1$, means $a+b \mod p$. +{\smath{\mbox{\bf addmod}\opLeftPren{}a, +\allowbreak{} b, \allowbreak{} p\opRightPren{}}, $0\le a, b
1$, +means $a+b \mod p$. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{airyAi}}\opLeftPren{}{\it complexDoubleFloat}\opRightPren{}% -\opand \mbox{\axiomFun{airyBi}}\opLeftPren{}{\it complexDoubleFloat}\opRightPren{}% +\opdata{{\mbox{\axiomFun{airyAi}}\opLeftPren{} +{\it complexDoubleFloat}\opRightPren{}% +\opand \mbox{\axiomFun{airyBi}}\opLeftPren{} +{\it complexDoubleFloat}\opRightPren{}% }% }% {1}{(Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} -{\smath{\mbox{\bf airyAi}\opLeftPren{}x\opRightPren{}} is the Airy function ${\rm Ai}(x)$ satisfying the +{\smath{\mbox{\bf airyAi}\opLeftPren{}x\opRightPren{}} +is the Airy function ${\rm Ai}(x)$ satisfying the differential equation ${\rm Ai}''(x) - x {\rm Ai}(x) = 0$. \newitem - \smath{\mbox{\bf airyBi}\opLeftPren{}x\opRightPren{}} is the Airy function ${\rm Bi}(x)$ satisfying the differential equation + \smath{\mbox{\bf airyBi}\opLeftPren{}x\opRightPren{}} +is the Airy function ${\rm Bi}(x)$ satisfying the differential equation ${\rm Bi}''(x) - x {\rm Bi}(x) = 0$. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{Aleph}}\opLeftPren{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{Aleph}}\opLeftPren{} +{\it nonNegativeInteger}\opRightPren{}% }% }% {1}{(NonNegativeInteger)->\$}{CardinalNumber} -{\smath{\mbox{\bf Aleph}\opLeftPren{}n\opRightPren{}} provides the named (infinite) cardinal number. +{\smath{\mbox{\bf Aleph}\opLeftPren{}n\opRightPren{}} provides the +named (infinite) cardinal number. } % ---------------------------------------------------------------------- @@ -68555,33 +69944,41 @@ ${\rm Bi}''(x) - x {\rm Bi}(x) = 0$. }% }% {1}{(\$)->Boolean}{ExtensionField} -{\smath{\mbox{\bf algebraic?}\opLeftPren{}a\opRightPren{}} tests whether an element \smath{a} is algebraic with respect to the ground field \smath{F}. +{\smath{\mbox{\bf algebraic?}\opLeftPren{}a\opRightPren{}} tests whether +an element \smath{a} is algebraic with respect to the ground field \smath{F}. } % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{alphabetic}}\opLeftPren{}\opRightPren{}% -\opand \mbox{\axiomFun{alphabetic?}}\opLeftPren{}{\it character}\opRightPren{}% +\opand \mbox{\axiomFun{alphabetic?}}\opLeftPren{} +{\it character}\opRightPren{}% }% }% {1}{(\$)->Boolean}{Character} -{\smath{\mbox{\bf alphabetic}\opLeftPren{}\opRightPren{}} returns the class of all characters +{\smath{\mbox{\bf alphabetic}\opLeftPren{}\opRightPren{}} +returns the class of all characters \smath{ch} for which \smath{\mbox{\bf alphabetic?}\opLeftPren{}ch\opRightPren{}} is \smath{true}. \newitem -\smath{\mbox{\bf alphabetic?}\opLeftPren{}ch\opRightPren{}} tests if \smath{ch} is an alphabetic +\smath{\mbox{\bf alphabetic?}\opLeftPren{}ch\opRightPren{}} +tests if \smath{ch} is an alphabetic character a$\ldots$z, A$\ldots$B. } % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{alphanumeric}}\opLeftPren{}\opRightPren{}% -\opand \mbox{\axiomFun{alphanumeric?}}\opLeftPren{}{\it character}\opRightPren{}% +\opand \mbox{\axiomFun{alphanumeric?}}\opLeftPren{} +{\it character}\opRightPren{}% }% }% {1}{(\$)->Boolean}{Character} -{\smath{\mbox{\bf alphanumeric}\opLeftPren{}\opRightPren{}} returns the class of all characters \smath{ch} for which +{\smath{\mbox{\bf alphanumeric}\opLeftPren{}\opRightPren{}} returns +the class of all characters \smath{ch} for which \smath{\mbox{\bf alphanumeric?}\opLeftPren{}ch\opRightPren{}} is \smath{true}. \newitem - \smath{\mbox{\bf alphanumeric?}\opLeftPren{}ch\opRightPren{}} tests if \smath{ch} is either an alphabetic character a$\ldots$z, A$\ldots$B or digit 0$\ldots$9. + \smath{\mbox{\bf alphanumeric?}\opLeftPren{}ch\opRightPren{}} +tests if \smath{ch} is either an alphabetic character a$\ldots$z, +A$\ldots$B or digit 0$\ldots$9. } % ---------------------------------------------------------------------- @@ -68589,17 +69986,20 @@ character a$\ldots$z, A$\ldots$B. }% }% { I -> SPOL RN}{}{} - {\smath{\mbox{\bf alternating}\opLeftPren{}n\opRightPren{}} is the cycle index of the + {\smath{\mbox{\bf alternating}\opLeftPren{}n\opRightPren{}} +is the cycle index of the alternating group of degree \smath{n}. See \axiomType{CycleIndicators} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{alternatingGroup}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{alternatingGroup}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% {1}{(List(Integer))->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf alternatingGroup}\opLeftPren{}li\opRightPren{}} constructs the alternating group +{\smath{\mbox{\bf alternatingGroup}\opLeftPren{}li\opRightPren{}} +constructs the alternating group acting on the integers in the list \smath{li}. If \smath{n} is odd, the generators are in general the \smath{(n-2)}-cycle \smath{(li.3, \ldots, li.n)} and the @@ -68608,7 +70008,8 @@ If \smath{n} is even, the generators are the product of the 2-cycle \smath{(li.1, li.2)} with \smath{(n-2)}-cycle \smath{(li.3, \ldots, li.n)} and the 3-cycle \smath{(li.1, li.2, li.3)}. Duplicates in the list will be removed. -\newitem\smath{\mbox{\bf alternatingGroup}\opLeftPren{}n\opRightPren{}} constructs the alternating +\newitem\smath{\mbox{\bf alternatingGroup}\opLeftPren{}n\opRightPren{}} +constructs the alternating group $A_n$ acting on the integers \smath{1, \ldots, n}. If \smath{n} is odd, the generators are in general the \smath{(n-2)}-cycle \smath{(3, \ldots, n)} and the 3-cycle \smath{(1, 2, 3)}. @@ -68622,14 +70023,16 @@ If \smath{n} is even, the generators are the product of the 2-cycle }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf alternative?}\opLeftPren{}\opRightPren{}\$F} tests if $2 \mbox{\bf associator}(a, a, b) = 0 = +{\smath{\mbox{\bf alternative?}\opLeftPren{}\opRightPren{}\$F} +tests if $2 \mbox{\bf associator}(a, a, b) = 0 = 2 \mbox{\bf associator}(a, b, b)$ for all \smath{a}, \smath{b} in the algebra \smath{F}. Note: in general, \smath{2 a=0} does not necessarily imply \smath{a=0}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{and}}\opLeftPren{}{\it boolean}, \allowbreak{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{and}}\opLeftPren{} +{\it boolean}, \allowbreak{}{\it boolean}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{BitAggregate} @@ -68644,21 +70047,25 @@ and} of the small integers \smath{si_1} and \smath{si_2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{approximants}}\opLeftPren{}{\it continuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{approximants}}\opLeftPren{} +{\it continuedFraction}\opRightPren{}% }% }% {1}{(\$)->Stream(Fraction(R))}{ContinuedFraction} -{\smath{\mbox{\bf approximants}\opLeftPren{}cf\opRightPren{}} returns the stream of approximants of the +{\smath{\mbox{\bf approximants}\opLeftPren{}cf\opRightPren{}} +returns the stream of approximants of the continued fraction \smath{cf}. If the continued fraction is finite, then the stream will be infinite and periodic with period 1.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{approximate}}\opLeftPren{}{\it series}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{approximate}}\opLeftPren{} +{\it series}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, Expon)->Coef}{UnivariatePowerSeriesCategory} -{\smath{\mbox{\bf approximate}\opLeftPren{}s, \allowbreak{} r\opRightPren{}} returns a truncated power series as an +{\smath{\mbox{\bf approximate}\opLeftPren{}s, +\allowbreak{} r\opRightPren{}} returns a truncated power series as an expression in the coefficient domain of the power series. For example, if \smath{R} is \axiomType{Fraction Polynomial Integer} and \smath{s} is a series over \smath{R}, then approximate(s, r) @@ -68667,21 +70074,25 @@ returns the power series \smath{s} truncated after the exponent } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{approximate}}\opLeftPren{}{\it pAdicInteger}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{approximate}}\opLeftPren{} +{\it pAdicInteger}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, Integer)->Integer}{PAdicIntegerCategory} -{\smath{\mbox{\bf approximate}\opLeftPren{}x, \allowbreak{} n\opRightPren{}}, \smath{x} a p-adic integer, returns an +{\smath{\mbox{\bf approximate}\opLeftPren{}x, +\allowbreak{} n\opRightPren{}}, \smath{x} a p-adic integer, returns an integer \smath{y} such that \smath{y = x \mod p^n} when \smath{n} is positive, and 0 otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{approxNthRoot}}\opLeftPren{}{\it integer}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{approxNthRoot}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(I, NonNegativeInteger)->I}{IntegerRoots} -{\smath{\mbox{\bf approxNthRoot}\opLeftPren{}n, \allowbreak{} p\opRightPren{}} returns an integer approximation +{\smath{\mbox{\bf approxNthRoot}\opLeftPren{}n, +\allowbreak{} p\opRightPren{}} returns an integer approximation \smath{i} to \smath{n^{1/p}} such that \smath{-1 < i - n^{1/p} < 1}. } @@ -68691,18 +70102,23 @@ is positive, and 0 otherwise. }% }% {2}{(I, NonNegativeInteger)->I}{IntegerRoots} -{\smath{\mbox{\bf approxSqrt}\opLeftPren{}n\opRightPren{}} returns an integer approximation \smath{i} +{\smath{\mbox{\bf approxSqrt}\opLeftPren{}n\opRightPren{}} returns an +integer approximation \smath{i} to \smath{\sqrt(n)} such that \smath{-1 < i - \sqrt(n) < 1}. A variable precision Newton iteration is used with running time \smath{O( \log(n)^2 )}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{areEquivalent?}}\opLeftPren{}{\it listOfMatrices}, \allowbreak{}{\it listOfMatrices}\allowbreak $\,[$ , \allowbreak{}{\it randomElements?}, \allowbreak{}{\it numberOfTries}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{areEquivalent?}}\opLeftPren{}{\it listOfMatrices}, +\allowbreak{}{\it listOfMatrices}\allowbreak $\,[$ , \allowbreak{} +{\it randomElements?}, \allowbreak{}{\it numberOfTries}$]$\opRightPren{}% }% }% -{4}{(List(Matrix(R)), List(Matrix(R)), Boolean, Integer)->Matrix(R)}{RepresentationPackage2} -{\smath{\mbox{\bf areEquivalent?}\opLeftPren{}lM, \allowbreak{} lM', \allowbreak{} b, \allowbreak{} numberOfTries\opRightPren{}} tests whether the +{4}{(List(Matrix(R)), List(Matrix(R)), Boolean, Integer)-> +Matrix(R)}{RepresentationPackage2} +{\smath{\mbox{\bf areEquivalent?}\opLeftPren{}lM, +\allowbreak{} lM', \allowbreak{} b, \allowbreak{} numberOfTries\opRightPren{}} tests whether the two lists of matrices, assumed of the same square shape, can be simultaneously conjugated by a non-singular matrix. If these matrices represent the same group generators, the @@ -68711,57 +70127,76 @@ The algorithm tries \smath{numberOfTries} times to create elements in the generated algebras in the same fashion. For details, consult HyperDoc. \newitem - \smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, \allowbreak{} aG1, \allowbreak{} numberOfTries\opRightPren{}} calls \smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, \allowbreak{} aG1, \allowbreak{} true, \allowbreak{} 25\opRightPren{}}. + \smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, +\allowbreak{} aG1, \allowbreak{} numberOfTries\opRightPren{}} calls +\smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, \allowbreak{} aG1, +\allowbreak{} true, \allowbreak{} 25\opRightPren{}}. \newitem - \smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, \allowbreak{} aG1\opRightPren{}} calls \smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, \allowbreak{} aG1, \allowbreak{} true, \allowbreak{} 25\opRightPren{}}. + \smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, +\allowbreak{} aG1\opRightPren{}} calls +\smath{\mbox{\bf areEquivalent?}\opLeftPren{}aG0, +\allowbreak{} aG1, \allowbreak{} true, \allowbreak{} 25\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{argscript}}\opLeftPren{}{\it symbol}, \allowbreak{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{argscript}}\opLeftPren{} +{\it symbol}, \allowbreak{}{\it listOfOutputForms}\opRightPren{}% }% }% {2}{(\$, List(OutputForm))->\$}{Symbol} -{\smath{\mbox{\bf argscript}\opLeftPren{}f, \allowbreak{} [o_1, \allowbreak{} \ldots, o_n]\opRightPren{}} returns a new symbol with \smath{f} with scripts +{\smath{\mbox{\bf argscript}\opLeftPren{}f, +\allowbreak{} [o_1, \allowbreak{} \ldots, o_n]\opRightPren{}} returns a new symbol +with \smath{f} with scripts \smath{o_1, \ldots, o_n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{argument}}\opLeftPren{}{\it complexExpression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{argument}}\opLeftPren{} +{\it complexExpression}\opRightPren{}% }% }% {1}{(\$)->R}{ComplexCategory} -{\smath{\mbox{\bf argument}\opLeftPren{}c\opRightPren{}} returns the angle made by complex expression \smath{c} with +{\smath{\mbox{\bf argument}\opLeftPren{}c\opRightPren{}} +returns the angle made by complex expression \smath{c} with the positive real axis. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{arity}}\opLeftPren{}{\it basicOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{arity}}\opLeftPren{} +{\it basicOperator}\opRightPren{}% }% }% {1}{(\$)->Union(NonNegativeInteger, \mbox{\tt "failed"})}{BasicOperator} -{\smath{\mbox{\bf arity}\opLeftPren{}op\opRightPren{}} returns \smath{n} if \smath{op} is +{\smath{\mbox{\bf arity}\opLeftPren{}op\opRightPren{}} +returns \smath{n} if \smath{op} is \smath{n}-ary, and \mbox{\tt "failed"} if \smath{op} has arbitrary arity. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{asec}}\opLeftPren{}{\it expression}\opRightPren{}% -\opand \mbox{\axiomFun{asecIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{asec}}\opLeftPren{} +{\it expression}\opRightPren{}% +\opand \mbox{\axiomFun{asecIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, \mbox{\tt "failed"})}{PartialTranscendentalFunctions} {\opkey{Argument \smath{x} can be a \axiomType{Complex}, \axiomType{Float}, \axiomType{DoubleFloat}, or \axiomType{Expression} value or a series.} -\newitem\smath{\mbox{\bf asec}\opLeftPren{}x\opRightPren{}} returns the arcsecant of \smath{x}. -\newitem\smath{\mbox{\bf asecIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf asec}\opLeftPren{}x\opRightPren{}} if possible, +\newitem\smath{\mbox{\bf asec}\opLeftPren{}x\opRightPren{}} +returns the arcsecant of \smath{x}. +\newitem\smath{\mbox{\bf asecIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf asec}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{asech}}\opLeftPren{}{\it expression}\opRightPren{}% -\opand \mbox{\axiomFun{asechIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{asech}}\opLeftPren{} +{\it expression}\opRightPren{}% +\opand \mbox{\axiomFun{asechIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, \mbox{\tt "failed"})}{PartialTranscendentalFunctions} @@ -68769,16 +70204,19 @@ and \mbox{\tt "failed"} otherwise. \axiomType{Float}, \axiomType{DoubleFloat}, or \axiomType{Expression} value or a series.} \newitem -\smath{\mbox{\bf asech}\opLeftPren{}x\opRightPren{}} returns the hyperbolic arcsecant of \smath{x}. +\smath{\mbox{\bf asech}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic arcsecant of \smath{x}. \newitem -\smath{\mbox{\bf asechIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf asech}\opLeftPren{}x\opRightPren{}} if possible, and +\smath{\mbox{\bf asechIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf asech}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{asin}}\opLeftPren{}{\it expression}\opRightPren{}% -\opand \mbox{\axiomFun{asinIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{asin}}\opLeftPren{} +{\it expression}\opRightPren{}% +\opand \mbox{\axiomFun{asinIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, \mbox{\tt "failed"})}{PartialTranscendentalFunctions} @@ -68786,14 +70224,18 @@ and \mbox{\tt "failed"} otherwise. \axiomType{Float}, \axiomType{DoubleFloat}, or \axiomType{Expression} value or a series. } -\newitem\smath{\mbox{\bf asin}\opLeftPren{}x\opRightPren{}} returns the arcsine of \smath{x}. -\newitem\smath{\mbox{\bf asinIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf asin}\opLeftPren{}x\opRightPren{}} if possible, +\newitem\smath{\mbox{\bf asin}\opLeftPren{}x\opRightPren{}} +returns the arcsine of \smath{x}. +\newitem\smath{\mbox{\bf asinIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf asin}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{asinh}}\opLeftPren{}{\it expression}\opRightPren{}% -\opand \mbox{\axiomFun{asinhIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{asinh}}\opLeftPren{} +{\it expression}\opRightPren{}% +\opand \mbox{\axiomFun{asinhIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, \mbox{\tt "failed"})}{PartialTranscendentalFunctions} @@ -68801,28 +70243,34 @@ and \mbox{\tt "failed"} otherwise. \axiomType{Float}, \axiomType{DoubleFloat}, or \axiomType{Expression} value or a series. } -\newitem\smath{\mbox{\bf asinh}\opLeftPren{}x\opRightPren{}} returns the hyperbolic arcsine of +\newitem\smath{\mbox{\bf asinh}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic arcsine of \smath{x}. -\newitem\smath{\mbox{\bf asinhIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf asinh}\opLeftPren{}x\opRightPren{}} if +\newitem\smath{\mbox{\bf asinhIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf asinh}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{assign}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{assign}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} - {\smath{\mbox{\bf assign}\opLeftPren{}f, \allowbreak{} g\opRightPren{}} creates an \axiomType{OutputForm} object + {\smath{\mbox{\bf assign}\opLeftPren{}f, \allowbreak{} g\opRightPren{}} +creates an \axiomType{OutputForm} object for the assignment \smath{f {\tt :=} g}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{associates?}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{associates?}}\opLeftPren{} +{\it element}, \allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{IntegralDomain} -{\smath{\mbox{\bf associates?}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} tests whether \smath{x} and \smath{y} are associates, that is, that +{\smath{\mbox{\bf associates?}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +tests whether \smath{x} and \smath{y} are associates, that is, that \smath{x} and \smath{y} differ by a unit factor. } @@ -68831,7 +70279,8 @@ for the assignment \smath{f {\tt :=} g}. }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf associative?}\opLeftPren{}\opRightPren{}\$F} tests if multiplication in \smath{F} is associative, where +{\smath{\mbox{\bf associative?}\opLeftPren{}\opRightPren{}\$F} tests +if multiplication in \smath{F} is associative, where \smath{F} is a \axiomType{FiniteRankNonAssociativeAlgebra}. } @@ -68840,43 +70289,54 @@ for the assignment \smath{f {\tt :=} g}. }% }% {0}{()->List(Vector(R))}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf associatorDependence}\opLeftPren{}\opRightPren{}\$F} computes associator identities for +{\smath{\mbox{\bf associatorDependence}\opLeftPren{}\opRightPren{}\$F} +computes associator identities for \smath{F}. \consultType{FiniteRankNonAssociativeAlgebra}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{associator}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{associator}}\opLeftPren{}{\it element}, +\allowbreak{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% }% }% {3}{(\$, \$, \$)->\$}{NonAssociativeRng} - {\smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} b, \allowbreak{} c\opRightPren{}} returns \smath{(ab)c-a(bc)}, + {\smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} b, +\allowbreak{} c\opRightPren{}} returns \smath{(ab)c-a(bc)}, where \smath{a}, \smath{b}, and \smath{c} are all members of a domain of category \axiomType{NonAssociateRng}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{assoc}}\opLeftPren{}{\it element}, \allowbreak{}{\it associationList}\opRightPren{}% +\opdata{{\mbox{\axiomFun{assoc}}\opLeftPren{}{\it element}, +\allowbreak{}{\it associationList}\opRightPren{}% }% }% -{2}{(Key, \$)->Union(Record(key:Key, entry:Entry), \mbox{\tt "failed"})}{AssociationListAggregate} -{\smath{\mbox{\bf assoc}\opLeftPren{}k, \allowbreak{} al\opRightPren{}} returns the element \smath{x} in the +{2}{(Key, \$)->Union(Record(key:Key, entry:Entry), +\mbox{\tt "failed"})}{AssociationListAggregate} +{\smath{\mbox{\bf assoc}\opLeftPren{}k, +\allowbreak{} al\opRightPren{}} returns the element \smath{x} in the \axiomType{AssociationList} \smath{al} stored under key \smath{k}, or \mbox{\tt "failed"} if no such element exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{atan}}\opLeftPren{}{\it expression\opt{, phase}}\opRightPren{}% - \opand \mbox{\axiomFun{atanIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{atan}}\opLeftPren{} +{\it expression\opt{, phase}}\opRightPren{}% + \opand \mbox{\axiomFun{atanIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(\$)->\$}{ArcTrigonometricFunctionCategory} {\opkey{Argument \smath{x} can be a \axiomType{Complex}, \axiomType{Float}, \axiomType{DoubleFloat}, or \axiomType{Expression} value or a series.} -\newitem\smath{\mbox{\bf atan}\opLeftPren{}x\opRightPren{}} returns the arctangent of \smath{x}. -\newitem\smath{\mbox{\bf atan}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} computes the arc tangent from \smath{x} +\newitem\smath{\mbox{\bf atan}\opLeftPren{}x\opRightPren{}} returns the +arctangent of \smath{x}. +\newitem\smath{\mbox{\bf atan}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +computes the arc tangent from \smath{x} with phase \smath{y}. -\newitem\smath{\mbox{\bf atanIfCan}\opLeftPren{}x\opRightPren{}} returns the \smath{\mbox{\bf atan}\opLeftPren{}x\opRightPren{}} if +\newitem\smath{\mbox{\bf atanIfCan}\opLeftPren{}x\opRightPren{}} +returns the \smath{\mbox{\bf atan}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } @@ -68889,9 +70349,11 @@ possible, and \mbox{\tt "failed"} otherwise. {\opkey{Argument \smath{x} can be a \axiomType{Complex}, \axiomType{Float}, \axiomType{DoubleFloat}, or \axiomType{Expression} value or a series.} -\newitem\smath{\mbox{\bf atanh}\opLeftPren{}x\opRightPren{}} returns the hyperbolic arctangent of +\newitem\smath{\mbox{\bf atanh}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic arctangent of \smath{x}. -\newitem\smath{\mbox{\bf atanhIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf atanh}\opLeftPren{}x\opRightPren{}} if +\newitem\smath{\mbox{\bf atanhIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf atanh}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } @@ -68901,35 +70363,43 @@ possible, and \mbox{\tt "failed"} otherwise. }% {1}{(\$)->Boolean}{SExpressionCategory} {\smath{\mbox{\bf atom?}\opLeftPren{}s\opRightPren{}} tests if \smath{x} is -atomic, where \smath{x} is an \axiomType{SExpression} or \axiomType{OutputForm}. +atomic, where \smath{x} is an \axiomType{SExpression} or +\axiomType{OutputForm}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{antiCommutator}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{antiCommutator}}\opLeftPren{} +{\it element}, \allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{NonAssociativeRng} - {\smath{\mbox{\bf antiCommutator}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns \smath{x y + y x}, + {\smath{\mbox{\bf antiCommutator}\opLeftPren{}x, +\allowbreak{} y\opRightPren{}} returns \smath{x y + y x}, where \smath{x} and \smath{y} are elements of a non-associative ring, possibly without identity. \seeType{NonAssociativeRng}} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{antisymmetric?}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{antisymmetric?}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% {1}{(\$)->Boolean}{MatrixCategory} -{\smath{\mbox{\bf antisymmetric?}\opLeftPren{}m\opRightPren{}} tests if the matrix \smath{m} is square +{\smath{\mbox{\bf antisymmetric?}\opLeftPren{}m\opRightPren{}} +tests if the matrix \smath{m} is square and antisymmetric, that is, \smath{m_{i, j} = -m_{j, i}} for all \smath{i} and \smath{j}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{antisymmetricTensors}}\opLeftPren{}{\it matrices}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{antisymmetricTensors}}\opLeftPren{} +{\it matrices}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% -{2}{(List(Matrix(R)), PositiveInteger)->List(Matrix(R))}{RepresentationPackage1} - {\smath{\mbox{\bf antisymmetricTensors}\opLeftPren{}A, \allowbreak{} n\opRightPren{}}, +{2}{(List(Matrix(R)), PositiveInteger)->List(Matrix(R))} +{RepresentationPackage1} + {\smath{\mbox{\bf antisymmetricTensors}\opLeftPren{}A, +\allowbreak{} n\opRightPren{}}, where \smath{A} is an \smath{m} by \smath{m} matrix, returns a matrix obtained by applying to \smath{A} the irreducible, polynomial representation of the @@ -68941,46 +70411,60 @@ with the sign representation of the symmetric group \smath{S_n}. The carrier spaces of the representation are the antisymmetric tensors of the \smath{n}-fold tensor product. - \newitem\smath{\mbox{\bf antisymmetricTensors}\opLeftPren{}lA, \allowbreak{} n\opRightPren{}}, + \newitem\smath{\mbox{\bf antisymmetricTensors}\opLeftPren{}lA, +\allowbreak{} n\opRightPren{}}, where \smath{lA} is a list of \smath{m} by \smath{m} matrices, similarly applies the representation of \smath{GL_m} to each matrix \smath{A} of \smath{lA}, returning a list of matrices. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{any?}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{any?}}\opLeftPren{}{\it predicate}, +\allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->Boolean}{HomogeneousAggregate} -{\smath{\mbox{\bf any?}\opLeftPren{}pred, \allowbreak{} a\opRightPren{}} tests if predicate \smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} is \smath{true} for any element \smath{x} of aggregate \smath{a}. Note: for collections, \code{any?(p, u) = reduce(or, map(p, u), false, true)}. +{\smath{\mbox{\bf any?}\opLeftPren{}pred, \allowbreak{} a\opRightPren{}} +tests if predicate \smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} +is \smath{true} for any element \smath{x} of aggregate \smath{a}. +Note: for collections, \code{any?(p, u) = reduce(or, map(p, u), false, true)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{any}}\opLeftPren{}{\it type}, \allowbreak{}{\it object}\opRightPren{}% +\opdata{{\mbox{\axiomFun{any}}\opLeftPren{}{\it type}, \allowbreak{} +{\it object}\opRightPren{}% }% }% {2}{(SExpression, None)->\$}{Any} -{\smath{\mbox{\bf any}\opLeftPren{}type, \allowbreak{} object\opRightPren{}} is a technical function for creating an \smath{object} of \axiomType{Any}. Argument \smath{type} is a \spadgloss{LISP} form for the \smath{type} of \smath{object}.} +{\smath{\mbox{\bf any}\opLeftPren{}type, \allowbreak{} object\opRightPren{}} +is a technical function for creating an \smath{object} of \axiomType{Any}. +Argument \smath{type} is a \spadgloss{LISP} form for the \smath{type} +of \smath{object}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{append}}\opLeftPren{}{\it list}, \allowbreak{}{\it list}\opRightPren{}% +\opdata{{\mbox{\axiomFun{append}}\opLeftPren{}{\it list}, \allowbreak{} +{\it list}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{List} -{\smath{\mbox{\bf append}\opLeftPren{}l_1, \allowbreak{} l_2\opRightPren{}} appends the elements of list \smath{l_1} +{\smath{\mbox{\bf append}\opLeftPren{}l_1, \allowbreak{} l_2\opRightPren{}} +appends the elements of list \smath{l_1} onto the front of list \smath{l_2}. See also \spadfun{concat}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{axesColorDefault}}\opLeftPren{}{\it \opt{palette}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{axesColorDefault}}\opLeftPren{} +{\it \opt{palette}}\opRightPren{}% }% }% {0}{()->Palette}{ViewDefaultsPackage} -{\smath{\mbox{\bf axesColorDefault}\opLeftPren{}p\opRightPren{}} sets the default color of the axes in a \twodim{} viewport to the palette \smath{p}. +{\smath{\mbox{\bf axesColorDefault}\opLeftPren{}p\opRightPren{}} sets +the default color of the axes in a \twodim{} viewport to the palette \smath{p}. \newitem - \smath{\mbox{\bf axesColorDefault}\opLeftPren{}\opRightPren{}} returns the default color of the axes in a \twodim{} viewport. + \smath{\mbox{\bf axesColorDefault}\opLeftPren{}\opRightPren{}} +returns the default color of the axes in a \twodim{} viewport. } @@ -68989,7 +70473,8 @@ See also \spadfun{concat}. }% }% {1}{(\$)->S}{QueueAggregate} -{\smath{\mbox{\bf back}\opLeftPren{}q\opRightPren{}} returns the element at the back of the queue, or calls +{\smath{\mbox{\bf back}\opLeftPren{}q\opRightPren{}} +returns the element at the back of the queue, or calls \spadfun{error} if \smath{q} is empty. } @@ -68998,15 +70483,18 @@ See also \spadfun{concat}. }% }% {0}{()->\$}{BagAggregate} -{\smath{\mbox{\bf bag}\opLeftPren{}\colx{x}{y}{z}\opRightPren{}} creates a bag with elements \smath{x}, \smath{y}, $\ldots$, \smath{z}. +{\smath{\mbox{\bf bag}\opLeftPren{}\colx{x}{y}{z}\opRightPren{}} +creates a bag with elements \smath{x}, \smath{y}, $\ldots$, \smath{z}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{balancedBinaryTree}}\opLeftPren{}{\it nonNegativeInteger}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{balancedBinaryTree}}\opLeftPren{} +{\it nonNegativeInteger}, \allowbreak{}{\it element}\opRightPren{}% }% }% {0}{()->\$}{BagAggregate} - {\smath{\mbox{\bf balancedBinaryTree}\opLeftPren{}n, \allowbreak{} s\opRightPren{}} creates a balanced binary tree with + {\smath{\mbox{\bf balancedBinaryTree}\opLeftPren{}n, +\allowbreak{} s\opRightPren{}} creates a balanced binary tree with \smath{n} nodes, each with value \smath{s}. } @@ -69015,14 +70503,16 @@ See also \spadfun{concat}. }% }% {0}{()->PositiveInteger}{FloatingPointSystem} -{\smath{\mbox{\bf base}\opLeftPren{}gp\opRightPren{}} returns a base for the group {\it gp}. \consultType{PermutationGroup} +{\smath{\mbox{\bf base}\opLeftPren{}gp\opRightPren{}} returns a +base for the group {\it gp}. \consultType{PermutationGroup} } % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{basis}}\opLeftPren{}\opRightPren{}% }% }% {0}{()->Vector(\$)}{FiniteAlgebraicExtensionField} -{\smath{\mbox{\bf basis}\opLeftPren{}\opRightPren{}\$R} returns a fixed basis of \smath{R} +{\smath{\mbox{\bf basis}\opLeftPren{}\opRightPren{}\$R} returns a +fixed basis of \smath{R} or a subspace of \smath{R}. See \axiomType{FiniteAlgebraicExtensionField}, \axiomType{FramedAlgebra}, @@ -69034,9 +70524,14 @@ See \axiomType{FiniteAlgebraicExtensionField}, }% }% {0}{()->List(A)}{AlgebraPackage} -{\smath{\mbox{\bf basisOfCenter}\opLeftPren{}\opRightPren{}\$R} returns a basis of the space of all -\smath{x} in \smath{R} satisfying \smath{\mbox{\bf commutator}\opLeftPren{}x, a) = 0} and -\smath{\mbox{\bf associator}\opLeftPren{}x, \allowbreak{} a, \allowbreak{} b\opRightPren{}} = \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} x, \allowbreak{} b\opRightPren{}} = \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} b, \allowbreak{} x\opRightPren{}} = +{\smath{\mbox{\bf basisOfCenter}\opLeftPren{}\opRightPren{}\$R} returns a +basis of the space of all +\smath{x} in \smath{R} satisfying \smath{\mbox{\bf commutator}\opLeftPren{}x, +a) = 0} and +\smath{\mbox{\bf associator}\opLeftPren{}x, \allowbreak{} a, \allowbreak{} b +\opRightPren{}} = \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} x, +\allowbreak{} b\opRightPren{}} = \smath{\mbox{\bf associator}\opLeftPren{}a, +\allowbreak{} b, \allowbreak{} x\opRightPren{}} = 0 for all \smath{a}, \smath{b} in \smath{R}. Domain \smath{R} is a domain of category \axiomType{FramedNonAssociativeAlgebra}. @@ -69047,7 +70542,8 @@ Domain \smath{R} is a domain of category }% }% {0}{()->List(Matrix(R))}{AlgebraPackage} -{\smath{\mbox{\bf basisOfCentroid}\opLeftPren{}\opRightPren{}\$R} returns a basis of the centroid of +{\smath{\mbox{\bf basisOfCentroid}\opLeftPren{}\opRightPren{}\$R} +returns a basis of the centroid of \smath{R}, that is, the endomorphism ring of \smath{R} considered as \smath{(R, R)}-bimodule. Domain \smath{R} is a domain of category @@ -69059,16 +70555,20 @@ Domain \smath{R} is a domain of category }% }% {0}{()->List(A)}{AlgebraPackage} -{\smath{\mbox{\bf basisOfCommutingElements}\opLeftPren{}\opRightPren{}\$R} returns a basis of the space -of all \smath{x} of \smath{R} satisfying \smath{\mbox{\bf commutator}\opLeftPren{}x, \allowbreak{} a\opRightPren{}} = 0 +{\smath{\mbox{\bf basisOfCommutingElements}\opLeftPren{}\opRightPren{}\$R} +returns a basis of the space +of all \smath{x} of \smath{R} satisfying +\smath{\mbox{\bf commutator}\opLeftPren{}x, \allowbreak{} a\opRightPren{}} = 0 for all \smath{a} in \smath{R}. Domain \smath{R} is a domain of category \axiomType{FramedNonAssociativeAlgebra}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{basisOfLeftAnnihilator}}\opLeftPren{}{\it element}\opRightPren{}% - \opand \mbox{\axiomFun{basisOfRightAnnihilator}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{basisOfLeftAnnihilator}}\opLeftPren{} +{\it element}\opRightPren{}% + \opand \mbox{\axiomFun{basisOfRightAnnihilator}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(A)->List(A)}{AlgebraPackage} @@ -69076,8 +70576,10 @@ Domain \smath{R} is a domain of category \smath{x} in \smath{R} of category \axiomType{FramedNonAssociativeAlgebra}, satisfying} \begin{simpleList} -\item\smath{\mbox{\bf basisOfLeftAnnihilator}\opLeftPren{}a\opRightPren{}}: \smath{0 = xa}. -\item\smath{\mbox{\bf basisOfRightAnnihilator}\opLeftPren{}a\opRightPren{}}: \smath{0 = ax}. +\item\smath{\mbox{\bf basisOfLeftAnnihilator}\opLeftPren{}a\opRightPren{}}: + \smath{0 = xa}. +\item\smath{\mbox{\bf basisOfRightAnnihilator}\opLeftPren{}a\opRightPren{}}: + \smath{0 = ax}. \end{simpleList} } @@ -69095,11 +70597,20 @@ Domain \smath{R} is a domain of category \smath{a} and \smath{b}:} \begin{simpleList} \item \smath{\mbox{\bf basisOfNucleus}\opLeftPren{}\opRightPren{}\$R}: -\smath{\mbox{\bf associator}\opLeftPren{}x, \allowbreak{} a, \allowbreak{} b\opRightPren{}} = \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} x, \allowbreak{} b\opRightPren{}} -= \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} b, \allowbreak{} x\opRightPren{}} = 0; -\item \smath{\mbox{\bf basisOfLeftNucleus}\opLeftPren{}\opRightPren{}\$R}: \smath{\mbox{\bf associator}\opLeftPren{}x, \allowbreak{} a, \allowbreak{} b\opRightPren{}} = 0; -\item \smath{\mbox{\bf basisOfMiddleNucleus}\opLeftPren{}\opRightPren{}\$R}: \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} x, \allowbreak{} b\opRightPren{}} = 0; -\item \smath{\mbox{\bf basisOfRightNucleus}\opLeftPren{}\opRightPren{}\$R}: \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} b, \allowbreak{} x\opRightPren{}} = 0. +\smath{\mbox{\bf associator}\opLeftPren{}x, \allowbreak{} a, +\allowbreak{} b\opRightPren{}} = \smath{\mbox{\bf associator}\opLeftPren{}a, +\allowbreak{} x, \allowbreak{} b\opRightPren{}} += \smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} b, +\allowbreak{} x\opRightPren{}} = 0; +\item \smath{\mbox{\bf basisOfLeftNucleus}\opLeftPren{}\opRightPren{}\$R}: +\smath{\mbox{\bf associator}\opLeftPren{}x, \allowbreak{} a, +\allowbreak{} b\opRightPren{}} = 0; +\item \smath{\mbox{\bf basisOfMiddleNucleus}\opLeftPren{}\opRightPren{}\$R}: +\smath{\mbox{\bf associator}\opLeftPren{}a, \allowbreak{} x, +\allowbreak{} b\opRightPren{}} = 0; +\item \smath{\mbox{\bf basisOfRightNucleus}\opLeftPren{}\opRightPren{}\$R}: +\smath{\mbox{\bf associator}\opLeftPren{}a, +\allowbreak{} b, \allowbreak{} x\opRightPren{}} = 0. \end{simpleList} } @@ -69113,8 +70624,10 @@ Domain \smath{R} is a domain of category endomorphisms of \smath{R}, a domain of category \axiomType{FramedNonAssociativeAlgebra}, considered as:} \begin{simpleList} - \item\smath{\mbox{\bf basisOfLeftNucloid}\opLeftPren{}\opRightPren{}}: a right module. - \item\smath{\mbox{\bf basisOfRightNucloid}\opLeftPren{}\opRightPren{}}: a left module. + \item\smath{\mbox{\bf basisOfLeftNucloid}\opLeftPren{}\opRightPren{}}: +a right module. + \item\smath{\mbox{\bf basisOfRightNucloid}\opLeftPren{}\opRightPren{}}: +a left module. \end{simpleList} Note: if \smath{R} has a unit, the left and right nucloid coincide with the left and right nucleus. @@ -69125,7 +70638,8 @@ the left and right nucleus. }% }% {1}{(BasicOperator)->Boolean}{ExpressionSpace} -{\smath{\mbox{\bf belong?}\opLeftPren{}op\opRightPren{}\$R} tests if \smath{op} is known as an +{\smath{\mbox{\bf belong?}\opLeftPren{}op\opRightPren{}\$R} tests +if \smath{op} is known as an operator to \smath{R}. For example, \smath{R} is an \axiomType{Expression} domain or \axiomType{AlgebraicNumber}. @@ -69136,34 +70650,44 @@ For example, \smath{R} is an \axiomType{Expression} domain or }% }% {1}{(Integer)->Fraction(Integer)}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf bernoulli}\opLeftPren{}n\opRightPren{}} returns the \eth{\smath{n}} Bernoulli +{\smath{\mbox{\bf bernoulli}\opLeftPren{}n\opRightPren{}} returns the +\eth{\smath{n}} Bernoulli number, that is, \smath{B(n, 0)} where \smath{B(n, x)} is the \eth{\smath{n}} Bernoulli polynomial. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{besselI}}\opLeftPren{}{\it complexDoubleFloat}, \allowbreak{}{\it complexDoubleFloat}\opRightPren{}% -\optand \mbox{\axiomFun{besselJ}}\opLeftPren{}{\it complexDoubleFloat}, \allowbreak{}{\it complexDoubleFloat}\opRightPren{}% -\optand \mbox{\axiomFun{besselK}}\opLeftPren{}{\it complexDoubleFloat}, \allowbreak{}{\it complexDoubleFloat}\opRightPren{}% -\opand \mbox{\axiomFun{besselY}}\opLeftPren{}{\it complexDoubleFloat}, \allowbreak{}{\it complexDoubleFloat}\opRightPren{}% +\opdata{{\mbox{\axiomFun{besselI}}\opLeftPren{}{\it complexDoubleFloat}, +\allowbreak{}{\it complexDoubleFloat}\opRightPren{}% +\optand \mbox{\axiomFun{besselJ}}\opLeftPren{}{\it complexDoubleFloat}, +\allowbreak{}{\it complexDoubleFloat}\opRightPren{}% +\optand \mbox{\axiomFun{besselK}}\opLeftPren{}{\it complexDoubleFloat}, +\allowbreak{}{\it complexDoubleFloat}\opRightPren{}% +\opand \mbox{\axiomFun{besselY}}\opLeftPren{}{\it complexDoubleFloat}, +\allowbreak{}{\it complexDoubleFloat}\opRightPren{}% }% }% -{2}{(Complex(DoubleFloat), Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} -{\smath{\mbox{\bf besselI}\opLeftPren{}v, \allowbreak{} x\opRightPren{}} is the modified Bessel function of the first +{2}{(Complex(DoubleFloat), +Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} +{\smath{\mbox{\bf besselI}\opLeftPren{}v, +\allowbreak{} x\opRightPren{}} is the modified Bessel function of the first kind, \smath{I(v, x)}, satisfying the differential equation \smath{x^2 {w''(x)} + x w'(x) - (x^2+v^2)w(x) = 0}. -\bigitem\smath{\mbox{\bf besselJ}\opLeftPren{}v, \allowbreak{} x\opRightPren{}} is the Bessel function of the second +\bigitem\smath{\mbox{\bf besselJ}\opLeftPren{}v, +\allowbreak{} x\opRightPren{}} is the Bessel function of the second kind, \smath{J(v, x)}, satisfying the differential equation \smath{x^2 {w''(x)} + x w'(x) + (x^2-v^2)w(x) = 0}. -\bigitem\smath{\mbox{\bf besselK}\opLeftPren{}v, \allowbreak{} x\opRightPren{}} is the modified Bessel function of +\bigitem\smath{\mbox{\bf besselK}\opLeftPren{}v, +\allowbreak{} x\opRightPren{}} is the modified Bessel function of the first kind, \smath{K(v, x)}, satisfying the differential equation \smath{x^2 {w''(x)} + x w'(x) - (x^2+v^2)w(x) = 0}. Note: The default implementation uses the relation \smath{K(v, x) = \pi/2(I(-v, x) - I(v, x))/\sin(v \pi)} so is not valid for integer values of \smath{v}. -\bigitem\smath{\mbox{\bf besselY}\opLeftPren{}v, \allowbreak{} x\opRightPren{}} is the Bessel function of the second +\bigitem\smath{\mbox{\bf besselY}\opLeftPren{}v, +\allowbreak{} x\opRightPren{}} is the Bessel function of the second kind, \smath{Y(v, x)}, satisfying the differential equation \smath{x^2 {w''(x)} + x w'(x) + (x^2-v^2)w(x) = 0}. @@ -69173,18 +70697,26 @@ integer values of \smath{v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{Beta}}\opLeftPren{}{\it complexDoubleFloat}, \allowbreak{}{\it complexDoubleFloat}\opRightPren{}% +\opdata{{\mbox{\axiomFun{Beta}}\opLeftPren{}{\it complexDoubleFloat}, +\allowbreak{}{\it complexDoubleFloat}\opRightPren{}% }% }% -{2}{(Complex(DoubleFloat), Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} -{\smath{\mbox{\bf Beta}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} is the Euler beta function, \smath{B(x, y)}, defined by -\smath{\mbox{\bf Beta}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} $\int_0^1{t^{x-1}(1-t)^{y-1} dt}$. +{2}{(Complex(DoubleFloat), +Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} +{\smath{\mbox{\bf Beta}\opLeftPren{}x, +\allowbreak{} y\opRightPren{}} +is the Euler beta function, \smath{B(x, y)}, defined by +\smath{\mbox{\bf Beta}\opLeftPren{}x, +\allowbreak{} y\opRightPren{}} $\int_0^1{t^{x-1}(1-t)^{y-1} dt}$. Note: this function is defined by -\smath{\mbox{\bf Beta}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} $= {{\Gamma(x)\Gamma(y)} \over {\Gamma(x + y)}}$. +\smath{\mbox{\bf Beta}\opLeftPren{}x, +\allowbreak{} y\opRightPren{}} $= +{{\Gamma(x)\Gamma(y)} \over {\Gamma(x + y)}}$. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{binaryTournament}}\opLeftPren{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{binaryTournament}}\opLeftPren{} +{\it listOfElements}\opRightPren{}% }% }% {1}{(Fraction(Integer))->\$}{BinaryExpansion} @@ -69198,8 +70730,10 @@ Note: this function is defined by }% }% {1}{(Fraction(Integer))->\$}{BinaryExpansion} - {\smath{\mbox{\bf binaryTree}\opLeftPren{}x\opRightPren{}} creates a binary tree -consisting of one node for which the \spadfunFrom{value}{BinaryTree} is \smath{x} and + {\smath{\mbox{\bf binaryTree}\opLeftPren{}x\opRightPren{}} creates +a binary tree +consisting of one node for which the \spadfunFrom{value}{BinaryTree} +is \smath{x} and the \spadfun{left} and \spadfun{right} subtrees are empty. } @@ -69208,19 +70742,24 @@ the \spadfun{left} and \spadfun{right} subtrees are empty. }% }% {1}{(Fraction(Integer))->\$}{BinaryExpansion} -{\smath{\mbox{\bf binary}\opLeftPren{}rn\opRightPren{}} converts rational number \smath{rn} to a binary expansion. +{\smath{\mbox{\bf binary}\opLeftPren{}rn\opRightPren{}} +converts rational number \smath{rn} to a binary expansion. \newitem - \smath{\mbox{\bf binary}\opLeftPren{}op, \allowbreak{} [a_1, \allowbreak{} \ldots, a_n]\opRightPren{}} returns the input form - corresponding to \smath{a_1 {\rm op} \ldots op a_n}, where - \smath{op} and the \smath{a_i}'s are of type \axiomType{InputForm}. + \smath{\mbox{\bf binary}\opLeftPren{}op, +\allowbreak{} [a_1, \allowbreak{} \ldots, a_n]\opRightPren{}} +returns the input form +corresponding to \smath{a_1 {\rm op} \ldots op a_n}, where +\smath{op} and the \smath{a_i}'s are of type \axiomType{InputForm}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{binomial}}\opLeftPren{}{\it integerNumber}, \allowbreak{}{\it integerNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{binomial}}\opLeftPren{} +{\it integerNumber}, \allowbreak{}{\it integerNumber}\opRightPren{}% }% }% {2}{(I, I)->I}{IntegerCombinatoricFunctions} -{\smath{\mbox{\bf binomial}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns the binomial coefficient +{\smath{\mbox{\bf binomial}\opLeftPren{}x, +\allowbreak{} y\opRightPren{}} returns the binomial coefficient \smath{C(x, y) = x!/(y! (x-y)!)}, where \texht{$x \geq y \geq 0$}{x >= y >= 0}, the number of combinations of \smath{x} objects taken @@ -69231,16 +70770,19 @@ Arguments \smath{x} and \smath{y} can come from any % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{bipolar}}\opLeftPren{}{\it x}\opRightPren{}% - \opand \mbox{\axiomFun{bipolarCylindrical}}\opLeftPren{}{\it x}\opRightPren{}% + \opand \mbox{\axiomFun{bipolarCylindrical}} +\opLeftPren{}{\it x}\opRightPren{}% }% }% {1}{(R)->(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf bipolar}\opLeftPren{}a\opRightPren{}} returns a function for transforming bipolar +{\smath{\mbox{\bf bipolar}\opLeftPren{}a\opRightPren{}} +returns a function for transforming bipolar coordinates to Cartesian coordinates; this function maps the point \smath{(u, v)} to \smath{(x = a \sinh(v)/(\cosh(v)-\cos(u)), y = a \sin(u)/(\cosh(v)-\cos(u)))}. - \newitem\smath{\mbox{\bf bipolarCylindrical}\opLeftPren{}a\opRightPren{}} returns a function for transforming + \newitem\smath{\mbox{\bf bipolarCylindrical} +\opLeftPren{}a\opRightPren{}} returns a function for transforming bipolar cylindrical coordinates to Cartesian coordinates; this function maps the point \smath{(u, v, z)} to \smath{(x = a \sinh(v)/(\cosh(v)-\cos(u)), y = a @@ -69248,7 +70790,8 @@ maps the point \smath{(u, v, z)} to } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{biRank}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{biRank}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(A)->NonNegativeInteger}{AlgebraPackage} @@ -69256,7 +70799,8 @@ maps the point \smath{(u, v, z)} to where \smath{R} is a domain of category \axiomType{FramedNonAssociativeAlgebra}, returns the number of linearly independent elements -among \smath{x}, \smath{x b_i}, \smath{b_i x}, \smath{b_i x b_j}, \smath{i, j=1, \ldots, n}, +among \smath{x}, \smath{x b_i}, \smath{b_i x}, +\smath{b_i x b_j}, \smath{i, j=1, \ldots, n}, where \smath{b=[b_1, \ldots, b_n]} is the fixed basis for \smath{R}. Note: if \smath{R} has a unit, then \spadfunFrom{doubleRank}{AlgebraPackage}, @@ -69265,11 +70809,14 @@ and \spadfunFrom{biRank}{AlgebraPackage} coincide. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{bit?}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{bit?}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{IntegerNumberSystem} -{\smath{\mbox{\bf bit?}\opLeftPren{}i, \allowbreak{} n\opRightPren{}} tests if the \eth{\smath{n}} bit of \smath{i} is a 1. +{\smath{\mbox{\bf bit?}\opLeftPren{}i, +\allowbreak{} n\opRightPren{}} tests if the +\eth{\smath{n}} bit of \smath{i} is a 1. } % ---------------------------------------------------------------------- @@ -69277,12 +70824,14 @@ and \spadfunFrom{biRank}{AlgebraPackage} coincide. }% }% {0}{()->PositiveInteger}{FloatingPointSystem} -{\smath{\mbox{\bf bits}\opLeftPren{}\opRightPren{}} returns the precision of floats in +{\smath{\mbox{\bf bits}\opLeftPren{}\opRightPren{}} +returns the precision of floats in bits. Also see \spadfun{precision}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{blankSeparate}}\opLeftPren{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{blankSeparate}}\opLeftPren{} +{\it listOfOutputForms}\opRightPren{}% }% }% {}{}{} @@ -69298,7 +70847,8 @@ of {\it lo} separated by blanks. \opdata{{\mbox{\axiomFun{blue}}\opLeftPren{}\opRightPren{}% }% }% -{0}{()->\$}{Color}{\smath{\mbox{\bf blue}\opLeftPren{}\opRightPren{}} returns the +{0}{()->\$}{Color}{\smath{\mbox{\bf blue}\opLeftPren{}\opRightPren{}} +returns the position of the blue hue from total hues.} % ---------------------------------------------------------------------- @@ -69306,7 +70856,8 @@ position of the blue hue from total hues.} }% }% {1}{(\$)->S}{DequeueAggregate} -{\smath{\mbox{\bf bottom!}\opLeftPren{}q\opRightPren{}} removes then returns the element at the bottom +{\smath{\mbox{\bf bottom!}\opLeftPren{}q\opRightPren{}} removes +then returns the element at the bottom (back) of the dequeue q. } @@ -69315,21 +70866,28 @@ position of the blue hue from total hues.} }% }% {1}{(List(\$))->\$}{ExpressionSpace} -{\smath{\mbox{\bf box}\opLeftPren{}e\opRightPren{}}, where \smath{e} is an expression, returns +{\smath{\mbox{\bf box}\opLeftPren{}e\opRightPren{}}, where \smath{e} +is an expression, returns \smath{e} with a box around it that prevents \smath{e} from being evaluated when operators are applied to it. -For example, \smath{\mbox{\bf log}\opLeftPren{}1\opRightPren{}} returns \smath{0}, but -\smath{\mbox{\bf log}\opLeftPren{}{\mbox{\bf box}}(1)\opRightPren{}} returns the formal kernel +For example, \smath{\mbox{\bf log}\opLeftPren{}1\opRightPren{}} +returns \smath{0}, but +\smath{\mbox{\bf log}\opLeftPren{}{\mbox{\bf box}}(1)\opRightPren{}} +returns the formal kernel \smath{\mbox{\bf log}\opLeftPren{}1\opRightPren{}}. \newitem -\smath{\mbox{\bf box}\opLeftPren{}f_1, \allowbreak{} \ldots, f_n\opRightPren{}}, where the \smath{f_i} are +\smath{\mbox{\bf box}\opLeftPren{}f_1, \allowbreak{} \ldots, f_n +\opRightPren{}}, where the \smath{f_i} are expressions, returns \smath{(f_1, \ldots, f_n)} with a box around them that prevents the \smath{f_i} from being evaluated when operators are applied to them, and makes them applicable to a unary operator. -For example, \smath{\mbox{\bf atan}\opLeftPren{}{\mbox {\bf box}} [x, \allowbreak{} 2]\opRightPren{}} returns the -formal kernel \smath{\mbox{\bf atan}\opLeftPren{}x, \allowbreak{} 2\opRightPren{}}. -\newitem\smath{\mbox{\bf box}\opLeftPren{}o\opRightPren{}}, where \smath{o} is an object of type +For example, \smath{\mbox{\bf atan}\opLeftPren{}{\mbox {\bf box}} +[x, \allowbreak{} 2]\opRightPren{}} returns the +formal kernel \smath{\mbox{\bf atan}\opLeftPren{}x, +\allowbreak{} 2\opRightPren{}}. +\newitem\smath{\mbox{\bf box}\opLeftPren{}o\opRightPren{}}, +where \smath{o} is an object of type \axiomType{OutputForm} (normally unexposed), returns an output form enclosing \smath{o} in a box. } @@ -69350,7 +70908,8 @@ returns an output form enclosing \smath{o} in braces. }% }% {}{}{} -{\smath{\mbox{\bf bracket}\opLeftPren{}o\opRightPren{}}, where \smath{o} is an object of type +{\smath{\mbox{\bf bracket}\opLeftPren{}o\opRightPren{}}, +where \smath{o} is an object of type \axiomType{OutputForm} (normally unexposed), returns an output form enclosing \smath{o} in brackets. } @@ -69358,14 +70917,18 @@ form enclosing \smath{o} in brackets. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{branchPoint}}\opLeftPren{}{\it element}\opRightPren{}% - \opand \mbox{\axiomFun{branchPointAtInfinity?}}\opLeftPren{}\opRightPren{}% +\opdata{{\mbox{\axiomFun{branchPoint}}\opLeftPren{} +{\it element}\opRightPren{}% + \opand \mbox{\axiomFun{branchPointAtInfinity?}} +\opLeftPren{}\opRightPren{}% }% }% {1}{(F)->Boolean}{FunctionFieldCategory} -{\smath{\mbox{\bf branchPoint?}\opLeftPren{}a\opRightPren{}\$F} tests if \smath{x = a} is a branch +{\smath{\mbox{\bf branchPoint?}\opLeftPren{}a\opRightPren{}\$F} +tests if \smath{x = a} is a branch point of the algebraic function field \smath{F}. -\newitem\smath{\mbox{\bf branchPointAtInfinity?}\opLeftPren{}\opRightPren{}\$F} tests if the +\newitem\smath{\mbox{\bf branchPointAtInfinity?} +\opLeftPren{}\opRightPren{}\$F} tests if the algebraic function field \smath{F} has a branch point at infinity. } @@ -69374,31 +70937,37 @@ algebraic function field \smath{F} has a branch point at infinity. }% }% {1}{(Color)->\$}{Palette} -{\smath{\mbox{\bf bright}\opLeftPren{}c\opRightPren{}} sets the shade of a hue, \smath{c}, above dim +{\smath{\mbox{\bf bright}\opLeftPren{}c\opRightPren{}} sets the shade +of a hue, \smath{c}, above dim but below pastel. \newitem -\smath{\mbox{\bf bright}\opLeftPren{}ls\opRightPren{}} sets the font property of a list of +\smath{\mbox{\bf bright}\opLeftPren{}ls\opRightPren{}} sets the font +property of a list of strings \smath{ls} to bold-face type. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cap}}\opLeftPren{}{\it symmetricPolynomial}, \allowbreak{}{\it symmetricPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cap}}\opLeftPren{}{\it symmetricPolynomial}, +\allowbreak{}{\it symmetricPolynomial}\opRightPren{}% }% }% { (SPOL RN, SPOL RN) -> RN}{}{} -{\smath{\mbox{\bf cap}\opLeftPren{}s_1, \allowbreak{} s_2\opRightPren{}}, introduced by Redfield, is the scalar +{\smath{\mbox{\bf cap}\opLeftPren{}s_1, +\allowbreak{} s_2\opRightPren{}}, introduced by Redfield, is the scalar product of two cycle indices, where the \smath{s_i} are \spadtype{SymmetricPolynomial}s with rational number coefficients. See also \spadfun{cup}. See \spadtype{CycleIndicators} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cardinality}}\opLeftPren{}{\it finiteSetAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cardinality}}\opLeftPren{} +{\it finiteSetAggregate}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{FiniteSetAggregate} -{\smath{\mbox{\bf cardinality}\opLeftPren{}u\opRightPren{}} returns the number of elements of +{\smath{\mbox{\bf cardinality}\opLeftPren{}u\opRightPren{}} +returns the number of elements of \smath{u}. Note: \code{cardinality(u) = \#u}. } @@ -69408,43 +70977,52 @@ Note: \code{cardinality(u) = \#u}. }% }% {1}{(\$)->\$}{SExpressionCategory} -{\smath{\mbox{\bf car}\opLeftPren{}se\opRightPren{}} returns \smath{a_1} when \smath{se} is the +{\smath{\mbox{\bf car}\opLeftPren{}se\opRightPren{}} +returns \smath{a_1} when \smath{se} is the \axiomType{SExpression} object \smath{(a_1, \ldots, a_n)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cdr}}\opLeftPren{}{\it sExpression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cdr}}\opLeftPren{} +{\it sExpression}\opRightPren{}% }% }% {1}{(\$)->\$}{SExpressionCategory} -{\smath{\mbox{\bf cdr}\opLeftPren{}se\opRightPren{}} returns \smath{(a_2, \ldots, a_n)} when \smath{se} +{\smath{\mbox{\bf cdr}\opLeftPren{}se\opRightPren{}} +returns \smath{(a_2, \ldots, a_n)} when \smath{se} is the \axiomType{SExpression} object \smath{(a_1, \ldots, a_n)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ceiling}}\opLeftPren{}{\it floatOrRationalNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ceiling}}\opLeftPren{} +{\it floatOrRationalNumber}\opRightPren{}% }% }% {1}{(\$)->D}{QuotientFieldCategory} {\opkey {Argument \smath{x} is a floating point number or fraction of numbers.} -\newitem\smath{\mbox{\bf ceiling}\opLeftPren{}x\opRightPren{}} returns the smallest integral element +\newitem\smath{\mbox{\bf ceiling}\opLeftPren{}x\opRightPren{}} +returns the smallest integral element above \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{center}}\opLeftPren{}{\it stringsOrSeries}\opRightPren{}% +\opdata{{\mbox{\axiomFun{center}}\opLeftPren{}{\it stringsOrSeries} +\opRightPren{}% }% }% {1}{(\$)->Coef}{UnivariatePowerSeriesCategory} -{\smath{\mbox{\bf center}\opLeftPren{}s\opRightPren{}} returns the point about which the series +{\smath{\mbox{\bf center}\opLeftPren{}s\opRightPren{}} returns the +point about which the series \smath{s} is expanded. -\newitem\smath{\mbox{\bf center}\opLeftPren{}ls, \allowbreak{} n, \allowbreak{} s\opRightPren{}} takes a list of strings \smath{ls}, +\newitem\smath{\mbox{\bf center}\opLeftPren{}ls, \allowbreak{} n, +\allowbreak{} s\opRightPren{}} takes a list of strings \smath{ls}, and centers them within a list of strings which is \smath{n} characters long. The remaining spaces are filled with strings composed of as many repetitions as possible of the last string parameter \smath{s}. -\newitem\smath{\mbox{\bf center}\opLeftPren{}s_1, \allowbreak{} n, \allowbreak{} s_2\opRightPren{}} is equivalent to +\newitem\smath{\mbox{\bf center}\opLeftPren{}s_1, \allowbreak{} n, +\allowbreak{} s_2\opRightPren{}} is equivalent to \smath{\mbox{\bf center}\opLeftPren{}[ s_1], n, s_2\opRightPren{}}. } @@ -69453,11 +71031,13 @@ repetitions as possible of the last string parameter \smath{s}. }% }% {1}{(Integer)->\$}{Character} -{\smath{\mbox{\bf char}\opLeftPren{}i\opRightPren{}} returns a \spadtype{Character} object with +{\smath{\mbox{\bf char}\opLeftPren{}i\opRightPren{}} returns a +\spadtype{Character} object with integer code \smath{i}. Note: {\bf ord(char({\it i})) \rm = \it i}. \newitem -\smath{\mbox{\bf char}\opLeftPren{}s\opRightPren{}} returns the unique character of a string +\smath{\mbox{\bf char}\opLeftPren{}s\opRightPren{}} returns the +unique character of a string \smath{s} of length one. } @@ -69467,29 +71047,35 @@ Note: {\bf ord(char({\it i})) \rm = \it i}. }% }% {0}{()->NonNegativeInteger}{NaiveRingInternalUseOnly} -{\smath{\mbox{\bf characteristic}\opLeftPren{}\opRightPren{}\$R} returns the characteristic of ring +{\smath{\mbox{\bf characteristic}\opLeftPren{}\opRightPren{}\$R} +returns the characteristic of ring \smath{R}: the smallest positive integer \smath{n} such that \smath{nx=0} for all \smath{x} in the ring, or zero if no such \smath{n} exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{characteristicPolynomial}}\opLeftPren{}{\it matrix\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{characteristicPolynomial}}\opLeftPren{} +{\it matrix\opt{, symbol}}\opRightPren{}% }% }% {1}{(\$)->UP}{FiniteRankAlgebra} -{\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}a\opRightPren{}} returns the characteristic +{\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}a\opRightPren{}} +returns the characteristic polynomial of the regular representation of \smath{a} with respect to any basis. \newitem -\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}m\opRightPren{}} returns the +\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}m\opRightPren{}} +returns the characteristic polynomial of the matrix \smath{m} expressed as polynomial with a new symbol as variable. \newitem -\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}m, \allowbreak{} sy\opRightPren{}} is similar except +\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}m, +\allowbreak{} sy\opRightPren{}} is similar except that the resulting polynomial has variable \smath{sy}. \newitem -\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}m, \allowbreak{} r\opRightPren{}}, where \smath{r} is +\smath{\mbox{\bf characteristicPolynomial}\opLeftPren{}m, +\allowbreak{} r\opRightPren{}}, where \smath{r} is a member of the coefficient domain of matrix \smath{m}, evaluates the characteristic polynomial at \smath{r}. In particular, if \smath{r} is the polynomial \smath{'x}, then it @@ -69502,10 +71088,12 @@ returns the characteristic polynomial expressed as a polynomial in }% }% {1}{(List(Character))->\$}{CharacterClass} -{\smath{\mbox{\bf charClass}\opLeftPren{}s\opRightPren{}} creates a character class containing exactly +{\smath{\mbox{\bf charClass}\opLeftPren{}s\opRightPren{}} creates +a character class containing exactly the characters given in the string \smath{s}. \newitem -\smath{\mbox{\bf charClass}\opLeftPren{}ls\opRightPren{}} creates a character class which contains +\smath{\mbox{\bf charClass}\opLeftPren{}ls\opRightPren{}} creates +a character class which contains exactly the characters given in the list \smath{ls} of strings. } @@ -69516,26 +71104,29 @@ exactly the characters given in the list \smath{ls} of strings. {1}{(\$)->Union(\$, "failed")}{PolynomialFactorizationExplicit} {\smath{\mbox{\bf charthRoot}\opLeftPren{}r\opRightPren{}}, where \smath{r} is an element of domain with \spadfun{characteristic} -$p \not= 0$, returns the \eth{\smath{p}} root of \smath{r}, or \mbox{\tt "failed"} if none exists in the domain. -\newitem\smath{\mbox{\bf charthRoot}\opLeftPren{}f\opRightPren{}\$R} takes the \eth{\smath{p}} root of finite field element \smath{f}, +$p \not= 0$, returns the \eth{\smath{p}} root of \smath{r}, or +\mbox{\tt "failed"} if none exists in the domain. +\newitem\smath{\mbox{\bf charthRoot}\opLeftPren{}f\opRightPren{}\$R} +takes the \eth{\smath{p}} root of finite field element \smath{f}, where \smath{p} is the characteristic of the finite field \smath{R}. Note: such a root is always defined in finite fields. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{chebyshevT}}\opLeftPren{}{\it positiveInteger}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{chebyshevT}}\opLeftPren{}{\it positiveInteger}, +\allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(NonNegativeInteger, R)->R}{OrthogonalPolynomialFunctions} -{\smath{\mbox{\bf chebyshevT}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} returns the \eth{\smath{n}} Chebyshev polynomial of the +{\smath{\mbox{\bf chebyshevT}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} +returns the \eth{\smath{n}} Chebyshev polynomial of the first kind, \smath{T_n(x)}, defined by \smath{(1-tx)/(1-2tx+t^2) = \sum\nolimits_{n=0}^\infty {T_n(x)\, t^n}}. } - - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{children}}\opLeftPren{}{\it recursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{children}}\opLeftPren{} +{\it recursiveAggregate}\opRightPren{}% }% }% {1}{(\$)->List(\$)}{RecursiveAggregate} @@ -69543,52 +71134,72 @@ first kind, \smath{T_n(x)}, defined by returns a list of the children of aggregate \smath{u}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{chineseRemainder}}\opLeftPren{}{\it listOfElements}, \allowbreak{}{\it listOfModuli}\opRightPren{}% -\opand \mbox{\axiomFun{chineseRemainder}}\opLeftPren{}{\it integer}, \allowbreak{}{\it modulus}, \allowbreak{}{\it integer}, \allowbreak{}{\it modulus}\opRightPren{}% +\opdata{{\mbox{\axiomFun{chineseRemainder}}\opLeftPren{} +{\it listOfElements}, \allowbreak{}{\it listOfModuli}\opRightPren{}% +\opand \mbox{\axiomFun{chineseRemainder}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it modulus}, \allowbreak{}{\it integer}, +\allowbreak{}{\it modulus}\opRightPren{}% }% }% -{4}{(Integer, Integer, Integer, Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf chineseRemainder}\opLeftPren{}lv, \allowbreak{} lm\opRightPren{}} where \smath{lv} is a list of +{4}{(Integer, Integer, Integer, Integer)->Integer} +{IntegerNumberTheoryFunctions} +{\smath{\mbox{\bf chineseRemainder}\opLeftPren{}lv, \allowbreak{} lm +\opRightPren{}} where \smath{lv} is a list of values \smath{[v_1, \ldots, v_n]} and \smath{lm} is a list of moduli \smath{[m_1, \ldots, m_n]}, returns \smath{m} such that \smath{m = n_i \bmod p_i}; the \smath{p_i} must be relatively prime. -\newitem\smath{\mbox{\bf chineseRemainder}\opLeftPren{}n_1, \allowbreak{} p_1, \allowbreak{} n_2, \allowbreak{} p_2\opRightPren{}} is equivalent to -\smath{\mbox{\bf chineseRemainder}\opLeftPren{}[n_1, \allowbreak{} n_2], \allowbreak{} [p_1, \allowbreak{} p_2]\opRightPren{}}, where all arguments +\newitem\smath{\mbox{\bf chineseRemainder}\opLeftPren{}n_1, +\allowbreak{} p_1, \allowbreak{} n_2, \allowbreak{} p_2\opRightPren{}} +is equivalent to +\smath{\mbox{\bf chineseRemainder}\opLeftPren{}[n_1, \allowbreak{} n_2], +\allowbreak{} [p_1, \allowbreak{} p_2]\opRightPren{}}, where all arguments are integers. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{clearDenominator}}\opLeftPren{}{\it fraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{clearDenominator}}\opLeftPren{} +{\it fraction}\opRightPren{}% }% }% {1}{(A)->A}{CommonDenominator} -{\smath{\mbox{\bf clearDenominator}\opLeftPren{}\col{q_1}{q_n}\opRightPren{}} returns \smath{\col{p_1}{p_n}} -such that \smath{q_i = p_i/d} where \smath{d} is a common denominator for the \smath{q_i}'s. -\newitem\smath{\mbox{\bf clearDenominator}\opLeftPren{}A\opRightPren{}}, where \smath{A} is a matrix of fractions, +{\smath{\mbox{\bf clearDenominator}\opLeftPren{}\col{q_1}{q_n}\opRightPren{}} +returns \smath{\col{p_1}{p_n}} +such that \smath{q_i = p_i/d} where \smath{d} is a common denominator for +the \smath{q_i}'s. +\newitem\smath{\mbox{\bf clearDenominator}\opLeftPren{}A\opRightPren{}}, +where \smath{A} is a matrix of fractions, returns matrix \smath{B} such that \smath{A = B/d} where \smath{d} is a common denominator for the elements of \smath{A}. -\newitem\smath{\mbox{\bf clearDenominator}\opLeftPren{}p\opRightPren{}} returns polynomial \smath{q} such that \smath{p = q/d} where \smath{d} is a common denominator for the coefficients of polynomial \smath{p}. +\newitem\smath{\mbox{\bf clearDenominator}\opLeftPren{}p\opRightPren{}} +returns polynomial \smath{q} such that \smath{p = q/d} where \smath{d} +is a common denominator for the coefficients of polynomial \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{clip}}\opLeftPren{}{\it rangeOrBoolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{clip}}\opLeftPren{} +{\it rangeOrBoolean}\opRightPren{}% }% }% {1}{(Boolean)->\$}{DrawOption} -{\smath{\mbox{\bf clip}\opLeftPren{}b\opRightPren{}} turns \twodim{} clipping on if \smath{b} is \smath{true}, and off if \smath{b} +{\smath{\mbox{\bf clip}\opLeftPren{}b\opRightPren{}} turns +\twodim{} clipping on if \smath{b} is \smath{true}, and off if \smath{b} is \smath{false}. \sayOption{clip}{b} \newitem -\smath{\mbox{\bf clip}\opLeftPren{}[ a..b]\opRightPren{}} defines the range for user-defined clipping. +\smath{\mbox{\bf clip}\opLeftPren{}[ a..b]\opRightPren{}} +defines the range for user-defined clipping. \sayOption{range}{[ a..b]} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{clipPointsDefault}}\opLeftPren{}{\it \opt{boolean}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{clipPointsDefault}}\opLeftPren{} +{\it \opt{boolean}}\opRightPren{}% }% }% {0}{()->Boolean}{GraphicsDefaults} -{\smath{\mbox{\bf clipPointsDefault}\opLeftPren{}\opRightPren{}} tests if automatic clipping is to be done. -\newitem\smath{\mbox{\bf clipPointsDefault}\opLeftPren{}b\opRightPren{}} turns on automatic clipping for \smath{b=true}, and +{\smath{\mbox{\bf clipPointsDefault}\opLeftPren{}\opRightPren{}} tests if +automatic clipping is to be done. +\newitem\smath{\mbox{\bf clipPointsDefault}\opLeftPren{}b\opRightPren{}} +turns on automatic clipping for \smath{b=true}, and off if \smath{b=false}. \sayOption{clip}{b} } @@ -69597,8 +71208,10 @@ off if \smath{b=false}. \sayOption{clip}{b} }% }% {1}{(\$)->\$}{FileCategory} -{\smath{\mbox{\bf close}\opLeftPren{}v\opRightPren{}} closes the viewport window of the given \twodim{} or -\threedim{} viewport \smath{v} and terminates the corresponding \Unix{} process. +{\smath{\mbox{\bf close}\opLeftPren{}v\opRightPren{}} +closes the viewport window of the given \twodim{} or +\threedim{} viewport \smath{v} and terminates the +corresponding Unix process. Argument \smath{v} is a member of domain \spadtype{TwoDimensionalViewport} or \spadtype{ThreeDimensionalViewport}. @@ -69609,43 +71222,57 @@ Argument \smath{v} is a member of domain }% }% {1}{(\$)->\$}{FileCategory} -{\smath{\mbox{\bf close!}\opLeftPren{}fn\opRightPren{}} returns the file \smath{fn} closed to input and output.} +{\smath{\mbox{\bf close!}\opLeftPren{}fn\opRightPren{}} +returns the file \smath{fn} closed to input and output.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{closedCurve?}}\opLeftPren{}{\it threeSpace}\opRightPren{}% +\opdata{{\mbox{\axiomFun{closedCurve?}}\opLeftPren{} +{\it threeSpace}\opRightPren{}% }% }% {1}{(\$)->Boolean}{ThreeSpace} -{\smath{\mbox{\bf closedCurve?}\opLeftPren{}sp\opRightPren{}} tests if the \spadtype{ThreeSpace} object +{\smath{\mbox{\bf closedCurve?}\opLeftPren{}sp\opRightPren{}} +tests if the \spadtype{ThreeSpace} object \smath{sp} contains a single closed curve component. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{closedCurve}}\opLeftPren{}{\it listsOfPoints\opt{, listOfPoints}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{closedCurve}}\opLeftPren{} +{\it listsOfPoints\opt{, listOfPoints}}\opRightPren{}% }% }% {1}{(List(Point(R)))->\$}{ThreeSpace} -{\smath{\mbox{\bf closedCurve}\opLeftPren{}lpt\opRightPren{}} returns a \spadtype{ThreeSpace} object -containing a single closed curve described by the list of points \smath{lpt} of the form +{\smath{\mbox{\bf closedCurve}\opLeftPren{}lpt\opRightPren{}} +returns a \spadtype{ThreeSpace} object +containing a single closed curve described by the list of points +\smath{lpt} of the form $[ p_0, p_1, \ldots, p_n, p_0]$. -\newitem\smath{\mbox{\bf closedCurve}\opLeftPren{}sp\opRightPren{}} returns a closed curve as a list of points, where +\newitem\smath{\mbox{\bf closedCurve}\opLeftPren{}sp\opRightPren{}} +returns a closed curve as a list of points, where \smath{sp} must be a \spadtype{ThreeSpace} object containing a single closed curve. -\newitem\smath{\mbox{\bf closedCurve}\opLeftPren{}sp, \allowbreak{} lpt\opRightPren{}} returns \spadtype{ThreeSpace} object with -the closed curve denoted by \smath{lpt} added. Argument \smath{lpt} is a list of points +\newitem\smath{\mbox{\bf closedCurve}\opLeftPren{}sp, +\allowbreak{} lpt\opRightPren{}} returns \spadtype{ThreeSpace} object with +the closed curve denoted by \smath{lpt} added. +Argument \smath{lpt} is a list of points of the form $[ p_0, p_1, \ldots, p_n, p_0]$. } - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coefficient}}\opLeftPren{}{\it polynomialOrSeries}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coefficient}}\opLeftPren{} +{\it polynomialOrSeries}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, E)->R}{AbelianMonoidRing} -{\smath{\mbox{\bf coefficient}\opLeftPren{}p, \allowbreak{} n\opRightPren{}} extracts the coefficient of the monomial with exponent \smath{n} from polynomial \smath{p}, or returns zero if exponent is not present. -\newitem\smath{\mbox{\bf coefficient}\opLeftPren{}u, \allowbreak{} x, \allowbreak{} n\opRightPren{}} returns the coefficient of variable \smath{x} to the power +{\smath{\mbox{\bf coefficient}\opLeftPren{}p, \allowbreak{} n\opRightPren{}} +extracts the coefficient of the monomial with exponent \smath{n} from +polynomial \smath{p}, or returns zero if exponent is not present. +\newitem\smath{\mbox{\bf coefficient}\opLeftPren{}u, \allowbreak{} x, +\allowbreak{} n\opRightPren{}} returns the coefficient of variable \smath{x} +to the power \smath{n} in \smath{u}, a multivariate polynomial or series. -\newitem\smath{\mbox{\bf coefficient}\opLeftPren{}u, \allowbreak{} \col{x_1}{x_k}, \col{n_1}{n_k}\opRightPren{}} +\newitem\smath{\mbox{\bf coefficient}\opLeftPren{}u, \allowbreak{} +\col{x_1}{x_k}, \col{n_1}{n_k}\opRightPren{}} returns the coefficient of \smath{x_1^{n_1} \cdots x_k^{n_k}} in \smath{u}, a multivariate series or polynomial. \newitem Also defined for domain \spadtype{CliffordAlgebra} and categories @@ -69653,62 +71280,78 @@ a multivariate series or polynomial. \spadtype{FreeAbelianCategory}, and \spadtype{MonogenicLinearOperator}. \newitem -{\smath{\mbox{\bf coefficient}\opLeftPren{}s, \allowbreak{} n\opRightPren{}} returns the terms of total degree \smath{n} of series +{\smath{\mbox{\bf coefficient}\opLeftPren{}s, +\allowbreak{} n\opRightPren{}} returns the terms of +total degree \smath{n} of series \smath{s} as a polynomial. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coefficients}}\opLeftPren{}{\it polynomialOrStream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coefficients}}\opLeftPren{} +{\it polynomialOrStream}\opRightPren{}% }% }% {1}{(\$)->List(R)}{FiniteAbelianMonoidRing} -{\smath{\mbox{\bf coefficients}\opLeftPren{}p\opRightPren{}} returns the list of non-zero coefficients of polynomial \smath{p} +{\smath{\mbox{\bf coefficients}\opLeftPren{}p\opRightPren{}} returns +the list of non-zero coefficients of polynomial \smath{p} starting with the coefficient of the maximum degree. -\newitem\smath{\mbox{\bf coefficients}\opLeftPren{}s\opRightPren{}} returns a stream of coefficients \smath{[ a_0, a_1, a_2, \ldots]} +\newitem\smath{\mbox{\bf coefficients}\opLeftPren{}s\opRightPren{}} +returns a stream of coefficients \smath{[ a_0, a_1, a_2, \ldots]} for the stream \smath{s}: \smath{a_0 + a_1 x + a_2 x^2 + \cdots}. Note: the entries of the stream may be zero.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coerceImages}}\opLeftPren{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coerceImages}}\opLeftPren{} +{\it listOfElements}\opRightPren{}% }% }% {1}{(List(S))->\$}{Permutation} -{\smath{\mbox{\bf coerceImages}\opLeftPren{}ls\opRightPren{}} coerces the list \smath{ls} to a +{\smath{\mbox{\bf coerceImages}\opLeftPren{}ls\opRightPren{}} +coerces the list \smath{ls} to a permutation whose image is given by \smath{ls} and whose preimage is fixed to be \smath{[ 1, \ldots, n]}. Note: -\smath{\mbox{\bf coerceImages}\opLeftPren{}ls\opRightPren{}}=\smath{coercePreimagesImages([ +\smath{\mbox{\bf coerceImages}\opLeftPren{}ls\opRightPren{}}= +\smath{coercePreimagesImages([ 1, \ldots, n ], ls)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coerceListOfPairs}}\opLeftPren{}{\it listOfPairsOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coerceListOfPairs}}\opLeftPren{} +{\it listOfPairsOfElements}\opRightPren{}% }% }% {1}{(List(List(S)))->\$}{Permutation} -{\smath{\mbox{\bf coerceListOfPairs}\opLeftPren{}lls\opRightPren{}} coerces a list of pairs +{\smath{\mbox{\bf coerceListOfPairs}\opLeftPren{}lls\opRightPren{}} +coerces a list of pairs \smath{lls} to a permutation, or calls \smath{error} if not consistent, that is, the set of the first elements coincides with the set of second elements. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coercePreimagesImages}}\opLeftPren{}{\it listOfListOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coercePreimagesImages}}\opLeftPren{} +{\it listOfListOfElements}\opRightPren{}% }% }% {1}{(List(List(S)))->\$}{Permutation} -{\smath{\mbox{\bf coercePreimagesImages}\opLeftPren{}lls\opRightPren{}} coerces the representation +{\smath{\mbox{\bf coercePreimagesImages}\opLeftPren{}lls\opRightPren{}} +coerces the representation \smath{lls} of a permutation as a list of preimages and images to a permutation.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coleman}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coleman}}\opLeftPren{} +{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}, \allowbreak{} +{\it listOfIntegers}\opRightPren{}% }% }% -{3}{(List(Integer), List(Integer), List(Integer))->Matrix(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf coleman}\opLeftPren{}alpha, \allowbreak{} beta, \allowbreak{} pi\opRightPren{}} generates the Coleman-matrix of a +{3}{(List(Integer), List(Integer), List(Integer))->Matrix(Integer)} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf coleman}\opLeftPren{}alpha, \allowbreak{} beta, +\allowbreak{} pi\opRightPren{}} generates the Coleman-matrix of a certain double coset of the symmetric group given by an representing element \smath{pi} and \smath{alpha} and \smath{beta}. @@ -69721,26 +71364,32 @@ column sums \smath{beta}. }% }% {1}{(Integer)->\$}{Color} -{\smath{\mbox{\bf color}\opLeftPren{}i\opRightPren{}} returns a color of the indicated hue \smath{i}. +{\smath{\mbox{\bf color}\opLeftPren{}i\opRightPren{}} returns a +color of the indicated hue \smath{i}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{colorDef}}\opLeftPren{}{\it viewPort}, \allowbreak{}{\it color}, \allowbreak{}{\it color}\opRightPren{}% +\opdata{{\mbox{\axiomFun{colorDef}}\opLeftPren{}{\it viewPort}, +\allowbreak{}{\it color}, \allowbreak{}{\it color}\opRightPren{}% }% }% {1}{(Integer)->\$}{Color} -{\smath{\mbox{\bf colorDef}\opLeftPren{}v, \allowbreak{} c_1, \allowbreak{} c_2\opRightPren{}} sets the range of colors along the colormap so +{\smath{\mbox{\bf colorDef}\opLeftPren{}v, +\allowbreak{} c_1, \allowbreak{} c_2\opRightPren{}} +sets the range of colors along the colormap so that the lower end of the colormap is defined by \smath{c_1} and the top end of the colormap is defined by \smath{c2} for the given \threedim{} viewport \smath{v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{colorFunction}}\opLeftPren{}{\it smallFloatFunction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{colorFunction}}\opLeftPren{} +{\it smallFloatFunction}\opRightPren{}% }% }% {1}{((DoubleFloat)->DoubleFloat)->\$}{DrawOption} -{\smath{\mbox{\bf colorFunction}\opLeftPren{}fn\opRightPren{}} specifies the color for +{\smath{\mbox{\bf colorFunction}\opLeftPren{}fn\opRightPren{}} +specifies the color for three-dimensional plots. Function \smath{fn} can take one to three \spadtype{DoubleFloat} arguments and always returns a \spadtype{DoubleFloat} value. @@ -69753,37 +71402,44 @@ If three arguments, the color is based on the \smath{x}, } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{column}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{column}}\opLeftPren{} +{\it matrix}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {2}{(\$, Integer)->Col}{RectangularMatrixCategory} -{\smath{\mbox{\bf column}\opLeftPren{}M, \allowbreak{} j\opRightPren{}} returns the \eth{\smath{j}} column of the +{\smath{\mbox{\bf column}\opLeftPren{}M, +\allowbreak{} j\opRightPren{}} returns the \eth{\smath{j}} column of the matrix or \spadtype{TwoDimensionalArrayCategory} object \smath{M}, or calls \spadfun{error} if the index is outside the proper range. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{commaSeparate}}\opLeftPren{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{commaSeparate}}\opLeftPren{} +{\it listOfOutputForms}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf commaSeparate}\opLeftPren{}lo\opRightPren{}}, where \smath{lo} is a list of objects +{\smath{\mbox{\bf commaSeparate}\opLeftPren{}lo\opRightPren{}}, +where \smath{lo} is a list of objects of type \spadtype{OutputForm} (normally unexposed), returns an output form which separates the elements of \smath{lo} by commas. } - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{commonDenominator}}\opLeftPren{}{\it fraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{commonDenominator}}\opLeftPren{} +{\it fraction}\opRightPren{}% }% }% {1}{(A)->R}{CommonDenominator} -{\smath{\mbox{\bf commonDenominator}\opLeftPren{}\col{q_1}{q_n}\opRightPren{}} returns a common +{\smath{\mbox{\bf commonDenominator}\opLeftPren{}\col{q_1}{q_n}\opRightPren{}} +returns a common denominator for the \smath{q_i}'s. -\newitem\smath{\mbox{\bf commonDenominator}\opLeftPren{}A\opRightPren{}}, where \smath{A} is a matrix +\newitem\smath{\mbox{\bf commonDenominator}\opLeftPren{}A\opRightPren{}}, +where \smath{A} is a matrix of fractions, returns a common denominator for the elements of \smath{A}. -\newitem\smath{\mbox{\bf commonDenominator}\opLeftPren{}p\opRightPren{}} returns a common denominator +\newitem\smath{\mbox{\bf commonDenominator}\opLeftPren{}p\opRightPren{}} +returns a common denominator for the coefficients of polynomial \smath{p}. } @@ -69792,46 +71448,60 @@ for the coefficients of polynomial \smath{p}. }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf commutative?}\opLeftPren{}\opRightPren{}\$R} tests if multiplication in the algebra \smath{R} is commutative. +{\smath{\mbox{\bf commutative?}\opLeftPren{}\opRightPren{}\$R} +tests if multiplication in the algebra \smath{R} is commutative. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{commutator}}\opLeftPren{}{\it groupElement}, \allowbreak{}{\it groupElement}\opRightPren{}% +\opdata{{\mbox{\axiomFun{commutator}}\opLeftPren{} +{\it groupElement}, \allowbreak{}{\it groupElement}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{Group} -{\smath{\mbox{\bf commutator}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} computes \smath{\mbox{\bf inv}\opLeftPren{}p) * {\mbox {\bf inv}}(q) * p * q} where \smath{p} and +{\smath{\mbox{\bf commutator}\opLeftPren{}p, +\allowbreak{} q\opRightPren{}} computes \smath{\mbox{\bf inv}\opLeftPren{}p) * +{\mbox {\bf inv}}(q) * p * q} where \smath{p} and \smath{q} are members of a \spadtype{Group} domain. \newitem -\smath{\mbox{\bf commutator}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns \smath{ab-ba} where \smath{a} +\smath{\mbox{\bf commutator}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns \smath{ab-ba} where \smath{a} and \smath{b} are members of a \spadtype{NonAssociativeRing} domain. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{compactFraction}}\opLeftPren{}{\it partialFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{compactFraction}}\opLeftPren{} +{\it partialFraction}\opRightPren{}% }% }% {1}{(\$)->\$}{PartialFraction} -{\smath{\mbox{\bf compactFraction}\opLeftPren{}u\opRightPren{}} normalizes the partial fraction \smath{u} to a compact representation where it has only one fractional term per prime in the denominator.} +{\smath{\mbox{\bf compactFraction}\opLeftPren{}u\opRightPren{}} +normalizes the partial fraction \smath{u} to a compact representation +where it has only one fractional term per prime in the denominator.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{comparison}}\opLeftPren{}{\it basicOperator}, \allowbreak{}{\it property}\opRightPren{}% +\opdata{{\mbox{\axiomFun{comparison}}\opLeftPren{} +{\it basicOperator}, \allowbreak{}{\it property}\opRightPren{}% }% }% {2}{(\$, (\$, \$)->Boolean)->\$}{BasicOperator} -{\smath{\mbox{\bf comparison}\opLeftPren{}op, \allowbreak{} p\opRightPren{}} attaches \smath{p} +{\smath{\mbox{\bf comparison}\opLeftPren{}op, +\allowbreak{} p\opRightPren{}} attaches \smath{p} as the \mbox{\tt "\%less?"} property to \smath{op}. If \smath{op1} and \smath{op2} have the same name, and one of them has a \mbox{\tt "\%less?"} property \smath{p}, then \smath{p(op1, op2)} is called to decide whether \smath{op1 < op2}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{compile}}\opLeftPren{}{\it symbol}, \allowbreak{}{\it listOfTypes}\opRightPren{}% +\opdata{{\mbox{\axiomFun{compile}}\opLeftPren{} +{\it symbol}, \allowbreak{}{\it listOfTypes}\opRightPren{}% }% }% { (Symbol, List $) -> Symbol}{}{} -{\smath{\mbox{\bf compile}\opLeftPren{}f, \allowbreak{} [T_1, \allowbreak{} \ldots, T_n]\opRightPren{}} forces the interpreter to compile -the function with name \smath{f} with signature \smath{(T_1, \ldots, T_n) -> T}, +{\smath{\mbox{\bf compile}\opLeftPren{}f, +\allowbreak{} [T_1, \allowbreak{} \ldots, T_n]\opRightPren{}} +forces the interpreter to compile +the function with name \smath{f} with signature +\smath{(T_1, \ldots, T_n) -> T}, where \smath{T} is a type determined by type analysis of the function body of \smath{f}. If the compilation is successful, the operation returns the name \smath{f}. @@ -69843,7 +71513,9 @@ See also \spadfun{function}, } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{compiledFunction}}\opLeftPren{}{\it expression}, \allowbreak{}{\it symbol }\allowbreak $\,[$ , \allowbreak{}{\it symbol}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{compiledFunction}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it symbol }\allowbreak $\,[$ , \allowbreak{} +{\it symbol}$]$\opRightPren{}% }% }% { (S, SY) -> (D -> I)}{}{} @@ -69852,14 +71524,16 @@ coercible to type \spadtype{InputForm} (most commonly used types). These functions must be package called to define the type of the function produced.} \newitem -\smath{\mbox{\bf compiledFunction}\opLeftPren{}expr, \allowbreak{} x\opRightPren{}\$P}, where \smath{P} is +\smath{\mbox{\bf compiledFunction}\opLeftPren{}expr, +\allowbreak{} x\opRightPren{}\$P}, where \smath{P} is \spadtype{MakeUnaryCompiledFunction(E, S, T)}, returns an anonymous function of type \smath{S}{T} defined by defined by \smath{x \mapsto {\rm expr}}. The anonymous function is compiled and directly applicable to objects of type \smath{S}. \newitem -\smath{\mbox{\bf compiledFunction}\opLeftPren{}expr, \allowbreak{} x, \allowbreak{} y\opRightPren{}\$P}, where \smath{P} is +\smath{\mbox{\bf compiledFunction}\opLeftPren{}expr, +\allowbreak{} x, \allowbreak{} y\opRightPren{}\$P}, where \smath{P} is \spadtype{MakeBinaryCompiledFunction(E, A, B, T)} returns an anonymous function of type \spadsig{(A, B)}{T} defined by \smath{(x, y) \mapsto {\rm expr}}. @@ -69871,33 +71545,42 @@ See also \spadfun{compile}, \spadfun{function}, and } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complement}}\opLeftPren{}{\it finiteSetElement}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complement}}\opLeftPren{} +{\it finiteSetElement}\opRightPren{}% }% }% {1}{(\$)->\$}{FiniteSetAggregate} -{\smath{\mbox{\bf complement}\opLeftPren{}u\opRightPren{}} returns the complement of the finite set \smath{u}, that is, the set of all values not in \smath{u}. +{\smath{\mbox{\bf complement}\opLeftPren{}u\opRightPren{}} +returns the complement of the finite set \smath{u}, that is, the +set of all values not in \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complementaryBasis}}\opLeftPren{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complementaryBasis}}\opLeftPren{} +{\it vector}\opRightPren{}% }% }% {1}{(Vector(\$))->Vector(\$)}{FunctionFieldCategory} -{\smath{\mbox{\bf complementaryBasis}\opLeftPren{}b_1, \allowbreak{} \ldots, b_n\opRightPren{}} returns the complementary -basis \smath{(b_1^{'}, \ldots, b_n^{'})} of \smath{(b_1, \ldots, b_n)} for a domain of category \spadtype{FunctionFieldCategory}. +{\smath{\mbox{\bf complementaryBasis}\opLeftPren{}b_1, +\allowbreak{} \ldots, b_n\opRightPren{}} returns the complementary +basis \smath{(b_1^{'}, \ldots, b_n^{'})} of \smath{(b_1, \ldots, b_n)} +for a domain of category \spadtype{FunctionFieldCategory}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complete}}\opLeftPren{}{\it streamOrInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complete}}\opLeftPren{} +{\it streamOrInteger}\opRightPren{}% }% }% {1}{(\$)->\$}{LazyStreamAggregate} -{\smath{\mbox{\bf complete}\opLeftPren{}u\opRightPren{}} causes all terms of a stream or continued +{\smath{\mbox{\bf complete}\opLeftPren{}u\opRightPren{}} +causes all terms of a stream or continued fraction \smath{u} to be computed. If not called on a finite stream or continued fraction, this function will compute until interrupted. \newitem -\smath{\mbox{\bf complete}\opLeftPren{}n\opRightPren{}} is the \eth{\smath{n}} complete +\smath{\mbox{\bf complete}\opLeftPren{}n\opRightPren{}} +is the \eth{\smath{n}} complete homogeneous symmetric function expressed in terms of power sums. Alternatively, it is the cycle index of the symmetric group of degree \smath{n}. @@ -69905,108 +71588,152 @@ See \spadtype{CycleIndicators} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{completeEchelonBasis}}\opLeftPren{}{\it vectorOfVectors}\opRightPren{}% +\opdata{{\mbox{\axiomFun{completeEchelonBasis}}\opLeftPren{} +{\it vectorOfVectors}\opRightPren{}% }% }% {1}{(Vector(Vector(R)))->Matrix(R)}{RepresentationPackage2} -{\smath{\mbox{\bf completeEchelonBasis}\opLeftPren{}vv\opRightPren{}} returns a completed basis from \smath{vv}, +{\smath{\mbox{\bf completeEchelonBasis}\opLeftPren{}vv\opRightPren{}} +returns a completed basis from \smath{vv}, a vector of vectors of domain elements. \seeDetails{RepresentationPackage2} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complex}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complex}}\opLeftPren{}{\it element}, +\allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(R, R)->\$}{ComplexCategory} -{\smath{\mbox{\bf complex}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} creates the complex expression \smath{x} + \%i*y.} +{\smath{\mbox{\bf complex}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +creates the complex expression \smath{x} + \%i*y.