dy
+ ,1 ,2
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+and the "divergence."
+
+ exteriorDifferential theta
+ (c (x,y,z) - b (x,y,z) + a (x,y,z))dx dy dz
+ ,1 ,2 ,3
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+Note that the De Rham complex is an algebra with unity. This element
+1 is the basis for elements for zero-forms, that is, functions in our
+space.
+
+ one : der := 1
+ 1
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+To convert a function to a function lying in the De Rham complex,
+multiply the function by "one."
+
+ g1 : der := a([x,t,y,u,v,z,e]) * one
+ a(x,t,y,u,v,z,e)
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+A current limitation of Axiom forces you to write functions with more
+than four arguments using square brackets in this way.
+
+ h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one
+ a(x,y,x,t,x,z,y,r,u,x)
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+Now note how the system keeps track of where your coordinate functions
+are located in expressions.
+
+ exteriorDifferential g1
+ a (x,t,y,u,v,z,e)dz + a (x,t,y,u,v,z,e)dy + a (x,t,y,u,v,z,e)dx
+ ,6 ,3 ,1
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+ exteriorDifferential h1
+ a (x,y,x,t,x,z,y,r,u,x)dz
+ ,6
+ +
+ (a (x,y,x,t,x,z,y,r,u,x) + a (x,y,x,t,x,z,y,r,u,x))dy
+ ,7 ,2
+ +
+ a (x,y,x,t,x,z,y,r,u,x) + a (x,y,x,t,x,z,y,r,u,x)
+ ,10 ,5
+ +
+ a (x,y,x,t,x,z,y,r,u,x) + a (x,y,x,t,x,z,y,r,u,x)
+ ,3 ,1
+ *
+ dx
+ Type: DeRhamComplex(Integer,[x,y,z])
+
+In this example of Euclidean three-space, the basis for the De Rham complex
+consists of the eight forms: 1, dx, dy, dz, dx*dy, dx*dz, dy*dz, and dx*dy*dz.
+
+ coefficient(gamma, dx*dy)
+ 2 3 2 2 4 2 5 3
+ (7z sin(x y ) - y z cos(z))cos(tan(x y z) + x y z) - 5x y z + x y z
+ Type: Expression Integer
+
+ coefficient(gamma, one)
+ 0
+ Type: Expression Integer
+
+ coefficient(g1,one)
+ a(x,t,y,u,v,z,e)
+ Type: Expression Integer
+
+See Also:
+o )help Operator
+o )show DeRhamComplex
+o $AXIOM/doc/src/algebra/derham.spad.dvi
+
+@
<<domain DERHAM DeRhamComplex>>=
)abbrev domain DERHAM DeRhamComplex
++ Author: Larry A. Lambe
diff --git a/src/algebra/gaussian.spad.pamphlet
b/src/algebra/gaussian.spad.pamphlet
index ccbd072..744488b 100644
--- a/src/algebra/gaussian.spad.pamphlet
+++ b/src/algebra/gaussian.spad.pamphlet
@@ -529,6 +529,286 @@ ComplexPatternMatch(R, S, CS) : C == T where
@
\section{domain COMPLEX Complex}
+<<Complex.input>>=
+-- gaussian.spad.pamphlet Complex.input
+)spool Complex.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 16
+a := complex(4/3,5/2)
+--R
+--R
+--R 4 5
+--R (1) - + - %i
+--R 3 2
+--R Type: Complex Fraction
Integer
+--E 1
+
+--S 2 of 16
+b := complex(4/3,-5/2)
+--R
+--R
+--R 4 5
+--R (2) - - - %i
+--R 3 2
+--R Type: Complex Fraction
Integer
+--E 2
+
+--S 3 of 16
+a + b
+--R
+--R
+--R 8
+--R (3) -
+--R 3
+--R Type: Complex Fraction
Integer
+--E 3
+
+--S 4 of 16
+a - b
+--R
+--R
+--R (4) 5%i
+--R Type: Complex Fraction
Integer
+--E 4
+
+--S 5 of 16
+a * b
+--R
+--R
+--R 289
+--R (5) ---
+--R 36
+--R Type: Complex Fraction
Integer
+--E 5
+
+--S 6 of 16
+a / b
+--R
+--R
+--R 161 240
+--R (6) - --- + --- %i
+--R 289 289
+--R Type: Complex Fraction
Integer
+--E 6
+
+--S 7 of 16
+% :: Fraction Complex Integer
+--R
+--R
+--R - 15 + 8%i
+--R (7) ----------
+--R 15 + 8%i
+--R Type: Fraction Complex
Integer
+--E 7
+
+--S 8 of 16
+3.4 + 6.7 * %i
+--R
+--R
+--R (8) 3.4 + 6.7 %i
+--R Type: Complex
Float
+--E 8
+
+--S 9 of 16
+conjugate a
+--R
+--R
+--R 4 5
+--R (9) - - - %i
+--R 3 2
+--R Type: Complex Fraction
Integer
+--E 9
+
+--S 10 of 16
+norm a
+--R
+--R
+--R 289
+--R (10) ---
+--R 36
+--R Type: Fraction
Integer
+--E 10
+
+--S 11 of 16
+real a
+--R
+--R
+--R 4
+--R (11) -
+--R 3
+--R Type: Fraction
Integer
+--E 11
+
+--S 12 of 16
+imag a
+--R
+--R
+--R 5
+--R (12) -
+--R 2
+--R Type: Fraction
Integer
+--E 12
+
+--S 13 of 16
+gcd(13 - 13*%i,31 + 27*%i)
+--R
+--R
+--R (13) 5 + %i
+--R Type: Complex
Integer
+--E 13
+
+--S 14 of 16
+lcm(13 - 13*%i,31 + 27*%i)
+--R
+--R
+--R (14) 143 - 39%i
+--R Type: Complex
Integer
+--E 14
+
+--S 15 of 16
+factor(13 - 13*%i)
+--R
+--R
+--R (15) - (1 + %i)(2 + 3%i)(3 + 2%i)
+--R Type: Factored Complex
Integer
+--E 15
+
+--S 16 of 16
+factor complex(2,0)
+--R
+--R
+--R 2
+--R (16) - %i (1 + %i)
+--R Type: Factored Complex
Integer
+--E 16
+)spool
+)lisp (bye)
+@
+<<Complex.help>>=
+====================================================================
+Complex
+====================================================================
+
+The Complex constructor implements complex objects over a commutative
+ring R. Typically, the ring R is Integer, Fraction Integer, Float or
+DoubleFloat. R can also be a symbolic type, like Polynomial Integer.
+
+Complex objects are created by the complex operation.
+
+ a := complex(4/3,5/2)
+ 4 5
+ - + - %i
+ 3 2
+ Type: Complex Fraction Integer
+
+ b := complex(4/3,-5/2)
+ 4 5
+ - - - %i
+ 3 2
+ Type: Complex Fraction Integer
+
+The standard arithmetic operations are available.
+
+ a + b
+ 8
+ -
+ 3
+ Type: Complex Fraction Integer
+
+ a - b
+ 5%i
+ Type: Complex Fraction Integer
+
+ a * b
+ 289
+ ---
+ 36
+ Type: Complex Fraction Integer
+
+If R is a field, you can also divide the complex objects.