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexEigenvalues}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it precision}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexEigenvalues}}\opLeftPren{}{\it matrix}, +\allowbreak{}{\it precision}\opRightPren{}% }% }% -{2}{(Matrix(Complex(Fraction(Integer))), Float)->List(Complex(Float))}{NumericComplexEigenPackage} -{\smath{\mbox{\bf complexEigenvalues}\opLeftPren{}m, \allowbreak{} eps\opRightPren{}} computes the eigenvalues of the matrix \smath{m} to precision \smath{eps}, chosen as a float or a rational number so as to agree with the type +{2}{(Matrix(Complex(Fraction(Integer))), Float)->List(Complex(Float))} +{NumericComplexEigenPackage} +{\smath{\mbox{\bf complexEigenvalues}\opLeftPren{}m, \allowbreak{} +eps\opRightPren{}} computes the eigenvalues of the matrix \smath{m} +to precision \smath{eps}, chosen as a float or a rational number +so as to agree with the type of the coefficients of the matrix \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexEigenvectors}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it precision}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexEigenvectors}}\opLeftPren{} +{\it matrix}, \allowbreak{}{\it precision}\opRightPren{}% }% }% -{2}{(Matrix(Complex(Fraction(Integer))), Float)->List(Record(floatval:Complex(Float), floatmult:Integer, floatvect:List(Matrix(Complex(Float)))))}{NumericComplexEigenPackage} -{\smath{\mbox{\bf complexEigenvectors}\opLeftPren{}m, \allowbreak{} eps\opRightPren{}} (\smath{m}, a matrix) returns a list of records, +{2}{(Matrix(Complex(Fraction(Integer))), Float)-> +List(Record(floatval:Complex(Float), floatmult:Integer, +floatvect:List(Matrix(Complex(Float)))))}{NumericComplexEigenPackage} +{\smath{\mbox{\bf complexEigenvectors}\opLeftPren{}m, +\allowbreak{} eps\opRightPren{}} (\smath{m}, a matrix) +returns a list of records, each containing a complex eigenvalue, its algebraic multiplicity, and a list of associated eigenvectors. -All results are expressed as complex floats or rationals with precision \smath{eps}. +All results are expressed as complex floats or rationals +with precision \smath{eps}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexElementary}}\opLeftPren{}{\it expression\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexElementary}}\opLeftPren{} +{\it expression\opt{, symbol}}\opRightPren{}% }% }% {1}{(F)->F}{TrigonometricManipulations} -{\smath{\mbox{\bf complexElementary}\opLeftPren{}e\opRightPren{}} rewrites \smath{e} in terms of the two fundamental complex transcendental elementary functions: \smath{log, exp}. -\newitem\smath{\mbox{\bf complexElementary}\opLeftPren{}e, \allowbreak{} x\opRightPren{}} does the same but only rewrites kernels of \smath{e} +{\smath{\mbox{\bf complexElementary}\opLeftPren{}e\opRightPren{}} +rewrites \smath{e} in terms of the two fundamental complex transcendental +elementary functions: \smath{log, exp}. +\newitem\smath{\mbox{\bf complexElementary}\opLeftPren{}e, +\allowbreak{} x\opRightPren{}} does the same but only rewrites +kernels of \smath{e} involving \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexExpand}}\opLeftPren{}{\it integrationResult}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexExpand}}\opLeftPren{} +{\it integrationResult}\opRightPren{}% }% }% {1}{(IntegrationResult(F))->F}{IntegrationResultToFunction} -{\smath{\mbox{\bf complexExpand}\opLeftPren{}ir\opRightPren{}}, where \smath{ir} is an \spadtype{IntegrationResult}, returns the expanded complex function corresponding to \smath{ir}. +{\smath{\mbox{\bf complexExpand}\opLeftPren{}ir\opRightPren{}}, +where \smath{ir} is an \spadtype{IntegrationResult}, +returns the expanded complex function corresponding to \smath{ir}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexIntegrate}}\opLeftPren{}{\it expression}, \allowbreak{}{\it variable}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexIntegrate}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it variable}\opRightPren{}% }% }% {2}{(F, Symbol)->F}{FunctionSpaceComplexIntegration} -{\smath{\mbox{\bf complexIntegrate}\opLeftPren{}f, \allowbreak{} x\opRightPren{}} returns $\int f(x)dx$ where \smath{x} is viewed as a complex variable. +{\smath{\mbox{\bf complexIntegrate}\opLeftPren{}f, +\allowbreak{} x\opRightPren{}} returns $\int f(x)dx$ where \smath{x} is viewed +as a complex variable. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexLimit}}\opLeftPren{}{\it expression}, \allowbreak{}{\it equation}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexLimit}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it equation}\opRightPren{}% }% }% -{2}{(Fraction(Polynomial(R)), Equation(Fraction(Polynomial(R))))->OnePointCompletion(Fraction(Polynomial(R)))}{RationalFunctionLimitPackage} -{\smath{\mbox{\bf complexLimit}\opLeftPren{}f(x), \allowbreak{} x = a\opRightPren{}} computes the complex limit of \smath{f} as its argument \smath{x} approaches \smath{a}. +{2}{(Fraction(Polynomial(R)), +Equation(Fraction(Polynomial(R))))-> +OnePointCompletion(Fraction(Polynomial(R)))}{RationalFunctionLimitPackage} +{\smath{\mbox{\bf complexLimit}\opLeftPren{}f(x), +\allowbreak{} x = a\opRightPren{}} computes the complex +limit of \smath{f} as its argument +\smath{x} approaches \smath{a}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexNormalize}}\opLeftPren{}{\it expression\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexNormalize}}\opLeftPren{} +{\it expression\opt{, symbol}}\opRightPren{}% }% }% {1}{(F)->F}{TrigonometricManipulations} -{\smath{\mbox{\bf complexNormalize}\opLeftPren{}e\opRightPren{}} rewrites \smath{e} using the least possible number of complex independent kernels. -\newitem\smath{\mbox{\bf complexNormalize}\opLeftPren{}e, \allowbreak{} x\opRightPren{}} rewrites \smath{e} using the least possible number of complex independent kernels involving \smath{x}. +{\smath{\mbox{\bf complexNormalize}\opLeftPren{}e\opRightPren{}} rewrites +\smath{e} using the least possible number of complex independent kernels. +\newitem\smath{\mbox{\bf complexNormalize}\opLeftPren{}e, +\allowbreak{} x\opRightPren{}} rewrites \smath{e} using the least possible +number of complex independent kernels involving \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexNumeric}}\opLeftPren{}{\it expression\opt{, positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexNumeric}}\opLeftPren{} +{\it expression\opt{, positiveInteger}}\opRightPren{}% }% }% {1}{(Expression(S))->Complex(Float)}{Numeric} -{\smath{\mbox{\bf complexNumeric}\opLeftPren{}u\opRightPren{}} returns a complex approximation of \smath{u}, +{\smath{\mbox{\bf complexNumeric}\opLeftPren{}u\opRightPren{}} +returns a complex approximation of \smath{u}, where \smath{u} is a polynomial or an expression. -\newitem\smath{\mbox{\bf complexNumeric}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} does the same but requires accuracy to be up to +\newitem\smath{\mbox{\bf complexNumeric}\opLeftPren{}u, +\allowbreak{} n\opRightPren{}} does the same but requires accuracy to be up to \smath{n} decimal places. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexRoots}}\opLeftPren{}{\it rationalFunctions\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexRoots}}\opLeftPren{} +{\it rationalFunctions\opt{, options}}\opRightPren{}% }% }% -{2}{(Fraction(Polynomial(Complex(Fraction(Integer)))), Par)->List(Complex(Par))}{FloatingComplexPackage} -{\smath{\mbox{\bf complexRoots}\opLeftPren{}rf, \allowbreak{} eps\opRightPren{}} finds all the complex solutions of a +{2}{(Fraction(Polynomial(Complex(Fraction(Integer)))), Par)-> +List(Complex(Par))}{FloatingComplexPackage} +{\smath{\mbox{\bf complexRoots}\opLeftPren{}rf, +\allowbreak{} eps\opRightPren{}} finds all the complex solutions of a univariate rational function with rational number coefficients with precision given by \smath{eps}. The complex solutions are returned either as rational numbers or floats depending on whether \smath{eps} is a rational number or a float. -\newitem\smath{\mbox{\bf complexRoots}\opLeftPren{}lrf, \allowbreak{} lv, \allowbreak{} eps\opRightPren{}} similarly finds all the complex solutions +\newitem\smath{\mbox{\bf complexRoots}\opLeftPren{}lrf, +\allowbreak{} lv, \allowbreak{} eps\opRightPren{}} similarly +finds all the complex solutions of a list of rational functions with rational number coefficients with respect the variables appearing in \smath{lv}. Solutions are computed to precision \smath{eps} and returned as @@ -70014,67 +71741,100 @@ a list of values corresponding to the order of variables in \smath{lv}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexSolve}}\opLeftPren{}{\it eq}, \allowbreak{}{\it x}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexSolve}}\opLeftPren{} +{\it eq}, \allowbreak{}{\it x}\opRightPren{}% }% }% -{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))}{TransSolvePackage} +{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))} +{TransSolvePackage} {See \smath{\mbox{\bf solve}\opLeftPren{}u, \allowbreak{} v\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{complexZeros}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it floatOrRationaNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{complexZeros}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it floatOrRationaNumber}\opRightPren{}% }% }% {2}{(UP, Par)->List(Complex(Par))}{ComplexRootPackage} -{\smath{\mbox{\bf complexZeros}\opLeftPren{}poly, \allowbreak{} eps\opRightPren{}} finds the complex zeros of the +{\smath{\mbox{\bf complexZeros}\opLeftPren{}poly, +\allowbreak{} eps\opRightPren{}} finds the complex zeros of the univariate polynomial \smath{poly} to precision \smath{eps}. Solutions are returned either as complex floats or rationals depending on the type of \smath{eps}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{components}}\opLeftPren{}{\it threeSpace}\opRightPren{}% +\opdata{{\mbox{\axiomFun{components}}\opLeftPren{} +{\it threeSpace}\opRightPren{}% }% }% {1}{(\$)->List(\$)}{ThreeSpace} -{\smath{\mbox{\bf components}\opLeftPren{}sp\opRightPren{}} takes the \spadtype{ThreeSpace} object \smath{sp}, and returns +{\smath{\mbox{\bf components}\opLeftPren{}sp\opRightPren{}} takes the +\spadtype{ThreeSpace} object \smath{sp}, and returns a list of \spadtype{ThreeSpace} objects, each having a single component. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{composite}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{composite}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it polynomial}\opRightPren{}% }% }% {1}{(List(\$))->\$}{ThreeSpace} -{\smath{\mbox{\bf composite}\opLeftPren{}p, \allowbreak{} q\opRightPren{}}, for polynomials \smath{p} and \smath{q}, -returns \smath{f} if \smath{p} = \smath{f(q)}, and \mbox{\tt "failed"} if no such \smath{f} exists. +{\smath{\mbox{\bf composite}\opLeftPren{}p, \allowbreak{} q\opRightPren{}}, +for polynomials \smath{p} and \smath{q}, +returns \smath{f} if \smath{p} = \smath{f(q)}, and \mbox{\tt "failed"} +if no such \smath{f} exists. \newitem -\smath{\mbox{\bf composite}\opLeftPren{}lsp\opRightPren{}}, where \smath{lsp} is a list +\smath{\mbox{\bf composite}\opLeftPren{}lsp\opRightPren{}}, +where \smath{lsp} is a list \smath{[ sp_1, sp_2, \ldots, sp_n]} of \spadtype{ThreeSpace} objects, returns a single \spadtype{ThreeSpace} object containing the union of all objects in the parameter list grouped as a single composite. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{composites}}\opLeftPren{}{\it threeSpace}\opRightPren{}% +\opdata{{\mbox{\axiomFun{composites}}\opLeftPren{} +{\it threeSpace}\opRightPren{}% }% }% {1}{(\$)->List(\$)}{ThreeSpace} -{\smath{\mbox{\bf composites}\opLeftPren{}sp\opRightPren{}} takes the \spadtype{ThreeSpace} object \smath{sp} -and returns a list of \spadtype{ThreeSpace} objects, one for each single composite -of \smath{sp}. If \smath{sp} has no defined composites (composites need to be explicitly created), the list returned is empty. Note that not all the components need to be part of a composite.} +{\smath{\mbox{\bf composites}\opLeftPren{}sp\opRightPren{}} +takes the \spadtype{ThreeSpace} object \smath{sp} +and returns a list of \spadtype{ThreeSpace} objects, +one for each single composite +of \smath{sp}. If \smath{sp} has no defined composites +(composites need to be explicitly created), the list returned is empty. +Note that not all the components need to be part of a composite.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{concat}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% -\opand \mbox{\axiomFun{concat!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{concat}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opand \mbox{\axiomFun{concat!}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{(\$, S)->\$}{ExtensibleLinearAggregate} -{\smath{\mbox{\bf concat}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} returns list \smath{u} with additional element \smath{x} at the end. Note: equivalent to \smath{\mbox{\bf concat}\opLeftPren{}u, \allowbreak{} [ x]\opRightPren{}}. -\newitem\smath{\mbox{\bf concat}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} returns an aggregate consisting of the elements of \smath{u} followed by the elements of \smath{v}. -\newitem\smath{\mbox{\bf concat}\opLeftPren{}u\opRightPren{}}, where \smath{u} is a list of aggregates \smath{[ a, b, $\ldots$, c]}, returns a single aggregate consisting of the elements of \smath{a} followed by those of \smath{b} followed $\ldots$ by the elements of \smath{c}. -\newitem\smath{\mbox{\bf concat!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}, where \smath{u} is extensible, destructively adds element \smath{x} to the end of aggregate \smath{u}; if \smath{u} is a stream, it must be finite. -\newitem\smath{\mbox{\bf concat!}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} destructively appends \smath{v} to the end of \smath{u}; if +{\smath{\mbox{\bf concat}\opLeftPren{}u, +\allowbreak{} x\opRightPren{}} returns list \smath{u} with additional element +\smath{x} at the end. Note: equivalent to +\smath{\mbox{\bf concat}\opLeftPren{}u, +\allowbreak{} [ x]\opRightPren{}}. +\newitem\smath{\mbox{\bf concat}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} returns an aggregate +consisting of the elements of \smath{u} +followed by the elements of \smath{v}. +\newitem\smath{\mbox{\bf concat}\opLeftPren{}u\opRightPren{}}, +where \smath{u} is a list of aggregates \smath{[ a, b, $\ldots$, c]}, +returns a single aggregate consisting of the elements of \smath{a} followed +by those of \smath{b} followed $\ldots$ by the elements of \smath{c}. +\newitem\smath{\mbox{\bf concat!}\opLeftPren{}u, +\allowbreak{} x\opRightPren{}}, where \smath{u} is +extensible, destructively adds +element \smath{x} to the end of aggregate \smath{u}; if \smath{u} is a stream, +it must be finite. +\newitem\smath{\mbox{\bf concat!}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} destructively appends \smath{v} +to the end of \smath{u}; if \smath{u} is a stream, it must be finite. } @@ -70083,7 +71843,8 @@ of \smath{sp}. If \smath{sp} has no defined composites (composites need to be ex }% }% {1}{(Matrix(\$))->Union(Vector(\$), "failed")}{FiniteFieldCategory} -{\smath{\mbox{\bf conditionP}\opLeftPren{}M\opRightPren{}}, given a matrix \smath{M} representing a +{\smath{\mbox{\bf conditionP}\opLeftPren{}M\opRightPren{}}, +given a matrix \smath{M} representing a homogeneous system of equations over a field \smath{F} with \spadfun{characteristic} \smath{p}, returns a non-zero vector whose \eth{\smath{p}} power is a non-trivial solution to these @@ -70091,7 +71852,8 @@ equations, or \mbox{\tt "failed"} if no such vector exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{conditionsForIdempotents}}\opLeftPren{}\opRightPren{}% +\opdata{{\mbox{\axiomFun{conditionsForIdempotents}} +\opLeftPren{}\opRightPren{}% }% }% {0}{()->List(Polynomial(R))}{FramedNonAssociativeAlgebra} @@ -70104,11 +71866,13 @@ definition} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{conical}}\opLeftPren{}{\it smallFloat}, \allowbreak{}{\it smallFloat}\opRightPren{}% +\opdata{{\mbox{\axiomFun{conical}}\opLeftPren{} +{\it smallFloat}, \allowbreak{}{\it smallFloat}\opRightPren{}% }% }% {2}{(R, R)->(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf conical}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns a function of two parameters for +{\smath{\mbox{\bf conical}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns a function of two parameters for mapping conical coordinates to Cartesian coordinates. The function maps the point \smath{(\lambda, \mu, \nu)} to \smath{x = \lambda\mu\nu/(ab)}, \smath{y = @@ -70116,48 +71880,61 @@ The function maps the point \smath{(\lambda, \mu, \nu)} to \smath{x \lambda/b\sqrt((mu^2-b^2)(nu^2-b^2)/(b^2-a^2))}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{conjugate}}\opLeftPren{}{\it element\opt{, element}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{conjugate}}\opLeftPren{} +{\it element\opt{, element}}\opRightPren{}% }% }% {1}{(\$)->\$}{ComplexCategory} -{\smath{\mbox{\bf conjugate}\opLeftPren{}u\opRightPren{}} returns the conjugate of a complex, -quaternion, or octonian expression \smath{u}. For example, if \smath{u} is the -complex expression \smath{x + \%i y}, \smath{\mbox{\bf conjugate}\opLeftPren{}u\opRightPren{}} +{\smath{\mbox{\bf conjugate}\opLeftPren{}u\opRightPren{}} +returns the conjugate of a complex, +quaternion, or octonian expression \smath{u}. For example, +if \smath{u} is the +complex expression \smath{x + \%i y}, +\smath{\mbox{\bf conjugate}\opLeftPren{}u\opRightPren{}} returns \smath{x - \% i y}. -\newitem\smath{\mbox{\bf conjugate}\opLeftPren{}pt\opRightPren{}} returns the conjugate of a partition \smath{pt}. +\newitem\smath{\mbox{\bf conjugate}\opLeftPren{}pt\opRightPren{}} +returns the conjugate of a partition \smath{pt}. \seeType{PartitionsAndPermutations} -\newitem\smath{\mbox{\bf conjugate}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns \smath{\mbox{\bf inv}\opLeftPren{}q) * p * q} +\newitem\smath{\mbox{\bf conjugate}\opLeftPren{}p, +\allowbreak{} q\opRightPren{}} returns +\smath{\mbox{\bf inv}\opLeftPren{}q) * p * q} for elements \smath{p} and \smath{q} of a group. Note: this operation is called {\it right action by conjugation}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{conjugates}}\opLeftPren{}{\it streamOfPartitions}\opRightPren{}% +\opdata{{\mbox{\axiomFun{conjugates}}\opLeftPren{} +{\it streamOfPartitions}\opRightPren{}% }% }% {1}{(Stream(List(Integer)))->Stream(List(Integer))}{PartitionsAndPermutations} -{\smath{\mbox{\bf conjugates}\opLeftPren{}lp\opRightPren{}} is the stream of conjugates of a stream of +{\smath{\mbox{\bf conjugates}\opLeftPren{}lp\opRightPren{}} +is the stream of conjugates of a stream of partitions \smath{lp}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{connect}}\opLeftPren{}{\it twoDimensionalViewport}, \allowbreak{}{\it positiveInteger}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{connect}}\opLeftPren{} +{\it twoDimensionalViewport}, \allowbreak{} +{\it positiveInteger}, \allowbreak{}{\it string}\opRightPren{}% }% }% {3}{(Kernel(S))->Union(R, "failed")}{KernelFunctions2} -{\smath{\mbox{\bf connect}\opLeftPren{}v, \allowbreak{} n, \allowbreak{} s\opRightPren{}} displays the lines connecting the graph points in +{\smath{\mbox{\bf connect}\opLeftPren{}v, +\allowbreak{} n, \allowbreak{} s\opRightPren{}} +displays the lines connecting the graph points in field \smath{n} of the \twodim{} viewport \smath{v} if \smath{s="on"}, and does not display the lines if \smath{s="off"}. } - - - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{constant}}\opLeftPren{}{\it variableOrfunction}\opRightPren{}% - \optand \mbox{\axiomFun{constantLeft}}\opLeftPren{}{\it function}, \allowbreak{}{\it element}\opRightPren{}% - \opand \mbox{\axiomFun{constantRight}}\opLeftPren{}{\it function}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{constant}}\opLeftPren{} +{\it variableOrfunction}\opRightPren{}% + \optand \mbox{\axiomFun{constantLeft}}\opLeftPren{} +{\it function}, \allowbreak{}{\it element}\opRightPren{}% + \opand \mbox{\axiomFun{constantRight}}\opLeftPren{} +{\it function}, \allowbreak{}{\it element}\opRightPren{}% }% }% { (()->C) -> (A ->C)}{}{} @@ -70165,86 +71942,115 @@ and does not display the lines if \smath{s="off"}. and must be package-called from package \smath{P} as indicated. See also \spadfun{curry}, \spadfun{curryLeft}, and \spadfun{curryRight}.} \newitem -\smath{\mbox{\bf constant}\opLeftPren{}f\opRightPren{}\$P} returns the function \smath{g} +\smath{\mbox{\bf constant}\opLeftPren{}f\opRightPren{}\$P} +returns the function \smath{g} such that \smath{g(a)= f()}, where function \smath{f} has type \spadsig{}{C} and \smath{a} has type \smath{A}. The function must be package-called from \smath{P =} \spadtype{MappingPackage2(A, C)}. \newitem -\smath{\mbox{\bf constantRight}\opLeftPren{}f\opRightPren{}\$P} returns the function \smath{g} +\smath{\mbox{\bf constantRight}\opLeftPren{}f\opRightPren{}\$P} +returns the function \smath{g} such that \smath{g(a, b)= f(a)}, where function \smath{f} has type \spadsig{A}{C} and \smath{b} has type \smath{B}. This function must be package-called from \smath{P =} \spadtype{MappingPackage3(A, B, C)}. \newitem -\smath{\mbox{\bf constantLeft}\opLeftPren{}f\opRightPren{}\$P} returns the function \smath{g} +\smath{\mbox{\bf constantLeft}\opLeftPren{}f\opRightPren{}\$P} +returns the function \smath{g} such that \smath{g(a, b)= f(b)}, where function \smath{f} has type \spadsig{B}{C} and \smath{a} has type \smath{A}. The function must be package-called from \smath{P =} \spadtype{MappingPackage3(A, B, C)}. \newitem -\smath{\mbox{\bf constant}\opLeftPren{}x\opRightPren{}} tells the pattern matcher that \smath{x} should match -the symbol \smath{'x} and no other quantity, or calls \spadfun{error} if \smath{x} is not a symbol. +\smath{\mbox{\bf constant}\opLeftPren{}x\opRightPren{}} +tells the pattern matcher that \smath{x} should match +the symbol \smath{'x} and no other quantity, or calls +\spadfun{error} if \smath{x} is not a symbol. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{constantOperator}}\opLeftPren{}{\it property}\opRightPren{}% +\opdata{{\mbox{\axiomFun{constantOperator}}\opLeftPren{} +{\it property}\opRightPren{}% \opand \mbox{\axiomFun{constantOpIfCan}}\opLeftPren{}{\it f}\opRightPren{}% }% }% {1}{(BasicOperator)->Union(A, "failed")}{BasicOperatorFunctions1} -{\smath{\mbox{\bf constantOperator}\opLeftPren{}f\opRightPren{}} returns a nullary operator op such that \smath{op()} always evaluate to \smath{f}. +{\smath{\mbox{\bf constantOperator}\opLeftPren{}f\opRightPren{}} +returns a nullary operator op such that \smath{op()} always +evaluate to \smath{f}. \newitem -\smath{\mbox{\bf constantOpIfCan}\opLeftPren{}op\opRightPren{}} returns \smath{f} if \smath{op} is the constant nullary operator always returning \smath{f}, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf constantOpIfCan}\opLeftPren{}op\opRightPren{}} +returns \smath{f} if \smath{op} is the constant nullary operator +always returning \smath{f}, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{construct}}\opLeftPren{}{\it element}, \allowbreak{}{\it ..}\opRightPren{}% +\opdata{{\mbox{\axiomFun{construct}}\opLeftPren{} +{\it element}, \allowbreak{}{\it ..}\opRightPren{}% }% }% {1}{(List(S))->\$}{Collection} -{\smath{\mbox{\bf construct}\opLeftPren{}x, \allowbreak{} y, \allowbreak{} \ldots, z\opRightPren{}\$R} returns the collection of elements \smath{x, y, \ldots, z} from domain \smath{R} ordered as given. This is equivalently written -as \smath{[ x, y, \ldots, z]}. The qualification \smath{R} may be omitted for domains +{\smath{\mbox{\bf construct}\opLeftPren{}x, \allowbreak{} y, +\allowbreak{} \ldots, z\opRightPren{}\$R} returns the collection of elements +\smath{x, y, \ldots, z} from domain \smath{R} ordered as given. +This is equivalently written +as \smath{[ x, y, \ldots, z]}. The qualification \smath{R} may be omitted +for domains of type \spadtype{List}. -Infinite tuples such as \smath{[ x_i \mbox{ \tt for } i \mbox{ \tt in } 1..]} are converted +Infinite tuples such as \smath{[ x_i \mbox{ \tt for } i \mbox{ \tt in } 1..]} +are converted to a \spadtype{Stream} object. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cons}}\opLeftPren{}{\it element}, \allowbreak{}{\it listOrStream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cons}}\opLeftPren{}{\it element}, \allowbreak{} +{\it listOrStream}\opRightPren{}% }% }% {2}{(S, \$)->\$}{List} -{\smath{\mbox{\bf cons}\opLeftPren{}x, \allowbreak{} u\opRightPren{}}, where u is a list or stream, +{\smath{\mbox{\bf cons}\opLeftPren{}x, \allowbreak{} u\opRightPren{}}, +where u is a list or stream, creates a new list or stream whose \spadfun{first} element is \smath{x} -and whose \spadfun{rest} is \smath{u}. Equivalent to \smath{\mbox{\bf concat}\opLeftPren{}x, \allowbreak{} u\opRightPren{}}. +and whose \spadfun{rest} is \smath{u}. Equivalent to +\smath{\mbox{\bf concat}\opLeftPren{}x, \allowbreak{} u\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{content}}\opLeftPren{}{\it polynomial\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{content}}\opLeftPren{} +{\it polynomial\opt{, symbol}}\opRightPren{}% }% }% {1}{(\$)->R}{FiniteAbelianMonoidRing} -{\smath{\mbox{\bf content}\opLeftPren{}p\opRightPren{}} returns the greatest common divisor (\spadfun{gcd}) of the coefficients of polynomial \smath{p}. -\newitem\smath{\mbox{\bf content}\opLeftPren{}p, \allowbreak{} v\opRightPren{}}, where \smath{p} is a multivariate polynomial type, returns -the \smath{gcd} of the coefficients of the polynomial \smath{p} viewed as a univariate polynomial with respect to the variable \smath{v}. +{\smath{\mbox{\bf content}\opLeftPren{}p\opRightPren{}} +returns the greatest common divisor (\spadfun{gcd}) of the +coefficients of polynomial \smath{p}. +\newitem\smath{\mbox{\bf content}\opLeftPren{}p, +\allowbreak{} v\opRightPren{}}, where \smath{p} is a multivariate +polynomial type, returns +the \smath{gcd} of the coefficients of the polynomial \smath{p} +viewed as a univariate polynomial with respect to the variable \smath{v}. For example, if \smath{p = 7x^2y + 14xy^2}, the \smath{gcd} of the coefficients with respect to \smath{x} is \smath{7y}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{continuedFraction}}\opLeftPren{}{\it fractionOrFloat\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{continuedFraction}}\opLeftPren{} +{\it fractionOrFloat\opt{, options}}\opRightPren{}% }% }% {1}{(F)->ContinuedFraction(Integer)}{NumericContinuedFraction} -{\smath{\mbox{\bf continuedFraction}\opLeftPren{}f\opRightPren{}} converts the floating point number \smath{f} +{\smath{\mbox{\bf continuedFraction}\opLeftPren{}f\opRightPren{}} +converts the floating point number \smath{f} to a reduced continued fraction. -\newitem\smath{\mbox{\bf continuedFraction}\opLeftPren{}r\opRightPren{}} converts the fraction +\newitem\smath{\mbox{\bf continuedFraction}\opLeftPren{}r\opRightPren{}} +converts the fraction \smath{r} with components of type \smath{R} to a continued fraction over \smath{R}. -\newitem\smath{\mbox{\bf continuedFraction}\opLeftPren{}r, \allowbreak{} s, \allowbreak{} s'\opRightPren{}}, where \smath{s} and +\newitem\smath{\mbox{\bf continuedFraction}\opLeftPren{}r, +\allowbreak{} s, \allowbreak{} s'\opRightPren{}}, where \smath{s} and \smath{s'} are streams over a domain \smath{R}, constructs a continued fraction in the following way: if \smath{s = [ a1, a2, $\ldots$]} and @@ -70253,19 +72059,24 @@ continued fraction \smath{r + a1/(b1 + a2/(b2 + $\ldots$))}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{contract}}\opLeftPren{}{\it idealOrTensors\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{contract}}\opLeftPren{} +{\it idealOrTensors\opt{, options}}\opRightPren{}% }% }% {3}{(\$, Integer, Integer)->\$}{CartesianTensor} -{\smath{\mbox{\bf contract}\opLeftPren{}I, \allowbreak{} lvar\opRightPren{}} contracts the ideal \smath{I} to the +{\smath{\mbox{\bf contract}\opLeftPren{}I, \allowbreak{} lvar\opRightPren{}} +contracts the ideal \smath{I} to the polynomial ring \smath{F[ lvar]}. -\newitem\smath{\mbox{\bf contract}\opLeftPren{}t, \allowbreak{} i, \allowbreak{} j\opRightPren{}} is the contraction of tensor +\newitem\smath{\mbox{\bf contract}\opLeftPren{}t, \allowbreak{} i, +\allowbreak{} j\opRightPren{}} is the contraction of tensor \smath{t} which sums along the \eth{\smath{i}} and \eth{\smath{j}} indices. For example, if \smath{r = contract(t, 1, 3)} for a rank 4 tensor \smath{t}, then \smath{r} is the rank 2 \smath{(= 4 - 2)} tensor given by \smath{r(i, j) = \sum\nolimits_{h=1}^{\rm dim}t(h, i, h, j)}. -\newitem\smath{\mbox{\bf contract}\opLeftPren{}t, \allowbreak{} i, \allowbreak{} s, \allowbreak{} j\opRightPren{}} is the inner product of tensors +\newitem\smath{\mbox{\bf contract}\opLeftPren{}t, +\allowbreak{} i, \allowbreak{} s, \allowbreak{} j\opRightPren{}} +is the inner product of tensors \smath{s} and \smath{t} which sums along the \smath{k_1}st index of \smath{t} and the \smath{k_2}st index of \smath{s}. For example, if \smath{r = contract(s, 2, t, 1)} for rank 3 tensors @@ -70275,11 +72086,14 @@ For example, if \smath{r = contract(s, 2, t, 1)} for rank 3 tensors } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{contractSolve}}\opLeftPren{}{\it equation}, \allowbreak{}{\it symbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{contractSolve}}\opLeftPren{} +{\it equation}, \allowbreak{}{\it symbol}\opRightPren{}% }% }% -{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))}{TransSolvePackage} -{\smath{\mbox{\bf contractSolve}\opLeftPren{}eq, \allowbreak{} x\opRightPren{}} finds the solutions expressed in +{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))} +{TransSolvePackage} +{\smath{\mbox{\bf contractSolve}\opLeftPren{}eq, +\allowbreak{} x\opRightPren{}} finds the solutions expressed in terms of radicals of the equation of rational functions \smath{eq} with respect to the symbol \smath{x}. The result contains new symbols for common subexpressions in order @@ -70288,47 +72102,53 @@ Alternatively, an expression \smath{u} may be given for \smath{eq} in which case the equation \smath{eq} is defined as \smath{u=0} } - - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{controlPanel}}\opLeftPren{}{\it viewport}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{controlPanel}}\opLeftPren{} +{\it viewport}, \allowbreak{}{\it string}\opRightPren{}% }% }% {1}{(\$)->Stream(Fraction(R))}{ContinuedFraction} -{\smath{\mbox{\bf controlPanel}\opLeftPren{}v, \allowbreak{} s\opRightPren{}} displays the control panel of the given +{\smath{\mbox{\bf controlPanel}\opLeftPren{}v, +\allowbreak{} s\opRightPren{}} displays the control panel of the given \twodim{} or \threedim{} viewport \smath{v} if \smath{s = "on"}, or hides the control panel if \smath{s = "off"}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{convergents}}\opLeftPren{}{\it continuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{convergents}}\opLeftPren{} +{\it continuedFraction}\opRightPren{}% }% }% {1}{(\$)->Stream(Fraction(R))}{ContinuedFraction} -{\smath{\mbox{\bf convergents}\opLeftPren{}cf\opRightPren{}} returns the stream of the convergents of +{\smath{\mbox{\bf convergents}\opLeftPren{}cf\opRightPren{}} +returns the stream of the convergents of the continued fraction \smath{cf}. If the continued fraction is finite, then the stream will be finite.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coordinate}}\opLeftPren{}{\it curveOrSurface}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coordinate}}\opLeftPren{} +{\it curveOrSurface}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->ComponentFunction}{ParametricPlaneCurve} -{\smath{\mbox{\bf coordinate}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} +{\smath{\mbox{\bf coordinate}\opLeftPren{}u, +\allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} coordinate function for the curve or surface \smath{u}. See \spadtype{ParametericPlaneCurve}, \spadtype{ParametricSpaceCurve}, and \spadtype{ParametericSurface}, using HyperDoc. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coordinates}}\opLeftPren{}{\it pointOrvector\opt{, basis}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coordinates}}\opLeftPren{} +{\it pointOrvector\opt{, basis}}\opRightPren{}% }% }% {2}{(\$, Vector(\$))->Vector(R)}{FiniteRankAlgebra} -{\smath{\mbox{\bf coordinates}\opLeftPren{}pt\opRightPren{}} specifies a change of coordinate systems +{\smath{\mbox{\bf coordinates}\opLeftPren{}pt\opRightPren{}} specifies a +change of coordinate systems of point \smath{pt}. This option is expressed in the form \smath{coordinates == pt}. \medbreak\opkey{The following operations return a matrix @@ -70337,14 +72157,16 @@ of the form \smath{[ v_1\ldots v_n]} with respect to the basis a domain \smath{R}. The coordinates of \smath{v_i} are contained in the \eth{\smath{i}} row of the matrix returned.} -\newitem\smath{\mbox{\bf coordinates}\opLeftPren{}v{, b}\opRightPren{}} returns the matrix +\newitem\smath{\mbox{\bf coordinates}\opLeftPren{}v{, b}\opRightPren{}} +returns the matrix representation with respect to the basis \smath{b} for vector \smath{v} of elements from domain \smath{R} of category \spadtype{FiniteRankNonAssociativeAlgebra} or \spadtype{FiniteRankAlgebra}. If a second argument is not given, the basis is taken to be the fixed basis of \smath{R}. -\newitem\smath{\mbox{\bf coordinates}\opLeftPren{}v\opRightPren{}\$R}, returns a matrix representation +\newitem\smath{\mbox{\bf coordinates}\opLeftPren{}v\opRightPren{}\$R}, +returns a matrix representation for \smath{v} with respect to a fixed basis for domain \smath{R} of category \spadtype{FiniteAlgebraicExtensionField}, \spadtype{FramedNonAssociativeAlgebra}, or @@ -70352,11 +72174,13 @@ of category \spadtype{FiniteAlgebraicExtensionField}, } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{copies}}\opLeftPren{}{\it integer}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{copies}}\opLeftPren{}{\it integer}, \allowbreak{} +{\it string}\opRightPren{}% }% }% {2}{(Integer, String)->String}{DisplayPackage} -{\smath{\mbox{\bf copies}\opLeftPren{}n, \allowbreak{} s\opRightPren{}} returns +{\smath{\mbox{\bf copies}\opLeftPren{}n, \allowbreak{} s\opRightPren{}} +returns a string composed of \smath{n} copies of string \smath{s}.} % ---------------------------------------------------------------------- @@ -70364,19 +72188,23 @@ a string composed of \smath{n} copies of string \smath{s}.} }% }% {1}{(\$)->\$}{Aggregate} -{\smath{\mbox{\bf copy}\opLeftPren{}u\opRightPren{}} returns a top-level (non-recursive) copy of an aggregate \smath{u}. +{\smath{\mbox{\bf copy}\opLeftPren{}u\opRightPren{}} returns a +top-level (non-recursive) copy of an aggregate \smath{u}. Note: for lists, \code{copy(u) == [x for x in u]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{copyInto!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it aggregate}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{copyInto!}}\opLeftPren{}{\it aggregate}, +\allowbreak{}{\it aggregate}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {3}{(\$, \$, Integer)->\$}{FiniteLinearAggregate} -{\smath{\mbox{\bf copyInto!}\opLeftPren{}u, \allowbreak{} v, \allowbreak{} p\opRightPren{}} returns linear aggregate \smath{u} with +{\smath{\mbox{\bf copyInto!}\opLeftPren{}u, \allowbreak{} v, +\allowbreak{} p\opRightPren{}} returns linear aggregate \smath{u} with elements of \smath{u} replaced by the successive elements of \smath{v} starting at index \smath{p}. -Arguments \smath{u} and \smath{v} can be elements of any \spadtype{FiniteLinearAggregate}. +Arguments \smath{u} and \smath{v} can be elements of any +\spadtype{FiniteLinearAggregate}. } % ---------------------------------------------------------------------- @@ -70387,8 +72215,10 @@ Arguments \smath{u} and \smath{v} can be elements of any \spadtype{FiniteLinearA {1}{(\$)->\$}{TrigonometricFunctionCategory} {\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. -} \newitem\smath{\mbox{\bf cos}\opLeftPren{}x\opRightPren{}} returns the cosine of \smath{x}. -\newitem\smath{\mbox{\bf cosIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf cos}\opLeftPren{}x\opRightPren{}} if possible, +} \newitem\smath{\mbox{\bf cos}\opLeftPren{}x\opRightPren{}} +returns the cosine of \smath{x}. +\newitem\smath{\mbox{\bf cosIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf cos}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } @@ -70397,15 +72227,18 @@ and \mbox{\tt "failed"} otherwise. }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf cos2sec}\opLeftPren{}e\opRightPren{}} converts every \smath{\mbox{\bf cos}\opLeftPren{}u\opRightPren{}} appearing in +{\smath{\mbox{\bf cos2sec}\opLeftPren{}e\opRightPren{}} converts +every \smath{\mbox{\bf cos}\opLeftPren{}u\opRightPren{}} appearing in \smath{e} into \smath{1/\sec(u)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cosh2sech}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cosh2sech}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf cosh2sech}\opLeftPren{}e\opRightPren{}} converts every \smath{\mbox{\bf cosh}\opLeftPren{}u\opRightPren{}} appearing in +{\smath{\mbox{\bf cosh2sech}\opLeftPren{}e\opRightPren{}} converts +every \smath{\mbox{\bf cosh}\opLeftPren{}u\opRightPren{}} appearing in \smath{e} into \smath{1/\mbox{\rm sech}(u)}.} % ---------------------------------------------------------------------- @@ -70417,9 +72250,11 @@ and \mbox{\tt "failed"} otherwise. {\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } -\newitem\smath{\mbox{\bf cosh}\opLeftPren{}x\opRightPren{}} returns the hyperbolic cosine of +\newitem\smath{\mbox{\bf cosh}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic cosine of \smath{x}. -\newitem\smath{\mbox{\bf coshIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf cosh}\opLeftPren{}x\opRightPren{}} if possible, +\newitem\smath{\mbox{\bf coshIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf cosh}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } @@ -70428,10 +72263,14 @@ and \mbox{\tt "failed"} otherwise. }% }% {1}{(\$)->\$}{TrigonometricFunctionCategory} -{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or +{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, +\spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } -\newitem\smath{\mbox{\bf cot}\opLeftPren{}x\opRightPren{}} returns the cotangent of \smath{x}. -\newitem\smath{\mbox{\bf cotIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf cot}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\newitem\smath{\mbox{\bf cot}\opLeftPren{}x\opRightPren{}} +returns the cotangent of \smath{x}. +\newitem\smath{\mbox{\bf cotIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf cot}\opLeftPren{}x\opRightPren{}} +if possible, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- @@ -70439,58 +72278,75 @@ and \mbox{\tt "failed"} otherwise. }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf cot2tan}\opLeftPren{}expression\opRightPren{}} converts every \smath{\cot(u)} +{\smath{\mbox{\bf cot2tan}\opLeftPren{}expression\opRightPren{}} +converts every \smath{\cot(u)} appearing in \smath{e} into \smath{1/\tan(u)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cot2trig}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cot2trig}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf cot2trig}\opLeftPren{}expression\opRightPren{}} converts every \smath{\cot(u)} +{\smath{\mbox{\bf cot2trig}\opLeftPren{}expression\opRightPren{}} +converts every \smath{\cot(u)} appearing in \smath{e} into \smath{\cos(u)/\sin(u)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coth}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{cothIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coth}}\opLeftPren{} +{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{cothIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} -{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or +{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, +\spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } -\newitem\smath{\mbox{\bf coth}\opLeftPren{}x\opRightPren{}} returns the hyperbolic cotangent of \smath{x}. +\newitem\smath{\mbox{\bf coth}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic cotangent of \smath{x}. \newitem -\smath{\mbox{\bf cothIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf coth}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf cothIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf coth}\opLeftPren{}x\opRightPren{}} if possible, +and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coth2tanh}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coth2tanh}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf coth2tanh}\opLeftPren{}expression\opRightPren{}} converts every \smath{\mbox{\rm +{\smath{\mbox{\bf coth2tanh}\opLeftPren{}expression\opRightPren{}} +converts every \smath{\mbox{\rm coth}(u)} appearing in \smath{e} into \smath{1/\mbox{\rm tanh}(u)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{coth2trigh}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{coth2trigh}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf coth2trigh}\opLeftPren{}expression\opRightPren{}} converts every \smath{\mbox{\rm +{\smath{\mbox{\bf coth2trigh}\opLeftPren{}expression\opRightPren{}} +converts every \smath{\mbox{\rm coth}(u)} appearing in \smath{e} into \smath{\mbox{\rm cosh}(u)/\mbox{\rm sinh}(u)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{count}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{count}}\opLeftPren{} +{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->NonNegativeInteger}{HomogeneousAggregate} -{\smath{\mbox{\bf count}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} returns the number of elements \smath{x} in -\smath{u} such that \smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} is \smath{true}. +{\smath{\mbox{\bf count}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} +returns the number of elements \smath{x} in +\smath{u} such that \smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} +is \smath{true}. For collections, \code{count(p, u) = reduce(+, [1 for x in u | p(x)], 0)}. -\newitem\smath{\mbox{\bf count}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} returns the number of occurrences of +\newitem\smath{\mbox{\bf count}\opLeftPren{}x, +\allowbreak{} u\opRightPren{}} returns the number of occurrences of \smath{x} in \smath{u}. For collections, \code{count(x, u) = reduce(+, [x=y for y in u], 0)}. } @@ -70500,7 +72356,8 @@ For collections, \code{count(x, u) = reduce(+, [x=y for y in u], 0)}. }% }% {1}{(\$)->Boolean}{CardinalNumber} -{\smath{\mbox{\bf countable?}\opLeftPren{}u\opRightPren{}} tests if the cardinal number \smath{u} is +{\smath{\mbox{\bf countable?}\opLeftPren{}u\opRightPren{}} +tests if the cardinal number \smath{u} is countable, that is, if $u \le $\smath{Aleph 0}.} % ---------------------------------------------------------------------- @@ -70508,33 +72365,39 @@ countable, that is, if $u \le $\smath{Aleph 0}.} }% }% {0}{()->\$}{ThreeSpace} -{\smath{\mbox{\bf createThreeSpace}\opLeftPren{}\opRightPren{}}{\bf \$}\spadtype{ThreeSpace(R)} creates a +{\smath{\mbox{\bf createThreeSpace}\opLeftPren{}\opRightPren{}} +{\bf \$}\spadtype{ThreeSpace(R)} creates a \spadtype{ThreeSpace} object capable of holding point, curve, mesh components or any combination of the three. The ring \smath{R} is usually \spadtype{DoubleFloat}. If you do not package call this function, \spadtype{DoubleFloat} is assumed. -\newitem\smath{\mbox{\bf createThreeSpace}\opLeftPren{}s\opRightPren{}} creates a \spadtype{ThreeSpace} +\newitem\smath{\mbox{\bf createThreeSpace}\opLeftPren{}s\opRightPren{}} +creates a \spadtype{ThreeSpace} object containing objects pre-defined within some \spadtype{SubSpace} \smath{s}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{createGenericMatrix}}\opLeftPren{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{createGenericMatrix}}\opLeftPren{} +{\it nonNegativeInteger}\opRightPren{}% }% }% {1}{(NonNegativeInteger)->Matrix(Polynomial(R))}{RepresentationPackage1} -{\smath{\mbox{\bf createGenericMatrix}\opLeftPren{}n\opRightPren{}} creates a square matrix of +{\smath{\mbox{\bf createGenericMatrix}\opLeftPren{}n\opRightPren{}} +creates a square matrix of dimension \smath{n} whose entry at the \smath{i}-th row and \smath{j}-th column is the indeterminate \smath{x_{i, j}} (double subscripted). \seeType{RepresentationPackage1}} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{createIrreduciblePoly}}\opLeftPren{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{createIrreduciblePoly}}\opLeftPren{} +{\it nonNegativeInteger}\opRightPren{}% }% }% {0}{()->\$}{FiniteAlgebraicExtensionField} -{\smath{\mbox{\bf createIrreduciblePoly}\opLeftPren{}n\opRightPren{}}{\bf \$}\spadtype{FFPOLY(GF)} generates +{\smath{\mbox{\bf createIrreduciblePoly}\opLeftPren{}n\opRightPren{}} +{\bf \$}\spadtype{FFPOLY(GF)} generates a monic irreducible polynomial of degree \smath{n} over the finite field \smath{GF}. } @@ -70544,7 +72407,8 @@ a monic irreducible polynomial of degree \smath{n} over the finite field }% }% {0}{()->\$}{FiniteAlgebraicExtensionField} -{\smath{\mbox{\bf createNormalElement}\opLeftPren{}\opRightPren{}\$F} computes a normal element over the ground field +{\smath{\mbox{\bf createNormalElement}\opLeftPren{}\opRightPren{}\$F} +computes a normal element over the ground field of a finite algebraic extension field \smath{F}, that is, an element \smath{a} such that \smath{a^{q^i}, 0 \leq i < {\mbox {\bf extensionDegree}}()\$F} is an \smath{F}-basis, where @@ -70552,11 +72416,13 @@ an element \smath{a} such that \smath{a^{q^i}, 0 \leq i < {\mbox } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{createNormalPrimitivePoly}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{createNormalPrimitivePoly}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {0}{()->\$}{FiniteAlgebraicExtensionField} -{\smath{\mbox{\bf createNormalPrimitivePoly}\opLeftPren{}n\opRightPren{}}{\bf \$}\spadtype{FFPOLY(GF)} +{\smath{\mbox{\bf createNormalPrimitivePoly}\opLeftPren{}n\opRightPren{}} +{\bf \$}\spadtype{FFPOLY(GF)} generates a normal and primitive polynomial of degree \smath{n} over the field \smath{GF}. } @@ -70567,16 +72433,19 @@ primitive polynomial of degree \smath{n} over the field \smath{GF}. }% }% {0}{()->\$}{FiniteFieldCategory} -{\smath{\mbox{\bf createPrimitiveElement}\opLeftPren{}\opRightPren{}\$F} computes a generator of the +{\smath{\mbox{\bf createPrimitiveElement}\opLeftPren{}\opRightPren{}\$F} +computes a generator of the (cyclic) multiplicative group of a finite field \smath{F}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{createRandomElement}}\opLeftPren{}{\it listOfMatrices}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{createRandomElement}}\opLeftPren{} +{\it listOfMatrices}, \allowbreak{}{\it matrix}\opRightPren{}% }% }% {2}{(List(Matrix(R)), Matrix(R))->Matrix(R)}{RepresentationPackage2} -{\smath{\mbox{\bf createRandomElement}\opLeftPren{}lm, \allowbreak{} m\opRightPren{}} creates a random element of +{\smath{\mbox{\bf createRandomElement}\opLeftPren{}lm, +\allowbreak{} m\opRightPren{}} creates a random element of the group algebra generated by \smath{lm}, where \smath{lm} is a list of matrices and \smath{m} is a matrix. \seeType{RepresentationPackage2} @@ -70587,33 +72456,41 @@ list of matrices and \smath{m} is a matrix. }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf csc2sin}\opLeftPren{}expression\opRightPren{}} converts every \smath{\mbox{\bf csc}\opLeftPren{}u\opRightPren{}} +{\smath{\mbox{\bf csc2sin}\opLeftPren{}expression\opRightPren{}} +converts every \smath{\mbox{\bf csc}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\tt sin}(u)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{csch2sinh}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{csch2sinh}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf csch2sinh}\opLeftPren{}expression\opRightPren{}} converts every \smath{\mbox{\bf csch}\opLeftPren{}u\opRightPren{}} +{\smath{\mbox{\bf csch2sinh}\opLeftPren{}expression\opRightPren{}} +converts every \smath{\mbox{\bf csch}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\sinh}(u)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{csch}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{cschIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{csch}}\opLeftPren{} +{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{cschIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} {\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or \spadtype{Expression} value or a series.} -\newitem\smath{\mbox{\bf csch}\opLeftPren{}x\opRightPren{}} returns the hyperbolic cosecant of \smath{x}. -\newitem\smath{\mbox{\bf cschIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf csch}\opLeftPren{}x\opRightPren{}} if possible, +\newitem\smath{\mbox{\bf csch}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic cosecant of \smath{x}. +\newitem\smath{\mbox{\bf cschIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf csch}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cscIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cscIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} @@ -70621,17 +72498,22 @@ and \mbox{\tt "failed"} otherwise. \spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } \newitem -\smath{\mbox{\bf csc}\opLeftPren{}x\opRightPren{}} returns the cosecant of \smath{x}. +\smath{\mbox{\bf csc}\opLeftPren{}x\opRightPren{}} +returns the cosecant of \smath{x}. \newitem -\smath{\mbox{\bf cscIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf csc}\opLeftPren{}x\opRightPren{}} if possible, +\smath{\mbox{\bf cscIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf csc}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cup}}\opLeftPren{}{\it symmetricPolynomial}, \allowbreak{}{\it symmetricPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cup}}\opLeftPren{} +{\it symmetricPolynomial}, \allowbreak{} +{\it symmetricPolynomial}\opRightPren{}% }% }% { (SPOL RN, SPOL RN) -> SPOL RN}{}{} -{\smath{\mbox{\bf cup}\opLeftPren{}s_1, \allowbreak{} s_2\opRightPren{}}, introduced by Redfield, +{\smath{\mbox{\bf cup}\opLeftPren{}s_1, +\allowbreak{} s_2\opRightPren{}}, introduced by Redfield, is the scalar product of two cycle indices, where the \smath{s_i} are of type \spadtype{SymmetricPolynomial} with rational number coefficients. @@ -70639,43 +72521,53 @@ See also \spadfun{cap}. See \spadtype{CycleIndicators} for details.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{curry}}\opLeftPren{}{\it function}\opRightPren{}% - \optand \mbox{\axiomFun{curryLeft}}\opLeftPren{}{\it function}, \allowbreak{}{\it element}\opRightPren{}% - \opand \mbox{\axiomFun{curryRight}}\opLeftPren{}{\it function}, \allowbreak{}{\it element}\opRightPren{}% + \optand \mbox{\axiomFun{curryLeft}}\opLeftPren{} +{\it function}, \allowbreak{}{\it element}\opRightPren{}% + \opand \mbox{\axiomFun{curryRight}}\opLeftPren{} +{\it function}, \allowbreak{}{\it element}\opRightPren{}% }% }% { ((A, B)->C, B) -> (A ->C)}{}{} {\opkey{These functions drop an argument from a function.} \newitem -\smath{\mbox{\bf curry}\opLeftPren{}f, \allowbreak{} a\opRightPren{}} returns the function \smath{g} +\smath{\mbox{\bf curry}\opLeftPren{}f, +\allowbreak{} a\opRightPren{}} returns the function \smath{g} such that \smath{g()= f(a)}, where function \smath{f} has type \spadsig{A}{C} and element \smath{a} has type \smath{A}. \newitem -\smath{\mbox{\bf curryRight}\opLeftPren{}f, \allowbreak{} b\opRightPren{}} returns the function \smath{g} such that +\smath{\mbox{\bf curryRight}\opLeftPren{}f, +\allowbreak{} b\opRightPren{}} returns the function \smath{g} such that \smath{g(a) = f(a, b)}, where function \smath{f} has type \spadsig{(A, B)}{C} and element \smath{b} has type \smath{B}. \newitem -\smath{\mbox{\bf curryLeft}\opLeftPren{}f, \allowbreak{} a\opRightPren{}} is the function \smath{g} +\smath{\mbox{\bf curryLeft}\opLeftPren{}f, +\allowbreak{} a\opRightPren{}} is the function \smath{g} such that \smath{g(b) = f(a, b)}, where function \smath{f} has type \spadsig{(A, B)}{C} and element \smath{a} has type \smath{A}. \newitem -See also \spadfun{constant}, \spadfun{constantLeft}, and \spadfun{constantRight}. +See also \spadfun{constant}, \spadfun{constantLeft}, +and \spadfun{constantRight}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{curve}}\opLeftPren{}{\it listOfPoints\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{curve}}\opLeftPren{} +{\it listOfPoints\opt{, options}}\opRightPren{}% }% }% {1}{(List(Point(R)))->\$}{ThreeSpace} -{\smath{\mbox{\bf curve}\opLeftPren{}[ p_0, p_1, $\ldots$, p_n]\opRightPren{}} creates a +{\smath{\mbox{\bf curve}\opLeftPren{} +[ p_0, p_1, $\ldots$, p_n]\opRightPren{}} creates a space curve defined by the list of points \smath{p_0} through \smath{p_n} and returns a \spadtype{ThreeSpace} object whose component is the curve. -\newitem\smath{\mbox{\bf curve}\opLeftPren{}sp\opRightPren{}} checks to see if the +\newitem\smath{\mbox{\bf curve}\opLeftPren{}sp\opRightPren{}} +checks to see if the \spadtype{ThreeSpace} object \smath{sp} is composed of a single curve defined by a list of points; if so, the list of points defining the curve is returned. Otherwise, the operation calls \spadfun{error}. -\newitem\smath{\mbox{\bf curve}\opLeftPren{}c_1, \allowbreak{} c_2\opRightPren{}} creates a plane curve from two +\newitem\smath{\mbox{\bf curve}\opLeftPren{}c_1, +\allowbreak{} c_2\opRightPren{}} creates a plane curve from two component functions \smath{c_1} and \smath{c_2}. \seeType{ComponentFunction} \newitem\smath{curve(sp, [ @@ -70683,14 +72575,18 @@ component functions \smath{c_1} and \smath{c_2}. p_n]])} adds a space curve defined by a list of points \smath{p_0} through \smath{p_n} to a \spadtype{ThreeSpace} object \smath{sp}. -Each \smath{p_i} is from a domain \smath{\mbox{\bf PointDomain}\opLeftPren{}m, \allowbreak{} R\opRightPren{}}, where +Each \smath{p_i} is from a domain +\smath{\mbox{\bf PointDomain}\opLeftPren{}m, +\allowbreak{} R\opRightPren{}}, where \smath{R} is the \spadtype{Ring} over which the point elements are defined and \smath{m} is the dimension of the points. -\newitem \smath{\mbox{\bf curve}\opLeftPren{}s, \allowbreak{} [ p_0, p_1, $\ldots$, p_n]\opRightPren{}} adds +\newitem \smath{\mbox{\bf curve}\opLeftPren{}s, +\allowbreak{} [ p_0, p_1, $\ldots$, p_n]\opRightPren{}} adds the space curve component designated by the list of points \smath{p_0} through \smath{p_n} to the \spadtype{ThreeSpace} object \smath{sp}. -\newline \smath{\mbox{\bf curve}\opLeftPren{}c_1, \allowbreak{} c_2, \allowbreak{} c_3\opRightPren{}} creates a space curve from +\newline \smath{\mbox{\bf curve}\opLeftPren{}c_1, \allowbreak{} c_2, +\allowbreak{} c_3\opRightPren{}} creates a space curve from three component functions \smath{c_1}, \smath{c_2}, and \smath{c_3}.} @@ -70699,7 +72595,8 @@ three component functions \smath{c_1}, \smath{c_2}, and }% }% {1}{(\$)->Boolean}{ThreeSpace} -{\smath{\mbox{\bf curve?}\opLeftPren{}sp\opRightPren{}} tests if the \spadtype{ThreeSpace} object \smath{sp} contains +{\smath{\mbox{\bf curve?}\opLeftPren{}sp\opRightPren{}} +tests if the \spadtype{ThreeSpace} object \smath{sp} contains a single curve object. } @@ -70708,78 +72605,97 @@ a single curve object. }% }% {1}{(Float)->\$}{DrawOption} -{\smath{\mbox{\bf curveColor}\opLeftPren{}p\opRightPren{}} specifies a color index for \twodim{} graph +{\smath{\mbox{\bf curveColor}\opLeftPren{}p\opRightPren{}} +specifies a color index for \twodim{} graph curves from the palette \smath{p}. This option is expressed in the form \smath{curveColor ==p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cycle}}\opLeftPren{}{\it listOfPermutations}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cycle}}\opLeftPren{} +{\it listOfPermutations}\opRightPren{}% }% }% {1}{(List(S))->\$}{PermutationCategory} -{\smath{\mbox{\bf cycle}\opLeftPren{}ls\opRightPren{}} converts a cycle \smath{ls}, a list with no repetitions, to +{\smath{\mbox{\bf cycle}\opLeftPren{}ls\opRightPren{}} +converts a cycle \smath{ls}, a list with no repetitions, to the permutation, which maps \smath{ls.i} to \smath{ls.(i+1)} (index modulo the length of the list). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cycleEntry}}\opLeftPren{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cycleEntry}}\opLeftPren{} +{\it aggregate}\opRightPren{}% }% }% {1}{(\$)->\$}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf cycleEntry}\opLeftPren{}u\opRightPren{}} returns the head of a top-level cycle -contained in aggregate \smath{u}, or \smath{\mbox{\bf empty}\opLeftPren{}\opRightPren{}} if none +{\smath{\mbox{\bf cycleEntry}\opLeftPren{}u\opRightPren{}} +returns the head of a top-level cycle +contained in aggregate \smath{u}, or +\smath{\mbox{\bf empty}\opLeftPren{}\opRightPren{}} if none exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cycleLength}}\opLeftPren{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cycleLength}}\opLeftPren{} +{\it aggregate}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf cycleLength}\opLeftPren{}u\opRightPren{}} returns the length of a top-level cycle +{\smath{\mbox{\bf cycleLength}\opLeftPren{}u\opRightPren{}} +returns the length of a top-level cycle contained in aggregate \smath{u}, or 0 if \smath{u} has no such cycle.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cyclePartition}}\opLeftPren{}{\it permutation}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cyclePartition}}\opLeftPren{} +{\it permutation}\opRightPren{}% }% }% {1}{(\$)->Partition}{Permutation} -{\smath{\mbox{\bf cyclePartition}\opLeftPren{}p\opRightPren{}} returns the cycle structure of a permutation \smath{p} including cycles of length 1. +{\smath{\mbox{\bf cyclePartition}\opLeftPren{}p\opRightPren{}} returns +the cycle structure of a permutation \smath{p} including cycles of length 1. The permutation is assumed to be a member of \spadtype{Permutation(S)} where \smath{S} is a finite set.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cycleRagits}}\opLeftPren{}{\it radixExpansion}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cycleRagits}}\opLeftPren{} +{\it radixExpansion}\opRightPren{}% }% }% {1}{(\$)->List(Integer)}{RadixExpansion} -{\smath{\mbox{\bf cycleRagits}\opLeftPren{}rx\opRightPren{}} returns the cyclic part of the ragits of the fractional part of a radix expansion. For example, if \smath{x = 3/28 = 0.10 714285 714285 \ldots}, +{\smath{\mbox{\bf cycleRagits}\opLeftPren{}rx\opRightPren{}} returns the +cyclic part of the ragits of the fractional part of a radix expansion. +For example, if \smath{x = 3/28 = 0.10 714285 714285 \ldots}, then \code{cycleRagits(x) = [7, 1, 4, 2, 8, 5]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cycleSplit!}}\opLeftPren{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cycleSplit!}}\opLeftPren{} +{\it aggregate}\opRightPren{}% }% }% {1}{(\$)->\$}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf cycleSplit!}\opLeftPren{}u\opRightPren{}} splits the recursive aggregate (for example, a list) \smath{u} into two +{\smath{\mbox{\bf cycleSplit!}\opLeftPren{}u\opRightPren{}} +splits the recursive aggregate (for example, a list) \smath{u} into two aggregates by dropping off the cycle. The value returned is the cycle entry, or \smath{nil} if none exists. -For example, if \smath{w = {\mbox {\bf concat}}(u, v)} is the cyclic list where \smath{v} is the -head of the cycle, \smath{\mbox{\bf cycleSplit!}\opLeftPren{}w\opRightPren{}} will drop \smath{v} off \smath{w}. Thus +For example, if \smath{w = {\mbox {\bf concat}}(u, v)} +is the cyclic list where \smath{v} is the +head of the cycle, \smath{\mbox{\bf cycleSplit!} +\opLeftPren{}w\opRightPren{}} will drop \smath{v} off \smath{w}. Thus \smath{w} is destructively changed to \smath{u}, and \smath{v} is returned. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cycles}}\opLeftPren{}{\it listOfListOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cycles}}\opLeftPren{} +{\it listOfListOfElements}\opRightPren{}% }% }% {1}{(List(List(S)))->\$}{PermutationCategory} -{\smath{\mbox{\bf cycles}\opLeftPren{}lls\opRightPren{}} coerces a list of list of cycles \smath{lls} +{\smath{\mbox{\bf cycles}\opLeftPren{}lls\opRightPren{}} +coerces a list of list of cycles \smath{lls} to a permutation. Each cycle, represented as a list \smath{ls} with no repetitions, is coerced to the permutation, which maps \smath{ls.i} to @@ -70788,12 +72704,15 @@ These permutations are then multiplied. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cycleTail}}\opLeftPren{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cycleTail}}\opLeftPren{} +{\it aggregate}\opRightPren{}% }% }% {1}{(\$)->\$}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf cycleTail}\opLeftPren{}u\opRightPren{}} returns the last node in the cycle of -a recursive aggregate (for example, a list) \smath{u}, or empty if none exists. +{\smath{\mbox{\bf cycleTail}\opLeftPren{}u\opRightPren{}} +returns the last node in the cycle of +a recursive aggregate (for example, a list) \smath{u}, +or empty if none exists. } % ---------------------------------------------------------------------- @@ -70801,13 +72720,15 @@ a recursive aggregate (for example, a list) \smath{u}, or empty if none exists. }% }% { I -> SPOL RN --cyclic group}{}{} -{\smath{\mbox{\bf cyclic}\opLeftPren{}n\opRightPren{}} returns the cycle index of the +{\smath{\mbox{\bf cyclic}\opLeftPren{}n\opRightPren{}} +returns the cycle index of the cyclic group of degree \smath{n}. \spadtype{CycleIndicators} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cyclic?}}\opLeftPren{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cyclic?}}\opLeftPren{} +{\it aggregate}\opRightPren{}% }% }% {1}{(\$)->Boolean}{RecursiveAggregate} @@ -70816,31 +72737,37 @@ recursive aggregate (for example, a list) \smath{u} has a cycle. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cyclicGroup}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cyclicGroup}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% {1}{(List(Integer))->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf cyclicGroup}\opLeftPren{}[ i_1, \ldots, i_k]\opRightPren{}} constructs +{\smath{\mbox{\bf cyclicGroup}\opLeftPren{} +[ i_1, \ldots, i_k]\opRightPren{}} constructs the cyclic group of order \smath{k} acting on the list of integers \smath{i_1}, \ldots, \smath{i_k}. Note: duplicates in the list will be removed. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cyclicGroup}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cyclicGroup}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(PositiveInteger)->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf cyclicGroup}\opLeftPren{}n\opRightPren{}} constructs the cyclic group of order +{\smath{\mbox{\bf cyclicGroup}\opLeftPren{}n\opRightPren{}} +constructs the cyclic group of order \smath{n} acting on the integers 1, \ldots, \smath{n}, \smath{n > 0}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{cyclicSubmodule}}\opLeftPren{}{\it listOfMatrices}, \allowbreak{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{cyclicSubmodule}}\opLeftPren{} +{\it listOfMatrices}, \allowbreak{}{\it vector}\opRightPren{}% }% }% {2}{(List(Matrix(R)), Vector(R))->Vector(Vector(R))}{RepresentationPackage2} -{\smath{\mbox{\bf cyclicSubmodule}\opLeftPren{}lm, \allowbreak{} v\opRightPren{}}, where \smath{lm} is a list of +{\smath{\mbox{\bf cyclicSubmodule}\opLeftPren{}lm, +\allowbreak{} v\opRightPren{}}, where \smath{lm} is a list of \smath{n} by \smath{n} square matrices and \smath{v} is a vector of size \smath{n}, generates a basis in echelon form. \seeDetails{RepresentationPackage2}} @@ -70850,44 +72777,55 @@ of size \smath{n}, generates a basis in echelon form. }% }% {1}{(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf cylindrical}\opLeftPren{}pt\opRightPren{}} transforms \smath{pt} from polar +{\smath{\mbox{\bf cylindrical}\opLeftPren{}pt\opRightPren{}} +transforms \smath{pt} from polar coordinates to Cartesian coordinates, by mapping the point \smath{(r, theta, z)} to \smath{x = r \cos(theta)}, \smath{y = r \sin(theta)}, \smath{z}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{D}}\opLeftPren{}{\it expression\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{D}}\opLeftPren{} +{\it expression\opt{, options}}\opRightPren{}% }% }% {1}{(\$)->\$}{DifferentialRing} -{\smath{\mbox{\bf D}\opLeftPren{}x\opRightPren{}} returns the derivative of \smath{x}. +{\smath{\mbox{\bf D}\opLeftPren{}x\opRightPren{}} returns the +derivative of \smath{x}. This function is a simple differential operator where no variable needs to be specified. -\newitem\smath{\mbox{\bf D}\opLeftPren{}x, \allowbreak{} [ s_1, \ldots s_n]\opRightPren{}} computes +\newitem\smath{\mbox{\bf D}\opLeftPren{}x, \allowbreak{} [ s_1, +\ldots s_n]\opRightPren{}} computes successive partial derivatives, that is, \smath{D(\ldots {\bf D}(x, s_1)\ldots, s_n)}. -\newitem\smath{\mbox{\bf D}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} computes the partial derivative of +\newitem\smath{\mbox{\bf D}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} +computes the partial derivative of \smath{u} with respect to \smath{x}. -\newitem\smath{\mbox{\bf D}\opLeftPren{}u, \allowbreak{} deriv\optinner{, n}\opRightPren{}} differentiates \smath{u} +\newitem\smath{\mbox{\bf D}\opLeftPren{}u, \allowbreak{} +deriv\optinner{, n}\opRightPren{}} differentiates \smath{u} \smath{n} times using a derivation which extends \smath{deriv} on \smath{R}. Argument \smath{n} defaults to 1. -\newitem\smath{\mbox{\bf D}\opLeftPren{}p, \allowbreak{} d, \allowbreak{} x'\opRightPren{}} extends the \smath{R}-derivation +\newitem\smath{\mbox{\bf D}\opLeftPren{}p, \allowbreak{} d, +\allowbreak{} x'\opRightPren{}} extends the \smath{R}-derivation \smath{d} to an extension \smath{R} in \smath{R[ x]} where \smath{Dx} is given by \smath{x'}, and returns \smath{Dp}. \newitem\smath{D(x, [ s_1, \ldots, s_n], [ n_1, \ldots, n_m])} computes multiple partial derivatives, -that is, \smath{\mbox{\bf D}\opLeftPren{}\ldots {\bf D}(x, s_1, n_1)\ldots, s_n, n_m\opRightPren{}}. -\newitem\smath{\mbox{\bf D}\opLeftPren{}u, \allowbreak{} x, \allowbreak{} n\opRightPren{}} computes multiple partial derivatives, +that is, \smath{\mbox{\bf D}\opLeftPren{}\ldots {\bf D}(x, s_1, n_1)\ldots, +s_n, n_m\opRightPren{}}. +\newitem\smath{\mbox{\bf D}\opLeftPren{}u, \allowbreak{} x, +\allowbreak{} n\opRightPren{}} computes multiple partial derivatives, that is, \eth{\smath{n}} derivative of \smath{u} with respect to \smath{x}. -\newitem \smath{\mbox{\bf D}\opLeftPren{}of\optinner{, n}\opRightPren{}}, where \smath{of} is an object +\newitem \smath{\mbox{\bf D}\opLeftPren{}of\optinner{, n}\opRightPren{}}, +where \smath{of} is an object of type \spadtype{OutputForm} (normally unexposed), returns an output form for the \eth{\smath{n}} derivative of \smath{f}, for example, \smath{f'}, \smath{f''}}, \smath{f'''}, \smath{f^{{\tt iv}}}, and so on. -\newitem \smath{\mbox{\bf D}\opLeftPren{}\opRightPren{}\$A} provides the operator corresponding to the +\newitem \smath{\mbox{\bf D}\opLeftPren{}\opRightPren{}\$A} +provides the operator corresponding to the derivation in the differential ring \smath{A}. } @@ -70896,40 +72834,52 @@ derivation in the differential ring \smath{A}. }% }% {1}{(Color)->\$}{Palette} -{\smath{\mbox{\bf dark}\opLeftPren{}color\opRightPren{}} returns the shade of the indicated hue of \smath{color} to its lowest value. +{\smath{\mbox{\bf dark}\opLeftPren{}color\opRightPren{}} returns +the shade of the indicated hue of \smath{color} to its lowest value. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ddFact}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it primeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ddFact}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it primeInteger}\opRightPren{}% }% }% -{2}{(U, Integer)->List(Record(factor:U, degree:Integer))}{ModularDistinctDegreeFactorizer} -{\smath{\mbox{\bf ddFact}\opLeftPren{}q, \allowbreak{} p\opRightPren{}} computes a distinct degree factorization of the polynomial \smath{q} modulo the prime \smath{p}, that is, such that each factor is a product of irreducibles of the same degrees. +{2}{(U, Integer)->List(Record(factor:U, degree:Integer))} +{ModularDistinctDegreeFactorizer} +{\smath{\mbox{\bf ddFact}\opLeftPren{}q, \allowbreak{} p\opRightPren{}} +computes a distinct degree factorization of the polynomial +\smath{q} modulo the prime \smath{p}, that is, such that each factor is a +product of irreducibles of the same degrees. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{decimal}}\opLeftPren{}{\it rationalNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{decimal}}\opLeftPren{} +{\it rationalNumber}\opRightPren{}% }% }% {1}{(Fraction(Integer))->\$}{DecimalExpansion} -{\smath{\mbox{\bf decimal}\opLeftPren{}rn\opRightPren{}} converts a rational number \smath{rn} to a decimal expansion.} +{\smath{\mbox{\bf decimal}\opLeftPren{}rn\opRightPren{}} +converts a rational number \smath{rn} to a decimal expansion.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{declare}}\opLeftPren{}{\it listOfInputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{declare}}\opLeftPren{} +{\it listOfInputForms}\opRightPren{}% }% }% { List $ -> Symbol}{}{} -{\smath{\mbox{\bf declare}\opLeftPren{}t\opRightPren{}} returns a name f such that f has been +{\smath{\mbox{\bf declare}\opLeftPren{}t\opRightPren{}} +returns a name f such that f has been declared to the interpreter to be of type t, but has not been assigned a value yet. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{decreasePrecision}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{decreasePrecision}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% {1}{(Integer)->PositiveInteger}{FloatingPointSystem} -{\smath{\mbox{\bf decreasePrecision}\opLeftPren{}n\opRightPren{}\$R} decreases the current +{\smath{\mbox{\bf decreasePrecision}\opLeftPren{}n\opRightPren{}\$R} +decreases the current \spadfunFrom{precision}{FloatingPointSystem} by \smath{n} decimal digits.} @@ -70940,89 +72890,129 @@ not been assigned a value yet. }% }% {0}{()->UP}{MonogenicAlgebra} -{\smath{\mbox{\bf definingPolynomial}\opLeftPren{}\opRightPren{}\$R} returns the minimal polynomial for -a \spadtype{MonogenicAlgebra} domain \smath{R}, that is, one which \smath{\mbox{\bf generator}\opLeftPren{}\opRightPren{}\$R} satisfies. -\newitem\smath{\mbox{\bf definingPolynomial}\opLeftPren{}x\opRightPren{}} returns an expression \smath{p} such that \smath{p(x) = 0}, where \smath{x} is an \spadtype{AlgebraicNumber} or an object of type \spadtype{Expression}. +{\smath{\mbox{\bf definingPolynomial}\opLeftPren{}\opRightPren{}\$R} +returns the minimal polynomial for +a \spadtype{MonogenicAlgebra} domain \smath{R}, that is, one which +\smath{\mbox{\bf generator}\opLeftPren{}\opRightPren{}\$R} satisfies. +\newitem\smath{\mbox{\bf definingPolynomial}\opLeftPren{}x\opRightPren{}} +returns an expression \smath{p} such that \smath{p(x) = 0}, +where \smath{x} is an \spadtype{AlgebraicNumber} +or an object of type \spadtype{Expression}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{degree}}\opLeftPren{}{\it polynomial\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{degree}}\opLeftPren{} +{\it polynomial\opt{, symbol}}\opRightPren{}% }% }% {1}{(\$)->E}{AbelianMonoidRing} -{\opkey{The meaning of degree(u\optinner{, s}) depends on the type of \smath{u}.} +{\opkey{The meaning of degree(u\optinner{, s}) depends on the type of +\smath{u}.} \begin{simpleList} \item if \smath{u} is a polynomial: -\smath{\mbox{\bf degree}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} returns the degree of polynomial \smath{u} with respect to the variable \smath{x}. -Similarly, \smath{\mbox{\bf degree}\opLeftPren{}u, \allowbreak{} lv\opRightPren{}}, where \smath{lv} is a list of variables, returns -a list of degrees of polynomial \smath{u} with respect to each of the variables +\smath{\mbox{\bf degree}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} +returns the degree of polynomial \smath{u} with respect to the variable +\smath{x}. +Similarly, \smath{\mbox{\bf degree}\opLeftPren{}u, +\allowbreak{} lv\opRightPren{}}, where \smath{lv} is +a list of variables, returns +a list of degrees of polynomial \smath{u} with respect +to each of the variables in \smath{lv}. \item if \smath{u} is an element of an \spadtype{AbelianMonoidRing} or \spadtype{GradedModule} domain: -\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} returns the maximum of the exponents of the terms of \smath{u}. +\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} +returns the maximum of the exponents of the terms of \smath{u}. \item if \smath{u} is a series: -\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} returns the degree of the leading term of \smath{u}. -\item if \smath{u} is an element of a domain of category \spadtype{ExtensionField}: \smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} -returns the degree of the minimal polynomial of \smath{u} if \smath{u} is algebraic +\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} +returns the degree of the leading term of \smath{u}. +\item if \smath{u} is an element of a domain of category +\spadtype{ExtensionField}: +\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} +returns the degree of the minimal polynomial of \smath{u} +if \smath{u} is algebraic with respect to the ground field \smath{F}, and {\tt \%infinity} otherwise. \item if \smath{u} is a permutation: -\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} returns the number of points moved by the permutation. +\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} +returns the number of points moved by the permutation. \item if \smath{u} is a permutation group: -\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} returns the number of points moved by all permutations of the group \smath{u}. +\smath{\mbox{\bf degree}\opLeftPren{}u\opRightPren{}} +returns the number of points moved by all permutations of the group \smath{u}. \seeOther{\spadfun{degree}} \end{simpleList} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{delete}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it integerOrSegment}\opRightPren{}% +\opdata{{\mbox{\axiomFun{delete}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it integerOrSegment}\opRightPren{}% }% }% {2}{(\$, Integer)->\$}{LinearAggregate} -{\smath{\mbox{\bf delete}\opLeftPren{}u, \allowbreak{} i\opRightPren{}} returns a copy of linear aggregate \smath{u} with +{\smath{\mbox{\bf delete}\opLeftPren{}u, +\allowbreak{} i\opRightPren{}} returns a copy of +linear aggregate \smath{u} with the \eth{\smath{i}} element deleted. Note: for lists, \code{delete(a, i) == {\tt concat}(a(0..i-1), a(i + 1, ..))}. -\newitem\smath{\mbox{\bf delete}\opLeftPren{}u, \allowbreak{} i..j\opRightPren{}} returns a copy of \smath{u} with the \eth{\smath{i}} +\newitem\smath{\mbox{\bf delete}\opLeftPren{}u, +\allowbreak{} i..j\opRightPren{}} returns a copy +of \smath{u} with the \eth{\smath{i}} through \eth{\smath{j}} element deleted. Note: for lists, \code{delete(a, i..j) = concat(a(0..i-1), a(j+1..))}. -\newitem\smath{\mbox{\bf delete!}\opLeftPren{}u, \allowbreak{} i\opRightPren{}} destructively deletes the \eth{\smath{i}} element of \smath{u}. -\newitem\smath{\mbox{\bf delete!}\opLeftPren{}u, \allowbreak{} i..j\opRightPren{}} destructively deletes elements \smath{u}.\smath{i} through \smath{u}.\smath{j} of \smath{u}. +\newitem\smath{\mbox{\bf delete!}\opLeftPren{}u, +\allowbreak{} i\opRightPren{}} destructively deletes +the \eth{\smath{i}} element of \smath{u}. +\newitem\smath{\mbox{\bf delete!}\opLeftPren{}u, +\allowbreak{} i..j\opRightPren{}} destructively deletes +elements \smath{u}.\smath{i} +through \smath{u}.\smath{j} of \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{deleteProperty}}\opLeftPren{}{\it basicOperator}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{deleteProperty}}\opLeftPren{} +{\it basicOperator}, \allowbreak{}{\it string}\opRightPren{}% }% }% {2}{(\$, String)->\$}{BasicOperator} -{\smath{\mbox{\bf deleteProperty}\opLeftPren{}op, \allowbreak{} s\opRightPren{}} destructively removes property \smath{s} from \smath{op}. +{\smath{\mbox{\bf deleteProperty}\opLeftPren{}op, \allowbreak{} s +\opRightPren{}} destructively removes property \smath{s} from \smath{op}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{denom}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{denominator}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{denom}}\opLeftPren{} +{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{denominator}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(\$)->D}{QuotientFieldCategory} -{\opkey{Argument x can be from domain \spadtype{Fraction(R)} for some domain \smath{R}, or of type \spadtype{Expression}} +{\opkey{Argument x can be from domain \spadtype{Fraction(R)} for +some domain \smath{R}, or of type \spadtype{Expression}} if the result is of type \smath{R}. \newitem -\smath{\mbox{\bf denom}\opLeftPren{}x\opRightPren{}} returns the denominator of \smath{x} as an object -of domain \smath{R}; if \smath{x} is of type \spadtype{Expression}, it returns +\smath{\mbox{\bf denom}\opLeftPren{}x\opRightPren{}} returns the +denominator of \smath{x} as an object +of domain \smath{R}; if \smath{x} is of type \spadtype{Expression}, +it returns an object of domain \spadtype{SMP(R, Kernel(Expression R))}. \newitem -\smath{\mbox{\bf denominator}\opLeftPren{}x\opRightPren{}} returns the denominator of \smath{x} as an element +\smath{\mbox{\bf denominator}\opLeftPren{}x\opRightPren{}} +returns the denominator of \smath{x} as an element of \spadtype{Fraction(R)}; if \smath{x} is of type \spadtype{Expression}, it returns an object of domain Expression(R). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{denominators}}\opLeftPren{}{\it fractionOrContinuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{denominators}}\opLeftPren{} +{\it fractionOrContinuedFraction}\opRightPren{}% }% }% {1}{(\$)->Stream(R)}{ContinuedFraction} -{\smath{\mbox{\bf denominator}\opLeftPren{}frac\opRightPren{}} is the denominator of the fraction \smath{frac}. -\newitem\smath{\mbox{\bf denominators}\opLeftPren{}cf\opRightPren{}} returns the stream of +{\smath{\mbox{\bf denominator}\opLeftPren{}frac\opRightPren{}} +is the denominator of the fraction \smath{frac}. +\newitem\smath{\mbox{\bf denominators}\opLeftPren{}cf\opRightPren{}} +returns the stream of denominators of the approximants of the continued fraction \smath{x}. If the continued fraction is finite, then the stream will be @@ -71034,7 +73024,8 @@ finite. }% }% {1}{(\$)->NonNegativeInteger}{StackAggregate} -{\smath{\mbox{\bf depth}\opLeftPren{}st\opRightPren{}} returns the number of elements of stack +{\smath{\mbox{\bf depth}\opLeftPren{}st\opRightPren{}} +returns the number of elements of stack \smath{st}.} % ---------------------------------------------------------------------- @@ -71043,11 +73034,13 @@ finite. }% }% {1}{(\$)->S}{QueueAggregate} -{\smath{\mbox{\bf dequeue}\opLeftPren{}[ x, y, \ldots, z]\opRightPren{}} creates a +{\smath{\mbox{\bf dequeue}\opLeftPren{}[ x, y, \ldots, z]\opRightPren{}} +creates a dequeue with first (top or front) element \smath{x}, second element \smath{y}, \smath{\ldots}, and last (bottom or back) element \smath{z}. -\newitem\smath{\mbox{\bf dequeue!}\opLeftPren{}q\opRightPren{}} destructively extracts the first (top) +\newitem\smath{\mbox{\bf dequeue!}\opLeftPren{}q\opRightPren{}} +destructively extracts the first (top) element from queue \smath{q}. The element previously second in the queue becomes the first element. @@ -71055,30 +73048,38 @@ A call to \spadfun{error} occurs if \smath{q} is empty. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{derivationCoordinates}}\opLeftPren{}{\it vectorOfElements}, \allowbreak{}{\it derivationFunction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{derivationCoordinates}}\opLeftPren{} +{\it vectorOfElements}, \allowbreak{}{\it derivationFunction}\opRightPren{}% }% }% {2}{(Vector(\$), (R)->R)->Matrix(R)}{MonogenicAlgebra} -{\smath{\mbox{\bf derivationCoordinates}\opLeftPren{}v, \allowbreak{} \quad{}'\opRightPren{}} returns a matrix \smath{M} such that \smath{v' = M v}. +{\smath{\mbox{\bf derivationCoordinates}\opLeftPren{}v, +\allowbreak{} \quad{}'\opRightPren{}} returns a matrix +\smath{M} such that \smath{v' = M v}. Argument \smath{v} is a vector of elements from \smath{R}, a domain of category \spadtype{MonogenicAlgebra} over a ring \smath{R}. Argument \smath{'} is a derivation function defined on \smath{R}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{derivative}}\opLeftPren{}{\it basicOperator\opt{, property}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{derivative}}\opLeftPren{} +{\it basicOperator\opt{, property}}\opRightPren{}% }% }% -{1}{(BasicOperator)->Union(List((List(A))->A), "failed")}{BasicOperatorFunctions1} -{\smath{\mbox{\bf derivative}\opLeftPren{}op\opRightPren{}} returns the value of the \mbox{\tt "\%diff"} +{1}{(BasicOperator)->Union(List((List(A))->A), "failed")} +{BasicOperatorFunctions1} +{\smath{\mbox{\bf derivative}\opLeftPren{}op\opRightPren{}} +returns the value of the \mbox{\tt "\%diff"} property of \smath{op} if it has one, and \mbox{\tt "failed"} otherwise. -\newitem\smath{\mbox{\bf derivative}\opLeftPren{}op, \allowbreak{} dprop\opRightPren{}} attaches \smath{dprop} as the +\newitem\smath{\mbox{\bf derivative}\opLeftPren{}op, +\allowbreak{} dprop\opRightPren{}} attaches \smath{dprop} as the \mbox{\tt "\%diff"} property of \smath{op}. Note: if \smath{op} has a \mbox{\tt "\%diff"} property \smath{f}, then applying a derivation \smath{D} to \smath{op}(a) returns \smath{f(a) D(a)}. Argument \smath{op} must be unary. -\newitem\smath{\mbox{\bf derivative}\opLeftPren{}op, \allowbreak{} [f_1, \allowbreak{} \ldots, f_n]\opRightPren{}} attaches +\newitem\smath{\mbox{\bf derivative}\opLeftPren{}op, +\allowbreak{} [f_1, \allowbreak{} \ldots, f_n]\opRightPren{}} attaches \smath{[ f_1, \ldots, f_n]} as the \mbox{\tt "\%diff"} property of \smath{op}. Note: if \smath{op} has such a \mbox{\tt "\%diff"} property, then applying @@ -71089,7 +73090,8 @@ D(a_n)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{destruct}}\opLeftPren{}{\it sExpression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{destruct}}\opLeftPren{} +{\it sExpression}\opRightPren{}% }% }% {1}{(\$)->List(\$)}{SExpressionCategory} @@ -71103,7 +73105,8 @@ the \smath{SExpression} \smath{(a_1, \ldots, a_n)}, returns the list }% }% {1}{(\$)->R}{MatrixCategory} -{\smath{\mbox{\bf determinant}\opLeftPren{}m\opRightPren{}} returns the determinant of the matrix \smath{m}, or calls +{\smath{\mbox{\bf determinant}\opLeftPren{}m\opRightPren{}} +returns the determinant of the matrix \smath{m}, or calls \spadfun{error} if the matrix is not square. Note: the underlying coefficient domain of \smath{m} is assumed to have a commutative \spadop{*}. @@ -71114,7 +73117,8 @@ have a commutative \spadop{*}. }% }% {1}{(List(R))->\$}{MatrixCategory} -{\smath{\mbox{\bf diagonal}\opLeftPren{}m\opRightPren{}}, where \smath{m} is a square matrix, returns a vector +{\smath{\mbox{\bf diagonal}\opLeftPren{}m\opRightPren{}}, +where \smath{m} is a square matrix, returns a vector consisting of the diagonal elements of \smath{m}. \newitem \smath{\mbox{\bf diagonal}\opLeftPren{}f\opRightPren{}}, @@ -71125,7 +73129,8 @@ See \spadtype{MappingPackage} for related functions. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{diagonal?}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{diagonal?}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% {1}{(\$)->Boolean}{MatrixCategory} @@ -71133,7 +73138,8 @@ See \spadtype{MappingPackage} for related functions. \smath{m} is square and diagonal.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{diagonalMatrix}}\opLeftPren{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{diagonalMatrix}}\opLeftPren{} +{\it listOfElements}\opRightPren{}% }% }% {1}{(List(R))->\$}{MatrixCategory} @@ -71141,56 +73147,74 @@ See \spadtype{MappingPackage} for related functions. where \smath{l} is a list or vector of elements, returns a (square) diagonal matrix with those elements of \smath{l} on the diagonal. -\newitem\smath{\mbox{\bf diagonalMatrix}\opLeftPren{}[ m_1, \ldots, m_k]\opRightPren{}} +\newitem\smath{\mbox{\bf diagonalMatrix}\opLeftPren{} +[ m_1, \ldots, m_k]\opRightPren{}} creates a block diagonal matrix \smath{M} with block matrices \smath{m_1}, \ldots, \smath{m_k} down the diagonal, with 0 block matrices elsewhere.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{diagonalProduct}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{diagonalProduct}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% {1}{(\$)->R}{SquareMatrixCategory} -{\smath{\mbox{\bf diagonalProduct}\opLeftPren{}m\opRightPren{}} returns the product of the elements on the diagonal of the matrix \smath{m}.} +{\smath{\mbox{\bf diagonalProduct}\opLeftPren{}m\opRightPren{}} +returns the product of the elements on the diagonal of the matrix \smath{m}.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{dictionary}}\opLeftPren{}\opRightPren{}% }% }% {0}{()->\$}{DictionaryOperations} -{\smath{\mbox{\bf dictionary}\opLeftPren{}\opRightPren{}}\$\smath{R} creates an empty dictionary of type \smath{R}. -\newitem\smath{\mbox{\bf dictionary}\opLeftPren{}[ x, y, \ldots, z]\opRightPren{}} creates a dictionary consisting of entries \smath{x, y, \ldots, z}. +{\smath{\mbox{\bf dictionary}\opLeftPren{}\opRightPren{}}\$\smath{R} +creates an empty dictionary of type \smath{R}. +\newitem\smath{\mbox{\bf dictionary}\opLeftPren{} +[ x, y, \ldots, z]\opRightPren{}} creates a +dictionary consisting of entries \smath{x, y, \ldots, z}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{difference}}\opLeftPren{}{\it setAggregate}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{difference}}\opLeftPren{} +{\it setAggregate}, \allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(\$, S)->\$}{SetAggregate} -{\smath{\mbox{\bf difference}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} returns the set aggregate \smath{u} with element +{\smath{\mbox{\bf difference}\opLeftPren{}u, +\allowbreak{} x\opRightPren{}} returns the set aggregate \smath{u} with element \smath{x} removed. -\newitem\smath{\mbox{\bf difference}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} returns the set aggregate \smath{w} consisting of elements in set aggregate \smath{u} but not in set aggregate \smath{v}. +\newitem\smath{\mbox{\bf difference}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} returns the set aggregate +\smath{w} consisting of elements +in set aggregate \smath{u} but not in set aggregate \smath{v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{differentialVariables}}\opLeftPren{}{\it differentialPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{differentialVariables}}\opLeftPren{} +{\it differentialPolynomial}\opRightPren{}% }% }% -{1}{(\$)->List(S)}{DifferentialPolynomialCategory}{\smath{\mbox{\bf differentialVariables}\opLeftPren{}p\opRightPren{}} returns a list of differential indeterminates occurring in a differential polynomial \smath{p}.} +{1}{(\$)->List(S)}{DifferentialPolynomialCategory} +{\smath{\mbox{\bf differentialVariables}\opLeftPren{}p\opRightPren{}} +returns a list of differential +indeterminates occurring in a differential polynomial \smath{p}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{differentiate}}\opLeftPren{}{\it expression\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{differentiate}}\opLeftPren{} +{\it expression\opt{, options}}\opRightPren{}% }% }% {1}{(\$)->\$}{DifferentialRing} {See \spadfun{D}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{digamma}}\opLeftPren{}{\it complexDoubleFloat}\opRightPren{}% +\opdata{{\mbox{\axiomFun{digamma}}\opLeftPren{} +{\it complexDoubleFloat}\opRightPren{}% }% }% {1}{(Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} -{\smath{\mbox{\bf digamma}\opLeftPren{}x\opRightPren{}} is the function, \smath{\psi(x)}, defined by +{\smath{\mbox{\bf digamma}\opLeftPren{}x\opRightPren{}} +is the function, \smath{\psi(x)}, defined by $\psi(x) = \Gamma'(x)/\Gamma(x).$ Argument x is either a small float or a complex small float. } @@ -71200,7 +73224,9 @@ Argument x is either a small float or a complex small float. }% }% {0}{()->\$}{CharacterClass} -{\smath{\mbox{\bf digit}\opLeftPren{}\opRightPren{}} returns the class of all characters for which \spadfunFrom{digit?}{Character} is \smath{true}.} +{\smath{\mbox{\bf digit}\opLeftPren{}\opRightPren{}} +returns the class of all characters for +which \spadfunFrom{digit?}{Character} is \smath{true}.} % ---------------------------------------------------------------------- @@ -71208,21 +73234,25 @@ Argument x is either a small float or a complex small float. }% }% {1}{(\$)->Boolean}{Character} -{\smath{\mbox{\bf digit?}\opLeftPren{}ch\opRightPren{}} tests if character \smath{c} is a digit character, that is, one of 0..9.} +{\smath{\mbox{\bf digit?}\opLeftPren{}ch\opRightPren{}} +tests if character \smath{c} is a digit character, that is, one of 0..9.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{digits}}\opLeftPren{}{\it \opt{positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{digits}}\opLeftPren{} +{\it \opt{positiveInteger}}\opRightPren{}% }% }% {1}{(PositiveInteger)->PositiveInteger}{FloatingPointSystem} -{\smath{\mbox{\bf digits}\opLeftPren{}\opRightPren{}} returns the current precision of floats in +{\smath{\mbox{\bf digits}\opLeftPren{}\opRightPren{}} +returns the current precision of floats in numbers of digits. \newitem \smath{\mbox{\bf digits}\opLeftPren{}n\opRightPren{}} set the \spadfunFrom{precision}{FloatingPointSystem} of floats to \smath{n} digits. \newitem -\smath{\mbox{\bf digits}\opLeftPren{}x\opRightPren{}} returns a stream of \smath{p}-adic +\smath{\mbox{\bf digits}\opLeftPren{}x\opRightPren{}} +returns a stream of \smath{p}-adic digits of p-adic integer \smath{n}. \seeType{PAdicInteger} @@ -71233,19 +73263,23 @@ digits of p-adic integer }% }% { I -> SPOL RN --dihedral group}{}{} -{\smath{\mbox{\bf dihedral}\opLeftPren{}n\opRightPren{}} is the cycle index of the +{\smath{\mbox{\bf dihedral}\opLeftPren{}n\opRightPren{}} +is the cycle index of the dihedral group of degree \smath{n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{dihedralGroup}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{dihedralGroup}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% {1}{(List(Integer))->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf dihedralGroup}\opLeftPren{}[ i_1, \ldots, i_k]\opRightPren{}} constructs +{\smath{\mbox{\bf dihedralGroup}\opLeftPren{} +[ i_1, \ldots, i_k]\opRightPren{}} constructs the dihedral group of order \smath{2k} acting on the integers \smath{i_1}, \ldots, \smath{i_k}. Note: duplicates in the list will be removed. -\newitem\smath{\mbox{\bf dihedralGroup}\opLeftPren{}n\opRightPren{}} constructs the dihedral group of order \smath{2n} +\newitem\smath{\mbox{\bf dihedralGroup}\opLeftPren{}n\opRightPren{}} +constructs the dihedral group of order \smath{2n} acting on integers \smath{1, \ldots, n}. } @@ -71254,25 +73288,31 @@ acting on integers \smath{1, \ldots, n}. }% }% {1}{(\$)->\$}{LiouvillianFunctionCategory} -{\smath{\mbox{\bf dilog}\opLeftPren{}x\opRightPren{}} returns the dilogarithm of \smath{x}, that is, \smath{\int {log(x) / (1 - x) dx}}.} +{\smath{\mbox{\bf dilog}\opLeftPren{}x\opRightPren{}} returns the +dilogarithm of \smath{x}, that is, \smath{\int {log(x) / (1 - x) dx}}.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{dim}}\opLeftPren{}{\it color}\opRightPren{}% }% }% {1}{(Color)->\$}{Palette} -{\smath{\mbox{\bf dim}\opLeftPren{}c\opRightPren{}} sets the shade of a hue \smath{c}, above dark but +{\smath{\mbox{\bf dim}\opLeftPren{}c\opRightPren{}} +sets the shade of a hue \smath{c}, above dark but below bright.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{dimension}}\opLeftPren{}{\it \opt{various}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{dimension}}\opLeftPren{} +{\it \opt{various}}\opRightPren{}% }% }% {0}{()->CardinalNumber}{VectorSpace} -{\smath{\mbox{\bf dimension}\opLeftPren{}\opRightPren{}\$R} returns the dimensionality of the vector space +{\smath{\mbox{\bf dimension}\opLeftPren{}\opRightPren{}\$R} +returns the dimensionality of the vector space or rank of Lie algebra \smath{R}. -\newitem\smath{\mbox{\bf dimension}\opLeftPren{}I\opRightPren{}} gives the dimension of the ideal \smath{I}. -\newitem\smath{\mbox{\bf dimension}\opLeftPren{}s\opRightPren{}} returns the dimension of the point category \smath{s}. +\newitem\smath{\mbox{\bf dimension}\opLeftPren{}I\opRightPren{}} +gives the dimension of the ideal \smath{I}. +\newitem\smath{\mbox{\bf dimension}\opLeftPren{}s\opRightPren{}} +returns the dimension of the point category \smath{s}. } @@ -71280,61 +73320,83 @@ or rank of Lie algebra \smath{R}. \opdata{{\mbox{\axiomFun{dioSolve}}\opLeftPren{}{\it equation}\opRightPren{}% }% }% -{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))}{TransSolvePackage} -{\smath{\mbox{\bf dioSolve}\opLeftPren{}eq\opRightPren{}} computes a basis of all minimal solutions +{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))} +{TransSolvePackage} +{\smath{\mbox{\bf dioSolve}\opLeftPren{}eq\opRightPren{}} +computes a basis of all minimal solutions for a linear homomogeneous Diophantine equation \smath{eq}, then all minimal solutions of the inhomogeneous equation. -Alternatively, an expression \smath{u} may be given for \smath{eq} in which case the equation +Alternatively, an expression \smath{u} may be given +for \smath{eq} in which case the equation \smath{eq} is defined as \smath{u=0}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{directory}}\opLeftPren{}{\it filename}\opRightPren{}% +\opdata{{\mbox{\axiomFun{directory}}\opLeftPren{} +{\it filename}\opRightPren{}% }% }% {1}{(\$)->String}{FileNameCategory} -{\smath{\mbox{\bf directory}\opLeftPren{}f\opRightPren{}} returns the directory part of the file name.} +{\smath{\mbox{\bf directory}\opLeftPren{}f\opRightPren{}} +returns the directory part of the file name.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{directProduct}}\opLeftPren{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{directProduct}}\opLeftPren{} +{\it vector}\opRightPren{}% }% }% {1}{(Vector(R))->\$}{DirectProductCategory} -{\smath{\mbox{\bf directProduct}\opLeftPren{}v\opRightPren{}} converts the vector \smath{v} to become a direct product +{\smath{\mbox{\bf directProduct}\opLeftPren{}v\opRightPren{}} +converts the vector \smath{v} to become a direct product } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{discreteLog}}\opLeftPren{}{\it finiteFieldElement}\opRightPren{}% +\opdata{{\mbox{\axiomFun{discreteLog}}\opLeftPren{} +{\it finiteFieldElement}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{FiniteFieldCategory} -{\smath{\mbox{\bf discreteLog}\opLeftPren{}a\opRightPren{}\$F} computes the discrete logarithm of \smath{a} with respect to \smath{\mbox{\bf primitiveElement}\opLeftPren{}\opRightPren{}\$F} of the field \smath{F}.} +{\smath{\mbox{\bf discreteLog}\opLeftPren{}a\opRightPren{}\$F} +computes the discrete logarithm of \smath{a} with respect to +\smath{\mbox{\bf primitiveElement}\opLeftPren{}\opRightPren{}\$F} +of the field \smath{F}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{discreteLog}}\opLeftPren{}{\it finiteFieldElement}, \allowbreak{}{\it finiteFieldElement}\opRightPren{}% +\opdata{{\mbox{\axiomFun{discreteLog}}\opLeftPren{} +{\it finiteFieldElement}, +\allowbreak{}{\it finiteFieldElement}\opRightPren{}% }% }% {2}{(\$, \$)->Union(NonNegativeInteger, "failed")}{FieldOfPrimeCharacteristic} -{\smath{\mbox{\bf discreteLog}\opLeftPren{}b, \allowbreak{} a\opRightPren{}} computes \smath{s} such that +{\smath{\mbox{\bf discreteLog}\opLeftPren{}b, +\allowbreak{} a\opRightPren{}} computes \smath{s} such that \smath{b^s = a} if such an \smath{s} exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{discriminant}}\opLeftPren{}{\it polynomial\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{discriminant}}\opLeftPren{} +{\it polynomial\opt{, symbol}}\opRightPren{}% }% }% {0}{()->R}{FramedAlgebra} -{\smath{\mbox{\bf discriminant}\opLeftPren{}p\optinner{, x}\opRightPren{}} returns the discriminant of the polynomial +{\smath{\mbox{\bf discriminant}\opLeftPren{}p\optinner{, x}\opRightPren{}} +returns the discriminant of the polynomial \smath{p} with respect to the variable \smath{x}. If \smath{x} is univariate, the second argument may be omitted. \newitem\smath{\mbox{\bf discriminant}\opLeftPren{}\opRightPren{}\$R} -returns \smath{\mbox{\bf determinant}\opLeftPren{}{\mbox {\bf traceMatrix}}()\$R\opRightPren{}} of a +returns \smath{\mbox{\bf determinant}\opLeftPren{} +{\mbox {\bf traceMatrix}}()\$R\opRightPren{}} of a \spadtype{FramedAlgebra} domain \smath{R}. -\newitem\smath{\mbox{\bf discriminant}\opLeftPren{}[ v_1, .., v_n]\opRightPren{}} returns \smath{\mbox{\bf determinant}\opLeftPren{}traceMatrix([ v_1, .., v_n])\opRightPren{}} where the \smath{v_i} each have \smath{n} elements. +\newitem\smath{\mbox{\bf discriminant}\opLeftPren{} +[ v_1, .., v_n]\opRightPren{}} returns \smath{\mbox{\bf determinant} +\opLeftPren{}traceMatrix([ v_1, .., v_n])\opRightPren{}} +where the \smath{v_i} each have +\smath{n} elements. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{display}}\opLeftPren{}{\it text\opt{, width}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{display}}\opLeftPren{} +{\it text\opt{, width}}\opRightPren{}% }% }% {1}{(\$)->Void}{ScriptFormulaFormat} @@ -71343,14 +73405,19 @@ where \smath{t} is either IBM SCRIPT Formula Format or \TeX{} text, outputs \smath{t} so that each line has length \smath{\leq w}. The default value of \smath{w} is that length set by the system command \spadsyscom{)set output length}. -\newitem\smath{\mbox{\bf display}\opLeftPren{}op, \allowbreak{} f\opRightPren{}} attaches \smath{f} as the \mbox{\tt "\%display"} property +\newitem\smath{\mbox{\bf display}\opLeftPren{}op, +\allowbreak{} f\opRightPren{}} attaches \smath{f} +as the \mbox{\tt "\%display"} property of \smath{op}. \newitem -\smath{\mbox{\bf display}\opLeftPren{}op\opRightPren{}} returns the \mbox{\tt "\%display"} property of \smath{op} if it has one attached, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf display}\opLeftPren{}op\opRightPren{}} +returns the \mbox{\tt "\%display"} property of \smath{op} if it has one +attached, and \mbox{\tt "failed"} otherwise. \newitem Value \smath{f} either has type \spadsig{OutputForm}{OutputForm} or else \spadsig{List(OutputForm)}{OutputForm}. Argument \smath{op} must be unary. -Note: if \smath{op} has a \mbox{\tt "\%display"} property \smath{f} of the former type, +Note: if \smath{op} has a \mbox{\tt "\%display"} +property \smath{f} of the former type, then \smath{op(a)} gets converted to \spadtype{OutputForm} as \smath{f(a)}. If \smath{f} has the latter type, then \smath{op(a_1, \ldots, a_n)} gets converted to @@ -71358,7 +73425,8 @@ then \smath{op(a_1, \ldots, a_n)} gets converted to } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{distance}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{distance}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{(\$, \$)->Integer}{RecursiveAggregate} @@ -71368,11 +73436,14 @@ are recursive aggregates (for example, lists) returns the path length (an integer) from node \smath{u} to \smath{v}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{distdfact}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{distdfact}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it boolean}\opRightPren{}% }% }% -{2}{(FP, Boolean)->Record(cont:F, factors:List(Record(irr:FP, pow:Integer)))}{DistinctDegreeFactorize} -{\smath{\mbox{\bf distdfact}\opLeftPren{}p, \allowbreak{} squareFreeFlag\opRightPren{}} produces the complete +{2}{(FP, Boolean)->Record(cont:F, +factors:List(Record(irr:FP, pow:Integer)))}{DistinctDegreeFactorize} +{\smath{\mbox{\bf distdfact}\opLeftPren{}p, +\allowbreak{} squareFreeFlag\opRightPren{}} produces the complete \typeout{check distdfact} factorization of the polynomial \smath{p} returning an internal data structure. @@ -71380,11 +73451,13 @@ If argument \smath{squareFreeFlag} is \smath{true}, the polynomial is assumed square free.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{distribute}}\opLeftPren{}{\it expression\opt{, f}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{distribute}}\opLeftPren{} +{\it expression\opt{, f}}\opRightPren{}% }% }% {1}{(\$)->\$}{ExpressionSpace} -{\smath{\mbox{\bf distribute}\opLeftPren{}f\optinner{, g}\opRightPren{}} expands all the kernels in +{\smath{\mbox{\bf distribute}\opLeftPren{}f\optinner{, g}\opRightPren{}} +expands all the kernels in \smath{f} that contain \smath{g} in their arguments and that are formally enclosed by a \spadfunFrom{box}{ExpressionSpace} or a \spadfunFrom{paren}{ExpressionSpace} expression. @@ -71392,11 +73465,13 @@ By default, \smath{g} is the list of all kernels in \smath{f}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{divide}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{divide}}\opLeftPren{}{\it element}, +\allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(\$, \$)->Record(quotient:\$, remainder:\$)}{EuclideanDomain} -{\smath{\mbox{\bf divide}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} divides \smath{x} by \smath{y} producing a +{\smath{\mbox{\bf divide}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +divides \smath{x} by \smath{y} producing a record containing a \smath{quotient} and \smath{remainder}, where the remainder is smaller (see \spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor @@ -71404,11 +73479,14 @@ the remainder is smaller (see % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{divideExponents}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{divideExponents}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% -{2}{(\$, NonNegativeInteger)->Union(\$, "failed")}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf divideExponents}\opLeftPren{}p, \allowbreak{} n\opRightPren{}} returns a new polynomial resulting +{2}{(\$, NonNegativeInteger)->Union(\$, "failed")} +{UnivariatePolynomialCategory} +{\smath{\mbox{\bf divideExponents}\opLeftPren{}p, +\allowbreak{} n\opRightPren{}} returns a new polynomial resulting from dividing all exponents of the polynomial \smath{p} by the non negative integer \smath{n}, or \mbox{\tt "failed"} if no exponent is exactly divisible by \smath{n}.} @@ -71418,38 +73496,46 @@ exactly divisible by \smath{n}.} }% }% {1}{(Integer)->List(Integer)}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf divisors}\opLeftPren{}i\opRightPren{}} returns a list of the divisors of integer +{\smath{\mbox{\bf divisors}\opLeftPren{}i\opRightPren{}} +returns a list of the divisors of integer \smath{i}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{domain}}\opLeftPren{}{\it typeAnyObject}\opRightPren{}% +\opdata{{\mbox{\axiomFun{domain}}\opLeftPren{} +{\it typeAnyObject}\opRightPren{}% }% }% {1}{(\$)->SExpression}{Any} -{\smath{\mbox{\bf domain}\opLeftPren{}a\opRightPren{}} returns the type of the original object that +{\smath{\mbox{\bf domain}\opLeftPren{}a\opRightPren{}} +returns the type of the original object that was converted to \spadtype{Any} as object of type \spadtype{SExpression} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{domainOf}}\opLeftPren{}{\it typeAnyObject}\opRightPren{}% +\opdata{{\mbox{\axiomFun{domainOf}}\opLeftPren{} +{\it typeAnyObject}\opRightPren{}% }% }% {1}{(\$)->OutputForm}{Any} -{\smath{\mbox{\bf domainOf}\opLeftPren{}a\opRightPren{}} returns a printable form of the type of the +{\smath{\mbox{\bf domainOf}\opLeftPren{}a\opRightPren{}} returns a +printable form of the type of the original type of \smath{a}, an object of type \spadtype{Any}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{dot}}\opLeftPren{}{\it vector}, \allowbreak{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{dot}}\opLeftPren{}{\it vector}, \allowbreak{} +{\it vector}\opRightPren{}% }% }% {2}{(\$, \$)->R}{DirectProductCategory} -{\smath{\mbox{\bf dot}\opLeftPren{}v_1, \allowbreak{} v_2\opRightPren{}} computes the inner product of the vectors +{\smath{\mbox{\bf dot}\opLeftPren{}v_1, \allowbreak{} v_2\opRightPren{}} +computes the inner product of the vectors \smath{v_1} and \smath{v_2}, or calls \spadfun{error} if \smath{x} and \smath{y} are not of the same length. -\newitem \smath{\mbox{\bf dot}\opLeftPren{}of\opRightPren{}}, where \smath{of} is an object of type +\newitem \smath{\mbox{\bf dot}\opLeftPren{}of\opRightPren{}}, +where \smath{of} is an object of type \spadtype{OutputForm} (normally unexposed), returns an output form with one dot overhead\texht{ (\.{x})}{}. } @@ -71463,7 +73549,8 @@ with one dot overhead\texht{ (\.{x})}{}. where \smath{x} is an element of a domain \smath{R} of category \spadtype{FramedNonAssociativeAlgebra}, determines the number of linearly independent elements in -\smath{b_1 x}, \ldots, \smath{b_n x}, where \smath{b=[ b_1, \ldots, b_n]} is the +\smath{b_1 x}, \ldots, \smath{b_n x}, where +\smath{b=[ b_1, \ldots, b_n]} is the fixed basis for \smath{R}. } @@ -71480,52 +73567,96 @@ a)} is irreducible over \smath{K(a)} where \smath{p(a) = 0}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{draw}}\opLeftPren{}{\it functionOrExpression}, \allowbreak{}{\it range}\allowbreak $\,[$ , \allowbreak{}{\it options}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{draw}}\opLeftPren{}{\it functionOrExpression}, +\allowbreak{}{\it range}\allowbreak $\,[$ , \allowbreak{} +{\it options}$]$\opRightPren{}% }% }% -{2}{((DoubleFloat)->DoubleFloat, Segment(Float))->TwoDimensionalViewport}{TopLevelDrawFunctionsForCompiledFunctions} +{2}{((DoubleFloat)->DoubleFloat, Segment(Float))-> +TwoDimensionalViewport}{TopLevelDrawFunctionsForCompiledFunctions} {\smath{f}, \smath{g}, and \smath{h} below denote user-defined functions which map one or more \spadtype{DoubleFloat} values to a \spadtype{DoubleFloat} value. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f, \allowbreak{} a..b\opRightPren{}} draws the \twodim{} graph of \smath{y = f(x)} as \smath{x} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f, +\allowbreak{} a..b\opRightPren{}} draws the \twodim{} graph of +\smath{y = f(x)} as \smath{x} +ranges from \smath{\mbox{\bf min}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f, g), \allowbreak{} a..b\opRightPren{}} draws the \twodim{} graph of +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f, g), +\allowbreak{} a..b\opRightPren{}} draws the \twodim{} graph of the parametric curve \smath{x = f(t), y = g(t)} as \smath{t} -ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f, \allowbreak{} a..b, \allowbreak{} c..d\opRightPren{}} draws the \threedim{} graph of -\smath{z = f(x, y)} as \smath{x} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to -\smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and \smath{y} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to +ranges from \smath{\mbox{\bf min}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}}. +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f, +\allowbreak{} a..b, \allowbreak{} c..d\opRightPren{}} +draws the \threedim{} graph of +\smath{z = f(x, y)} as \smath{x} ranges from +\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to +\smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +and \smath{y} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, +\allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f, g, h), \allowbreak{} a..b\opRightPren{}} draws a \threedim{} graph +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f, g, h), +\allowbreak{} a..b\opRightPren{}} draws a \threedim{} graph of the parametric curve \smath{x = f(t), y = g(t), z = h(t)} as -\smath{t} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}surface(f, g, h), \allowbreak{} a..b, \allowbreak{} c..d\opRightPren{}} draws the +\smath{t} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}}. +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}surface(f, g, h), +\allowbreak{} a..b, \allowbreak{} c..d\opRightPren{}} draws the \threedim{} graph of the parametric -surface \smath{x = f(u, v)}, \smath{y = g(u, v)}, \smath{z = h(u, v)} as \smath{u} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and \smath{v} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. +surface \smath{x = f(u, v)}, \smath{y = g(u, v)}, +\smath{z = h(u, v)} as \smath{u} ranges from +\smath{\mbox{\bf min}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} to +\smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +and \smath{v} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, +\allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, +\allowbreak{} d\opRightPren{}}. \medbreak\bigopkey{Arguments \smath{f}, \smath{g}, and \smath{h} below denote an \spadtype{Expression} involving the variables indicated as arguments. For example, \smath{f(x, y)} denotes an expression involving the variables \smath{x} and \smath{y}.} -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f(x), \allowbreak{} x = a..b\opRightPren{}} draws the \twodim{} graph of +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f(x), +\allowbreak{} x = a..b\opRightPren{}} draws the \twodim{} graph of \smath{y = f(x)} as -\smath{x} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f(t), g(t)), \allowbreak{} t = a..b\opRightPren{}} draws the \twodim{} graph of the -parametric curve \smath{x = f(t), y = g(t)} as \smath{t} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f(x, y), \allowbreak{} x = a..b, \allowbreak{} y = c..d\opRightPren{}} draws the +\smath{x} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}}. +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f(t), g(t)), +\allowbreak{} t = a..b\opRightPren{}} draws the \twodim{} graph of the +parametric curve \smath{x = f(t), y = g(t)} as \smath{t} ranges from +\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to +\smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}f(x, y), \allowbreak{} x = a..b, +\allowbreak{} y = c..d\opRightPren{}} draws the \threedim{} graph of \smath{z = f(x, y)} as \smath{x} ranges from -\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and \smath{y} ranges from -\smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. -\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f(t), g(t), h(t)), \allowbreak{} t = a..b\opRightPren{}} draws the +\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to +\smath{\mbox{\bf max}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} and \smath{y} +ranges from +\smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to +\smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. +\bigitem\smath{\mbox{\bf draw}\opLeftPren{}curve(f(t), g(t), h(t)), +\allowbreak{} t = a..b\opRightPren{}} draws the \threedim{} graph of the parametric curve \smath{x = f(t)}, \smath{y = g(t)}, \smath{z = h(t)} as \smath{t} ranges from -\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. +\smath{\mbox{\bf min}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}}. \bigitem\smath{{\bf draw}(surface(f(u, v), g(u, v), h(u, v)), u = a..b, v = c..d)} draws the \threedim{} graph of the parametric surface \smath{x = f(u, v)}, \smath{y = g(u, v)}, \smath{z = h(u, v)} as -\smath{u} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and -\smath{v} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. +\smath{u} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} and +\smath{v} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, +\allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, +\allowbreak{} d\opRightPren{}}. \medbreak Each of the \spadfun{draw} operations optionally take options given as extra arguments. @@ -71547,25 +73678,32 @@ coordinate systems of point \smath{p}: \smath{prolateSpheroidal}, \smath{spherical}, and \smath{toroidal} -\smallbreak\opoption{curveColor}{p} specifies a color index for \twodim{} graph curves from the pallete \smath{p}. -\smallbreak\opoption{pointColor}{p} specifies a color index for \twodim{} graph points +\smallbreak\opoption{curveColor}{p} specifies a color index +for \twodim{} graph curves from the pallete \smath{p}. +\smallbreak\opoption{pointColor}{p} specifies a color index +for \twodim{} graph points from the palette \smath{p}. \smallbreak\opoption{range}{[ a..b]} provides a user-specified range for implicit curve plots. -\smallbreak\opoption{space}{sp} adds the current graph to \spadtype{ThreeSpace} object +\smallbreak\opoption{space}{sp} adds the current graph to +\spadtype{ThreeSpace} object \smath{sp}. \smallbreak\opoption{style}{s} specifies the drawing style in which the graph will be plotted: \smath{wire}, \smath{solid}, \smath{shade}, \smath{smooth}. \smallbreak\opoption{title}{s} titles the graph with string \smath{s}. \smallbreak\opoption{toScale}{true} causes the graph to be drawn to scale. -\smallbreak\opoption{tubePoints}{n} specifies the number of points \smath{n} defining -the circle which creates the tube around a \threedim{} curve. The default value is 6. -\smallbreak\opoption{tubeRadius}{r} specifies a \spadtype{Float} radius \smath{r} +\smallbreak\opoption{tubePoints}{n} specifies the number of +points \smath{n} defining +the circle which creates the tube around a \threedim{} curve. +The default value is 6. +\smallbreak\opoption{tubeRadius}{r} specifies a \spadtype{Float} +radius \smath{r} for a tube plot around a \threedim{} curve. \smallbreak\opoption{unit}{[ a, b]} marks off the units of a \twodim{} graph in increments \smath{a} along the x-axis, \smath{b} along the y-axis. -\smallbreak\opoption{var1Steps}{n} indicates the number of subdivisions \smath{n} +\smallbreak\opoption{var1Steps}{n} indicates the number of +subdivisions \smath{n} of the first range variable. \smallbreak\opoption{var2Steps}{n} indicates the number of subdivisions \smath{n} @@ -71573,39 +73711,49 @@ of the second range variable. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{drawToScale}}\opLeftPren{}{\optArg{boolean}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{drawToScale}}\opLeftPren{} +{\optArg{boolean}}\opRightPren{}% }% }% {0}{()->Boolean}{GraphicsDefaults} -{\smath{\mbox{\bf drawToScale}\opLeftPren{}\opRightPren{}} tests if plots are currently to be drawn to scale. +{\smath{\mbox{\bf drawToScale}\opLeftPren{}\opRightPren{}} +tests if plots are currently to be drawn to scale. \newitem\ -\smath{\mbox{\bf drawToScale}\opLeftPren{}true\opRightPren{}} causes plots to be drawn to scale. -\smath{\mbox{\bf drawToScale}\opLeftPren{}false\opRightPren{}} causes plots to be drawn to fill up the +\smath{\mbox{\bf drawToScale}\opLeftPren{}true\opRightPren{}} +causes plots to be drawn to scale. +\smath{\mbox{\bf drawToScale}\opLeftPren{}false\opRightPren{}} +causes plots to be drawn to fill up the viewport window. The default setting is \smath{false}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{duplicates}}\opLeftPren{}{\it dictionary}\opRightPren{}% +\opdata{{\mbox{\axiomFun{duplicates}}\opLeftPren{} +{\it dictionary}\opRightPren{}% }% }% {1}{(\$)->List(Record(entry:S, count:NonNegativeInteger))}{MultiDictionary} -{\smath{\mbox{\bf duplicates}\opLeftPren{}d\opRightPren{}} returns a list of values which have +{\smath{\mbox{\bf duplicates}\opLeftPren{}d\opRightPren{}} +returns a list of values which have duplicates in \smath{d}} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{Ei}}\opLeftPren{}{\it variable}\opRightPren{}% }% }% {1}{(\$)->\$}{LiouvillianFunctionCategory} -{\smath{\mbox{\bf Ei}\opLeftPren{}x\opRightPren{}} returns the exponential integral of \smath{x}: +{\smath{\mbox{\bf Ei}\opLeftPren{}x\opRightPren{}} +returns the exponential integral of \smath{x}: \smath{\int exp(x)/x {\rm dx}}.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{eigenMatrix}}\opLeftPren{}{\it matrix}\opRightPren{}% }% }% -{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))->Union(Matrix(Expression(Fraction(Integer))), \mbox{\tt "failed"})}{RadicalEigenPackage} -{\smath{\mbox{\bf eigenMatrix}\opLeftPren{}A\opRightPren{}} returns the matrix \smath{B} such that +{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +Union(Matrix(Expression(Fraction(Integer))), +\mbox{\tt "failed"})}{RadicalEigenPackage} +{\smath{\mbox{\bf eigenMatrix}\opLeftPren{}A\opRightPren{}} +returns the matrix \smath{B} such that \smath{BA(\mbox{\bf inverse } B)} is diagonal, or \mbox{\tt "failed"} if no such \smath{B} exists.} @@ -71614,26 +73762,39 @@ duplicates in \smath{d}} \opdata{{\mbox{\axiomFun{eigenvalues}}\opLeftPren{}{\it matrix}\opRightPren{}% }% }% -{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))->List(Fraction(Polynomial(Fraction(Integer))))}{EigenPackage} +{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +List(Fraction(Polynomial(Fraction(Integer))))}{EigenPackage} {\smath{\mbox{\bf eigenvalues}\opLeftPren{}A\opRightPren{}}, where \smath{A} is a matrix with rational function coefficients, returns the eigenvalues of the matrix \smath{A} which are expressible as rational functions over the rational numbers.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{eigenvector}}\opLeftPren{}{\it eigenvalue}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{eigenvector}}\opLeftPren{} +{\it eigenvalue}, \allowbreak{}{\it matrix}\opRightPren{}% }% }% -{2}{(Fraction(Polynomial(Fraction(Integer))), Matrix(Fraction(Polynomial(Fraction(Integer)))))->List(Matrix(Fraction(Polynomial(Fraction(Integer)))))}{EigenPackage} -{\smath{\mbox{\bf eigenvector}\opLeftPren{}eigval, \allowbreak{} A\opRightPren{}} returns the eigenvectors belonging +{2}{(Fraction(Polynomial(Fraction(Integer))), +Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +List(Matrix(Fraction(Polynomial(Fraction(Integer)))))} +{EigenPackage} +{\smath{\mbox{\bf eigenvector}\opLeftPren{}eigval, +\allowbreak{} A\opRightPren{}} returns the eigenvectors belonging to the eigenvalue \smath{eigval} for the matrix \smath{A}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{eigenvectors}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{eigenvectors}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% -{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))->List(Union(Record(algrel:Fraction(Polynomial(Fraction(Integer))), algmult:Integer, algvec:List(Matrix(Fraction(Polynomial(Fraction(Integer)))))), Record(eigval:Fraction(Polynomial(Fraction(Integer))), eigmult:Integer, eigvec:List(Matrix(Fraction(Polynomial(Fraction(Integer))))))))}{EigenPackage} -{\smath{\mbox{\bf eigenvectors}\opLeftPren{}A\opRightPren{}} returns the eigenvalues and eigenvectors +{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +List(Union(Record(algrel:Fraction(Polynomial(Fraction(Integer))), +algmult:Integer, +algvec:List(Matrix(Fraction(Polynomial(Fraction(Integer)))))), +Record(eigval:Fraction(Polynomial(Fraction(Integer))), eigmult:Integer, +eigvec:List(Matrix(Fraction(Polynomial(Fraction(Integer))))))))}{EigenPackage} +{\smath{\mbox{\bf eigenvectors}\opLeftPren{}A\opRightPren{}} +returns the eigenvalues and eigenvectors for the matrix \smath{A}. The rational eigenvalues and the corresponding eigenvectors are explicitly computed. @@ -71643,11 +73804,13 @@ Their corresponding eigenvectors are expressed in terms of a ``generic'' root of this polynomial.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{element?}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it ideal}\opRightPren{}% +\opdata{{\mbox{\axiomFun{element?}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it ideal}\opRightPren{}% }% }% {2}{(DPoly, \$)->Boolean}{PolynomialIdeals} -{\smath{\mbox{\bf element?}\opLeftPren{}f, \allowbreak{} I\opRightPren{}} tests if the polynomial \smath{f} belongs +{\smath{\mbox{\bf element?}\opLeftPren{}f, \allowbreak{} I\opRightPren{}} +tests if the polynomial \smath{f} belongs to the ideal \smath{I}.} % ---------------------------------------------------------------------- @@ -71655,28 +73818,32 @@ to the ideal \smath{I}.} }% }% { I -> SPOL RN}{}{} -{\smath{\mbox{\bf elementary}\opLeftPren{}n\opRightPren{}} is the \eth{\smath{n}} elementary symmetric +{\smath{\mbox{\bf elementary}\opLeftPren{}n\opRightPren{}} +is the \eth{\smath{n}} elementary symmetric function expressed in terms of power sums. See \axiomType{CycleIndicators} for details. } - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{elliptic}}\opLeftPren{}{\it scaleFactor}\opRightPren{}% +\opdata{{\mbox{\axiomFun{elliptic}}\opLeftPren{} +{\it scaleFactor}\opRightPren{}% }% }% {1}{(R)->(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf elliptic}\opLeftPren{}r\opRightPren{}} returns a function for transforming elliptic +{\smath{\mbox{\bf elliptic}\opLeftPren{}r\opRightPren{}} +returns a function for transforming elliptic coordinates to Cartesian coordinates. The function returned will map the point \smath{(u, v)} to \smath{x = r \cosh(u) \cos(v)}, \smath{y = r \sinh(u) \sin(v)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ellipticCylindrical}}\opLeftPren{}{\it scaleFactor}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ellipticCylindrical}}\opLeftPren{} +{\it scaleFactor}\opRightPren{}% }% }% {1}{(R)->(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf ellipticCylindrical}\opLeftPren{}r\opRightPren{}} returns a function for +{\smath{\mbox{\bf ellipticCylindrical}\opLeftPren{}r\opRightPren{}} +returns a function for transforming elliptic cylindrical coordinates to Cartesian coordinates as a function of the scale factor \smath{r}. The function returned will map the point \smath{(u, v, z)} to @@ -71685,15 +73852,19 @@ The function returned will map the point \smath{(u, v, z)} to % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{elt}}\opLeftPren{}{\it structure}, \allowbreak{}{\it various}\allowbreak $\,[$ , \allowbreak{}{\it \ldots}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{elt}}\opLeftPren{}{\it structure}, \allowbreak{} +{\it various}\allowbreak $\,[$ , \allowbreak{} +{\it \ldots}$]$\opRightPren{}% }% }% {1}{(\$)->R}{CartesianTensor} -{\opkey{\smath{\mbox{\bf elt}\opLeftPren{}u, \allowbreak{} v\opRightPren{}}, usually written as \smath{u.v} or +{\opkey{\smath{\mbox{\bf elt}\opLeftPren{}u, \allowbreak{} v\opRightPren{}}, +usually written as \smath{u.v} or \smath{u(v)}, regards the structure \smath{u} as a function and applies structure \smath{u} to argument \smath{v}. Many types export \spadfun{elt} with multiple arguments; -\smath{\mbox{\bf elt}\opLeftPren{}u, \allowbreak{} v, \allowbreak{} w\ldots\opRightPren{}} is generally written +\smath{\mbox{\bf elt}\opLeftPren{}u, \allowbreak{} v, +\allowbreak{} w\ldots\opRightPren{}} is generally written \smath{u(v, w\ldots)}. The interpretation of \smath{u} depends on its type. If \smath{u} is:} @@ -71721,15 +73892,19 @@ apply the binary, ternary, or 4-ary operator \smath{u} to arguments. the distinguished variable replaced by the value of \smath{y}; this value may either be another rational function or polynomial or a member of the underlying coefficient domain. -\item a list: \smath{u.first} is equivalent to \smath{\mbox{\bf first}\opLeftPren{}u\opRightPren{}} and returns +\item a list: \smath{u.first} is equivalent to +\smath{\mbox{\bf first}\opLeftPren{}u\opRightPren{}} and returns the first element of list \smath{u}. -Also, \smath{u.last} is equivalent to \smath{\mbox{\bf last}\opLeftPren{}u\opRightPren{}} and returns +Also, \smath{u.last} is equivalent to +\smath{\mbox{\bf last}\opLeftPren{}u\opRightPren{}} and returns the last element of list \smath{u}. Both of these call \spadfun{error} if \smath{u} is the empty list. -Similarly, \smath{u.rest} is equivalent to \smath{\mbox{\bf rest}\opLeftPren{}u\opRightPren{}} and +Similarly, \smath{u.rest} is equivalent to +\smath{\mbox{\bf rest}\opLeftPren{}u\opRightPren{}} and returns the list \smath{u} -beginning at its second element, or calls \spadfun{error} if \smath{u} has less than +beginning at its second element, or calls \spadfun{error} +if \smath{u} has less than two elements. \item a library: \smath{u(name)} returns the entry in the library stored under the @@ -71738,26 +73913,36 @@ key \smath{name}. \smath{u(x)} applies the differential operator \smath{u} to the value \smath{x}. \item a matrix or two-dimensional array: -\smath{u(i, j\optinner{, x})}, \smath{1 \leq i \leq nrows(u), 1 \leq j \leq ncols(m)}, returns the -element in the \eth{\smath{i}} row and \eth{\smath{j}} column of the matrix \smath{m}. -If the indices are out of range and an extra argument \smath{x} is provided, +\smath{u(i, j\optinner{, x})}, \smath{1 \leq i \leq nrows(u), +1 \leq j \leq ncols(m)}, returns the +element in the \eth{\smath{i}} row and \eth{\smath{j}} +column of the matrix \smath{m}. +If the indices are out of range and an extra argument +\smath{x} is provided, then \smath{x} is returned; otherwise, \spadfun{error} is called. Also, \smath{u([i_1, \ldots, i_m], [j_1, \ldots, j_m])} returns -the \smath{m}-by-\smath{n} matrix consisting of elements \smath{u(i_k, j_l)} of \smath{u}. +the \smath{m}-by-\smath{n} matrix consisting of elements +\smath{u(i_k, j_l)} of \smath{u}. \item a permutation group: \smath{u(i)} returns the \smath{i}-th generator of the group \smath{u}. -\item a point: \smath{u.i} returns the \eth{\smath{i}} component of the point \smath{u}. +\item a point: \smath{u.i} returns the \eth{\smath{i}} component +of the point \smath{u}. \item a rewrite rule: -\smath{u(f\optinner{, n})} applies rewrite rule \smath{u} to expression \smath{f} +\smath{u(f\optinner{, n})} applies rewrite rule \smath{u} to +expression \smath{f} at most \smath{n} times, where \smath{n=\infty} by default. When the left-hand side of \smath{u} matches a subexpression of \smath{f}, -the subexpression is replaced by the right-hand side of \smath{u} producing a new \smath{f}. -After \smath{n} iterations or when no further match occurs, the transformed \smath{f} +the subexpression is replaced by the right-hand side of \smath{u} +producing a new \smath{f}. +After \smath{n} iterations or when no further match occurs, the +transformed \smath{f} is returned. \item a ruleset: -\smath{u(f\optinner{, n})} applies ruleset \smath{u} to expression \smath{f} at most \smath{n} +\smath{u(f\optinner{, n})} applies ruleset \smath{u} to expression +\smath{f} at most \smath{n} times, where \smath{n=\infty} by default. -Similar to last case, except that on each iteration, each rule in the ruleset is +Similar to last case, except that on each iteration, each rule in +the ruleset is applied in turn in attempt to find a match. \item an \axiomType{SExpression} \smath{(a_1, \ldots, a_n\quad{}.\quad{}b)} (where \smath{b} denotes the \spadfun{cdr} of the last node): @@ -71787,7 +73972,8 @@ tensors respectively. }% }% {0}{()->\$}{Aggregate} -{\smath{\mbox{\bf empty}\opLeftPren{}\opRightPren{}\$R} creates an aggregate of type \smath{R} with 0 +{\smath{\mbox{\bf empty}\opLeftPren{}\opRightPren{}\$R} creates an +aggregate of type \smath{R} with 0 elements.} % ---------------------------------------------------------------------- @@ -71795,38 +73981,46 @@ elements.} }% }% {1}{(\$)->Boolean}{Aggregate} -{\smath{\mbox{\bf empty?}\opLeftPren{}u\opRightPren{}} tests if aggregate \smath{u} has 0 elements.} +{\smath{\mbox{\bf empty?}\opLeftPren{}u\opRightPren{}} tests if aggregate +\smath{u} has 0 elements.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{endOfFile?}}\opLeftPren{}{\it file}\opRightPren{}% }% }% {1}{(\$)->Boolean}{TextFile} -{\smath{\mbox{\bf endOfFile?}\opLeftPren{}f\opRightPren{}} tests whether the file \smath{f} is +{\smath{\mbox{\bf endOfFile?}\opLeftPren{}f\opRightPren{}} tests whether +the file \smath{f} is positioned after the end of all text. If the file is open for output, then this test always returns \smath{true}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{enqueue!}}\opLeftPren{}{\it value}, \allowbreak{}{\it queue}\opRightPren{}% +\opdata{{\mbox{\axiomFun{enqueue!}}\opLeftPren{}{\it value}, \allowbreak{} +{\it queue}\opRightPren{}% }% }% {2}{(S, \$)->S}{QueueAggregate} -{\smath{\mbox{\bf enqueue!}\opLeftPren{}x, \allowbreak{} q\opRightPren{}} inserts \smath{x} into the queue \smath{q} +{\smath{\mbox{\bf enqueue!}\opLeftPren{}x, \allowbreak{} q\opRightPren{}} +inserts \smath{x} into the queue \smath{q} at the back end.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{enterPointData}}\opLeftPren{}{\it space}, \allowbreak{}{\it listOfPoints}\opRightPren{}% +\opdata{{\mbox{\axiomFun{enterPointData}}\opLeftPren{}{\it space}, +\allowbreak{}{\it listOfPoints}\opRightPren{}% }% }% {2}{(\$, List(Point(R)))->NonNegativeInteger}{ThreeSpace} -{\smath{\mbox{\bf enterPointData}\opLeftPren{}s, \allowbreak{} [p_0, \allowbreak{} p_1, \allowbreak{} \ldots, p_n]\opRightPren{}} adds a list of points +{\smath{\mbox{\bf enterPointData}\opLeftPren{}s, \allowbreak{} +[p_0, \allowbreak{} p_1, \allowbreak{} \ldots, p_n]\opRightPren{}} +adds a list of points from \smath{p_0} through \smath{p_n} to the \axiomType{ThreeSpace} \smath{s}, and returns the index of the start of the list.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{entry?}}\opLeftPren{}{\it value}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{entry?}}\opLeftPren{}{\it value}, \allowbreak{} +{\it aggregate}\opRightPren{}% }% }% {2}{(Entry, \$)->Boolean}{IndexedAggregate} @@ -71836,15 +74030,18 @@ where \smath{u} is an indexed aggregate tests if \smath{x} equals \smath{u . i} for some index \smath{i}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{epilogue}}\opLeftPren{}{\it formattedObject}\opRightPren{}% +\opdata{{\mbox{\axiomFun{epilogue}}\opLeftPren{} +{\it formattedObject}\opRightPren{}% }% }% {1}{(\$)->List(String)}{ScriptFormulaFormat} -{\smath{\mbox{\bf epilogue}\opLeftPren{}t\opRightPren{}} extracts the epilogue section of an IBM +{\smath{\mbox{\bf epilogue}\opLeftPren{}t\opRightPren{}} extracts the +epilogue section of an IBM SCRIPT Formula Format or \TeX{} formatted object \smath{t}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{eq}}\opLeftPren{}{\it sExpression}, \allowbreak{}{\it sExpression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{eq}}\opLeftPren{}{\it sExpression}, \allowbreak{} +{\it sExpression}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{SExpressionCategory} @@ -71853,34 +74050,43 @@ SCRIPT Formula Format or \TeX{} formatted object \smath{t}.} is \smath{true} in \Lisp{}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{eq?}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{eq?}}\opLeftPren{}{\it aggregate}, \allowbreak{} +{\it aggregate}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{Aggregate} -{\smath{\mbox{\bf eq?}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} tests if two aggregates \smath{u} and \smath{v} +{\smath{\mbox{\bf eq?}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} +tests if two aggregates \smath{u} and \smath{v} are same objects -in the \Language{} store.} +in the Axiom store.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{equality}}\opLeftPren{}{\it operator}, \allowbreak{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{equality}}\opLeftPren{} +{\it operator}, \allowbreak{}{\it function}\opRightPren{}% }% }% {2}{(\$, (\$, \$)->Boolean)->\$}{BasicOperator} -{\smath{\mbox{\bf equality}\opLeftPren{}op, \allowbreak{} f\opRightPren{}} attaches \smath{f} as the \mbox{\tt "\%equal?"} property to +{\smath{\mbox{\bf equality}\opLeftPren{}op, +\allowbreak{} f\opRightPren{}} attaches \smath{f} as +the \mbox{\tt "\%equal?"} property to \smath{op}. Argument \smath{f} must be a boolean-valued ``equality'' function defined on \axiomType{BasicOperator} objects. If \smath{op1} and \smath{op2} have the same name, and one of them has -an \mbox{\tt "\%equal?"} property \smath{f}, then \smath{f(op1, op2)} is called to +an \mbox{\tt "\%equal?"} property \smath{f}, then +\smath{f(op1, op2)} is called to decide whether \smath{op1} and \smath{op2} should be considered equal.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{equation}}\opLeftPren{}{\it expression}, \allowbreak{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{equation}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it expression}\opRightPren{}% }% }% {2}{(S, S)->\$}{Equation} -{\smath{\mbox{\bf equation}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} creates the equation \smath{a = b}. -\newitem\smath{\mbox{\bf equation}\opLeftPren{}v, \allowbreak{} a..b\opRightPren{}}, +{\smath{\mbox{\bf equation}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} creates the equation \smath{a = b}. +\newitem\smath{\mbox{\bf equation}\opLeftPren{}v, +\allowbreak{} a..b\opRightPren{}}, also written: \smath{v=a..b}, creates a segment binding value with variable \smath{v} and segment \smath{a..b}. @@ -71891,101 +74097,122 @@ segment \smath{a..b}. }% }% {1}{(\$)->\$}{LiouvillianFunctionCategory} -{\smath{\mbox{\bf erf}\opLeftPren{}x\opRightPren{}} returns the error function of \smath{x}: +{\smath{\mbox{\bf erf}\opLeftPren{}x\opRightPren{}} +returns the error function of \smath{x}: \smath{{2 \over \sqrt(\pi)}\int {exp^{-x^2} dx}}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{error}}\opLeftPren{}{\it string\opt{, string}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{error}}\opLeftPren{} +{\it string\opt{, string}}\opRightPren{}% }% }% {1}{(List(String))->Exit}{ErrorFunctions} -{\smath{\mbox{\bf error}\opLeftPren{}msg\opRightPren{}} displays error message \smath{msg} and terminates. +{\smath{\mbox{\bf error}\opLeftPren{}msg\opRightPren{}} +displays error message \smath{msg} and terminates. Argument \smath{msg} is either a string or a list of strings. -\newitem\smath{\mbox{\bf error}\opLeftPren{}name, \allowbreak{} msg\opRightPren{}} is similar except that +\newitem\smath{\mbox{\bf error}\opLeftPren{}name, +\allowbreak{} msg\opRightPren{}} is similar except that the error message is preceded by a message saying that the error occured in a function named \smath{name}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{euclideanGroebner}}\opLeftPren{}{\it ideal\opt{, string, string}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{euclideanGroebner}}\opLeftPren{} +{\it ideal\opt{, string, string}}\opRightPren{}% }% }% {1}{(List(Dpol))->List(Dpol)}{EuclideanGroebnerBasisPackage} -{\smath{\mbox{\bf euclideanGroebner}\opLeftPren{}lp\optinner{, "info", "redcrit}\opRightPren{}} computes a Gr\"obner basis for a polynomial ideal over a Euclidean domain generated by the list of polynomials \smath{lp}. +{\smath{\mbox{\bf euclideanGroebner}\opLeftPren{}lp\optinner{, +"info", "redcrit}\opRightPren{}} computes a Gr\"obner basis for a polynomial +ideal over a Euclidean domain generated by the list of polynomials \smath{lp}. If the string \mbox{\tt "info"} is given as a second argument, a summary is given of the critical pairs. If the string \mbox{\tt "redcrit"} is given as a third argument, the critical pairs are printed.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{euclideanNormalForm}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it groebnerBasis}\opRightPren{}% +\opdata{{\mbox{\axiomFun{euclideanNormalForm}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it groebnerBasis}\opRightPren{}% }% }% {2}{(Dpol, List(Dpol))->Dpol}{EuclideanGroebnerBasisPackage} -{\smath{\mbox{\bf euclideanNormalForm}\opLeftPren{}poly, \allowbreak{} gb\opRightPren{}} reduces the polynomial +{\smath{\mbox{\bf euclideanNormalForm}\opLeftPren{}poly, +\allowbreak{} gb\opRightPren{}} reduces the polynomial \smath{poly} modulo the precomputed Gr\"obner basis \smath{gb} giving a canonical representative of the residue class.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{euclideanSize}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{euclideanSize}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{EuclideanDomain} -{\smath{\mbox{\bf euclideanSize}\opLeftPren{}x\opRightPren{}} returns the Euclidean size of the +{\smath{\mbox{\bf euclideanSize}\opLeftPren{}x\opRightPren{}} +returns the Euclidean size of the element \smath{x}, or calls \spadfun{error} if \smath{x} is zero. } - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{eulerPhi}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{eulerPhi}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf eulerPhi}\opLeftPren{}n\opRightPren{}} returns the number of integers between 1 and +{\smath{\mbox{\bf eulerPhi}\opLeftPren{}n\opRightPren{}} +returns the number of integers between 1 and \smath{n} (including 1) which are relatively prime to \smath{n}. This is the Euler phi function \smath{\phi(n)}, also called the totient function.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{euler}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{euler}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf euler}\opLeftPren{}n\opRightPren{}} returns the \eth{\smath{n}} Euler number. +{\smath{\mbox{\bf euler}\opLeftPren{}n\opRightPren{}} +returns the \eth{\smath{n}} Euler number. This is \smath{2^n E(n, 1/2)}, where \smath{E(n, x)} is the \eth{\smath{n}} Euler polynomial.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{eval}}\opLeftPren{}{\it expression\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{eval}}\opLeftPren{} +{\it expression\opt{, options}}\opRightPren{}% }% }% {1}{(\$)->\$}{FunctionSpace} {\opkey{Many domains have forms of the \spadfun{eval} defined. Here are some the most common forms.} \newitem -\smath{\mbox{\bf eval}\opLeftPren{}f\opRightPren{}} unquotes all the quoted operators in \smath{f}. +\smath{\mbox{\bf eval}\opLeftPren{}f\opRightPren{}} unquotes all the +quoted operators in \smath{f}. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} x = v\opRightPren{}} replaces symbol or expression \smath{x} by +\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} x = v\opRightPren{}} +replaces symbol or expression \smath{x} by \smath{v} in \smath{f}; if \smath{x} is an expression, it must be retractable to a single \axiomType{Kernel}. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} [x_1 = v_1, \allowbreak{} \ldots, x_n = v_n]\opRightPren{}} returns \smath{f} +\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} [x_1 = v_1, +\allowbreak{} \ldots, x_n = v_n]\opRightPren{}} returns \smath{f} with symbols or expressions \smath{x_i} replaced by \smath{v_i} in parallel; if \smath{x_i} is an expression, it must be retractable to a single \axiomType{Kernel}. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} [x_1, \allowbreak{} \ldots, x_n]\opRightPren{}} unquotes all +\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} [x_1, \allowbreak{} +\ldots, x_n]\opRightPren{}} unquotes all the quoted operations in \smath{f} whose name is one of the \smath{x_i}.'s. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} x\opRightPren{}} unquotes all quoted operators in \smath{f} +\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} x\opRightPren{}} +unquotes all quoted operators in \smath{f} whose name is \smath{x}. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}e, \allowbreak{} s, \allowbreak{} f\opRightPren{}} +\smath{\mbox{\bf eval}\opLeftPren{}e, \allowbreak{} s, +\allowbreak{} f\opRightPren{}} replaces every subexpression of \smath{e} of the form \smath{s(a_1, \dots, a_n)} by \smath{f(a_1, \ldots, a_n)}. The function \smath{f} can have type @@ -71993,7 +74220,8 @@ The function \smath{f} can have type otherwise \smath{f} must have signature \spadsig{List(Expression)}{Expression}. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}e, \allowbreak{} [s_1, \allowbreak{} \ldots, s_n], [f_1, \ldots, f_n]\opRightPren{}}, +\smath{\mbox{\bf eval}\opLeftPren{}e, \allowbreak{} [s_1, +\allowbreak{} \ldots, s_n], [f_1, \ldots, f_n]\opRightPren{}}, replaces every subexpression of \smath{e} of the form \smath{s_i(a_1, \dots, a_{n_i})} by \smath{f_i(a_1, \ldots, a_{n_i})}. If all the \smath{s_i}'s are unary operators, the functions @@ -72001,14 +74229,19 @@ If all the \smath{s_i}'s are unary operators, the functions otherwise, the \smath{f_i} must have signature \spadsig{List(Expression)}{Expression}. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}p, \allowbreak{} el\opRightPren{}}, where \smath{p} is a permutation, +\smath{\mbox{\bf eval}\opLeftPren{}p, \allowbreak{} el\opRightPren{}}, +where \smath{p} is a permutation, returns the image of element {\it el} under \smath{p}. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}s\opRightPren{}}, where \spad{s} is of type \axiomType{SymmetricPolynomial} with -rational number coefficients, returns the sum of the coefficients of a cycle index. +\smath{\mbox{\bf eval}\opLeftPren{}s\opRightPren{}}, +where \spad{s} is of type \axiomType{SymmetricPolynomial} with +rational number coefficients, returns the sum of the +coefficients of a cycle index. See \axiomType{CycleIndicators} for details. \newitem -\smath{\mbox{\bf eval}\opLeftPren{}f, \allowbreak{} s\opRightPren{}}, where \spad{s} is of type \axiomType{SymmetricPolynomial} +\smath{\mbox{\bf eval}\opLeftPren{}f, +\allowbreak{} s\opRightPren{}}, where \spad{s} is of type +\axiomType{SymmetricPolynomial} with rational number coefficients and \spad{f} is a function of type \spadsig{Integer}{Algebra Fraction Integer}, evaluates the cycle index s by applying @@ -72018,15 +74251,18 @@ See \axiomType{EvaluateCycleIndicators} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{evaluate}}\opLeftPren{}{\it operator}, \allowbreak{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{evaluate}}\opLeftPren{} +{\it operator}, \allowbreak{}{\it function}\opRightPren{}% }% }% {2}{(BasicOperator, (A)->A)->BasicOperator}{BasicOperatorFunctions1} -{\smath{\mbox{\bf evaluate}\opLeftPren{}op\opRightPren{}} returns the value of the \mbox{\tt "\%eval"} property +{\smath{\mbox{\bf evaluate}\opLeftPren{}op\opRightPren{}} +returns the value of the \mbox{\tt "\%eval"} property of \axiomType{BasicOperator} object \smath{op} if it has one, and \mbox{\tt "failed"} otherwise. \newitem -\smath{\mbox{\bf evaluate}\opLeftPren{}op, \allowbreak{} f\opRightPren{}} attaches \smath{f} as the \mbox{\tt "\%eval"} +\smath{\mbox{\bf evaluate}\opLeftPren{}op, +\allowbreak{} f\opRightPren{}} attaches \smath{f} as the \mbox{\tt "\%eval"} property of \smath{op}. If \smath{op} has an \mbox{\tt "\%eval"} property \smath{f}, then applying \smath{op} to a @@ -72047,23 +74283,28 @@ in \smath{R} and integer \smath{n > 0}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{even?}}\opLeftPren{}{\it integerNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{even?}}\opLeftPren{} +{\it integerNumber}\opRightPren{}% }% }% {1}{(\$)->Boolean}{IntegerNumberSystem} -{\smath{\mbox{\bf even?}\opLeftPren{}n\opRightPren{}} tests if integer \smath{n} is even. +{\smath{\mbox{\bf even?}\opLeftPren{}n\opRightPren{}} +tests if integer \smath{n} is even. \newitem -\smath{\mbox{\bf even?}\opLeftPren{}p\opRightPren{}} tests if permutation \smath{p} is an even +\smath{\mbox{\bf even?}\opLeftPren{}p\opRightPren{}} +tests if permutation \smath{p} is an even permutation, that is, that the \smath{\mbox{\bf sign}\opLeftPren{}p) = 1}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{every?}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{every?}}\opLeftPren{}{\it predicate}, +\allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->Boolean}{HomogeneousAggregate} -{\smath{\mbox{\bf every?}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} tests if {\it pred(x)} is \smath{true} +{\smath{\mbox{\bf every?}\opLeftPren{}pred, +\allowbreak{} u\opRightPren{}} tests if {\it pred(x)} is \smath{true} for all elements \smath{x} of \smath{u}.} % ---------------------------------------------------------------------- @@ -72071,7 +74312,8 @@ for all elements \smath{x} of \smath{u}.} }% }% {1}{(\$)->Boolean}{FileNameCategory} -{\smath{\mbox{\bf exists?}\opLeftPren{}f\opRightPren{}} tests if the file \smath{f} exists in the file +{\smath{\mbox{\bf exists?}\opLeftPren{}f\opRightPren{}} +tests if the file \smath{f} exists in the file system.} % ---------------------------------------------------------------------- @@ -72080,9 +74322,11 @@ system.} }% }% {1}{(\$)->\$}{ElementaryFunctionCategory} -{\smath{\mbox{\bf exp}\opLeftPren{}x\opRightPren{}} returns {\tt \%e} to the power \smath{x}. +{\smath{\mbox{\bf exp}\opLeftPren{}x\opRightPren{}} +returns {\tt \%e} to the power \smath{x}. \newitem -\smath{\mbox{\bf expIfCan}\opLeftPren{}z\opRightPren{}} returns exp(\smath{z}) if possible, and +\smath{\mbox{\bf expIfCan}\opLeftPren{}z\opRightPren{}} +returns exp(\smath{z}) if possible, and \mbox{\tt "failed"} otherwise. } @@ -72091,7 +74335,8 @@ system.} }% }% {0}{()->\$}{Float} -{\smath{\mbox{\bf exp1}\opLeftPren{}\opRightPren{}\$R} returns exp 1: \smath{2.7182818284\ldots} either +{\smath{\mbox{\bf exp1}\opLeftPren{}\opRightPren{}\$R} returns exp 1: +\smath{2.7182818284\ldots} either a float or a small float according to whether \smath{R=} \axiomType{Float} or \smath{R=} \axiomType{DoubleFloat}. } @@ -72101,7 +74346,8 @@ or \smath{R=} \axiomType{DoubleFloat}. }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf expand}\opLeftPren{}f\opRightPren{}} performs the following expansions +{\smath{\mbox{\bf expand}\opLeftPren{}f\opRightPren{}} performs the +following expansions on \axiomType{Expression} \smath{f:} \begin{simpleList} \item Logs of products are expanded into sums of logs. @@ -72121,58 +74367,72 @@ where \smath{lseg} is a list of segments, returns a list with all segments expanded. For example, \code{expand [1..4, 7..9] = [1, 2, 3, 4, 7, 8, 9]}. \newitem -\smath{\mbox{\bf expand}\opLeftPren{}l..h \mbox{ \tt by } k\opRightPren{}} returns +\smath{\mbox{\bf expand}\opLeftPren{}l..h \mbox{ \tt by } k\opRightPren{}} +returns a list of explicit elements. For example, \code{expand(1..5 by 2) = [1, 3, 5]}. \newitem -\smath{\mbox{\bf expand}\opLeftPren{}f\opRightPren{}} returns an unfactored form of factored object \smath{f}. +\smath{\mbox{\bf expand}\opLeftPren{}f\opRightPren{}} +returns an unfactored form of factored object \smath{f}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{expandLog}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{expandLog}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf expandLog}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf log}\opLeftPren{}a/b\opRightPren{}} +{\smath{\mbox{\bf expandLog}\opLeftPren{}f\opRightPren{}} +converts every \smath{\mbox{\bf log}\opLeftPren{}a/b\opRightPren{}} appearing in \axiomType{Expression} \smath{f} into \smath{\log(a) - \log(b)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{expandPower}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{expandPower}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf expandPower}\opLeftPren{}f\opRightPren{}} converts every power \smath{(a/b)^c} appearing +{\smath{\mbox{\bf expandPower}\opLeftPren{}f\opRightPren{}} +converts every power \smath{(a/b)^c} appearing in \axiomType{Expression} \smath{f} into \smath{a^c b^{-c}}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{explicitEntries?}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{explicitEntries?}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(\$)->Boolean}{LazyStreamAggregate} -{\smath{\mbox{\bf explicitEntries?}\opLeftPren{}s\opRightPren{}} tests if the stream \smath{s} has +{\smath{\mbox{\bf explicitEntries?}\opLeftPren{}s\opRightPren{}} +tests if the stream \smath{s} has explicitly computed entries. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{explicitlyEmpty?}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{explicitlyEmpty?}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(\$)->Boolean}{LazyStreamAggregate} -{\smath{\mbox{\bf explicitlyEmpty?}\opLeftPren{}s\opRightPren{}} tests if the stream is an (explicitly) empty stream. +{\smath{\mbox{\bf explicitlyEmpty?}\opLeftPren{}s\opRightPren{}} +tests if the stream is an (explicitly) empty stream. Note: this is a null test which will not cause lazy evaluation.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{explicitlyFinite?}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{explicitlyFinite?}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(\$)->Boolean}{StreamAggregate} -{\smath{\mbox{\bf explicitlyFinite?}\opLeftPren{}s\opRightPren{}} tests if the stream \smath{s} has a finite -number of elements. Note: for many datatypes, \code{explicitlyFinite?(s) = not possiblyInfinite?(s)}.} +{\smath{\mbox{\bf explicitlyFinite?}\opLeftPren{}s\opRightPren{}} +tests if the stream \smath{s} has a finite +number of elements. Note: for many datatypes, +\code{explicitlyFinite?(s) = not possiblyInfinite?(s)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{exponent}}\opLeftPren{}{\it floatOrFactored}\opRightPren{}% +\opdata{{\mbox{\axiomFun{exponent}}\opLeftPren{} +{\it floatOrFactored}\opRightPren{}% }% }% {1}{(\$)->Integer}{FloatingPointSystem} @@ -72180,52 +74440,69 @@ number of elements. Note: for many datatypes, \code{explicitlyFinite?(s) = not p \spadfunFrom{exponent}{FloatingPointSystem} part of a float or small float \smath{fl}. \newitem -\smath{\mbox{\bf exponent}\opLeftPren{}u\opRightPren{}}, where \smath{u} is a factored object, returns the +\smath{\mbox{\bf exponent}\opLeftPren{}u\opRightPren{}}, +where \smath{u} is a factored object, returns the exponent of the -first factor of \smath{u}, or 0 if the factored object consists solely of a unit. +first factor of \smath{u}, or 0 if the factored object +consists solely of a unit. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{expressIdealMember}}\opLeftPren{}{\it listOfIdeals}, \allowbreak{}{\it ideal}\opRightPren{}% +\opdata{{\mbox{\axiomFun{expressIdealMember}}\opLeftPren{} +{\it listOfIdeals}, \allowbreak{}{\it ideal}\opRightPren{}% }% }% {2}{(List(\$), \$)->Union(List(\$), "failed")}{PrincipalIdealDomain} -{\smath{\mbox{\bf expressIdealMember}\opLeftPren{}[f_1, \allowbreak{} \ldots, f_n], h\opRightPren{}} returns a representation of ideal \smath{h} as a linear combination of the ideals \smath{f_i} or \mbox{\tt "failed"} if \smath{h} is not in the ideal generated by the \smath{f_i}.} +{\smath{\mbox{\bf expressIdealMember}\opLeftPren{}[f_1, \allowbreak{} +\ldots, f_n], h\opRightPren{}} returns a representation of ideal \smath{h} +as a linear combination of the ideals \smath{f_i} or \mbox{\tt "failed"} +if \smath{h} is not in the ideal generated by the \smath{f_i}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{exptMod}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it nonNegativeInteger}, \allowbreak{}{\it polynomial}\allowbreak $\,[$ , \allowbreak{}{\it prime}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{exptMod}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it nonNegativeInteger}, +\allowbreak{}{\it polynomial} +\allowbreak $\,[$ , \allowbreak{}{\it prime}$]$\opRightPren{}% }% }% {3}{(FP, NonNegativeInteger, FP)->FP}{DistinctDegreeFactorize} -{\smath{\mbox{\bf exptMod}\opLeftPren{}u, \allowbreak{} k, \allowbreak{} v\optinner{, p}\opRightPren{}} -raises the polynomial \smath{u} to the \eth{\smath{k}} power modulo the polynomial \smath{v}. +{\smath{\mbox{\bf exptMod}\opLeftPren{}u, \allowbreak{} k, +\allowbreak{} v\optinner{, p}\opRightPren{}} +raises the polynomial \smath{u} to the \eth{\smath{k}} +power modulo the polynomial \smath{v}. If a prime \smath{p} is given, the power is also computed modulo that prime. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{exquo}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{exquo}}\opLeftPren{} +{\it element}, \allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(\$, R)->Union(\$, "failed")}{ComplexCategory} -{\smath{\mbox{\bf exquo}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} either returns an element \smath{c} such that +{\smath{\mbox{\bf exquo}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} either returns an element \smath{c} such that \smath{cb=a} or \mbox{\tt "failed"} if no such element can be found. Values \smath{a} and \smath{b} are members of a domain of category \axiomType{IntegralDomain}. -\newitem\smath{\mbox{\bf exquo}\opLeftPren{}A, \allowbreak{} r\opRightPren{}} returns the exact quotient of the +\newitem\smath{\mbox{\bf exquo}\opLeftPren{}A, +\allowbreak{} r\opRightPren{}} returns the exact quotient of the elements of matrix \smath{A} by coefficient \smath{r}, or calls \spadfun{error} if this is not possible. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{extend}}\opLeftPren{}{\it stream}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{extend}}\opLeftPren{} +{\it stream}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, Integer)->\$}{LazyStreamAggregate} -{\smath{\mbox{\bf extend}\opLeftPren{}ps, \allowbreak{} n\opRightPren{}}, where \smath{ps} is a power series, +{\smath{\mbox{\bf extend}\opLeftPren{}ps, +\allowbreak{} n\opRightPren{}}, where \smath{ps} is a power series, causes all terms of \smath{ps} of degree \smath{\leq n} to be computed. -\newitem\smath{\mbox{\bf extend}\opLeftPren{}st, \allowbreak{} n\opRightPren{}}, where \smath{st} is a stream, causes +\newitem\smath{\mbox{\bf extend}\opLeftPren{}st, +\allowbreak{} n\opRightPren{}}, where \smath{st} is a stream, causes entries to be computed so that \smath{st} has at least \smath{n} explicit entries, or so that all entries of \smath{st} are finite with @@ -72233,13 +74510,16 @@ length \smath{\leq n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{extendedEuclidean}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\allowbreak $\,[$ , \allowbreak{}{\it element}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{extendedEuclidean}}\opLeftPren{} +{\it element}, \allowbreak{}{\it element}\allowbreak $\,[$ , \allowbreak{} +{\it element}$]$\opRightPren{}% }% }% {2}{(\$, \$)->Record(coef1:\$, coef2:\$, generator:\$)}{EuclideanDomain} {\opkey{Argments \smath{x}, \smath{y}, and \smath{z} are members of a domain of category \axiomType{EuclideanDomain}.} -\newitem\smath{\mbox{\bf extendedEuclidean}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns a record +\newitem\smath{\mbox{\bf extendedEuclidean}\opLeftPren{}x, +\allowbreak{} y\opRightPren{}} returns a record \smath{rec} containing three fields: \smath{coef1}, \smath{coef2}, and \smath{generator} where \smath{rec.coef1*x+rec.coef2*y = rec.generator} and \smath{rec.generator} is a \smath{gcd} of @@ -72248,7 +74528,8 @@ The \smath{gcd} is unique only up to associates if \spadatt{canonicalUnitNormal} is not asserted. Note: See \spadfun{principalIdeal} for a version of this operation which accepts an arbitrary length list of arguments. -\newitem\smath{\mbox{\bf extendedEuclidean}\opLeftPren{}x, \allowbreak{} y, \allowbreak{} z\opRightPren{}} either returns a record +\newitem\smath{\mbox{\bf extendedEuclidean}\opLeftPren{}x, +\allowbreak{} y, \allowbreak{} z\opRightPren{}} either returns a record \smath{rec} of two fields \smath{coef1} and \smath{coef2} where \smath{rec.coef1*x+rec.coef2*y=z}, and \mbox{\tt "failed"} if \smath{z} cannot be expressed as such a linear combination of \smath{x} and @@ -72256,11 +74537,17 @@ cannot be expressed as such a linear combination of \smath{x} and } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{extendedIntegrate}}\opLeftPren{}{\it rationalFnct}, \allowbreak{}{\it symbol}, \allowbreak{}{\it rationalFnct}\opRightPren{}% +\opdata{{\mbox{\axiomFun{extendedIntegrate}}\opLeftPren{} +{\it rationalFnct}, \allowbreak{}{\it symbol}, \allowbreak{}{\it +rationalFnct}\opRightPren{}% }% }% -{3}{(Fraction(Polynomial(F)), Symbol, Fraction(Polynomial(F)))->Union(Record(ratpart:Fraction(Polynomial(F)), coeff:Fraction(Polynomial(F))), "failed")}{RationalFunctionIntegration} -{\smath{\mbox{\bf extendedIntegrate}\opLeftPren{}f, \allowbreak{} x, \allowbreak{} g\opRightPren{}} returns fractions \smath{[h, +{3}{(Fraction(Polynomial(F)), Symbol, Fraction(Polynomial(F)))-> +Union(Record(ratpart:Fraction(Polynomial(F)), +coeff:Fraction(Polynomial(F))), "failed")} +{RationalFunctionIntegration} +{\smath{\mbox{\bf extendedIntegrate}\opLeftPren{}f, \allowbreak{} x, +\allowbreak{} g\opRightPren{}} returns fractions \smath{[h, c]} such that \smath{dc/dx = 0} and \smath{dh/dx = f - cg} if \smath{(h, c)} exist, and \mbox{\tt "failed"} otherwise.} @@ -72269,7 +74556,8 @@ c]} such that \smath{dc/dx = 0} and \smath{dh/dx = f - cg} if }% }% {0}{()->OnePointCompletion(PositiveInteger)}{ExtensionField} -{\smath{\mbox{\bf extensionDegree}\opLeftPren{}\opRightPren{}\$F} returns the degree of the field extension \smath{F} +{\smath{\mbox{\bf extensionDegree}\opLeftPren{}\opRightPren{}\$F} +returns the degree of the field extension \smath{F} if the extension is algebraic, and {\tt infinity} if it is not.} % ---------------------------------------------------------------------- @@ -72277,7 +74565,8 @@ if the extension is algebraic, and {\tt infinity} if it is not.} }% }% {1}{(\$)->String}{FileNameCategory} -{\smath{\mbox{\bf extension}\opLeftPren{}fn\opRightPren{}} returns the type part of the file name +{\smath{\mbox{\bf extension}\opLeftPren{}fn\opRightPren{}} +returns the type part of the file name \smath{fn} as a string.} % ---------------------------------------------------------------------- @@ -72285,15 +74574,18 @@ if the extension is algebraic, and {\tt infinity} if it is not.} }% }% {1}{(\$)->S}{BagAggregate} -{\smath{\mbox{\bf extract!}\opLeftPren{}bg\opRightPren{}} destructively removes a (random) item from +{\smath{\mbox{\bf extract!}\opLeftPren{}bg\opRightPren{}} +destructively removes a (random) item from bag \smath{bg}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{extractBottom!}}\opLeftPren{}{\it dequeue}\opRightPren{}% +\opdata{{\mbox{\axiomFun{extractBottom!}}\opLeftPren{} +{\it dequeue}\opRightPren{}% }% }% {1}{(\$)->S}{DequeueAggregate} -{\smath{\mbox{\bf extractBottom!}\opLeftPren{}d\opRightPren{}} destructively extracts the bottom +{\smath{\mbox{\bf extractBottom!}\opLeftPren{}d\opRightPren{}} +destructively extracts the bottom (back) element from the dequeue \smath{d}, or calls \spadfun{error} if \smath{d} is empty.} @@ -72302,7 +74594,8 @@ bag \smath{bg}.} }% }% {1}{(\$)->S}{DequeueAggregate} -{\smath{\mbox{\bf extractTop!}\opLeftPren{}d\opRightPren{}} destructively extracts the top (front) +{\smath{\mbox{\bf extractTop!}\opLeftPren{}d\opRightPren{}} +destructively extracts the top (front) element from the dequeue \smath{d}, or calls \spadfun{error} if \smath{d} is empty.} @@ -72311,22 +74604,27 @@ dequeue \smath{d}, or calls \spadfun{error} if \smath{d} is empty.} }% }% {1}{(PositiveInteger)->\$}{CliffordAlgebra} -{\smath{\mbox{\bf e}\opLeftPren{}n\opRightPren{}} produces the appropriate unit element of a +{\smath{\mbox{\bf e}\opLeftPren{}n\opRightPren{}} +produces the appropriate unit element of a \axiomType{CliffordAlgebra}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factor}}\opLeftPren{}{\it polynomial\opt{, numbers}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factor}}\opLeftPren{} +{\it polynomial\opt{, numbers}}\opRightPren{}% }% }% {1}{(\$)->Factored(\$)}{UniqueFactorizationDomain} -{\smath{\mbox{\bf factor}\opLeftPren{}x\opRightPren{}} returns the factorization of \smath{x} into +{\smath{\mbox{\bf factor}\opLeftPren{}x\opRightPren{}} +returns the factorization of \smath{x} into irreducibles, where \smath{x} is a member of any domain of category \axiomType{UniqueFactorizationDomain}. -\newitem\smath{\mbox{\bf factor}\opLeftPren{}p, \allowbreak{} lan\opRightPren{}}, where \smath{p} is a polynomial +\newitem\smath{\mbox{\bf factor}\opLeftPren{}p, +\allowbreak{} lan\opRightPren{}}, where \smath{p} is a polynomial and \smath{lan} is a list of algebraic numbers, factors \smath{p} over the extension generated by the algebraic numbers given by the list \smath{lan}. -\newitem\smath{\mbox{\bf factor}\opLeftPren{}upoly, \allowbreak{} prime\opRightPren{}}, where \smath{upoly} is a +\newitem\smath{\mbox{\bf factor}\opLeftPren{}upoly, +\allowbreak{} prime\opRightPren{}}, where \smath{upoly} is a univariate polynomial and \smath{prime} is a prime integer, returns the list of factors of \smath{upoly} modulo the integer prime \smath{p}, or calls \spadfun{error} if \smath{upoly} is not @@ -72334,58 +74632,70 @@ square-free modulo \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factorFraction}}\opLeftPren{}{\it fraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factorFraction}}\opLeftPren{} +{\it fraction}\opRightPren{}% }% }% -{1}{(Fraction(Polynomial(R)))->Fraction(Factored(Polynomial(R)))}{RationalFunctionFactorizer} -{\smath{\mbox{\bf factorFraction}\opLeftPren{}r\opRightPren{}} factors the numerator and the +{1}{(Fraction(Polynomial(R)))->Fraction(Factored(Polynomial(R)))} +{RationalFunctionFactorizer} +{\smath{\mbox{\bf factorFraction}\opLeftPren{}r\opRightPren{}} +factors the numerator and the denominator of the polynomial fraction \smath{r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factorGroebnerBasis}}\opLeftPren{}{\it listOfPolynomials\opt{, boolean}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factorGroebnerBasis}}\opLeftPren{} +{\it listOfPolynomials\opt{, boolean}}\opRightPren{}% }% }% {1}{(List(Dpol))->List(List(Dpol))}{GroebnerFactorizationPackage} -{\smath{\mbox{\bf factorGroebnerBasis}\opLeftPren{}basis\optinner{, flag}\opRightPren{}} checks +{\smath{\mbox{\bf factorGroebnerBasis}\opLeftPren{}basis +\optinner{, flag}\opRightPren{}} checks whether the \smath{basis} contains reducible polynomials and uses these to split the \smath{basis}. Information about partial results is given if a second argument of \smath{true} is given. } - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factorials}}\opLeftPren{}{\it expression\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factorials}}\opLeftPren{} +{\it expression\opt{, symbol}}\opRightPren{}% }% }% {1}{(\$)->\$}{CombinatorialOpsCategory} -{\smath{\mbox{\bf factorials}\opLeftPren{}f\optinner{, x}\opRightPren{}} rewrites the permutations and +{\smath{\mbox{\bf factorials}\opLeftPren{}f +\optinner{, x}\opRightPren{}} rewrites the permutations and binomials in \smath{f} in terms of factorials. If a symbol \smath{x} is given as a second argument, the operation rewrites only those terms involving \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factorial}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factorial}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(I)->I}{IntegerCombinatoricFunctions} -{\smath{\mbox{\bf factorial}\opLeftPren{}n\opRightPren{}}, where \smath{n} is an integer, returns the +{\smath{\mbox{\bf factorial}\opLeftPren{}n\opRightPren{}}, +where \smath{n} is an integer, returns the integer value of \smath{n! = \prod\nolimits_1^n{i}}. -\newitem \smath{\mbox{\bf factorial}\opLeftPren{}n\opRightPren{}}, where n is an expression, returns a +\newitem \smath{\mbox{\bf factorial}\opLeftPren{}n\opRightPren{}}, +where n is an expression, returns a formal expression denoting \smath{n!} Note: \smath{n! = n (n-1)!} when \smath{n > 0}; also, \smath{0! = 1}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factorList}}\opLeftPren{}{\it factoredForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factorList}}\opLeftPren{} +{\it factoredForm}\opRightPren{}% }% }% -{1}{(\$)->List(Record(flg:Union("nil", "sqfr", "irred", "prime"), fctr:R, xpnt:Integer))}{Factored} -{\smath{\mbox{\bf factorList}\opLeftPren{}f\opRightPren{}}, for a factored form \smath{f}, returns +{1}{(\$)->List(Record(flg:Union("nil", "sqfr", "irred", "prime"), +fctr:R, xpnt:Integer))}{Factored} +{\smath{\mbox{\bf factorList}\opLeftPren{}f\opRightPren{}}, +for a factored form \smath{f}, returns list of records. Each record corresponds to a factor of \smath{f} and has three fields: \smath{flg}, \smath{fctr}, and \smath{xpnt}. @@ -72395,19 +74705,24 @@ The \smath{flg} is one of the strings: \mbox{\tt "nil"}, \mbox{\tt "sqfr"}, } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factorPolynomial}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factorPolynomial}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% -{1}{(SparseUnivariatePolynomial(\$))->Factored(SparseUnivariatePolynomial(\$))}{PolynomialFactorizationExplicit} -{\smath{\mbox{\bf factorPolynomial}\opLeftPren{}p\opRightPren{}} returns the factorization +{1}{(SparseUnivariatePolynomial(\$))-> +Factored(SparseUnivariatePolynomial(\$))}{PolynomialFactorizationExplicit} +{\smath{\mbox{\bf factorPolynomial}\opLeftPren{}p\opRightPren{}} +returns the factorization of a sparse univariate polynomial \smath{p} as a factored form.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factors}}\opLeftPren{}{\it factoredForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factors}}\opLeftPren{} +{\it factoredForm}\opRightPren{}% }% }% {1}{(\$)->List(Record(factor:R, exponent:Integer))}{Factored} -{\smath{\mbox{\bf factors}\opLeftPren{}u\opRightPren{}} returns a list of the factors of a factored +{\smath{\mbox{\bf factors}\opLeftPren{}u\opRightPren{}} returns a list of +the factors of a factored form \smath{u} in a form as a list suitable for iteration. Each element in the list is a record containing both a \smath{factor} and \smath{exponent} field. @@ -72418,24 +74733,30 @@ Each element in the list is a record containing both a }% }% {0}{()->List(Record(factor:Integer, exponent:Integer))}{FiniteFieldCategory} -{\smath{\mbox{\bf factorsOfCyclicGroupSize}\opLeftPren{}\opRightPren{}} returns the factorization of +{\smath{\mbox{\bf factorsOfCyclicGroupSize}\opLeftPren{}\opRightPren{}} +returns the factorization of \smath{\mbox{\bf size}\opLeftPren{})-1}} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{factorSquareFreePolynomial}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{factorSquareFreePolynomial}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% -{1}{(SparseUnivariatePolynomial(\$))->Factored(SparseUnivariatePolynomial(\$))}{PolynomialFactorizationExplicit} -{\smath{\mbox{\bf factorSquareFreePolynomial}\opLeftPren{}p\opRightPren{}} factors the univariate +{1}{(SparseUnivariatePolynomial(\$))-> +Factored(SparseUnivariatePolynomial(\$))}{PolynomialFactorizationExplicit} +{\smath{\mbox{\bf factorSquareFreePolynomial}\opLeftPren{}p\opRightPren{}} +factors the univariate polynomial \smath{p} into irreducibles, where \smath{p} is known to be square free and primitive with respect to its main variable.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fibonacci}}\opLeftPren{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fibonacci}}\opLeftPren{} +{\it nonNegativeInteger}\opRightPren{}% }% }% {1}{(Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf fibonacci}\opLeftPren{}n\opRightPren{}} returns the \eth{\smath{n}} Fibonacci +{\smath{\mbox{\bf fibonacci}\opLeftPren{}n\opRightPren{}} returns +the \eth{\smath{n}} Fibonacci number. The Fibonacci numbers \smath{F[n]} are defined by \smath{F[0] = F[1] = 1} and \smath{F[n] = F[n-1] + F[n-2]}. @@ -72443,64 +74764,81 @@ The algorithm has running time \smath{O(\log(n)^3)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{filename}}\opLeftPren{}{\it directory}, \allowbreak{}{\it name}, \allowbreak{}{\it extension}\opRightPren{}% +\opdata{{\mbox{\axiomFun{filename}}\opLeftPren{}{\it directory}, +\allowbreak{}{\it name}, \allowbreak{}{\it extension}\opRightPren{}% }% }% {3}{(String, String, String)->\$}{FileNameCategory} -{\smath{\mbox{\bf filename}\opLeftPren{}d, \allowbreak{} n, \allowbreak{} e\opRightPren{}} creates a file name with string +{\smath{\mbox{\bf filename}\opLeftPren{}d, \allowbreak{} n, +\allowbreak{} e\opRightPren{}} creates a file name with string \smath{d} as its directory, string \smath{n} as its name and string \smath{e} as its extension. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fill!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it value}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fill!}}\opLeftPren{}{\it aggregate}, +\allowbreak{}{\it value}\opRightPren{}% }% }% {2}{(\$, Entry)->\$}{IndexedAggregate} -{\smath{\mbox{\bf fill!}\opLeftPren{}a, \allowbreak{} x\opRightPren{}} replaces each entry in aggregate \smath{a} by +{\smath{\mbox{\bf fill!}\opLeftPren{}a, \allowbreak{} x\opRightPren{}} +replaces each entry in aggregate \smath{a} by \smath{x}. The modified \smath{a} is returned. If \smath{a} is a domain of category \smath{TwoDimensionalArrayCategory} such as a matrix, -\smath{\mbox{\bf fill!}\opLeftPren{}a, \allowbreak{} x\opRightPren{}} sets every element of \smath{a} to \smath{x}. +\smath{\mbox{\bf fill!}\opLeftPren{}a, \allowbreak{} x\opRightPren{}} +sets every element of \smath{a} to \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{filterUntil}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{filterUntil}}\opLeftPren{} +{\it predicate}, \allowbreak{}{\it stream}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->\$}{InfiniteTuple} -{\smath{\mbox{\bf filterUntil}\opLeftPren{}p, \allowbreak{} s\opRightPren{}} returns \smath{[x_0, x_1, \ldots, x_n]}, -where stream \smath{s = [x_0, x_1, x_2, ..]} and \smath{n} is the smallest index +{\smath{\mbox{\bf filterUntil}\opLeftPren{}p, +\allowbreak{} s\opRightPren{}} returns \smath{[x_0, x_1, \ldots, x_n]}, +where stream \smath{s = [x_0, x_1, x_2, ..]} and +\smath{n} is the smallest index such that \smath{p(x_n) = true}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{filterWhile}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{filterWhile}}\opLeftPren{} +{\it predicate}, \allowbreak{}{\it stream}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->\$}{Stream} -{\smath{\mbox{\bf filterWhile}\opLeftPren{}pred, \allowbreak{} s\opRightPren{}} returns \smath{[x_0, x_1, \ldots, +{\smath{\mbox{\bf filterWhile}\opLeftPren{}pred, +\allowbreak{} s\opRightPren{}} returns \smath{[x_0, x_1, \ldots, x_{(n-1)}]} where \smath{s = [x_0, x_1, x_2, ..]} and \smath{n} is the smallest index such that \smath{p(x_n) = false}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{find}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{find}}\opLeftPren{}{\it predicate}, +\allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->Union(S, "failed")}{Collection} -{\smath{\mbox{\bf find}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} returns the first \smath{x} in \smath{u} -such that \smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} is \smath{true}, and \mbox{\tt "failed"} +{\smath{\mbox{\bf find}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} +returns the first \smath{x} in \smath{u} +such that \smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} +is \smath{true}, and \mbox{\tt "failed"} if no such \smath{x} exists.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{findCycle}}\opLeftPren{}{\it nonNegativeInteger}, \allowbreak{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{findCycle}}\opLeftPren{} +{\it nonNegativeInteger}, \allowbreak{}{\it stream}\opRightPren{}% }% }% -{2}{(NonNegativeInteger, \$)->Record(cycle?:Boolean, prefix:NonNegativeInteger, period:NonNegativeInteger)}{Stream} -{\smath{\mbox{\bf findCycle}\opLeftPren{}n, \allowbreak{} st\opRightPren{}} determines if stream \smath{st} is +{2}{(NonNegativeInteger, \$)-> +Record(cycle?:Boolean, prefix:NonNegativeInteger, +period:NonNegativeInteger)}{Stream} +{\smath{\mbox{\bf findCycle}\opLeftPren{}n, +\allowbreak{} st\opRightPren{}} determines if stream \smath{st} is periodic within \smath{n} terms. The operation returns a record with three fields: \smath{cycle?}, @@ -72512,46 +74850,58 @@ the cycle begins. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{finite?}}\opLeftPren{}{\it cardinalNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{finite?}}\opLeftPren{} +{\it cardinalNumber}\opRightPren{}% }% }% {1}{(\$)->Boolean}{CardinalNumber} -{\smath{\mbox{\bf finite?}\opLeftPren{}f\opRightPren{}} tests if expression f is finite. +{\smath{\mbox{\bf finite?}\opLeftPren{}f\opRightPren{}} +tests if expression f is finite. \newitem -\smath{\mbox{\bf finite?}\opLeftPren{}a\opRightPren{}} tests if cardinal number \smath{a} is a +\smath{\mbox{\bf finite?}\opLeftPren{}a\opRightPren{}} +tests if cardinal number \smath{a} is a finite cardinal, that is, an integer. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fintegrate}}\opLeftPren{}{\it taylorSeries}, \allowbreak{}{\it symbol}, \allowbreak{}{\it coefficient}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fintegrate}}\opLeftPren{} +{\it taylorSeries}, \allowbreak{}{\it symbol}, \allowbreak{} +{\it coefficient}\opRightPren{}% }% }% {3}{(()->\$, Symbol, Coef)->\$}{TaylorSeries} -{\smath{\mbox{\bf fintegrate}\opLeftPren{}s, \allowbreak{} v, \allowbreak{} c\opRightPren{}} integrates the series \smath{s} with -respect to variable \smath{v} and having \smath{c} as the constant of integration. +{\smath{\mbox{\bf fintegrate}\opLeftPren{}s, +\allowbreak{} v, \allowbreak{} c\opRightPren{}} integrates the series \smath{s} with +respect to variable \smath{v} and having \smath{c} +as the constant of integration. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{first}}\opLeftPren{}{\it aggregate\opt{, nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{first}}\opLeftPren{} +{\it aggregate\opt{, nonNegativeInteger}}\opRightPren{}% }% }% {1}{(\$)->Entry}{IndexedAggregate} -{\smath{\mbox{\bf first}\opLeftPren{}u\opRightPren{}} returns the first element \smath{x} of aggregate \smath{u}. -\newitem\smath{\mbox{\bf first}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns a copy of the first \smath{n} +{\smath{\mbox{\bf first}\opLeftPren{}u\opRightPren{}} +returns the first element \smath{x} of aggregate \smath{u}. +\newitem\smath{\mbox{\bf first}\opLeftPren{}u, +\allowbreak{} n\opRightPren{}} returns a copy of the first \smath{n} elements of \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fixedPoint}}\opLeftPren{}{\it function\opt{, positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fixedPoint}}\opLeftPren{} +{\it function\opt{, positiveInteger}}\opRightPren{}% }% }% { (A->A) -> A}{}{} {\smath{\mbox{\bf fixedPoint}\opLeftPren{}f\opRightPren{}}, a function of type \spadsig{A}{A}, is the fixed point of function \smath{f}. -That is, \smath{\mbox{\bf fixedPoint}\opLeftPren{}f) = f(\mbox{\bf fixedPoint}(f))}. +That is, \smath{\mbox{\bf fixedPoint}\opLeftPren{}f) = +f(\mbox{\bf fixedPoint}(f))}. \newitem \smath{\mbox{\bf fixedPoint}\opLeftPren{}f, \allowbreak{} n\opRightPren{}}, where \smath{f} is a function of type \spadsig{List(A)}{List(A)} @@ -72561,18 +74911,25 @@ and \smath{n} is a positive integer, is the fixed point of function } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fixedPoints}}\opLeftPren{}{\it permutation}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fixedPoints}}\opLeftPren{} +{\it permutation}\opRightPren{}% }% }% {1}{(\$)->Set(S)}{Permutation} -{\smath{\mbox{\bf fixedPoints}\opLeftPren{}p\opRightPren{}} returns the points fixed by the permutation \smath{p}.