+
+ a / b
+ 161 240
+ - --- + --- %i
+ 289 289
+ Type: Complex Fraction Integer
+
+
+We can view the last object as a fraction of complex integers.
+
+ % :: Fraction Complex Integer
+ - 15 + 8%i
+ ----------
+ 15 + 8%i
+ Type: Fraction Complex Integer
+
+The predefined macro %i is defined to be complex(0,1).
+
+ 3.4 + 6.7 * %i
+ 3.4 + 6.7 %i
+ Type: Complex Float
+
+You can also compute the conjugate and norm of a complex number.
+
+ conjugate a
+ 4 5
+ - - - %i
+ 3 2
+ Type: Complex Fraction Integer
+
+ norm a
+ 289
+ ---
+ 36
+ Type: Fraction Integer
+
+The real and imag operations are provided to extract the real and
+imaginary parts, respectively.
+
+ real a
+ 4
+ -
+ 3
+ Type: Fraction Integer
+
+ imag a
+ 5
+ -
+ 2
+ Type: Fraction Integer
+
+The domain Complex Integer is also called the Gaussian integers. If R
+is the integers (or, more generally, a EuclideanDomain), you can compute
+greatest common divisors.
+
+ gcd(13 - 13*%i,31 + 27*%i)
+ 5 + %i
+ Type: Complex Integer
+
+You can also compute least common multiples.
+
+ lcm(13 - 13*%i,31 + 27*%i)
+ 143 - 39%i
+ Type: Complex Integer
+
+You can factor Gaussian integers.
+
+ factor(13 - 13*%i)
+ - (1 + %i)(2 + 3%i)(3 + 2%i)
+ Type: Factored Complex Integer
+
+ factor complex(2,0)
+ 2
+ - %i (1 + %i)
+ Type: Factored Complex Integer
+
+See Also
+o )show Complex
+o $AXIOM/doc/src/algebra/gaussian.spad.dvi
+
+@
<<domain COMPLEX Complex>>=
)abbrev domain COMPLEX Complex
++ Author:
diff --git a/src/algebra/list.spad.pamphlet b/src/algebra/list.spad.pamphlet
index d773bb6..1c534e9 100644
--- a/src/algebra/list.spad.pamphlet
+++ b/src/algebra/list.spad.pamphlet
@@ -671,6 +671,194 @@ ListToMap(A:SetCategory, B:Type): Exports ==
Implementation where
@
\section{domain ALIST AssociationList}
+<<AssociationList.input>>=
+-- list.spad.pamphlet AssociationList.input
+)spool AssociationList.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 10
+Data := Record(monthsOld : Integer, gender : String)
+--R
+--R
+--R (1) Record(monthsOld: Integer,gender: String)
+--R Type:
Domain
+--E 1
+
+--S 2 of 10
+al : AssociationList(String,Data)
+--R
+--R Type:
Void
+--E 2
+
+--S 3 of 10
+al := table()
+--R
+--R
+--R (3) table()
+--R Type: AssociationList(String,Record(monthsOld: Integer,gender:
String))
+--E 3
+
+--S 4 of 10
+al."bob" := [407,"male"]$Data
+--R
+--R
+--R (4) [monthsOld= 407,gender= "male"]
+--R Type: Record(monthsOld: Integer,gender:
String)
+--E 4
+
+--S 5 of 10
+al."judith" := [366,"female"]$Data
+--R
+--R
+--R (5) [monthsOld= 366,gender= "female"]
+--R Type: Record(monthsOld: Integer,gender:
String)
+--E 5
+
+--S 6 of 10
+al."katie" := [24,"female"]$Data
+--R
+--R
+--R (6) [monthsOld= 24,gender= "female"]
+--R Type: Record(monthsOld: Integer,gender:
String)
+--E 6
+
+--S 7 of 10
+al."smokie" := [200,"female"]$Data
+--R
+--R
+--R (7) [monthsOld= 200,gender= "female"]
+--R Type: Record(monthsOld: Integer,gender:
String)
+--E 7
+
+--S 8 of 10
+al
+--R
+--R
+--R (8)
+--R table
+--R "smokie"= [monthsOld= 200,gender= "female"]
+--R ,
+--R "katie"= [monthsOld= 24,gender= "female"]
+--R ,
+--R "judith"= [monthsOld= 366,gender= "female"]
+--R ,
+--R "bob"= [monthsOld= 407,gender= "male"]
+--R Type: AssociationList(String,Record(monthsOld: Integer,gender:
String))
+--E 8
+
+--S 9 of 10
+al."katie" := [23,"female"]$Data
+--R
+--R
+--R (9) [monthsOld= 23,gender= "female"]
+--R Type: Record(monthsOld: Integer,gender:
String)
+--E 9
+
+--S 10 of 10
+delete!(al,1)
+--R
+--R
+--R (10)
+--R table
+--R "katie"= [monthsOld= 23,gender= "female"]
+--R ,
+--R "judith"= [monthsOld= 366,gender= "female"]
+--R ,
+--R "bob"= [monthsOld= 407,gender= "male"]
+--R Type: AssociationList(String,Record(monthsOld: Integer,gender:
String))
+--E 10
+)spool
+)lisp (bye)
+@
+<<AssociationList.help>>=
+====================================================================
+AssociationList examples
+====================================================================
+
+The AssociationList constructor provides a general structure for
+associative storage. This type provides association lists in which
+data objects can be saved according to keys of any type. For a given
+association list, specific types must be chosen for the keys and
+entries. You can think of the representation of an association list
+as a list of records with key and entry fields.
+
+Association lists are a form of table and so most of the operations
+available for Table are also available for AssociationList. They can
+also be viewed as lists and can be manipulated accordingly.
+
+This is a Record type with age and gender fields.
+
+ Data := Record(monthsOld : Integer, gender : String)
+ Record(monthsOld: Integer,gender: String)
+ Type: Domain
+
+In this expression, al is declared to be an association
+list whose keys are strings and whose entries are the above records.
+
+ al : AssociationList(String,Data)
+ Type: Void
+
+The table operation is used to create an empty association list.
+
+ al := table()
+ table()
+ Type: AssociationList(String,Record(monthsOld: Integer,gender: String))
+
+You can use assignment syntax to add things to the association list.
+
+ al."bob" := [407,"male"]$Data
+ [monthsOld=407, gender= "male"]
+ Type: Record(monthsOld: Integer,gender: String)
+
+ al."judith" := [366,"female"]$Data
+ [monthsOld=366, gender= "female"]
+ Type: Record(monthsOld: Integer,gender: String)
+
+ al."katie" := [24,"female"]$Data
+ [monthsOld=24, gender= "female"]
+ Type: Record(monthsOld: Integer,gender: String)
+
+Perhaps we should have included a species field.
+
+ al."smokie" := [200,"female"]$Data
+ [monthsOld=200, gender= "female"]
+ Type: Record(monthsOld: Integer,gender: String)
+
+Now look at what is in the association list. Note that the last-added
+(key, entry) pair is at the beginning of the list.
+
+ al
+ table("smokie" = [monthsOld=200, gender= "female"],
+ "katie" = [monthsOld=24, gender= "female"],
+ "judith" = [monthsOld=366, gender= "female"],
+ "bob" = [monthsOld=407, gender= "male"])
+ Type: AssociationList(String,Record(monthsOld: Integer,gender: String))
+
+You can reset the entry for an existing key.
+
+ al."katie" := [23,"female"]$Data
+ [monthsOld=23, gender= "female"]
+ Type: Record(monthsOld: Integer,gender: String)
+
+Use delete! to destructively remove an element of the association
+list. Use delete to return a copy of the association list with the
+element deleted. The second argument is the index of the element to
+delete.
+
+ delete!(al,1)
+ table("katie" = [monthsOld=23, gender= "female"],
+ "judith" = [monthsOld=366, gender= "female"],
+ "bob" = [monthsOld=407, gender= "male"])
+ Type: AssociationList(String,Record(monthsOld: Integer,gender: String))
+
+See Also:
+o )help Table
+o )help List
+o )show AssociationList
+o $AXIOM/doc/src/algebra/list.spad.dvi
+
+@
<<domain ALIST AssociationList>>=
)abbrev domain ALIST AssociationList
++ Author:
diff --git a/src/algebra/op.spad.pamphlet b/src/algebra/op.spad.pamphlet
index 5b93a6a..e443d3c 100644
--- a/src/algebra/op.spad.pamphlet
+++ b/src/algebra/op.spad.pamphlet
@@ -10,6 +10,310 @@
\tableofcontents
\eject
\section{domain BOP BasicOperator}
+<<BasicOperator.input>>=
+-- op.spad.pamphlet BasicOperator.input
+)spool BasicOperator.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 18
+y := operator 'y
+--R
+--R
+--R (1) y
+--R Type:
BasicOperator
+--E 1
+
+--S 2 of 18
+deq := D(y x, x, 2) + D(y x, x) + y x = 0
+--R
+--R
+--R ,, ,
+--R (2) y (x) + y (x) + y(x)= 0
+--R
+--R Type: Equation Expression
Integer
+--E 2
+
+--S 3 of 18
+solve(deq, y, x)
+--R
+--R
+--R x x
+--R +-+ - - - - +-+
+--R x\|3 2 2 x\|3
+--R (3) [particular= 0,basis= [cos(-----)%e ,%e sin(-----)]]
+--R 2 2
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
+--E 3
+
+--S 4 of 18
+nary? y
+--R
+--R
+--R (4) true
+--R Type:
Boolean
+--E 4
+
+--S 5 of 18
+unary? y
+--R
+--R
+--R (5) false
+--R Type:
Boolean
+--E 5
+
+--S 6 of 18
+opOne := operator('opOne, 1)
+--R
+--R
+--R (6) opOne
+--R Type:
BasicOperator
+--E 6
+
+--S 7 of 18
+nary? opOne
+--R
+--R
+--R (7) false
+--R Type:
Boolean
+--E 7
+
+--S 8 of 18
+unary? opOne
+--R
+--R
+--R (8) true
+--R Type:
Boolean
+--E 8
+
+--S 9 of 18
+arity opOne
+--R
+--R
+--R (9) 1
+--R Type:
Union(NonNegativeInteger,...)
+--E 9
+
+--S 10 of 18
+name opOne
+--R
+--R
+--R (10) opOne
+--R Type:
Symbol
+--E 10
+
+--S 11 of 18
+is?(opOne, 'z2)
+--R
+--R
+--R (11) false
+--R Type:
Boolean
+--E 11
+
+--S 12 of 18
+is?(opOne, "opOne")
+--R
+--R
+--R (12) true
+--R Type:
Boolean
+--E 12
+
+--S 13 of 18
+properties y
+--R
+--R
+--R (13) table()
+--R Type:
AssociationList(String,None)
+--E 13
+
+--S 14 of 18
+setProperty(y, "use", "unknown function" :: None )
+--R
+--R
+--R (14) y
+--R Type:
BasicOperator
+--E 14
+
+--S 15 of 18
+properties y
+--R
+--R
+--R (15) table("use"= NONE)
+--R Type:
AssociationList(String,None)
+--E 15
+
+--S 16 of 18
+property(y, "use") :: None pretend String
+--R
+--R
+--R (16) "unknown function"
+--R Type:
String
+--E 16
+
+--S 17 of 18
+deleteProperty!(y, "use")
+--R
+--R
+--R (17) y
+--R Type:
BasicOperator
+--E 17
+
+--S 18 of 18
+properties y
+--R
+--R
+--R (18) table()
+--R Type:
AssociationList(String,None)
+--E 18
+)spool
+)lisp (bye)
+@
+<<BasicOperator.help>>=
+====================================================================
+BasicOperator examples
+====================================================================
+
+A basic operator is an object that can be symbolically applied to a
+list of arguments from a set, the result being a kernel over that set
+or an expression.
+
+You create an object of type BasicOperator by using the operator
+operation. This first form of this operation has one argument and it
+must be a symbol. The symbol should be quoted in case the name has
+been used as an identifier to which a value has been assigned.
+
+A frequent application of BasicOperator is the creation of an operator
+to represent the unknown function when solving a differential equation.
+
+Let y be the unknown function in terms of x.
+
+ y := operator 'y
+ y
+ Type: BasicOperator
+
+This is how you enter the equation y'' + y' + y = 0.
+
+ deq := D(y x, x, 2) + D(y x, x) + y x = 0
+ ,, ,
+ y (x) + y (x) + y(x) = 0
+ Type: Equation Expression Integer
+
+To solve the above equation, enter this.
+
+ solve(deq, y, x)
+ x x
+ +-+ - - - - +-+
+ x\|3 2 2 x\|3
+ [particular= 0,basis= [cos(-----)%e ,%e sin(-----)]]
+ 2 2
+ Type: Union(Record(particular: Expression Integer,
+ basis: List Expression Integer),...)
+
+Use the single argument form of BasicOperator (as above) when you
+intend to use the operator to create functional expressions with an
+arbitrary number of arguments
+
+Nary means an arbitrary number of arguments can be used in the
+functional expressions.
+
+ nary? y
+ true
+ Type: Boolean
+
+ unary? y
+ false
+ Type: Boolean
+
+Use the two-argument form when you want to restrict the number of
+arguments in the functional expressions created with the operator.
+
+This operator can only be used to create functional expressions
+with one argument.
+
+ opOne := operator('opOne, 1)
+ opOne
+ Type: BasicOperator
+
+ nary? opOne
+ false
+ Type: Boolean
+
+ unary? opOne
+ true
+ Type: Boolean
+
+Use arity to learn the number of arguments that can be used. It
+returns "false" if the operator is nary.
+
+ arity opOne
+ 1
+ Type: Union(NonNegativeInteger,...)
+
+Use name to learn the name of an operator.
+
+ name opOne
+ opOne
+ Type: Symbol
+
+Use is? to learn if an operator has a particular name.
+
+ is?(opOne, 'z2)
+ false
+ Type: Boolean
+
+You can also use a string as the name to be tested against.
+
+ is?(opOne, "opOne")
+ true
+ Type: Boolean
+
+You can attached named properties to an operator. These are rarely
+used at the top-level of the Axiom interactive environment but are
+used with Axiom library source code.
+
+By default, an operator has no properties.
+
+ properties y
+ table()
+ Type: AssociationList(String,None)
+
+The interface for setting and getting properties is somewhat awkward
+because the property values are stored as values of type None.
+
+Attach a property by using setProperty.
+
+ setProperty(y, "use", "unknown function" :: None )
+ y
+ Type: BasicOperator
+
+ properties y
+ table("use"=NONE)
+ Type: AssociationList(String,None)
+
+We know the property value has type String.