} +{\smath{\mbox{\bf fixedPoints}\opLeftPren{}p\opRightPren{}} +returns the points fixed by the permutation \smath{p}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{flagFactor}}\opLeftPren{}{\it base}, \allowbreak{}{\it exponent}, \allowbreak{}{\it flag}\opRightPren{}% +\opdata{{\mbox{\axiomFun{flagFactor}}\opLeftPren{}{\it base}, +\allowbreak{}{\it exponent}, \allowbreak{}{\it flag}\opRightPren{}% }% }% {3}{(R, Integer, Union("nil", "sqfr", "irred", "prime"))->\$}{Factored} -{\smath{\mbox{\bf flagFactor}\opLeftPren{}base, \allowbreak{} exponent, \allowbreak{} flag\opRightPren{}} creates a factored object with a single factor whose \smath{base} is asserted to be properly described by the information \smath{flag}: +{\smath{\mbox{\bf flagFactor}\opLeftPren{}base, +\allowbreak{} exponent, \allowbreak{} flag\opRightPren{}} +creates a factored object +with a single factor whose \smath{base} is asserted to be properly described +by the information \smath{flag}: one of the strings \mbox{\tt "nil"}, \mbox{\tt "sqfr"}, \mbox{\tt "irred"}, and \mbox{\tt "prime"}. } @@ -72581,7 +74938,8 @@ one of the strings \mbox{\tt "nil"}, \mbox{\tt "sqfr"}, \mbox{\tt "irred"}, and }% }% { $ -> $}{}{} -{\smath{\mbox{\bf flatten}\opLeftPren{}s\opRightPren{}} returns an input form corresponding to \smath{s} with +{\smath{\mbox{\bf flatten}\opLeftPren{}s\opRightPren{}} +returns an input form corresponding to \smath{s} with all the nested operations flattened to triples using new local variables. This operation is used to optimize compiled code. @@ -72591,7 +74949,8 @@ This operation is used to optimize compiled code. }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf flexible?}\opLeftPren{}\opRightPren{}\$R} tests if \smath{2 \mbox{\bf +{\smath{\mbox{\bf flexible?}\opLeftPren{}\opRightPren{}\$R} +tests if \smath{2 \mbox{\bf associator}(a, b, a) = 0} for all \smath{a}, \smath{b} in a domain \smath{R} of category \axiomType{FiniteRankNonAssociativeAlgebra}. @@ -72599,11 +74958,13 @@ Note: only this can be tested since, in general, it is not known whether \smath{2a=0} implies \smath{a=0}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{flexibleArray}}\opLeftPren{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{flexibleArray}}\opLeftPren{} +{\it listOfElements}\opRightPren{}% }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf flexibleArray}\opLeftPren{}ls\opRightPren{}} creates a flexible array from a list of +{\smath{\mbox{\bf flexibleArray}\opLeftPren{}ls +\opRightPren{}} creates a flexible array from a list of elements \smath{ls}. } @@ -72611,74 +74972,93 @@ elements \smath{ls}. \opdata{{\mbox{\axiomFun{float?}}\opLeftPren{}{\it sExpression}\opRightPren{}% }% }% -{1}{(\$)->Boolean}{SExpressionCategory}{\smath{\mbox{\bf float?}\opLeftPren{}s\opRightPren{}} is \smath{true} if \smath{s} is an atom and belongs o \smath{Flt}.} +{1}{(\$)->Boolean}{SExpressionCategory}{\smath{ +\mbox{\bf float?}\opLeftPren{}s\opRightPren{}} is \smath{true} if \smath{s} is +an atom and belongs o \smath{Flt}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{float}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\allowbreak $\,[$ , \allowbreak{}{\it positiveinteger}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{float}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\allowbreak $\,[$ , +\allowbreak{}{\it positiveinteger}$]$\opRightPren{}% }% }% {2}{(Integer, Integer)->\$}{FloatingPointSystem} -{\smath{\mbox{\bf float}\opLeftPren{}a, \allowbreak{} e\opRightPren{}} returns \smath{a {\tt base()}^e} as a float. -\newitem\smath{\mbox{\bf float}\opLeftPren{}a, \allowbreak{} e, \allowbreak{} b\opRightPren{}} returns \smath{a b ^ e} as a float.} +{\smath{\mbox{\bf float}\opLeftPren{}a, \allowbreak{} e\opRightPren{}} +returns \smath{a {\tt base()}^e} as a float. +\newitem\smath{\mbox{\bf float}\opLeftPren{}a, \allowbreak{} e, +\allowbreak{} b\opRightPren{}} returns \smath{a b ^ e} as a float.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{floor}}\opLeftPren{}{\it rationalNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{floor}}\opLeftPren{} +{\it rationalNumber}\opRightPren{}% }% }% {1}{(\$)->D}{QuotientFieldCategory} {\smath{\mbox{\bf floor}\opLeftPren{}fr\opRightPren{}}, where \smath{fr} is a fraction, returns the largest integral element below \smath{fr}. -\newitem \smath{\mbox{\bf floor}\opLeftPren{}fl\opRightPren{}}, where \smath{fl} is a float, +\newitem \smath{\mbox{\bf floor}\opLeftPren{}fl\opRightPren{}}, +where \smath{fl} is a float, returns the largest integer \smath{<= fl}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{formula}}\opLeftPren{}{\it formulaFormat}\opRightPren{}% +\opdata{{\mbox{\axiomFun{formula}}\opLeftPren{} +{\it formulaFormat}\opRightPren{}% }% }% {1}{(\$)->List(String)}{ScriptFormulaFormat} -{\smath{\mbox{\bf formula}\opLeftPren{}t\opRightPren{}} extracts the formula section of an IBM +{\smath{\mbox{\bf formula}\opLeftPren{}t\opRightPren{}} +extracts the formula section of an IBM SCRIPT Formula formatted object \smath{t}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fractionPart}}\opLeftPren{}{\it fraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fractionPart}}\opLeftPren{} +{\it fraction}\opRightPren{}% }% }% {1}{(\$)->\$}{QuotientFieldCategory} -{\smath{\mbox{\bf fractionPart}\opLeftPren{}x\opRightPren{}} returns the fractional part of \smath{x}. +{\smath{\mbox{\bf fractionPart}\opLeftPren{}x\opRightPren{}} +returns the fractional part of \smath{x}. Argument \smath{x} can be a fraction, a radix (binary, decimal, or hexadecimal) expansion, or a float. Note: \smath{x} = whole(\smath{x}) + fractionPart(\smath{x}). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fractRadix}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fractRadix}}\opLeftPren{} +{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% }% }% {2}{(List(Integer), List(Integer))->\$}{RadixExpansion} -{\smath{\mbox{\bf fractRadix}\opLeftPren{}pre, \allowbreak{} cyc\opRightPren{}} creates a fractional radix expansion +{\smath{\mbox{\bf fractRadix}\opLeftPren{}pre, +\allowbreak{} cyc\opRightPren{}} creates a fractional radix expansion from a list of prefix ragits and a list of cyclic ragits. -For example, \smath{\mbox{\bf fractRadix}\opLeftPren{}[1], \allowbreak{} [6]\opRightPren{}} will return +For example, \smath{\mbox{\bf fractRadix}\opLeftPren{}[1], +\allowbreak{} [6]\opRightPren{}} will return \smath{0.16666666\ldots}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{fractRagits}}\opLeftPren{}{\it radixExpansion}\opRightPren{}% +\opdata{{\mbox{\axiomFun{fractRagits}}\opLeftPren{} +{\it radixExpansion}\opRightPren{}% }% }% {1}{(\$)->Stream(Integer)}{RadixExpansion} -{\smath{\mbox{\bf fractRagits}\opLeftPren{}rx\opRightPren{}} returns the ragits of the fractional +{\smath{\mbox{\bf fractRagits}\opLeftPren{}rx\opRightPren{}} +returns the ragits of the fractional part of a radix expansion as a stream of integers.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{freeOf?}}\opLeftPren{}{\it expression}, \allowbreak{}{\it kernel}\opRightPren{}% +\opdata{{\mbox{\axiomFun{freeOf?}}\opLeftPren{}{\it expression}, +\allowbreak{}{\it kernel}\opRightPren{}% }% }% {2}{(\$, Symbol)->Boolean}{ExpressionSpace} -{\smath{\mbox{\bf freeOf?}\opLeftPren{}x, \allowbreak{} k\opRightPren{}} tests if expression \smath{x} does not +{\smath{\mbox{\bf freeOf?}\opLeftPren{}x, \allowbreak{} k\opRightPren{}} +tests if expression \smath{x} does not contain any operator whose name is the symbol or kernel \smath{k}.} @@ -72687,15 +75067,18 @@ any operator whose name is the symbol or kernel \smath{k}.} }% }% {1}{(\$)->\$}{ExtensionField} -{\smath{\mbox{\bf Frobenius}\opLeftPren{}a\opRightPren{}\$F} returns \smath{a^q} where \smath{q} is -the \smath{\mbox{\bf size}\opLeftPren{}\opRightPren{}\$F} of extension field \smath{F}.} +{\smath{\mbox{\bf Frobenius}\opLeftPren{}a\opRightPren{}\$F} +returns \smath{a^q} where \smath{q} is +the \smath{\mbox{\bf size}\opLeftPren{}\opRightPren{}\$F} +of extension field \smath{F}.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{front}}\opLeftPren{}{\it queue}\opRightPren{}% }% }% {1}{(\$)->S}{QueueAggregate} -{\smath{\mbox{\bf front}\opLeftPren{}q\opRightPren{}} returns the element at the front of the queue, or +{\smath{\mbox{\bf front}\opLeftPren{}q\opRightPren{}} +returns the element at the front of the queue, or calls \spadfun{error} if \smath{q} is empty.} % ---------------------------------------------------------------------- @@ -72703,12 +75086,15 @@ calls \spadfun{error} if \smath{q} is empty.} }% }% {1}{(\$)->S}{LazyStreamAggregate} -{\smath{\mbox{\bf frst}\opLeftPren{}s\opRightPren{}} returns the first element of stream \smath{s}. +{\smath{\mbox{\bf frst}\opLeftPren{}s\opRightPren{}} +returns the first element of stream \smath{s}. Warning: this function should only be called after a \smath{empty?} test has been made since there is no error check.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{function}}\opLeftPren{}{\it expression}, \allowbreak{}{\it name}\allowbreak $\,[$ , \allowbreak{}{\it options}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{function}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it name}\allowbreak $\,[$ , \allowbreak{} +{\it options}$]$\opRightPren{}% }% }% {2}{(S, Symbol)->Symbol}{MakeFunction} @@ -72716,11 +75102,19 @@ Warning: this function should only be called after a objects to type \axiomType{InputForm}. Argument \smath{e} below denotes an object from such a domain. These operations create user-functions from already computed results.} -\newitem\smath{\mbox{\bf function}\opLeftPren{}e, \allowbreak{} f\opRightPren{}} creates a function \smath{f() == e}. -\newitem\smath{\mbox{\bf function}\opLeftPren{}e, \allowbreak{} f, \allowbreak{} [x_1, \allowbreak{} \ldots, x_n]\opRightPren{}} creates a function \smath{f(x_1, \ldots, x_n) == e}. -\newitem\smath{\mbox{\bf function}\opLeftPren{}e, \allowbreak{} f, \allowbreak{} x\opRightPren{}} creates a function \smath{f(x) == e}. -\newitem\smath{\mbox{\bf function}\opLeftPren{}e, \allowbreak{} f, \allowbreak{} x, \allowbreak{} y\opRightPren{}} creates a function \smath{f(x, y) == e}. -\newitem\smath{\mbox{\bf function}\opLeftPren{}expr, \allowbreak{} [x_1, \allowbreak{} \ldots, x_n], f\opRightPren{}}, +\newitem\smath{\mbox{\bf function}\opLeftPren{}e, \allowbreak{} f +\opRightPren{}} creates a function \smath{f() == e}. +\newitem\smath{\mbox{\bf function}\opLeftPren{}e, \allowbreak{} f, +\allowbreak{} [x_1, \allowbreak{} \ldots, x_n]\opRightPren{}} +creates a function +\smath{f(x_1, \ldots, x_n) == e}. +\newitem\smath{\mbox{\bf function}\opLeftPren{}e, \allowbreak{} f, +\allowbreak{} x\opRightPren{}} creates a function \smath{f(x) == e}. +\newitem\smath{\mbox{\bf function}\opLeftPren{}e, +\allowbreak{} f, \allowbreak{} x, \allowbreak{} y\opRightPren{}} +creates a function \smath{f(x, y) == e}. +\newitem\smath{\mbox{\bf function}\opLeftPren{}expr, +\allowbreak{} [x_1, \allowbreak{} \ldots, x_n], f\opRightPren{}}, where \smath{expr} is an input form and where \smath{f} and the \smath{x_i}'s are symbols, returns the input form @@ -72733,68 +75127,86 @@ See also \spadfun{unparse}. }% }% {1}{(Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} -{\smath{\mbox{\bf Gamma}\opLeftPren{}x\opRightPren{}} is the Euler gamma function, \smath{\mbox{\bf Gamma}\opLeftPren{}x\opRightPren{}}, +{\smath{\mbox{\bf Gamma}\opLeftPren{}x\opRightPren{}} is the Euler +gamma function, \smath{\mbox{\bf Gamma}\opLeftPren{}x\opRightPren{}}, defined by \smath{\Gamma(x) = \int\nolimits_0^\infty {t^{(x-1)}*exp(-t) dt}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{gcdPolynomial}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{gcdPolynomial}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it polynomial}\opRightPren{}% }% }% -{2}{(SparseUnivariatePolynomial(\$), SparseUnivariatePolynomial(\$))->SparseUnivariatePolynomial(\$)}{PolynomialFactorizationExplicit} +{2}{(SparseUnivariatePolynomial(\$), SparseUnivariatePolynomial(\$))-> +SparseUnivariatePolynomial(\$)}{PolynomialFactorizationExplicit} {\smath{\mbox{\bf gcdPolynomial}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns the \spadfun{gcd} of the univariate polynomials \smath{p} and \smath{q}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{gcd}}\opLeftPren{}{\it element\opt{, element, element}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{gcd}}\opLeftPren{} +{\it element\opt{, element, element}}\opRightPren{}% }% }% {1}{(List(\$))->\$}{GcdDomain} -{\smath{\mbox{\bf gcd}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns the greatest common divisor of \smath{x} and \smath{y}. +{\smath{\mbox{\bf gcd}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +returns the greatest common divisor of \smath{x} and \smath{y}. Arguments \smath{x} and \smath{y} are elements of a domain of category \smath{GcdDomain}. -\newitem\smath{\mbox{\bf gcd}\opLeftPren{}[x_1, \allowbreak{} \ldots, x_n]\opRightPren{}} returns the +\newitem\smath{\mbox{\bf gcd}\opLeftPren{}[x_1, +\allowbreak{} \ldots, x_n]\opRightPren{}} returns the common \smath{gcd} of the elements of the list of \smath{x_i}. -\newitem\smath{\mbox{\bf gcd}\opLeftPren{}p_1, \allowbreak{} p_2, \allowbreak{} prime\opRightPren{}} computes the \smath{gcd} of the univariate -polynomials \smath{p_1} and \smath{p_2} modulo the prime integer \smath{prime}.} +\newitem\smath{\mbox{\bf gcd}\opLeftPren{}p_1, \allowbreak{} p_2, +\allowbreak{} prime\opRightPren{}} computes the \smath{gcd} of the univariate +polynomials \smath{p_1} and \smath{p_2} modulo the prime integer +\smath{prime}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{generalizedContinuumHypothesisAssumed?}}\opLeftPren{}{\it \opt{bool}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{generalizedContinuumHypothesisAssumed?}} +\opLeftPren{}{\it \opt{bool}}\opRightPren{}% }% }% {0}{()->Boolean}{CardinalNumber} -{\smath{\mbox{\bf generalizedContinuumHypothesisAssumed?}\opLeftPren{}\opRightPren{}} tests if the hypothesis is currently assumed. +{\smath{\mbox{\bf generalizedContinuumHypothesisAssumed?} +\opLeftPren{}\opRightPren{}} tests if the hypothesis is currently assumed. \newitem -\smath{\mbox{\bf generalizedContinuumHypothesisAssumed}\opLeftPren{}bool\opRightPren{}} dictates +\smath{\mbox{\bf generalizedContinuumHypothesisAssumed} +\opLeftPren{}bool\opRightPren{}} dictates that the hypothesis is or is not to be assumed, according to whether \smath{bool} is true or false.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{generalPosition}}\opLeftPren{}{\it ideal}, \allowbreak{}{\it listOfVariables}\opRightPren{}% +\opdata{{\mbox{\axiomFun{generalPosition}}\opLeftPren{} +{\it ideal}, \allowbreak{}{\it listOfVariables}\opRightPren{}% }% }% -{2}{(\$, List(OrderedVariableList(vl)))->Record(mval:Matrix(F), invmval:Matrix(F), genIdeal:\$)}{PolynomialIdeals} -{\smath{\mbox{\bf generalPosition}\opLeftPren{}I, \allowbreak{} listvar\opRightPren{}} performs a random linear +{2}{(\$, List(OrderedVariableList(vl)))-> +Record(mval:Matrix(F), invmval:Matrix(F), genIdeal:\$)}{PolynomialIdeals} +{\smath{\mbox{\bf generalPosition}\opLeftPren{}I, +\allowbreak{} listvar\opRightPren{}} performs a random linear transformation on the variables in \smath{listvar} and returns the transformed ideal \smath{I} along with the change of basis matrix. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{generate}}\opLeftPren{}{\it function\opt{, element}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{generate}}\opLeftPren{} +{\it function\opt{, element}}\opRightPren{}% }% }% {1}{(()->S)->\$}{Stream} {\smath{\mbox{\bf generate}\opLeftPren{}f\opRightPren{}}, where \smath{f} is a function of no arguments, creates an infinite stream all of whose elements are equal to -the value of \smath{f()}. Note: \smath{\mbox{\bf generate}\opLeftPren{}f) = [f(), f(), f(), \ldots]}. -\newitem\smath{\mbox{\bf generate}\opLeftPren{}f, \allowbreak{} x\opRightPren{}}, +the value of \smath{f()}. Note: \smath{\mbox{\bf generate} +\opLeftPren{}f) = [f(), f(), f(), \ldots]}. +\newitem\smath{\mbox{\bf generate}\opLeftPren{}f, +\allowbreak{} x\opRightPren{}}, where \smath{f} is a function of one argument, creates an infinite stream whose first element is \smath{x} and whose \eth{\smath{n}} element (\smath{n > 1}) is \smath{f} applied to the previous element. -Note: \smath{\mbox{\bf generate}\opLeftPren{}f, x) = [x, f(x), f(f(x)), \ldots]}. +Note: \smath{\mbox{\bf generate}\opLeftPren{}f, x) = +[x, f(x), f(f(x)), \ldots]}. \newitem See also \axiomType{HallBasis}. } @@ -72803,7 +75215,8 @@ Note: \smath{\mbox{\bf generate}\opLeftPren{}f, x) = [x, f(x), f(f(x)), \ldots]} }% }% {0}{()->\$}{FiniteAlgebraicExtensionField} -{\smath{\mbox{\bf generator}\opLeftPren{}\opRightPren{}\$R} returns a root of the defining polynomial +{\smath{\mbox{\bf generator}\opLeftPren{}\opRightPren{}\$R} returns +a root of the defining polynomial of a domain of category \axiomType{FiniteAlgebraicExtensionField} \smath{R}. This element generates the field as an algebra over the ground field. \newitem @@ -72815,8 +75228,10 @@ See also \axiomType{MonogenicAlgebra} and \axiomType{FreeNilpotentLie}. }% }% {1}{(\$)->List(DPoly)}{PolynomialIdeals} -{\smath{\mbox{\bf generators}\opLeftPren{}I\opRightPren{}} returns a list of generators for the ideal \smath{I}. -\newitem\smath{\mbox{\bf generators}\opLeftPren{}gp\opRightPren{}} returns the generators of a permutation +{\smath{\mbox{\bf generators}\opLeftPren{}I\opRightPren{}} returns +a list of generators for the ideal \smath{I}. +\newitem\smath{\mbox{\bf generators}\opLeftPren{}gp\opRightPren{}} +returns the generators of a permutation group {\it gp}. } @@ -72825,7 +75240,8 @@ group {\it gp}. }% }% {0}{()->NonNegativeInteger}{FunctionFieldCategory} -{\smath{\mbox{\bf genus}\opLeftPren{}\opRightPren{}}{\bf \$}\smath{R} returns the genus of +{\smath{\mbox{\bf genus}\opLeftPren{}\opRightPren{}}{\bf \$}\smath{R} +returns the genus of the algebraic function field \smath{R}. If \smath{R} has several absolutely irreducible components, then the genus of one of them is @@ -72838,13 +75254,15 @@ returned. }% }% {0}{()->Matrix(PrimeField(p))}{FiniteFieldNormalBasis} -{\smath{\mbox{\bf getMultiplicationMatrix}\opLeftPren{}\opRightPren{}\$R} returns a matrix multiplication table +{\smath{\mbox{\bf getMultiplicationMatrix}\opLeftPren{}\opRightPren{}\$R} +returns a matrix multiplication table for domain \axiomType{FiniteFieldNormalBasis(p, n)}, a finite extension field of degree \smath{n} over the domain \axiomType{PrimeField(p)} with \smath{p} elements. Each element of the matrix is a member of the underlying prime field. \newitem -\smath{\mbox{\bf getMultiplicationTable}\opLeftPren{}\opRightPren{}\$R} is similar except that the +\smath{\mbox{\bf getMultiplicationTable}\opLeftPren{}\opRightPren{}\$R} +is similar except that the multiplication table for the normal basis of the field is represented by a vector of lists of records, each record having two fields: \smath{value}, an element of the prime field over @@ -72856,17 +75274,21 @@ This table is used to perform multiplications between field elements. \opdata{{\mbox{\axiomFun{getVariableOrder}}\opLeftPren{}\opRightPren{}% }% }% -{0}{()->Record(high:List(Symbol), low:List(Symbol))}{UserDefinedVariableOrdering} -{\smath{\mbox{\bf getVariableOrder}\opLeftPren{}\opRightPren{}} returns \smath{[[b_1, \ldots, b_m], +{0}{()->Record(high:List(Symbol), low:List(Symbol))} +{UserDefinedVariableOrdering} +{\smath{\mbox{\bf getVariableOrder}\opLeftPren{}\opRightPren{}} +returns \smath{[[b_1, \ldots, b_m], [a_1, \ldots, a_n] ]} such that the ordering on the variables was given -by \smath{\mbox{\bf setVariableOrder}\opLeftPren{}[b_1, \allowbreak{} \ldots, b_m], [a_1, \ldots, a_n]\opRightPren{}}.} +by \smath{\mbox{\bf setVariableOrder}\opLeftPren{}[b_1, +\allowbreak{} \ldots, b_m], [a_1, \ldots, a_n]\opRightPren{}}.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{getZechTable}}\opLeftPren{}\opRightPren{}% }% }% {0}{()->PrimitiveArray(SingleInteger)}{FiniteFieldCyclicGroup} -{\smath{\mbox{\bf getZechTable}\opLeftPren{}\opRightPren{}\$F} returns the Zech logarithm table of the +{\smath{\mbox{\bf getZechTable}\opLeftPren{}\opRightPren{}\$F} +returns the Zech logarithm table of the \index{Zech logarithm} field \smath{F} where \smath{F} is some domain @@ -72874,14 +75296,17 @@ field This table is used to perform additions in the field quickly.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{gramschmidt}}\opLeftPren{}{\it listOfMatrices}\opRightPren{}% +\opdata{{\mbox{\axiomFun{gramschmidt}}\opLeftPren{} +{\it listOfMatrices}\opRightPren{}% }% }% -{1}{(List(Matrix(Expression(Fraction(Integer)))))->List(Matrix(Expression(Fraction(Integer))))}{RadicalEigenPackage} +{1}{(List(Matrix(Expression(Fraction(Integer)))))-> +List(Matrix(Expression(Fraction(Integer))))}{RadicalEigenPackage} {\opkey{Argument \smath{lv} has the form of a list of matrices of elements of type \axiomType{Expression}.} \newitem -\smath{\mbox{\bf gramschmidt}\opLeftPren{}lv\opRightPren{}} converts the list of column vectors +\smath{\mbox{\bf gramschmidt}\opLeftPren{}lv\opRightPren{}} +converts the list of column vectors \smath{lv} into a set of orthogonal column vectors of Euclidean length 1 using the Gram-Schmidt algorithm. \index{Gram-Schmidt algorithm} @@ -72892,7 +75317,8 @@ length 1 using the Gram-Schmidt algorithm. }% }% { I -> SPOL RN}{}{} -{\smath{\mbox{\bf graphs}\opLeftPren{}n\opRightPren{}} is the cycle index of the group induced on +{\smath{\mbox{\bf graphs}\opLeftPren{}n\opRightPren{}} +is the cycle index of the group induced on the edges of a graph by applying the symmetric function to the \smath{n} nodes. See \axiomType{CycleIndicators} for details. @@ -72903,28 +75329,35 @@ See \axiomType{CycleIndicators} for details. }% }% {0}{()->\$}{Color} -{\smath{\mbox{\bf green}\opLeftPren{}\opRightPren{}} returns the position of the green hue from total hues.} +{\smath{\mbox{\bf green}\opLeftPren{}\opRightPren{}} +returns the position of the green hue from total hues.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{groebner}}\opLeftPren{}{\it listOfPolynomials}\opRightPren{}% +\opdata{{\mbox{\axiomFun{groebner}}\opLeftPren{} +{\it listOfPolynomials}\opRightPren{}% }% }% {1}{(List(Dpol))->List(Dpol)}{GroebnerPackage} -{\smath{\mbox{\bf groebner}\opLeftPren{}lp\opRightPren{}} computes a Gr\"obner basis for a polynomial +{\smath{\mbox{\bf groebner}\opLeftPren{}lp\opRightPren{}} +computes a Gr\"obner basis for a polynomial ideal generated by the list of polynomials \smath{lp}. -\newitem \smath{\mbox{\bf groebner}\opLeftPren{}I\opRightPren{}} returns a set of generators of ideal +\newitem \smath{\mbox{\bf groebner}\opLeftPren{}I\opRightPren{}} +returns a set of generators of ideal \smath{I} that are a Gr\"obner basis for \smath{I}. -\newitem \smath{\mbox{\bf groebner}\opLeftPren{}lp, \allowbreak{} infoflag\opRightPren{}} computes a Gr\"obner basis +\newitem \smath{\mbox{\bf groebner}\opLeftPren{}lp, \allowbreak{} +infoflag\opRightPren{}} computes a Gr\"obner basis for a polynomial ideal generated by the list of polynomials \smath{lp}. Argument \smath{infoflag} is used to get information on the computation. -If \smath{infoflag} is \mbox{\tt "info"}, then summary information is displayed for +If \smath{infoflag} is \mbox{\tt "info"}, then summary information +is displayed for each s-polynomial generated. If \smath{infoflag} is \mbox{\tt "redcrit"}, the reduced critical pairs are displayed. To get the display of both kinds of information, use -\smath{\mbox{\bf groebner}\opLeftPren{}lp, \allowbreak{} "info", \allowbreak{} "redcrit"\opRightPren{}}. +\smath{\mbox{\bf groebner}\opLeftPren{}lp, \allowbreak{} "info", +\allowbreak{} "redcrit"\opRightPren{}}. } % ---------------------------------------------------------------------- @@ -72932,32 +75365,38 @@ To get the display of both kinds of information, use }% }% {1}{(\$)->Boolean}{PolynomialIdeals} -{\smath{\mbox{\bf groebner?}\opLeftPren{}I\opRightPren{}} tests if the generators of the ideal +{\smath{\mbox{\bf groebner?}\opLeftPren{}I\opRightPren{}} +tests if the generators of the ideal \smath{I} are a Gr\"obner basis.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{groebnerIdeal}}\opLeftPren{}{\it listOfPolynomials}\opRightPren{}% +\opdata{{\mbox{\axiomFun{groebnerIdeal}}\opLeftPren{} +{\it listOfPolynomials}\opRightPren{}% }% }% {1}{(List(DPoly))->\$}{PolynomialIdeals} -{\smath{\mbox{\bf groebnerIdeal}\opLeftPren{}lp\opRightPren{}} constructs the ideal generated by the +{\smath{\mbox{\bf groebnerIdeal}\opLeftPren{}lp\opRightPren{}} +constructs the ideal generated by the list of polynomials \smath{lp} assumed to be a Gr\"obner basis. Note: this operation avoids a Gr\"obner basis computation.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{groebnerFactorize}}\opLeftPren{}{\it listOfPolynomials\opt{options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{groebnerFactorize}}\opLeftPren{} +{\it listOfPolynomials\opt{options}}\opRightPren{}% }% }% {1}{(List(Dpol))->List(List(Dpol))}{GroebnerFactorizationPackage} -{\smath{\mbox{\bf groebnerFactorize}\opLeftPren{}lp\optinner{, bool}\opRightPren{}} returns +{\smath{\mbox{\bf groebnerFactorize}\opLeftPren{}lp +\optinner{, bool}\opRightPren{}} returns a list of list of polynomials, each inner list denoting a Gr\"obner basis. The union of the solutions of the bases is the solution of the system of equations given by \smath{lp}. Information about partial results is printed if a second argument is given with value \smath{true}. \newitem -\smath{\mbox{\bf groebnerFactorize}\opLeftPren{}lp, \allowbreak{} nonZeroRestrictions\optinner{, bool}\opRightPren{}}, +\smath{\mbox{\bf groebnerFactorize}\opLeftPren{}lp, +\allowbreak{} nonZeroRestrictions\optinner{, bool}\opRightPren{}}, where \smath{nonZeroRestrictions} is a list of polynomials, is similar. Here, however, the solutions to the system of equations are computed under the restriction that the polynomials in the second argument @@ -72973,24 +75412,30 @@ third argument with value \smath{true} is given. }% }% {1}{(\$)->Boolean}{FiniteAbelianMonoidRing} -{\smath{\mbox{\bf ground}\opLeftPren{}p\opRightPren{}} retracts expression polynomial \smath{p} to the coefficient ring, or +{\smath{\mbox{\bf ground}\opLeftPren{}p\opRightPren{}} +retracts expression polynomial \smath{p} to the coefficient ring, or calls \spadfun{error} if such a retraction is not possible. \newitem -\smath{\mbox{\bf ground?}\opLeftPren{}p\opRightPren{}} tests if an expression or polynomial \smath{p} is a member of the coefficient ring. +\smath{\mbox{\bf ground?}\opLeftPren{}p\opRightPren{}} +tests if an expression or polynomial \smath{p} is a member +of the coefficient ring. \seeAlso{\spadfun{ground?}} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{harmonic}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{harmonic}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(Integer)->Fraction(Integer)}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf harmonic}\opLeftPren{}n\opRightPren{}} returns the \eth{\smath{n}} harmonic number, defined by +{\smath{\mbox{\bf harmonic}\opLeftPren{}n\opRightPren{}} +returns the \eth{\smath{n}} harmonic number, defined by \smath{H[n] = \sum\nolimits_{k=1}^n {1/k}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{has}}\opLeftPren{}{\it domain}, \allowbreak{}{\it property}\opRightPren{}% +\opdata{{\mbox{\axiomFun{has}}\opLeftPren{} +{\it domain}, \allowbreak{}{\it property}\opRightPren{}% }% }% {}{}{} @@ -73003,11 +75448,13 @@ and \code{Integer has commutative("*")}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{has?}}\opLeftPren{}{\it operation}, \allowbreak{}{\it property}\opRightPren{}% +\opdata{{\mbox{\axiomFun{has?}}\opLeftPren{} +{\it operation}, \allowbreak{}{\it property}\opRightPren{}% }% }% {2}{(\$, String)->Boolean}{BasicOperator} -{\smath{\mbox{\bf has?}\opLeftPren{}op, \allowbreak{} s\opRightPren{}} tests if property \smath{s} is attached to +{\smath{\mbox{\bf has?}\opLeftPren{}op, +\allowbreak{} s\opRightPren{}} tests if property \smath{s} is attached to \smath{op}.} % ---------------------------------------------------------------------- @@ -73015,33 +75462,41 @@ and \code{Integer has commutative("*")}. }% }% {1}{(\$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf hash}\opLeftPren{}n\opRightPren{}} returns the hash code for \smath{n}, +{\smath{\mbox{\bf hash}\opLeftPren{}n\opRightPren{}} +returns the hash code for \smath{n}, an integer or a float.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{hasHi}}\opLeftPren{}{\it segment}\opRightPren{}% +\opdata{{\mbox{\axiomFun{hasHi}}\opLeftPren{} +{\it segment}\opRightPren{}% }% }% {1}{(\$)->Boolean}{UniversalSegment} -{\smath{\mbox{\bf hasHi}\opLeftPren{}seg\opRightPren{}} tests whether the segment \smath{seg} has an +{\smath{\mbox{\bf hasHi}\opLeftPren{}seg\opRightPren{}} +tests whether the segment \smath{seg} has an upper bound. For example, \smath{\mbox{\bf hasHi}\opLeftPren{}1..) = false}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{hasSolution?}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{hasSolution?}}\opLeftPren{} +{\it matrix}, \allowbreak{}{\it vector}\opRightPren{}% }% }% {2}{(M, Col)->Boolean}{LinearSystemMatrixPackage} -{\smath{\mbox{\bf hasSolution?}\opLeftPren{}A, \allowbreak{} B\opRightPren{}} tests if the linear system \smath{AX = B} has a solution, +{\smath{\mbox{\bf hasSolution?}\opLeftPren{}A, +\allowbreak{} B\opRightPren{}} tests if the linear +system \smath{AX = B} has a solution, where \smath{A} is a matrix and \smath{B} is a (column) vector.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{hconcat}}\opLeftPren{}{\it outputForms\opt{, outputForm}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{hconcat}}\opLeftPren{} +{\it outputForms\opt{, outputForm}}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf hconcat}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, where \smath{o_1} and \smath{o_2} are +{\smath{\mbox{\bf hconcat}\opLeftPren{}o_1, +\allowbreak{} o_2\opRightPren{}}, where \smath{o_1} and \smath{o_2} are objects of type \axiomType{OutputForm} (normally unexposed), returns an output form for the horizontal concatenation of forms \smath{o_1} and \smath{o_2}. @@ -73056,19 +75511,23 @@ elements of \smath{lof}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{heap}}\opLeftPren{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{heap}}\opLeftPren{} +{\it listOfElements}\opRightPren{}% }% }% {2}{((S, S)->Boolean, V)->V}{FiniteLinearAggregateSort} -{\smath{\mbox{\bf heap}\opLeftPren{}ls\opRightPren{}} creates a \axiomType{Heap} of elements consisting +{\smath{\mbox{\bf heap}\opLeftPren{}ls\opRightPren{}} +creates a \axiomType{Heap} of elements consisting of the elements of \smath{ls}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{heapSort}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{heapSort}}\opLeftPren{} +{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S, S)->Boolean, V)->V}{FiniteLinearAggregateSort} -{\smath{\mbox{\bf heapSort}\opLeftPren{}pred, \allowbreak{} agg\opRightPren{}} sorts the aggregate agg with the +{\smath{\mbox{\bf heapSort}\opLeftPren{}pred, +\allowbreak{} agg\opRightPren{}} sorts the aggregate agg with the ordering function \smath{pred} using the heapsort algorithm.} % ---------------------------------------------------------------------- @@ -73076,7 +75535,8 @@ ordering function \smath{pred} using the heapsort algorithm.} }% }% {1}{(\$)->NonNegativeInteger}{ExpressionSpace} -{\smath{\mbox{\bf height}\opLeftPren{}f\opRightPren{}}, where \smath{f} is an expression, returns the +{\smath{\mbox{\bf height}\opLeftPren{}f\opRightPren{}}, +where \smath{f} is an expression, returns the highest nesting level appearing in \smath{f}. Constants have height 0. Symbols have height 1. @@ -73084,19 +75544,23 @@ For any operator \smath{op} and expressions \smath{f_1}, \ldots, \smath{f_n}, \smath{op(f_1, \ldots, f_n)} has height equal to \smath{1 + max(height(f_1), \ldots, height(f_n))}. -\newitem\smath{\mbox{\bf height}\opLeftPren{}d\opRightPren{}} returns the number of elements in +\newitem\smath{\mbox{\bf height}\opLeftPren{}d\opRightPren{}} +returns the number of elements in dequeue \smath{d}. Note: \smath{\mbox{\bf height}\opLeftPren{}d) = \# d}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{hermiteH}}\opLeftPren{}{\it nonNegativeInteger}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{hermiteH}}\opLeftPren{} +{\it nonNegativeInteger}, \allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(NonNegativeInteger, R)->R}{OrthogonalPolynomialFunctions} -{\smath{\mbox{\bf hermiteH}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} is the \eth{\smath{n}} Hermite polynomial, +{\smath{\mbox{\bf hermiteH}\opLeftPren{}n, +\allowbreak{} x\opRightPren{}} is the \eth{\smath{n}} Hermite polynomial, \smath{H[n](x)}, defined by -\smath{\mbox{\bf exp}\opLeftPren{}2 t x-t^2) = \sum\nolimits_{n=0}^\infty{H[n](x)t^n/n!}}. +\smath{\mbox{\bf exp}\opLeftPren{}2 t x-t^2) = \sum +\nolimits_{n=0}^\infty{H[n](x)t^n/n!}}. } % ---------------------------------------------------------------------- @@ -73104,15 +75568,18 @@ Note: \smath{\mbox{\bf height}\opLeftPren{}d) = \# d}. }% }% {0}{()->\$}{CharacterClass} -{\smath{\mbox{\bf hexDigit}\opLeftPren{}\opRightPren{}} returns the class of all characters for which +{\smath{\mbox{\bf hexDigit}\opLeftPren{}\opRightPren{}} +returns the class of all characters for which \spadfunFrom{hexDigit?}{Character} is \smath{true}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{hexDigit?}}\opLeftPren{}{\it character}\opRightPren{}% +\opdata{{\mbox{\axiomFun{hexDigit?}}\opLeftPren{} +{\it character}\opRightPren{}% }% }% {1}{(\$)->Boolean}{Character} -{\smath{\mbox{\bf hexDigit?}\opLeftPren{}c\opRightPren{}} tests if \smath{c} is a hexadecimal numeral, +{\smath{\mbox{\bf hexDigit?}\opLeftPren{}c\opRightPren{}} +tests if \smath{c} is a hexadecimal numeral, that is, one of 0..9, a..\smath{f} or A..\smath{F}.} % ---------------------------------------------------------------------- @@ -73120,7 +75587,8 @@ that is, one of 0..9, a..\smath{f} or A..\smath{F}.} }% }% {1}{(Fraction(Integer))->\$}{HexadecimalExpansion} -{\smath{\mbox{\bf hex}\opLeftPren{}r\opRightPren{}} converts a rational number to a hexadecimal +{\smath{\mbox{\bf hex}\opLeftPren{}r\opRightPren{}} +converts a rational number to a hexadecimal expansion.} % ---------------------------------------------------------------------- @@ -73128,48 +75596,59 @@ expansion.} }% }% {1}{(\$)->S}{SegmentCategory} -{\smath{\mbox{\bf hi}\opLeftPren{}s\opRightPren{}} returns the second endpoint of segment \smath{s}. +{\smath{\mbox{\bf hi}\opLeftPren{}s\opRightPren{}} +returns the second endpoint of segment \smath{s}. For example, \smath{\mbox{\bf hi}\opLeftPren{}l..h) = h}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{horizConcat}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{horizConcat}}\opLeftPren{} +{\it matrix}, \allowbreak{}{\it matrix}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{MatrixCategory} -{\smath{\mbox{\bf horizConcat}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} horizontally concatenates two matrices +{\smath{\mbox{\bf horizConcat}\opLeftPren{}x, +\allowbreak{} y\opRightPren{}} horizontally concatenates two matrices with an equal number of rows. The entries of \smath{y} appear to the right of the entries of \smath{x}. -The operation calls \spadfun{error} if the matrices do not have the same number of rows.} +The operation calls \spadfun{error} if the matrices do not have the +same number of rows.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{htrigs}}\opLeftPren{}{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf htrigs}\opLeftPren{}f\opRightPren{}} converts all the exponentials in expression \smath{f} into hyperbolic sines and cosines.} +{\smath{\mbox{\bf htrigs}\opLeftPren{}f\opRightPren{}} converts all +the exponentials in expression \smath{f} into hyperbolic sines and cosines.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{hue}}\opLeftPren{}{\it palette}\opRightPren{}% }% }% {1}{(\$)->Color}{Palette} -{\smath{\mbox{\bf hue}\opLeftPren{}p\opRightPren{}} returns the hue field of the indicated palette \smath{p}.} +{\smath{\mbox{\bf hue}\opLeftPren{}p\opRightPren{}} returns +the hue field of the indicated palette \smath{p}.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{hue}}\opLeftPren{}{\it color}\opRightPren{}% }% }% {1}{(\$)->Integer}{Color} -{\smath{\mbox{\bf hue}\opLeftPren{}c\opRightPren{}} returns the hue index of the indicated color \smath{c}.} +{\smath{\mbox{\bf hue}\opLeftPren{}c\opRightPren{}} returns +the hue index of the indicated color \smath{c}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{hypergeometric0F1}}\opLeftPren{}{\it complexDF}, \allowbreak{}{\it complexSF}\opRightPren{}% +\opdata{{\mbox{\axiomFun{hypergeometric0F1}}\opLeftPren{} +{\it complexDF}, \allowbreak{}{\it complexSF}\opRightPren{}% }% }% -{2}{(Complex(DoubleFloat), Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} -{\smath{\mbox{\bf hypergeometric0F1}\opLeftPren{}c, \allowbreak{} z\opRightPren{}} is the hypergeometric function \smath{0F1(c; z)}. +{2}{(Complex(DoubleFloat), Complex(DoubleFloat))->Complex(DoubleFloat)} +{DoubleFloatSpecialFunctions} +{\smath{\mbox{\bf hypergeometric0F1}\opLeftPren{}c, +\allowbreak{} z\opRightPren{}} is the hypergeometric function +\smath{0F1(c; z)}. Arguments \smath{c} and \smath{z} are both either small floats or complex small floats.} % ---------------------------------------------------------------------- @@ -73177,7 +75656,8 @@ or complex small floats.} }% }% {1}{(List(DPoly))->\$}{PolynomialIdeals} -{\smath{\mbox{\bf ideal}\opLeftPren{}polyList\opRightPren{}} constructs the ideal generated by the +{\smath{\mbox{\bf ideal}\opLeftPren{}polyList\opRightPren{}} +constructs the ideal generated by the list of polynomials \smath{polyList}. } @@ -73185,45 +75665,57 @@ list of polynomials \smath{polyList}. % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{imag}}\opLeftPren{}{\it expression}\opRightPren{}% - \optand \mbox{\axiomFun{imagi}}\opLeftPren{}{\it quaternionOrOctonion}\opRightPren{}% + \optand \mbox{\axiomFun{imagi}}\opLeftPren{} +{\it quaternionOrOctonion}\opRightPren{}% \opand \mbox{\axiomFun{imagI}}\opLeftPren{}{\it octonion}\opRightPren{}% }% }% {1}{(\$)->R}{OctonionCategory} -{\smath{\mbox{\bf imag}\opLeftPren{}x\opRightPren{}} extracts the imaginary part of a complex value or +{\smath{\mbox{\bf imag}\opLeftPren{}x\opRightPren{}} extracts the +imaginary part of a complex value or expression \smath{x}. \newitem -\smath{\mbox{\bf imagI}\opLeftPren{}q\opRightPren{}} extracts the \smath{i} part of quaternion \smath{q}. +\smath{\mbox{\bf imagI}\opLeftPren{}q\opRightPren{}} extracts the +\smath{i} part of quaternion \smath{q}. Similarly, operations \spadfun{imagJ}, and \spadfun{imagK} are used to extract the \smath{j} and \smath{k} parts. \newitem -\smath{\mbox{\bf imagi}\opLeftPren{}o\opRightPren{}} extracts the \smath{i} part of octonion \smath{o}. +\smath{\mbox{\bf imagi}\opLeftPren{}o\opRightPren{}} extracts the +\smath{i} part of octonion \smath{o}. Similarly, \spadfun{imagj}, \spadfun{imagk}, \spadfun{imagE}, -\spadfun{imagI}, \spadfun{imagJ}, and \spadfun{imagK} are used to extract other parts. +\spadfun{imagI}, \spadfun{imagJ}, and \spadfun{imagK} are used to extract +other parts. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{implies}}\opLeftPren{}{\it boolean}, \allowbreak{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{implies}}\opLeftPren{}{\it boolean}, \allowbreak{} +{\it boolean}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{Boolean} -{\smath{\mbox{\bf implies}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} tests if boolean value \smath{a} +{\smath{\mbox{\bf implies}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +tests if boolean value \smath{a} implies boolean value \smath{b}. The result is \smath{true} except when \smath{a} is \smath{true} and \smath{b} is \smath{false}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{in?}}\opLeftPren{}{\it ideal}, \allowbreak{}{\it ideal}\opRightPren{}% +\opdata{{\mbox{\axiomFun{in?}}\opLeftPren{}{\it ideal}, \allowbreak{} +{\it ideal}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{PolynomialIdeals} -{\smath{\mbox{\bf in?}\opLeftPren{}I, \allowbreak{} J\opRightPren{}} tests if the ideal \smath{I} is contained in the ideal \smath{J}. +{\smath{\mbox{\bf in?}\opLeftPren{}I, \allowbreak{} J\opRightPren{}} +tests if the ideal \smath{I} is contained in the ideal \smath{J}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{inHallBasis}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{inHallBasis}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}, \allowbreak{} +{\it integer}, \allowbreak{} +{\it integer}\opRightPren{}% }% }% {4}{(Integer, Integer, Integer, Integer)->Boolean}{HallBasis} @@ -73237,7 +75729,8 @@ the basis if in the unique factorization of \smath{leftOfRight <= leftCandidate}} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{increasePrecision}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{increasePrecision}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% {1}{(Integer)->PositiveInteger}{FloatingPointSystem} @@ -73246,11 +75739,13 @@ increases the current \spadfunFrom{precision}{FloatingPointSystem} by \smath{n} decimal digits.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{index}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{index}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(PositiveInteger)->\$}{Finite} -{\smath{\mbox{\bf index}\opLeftPren{}i\opRightPren{}} takes a positive integer \smath{i} less than or +{\smath{\mbox{\bf index}\opLeftPren{}i\opRightPren{}} takes a +positive integer \smath{i} less than or equal to \smath{\mbox{\bf size}\opLeftPren{}\opRightPren{}} and returns the \eth{\smath{i}} element of the set. This operation establishes a bijection between @@ -73258,11 +75753,13 @@ the elements of the finite set and \smath{1..{\mbox {\bf size}}()}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{index?}}\opLeftPren{}{\it index}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{index?}}\opLeftPren{}{\it index}, +\allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{(Index, \$)->Boolean}{IndexedAggregate} -{\smath{\mbox{\bf index?}\opLeftPren{}i, \allowbreak{} u\opRightPren{}} tests if \smath{i} is an index of +{\smath{\mbox{\bf index?}\opLeftPren{}i, \allowbreak{} u\opRightPren{}} +tests if \smath{i} is an index of aggregate \smath{u}. For example, \code{index?(2, [1, 2, 3])} is \smath{true} but \code{index?(4, [1, 2, 3])} is \smath{false}. @@ -73270,22 +75767,27 @@ but \code{index?(4, [1, 2, 3])} is \smath{false}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{infieldIntegrate}}\opLeftPren{}{\it rationalFunction}, \allowbreak{}{\it symbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{infieldIntegrate}}\opLeftPren{} +{\it rationalFunction}, \allowbreak{}{\it symbol}\opRightPren{}% }% }% -{2}{(Fraction(Polynomial(F)), Symbol)->Union(Fraction(Polynomial(F)), "failed")}{RationalFunctionIntegration} -{\smath{\mbox{\bf infieldIntegrate}\opLeftPren{}f, \allowbreak{} x\opRightPren{}}, +{2}{(Fraction(Polynomial(F)), Symbol)-> +Union(Fraction(Polynomial(F)), "failed")}{RationalFunctionIntegration} +{\smath{\mbox{\bf infieldIntegrate}\opLeftPren{}f, +\allowbreak{} x\opRightPren{}}, where \smath{f} is a fraction of polynomials, returns a fraction \smath{g} such that \smath{{dg\over dx} = f} if \smath{g} exists, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{infinite?}}\opLeftPren{}{\it orderedCompletion}\opRightPren{}% +\opdata{{\mbox{\axiomFun{infinite?}}\opLeftPren{} +{\it orderedCompletion}\opRightPren{}% }% }% {1}{(\$)->Boolean}{OrderedCompletion} -{\smath{\mbox{\bf infinite?}\opLeftPren{}x\opRightPren{}} tests if \smath{x} is infinite, +{\smath{\mbox{\bf infinite?}\opLeftPren{}x\opRightPren{}} tests +if \smath{x} is infinite, where \smath{x} is a member of the ordered completion of a domain. \seeType{OrderedCompletion}} @@ -73295,14 +75797,17 @@ completion of a domain. }% }% {0}{()->OnePointCompletion(Integer)}{Infinity} -{\smath{\mbox{\bf infinity}\opLeftPren{}\opRightPren{}} returns \smath{infinity} +{\smath{\mbox{\bf infinity}\opLeftPren{}\opRightPren{}} returns +\smath{infinity} denoting \smath{+\infty} as a one point completion of the integers. \seeType{OnePointCompletion} See also \spadfun{minusInfinity} and \spadfun{plusInfinitity}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{infix}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForms}\allowbreak $\,[$ , \allowbreak{}{\it OutputForm}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{infix}}\opLeftPren{}{\it outputForm}, +\allowbreak{}{\it outputForms}\allowbreak $\,[$ , +\allowbreak{}{\it OutputForm}$]$\opRightPren{}% }% }% {}{}{} @@ -73312,114 +75817,146 @@ and \smath{lo} is a list of objects of type \spadtype{OutputForm}, creates a form depicting the \smath{n}ary application of infix operation \smath{o} to a tuple of arguments \smath{lo}. \newitem -\smath{\mbox{\bf infix}\opLeftPren{}o, \allowbreak{} a, \allowbreak{} b\opRightPren{}}, -where \smath{o}, \smath{a}, and \smath{b} are objects of type \spadtype{OutputForm} (normally unexposed), +\smath{\mbox{\bf infix}\opLeftPren{}o, \allowbreak{} a, +\allowbreak{} b\opRightPren{}}, +where \smath{o}, \smath{a}, and \smath{b} are objects of type +\spadtype{OutputForm} (normally unexposed), creates an output form which displays as: \smath{a\ {\rm op}\ b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{initial}}\opLeftPren{}{\it differentialPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{initial}}\opLeftPren{} +{\it differentialPolynomial}\opRightPren{}% }% }% {1}{(\$)->\$}{DifferentialPolynomialCategory} -{\smath{\mbox{\bf initial}\opLeftPren{}p\opRightPren{}} returns the leading coefficient of +{\smath{\mbox{\bf initial}\opLeftPren{}p\opRightPren{}} returns the +leading coefficient of differential polynomial \smath{p} expressed as a univariate polynomial in its leader.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{initializeGroupForWordProblem}}\opLeftPren{}{\it group\opt{, integer, integer}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{initializeGroupForWordProblem}}\opLeftPren{} +{\it group\opt{, integer, integer}}\opRightPren{}% }% }% {1}{(\$)->Void}{PermutationGroup} -{\smath{\mbox{\bf initializeGroupForWordProblem}\opLeftPren{}gp \optinner{, n, m}\opRightPren{}} +{\smath{\mbox{\bf initializeGroupForWordProblem} +\opLeftPren{}gp \optinner{, n, m}\opRightPren{}} initializes the group {\it gp} for the word problem. \seeDetails{PermutationGroup} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{input}}\opLeftPren{}{\it operator\opt{, function}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{input}}\opLeftPren{} +{\it operator\opt{, function}}\opRightPren{}% }% }% {1}{(\$)->Union((List(InputForm))->InputForm, "failed")}{BasicOperator} -{\smath{\mbox{\bf input}\opLeftPren{}op\opRightPren{}} returns the \mbox{\tt "\%input"} property of \smath{op} +{\smath{\mbox{\bf input}\opLeftPren{}op\opRightPren{}} returns the +\mbox{\tt "\%input"} property of \smath{op} if it has one attached, and \mbox{\tt "failed"} otherwise. \newitem -\smath{\mbox{\bf input}\opLeftPren{}op, \allowbreak{} f\opRightPren{}} attaches \smath{f} as the \mbox{\tt "\%input"} property +\smath{\mbox{\bf input}\opLeftPren{}op, \allowbreak{} f\opRightPren{}} +attaches \smath{f} as the \mbox{\tt "\%input"} property of \smath{op}. -If \smath{op} has a \mbox{\tt "\%input"} property \smath{f}, then \smath{op(a1, \ldots, an)} is +If \smath{op} has a \mbox{\tt "\%input"} property \smath{f}, then +\smath{op(a1, \ldots, an)} is converted to InputForm using \smath{f(a1, \ldots, an)}. Argument f must be a function with signature \spadsig{List(InputForm)}{InputForm}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{inRadical?}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it ideal}\opRightPren{}% +\opdata{{\mbox{\axiomFun{inRadical?}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it ideal}\opRightPren{}% }% }% {2}{(DPoly, \$)->Boolean}{PolynomialIdeals} -{\smath{\mbox{\bf inRadical?}\opLeftPren{}f, \allowbreak{} I\opRightPren{}} tests if some power of the polynomial \smath{f} belongs to the ideal \smath{I}. +{\smath{\mbox{\bf inRadical?}\opLeftPren{}f, \allowbreak{} I\opRightPren{}} +tests if some power of the polynomial \smath{f} belongs to the ideal \smath{I}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{insert}}\opLeftPren{}{\it x}, \allowbreak{}{\it aggregate}\allowbreak $\,[$ , \allowbreak{}{\it integer}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{insert}}\opLeftPren{}{\it x}, \allowbreak{} +{\it aggregate}\allowbreak $\,[$ , +\allowbreak{}{\it integer}$]$\opRightPren{}% }% }% {3}{(S, \$, Integer)->\$}{LinearAggregate} -{\smath{\mbox{\bf insert}\opLeftPren{}x, \allowbreak{} u, \allowbreak{} i\opRightPren{}} returns a copy of \smath{u} having +{\smath{\mbox{\bf insert}\opLeftPren{}x, \allowbreak{} u, +\allowbreak{} i\opRightPren{}} returns a copy of \smath{u} having \smath{x} as its \eth{\smath{i}} element. -\newitem\smath{\mbox{\bf insert}\opLeftPren{}v, \allowbreak{} u, \allowbreak{} k\opRightPren{}} returns a copy of \smath{u} having \smath{v} +\newitem\smath{\mbox{\bf insert}\opLeftPren{}v, \allowbreak{} u, +\allowbreak{} k\opRightPren{}} returns a copy of \smath{u} having \smath{v} inserted beginning at the \eth{\smath{i}} element. \newitem -\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} destructively inserts item \smath{x} into +\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} +destructively inserts item \smath{x} into bag \smath{u}. \newitem -\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} destructively inserts item \smath{x} as a +\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} +destructively inserts item \smath{x} as a leaf into binary search tree or binary tournament \spad{u}. \newitem -\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} u, \allowbreak{} i\opRightPren{}} destructively inserts \smath{x} into +\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} u, +\allowbreak{} i\opRightPren{}} destructively inserts \smath{x} into aggregate \smath{u} at position \smath{i}. \newitem -\smath{\mbox{\bf insert!}\opLeftPren{}v, \allowbreak{} u, \allowbreak{} i\opRightPren{}} destructively inserts aggregate \smath{v} +\smath{\mbox{\bf insert!}\opLeftPren{}v, \allowbreak{} u, +\allowbreak{} i\opRightPren{}} destructively inserts aggregate \smath{v} into \smath{u} at position \smath{i}. \newitem -\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} d, \allowbreak{} n\opRightPren{}} destructively inserts \smath{n} copies of +\smath{\mbox{\bf insert!}\opLeftPren{}x, \allowbreak{} d, +\allowbreak{} n\opRightPren{}} destructively inserts \smath{n} copies of \smath{x} into dictionary \smath{d}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{insertBottom!}}\opLeftPren{}{\it element}, \allowbreak{}{\it queue}\opRightPren{}% +\opdata{{\mbox{\axiomFun{insertBottom!}}\opLeftPren{} +{\it element}, \allowbreak{}{\it queue}\opRightPren{}% }% }% {2}{(S, \$)->S}{DequeueAggregate} -{\smath{\mbox{\bf insertBottom!}\opLeftPren{}x, \allowbreak{} d\opRightPren{}} destructively inserts \smath{x} into the dequeue \smath{d} at the bottom (back) of the dequeue. +{\smath{\mbox{\bf insertBottom!}\opLeftPren{}x, +\allowbreak{} d\opRightPren{}} destructively inserts \smath{x} +into the dequeue \smath{d} at the +bottom (back) of the dequeue. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{insertTop!}}\opLeftPren{}{\it element}, \allowbreak{}{\it dequeue}\opRightPren{}% +\opdata{{\mbox{\axiomFun{insertTop!}}\opLeftPren{}{\it element}, +\allowbreak{}{\it dequeue}\opRightPren{}% }% }% {2}{(S, \$)->S}{DequeueAggregate} -{\smath{\mbox{\bf insertTop!}\opLeftPren{}x, \allowbreak{} d\opRightPren{}} destructively inserts \smath{x} into the dequeue \smath{d} -at the top (front). The element previously at the top of the dequeue becomes the second in the dequeue, and so on. +{\smath{\mbox{\bf insertTop!}\opLeftPren{}x, \allowbreak{} d\opRightPren{}} +destructively inserts \smath{x} into the dequeue \smath{d} +at the top (front). The element previously at the top of the +dequeue becomes the second in the dequeue, and so on. } % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{integer}}\opLeftPren{}{\it expression}\opRightPren{}% \optand \mbox{\axiomFun{integer?}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{integerIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{integerIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(S)->Boolean}{IntegerRetractions} -{\smath{\mbox{\bf integer}\opLeftPren{}x\opRightPren{}} returns \smath{x} as an integer, or calls \spadfun{error} +{\smath{\mbox{\bf integer}\opLeftPren{}x\opRightPren{}} returns +\smath{x} as an integer, or calls \spadfun{error} if this is not possible. \newitem -\smath{\mbox{\bf integer?}\opLeftPren{}x\opRightPren{}} tests if expression \smath{x} is an integer. +\smath{\mbox{\bf integer?}\opLeftPren{}x\opRightPren{}} tests +if expression \smath{x} is an integer. \newitem -\smath{\mbox{\bf integerIfCan}\opLeftPren{}x\opRightPren{}} returns expression x as of type \spadtype{Integer} +\smath{\mbox{\bf integerIfCan}\opLeftPren{}x\opRightPren{}} +returns expression x as of type \spadtype{Integer} or else \mbox{\tt "failed"} if it cannot. } @@ -73428,21 +75965,26 @@ or else \mbox{\tt "failed"} if it cannot. }% }% {1}{(\$)->Integer}{RealNumberSystem} -{\smath{\mbox{\bf integerPart}\opLeftPren{}fl\opRightPren{}} returns the integer part of the +{\smath{\mbox{\bf integerPart}\opLeftPren{}fl\opRightPren{}} +returns the integer part of the mantissa of float \smath{fl}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{integral}}\opLeftPren{}{\it expression}, \allowbreak{}{\it symbol}\opRightPren{}% - \opand \mbox{\axiomFun{integral}}\opLeftPren{}{\it expression}, \allowbreak{}{\it segmentBinding}\opRightPren{}% +\opdata{{\mbox{\axiomFun{integral}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it symbol}\opRightPren{}% + \opand \mbox{\axiomFun{integral}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it segmentBinding}\opRightPren{}% }% }% {2}{(\$, SegmentBinding(\$))->\$}{PrimitiveFunctionCategory} -{\smath{\mbox{\bf integral}\opLeftPren{}f, \allowbreak{} x\opRightPren{}} returns the formal integral \smath{\int {f dx}}. +{\smath{\mbox{\bf integral}\opLeftPren{}f, +\allowbreak{} x\opRightPren{}} returns the formal integral \smath{\int {f dx}}. \newitem -\smath{\mbox{\bf integral}\opLeftPren{}f, \allowbreak{} x = a..b\opRightPren{}} returns the formal definite integral +\smath{\mbox{\bf integral}\opLeftPren{}f, +\allowbreak{} x = a..b\opRightPren{}} returns the formal definite integral \smath{\int\nolimits_a^b{f(x) dx}}. } @@ -73454,37 +75996,45 @@ mantissa of float \smath{fl}.} {0}{()->Vector(\$)}{FunctionFieldCategory} {\opkey{Domain \smath{F} is the domain of functions on a fixed curve. \seeType{FunctionFieldCategory}} -\newitem\smath{\mbox{\bf integralBasisAtInfinity}\opLeftPren{}\opRightPren{}\$F} returns the local +\newitem\smath{\mbox{\bf integralBasisAtInfinity}\opLeftPren{} +\opRightPren{}\$F} returns the local integral basis at infinity. -\newitem\smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F} returns the integral basis for +\newitem\smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F} +returns the integral basis for the curve. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{integralCoordinates}}\opLeftPren{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{integralCoordinates}}\opLeftPren{} +{\it function}\opRightPren{}% }% }% {1}{(\$)->Record(num:Vector(UP), den:UP)}{FunctionFieldCategory} {\smath{\mbox{\bf integralCoordinates}\opLeftPren{}f\opRightPren{}}, where \smath{f} is a function on a curve defined by domain \smath{F}, -returns the coordinates of \smath{f} with respect to the \smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F} +returns the coordinates of \smath{f} with respect to the +\smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F} as polynomials \smath{A_i} together with a common denominator \smath{d}. Specifically, the operation returns a record having selector \smath{num} with value \smath{[A_1, \ldots, A_n]} and selector \smath{den} with value \smath{d} such that -\smath{f = (A_1 w_1 +\ldots+ A_n w_n) / d} where \smath{(w_1, \ldots, w_n)} is the +\smath{f = (A_1 w_1 +\ldots+ A_n w_n) / d} where +\smath{(w_1, \ldots, w_n)} is the integral basis. \seeType{FunctionFieldCategory} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{integralDerivationMatrix}}\opLeftPren{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{integralDerivationMatrix}}\opLeftPren{} +{\it function}\opRightPren{}% }% }% {1}{((UP)->UP)->Record(num:Matrix(UP), den:UP)}{FunctionFieldCategory} -{\smath{\mbox{\bf integralDerivationMatrix}\opLeftPren{}d\opRightPren{}} extends the derivation \smath{d} and +{\smath{\mbox{\bf integralDerivationMatrix}\opLeftPren{}d\opRightPren{}} +extends the derivation \smath{d} and returns the coordinates of the derivative of \smath{f} with respect to -the \smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F} as a matrix of polynomials and a common denominator \smath{Q}. +the \smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F} +as a matrix of polynomials and a common denominator \smath{Q}. Specifically, the operation returns a record having selector \smath{num} with value \smath{M} and selector \smath{den} with value \smath{Q} such that @@ -73501,28 +76051,34 @@ coordinates of \smath{f} with respect to integral basis }% {0}{()->Matrix(Fraction(UP))}{FunctionFieldCategory} {\opkey{Domain \smath{F} is a domain of functions on a fixed curve. These -operations return a matrix which transform the natural basis to an integral basis. +operations return a matrix which transform the natural basis to an +integral basis. \seeType{FunctionFieldCategory} } \newitem -\smath{\mbox{\bf integralMatrix}\opLeftPren{}\opRightPren{}} returns \smath{M} such that +\smath{\mbox{\bf integralMatrix}\opLeftPren{}\opRightPren{}} returns +\smath{M} such that \smath{(w_1, \ldots, w_n) = M (1, y, \ldots, y^{n-1})}, where \smath{(w_1, \ldots, w_n)} is the integral basis returned by \smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F}. \newitem -\smath{\mbox{\bf integralMatrixAtInfinity}\opLeftPren{}\opRightPren{}\$F} returns matrix \smath{M} +\smath{\mbox{\bf integralMatrixAtInfinity}\opLeftPren{}\opRightPren{}\$F} +returns matrix \smath{M} which transforms the natural basis such that \smath{(v_1, \ldots, v_n) = M (1, y, \ldots, y^{n-1})} where \smath{(v_1, \ldots, v_n)} is the local integral basis at -infinity returned by \smath{\mbox{\bf integralBasisAtInfinity}\opLeftPren{}\opRightPren{}\$F}. +infinity returned by \smath{\mbox{\bf integralBasisAtInfinity} +\opLeftPren{}\opRightPren{}\$F}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{integralRepresents}}\opLeftPren{}{\it vector}, \allowbreak{}{\it commonDenominator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{integralRepresents}} +\opLeftPren{}{\it vector}, \allowbreak{}{\it commonDenominator}\opRightPren{}% }% }% {2}{(Vector(UP), UP)->\$}{FunctionFieldCategory} -{\smath{\mbox{\bf integralRepresents}\opLeftPren{}[A_1, \allowbreak{} \ldots, A_n], d\opRightPren{}} is the inverse +{\smath{\mbox{\bf integralRepresents}\opLeftPren{} +[A_1, \allowbreak{} \ldots, A_n], d\opRightPren{}} is the inverse of the operation \spadfunFrom{integralCoordinates}{FunctionFieldCategory} defined for domain \smath{F}, a domain of functions on a fixed curve. Given @@ -73530,20 +76086,28 @@ the coordinates as polynomials \smath{[A_1, \ldots, A_n]} over a common denominator \smath{d}, this operation returns the function represented as\smath{(A_1 w_1+\ldots+A_n w_n)/d} where \smath{(w_1, \ldots, w_n)} is the integral basisreturned by -\smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F}.\seeType{FunctionFieldCategory}} +\smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F}. +\seeType{FunctionFieldCategory}} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{integrate}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{integrate}}\opLeftPren{}{\it expression}, \allowbreak{}{\it variable }\allowbreak $\,[$ , \allowbreak{}{\it options}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{integrate}}\opLeftPren{} +{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{integrate}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it variable } +\allowbreak $\,[$ , \allowbreak{} +{\it options}$]$\opRightPren{}% }% }% {1}{(\$)->\$}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf integrate}\opLeftPren{}f\opRightPren{}} returns the integral of a +{\smath{\mbox{\bf integrate}\opLeftPren{}f\opRightPren{}} +returns the integral of a univariate polynomial or power series \smath{f} with respect to its distinguished variable. -\newitem\smath{\mbox{\bf integrate}\opLeftPren{}f, \allowbreak{} x\opRightPren{}} returns the integral of +\newitem\smath{\mbox{\bf integrate}\opLeftPren{}f, +\allowbreak{} x\opRightPren{}} returns the integral of \smath{f(x)dx}, where \smath{x} is viewed as a real variable. -\newitem\smath{\mbox{\bf integrate}\opLeftPren{}f, \allowbreak{} x = a..b\optinner{, "noPole"}\opRightPren{}} +\newitem\smath{\mbox{\bf integrate}\opLeftPren{}f, +\allowbreak{} x = a..b\optinner{, "noPole"}\opRightPren{}} returns the integral of \smath{f(x)dx} from \smath{a} to \smath{b}. If it is not possible to check whether \smath{f} has a pole for \smath{x} between \smath{a} and \smath{b}, then a third argument @@ -73552,15 +76116,13 @@ has no such pole.This operation calls \spadfun{error} if \smath{f} has a pole for \smath{x} between \smath{a} and \smath{b} or if a third argument different from \mbox{\tt "noPole"} is given.} - - - % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{interpret}}\opLeftPren{}{\it inputForm}\opRightPren{}% }% }% { $ -> Any}{}{} -{\smath{\mbox{\bf interpret}\opLeftPren{}f\opRightPren{}} passes f of type \spadtype{InputForm} to the interpreter. +{\smath{\mbox{\bf interpret}\opLeftPren{}f\opRightPren{}} +passes f of type \spadtype{InputForm} to the interpreter. \newitem \smath{\mbox{\bf interpret}\opLeftPren{}f\opRightPren{}\$P}, where \smath{P} is the package \spadtype{InputFormFunctions1(R)} for @@ -73570,7 +76132,8 @@ and transforms the result into an object of type \smath{R}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{intersect}}\opLeftPren{}{\it elements\opt{, element}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{intersect}}\opLeftPren{} +{\it elements\opt{, element}}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{SetAggregate} @@ -73594,7 +76157,8 @@ computes the intersection of the ideals \smath{I} and \smath{J}. }% }% {1}{(\$)->\$}{DivisionRing} -{\smath{\mbox{\bf inv}\opLeftPren{}x\opRightPren{}} returns the multiplicative inverse of \smath{x}, +{\smath{\mbox{\bf inv}\opLeftPren{}x\opRightPren{}} +returns the multiplicative inverse of \smath{x}, where \smath{x} is an element of a domain of category \spadtype{Group} or \spadtype{DivisionRing}, or calls \spadfun{error} if \smath{x} is 0. @@ -73605,17 +76169,23 @@ where \smath{x} is an element of a domain of category \spadtype{Group} or }% }% {1}{(\$)->Union(\$, "failed")}{MatrixCategory} -{\smath{\mbox{\bf inverse}\opLeftPren{}A\opRightPren{}} returns the inverse of the matrix \smath{A}, or -\mbox{\tt "failed"} if the matrix is not invertible, or calls \spadfun{error} if +{\smath{\mbox{\bf inverse}\opLeftPren{}A\opRightPren{}} +returns the inverse of the matrix \smath{A}, or +\mbox{\tt "failed"} if the matrix is not invertible, +or calls \spadfun{error} if the matrix is not square. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{inverseColeman}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{inverseColeman}}\opLeftPren{} +{\it listOfIntegers}, \allowbreak{} +{\it listOfIntegers}, \allowbreak{}{\it matrix}\opRightPren{}% }% }% -{3}{(List(Integer), List(Integer), Matrix(Integer))->List(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf inverseColeman}\opLeftPren{}alpha, \allowbreak{} beta, \allowbreak{} C\opRightPren{}} +{3}{(List(Integer), List(Integer), Matrix(Integer))-> +List(Integer)}{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf inverseColeman}\opLeftPren{}alpha, +\allowbreak{} beta, \allowbreak{} C\opRightPren{}} returns the lexicographically smallest permutation in a double coset of the symmetric group corresponding to a non-negative Coleman-matrix. @@ -73623,8 +76193,10 @@ corresponding to a non-negative Coleman-matrix. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{inverseIntegralMatrix}}\opLeftPren{}\opRightPren{}% - \opand \mbox{\axiomFun{inverseIntegralMatrixAtInfinity}}\opLeftPren{}\opRightPren{}% +\opdata{{\mbox{\axiomFun{inverseIntegralMatrix}}\opLeftPren{} +\opRightPren{} +\opand \mbox{\axiomFun{inverseIntegralMatrixAtInfinity}} +\opLeftPren{}\opRightPren{}% }% }% {0}{()->Matrix(Fraction(UP))}{FunctionFieldCategory} @@ -73633,12 +76205,14 @@ curve. These operations return a matrix which transform an integral basis to a natural basis. \seeType{FunctionFieldCategory}} -\newitem\smath{\mbox{\bf inverseIntegralMatrix}\opLeftPren{}\opRightPren{}\$F} returns \smath{M} such +\newitem\smath{\mbox{\bf inverseIntegralMatrix}\opLeftPren{} +\opRightPren{}\$F} returns \smath{M} such that \smath{M (w_1, \ldots, w_n) = (1, y, \ldots, y^{n-1})} where \smath{(w_1, \ldots, w_n)} is the integral basis returned by \smath{\mbox{\bf integralBasis}\opLeftPren{}\opRightPren{}\$F}. See also \spadfunFrom{integralMatrix}{FunctionFieldCategory}. -\newitem \smath{\mbox{\bf inverseIntegralMatrixAtInfinity}\opLeftPren{}\opRightPren{}} returns +\newitem \smath{\mbox{\bf inverseIntegralMatrixAtInfinity}\opLeftPren{} +\opRightPren{}} returns \smath{M} such that \smath{M (v_1, \ldots, v_n) = (1, y, \ldots, y^(n-1))} where \smath{(v_1, \ldots, v_n)} is the local integral basis at infinity returned by @@ -73648,21 +76222,25 @@ See also } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{inverseLaplace}}\opLeftPren{}{\it expression}, \allowbreak{}{\it symbol}, \allowbreak{}{\it symbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{inverseLaplace}}\opLeftPren{}{\it expression}, +\allowbreak{}{\it symbol}, \allowbreak{}{\it symbol}\opRightPren{}% }% }% {3}{(F, Symbol, Symbol)->Union(F, "failed")}{InverseLaplaceTransform} -{\smath{\mbox{\bf inverseLaplace}\opLeftPren{}f, \allowbreak{} s, \allowbreak{} t\opRightPren{}} returns the Inverse Laplace +{\smath{\mbox{\bf inverseLaplace}\opLeftPren{}f, \allowbreak{} s, +\allowbreak{} t\opRightPren{}} returns the Inverse Laplace transform of \smath{f(s)} using \smath{t} as the new variable, or \mbox{\tt "failed"} if unable to find a closed form. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{invmod}}\opLeftPren{}{\it positiveInteger}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{invmod}}\opLeftPren{}{\it positiveInteger}, +\allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf invmod}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}, for relatively prime positive integers \smath{a} and \smath{b} +{\smath{\mbox{\bf invmod}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}, +for relatively prime positive integers \smath{a} and \smath{b} such that \smath{a < b}, returns \smath{1/a \mod b}. } @@ -73672,36 +76250,42 @@ such that \smath{a < b}, returns \smath{1/a \mod b}. }% }% {1}{(\$)->String}{FileCategory} -{\smath{\mbox{\bf iomode}\opLeftPren{}f\opRightPren{}} returns the status of the file \smath{f} +{\smath{\mbox{\bf iomode}\opLeftPren{}f\opRightPren{}} returns the +status of the file \smath{f} as one of the following strings: {\tt "input", "output"} or {\tt "closed".} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{irreducible?}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{irreducible?}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(FP)->Boolean}{DistinctDegreeFactorize} -{\smath{\mbox{\bf irreducible?}\opLeftPren{}p\opRightPren{}} tests whether the polynomial \smath{p} is irreducible. +{\smath{\mbox{\bf irreducible?}\opLeftPren{}p\opRightPren{}} +tests whether the polynomial \smath{p} is irreducible. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{irreducibleFactor}}\opLeftPren{}{\it element}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{irreducibleFactor}}\opLeftPren{} +{\it element}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(R, Integer)->\$}{Factored} -{\smath{\mbox{\bf irreducibleFactor}\opLeftPren{}base, \allowbreak{} exponent\opRightPren{}} creates a factored +{\smath{\mbox{\bf irreducibleFactor}\opLeftPren{}base, \allowbreak{} +exponent\opRightPren{}} creates a factored object with a single factor whose \smath{base} is asserted to be irreducible (flag = \mbox{\tt "irred"}). } - - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{irreducibleRepresentation}}\opLeftPren{}{\it listOfIntegers\opt{, permutations}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{irreducibleRepresentation}}\opLeftPren{} +{\it listOfIntegers\opt{, permutations}}\opRightPren{}% }% }% -{2}{(List(Integer), Permutation(Integer))->Matrix(Integer)}{irreducibleRepresentationPackage} -{\smath{\mbox{\bf irreducibleRepresentation}\opLeftPren{}lambda\optinner{, pi}\opRightPren{}} returns +{2}{(List(Integer), Permutation(Integer))->Matrix(Integer)} +{irreducibleRepresentationPackage} +{\smath{\mbox{\bf irreducibleRepresentation}\opLeftPren{}lambda +\optinner{, pi}\opRightPren{}} returns a matrix giving the irreducible representation corresponding to partition \smath{lambda}, represented as a list of integers, in Young's natural form of the permutation \smath{pi} in the @@ -73713,33 +76297,39 @@ the following two generators of the symmetric group, namely } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{is?}}\opLeftPren{}{\it expression}, \allowbreak{}{\it pattern}\opRightPren{}% +\opdata{{\mbox{\axiomFun{is?}}\opLeftPren{}{\it expression}, \allowbreak{} +{\it pattern}\opRightPren{}% }% }% {2}{(\$, BasicOperator)->Boolean}{ExpressionSpace} -{\smath{\mbox{\bf is?}\opLeftPren{}expr, \allowbreak{} pat\opRightPren{}} tests if the expression \smath{expr} matches the pattern \smath{pat}. +{\smath{\mbox{\bf is?}\opLeftPren{}expr, \allowbreak{} pat\opRightPren{}} +tests if the expression \smath{expr} matches the pattern \smath{pat}. \newitem -\smath{\mbox{\bf is?}\opLeftPren{}expression, \allowbreak{} op\opRightPren{}} tests if \smath{expression} is a kernel and is its operator is op. +\smath{\mbox{\bf is?}\opLeftPren{}expression, \allowbreak{} op\opRightPren{}} +tests if \smath{expression} is a kernel and is its operator is op. } - - % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{isAbsolutelyIrreducible?}}\opLeftPren{}{\it listOfMatrices}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{isAbsolutelyIrreducible?}}\opLeftPren{} +{\it listOfMatrices}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(List(Matrix(R)), Integer)->Boolean}{RepresentationPackage2} -{\smath{\mbox{\bf isAbsolutelyIrreducible?}\opLeftPren{}aG, \allowbreak{} numberOfTries\opRightPren{}} uses Norton's irreducibility +{\smath{\mbox{\bf isAbsolutelyIrreducible?}\opLeftPren{}aG, \allowbreak{} +numberOfTries\opRightPren{}} uses Norton's irreducibility test to check for absolute irreduciblity. \seeDetails{RepresentationPackage2} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{isExpt}}\opLeftPren{}{\it expression\opt{, operator}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{isExpt}}\opLeftPren{} +{\it expression\opt{, operator}}\opRightPren{}% }% }% -{1}{(\$)->Union(Record(var:Kernel(\$), exponent:Integer), "failed")}{FunctionSpace} -{\smath{\mbox{\bf isExpt}\opLeftPren{}p\optinner{, op}\opRightPren{}} returns a record with two fields: +{1}{(\$)->Union(Record(var:Kernel(\$), exponent:Integer), "failed")} +{FunctionSpace} +{\smath{\mbox{\bf isExpt}\opLeftPren{}p\optinner{, op}\opRightPren{}} +returns a record with two fields: \smath{var} denoting a kernel \smath{x}, and \smath{exponent} denoting an integer \smath{n}, if expression \smath{p} @@ -73762,19 +76352,24 @@ and \mbox{\tt "failed"} if this is not possible. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{isobaric?}}\opLeftPren{}{\it differentialPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{isobaric?}}\opLeftPren{} +{\it differentialPolynomial}\opRightPren{}% }% }% {1}{(\$)->Boolean}{DifferentialPolynomialCategory} -{\smath{\mbox{\bf isobaric?}\opLeftPren{}p\opRightPren{}} tests if every differential monomial appearing in the differential polynomial \smath{p} has the same weight.} +{\smath{\mbox{\bf isobaric?}\opLeftPren{}p\opRightPren{}} +tests if every differential monomial appearing in the differential polynomial +\smath{p} has the same weight.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{isPlus}}\opLeftPren{}{\it expression}\opRightPren{}% }% }% {1}{(\$)->Union(List(\$), "failed")}{FunctionSpace} -{\smath{\mbox{\bf isPlus}\opLeftPren{}p\opRightPren{}} returns \smath{[m_1, \ldots, m_n]} if -\smath{p} has the form \smath{m_1 +\ldots+ m_n} for \smath{n > 1} and \smath{m_i \not= 0}, +{\smath{\mbox{\bf isPlus}\opLeftPren{}p\opRightPren{}} +returns \smath{[m_1, \ldots, m_n]} if +\smath{p} has the form \smath{m_1 +\ldots+ m_n} for +\smath{n > 1} and \smath{m_i \not= 0}, and \mbox{\tt "failed"} if this is not possible. } @@ -73783,7 +76378,8 @@ and \mbox{\tt "failed"} if this is not possible. }% }% {1}{(\$)->Union(List(\$), "failed")}{FunctionSpace} -{\smath{\mbox{\bf isTimes}\opLeftPren{}p\opRightPren{}} returns \smath{[a_1, \ldots, a_n]} if +{\smath{\mbox{\bf isTimes}\opLeftPren{}p\opRightPren{}} +returns \smath{[a_1, \ldots, a_n]} if \smath{p} has the form \smath{a_1*\ldots*a_n} for \smath{n > 1} and \smath{m_i \not= 1}, and \mbox{\tt "failed"} if this is not possible. @@ -73791,10 +76387,12 @@ and \mbox{\tt "failed"} if this is not possible. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{Is}}\opLeftPren{}{\it subject}, \allowbreak{}{\it pattern}\opRightPren{}% +\opdata{{\mbox{\axiomFun{Is}}\opLeftPren{} +{\it subject}, \allowbreak{}{\it pattern}\opRightPren{}% }% }% -{2}{(List(Subject), Pat)->PatternMatchListResult(Base, Subject, List(Subject))}{PatternMatch} +{2}{(List(Subject), Pat)-> +PatternMatchListResult(Base, Subject, List(Subject))}{PatternMatch} {\smath{Is(expr, pat)} matches the pattern \smath{pat} on the expression \smath{expr} and returns a list of matches \smath{[v_1 = e_1, \ldots, v_n = e_n]} or \mbox{\tt "failed"} if matching @@ -73804,15 +76402,18 @@ to \smath{pat} or if \smath{pat} does not match \smath{expr}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{jacobi}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{jacobi}}\opLeftPren{}{\it integer}, +\allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(Integer, Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf jacobi}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns the Jacobi symbol \smath{J(a/b)}. +{\smath{\mbox{\bf jacobi}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns the Jacobi symbol \smath{J(a/b)}. \index{Jacobi symbol} When \smath{b} is odd, \smath{J(a/b) = \prod\nolimits_{p \in {\bf factors}(b)}{L(a/p)}}. -Note: by convention, 0 is returned if \smath{\mbox{\bf gcd}\opLeftPren{}a, b) \not= 1}. +Note: by convention, 0 is returned if +\smath{\mbox{\bf gcd}\opLeftPren{}a, b) \not= 1}. } % ---------------------------------------------------------------------- @@ -73820,20 +76421,24 @@ Note: by convention, 0 is returned if \smath{\mbox{\bf gcd}\opLeftPren{}a, b) \n }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf jacobiIdentity?}\opLeftPren{}\opRightPren{}} tests if \smath{(a b) c + (b c) a + (c a) b = 0} +{\smath{\mbox{\bf jacobiIdentity?}\opLeftPren{}\opRightPren{}} +tests if \smath{(a b) c + (b c) a + (c a) b = 0} for all \smath{a}, \smath{b}, \smath{c} in a domain of \spadtype{FiniteRankNonAssociativeAlgebra}. For example, this relation holds for crossed products of three-dimensional vectors. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{janko2}}\opLeftPren{}{\it \optinit{listOfIntegers}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{janko2}}\opLeftPren{} +{\it \optinit{listOfIntegers}}\opRightPren{}% }% }% {1}{(List(Integer))->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf janko2}\opLeftPren{}\opRightPren{}} constructs the janko group acting on the integers \smath{1, \ldots, 100}. +{\smath{\mbox{\bf janko2}\opLeftPren{}\opRightPren{}} constructs the janko +group acting on the integers \smath{1, \ldots, 100}. \newitem -\smath{\mbox{\bf janko2}\opLeftPren{}\optinner{li}\opRightPren{}} constructs the janko group acting on the 100 integers given in the list \smath{li}. +\smath{\mbox{\bf janko2}\opLeftPren{}\optinner{li}\opRightPren{}} constructs +the janko group acting on the 100 integers given in the list \smath{li}. The default value of \smath{li} is \smath{[1, \ldots, 100]}. This operation removes duplicates in the list and calls \spadfun{error} if \smath{li} does not have exactly 100 distinct entries. @@ -73851,10 +76456,12 @@ where \smath{F} is a member of a commutative ring \smath{R}, tests if 2 is invertible in \smath{R} and if the algebra defined -by \smath{\{a, b\}} defined by \smath{(1/2)(a b+b a)} is a Jordan algebra, that is, +by \smath{\{a, b\}} defined by \smath{(1/2)(a b+b a)} is a Jordan +algebra, that is, satisfies the Jordan identity. \newitem -\smath{\mbox{\bf jordanAlgebra?}\opLeftPren{}\opRightPren{}\$F} tests if the algebra is commutative, +\smath{\mbox{\bf jordanAlgebra?}\opLeftPren{}\opRightPren{}\$F} +tests if the algebra is commutative, that \smath{\mbox{\bf characteristic}\opLeftPren{})\$F \not= 2}, and \smath{(a b) a^2 - a (b a^2) = 0} for all \smath{a} and \smath{b} in the algebra (Jordan identity). Example: for every associative algebra @@ -73863,13 +76470,16 @@ where \smath{a*b := (a@b+b@a)/2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{kernel}}\opLeftPren{}{\it operator}, \allowbreak{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{kernel}}\opLeftPren{}{\it operator}, +\allowbreak{}{\it expression}\opRightPren{}% }% }% {2}{(BasicOperator, \$)->\$}{ExpressionSpace} -{\smath{\mbox{\bf kernel}\opLeftPren{}op, \allowbreak{} x\opRightPren{}} constructs \smath{op}(\smath{x}) without evaluating it. +{\smath{\mbox{\bf kernel}\opLeftPren{}op, \allowbreak{} x\opRightPren{}} +constructs \smath{op}(\smath{x}) without evaluating it. \newitem -\smath{\mbox{\bf kernel}\opLeftPren{}op, \allowbreak{} [f_1, \allowbreak{} \ldots, f_n]\opRightPren{}} constructs \smath{op(f1, \ldots, fn)} +\smath{\mbox{\bf kernel}\opLeftPren{}op, \allowbreak{} [f_1, +\allowbreak{} \ldots, f_n]\opRightPren{}} constructs \smath{op(f1, \ldots, fn)} without evaluating it. } @@ -73878,40 +76488,49 @@ without evaluating it. }% }% {1}{(\$)->List(Kernel(\$))}{ExpressionSpace} -{\smath{\mbox{\bf kernels}\opLeftPren{}f\opRightPren{}} returns the list of all the top-level kernels appearing in +{\smath{\mbox{\bf kernels}\opLeftPren{}f\opRightPren{}} returns the +list of all the top-level kernels appearing in expression \smath{f}, but not the ones appearing in the arguments of the top-level kernels. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{key?}}\opLeftPren{}{\it key}, \allowbreak{}{\it dictionary}\opRightPren{}% +\opdata{{\mbox{\axiomFun{key?}}\opLeftPren{}{\it key}, \allowbreak{} +{\it dictionary}\opRightPren{}% \opand \mbox{\axiomFun{keys}}\opLeftPren{}{\it dictionary}\opRightPren{}% }% }% {2}{(Key, \$)->Boolean}{KeyedDictionary} -{\smath{\mbox{\bf key?}\opLeftPren{}k, \allowbreak{} d\opRightPren{}} tests if \smath{k} is a key in dictionary \smath{d}. +{\smath{\mbox{\bf key?}\opLeftPren{}k, \allowbreak{} d\opRightPren{}} +tests if \smath{k} is a key in dictionary \smath{d}. Dictionary \smath{d} is an element of a domain of category \spadtype{KeyedDictionary(K, E)}, where \smath{K} and \smath{E} denote the domains of keys and entries. \newitem -\smath{\mbox{\bf keys}\opLeftPren{}d\opRightPren{}} returns the list the keys in table \smath{d}. +\smath{\mbox{\bf keys}\opLeftPren{}d\opRightPren{}} +returns the list the keys in table \smath{d}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{kroneckerDelta}}\opLeftPren{}{\it [integer}, \allowbreak{}{\it integer]}\opRightPren{}% +\opdata{{\mbox{\axiomFun{kroneckerDelta}}\opLeftPren{} +{\it [integer}, \allowbreak{}{\it integer]}\opRightPren{}% }% }% {0}{()->\$}{CartesianTensor} -{\smath{\mbox{\bf kroneckerDelta}\opLeftPren{}\opRightPren{}} is the rank 2 tensor defined by -\smath{\mbox{\bf kroneckerDelta}\opLeftPren{}i, j) = 1} if \smath{i=j}, and 0 otherwise. +{\smath{\mbox{\bf kroneckerDelta}\opLeftPren{}\opRightPren{}} +is the rank 2 tensor defined by +\smath{\mbox{\bf kroneckerDelta}\opLeftPren{}i, j) = 1} +if \smath{i=j}, and 0 otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{label}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{label}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf label}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, where \smath{o_1} and \smath{o_2} are +{\smath{\mbox{\bf label}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, +where \smath{o_1} and \smath{o_2} are objects of type \spadtype{OutputForm} (normally unexposed), returns an output form displaying equation \smath{o_2} with label @@ -73920,91 +76539,114 @@ an output form displaying equation \smath{o_2} with label % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{laguerreL}}\opLeftPren{}{\it nonNegativeInteger}, \allowbreak{}{\it x}\opRightPren{}% - \opand \mbox{\axiomFun{laguerreL}}\opLeftPren{}{\it nonNegativeInteger}, \allowbreak{}{\it nonNegativeInteger}, \allowbreak{}{\it x}\opRightPren{}% +\opdata{{\mbox{\axiomFun{laguerreL}}\opLeftPren{}{\it nonNegativeInteger}, +\allowbreak{}{\it x}\opRightPren{}% + \opand \mbox{\axiomFun{laguerreL}}\opLeftPren{}{\it nonNegativeInteger}, +\allowbreak{}{\it nonNegativeInteger}, \allowbreak{}{\it x}\opRightPren{}% }% }% {2}{(NonNegativeInteger, R)->R}{OrthogonalPolynomialFunctions} -{\smath{\mbox{\bf laguerreL}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} is the \eth{\smath{n}} Laguerre +{\smath{\mbox{\bf laguerreL}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} +is the \eth{\smath{n}} Laguerre polynomial, \smath{L[n](x)}, defined by \smath{{exp({{-t x}\over{1-t}})/(1-t)} = \sum\nolimits_{n=0}^\infty {L[n](x) t^n/n!}}. -\newitem \smath{\mbox{\bf laguerreL}\opLeftPren{}m, \allowbreak{} n, \allowbreak{} x\opRightPren{}} is the associated Laguerre +\newitem \smath{\mbox{\bf laguerreL}\opLeftPren{}m, +\allowbreak{} n, \allowbreak{} x\opRightPren{}} is the associated Laguerre polynomial, \smath{L_m[n](x)}, defined as the \eth{\smath{m}} derivative of \smath{L[n](x)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{lambda}}\opLeftPren{}{\it inputForm}, \allowbreak{}{\it listOfSymbols}\opRightPren{}% +\opdata{{\mbox{\axiomFun{lambda}}\opLeftPren{}{\it inputForm}, +\allowbreak{}{\it listOfSymbols}\opRightPren{}% }% }% { ($, List Symbol) -> $}{}{} -{\smath{\mbox{\bf lambda}\opLeftPren{}i, \allowbreak{} [x_1, \allowbreak{} \ldots x_n]\opRightPren{}} returns the input form +{\smath{\mbox{\bf lambda}\opLeftPren{}i, \allowbreak{} [x_1, +\allowbreak{} \ldots x_n]\opRightPren{}} returns the input form corresponding to \smath{(x_1, \ldots, x_n) \mapsto i} if \smath{n > 1}. See also \spadfun{compiledFunction}, \spadfun{flatten}, and \spadfun{unparse}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{laplace}}\opLeftPren{}{\it expression}, \allowbreak{}{\it symbol}, \allowbreak{}{\it symbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{laplace}}\opLeftPren{}{\it expression}, +\allowbreak{}{\it symbol}, \allowbreak{}{\it symbol}\opRightPren{}% }% }% {3}{(F, Symbol, Symbol)->F}{LaplaceTransform} -{\smath{\mbox{\bf laplace}\opLeftPren{}f, \allowbreak{} t, \allowbreak{} s\opRightPren{}} returns the Laplace transform of \smath{f(t)}, +{\smath{\mbox{\bf laplace}\opLeftPren{}f, \allowbreak{} t, +\allowbreak{} s\opRightPren{}} returns the Laplace transform of \smath{f(t)}, defined by \smath{\int_{t=0}^\infty{exp(-s t)f(t) {\rm dt}}}. If the transform cannot be computed, the -formal object \smath{\mbox{\bf laplace}\opLeftPren{}f, \allowbreak{} t, \allowbreak{} s\opRightPren{}} is returned. +formal object \smath{\mbox{\bf laplace}\opLeftPren{}f, +\allowbreak{} t, \allowbreak{} s\opRightPren{}} is returned. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{last}}\opLeftPren{}{\it indexedAggregate\opt{, nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{last}}\opLeftPren{}{\it indexedAggregate +\opt{, nonNegativeInteger}}\opRightPren{}% }% }% {1}{(\$)->S}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf last}\opLeftPren{}u\opRightPren{}} returns the last element of \smath{u}. +{\smath{\mbox{\bf last}\opLeftPren{}u\opRightPren{}} returns the last +element of \smath{u}. \newitem -\smath{\mbox{\bf last}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns a copy of the last \smath{n} (\smath{n +\smath{\mbox{\bf last}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} +returns a copy of the last \smath{n} (\smath{n \geq 0}) elements of \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{laurent}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{laurentIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{laurent}}\opLeftPren{} +{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{laurentIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(\$)->ULS}{UnivariatePuiseuxSeriesConstructorCategory} -{\smath{\mbox{\bf laurent}\opLeftPren{}u\opRightPren{}} converts \smath{u} to a Laurent series, or +{\smath{\mbox{\bf laurent}\opLeftPren{}u\opRightPren{}} converts +\smath{u} to a Laurent series, or calls \spadfun{error} if this is not possible. \newitem -\smath{\mbox{\bf laurentIfCan}\opLeftPren{}u\opRightPren{}} converts the Puiseux series \smath{u} +\smath{\mbox{\bf laurentIfCan}\opLeftPren{}u\opRightPren{}} converts the +Puiseux series \smath{u} to a Laurent series, or returns \mbox{\tt "failed"} if this is not possible. \newitem -\smath{\mbox{\bf laurent}\opLeftPren{}f, \allowbreak{} x = a\opRightPren{}} expands the expression \smath{f} as a +\smath{\mbox{\bf laurent}\opLeftPren{}f, \allowbreak{} x = a\opRightPren{}} +expands the expression \smath{f} as a Laurent series in powers of \smath{(x - a)}. \newitem -\smath{\mbox{\bf laurent}\opLeftPren{}f, \allowbreak{} n\opRightPren{}} expands the expression \smath{f} as a Laurent series in +\smath{\mbox{\bf laurent}\opLeftPren{}f, \allowbreak{} n\opRightPren{}} +expands the expression \smath{f} as a Laurent series in powers of \smath{x}; at least \smath{n} terms are computed. \newitem -\smath{\mbox{\bf laurent}\opLeftPren{}n \mapsto a_n, x = a, n_0..