+
+ property(y, "use") :: None pretend String
+ "unknown function"
+ Type: String
+
+Use deleteProperty! to destructively remove a property.
+
+ deleteProperty!(y, "use")
+ y
+ Type: BasicOperator
+
+ properties y
+ table()
+ Type: AssociationList(String,None)
+
+
+See Also
+o )help Expression
+o )help Kernel
+o )show BasicOperator
+o $AXIOM/doc/src/algebra/op.spad.dvi
+
+@
<<domain BOP BasicOperator>>=
)abbrev domain BOP BasicOperator
++ Basic system operators
diff --git a/src/algebra/radix.spad.pamphlet b/src/algebra/radix.spad.pamphlet
index d26f7b5..f35780e 100644
--- a/src/algebra/radix.spad.pamphlet
+++ b/src/algebra/radix.spad.pamphlet
@@ -255,6 +255,168 @@ RadixExpansion(bb): Exports == Implementation where
@
\section{domain BINARY BinaryExpansion}
+<<BinaryExpansion.input>>=
+-- radix.spad.pamphlet BinaryExpansion.input
+)spool BinaryExpansion.output
+)set message test on
+)set message auto off
+)clear all
+--S 1
+r := binary(22/7)
+--R
+--R
+--R ___
+--R (1) 11.001
+--R Type:
BinaryExpansion
+--E 1
+
+--S 2
+r + binary(6/7)
+--R
+--R
+--R (2) 100
+--R Type:
BinaryExpansion
+--E 2
+
+--S 3
+[binary(1/i) for i in 102..106]
+--R
+--R
+--R (3)
+--R ________ ___________________________________________________
+--R [0.000000101, 0.000000100111110001000101100101111001110010010101001,
+--R ____________ ____________
+--R 0.000000100111011, 0.000000100111,
+--R ____________________________________________________
+--R 0.00000010011010100100001110011111011001010110111100011]
+--R Type: List
BinaryExpansion
+--E 3
+
+--S 4
+binary(1/1007)
+--R
+--R
+--R (4)
+--R 0.
+--R OVERBAR
+--R
00000000010000010001010010010111100000111111000010111111001011000111110
+--R
100010011100100110011000110010010101011110110100110000000011000011001
+--R
111011100011010001011110100100011110110000101011101110011101010111001
+--R
100101001011100000001110001111001000000100100100110111001010100111010
+--R
001101110110101110001001000001100101101100000010110010111110001010000
+--R
010101010110101100000110110111010010101111111010111010100110010000101
+--R 0011011000100110001000100001000011000111010011110001
+--R Type:
BinaryExpansion
+--E 4
+
+--S 5
+p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9)
+--R
+--R
+--R 2 __ ______
+--R (5) 0.01x + 0.10x + 0.011100
+--R Type: Polynomial
BinaryExpansion
+--E 5
+
+--S 6
+q := D(p, x)
+--R
+--R
+--R __
+--R (6) 0.1x + 0.10
+--R Type: Polynomial
BinaryExpansion
+--E 6
+
+--S 7
+g := gcd(p, q)
+--R
+--R
+--R __
+--R (7) x + 1.01
+--R Type: Polynomial
BinaryExpansion
+--E 7
+)spool
+)lisp (bye)
+@
+<<BinaryExpansion.help>>=
+====================================================================
+BinaryExpansion examples
+====================================================================
+All rational numbers have repeating binary expansions. Operations to
+access the individual bits of a binary expansion can be obtained by
+converting the value to RadixExpansion(2). More examples of
+expansions are available with
+
+The expansion (of type BinaryExpansion) of a rational number is
+returned by the binary operation.
+
+ r := binary(22/7)
+ ___
+ 11.001
+ Type: BinaryExpansion
+
+Arithmetic is exact.
+
+ r + binary(6/7)
+ 100
+ Type: BinaryExpansion
+
+The period of the expansion can be short or long.
+
+ [binary(1/i) for i in 102..106]
+ ________
+ [0.00000101,
+ ___________________________________________________
+ 0.000000100111110001000101100101111001110010010101001,
+ ____________ ____________
+ 0.000000100111011, 0.000000100111,
+ ____________________________________________________
+ 0.00000010011010100100001110011111011001010110111100011]
+ Type: List BinaryExpansion
+
+or very long.
+
+ binary(1/1007)
+ ________________________________________________________________________
+ 0.000000000100000100010100100101111000001111110000101111110010110001111101
+ ________________________________________________________________________
+ 000100111001001100110001100100101010111101101001100000000110000110011110
+ ________________________________________________________________________
+ 111000110100010111101001000111101100001010111011100111010101110011001010
+ ________________________________________________________________________
+ 010111000000011100011110010000001001001001101110010101001110100011011101
+ ________________________________________________________________________
+ 101011100010010000011001011011000000101100101111100010100000101010101101
+ ________________________________________________________________________
+ 011000001101101110100101011111110101110101001100100001010011011000100110
+ ____________________________________
+ 001000100001000011000111010011110001
+ Type: BinaryExpansion
+
+These numbers are bona fide algebraic objects.
+
+ p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9)
+ __ ______
+ 0.01 x^2 +0.10 x + 0.011100
+ Type: Polynomial BinaryExpansion
+
+ q := D(p, x)
+ __
+ 0.1 x + 0.10
+ Type: Polynomial BinaryExpansion
+
+ g := gcd(p, q)
+ __
+ x+1.01
+ Type: Polynomial BinaryExpansion
+
+See Also:
+o )help DecimalExpansion
+o )help HexadecimalExpansion
+o )show BinaryExpansion
+o $AXIOM/doc/src/algebra/radix.spad.dvi
+
+@
<<domain BINARY BinaryExpansion>>=
)abbrev domain BINARY BinaryExpansion
++ Author: Clifton J. Williamson
@@ -288,6 +450,157 @@ BinaryExpansion(): Exports == Implementation where
@
\section{domain DECIMAL DecimalExpansion}
+<<DecimalExpansion.input>>=
+-- radix.spad.pamphlet DecimalExpansion.input
+)spool DecimalExpansion.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 7
+r := decimal(22/7)
+--R
+--R
+--R ______
+--R (1) 3.142857
+--R Type:
DecimalExpansion
+--E 1
+
+--S 2 of 7
+r + decimal(6/7)
+--R
+--R
+--R (2) 4
+--R Type:
DecimalExpansion
+--E 2
+
+--S 3 of 7
+[decimal(1/i) for i in 350..354]
+--R
+--R
+--R (3)
+--R ______ ______ __ ________________________________
+--R [0.00285714, 0.002849, 0.0028409, 0.00283286118980169971671388101983,
+--R __________________________________________________________
+--R 0.00282485875706214689265536723163841807909604519774011299435]
+--R Type: List
DecimalExpansion
+--E 3
+
+--S 4 of 7
+decimal(1/2049)
+--R
+--R
+--R (4)
+--R 0.
+--R OVERBAR
+--R
00048804294777940458760370912640312347486578818936066373840897999023914
+--R
104441190824792581747193753050268423621278672523182040019521717911176
+--R
183504148365056124938994631527574426549536359199609565641776476329917
+--R
032698877501220107369448511469009272816007808687164470473401659346022
+--R 449975597852611029770619814543679843826256710590531966813079551
+--R Type:
DecimalExpansion
+--E 4
+
+--S 5 of 7
+p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9)
+--R
+--R
+--R 2 _ _
+--R (5) 0.25x + 0.6x + 0.4
+--R Type: Polynomial
DecimalExpansion
+--E 5
+
+--S 6 of 7
+q := differentiate(p, x)
+--R
+--R
+--R _
+--R (6) 0.5x + 0.6
+--R Type: Polynomial
DecimalExpansion
+--E 6
+
+--S 7 of 7
+g := gcd(p, q)
+--R
+--R
+--R _
+--R (7) x + 1.3
+--R Type: Polynomial
DecimalExpansion
+--E 7
+)spool
+)lisp (bye)
+@
+<<DecimalExpansion.help>>=
+====================================================================
+DecimalExpansion examples
+====================================================================
+
+All rationals have repeating decimal expansions. Operations to access
+the individual digits of a decimal expansion can be obtained by
+converting the value to RadixExpansion(10).
+
+The operation decimal is used to create this expansion of type
+DecimalExpansion.
+
+ r := decimal(22/7)
+ ______
+ 3.142857
+ Type: DecimalExpansion
+
+Arithmetic is exact.
+
+ r + decimal(6/7)
+ 4
+ Type: DecimalExpansion
+
+The period of the expansion can be short or long ...
+
+ [decimal(1/i) for i in 350..354]
+ ______ ______ __ ________________________________
+ [0.00285714, 0.002849, 0.0028409, 0.00283286118980169971671388101983,
+ __________________________________________________________
+ 0.00282485875706214689265536723163841807909604519774011299435]
+ Type: List DecimalExpansion
+
+or very long.
+
+ decimal(1/2049)
+ _______________________________________________________________________
+ 0.00048804294777940458760370912640312347486578818936066373840897999023914
+ _____________________________________________________________________
+ 104441190824792581747193753050268423621278672523182040019521717911176
+ _____________________________________________________________________
+ 183504148365056124938994631527574426549536359199609565641776476329917
+ _____________________________________________________________________
+ 032698877501220107369448511469009272816007808687164470473401659346022
+ _______________________________________________________________
+ 449975597852611029770619814543679843826256710590531966813079551
+ Type: DecimalExpansion
+
+These numbers are bona fide algebraic objects.
+
+ p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9)
+ 2 _ _
+ 0.25x + 0.6x + 0.4
+ Type: Polynomial DecimalExpansion
+
+ q := differentiate(p, x)
+ _
+ 0.5x + 0.6
+ Type: Polynomial DecimalExpansion
+
+ g := gcd(p, q)
+ _
+ x + 1.3
+ Type: Polynomial DecimalExpansion
+
+See Also:
+o )help RadixExpansion
+o )help BinaryExpansion
+o )help HexadecimalExpansion
+o )show DecimalExpansion
+o $AXIOM/doc/src/algebra/radix.spad.dvi
+
+@
<<domain DECIMAL DecimalExpansion>>=
)abbrev domain DECIMAL DecimalExpansion
++ Author: Stephen M. Watt
diff --git a/src/algebra/string.spad.pamphlet b/src/algebra/string.spad.pamphlet
index 6fb42a5..fe270fc 100644
--- a/src/algebra/string.spad.pamphlet
+++ b/src/algebra/string.spad.pamphlet
@@ -10,6 +10,208 @@
\tableofcontents
\eject
\section{domain CHAR Character}
+<<Character.input>>=
+-- string.spad.pamphlet Character.input
+)spool Character.output
+)set message test on
+)set message auto off
+)clear all
+--S 1
+chars := [char "a", char "A", char "X", char "8", char "+"]
+--R
+--R
+--R (1) [a,A,X,8,+]
+--R Type: List
Character
+--E 1
+
+--S 2
+space()
+--R
+--R
+--R (2)
+--R Type:
Character
+--E 2
+
+--S 3
+quote()
+--R
+--R
+--R (3) "
+--R Type:
Character
+--E 3
+
+--S 4
+escape()
+--R
+--R
+--R (4) _
+--R Type:
Character
+--E 4
+
+--S 5
+[ord c for c in chars]
+--R
+--R
+--R (5) [97,65,88,56,43]
+--R Type: List
Integer
+--E 5
+
+--S 6
+[upperCase c for c in chars]
+--R
+--R
+--R (6) [A,A,X,8,+]
+--R Type: List
Character
+--E 6
+
+--S 7
+[lowerCase c for c in chars]
+--R
+--R
+--R (7) [a,a,x,8,+]
+--R Type: List
Character
+--E 7
+
+--S 8
+[alphabetic? c for c in chars]
+--R
+--R
+--R (8) [true,true,true,false,false]
+--R Type: List
Boolean
+--E 8
+
+--S 9
+[upperCase? c for c in chars]
+--R
+--R
+--R (9) [false,true,true,false,false]
+--R Type: List
Boolean
+--E 9
+
+--S 10
+[lowerCase? c for c in chars]
+--R
+--R
+--R (10) [true,false,false,false,false]
+--R Type: List
Boolean
+--E 10
+
+--S 11
+[digit? c for c in chars]
+--R
+--R
+--R (11) [false,false,false,true,false]
+--R Type: List
Boolean
+--E 11
+
+--S 12
+[hexDigit? c for c in chars]
+--R
+--R
+--R (12) [true,true,false,true,false]
+--R Type: List
Boolean
+--E 12
+
+--S 13
+[alphanumeric? c for c in chars]
+--R
+--R
+--R (13) [true,true,true,true,false]
+--R Type: List
Boolean
+--E 13
+)spool
+)lisp (bye)
+@
+<<Character.help>>=
+====================================================================
+Character examples
+====================================================================
+
+The members of the domain Character are values representing letters,
+numerals and other text elements.
+
+Characters can be obtained using String notation.
+
+ chars := [char "a", char "A", char "X", char "8", char "+"]
+ [a,A,X,8,+]
+ Type: List Character
+
+Certain characters are available by name. This is the blank character.
+
+ space()
+
+ Type: Character
+
+This is the quote that is used in strings.
+
+ quote()
+ "
+ Type: Character
+
+This is the escape character that allows quotes and other characters
+within strings.
+
+ escape()
+ _
+ Type: Character
+
+Characters are represented as integers in a machine-dependent way.
+The integer value can be obtained using the ord operation. It is
+always true that char(ord c) = c and ord(char i) = i, provided that i
+is in the range 0..size()$Character-1.
+
+ [ord c for c in chars]
+ [97,65,88,56,43]
+ Type: List Integer
+
+The lowerCase operation converts an upper case letter to the
+corresponding lower case letter. If the argument is not an upper case
+letter, then it is returned unchanged.
+
+ [upperCase c for c in chars]
+ [A,A,X,8,+]
+ Type: List Character
+
+The upperCase operation converts lower case letters to upper case.
+
+ [lowerCase c for c in chars]
+ [a,a,x,8,+]
+ Type: List Character
+
+A number of tests are available to determine whether characters
+belong to certain families.