\optinner{n_1}\opRightPren{}} returns +\smath{\mbox{\bf laurent}\opLeftPren{}n \mapsto a_n, x = a, +n_0..\optinner{n_1}\opRightPren{}} returns a Laurent series defined by \smath{\sum\nolimits_{n = n_0}^{n_1}{a_n (x - a)^n}}, where \smath{n_1} is \smath{\infty} by default. \newitem -\smath{\mbox{\bf laurent}\opLeftPren{}a_n, \allowbreak{} n, \allowbreak{} x=a, \allowbreak{} n_0..\optinner{n_1}\opRightPren{}} returns +\smath{\mbox{\bf laurent}\opLeftPren{}a_n, \allowbreak{} n, +\allowbreak{} x=a, \allowbreak{} n_0..\optinner{n_1}\opRightPren{}} returns a Laurent series defined by \smath{\sum\nolimits_{n=n_0}^{n_1}{a_n(x - a)^n}}, where \smath{n_1} is \smath{\infty} by default. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{laurentRep}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{laurentRep}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(\$)->ULS}{UnivariatePuiseuxSeriesConstructorCategory} -{\smath{\mbox{\bf laurentRep}\opLeftPren{}f(x)\opRightPren{}} returns \smath{g(x)} -where the Puiseux series \smath{f(x) = g(x^r)} is represented by \smath{[r, g(x)]}. +{\smath{\mbox{\bf laurentRep}\opLeftPren{}f(x)\opRightPren{}} +returns \smath{g(x)} +where the Puiseux series \smath{f(x) = g(x^r)} is +represented by \smath{[r, g(x)]}. } % ---------------------------------------------------------------------- @@ -74013,15 +76655,18 @@ where the Puiseux series \smath{f(x) = g(x^r)} is represented by \smath{[r, g(x) }% {1}{(\$)->Boolean}{LazyStreamAggregate} {\smath{\mbox{\bf lazy?}\opLeftPren{}s\opRightPren{}} tests -if the first node of the stream \smath{s} is a lazy evaluation mechanism which could produce an additional entry to \smath{s}. +if the first node of the stream \smath{s} is a lazy +evaluation mechanism which could produce an additional entry to \smath{s}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{lazyEvaluate}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{lazyEvaluate}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(\$)->\$}{LazyStreamAggregate} -{\smath{\mbox{\bf lazyEvaluate}\opLeftPren{}s\opRightPren{}} causes one lazy evaluation of stream +{\smath{\mbox{\bf lazyEvaluate}\opLeftPren{}s\opRightPren{}} +causes one lazy evaluation of stream \smath{s}. Caution: \smath{s} must be a ``lazy node'' satisfying \smath{\mbox{\bf lazy?}\opLeftPren{}s) = true}, as there is no error check. @@ -74030,57 +76675,74 @@ entry. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{lcm}}\opLeftPren{}{\it elements\opt{, element}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{lcm}}\opLeftPren{} +{\it elements\opt{, element}}\opRightPren{}% }% }% {1}{(List(\$))->\$}{GcdDomain} -{\smath{\mbox{\bf lcm}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns the least common multiple of \smath{x} and \smath{y}. +{\smath{\mbox{\bf lcm}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +returns the least common multiple of \smath{x} and \smath{y}. \newitem -\smath{\mbox{\bf lcm}\opLeftPren{}lx\opRightPren{}} returns the least common multiple of the elements of +\smath{\mbox{\bf lcm}\opLeftPren{}lx\opRightPren{}} returns the +least common multiple of the elements of the list \smath{lx}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ldexquo}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ldexquo}}\opLeftPren{}{\it lodOperator}, +\allowbreak{}{\it lodOperator}\opRightPren{}% }% }% {2}{(\$, \$)->Union(\$, "failed")}{LinearOrdinaryDifferentialOperator} -{\smath{\mbox{\bf ldexquo}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns -\smath{q} such that \smath{a = b*q}, or \mbox{\tt "failed"} if no such \smath{q} exists. +{\smath{\mbox{\bf ldexquo}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +returns +\smath{q} such that \smath{a = b*q}, or \mbox{\tt "failed"} +if no such \smath{q} exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftDivide}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% -\optand \mbox{\axiomFun{leftQuotient}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% -\opand \mbox{\axiomFun{leftRemainder}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftDivide}}\opLeftPren{} +{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\optand \mbox{\axiomFun{leftQuotient}}\opLeftPren{} +{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opand \mbox{\axiomFun{leftRemainder}}\opLeftPren{} +{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% }% }% -{2}{(\$, \$)->Record(quotient:\$, remainder:\$)}{LinearOrdinaryDifferentialOperator} -{\smath{\mbox{\bf leftDivide}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns +{2}{(\$, \$)->Record(quotient:\$, remainder:\$)} +{LinearOrdinaryDifferentialOperator} +{\smath{\mbox{\bf leftDivide}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns a record with two fields: ``quotient'' \smath{q} and ``remainder'' \smath{r} such that \smath{a = b q + r} and the degree of \smath{r} is less than the degree of \smath{b}. This operation is called ``left division.'' -Operation \smath{\mbox{\bf leftQuotient}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns \smath{q}, and -\smath{\mbox{\bf leftRemainder}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns \smath{r}. +Operation \smath{\mbox{\bf leftQuotient}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns \smath{q}, and +\smath{\mbox{\bf leftRemainder}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns \smath{r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leader}}\opLeftPren{}{\it differentialPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leader}}\opLeftPren{} +{\it differentialPolynomial}\opRightPren{}% }% }% {1}{(\$)->V}{DifferentialPolynomialCategory} -{\smath{\mbox{\bf leader}\opLeftPren{}p\opRightPren{}} returns the derivative of the highest rank +{\smath{\mbox{\bf leader}\opLeftPren{}p\opRightPren{}} +returns the derivative of the highest rank appearing in the differential polynomial \smath{p}, or calls \spadfun{error} if \smath{p} is in the ground ring.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leadingCoefficient}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leadingCoefficient}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->R}{AbelianMonoidRing} -{\smath{\mbox{\bf leadingCoefficient}\opLeftPren{}p\opRightPren{}} returns the coefficient of the highest degree +{\smath{\mbox{\bf leadingCoefficient}\opLeftPren{}p\opRightPren{}} +returns the coefficient of the highest degree term of polynomial \smath{p}. See also \spadtype{IndexedDirectProductCategory} and \spadtype{MonogenicLinearOperator}. @@ -74091,15 +76753,18 @@ See also \spadtype{IndexedDirectProductCategory} and }% }% {1}{(\$)->\$}{PolynomialIdeals} -{\smath{\mbox{\bf leadingIdeal}\opLeftPren{}I\opRightPren{}} is the ideal generated by the leading terms of the elements of the ideal \smath{I}. +{\smath{\mbox{\bf leadingIdeal}\opLeftPren{}I\opRightPren{}} is the +ideal generated by the leading terms of the elements of the ideal \smath{I}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leadingMonomial}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leadingMonomial}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->\$}{AbelianMonoidRing} -{\smath{\mbox{\bf leadingMonomial}\opLeftPren{}p\opRightPren{}} returns the monomial of polynomial \smath{p} with the highest degree. +{\smath{\mbox{\bf leadingMonomial}\opLeftPren{}p\opRightPren{}} +returns the monomial of polynomial \smath{p} with the highest degree. } @@ -74113,16 +76778,20 @@ See also \spadtype{IndexedDirectProductCategory} and {\opkey{These operations apply to a recursive aggregate \smath{a}. See, for example, \spadtype{BinaryTree}.} -\newitem\smath{\mbox{\bf leaf?}\opLeftPren{}a\opRightPren{}} tests if \smath{a} is a terminal node. -\newitem\smath{\mbox{\bf leaves}\opLeftPren{}a\opRightPren{}} returns the list of values at the leaf nodes +\newitem\smath{\mbox{\bf leaf?}\opLeftPren{}a\opRightPren{}} +tests if \smath{a} is a terminal node. +\newitem\smath{\mbox{\bf leaves}\opLeftPren{}a\opRightPren{}} +returns the list of values at the leaf nodes in left-to-right order.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{left}}\opLeftPren{}{\it binaryRecursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{left}}\opLeftPren{} +{\it binaryRecursiveAggregate}\opRightPren{}% }% }% {1}{(\$)->\$}{BinaryRecursiveAggregate} -{\smath{\mbox{\bf left}\opLeftPren{}a\opRightPren{}} returns the left child of binary aggregate \smath{a}.} +{\smath{\mbox{\bf left}\opLeftPren{}a\opRightPren{}} +returns the left child of binary aggregate \smath{a}.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{leftAlternative?}}\opLeftPren{}\opRightPren{}% @@ -74137,11 +76806,13 @@ Note: in general, you do not know whether \smath{2*a=0} implies \smath{a=0}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftCharacteristicPolynomial}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftCharacteristicPolynomial}} +\opLeftPren{}{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->SparseUnivariatePolynomial(R)}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf leftCharacteristicPolynomial}\opLeftPren{}p\opRightPren{}\$F} returns the +{\smath{\mbox{\bf leftCharacteristicPolynomial} +\opLeftPren{}p\opRightPren{}\$F} returns the characteristic polynomial of the left regular representation of \smath{p} of domain \smath{F} with respect to any basis. Argument \smath{p} is a member of a domain of category @@ -74150,11 +76821,13 @@ a commutative ring. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftDiscriminant}}\opLeftPren{}{\it \optinit{listOfVectors}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftDiscriminant}}\opLeftPren{} +{\it \optinit{listOfVectors}}\opRightPren{}% }% }% {1}{(Vector(\$))->R}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf leftDiscriminant}\opLeftPren{}[v_1, \allowbreak{} \ldots, v_n]\opRightPren{}\$F} +{\smath{\mbox{\bf leftDiscriminant}\opLeftPren{}[v_1, +\allowbreak{} \ldots, v_n]\opRightPren{}\$F} where \smath{F} is a domain of category \spadtype{FramedNonAssociativeAlgebra} over a commutative ring \smath{R}, @@ -74163,38 +76836,46 @@ of the \smath{n}-by-\smath{n} matrix whose element at the \eth{\smath{i}} row and \eth{\smath{j}} column is given by the left trace of the product \smath{v_i*v_j}. -Same as \spadfun{determinant}(\smath{\mbox{\bf leftTraceMatrix}\opLeftPren{}[v_1, \allowbreak{} \ldots, v_n]\opRightPren{}}). +Same as \spadfun{determinant}(\smath{\mbox{\bf leftTraceMatrix} +\opLeftPren{}[v_1, \allowbreak{} \ldots, v_n]\opRightPren{}}). If no argument is given, \smath{v_1, \ldots, v_n} are taken to elements of the fixed \smath{R}-basis. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftGcd}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftGcd}}\opLeftPren{}{\it lodOperator}, +\allowbreak{}{\it lodOperator}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{LinearOrdinaryDifferentialOperator} -{\smath{\mbox{\bf leftGcd}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} computes the value \smath{g} of highest degree such that -\smath{a = aa*g} and \smath{b = bb*g} for some values \smath{aa} and \smath{bb}. +{\smath{\mbox{\bf leftGcd}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +computes the value \smath{g} of highest degree such that +\smath{a = aa*g} and \smath{b = bb*g} for some values \smath{aa} and +\smath{bb}. The value \smath{g} is computed using left-division. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftLcm}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftLcm}}\opLeftPren{}{\it lodOperator}, +\allowbreak{}{\it lodOperator}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{LinearOrdinaryDifferentialOperator} -{\smath{\mbox{\bf leftLcm}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} computes the value \smath{m} of lowest +{\smath{\mbox{\bf leftLcm}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +computes the value \smath{m} of lowest degree such that \smath{m = a*aa = b*bb} for some values \smath{aa} and \smath{bb}. The value \smath{m} is computed using left-division. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftMinimalPolynomial}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftMinimalPolynomial}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->SparseUnivariatePolynomial(R)}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf leftMinimalPolynomial}\opLeftPren{}a\opRightPren{}} returns the polynomial +{\smath{\mbox{\bf leftMinimalPolynomial}\opLeftPren{}a\opRightPren{}} +returns the polynomial determined by the smallest non-trivial linear combination of left powers of \smath{a}, an element of a domain of category \spadtype{FiniteRankNonAssociativeAlgebra}. @@ -74207,18 +76888,23 @@ algebra has no unit. }% }% {1}{(\$)->R}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf leftNorm}\opLeftPren{}a\opRightPren{}} returns the determinant of the left regular representation of \smath{a}, +{\smath{\mbox{\bf leftNorm}\opLeftPren{}a\opRightPren{}} +returns the determinant of the left regular representation of \smath{a}, an element of a domain of category \spadtype{FiniteRankNonAssociativeAlgebra}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftPower}}\opLeftPren{}{\it monad}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftPower}}\opLeftPren{} +{\it monad}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->\$}{MonadWithUnit} -{\smath{\mbox{\bf leftPower}\opLeftPren{}a, \allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} left power of monad \smath{a}, -that is, \smath{\mbox{\bf leftPower}\opLeftPren{}a, n) := a {\mbox{\bf leftPower}}(a, n-1)}. -If the monad has a unit then \smath{\mbox{\bf leftPower}\opLeftPren{}a, 0) := 1}. +{\smath{\mbox{\bf leftPower}\opLeftPren{}a, \allowbreak{} n +\opRightPren{}} returns the \eth{\smath{n}} left power of monad \smath{a}, +that is, \smath{\mbox{\bf leftPower}\opLeftPren{}a, n) := +a {\mbox{\bf leftPower}}(a, n-1)}. +If the monad has a unit then \smath{\mbox{\bf leftPower} +\opLeftPren{}a, 0) := 1}. Otherwise, define \smath{\mbox{\bf leftPower}\opLeftPren{}a, 1) = a} See \spadtype{Monad} and \spadtype{MonadWithUnit} for details. See also \spadfun{leftRecip}.} @@ -74228,7 +76914,8 @@ See also \spadfun{leftRecip}.} }% }% {0}{()->SparseUnivariatePolynomial(Polynomial(R))}{FramedNonAssociativeAlgebra} -{\smath{\mbox{\bf leftRankPolynomial}\opLeftPren{}\opRightPren{}\$F} calculates the left minimal +{\smath{\mbox{\bf leftRankPolynomial}\opLeftPren{}\opRightPren{}\$F} +calculates the left minimal polynomial of a generic element of an algebra of domain \smath{F}, a domain of category \spadtype{FramedNonAssociativeAlgebra} over a commutative ring \smath{R}. @@ -74242,7 +76929,8 @@ coefficients with respect to the fixed basis. }% }% {1}{(A)->NonNegativeInteger}{AlgebraPackage} -{\smath{\mbox{\bf leftRank}\opLeftPren{}x\opRightPren{}} returns the number of linearly independent +{\smath{\mbox{\bf leftRank}\opLeftPren{}x\opRightPren{}} +returns the number of linearly independent elements in \smath{x b_1}, \ldots, \smath{x b_n}, where \smath{b=[b_1, \ldots, b_n]} is a basis. Argument \smath{x} is an element of a domain of category @@ -74255,7 +76943,8 @@ Argument \smath{x} is an element of a domain of category }% }% {1}{(\$)->Union(\$, "failed")}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf leftRecip}\opLeftPren{}a\opRightPren{}} returns an element that is a left inverse +{\smath{\mbox{\bf leftRecip}\opLeftPren{}a\opRightPren{}} +returns an element that is a left inverse of \smath{a}, or \mbox{\tt "failed"}, if there is no unit element, such an element does not exist, or the left reciprocal cannot be determined (see \spadatt{unitsKnown}).} @@ -74265,14 +76954,16 @@ determined (see \spadatt{unitsKnown}).} }% }% {1}{(\$)->Union(\$, "failed")}{MonadWithUnit} -{\smath{\mbox{\bf leftRecip}\opLeftPren{}a\opRightPren{}} returns an element, which is a left inverse +{\smath{\mbox{\bf leftRecip}\opLeftPren{}a\opRightPren{}} +returns an element, which is a left inverse of \smath{a}, or \mbox{\tt "failed"} if such an element doesn't exist or cannot be determined (see \spadatt{unitsKnown}). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftRegularRepresentation}}\opLeftPren{}{\it element\opt{, vectorOfElements}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftRegularRepresentation}}\opLeftPren{} +{\it element\opt{, vectorOfElements}}\opRightPren{}% }% }% {2}{(\$, Vector(\$))->Matrix(R)}{FiniteRankNonAssociativeAlgebra} @@ -74287,39 +76978,48 @@ fixed basis for \smath{F}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftTraceMatrix}}\opLeftPren{}{\it \opt{vectorOfElements}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftTraceMatrix}}\opLeftPren{} +{\it \opt{vectorOfElements}}\opRightPren{}% }% }% {1}{(Vector(\$))->Matrix(R)}{FiniteRankNonAssociativeAlgebra} -{\opkey{This operation is defined on a domain \smath{F} of category \spadtype{NonAssociativeAlgebra}.} -\newline\smath{\mbox{\bf leftTraceMatrix}\opLeftPren{}\optinner{v}\opRightPren{}}, where \smath{v} +{\opkey{This operation is defined on a domain \smath{F} +of category \spadtype{NonAssociativeAlgebra}.} +\newline\smath{\mbox{\bf leftTraceMatrix}\opLeftPren{} +\optinner{v}\opRightPren{}}, where \smath{v} is an optional vector \smath{[v_1, \ldots, v_n]}, returns the \smath{n}-by-\smath{n} matrix \smath{M} such that \smath{M_{i, j}} is the left trace of the product \smath{v_i*v_j} of elements from the basis \smath{[v_1, \ldots, v_n]}. -If the argument is missing, the basis is taken to be the fixed basis for \smath{F}. +If the argument is missing, the basis is taken to be the +fixed basis for \smath{F}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftTrace}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftTrace}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->R}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf leftTrace}\opLeftPren{}a\opRightPren{}} returns the trace of the left regular representation of \smath{a}, +{\smath{\mbox{\bf leftTrace}\opLeftPren{}a\opRightPren{}} returns +the trace of the left regular representation of \smath{a}, an element of a domain of category \spadtype{FiniteRankNonAssociativeAlgebra}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{leftTrim}}\opLeftPren{}{\it string}, \allowbreak{}{\it various}\opRightPren{}% +\opdata{{\mbox{\axiomFun{leftTrim}}\opLeftPren{}{\it string}, +\allowbreak{}{\it various}\opRightPren{}% }% }% {2}{(\$, Character)->\$}{StringAggregate} -{\smath{\mbox{\bf leftTrim}\opLeftPren{}s, \allowbreak{} c\opRightPren{}} returns string \smath{s} with all leading characters +{\smath{\mbox{\bf leftTrim}\opLeftPren{}s, \allowbreak{} c\opRightPren{}} +returns string \smath{s} with all leading characters \smath{c} deleted. For example, \spadfun{leftTrim}{\tt (" abc ", " ")} returns {\tt "abc "}. \newitem -\smath{\mbox{\bf leftTrim}\opLeftPren{}s, \allowbreak{} cc\opRightPren{}} returns \smath{s} with all leading characters in +\smath{\mbox{\bf leftTrim}\opLeftPren{}s, \allowbreak{} cc\opRightPren{}} +returns \smath{s} with all leading characters in \smath{cc} deleted. For example, \code{leftTrim("(abc)", charClass "()")} returns \code{"abc"}. } @@ -74329,12 +77029,15 @@ returns \code{"abc"}. \opand \mbox{\axiomFun{leftUnits}}\opLeftPren{}\opRightPren{}% }% }% -{0}{()->Union(Record(particular:\$, basis:List(\$)), "failed")}{FiniteRankNonAssociativeAlgebra} +{0}{()->Union(Record(particular:\$, basis:List(\$)), "failed")} +{FiniteRankNonAssociativeAlgebra} {\opkey{These operations are defined on a domain \smath{F} of category \spadtype{NonAssociativeAlgebra}.} -\newitem\smath{\mbox{\bf leftUnit}\opLeftPren{}\opRightPren{}\$F} returns a left unit of the algebra (not +\newitem\smath{\mbox{\bf leftUnit}\opLeftPren{}\opRightPren{}\$F} +returns a left unit of the algebra (not necessarily unique), or \mbox{\tt "failed"} if there is none. -\newitem\smath{\mbox{\bf leftUnits}\opLeftPren{}\opRightPren{}\$F} returns the affine space of all +\newitem\smath{\mbox{\bf leftUnits}\opLeftPren{}\opRightPren{}\$F} +returns the affine space of all left units of an algebra \smath{F}, or \mbox{\tt "failed"} if there is none, where \smath{F} is a domain of category \spadtype{FiniteRankNonAssociativeAlgebra}. @@ -74344,25 +77047,30 @@ for a list of elements of \smath{F}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{legendreSymbol}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{legendreSymbol}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(Integer, Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf legendreSymbol}\opLeftPren{}a, \allowbreak{} p\opRightPren{}} returns the Legendre symbol +{\smath{\mbox{\bf legendreSymbol}\opLeftPren{}a, +\allowbreak{} p\opRightPren{}} returns the Legendre symbol \smath{L(a/p)}, \smath{L(a/p) = (-1)^{(p-1)/2} \mod p} for prime \smath{p}. This is 0 if \smath{a = 0}, 1 if \smath{a} is a quadratic residue \smath{\mod p}, and \smath{-1} otherwise. Note: because the primality test is expensive, use -\smath{\mbox{\bf jacobi}\opLeftPren{}a, \allowbreak{} p\opRightPren{}} if you know that \smath{p} is prime. +\smath{\mbox{\bf jacobi}\opLeftPren{}a, \allowbreak{} p\opRightPren{}} +if you know that \smath{p} is prime. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{LegendreP}}\opLeftPren{}{\it nonNegativeInteger}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{LegendreP}}\opLeftPren{}{\it nonNegativeInteger}, +\allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(NonNegativeInteger, R)->R}{OrthogonalPolynomialFunctions} -{\smath{\mbox{\bf LegendreP}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} is the \eth{\smath{n}} Legendre +{\smath{\mbox{\bf LegendreP}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} +is the \eth{\smath{n}} Legendre polynomial, \smath{P[n](x)}, defined by \smath{{1\over \sqrt(1-2xt+t^2)} ={ \sum\nolimits_{n=0}^\infty {P[n](x) t^n}}}. } @@ -74372,14 +77080,17 @@ polynomial, \smath{P[n](x)}, defined by \smath{{1\over }% }% {1}{(\$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf length}\opLeftPren{}a\opRightPren{}} returns the length of integer \smath{a} in digits.} +{\smath{\mbox{\bf length}\opLeftPren{}a\opRightPren{}} +returns the length of integer \smath{a} in digits.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{less?}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{less?}}\opLeftPren{}{\it aggregate}, +\allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->Boolean}{Aggregate} -{\smath{\mbox{\bf less?}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} tests if \smath{u} has less than \smath{n} elements. +{\smath{\mbox{\bf less?}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} +tests if \smath{u} has less than \smath{n} elements. } % ---------------------------------------------------------------------- @@ -74387,8 +77098,10 @@ polynomial, \smath{P[n](x)}, defined by \smath{{1\over }% }% {0}{()->\$}{CartesianTensor} -{\smath{\mbox{\bf leviCivitaSymbol}\opLeftPren{}\opRightPren{}} is the rank \smath{dim} tensor defined -by \smath{\mbox{\bf leviCivitaSymbol}\opLeftPren{})(i_1, \ldots i_{\rm dim})}, which is +{\smath{\mbox{\bf leviCivitaSymbol}\opLeftPren{}\opRightPren{}} +is the rank \smath{dim} tensor defined +by \smath{\mbox{\bf leviCivitaSymbol}\opLeftPren{}) +(i_1, \ldots i_{\rm dim})}, which is \smath{+1}, \smath{-1} or \smath{0} according to whether the permutation \smath{i_1, \ldots, i_{\rm dim}} is an even permutation, an odd permutation, or not a permutation of @@ -74397,11 +77110,13 @@ permutation, an odd permutation, or not a permutation of } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{lexGroebner}}\opLeftPren{}{\it listOfPolynomials}, \allowbreak{}{\it listOfSymbols}\opRightPren{}% +\opdata{{\mbox{\axiomFun{lexGroebner}}\opLeftPren{} +{\it listOfPolynomials}, \allowbreak{}{\it listOfSymbols}\opRightPren{}% }% }% {2}{(List(Polynomial(F)), List(Symbol))->List(Polynomial(F))}{PolyGroebner} -{\smath{\mbox{\bf lexGroebner}\opLeftPren{}lp, \allowbreak{} lv\opRightPren{}} computes a Gr\"obner basis for the +{\smath{\mbox{\bf lexGroebner}\opLeftPren{}lp, +\allowbreak{} lv\opRightPren{}} computes a Gr\"obner basis for the list of polynomials \smath{lp} in lexicographic order. The variables \smath{lv} are ordered by their position in the list \smath{lp}. @@ -74409,11 +77124,13 @@ The variables \smath{lv} are ordered by their position in the list % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{lhs}}\opLeftPren{}{\it equationOrRewriteRule}\opRightPren{}% +\opdata{{\mbox{\axiomFun{lhs}}\opLeftPren{} +{\it equationOrRewriteRule}\opRightPren{}% }% }% {1}{(\$)->F}{RewriteRule} -{\smath{\mbox{\bf lhs}\opLeftPren{}x\opRightPren{}} returns the left hand side of an equation or rewrite-rule. +{\smath{\mbox{\bf lhs}\opLeftPren{}x\opRightPren{}} +returns the left hand side of an equation or rewrite-rule. } % ---------------------------------------------------------------------- @@ -74421,7 +77138,8 @@ The variables \smath{lv} are ordered by their position in the list }% }% {1}{(FileName)->\$}{Library} -{\smath{\mbox{\bf library}\opLeftPren{}name\opRightPren{}} creates a new library file with filename +{\smath{\mbox{\bf library}\opLeftPren{}name\opRightPren{}} +creates a new library file with filename \smath{name}. } @@ -74430,7 +77148,8 @@ The variables \smath{lv} are ordered by their position in the list }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf lieAdmissible?}\opLeftPren{}\opRightPren{}\$F} tests if the algebra defined by the commutators +{\smath{\mbox{\bf lieAdmissible?}\opLeftPren{}\opRightPren{}\$F} +tests if the algebra defined by the commutators is a Lie algebra. The domain \smath{F} is a member of the category \spadtype{FiniteRankNonAssociativeAlgebra(R)}. @@ -74442,7 +77161,8 @@ The property of anticommutativity follows from the definition. }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf lieAlgebra?}\opLeftPren{}\opRightPren{}\$F} tests if the algebra of +{\smath{\mbox{\bf lieAlgebra?}\opLeftPren{}\opRightPren{}\$F} +tests if the algebra of \smath{F} is anticommutative and that the Jacobi identity \smath{(a*b)*c + (b*c)*a + (c*a)*b = 0} @@ -74455,26 +77175,50 @@ is satisfied for all \smath{a}, \smath{b}, \smath{c} in \smath{F}. }% }% {1}{(Color)->\$}{Palette} -{\smath{\mbox{\bf light}\opLeftPren{}c\opRightPren{}} sets the shade of a hue \smath{c} to its highest value.} +{\smath{\mbox{\bf light}\opLeftPren{}c\opRightPren{}} +sets the shade of a hue \smath{c} to its highest value.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{limit}}\opLeftPren{}{\it expression}, \allowbreak{}{\it equation}\allowbreak $\,[$ , \allowbreak{}{\it direction}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{limit}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it equation}\allowbreak $\,[$ , \allowbreak{} +{\it direction}$]$\opRightPren{}% }% }% -{2}{(Fraction(Polynomial(R)), Equation(Fraction(Polynomial(R))))->Union(OrderedCompletion(Fraction(Polynomial(R))), Record(leftHandLimit:Union(OrderedCompletion(Fraction(Polynomial(R))), "failed"), rightHandLimit:Union(OrderedCompletion(Fraction(Polynomial(R))), "failed")), "failed")}{RationalFunctionLimitPackage} -{\smath{\mbox{\bf limit}\opLeftPren{}f(x), \allowbreak{} x = a\opRightPren{}} computes the real two-sided limit of \smath{f} as its argument \smath{x} approaches \smath{a}. +{2}{(Fraction(Polynomial(R)), Equation(Fraction(Polynomial(R))))-> +Union(OrderedCompletion(Fraction(Polynomial(R))), +Record(leftHandLimit:Union(OrderedCompletion(Fraction(Polynomial(R))), +"failed"), +rightHandLimit:Union(OrderedCompletion(Fraction(Polynomial(R))), "failed")), +"failed")} +{RationalFunctionLimitPackage} +{\smath{\mbox{\bf limit}\opLeftPren{}f(x), +\allowbreak{} x = a\opRightPren{}} computes +the real two-sided limit of \smath{f} as its argument +\smath{x} approaches \smath{a}. \newitem -\smath{\mbox{\bf limit}\opLeftPren{}f(x), \allowbreak{} x=a, \allowbreak{} "left"\opRightPren{}} computes the real limit of \smath{f} as its argument \smath{x} approaches \smath{a} from the left. +\smath{\mbox{\bf limit}\opLeftPren{}f(x), \allowbreak{} x=a, +\allowbreak{} "left"\opRightPren{}} computes the real limit +of \smath{f} as its +argument \smath{x} approaches \smath{a} from the left. \newitem -\smath{\mbox{\bf limit}\opLeftPren{}f(x), \allowbreak{} x=a, \allowbreak{} "right"\opRightPren{}} computes the corresponding limit as \smath{x} approaches \smath{a} from the right. +\smath{\mbox{\bf limit}\opLeftPren{}f(x), \allowbreak{} x=a, +\allowbreak{} "right"\opRightPren{}} computes the corresponding limit as +\smath{x} approaches \smath{a} from the right. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{limitedIntegrate}}\opLeftPren{}{\it rationalFunction}, \allowbreak{}{\it symbol}, \allowbreak{}{\it listOfRationalFunctions}\opRightPren{}% +\opdata{{\mbox{\axiomFun{limitedIntegrate}}\opLeftPren{} +{\it rationalFunction}, \allowbreak{}{\it symbol}, \allowbreak{} +{\it listOfRationalFunctions}\opRightPren{}% }% }% -{3}{(Fraction(Polynomial(F)), Symbol, List(Fraction(Polynomial(F))))->Union(Record(mainpart:Fraction(Polynomial(F)), limitedlogs:List(Record(coeff:Fraction(Polynomial(F)), logand:Fraction(Polynomial(F))))), "failed")}{RationalFunctionIntegration} -{\smath{\mbox{\bf limitedIntegrate}\opLeftPren{}f, \allowbreak{} x, \allowbreak{} [g_1, \allowbreak{} \ldots, g_n]\opRightPren{}} returns +{3}{(Fraction(Polynomial(F)), Symbol, +List(Fraction(Polynomial(F))))->Union(Record(mainpart:Fraction(Polynomial(F)), +limitedlogs:List(Record(coeff:Fraction(Polynomial(F)), +logand:Fraction(Polynomial(F))))), "failed")} +{RationalFunctionIntegration} +{\smath{\mbox{\bf limitedIntegrate}\opLeftPren{}f, \allowbreak{} x, +\allowbreak{} [g_1, \allowbreak{} \ldots, g_n]\opRightPren{}} returns fractions \smath{[h, [c_i, g_i]]} such that the \smath{g_i}'s are among \smath{[g_1, \ldots, g_n]}, \smath{dc_i/dx = 0}, @@ -74483,12 +77227,15 @@ possible, \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{linearDependenceOverZ}}\opLeftPren{}{\it vector}\opRightPren{}% - \opand \mbox{\axiomFun{linearlyDependentOverZ?}}\opLeftPren{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{linearDependenceOverZ}}\opLeftPren{} +{\it vector}\opRightPren{}% + \opand \mbox{\axiomFun{linearlyDependentOverZ?}}\opLeftPren{} +{\it vector}\opRightPren{}% }% }% {1}{(Vector(R))->Union(Vector(Integer), "failed")}{IntegerLinearDependence} -{\smath{\mbox{\bf linearlyDependenceOverZ}\opLeftPren{}[v_1, \allowbreak{} \ldots, v_n]\opRightPren{}} tests +{\smath{\mbox{\bf linearlyDependenceOverZ}\opLeftPren{} +[v_1, \allowbreak{} \ldots, v_n]\opRightPren{}} tests if the elements \smath{v_i} of a ring (typically algebraic numbers or \spadtype{Expression}s) are linearly dependent over the integers. If so, the operation returns @@ -74498,29 +77245,38 @@ If so, the operation returns If linearly independent over the integers, \mbox{\tt "failed"} is returned. \newline -\smath{\mbox{\bf linearlyDependentOverZ?}\opLeftPren{}[v1, \allowbreak{} \ldots, vn]\opRightPren{}} returns \smath{true} -if the \smath{vi}'s are linearly dependent over the integers, and \smath{false} otherwise. +\smath{\mbox{\bf linearlyDependentOverZ?}\opLeftPren{}[v1, +\allowbreak{} \ldots, vn]\opRightPren{}} returns \smath{true} +if the \smath{vi}'s are linearly dependent over the integers, +and \smath{false} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{lineColorDefault}}\opLeftPren{}{\it \opt{palette}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{lineColorDefault}}\opLeftPren{} +{\it \opt{palette}}\opRightPren{}% }% }% {0}{()->Palette}{ViewDefaultsPackage} -{\smath{\mbox{\bf lineColorDefault}\opLeftPren{}\opRightPren{}} returns the default color of lines +{\smath{\mbox{\bf lineColorDefault}\opLeftPren{}\opRightPren{}} +returns the default color of lines connecting points in a \twodim{} viewport. \newline -\smath{\mbox{\bf lineColorDefault}\opLeftPren{}p\opRightPren{}} sets the default color of lines +\smath{\mbox{\bf lineColorDefault}\opLeftPren{}p\opRightPren{}} +sets the default color of lines connecting points in a \twodim{} viewport to the palette \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{linSolve}}\opLeftPren{}{\it listOfPolynomials}, \allowbreak{}{\it listOfVariables}\opRightPren{}% +\opdata{{\mbox{\axiomFun{linSolve}}\opLeftPren{} +{\it listOfPolynomials}, \allowbreak{}{\it listOfVariables}\opRightPren{}% }% }% -{2}{(List(P), List(OV))->Record(particular:Union(Vector(F), "failed"), basis:List(Vector(F)))}{LinearSystemPolynomialPackage} -{\smath{\mbox{\bf linSolve}\opLeftPren{}lp, \allowbreak{} lvar\opRightPren{}} finds the solutions of the linear system of polynomials \smath{lp} = 0 with respect to the list of symbols \smath{lvar}. +{2}{(List(P), List(OV))->Record(particular:Union(Vector(F), "failed"), +basis:List(Vector(F)))}{LinearSystemPolynomialPackage} +{\smath{\mbox{\bf linSolve}\opLeftPren{}lp, \allowbreak{} +lvar\opRightPren{}} finds the solutions of the linear system of polynomials +\smath{lp} = 0 with respect to the list of symbols \smath{lvar}. } % ---------------------------------------------------------------------- @@ -74537,34 +77293,41 @@ defined by, \smath{\int {dx\over log(x)}}. }% }% {1}{(S)->\$}{ListAggregate} -{\smath{\mbox{\bf list}\opLeftPren{}x\opRightPren{}} creates a list consisting of the one element \smath{x}. +{\smath{\mbox{\bf list}\opLeftPren{}x\opRightPren{}} +creates a list consisting of the one element \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{list?}}\opLeftPren{}{\it sExpression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{list?}}\opLeftPren{} +{\it sExpression}\opRightPren{}% }% }% {1}{(\$)->Boolean}{SExpressionCategory} -{\smath{\mbox{\bf list?}\opLeftPren{}s\opRightPren{}} tests if \spadtype{SExpression} value \smath{s} +{\smath{\mbox{\bf list?}\opLeftPren{}s\opRightPren{}} +tests if \spadtype{SExpression} value \smath{s} is a Lisp list, possibly the null list. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{listBranches}}\opLeftPren{}{\it listOfListsOfPoints}\opRightPren{}% +\opdata{{\mbox{\axiomFun{listBranches}}\opLeftPren{} +{\it listOfListsOfPoints}\opRightPren{}% }% }% {1}{(\$)->List(List(Point(DoubleFloat)))}{PlottablePlaneCurveCategory} -{\smath{\mbox{\bf listBranches}\opLeftPren{}c\opRightPren{}} returns a list of lists of points +{\smath{\mbox{\bf listBranches}\opLeftPren{}c\opRightPren{}} +returns a list of lists of points representing the branches of the curve \smath{c}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{listRepresentation}}\opLeftPren{}{\it permutation}\opRightPren{}% +\opdata{{\mbox{\axiomFun{listRepresentation}}\opLeftPren{} +{\it permutation}\opRightPren{}% }% }% {1}{(\$)->Record(preimage:List(S), image:List(S))}{Permutation} -{\smath{\mbox{\bf listRepresentation}\opLeftPren{}p\opRightPren{}} produces a representation {\it rep} +{\smath{\mbox{\bf listRepresentation}\opLeftPren{}p\opRightPren{}} +produces a representation {\it rep} of the permutation \smath{p} as a list of preimages and images \smath{i}, that is, permutation \smath{p} maps \smath{(rep.{\tt preimage}).k} to \smath{(rep.{\tt image}).k} for all indices @@ -74572,11 +77335,14 @@ preimage}).k} to \smath{(rep.{\tt image}).k} for all indices } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{listYoungTableaus}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{listYoungTableaus}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% -{1}{(List(Integer))->List(Matrix(Integer))}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf listYoungTableaus}\opLeftPren{}lambda\opRightPren{}}, where {\it lambda} is a proper +{1}{(List(Integer))->List(Matrix(Integer))} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf listYoungTableaus}\opLeftPren{}lambda\opRightPren{}}, +where {\it lambda} is a proper partition, generates the list of all standard tableaus of shape {\it lambda} by means of lattice permutations. The numbers of the lattice permutation are interpreted as column @@ -74585,19 +77351,25 @@ labels. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{listOfComponents}}\opLeftPren{}{\it threeSpace}\opRightPren{}% +\opdata{{\mbox{\axiomFun{listOfComponents}}\opLeftPren{} +{\it threeSpace}\opRightPren{}% }% }% {1}{(\$)->List(List(List(Point(R))))}{ThreeSpace} -{\smath{\mbox{\bf listOfComponents}\opLeftPren{}sp\opRightPren{}} returns a list of list of list of points for threeSpace -object \smath{sp} assumed to be composed of a list of components, each a list of +{\smath{\mbox{\bf listOfComponents}\opLeftPren{}sp\opRightPren{}} +returns a list of list of list of points for threeSpace +object \smath{sp} assumed to be composed of a list of components, +each a list of curves, which in turn is each a list of points, or calls \spadfun{error} if this is not possible. } -{\smath{\mbox{\bf listOfCurves}\opLeftPren{}sp\opRightPren{}} returns a list of list of subspace component properties -for threeSpace object \smath{sp} assumed to be a list of curves, each of which is -a list of subspace components, or calls \spadfun{error} if this is not possible. +{\smath{\mbox{\bf listOfCurves}\opLeftPren{}sp\opRightPren{}} +returns a list of list of subspace component properties +for threeSpace object \smath{sp} assumed to be a list of +curves, each of which is +a list of subspace components, or calls \spadfun{error} +if this is not possible. } % ---------------------------------------------------------------------- @@ -74605,19 +77377,24 @@ a list of subspace components, or calls \spadfun{error} if this is not possible. }% }% {1}{(\$)->S}{SegmentCategory} -{\smath{\mbox{\bf lo}\opLeftPren{}s\opRightPren{}} returns the first endpoint of \smath{s}. +{\smath{\mbox{\bf lo}\opLeftPren{}s\opRightPren{}} +returns the first endpoint of \smath{s}. For example, \code{lo(l..h) = l}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{log}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{logIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{log}}\opLeftPren{} +{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{logIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} -{\smath{\mbox{\bf log}\opLeftPren{}x\opRightPren{}} returns the natural logarithm of \smath{x}. +{\smath{\mbox{\bf log}\opLeftPren{}x\opRightPren{}} +returns the natural logarithm of \smath{x}. \newitem -\smath{\mbox{\bf logIfCan}\opLeftPren{}z\opRightPren{}} returns \smath{\mbox{\bf log}\opLeftPren{}z\opRightPren{}} if possible, and +\smath{\mbox{\bf logIfCan}\opLeftPren{}z\opRightPren{}} +returns \smath{\mbox{\bf log}\opLeftPren{}z\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. } @@ -74626,9 +77403,11 @@ For example, \code{lo(l..h) = l}. }% }% {0}{()->\$}{Float} -{\smath{\mbox{\bf log2}\opLeftPren{}\opRightPren{}} returns \smath{ln(2) = 0.6931471805\ldots}. +{\smath{\mbox{\bf log2}\opLeftPren{}\opRightPren{}} +returns \smath{ln(2) = 0.6931471805\ldots}. \newitem -\smath{\mbox{\bf log2}\opLeftPren{}x\opRightPren{}} computes the base 2 logarithm for \smath{x}. +\smath{\mbox{\bf log2}\opLeftPren{}x\opRightPren{}} +computes the base 2 logarithm for \smath{x}. } % ---------------------------------------------------------------------- @@ -74636,9 +77415,11 @@ For example, \code{lo(l..h) = l}. }% }% {0}{()->\$}{Float} -{\smath{\mbox{\bf log10}\opLeftPren{}\opRightPren{}} returns \smath{ln(10) = 2.3025809299\ldots}. +{\smath{\mbox{\bf log10}\opLeftPren{}\opRightPren{}} +returns \smath{ln(10) = 2.3025809299\ldots}. \newitem -\smath{\mbox{\bf log10}\opLeftPren{}x\opRightPren{}} computes the base 10 logarithm for \smath{x}. +\smath{\mbox{\bf log10}\opLeftPren{}x\opRightPren{}} +computes the base 10 logarithm for \smath{x}. } % ---------------------------------------------------------------------- @@ -74646,42 +77427,52 @@ For example, \code{lo(l..h) = l}. }% }% {1}{(Complex(DoubleFloat))->Complex(DoubleFloat)}{DoubleFloatSpecialFunctions} -{\smath{\mbox{\bf logGamma}\opLeftPren{}x\opRightPren{}} is the natural log of $\Gamma(x).$ +{\smath{\mbox{\bf logGamma}\opLeftPren{}x\opRightPren{}} +is the natural log of $\Gamma(x).$ Note: this can often be computed even if $\Gamma(x)$ cannot. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{lowerCase}}\opLeftPren{}{\it \opt{string}}\opRightPren{}% - \opand \mbox{\axiomFun{lowerCase?}}\opLeftPren{}{\it character}\opRightPren{}% +\opdata{{\mbox{\axiomFun{lowerCase}}\opLeftPren{} +{\it \opt{string}}\opRightPren{}% + \opand \mbox{\axiomFun{lowerCase?}}\opLeftPren{} +{\it character}\opRightPren{}% }% }% {1}{(\$)->Boolean}{Character} -{\smath{\mbox{\bf lowerCase}\opLeftPren{}\opRightPren{}} returns the class of all characters for which \spadfunFrom{lowerCase?}{Character} is \smath{true}. +{\smath{\mbox{\bf lowerCase}\opLeftPren{}\opRightPren{}} +returns the class of all characters for which +\spadfunFrom{lowerCase?}{Character} is \smath{true}. \newitem \smath{\mbox{\bf lowerCase}\opLeftPren{}c\opRightPren{}} returns a corresponding lower case alphabetic character \smath{c} if \smath{c} is an upper case alphabetic character, and \smath{c} otherwise. \newitem -\smath{\mbox{\bf lowerCase}\opLeftPren{}s\opRightPren{}} returns the string with all characters in lower case. +\smath{\mbox{\bf lowerCase}\opLeftPren{}s\opRightPren{}} +returns the string with all characters in lower case. \newitem -\smath{\mbox{\bf lowerCase?}\opLeftPren{}c\opRightPren{}} tests if character \smath{c} is an lower case letter, that is, +\smath{\mbox{\bf lowerCase?}\opLeftPren{}c\opRightPren{}} +tests if character \smath{c} is an lower case letter, that is, one of \smath{a}\ldots\smath{z}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{listOfProperties}}\opLeftPren{}{\it threeSpace}\opRightPren{}% +\opdata{{\mbox{\axiomFun{listOfProperties}}\opLeftPren{} +{\it threeSpace}\opRightPren{}% }% }% {1}{(\$)->List(SubSpaceComponentProperty)}{ThreeSpace} -{\smath{\mbox{\bf listOfProperties}\opLeftPren{}sp\opRightPren{}} returns a list of subspace +{\smath{\mbox{\bf listOfProperties}\opLeftPren{}sp\opRightPren{}} +returns a list of subspace component properties for \smath{sp} of type \spadtype{ThreeSpace}, or calls \spadfun{error} if this is not possible. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{listOfPoints}}\opLeftPren{}{\it threeSpace}\opRightPren{}% +\opdata{{\mbox{\axiomFun{listOfPoints}}\opLeftPren{} +{\it threeSpace}\opRightPren{}% }% }% {1}{(\$)->List(Point(R))}{ThreeSpace} @@ -74691,28 +77482,34 @@ returns the list of points component contained in \smath{sp}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mainKernel}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mainKernel}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(\$)->Union(Kernel(\$), "failed")}{ExpressionSpace} -{\smath{\mbox{\bf mainKernel}\opLeftPren{}f\opRightPren{}} returns a kernel of \smath{f} with maximum +{\smath{\mbox{\bf mainKernel}\opLeftPren{}f\opRightPren{}} +returns a kernel of \smath{f} with maximum nesting level, or \mbox{\tt "failed"} if \smath{f} has no kernels (that is, \smath{f} is a constant). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mainVariable}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mainVariable}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->Union(VarSet, "failed")}{PolynomialCategory} -{\smath{\mbox{\bf mainVariable}\opLeftPren{}u\opRightPren{}} returns the variable of highest ordering +{\smath{\mbox{\bf mainVariable}\opLeftPren{}u\opRightPren{}} +returns the variable of highest ordering that actually occurs in the polynomial \smath{p}, or \mbox{\tt "failed"} if no variables are present. Argument u can be either a polynomial or a rational function. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{makeFloatFunction}}\opLeftPren{}{\it expression}, \allowbreak{}{\it symbol}\allowbreak $\,[$ , \allowbreak{}{\it symbol}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{makeFloatFunction}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it symbol}\allowbreak $\,[$ , \allowbreak{} +{\it symbol}$]$\opRightPren{}% }% }% {2}{(S, Symbol)->(DoubleFloat)->DoubleFloat}{MakeFloatCompiledFunction} @@ -74720,11 +77517,13 @@ Argument u can be either a polynomial or a rational function. type \spadtype{InputForm} (objects of the most common types can be so coerced).} \newitem -\smath{\mbox{\bf makeFloatFunction}\opLeftPren{}expr, \allowbreak{} x\opRightPren{}} returns an anonymous function +\smath{\mbox{\bf makeFloatFunction}\opLeftPren{}expr, +\allowbreak{} x\opRightPren{}} returns an anonymous function of type \spadsig{Float}{Float} defined by \smath{x \mapsto {\rm expr}}. \newitem -\smath{\mbox{\bf makeFloatFunction}\opLeftPren{}expr, \allowbreak{} x, \allowbreak{} y\opRightPren{}} returns an anonymous function +\smath{\mbox{\bf makeFloatFunction}\opLeftPren{}expr, +\allowbreak{} x, \allowbreak{} y\opRightPren{}} returns an anonymous function of type \spadsig{(Float, Float)}{Float} defined by \smath{(x, y) \mapsto {\rm expr}}. } @@ -74732,25 +77531,31 @@ of type \spadsig{(Float, Float)}{Float} defined by % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{makeVariable}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{makeVariable}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(S)->(NonNegativeInteger)->\$}{DifferentialPolynomialCategory} -{\smath{\mbox{\bf makeVariable}\opLeftPren{}s\opRightPren{}}, where \smath{s} is a symbol, +{\smath{\mbox{\bf makeVariable}\opLeftPren{}s\opRightPren{}}, +where \smath{s} is a symbol, differential indeterminate, or a differential polynomial, returns a function \smath{f} defined on the non-negative integers such that \smath{f(n)} returns the \eth{\smath{n}} derivative of \smath{s}. \newitem -\smath{\mbox{\bf makeVariable}\opLeftPren{}s, \allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} derivative of a +\smath{\mbox{\bf makeVariable}\opLeftPren{}s, +\allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} derivative of a differential indeterminate \smath{s} as an algebraic indeterminate. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{makeObject}}\opLeftPren{}{\it functions}, \allowbreak{}{\it range}\allowbreak $\,[$ , \allowbreak{}{\it range}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{makeObject}}\opLeftPren{}{\it functions}, +\allowbreak{}{\it range}\allowbreak $\,[$ , \allowbreak{} +{\it range}$]$\opRightPren{}% }% }% -{2}{(ParametricSpaceCurve((DoubleFloat)->DoubleFloat), Segment(Float))->ThreeSpace(DoubleFloat)}{TopLevelDrawFunctionsForCompiledFunctions} +{2}{(ParametricSpaceCurve((DoubleFloat)->DoubleFloat), Segment(Float))-> +ThreeSpace(DoubleFloat)}{TopLevelDrawFunctionsForCompiledFunctions} {Arguments \smath{f}, \smath{g}, and \smath{h} appearing below with arguments (for example, \smath{f(x, y)}) denote symbolic expressions involving those arguments. @@ -74760,47 +77565,69 @@ functions which map one or more \spadtype{DoubleFloat} values to \spadtype{DoubleFloat} values. \largerbreak Values \smath{a}, \smath{b}, \smath{c}, and \smath{d} denote numerical values. -\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}curve(f, g, h), \allowbreak{} a..b\opRightPren{}} returns the space +\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}curve(f, g, h), +\allowbreak{} a..b\opRightPren{}} returns the space \smath{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of the parametric curve \smath{x = f(t)}, \smath{y = -g(t)}, \smath{z = h(t)} as \smath{t} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +g(t)}, \smath{z = h(t)} as \smath{t} ranges from +\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. -\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}curve(f(t), g(t), h(t)), \allowbreak{} t = a..b\opRightPren{}} +\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}curve(f(t), g(t), h(t)), +\allowbreak{} t = a..b\opRightPren{}} returns the space \smath{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of the parametric curve \smath{x = f(t)}, \smath{y = g(t)}, \smath{z = h(t)} as \smath{t} ranges from -\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. -\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}f, \allowbreak{} a..b, \allowbreak{} c..d\opRightPren{}} returns the space +\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}. +\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}f, \allowbreak{} a..b, +\allowbreak{} c..d\opRightPren{}} returns the space \smath{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of \smath{z = f(x, y)} as \smath{x} ranges from -\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and \smath{y} ranges from -\smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. -\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}f(x, y), \allowbreak{} x = a..b, \allowbreak{} y = c..d\opRightPren{}} returns +\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +and \smath{y} ranges from +\smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} +to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. +\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}f(x, y), +\allowbreak{} x = a..b, \allowbreak{} y = c..d\opRightPren{}} returns the space \smath{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of \smath{z = f(x, y)} as \smath{x} ranges -from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and \smath{y} ranges -from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. -\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}surface(f, g, h), \allowbreak{} a..b, \allowbreak{} c..d\opRightPren{}} returns +from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and +\smath{y} ranges +from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} +to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. +\bigitem\smath{\mbox{\bf makeObject}\opLeftPren{}surface(f, g, h), +\allowbreak{} a..b, \allowbreak{} c..d\opRightPren{}} returns the space \smath{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of the parametric surface \smath{x = f(u, v)}, \smath{y = g(u, v)}, \smath{z = h(u, v)} as \smath{u} ranges -from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and \smath{v} ranges -from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. +from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +to \smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +and \smath{v} ranges +from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} +to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. \bigitem \smath{makeObject(surface(f(u, v), g(u, v), h(u, v)), u = a..b, v = c..d)} returns the space \smath{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of the parametric surface \smath{x = f(u, v)}, \smath{y = g(u, v)}, -\smath{z = h(u, v)} as \smath{u} ranges from \smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to -\smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} and \smath{v} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, \allowbreak{} d\opRightPren{}} to +\smath{z = h(u, v)} as \smath{u} ranges from +\smath{\mbox{\bf min}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} to +\smath{\mbox{\bf max}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +and \smath{v} ranges from \smath{\mbox{\bf min}\opLeftPren{}c, +\allowbreak{} d\opRightPren{}} to \smath{\mbox{\bf max}\opLeftPren{}c, \allowbreak{} d\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{makeYoungTableau}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{makeYoungTableau}}\opLeftPren{} +{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% }% }% -{2}{(List(Integer), List(Integer))->Matrix(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf makeYoungTableau}\opLeftPren{}lambda, \allowbreak{} gitter\opRightPren{}} computes for a given +{2}{(List(Integer), List(Integer))->Matrix(Integer)} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf makeYoungTableau}\opLeftPren{}lambda, +\allowbreak{} gitter\opRightPren{}} computes for a given lattice permutation {\it gitter} and for an improper partition {\it lambda} the corresponding standard tableau of shape {\it lambda}. @@ -74812,16 +77639,21 @@ See \spadfun{listYoungTableaus}. }% }% {1}{(\$)->Integer}{FloatingPointSystem} -{\smath{\mbox{\bf mantissa}\opLeftPren{}x\opRightPren{}} returns the mantissa part of \smath{x}. +{\smath{\mbox{\bf mantissa}\opLeftPren{}x\opRightPren{}} +returns the mantissa part of \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{map}}\opLeftPren{}{\it function}, \allowbreak{}{\it structure}\allowbreak $\,[$ , \allowbreak{}{\it structure}$]$\opRightPren{}% - \opand \mbox{\axiomFun{map!}}\opLeftPren{}{\it function}, \allowbreak{}{\it structure}\opRightPren{}% +\opdata{{\mbox{\axiomFun{map}}\opLeftPren{}{\it function}, \allowbreak{} +{\it structure}\allowbreak $\,[$ , \allowbreak{} +{\it structure}$]$\opRightPren{}% + \opand \mbox{\axiomFun{map!}}\opLeftPren{}{\it function}, \allowbreak{} +{\it structure}\opRightPren{}% }% }% {2}{((R)->R, \$)->\$}{AbelianMonoidRing} -{\smath{\mbox{\bf map}\opLeftPren{}fn, \allowbreak{} u\opRightPren{}} maps the one-argument function \smath{fn} onto +{\smath{\mbox{\bf map}\opLeftPren{}fn, \allowbreak{} u\opRightPren{}} +maps the one-argument function \smath{fn} onto the components of a structure, returning a new structure. Most structures allow \smath{f} to have different source and target domains. @@ -74842,34 +77674,41 @@ coefficients of the series. \item a segment \smath{a..b} or a segment binding of the form \smath{x=a..b}: each of the elements from \smath{a} to \smath{b}. \item an equation: both sides of the equation. -\end{simpleList} \smath{\mbox{\bf map}\opLeftPren{}fn, \allowbreak{} u, \allowbreak{} v\opRightPren{}} maps the two argument +\end{simpleList} \smath{\mbox{\bf map}\opLeftPren{}fn, +\allowbreak{} u, \allowbreak{} v\opRightPren{}} maps the two argument function \smath{fn} onto the components of a structure, returning a new structure. Arguments \smath{u} and \smath{v} can be matrices, finite aggregates such as lists, tables, and vectors, and infinite aggregates such as streams and series. -\bigitem \smath{\mbox{\bf map!}\opLeftPren{}f, \allowbreak{} u\opRightPren{}}, where \smath{u} is homogeneous +\bigitem \smath{\mbox{\bf map!}\opLeftPren{}f, +\allowbreak{} u\opRightPren{}}, where \smath{u} is homogeneous aggregate, destructively replaces each element \smath{x} of \smath{u} by \smath{f(x)}. \bigitem \seeAlso{\spadfun{match}} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mapCoef}}\opLeftPren{}{\it function}, \allowbreak{}{\it freeAbelianMonoid}\opRightPren{}% - \opand \mbox{\axiomFun{mapGen}}\opLeftPren{}{\it function}, \allowbreak{}{\it freeAbelianMonoid}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mapCoef}}\opLeftPren{} +{\it function}, \allowbreak{}{\it freeAbelianMonoid}\opRightPren{}% + \opand \mbox{\axiomFun{mapGen}}\opLeftPren{} +{\it function}, \allowbreak{}{\it freeAbelianMonoid}\opRightPren{}% }% }% {2}{((E)->E, \$)->\$}{FreeAbelianMonoidCategory} -{\smath{\mbox{\bf mapCoeff}\opLeftPren{}f, \allowbreak{} m\opRightPren{}} maps unary function \smath{f} onto the +{\smath{\mbox{\bf mapCoeff}\opLeftPren{}f, +\allowbreak{} m\opRightPren{}} maps unary function \smath{f} onto the coefficients of a free abelian monoid of the form \smath{e_1 a_1 +\ldots+ e_n a_n} returning \smath{f(e_1) a_1 +\ldots+ f(e_n) a_n}. -\newitem\smath{\mbox{\bf mapGen}\opLeftPren{}fn, \allowbreak{} m\opRightPren{}} similarly returns +\newitem\smath{\mbox{\bf mapGen}\opLeftPren{}fn, +\allowbreak{} m\opRightPren{}} similarly returns \smath{e_1 f(a_1) +\ldots+ e_n f(a_n)}. \seeType{FreeAbelianMonoidCategory} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mapDown!}}\opLeftPren{}{\it tree}, \allowbreak{}{\it value}, \allowbreak{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mapDown!}}\opLeftPren{}{\it tree}, +\allowbreak{}{\it value}, \allowbreak{}{\it function}\opRightPren{}% }% }% {2}{((E)->E, \$)->\$}{FreeAbelianMonoidCategory} @@ -74880,14 +77719,16 @@ then right branch) of a tree \smath{t} of type type \smath{S} from the root to the leaves of the tree, then returning the modified tree as value; \smath{p} is a value of type \smath{S}.} -\newitem\smath{\mbox{\bf mapDown!}\opLeftPren{}t, \allowbreak{} p, \allowbreak{} f\opRightPren{}}, where \smath{f} is a function +\newitem\smath{\mbox{\bf mapDown!}\opLeftPren{}t, +\allowbreak{} p, \allowbreak{} f\opRightPren{}}, where \smath{f} is a function of type \spadsig{(S, S)}{S}, replaces the successive interior nodes of \smath{t} as follows. The root value \smath{x} is replaced by \smath{q = f(x, p)}. Then \spadfun{mapDown!} is recursively applied to \smath{(l, q, f)} and \smath{(r, q, f)} where \smath{l} and \smath{r} are respectively the left and right subtrees of \smath{t}. -\newitem\smath{\mbox{\bf mapDown!}\opLeftPren{}t, \allowbreak{} p, \allowbreak{} f\opRightPren{}}, where \smath{f} is a function +\newitem\smath{\mbox{\bf mapDown!}\opLeftPren{}t, +\allowbreak{} p, \allowbreak{} f\opRightPren{}}, where \smath{f} is a function of type \spadsig{(S, S, S)}{List S}, is similar. The root value of \smath{t} is first replaced by \smath{p}. Then \smath{f} is applied to three values: the value at the @@ -74898,16 +77739,19 @@ the left and right subtrees. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mapExponents}}\opLeftPren{}{\it function}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mapExponents}}\opLeftPren{} +{\it function}, \allowbreak{}{\it polynomial}\opRightPren{}% }% }% {2}{((E)->E, \$)->\$}{FiniteAbelianMonoidRing} -{\smath{\mbox{\bf mapExponents}\opLeftPren{}fn, \allowbreak{} u\opRightPren{}} maps function \smath{fn} onto the +{\smath{\mbox{\bf mapExponents}\opLeftPren{}fn, +\allowbreak{} u\opRightPren{}} maps function \smath{fn} onto the exponents of the non-zero monomials of polynomial \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mapUp!}}\opLeftPren{}{\it \optinit{tree, }tree, function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mapUp!}}\opLeftPren{} +{\it \optinit{tree, }tree, function}\opRightPren{}% }% }% {2}{((E)->E, \$)->\$}{FreeAbelianMonoidCategory} @@ -74917,11 +77761,14 @@ right branch then node) of a tree \smath{t} of type type \smath{S} from the leaves to the root of the tree, then returning the modified tree as value; \smath{p} is a value of type \smath{S}.} -\newitem\smath{\mbox{\bf mapUp!}\opLeftPren{}t, \allowbreak{} f\opRightPren{}}, where \smath{f} has type +\newitem\smath{\mbox{\bf mapUp!}\opLeftPren{}t, +\allowbreak{} f\opRightPren{}}, where \smath{f} has type \spadsig{(S, S)}{S}, replaces the value at each interior node by \smath{f(l, r)}, where \smath{l} and \smath{r} are the values at the immediate left and right nodes. -\newitem\smath{\mbox{\bf mapUp!}\opLeftPren{}t, \allowbreak{} t_1, \allowbreak{} f\opRightPren{}} makes an endorder traversal of +\newitem\smath{\mbox{\bf mapUp!}\opLeftPren{}t, +\allowbreak{} t_1, \allowbreak{} f\opRightPren{}} +makes an endorder traversal of both \smath{t} and \smath{t_1} (of identical shape) in parallel. The value at each successive interior node of \smath{t} is replaced by \smath{f(l, r, l_1, r_1)}, where \smath{l} and @@ -74935,48 +77782,63 @@ values of \smath{t_1}. }% }% {1}{(\$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf mask}\opLeftPren{}n\opRightPren{}} returns \smath{2^n-1} (an \smath{n}-bit mask). +{\smath{\mbox{\bf mask}\opLeftPren{}n\opRightPren{}} returns +\smath{2^n-1} (an \smath{n}-bit mask). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{match?}}\opLeftPren{}{\it string}, \allowbreak{}{\it string}, \allowbreak{}{\it character}\opRightPren{}% +\opdata{{\mbox{\axiomFun{match?}}\opLeftPren{}{\it string}, \allowbreak{} +{\it string}, \allowbreak{}{\it character}\opRightPren{}% }% }% {3}{(\$, \$, Character)->Boolean}{StringAggregate} -{\smath{\mbox{\bf match?}\opLeftPren{}s, \allowbreak{} t, \allowbreak{} char\opRightPren{}} tests if \smath{s} matches \smath{t} except perhaps for multiple and consecutive occurrences of character \smath{char}. Typically \smath{char} is the blank character. +{\smath{\mbox{\bf match?}\opLeftPren{}s, \allowbreak{} t, +\allowbreak{} char\opRightPren{}} tests if \smath{s} matches \smath{t} except +perhaps for multiple and consecutive occurrences of character \smath{char}. +Typically \smath{char} is the blank character. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{match}}\opLeftPren{}{\it list}, \allowbreak{}{\it list}\allowbreak $\,[$ , \allowbreak{}{\it option}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{match}}\opLeftPren{}{\it list}, \allowbreak{} +{\it list}\allowbreak $\,[$ , \allowbreak{}{\it option}$]$\opRightPren{}% }% }% {2}{(List(A), List(B))->(A)->B}{ListToMap} -{\smath{\mbox{\bf match}\opLeftPren{}la, \allowbreak{} lb\optinner{, u}\opRightPren{}}, +{\smath{\mbox{\bf match}\opLeftPren{}la, +\allowbreak{} lb\optinner{, u}\opRightPren{}}, where \smath{la} and \smath{lb} are lists of equal length, creates a function that can be used by \spadfun{map}. The target of a source value \smath{x} in \smath{la} is the value \smath{y} with the corresponding index in \smath{lb}. Optional argument \smath{u} defines the target for a source value \smath{a} which is not in \smath{la}. -If \smath{u} is a value of the source domain, then \smath{a} is replaced by \smath{u}, +If \smath{u} is a value of the source domain, then \smath{a} is +replaced by \smath{u}, which must be a member of \smath{la}. -If \smath{u} is a value of the target domain, the value returned by the map for +If \smath{u} is a value of the target domain, the value returned +by the map for \smath{a} is \smath{u}. If \smath{u} is a function \smath{f}, then the value returned is \smath{f(a)}. If no third argument is given, an error occurs when such a \smath{a} is found. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mathieu11}}\opLeftPren{}{\it \opt{listOfIntegers}}\opRightPren{}% - \optand \mbox{\axiomFun{mathieu12}}\opLeftPren{}{\it \opt{listOfIntegers}}\opRightPren{}% - \optand \mbox{\axiomFun{mathieu22}}\opLeftPren{}{\it \opt{listOfIntegers}}\opRightPren{}% - \optand \mbox{\axiomFun{mathieu23}}\opLeftPren{}{\it \opt{listOfIntegers}}\opRightPren{}% - \opand \mbox{\axiomFun{mathieu24}}\opLeftPren{}{\it \opt{listOfIntegers}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mathieu11}}\opLeftPren{} +{\it \opt{listOfIntegers}}\opRightPren{}% + \optand \mbox{\axiomFun{mathieu12}}\opLeftPren{} +{\it \opt{listOfIntegers}}\opRightPren{}% + \optand \mbox{\axiomFun{mathieu22}}\opLeftPren{} +{\it \opt{listOfIntegers}}\opRightPren{}% + \optand \mbox{\axiomFun{mathieu23}}\opLeftPren{} +{\it \opt{listOfIntegers}}\opRightPren{}% + \opand \mbox{\axiomFun{mathieu24}}\opLeftPren{} +{\it \opt{listOfIntegers}}\opRightPren{}% }% }% % {0}{()->PermutationGroup(Integer)}{PermutationGroupExamples}{% -\smath{\mbox{\bf mathieu11}\opLeftPren{}\optinner{li}\opRightPren{}} constructs the +\smath{\mbox{\bf mathieu11}\opLeftPren{}\optinner{li}\opRightPren{}} +constructs the mathieu group acting on the eleven integers given in the list \smath{li}. Duplicates in the list will be removed and \spadfun{error} will be called @@ -74985,14 +77847,15 @@ The default value of \smath{li} is \smath{[1, \ldots, 11]}. Operations \smath{mathieu12}, \smath{mathieu22}, and \smath{mathieu23} and \smath{mathieu24} are similar. -These operations provide examples of permutation groups in \Language{}.} +These operations provide examples of permutation groups in Axiom.} % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{matrix}}\opLeftPren{}{\it listOfLists}\opRightPren{}% }% }% {1}{(List(List(R)))->\$}{RectangularMatrixCategory} -{\smath{\mbox{\bf matrix}\opLeftPren{}l\opRightPren{}} converts the list of lists \smath{l} to a +{\smath{\mbox{\bf matrix}\opLeftPren{}l\opRightPren{}} +converts the list of lists \smath{l} to a matrix, where the list of lists is viewed as a list of the rows of the matrix. \newitem\smath{\mbox{\bf matrix}\opLeftPren{}llo\opRightPren{}}, @@ -75008,10 +77871,13 @@ returns an output form displaying \smath{llo} as a matrix. }% }% {0}{()->\$}{SingleInteger} -{\smath{\mbox{\bf max}\opLeftPren{}\opRightPren{}} returns the largest small integer. -\newline\smath{\mbox{\bf max}\opLeftPren{}u\opRightPren{}} returns the largest element of aggregate +{\smath{\mbox{\bf max}\opLeftPren{}\opRightPren{}} returns +the largest small integer. +\newline\smath{\mbox{\bf max}\opLeftPren{}u\opRightPren{}} +returns the largest element of aggregate \smath{u}. -\newline\smath{\mbox{\bf max}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns the maximum of \smath{x} and +\newline\smath{\mbox{\bf max}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +returns the maximum of \smath{x} and \smath{y} relative to a total ordering \axiomOp{<}. } @@ -75020,7 +77886,8 @@ returns an output form displaying \smath{llo} as a matrix. }% }% {1}{(\$)->Integer}{RectangularMatrixCategory} -{\smath{\mbox{\bf maxColIndex}\opLeftPren{}m\opRightPren{}} returns the index of the last column of the matrix +{\smath{\mbox{\bf maxColIndex}\opLeftPren{}m\opRightPren{}} +returns the index of the last column of the matrix or two-dimensional array \smath{m}. } @@ -75030,9 +77897,11 @@ or two-dimensional array \smath{m}. }% }% {1}{(\$)->Index}{IndexedAggregate} -{\smath{\mbox{\bf maxIndex}\opLeftPren{}u\opRightPren{}} returns the maximum index \smath{i} of indexed +{\smath{\mbox{\bf maxIndex}\opLeftPren{}u\opRightPren{}} +returns the maximum index \smath{i} of indexed aggregate \smath{u}. For most indexed aggregates (vectors, strings, lists), -\smath{\mbox{\bf maxIndex}\opLeftPren{}u\opRightPren{}} is equivalent to \smath{\# u.} +\smath{\mbox{\bf maxIndex}\opLeftPren{}u\opRightPren{}} is +equivalent to \smath{\# u.} } @@ -75043,30 +77912,39 @@ aggregate \smath{u}. For most indexed aggregates (vectors, strings, lists), }% }% {1}{(\$)->Integer}{RectangularMatrixCategory} -{\smath{\mbox{\bf maxRowIndex}\opLeftPren{}m\opRightPren{}} returns the index of the ``last'' row of the matrix +{\smath{\mbox{\bf maxRowIndex}\opLeftPren{}m\opRightPren{}} +returns the index of the ``last'' row of the matrix or two-dimensional array \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{meatAxe}}\opLeftPren{}{\it listOfListsOfMatrices \opt{, boolean, integer, integer}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{meatAxe}}\opLeftPren{} +{\it listOfListsOfMatrices \opt{, boolean, integer, integer}}\opRightPren{}% }% }% -{4}{(List(Matrix(R)), Boolean, Integer, Integer)->List(List(Matrix(R)))}{RepresentationPackage2} -{\smath{\mbox{\bf meatAxe}\opLeftPren{}aG \optinner{, randomElts, \allowbreak numOfTries, \allowbreak maxTests}\opRightPren{}} tries to +{4}{(List(Matrix(R)), Boolean, Integer, Integer)-> +List(List(Matrix(R)))}{RepresentationPackage2} +{\smath{\mbox{\bf meatAxe}\opLeftPren{}aG \optinner{, randomElts, +\allowbreak numOfTries, \allowbreak maxTests}\opRightPren{}} tries to split the representation given by \smath{aG} and returns a 2-list of -representations. All matrices of argument \smath{aG} are assumed to be square and of equal size. +representations. All matrices of argument \smath{aG} are assumed to be +square and of equal size. The default values of arguments \smath{randomElts}, -\smath{numOfTries} and \smath{maxTests} are \smath{false}, 25, and 7, respectively. +\smath{numOfTries} and \smath{maxTests} are \smath{false}, 25, and 7, +respectively. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{member?}}\opLeftPren{}{\it element}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{member?}}\opLeftPren{}{\it element}, \allowbreak{} +{\it aggregate}\opRightPren{}% }% }% {2}{(S, \$)->Boolean}{HomogeneousAggregate} -{\smath{\mbox{\bf member?}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} tests if \smath{x} is a member of \smath{u}. +{\smath{\mbox{\bf member?}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} +tests if \smath{x} is a member of \smath{u}. \newitem -\smath{\mbox{\bf member?}\opLeftPren{}pp, \allowbreak{} gp\opRightPren{}}, where \smath{pp} is a permutation and +\smath{\mbox{\bf member?}\opLeftPren{}pp, \allowbreak{} gp\opRightPren{}}, +where \smath{pp} is a permutation and \smath{gp} is a group, tests whether {\it pp} is in the group {\it gp}. } @@ -75078,16 +77956,19 @@ whether {\it pp} is in the group {\it gp}. }% }% {2}{(\$, \$)->\$}{ExtensibleLinearAggregate} -{\smath{\mbox{\bf merge}\opLeftPren{}[s1, \allowbreak{} s2, \allowbreak{} \ldots, sn]\opRightPren{}} will create a new +{\smath{\mbox{\bf merge}\opLeftPren{}[s1, \allowbreak{} s2, \allowbreak{} +\ldots, sn]\opRightPren{}} will create a new \spadtype{ThreeSpace} object that has the components of all the ones in the list; groupings of components into composites are maintained. \newitem -\smath{\mbox{\bf merge}\opLeftPren{}s1, \allowbreak{} s2\opRightPren{}} will create a new \spadtype{ThreeSpace} +\smath{\mbox{\bf merge}\opLeftPren{}s1, \allowbreak{} s2\opRightPren{}} +will create a new \spadtype{ThreeSpace} object that has the components of \smath{s1} and \smath{s2}; groupings of components into composites are maintained. \newitem -\smath{\mbox{\bf merge}\opLeftPren{}\optfirst{p, }a, b\opRightPren{}} returns an aggregate \smath{c} which +\smath{\mbox{\bf merge}\opLeftPren{}\optfirst{p, }a, b\opRightPren{}} +returns an aggregate \smath{c} which merges \smath{a} and \smath{b}. The result is produced by examining each element \smath{x} of \smath{a} and \smath{y} of \smath{b} successively. @@ -75095,16 +77976,21 @@ If \smath{p(x, y)} is \smath{true}, then \smath{x} is inserted into the result. Otherwise \smath{y} is inserted. If \smath{x} is chosen, the next element of \smath{a} is examined, and so on. -When all the elements of one aggregate are examined, the remaining elements of the other are appended. For example, \smath{\mbox{\bf merge}\opLeftPren{}<, \allowbreak{} [1, \allowbreak{} 3], \allowbreak{} [2, \allowbreak{} 7, \allowbreak{} 5]\opRightPren{}} returns \smath{[1, 2, 3, 7, 5]}. +When all the elements of one aggregate are examined, the remaining elements +of the other are appended. For example, \smath{\mbox{\bf merge}\opLeftPren{}<, +\allowbreak{} [1, \allowbreak{} 3], \allowbreak{} [2, \allowbreak{} 7, +\allowbreak{} 5]\opRightPren{}} returns \smath{[1, 2, 3, 7, 5]}. By default, function \smath{p} is \smath{\leq}. \newitem -\smath{\mbox{\bf merge!}\opLeftPren{}\optfirst{p}, u, v\opRightPren{}} destructively merges the elements \smath{u} and \smath{v} into +\smath{\mbox{\bf merge!}\opLeftPren{}\optfirst{p}, u, v\opRightPren{}} +destructively merges the elements \smath{u} and \smath{v} into \smath{u} using comparison function \smath{p}. Function \smath{p} is \smath{\leq} by default. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mesh}}\opLeftPren{}{\it u\opt{, v, w, x}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mesh}}\opLeftPren{} +{\it u\opt{, v, w, x}}\opRightPren{}% }% }% {4}{(\$, List(List(List(R))), Boolean, Boolean)->\$}{ThreeSpace} @@ -75113,11 +77999,14 @@ object \smath{sp}. Argument \smath{lc} is a list of curves. Each curve is either a list of points (objects of type \spadtype{Point}) or else a list of lists of small floats.} -\newitem\smath{\mbox{\bf mesh}\opLeftPren{}lc\opRightPren{}} returns a \spadtype{ThreeSpace} object +\newitem\smath{\mbox{\bf mesh}\opLeftPren{}lc\opRightPren{}} +returns a \spadtype{ThreeSpace} object defined by \smath{lc}. -\newitem\smath{\mbox{\bf mesh}\opLeftPren{}sp\opRightPren{}} returns the list of curves contained in +\newitem\smath{\mbox{\bf mesh}\opLeftPren{}sp\opRightPren{}} +returns the list of curves contained in space \smath{sp}. -\newitem\smath{\mbox{\bf mesh}\opLeftPren{}\optfirst{sp, }, lc, close1, close2\opRightPren{}} adds the +\newitem\smath{\mbox{\bf mesh}\opLeftPren{} +\optfirst{sp, }, lc, close1, close2\opRightPren{}} adds the list of curves \smath{lc} to the \spadtype{ThreeSpace} object \smath{sp}. Boolean arguments \smath{close1} and \smath{close2} tell how the @@ -75131,14 +78020,19 @@ By default, the argument \smath{sp} is empty. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{midpoints}}\opLeftPren{}{\it listOfIntervals}\opRightPren{}% +\opdata{{\mbox{\axiomFun{midpoints}}\opLeftPren{} +{\it listOfIntervals}\opRightPren{}% }% }% -{1}{(List(Record(left:Fraction(Integer), right:Fraction(Integer))))->List(Fraction(Integer))}{RealZeroPackage} +{1}{(List(Record(left:Fraction(Integer), right:Fraction(Integer))))-> +List(Fraction(Integer))}{RealZeroPackage} {\opkey{These operations are defined on ``intervals'' represented -by records with keys \smath{right} and \smath{left}, and rational number values.} -\newitem\smath{\mbox{\bf midpoints}\opLeftPren{}isolist\opRightPren{}} returns the list of midpoints for the list of intervals \smath{isolist}. -\newitem\smath{\mbox{\bf midpoint}\opLeftPren{}int\opRightPren{}} returns the midpoint of the interval \smath{int}. +by records with keys \smath{right} and \smath{left}, and rational +number values.} +\newitem\smath{\mbox{\bf midpoints}\opLeftPren{}isolist\opRightPren{}} +returns the list of midpoints for the list of intervals \smath{isolist}. +\newitem\smath{\mbox{\bf midpoint}\opLeftPren{}int\opRightPren{}} +returns the midpoint of the interval \smath{int}. } % ---------------------------------------------------------------------- @@ -75146,11 +78040,14 @@ by records with keys \smath{right} and \smath{left}, and rational number values. }% }% {0}{()->\$}{SingleInteger} -{\smath{\mbox{\bf min}\opLeftPren{}\opRightPren{}} returns the element of type \spadtype{SingleInteger}. +{\smath{\mbox{\bf min}\opLeftPren{}\opRightPren{}} returns the +element of type \spadtype{SingleInteger}. \newitem -\smath{\mbox{\bf min}\opLeftPren{}u\opRightPren{}} returns the smallest element of aggregate \smath{u}. +\smath{\mbox{\bf min}\opLeftPren{}u\opRightPren{}} returns the +smallest element of aggregate \smath{u}. \newitem -\smath{\mbox{\bf min}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns the minimum of \smath{x} and \smath{y} relative to +\smath{\mbox{\bf min}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +returns the minimum of \smath{x} and \smath{y} relative to total ordering \smath{<}. } @@ -75159,41 +78056,56 @@ total ordering \smath{<}. }% }% {1}{(\$)->Integer}{RectangularMatrixCategory} -{\smath{\mbox{\bf minColIndex}\opLeftPren{}m\opRightPren{}} returns the index of the ``first'' column +{\smath{\mbox{\bf minColIndex}\opLeftPren{}m\opRightPren{}} +returns the index of the ``first'' column of the matrix or two-dimensional array \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{minimalPolynomial}}\opLeftPren{}{\it element}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{minimalPolynomial}}\opLeftPren{} +{\it element}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {1}{(\$)->SparseUnivariatePolynomial(F)}{FiniteAlgebraicExtensionField} -{\smath{\mbox{\bf minimalPolynomial}\opLeftPren{}x\optinner{, n}\opRightPren{}} computes the minimal polynomial of \smath{x} over the field of extension degree \smath{n} over the ground field \smath{F}. +{\smath{\mbox{\bf minimalPolynomial}\opLeftPren{}x +\optinner{, n}\opRightPren{}} computes the minimal +polynomial of \smath{x} over the +field of extension degree \smath{n} over the ground field \smath{F}. The default value of \smath{n} is 1. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{minimalPolynomial}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{minimalPolynomial}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->UP}{FiniteRankAlgebra} -{\smath{\mbox{\bf minimalPolynomial}\opLeftPren{}a\opRightPren{}} returns the minimal polynomial of element \smath{a} +{\smath{\mbox{\bf minimalPolynomial}\opLeftPren{}a\opRightPren{}} +returns the minimal polynomial of element \smath{a} of a finite rank algebra. \seeType{FiniteRankAlgebra} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{minimumDegree}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it variable}\opRightPren{}% +\opdata{{\mbox{\axiomFun{minimumDegree}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it variable}\opRightPren{}% }% }% {1}{(\$)->E}{FiniteAbelianMonoidRing} -{\smath{\mbox{\bf minimumDegree}\opLeftPren{}p, \allowbreak{} v\opRightPren{}} gives the minimum degree of polynomial \smath{p} with respect to \smath{v}, that is, viewed as a univariate polynomial in \smath{v}. +{\smath{\mbox{\bf minimumDegree}\opLeftPren{}p, +\allowbreak{} v\opRightPren{}} gives the minimum degree of +polynomial \smath{p} with +respect to \smath{v}, that is, viewed as a univariate polynomial in \smath{v}. \newitem -\smath{\mbox{\bf minimumDegree}\opLeftPren{}p, \allowbreak{} lv\opRightPren{}} gives the list of minimum degrees of the polynomial \smath{p} with respect to each of the variables in the list \smath{lv}. +\smath{\mbox{\bf minimumDegree}\opLeftPren{}p, +\allowbreak{} lv\opRightPren{}} gives the list of +minimum degrees of the polynomial \smath{p} +with respect to each of the variables in the list \smath{lv}. \newitem -\seeAlso{\spadtype{FiniteAbelianMonoidRing} and \spadtype{MonogenicLinearOperator}} +\seeAlso{\spadtype{FiniteAbelianMonoidRing} and +\spadtype{MonogenicLinearOperator}} } % ---------------------------------------------------------------------- @@ -75201,7 +78113,8 @@ of a finite rank algebra. }% }% {1}{(\$)->Index}{IndexedAggregate} -{\smath{\mbox{\bf minIndex}\opLeftPren{}aggregate\opRightPren{}} returns the minimum index \smath{i} of aggregate \smath{u}. +{\smath{\mbox{\bf minIndex}\opLeftPren{}aggregate\opRightPren{}} +returns the minimum index \smath{i} of aggregate \smath{u}. Note: the \axiomFun{minIndex} of most system-defined indexed aggregates is 1. See also \spadtype{PointCategory}. } @@ -75212,7 +78125,8 @@ See also \spadtype{PointCategory}. }% }% {1}{(\$)->R}{MatrixCategory} -{\smath{\mbox{\bf minordet}\opLeftPren{}m\opRightPren{}} computes the determinant of the matrix \smath{m} using minors, +{\smath{\mbox{\bf minordet}\opLeftPren{}m\opRightPren{}} +computes the determinant of the matrix \smath{m} using minors, or calls \spadfun{error} if the matrix is not square. } @@ -75223,7 +78137,8 @@ or calls \spadfun{error} if the matrix is not square. }% }% {1}{(Kernel(\$))->SparseUnivariatePolynomial(\$)}{ExpressionSpace} -{\smath{\mbox{\bf minPoly}\opLeftPren{}k\opRightPren{}} returns polynomial \smath{p} such that \smath{p(k) = 0}. +{\smath{\mbox{\bf minPoly}\opLeftPren{}k\opRightPren{}} +returns polynomial \smath{p} such that \smath{p(k) = 0}. } % ---------------------------------------------------------------------- @@ -75231,7 +78146,8 @@ or calls \spadfun{error} if the matrix is not square. }% }% {1}{(\$)->Integer}{RectangularMatrixCategory} -{\smath{\mbox{\bf minRowIndex}\opLeftPren{}m\opRightPren{}} returns the index of the ``first'' row of the matrix +{\smath{\mbox{\bf minRowIndex}\opLeftPren{}m\opRightPren{}} +returns the index of the ``first'' row of the matrix or two-dimensional array \smath{m}. } @@ -75240,16 +78156,21 @@ or two-dimensional array \smath{m}. }% }% {0}{()->OrderedCompletion(Integer)}{Infinity} -{\smath{\mbox{\bf minusInfinity}\opLeftPren{}\opRightPren{}} returns {\tt \%minusInfinity}, -the \Language{} name for \smath{-\infty}. +{\smath{\mbox{\bf minusInfinity}\opLeftPren{}\opRightPren{}} +returns {\tt \%minusInfinity}, +the Axiom name for \smath{-\infty}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{modifyPointData}}\opLeftPren{}{\it space}, \allowbreak{}{\it nonNegativeInteger}, \allowbreak{}{\it point}\opRightPren{}% +\opdata{{\mbox{\axiomFun{modifyPointData}}\opLeftPren{}{\it space}, +\allowbreak{}{\it nonNegativeInteger}, \allowbreak{} +{\it point}\opRightPren{}% }% }% {3}{(\$, NonNegativeInteger, Point(R))->\$}{ThreeSpace} -{\smath{\mbox{\bf modifyPointData}\opLeftPren{}sp, \allowbreak{} i, \allowbreak{} p\opRightPren{}} changes the point at the indexed location \smath{i} in the \spadtype{ThreeSpace} object \smath{sp} to \smath{p}. +{\smath{\mbox{\bf modifyPointData}\opLeftPren{}sp, \allowbreak{} i, +\allowbreak{} p\opRightPren{}} changes the point at the indexed location +\smath{i} in the \spadtype{ThreeSpace} object \smath{sp} to \smath{p}. This operation is useful for making changes to existing data. } @@ -75258,7 +78179,8 @@ This operation is useful for making changes to existing data. }% }% {1}{(\$)->Integer}{PAdicIntegerCategory} -{\smath{\mbox{\bf moduloP}\opLeftPren{}x\opRightPren{}}, such that \smath{p = \mbox{\bf modulus}()}, +{\smath{\mbox{\bf moduloP}\opLeftPren{}x\opRightPren{}}, +such that \smath{p = \mbox{\bf modulus}()}, returns a, where \smath{x = a + b p} where \smath{x} is a \smath{p}-adic integer. \seeType{PAdicIntegerCategory} @@ -75269,7 +78191,8 @@ where \smath{x} is a \smath{p}-adic integer. }% }% {0}{()->Integer}{PAdicIntegerCategory} -{\smath{\mbox{\bf modulus}\opLeftPren{}\opRightPren{}\$R} returns the value of the modulus \smath{p} +{\smath{\mbox{\bf modulus}\opLeftPren{}\opRightPren{}\$R} +returns the value of the modulus \smath{p} of a p-adic integer domain \smath{R}. \seeType{PAdicIntegerCategory} } @@ -75279,7 +78202,8 @@ of a p-adic integer domain \smath{R}. }% }% {1}{(Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf moebiusMu}\opLeftPren{}n\opRightPren{}} returns the Moebius function \smath{\mu(n)}, +{\smath{\mbox{\bf moebiusMu}\opLeftPren{}n\opRightPren{}} +returns the Moebius function \smath{\mu(n)}, defined as \smath{-1}, \smath{0} or \smath{1} as follows: \smath{\mu(n) = 0} if \smath{n} is divisible by a square \smath{ > 1}, and \smath{(-1)^k} if \smath{n} is square-free and @@ -75287,11 +78211,14 @@ has \smath{k} distinct prime divisors. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{monicDivide}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\allowbreak $\,[$ , \allowbreak{}{\it variable}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{monicDivide}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it polynomial}\allowbreak $\,[$ , +\allowbreak{}{\it variable}$]$\opRightPren{}% }% }% {2}{(\$, \$)->Record(quotient:\$, remainder:\$)}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf monicDivide}\opLeftPren{}p, \allowbreak{} q\optinner{, v}\opRightPren{}} divides the polynomial +{\smath{\mbox{\bf monicDivide}\opLeftPren{}p, +\allowbreak{} q\optinner{, v}\opRightPren{}} divides the polynomial \smath{p} by the monic polynomial \smath{q}, returning the record containing a \smath{quotient} and \smath{remainder}. For multivariate polynomials, the polynomials are viewed as a @@ -75303,52 +78230,64 @@ respect to \smath{v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{monomial}}\opLeftPren{}{\it coefficient}, \allowbreak{}{\it exponent}\allowbreak $\,[$ , \allowbreak{}{\it option}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{monomial}}\opLeftPren{}{\it coefficient}, +\allowbreak{}{\it exponent}\allowbreak $\,[$ , +\allowbreak{}{\it option}$]$\opRightPren{}% }% }% {2}{(R, E)->\$}{AbelianMonoidRing} -{\smath{\mbox{\bf monomial}\opLeftPren{}coef, \allowbreak{} exp\opRightPren{}} creates a term of +{\smath{\mbox{\bf monomial}\opLeftPren{}coef, +\allowbreak{} exp\opRightPren{}} creates a term of a univariate polynomial or series object from a coefficient \smath{coef} and exponent \smath{exp}. The variable name must be given by context (as through a declaration for the result). \newitem -\smath{\mbox{\bf monomial}\opLeftPren{}c, \allowbreak{} [x_1, \allowbreak{} \ldots, x_k], [n_1, \ldots, n_k]\opRightPren{}} creates a term +\smath{\mbox{\bf monomial}\opLeftPren{}c, +\allowbreak{} [x_1, \allowbreak{} \ldots, x_k], +[n_1, \ldots, n_k]\opRightPren{}} creates a term \smath{c x_1^{n_1}\ldots x_k^{n_k}} of a multivariate power series or polynomial from coefficient \smath{c}, variables \smath{x_j} and exponents \smath{n_j}. \newitem -\smath{\mbox{\bf monomial}\opLeftPren{}c, \allowbreak{} x, \allowbreak{} n\opRightPren{}} creates a term \smath{c x^n} +\smath{\mbox{\bf monomial}\opLeftPren{}c, +\allowbreak{} x, \allowbreak{} n\opRightPren{}} creates a term \smath{c x^n} of a polynomial or series from a coefficient \smath{c}, variable \smath{x}, and exponent \smath{n}. \newitem -\smath{\mbox{\bf monomial}\opLeftPren{}c, \allowbreak{} [n_1, \allowbreak{} \ldots, n_k]\opRightPren{}} +\smath{\mbox{\bf monomial}\opLeftPren{}c, +\allowbreak{} [n_1, \allowbreak{} \ldots, n_k]\opRightPren{}} creates a \spadtype{CliffordAlgebra} element \smath{c e(n_1), \ldots, c e(n_k)} from a coefficient \smath{c} and basis elements \smath{c(i_j)} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{monomial?}}\opLeftPren{}{\it polynomialOrSeries}\opRightPren{}% +\opdata{{\mbox{\axiomFun{monomial?}}\opLeftPren{} +{\it polynomialOrSeries}\opRightPren{}% }% }% {1}{(\$)->Boolean}{AbelianMonoidRing} -{\smath{\mbox{\bf monomial?}\opLeftPren{}p\opRightPren{}} tests if polynomial or series \smath{p} is a +{\smath{\mbox{\bf monomial?}\opLeftPren{}p\opRightPren{}} +tests if polynomial or series \smath{p} is a single monomial. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{monomials}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{monomials}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->List(\$)}{PolynomialCategory} -{\smath{\mbox{\bf monomials}\opLeftPren{}p\opRightPren{}} returns the list of non-zero monomials of +{\smath{\mbox{\bf monomials}\opLeftPren{}p\opRightPren{}} +returns the list of non-zero monomials of polynomial \smath{p}, \smath{[a_1 X^{(1)}, \ldots, a_n X^{(n)}]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{more?}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{more?}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->Boolean}{Aggregate} @@ -75357,31 +78296,38 @@ tests if \smath{u} has greater than \smath{n} elements. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{movedPoints}}\opLeftPren{}{\it permutation}\opRightPren{}% +\opdata{{\mbox{\axiomFun{movedPoints}}\opLeftPren{} +{\it permutation}\opRightPren{}% }% }% {1}{(\$)->Set(S)}{Permutation} -{\smath{\mbox{\bf movedPoints}\opLeftPren{}p\opRightPren{}} returns the set of points moved by the permutation \smath{p}. +{\smath{\mbox{\bf movedPoints}\opLeftPren{}p\opRightPren{}} +returns the set of points moved by the permutation \smath{p}. \newitem -\smath{\mbox{\bf movedPoints}\opLeftPren{}gp\opRightPren{}} returns the points moved by the group {\it gp}. +\smath{\mbox{\bf movedPoints}\opLeftPren{}gp\opRightPren{}} +returns the points moved by the group {\it gp}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{mulmod}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{mulmod}}\opLeftPren{}{\it integer}, +\allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {3}{(\$, \$, \$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf mulmod}\opLeftPren{}a, \allowbreak{} b, \allowbreak{} p\opRightPren{}}, where \smath{a}, +{\smath{\mbox{\bf mulmod}\opLeftPren{}a, \allowbreak{} b, +\allowbreak{} p\opRightPren{}}, where \smath{a}, \smath{b} are non-negative integers both \smath{<} integer \smath{p}, returns \smath{a b \mod p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{multiEuclidean}}\opLeftPren{}{\it listOfElements}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{multiEuclidean}}\opLeftPren{} +{\it listOfElements}, \allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(List(\$), \$)->Union(List(\$), "failed")}{EuclideanDomain} -{\smath{\mbox{\bf multiEuclidean}\opLeftPren{}[f_1, \allowbreak{} \ldots, f_n], z\opRightPren{}} returns a list of +{\smath{\mbox{\bf multiEuclidean}\opLeftPren{}[f_1, \allowbreak{} +\ldots, f_n], z\opRightPren{}} returns a list of coefficients \smath{[a_1, \ldots, a_n]} such that \smath{z/\prod_{i=1}^n f_i = \sum\nolimits_{j=1}^n {a_j/f_j}}. @@ -75390,20 +78336,24 @@ returned. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{multinomial}}\opLeftPren{}{\it integer}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{multinomial}}\opLeftPren{}{\it integer}, +\allowbreak{}{\it listOfIntegers}\opRightPren{}% }% }% {2}{(I, List(I))->I}{IntegerCombinatoricFunctions} -{\smath{\mbox{\bf multinomial}\opLeftPren{}n, \allowbreak{} [m_1, \allowbreak{} m_2, \allowbreak{} \ldots, m_k]\opRightPren{}} returns the +{\smath{\mbox{\bf multinomial}\opLeftPren{}n, \allowbreak{} +[m_1, \allowbreak{} m_2, \allowbreak{} \ldots, m_k]\opRightPren{}} returns the multinomial coefficient \smath{n!/(m_1! m_2! \ldots m_k!)}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{multiple}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{multiple}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{FunctionSpaceAssertions} -{\smath{\mbox{\bf multiple}\opLeftPren{}x\opRightPren{}} directs the pattern matcher that \smath{x} +{\smath{\mbox{\bf multiple}\opLeftPren{}x\opRightPren{}} +directs the pattern matcher that \smath{x} should preferably match a multi-term quantity in a sum or product. For matching on lists, multiple(\smath{x}) tells the pattern matcher that \smath{x} should match a list instead of an element @@ -75412,21 +78362,25 @@ This operation calls \spadfun{error} if \smath{x} is not a symbol. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{multiplyCoefficients}}\opLeftPren{}{\it function}, \allowbreak{}{\it series}\opRightPren{}% +\opdata{{\mbox{\axiomFun{multiplyCoefficients}}\opLeftPren{} +{\it function}, \allowbreak{}{\it series}\opRightPren{}% }% }% {2}{((Integer)->Coef, \$)->\$}{UnivariateTaylorSeriesCategory} -{\smath{\mbox{\bf multiplyCoefficients}\opLeftPren{}f, \allowbreak{} s\opRightPren{}} returns +{\smath{\mbox{\bf multiplyCoefficients}\opLeftPren{}f, +\allowbreak{} s\opRightPren{}} returns \smath{\sum\nolimits_{n = 0}^\infty {f(n) a_n x^n}} where \smath{s} is the series \smath{\sum\nolimits_{n = 0}^\infty{a_n x^n}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{multiplyExponents}}\opLeftPren{}{\it various}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{multiplyExponents}}\opLeftPren{} +{\it various}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->\$}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf multiplyExponents}\opLeftPren{}p, \allowbreak{} n\opRightPren{}}, +{\smath{\mbox{\bf multiplyExponents}\opLeftPren{}p, +\allowbreak{} n\opRightPren{}}, where \smath{p} is a univariate polynomial or series, returns a new polynomial or series resulting from multiplying all exponents by the non @@ -75434,19 +78388,23 @@ negative integer \smath{n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{multiset}}\opLeftPren{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{multiset}}\opLeftPren{} +{\it listOfElements}\opRightPren{}% }% }% {2}{(SparseUnivariatePolynomial(R), VarSet)->\$}{PolynomialCategory} -{\smath{\mbox{\bf multiset}\opLeftPren{}ls\opRightPren{}} creates a multiset with elements from \smath{ls}. +{\smath{\mbox{\bf multiset}\opLeftPren{}ls\opRightPren{}} +creates a multiset with elements from \smath{ls}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{multivariate}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it symbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{multivariate}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it symbol}\opRightPren{}% }% }% {2}{(SparseUnivariatePolynomial(R), VarSet)->\$}{PolynomialCategory} -{\smath{\mbox{\bf multivariate}\opLeftPren{}p, \allowbreak{} v\opRightPren{}} converts an anonymous univariate +{\smath{\mbox{\bf multivariate}\opLeftPren{}p, \allowbreak{} v\opRightPren{}} +converts an anonymous univariate polynomial \smath{p} to a polynomial in the variable \smath{v}. } % ---------------------------------------------------------------------- @@ -75454,9 +78412,12 @@ polynomial \smath{p} to a polynomial in the variable \smath{v}. }% }% {1}{(\$)->String}{FileNameCategory} -{\smath{\mbox{\bf name}\opLeftPren{}f\opRightPren{}} returns the name part of the file name for file \smath{f}. -\newitem\smath{\mbox{\bf name}\opLeftPren{}op\opRightPren{}} returns the name of basic operator \smath{op}. -\newitem\smath{\mbox{\bf name}\opLeftPren{}s\opRightPren{}} returns symbol \smath{s} without its scripts. +{\smath{\mbox{\bf name}\opLeftPren{}f\opRightPren{}} returns +the name part of the file name for file \smath{f}. +\newitem\smath{\mbox{\bf name}\opLeftPren{}op\opRightPren{}} +returns the name of basic operator \smath{op}. +\newitem\smath{\mbox{\bf name}\opLeftPren{}s\opRightPren{}} +returns symbol \smath{s} without its scripts. } @@ -75465,7 +78426,8 @@ polynomial \smath{p} to a polynomial in the variable \smath{v}. }% }% {2}{(\$, \$)->\$}{BitAggregate} -{\smath{\mbox{\bf nand}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns the logical negation of +{\smath{\mbox{\bf nand}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +returns the logical negation of \smath{a} and \smath{b}, either booleans or bit aggregates. Note: \smath{\mbox{\bf nand}\opLeftPren{}a, b) = true} if and only if one of \smath{a} and \smath{b} is \smath{false}. @@ -75476,7 +78438,8 @@ Note: \smath{\mbox{\bf nand}\opLeftPren{}a, b) = true} if and only if one of }% }% {1}{(\$)->Boolean}{BasicOperator} -{\smath{\mbox{\bf nary?}\opLeftPren{}op\opRightPren{}} tests if \smath{op} accepts an arbitrary number +{\smath{\mbox{\bf nary?}\opLeftPren{}op\opRightPren{}} +tests if \smath{op} accepts an arbitrary number of arguments. } @@ -75485,7 +78448,8 @@ of arguments. }% }% {1}{(\$)->NonNegativeInteger}{RectangularMatrixCategory} -{\smath{\mbox{\bf ncols}\opLeftPren{}m\opRightPren{}} returns the number of columns in the matrix +{\smath{\mbox{\bf ncols}\opLeftPren{}m\opRightPren{}} +returns the number of columns in the matrix or two-dimensional array \smath{m}. } @@ -75502,11 +78466,15 @@ Other variations are as follows: \begin{simpleList} \item\smath{\mbox{\bf new}\opLeftPren{}s\opRightPren{}}, where s is a symbol, returns a new symbol whose name starts with \%\smath{s}. -\item\smath{\mbox{\bf new}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} returns \smath{\mbox{\bf fill!}\opLeftPren{}new(n), \allowbreak{} x\opRightPren{}}, an -aggregate of \smath{n} elements, each with value \smath{x}. -\item\smath{\mbox{\bf new}\opLeftPren{}m, \allowbreak{} n, \allowbreak{} r\opRightPren{}\$R} creates an \smath{m}-by-\smath{n} array +\item\smath{\mbox{\bf new}\opLeftPren{}n, \allowbreak{} x\opRightPren{}} +returns \smath{\mbox{\bf fill!}\opLeftPren{}new(n), +\allowbreak{} x\opRightPren{}}, +an aggregate of \smath{n} elements, each with value \smath{x}. +\item\smath{\mbox{\bf new}\opLeftPren{}m, \allowbreak{} n, +\allowbreak{} r\opRightPren{}\$R} creates an \smath{m}-by-\smath{n} array or matrix of type \smath{R} all of whose entries are \smath{r}. -\item\smath{\mbox{\bf new}\opLeftPren{}d, \allowbreak{} pre, \allowbreak{} e\opRightPren{}}, +\item\smath{\mbox{\bf new}\opLeftPren{}d, \allowbreak{} pre, +\allowbreak{} e\opRightPren{}}, where \smath{d}, smath{pre}, and smath{e} are strings, constructs the name of a new writable file with \smath{d} as its directory, \smath{pre} as a prefix of its name and \smath{e} @@ -75522,47 +78490,64 @@ written in the given directory. }% }% {0}{()->String}{DisplayPackage} -{\smath{\mbox{\bf newLine}\opLeftPren{}\opRightPren{}} sends a new line command to output. +{\smath{\mbox{\bf newLine}\opLeftPren{}\opRightPren{}} sends a new line +command to output. See \spadtype{DisplayPackage}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nextColeman}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nextColeman}}\opLeftPren{}{\it listOfIntegers}, +\allowbreak{}{\it listOfIntegers}, \allowbreak{}{\it matrix}\opRightPren{}% }% }% -{3}{(List(Integer), List(Integer), Matrix(Integer))->Matrix(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf nextColeman}\opLeftPren{}alpha, \allowbreak{} beta, \allowbreak{} C\opRightPren{}} generates the next +{3}{(List(Integer), List(Integer), Matrix(Integer))->Matrix(Integer)} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf nextColeman}\opLeftPren{}alpha, \allowbreak{} beta, +\allowbreak{} C\opRightPren{}} generates the next Coleman-matrix of column sums {\it alpha} and row sums {\it beta} according to the lexicographical order from bottom-to-top. The first Coleman matrix is created using \smath{C = {\bf new}(1, 1, 0)}. -Also, \smath{\mbox{\bf new}\opLeftPren{}1, \allowbreak{} 1, \allowbreak{} 0\opRightPren{}} indicates that \smath{C} is the last Coleman matrix. +Also, \smath{\mbox{\bf new}\opLeftPren{}1, \allowbreak{} 1, +\allowbreak{} 0\opRightPren{}} indicates that \smath{C} is the +last Coleman matrix. See \spadtype{SymmetricGroupCombinatoricFunctions} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nextLatticePermutation}}\opLeftPren{}{\it integers}, \allowbreak{}{\it integers}, \allowbreak{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nextLatticePermutation}}\opLeftPren{} +{\it integers}, \allowbreak{}{\it integers}, \allowbreak{} +{\it boolean}\opRightPren{}% }% }% -{3}{(List(Integer), List(Integer), Boolean)->List(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf nextLatticePermutation}\opLeftPren{}lambda, \allowbreak{} lattP, \allowbreak{} constructNotFirst\opRightPren{}} +{3}{(List(Integer), List(Integer), Boolean)->List(Integer)} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf nextLatticePermutation}\opLeftPren{}lambda, +\allowbreak{} lattP, \allowbreak{} constructNotFirst\opRightPren{}} generates the -lattice permutation according to the proper partition \smath{lambda} succeeding +lattice permutation according to the proper partition +\smath{lambda} succeeding the lattice permutation \smath{lattP} in lexicographical order as long as \smath{constructNotFirst} is \smath{true}. If \smath{constructNotFirst} is \smath{false}, the first -lattice permutation is returned. The result \smath{nil} indicates that \smath{lattP} has no successor. +lattice permutation is returned. The result \smath{nil} +indicates that \smath{lattP} has no successor. See \spadtype{SymmetricGroupCombinatoricFunctions} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nextPartition}}\opLeftPren{}{\it vectorOfIntegers}, \allowbreak{}{\it vectorOfIntegers}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nextPartition}}\opLeftPren{} +{\it vectorOfIntegers}, \allowbreak{}{\it vectorOfIntegers}, \allowbreak{} +{\it integer}\opRightPren{}% }% }% -{3}{(List(Integer), Vector(Integer), Integer)->Vector(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf nextPartition}\opLeftPren{}gamma, \allowbreak{} part, \allowbreak{} number\opRightPren{}} generates the partition of +{3}{(List(Integer), Vector(Integer), Integer)-> +Vector(Integer)}{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf nextPartition}\opLeftPren{}gamma, +\allowbreak{} part, \allowbreak{} number\opRightPren{}} +generates the partition of \smath{number} which follows \smath{part} according to the right-to-left lexicographical order. The partition has the property that its components do not exceed @@ -75573,11 +78558,13 @@ See \spadtype{SymmetricGroupCombinatoricFunctions} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nextPrime}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nextPrime}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(I)->I}{IntegerPrimesPackage} -{\smath{\mbox{\bf nextPrime}\opLeftPren{}n\opRightPren{}} returns the smallest prime strictly larger +{\smath{\mbox{\bf nextPrime}\opLeftPren{}n\opRightPren{}} +returns the smallest prime strictly larger than \smath{n}. } @@ -75587,74 +78574,91 @@ than \smath{n}. }% }% {0}{()->\$}{List} -{\smath{\mbox{\bf nil}\opLeftPren{}\opRightPren{}\$R} returns the empty list of type \smath{R}. +{\smath{\mbox{\bf nil}\opLeftPren{}\opRightPren{}\$R} +returns the empty list of type \smath{R}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nilFactor}}\opLeftPren{}{\it element}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nilFactor}}\opLeftPren{} +{\it element}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(R, Integer)->\$}{Factored} -{\smath{\mbox{\bf nilFactor}\opLeftPren{}base, \allowbreak{} exponent\opRightPren{}} creates a factored object with a single factor +{\smath{\mbox{\bf nilFactor}\opLeftPren{}base, +\allowbreak{} exponent\opRightPren{}} creates a +factored object with a single factor with no information about the kind of \smath{base}. See \spadtype{Factored} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{node?}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{node?}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{RecursiveAggregate} -{\smath{\mbox{\bf node?}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} tests if node \smath{u} is contained in node +{\smath{\mbox{\bf node?}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} +tests if node \smath{u} is contained in node \smath{v} (either as a child, a child of a child, etc.). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nodes}}\opLeftPren{}{\it recursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nodes}}\opLeftPren{} +{\it recursiveAggregate}\opRightPren{}% }% }% {1}{(\$)->List(\$)}{RecursiveAggregate} -{\smath{\mbox{\bf nodes}\opLeftPren{}a\opRightPren{}} returns a list of all the nodes of aggregate +{\smath{\mbox{\bf nodes}\opLeftPren{}a\opRightPren{}} +returns a list of all the nodes of aggregate \smath{a}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{noncommutativeJordanAlgebra?}}\opLeftPren{}\opRightPren{}% +\opdata{{\mbox{\axiomFun{noncommutativeJordanAlgebra?}} +\opLeftPren{}\opRightPren{}% }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf noncommutativeJordanAlgebra?}\opLeftPren{}\opRightPren{}\$F} tests if the algebra +{\smath{\mbox{\bf noncommutativeJordanAlgebra?} +\opLeftPren{}\opRightPren{}\$F} tests if the algebra \smath{F} is flexible and Jordan admissible. See \spadtype{FiniteRankNonAssociativeAlgebra}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nor}}\opLeftPren{}{\it boolean}, \allowbreak{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nor}}\opLeftPren{} +{\it boolean}, \allowbreak{}{\it boolean}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{BitAggregate} -{\smath{\mbox{\bf nor}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns the logical \smath{nor} of booleans or +{\smath{\mbox{\bf nor}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns the logical \smath{nor} of booleans or bit aggregates \smath{a} and \smath{b}. -Note: \smath{\mbox{\bf nor}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} = true if and only if both \smath{a} and +Note: \smath{\mbox{\bf nor}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} = true if and only if both \smath{a} and \smath{b} are \smath{false}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{norm}}\opLeftPren{}{\it element\opt{, option}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{norm}}\opLeftPren{} +{\it element\opt{, option}}\opRightPren{}% }% }% {1}{(\$)->R}{ComplexCategory} {\smath{\mbox{\bf norm}\opLeftPren{}x\opRightPren{}} returns: \begin{simpleList} -\item for complex \smath{x}: \smath{\mbox{\bf conjugate}\opLeftPren{}x\opRightPren{}} +\item for complex \smath{x}: +\smath{\mbox{\bf conjugate}\opLeftPren{}x\opRightPren{}} . \item for floats: the absolute value. \item for quaternions or octonions: the sum of the squares of its coefficients. \item for a domain of category \spadtype{FiniteRankAlgebra}: -the determinant of the regular representation of \smath{x} with respect to any basis. +the determinant of the regular representation of \smath{x} with +respect to any basis. \end{simpleList} -\smath{\mbox{\bf norm}\opLeftPren{}x\optinner{, p}\opRightPren{}}, where \smath{p} is a +\smath{\mbox{\bf norm}\opLeftPren{}x\optinner{, p}\opRightPren{}}, +where \smath{p} is a positiveInteger and \smath{x} is an element of a domain of category \spadtype{FiniteAlgebraExtensionField} over ground field \smath{F}, @@ -75676,7 +78680,8 @@ where \smath{a} is a member of a domain of category \smath{FiniteAlgebraicExtensionField} over a field \smath{F}, tests whether the element \smath{a} is normal over the ground field \smath{F}, that is, if -\smath{a^{q^i}, 0 \leq i \leq {\bf extensionDegree}()-1} is an \smath{F}-basis, +\smath{a^{q^i}, 0 \leq i \leq {\bf extensionDegree}()-1} is +an \smath{F}-basis, where \smath{q = {\bf size}()}. } @@ -75689,7 +78694,8 @@ where \smath{q = {\bf size}()}. where \smath{R} is a domain of category \smath{FiniteAlgebraicExtensionField} over a field \smath{F}, returns a element, normal over the ground field \smath{F}, -that is, \smath{a^{q^i}, 0 \leq i < {\bf extensionDegree}()} is an \smath{F}-basis, +that is, \smath{a^{q^i}, 0 \leq i < {\bf extensionDegree}()} +is an \smath{F}-basis, where \smath{q = {\bf size}()}. At the first call, the element is computed by \spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} @@ -75699,11 +78705,13 @@ global variable. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{normalForm}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it listOfpolynomials}\opRightPren{}% +\opdata{{\mbox{\axiomFun{normalForm}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it listOfpolynomials}\opRightPren{}% }% }% {2}{(Dpol, List(Dpol))->Dpol}{GroebnerPackage} -{\smath{\mbox{\bf normalForm}\opLeftPren{}poly, \allowbreak{} gb\opRightPren{}} reduces the polynomial \smath{poly} +{\smath{\mbox{\bf normalForm}\opLeftPren{}poly, +\allowbreak{} gb\opRightPren{}} reduces the polynomial \smath{poly} modulo the precomputed Gr\"obner basis \smath{gb} giving a canonical representative of the residue class. } @@ -75712,19 +78720,25 @@ canonical representative of the residue class. \opdata{{\mbox{\axiomFun{normalise}}\opLeftPren{}{\it element}\opRightPren{}% }% }% -{1}{(Matrix(Expression(Fraction(Integer))))->Matrix(Expression(Fraction(Integer)))}{RadicalEigenPackage} -{\smath{\mbox{\bf normalise}\opLeftPren{}v\opRightPren{}} returns the column vector \smath{v} divided +{1}{(Matrix(Expression(Fraction(Integer))))-> +Matrix(Expression(Fraction(Integer)))}{RadicalEigenPackage} +{\smath{\mbox{\bf normalise}\opLeftPren{}v\opRightPren{}} +returns the column vector \smath{v} divided by its Euclidean norm; when possible, the vector \smath{v} is expressed in terms of radicals.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{normalize}}\opLeftPren{}{\it element \opt, option}\opRightPren{}% +\opdata{{\mbox{\axiomFun{normalize}}\opLeftPren{} +{\it element \opt, option}\opRightPren{}% }% }% -{1}{(Matrix(Expression(Fraction(Integer))))->Matrix(Expression(Fraction(Integer)))}{RadicalEigenPackage} -{\smath{\mbox{\bf normalize}\opLeftPren{}flt\opRightPren{}} normalizes float \smath{flt} at current precision. +{1}{(Matrix(Expression(Fraction(Integer))))-> +Matrix(Expression(Fraction(Integer)))}{RadicalEigenPackage} +{\smath{\mbox{\bf normalize}\opLeftPren{}flt\opRightPren{}} +normalizes float \smath{flt} at current precision. \newitem -\smath{\mbox{\bf normalize}\opLeftPren{}f\optinner{, x}\opRightPren{}} rewrites \smath{f} using the +\smath{\mbox{\bf normalize}\opLeftPren{}f\optinner{, x}\opRightPren{}} +rewrites \smath{f} using the least possible number of real algebraically independent kernels involving symbol \smath{x}. If no symbol \smath{x} is given, the operation @@ -75733,11 +78747,13 @@ algebraically independent kernels. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{normalizeAtInfinity}}\opLeftPren{}{\it vectorOfFunctions}\opRightPren{}% +\opdata{{\mbox{\axiomFun{normalizeAtInfinity}}\opLeftPren{} +{\it vectorOfFunctions}\opRightPren{}% }% }% {1}{(Vector(\$))->Vector(\$)}{FunctionFieldCategory} -{\smath{\mbox{\bf normalizeAtInfinity}\opLeftPren{}v\opRightPren{}} makes \smath{v} normal at infinity, +{\smath{\mbox{\bf normalizeAtInfinity}\opLeftPren{}v\opRightPren{}} +makes \smath{v} normal at infinity, where \smath{v} is a vector of functions defined on a curve. } @@ -75748,10 +78764,12 @@ where }% }% {1}{(\$)->\$}{Boolean} -{\smath{\mbox{\bf not}\opLeftPren{}n\opRightPren{}} returns the negation of boolean or bit aggregate +{\smath{\mbox{\bf not}\opLeftPren{}n\opRightPren{}} +returns the negation of boolean or bit aggregate \smath{n}. \newitem -\smath{\mbox{\bf not}\opLeftPren{}n\opRightPren{}} returns the bit-by-bit logical \smath{not} of the +\smath{\mbox{\bf not}\opLeftPren{}n\opRightPren{}} +returns the bit-by-bit logical \smath{not} of the small integer \smath{n}. } @@ -75761,67 +78779,80 @@ small integer \smath{n}. }% }% {1}{(\$)->NonNegativeInteger}{RectangularMatrixCategory} -{\smath{\mbox{\bf nrows}\opLeftPren{}m\opRightPren{}} returns the number of rows in the matrix +{\smath{\mbox{\bf nrows}\opLeftPren{}m\opRightPren{}} +returns the number of rows in the matrix or two-dimensional array \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nthExponent}}\opLeftPren{}{\it factored}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nthExponent}}\opLeftPren{} +{\it factored}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {2}{(\$, Integer)->Integer}{Factored} -{\smath{\mbox{\bf nthExponent}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns the exponent of +{\smath{\mbox{\bf nthExponent}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} +returns the exponent of the \eth{\smath{n}} factor of \smath{u}, or 0 if \smath{u} has no such factor. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nthFactor}}\opLeftPren{}{\it factor}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nthFactor}}\opLeftPren{} +{\it factor}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {2}{(\$, Integer)->R}{Factored} -{\smath{\mbox{\bf nthFactor}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns the base of the \eth{\smath{n}} +{\smath{\mbox{\bf nthFactor}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} +returns the base of the \eth{\smath{n}} factor of \smath{u}, or 1 if \smath{n} is not a valid index for a factor. If \smath{u} consists only of a unit, the unit is returned. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nthFlag}}\opLeftPren{}{\it factored}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nthFlag}}\opLeftPren{} +{\it factored}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {2}{(\$, Integer)->Union("nil", "sqfr", "irred", "prime")}{Factored} -{\smath{\mbox{\bf nthFlag}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns the information flag of the +{\smath{\mbox{\bf nthFlag}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} +returns the information flag of the \eth{\smath{n}} factor of \smath{u}, \mbox{\tt "nil"} if \smath{n} is not a valid index for a factor. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nthFractionalTerm}}\opLeftPren{}{\it partialFraction}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nthFractionalTerm}}\opLeftPren{} +{\it partialFraction}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, Integer)->\$}{PartialFraction} -{\smath{\mbox{\bf nthFractionalTerm}\opLeftPren{}p, \allowbreak{} n\opRightPren{}} extracts the \eth{\smath{n}} +{\smath{\mbox{\bf nthFractionalTerm}\opLeftPren{}p, +\allowbreak{} n\opRightPren{}} extracts the \eth{\smath{n}} fractional term from the partial fraction \smath{p}, or 0 if the index \smath{n} is out of range. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nthRoot}}\opLeftPren{}{\it expression}, \allowbreak{}{\it integer}\opRightPren{}% - \opand \mbox{\axiomFun{nthRootIfCan}}\opLeftPren{}{\it expression}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nthRoot}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it integer}\opRightPren{}% + \opand \mbox{\axiomFun{nthRootIfCan}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, Integer)->\$}{RadicalCategory} {\opkey{Argument \smath{x} can be of type \spadtype{Expression}, \spadtype{Complex}, \spadtype{Float} and \spadtype{DoubleFloat}, or a series.} -\newitem \smath{\mbox{\bf nthRoot}\opLeftPren{}x, \allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} root of +\newitem \smath{\mbox{\bf nthRoot}\opLeftPren{}x, +\allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} root of \smath{x}. If x is not an expression, the operation calls \spadfun{error} if this is not possible. -\newitem \smath{\mbox{\bf nthRootIfCan}\opLeftPren{}z, \allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} +\newitem \smath{\mbox{\bf nthRootIfCan}\opLeftPren{}z, +\allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} root of \smath{z} if possible, and \mbox{\tt "failed"} otherwise. } @@ -75830,7 +78861,8 @@ root of \smath{z} if possible, and \mbox{\tt "failed"} otherwise. }% }% {1}{(\$)->Boolean}{SExpressionCategory} -{\smath{\mbox{\bf null?}\opLeftPren{}s\opRightPren{}} is \smath{true} if \smath{s} is the \spadtype{SExpression} +{\smath{\mbox{\bf null?}\opLeftPren{}s\opRightPren{}} +is \smath{true} if \smath{s} is the \spadtype{SExpression} object \smath{()}. } @@ -75839,18 +78871,21 @@ object \smath{()}. }% }% { C -> (()->C)}{}{} -{\smath{\mbox{\bf nullary}\opLeftPren{}x\opRightPren{}}, where \smath{x} has type \smath{R}, returns a function +{\smath{\mbox{\bf nullary}\opLeftPren{}x\opRightPren{}}, +where \smath{x} has type \smath{R}, returns a function \smath{f} of type \spadsig{}{R} such that such that \smath{f()} returns the value \smath{c}. See also \spadfun{constant} for a similar operation. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nullary?}}\opLeftPren{}{\it basicOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nullary?}}\opLeftPren{} +{\it basicOperator}\opRightPren{}% }% }% {1}{(\$)->Boolean}{BasicOperator} -{\smath{\mbox{\bf nullary?}\opLeftPren{}op\opRightPren{}} tests if basic operator \smath{op} is nullary. +{\smath{\mbox{\bf nullary?}\opLeftPren{}op\opRightPren{}} +tests if basic operator \smath{op} is nullary. } % ---------------------------------------------------------------------- @@ -75858,72 +78893,88 @@ See also \spadfun{constant} for a similar operation. }% }% {1}{(\$)->NonNegativeInteger}{MatrixCategory} -{\smath{\mbox{\bf nullity}\opLeftPren{}m\opRightPren{}} returns the dimension of the null space of the matrix \smath{m}. +{\smath{\mbox{\bf nullity}\opLeftPren{}m\opRightPren{}} +returns the dimension of the null space of the matrix \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{nullSpace}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{nullSpace}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% {1}{(\$)->List(Col)}{MatrixCategory} -{\smath{\mbox{\bf nullSpace}\opLeftPren{}m\opRightPren{}} returns a basis for the null space of the matrix \smath{m}. +{\smath{\mbox{\bf nullSpace}\opLeftPren{}m\opRightPren{}} +returns a basis for the null space of the matrix \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfComponents}}\opLeftPren{}{\it \opt{threeSpace}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfComponents}}\opLeftPren{} +{\it \opt{threeSpace}}\opRightPren{}% }% }% {0}{()->NonNegativeInteger}{FunctionFieldCategory} -{\smath{\mbox{\bf numberOfComponents}\opLeftPren{}\opRightPren{}\$F} returns the number of absolutely irreducible components for +{\smath{\mbox{\bf numberOfComponents}\opLeftPren{}\opRightPren{}\$F} +returns the number of absolutely irreducible components for a domain \smath{F} of functions defined over a curve. \newitem -\smath{\mbox{\bf numberOfComponents}\opLeftPren{}sp\opRightPren{}} returns the number of distinct +\smath{\mbox{\bf numberOfComponents}\opLeftPren{}sp\opRightPren{}} +returns the number of distinct object components in the \spadtype{ThreeSpace} object \smath{s} such as points, curves, and polygons. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfComputedEntries}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfComputedEntries}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{LazyStreamAggregate} -{\smath{\mbox{\bf numberOfComputedEntries}\opLeftPren{}st\opRightPren{}} returns the number of explicitly computed entries of stream \smath{st}. +{\smath{\mbox{\bf numberOfComputedEntries}\opLeftPren{}st\opRightPren{}} +returns the number of explicitly computed entries of stream \smath{st}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfCycles}}\opLeftPren{}{\it permutation}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfCycles}}\opLeftPren{} +{\it permutation}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{Permutation} -{\smath{\mbox{\bf numberOfCycles}\opLeftPren{}p\opRightPren{}} returns the number of non-trivial cycles of the permutation \smath{p}. +{\smath{\mbox{\bf numberOfCycles}\opLeftPren{}p\opRightPren{}} +returns the number of non-trivial cycles of the permutation \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfDivisors}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfDivisors}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% {1}{(Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf numberOfDivisors}\opLeftPren{}n\opRightPren{}} returns the number of integers +{\smath{\mbox{\bf numberOfDivisors}\opLeftPren{}n\opRightPren{}} +returns the number of integers between 1 and \smath{n} inclusive which divide \smath{n}. The number of divisors of \smath{n} is often denoted by \smath{\tau(n)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfFactors}}\opLeftPren{}{\it factored}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfFactors}}\opLeftPren{} +{\it factored}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{Factored} -{\smath{\mbox{\bf numberOfFactors}\opLeftPren{}u\opRightPren{}} returns the number of factors in factored form \smath{u}. +{\smath{\mbox{\bf numberOfFactors}\opLeftPren{}u\opRightPren{}} +returns the number of factors in factored form \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfFractionalTerms}}\opLeftPren{}{\it partialFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfFractionalTerms}}\opLeftPren{} +{\it partialFraction}\opRightPren{}% }% }% {1}{(\$)->Integer}{PartialFraction} -{\smath{\mbox{\bf numberOfFractionalTerms}\opLeftPren{}p\opRightPren{}} computes the number of +{\smath{\mbox{\bf numberOfFractionalTerms}\opLeftPren{}p\opRightPren{}} +computes the number of fractional terms in \smath{p}, or 0 if there is no fractional part. } @@ -75932,30 +78983,36 @@ fractional terms in \smath{p}, or 0 if there is no fractional part. }% }% {0}{()->PositiveInteger}{Color} -{\smath{\mbox{\bf numberOfHues}\opLeftPren{}\opRightPren{}} returns the number of total hues. +{\smath{\mbox{\bf numberOfHues}\opLeftPren{}\opRightPren{}} +returns the number of total hues. See also \spadfun{totalHues}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfImproperPartitions}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfImproperPartitions}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(Integer, Integer)->Integer}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf numberOfImproperPartitions}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} computes the number of +{\smath{\mbox{\bf numberOfImproperPartitions}\opLeftPren{}n, +\allowbreak{} m\opRightPren{}} computes the number of partitions of the nonnegative integer \smath{n} in \smath{m} nonnegative parts with regarding the order (improper partitions). Example: \code{numberOfImproperPartitions (3, 3)} is 10, since -\code{[0, 0, 3], [0, 1, 2], [0, 2, 1], [0, 3, 0], [1, 0, 2], [1, 1, 1], [1, 2, 0], [2, 0, 1], [2, 1, 0], [3, 0, 0]} +\code{[0, 0, 3], [0, 1, 2], [0, 2, 1], [0, 3, 0], [1, 0, 2], +[1, 1, 1], [1, 2, 0], [2, 0, 1], [2, 1, 0], [3, 0, 0]} are the possibilities. Note: this operation has a recursive implementation. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numberOfMonomials}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numberOfMonomials}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->NonNegativeInteger}{FiniteAbelianMonoidRing} -{\smath{\mbox{\bf numberOfMonomials}\opLeftPren{}p\opRightPren{}} gives the number of non-zero monomials in polynomial \smath{p}. +{\smath{\mbox{\bf numberOfMonomials}\opLeftPren{}p\opRightPren{}} +gives the number of non-zero monomials in polynomial \smath{p}. } % ---------------------------------------------------------------------- @@ -75964,24 +79021,30 @@ are the possibilities. Note: this operation has a recursive implementation. }% }% {1}{(\$)->D}{QuotientFieldCategory} -{\opkey{Argument x is from domain \spadtype{Fraction(R)} for some domain \smath{R}, +{\opkey{Argument x is from domain \spadtype{Fraction(R)} for some +domain \smath{R}, or of type \spadtype{Expression}} \newitem -\smath{\mbox{\bf numer}\opLeftPren{}x\opRightPren{}} returns the numerator of \smath{x} as an object -of domain \smath{R}; if \smath{x} is of type \spadtype{Expression}, it returns +\smath{\mbox{\bf numer}\opLeftPren{}x\opRightPren{}} returns the +numerator of \smath{x} as an object +of domain \smath{R}; if \smath{x} is of type \spadtype{Expression}, +it returns an object of domain \spadtype{SMP(D, Kernel(Expression R))}. \newitem -\smath{\mbox{\bf numerator}\opLeftPren{}x\opRightPren{}} returns the numerator of \smath{x} as an element +\smath{\mbox{\bf numerator}\opLeftPren{}x\opRightPren{}} returns +the numerator of \smath{x} as an element of \spadtype{Fraction(R)}; if \smath{x} if of type \spadtype{Expression}, it returns an object of domain \spadtype{Expression}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numerators}}\opLeftPren{}{\it continuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numerators}}\opLeftPren{} +{\it continuedFraction}\opRightPren{}% }% }% {1}{(\$)->Stream(R)}{ContinuedFraction} -{\smath{\mbox{\bf numerators}\opLeftPren{}cf\opRightPren{}} returns the stream of numerators of the approximants of +{\smath{\mbox{\bf numerators}\opLeftPren{}cf\opRightPren{}} +returns the stream of numerators of the approximants of the continued fraction \smath{cf}. If the continued fraction is finite, then the stream will be finite. } @@ -75989,21 +79052,25 @@ If the continued fraction is finite, then the stream will be finite. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{numeric}}\opLeftPren{}{\it expression\opt{, n}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{numeric}}\opLeftPren{} +{\it expression\opt{, n}}\opRightPren{}% }% }% {1}{(Expression(S))->Float}{Numeric} -{\smath{\mbox{\bf numeric}\opLeftPren{}x, \allowbreak{} n\opRightPren{}} returns a float approximation of expression \smath{x} +{\smath{\mbox{\bf numeric}\opLeftPren{}x, \allowbreak{} n\opRightPren{}} +returns a float approximation of expression \smath{x} to \smath{n} decimal digits accurary. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{objectOf}}\opLeftPren{}{\it typeAnyObject}\opRightPren{}% +\opdata{{\mbox{\axiomFun{objectOf}}\opLeftPren{} +{\it typeAnyObject}\opRightPren{}% }% }% {1}{(\$)->OutputForm}{Any} -{\smath{\mbox{\bf objectOf}\opLeftPren{}a\opRightPren{}} returns a printable form of an object +{\smath{\mbox{\bf objectOf}\opLeftPren{}a\opRightPren{}} returns a +printable form of an object of type \spadtype{Any}. } @@ -76012,8 +79079,10 @@ of type \spadtype{Any}. }% }% {1}{(\$)->Record(points:NonNegativeInteger, -curves:NonNegativeInteger, polygons:NonNegativeInteger, constructs:NonNegativeInteger)}{ThreeSpace} -{\smath{\mbox{\bf objects}\opLeftPren{}sp\opRightPren{}} returns the \spadtype{ThreeSpace} object \smath{sp}. +curves:NonNegativeInteger, polygons:NonNegativeInteger, +constructs:NonNegativeInteger)}{ThreeSpace} +{\smath{\mbox{\bf objects}\opLeftPren{}sp\opRightPren{}} +returns the \spadtype{ThreeSpace} object \smath{sp}. The result is returned as record with fields: \smath{points}, the number of points; \smath{curves}, the number of curves; @@ -76023,7 +79092,8 @@ The result is returned as record with fields: % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{oblateSpheroidal}}\opLeftPren{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{oblateSpheroidal}}\opLeftPren{} +{\it function}\opRightPren{}% }% }% {1}{(R)->(Point(R))->Point(R)}{CoordinateSystems} @@ -76035,16 +79105,21 @@ sinh(\xi) sin(\eta) sin(\phi)}, \smath{z = a cosh(\xi) cos(\eta)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{octon}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\allowbreak $\,[$ , \allowbreak{}{\it elements}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{octon}}\opLeftPren{} +{\it element}, \allowbreak{}{\it element} +\allowbreak $\,[$ , \allowbreak{}{\it elements}$]$\opRightPren{}% }% }% {2}{(Quaternion(R), Quaternion(R)) ->\$}{Octonion} -{\smath{\mbox{\bf octon}\opLeftPren{}q_e, \allowbreak{} q_E\opRightPren{}} constructs an octonion +{\smath{\mbox{\bf octon}\opLeftPren{}q_e, \allowbreak{} q_E\opRightPren{}} +constructs an octonion whose first 4 components are given by a quaternion \smath{q_e} and whose last 4 components are given by a quaternion \smath{q_E}. \newitem -\smath{\mbox{\bf octon}\opLeftPren{}r_e, \allowbreak{} r_i, \allowbreak{} r_j, \allowbreak{} r_k, \allowbreak{} r_E, \allowbreak{} r_I, \allowbreak{} r_J, \allowbreak{} r_K\opRightPren{}} constructs an octonion from scalars. +\smath{\mbox{\bf octon}\opLeftPren{}r_e, \allowbreak{} r_i, \allowbreak{} r_j, +\allowbreak{} r_k, \allowbreak{} r_E, \allowbreak{} r_I, \allowbreak{} r_J, +\allowbreak{} r_K\opRightPren{}} constructs an octonion from scalars. } % ---------------------------------------------------------------------- @@ -76052,20 +79127,26 @@ last 4 components are given by a quaternion \smath{q_E}. }% }% {1}{(\$)->Boolean}{IntegerNumberSystem} -{\smath{\mbox{\bf odd?}\opLeftPren{}n\opRightPren{}} tests if integer \smath{n} is odd. +{\smath{\mbox{\bf odd?}\opLeftPren{}n\opRightPren{}} tests +if integer \smath{n} is odd. \newitem -\smath{\mbox{\bf odd?}\opLeftPren{}p\opRightPren{}} tests if \smath{p} is an odd permutation, that is, \smath{\mbox{\bf sign}\opLeftPren{}p\opRightPren{}} +\smath{\mbox{\bf odd?}\opLeftPren{}p\opRightPren{}} tests +if \smath{p} is an odd permutation, that is, +\smath{\mbox{\bf sign}\opLeftPren{}p\opRightPren{}} is \smath{-1}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{oneDimensionalArray}}\opLeftPren{}{\it \optinit{integer, }elements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{oneDimensionalArray}}\opLeftPren{} +{\it \optinit{integer, }elements}\opRightPren{}% }% }% {1}{(\$)->Boolean}{IntegerNumberSystem} -{\smath{\mbox{\bf oneDimensionalArray}\opLeftPren{}ls\opRightPren{}} creates a one-dimensional array consisting of +{\smath{\mbox{\bf oneDimensionalArray}\opLeftPren{}ls\opRightPren{}} +creates a one-dimensional array consisting of the elements of list \smath{ls}. -\newitem\smath{\mbox{\bf oneDimensionalArray}\opLeftPren{}n, \allowbreak{} s\opRightPren{}} creates a one-dimensional array of \smath{n} +\newitem\smath{\mbox{\bf oneDimensionalArray}\opLeftPren{}n, +\allowbreak{} s\opRightPren{}} creates a one-dimensional array of \smath{n} elements, each with value \smath{s}. } @@ -76074,36 +79155,43 @@ elements, each with value \smath{s}. }% }% {1}{(\$)->Boolean}{MonadWithUnit} -{\smath{\mbox{\bf one?}\opLeftPren{}a\opRightPren{}} tests whether \smath{a} is the unit 1. +{\smath{\mbox{\bf one?}\opLeftPren{}a\opRightPren{}} +tests whether \smath{a} is the unit 1. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{open}}\opLeftPren{}{\it file\opt{, string}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{open}}\opLeftPren{} +{\it file\opt{, string}}\opRightPren{}% }% }% {1}{(Name)->\$}{FileCategory} -{\smath{\mbox{\bf open}\opLeftPren{}s\optinner{, mode}\opRightPren{}} returns the file \smath{s} open +{\smath{\mbox{\bf open}\opLeftPren{}s\optinner{, mode}\opRightPren{}} +returns the file \smath{s} open in the indicated mode: \mbox{\tt "input"} or \mbox{\tt "output"}. Argument \smath{mode} is \mbox{\tt "output"} by default. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{operator}}\opLeftPren{}{\it symbol\opt{, nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{operator}}\opLeftPren{} +{\it symbol\opt{, nonNegativeInteger}}\opRightPren{}% }% }% {1}{(BasicOperator)->BasicOperator}{ExpressionSpace} -{\smath{\mbox{\bf operator}\opLeftPren{}f, \allowbreak{} n\opRightPren{}} makes \smath{f} into an \smath{n}-ary operator. +{\smath{\mbox{\bf operator}\opLeftPren{}f, +\allowbreak{} n\opRightPren{}} makes \smath{f} into an \smath{n}-ary operator. If the second argument \smath{n} is omitted, \smath{f} has arbitary \spadgloss{arity}, that is, \smath{f} takes an arbitrary number of arguments. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{operators}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{operators}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(\$)->List(BasicOperator)}{ExpressionSpace} -{\smath{\mbox{\bf operators}\opLeftPren{}f\opRightPren{}} returns a list of all basic operators in \smath{f}, +{\smath{\mbox{\bf operators}\opLeftPren{}f\opRightPren{}} returns a +list of all basic operators in \smath{f}, regardless of level. } @@ -76112,28 +79200,35 @@ regardless of level. }% }% {1}{(F)->F}{FunctionSpaceAssertions} -{\smath{\mbox{\bf optional}\opLeftPren{}x\opRightPren{}} tells the pattern matcher that \smath{x} can +{\smath{\mbox{\bf optional}\opLeftPren{}x\opRightPren{}} tells the +pattern matcher that \smath{x} can match an identity (0 in a sum, 1 in a product or exponentiation), or calls \spadfun{error} if \smath{x} is not a symbol. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{or}}\opLeftPren{}{\it boolean}, \allowbreak{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{or}}\opLeftPren{}{\it boolean}, \allowbreak{} +{\it boolean}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{BitAggregate} -{\smath{a \or b} returns the logical \smath{or} of booleans or bit aggregates \smath{a} and \smath{b}. +{\smath{a \or b} returns the logical \smath{or} of booleans or bit +aggregates \smath{a} and \smath{b}. \newitem -\smath{n \or m} returns the bit-by-bit logical \smath{or} of the small integers \smath{n} and \smath{m}. +\smath{n \or m} returns the bit-by-bit logical \smath{or} of the small +integers \smath{n} and \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{orbit}}\opLeftPren{}{\it group}, \allowbreak{}{\it elements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{orbit}}\opLeftPren{}{\it group}, \allowbreak{} +{\it elements}\opRightPren{}% }% }% {2}{(\$, List(S))->Set(List(S))}{PermutationGroup} -{\smath{\mbox{\bf orbit}\opLeftPren{}gp, \allowbreak{} el\opRightPren{}} returns the orbit of the element \smath{el} under -the permutation group \smath{gp}, that is, the set of all points gained by applying each group element to \smath{el}. +{\smath{\mbox{\bf orbit}\opLeftPren{}gp, \allowbreak{} el\opRightPren{}} +returns the orbit of the element \smath{el} under +the permutation group \smath{gp}, that is, the set of all +points gained by applying each group element to \smath{el}. \newitem \smath{\mbox{\bf orbit}\opLeftPren{}gp, \allowbreak{} ls\opRightPren{}}, where \smath{ls} is a list or @@ -76148,7 +79243,8 @@ returns the orbit of }% }% {1}{(\$)->Set(Set(S))}{PermutationGroup} -{\smath{\mbox{\bf orbits}\opLeftPren{}gp\opRightPren{}} returns the orbits of the permutation group \smath{gp}. +{\smath{\mbox{\bf orbits}\opLeftPren{}gp\opRightPren{}} +returns the orbits of the permutation group \smath{gp}. } % ---------------------------------------------------------------------- @@ -76156,8 +79252,8 @@ returns the orbit of }% }% {1}{(\$)->Integer}{Character} -{\smath{\mbox{\bf ord}\opLeftPren{}c\opRightPren{}} returns an integer code corresponding to the -character \smath{c}. +{\smath{\mbox{\bf ord}\opLeftPren{}c\opRightPren{}} returns an integer code +corresponding to the character \smath{c}. } % ---------------------------------------------------------------------- @@ -76190,26 +79286,32 @@ exponent of the highest power of \smath{q} dividing \smath{p} \smath{p} as a group element. \item if \smath{p} is permutation group: the order of the group. \end{simpleList} -\smath{\mbox{\bf order}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns the order of the differential +\smath{\mbox{\bf order}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} +returns the order of the differential polynomial \smath{p} in differential indeterminate \smath{q}. \newitem -\smath{\mbox{\bf order}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns the order of multivariate series +\smath{\mbox{\bf order}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} +returns the order of multivariate series \smath{p} viewed as a series in \smath{q} (this operation results in an infinite loop if \smath{f} has no non-zero terms). \newitem -\smath{\mbox{\bf order}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns the largest \smath{n} such that +\smath{\mbox{\bf order}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} +returns the largest \smath{n} such that \smath{q^n} divides polynomial \smath{p}, that is, the order of \smath{p(x)} at \smath{q(x)=0}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{orthonormalBasis}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{orthonormalBasis}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% -{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))->List(Matrix(Expression(Fraction(Integer))))}{RadicalEigenPackage} -{\smath{\mbox{\bf orthonormalBasis}\opLeftPren{}M\opRightPren{}} returns the orthogonal matrix +{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +List(Matrix(Expression(Fraction(Integer))))}{RadicalEigenPackage} +{\smath{\mbox{\bf orthonormalBasis}\opLeftPren{}M\opRightPren{}} +returns the orthogonal matrix \smath{B} such that \smath{B M B^{-1}} is diagonal, or calls \spadfun{error} if \smath{M} is not a symmetric matrix. @@ -76220,12 +79322,14 @@ calls \spadfun{error} if \smath{M} is not a symmetric matrix. }% }% {1}{(OutputForm)->Void}{OutputPackage} -{\smath{\mbox{\bf output}\opLeftPren{}x\opRightPren{}} displays \smath{x} on the ``algebra output'' stream +{\smath{\mbox{\bf output}\opLeftPren{}x\opRightPren{}} +displays \smath{x} on the ``algebra output'' stream defined by \spadsyscom{)set output algebra}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{outputAsFortran}}\opLeftPren{}{\it outputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{outputAsFortran}}\opLeftPren{} +{\it outputForms}\opRightPren{}% }% }% {1}{(List(OutputForm))->Void}{SpecialOutputPackage} @@ -76236,12 +79340,14 @@ the system command \spadsyscom{)set output fortran}. If \smath{f} is a list of \spadtype{OutputForm} objects, each expression in \smath{f} is output in order. \newitem -\smath{\mbox{\bf outputAsFortran}\opLeftPren{}s, \allowbreak{} f\opRightPren{}}, where \smath{s} is a string, +\smath{\mbox{\bf outputAsFortran}\opLeftPren{}s, +\allowbreak{} f\opRightPren{}}, where \smath{s} is a string, outputs \smath{s = f}, but is otherwise identical. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{outputAsTex}}\opLeftPren{}{\it outputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{outputAsTex}}\opLeftPren{} +{\it outputForms}\opRightPren{}% }% }% {1}{(List(OutputForm))->Void}{SpecialOutputPackage} @@ -76256,11 +79362,13 @@ each expression in \smath{f} is output in order. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{outputFixed}}\opLeftPren{}{\it \opt{nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{outputFixed}}\opLeftPren{} +{\it \opt{nonNegativeInteger}}\opRightPren{}% }% }% {1}{(NonNegativeInteger)->Void}{Float} -{\smath{\mbox{\bf outputFixed}\opLeftPren{}\optinner{n}\opRightPren{}} sets the output mode of +{\smath{\mbox{\bf outputFixed}\opLeftPren{}\optinner{n}\opRightPren{}} +sets the output mode of floats to fixed point notation, that is, as an integer, a decimal point, and a number of digits. If \smath{n} is given, then @@ -76268,11 +79376,13 @@ If \smath{n} is given, then } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{outputFloating}}\opLeftPren{}{\it \opt{nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{outputFloating}}\opLeftPren{} +{\it \opt{nonNegativeInteger}}\opRightPren{}% }% }% {1}{(NonNegativeInteger)->Void}{Float} -{\smath{\mbox{\bf outputFloating}\opLeftPren{}\optinner{n}\opRightPren{}} sets the output mode to +{\smath{\mbox{\bf outputFloating}\opLeftPren{}\optinner{n}\opRightPren{}} +sets the output mode to floating (scientific) notation, that is, \smath{m 10^e} is displayed as \code{ @@ -76286,18 +79396,21 @@ the decimal point. }% }% {}{}{} -{\smath{\mbox{\bf outputForm}\opLeftPren{}x\opRightPren{}} creates an object of type +{\smath{\mbox{\bf outputForm}\opLeftPren{}x\opRightPren{}} +creates an object of type \spadtype{OutputForm} from \smath{x}, an object of type \smath{Integer}, \smath{DoubleFloat}, \smath{String}, or \smath{Symbol}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{outputGeneral}}\opLeftPren{}{\it \opt{nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{outputGeneral}}\opLeftPren{} +{\it \opt{nonNegativeInteger}}\opRightPren{}% }% }% {0}{()->Void}{Float} -{\smath{\mbox{\bf outputGeneral}\opLeftPren{}\optinner{n}\opRightPren{}} sets the output +{\smath{\mbox{\bf outputGeneral}\opLeftPren{}\optinner{n}\opRightPren{}} +sets the output mode (default mode) to general notation, that is, numbers will be displayed in either fixed or floating (scientific) notation depending on the magnitude. @@ -76306,32 +79419,40 @@ the decimal point. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{outputSpacing}}\opLeftPren{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{outputSpacing}}\opLeftPren{} +{\it nonNegativeInteger}\opRightPren{}% }% }% {1}{(NonNegativeInteger)->Void}{Float} -{\smath{\mbox{\bf outputSpacing}\opLeftPren{}n\opRightPren{}} inserts a space after \smath{n} digits on output. -\smath{\mbox{\bf outputSpacing}\opLeftPren{}0\opRightPren{}} means no spaces are inserted. +{\smath{\mbox{\bf outputSpacing}\opLeftPren{}n\opRightPren{}} +inserts a space after \smath{n} digits on output. +\smath{\mbox{\bf outputSpacing}\opLeftPren{}0\opRightPren{}} +means no spaces are inserted. By default, \smath{n = 10}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{over}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{over}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf over}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, -where \smath{o_1} and \smath{o_2} are objects of type \spadtype{OutputForm} (normally unexposed), -creates an output form for the vertical fraction of \smath{o_1} over \smath{o_2}. +where \smath{o_1} and \smath{o_2} are objects of type +\spadtype{OutputForm} (normally unexposed), +creates an output form for the vertical fraction of +\smath{o_1} over \smath{o_2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{overbar}}\opLeftPren{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{overbar}}\opLeftPren{} +{\it outputForm}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf overbar}\opLeftPren{}o\opRightPren{}}, -where \smath{o} is an object of type \spadtype{OutputForm} (normally unexposed), +where \smath{o} is an object of type +\spadtype{OutputForm} (normally unexposed), creates the output form \smath{o} with an overbar. } @@ -76342,7 +79463,8 @@ creates the output form \smath{o} with an overbar. }% }% {1}{(\$)->\$}{KeyedAccessFile} -{\smath{\mbox{\bf pack!}\opLeftPren{}f\opRightPren{}} reorganizes the file \smath{f} on disk to recover unused space. +{\smath{\mbox{\bf pack!}\opLeftPren{}f\opRightPren{}} +reorganizes the file \smath{f} on disk to recover unused space. } % ---------------------------------------------------------------------- @@ -76358,11 +79480,17 @@ See also \spadfun{interpret}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pade}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it series}\allowbreak $\,[$ , \allowbreak{}{\it series}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{pade}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it series} +\allowbreak $\,[$ , \allowbreak{}{\it series}$]$\opRightPren{}% }% }% -{3}{(NonNegativeInteger, NonNegativeInteger, UnivariateTaylorSeries(R, x, pt))->Union(Fraction(UnivariatePolynomial(x, R)), "failed")}{PadeApproximantPackage} -{\smath{\mbox{\bf pade}\opLeftPren{}nd, \allowbreak{} dd, \allowbreak{} s \optinner{, ds}\opRightPren{}} computes the quotient of +{3}{(NonNegativeInteger, NonNegativeInteger, +UnivariateTaylorSeries(R, x, pt))-> +Union(Fraction(UnivariatePolynomial(x, R)), "failed")} +{PadeApproximantPackage} +{\smath{\mbox{\bf pade}\opLeftPren{}nd, \allowbreak{} dd, +\allowbreak{} s \optinner{, ds}\opRightPren{}} computes the quotient of polynomials (if it exists) with numerator degree at most \smath{nd} and denominator degree at most \smath{dd}. If a single univariate Taylor series \smath{s} is given, the quotient @@ -76375,13 +79503,16 @@ series. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{padicFraction}}\opLeftPren{}{\it partialFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{padicFraction}}\opLeftPren{} +{\it partialFraction}\opRightPren{}% }% }% {1}{(\$)->\$}{PartialFraction} -{\smath{\mbox{\bf padicFraction}\opLeftPren{}q\opRightPren{}} expands the fraction \smath{p}-adically in +{\smath{\mbox{\bf padicFraction}\opLeftPren{}q\opRightPren{}} +expands the fraction \smath{p}-adically in the primes \smath{p} in the denominator of \smath{q}. -For example, \smath{\mbox{\bf padicFraction}\opLeftPren{}3/(2^2)) = 1/2 + 1/(2^2)}. +For example, \smath{\mbox{\bf padicFraction} +\opLeftPren{}3/(2^2)) = 1/2 + 1/(2^2)}. Use \spadfunFrom{compactFraction}{PartialFraction} to return to compact form. } @@ -76390,7 +79521,8 @@ Use \spadfunFrom{compactFraction}{PartialFraction} to return to compact form. }% }% {1}{(\$)->Boolean}{SExpressionCategory} -{\smath{\mbox{\bf pair?}\opLeftPren{}s\opRightPren{}} tests if \smath{SExpression} object +{\smath{\mbox{\bf pair?}\opLeftPren{}s\opRightPren{}} tests if +\smath{SExpression} object is a non-null Lisp object. } @@ -76399,18 +79531,21 @@ is a non-null Lisp object. }% }% {1}{(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf parabolic}\opLeftPren{}pt\opRightPren{}} transforms \smath{pt} from parabolic +{\smath{\mbox{\bf parabolic}\opLeftPren{}pt\opRightPren{}} +transforms \smath{pt} from parabolic coordinates to Cartesian coordinates: the function produced will map the point \smath{(u, v)} to \smath{x = 1/2 (u^2 - v^2)}, \smath{y = u v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{parabolicCylindrical}}\opLeftPren{}{\it point}\opRightPren{}% +\opdata{{\mbox{\axiomFun{parabolicCylindrical}}\opLeftPren{} +{\it point}\opRightPren{}% }% }% {1}{(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf parabolicCylindrical}\opLeftPren{}pt\opRightPren{}} transforms \smath{pt} from +{\smath{\mbox{\bf parabolicCylindrical}\opLeftPren{}pt\opRightPren{}} +transforms \smath{pt} from parabolic cylindrical coordinates to Cartesian coordinates: the function produced will map the point \smath{(u, v, z)} to \smath{x = 1/2(u^2 - v^2)}, \smath{y = u v}, \smath{z}. @@ -76421,7 +79556,8 @@ function produced will map the point \smath{(u, v, z)} to \smath{x = }% }% {1}{(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf paraboloidal}\opLeftPren{}pt\opRightPren{}} transforms \smath{pt} from paraboloidal +{\smath{\mbox{\bf paraboloidal}\opLeftPren{}pt\opRightPren{}} +transforms \smath{pt} from paraboloidal coordinates to Cartesian coordinates: the function produced will map the point \smath{(u, v, phi)} to \smath{x = u v {\bf cos}(\phi)}, \smath{y = u v {\bf sin}(\phi)}, \smath{z = 1/2 (u^2 @@ -76429,7 +79565,8 @@ cos}(\phi)}, \smath{y = u v {\bf sin}(\phi)}, \smath{z = 1/2 (u^2 } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{paren}}\opLeftPren{}{\it expressions}\opRightPren{}% +\opdata{{\mbox{\axiomFun{paren}}\opLeftPren{} +{\it expressions}\opRightPren{}% }% }% {1}{(\$)->\$}{ExpressionSpace} @@ -76440,61 +79577,77 @@ This prevents \smath{f} or the constituent \smath{f_i} from being evaluated when operators are applied to it. For example, \code{log(1)} returns 0, but \code{log(paren 1)} returns the formal kernel \code{log((1))}. -Also, {\bf atan}{\it ({\bf paren [}x, 2{\bf ]})} returns the formal kernel {\bf atan}{\it ((x, 2))}. +Also, {\bf atan}{\it ({\bf paren [}x, 2{\bf ]})} returns the +formal kernel {\bf atan}{\it ((x, 2))}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{partialDenominators}}\opLeftPren{}{\it continuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{partialDenominators}}\opLeftPren{} +{\it continuedFraction}\opRightPren{}% }% }% {1}{(\$)->Stream(R)}{ContinuedFraction} -{\smath{\mbox{\bf partialDenominators}\opLeftPren{}x\opRightPren{}} extracts the denominators +{\smath{\mbox{\bf partialDenominators}\opLeftPren{}x\opRightPren{}} +extracts the denominators in \smath{x}. -If \smath{x = {\mbox {\bf continuedFraction}}(b_0, [a_1, \ldots], [b_1, \ldots])}, +If \smath{x = {\mbox {\bf continuedFraction}} +(b_0, [a_1, \ldots], [b_1, \ldots])}, then \smath{\mbox{\bf partialDenominators}\opLeftPren{}x) = [b_1, b_2\ldots]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{partialFraction}}\opLeftPren{}{\it element}, \allowbreak{}{\it factored}\opRightPren{}% +\opdata{{\mbox{\axiomFun{partialFraction}}\opLeftPren{} +{\it element}, \allowbreak{}{\it factored}\opRightPren{}% }% }% {2}{(R, Factored(R))->\$}{PartialFraction} -{\smath{\mbox{\bf partialFraction}\opLeftPren{}numer, \allowbreak{} denom\opRightPren{}} is the main function for +{\smath{\mbox{\bf partialFraction}\opLeftPren{}numer, +\allowbreak{} denom\opRightPren{}} is the main function for constructing partial fractions. The second argument \smath{denom} is the denominator and should be factored. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{partialNumerators}}\opLeftPren{}{\it continuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{partialNumerators}}\opLeftPren{} +{\it continuedFraction}\opRightPren{}% }% }% {1}{(\$)->Stream(R)}{ContinuedFraction} -{\smath{\mbox{\bf partialNumerators}\opLeftPren{}x\opRightPren{}} extracts the numerators in +{\smath{\mbox{\bf partialNumerators}\opLeftPren{}x\opRightPren{}} +extracts the numerators in \smath{x}, -if \smath{x = {\mbox {\bf continuedFraction}}(b_0, [a_1, \ldots], [b_1, \ldots], \ldots)}, +if \smath{x = {\mbox {\bf continuedFraction}} +(b_0, [a_1, \ldots], [b_1, \ldots], \ldots)}, then \smath{\mbox{\bf partialNumerators}\opLeftPren{}x) = [a_1, \ldots]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{partialQuotients}}\opLeftPren{}{\it continuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{partialQuotients}}\opLeftPren{} +{\it continuedFraction}\opRightPren{}% }% }% {1}{(\$)->Stream(R)}{ContinuedFraction} -{\smath{\mbox{\bf partialQuotients}\opLeftPren{}x\opRightPren{}} extracts the partial quotients in +{\smath{\mbox{\bf partialQuotients}\opLeftPren{}x\opRightPren{}} +extracts the partial quotients in \smath{x}, if -\smath{x = {\mbox {\bf continuedFraction}}(b_0, [a_1, \dots], [b_1, \ldots], \ldots)}, +\smath{x = {\mbox {\bf continuedFraction}} +(b_0, [a_1, \dots], [b_1, \ldots], \ldots)}, then \smath{\mbox{\bf partialQuotients}\opLeftPren{}x) = [b_0, b_1, \ldots]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{particularSolution}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{particularSolution}}\opLeftPren{} +{\it matrix}, \allowbreak{}{\it vector}\opRightPren{}% }% }% {2}{(M, Col)->Union(Col, \mbox{\tt "failed"})}{LinearSystemMatrixPackage} -{\smath{\mbox{\bf aSolution}\opLeftPren{}M, \allowbreak{} v\opRightPren{}} finds a particular solution \smath{x} of the linear system \smath{Mx = v}. -The result \smath{x} is returned as a vector, or \mbox{\tt "failed"} if no solution exists. +{\smath{\mbox{\bf aSolution}\opLeftPren{}M, +\allowbreak{} v\opRightPren{}} finds a particular solution +\smath{x} of the linear system \smath{Mx = v}. +The result \smath{x} is returned as a vector, or +\mbox{\tt "failed"} if no solution exists. } @@ -76503,25 +79656,30 @@ The result \smath{x} is returned as a vector, or \mbox{\tt "failed"} if no solut }% }% {1}{(I)->I}{IntegerCombinatoricFunctions} -{\smath{\mbox{\bf partition}\opLeftPren{}n\opRightPren{}} returns the number of partitions of the +{\smath{\mbox{\bf partition}\opLeftPren{}n\opRightPren{}} +returns the number of partitions of the integer \smath{n}. This is the number of distinct ways that \smath{n} can be written as a sum of positive integers. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{partitions}}\opLeftPren{}{\it integer\opt{, integer, integer}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{partitions}}\opLeftPren{} +{\it integer\opt{, integer, integer}}\opRightPren{}% }% }% {1}{(Integer)->Stream(List(Integer))}{PartitionsAndPermutations} -{\smath{\mbox{\bf partitions}\opLeftPren{}i, \allowbreak{} j\opRightPren{}} is the stream of all partitions whose +{\smath{\mbox{\bf partitions}\opLeftPren{}i, +\allowbreak{} j\opRightPren{}} is the stream of all partitions whose number of parts and largest part are no greater than \smath{i} and \smath{j}. \newitem -\smath{\mbox{\bf partitions}\opLeftPren{}n\opRightPren{}} is the stream of all partitions of integer +\smath{\mbox{\bf partitions}\opLeftPren{}n\opRightPren{}} is +the stream of all partitions of integer \smath{n}. \newitem -\smath{\mbox{\bf partitions}\opLeftPren{}p, \allowbreak{} l, \allowbreak{} n\opRightPren{}} is the stream of partitions of \smath{n} +\smath{\mbox{\bf partitions}\opLeftPren{}p, \allowbreak{} l, +\allowbreak{} n\opRightPren{}} is the stream of partitions of \smath{n} whose number of parts is no greater than \smath{p} and whose largest part is no greater than \smath{l}. } @@ -76531,7 +79689,8 @@ largest part is no greater than \smath{l}. }% }% {1}{(\$)->List(R)}{TwoDimensionalArrayCategory} -{\smath{\mbox{\bf parts}\opLeftPren{}u\opRightPren{}} returns a list of the consecutive elements of \smath{u}. +{\smath{\mbox{\bf parts}\opLeftPren{}u\opRightPren{}} +returns a list of the consecutive elements of \smath{u}. Note: if \smath{u} is a list, \smath{\mbox{\bf parts}\opLeftPren{}u) = u}. } @@ -76540,50 +79699,64 @@ Note: if \smath{u} is a list, \smath{\mbox{\bf parts}\opLeftPren{}u) = u}. }% }% {1}{(Color)->\$}{Palette} -{\smath{\mbox{\bf pastel}\opLeftPren{}c\opRightPren{}} sets the shade of a hue \smath{c} above ``bright'' but below ``light''. +{\smath{\mbox{\bf pastel}\opLeftPren{}c\opRightPren{}} +sets the shade of a hue \smath{c} above ``bright'' but below ``light''. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pattern}}\opLeftPren{}{\it rewriteRule}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pattern}}\opLeftPren{} +{\it rewriteRule}\opRightPren{}% }% }% {1}{(\$)->Pattern(Base)}{RewriteRule} -{\smath{\mbox{\bf pattern}\opLeftPren{}r\opRightPren{}} returns the pattern corresponding to the left hand +{\smath{\mbox{\bf pattern}\opLeftPren{}r\opRightPren{}} +returns the pattern corresponding to the left hand side of the rewrite rule \smath{r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{patternMatch}}\opLeftPren{}{\it expression}, \allowbreak{}{\it expression}, \allowbreak{}{\it patternMatchResult}\opRightPren{}% +\opdata{{\mbox{\axiomFun{patternMatch}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it expression}, \allowbreak{} +{\it patternMatchResult}\opRightPren{}% }% }% -{3}{(\$, Pattern(S), PatternMatchResult(S, \$))->PatternMatchResult(S, \$)}{PatternMatchable} -{\smath{\mbox{\bf patternMatch}\opLeftPren{}expr, \allowbreak{} pat, \allowbreak{} res\opRightPren{}} matches the pattern \smath{pat} to the expression \smath{expr}. +{3}{(\$, Pattern(S), PatternMatchResult(S, \$))-> +PatternMatchResult(S, \$)}{PatternMatchable} +{\smath{\mbox{\bf patternMatch}\opLeftPren{}expr, \allowbreak{} pat, +\allowbreak{} res\opRightPren{}} matches the pattern +\smath{pat} to the expression +\smath{expr}. The argument \smath{res} contains the variables of \smath{pat} which are already matched and their matches. Initially, \smath{res} is the result of \spadfun{new()}, an empty list of matches. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{perfectNthPower?}}\opLeftPren{}{\it integer}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{perfectNthPower?}}\opLeftPren{}{\it integer}, +\allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(I, NonNegativeInteger)->Boolean}{IntegerRoots} -{\smath{\mbox{\bf perfectNthPower?}\opLeftPren{}n, \allowbreak{} r\opRightPren{}} tests if \smath{n} is an +{\smath{\mbox{\bf perfectNthPower?}\opLeftPren{}n, +\allowbreak{} r\opRightPren{}} tests if \smath{n} is an \eth{\smath{r}} power. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{perfectNthRoot}}\opLeftPren{}{\it integer\opt{, nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{perfectNthRoot}}\opLeftPren{} +{\it integer\opt{, nonNegativeInteger}}\opRightPren{}% }% }% {2}{(I, NonNegativeInteger)->Union(I, "failed")}{IntegerRoots} -{\smath{\mbox{\bf perfectNthRoot}\opLeftPren{}n\opRightPren{}} returns a record +{\smath{\mbox{\bf perfectNthRoot}\opLeftPren{}n\opRightPren{}} +returns a record with fields ``base'' \smath{x} and ``exponent'' \smath{r} such that \smath{n = x^r} and \smath{r} is the largest integer such that \smath{n} is a perfect \eth{\smath{r}} power. \newitem -\smath{\mbox{\bf perfectNthRoot}\opLeftPren{}n, \allowbreak{} r\opRightPren{}} returns the \eth{\smath{r}} root of +\smath{\mbox{\bf perfectNthRoot}\opLeftPren{}n, +\allowbreak{} r\opRightPren{}} returns the \eth{\smath{r}} root of \smath{n} if \smath{n} is an \eth{\smath{r}} power, and \mbox{\tt "failed"} otherwise. } @@ -76593,16 +79766,19 @@ otherwise. }% }% {1}{(I)->Union(I, "failed")}{IntegerRoots} -{\smath{\mbox{\bf perfectSqrt}\opLeftPren{}n\opRightPren{}} returns the square root of \smath{n} if +{\smath{\mbox{\bf perfectSqrt}\opLeftPren{}n\opRightPren{}} +returns the square root of \smath{n} if \smath{n} is a perfect square, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{perfectSquare?}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{perfectSquare?}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% {1}{(I)->Boolean}{IntegerRoots} -{\smath{\mbox{\bf perfectSquare?}\opLeftPren{}n\opRightPren{}} tests if \smath{n} is a perfect square. +{\smath{\mbox{\bf perfectSquare?}\opLeftPren{}n\opRightPren{}} +tests if \smath{n} is a perfect square. } % ---------------------------------------------------------------------- @@ -76610,41 +79786,50 @@ otherwise. }% }% {1}{(SquareMatrix(n, R))->R}{Permanent} -{\smath{\mbox{\bf permanent}\opLeftPren{}x\opRightPren{}} returns the permanent of a square matrix \smath{x}, -equivalent to the \spadfun{determinant} except that coefficients have no change of sign. +{\smath{\mbox{\bf permanent}\opLeftPren{}x\opRightPren{}} +returns the permanent of a square matrix \smath{x}, +equivalent to the \spadfun{determinant} except that +coefficients have no change of sign. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{permutation}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{permutation}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(I, I)->I}{IntegerCombinatoricFunctions} -{\smath{\mbox{\bf permutation}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} returns the number of permutations of +{\smath{\mbox{\bf permutation}\opLeftPren{}n, +\allowbreak{} m\opRightPren{}} returns the number of permutations of \smath{n} objects taken \smath{m} at a time. Note: \smath{\mbox{\bf permutation}\opLeftPren{}n, m) = n!/(n-m)!}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{permutationGroup}}\opLeftPren{}{\it listPermutations}\opRightPren{}% +\opdata{{\mbox{\axiomFun{permutationGroup}}\opLeftPren{} +{\it listPermutations}\opRightPren{}% }% }% {1}{(List(Permutation(S)))->\$}{PermutationGroup} -{\smath{\mbox{\bf permutationGroup}\opLeftPren{}ls\opRightPren{}} coerces a list of permutations \smath{ls} to the group generated by this list. +{\smath{\mbox{\bf permutationGroup}\opLeftPren{}ls\opRightPren{}} +coerces a list of permutations \smath{ls} to the group generated by this list. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{permutationRepresentation}}\opLeftPren{}{\it permutations\opt{, n}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{permutationRepresentation}}\opLeftPren{} +{\it permutations\opt{, n}}\opRightPren{}% }% }% {1}{(List(Integer))->Matrix(Integer)}{RepresentationPackage1} -{\smath{\mbox{\bf permutationRepresentation}\opLeftPren{}pi, \allowbreak{} n\opRightPren{}} returns the matrix +{\smath{\mbox{\bf permutationRepresentation}\opLeftPren{}pi, +\allowbreak{} n\opRightPren{}} returns the matrix \smath{\delta_{i, pi(i)}} (Kronecker delta) if the permutation \smath{pi} is in list notation and permutes \smath{{1, 2, \ldots, n}}. Argument \smath{pi} may either be permutation or a list of integers describing a permutation by list notation. \newitem -\smath{\mbox{\bf permutationRepresentation}\opLeftPren{}[pi_1, \allowbreak{} \ldots, pi_k], n\opRightPren{}} returns +\smath{\mbox{\bf permutationRepresentation}\opLeftPren{} +[pi_1, \allowbreak{} \ldots, pi_k], n\opRightPren{}} returns the list of matrices \smath{[(\delta_{i, pi_1}(i)), \ldots, (\delta_{i, pi_k(i)})]} (Kronecker delta) for permutations @@ -76652,17 +79837,21 @@ the list of matrices } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{permutations}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{permutations}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% {1}{(Integer)->Stream(List(Integer))}{PartitionsAndPermutations} -{\smath{\mbox{\bf permutations}\opLeftPren{}n\opRightPren{}} returns the stream of permutations +{\smath{\mbox{\bf permutations}\opLeftPren{}n\opRightPren{}} +returns the stream of permutations formed from \smath{1, 2, \ldots, n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{physicalLength}}\opLeftPren{}{\it flexibleArray}\opRightPren{}% -\opand \mbox{\axiomFun{physicalLength!}}\opLeftPren{}{\it flexibleArray}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{physicalLength}}\opLeftPren{} +{\it flexibleArray}\opRightPren{}% +\opand \mbox{\axiomFun{physicalLength!}}\opLeftPren{} +{\it flexibleArray}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {1}{(Integer)->Stream(List(Integer))}{PartitionsAndPermutations} @@ -76675,12 +79864,15 @@ of elements of \smath{a} beyond this number, new storage is allocated (generally to be about 50\% larger than current storage allocation) and the elements from the old storage are copied over to the new storage area.} -\newitem\smath{\mbox{\bf physicalLength}\opLeftPren{}a\opRightPren{}} returns the physical length of \smath{a}. -\newitem\smath{\mbox{\bf physicalLength!}\opLeftPren{}a, \allowbreak{} n\opRightPren{}} causes +\newitem\smath{\mbox{\bf physicalLength}\opLeftPren{}a\opRightPren{}} +returns the physical length of \smath{a}. +\newitem\smath{\mbox{\bf physicalLength!}\opLeftPren{}a, +\allowbreak{} n\opRightPren{}} causes new storage to be allocated for the elements of \smath{a} with a physical length of \smath{n}. The \spadfun{maxIndex} elements from the old storage area are copied. -An \spadfun{error} is called if \smath{n} is less than \spadfun{maxIndex}\smath{(a)}.} +An \spadfun{error} is called if \smath{n} is less than +\spadfun{maxIndex}\smath{(a)}.} % ---------------------------------------------------------------------- @@ -76692,26 +79884,32 @@ An \spadfun{error} is called if \smath{n} is less than \spadfun{maxIndex}\smath{ also denoted by the special symbol \code{\%pi}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pile}}\opLeftPren{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pile}}\opLeftPren{} +{\it listOfOutputForms}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf pile}\opLeftPren{}lo\opRightPren{}}, -where \smath{lo} is a list of objects of type \spadtype{OutputForm} (normally unexposed), +where \smath{lo} is a list of objects of type \spadtype{OutputForm} +(normally unexposed), creates the output form consisting of the elements of \smath{lo} displayed as a pile, that is, each element begins on a new line and is indented right to the same margin. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{plenaryPower}}\opLeftPren{}{\it element}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{plenaryPower}}\opLeftPren{} +{\it element}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% {2}{(\$, PositiveInteger)->\$}{NonAssociativeAlgebra} -{\opkey{Argument \smath{a} is a member of a domain of category \spadtype{NonAssociativeAlgebra}} +{\opkey{Argument \smath{a} is a member of a domain of +category \spadtype{NonAssociativeAlgebra}} \newitem -\smath{\mbox{\bf plenaryPower}\opLeftPren{}a, \allowbreak{} n\opRightPren{}} is recursively defined to be -\smath{\mbox{\bf plenaryPower}\opLeftPren{}a, n-1) * {\bf plenaryPower}(a, n-1)} for \smath{n>1} and \smath{a} for \smath{n=1}. +\smath{\mbox{\bf plenaryPower}\opLeftPren{}a, +\allowbreak{} n\opRightPren{}} is recursively defined to be +\smath{\mbox{\bf plenaryPower}\opLeftPren{}a, n-1) * +{\bf plenaryPower}(a, n-1)} for \smath{n>1} and \smath{a} for \smath{n=1}. } % ---------------------------------------------------------------------- @@ -76719,31 +79917,41 @@ line and is indented right to the same margin. }% }% {0}{()->OrderedCompletion(Integer)}{Infinity} -{\smath{\mbox{\bf plusInfinity}\opLeftPren{}\opRightPren{}} returns the constant {\tt \%plusInfinity} +{\smath{\mbox{\bf plusInfinity}\opLeftPren{}\opRightPren{}} +returns the constant {\tt \%plusInfinity} denoting $+\infty$. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{point}}\opLeftPren{}{\it u\opt{, option}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{point}}\opLeftPren{} +{\it u\opt{, option}}\opRightPren{}% }% }% {1}{(List(R))->\$}{PointCategory} -{\smath{\mbox{\bf point}\opLeftPren{}p\opRightPren{}} returns a \spadtype{ThreeSpace} object which is composed of one component, the point \smath{p}. -\smath{\mbox{\bf point}\opLeftPren{}l\opRightPren{}} creates a point defined by a list \smath{l}. +{\smath{\mbox{\bf point}\opLeftPren{}p\opRightPren{}} returns a +\spadtype{ThreeSpace} object which is composed of one component, +the point \smath{p}. +\smath{\mbox{\bf point}\opLeftPren{}l\opRightPren{}} creates a point +defined by a list \smath{l}. \newitem \newitem -\smath{\mbox{\bf point}\opLeftPren{}sp\opRightPren{}} checks to see if the \spadtype{ThreeSpace} +\smath{\mbox{\bf point}\opLeftPren{}sp\opRightPren{}} checks to see if +the \spadtype{ThreeSpace} object \smath{sp} is composed of only a single point and, if so, returns the point, or calls \spadfun{error} if \smath{sp} has more than one point. \newitem -\smath{\mbox{\bf point}\opLeftPren{}sp, \allowbreak{} l\opRightPren{}} adds a point component defined by a list +\smath{\mbox{\bf point}\opLeftPren{}sp, \allowbreak{} l\opRightPren{}} +adds a point component defined by a list \smath{l} to the \spadtype{ThreeSpace} object \smath{sp}. \newitem -\smath{\mbox{\bf point}\opLeftPren{}sp, \allowbreak{} i\opRightPren{}} adds a point component into a component -list of the \spadtype{ThreeSpace} object \smath{sp} at the index given by \smath{i}. +\smath{\mbox{\bf point}\opLeftPren{}sp, \allowbreak{} i\opRightPren{}} +adds a point component into a component +list of the \spadtype{ThreeSpace} object \smath{sp} at the index given +by \smath{i}. \newitem -\smath{\mbox{\bf point}\opLeftPren{}sp, \allowbreak{} p\opRightPren{}} adds a point component defined by the point \smath{p} +\smath{\mbox{\bf point}\opLeftPren{}sp, \allowbreak{} p\opRightPren{}} +adds a point component defined by the point \smath{p} described as a list, to the \spadtype{ThreeSpace} object \smath{sp}. } @@ -76753,40 +79961,49 @@ described as a list, to the \spadtype{ThreeSpace} object \smath{sp}. }% }% {1}{(\$)->Boolean}{ThreeSpace} -{\smath{\mbox{\bf point?}\opLeftPren{}sp\opRightPren{}} queries whether the \spadtype{ThreeSpace} object \smath{sp}, +{\smath{\mbox{\bf point?}\opLeftPren{}sp\opRightPren{}} +queries whether the \spadtype{ThreeSpace} object \smath{sp}, is composed of a single component which is a point. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pointColor}}\opLeftPren{}{\it palette}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pointColor}}\opLeftPren{} +{\it palette}\opRightPren{}% }% }% {1}{(Float)->\$}{DrawOption} -{\smath{\mbox{\bf pointColor}\opLeftPren{}v\opRightPren{}} specifies a color \smath{v} for \twodim{} graph points. +{\smath{\mbox{\bf pointColor}\opLeftPren{}v\opRightPren{}} +specifies a color \smath{v} for \twodim{} graph points. This option is expressed in the form \code{pointColor == v} in the \spadfun{draw} command. Argument \smath{p} is either a palette or a float. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pointColorDefault}}\opLeftPren{}{\it \opt{palette}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pointColorDefault}}\opLeftPren{} +{\it \opt{palette}}\opRightPren{}% }% }% {0}{()->Palette}{ViewDefaultsPackage} -{\smath{\mbox{\bf pointColorDefault}\opLeftPren{}\opRightPren{}} returns the default color of points in a \twodim{} viewport. +{\smath{\mbox{\bf pointColorDefault}\opLeftPren{}\opRightPren{}} +returns the default color of points in a \twodim{} viewport. \newitem -\smath{\mbox{\bf pointColorDefault}\opLeftPren{}p\opRightPren{}} sets the default color of points in a \twodim{} viewport to the palette \smath{p}. +\smath{\mbox{\bf pointColorDefault}\opLeftPren{}p\opRightPren{}} sets +the default color of points in a \twodim{} viewport to the palette \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pointSizeDefault}}\opLeftPren{}{\it \opt{positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pointSizeDefault}}\opLeftPren{} +{\it \opt{positiveInteger}}\opRightPren{}% }% }% {0}{()->PositiveInteger}{ViewDefaultsPackage} -{\smath{\mbox{\bf pointSizeDefault}\opLeftPren{}\opRightPren{}} returns the default size of the points +{\smath{\mbox{\bf pointSizeDefault}\opLeftPren{}\opRightPren{}} +returns the default size of the points in a \twodim{} viewport. \newitem -\smath{\mbox{\bf pointSizeDefault}\opLeftPren{}i\opRightPren{}} sets the default size of the points in +\smath{\mbox{\bf pointSizeDefault}\opLeftPren{}i\opRightPren{}} +sets the default size of the points in a \twodim{} viewport to \smath{i}. } @@ -76804,9 +80021,11 @@ re^{i\phi}.}} }% }% {1}{(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf polar}\opLeftPren{}pt\opRightPren{}} transforms point \smath{pt} from polar coordinates to +{\smath{\mbox{\bf polar}\opLeftPren{}pt\opRightPren{}} +transforms point \smath{pt} from polar coordinates to Cartesian coordinates. The function produced will map the point -\smath{(r, \theta)} to \smath{x = r {\bf cos}(\theta)} , \smath{y = r {\bf sin}(\theta)}. +\smath{(r, \theta)} to \smath{x = r {\bf cos}(\theta)} , +\smath{y = r {\bf sin}(\theta)}. } % ---------------------------------------------------------------------- @@ -76814,49 +80033,61 @@ Cartesian coordinates. The function produced will map the point }% }% {1}{(\$)->Boolean}{PowerSeriesCategory} -{\smath{\mbox{\bf pole?}\opLeftPren{}f\opRightPren{}} tests if the power series \smath{f} has a pole. +{\smath{\mbox{\bf pole?}\opLeftPren{}f\opRightPren{}} tests if the power +series \smath{f} has a pole. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{polygamma}}\opLeftPren{}{\it k}, \allowbreak{}{\it x}\opRightPren{}% +\opdata{{\mbox{\axiomFun{polygamma}}\opLeftPren{}{\it k}, \allowbreak{} +{\it x}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{SpecialFunctionCategory} -{\smath{\mbox{\bf polygamma}\opLeftPren{}k, \allowbreak{} x\opRightPren{}} is the \eth{\smath{k}} derivative of -\smath{\mbox{\bf digamma}\opLeftPren{}x\opRightPren{}}, often written \smath{\psi(k, x)} in the +{\smath{\mbox{\bf polygamma}\opLeftPren{}k, \allowbreak{} x\opRightPren{}} +is the \eth{\smath{k}} derivative of +\smath{\mbox{\bf digamma}\opLeftPren{}x\opRightPren{}}, +often written \smath{\psi(k, x)} in the literature. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{polygon}}\opLeftPren{}{\it \optinit{sp, }listOfPoints}\opRightPren{}% +\opdata{{\mbox{\axiomFun{polygon}}\opLeftPren{} +{\it \optinit{sp, }listOfPoints}\opRightPren{}% \opand \mbox{\axiomFun{polygon?}}\opLeftPren{}{\it space}\opRightPren{}% }% }% {1}{(List(Point(R)))->\$}{ThreeSpace} -{\smath{\mbox{\bf polygon}\opLeftPren{}\optfirst{sp, }lp\opRightPren{}} adds a polygon defined +{\smath{\mbox{\bf polygon}\opLeftPren{}\optfirst{sp, }lp\opRightPren{}} +adds a polygon defined by \smath{lp} to the \spadtype{ThreeSpace} object \smath{sp}. Each \smath{lp} is either a list of points (objects of type \spadtype{Point}) or else a list of small floats. If \smath{sp} is omitted, it is understood to be empty. \newitem -\smath{\mbox{\bf polygon}\opLeftPren{}sp\opRightPren{}} returns \spadtype{ThreeSpace} object \smath{sp} as a list +\smath{\mbox{\bf polygon}\opLeftPren{}sp\opRightPren{}} returns +\spadtype{ThreeSpace} object \smath{sp} as a list of polygons, or an error if \smath{sp} is not composed of a single polygon. \newitem -\smath{\mbox{\bf polygon?}\opLeftPren{}sp\opRightPren{}} tests if the \spadtype{ThreeSpace} object \smath{sp} +\smath{\mbox{\bf polygon?}\opLeftPren{}sp\opRightPren{}} tests +if the \spadtype{ThreeSpace} object \smath{sp} contains a single polygon component. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{polynomial}}\opLeftPren{}{\it series}, \allowbreak{}{\it integer}\allowbreak $\,[$ , \allowbreak{}{\it integer}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{polynomial}}\opLeftPren{}{\it series}, +\allowbreak{}{\it integer}\allowbreak $\,[$ , \allowbreak{} +{\it integer}$]$\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->Polynomial(Coef)}{UnivariateTaylorSeriesCategory} -{\smath{\mbox{\bf polynomial}\opLeftPren{}s, \allowbreak{} k\opRightPren{}} returns a polynomial consisting of the +{\smath{\mbox{\bf polynomial}\opLeftPren{}s, \allowbreak{} k\opRightPren{}} +returns a polynomial consisting of the sum of all terms of series \smath{s} of degree \smath{\leq k} and greater than or equal to 0. \newitem -\smath{\mbox{\bf polynomial}\opLeftPren{}s, \allowbreak{} k_1, \allowbreak{} k_2\opRightPren{}} returns a polynomial consisting of the sum +\smath{\mbox{\bf polynomial}\opLeftPren{}s, \allowbreak{} k_1, +\allowbreak{} k_2\opRightPren{}} returns a polynomial consisting of the sum of all terms of Taylor series \smath{s} of degree \smath{d} with \smath{0 \leq k_1 \leq d \leq k_2}. } @@ -76867,57 +80098,72 @@ with \smath{0 \leq k_1 \leq d \leq k_2}. }% }% {1}{(\$)->S}{StackAggregate} -{\smath{\mbox{\bf pop!}\opLeftPren{}s\opRightPren{}} returns the top element \smath{x} +{\smath{\mbox{\bf pop!}\opLeftPren{}s\opRightPren{}} returns +the top element \smath{x} from stack \smath{s}, destructively removing it from \smath{s}, or calls \spadfun{error} if \smath{s} is empty. -Note: Use \smath{\mbox{\bf top}\opLeftPren{}s\opRightPren{}} to obtain \smath{x} without removing it from \smath{s}. +Note: Use \smath{\mbox{\bf top}\opLeftPren{}s\opRightPren{}} +to obtain \smath{x} without removing it from \smath{s}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{position}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it aggregate}\allowbreak $\,[$ , \allowbreak{}{\it index}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{position}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it aggregate}\allowbreak $\,[$ , \allowbreak{} +{\it index}$]$\opRightPren{}% }% }% {2}{(S, \$)->Integer}{FiniteLinearAggregate} -{\smath{\mbox{\bf position}\opLeftPren{}x, \allowbreak{} a\optinner{, n}\opRightPren{}} returns the index \smath{i} +{\smath{\mbox{\bf position}\opLeftPren{}x, +\allowbreak{} a\optinner{, n}\opRightPren{}} returns the index \smath{i} of the first occurrence of \smath{x} in \smath{a} where \smath{i -\geq n}, and \smath{\mbox{\bf minIndex}\opLeftPren{}a) - 1} if no such \smath{x} is +\geq n}, and \smath{\mbox{\bf minIndex}\opLeftPren{}a) - 1} +if no such \smath{x} is found. The default value of \smath{n} is 1. -\newitem \smath{\mbox{\bf position}\opLeftPren{}cc, \allowbreak{} t, \allowbreak{} i\opRightPren{}} returns the position \smath{j +\newitem \smath{\mbox{\bf position}\opLeftPren{}cc, \allowbreak{} t, +\allowbreak{} i\opRightPren{}} returns the position \smath{j >= i} in \smath{t} of the first character belonging to character class \smath{cc}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{positive?}}\opLeftPren{}{\it orderedSetElement}\opRightPren{}% +\opdata{{\mbox{\axiomFun{positive?}}\opLeftPren{} +{\it orderedSetElement}\opRightPren{}% }% }% {1}{(\$)->Boolean}{OrderedRing} -{\smath{\mbox{\bf positive?}\opLeftPren{}x\opRightPren{}} tests if \smath{x} is strictly greater than 0. +{\smath{\mbox{\bf positive?}\opLeftPren{}x\opRightPren{}} +tests if \smath{x} is strictly greater than 0. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{positiveRemainder}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{positiveRemainder}}\opLeftPren{}{\it integer}, +\allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf positiveRemainder}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}, where \smath{b > 1}, yields +{\smath{\mbox{\bf positiveRemainder}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}}, where \smath{b > 1}, yields \smath{r} where \smath{0 \leq r < b} and \smath{r = a \rem b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{possiblyInfinite?}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{possiblyInfinite?}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(\$)->Boolean}{StreamAggregate} -{\smath{\mbox{\bf possiblyInfinite?}\opLeftPren{}s\opRightPren{}} tests if the stream \smath{s} could +{\smath{\mbox{\bf possiblyInfinite?}\opLeftPren{}s\opRightPren{}} +tests if the stream \smath{s} could possibly have an infinite number of elements. Note: for many datatypes, -\smath{\mbox{\bf possiblyInfinite?