+
+ [alphabetic? c for c in chars]
+ [true,true,true,false,false]
+ Type: List Boolean
+
+ [upperCase? c for c in chars]
+ [false,true,true,false,false]
+ Type: List Boolean
+
+ [lowerCase? c for c in chars]
+ [true,false,false,false,false]
+ Type: List Boolean
+
+ [digit? c for c in chars]
+ [false,false,false,true,false]
+ Type: List Boolean
+
+ [hexDigit? c for c in chars]
+ [true,true,false,true,false]
+ Type: List Boolean
+
+ [alphanumeric? c for c in chars]
+ [true,true,true,true,false]
+ Type: List Boolean
+
+See Also:
+o )help CharacterClass
+o )help String
+o )show Character
+o $AXIOM/doc/src/algebra/string.spad.dvi
+
+@
<<domain CHAR Character>>=
)abbrev domain CHAR Character
++ Author: Stephen M. Watt
@@ -202,6 +404,233 @@ Note that this code is not included in the generated
catdef.spad file.
(MAKEPROP (QUOTE |Character|) (QUOTE NILADIC) T)
@
\section{domain CCLASS CharacterClass}
+<<CharacterClass.input>>=
+-- string.spad.pamphlet CharacterClass.input
+)spool CharacterClass.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 16
+cl1:=charClass[char "a",char "e",char "i",char "o",char "u",char "y"]
+--R
+--R
+--R (1) "aeiouy"
+--R Type:
CharacterClass
+--E 1
+
+--S 2 of 16
+cl2 := charClass "bcdfghjklmnpqrstvwxyz"
+--R
+--R
+--R (2) "bcdfghjklmnpqrstvwxyz"
+--R Type:
CharacterClass
+--E 2
+
+--S 3 of 16
+digit()
+--R
+--R
+--R (3) "0123456789"
+--R Type:
CharacterClass
+--E 3
+
+--S 4 of 16
+hexDigit()
+--R
+--R
+--R (4) "0123456789ABCDEFabcdef"
+--R Type:
CharacterClass
+--E 4
+
+--S 5 of 16
+upperCase()
+--R
+--R
+--R (5) "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+--R Type:
CharacterClass
+--E 5
+
+--S 6 of 16
+lowerCase()
+--R
+--R
+--R (6) "abcdefghijklmnopqrstuvwxyz"
+--R Type:
CharacterClass
+--E 6
+
+--S 7 of 16
+alphabetic()
+--R
+--R
+--R (7) "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
+--R Type:
CharacterClass
+--E 7
+
+--S 8 of 16
+alphanumeric()
+--R
+--R
+--R (8) "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
+--R Type:
CharacterClass
+--E 8
+
+--S 9 of 16
+member?(char "a", cl1)
+--R
+--R
+--R (9) true
+--R Type:
Boolean
+--E 9
+
+--S 10 of 16
+member?(char "a", cl2)
+--R
+--R
+--R (10) false
+--R Type:
Boolean
+--E 10
+
+--S 11 of 16
+intersect(cl1, cl2)
+--R
+--R
+--R (11) "y"
+--R Type:
CharacterClass
+--E 11
+
+--S 12 of 16
+union(cl1,cl2)
+--R
+--R
+--R (12) "abcdefghijklmnopqrstuvwxyz"
+--R Type:
CharacterClass
+--E 12
+
+--S 13 of 16
+difference(cl1,cl2)
+--R
+--R
+--R (13) "aeiou"
+--R Type:
CharacterClass
+--E 13
+
+--S 14 of 16
+intersect(complement(cl1),cl2)
+--R
+--R
+--R (14) "bcdfghjklmnpqrstvwxz"
+--R Type:
CharacterClass
+--E 14
+
+--S 15 of 16
+insert!(char "a", cl2)
+--R
+--R
+--R (15) "abcdfghjklmnpqrstvwxyz"
+--R Type:
CharacterClass
+--E 15
+
+--S 16 of 16
+remove!(char "b", cl2)
+--R
+--R
+--R (16) "acdfghjklmnpqrstvwxyz"
+--R Type:
CharacterClass
+--E 16
+)spool
+)lisp (bye)
+@
+<<CharacterClass.help>>=
+====================================================================
+CharacterClass examples
+====================================================================
+
+The CharacterClass domain allows classes of characters to be defined
+and manipulated efficiently.
+
+Character classes can be created by giving either a string or a list
+of characters.
+
+ cl1:=charClass[char "a",char "e",char "i",char "o",char "u",char "y"]
+ "aeiouy"
+ Type: CharacterClass
+
+ cl2 := charClass "bcdfghjklmnpqrstvwxyz"
+ "bcdfghjklmnpqrstvwxyz"
+ Type: CharacterClass
+
+A number of character classes are predefined for convenience.
+
+ digit()
+ "0123456789"
+ Type: CharacterClass
+
+ hexDigit()
+ "0123456789ABCDEFabcdef"
+ Type: CharacterClass
+
+ upperCase()
+ "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+ Type: CharacterClass
+
+ lowerCase()
+ "abcdefghijklmnopqrstuvwxyz"
+ Type: CharacterClass
+
+ alphabetic()
+ "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
+ Type: CharacterClass
+
+ alphanumeric()
+ "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
+ Type: CharacterClass
+
+You can quickly test whether a character belongs to a class.
+
+ member?(char "a", cl1)
+ true
+ Type: Boolean
+
+ member?(char "a", cl2)
+ false
+ Type: Boolean
+
+Classes have the usual set operations because the CharacterClass
+domain belongs to the category FiniteSetAggregate(Character).
+
+ intersect(cl1, cl2)
+ "y"
+ Type: CharacterClass
+
+ union(cl1,cl2)
+ "abcdefghijklmnopqrstuvwxyz"
+ Type: CharacterClass
+
+ difference(cl1,cl2)
+ "aeiou"
+ Type: CharacterClass
+
+ intersect(complement(cl1),cl2)
+ "bcdfghjklmnpqrstvwxz"
+ Type: CharacterClass
+
+You can modify character classes by adding or removing characters.
+
+ insert!(char "a", cl2)
+ "abcdfghjklmnpqrstvwxyz"
+ Type: CharacterClass
+
+ remove!(char "b", cl2)
+ "acdfghjklmnpqrstvwxyz"
+ Type: CharacterClass
+
+See Also:
+o )help Character
+o )help String
+o )show CharacterClass
+o $AXIOM/doc/src/algebra/string.spad.dvi
+
+@
<<domain CCLASS CharacterClass>>=
)abbrev domain CCLASS CharacterClass
++ Author: Stephen M. Watt
diff --git a/src/algebra/tree.spad.pamphlet b/src/algebra/tree.spad.pamphlet
index ff8ba34..573f7f8 100644
--- a/src/algebra/tree.spad.pamphlet
+++ b/src/algebra/tree.spad.pamphlet
@@ -411,6 +411,210 @@ BinaryTree(S: SetCategory): Exports == Implementation
where
@
\section{domain BSTREE BinarySearchTree}
+<<BinarySearchTree.input>>=
+-- tree.spad.pamphlet BinarySearchTree.input
+)spool BinarySearchTree.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 12
+lv := [8,3,5,4,6,2,1,5,7]
+--R
+--R
+--R (1) [8,3,5,4,6,2,1,5,7]
+--R Type: List
PositiveInteger
+--E 1
+
+--S 2 of 12
+t := binarySearchTree lv
+--R
+--R
+--R (2) [[[1,2,.],3,[4,5,[5,6,7]]],8,.]
+--R Type: BinarySearchTree
PositiveInteger
+--E 2
+
+--S 3 of 12
+emptybst := empty()$BSTREE(INT)
+--R
+--R
+--R (3) []
+--R Type: BinarySearchTree
Integer
+--E 3
+
+--S 4 of 12
+t1 := insert!(8,emptybst)
+--R
+--R
+--R (4) 8
+--R Type: BinarySearchTree
Integer
+--E 4
+
+--S 5 of 12
+insert!(3,t1)
+--R
+--R
+--R (5) [3,8,.]
+--R Type: BinarySearchTree
Integer
+--E 5
+
+--S 6 of 12
+leaves t
+--R
+--R
+--R (6) [1,4,5,7]
+--R Type: List
PositiveInteger
+--E 6
+
+--S 7 of 12
+split(3,t)
+--R
+--R
+--R (7) [less= [1,2,.],greater= [[.,3,[4,5,[5,6,7]]],8,.]]
+--RType: Record(less: BinarySearchTree PositiveInteger,greater:
BinarySearchTree PositiveInteger)
+--E 7
+
+--S 8 of 12
+insertRoot: (INT,BSTREE INT) -> BSTREE INT
+--R
+--R Type:
Void
+--E 8
+
+--S 9 of 12
+insertRoot(x, t) ==
+ a := split(x, t)
+ node(a.less, x, a.greater)
+--R
+--R Type:
Void
+--E 9
+
+--S 10 of 12
+buildFromRoot ls == reduce(insertRoot,ls,emptybst)
+--R
+--R Type:
Void
+--E 10
+
+--S 11 of 12
+rt := buildFromRoot reverse lv
+--R
+--R Compiling function buildFromRoot with type List PositiveInteger ->
+--R BinarySearchTree Integer
+--R Compiling function insertRoot with type (Integer,BinarySearchTree
+--R Integer) -> BinarySearchTree Integer
+--R
+--R (11) [[[1,2,.],3,[4,5,[5,6,7]]],8,.]
+--R Type: BinarySearchTree
Integer
+--E 11
+
+--S 12 of 12
+(t = rt)@Boolean
+--R
+--R
+--R (12) true
+--R Type:
Boolean
+--E 12
+)spool
+)lisp (bye)
+@
+<<BinarySearchTree.help>>=
+====================================================================
+BinarySearchTree examples
+====================================================================
+
+BinarySearchTree(R) is the domain of binary trees with elements of
+type R, ordered across the nodes of the tree. A non-empty binary
+search tree has a value of type R, and right and left binary search
+subtrees. If a subtree is empty, it is displayed as a period (".").
+
+Define a list of values to be placed across the tree. The resulting
+tree has 8 at the root; all other elements are in the left subtree.
+
+ lv := [8,3,5,4,6,2,1,5,7]
+ [8, 3, 5, 4, 6, 2, 1, 5, 7]
+ Type: List PositiveInteger
+
+A convenient way to create a binary search tree is to apply the
+operation binarySearchTree to a list of elements.
+
+ t := binarySearchTree lv
+ [[[1, 2, .], 3, [4, 5, [5, 6, 7]]], 8, .]
+ Type: BinarySearchTree PositiveInteger
+
+Another approach is to first create an empty binary search tree of integers.
+
+ emptybst := empty()$BSTREE(INT)
+ []
+ Type: BinarySearchTree Integer
+
+Insert the value 8. This establishes 8 as the root of the binary
+search tree. Values inserted later that are less than 8 get stored in
+the left subtree, others in the right subtree.
+
+ t1 := insert!(8,emptybst)
+ 8
+ Type: BinarySearchTree Integer
+
+Insert the value 3. This number becomes the root of the left subtree
+of t1. For optimal retrieval, it is thus important to insert the
+middle elements first.
+
+ insert!(3,t1)
+ [3, 8, .]
+ Type: BinarySearchTree Integer
+
+We go back to the original tree t. The leaves of the binary search
+tree are those which have empty left and right subtrees.
+
+ leaves t
+ [1, 4, 5, 7]
+ Type: List PositiveInteger
+
+The operation split(k,t) returns a record containing the two subtrees:
+one with all elements "less" than k, another with elements "greater"
+than k.
+
+ split(3,t)
+ [less=[1, 2, .], greater=[[., 3, [4, 5, [5, 6, 7]]], 8, .]]
+ Type: Record(less: BinarySearchTree PositiveInteger,greater:
+ BinarySearchTree PositiveInteger)
+
+Define insertRoot to insert new elements by creating a new node.
+
+ insertRoot: (INT,BSTREE INT) -> BSTREE INT
+ Type: Void
+
+The new node puts the inserted value between its "less" tree and
+"greater" tree.
+
+
+ insertRoot(x, t) ==
+ a := split(x, t)
+ node(a.less, x, a.greater)
+ Type: Void
+
+
+Function buildFromRoot builds a binary search tree from a list
+of elements ls and the empty tree emptybst.
+
+ buildFromRoot ls == reduce(insertRoot,ls,emptybst)
+ Type: Void
+
+Apply this to the reverse of the list lv.
+
+ rt := buildFromRoot reverse lv
+ [[[1, 2, . ], 3, [4, 5, [5, 6, 7]]], 8, .]
+ Type: BinarySearchTree Integer
+
+Have Axiom check that these are equal.
+
+ (t = rt)@Boolean
+ true
+ Type: Boolean
+
+See Also:
+o )show BinarySearchTree
+o $AXIOM/doc/src/algebra/tree.spad.dvi
+
+@
<<domain BSTREE BinarySearchTree>>=
)abbrev domain BSTREE BinarySearchTree
++ Description: BinarySearchTree(S) is the domain of
@@ -492,6 +696,198 @@ BinaryTournament(S: OrderedSet): Exports ==
Implementation where
@
\section{domain BBTREE BalancedBinaryTree}
+<<BalancedBinaryTree.input>>=
+-- tree.spad.pamphlet BalancedBinaryTree.input
+)spool BalancedBinaryTree.output
+)set message test on
+)set message auto off
+)clear all
+--S 1
+lm := [3,5,7,11]
+--R
+--R
+--R (1) [3,5,7,11]
+--R Type: List
PositiveInteger
+--E 1
+
+--S 2
+modTree(12,lm)
+--R
+--R
+--R (2) [0,2,5,1]
+--R Type: List
Integer
+--E 2
+
+--S 3
+t := balancedBinaryTree(#lm, 0)
+--R
+--R
+--R (3) [[0,0,0],0,[0,0,0]]
+--R Type: BalancedBinaryTree
NonNegativeInteger
+--E 3
+
+--S 4
+setleaves!(t,lm)
+--R
+--R
+--R (4) [[3,0,5],0,[7,0,11]]
+--R Type: BalancedBinaryTree
NonNegativeInteger
+--E 4
+
+--S 5
+mapUp!(t,_*)
+--R
+--R
+--R (5) 1155
+--R Type:
PositiveInteger
+--E 5
+
+--S 6
+t
+--R
+--R
+--R (6) [[3,15,5],1155,[7,77,11]]
+--R Type: BalancedBinaryTree
NonNegativeInteger
+--E 6
+
+--S 7
+mapDown!(t,12,_rem)
+--R
+--R
+--R (7) [[0,12,2],12,[5,12,1]]
+--R Type: BalancedBinaryTree
NonNegativeInteger
+--E 7
+
+--S 8
+leaves %
+--R
+--R
+--R (8) [0,2,5,1]
+--R Type: List
NonNegativeInteger
+--E 8
+
+--S 9
+squares := [x**2 rem m for x in % for m in lm]
+--R
+--R
+--R (9) [0,4,4,1]
+--R Type: List
NonNegativeInteger
+--E 9
+
+--S 10
+chineseRemainder(%,lm)
+--R
+--R
+--R (10) 144
+--R Type:
PositiveInteger
+--E 10
+)spool
+)lisp (bye)
+@
+<<BalancedBinaryTree.help>>=
+====================================================================
+BalancedBinaryTree examples
+====================================================================
+
+BalancedBinaryTrees(S) is the domain of balanced binary trees with
+elements of type S at the nodes. A binary tree is either empty or
+else consists of a node having a value and two branches, each branch a
+binary tree. A balanced binary tree is one that is balanced with
+respect its leaves. One with 2^k leaves is perfectly "balanced": the
+tree has minimum depth, and the left and right branch of every
+interior node is identical in shape.
+
+Balanced binary trees are useful in algebraic computation for
+so-called "divide-and-conquer" algorithms. Conceptually, the data
+for a problem is initially placed at the root of the tree. The
+original data is then split into two subproblems, one for each
+subtree. And so on. Eventually, the problem is solved at the leaves
+of the tree. A solution to the original problem is obtained by some
+mechanism that can reassemble the pieces. In fact, an implementation
+of the Chinese Remainder Algorithm using balanced binary trees was
+first proposed by David Y. Y. Yun at the IBM T. J. Watson Research
+Center in Yorktown Heights, New York, in 1978. It served as the
+prototype for polymorphic algorithms in Axiom.
+
+In what follows, rather than perform a series of computations with a
+single expression, the expression is reduced modulo a number of
+integer primes, a computation is done with modular arithmetic for each
+prime, and the Chinese Remainder Algorithm is used to obtain the
+answer to the original problem. We illustrate this principle with the
+computation of 12^2 = 144.
+
+A list of moduli:
+
+ lm := [3,5,7,11]
+ [3,5,7,11]
+ Type: PositiveInteger
+
+The expression modTree(n, lm) creates a balanced binary tree with leaf
+values n mod m for each modulus m in lm.
+
+ modTree(12,lm)
+ [0, 2, 5, 1]
+ Type: List Integer
+
+Operation modTree does this using operations on balanced binary trees.
+We trace its steps. Create a balanced binary tree t of zeros with
+four leaves.
+
+ t := balancedBinaryTree(#lm, 0)
+ [[0, 0, 0], 0, [0, 0, 0]]
+ Type: BalancedBinaryTree NonNegativeInteger
+
+The leaves of the tree are set to the individual moduli.
+
+ setleaves!(t,lm)
+ [[3, 0, 5], 0, [7, 0, 11]]
+ Type: BalancedBinaryTree NonNegativeInteger
+
+mapUp! to do a bottom-up traversal of t, setting each interior node to
+the product of the values at the nodes of its children.
+
+ mapUp!(t,_*)
+ 1155
+ Type: PositiveInteger
+
+The value at the node of every subtree is the product of the moduli
+of the leaves of the subtree.
+
+ t
+ [[3, 15, 5], 1155, [7, 77, 11]]
+ Type: BalancedBinaryTree NonNegativeInteger
+
+Operation mapDown!(t,a,fn) replaces the value v at each node of t by
+fn(a,v).
+
+ mapDown!(t,12,_rem)
+ [[0, 12, 2], 12, [5, 12, 1]]
+ Type: BalancedBinaryTree NonNegativeInteger
+
+The operation leaves returns the leaves of the resulting tree. In
+this case, it returns the list of 12 mod m for each modulus m.
+
+ leaves %
+ [0, 2, 5, 1]
+ Type: List NonNegativeInteger
+
+Compute the square of the images of 12 modulo each m.
+
+ squares := [x**2 rem m for x in % for m in lm]
+ [0, 4, 4, 1]
+ Type: List NonNegativeInteger
+
+Call the Chinese Remainder Algorithm to get the answer for 12^2.
+
+ chineseRemainder(%,lm)
+ 144
+ Type: PositiveInteger
+
+See Also:
+o )show BalancedBinaryTree
+o $AXIOM/doc/src/algebra/tree.spad.dvi
+
+@
<<domain BBTREE BalancedBinaryTree>>=
)abbrev domain BBTREE BalancedBinaryTree
++ Description: \spadtype{BalancedBinaryTree(S)} is the domain of balanced
diff --git a/src/doc/Makefile.pamphlet b/src/doc/Makefile.pamphlet
index 7556528..cd17c11 100644
--- a/src/doc/Makefile.pamphlet
+++ b/src/doc/Makefile.pamphlet
@@ -146,7 +146,6 @@ syntax trace undo what while
<<spadhelp.files>>=
${DVI}/spadhelp/spadhelp.files: ${IN}/spadhelp.pamphlet
@echo 9 making ${DVI}/spadhelp from ${IN}/spadhelp.pamphlet
- @mkdir ${DVI}/spadhelp
@(cd ${DVI}/spadhelp ; \
for i in ${SPADHELP} ; do \
${TANGLE} -R"$$i" ${IN}/spadhelp.pamphlet >$$i.help ; \
diff --git a/src/doc/book.pamphlet b/src/doc/book.pamphlet
index 7f74c97..4930c49 100644
--- a/src/doc/book.pamphlet
+++ b/src/doc/book.pamphlet
@@ -25004,7 +25004,7 @@ $$
$$
\returnType{Type: Record(monthsOld: Integer,gender: String)}
-Use \spadfunFrom{delete}{AssociationList} to destructively remove an
+Use \spadfunFrom{delete!}{AssociationList} to destructively remove an
element of the association list. Use
\spadfunFrom{delete}{AssociationList} to return a copy of the
association list with the element deleted. The second argument is the
diff --git a/src/doc/spadhelp.pamphlet b/src/doc/spadhelp.pamphlet
index 54ea77c..f0e163d 100644
--- a/src/doc/spadhelp.pamphlet
+++ b/src/doc/spadhelp.pamphlet
@@ -1480,6 +1480,15 @@ parallel pquit quit read repeat
savesystem
set show spool suchthat synonym system
syntax trace undo what while
+Available algebra help topics are:
+
+AssociationList BalancedBinaryTree BasicOperator
+BinaryExpansion BinarySearchTree CardinalNumber
+CartesianTensor Character CharacterClass
+CliffordAlgebra Complex ContinuedFraction
+CycleIndicators DeRhamComplex DecimalExpansion
+
+
@
\section{command history}
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 34850f2..6a8e7e0 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -361,7 +361,12 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
IN= ${SRC}/input
MID= ${INT}/input
-all: ${OUTS} ${ALGEBRA} ${REGRES}
+all: ${OUTS} ${ALGEBRA} ${REGRES}
+ @ echo 0 starting algebra regression testing
+ @ (cd ${MID} ; \
+ ${TANGLE} -t8 -R"algebra.regress" ${SRC}/algebra/Makefile.pamphlet \
+ >Makefile.algebra ; \
+ make -f Makefile.algebra )
@ echo 1 finished ${INT}/input
%.input: ${IN}/%.input.pamphlet
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