}\opLeftPren{}s\opRightPren{}} \smath{= \mbox{\bf not } \mbox{\bf explictlyFinite?}(s)}. +\smath{\mbox{\bf possiblyInfinite?}\opLeftPren{}s\opRightPren{}} +\smath{= \mbox{\bf not } \mbox{\bf explictlyFinite?}(s)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{postfix}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{postfix}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} @@ -76932,45 +80178,54 @@ creates an output form which prints as: \smath{a\quad{}{\rm op}}. }% }% {0}{()->Boolean}{FiniteRankNonAssociativeAlgebra} -{\smath{\mbox{\bf powerAssociative?}\opLeftPren{}\opRightPren{}\$F}, where \spad{F} is +{\smath{\mbox{\bf powerAssociative?}\opLeftPren{}\opRightPren{}\$F}, +where \spad{F} is a domain of category \spadtype{FiniteRankNonAssociativeAlgebra}, tests if all subalgebras generated by a single element are associative. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{powerSum}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{powerSum}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% { I -> SPOL RN}{}{} -{\smath{\mbox{\bf powerSum}\opLeftPren{}n\opRightPren{}} is the \smath{n} th power sum symmetric +{\smath{\mbox{\bf powerSum}\opLeftPren{}n\opRightPren{}} +is the \smath{n} th power sum symmetric function. See \spadtype{CycleIndicators} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{powmod}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{powmod}}\opLeftPren{}{\it integer}, +\allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {3}{(\$, \$, \$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf powmod}\opLeftPren{}a, \allowbreak{} b, \allowbreak{} p\opRightPren{}}, +{\smath{\mbox{\bf powmod}\opLeftPren{}a, \allowbreak{} b, +\allowbreak{} p\opRightPren{}}, where \smath{a} and \smath{b} are non-negative integers, each \smath{< p}, returns \smath{a^b\mod p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{precision}}\opLeftPren{}{\it \opt{positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{precision}}\opLeftPren{} +{\it \opt{positiveInteger}}\opRightPren{}% }% }% {0}{()->PositiveInteger}{FloatingPointSystem} -{\smath{\mbox{\bf precision}\opLeftPren{}\opRightPren{}} returns the precision of \spadtype{Float} values +{\smath{\mbox{\bf precision}\opLeftPren{}\opRightPren{}} +returns the precision of \spadtype{Float} values in decimal digits. \newitem -\smath{\mbox{\bf precision}\opLeftPren{}n\opRightPren{}} set the precision in the base to \smath{n} +\smath{\mbox{\bf precision}\opLeftPren{}n\opRightPren{}} +set the precision in the base to \smath{n} decimal digits. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{prefix}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{prefix}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it listOfOutputForms}\opRightPren{}% }% }% {}{}{} @@ -76983,19 +80238,23 @@ of \smath{o} to a tuple of arguments given by list \smath{lo}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{prefix?}}\opLeftPren{}{\it string}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{prefix?}}\opLeftPren{}{\it string}, \allowbreak{} +{\it string}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{StringAggregate} -{\smath{\mbox{\bf prefix?}\opLeftPren{}s, \allowbreak{} t\opRightPren{}} tests if the string \smath{s} is the initial +{\smath{\mbox{\bf prefix?}\opLeftPren{}s, \allowbreak{} t\opRightPren{}} +tests if the string \smath{s} is the initial substring of \smath{t}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{prefixRagits}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{prefixRagits}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% {1}{(\$)->List(Integer)}{RadixExpansion} -{\smath{\mbox{\bf prefixRagits}\opLeftPren{}rx\opRightPren{}} returns the non-cyclic part of the +{\smath{\mbox{\bf prefixRagits}\opLeftPren{}rx\opRightPren{}} +returns the non-cyclic part of the ragits of the fractional part of a radix expansion. For example, if \smath{x = 3/28 = 0.10 714285 714285 \ldots}, then \smath{\mbox{\bf prefixRagits}\opLeftPren{}x)=[1, 0]}. @@ -77003,43 +80262,58 @@ For example, if \smath{x = 3/28 = 0.10 714285 714285 \ldots}, then % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{presub}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{presub}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf presub}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, -where \smath{o_1} and \smath{o_2} are objects of type \spadtype{OutputForm} (normally unexposed), +where \smath{o_1} and \smath{o_2} are objects of type +\spadtype{OutputForm} (normally unexposed), creates an output form for \smath{o_1} presubscripted by \smath{o_2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{presuper}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{presuper}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf presuper}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, -where \smath{o_1} and \smath{o_2} are objects of type \spadtype{OutputForm} (normally unexposed), +{\smath{\mbox{\bf presuper}\opLeftPren{}o_1, +\allowbreak{} o_2\opRightPren{}}, +where \smath{o_1} and \smath{o_2} are objects of type +\spadtype{OutputForm} (normally unexposed), creates an output form for \smath{o_1} presuperscripted by \smath{o_2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primaryDecomp}}\opLeftPren{}{\it ideal}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primaryDecomp}}\opLeftPren{} +{\it ideal}\opRightPren{}% }% }% -{1}{(PolynomialIdeals(Fraction(Integer), DirectProduct(nv, NonNegativeInteger), vl, DistributedMultivariatePolynomial(vl, Fraction(Integer))))->List(PolynomialIdeals(Fraction(Integer), DirectProduct(nv, NonNegativeInteger), vl, DistributedMultivariatePolynomial(vl, Fraction(Integer))))}{IdealDecompositionPackage} -{\smath{\mbox{\bf primaryDecomp}\opLeftPren{}I\opRightPren{}} returns a list of primary ideals such +{1}{(PolynomialIdeals(Fraction(Integer), +DirectProduct(nv, NonNegativeInteger), vl, +DistributedMultivariatePolynomial(vl, Fraction(Integer))))-> +List(PolynomialIdeals(Fraction(Integer), +DirectProduct(nv, NonNegativeInteger), vl, +DistributedMultivariatePolynomial(vl, Fraction(Integer))))} +{IdealDecompositionPackage} +{\smath{\mbox{\bf primaryDecomp}\opLeftPren{}I\opRightPren{}} +returns a list of primary ideals such that their intersection is the ideal \smath{I}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{prime}}\opLeftPren{}{\it outputForm\opt{, positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{prime}}\opLeftPren{} +{\it outputForm\opt{, positiveInteger}}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf prime}\opLeftPren{}o\optinner{, n}\opRightPren{}}, -where \smath{o} is an object of type \spadtype{OutputForm} (normally unexposed), +where \smath{o} is an object of type \spadtype{OutputForm} +(normally unexposed), creates an output form for \smath{o} following by \smath{n} primes (that is, a prime like `` ' ''). By default, \smath{n = 1}. @@ -77050,7 +80324,8 @@ By default, \smath{n = 1}. }% }% {1}{(\$)->Boolean}{UniqueFactorizationDomain} -{\smath{\mbox{\bf prime?}\opLeftPren{}x\opRightPren{}} tests if \smath{x} cannot be written as the +{\smath{\mbox{\bf prime?}\opLeftPren{}x\opRightPren{}} +tests if \smath{x} cannot be written as the product of two non-units, that is, \smath{x} is an irreducible element. Argument \smath{x} may be an integer, a polynomial, an ideal, or, @@ -77059,51 +80334,63 @@ in general, any element of a domain of category } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primeFactor}}\opLeftPren{}{\it element}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primeFactor}}\opLeftPren{} +{\it element}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(R, Integer)->\$}{Factored} -{\smath{\mbox{\bf primeFactor}\opLeftPren{}base, \allowbreak{} exponent\opRightPren{}} creates a factored object with +{\smath{\mbox{\bf primeFactor}\opLeftPren{}base, +\allowbreak{} exponent\opRightPren{}} creates a factored object with a single factor whose \smath{base} is asserted to be prime (flag = \mbox{\tt "prime"}). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primeFrobenius}}\opLeftPren{}{\it finiteFieldElement\opt{, nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primeFrobenius}}\opLeftPren{} +{\it finiteFieldElement\opt{, nonNegativeInteger}}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->\$}{FieldOfPrimeCharacteristic} {\opkey{Argument \smath{a} is a member of a domain of category \spadtype{FieldOfPrimeCharacteristic(p)}.} \newitem -\smath{\mbox{\bf primeFrobenius}\opLeftPren{}a\optinner{, s}\opRightPren{}} returns \smath{a^{p^s}}. +\smath{\mbox{\bf primeFrobenius}\opLeftPren{}a\optinner{, s}\opRightPren{}} +returns \smath{a^{p^s}}. The default value of \smath{s} is 1. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primes}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primes}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(I, I)->List(I)}{IntegerPrimesPackage} -{\smath{\mbox{\bf primes}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns a list of all primes \smath{p} with \smath{a \leq p \leq b}. +{\smath{\mbox{\bf primes}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +returns a list of all primes \smath{p} with \smath{a \leq p \leq b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primitive?}}\opLeftPren{}{\it finiteFieldElement}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primitive?}}\opLeftPren{} +{\it finiteFieldElement}\opRightPren{}% }% }% {1}{(\$)->Boolean}{FiniteFieldCategory} -{\smath{\mbox{\bf primitive?}\opLeftPren{}b\opRightPren{}} tests whether the element \smath{b} +{\smath{\mbox{\bf primitive?}\opLeftPren{}b\opRightPren{}} +tests whether the element \smath{b} of a finite field is a generator of the (cyclic) multiplicative group of the field, that is, is a primitive element. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primitiveElement}}\opLeftPren{}{\it expressions\opt{, expression}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primitiveElement}}\opLeftPren{} +{\it expressions\opt{, expression}}\opRightPren{}% }% }% -{2}{(F, F)->Record(primelt:F, pol1:SparseUnivariatePolynomial(F), pol2:SparseUnivariatePolynomial(F), prim:SparseUnivariatePolynomial(F))}{FunctionSpacePrimitiveElement} -{\smath{\mbox{\bf primitiveElement}\opLeftPren{}a_1, \allowbreak{} a_2\opRightPren{}} returns a record with four +{2}{(F, F)->Record(primelt:F, pol1:SparseUnivariatePolynomial(F), +pol2:SparseUnivariatePolynomial(F), prim:SparseUnivariatePolynomial(F))} +{FunctionSpacePrimitiveElement} +{\smath{\mbox{\bf primitiveElement}\opLeftPren{}a_1, +\allowbreak{} a_2\opRightPren{}} returns a record with four components: a primitive element \smath{a} with selector \smath{primelt}, and three polynomials \smath{q_1}, \smath{q_2}, and \smath{q} with selectors \smath{pol1}, \smath{pol2}, and @@ -77116,32 +80403,41 @@ the minimal polynomial for \smath{a_1} may not involve \smath{a_2}. This operations uses \spadfun{resultant}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primitiveMonomials}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primitiveMonomials}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->List(\$)}{PolynomialCategory} -{\smath{\mbox{\bf primitiveMonomials}\opLeftPren{}p\opRightPren{}} gives the list of monomials of the +{\smath{\mbox{\bf primitiveMonomials}\opLeftPren{}p\opRightPren{}} +gives the list of monomials of the polynomial \smath{p} with their coefficients removed. -Note: \smath{\mbox{\bf primitiveMonomials}\opLeftPren{}\sum {a_i X^{(i)}}) = [X^{(1)}, \ldots, X^{(n)}]}. +Note: \smath{\mbox{\bf primitiveMonomials}\opLeftPren{} +\sum {a_i X^{(i)}}) = [X^{(1)}, \ldots, X^{(n)}]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{primitivePart}}\opLeftPren{}{\it polynomial\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{primitivePart}}\opLeftPren{} +{\it polynomial\opt{, symbol}}\opRightPren{}% }% }% {1}{(\$)->\$}{FiniteAbelianMonoidRing} -{\smath{\mbox{\bf primitivePart}\opLeftPren{}p\optinner{, v}\opRightPren{}} returns -the unit normalized form of polynomial \smath{p} divided by the \spadfun{content} of \smath{p} with respect to variable \smath{v}. +{\smath{\mbox{\bf primitivePart}\opLeftPren{}p\optinner{, v} +\opRightPren{}} returns +the unit normalized form of polynomial \smath{p} divided +by the \spadfun{content} of \smath{p} with respect to variable \smath{v}. If no \smath{v} is given, the content is removed with respect to all variables. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{principalIdeal}}\opLeftPren{}{\it listOfPolynomials}\opRightPren{}% +\opdata{{\mbox{\axiomFun{principalIdeal}}\opLeftPren{} +{\it listOfPolynomials}\opRightPren{}% }% }% -{1}{(List(\$))->Record(coef:List(\$), generator:\$)}{PrincipalIdealDomain} -{\smath{\mbox{\bf principalIdeal}\opLeftPren{}[f_1, \allowbreak{} \ldots, f_n]\opRightPren{}} returns a record whose +{1}{(List(\$))->Record(coef:List(\$), generator:\$)} +{PrincipalIdealDomain} +{\smath{\mbox{\bf principalIdeal}\opLeftPren{}[f_1, +\allowbreak{} \ldots, f_n]\opRightPren{}} returns a record whose ``generator'' component is a generator of the ideal generated by \smath{[f_1, \ldots, f_n]} whose ``coef'' component is a list of coefficients \smath{[c_1, \ldots, c_n]} such that @@ -77152,32 +80448,39 @@ coefficients \smath{[c_1, \ldots, c_n]} such that }% }% {1}{(OutputForm)->Void}{PrintPackage} -{\smath{\mbox{\bf print}\opLeftPren{}o\opRightPren{}} writes the output form \smath{o} on standard +{\smath{\mbox{\bf print}\opLeftPren{}o\opRightPren{}} +writes the output form \smath{o} on standard output using the two-dimensional formatter. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{product}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{product}}\opLeftPren{} +{\it element}, \allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{GradedAlgebra} -{\smath{\mbox{\bf product}\opLeftPren{}f(n), \allowbreak{} n = a..b\opRightPren{}} returns +{\smath{\mbox{\bf product}\opLeftPren{}f(n), +\allowbreak{} n = a..b\opRightPren{}} returns \smath{\prod\nolimits_{n=a}^b{f(n)}} as a formal product. -\newitem\smath{\mbox{\bf product}\opLeftPren{}f(n), \allowbreak{} n\opRightPren{}} returns the formal product +\newitem\smath{\mbox{\bf product}\opLeftPren{}f(n), +\allowbreak{} n\opRightPren{}} returns the formal product \smath{P(n)} verifying \smath{{P(n+1)/P(n)} = f(n)}. -\newitem\smath{\mbox{\bf product}\opLeftPren{}s, \allowbreak{} t\opRightPren{}}, where \smath{s} and \smath{t} are +\newitem\smath{\mbox{\bf product}\opLeftPren{}s, +\allowbreak{} t\opRightPren{}}, where \smath{s} and \smath{t} are cartesian tensors, returns the outer product of \smath{s} and \smath{t}. For example, if \smath{r = {\bf product}(s, t)} for rank 2 tensors \smath{s} and \smath{t}, then \smath{r} is a rank 4 tensor given by \smath{r_{i, j, k, l} = s_{i, j} t_{k, l}}. -\newitem\smath{\mbox{\bf product}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}, where \smath{a} and \smath{b} are +\newitem\smath{\mbox{\bf product}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}}, where \smath{a} and \smath{b} are elements of a graded algebra returns the degree-preserving linear product. See \spadtype{GradedAlgebra} for details.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{prolateSpheroidal}}\opLeftPren{}{\it smallFloat}\opRightPren{}% +\opdata{{\mbox{\axiomFun{prolateSpheroidal}}\opLeftPren{} +{\it smallFloat}\opRightPren{}% }% }% {1}{(R)->(Point(R))->Point(R)}{CoordinateSystems} @@ -77195,50 +80498,61 @@ This function will map the point }% }% {1}{(\$)->List(String)}{ScriptFormulaFormat} -{\smath{\mbox{\bf prologue}\opLeftPren{}t\opRightPren{}} extracts the prologue section of a +{\smath{\mbox{\bf prologue}\opLeftPren{}t\opRightPren{}} +extracts the prologue section of a IBM SCRIPT Formula Formatter or \TeX{} formatted object \smath{t}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{properties}}\opLeftPren{}{\it basicOperator\opt{, prop}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{properties}}\opLeftPren{} +{\it basicOperator\opt{, prop}}\opRightPren{}% }% }% {1}{(\$)->AssociationList(String, None)}{BasicOperator} -{\smath{\mbox{\bf properties}\opLeftPren{}op\opRightPren{}} returns the list of all the properties currently attached to \smath{op}. +{\smath{\mbox{\bf properties}\opLeftPren{}op\opRightPren{}} +returns the list of all the properties currently attached to \smath{op}. \newitem -\smath{\mbox{\bf property}\opLeftPren{}op, \allowbreak{} s\opRightPren{}} returns the value of property \smath{s} if +\smath{\mbox{\bf property}\opLeftPren{}op, \allowbreak{} s\opRightPren{}} +returns the value of property \smath{s} if it is attached to \smath{op}, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pseudoDivide}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pseudoDivide}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% }% }% -{2}{(\$, \$)->Record(coef:R, quotient:\$, remainder:\$)}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf pseudoDivide}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns \smath{(c, q, r)}, when +{2}{(\$, \$)->Record(coef:R, quotient:\$, remainder:\$)} +{UnivariatePolynomialCategory} +{\smath{\mbox{\bf pseudoDivide}\opLeftPren{}p, +\allowbreak{} q\opRightPren{}} returns \smath{(c, q, r)}, when \smath{p' := p \mbox{ \bf leadingCoefficient}(q)^{{\rm deg}(p) - {\rm deg}(q) + 1}= c p} is pseudo right-divided by \smath{q}, that is, \smath{p' = s q + r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pseudoQuotient}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pseudoQuotient}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf pseudoQuotient}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns \smath{r}, the quotient when +{\smath{\mbox{\bf pseudoQuotient}\opLeftPren{}p, +\allowbreak{} q\opRightPren{}} returns \smath{r}, the quotient when \smath{p' := p {\rm leadingCoefficient}(q)^{{\rm deg} p - {\rm deg} q + 1}} is pseudo right-divided by \smath{q}, that is, \smath{p' = s q + r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pseudoRemainder}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pseudoRemainder}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf pseudoRemainder}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} = \smath{r}, for polynomials +{\smath{\mbox{\bf pseudoRemainder}\opLeftPren{}p, +\allowbreak{} q\opRightPren{}} = \smath{r}, for polynomials \smath{p} and \smath{q}, returns the remainder when \smath{p' := p {\rm leadingCoefficient}(q)^{{\rm deg} p - {\rm deg} q + 1}} is @@ -77246,34 +80560,44 @@ pseudo right-divided by \smath{q}, that is, \smath{p' = s q + r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{puiseux}}\opLeftPren{}{\it expression\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{puiseux}}\opLeftPren{} +{\it expression\opt{, options}}\opRightPren{}% }% }% {1}{(FE)->Any}{ExpressionToUnivariatePowerSeries} -{\smath{\mbox{\bf puiseux}\opLeftPren{}f\opRightPren{}} returns a Puiseux expansion of the expression +{\smath{\mbox{\bf puiseux}\opLeftPren{}f\opRightPren{}} +returns a Puiseux expansion of the expression \smath{f}. Note: \smath{f} should have only one variable; the series will be expanded in powers of that variable. Also, if \smath{x} is a symbol, -\smath{\mbox{\bf puiseux}\opLeftPren{}x\opRightPren{}} returns \smath{x} as a Puiseux series. +\smath{\mbox{\bf puiseux}\opLeftPren{}x\opRightPren{}} returns \smath{x} +as a Puiseux series. \newitem -\smath{\mbox{\bf puiseux}\opLeftPren{}f, \allowbreak{} x = a\opRightPren{}} expands the expression \smath{f} as a +\smath{\mbox{\bf puiseux}\opLeftPren{}f, \allowbreak{} x = a\opRightPren{}} +expands the expression \smath{f} as a Puiseux series in powers of \smath{(x - a)}. \newitem -\smath{\mbox{\bf puiseux}\opLeftPren{}f, \allowbreak{} n\opRightPren{}} returns a Puiseux expansion of the expression +\smath{\mbox{\bf puiseux}\opLeftPren{}f, \allowbreak{} n\opRightPren{}} +returns a Puiseux expansion of the expression \smath{f}. Note: \smath{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \smath{n}. \newitem -\smath{\mbox{\bf puiseux}\opLeftPren{}f, \allowbreak{} x = a, \allowbreak{} n\opRightPren{}} expands the expression \smath{f} as a +\smath{\mbox{\bf puiseux}\opLeftPren{}f, \allowbreak{} x = a, +\allowbreak{} n\opRightPren{}} expands the expression \smath{f} as a Puiseux series in powers of \smath{(x - a)}; terms will be computed up to order at least \smath{n}. \newitem -\smath{\mbox{\bf puiseux}\opLeftPren{}n {\tt +->} a(n), \allowbreak{} x = a, \allowbreak{} r_0.., \allowbreak{} r\opRightPren{}} returns +\smath{\mbox{\bf puiseux}\opLeftPren{}n {\tt +->} a(n), +\allowbreak{} x = a, \allowbreak{} r_0.., \allowbreak{} r\opRightPren{}} +returns \smath{\sum\nolimits_{n = r_0, r_0 + r, r_0 + 2 r, \ldots} a(n) (x - a)^n}. \newitem -\smath{\mbox{\bf puiseux}\opLeftPren{}a(n), \allowbreak{} n, \allowbreak{} x = a, \allowbreak{} r_0.., \allowbreak{} r\opRightPren{}} returns +\smath{\mbox{\bf puiseux}\opLeftPren{}a(n), +\allowbreak{} n, \allowbreak{} x = a, \allowbreak{} r_0.., +\allowbreak{} r\opRightPren{}} returns \smath{\sum\nolimits_{n = r_0, r_0 + r, r_0 + 2 r, \ldots} a(n) (x - a)^n}. \newitem Note: Each of the last two commands have alternate forms whose third argument is the finite segment \smath{r_0..r_1} @@ -77283,119 +80607,148 @@ a finite number of terms. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{push!}}\opLeftPren{}{\it element}, \allowbreak{}{\it stack}\opRightPren{}% +\opdata{{\mbox{\axiomFun{push!}}\opLeftPren{} +{\it element}, \allowbreak{}{\it stack}\opRightPren{}% }% }% {2}{(S, \$)->S}{StackAggregate} -{\smath{\mbox{\bf push!}\opLeftPren{}x, \allowbreak{} s\opRightPren{}} pushes \smath{x} onto stack \smath{s}, that +{\smath{\mbox{\bf push!}\opLeftPren{}x, +\allowbreak{} s\opRightPren{}} pushes \smath{x} onto stack \smath{s}, that is, destructively changing \smath{s} so as to have a new first (top) element \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pushdown}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it symbol}\opRightPren{}% - \opand \mbox{\axiomFun{pushdterm}}\opLeftPren{}{\it monomial}, \allowbreak{}{\it symbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pushdown}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it symbol}\opRightPren{}% + \opand \mbox{\axiomFun{pushdterm}}\opLeftPren{} +{\it monomial}, \allowbreak{}{\it symbol}\opRightPren{}% }% }% {2}{(PRF, OV)->PRF}{MPolyCatRationalFunctionFactorizer} -{\smath{\mbox{\bf pushdown}\opLeftPren{}prf, \allowbreak{} var\opRightPren{}} pushes all top level occurences of the +{\smath{\mbox{\bf pushdown}\opLeftPren{}prf, +\allowbreak{} var\opRightPren{}} pushes all top level occurences of the variable \smath{var} into the coefficient domain for the polynomial \smath{prf}. -\newitem\smath{\mbox{\bf pushdterm}\opLeftPren{}monom, \allowbreak{} var\opRightPren{}} pushes all top level +\newitem\smath{\mbox{\bf pushdterm}\opLeftPren{}monom, +\allowbreak{} var\opRightPren{}} pushes all top level occurences of the variable \smath{var} into the coefficient domain for the monomial \smath{monom}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pushucoef}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it variable}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pushucoef}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it variable}\opRightPren{}% }% }% -{2}{(SparseUnivariatePolynomial(Polynomial(R)), OV)->PRF}{MPolyCatRationalFunctionFactorizer} -{\smath{\mbox{\bf pushucoef}\opLeftPren{}upoly, \allowbreak{} var\opRightPren{}} converts the anonymous univariate +{2}{(SparseUnivariatePolynomial(Polynomial(R)), OV)->PRF} +{MPolyCatRationalFunctionFactorizer} +{\smath{\mbox{\bf pushucoef}\opLeftPren{}upoly, +\allowbreak{} var\opRightPren{}} converts the anonymous univariate polynomial \smath{upoly} to a polynomial in \smath{var} over rational functions.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pushuconst}}\opLeftPren{}{\it rationalFunction}, \allowbreak{}{\it variable}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pushuconst}}\opLeftPren{} +{\it rationalFunction}, \allowbreak{}{\it variable}\opRightPren{}% }% }% {2}{(Fraction(Polynomial(R)), OV)->PRF}{MPolyCatRationalFunctionFactorizer} -{\smath{\mbox{\bf pushuconst}\opLeftPren{}r, \allowbreak{} var\opRightPren{}} takes a rational function and raises +{\smath{\mbox{\bf pushuconst}\opLeftPren{}r, \allowbreak{} var\opRightPren{}} +takes a rational function and raises all occurences of the variable \smath{var} to the polynomial level. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{pushup}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it variable}\opRightPren{}% +\opdata{{\mbox{\axiomFun{pushup}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it variable}\opRightPren{}% }% }% {2}{(PRF, OV)->PRF}{MPolyCatRationalFunctionFactorizer} -{\smath{\mbox{\bf pushup}\opLeftPren{}prf, \allowbreak{} var\opRightPren{}} raises all occurences of the variable +{\smath{\mbox{\bf pushup}\opLeftPren{}prf, \allowbreak{} var\opRightPren{}} +raises all occurences of the variable \smath{var} in the coefficients of the polynomial \smath{prf} back to the polynomial level. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{qelt}}\opLeftPren{}{\it u\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{qelt}}\opLeftPren{}{\it u\opt{, options}} +\opRightPren{}% }% }% {2}{(\$, Dom)->Im}{EltableAggregate} -{\smath{\mbox{\bf qelt}\opLeftPren{}u, \allowbreak{} p\optinner{, options}\opRightPren{}} is equivalent to +{\smath{\mbox{\bf qelt}\opLeftPren{}u, \allowbreak{} p\optinner{, options} +\opRightPren{}} is equivalent to a corresponding \spadfun{elt} form except that it performs no check that indicies are in range. Use HyperDoc to discover if a given domain has this alternative operation. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{qsetelt!}}\opLeftPren{}{\it u}, \allowbreak{}{\it x}, \allowbreak{}{\it y}\allowbreak $\,[$ , \allowbreak{}{\it z}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{qsetelt!}}\opLeftPren{}{\it u}, \allowbreak{} +{\it x}, \allowbreak{}{\it y}\allowbreak $\,[$ , \allowbreak{} +{\it z}$]$\opRightPren{}% }% }% {3}{(\$, Dom, Im)->Im}{EltableAggregate} -{\smath{\mbox{\bf qsetelt!}\opLeftPren{}u, \allowbreak{} x, \allowbreak{} y\optinner{, z}\opRightPren{}} is equivalent to +{\smath{\mbox{\bf qsetelt!}\opLeftPren{}u, \allowbreak{} x, +\allowbreak{} y\optinner{, z}\opRightPren{}} is equivalent to a corresponding \spadfun{setelt} form except that it performs no check that indicies are in range. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{quadraticForm}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{quadraticForm}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% {1}{(SquareMatrix(n, K))->\$}{QuadraticForm} -{\smath{\mbox{\bf quadraticForm}\opLeftPren{}m\opRightPren{}} creates a quadratic form from a symmetric, square matrix \smath{m}. +{\smath{\mbox{\bf quadraticForm}\opLeftPren{}m\opRightPren{}} +creates a quadratic form from a symmetric, square matrix \smath{m}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{quatern}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}, \allowbreak{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{quatern}}\opLeftPren{}{\it element}, \allowbreak{} +{\it element}, \allowbreak{}{\it element}, \allowbreak{} +{\it element}\opRightPren{}% }% }% {4}{(R, R, R, R)->\$}{QuaternionCategory} -{\smath{\mbox{\bf quatern}\opLeftPren{}r, \allowbreak{} i, \allowbreak{} j, \allowbreak{} k\opRightPren{}} constructs a quaternion from scalars. +{\smath{\mbox{\bf quatern}\opLeftPren{}r, \allowbreak{} i, \allowbreak{} j, +\allowbreak{} k\opRightPren{}} constructs a quaternion from scalars. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{queue}}\opLeftPren{}{\it \opt{listOfElements}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{queue}}\opLeftPren{} +{\it \opt{listOfElements}}\opRightPren{}% }% }% {0}{()->\$}{QueueAggregate} -{\smath{\mbox{\bf queue}\opLeftPren{}\opRightPren{}}\$\smath{R} returns an empty queue of type +{\smath{\mbox{\bf queue}\opLeftPren{}\opRightPren{}}\$\smath{R} +returns an empty queue of type \smath{R}. \newitem -\smath{\mbox{\bf queue}\opLeftPren{}[x, \allowbreak{} y, \allowbreak{} \ldots, z]\opRightPren{}} creates a queue with first (top) +\smath{\mbox{\bf queue}\opLeftPren{}[x, \allowbreak{} y, +\allowbreak{} \ldots, z]\opRightPren{}} creates a queue with first (top) element \smath{x}, second element \smath{y}, \ldots, and last (bottom) element \smath{z}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{quickSort}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{quickSort}}\opLeftPren{} +{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S, S)->Boolean, V)->V}{FiniteLinearAggregateSort} -{\smath{\mbox{\bf quickSort}\opLeftPren{}f, \allowbreak{} agg\opRightPren{}} sorts the aggregate agg with the +{\smath{\mbox{\bf quickSort}\opLeftPren{}f, +\allowbreak{} agg\opRightPren{}} sorts the aggregate agg with the ordering predicate \smath{f} using the quicksort algorithm. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{quo}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{quo}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{NonNegativeInteger} @@ -77407,7 +80760,8 @@ discarding the remainder.} }% }% {1}{(\$)->\$}{UnivariateTaylorSeriesCategory} -{\smath{\mbox{\bf quoByVar}\opLeftPren{}a_0 + a_1 x + a_2 x^2 + \cdots\opRightPren{}} +{\smath{\mbox{\bf quoByVar} +\opLeftPren{}a_0 + a_1 x + a_2 x^2 + \cdots\opRightPren{}} returns \smath{a_1 + a_2 x + a_3 x^2 + \cdots} Thus, this function subtracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a @@ -77419,18 +80773,21 @@ Taylor series and an order. }% }% {}{}{} -{\smath{\mbox{\bf quote}\opLeftPren{}o\opRightPren{}}, where \smath{o} is an object of type +{\smath{\mbox{\bf quote}\opLeftPren{}o\opRightPren{}}, +where \smath{o} is an object of type \spadtype{OutputForm} (normally unexposed), creates an output form \smath{o} with a prefix quote. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{quotedOperators}}\opLeftPren{}{\it rewriteRule}\opRightPren{}% +\opdata{{\mbox{\axiomFun{quotedOperators}}\opLeftPren{} +{\it rewriteRule}\opRightPren{}% }% }% {1}{(\$)->List(Symbol)}{RewriteRule} -{\smath{\mbox{\bf quotedOperators}\opLeftPren{}r\opRightPren{}}, where \smath{r} is a rewrite rule, +{\smath{\mbox{\bf quotedOperators}\opLeftPren{}r\opRightPren{}}, +where \smath{r} is a rewrite rule, returns the list of operators on the right-hand side of \smath{r} that are considered quoted, that is, they are not evaluated during any rewrite, but applied formally to their arguments. @@ -77439,14 +80796,17 @@ any rewrite, but applied formally to their arguments. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{quotient}}\opLeftPren{}{\it ideal}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{quotient}}\opLeftPren{}{\it ideal}, +\allowbreak{}{\it polynomial}\opRightPren{}% }% }% {2}{(\$, DPoly)->\$}{PolynomialIdeals} -{\smath{\mbox{\bf quotient}\opLeftPren{}I, \allowbreak{} f\opRightPren{}} computes the quotient of the ideal +{\smath{\mbox{\bf quotient}\opLeftPren{}I, +\allowbreak{} f\opRightPren{}} computes the quotient of the ideal \smath{I} by the principal ideal generated by the polynomial \smath{f}, \smath{(I:(f))}. -\newitem\smath{\mbox{\bf quotient}\opLeftPren{}I, \allowbreak{} J\opRightPren{}} computes the quotient of the ideals +\newitem\smath{\mbox{\bf quotient}\opLeftPren{}I, +\allowbreak{} J\opRightPren{}} computes the quotient of the ideals \smath{I} and \smath{J}, \smath{(I:J)}. } @@ -77454,36 +80814,53 @@ any rewrite, but applied formally to their arguments. \opdata{{\mbox{\axiomFun{radical}}\opLeftPren{}{\it ideal}\opRightPren{}% }% }% -{1}{(PolynomialIdeals(Fraction(Integer), DirectProduct(nv, NonNegativeInteger), vl, DistributedMultivariatePolynomial(vl, Fraction(Integer))))->PolynomialIdeals(Fraction(Integer), DirectProduct(nv, NonNegativeInteger), vl, DistributedMultivariatePolynomial(vl, Fraction(Integer)))}{IdealDecompositionPackage} -{\smath{\mbox{\bf radical}\opLeftPren{}I\opRightPren{}} returns the radical of the ideal \smath{I}. +{1}{(PolynomialIdeals(Fraction(Integer), +DirectProduct(nv, NonNegativeInteger), vl, +DistributedMultivariatePolynomial(vl, Fraction(Integer))))-> +PolynomialIdeals(Fraction(Integer), DirectProduct(nv, NonNegativeInteger), +vl, DistributedMultivariatePolynomial(vl, Fraction(Integer)))} +{IdealDecompositionPackage} +{\smath{\mbox{\bf radical}\opLeftPren{}I\opRightPren{}} +returns the radical of the ideal \smath{I}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{radicalEigenvalues}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{radicalEigenvalues}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% -{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))->List(Expression(Fraction(Integer)))}{RadicalEigenPackage} -{\smath{\mbox{\bf radicalEigenvalues}\opLeftPren{}m\opRightPren{}} computes the eigenvalues of the +{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +List(Expression(Fraction(Integer)))}{RadicalEigenPackage} +{\smath{\mbox{\bf radicalEigenvalues}\opLeftPren{}m\opRightPren{}} +computes the eigenvalues of the matrix \smath{m}; when possible, the eigenvalues are expressed in terms of radicals. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{radicalEigenvectors}}\opLeftPren{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{radicalEigenvectors}}\opLeftPren{} +{\it matrix}\opRightPren{}% }% }% -{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))->List(Record(radval:Expression(Fraction(Integer)), radmult:Integer, radvect:List(Matrix(Expression(Fraction(Integer))))))}{RadicalEigenPackage} -{\smath{\mbox{\bf radicalEigenvectors}\opLeftPren{}m\opRightPren{}} computes the eigenvalues and the +{1}{(Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +List(Record(radval:Expression(Fraction(Integer)), radmult:Integer, +radvect:List(Matrix(Expression(Fraction(Integer))))))}{RadicalEigenPackage} +{\smath{\mbox{\bf radicalEigenvectors}\opLeftPren{}m\opRightPren{}} +computes the eigenvalues and the corresponding eigenvectors of the matrix \smath{m}; when possible, values are expressed in terms of radicals. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{radicalEigenvector}}\opLeftPren{}{\it eigenvalue}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{radicalEigenvector}}\opLeftPren{} +{\it eigenvalue}, \allowbreak{}{\it matrix}\opRightPren{}% }% }% -{2}{(Expression(Fraction(Integer)), Matrix(Fraction(Polynomial(Fraction(Integer)))))->List(Matrix(Expression(Fraction(Integer))))}{RadicalEigenPackage} -{\smath{\mbox{\bf radicalEigenvector}\opLeftPren{}c, \allowbreak{} m\opRightPren{}} computes the +{2}{(Expression(Fraction(Integer)), +Matrix(Fraction(Polynomial(Fraction(Integer)))))-> +List(Matrix(Expression(Fraction(Integer))))}{RadicalEigenPackage} +{\smath{\mbox{\bf radicalEigenvector}\opLeftPren{}c, +\allowbreak{} m\opRightPren{}} computes the eigenvector(\smath{s}) of the matrix \smath{m} corresponding to the eigenvalue \smath{c}; when possible, values are expressed in terms of radicals. @@ -77494,55 +80871,69 @@ terms of radicals. }% }% {0}{()->List(A)}{AlgebraPackage} -{\smath{\mbox{\bf radicalOfLeftTraceForm}\opLeftPren{}\opRightPren{}\$F} returns the basis for the +{\smath{\mbox{\bf radicalOfLeftTraceForm}\opLeftPren{}\opRightPren{}\$F} +returns the basis for the null space of \smath{\mbox{\bf leftTraceMatrix}\opLeftPren{}\opRightPren{}\$F}, -where \smath{F} is a domain of category \spadtype{FramedNonAssociativeAlgebra}. +where \smath{F} is a domain of category +\spadtype{FramedNonAssociativeAlgebra}. If the algebra is associative, alternative or a Jordan algebra, then this space equals the radical (maximal nil ideal) of the algebra. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{radicalRoots}}\opLeftPren{}{\it fractions}\opRightPren{}% +\opdata{{\mbox{\axiomFun{radicalRoots}}\opLeftPren{} +{\it fractions}\opRightPren{}% }% }% -{2}{(Fraction(Polynomial(R)), Symbol)->List(Expression(R))}{RadicalSolvePackage} -{\smath{\mbox{\bf radicalRoots}\opLeftPren{}rf, \allowbreak{} v\opRightPren{}} finds the roots expressed in terms of +{2}{(Fraction(Polynomial(R)), Symbol)-> +List(Expression(R))}{RadicalSolvePackage} +{\smath{\mbox{\bf radicalRoots}\opLeftPren{}rf, +\allowbreak{} v\opRightPren{}} finds the roots expressed in terms of radicals of the rational function \smath{rf} with respect to the symbol \smath{v}. \newitem -\smath{\mbox{\bf radicalRoots}\opLeftPren{}lrf, \allowbreak{} lv\opRightPren{}} finds the roots expressed in terms of +\smath{\mbox{\bf radicalRoots}\opLeftPren{}lrf, +\allowbreak{} lv\opRightPren{}} finds the roots expressed in terms of radicals of the list of rational functions \smath{lrf} with respect to the list of symbols \smath{lv}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{radicalSolve}}\opLeftPren{}{\it eq}, \allowbreak{}{\it x}\opRightPren{}% +\opdata{{\mbox{\axiomFun{radicalSolve}}\opLeftPren{} +{\it eq}, \allowbreak{}{\it x}\opRightPren{}% }% }% -{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))}{TransSolvePackage}{See \smath{\mbox{\bf solve}\opLeftPren{}u, \allowbreak{} v\opRightPren{}}. +{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))} +{TransSolvePackage} +{See \smath{\mbox{\bf solve}\opLeftPren{}u, \allowbreak{} v\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{radix}}\opLeftPren{}{\it rationalNumber}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{radix}}\opLeftPren{}{\it rationalNumber}, +\allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(Fraction(Integer), Integer)->Any}{RadixUtilities} -{\smath{\mbox{\bf radix}\opLeftPren{}rn, \allowbreak{} b\opRightPren{}} converts rational number \smath{rn} to a +{\smath{\mbox{\bf radix}\opLeftPren{}rn, \allowbreak{} b\opRightPren{}} +converts rational number \smath{rn} to a radix expansion in base \smath{b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ramified?}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ramified?}}\opLeftPren{} +{\it polynomial}\opRightPren{}% \opand \mbox{\axiomFun{ramifiedAtInfinity?}}\opLeftPren{}\opRightPren{}% }% }% {1}{(UP)->Boolean}{FunctionFieldCategory} {\opkey{Domain \smath{F} is a domain of functions on a fixed curve.} -\newitem\smath{\mbox{\bf ramified?}\opLeftPren{}p\opRightPren{}\$F} tests whether \smath{p(x) = 0} is ramified. -\newitem\smath{\mbox{\bf ramifiedAtInfinity?}\opLeftPren{}\opRightPren{}} tests if infinity is +\newitem\smath{\mbox{\bf ramified?}\opLeftPren{}p\opRightPren{}\$F} +tests whether \smath{p(x) = 0} is ramified. +\newitem\smath{\mbox{\bf ramifiedAtInfinity?}\opLeftPren{}\opRightPren{}} +tests if infinity is ramified. } @@ -77551,16 +80942,19 @@ ramified. }% }% {0}{()->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf random}\opLeftPren{}\opRightPren{}\$R} creates a random element from +{\smath{\mbox{\bf random}\opLeftPren{}\opRightPren{}\$R} creates a +random element from domain \smath{D}. \newitem -\smath{\mbox{\bf random}\opLeftPren{}gp\optinner{, i}\opRightPren{}} returns a random product +\smath{\mbox{\bf random}\opLeftPren{}gp\optinner{, i}\opRightPren{}} +returns a random product of maximal \smath{i} generators of the permutation group {\it gp}. The value of \smath{i} is 20 by default. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{range}}\opLeftPren{}{\it listOfSegments}\opRightPren{}% +\opdata{{\mbox{\axiomFun{range}}\opLeftPren{} +{\it listOfSegments}\opRightPren{}% }% }% {1}{(List(Segment(Float)))->\$}{DrawOption} @@ -77577,11 +80971,13 @@ as floats or rational numbers. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ranges}}\opLeftPren{}{\it listOfSegments}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ranges}}\opLeftPren{} +{\it listOfSegments}\opRightPren{}% }% }% {1}{(List(Segment(Float)))->\$}{DrawOption} -{\smath{\mbox{\bf ranges}\opLeftPren{}l\opRightPren{}} provides a list of user-specified ranges +{\smath{\mbox{\bf ranges}\opLeftPren{}l\opRightPren{}} provides a +list of user-specified ranges for the \spadfun{draw} command. This command may also be expressed as an option to the \spadfun{draw} command in the form \smath{{\bf ranges} == l}. @@ -77592,9 +80988,11 @@ command in the form \smath{{\bf ranges} == l}. }% }% {1}{(M)->NonNegativeInteger}{MatrixLinearAlgebraFunctions} -{\smath{\mbox{\bf rank}\opLeftPren{}m\opRightPren{}} returns the rank of the matrix \smath{m}. Also: +{\smath{\mbox{\bf rank}\opLeftPren{}m\opRightPren{}} returns the +rank of the matrix \smath{m}. Also: \newitem -\smath{\mbox{\bf rank}\opLeftPren{}A, \allowbreak{} B\opRightPren{}} computes the rank of the complete matrix +\smath{\mbox{\bf rank}\opLeftPren{}A, \allowbreak{} B\opRightPren{}} +computes the rank of the complete matrix \smath{(A|B)} of the linear system \smath{AX = B}. \newitem \smath{\mbox{\bf rank}\opLeftPren{}t\opRightPren{}}, @@ -77607,7 +81005,8 @@ indices). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rarrow}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rarrow}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} @@ -77619,7 +81018,8 @@ creates a form for the mapping \smath{o_1 \rightarrow o_2}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ratDenom}}\opLeftPren{}{\it expression\opt{, option}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ratDenom}}\opLeftPren{} +{\it expression\opt{, option}}\opRightPren{}% }% }% {2}{(F, F)->F}{AlgebraicManipulations} @@ -77636,29 +81036,37 @@ Similarly, if \smath{u} is a list of algebraic kernels % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{rational?}}\opLeftPren{}{\it element}\opRightPren{}% - \optand \mbox{\axiomFun{rationalIfCan}}\opLeftPren{}{\it element}\opRightPren{}% +\optand \mbox{\axiomFun{rationalIfCan}}\opLeftPren{} +{\it element}\opRightPren{}% \opand \mbox{\axiomFun{rational}}\opLeftPren{}{\it element}\opRightPren{}% }% }% {1}{(\$)->Boolean}{ComplexCategory} -{\smath{\mbox{\bf rational?}\opLeftPren{}x\opRightPren{}} tests if \smath{x} is a rational number, +{\smath{\mbox{\bf rational?}\opLeftPren{}x\opRightPren{}} +tests if \smath{x} is a rational number, that is, that it can be converted to type \spadtype{Fraction(Integer)}. Specifically, if \smath{x} is complex, a quaternion, or an octonion, it tests that all imaginary parts are 0. \newitem -\smath{\mbox{\bf rationalIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{x} as a rational number +\smath{\mbox{\bf rationalIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{x} as a rational number if possible, and \mbox{\tt "failed"} if it is not. \newitem -\smath{\mbox{\bf rational}\opLeftPren{}x\opRightPren{}} returns \smath{x} as a rational number if +\smath{\mbox{\bf rational}\opLeftPren{}x\opRightPren{}} +returns \smath{x} as a rational number if possible, and calls \spadfun{error} if it is not. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rationalApproximation}}\opLeftPren{}{\it float}, \allowbreak{}{\it nonNegativeInteger}\allowbreak $\,[$ , \allowbreak{}{\it positiveInteger}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{rationalApproximation}}\opLeftPren{} +{\it float}, \allowbreak{} +{\it nonNegativeInteger}\allowbreak $\,[$ , \allowbreak{} +{\it positiveInteger}$]$\opRightPren{}% }% }% {3}{(\$, NonNegativeInteger, NonNegativeInteger)->Fraction(Integer)}{Float} -{\smath{\mbox{\bf rationalApproximation}\opLeftPren{}f, \allowbreak{} n\optinner{, b}\opRightPren{}} computes a +{\smath{\mbox{\bf rationalApproximation}\opLeftPren{}f, +\allowbreak{} n\optinner{, b}\opRightPren{}} computes a rational approximation \smath{r} to \smath{f} with relative error \smath{< b^{-n}}, that is \smath{|(r-f)/f| < b^{-n}}, for some positive integer base \smath{b}. @@ -77668,22 +81076,28 @@ The first argument \smath{f} is either a float or small float. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rationalFunction}}\opLeftPren{}{\it series}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rationalFunction}}\opLeftPren{} +{\it series}, \allowbreak{}{\it integer}, \allowbreak{} +{\it integer}\opRightPren{}% }% }% -{3}{(\$, Integer, Integer)->Fraction(Polynomial(Coef))}{UnivariateLaurentSeriesCategory} -{\smath{\mbox{\bf rationalFunction}\opLeftPren{}f, \allowbreak{} m, \allowbreak{} n\opRightPren{}} returns a rational function +{3}{(\$, Integer, Integer)->Fraction(Polynomial(Coef))} +{UnivariateLaurentSeriesCategory} +{\smath{\mbox{\bf rationalFunction}\opLeftPren{}f, +\allowbreak{} m, \allowbreak{} n\opRightPren{}} returns a rational function consisting of the sum of all terms of \smath{f} of degree \smath{d} with \smath{m \leq d \leq n}. By default, \smath{n} is the maximum degree of \smath{f}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rationalPoint?}}\opLeftPren{}{\it value}, \allowbreak{}{\it value}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rationalPoint?}}\opLeftPren{} +{\it value}, \allowbreak{}{\it value}\opRightPren{}% }% }% {2}{(F, F)->Boolean}{FunctionFieldCategory} -{\smath{\mbox{\bf rationalPoint?}\opLeftPren{}a, \allowbreak{} b\opRightPren{}\$F} tests if +{\smath{\mbox{\bf rationalPoint?}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}\$F} tests if \smath{(x=a, y=b)} is on the curve defining function field \smath{F}. See \spadtype{FunctionFieldCategory}. @@ -77701,11 +81115,13 @@ See \spadtype{FunctionFieldCategory}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rationalPower}}\opLeftPren{}{\it puiseuxSeries}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rationalPower}}\opLeftPren{} +{\it puiseuxSeries}\opRightPren{}% }% }% {1}{(\$)->Fraction(Integer)}{UnivariatePuiseuxSeriesConstructorCategory} -{\smath{\mbox{\bf rationalPower}\opLeftPren{}f(x)\opRightPren{}} returns \smath{r} where the Puiseux series \smath{f(x) = g(x^r)}. +{\smath{\mbox{\bf rationalPower}\opLeftPren{}f(x)\opRightPren{}} +returns \smath{r} where the Puiseux series \smath{f(x) = g(x^r)}. } % ---------------------------------------------------------------------- @@ -77713,7 +81129,9 @@ See \spadtype{FunctionFieldCategory}. }% }% {1}{(F)->SparseUnivariatePolynomial(F)}{AlgebraicManipulations} -{\smath{\mbox{\bf ratPoly}\opLeftPren{}f\opRightPren{}} returns a polynomial \smath{p} such that \smath{p} has no algebraic coefficients, and \smath{p(f) = 0}. +{\smath{\mbox{\bf ratPoly}\opLeftPren{}f\opRightPren{}} +returns a polynomial \smath{p} such that \smath{p} has no algebraic +coefficients, and \smath{p(f) = 0}. } @@ -77729,20 +81147,27 @@ returns \smath{q} such that } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightDivide}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% -\optand \mbox{\axiomFun{rightQuotient}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% -\opand \mbox{\axiomFun{rightRemainder}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightDivide}}\opLeftPren{} +{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\optand \mbox{\axiomFun{rightQuotient}}\opLeftPren{} +{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opand \mbox{\axiomFun{rightRemainder}}\opLeftPren{} +{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% }% }% -{2}{(\$, \$)->Record(quotient:\$, remainder:\$)}{LinearOrdinaryDifferentialOperator} -{\smath{\mbox{\bf rightDivide}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns the pair \smath{q, r} such that +{2}{(\$, \$)->Record(quotient:\$, remainder:\$)} +{LinearOrdinaryDifferentialOperator} +{\smath{\mbox{\bf rightDivide}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns the pair \smath{q, r} such that \smath{a = qb + r} and the degree of \smath{r} is less than the degree of \smath{b}. The pair is returned as a record with fields \smath{quotient} and \smath{remainder}. This process is called ``right division''. Also: -\smath{\mbox{\bf rightQuotient}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns only \smath{q}. -\smath{\mbox{\bf rightRemainder}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns only \smath{r}. +\smath{\mbox{\bf rightQuotient}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns only \smath{q}. +\smath{\mbox{\bf rightRemainder}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} returns only \smath{r}. } @@ -77752,9 +81177,12 @@ This process is called ``right division''. Also: }% }% {1}{(\$)->S}{FileCategory} -{\smath{\mbox{\bf read!}\opLeftPren{}f\opRightPren{}} extracts a value from file \smath{f}. The state of \smath{f} is modified so a subsequent call to \spadfun{read!} will return the next element. +{\smath{\mbox{\bf read!}\opLeftPren{}f\opRightPren{}} extracts a +value from file \smath{f}. The state of \smath{f} is modified so a subsequent +call to \spadfun{read!} will return the next element. \newitem -\smath{\mbox{\bf readIfCan!}\opLeftPren{}f\opRightPren{}} returns a value from the file \smath{f} +\smath{\mbox{\bf readIfCan!}\opLeftPren{}f\opRightPren{}} returns +a value from the file \smath{f} or \mbox{\tt "failed"} if this is not possible (that is, either \smath{f} is not open for reading, or \smath{f} is at the end of the file). @@ -77765,7 +81193,8 @@ end of the file). }% }% {1}{(\$)->Boolean}{FileNameCategory} -{\smath{\mbox{\bf readable?}\opLeftPren{}f\opRightPren{}} tests if the named file exists and can be opened for reading. +{\smath{\mbox{\bf readable?}\opLeftPren{}f\opRightPren{}} tests +if the named file exists and can be opened for reading. } % ---------------------------------------------------------------------- @@ -77774,12 +81203,15 @@ end of the file). }% }% {1}{(\$)->String}{TextFile} -{\smath{\mbox{\bf readLineIfCan!}\opLeftPren{}f\opRightPren{}} returns a string of the contents +{\smath{\mbox{\bf readLineIfCan!}\opLeftPren{}f\opRightPren{}} +returns a string of the contents of a line from file \smath{f}, or \mbox{\tt "failed"} if this is not possible (if \smath{f} is not readable or is positioned at the end of file). \newitem -\smath{\mbox{\bf readLine!}\opLeftPren{}f\opRightPren{}} returns a string of the contents of a line from the file \smath{f}, and calls \spadfun{error} if this is not possible. +\smath{\mbox{\bf readLine!}\opLeftPren{}f\opRightPren{}} +returns a string of the contents of a line from the file \smath{f}, and calls +\spadfun{error} if this is not possible. } % ---------------------------------------------------------------------- @@ -77788,19 +81220,25 @@ possible (if }% }% {1}{(\$)->R}{ComplexCategory} -{\smath{\mbox{\bf real}\opLeftPren{}x\opRightPren{}} returns real part of \smath{x}. +{\smath{\mbox{\bf real}\opLeftPren{}x\opRightPren{}} returns +real part of \smath{x}. Argument \smath{x} can be an expression or a complex value, quaternion, or octonion. \newitem -\smath{\mbox{\bf real?}\opLeftPren{}f\opRightPren{}} tests if expression \smath{f = real(f)}. +\smath{\mbox{\bf real?}\opLeftPren{}f\opRightPren{}} tests +if expression \smath{f = real(f)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{realEigenvectors}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it float}\opRightPren{}% +\opdata{{\mbox{\axiomFun{realEigenvectors}}\opLeftPren{} +{\it matrix}, \allowbreak{}{\it float}\opRightPren{}% }% }% -{2}{(Matrix(Fraction(Integer)), Par)->List(Record(outval:Par, outmult:Integer, outvect:List(Matrix(Par))))}{NumericRealEigenPackage} -{\smath{\mbox{\bf realEigenvectors}\opLeftPren{}m, \allowbreak{} eps\opRightPren{}} returns a list of records, each +{2}{(Matrix(Fraction(Integer)), Par)-> +List(Record(outval:Par, outmult:Integer, outvect:List(Matrix(Par))))} +{NumericRealEigenPackage} +{\smath{\mbox{\bf realEigenvectors}\opLeftPren{}m, +\allowbreak{} eps\opRightPren{}} returns a list of records, each containing a real eigenvalue, its algebraic multiplicity, and a list of associated eigenvectors. All these results are computed to precision \smath{eps} as floats @@ -77809,40 +81247,62 @@ Argument \smath{m} is a matrix of rational functions. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{realElementary}}\opLeftPren{}{\it expression\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{realElementary}}\opLeftPren{} +{\it expression\opt{, symbol}}\opRightPren{}% }% }% {2}{(F, Symbol)->F}{ElementaryFunctionStructurePackage} -{\smath{\mbox{\bf realElementary}\opLeftPren{}f, \allowbreak{} sy\opRightPren{}} rewrites the kernels of \smath{f} -involving \smath{sy} in terms of the 4 fundamental real transcendental elementary functions: \smath{log, exp, tan, atan}. If \smath{sy} +{\smath{\mbox{\bf realElementary}\opLeftPren{}f, +\allowbreak{} sy\opRightPren{}} rewrites the kernels of \smath{f} +involving \smath{sy} in terms of the 4 fundamental +real transcendental elementary functions: \smath{log, exp, tan, atan}. +If \smath{sy} is omitted, all kernels of \smath{f} are rewritten. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{realRoots}}\opLeftPren{}{\it rationalfunctions}, \allowbreak{}{\it v}\allowbreak $\,[$ , \allowbreak{}{\it w}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{realRoots}}\opLeftPren{} +{\it rationalfunctions}, \allowbreak{}{\it v}\allowbreak $\,[$ , \allowbreak{} +{\it w}$]$\opRightPren{}% }% }% -{3}{(List(Fraction(Polynomial(Fraction(Integer)))), List(Symbol), Par)->List(List(Par))}{FloatingRealPackage} -{\smath{\mbox{\bf realRoots}\opLeftPren{}rf, \allowbreak{} eps\opRightPren{}} finds the real zeros of a univariate rational function +{3}{(List(Fraction(Polynomial(Fraction(Integer)))), List(Symbol), Par)-> +List(List(Par))}{FloatingRealPackage} +{\smath{\mbox{\bf realRoots}\opLeftPren{}rf, +\allowbreak{} eps\opRightPren{}} finds the real zeros of a +univariate rational function \smath{rf} with precision given by eps. \newitem -\smath{\mbox{\bf realRoots}\opLeftPren{}lp, \allowbreak{} lv, \allowbreak{} eps\opRightPren{}} computes the list of the real solutions of the list \smath{lp} of rational functions with rational coefficients with respect to the variables in \smath{lv}, with precision \smath{eps}. Each solution is expressed as a list of numbers in order corresponding to the variables in \smath{lv}. +\smath{\mbox{\bf realRoots}\opLeftPren{}lp, \allowbreak{} lv, +\allowbreak{} eps\opRightPren{}} computes the list of the real solutions +of the list \smath{lp} of rational functions with rational +coefficients with respect to the +variables in \smath{lv}, with precision \smath{eps}. +Each solution is expressed +as a list of numbers in order corresponding to the variables in \smath{lv}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{realZeros}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it rationalNumber}\allowbreak $\,[$ , \allowbreak{}{\it option}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{realZeros}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it rationalNumber}\allowbreak $\,[$ , +\allowbreak{}{\it option}$]$\opRightPren{}% }% }% -{2}{(Pol, Fraction(Integer))->List(Record(left:Fraction(Integer), right:Fraction(Integer)))}{RealZeroPackageQ} -{\smath{\mbox{\bf realZeros}\opLeftPren{}pol\opRightPren{}} returns a list of isolating intervals for all the real zeros of the univariate polynomial \smath{pol}. +{2}{(Pol, Fraction(Integer))->List(Record(left:Fraction(Integer), +right:Fraction(Integer)))}{RealZeroPackageQ} +{\smath{\mbox{\bf realZeros}\opLeftPren{}pol\opRightPren{}} +returns a list of isolating intervals for all the real zeros of the univariate +polynomial \smath{pol}. \newitem -\smath{\mbox{\bf realZeros}\opLeftPren{}pol\optinner{, eps}\opRightPren{}} returns a list of intervals +\smath{\mbox{\bf realZeros}\opLeftPren{}pol\optinner{, eps}\opRightPren{}} +returns a list of intervals of length less than the rational number \smath{eps} for all the real roots of the polynomial \smath{pol}. The default value of \smath{eps} is ???. \newitem -\smath{\mbox{\bf realZeros}\opLeftPren{}pol, \allowbreak{} int\optinner{, eps}\opRightPren{}} returns a list of intervals of +\smath{\mbox{\bf realZeros}\opLeftPren{}pol, +\allowbreak{} int\optinner{, eps}\opRightPren{}} returns a list of intervals of length less than the rational number \smath{eps} for all the real roots of the polynomial \smath{pol} which lie in the interval expressed by the record \smath{int}. @@ -77854,9 +81314,11 @@ The default value of \smath{eps} is ???. }% }% {1}{(\$)->Union(\$, "failed")}{Monoid} -{\smath{\mbox{\bf recip}\opLeftPren{}x\opRightPren{}} returns the multiplicative inverse for \smath{x}, +{\smath{\mbox{\bf recip}\opLeftPren{}x\opRightPren{}} +returns the multiplicative inverse for \smath{x}, or \mbox{\tt "failed"} if no inverse can be found. -\seeAlso{\spadtype{FiniteRankNonAssociativeAlgebra} and\spadtype{MonadWithUnit}} +\seeAlso{\spadtype{FiniteRankNonAssociativeAlgebra} and +\spadtype{MonadWithUnit}} } % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{recur}}\opLeftPren{}{\it function}\opRightPren{}% @@ -77876,18 +81338,23 @@ For related functions, see \spadtype{MappingPackage}. }% }% {0}{()->\$}{Color} -{\smath{\mbox{\bf red}\opLeftPren{}\opRightPren{}} returns the position of the red hue from total hues. +{\smath{\mbox{\bf red}\opLeftPren{}\opRightPren{}} returns +the position of the red hue from total hues. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reduce}}\opLeftPren{}{\it op}, \allowbreak{}{\it aggregate}\allowbreak $\,[$ , \allowbreak{}{\it identity}, \allowbreak{}{\it element}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{reduce}}\opLeftPren{}{\it op}, +\allowbreak{}{\it aggregate}\allowbreak $\,[$ , \allowbreak{}{\it identity}, +\allowbreak{}{\it element}$]$\opRightPren{}% }% }% {2}{((S, S)->S, \$)->S}{Collection} -{\smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} u\optinner{, ident, a}\opRightPren{}} reduces the binary +{\smath{\mbox{\bf reduce}\opLeftPren{}f, +\allowbreak{} u\optinner{, ident, a}\opRightPren{}} reduces the binary operation \smath{f} across \smath{u}. For example, if \smath{u} is \smath{[x_1, x_2, \ldots, x_n]} then -\smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} u\opRightPren{}} returns \smath{f(\ldots f(x_1, x_2), \ldots, x_n)}. +\smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} u\opRightPren{}} +returns \smath{f(\ldots f(x_1, x_2), \ldots, x_n)}. \medbreak An optional identity element of \smath{f} provided as a third argument affects the result @@ -77898,28 +81365,39 @@ otherwise. If \smath{u} has one element and the third argument is given, the value returned is \smath{f(ident, x_1)}. Otherwise \smath{x_1} is returned. -Thus both \smath{\mbox{\bf reduce}\opLeftPren{}+, \allowbreak{} u\opRightPren{}} and \smath{\mbox{\bf reduce}\opLeftPren{}+, \allowbreak{} u, \allowbreak{} 0\opRightPren{}} +Thus both \smath{\mbox{\bf reduce}\opLeftPren{}+, +\allowbreak{} u\opRightPren{}} and \smath{\mbox{\bf reduce}\opLeftPren{}+, +\allowbreak{} u, \allowbreak{} 0\opRightPren{}} return \smath{\sum\nolimits_{i=1}^n{x_i}}. -Similarly, \smath{\mbox{\bf reduce}\opLeftPren{}*, \allowbreak{} u\opRightPren{}} and \smath{\mbox{\bf reduce}\opLeftPren{}*, \allowbreak{} u, \allowbreak{} 1\opRightPren{}} +Similarly, \smath{\mbox{\bf reduce}\opLeftPren{}*, +\allowbreak{} u\opRightPren{}} and +\smath{\mbox{\bf reduce}\opLeftPren{}*, \allowbreak{} u, +\allowbreak{} 1\opRightPren{}} return \smath{\prod\nolimits_{i=1}^n{x_i}}. \medbreak An optional fourth argument \smath{z} acts as an ``absorbing element''. -\smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} u, \allowbreak{} x, \allowbreak{} z\opRightPren{}} reduces the binary operation \smath{f} across \smath{u}, +\smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} u, \allowbreak{} x, +\allowbreak{} z\opRightPren{}} reduces the binary operation \smath{f} across +\smath{u}, stopping when an ``absorbing element'' \smath{z} is encountered. -For example \smath{\mbox{\bf reduce}\opLeftPren{}or, \allowbreak{} u, \allowbreak{} false, \allowbreak{} true\opRightPren{}} will +For example \smath{\mbox{\bf reduce}\opLeftPren{}or, \allowbreak{} u, +\allowbreak{} false, \allowbreak{} true\opRightPren{}} will stop iterating across \smath{u} returning \smath{true} as soon as an \smath{x_i = true} is found. Note: if \smath{u} has one element \smath{x}, -\smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} u\opRightPren{}} returns \smath{x}, or calls \spadfun{error} if \smath{u} is empty. +\smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} u\opRightPren{}} +returns \smath{x}, or calls \spadfun{error} if \smath{u} is empty. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reduceBasisAtInfinity}}\opLeftPren{}{\it basis}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reduceBasisAtInfinity}}\opLeftPren{} +{\it basis}\opRightPren{}% }% }% {1}{(Vector(\$))->Vector(\$)}{FunctionFieldCategory} -{\smath{\mbox{\bf reduceBasisAtInfinity}\opLeftPren{}b_1, \allowbreak{} \ldots, b_n\opRightPren{}}, +{\smath{\mbox{\bf reduceBasisAtInfinity}\opLeftPren{}b_1, +\allowbreak{} \ldots, b_n\opRightPren{}}, where the \smath{b_i} are functions on a fixed curve, returns \smath{(x^i \, b_j)} for all \smath{i}, \smath{j} such that \smath{x^i \, b_j} is locally @@ -77928,55 +81406,76 @@ integral at infinity. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reducedContinuedFraction}}\opLeftPren{}{\it element}, \allowbreak{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reducedContinuedFraction}}\opLeftPren{} +{\it element}, \allowbreak{}{\it stream}\opRightPren{}% }% }% {2}{(R, Stream(R))->\$}{ContinuedFraction} -{\smath{\mbox{\bf reducedContinuedFraction}\opLeftPren{}b_0, \allowbreak{} b\opRightPren{}} returns a +{\smath{\mbox{\bf reducedContinuedFraction}\opLeftPren{}b_0, +\allowbreak{} b\opRightPren{}} returns a continued fraction constructed as follows. If \smath{b = [b_1, b_2, \ldots]} then the result is the continued fraction \smath{b_0 + 1/(b_1 + 1/(b_2 + \cdots))}. That is, the result is the same as -\smath{\mbox{\bf continuedFraction}\opLeftPren{}b_0, \allowbreak{} [1, \allowbreak{} 1, \allowbreak{} 1, \allowbreak{} \ldots], [b_1, b_2, b_3, \ldots]\opRightPren{}}. +\smath{\mbox{\bf continuedFraction}\opLeftPren{}b_0, \allowbreak{} +[1, \allowbreak{} 1, \allowbreak{} 1, \allowbreak{} \ldots], +[b_1, b_2, b_3, \ldots]\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reducedForm}}\opLeftPren{}{\it continuedFraction}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reducedForm}}\opLeftPren{} +{\it continuedFraction}\opRightPren{}% }% }% {1}{(\$)->\$}{ContinuedFraction} -{\smath{\mbox{\bf reducedForm}\opLeftPren{}x\opRightPren{}} puts the continued fraction \smath{x} +{\smath{\mbox{\bf reducedForm}\opLeftPren{}x\opRightPren{}} +puts the continued fraction \smath{x} in reduced form, that is, the function returns an equivalent continued fraction of the -form \smath{\mbox{\bf continuedFraction}\opLeftPren{}b_0, \allowbreak{} [1, \allowbreak{} 1, \allowbreak{} 1, \allowbreak{} \ldots], [b_1, b_2, b_3, \ldots]\opRightPren{}}. +form \smath{\mbox{\bf continuedFraction}\opLeftPren{}b_0, +\allowbreak{} [1, \allowbreak{} 1, \allowbreak{} 1, \allowbreak{} \ldots], +[b_1, b_2, b_3, \ldots]\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reducedSystem}}\opLeftPren{}{\it matrix\opt{, vector}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reducedSystem}}\opLeftPren{} +{\it matrix\opt{, vector}}\opRightPren{}% }% }% {1}{(Matrix(\$))->Matrix(R)}{LinearlyExplicitRingOver} -{\smath{\mbox{\bf reducedSystem}\opLeftPren{}A, \allowbreak{} v\opRightPren{}} returns a matrix \smath{B} such that \smath{A x = v} and \smath{B x = v} have the same solutions. +{\smath{\mbox{\bf reducedSystem}\opLeftPren{}A, +\allowbreak{} v\opRightPren{}} returns a matrix \smath{B} +such that \smath{A x = v} +and \smath{B x = v} have the same solutions. By default, \smath{v = 0}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reductum}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reductum}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(\$)->\$}{AbelianMonoidRing} -{\smath{\mbox{\bf reductum}\opLeftPren{}p\opRightPren{}} returns polynomial \smath{p} minus its leading +{\smath{\mbox{\bf reductum}\opLeftPren{}p\opRightPren{}} +returns polynomial \smath{p} minus its leading monomial, or zero if handed the zero element. -\seeAlso{\spadtype{IndexedDirectProdcutCategory} and \spadtype{MonogenicLinearOperator}} +\seeAlso{\spadtype{IndexedDirectProdcutCategory} and +\spadtype{MonogenicLinearOperator}} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{refine}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it interval}, \allowbreak{}{\it precision}\opRightPren{}% +\opdata{{\mbox{\axiomFun{refine}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it interval}, \allowbreak{}{\it precision}\opRightPren{}% }% }% -{3}{(Pol, Record(left:Fraction(Integer), right:Fraction(Integer)), Fraction(Integer))->Record(left:Fraction(Integer), right:Fraction(Integer))}{RealZeroPackageQ} -{\smath{\mbox{\bf refine}\opLeftPren{}pol, \allowbreak{} int, \allowbreak{} tolerance\opRightPren{}} refines the interval \smath{int} containing exactly one root of the univariate polynomial \smath{pol} to size less than +{3}{(Pol, Record(left:Fraction(Integer), right:Fraction(Integer)), +Fraction(Integer))->Record(left:Fraction(Integer), right:Fraction(Integer))} +{RealZeroPackageQ} +{\smath{\mbox{\bf refine}\opLeftPren{}pol, \allowbreak{} int, +\allowbreak{} tolerance\opRightPren{}} refines the interval +\smath{int} containing +exactly one root of the univariate polynomial \smath{pol} to size less than the indicated \smath{tolerance}. Argument \smath{int} is an interval denoted by a record with selectors \smath{left} and \smath{right}, each with rational number values. @@ -77986,11 +81485,15 @@ isolating interval exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{regularRepresentation}}\opLeftPren{}{\it element}, \allowbreak{}{\it basis}\opRightPren{}% +\opdata{{\mbox{\axiomFun{regularRepresentation}}\opLeftPren{} +{\it element}, \allowbreak{}{\it basis}\opRightPren{}% }% }% {2}{(\$, Vector(\$))->Matrix(R)}{FiniteRankAlgebra} -{\smath{\mbox{\bf regularRepresentation}\opLeftPren{}a, \allowbreak{} basis\opRightPren{}} returns the matrix of the linear map defined by left multiplication by \smath{a} with respect to basis \smath{basis}. +{\smath{\mbox{\bf regularRepresentation}\opLeftPren{}a, +\allowbreak{} basis\opRightPren{}} returns the matrix of the linear map +defined by left multiplication by \smath{a} with respect +to basis \smath{basis}. Element \smath{a} is a complex element or an element of a domain \smath{R} of category \spadtype{FramedAlgebra}. @@ -77998,11 +81501,14 @@ The second argument may be omitted when a fixed basis is defined for \smath{R}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reindex}}\opLeftPren{}{\it cartesianTensor}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reindex}}\opLeftPren{} +{\it cartesianTensor}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% }% }% {2}{(\$, List(Integer))->\$}{CartesianTensor} -{\smath{\mbox{\bf reindex}\opLeftPren{}t, \allowbreak{} [i_1, \allowbreak{} \ldots, i_{\rm dim}]\opRightPren{}} permutes the indices of +{\smath{\mbox{\bf reindex}\opLeftPren{}t, \allowbreak{} +[i_1, \allowbreak{} \ldots, i_{\rm dim}]\opRightPren{}} +permutes the indices of cartesian tensor \smath{t}. For example, if \smath{r = {\bf reindex}(t, [4, 1, 2, 3])} for a rank 4 tensor \smath{t}, then \smath{r} is the rank 4 tensor @@ -78010,15 +81516,19 @@ given by \smath{r(i, j, k, l) = t(l, i, j, k)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{relationsIdeal}}\opLeftPren{}{\it listOfPolynomials}\opRightPren{}% +\opdata{{\mbox{\axiomFun{relationsIdeal}}\opLeftPren{} +{\it listOfPolynomials}\opRightPren{}% }% }% -{1}{(List(DPoly))->SuchThat(List(Polynomial(F)), List(Equation(Polynomial(F))))}{PolynomialIdeals} -{\smath{\mbox{\bf relationsIdeal}\opLeftPren{}polyList\opRightPren{}} returns the ideal of relations among the polynomials in \smath{polyList}. +{1}{(List(DPoly))->SuchThat(List(Polynomial(F)), +List(Equation(Polynomial(F))))}{PolynomialIdeals} +{\smath{\mbox{\bf relationsIdeal}\opLeftPren{}polyList\opRightPren{}} +returns the ideal of relations among the polynomials in \smath{polyList}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{relerror}}\opLeftPren{}{\it float}, \allowbreak{}{\it float}\opRightPren{}% +\opdata{{\mbox{\axiomFun{relerror}}\opLeftPren{}{\it float}, \allowbreak{} +{\it float}\opRightPren{}% }% }% {2}{(\$, \$)->Integer}{Float} @@ -78029,7 +81539,8 @@ when \smath{y \not= 0}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rem}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rem}}\opLeftPren{}{\it element}, \allowbreak{} +{\it element}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{NonNegativeInteger} @@ -78038,21 +81549,27 @@ when \smath{y \not= 0}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{remove}}\opLeftPren{}{\it predicate}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{remove}}\opLeftPren{}{\it predicate}, +\allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->\$}{ExtensibleLinearAggregate} {\opkey{Argument \smath{u} is any extensible aggregate such as a list.} \newitem -\smath{\mbox{\bf remove}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} returns a copy of \smath{u} removing all elements \smath{x} such that \smath{p(x)} is \smath{true}. +\smath{\mbox{\bf remove}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} +returns a copy of \smath{u} removing all elements \smath{x} such that +\smath{p(x)} is \smath{true}. Argument \smath{u} may be any homogeneous aggregate including infinite streams. Note: for lists and streams, \code{remove(p, u) == [x for x in u | not p(x)]}. \newitem -\smath{\mbox{\bf remove!}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} destructively removes all elements \smath{x} of \smath{u} such that \smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} is \smath{true}. +\smath{\mbox{\bf remove!}\opLeftPren{}pred, \allowbreak{} u\opRightPren{}} +destructively removes all elements \smath{x} of \smath{u} such that +\smath{\mbox{\bf pred}\opLeftPren{}x\opRightPren{}} is \smath{true}. The value of \smath{u} after all such elements are removed is returned. \newitem -\smath{\mbox{\bf remove!}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} destructively removes all values \smath{x} from \smath{u}. +\smath{\mbox{\bf remove!}\opLeftPren{}x, \allowbreak{} u\opRightPren{}} +destructively removes all values \smath{x} from \smath{u}. \newitem \smath{\mbox{\bf remove!}\opLeftPren{}k, \allowbreak{} t\opRightPren{}}, where \smath{t} is a keyed dictionary, @@ -78062,46 +81579,63 @@ If \smath{t} has no such key, it returns \mbox{\tt "failed"}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{removeCoshSq}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{removeCoshSq}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf removeCoshSq}\opLeftPren{}f\opRightPren{}} converts every \smath{\cosh(u)^2} -appearing in \smath{f} into \smath{1 - \sinh(x)^2}, and also reduces higher powers of \smath{\mbox{\bf cosh}\opLeftPren{}u\opRightPren{}} with that formula. +{\smath{\mbox{\bf removeCoshSq}\opLeftPren{}f\opRightPren{}} +converts every \smath{\cosh(u)^2} +appearing in \smath{f} into \smath{1 - \sinh(x)^2}, and also +reduces higher powers of \smath{\mbox{\bf cosh}\opLeftPren{}u\opRightPren{}} +with that formula. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{removeDuplicates}}\opLeftPren{}{\it aggregate}\opRightPren{}% - \opand \mbox{\axiomFun{removeDuplicates!}}\opLeftPren{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{removeDuplicates}}\opLeftPren{} +{\it aggregate}\opRightPren{}% + \opand \mbox{\axiomFun{removeDuplicates!}}\opLeftPren{} +{\it aggregate}\opRightPren{}% }% }% {1}{(\$)->\$}{Collection} -{\smath{\mbox{\bf removeDuplicates}\opLeftPren{}u\opRightPren{}} returns a copy of \smath{u} with all duplicates removed. +{\smath{\mbox{\bf removeDuplicates}\opLeftPren{}u\opRightPren{}} +returns a copy of \smath{u} with all duplicates removed. \newitem -\smath{\mbox{\bf removeDuplicates!}\opLeftPren{}u\opRightPren{}} destructively removes duplicates from \smath{u}. +\smath{\mbox{\bf removeDuplicates!}\opLeftPren{}u\opRightPren{}} +destructively removes duplicates from \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{removeSinhSq}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{removeSinhSq}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf removeSinhSq}\opLeftPren{}f\opRightPren{}} converts every \smath{\sinh(u)^2} -appearing in \smath{f} into \smath{1 - \cosh(x)^2}, and also reduces higher powers of \smath{\mbox{\bf sinh}\opLeftPren{}u\opRightPren{}} with that formula. +{\smath{\mbox{\bf removeSinhSq}\opLeftPren{}f\opRightPren{}} +converts every \smath{\sinh(u)^2} +appearing in \smath{f} into \smath{1 - \cosh(x)^2}, and also +reduces higher powers of \smath{\mbox{\bf sinh}\opLeftPren{}u\opRightPren{}} +with that formula. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{removeSinSq}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{removeSinSq}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf removeSinSq}\opLeftPren{}f\opRightPren{}} converts every \smath{\sin(u)^2} -appearing in \smath{f} into \smath{1 - \cos(x)^2}, and also reduces higher powers of \smath{\mbox{\bf sin}\opLeftPren{}u\opRightPren{}} with that formula. +{\smath{\mbox{\bf removeSinSq}\opLeftPren{}f\opRightPren{}} +converts every \smath{\sin(u)^2} +appearing in \smath{f} into \smath{1 - \cos(x)^2}, and also +reduces higher powers of \smath{\mbox{\bf sin}\opLeftPren{}u\opRightPren{}} +with that formula. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{removeZeroes}}\opLeftPren{}{\it \optinit{integer, }laurentSeries}\opRightPren{}% +\opdata{{\mbox{\axiomFun{removeZeroes}}\opLeftPren{} +{\it \optinit{integer, }laurentSeries}\opRightPren{}% }% }% {2}{(Integer, \$)->\$}{UnivariateLaurentSeriesConstructorCategory} @@ -78111,44 +81645,56 @@ If no integer \smath{n} is given, all leading zeroes are removed. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reopen!}}\opLeftPren{}{\it file}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reopen!}}\opLeftPren{} +{\it file}, \allowbreak{}{\it string}\opRightPren{}% }% }% {2}{(\$, String)->\$}{FileCategory} -{\smath{\mbox{\bf reopen!}\opLeftPren{}f, \allowbreak{} mode\opRightPren{}} returns a file \smath{f} -reopened for operation in the indicated mode: \mbox{\tt "input"} or \mbox{\tt "output"}. +{\smath{\mbox{\bf reopen!}\opLeftPren{}f, \allowbreak{} mode\opRightPren{}} +returns a file \smath{f} +reopened for operation in the indicated mode: \mbox{\tt "input"} or +\mbox{\tt "output"}. For example, -\smath{\mbox{\bf reopen!}\opLeftPren{}f, \allowbreak{} \mbox{\tt "input"}\opRightPren{}} will reopen the file \smath{f} for input. +\smath{\mbox{\bf reopen!}\opLeftPren{}f, \allowbreak{} +\mbox{\tt "input"}\opRightPren{}} will reopen the file \smath{f} for input. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{repeating}}\opLeftPren{}{\it listOfElements\opt{, stream}}\opRightPren{}% - \opand \mbox{\axiomFun{repeating?}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{repeating}}\opLeftPren{} +{\it listOfElements\opt{, stream}}\opRightPren{}% + \opand \mbox{\axiomFun{repeating?}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(List(S))->\$}{Stream} -{\smath{\mbox{\bf repeating}\opLeftPren{}l\opRightPren{}} is a repeating stream whose period is the list \smath{l}. +{\smath{\mbox{\bf repeating}\opLeftPren{}l\opRightPren{}} +is a repeating stream whose period is the list \smath{l}. \newitem -\smath{\mbox{\bf repeating?}\opLeftPren{}l, \allowbreak{} s\opRightPren{}} tests if a stream \smath{s} is periodic +\smath{\mbox{\bf repeating?}\opLeftPren{}l, +\allowbreak{} s\opRightPren{}} tests if a stream \smath{s} is periodic with period \smath{l}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{replace}}\opLeftPren{}{\it string}, \allowbreak{}{\it segment}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{replace}}\opLeftPren{}{\it string}, +\allowbreak{}{\it segment}, \allowbreak{}{\it string}\opRightPren{}% }% }% {3}{(\$, UniversalSegment(Integer), \$)->\$}{StringAggregate} -{\smath{\mbox{\bf replace}\opLeftPren{}s, \allowbreak{} i..j, \allowbreak{} t\opRightPren{}} replaces the substring \smath{s(i..j)} +{\smath{\mbox{\bf replace}\opLeftPren{}s, \allowbreak{} i..j, +\allowbreak{} t\opRightPren{}} replaces the substring \smath{s(i..j)} of \smath{s} by string \smath{t}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{represents}}\opLeftPren{}{\it listOfElements\opt{, listOfBasisElements}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{represents}}\opLeftPren{} +{\it listOfElements\opt{, listOfBasisElements}}\opRightPren{}% }% }% {2}{(Vector(R), Vector(\$))->\$}{FiniteRankAlgebra} -{\smath{\mbox{\bf represents}\opLeftPren{}[a^1, \allowbreak{} .., \allowbreak{} a^n]\optinner{, [v^1, .., v^n]}\opRightPren{}} +{\smath{\mbox{\bf represents}\opLeftPren{}[a^1, \allowbreak{} .., +\allowbreak{} a^n]\optinner{, [v^1, .., v^n]}\opRightPren{}} returns \smath{a^1 v^1 + \cdots + a^n v^n}. Arguments \smath{v_i} are elements of a domain of category \spadtype{FiniteRankAlgebra} or @@ -78168,7 +81714,8 @@ and \spadtype{FiniteAlgebraicExtensionField}. }% }% {0}{()->Void}{Symbol} -{\smath{\mbox{\bf resetNew}\opLeftPren{}\opRightPren{}} resets the internal counter that \smath{\mbox{\bf new}\opLeftPren{}\opRightPren{}} uses. +{\smath{\mbox{\bf resetNew}\opLeftPren{}\opRightPren{}} resets the +internal counter that \smath{\mbox{\bf new}\opLeftPren{}\opRightPren{}} uses. } % ---------------------------------------------------------------------- @@ -78176,27 +81723,37 @@ and \spadtype{FiniteAlgebraicExtensionField}. }% }% {0}{()->Void}{UserDefinedVariableOrdering} -{\smath{\mbox{\bf resetVariableOrder}\opLeftPren{}\opRightPren{}} cancels any previous use of \spadfun{setVariableOrder} +{\smath{\mbox{\bf resetVariableOrder}\opLeftPren{}\opRightPren{}} +cancels any previous use of \spadfun{setVariableOrder} and returns to the default system ordering. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rest}}\opLeftPren{}{\it aggregate\opt{, nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rest}}\opLeftPren{} +{\it aggregate\opt{, nonNegativeInteger}}\opRightPren{}% }% }% {1}{(\$)->\$}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf rest}\opLeftPren{}u\opRightPren{}} returns an aggregate consisting of all but the first element of \smath{u} (equivalently, the next node of \smath{u}). +{\smath{\mbox{\bf rest}\opLeftPren{}u\opRightPren{}} returns an aggregate +consisting of all but the first element of \smath{u} (equivalently, the +next node of \smath{u}). \newitem -\smath{\mbox{\bf rest}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns the \eth{\smath{n}} node of \smath{u}. +\smath{\mbox{\bf rest}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} +returns the \eth{\smath{n}} node of \smath{u}. Note: \smath{\mbox{\bf rest}\opLeftPren{}u, 0) = u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{resultant}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynoial}\allowbreak $\,[$ , \allowbreak{}{\it variable}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{resultant}}\opLeftPren{} +{\it polynomial}, \allowbreak{} +{\it polynoial}\allowbreak $\,[$ , \allowbreak{} +{\it variable}$]$\opRightPren{}% }% }% {3}{(\$, \$, VarSet)->\$}{PolynomialCategory} -{\smath{\mbox{\bf resultant}\opLeftPren{}p, \allowbreak{} q, \allowbreak{} v\opRightPren{}} returns the resultant of the polynomials \smath{p} and \smath{q} with respect to the variable \smath{v}. +{\smath{\mbox{\bf resultant}\opLeftPren{}p, \allowbreak{} q, +\allowbreak{} v\opRightPren{}} returns the resultant of the polynomials +\smath{p} and \smath{q} with respect to the variable \smath{v}. If \smath{p} and \smath{q} are univariate polynomials, the variable \smath{v} defaults to the unique variable. } @@ -78207,36 +81764,45 @@ variable \smath{v} defaults to the unique variable. }% }% {1}{(\$)->Union(S, "failed")}{RetractableTo} -{\smath{\mbox{\bf retractIfCan}\opLeftPren{}a\opRightPren{}}\smath{@S} returns \smath{a} as an object +{\smath{\mbox{\bf retractIfCan}\opLeftPren{}a\opRightPren{}}\smath{@S} +returns \smath{a} as an object of type \smath{S}, or \mbox{\tt "failed"} if this is not possible. \newitem -\smath{\mbox{\bf retract}\opLeftPren{}a\opRightPren{}}\smath{@S} transforms \smath{a} into an element of \smath{S}, +\smath{\mbox{\bf retract}\opLeftPren{}a\opRightPren{}}\smath{@S} +transforms \smath{a} into an element of \smath{S}, or calls \spadfun{error} if this is not possible. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{retractable?}}\opLeftPren{}{\it typeAnyObject}\opRightPren{}% +\opdata{{\mbox{\axiomFun{retractable?}}\opLeftPren{} +{\it typeAnyObject}\opRightPren{}% }% }% {1}{(Any)->Boolean}{AnyFunctions1} -{\smath{\mbox{\bf retractable?}\opLeftPren{}a\opRightPren{}\$S} tests if object \smath{a} +{\smath{\mbox{\bf retractable?}\opLeftPren{}a\opRightPren{}\$S} +tests if object \smath{a} of type \spadtype{Any} can be converted into an object of type \smath{S}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{reverse}}\opLeftPren{}{\it linearAggregate}\opRightPren{}% - \opand \mbox{\axiomFun{reverse!}}\opLeftPren{}{\it linearAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{reverse}}\opLeftPren{} +{\it linearAggregate}\opRightPren{}% + \opand \mbox{\axiomFun{reverse!}}\opLeftPren{} +{\it linearAggregate}\opRightPren{}% }% }% {1}{(\$)->\$}{FiniteLinearAggregate} -{\smath{\mbox{\bf reverse}\opLeftPren{}a\opRightPren{}} returns a copy of linear aggregate \smath{a} with elements in reverse order. +{\smath{\mbox{\bf reverse}\opLeftPren{}a\opRightPren{}} returns a +copy of linear aggregate \smath{a} with elements in reverse order. \newitem -\smath{\mbox{\bf reverse!}\opLeftPren{}a\opRightPren{}} destructively puts the elements of +\smath{\mbox{\bf reverse!}\opLeftPren{}a\opRightPren{}} destructively +puts the elements of linear aggregate \smath{a} in reverse order. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightGcd}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightGcd}}\opLeftPren{}{\it lodOperator}, +\allowbreak{}{\it lodOperator}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{LinearOrdinaryDifferentialOperator} @@ -78248,16 +81814,19 @@ and \smath{bb}. The value \smath{g} is computed using right-division. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rhs}}\opLeftPren{}{\it rewriteRuleOrEquation}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rhs}}\opLeftPren{} +{\it rewriteRuleOrEquation}\opRightPren{}% }% }% {1}{(\$)->F}{RewriteRule} -{\smath{\mbox{\bf rhs}\opLeftPren{}u\opRightPren{}} returns the right-hand side of the rewrite rule +{\smath{\mbox{\bf rhs}\opLeftPren{}u\opRightPren{}} +returns the right-hand side of the rewrite rule or equation \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{right}}\opLeftPren{}{\it binaryRecursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{right}}\opLeftPren{} +{\it binaryRecursiveAggregate}\opRightPren{}% }% }% {1}{(\$)->\$}{BinaryRecursiveAggregate} @@ -78272,35 +81841,40 @@ or equation \smath{u}. {See \spadfun{leftAlternative?}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightCharacteristicPolynomial}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightCharacteristicPolynomial}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->SparseUnivariatePolynomial(R)}{FiniteRankNonAssociativeAlgebra} {See \spadfun{leftCharacteristicPolynomial}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightDiscriminant}}\opLeftPren{}{\it basis}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightDiscriminant}}\opLeftPren{} +{\it basis}\opRightPren{}% }% }% {0}{()->R}{FramedNonAssociativeAlgebra} {See \spadfun{leftDiscriminant}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightMinimalPolynomial}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightMinimalPolynomial}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->SparseUnivariatePolynomial(R)}{FiniteRankNonAssociativeAlgebra} {See \spadfun{leftMinimalPolynomial}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightNorm}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightNorm}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->R}{FiniteRankNonAssociativeAlgebra} {See \spadfun{leftNorm}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightPower}}\opLeftPren{}{\it monad}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightPower}}\opLeftPren{} +{\it monad}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->\$}{MonadWithUnit} @@ -78311,7 +81885,8 @@ or equation \smath{u}. \opdata{{\mbox{\axiomFun{rightRankPolynomial}}\opLeftPren{}\opRightPren{}% }% }% -{0}{()->SparseUnivariatePolynomial(Polynomial(R))}{FramedNonAssociativeAlgebra} +{0}{()->SparseUnivariatePolynomial(Polynomial(R))} +{FramedNonAssociativeAlgebra} {See \spadfun{leftRankPolynomial}.} % ---------------------------------------------------------------------- @@ -78322,28 +81897,32 @@ or equation \smath{u}. {See \spadfun{leftRank}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightRecip}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightRecip}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->Union(\$, "failed")}{MonadWithUnit} {See \spadfun{leftRecip}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightRegularRepresentation}}\opLeftPren{}{\it element\opt{, basis}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightRegularRepresentation}}\opLeftPren{} +{\it element\opt{, basis}}\opRightPren{}% }% }% {1}{(\$)->Matrix(R)}{FramedNonAssociativeAlgebra} {See \spadfun{leftRegularRepresentation}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightTraceMatrix}}\opLeftPren{}{\it \opt{basis}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightTraceMatrix}}\opLeftPren{} +{\it \opt{basis}}\opRightPren{}% }% }% {1}{(Vector(\$))->Matrix(R)}{FiniteRankNonAssociativeAlgebra} {See \spadfun{leftTraceMatrix}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightTrim}}\opLeftPren{}{\it string}, \allowbreak{}{\it various}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightTrim}}\opLeftPren{}{\it string}, +\allowbreak{}{\it various}\opRightPren{}% }% }% {2}{(\$, Character)->\$}{StringAggregate} @@ -78357,11 +81936,14 @@ or equation \smath{u}. {See \spadfun{leftUnits}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rischNormalize}}\opLeftPren{}{\it expression}, \allowbreak{}{\it x}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rischNormalize}}\opLeftPren{}{\it expression}, +\allowbreak{}{\it x}\opRightPren{}% }% }% -{2}{(F, Symbol)->Record(func:F, kers:List(Kernel(F)), vals:List(F))}{ElementaryFunctionStructurePackage} -{\smath{\mbox{\bf rischNormalize}\opLeftPren{}f, \allowbreak{} x\opRightPren{}} returns \smath{[g, [k_1, \ldots, +{2}{(F, Symbol)->Record(func:F, kers:List(Kernel(F)), vals:List(F))} +{ElementaryFunctionStructurePackage} +{\smath{\mbox{\bf rischNormalize}\opLeftPren{}f, +\allowbreak{} x\opRightPren{}} returns \smath{[g, [k_1, \ldots, k_n], [h_1, \ldots, h_n]]} such that \smath{g = {\bf normalize}(f, x)} and each \smath{k_i} was rewritten as \smath{h_i} during the normalization. @@ -78369,32 +81951,44 @@ normalization. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rightLcm}}\opLeftPren{}{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rightLcm}}\opLeftPren{} +{\it lodOperator}, \allowbreak{}{\it lodOperator}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{LinearOrdinaryDifferentialOperator} {\smath{\mbox{\bf rightLcm}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}, where \smath{a} and \smath{b} are linear ordinary differential operators, -computes the value \smath{m} of lowest degree such that \smath{m = aa*a = bb*b} -for some values \smath{aa} and \smath{bb}. The value \smath{m} is computed using right-division. +computes the value \smath{m} of lowest degree +such that \smath{m = aa*a = bb*b} +for some values \smath{aa} and \smath{bb}. +The value \smath{m} is computed using right-division. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{roman}}\opLeftPren{}{\it integerOrSymbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{roman}}\opLeftPren{} +{\it integerOrSymbol}\opRightPren{}% }% }% {1}{(Integer)->\$}{RomanNumeral} -{\smath{\mbox{\bf roman}\opLeftPren{}x\opRightPren{}} creates a roman numeral for integer or symbol \smath{x}. +{\smath{\mbox{\bf roman}\opLeftPren{}x\opRightPren{}} creates a roman +numeral for integer or symbol \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{romberg}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it threeIntegers}\opRightPren{}% -\optand \mbox{\axiomFun{rombergOpen}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% - \opand \mbox{\axiomFun{rombergClose}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{romberg}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it threeIntegers} +\opRightPren{}% +\optand \mbox{\axiomFun{rombergOpen}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% + \opand \mbox{\axiomFun{rombergClose}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf romberg}\opLeftPren{}fn, \allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, \allowbreak{} nmin, \allowbreak{} nmax, \allowbreak{} nint\opRightPren{}} +{\smath{\mbox{\bf romberg}\opLeftPren{}fn, \allowbreak{} a, +\allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, +\allowbreak{} nmin, +\allowbreak{} nmax, \allowbreak{} nint\opRightPren{}} uses an adaptive romberg method to numerically integrate function \smath{fn} over the closed interval from \smath{a} to \smath{b}, with relative accuracy \smath{epsrel} and absolute accuracy @@ -78418,19 +82012,24 @@ function evaluations, and {\tt success}, a boolean value that is \smath{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details. -\bigitem\smath{\mbox{\bf rombergClosed}\opLeftPren{}fn, \allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, \allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} +\bigitem\smath{\mbox{\bf rombergClosed}\opLeftPren{}fn, +\allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, +\allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} similarly uses the Romberg method to numerically integrate function \smath{fn} over the closed interval \smath{a} to \smath{b}, but is not adaptive. -\bigitem\smath{\mbox{\bf rombergOpen}\opLeftPren{}fn, \allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, \allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} is similar +\bigitem\smath{\mbox{\bf rombergOpen}\opLeftPren{}fn, +\allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, +\allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} is similar to \spadfun{rombergClosed}, except that it integrates function \smath{fn} over the open interval from \smath{a} to \smath{b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{root}}\opLeftPren{}{\it outputForm\opt{, positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{root}}\opLeftPren{} +{\it outputForm\opt{, positiveInteger}}\opRightPren{}% }% }% {}{}{} @@ -78444,22 +82043,26 @@ By default, \smath{n = 2}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rootOfIrreduciblePoly}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rootOfIrreduciblePoly}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% {1}{(SparseUnivariatePolynomial(GF))->F}{FiniteFieldPolynomialPackage2} -{\smath{\mbox{\bf rootOfIrreduciblePoly}\opLeftPren{}f\opRightPren{}} computes one root of the monic, +{\smath{\mbox{\bf rootOfIrreduciblePoly}\opLeftPren{}f\opRightPren{}} +computes one root of the monic, irreducible polynomial \smath{f}, whose degree must divide the extension degree of \smath{F} over \smath{GF}. That is, \smath{f} splits into linear factors over \smath{F}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rootOf}}\opLeftPren{}{\it polynomial\opt{, variable}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rootOf}}\opLeftPren{} +{\it polynomial\opt{, variable}}\opRightPren{}% }% }% {1}{(Polynomial(\$))->\$}{AlgebraicallyClosedField} -{\smath{\mbox{\bf rootOf}\opLeftPren{}p\optinner{, y}\opRightPren{}} returns \smath{y} such that +{\smath{\mbox{\bf rootOf}\opLeftPren{}p\optinner{, y}\opRightPren{}} +returns \smath{y} such that \smath{p(y) = 0}. The object returned displays as \smath{'y}. The second argument may be omitted when \smath{p} is a polynomial in a unique @@ -78468,21 +82071,25 @@ variable \smath{y}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rootSimp}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rootSimp}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{AlgebraicManipulations} -{\smath{\mbox{\bf rootSimp}\opLeftPren{}f\opRightPren{}} transforms every radical of the form +{\smath{\mbox{\bf rootSimp}\opLeftPren{}f\opRightPren{}} +transforms every radical of the form \smath{(a b^{q n+r})^{1/n}} appearing in expression \smath{f} into \smath{b^q (a b^r)^{1/n}}. This transformation is not in general valid for all complex numbers \smath{b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rootsOf}}\opLeftPren{}{\it polynomialOrExpression \opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rootsOf}}\opLeftPren{} +{\it polynomialOrExpression \opt{, symbol}}\opRightPren{}% }% }% -{2}{(SparseUnivariatePolynomial(\$), Symbol)->List(\$)}{AlgebraicallyClosedField} +{2}{(SparseUnivariatePolynomial(\$), Symbol)->List(\$)} +{AlgebraicallyClosedField} {\smath{\mbox{\bf rootsOf}\opLeftPren{}p \optinner{, y}\opRightPren{}} returns the value of \smath{[y_1, \ldots, y_n]} such that \smath{p(y_i) = 0}. The \smath{y_i} are symbols of the form \%\smath{y} with a suffix number @@ -78493,11 +82100,13 @@ exactly one symbol. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rootSplit}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rootSplit}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{AlgebraicManipulations} -{\smath{\mbox{\bf rootSplit}\opLeftPren{}f\opRightPren{}} transforms every radical of the form +{\smath{\mbox{\bf rootSplit}\opLeftPren{}f\opRightPren{}} +transforms every radical of the form \smath{(a/b)^{1/n}} appearing in \smath{f} into \smath{a^{1/n} / b^{1/n}}. This transformation is @@ -78509,7 +82118,8 @@ not in general valid for all complex numbers \smath{a} and \smath{b}. }% }% {1}{(\$)->\$}{QueueAggregate} -{\smath{\mbox{\bf rotate!}\opLeftPren{}q\opRightPren{}} rotates queue \smath{q} so that the element at +{\smath{\mbox{\bf rotate!}\opLeftPren{}q\opRightPren{}} rotates +queue \smath{q} so that the element at the front of the queue goes to the back of the queue. } @@ -78518,15 +82128,18 @@ the front of the queue goes to the back of the queue. }% }% {1}{(\$)->\$}{RealNumberSystem} -{\smath{\mbox{\bf round}\opLeftPren{}x\opRightPren{}} computes the integer closest to \smath{x}. +{\smath{\mbox{\bf round}\opLeftPren{}x\opRightPren{}} computes the +integer closest to \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{row}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{row}}\opLeftPren{}{\it matrix}, \allowbreak{} +{\it positiveInteger}\opRightPren{}% }% }% {2}{(\$, Integer)->Row}{RectangularMatrixCategory} -{\smath{\mbox{\bf row}\opLeftPren{}m, \allowbreak{} i\opRightPren{}} returns the \eth{\smath{i}} row of the matrix +{\smath{\mbox{\bf row}\opLeftPren{}m, \allowbreak{} i\opRightPren{}} +returns the \eth{\smath{i}} row of the matrix or two-dimensional array \smath{m}. } @@ -78536,7 +82149,8 @@ two-dimensional array \smath{m}. }% }% {1}{(\$)->\$}{MatrixCategory} -{\smath{\mbox{\bf rowEchelon}\opLeftPren{}m\opRightPren{}} returns the row echelon form of the matrix +{\smath{\mbox{\bf rowEchelon}\opLeftPren{}m\opRightPren{}} +returns the row echelon form of the matrix \smath{m}. } @@ -78545,7 +82159,8 @@ two-dimensional array \smath{m}. }% }% {1}{(\$)->\$}{LazyStreamAggregate} -{\smath{\mbox{\bf rst}\opLeftPren{}s\opRightPren{}} returns a pointer to the next node of stream +{\smath{\mbox{\bf rst}\opLeftPren{}s\opRightPren{}} +returns a pointer to the next node of stream \smath{s}. Caution: this function should only be called after a \axiomFun{empty?} test returns @@ -78557,14 +82172,18 @@ Caution: this function should only be called after a }% }% {0}{()->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf rubiksGroup}\opLeftPren{}\opRightPren{}} constructs the permutation group representing Rubic's Cube +{\smath{\mbox{\bf rubiksGroup}\opLeftPren{}\opRightPren{}} +constructs the permutation group representing Rubic's Cube acting on integers \smath{10i+j} for \smath{1 \leq i \leq 6, 1 \leq j \leq 8}. -The faces of Rubik's Cube are labelled: Front, Right, Up, Down, Left, Back and numbered from 1 to 6. -The pieces on each face (except the unmoveable center piece) are clockwise numbered +The faces of Rubik's Cube are labelled: Front, Right, Up, +Down, Left, Back and numbered from 1 to 6. +The pieces on each face (except the unmoveable center piece) +are clockwise numbered from 1 to 8 starting with the piece in the upper left corner. The moves of the cube are represented as permutations on these pieces, -represented as a two digit integer \smath{ij} where \smath{i} is the number of the face +represented as a two digit integer \smath{ij} where +\smath{i} is the number of the face and \smath{j} is the number of the piece on this face. The remaining ambiguities are resolved by looking at the 6 generators representing 90-degree turns of the faces. @@ -78583,27 +82202,40 @@ representing 90-degree turns of the faces. }% }% {1}{(\$)->List(RewriteRule(Base, R, F))}{Ruleset} -{\smath{\mbox{\bf rules}\opLeftPren{}r\opRightPren{}} returns the list of rewrite rules contained in ruleset \smath{r}. +{\smath{\mbox{\bf rules}\opLeftPren{}r\opRightPren{}} +returns the list of rewrite rules contained in ruleset \smath{r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{ruleset}}\opLeftPren{}{\it listOfRules}\opRightPren{}% +\opdata{{\mbox{\axiomFun{ruleset}}\opLeftPren{} +{\it listOfRules}\opRightPren{}% }% }% {1}{(List(RewriteRule(Base, R, F)))->\$}{Ruleset} -{\smath{\mbox{\bf ruleset}\opLeftPren{}[r1, \allowbreak{} \ldots, rn]\opRightPren{}} creates a ruleset +{\smath{\mbox{\bf ruleset}\opLeftPren{}[r1, \allowbreak{} +\ldots, rn]\opRightPren{}} creates a ruleset from a list of rewrite rules \smath{r_1}, \ldots, \smath{r_n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{rungaKutta}}\opLeftPren{}{\it vector}, \allowbreak{}{\it integer}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it integer}, \allowbreak{}{\it function}\opRightPren{}% -\opand \mbox{\axiomFun{rungaKuttaFixed}}\opLeftPren{}{\it vector}, \allowbreak{}{\it integer}, \allowbreak{}{\it float}, \allowbreak{}{\it float}, \allowbreak{}{\it integer}, \allowbreak{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{rungaKutta}}\opLeftPren{} +{\it vector}, \allowbreak{}{\it integer}, \allowbreak{} +{\it fourFloats}, \allowbreak{} +{\it integer}, \allowbreak{}{\it function}\opRightPren{}% +\opand \mbox{\axiomFun{rungaKuttaFixed}}\opLeftPren{} +{\it vector}, \allowbreak{}{\it integer}, \allowbreak{} +{\it float}, \allowbreak{} +{\it float}, \allowbreak{}{\it integer}, \allowbreak{} +{\it function}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf rungaKutta}\opLeftPren{}y, \allowbreak{} n, \allowbreak{} a, \allowbreak{} b, \allowbreak{} eps, \allowbreak{} h, \allowbreak{} ncalls, \allowbreak{} derivs\opRightPren{}} +{\smath{\mbox{\bf rungaKutta}\opLeftPren{}y, \allowbreak{} n, +\allowbreak{} a, \allowbreak{} b, \allowbreak{} eps, \allowbreak{} h, +\allowbreak{} ncalls, \allowbreak{} derivs\opRightPren{}} uses a \smath{4}--th order Runga-Kutta method to numerically -integrate the ordinary differential equation $dy/dx = f(y, x)$ from \smath{x_1} to \smath{x_2}, +integrate the ordinary differential equation $dy/dx = f(y, x)$ +from \smath{x_1} to \smath{x_2}, where \smath{y} is an \smath{n}--vector of \smath{n} variables. Initial and final values are provided by solution vector \smath{y}. @@ -78618,18 +82250,23 @@ function of type \spadsig{(Vector Float, Vector Float, Float)}{Void}, which computes the right-hand side of the ordinary differential equation, then replaces the elements of the first argument by updated elements. -\bigitem\smath{\mbox{\bf rungaKuttaFixed}\opLeftPren{}y, \allowbreak{} n, \allowbreak{} x_1, \allowbreak{} x_2, \allowbreak{} ns, \allowbreak{} derivs\opRightPren{}} is similar to +\bigitem\smath{\mbox{\bf rungaKuttaFixed}\opLeftPren{}y, +\allowbreak{} n, \allowbreak{} x_1, \allowbreak{} x_2, \allowbreak{} ns, +\allowbreak{} derivs\opRightPren{}} is similar to \spadfun{rungaKutta} except that it uses \smath{ns} fixed steps to integrate the solution vector \smath{y} from \smath{x_1} to \smath{x_2}, returning the values in \smath{y}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{saturate}}\opLeftPren{}{\it ideal}, \allowbreak{}{\it polynomial}\allowbreak $\,[$ , \allowbreak{}{\it listOfVariables}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{saturate}}\opLeftPren{}{\it ideal}, +\allowbreak{}{\it polynomial}\allowbreak $\,[$ , \allowbreak{} +{\it listOfVariables}$]$\opRightPren{}% }% }% {2}{(\$, DPoly)->\$}{PolynomialIdeals} -{\smath{\mbox{\bf saturate}\opLeftPren{}I, \allowbreak{} f\optinner{, lvar}\opRightPren{}} is the saturation of the +{\smath{\mbox{\bf saturate}\opLeftPren{}I, +\allowbreak{} f\optinner{, lvar}\opRightPren{}} is the saturation of the ideal \smath{I} with respect to the multiplicative set generated by the polynomial \smath{f} in the variables given by \smath{lvar}, a list of variables. @@ -78642,53 +82279,67 @@ taken to be the list of all variables appearing in \smath{f}. }% }% {1}{(List(String))->Void}{DisplayPackage} -{\smath{\mbox{\bf say}\opLeftPren{}u\opRightPren{}} sends a string or a list of strings \smath{u} to +{\smath{\mbox{\bf say}\opLeftPren{}u\opRightPren{}} +sends a string or a list of strings \smath{u} to output. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sayLength}}\opLeftPren{}{\it listOfStrings}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sayLength}}\opLeftPren{} +{\it listOfStrings}\opRightPren{}% }% }% {1}{(List(String))->Integer}{DisplayPackage} -{\smath{\mbox{\bf sayLength}\opLeftPren{}ls\opRightPren{}} returns the total number of characters +{\smath{\mbox{\bf sayLength}\opLeftPren{}ls\opRightPren{}} +returns the total number of characters in the list of strings \smath{ls}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{scalarMatrix}}\opLeftPren{}{\it scalar\opt{, dimension}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{scalarMatrix}}\opLeftPren{} +{\it scalar\opt{, dimension}}\opRightPren{}% }% }% {1}{(R)->\$}{SquareMatrixCategory} -{\smath{\mbox{\bf scalarMatrix}\opLeftPren{}r\optinner{, n}\opRightPren{}} returns an +{\smath{\mbox{\bf scalarMatrix}\opLeftPren{}r\optinner{, n}\opRightPren{}} +returns an \smath{n}-by-\smath{n} matrix with scalar \smath{r} on the diagonal and zero elsewhere. The dimension may be omitted if the result is to be an object -of type \smath{\mbox{\bf SquareMatrix}\opLeftPren{}n, \allowbreak{} R\opRightPren{}} for some \smath{n}. +of type \smath{\mbox{\bf SquareMatrix}\opLeftPren{}n, +\allowbreak{} R\opRightPren{}} for some \smath{n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{scan}}\opLeftPren{}{\it binaryFunction}, \allowbreak{}{\it aggregate}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{scan}}\opLeftPren{}{\it binaryFunction}, +\allowbreak{}{\it aggregate}, \allowbreak{}{\it element}\opRightPren{}% }% }% {3}{((S, R)->R, A, R)->B}{FiniteLinearAggregateFunctions2} -{\smath{\mbox{\bf scan}\opLeftPren{}f, \allowbreak{} a, \allowbreak{} r\opRightPren{}} successively applies \smath{\mbox{\bf reduce}\opLeftPren{}f, \allowbreak{} x, \allowbreak{} r\opRightPren{}} to +{\smath{\mbox{\bf scan}\opLeftPren{}f, \allowbreak{} a, +\allowbreak{} r\opRightPren{}} successively applies +\smath{\mbox{\bf reduce}\opLeftPren{}f, +\allowbreak{} x, \allowbreak{} r\opRightPren{}} to more and more leading sub-aggregates \smath{x} of aggregrate \smath{a}. More precisely, if \smath{a} is \smath{[a1, a2, \ldots]}, then -\smath{\mbox{\bf scan}\opLeftPren{}f, \allowbreak{} a, \allowbreak{} r\opRightPren{}} returns +\smath{\mbox{\bf scan}\opLeftPren{}f, \allowbreak{} a, +\allowbreak{} r\opRightPren{}} returns \smath{[reduce(f, [a1], r), reduce(f, [a1, a2], r), \ldots]}. Argument \smath{a} can be any linear aggregate including streams. -For example, if \smath{a} is a list or an infinite stream of the form \smath{[x_1, x_2, \ldots]}, +For example, if \smath{a} is a list or an infinite stream of +the form \smath{[x_1, x_2, \ldots]}, then \code{scan(+, a, 0)} returns a list or stream of the form \smath{[x_1, x_1 + x_2, \ldots]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{scanOneDimSubspaces}}\opLeftPren{}{\it listOfVectors}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{scanOneDimSubspaces}}\opLeftPren{} +{\it listOfVectors}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(List(Vector(R)), Integer)->Vector(R)}{RepresentationPackage2} -{\smath{\mbox{\bf scanOneDimSubspaces}\opLeftPren{}basis, \allowbreak{} n\opRightPren{}} gives a canonical +{\smath{\mbox{\bf scanOneDimSubspaces}\opLeftPren{}basis, +\allowbreak{} n\opRightPren{}} gives a canonical representative of the \eth{\smath{n}} one-dimensional subspace of the vector space generated by the elements of \smath{basis}. @@ -78698,17 +82349,25 @@ Consult \spadtype{RepresentationPackage2} using details. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{script}}\opLeftPren{}{\it symbol}, \allowbreak{}{\it listOfListsOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{script}}\opLeftPren{}{\it symbol}, +\allowbreak{}{\it listOfListsOfOutputForms}\opRightPren{}% }% }% {2}{(\$, List(List(OutputForm)))->\$}{Symbol} -{\smath{\mbox{\bf script}\opLeftPren{}sy, \allowbreak{} [a, \allowbreak{} b, \allowbreak{} c, \allowbreak{} d, \allowbreak{} e]\opRightPren{}} returns \smath{sy} with +{\smath{\mbox{\bf script}\opLeftPren{}sy, \allowbreak{} +[a, \allowbreak{} b, \allowbreak{} c, \allowbreak{} d, +\allowbreak{} e]\opRightPren{}} +returns \smath{sy} with subscripts \smath{a}, superscripts \smath{b}, pre-superscripts \smath{c}, pre-subscripts \smath{d}, and argument-scripts \smath{e}. Omitted components are taken to be empty. -For example, \smath{\mbox{\bf script}\opLeftPren{}s, \allowbreak{} {\tt [}a, \allowbreak{} b, \allowbreak{} c{\tt ]}\opRightPren{}} is equivalent -to \smath{\mbox{\bf script}\opLeftPren{}s, \allowbreak{} {\tt [}a, \allowbreak{} b, \allowbreak{} c, \allowbreak{} {\tt []}, \allowbreak{} {\tt []]}\opRightPren{}}. +For example, \smath{\mbox{\bf script}\opLeftPren{}s, +\allowbreak{} {\tt [}a, \allowbreak{} b, +\allowbreak{} c{\tt ]}\opRightPren{}} is equivalent +to \smath{\mbox{\bf script}\opLeftPren{}s, \allowbreak{} +{\tt [}a, \allowbreak{} b, \allowbreak{} c, \allowbreak{} {\tt []}, +\allowbreak{} {\tt []]}\opRightPren{}}. } % ---------------------------------------------------------------------- @@ -78716,31 +82375,38 @@ to \smath{\mbox{\bf script}\opLeftPren{}s, \allowbreak{} {\tt [}a, \allowbreak{} }% }% {1}{(\$)->Boolean}{Symbol} -{\smath{\mbox{\bf scripted?}\opLeftPren{}sy\opRightPren{}} tests if \smath{sy} has been given any scripts. +{\smath{\mbox{\bf scripted?}\opLeftPren{}sy\opRightPren{}} tests +if \smath{sy} has been given any scripts. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{scripts}}\opLeftPren{}{\it symbolOrOutputForm\opt{, listOfOutputForms}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{scripts}}\opLeftPren{} +{\it symbolOrOutputForm\opt{, listOfOutputForms}}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf scripts}\opLeftPren{}o, \allowbreak{} lo\opRightPren{}}, -where \smath{o} is an object of type \spadtype{OutputForm} (normally unexposed) +where \smath{o} is an object of type \spadtype{OutputForm} +(normally unexposed) and \smath{lo} is a list \smath{[sub, super, presuper, presub]} of four objects of type \spadtype{OutputForm} (normally unexposed), creates a form for \smath{o} with scripts on all four corners. -\newline\smath{\mbox{\bf scripts}\opLeftPren{}s\opRightPren{}} returns all the scripts of \smath{s} as a record with -selectors \smath{sub}, \smath{sup}, \smath{presup}, \smath{presub}, and \smath{args}, +\newline\smath{\mbox{\bf scripts}\opLeftPren{}s\opRightPren{}} +returns all the scripts of \smath{s} as a record with +selectors \smath{sub}, \smath{sup}, \smath{presup}, \smath{presub}, +and \smath{args}, each with a list of output forms as a value. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{search}}\opLeftPren{}{\it key}, \allowbreak{}{\it table}\opRightPren{}% +\opdata{{\mbox{\axiomFun{search}}\opLeftPren{}{\it key}, \allowbreak{} +{\it table}\opRightPren{}% }% }% {2}{(Key, \$)->Union(Entry, "failed")}{KeyedDictionary} -{\smath{\mbox{\bf search}\opLeftPren{}k, \allowbreak{} t\opRightPren{}} searches the table \smath{t} for the key +{\smath{\mbox{\bf search}\opLeftPren{}k, \allowbreak{} t\opRightPren{}} +searches the table \smath{t} for the key \smath{k}, returning the entry stored in \smath{t} for key \smath{k}, or \mbox{\tt "failed"} if \smath{t} has no such key. @@ -78752,9 +82418,12 @@ or \mbox{\tt "failed"} if \smath{t} has no such key. }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} -{\smath{\mbox{\bf sec}\opLeftPren{}x\opRightPren{}} returns the secant of \smath{x}. +{\smath{\mbox{\bf sec}\opLeftPren{}x\opRightPren{}} +returns the secant of \smath{x}. \newitem -\smath{\mbox{\bf secIfCan}\opLeftPren{}z\opRightPren{}} returns \smath{\mbox{\bf sec}\opLeftPren{}z\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf secIfCan}\opLeftPren{}z\opRightPren{}} +returns \smath{\mbox{\bf sec}\opLeftPren{}z\opRightPren{}} if possible, +and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- @@ -78762,26 +82431,35 @@ or \mbox{\tt "failed"} if \smath{t} has no such key. }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf sec2cos}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf sec}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\rm cos}(u)}. +{\smath{\mbox{\bf sec2cos}\opLeftPren{}f\opRightPren{}} +converts every \smath{\mbox{\bf sec}\opLeftPren{}u\opRightPren{}} +appearing in \smath{f} into \smath{1/{\rm cos}(u)}. } % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{sech}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{sechIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{sechIfCan}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} -{\smath{\mbox{\bf sech}\opLeftPren{}x\opRightPren{}} returns the hyperbolic secant of \smath{x}. +{\smath{\mbox{\bf sech}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic secant of \smath{x}. \newitem -\smath{\mbox{\bf sechIfCan}\opLeftPren{}z\opRightPren{}} returns \smath{\mbox{\bf sech}\opLeftPren{}z\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf sechIfCan}\opLeftPren{}z\opRightPren{}} +returns \smath{\mbox{\bf sech}\opLeftPren{}z\opRightPren{}} if possible, +and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sech2cosh}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sech2cosh}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf sech2cosh}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf sech}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\rm cosh}(u)}. +{\smath{\mbox{\bf sech2cosh}\opLeftPren{}f\opRightPren{}} +converts every \smath{\mbox{\bf sech}\opLeftPren{}u\opRightPren{}} +appearing in \smath{f} into \smath{1/{\rm cosh}(u)}. } % ---------------------------------------------------------------------- @@ -78789,12 +82467,14 @@ or \mbox{\tt "failed"} if \smath{t} has no such key. }% }% {1}{(\$)->S}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf second}\opLeftPren{}u\opRightPren{}} returns the second element of recursive aggregate \smath{u}. +{\smath{\mbox{\bf second}\opLeftPren{}u\opRightPren{}} +returns the second element of recursive aggregate \smath{u}. Note: \smath{\mbox{\bf second}\opLeftPren{}u) = {\bf first}({\bf rest}(u))}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{segment}}\opLeftPren{}{\it integer\opt{, integer}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{segment}}\opLeftPren{} +{\it integer\opt{, integer}}\opRightPren{}% }% }% {1}{(S)->\$}{UniversalSegment} @@ -78803,59 +82483,72 @@ the segment \smath{i..j}. If not qualified by a {\bf by} clause, this notation for integers \smath{i} and \smath{j} denotes the tuple of integers \smath{i}, \smath{i+1}, \ldots, \smath{j}. -When \smath{j} is omitted, \smath{\mbox{\bf segment}\opLeftPren{}i\opRightPren{}} +When \smath{j} is omitted, +\smath{\mbox{\bf segment}\opLeftPren{}i\opRightPren{}} denotes the half open segment \smath{i..}, that is, a segment with no upper bound. \newline \smath{\mbox{\bf segment}\opLeftPren{}x = bd\opRightPren{}}, where \smath{bd} is a binding, returns \smath{bd}. -For example, \smath{\mbox{\bf segment}\opLeftPren{}x = a..b\opRightPren{}} returns +For example, \smath{\mbox{\bf segment}\opLeftPren{}x = a..b\opRightPren{}} +returns \smath{a..b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{select}}\opLeftPren{}{\it pred}, \allowbreak{}{\it aggregate}\opRightPren{}% - \opand \mbox{\axiomFun{select!}}\opLeftPren{}{\it pred}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{select}}\opLeftPren{}{\it pred}, \allowbreak{} +{\it aggregate}\opRightPren{}% + \opand \mbox{\axiomFun{select!}}\opLeftPren{}{\it pred}, \allowbreak{} +{\it aggregate}\opRightPren{}% }% }% {2}{((S)->Boolean, \$)->\$}{Collection} -{\smath{\mbox{\bf select}\opLeftPren{}p, \allowbreak{} u\opRightPren{}} returns a copy of \smath{u} containing only +{\smath{\mbox{\bf select}\opLeftPren{}p, +\allowbreak{} u\opRightPren{}} returns a copy of \smath{u} containing only those elements \smath{x} such that \smath{p(x)} is \smath{true}. For a list \smath{l}, \smath{select(p, l) == [x {\mbox{ \tt for } } x {\mbox{ \tt in }} l {\tt |} p(x)]}. Argument \smath{u} may be any finite aggregate or infinite stream. -\newitem \smath{\mbox{\bf select!}\opLeftPren{}p, \allowbreak{} u\opRightPren{}} destructively changes \smath{u} by +\newitem \smath{\mbox{\bf select!}\opLeftPren{}p, +\allowbreak{} u\opRightPren{}} destructively changes \smath{u} by keeping only values \smath{x} such that \smath{p(x)} is true. Argument \smath{u} can be any extensible linear aggregate or dictionary. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{semicolonSeparate}}\opLeftPren{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{semicolonSeparate}}\opLeftPren{} +{\it listOfOutputForms}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf semicolonSeparate}\opLeftPren{}lo\opRightPren{}}, where \smath{lo} is a list of +{\smath{\mbox{\bf semicolonSeparate}\opLeftPren{}lo\opRightPren{}}, +where \smath{lo} is a list of objects of type \spadtype{OutputForm} (normally unexposed), returns an output form which separates the elements of \smath{lo} by semicolons. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{separant}}\opLeftPren{}{\it differentialPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{separant}}\opLeftPren{} +{\it differentialPolynomial}\opRightPren{}% }% }% {1}{(\$)->\$}{DifferentialPolynomialCategory} -{\smath{\mbox{\bf separant}\opLeftPren{}polynomial\opRightPren{}} returns the partial derivative of +{\smath{\mbox{\bf separant}\opLeftPren{}polynomial\opRightPren{}} +returns the partial derivative of the differential polynomial \smath{p} with respect to its leader. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{separate}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{separate}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% }% }% -{2}{(\$, \$)->Record(primePart:\$, commonPart:\$)}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf separate}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} returns \spad({a, b)} such that polynomial +{2}{(\$, \$)->Record(primePart:\$, commonPart:\$)} +{UnivariatePolynomialCategory} +{\smath{\mbox{\bf separate}\opLeftPren{}p, +\allowbreak{} q\opRightPren{}} returns \spad({a, b)} such that polynomial \smath{p = a b} and \smath{a} is relatively prime to \smath{q}. The result produced is a record with selectors \smath{primePart} and \smath{commonPart} @@ -78863,109 +82556,153 @@ with value \smath{a} and \smath{b} respectively. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{separateDegrees}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{separateDegrees}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% -{1}{(FP)->List(Record(deg:NonNegativeInteger, prod:FP))}{DistinctDegreeFactorize} -{\smath{\mbox{\bf separateDegrees}\opLeftPren{}p\opRightPren{}} splits the polynomial \smath{p} into factors. +{1}{(FP)->List(Record(deg:NonNegativeInteger, prod:FP))} +{DistinctDegreeFactorize} +{\smath{\mbox{\bf separateDegrees}\opLeftPren{}p\opRightPren{}} +splits the polynomial \smath{p} into factors. Each factor is a record with selector \smath{deg}, a non-negative integer, and \smath{prod}, a product of irreducible polynomials of degree \smath{deg}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{separateFactors}}\opLeftPren{}{\it listOfRecords}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{separateFactors}}\opLeftPren{} +{\it listOfRecords}, \allowbreak{}{\it polynomial}\opRightPren{}% }% }% -{2}{(List(Record(deg:NonNegativeInteger, prod:FP)), FP)->List(FP)}{DistinctDegreeFactorize} -{\smath{\mbox{\bf separateFactors}\opLeftPren{}lfact, \allowbreak{} p\opRightPren{}} takes the list produced by +{2}{(List(Record(deg:NonNegativeInteger, prod:FP)), FP)-> +List(FP)}{DistinctDegreeFactorize} +{\smath{\mbox{\bf separateFactors}\opLeftPren{}lfact, +\allowbreak{} p\opRightPren{}} takes the list produced by \spadfunFrom{separateDegrees}{DistinctDegreeFactorization} along -with the original polynomial \smath{p}, and produces the complete list of factors. +with the original polynomial \smath{p}, and produces the +complete list of factors. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{separateFactors}}\opLeftPren{}{\it listOfRecords}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{separateFactors}}\opLeftPren{} +{\it listOfRecords}, \allowbreak{}{\it integer}\opRightPren{}% }% }% -{2}{(List(Record(factor:U, degree:Integer)), Integer)->List(U)}{ModularDistinctDegreeFactorizer} -{\smath{\mbox{\bf separateFactors}\opLeftPren{}ddl, \allowbreak{} p\opRightPren{}} refines the distinct degree factorization produced by \spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} to give a complete list of factors. +{2}{(List(Record(factor:U, degree:Integer)), +Integer)->List(U)}{ModularDistinctDegreeFactorizer} +{\smath{\mbox{\bf separateFactors}\opLeftPren{}ddl, +\allowbreak{} p\opRightPren{}} refines the distinct degree factorization +produced by \spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} to give a +complete list of factors. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sequences}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% -\opand \mbox{\axiomFun{sequences}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sequences}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% +\opand \mbox{\axiomFun{sequences}}\opLeftPren{}{\it listOfIntegers}, +\allowbreak{}{\it listOfIntegers}\opRightPren{}% }% }% -{2}{(List(Integer), List(Integer))->Stream(List(Integer))}{PartitionsAndPermutations} -{\smath{\mbox{\bf sequences}\opLeftPren{}[l_0, \allowbreak{} l_1, \allowbreak{} l_2, \allowbreak{} .., \allowbreak{} l_n]\opRightPren{}} is the set of all sequences -formed from \smath{l_0} 0's, \smath{l_1} 1's, \smath{l_2} 2's, \ldots, \smath{l_n} \smath{n}'s. +{2}{(List(Integer), List(Integer))->Stream(List(Integer))} +{PartitionsAndPermutations} +{\smath{\mbox{\bf sequences}\opLeftPren{}[l_0, \allowbreak{} l_1, +\allowbreak{} l_2, \allowbreak{} .., \allowbreak{} l_n]\opRightPren{}} +is the set of +all sequences +formed from \smath{l_0} 0's, \smath{l_1} 1's, \smath{l_2} 2's, \ldots, +\smath{l_n} \smath{n}'s. \newitem -\smath{\mbox{\bf sequences}\opLeftPren{}l1, \allowbreak{} l2\opRightPren{}} is the stream of all sequences that can be composed from +\smath{\mbox{\bf sequences}\opLeftPren{}l1, \allowbreak{} l2\opRightPren{}} +is the stream of all sequences that can be composed from the multiset defined from two lists of integers \smath{l1} and \smath{l2}. For example, the pair \smath{([1, 2, 4], [2, 3, 5])} represents multiset with 1 \smath{2}, 2 \smath{3}'s, and 4 \smath{5}'s. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{series}}\opLeftPren{}{\it specifications\opt{, \ldots}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{series}}\opLeftPren{} +{\it specifications\opt{, \ldots}}\opRightPren{}% }% }% {1}{(Stream(Coef))->\$}{UnivariateTaylorSeriesCategory} -{\smath{\mbox{\bf series}\opLeftPren{}expression\opRightPren{}} returns a series expansion of the expression \smath{f}. -Note: \smath{f} must have only one variable. The series will be expanded in powers +{\smath{\mbox{\bf series}\opLeftPren{}expression\opRightPren{}} +returns a series expansion of the expression \smath{f}. +Note: \smath{f} must have only one variable. The series will be +expanded in powers of that variable. \newitem -\smath{\mbox{\bf series}\opLeftPren{}sy\opRightPren{}}, where \smath{sy} is a symbol, returns \smath{sy} as a series. +\smath{\mbox{\bf series}\opLeftPren{}sy\opRightPren{}}, +where \smath{sy} is a symbol, returns \smath{sy} as a series. \newitem \smath{\mbox{\bf series}\opLeftPren{}st\opRightPren{}}, -where \smath{t} is a stream \smath{[a_0, a_1, a_2, \ldots]} of coefficients \smath{a_i} +where \smath{t} is a stream \smath{[a_0, a_1, a_2, \ldots]} +of coefficients \smath{a_i} from some ring, creates the Taylor series \smath{a_0 + a_1 x + a_2 x^2 +\ldots}. -Also, if \smath{st} is a stream of elements of type \spadtype{Record(k:NonNegativeInteger, c:R)}, -where \smath{k} denotes an exponent and \smath{c}, a non-zero coefficient from some ring \smath{R}, +Also, if \smath{st} is a stream of elements of type +\spadtype{Record(k:NonNegativeInteger, c:R)}, +where \smath{k} denotes an exponent and \smath{c}, +a non-zero coefficient from some ring \smath{R}, it creates a stream of non-zero terms. The terms in \smath{st} must be ordered by increasing order of exponents. \newitem -\smath{\mbox{\bf series}\opLeftPren{}f, \allowbreak{} x = a\optinner{, n}\opRightPren{}} expands the expression \smath{f} as a +\smath{\mbox{\bf series}\opLeftPren{}f, +\allowbreak{} x = a\optinner{, n}\opRightPren{}} +expands the expression \smath{f} as a series in powers of \smath{(x - a)} with \smath{n} terms. If \smath{n} is missing, the number of terms is governed by the value set by the system command \spadsyscom{)set streams calculate}. \newitem -\smath{\mbox{\bf series}\opLeftPren{}f, \allowbreak{} n\opRightPren{}} returns a series expansion of the expression +\smath{\mbox{\bf series}\opLeftPren{}f, \allowbreak{} n\opRightPren{}} +returns a series expansion of the expression \smath{f}. Note: \smath{f} should have only one variable; the series will be expanded in powers of that variable and terms will be computed up to order at least \smath{n}. \newitem -\smath{\mbox{\bf series}\opLeftPren{}i {\tt +->} a(i), \allowbreak{} x = a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} creates the series +\smath{\mbox{\bf series}\opLeftPren{}i {\tt +->} a(i), +\allowbreak{} x = a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} +creates the series \smath{\sum\nolimits_{i = m..n {\ \tt by \ } k}{a(i) (x - a)^i}}. Here \smath{m}, \smath{n}, and \smath{k} are rational numbers. -Upper-limit \smath{n} and stepsize \smath{k} are optional and have default values +Upper-limit \smath{n} and stepsize \smath{k} are optional and +have default values \smath{n = \infty} and \smath{k = 1}. \newitem -\smath{\mbox{\bf series}\opLeftPren{}a(i), \allowbreak{} i, \allowbreak{} x=a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} returns +\smath{\mbox{\bf series}\opLeftPren{}a(i), \allowbreak{} i, +\allowbreak{} x=a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} returns \smath{\sum\nolimits_{i = m..n {\bf by} k}{a(n) (x - a)^n}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{seriesSolve}}\opLeftPren{}{\it eq}, \allowbreak{}{\it y}, \allowbreak{}{\it x}, \allowbreak{}{\it c}\opRightPren{}% +\opdata{{\mbox{\axiomFun{seriesSolve}}\opLeftPren{}{\it eq}, +\allowbreak{}{\it y}, \allowbreak{}{\it x}, \allowbreak{} +{\it c}\opRightPren{}% }% }% -{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))}{TransSolvePackage} +{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))} +{TransSolvePackage} {\bigopkey{\smath{eq} denotes an equation to be solved; -alternatively, an expression \smath{u} may be given for \smath{eq} in which case the equation +alternatively, an expression \smath{u} may be given for \smath{eq} +in which case the equation \smath{eq} is defined as \smath{u=0}. -\largerbreak\smath{leq} denotes a list \smath{[ eq_1\ldots eq_n]} of equations; alternatively, +\largerbreak\smath{leq} denotes a list \smath{[ eq_1\ldots eq_n]} of +equations; alternatively, a list of expressions \smath{[ u_1\ldots u_n]} may be given of \smath{leq} in which case the equations \smath{eq_i} are defined by \smath{u_i=0}. } -\bigitem\smath{\mbox{\bf seriesSolve}\opLeftPren{}eq, \allowbreak{} y, \allowbreak{} x=a, \allowbreak{} \optinner{y(a)=}b\opRightPren{}} returns a Taylor series solution of +\bigitem\smath{\mbox{\bf seriesSolve}\opLeftPren{}eq, \allowbreak{} y, +\allowbreak{} x=a, \allowbreak{} \optinner{y(a)=}b\opRightPren{}} +returns a Taylor series solution of \smath{eq} around \smath{x=a} with initial condition \smath{y(a)=b}. Note: \smath{eq} must be of the form \smath{f(x, y)y'(x) + g(x, y) = h(x, y)}. -\bigitem\smath{\mbox{\bf seriesSolve}\opLeftPren{}eq, \allowbreak{} y, \allowbreak{} x=a, \allowbreak{} [ b_0, \ldots, b_{(n-1)}]\opRightPren{}} returns -a Taylor series solution of \smath{eq} around \smath{x=a} with initial conditions +\bigitem\smath{\mbox{\bf seriesSolve}\opLeftPren{}eq, +\allowbreak{} y, \allowbreak{} x=a, \allowbreak{} +[ b_0, \ldots, b_{(n-1)}]\opRightPren{}} returns +a Taylor series solution of \smath{eq} around +\smath{x=a} with initial conditions \smath{y(a)=b_0}, \smath{y'(a)=b_1}, $\ldots$ @@ -78988,11 +82725,13 @@ fourth argument \smath{[ y_1(a)=b_1, \ldots, y_n(a)=b_n]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setchildren!}}\opLeftPren{}{\it recursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setchildren!}}\opLeftPren{} +{\it recursiveAggregate}\opRightPren{}% }% }% {2}{(\$, List(\$))->\$}{RecursiveAggregate} -{\smath{\mbox{\bf setchildren!}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} replaces the current children of node +{\smath{\mbox{\bf setchildren!}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} replaces the current children of node \smath{u} with the members of \smath{v} in left-to-right order. } @@ -79001,61 +82740,81 @@ fourth argument \smath{[ y_1(a)=b_1, \ldots, y_n(a)=b_n]}. }% }% {3}{(\$, Integer, Col)->\$}{TwoDimensionalArrayCategory} -{\smath{\mbox{\bf setColumn!}\opLeftPren{}m, \allowbreak{} j, \allowbreak{} v\opRightPren{}} sets the \eth{\smath{j}} column of +{\smath{\mbox{\bf setColumn!}\opLeftPren{}m, \allowbreak{} j, +\allowbreak{} v\opRightPren{}} sets the \eth{\smath{j}} column of matrix or two-dimensional array \smath{m} to \smath{v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setDifference}}\opLeftPren{}{\it list}, \allowbreak{}{\it list}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setDifference}}\opLeftPren{} +{\it list}, \allowbreak{}{\it list}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{List} -{\smath{\mbox{\bf setDifference}\opLeftPren{}l_1, \allowbreak{} l_2\opRightPren{}} returns a list of the elements of \smath{l_1} that are +{\smath{\mbox{\bf setDifference}\opLeftPren{}l_1, +\allowbreak{} l_2\opRightPren{}} returns a list of the +elements of \smath{l_1} that are not also in \smath{l_2}. The order of elements in the resulting list is unspecified. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setelt}}\opLeftPren{}{\it structure}, \allowbreak{}{\it index}, \allowbreak{}{\it value}\allowbreak $\,[$ , \allowbreak{}{\it option}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{setelt}}\opLeftPren{} +{\it structure}, \allowbreak{}{\it index}, \allowbreak{} +{\it value}\allowbreak $\,[$ , \allowbreak{}{\it option}$]$\opRightPren{}% }% }% {3}{(\$, Dom, Im)->Im}{EltableAggregate} -{\smath{\mbox{\bf setelt}\opLeftPren{}u, \allowbreak{} x, \allowbreak{} y\opRightPren{}}, also written \smath{u.x := y}, +{\smath{\mbox{\bf setelt}\opLeftPren{}u, +\allowbreak{} x, \allowbreak{} y\opRightPren{}}, also written \smath{u.x := y}, sets the image of \smath{x} to be \smath{y} under \smath{u}, regarded as a function mapping values from the domain of \smath{x} to the domain of \smath{y}. Specifically, if \smath{u} is: \begin{simpleList} \item a list: \smath{u.first := x} is equivalent to -\smath{\mbox{\bf setfirst!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}. +\smath{\mbox{\bf setfirst!}\opLeftPren{}u, +\allowbreak{} x\opRightPren{}}. Also, -\smath{u.rest := x} is equivalent to \smath{\mbox{\bf setrest!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}, and -\smath{u.last := x} is equivalent to \smath{\mbox{\bf setlast!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}. +\smath{u.rest := x} is equivalent to +\smath{\mbox{\bf setrest!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}, and +\smath{u.last := x} is equivalent to +\smath{\mbox{\bf setlast!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}. \item a linear aggregate, \smath{u(i..j) := x} destructively replaces each element in the segment \smath{u(i..j)} by \smath{x}. The value \smath{x} is returned. -Note: This function has the same effect as \code{for k in i..j repeat u.k := x; x}. +Note: This function has the same effect as +\code{for k in i..j repeat u.k := x; x}. The length of \smath{u} is unchanged. \item a recursive aggregate, -\smath{u.value := x} is equivalent to \smath{\mbox{\bf setvalue!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} and sets the value part of node \spad{u} to \spad{x}. +\smath{u.value := x} is equivalent to +\smath{\mbox{\bf setvalue!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} and +sets the value part of node \spad{u} to \spad{x}. Also, if \spad{u} is a \spadtype{BinaryTreeAggregate}, -\smath{u.left := x} is equivalent to \smath{\mbox{\bf setleft!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} and sets the left child +\smath{u.left := x} is equivalent to +\smath{\mbox{\bf setleft!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} +and sets the left child of \spad{u} to \spad{x}. Simiarly, -\smath{u.right := x} is equivalent to \smath{\mbox{\bf setright!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}. +\smath{u.right := x} is equivalent to +\smath{\mbox{\bf setright!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}}. See also \spadfun{setchildren!}. \item a table of category TableAggregate(Key, Entry): -u(k) := e is equivalent to \smath{({\bf insert}([k, e], t); e)}, where \smath{k} is +u(k) := e is equivalent to \smath{({\bf insert}([k, e], t); e)}, +where \smath{k} is a key and \smath{e} is an entry. \item a library: \smath{u.k := v} saves the value \smath{v} in the library \smath{u}, so that it can later be extracted by \smath{u.k}. \end{simpleList} -\smath{\mbox{\bf setelt}\opLeftPren{}u, \allowbreak{} i, \allowbreak{} j, \allowbreak{} r\opRightPren{}}, also written, \smath{u(i, j) := r}, sets +\smath{\mbox{\bf setelt}\opLeftPren{}u, +\allowbreak{} i, \allowbreak{} j, +\allowbreak{} r\opRightPren{}}, also written, \smath{u(i, j) := r}, sets the element in the \eth{\smath{i}} row and \eth{\smath{j}} column of matrix or two-dimensional array \smath{u} to \smath{r}. \newitem -\smath{\mbox{\bf setelt}\opLeftPren{}u, \allowbreak{} rowList, \allowbreak{} colList, \allowbreak{} r\opRightPren{}}, +\smath{\mbox{\bf setelt}\opLeftPren{}u, +\allowbreak{} rowList, \allowbreak{} colList, \allowbreak{} r\opRightPren{}}, also written \smath{u([i_1, i_2, \ldots, i_m], [j_1, j_2, \ldots, j_n]) := r}, where \smath{u} is a matrix or two-dimensional array and \smath{r} is another \smath{m} by \smath{n} matrix or array, @@ -79064,64 +82823,84 @@ the \smath{x_{i_k, j_l}} is set to \smath{r(k, l)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setEpilogue!}}\opLeftPren{}{\it formattedObject}, \allowbreak{}{\it listOfStrings}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setEpilogue!}}\opLeftPren{} +{\it formattedObject}, \allowbreak{}{\it listOfStrings}\opRightPren{}% }% }% {2}{(\$, List(String))->List(String)}{ScriptFormulaFormat} -{\smath{\mbox{\bf setEpilogue!}\opLeftPren{}t, \allowbreak{} strings\opRightPren{}} sets the epilogue section of a +{\smath{\mbox{\bf setEpilogue!}\opLeftPren{}t, +\allowbreak{} strings\opRightPren{}} sets the epilogue section of a formatted object \smath{t} to \smath{strings}. -Argument \smath{t} is either an IBM SCRIPT Formula Formatted or \TeX{} formatted object. +Argument \smath{t} is either an IBM SCRIPT Formula +Formatted or \TeX{} formatted object. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setfirst!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it value}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setfirst!}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it value}\opRightPren{}% }% }% {2}{(\$, S)->S}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf setfirst!}\opLeftPren{}a, \allowbreak{} x\opRightPren{}} destructively changes the first element of recursive +{\smath{\mbox{\bf setfirst!}\opLeftPren{}a, +\allowbreak{} x\opRightPren{}} destructively +changes the first element of recursive aggregate \smath{a} to \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setFormula!}}\opLeftPren{}{\it formattedObject}, \allowbreak{}{\it listOfStrings}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setFormula!}}\opLeftPren{} +{\it formattedObject}, \allowbreak{}{\it listOfStrings}\opRightPren{}% }% }% -{2}{(\$, List(String))->List(String)}{ScriptFormulaFormat} {\smath{\mbox{\bf setFormula!}\opLeftPren{}t, \allowbreak{} strings\opRightPren{}} sets the formula section of a formatted object \smath{t} to \smath{strings}.} +{2}{(\$, List(String))->List(String)}{ScriptFormulaFormat} +{\smath{\mbox{\bf setFormula!}\opLeftPren{}t, +\allowbreak{} strings\opRightPren{}} sets the formula section of a formatted +object \smath{t} to \smath{strings}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setIntersection}}\opLeftPren{}{\it list}, \allowbreak{}{\it list}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setIntersection}}\opLeftPren{} +{\it list}, \allowbreak{}{\it list}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{List} -{\smath{\mbox{\bf setIntersection}\opLeftPren{}l_1, \allowbreak{} l_2\opRightPren{}} returns a list of the elements that lists \smath{l_1} -and \smath{l_2} have in common. The order of elements in the resulting list is unspecified. +{\smath{\mbox{\bf setIntersection}\opLeftPren{}l_1, +\allowbreak{} l_2\opRightPren{}} returns a list of the +elements that lists \smath{l_1} +and \smath{l_2} have in common. The order of elements +in the resulting list is unspecified. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setlast!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it value}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setlast!}}\opLeftPren{} +{\it aggregate}, \allowbreak{}{\it value}\opRightPren{}% }% }% {2}{(\$, S)->S}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf setlast!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} destructively changes the last element of +{\smath{\mbox{\bf setlast!}\opLeftPren{}u, +\allowbreak{} x\opRightPren{}} destructively changes the last element of \smath{u} to \smath{x}. Note: \smath{u.last := x} is equivalent. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setleaves!}}\opLeftPren{}{\it balancedBinaryTree}, \allowbreak{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setleaves!}}\opLeftPren{} +{\it balancedBinaryTree}, \allowbreak{}{\it listOfElements}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{BinaryRecursiveAggregate} -{\smath{\mbox{\bf setleaves!}\opLeftPren{}t, \allowbreak{} ls\opRightPren{}} sets the leaves of balanced binary tree +{\smath{\mbox{\bf setleaves!}\opLeftPren{}t, +\allowbreak{} ls\opRightPren{}} sets the leaves of balanced binary tree \smath{t} in left-to-right order to the elements of \smath{ls}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setleft!}}\opLeftPren{}{\it binaryRecursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setleft!}}\opLeftPren{} +{\it binaryRecursiveAggregate}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{BinaryRecursiveAggregate} -{\smath{\mbox{\bf setleft!}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} sets the left child of \smath{a} to be +{\smath{\mbox{\bf setleft!}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}} sets the left child of \smath{a} to be \smath{b}. } @@ -79129,124 +82908,163 @@ Note: \smath{u.last := x} is equivalent. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setPrologue!}}\opLeftPren{}{\it formattedObject}, \allowbreak{}{\it listOfStrings}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setPrologue!}}\opLeftPren{} +{\it formattedObject}, \allowbreak{}{\it listOfStrings}\opRightPren{}% }% }% {2}{(\$, List(String))->List(String)}{ScriptFormulaFormat} -{\smath{\mbox{\bf setPrologue!}\opLeftPren{}t, \allowbreak{} strings\opRightPren{}} sets the prologue section of a formatted object \smath{t} to \smath{strings}. -Argument \smath{t} is either an IBM SCRIPT Formula Formatted or \TeX{} formatted object. +{\smath{\mbox{\bf setPrologue!}\opLeftPren{}t, +\allowbreak{} strings\opRightPren{}} sets the prologue +section of a formatted object \smath{t} +to \smath{strings}. +Argument \smath{t} is either an IBM SCRIPT Formula Formatted or +\TeX{} formatted object. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setProperties!}}\opLeftPren{}{\it basicOperator}, \allowbreak{}{\it associationList}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setProperties!}}\opLeftPren{} +{\it basicOperator}, \allowbreak{}{\it associationList}\opRightPren{}% }% }% {2}{(\$, AssociationList(String, None))->\$}{BasicOperator} -{\smath{\mbox{\bf setProperties!}\opLeftPren{}op, \allowbreak{} al\opRightPren{}} sets the property list of basic operator \smath{op} to +{\smath{\mbox{\bf setProperties!}\opLeftPren{}op, +\allowbreak{} al\opRightPren{}} sets the property +list of basic operator \smath{op} to association list \smath{l}. Note: argument \smath{op} is modified ``in place'', that is, no copy is made. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setProperty!}}\opLeftPren{}{\it basicOperator}, \allowbreak{}{\it string}, \allowbreak{}{\it value}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setProperty!}}\opLeftPren{}{\it basicOperator}, +\allowbreak{}{\it string}, \allowbreak{}{\it value}\opRightPren{}% }% }% {3}{(\$, String, None)->\$}{BasicOperator} -{\smath{\mbox{\bf setProperty!}\opLeftPren{}op, \allowbreak{} s, \allowbreak{} v\opRightPren{}} attaches property \smath{s} to +{\smath{\mbox{\bf setProperty!}\opLeftPren{}op, +\allowbreak{} s, \allowbreak{} v\opRightPren{}} attaches property \smath{s} to \smath{op}, and sets its value to \smath{v}. Argument \smath{op} is modified ``in place'', that is, no copy is made. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setrest!}}\opLeftPren{}{\it aggregate\optinit{, integer}, aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setrest!}}\opLeftPren{} +{\it aggregate\optinit{, integer}, aggregate}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{UnaryRecursiveAggregate} -{\opkey{Arguments \smath{u} and \smath{v} are finite or infinite aggregates of the same type.} +{\opkey{Arguments \smath{u} and \smath{v} +are finite or infinite aggregates of the same type.} \newitem -\smath{\mbox{\bf setrest!}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} destructively changes the rest of \smath{u} to \smath{v}. +\smath{\mbox{\bf setrest!}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} destructively +changes the rest of \smath{u} to \smath{v}. \newitem -\smath{\mbox{\bf setrest!}\opLeftPren{}x, \allowbreak{} n, \allowbreak{} y\opRightPren{}} destructively changes \smath{x} -so that \smath{\mbox{\bf rest}\opLeftPren{}x, \allowbreak{} n\opRightPren{}}, that is, \smath{x} after the \eth{\smath{n}} element, equals \smath{y}. +\smath{\mbox{\bf setrest!}\opLeftPren{}x, \allowbreak{} n, +\allowbreak{} y\opRightPren{}} destructively changes \smath{x} +so that \smath{\mbox{\bf rest}\opLeftPren{}x, +\allowbreak{} n\opRightPren{}}, that is, \smath{x} +after the \eth{\smath{n}} element, equals \smath{y}. The function will expand cycles if necessary. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setright!}}\opLeftPren{}{\it binaryRecursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setright!}}\opLeftPren{} +{\it binaryRecursiveAggregate}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{BinaryRecursiveAggregate} -{\smath{\mbox{\bf setright!}\opLeftPren{}a, \allowbreak{} x\opRightPren{}} sets the right child of \smath{t} to be +{\smath{\mbox{\bf setright!}\opLeftPren{}a, +\allowbreak{} x\opRightPren{}} sets the right child of \smath{t} to be \smath{x}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setRow!}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it integer}, \allowbreak{}{\it row}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setRow!}}\opLeftPren{} +{\it matrix}, \allowbreak{}{\it integer}, \allowbreak{} +{\it row}\opRightPren{}% }% }% {3}{(\$, Integer, Row)->\$}{TwoDimensionalArrayCategory} -{\smath{\mbox{\bf setRow!}\opLeftPren{}m, \allowbreak{} i, \allowbreak{} v\opRightPren{}} sets the \eth{\smath{i}} row of matrix or +{\smath{\mbox{\bf setRow!}\opLeftPren{}m, \allowbreak{} i, +\allowbreak{} v\opRightPren{}} sets the \eth{\smath{i}} row of matrix or two-dimensional array \smath{m} to \smath{v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setsubMatrix!}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setsubMatrix!}}\opLeftPren{}{\it matrix}, +\allowbreak{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{} +{\it matrix}\opRightPren{}% }% }% {4}{(\$, Integer, Integer, \$)->\$}{MatrixCategory} -{\smath{\mbox{\bf setsubMatrix}\opLeftPren{}x, \allowbreak{} i_1, \allowbreak{} j_1, \allowbreak{} y\opRightPren{}} destructively alters the matrix \smath{x}. +{\smath{\mbox{\bf setsubMatrix}\opLeftPren{}x, \allowbreak{} i_1, +\allowbreak{} j_1, \allowbreak{} y\opRightPren{}} destructively +alters the matrix \smath{x}. Here \smath{x(i, j)} is set to \smath{y(i-i_1+1, j-j_1+1)} for \smath{i = i_1, \ldots, i_1-1+{\bf nrows}(y)} and \smath{j = j_1, \ldots, j_1-1+{\bf ncols}(y)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setTex!}}\opLeftPren{}{\it text}, \allowbreak{}{\it listOfStrings}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setTex!}}\opLeftPren{} +{\it text}, \allowbreak{}{\it listOfStrings}\opRightPren{}% }% }% {2}{(\$, List(String))->List(String)}{TexFormat} -{\smath{\mbox{\bf setTex!}\opLeftPren{}t, \allowbreak{} strings\opRightPren{}} sets the TeX section of a TeX form +{\smath{\mbox{\bf setTex!}\opLeftPren{}t, +\allowbreak{} strings\opRightPren{}} sets the TeX section of a TeX form \smath{t} to \smath{strings}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setUnion}}\opLeftPren{}{\it list}, \allowbreak{}{\it list}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setUnion}}\opLeftPren{} +{\it list}, \allowbreak{}{\it list}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{List} -{\smath{\mbox{\bf setUnion}\opLeftPren{}l_1, \allowbreak{} l_2\opRightPren{}} appends the two lists \smath{l_1} and +{\smath{\mbox{\bf setUnion}\opLeftPren{}l_1, +\allowbreak{} l_2\opRightPren{}} appends the two lists \smath{l_1} and \smath{l_2}, then removes all duplicates. The order of elements in the resulting list is unspecified. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setvalue!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it value}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setvalue!}}\opLeftPren{}{\it aggregate}, +\allowbreak{}{\it value}\opRightPren{}% }% }% {2}{(\$, S)->S}{RecursiveAggregate} -{\smath{\mbox{\bf setvalue!}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} destructively changes the value of node \smath{u} to \smath{x}. +{\smath{\mbox{\bf setvalue!}\opLeftPren{}u, +\allowbreak{} x\opRightPren{}} destructively changes the value of +node \smath{u} to \smath{x}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{setVariableOrder}}\opLeftPren{}{\it listOfSymbols\opt{, listOfSymbols}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{setVariableOrder}}\opLeftPren{} +{\it listOfSymbols\opt{, listOfSymbols}}\opRightPren{}% }% }% {1}{(List(Symbol))->Void}{UserDefinedVariableOrdering} -{\smath{\mbox{\bf setVariableOrder}\opLeftPren{}[a_1, \allowbreak{} \ldots, a_m], [b_1, \ldots, b_n]\opRightPren{}} defines an ordering +{\smath{\mbox{\bf setVariableOrder}\opLeftPren{} +[a_1, \allowbreak{} \ldots, a_m], [b_1, \ldots, b_n]\opRightPren{}} +defines an ordering on the variables given by \smath{a_1 > a_2 > \ldots > a_m >} other variables \smath{b_1 > b_2 > \ldots > b_n}. \newitem -\smath{\mbox{\bf setVariableOrder}\opLeftPren{}[a1, \allowbreak{} \ldots, an]\opRightPren{}} defines an ordering given by +\smath{\mbox{\bf setVariableOrder}\opLeftPren{} +[a1, \allowbreak{} \ldots, an]\opRightPren{}} defines an ordering given by \smath{a_1 > a_2 > \ldots > a_n >} all other variables. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sFunction}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sFunction}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% {L I -> SPOL RN}{}{} -{\smath{\mbox{\bf sFunction}\opLeftPren{}li\opRightPren{}} is the S-function of the partition given by list of +{\smath{\mbox{\bf sFunction}\opLeftPren{}li\opRightPren{}} +is the S-function of the partition given by list of linteger \smath{li}, expressed in terms of power sum symmetric functions. See \spadtype{CycleIndicators} for details. @@ -79257,15 +83075,18 @@ See \spadtype{CycleIndicators} for details. }% }% {1}{(\$)->Integer}{Palette} -{\smath{\mbox{\bf shade}\opLeftPren{}p\opRightPren{}} returns the shade index of the indicated palette \smath{p}. +{\smath{\mbox{\bf shade}\opLeftPren{}p\opRightPren{}} +returns the shade index of the indicated palette \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{shellSort}}\opLeftPren{}{\it sortingFunction}, \allowbreak{}{\it aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{shellSort}}\opLeftPren{}{\it sortingFunction}, +\allowbreak{}{\it aggregate}\opRightPren{}% }% }% {2}{((S, S)->Boolean, V)->V}{FiniteLinearAggregateSort} -{\smath{\mbox{\bf shellSort}\opLeftPren{}f, \allowbreak{} a\opRightPren{}} sorts the aggregate \smath{a} +{\smath{\mbox{\bf shellSort}\opLeftPren{}f, \allowbreak{} a\opRightPren{}} +sorts the aggregate \smath{a} using the shellSort algorithm with sorting function \smath{f}. Aggregate \smath{a} can be any finite linear aggregate which is mutable (for example, @@ -79275,11 +83096,13 @@ The sorting function \smath{f} has type \spadsig{(R, R)}{Boolean} where } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{shift}}\opLeftPren{}{\it integerNumber}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{shift}}\opLeftPren{}{\it integerNumber}, +\allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf shift}\opLeftPren{}a, \allowbreak{} i\opRightPren{}} shifts integer number or float \smath{a} by \smath{i} digits. +{\smath{\mbox{\bf shift}\opLeftPren{}a, \allowbreak{} i\opRightPren{}} +shifts integer number or float \smath{a} by \smath{i} digits. } @@ -79288,29 +83111,36 @@ The sorting function \smath{f} has type \spadsig{(R, R)}{Boolean} where }% }% {0}{()->Boolean}{Stream} -{\smath{\mbox{\bf showAll?}\opLeftPren{}\opRightPren{}} tests if all computed entries of streams will be displayed +{\smath{\mbox{\bf showAll?}\opLeftPren{}\opRightPren{}} +tests if all computed entries of streams will be displayed according to system command \spadsyscom{)set streams showall}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{showAllElements}}\opLeftPren{}{\it stream}\opRightPren{}% +\opdata{{\mbox{\axiomFun{showAllElements}}\opLeftPren{} +{\it stream}\opRightPren{}% }% }% {1}{(\$)->OutputForm}{Stream} -{\smath{\mbox{\bf showAllElements}\opLeftPren{}s\opRightPren{}} creates an output form displaying all the already +{\smath{\mbox{\bf showAllElements}\opLeftPren{}s\opRightPren{}} +creates an output form displaying all the already computed elements of stream \smath{s}. This command will not result in any further computation of elements of \smath{s}. Also, the command has no effect if the user has previously entered \spadsyscom{)set streams showall true}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{showTypeInOutput}}\opLeftPren{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{showTypeInOutput}}\opLeftPren{} +{\it boolean}\opRightPren{}% }% }% {1}{(Boolean)->String}{Any} -{\smath{\mbox{\bf showTypeInOutput}\opLeftPren{}bool\opRightPren{}} affects the way objects of \spadtype{Any} -are displayed. If \smath{bool} is \smath{true}, the type of the original object -that was converted to \spadtype{Any} will be printed. If \smath{bool} is \smath{false}, +{\smath{\mbox{\bf showTypeInOutput}\opLeftPren{}bool\opRightPren{}} +affects the way objects of \spadtype{Any} +are displayed. If \smath{bool} is \smath{true}, the type of the original +object +that was converted to \spadtype{Any} will be printed. +If \smath{bool} is \smath{false}, it will not be printed. } @@ -79319,44 +83149,63 @@ it will not be printed. }% }% {1}{(Boolean)->String}{Any} -{\smath{\mbox{\bf shrinkable}\opLeftPren{}b\opRightPren{}\$R} tells \Language{} that flexible arrays of domain \smath{R} +{\smath{\mbox{\bf shrinkable}\opLeftPren{}b\opRightPren{}\$R} +tells Axiom that flexible arrays of domain \smath{R} are or are not allowed to shrink (reduce their \spadfun{physicalLength}) according to whether \smath{b} is \smath{true} or \smath{false}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{shufflein}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it streamOfListsOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{shufflein}}\opLeftPren{} +{\it listOfIntegers}, \allowbreak{}{\it streamOfListsOfIntegers} +\opRightPren{}% }% }% -{2}{(List(Integer), Stream(List(Integer)))->Stream(List(Integer))}{PartitionsAndPermutations} -{\smath{\mbox{\bf shufflein}\opLeftPren{}li, \allowbreak{} sli\opRightPren{}} maps \smath{\mbox{\bf shuffle}\opLeftPren{}li, \allowbreak{} u\opRightPren{}} onto all members \smath{u} +{2}{(List(Integer), Stream(List(Integer)))->Stream(List(Integer))} +{PartitionsAndPermutations} +{\smath{\mbox{\bf shufflein}\opLeftPren{}li, +\allowbreak{} sli\opRightPren{}} +maps \smath{\mbox{\bf shuffle}\opLeftPren{}li, +\allowbreak{} u\opRightPren{}} onto all members \smath{u} of \smath{sli}, concatenating the results. See \spadtype{PartitionsAndPermutations}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{shuffle}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{shuffle}}\opLeftPren{} +{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% }% }% -{2}{(List(Integer), List(Integer))->Stream(List(Integer))}{PartitionsAndPermutations} -{\smath{\mbox{\bf shuffle}\opLeftPren{}l1, \allowbreak{} l2\opRightPren{}} forms the stream -of all shuffles of \smath{l1} and \smath{l2}, that is, all sequences that can +{2}{(List(Integer), List(Integer))->Stream(List(Integer))} +{PartitionsAndPermutations} +{\smath{\mbox{\bf shuffle}\opLeftPren{}l1, +\allowbreak{} l2\opRightPren{}} forms the stream +of all shuffles of \smath{l1} and \smath{l2}, that is, +all sequences that can be formed from merging \smath{l1} and \smath{l2}. See \spadtype{PartitionsAndPermutations}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sign}}\opLeftPren{}{\it various\opt{, \ldots}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sign}}\opLeftPren{} +{\it various\opt{, \ldots}}\opRightPren{}% }% }% -{4}{(Fraction(Polynomial(R)), Symbol, Fraction(Polynomial(R)), String)->Union(Integer, "failed")}{RationalFunctionSign} -{\smath{\mbox{\bf sign}\opLeftPren{}x\opRightPren{}}, where \smath{x} is an element of an ordered ring, -returns 1 if \smath{x} is positive, \smath{-1} if \smath{x} is negative, 0 if \smath{x} equals 0. +{4}{(Fraction(Polynomial(R)), Symbol, +Fraction(Polynomial(R)), String)->Union(Integer, "failed")} +{RationalFunctionSign} +{\smath{\mbox{\bf sign}\opLeftPren{}x\opRightPren{}}, +where \smath{x} is an element of an ordered ring, +returns 1 if \smath{x} is positive, \smath{-1} +if \smath{x} is negative, 0 if \smath{x} equals 0. \newitem -\smath{\mbox{\bf sign}\opLeftPren{}p\opRightPren{}}, where \smath{p} is a permutation, returns +\smath{\mbox{\bf sign}\opLeftPren{}p\opRightPren{}}, +where \smath{p} is a permutation, returns \smath{1}, if \smath{p} is an even permutation, or \smath{-1}, if it is odd. \newitem -\smath{\mbox{\bf sign}\opLeftPren{}f, \allowbreak{} x, \allowbreak{} a, \allowbreak{} s\opRightPren{}} returns the sign of rational function \smath{f} as +\smath{\mbox{\bf sign}\opLeftPren{}f, \allowbreak{} x, +\allowbreak{} a, \allowbreak{} s\opRightPren{}} returns +the sign of rational function \smath{f} as symbol \smath{x} nears \smath{a}, a real value represented by either a rational function or one of the values {\tt \%plusInfinity} @@ -79370,36 +83219,48 @@ If \smath{s} is: } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{simplify}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{simplify}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf simplify}\opLeftPren{}f\opRightPren{}} performs the following simplifications on \smath{f:} +{\smath{\mbox{\bf simplify}\opLeftPren{}f\opRightPren{}} +performs the following simplifications on \smath{f:} \begin{simpleList} \item rewrites trigs and hyperbolic trigs in terms of \smath{sin}, \smath{cos}, \smath{sinh}, \smath{cosh}. -\item rewrites \smath{sin^2} and \smath{sinh^2} in terms of \smath{cos} and \smath{cosh}. +\item rewrites \smath{sin^2} and \smath{sinh^2} in terms of +\smath{cos} and \smath{cosh}. \item rewrites \smath{e^a e^b} as \smath{e^{a+b}}. \end{simpleList} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{simplifyExp}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{simplifyExp}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf simplifyExp}\opLeftPren{}f\opRightPren{}} converts every product \smath{e^a +{\smath{\mbox{\bf simplifyExp}\opLeftPren{}f\opRightPren{}} +converts every product \smath{e^a e^b} appearing in \smath{f} into \smath{e^{a+b}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{simpson}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it threeIntegers}\opRightPren{}% -\optand \mbox{\axiomFun{simpsonClosed}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% -\opand \mbox{\axiomFun{simpsonOpen}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{simpson}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it threeIntegers} +\opRightPren{}% +\optand \mbox{\axiomFun{simpsonClosed}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% +\opand \mbox{\axiomFun{simpsonOpen}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf simpson}\opLeftPren{}fn, \allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, \allowbreak{} nmin, \allowbreak{} nmax, \allowbreak{} nint\opRightPren{}} uses the +{\smath{\mbox{\bf simpson}\opLeftPren{}fn, \allowbreak{} a, +\allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, +\allowbreak{} nmin, +\allowbreak{} nmax, \allowbreak{} nint\opRightPren{}} uses the adaptive simpson method to numerically integrate function \smath{fn} over the closed interval from \smath{a} to \smath{b}, with relative accuracy \smath{epsrel} and absolute accuracy \smath{epsabs}; @@ -79422,12 +83283,16 @@ function evaluations, and {\tt success}, a boolean value which is \smath{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details. -\bigitem\smath{\mbox{\bf simpsonClosed}\opLeftPren{}fn, \allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, \allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} similarly uses +\bigitem\smath{\mbox{\bf simpsonClosed}\opLeftPren{}fn, +\allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, +\allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} similarly uses the Simpson method to numerically integrate function \smath{fn} \index{Simpson's method} over the closed interval \smath{a} to \smath{b}, but is not adaptive. -\bigitem\smath{\mbox{\bf simpsonOpen}\opLeftPren{}fn, \allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, \allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} is similar +\bigitem\smath{\mbox{\bf simpsonOpen}\opLeftPren{}fn, +\allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, +\allowbreak{} nmin, \allowbreak{} nmax\opRightPren{}} is similar to \spadfun{simpsonClosed}, except that it integrates function \smath{fn} over the open interval from \smath{a} to \smath{b}. @@ -79438,13 +83303,17 @@ the open interval from \smath{a} to \smath{b}. }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} -{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or +{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, +\spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } \newitem -\smath{\mbox{\bf sin}\opLeftPren{}x\opRightPren{}} returns the sine of \smath{x} if possible, and +\smath{\mbox{\bf sin}\opLeftPren{}x\opRightPren{}} returns +the sine of \smath{x} if possible, and calls \spadfun{error} otherwise. \newitem -\smath{\mbox{\bf sinIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf sin}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf sinIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf sin}\opLeftPren{}x\opRightPren{}} if possible, +and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- @@ -79452,45 +83321,60 @@ calls \spadfun{error} otherwise. }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf sin2csc}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf sin}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\rm csc}(u)}. +{\smath{\mbox{\bf sin2csc}\opLeftPren{}f\opRightPren{}} +converts every \smath{\mbox{\bf sin}\opLeftPren{}u\opRightPren{}} +appearing in \smath{f} into \smath{1/{\rm csc}(u)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{singular?}}\opLeftPren{}{\it polynomialOrFunctionField}\opRightPren{}% +\opdata{{\mbox{\axiomFun{singular?}}\opLeftPren{} +{\it polynomialOrFunctionField}\opRightPren{}% \opand \mbox{\axiomFun{singularAtInfinity?}}\opLeftPren{}\opRightPren{}% }% }% {1}{(F)->Boolean}{FunctionFieldCategory} -{\smath{\mbox{\bf singular?}\opLeftPren{}p\opRightPren{}} tests whether \smath{p(x) = 0} is singular. +{\smath{\mbox{\bf singular?}\opLeftPren{}p\opRightPren{}} +tests whether \smath{p(x) = 0} is singular. \newitem -\smath{\mbox{\bf singular?}\opLeftPren{}a\opRightPren{}\$F} tests if \smath{x = a} is a singularity of the -algebraic function field \smath{F} (a domain of \spadtype{FunctionFieldCategory}). +\smath{\mbox{\bf singular?}\opLeftPren{}a\opRightPren{}\$F} +tests if \smath{x = a} is a singularity of the +algebraic function field \smath{F} (a domain of +\spadtype{FunctionFieldCategory}). \newitem -\smath{\mbox{\bf singularAtInfinity?}\opLeftPren{}\opRightPren{}\$F} tests if the +\smath{\mbox{\bf singularAtInfinity?}\opLeftPren{}\opRightPren{}\$F} +tests if the algebraic function field \smath{F} has a singularity at infinity. } % ---------------------------------------------------------------------- \opdata{{\mbox{\axiomFun{sinh}}\opLeftPren{}{\it expression}\opRightPren{}% - \opand \mbox{\axiomFun{sinhIfCan}}\opLeftPren{}{\it expression}\opRightPren{}% + \opand \mbox{\axiomFun{sinhIfCan}}\opLeftPren{}{\it expression} +\opRightPren{}% }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} -{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or +{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, +\spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } \newitem -\smath{\mbox{\bf sinh}\opLeftPren{}x\opRightPren{}} returns the hyperbolic sine of \smath{x} if possible, +\smath{\mbox{\bf sinh}\opLeftPren{}x\opRightPren{}} returns the +hyperbolic sine of \smath{x} if possible, and calls \spadfun{error} otherwise. \newitem -\smath{\mbox{\bf sinhIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf sinh}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf sinhIfCan}\opLeftPren{}x\opRightPren{}} returns +\smath{\mbox{\bf sinh}\opLeftPren{}x\opRightPren{}} if possible, and +\mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sinh2csch}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sinh2csch}}\opLeftPren{}{\it expression} +\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf sinh2csch}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf sinh}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\rm csch}(u)}. +{\smath{\mbox{\bf sinh2csch}\opLeftPren{}f\opRightPren{}} converts +every \smath{\mbox{\bf sinh}\opLeftPren{}u\opRightPren{}} appearing in +\smath{f} into \smath{1/{\rm csch}(u)}. } % ---------------------------------------------------------------------- @@ -79498,26 +83382,32 @@ and calls \spadfun{error} otherwise. }% }% {0}{()->NonNegativeInteger}{Finite} -{\smath{\mbox{\bf size}\opLeftPren{}\opRightPren{}\$F} returns the number of elements in the domain of +{\smath{\mbox{\bf size}\opLeftPren{}\opRightPren{}\$F} returns the +number of elements in the domain of category \spadtype{Finite}. By definition, each such domain must have a finite number of elements. \seeAlso{\spadtype{FreeAbelianMonoidCategory}} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{size?}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{size?}}\opLeftPren{}{\it aggregate}, +\allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->Boolean}{Aggregate} -{\smath{\mbox{\bf size?}\opLeftPren{}a, \allowbreak{} n\opRightPren{}} tests if aggregate \smath{a} has exactly \smath{n} elements. +{\smath{\mbox{\bf size?}\opLeftPren{}a, \allowbreak{} n\opRightPren{}} +tests if aggregate \smath{a} has exactly \smath{n} elements. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sizeLess?}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sizeLess?}}\opLeftPren{}{\it element}, +\allowbreak{}{\it element}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{EuclideanDomain} -{\smath{\mbox{\bf sizeLess?}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} tests whether \smath{x} is strictly smaller than \smath{y} with respect to the \spadfunFrom{euclideanSize}{EuclideanDomain}. +{\smath{\mbox{\bf sizeLess?}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +tests whether \smath{x} is strictly smaller than \smath{y} with respect +to the \spadfunFrom{euclideanSize}{EuclideanDomain}. } % ---------------------------------------------------------------------- @@ -79525,95 +83415,115 @@ a finite number of elements. }% }% {0}{()->NonNegativeInteger}{FiniteFieldNormalBasis} -{\smath{\mbox{\bf sizeMultiplication}\opLeftPren{}\opRightPren{}\$F} returns the number of entries in the +{\smath{\mbox{\bf sizeMultiplication}\opLeftPren{}\opRightPren{}\$F} +returns the number of entries in the multiplication table of the field. Note: The time of multiplication of field elements depends on this size. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{skewSFunction}}\opLeftPren{}{\it listOfIntegers}, \allowbreak{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{skewSFunction}}\opLeftPren{}{\it listOfIntegers}, +\allowbreak{}{\it listOfIntegers}\opRightPren{}% }% }% {(L I, L I) -> SPOL RN}{}{} -{\smath{\mbox{\bf skewSFunction}\opLeftPren{}li_1, \allowbreak{} li_2\opRightPren{}} is the S-function +{\smath{\mbox{\bf skewSFunction}\opLeftPren{}li_1, \allowbreak{} li_2 +\opRightPren{}} is the S-function of the partition difference \smath{li_1 - li_2}, expressed in terms of power sum symmetric functions. See \spadtype{CycleIndicators} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{solve}}\opLeftPren{}{\it u}, \allowbreak{}{\it v}\allowbreak $\,[$ , \allowbreak{}{\it w}$]$\opRightPren{}% +\opdata{{\mbox{\axiomFun{solve}}\opLeftPren{}{\it u}, \allowbreak{} +{\it v}\allowbreak $\,[$ , \allowbreak{}{\it w}$]$\opRightPren{}% }% }% -{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))}{TransSolvePackage} +{1}{(Equation(Expression(R)))->List(Equation(Expression(R)))} +{TransSolvePackage} {\bigopkey{\smath{eq} denotes an equation to be solved; alternatively, -an expression \smath{u} may be given for \smath{eq} in which case the equation -\smath{eq} is defined as \smath{u=0}. -\largerbreak\smath{leq} denotes a list \smath{[ eq_1\ldots eq_n]} of equations; alternatively, +an expression \smath{u} may be given for \smath{eq} in which case the +equation \smath{eq} is defined as \smath{u=0}. +\largerbreak\smath{leq} denotes a list \smath{[ eq_1\ldots eq_n]} of +equations; alternatively, a list of expressions \smath{[ u_1\ldots u_n]} may be given for \smath{leq} in which case the equations \smath{eq_i} are defined by \smath{u_i=0}. \largerbreak\smath{epsilon} is either a rational number or a float. } -\bigitem\smath{\mbox{\bf complexSolve}\opLeftPren{}eq, \allowbreak{} epsilon\opRightPren{}} finds all the real solutions +\bigitem\smath{\mbox{\bf complexSolve}\opLeftPren{}eq, \allowbreak{} +epsilon\opRightPren{}} finds all the real solutions to precision \smath{epsilon} of the univariate equation \smath{eq} of rational functions with respect to the unique variable appearing in \smath{eq}. The complex solutions are either expressed as rational numbers or floats depending on the type of \smath{epsilon}. -\bigitem\smath{\mbox{\bf complexSolve}\opLeftPren{}[ eq_1\ldots eq_n], epsilon\opRightPren{}} +\bigitem\smath{\mbox{\bf complexSolve}\opLeftPren{}[ eq_1\ldots eq_n], +epsilon\opRightPren{}} computes the real solutions to precision \smath{epsilon} of a system of equations \smath{eq_i} involving rational functions. The complex solutions are either expressed as rational numbers or floats depending on the type of \smath{epsilon}. -\bigitem\smath{\mbox{\bf radicalSolve}\opLeftPren{}eq\optinner{, x}\opRightPren{}} finds solutions expressed in terms of +\bigitem\smath{\mbox{\bf radicalSolve}\opLeftPren{}eq\optinner{, x} +\opRightPren{}} finds solutions expressed in terms of radicals of the equation \smath{eq} involving rational functions. Solutions will be found with respect to a \spadtype{Symbol} given as a second argument to the operation. This second argument may be omitted when \smath{eq} contains a unique symbol. -\bigitem\smath{\mbox{\bf radicalSolve}\opLeftPren{}leq{, lv}\opRightPren{}} finds solutions +\bigitem\smath{\mbox{\bf radicalSolve}\opLeftPren{}leq{, lv}\opRightPren{}} +finds solutions expressed in terms of radicals of the system of equations \smath{leq} involving rational functions. Solutions are found with respect to a list \smath{lv} of \spadtype{Symbol}s, or with respect to all variables appearing in the equations, if no second argument is given. -\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq\optinner{, x}\opRightPren{}} finds exact symbolic solutions to equation +\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq\optinner{, x}\opRightPren{}} +finds exact symbolic solutions to equation \smath{eq} involving either rational functions or expressions of type \spadtype{Expression(R)}. Solutions will be found with respect to a \spadtype{Symbol} given as a second argument to the operation. The second argument may be omitted when \smath{eq} contains a unique symbol. -\bigitem\smath{\mbox{\bf solve}\opLeftPren{}leq{, lv}\opRightPren{}} finds exact solutions -to a system of equations \smath{leq} involving rational functions or expressions +\bigitem\smath{\mbox{\bf solve}\opLeftPren{}leq{, lv}\opRightPren{}} +finds exact solutions +to a system of equations \smath{leq} involving rational functions +or expressions of type \smath{\mbox{\bf Expression}\opLeftPren{}R\opRightPren{}}. -Solutions are found with respect to a list of \smath{lv} of \spadtype{Symbol}s, +Solutions are found with respect to a list of \smath{lv} of +\spadtype{Symbol}s, or with respect to all variables appearing in the equations if no second argument is given. -\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq, \allowbreak{} epsilon\opRightPren{}} finds all the real solutions +\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq, \allowbreak{} +epsilon\opRightPren{}} finds all the real solutions to precision \smath{epsilon} of the univariate equation \smath{eq} of rational functions with respect to the unique variable appearing in \smath{eq}. The real solutions are either expressed as rational numbers or floats depending on the type of \smath{epsilon}. -\bigitem\smath{\mbox{\bf solve}\opLeftPren{}[ eq_1\ldots eq_n], epsilon\opRightPren{}} +\bigitem\smath{\mbox{\bf solve}\opLeftPren{}[ eq_1\ldots eq_n], +epsilon\opRightPren{}} computes the real solutions to precision \smath{epsilon} of a system of equations \smath{eq_i} involving rational functions. The real solutions are either expressed as rational numbers or floats depending on the type of \smath{epsilon}. -\bigitem\smath{\mbox{\bf solve}\opLeftPren{}M, \allowbreak{} v\opRightPren{}}, where \smath{M} is a matrix and +\bigitem\smath{\mbox{\bf solve}\opLeftPren{}M, +\allowbreak{} v\opRightPren{}}, where \smath{M} is a matrix and \smath{v} is a \spadtype{Vector} of coefficients, finds a particular -solution of the system \smath{Mx=v} and a basis of the associated homogeneous system +solution of the system \smath{Mx=v} and a basis of the associated +homogeneous system \smath{MX=0}. -\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq, \allowbreak{} y, \allowbreak{} x=a, \allowbreak{} [ y_0\ldots y_m]\opRightPren{}} returns either +\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq, \allowbreak{} y, +\allowbreak{} x=a, \allowbreak{} [ y_0\ldots y_m]\opRightPren{}} returns either the solution of the initial value problem \smath{eq}, \smath{y(a) = y_0}, \smath{y'(a)=a_1}, \smath{\ldots} or \mbox{\tt "failed"} if no solution can be found. Note: an error occurs if the equation \smath{eq} is not a linear ordinary equation or of the form \smath{dy/dx = f(x, y)}. -\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq, \allowbreak{} y, \allowbreak{} x\opRightPren{}} returns either a solution of the +\bigitem\smath{\mbox{\bf solve}\opLeftPren{}eq, \allowbreak{} y, +\allowbreak{} x\opRightPren{}} returns either a solution of the ordinary diffential equation \smath{eq} or \mbox{\tt "failed"} if no non-trivial solution can be found. If \smath{eq} is a linear ordinary differential equation, a solution is of the form @@ -79632,11 +83542,15 @@ always a full basis). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{solveLinearlyOverQ}}\opLeftPren{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{solveLinearlyOverQ}}\opLeftPren{} +{\it vector}\opRightPren{}% }% }% -{2}{(Vector(R), R)->Union(Vector(Fraction(Integer)), "failed")}{IntegerLinearDependence} -{\smath{\mbox{\bf solveLinearlyOverQ}\opLeftPren{}[v_1, \allowbreak{} \ldots, v_n], u\opRightPren{}} returns \smath{[c_1, \ldots, c_n]} such that +{2}{(Vector(R), R)->Union(Vector(Fraction(Integer)), "failed")} +{IntegerLinearDependence} +{\smath{\mbox{\bf solveLinearlyOverQ}\opLeftPren{}[v_1, +\allowbreak{} \ldots, v_n], u\opRightPren{}} +returns \smath{[c_1, \ldots, c_n]} such that \smath{c_1v_1 + \cdots + c_n v_n = u}, or \mbox{\tt "failed"} if no such rational numbers \smath{c_i} exist. The elements of the \smath{v_i} and \smath{u} can @@ -79647,38 +83561,52 @@ See \spadtype{LinearExplicitRingOver}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{solveLinearPolynomialEquation}}\opLeftPren{}{\it listOfPolys}, \allowbreak{}{\it poly}\opRightPren{}% +\opdata{{\mbox{\axiomFun{solveLinearPolynomialEquation}}\opLeftPren{} +{\it listOfPolys}, \allowbreak{}{\it poly}\opRightPren{}% }% }% -{2}{(List(SparseUnivariatePolynomial(\$)), SparseUnivariatePolynomial(\$))->Union(List(SparseUnivariatePolynomial(\$)), "failed")}{PolynomialFactorizationExplicit} -{\smath{\mbox{\bf solveLinearPolynomialEquation}\opLeftPren{}[f_1, \allowbreak{} \ldots, f_n], g\opRightPren{}}, -where \smath{g} is a polynomial and the \smath{f_i} are polynomials relatively prime to +{2}{(List(SparseUnivariatePolynomial(\$)), +SparseUnivariatePolynomial(\$))->Union(List(SparseUnivariatePolynomial(\$)), + "failed")} +{PolynomialFactorizationExplicit} +{\smath{\mbox{\bf solveLinearPolynomialEquation}\opLeftPren{} +[f_1, \allowbreak{} \ldots, f_n], g\opRightPren{}}, +where \smath{g} is a polynomial and the \smath{f_i} +are polynomials relatively prime to one another, -returns a list of polynomials \smath{a_i} such that \smath{g/{\prod_i{f_i}} = \sum_i{ai/fi}}, +returns a list of polynomials \smath{a_i} such that +\smath{g/{\prod_i{f_i}} = \sum_i{ai/fi}}, or \mbox{\tt "failed"} if no such list of \smath{a_i}'s exists. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sort}}\opLeftPren{}{\it \optinit{predicate, }aggregate}\opRightPren{}% - \opand \mbox{\axiomFun{sort!}}\opLeftPren{}{\it \optinit{predicate, }aggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sort}}\opLeftPren{}{\it +\optinit{predicate, }aggregate}\opRightPren{}% + \opand \mbox{\axiomFun{sort!}}\opLeftPren{}{\it +\optinit{predicate, }aggregate}\opRightPren{}% }% }% {2}{((S, S)->Boolean, \$)->\$}{FiniteLinearAggregate} -{\smath{\mbox{\bf sort}\opLeftPren{}\optfirst{p, }a\opRightPren{}} returns a copy of +{\smath{\mbox{\bf sort}\opLeftPren{}\optfirst{p, }a\opRightPren{}} +returns a copy of \smath{a} sorted using total ordering predicate \smath{p}. \newitem -\smath{\mbox{\bf sort!}\opLeftPren{}\optfirst{p, }u\opRightPren{}} returns \smath{u} destructively changed with its +\smath{\mbox{\bf sort!}\opLeftPren{}\optfirst{p, }u\opRightPren{}} +returns \smath{u} destructively changed with its elements ordered by comparison function \smath{p}. \newitem By default, \smath{p} is the operation \smath{\leq}. -Thus both \smath{\mbox{\bf sort}\opLeftPren{}u\opRightPren{}} and \smath{\mbox{\bf sort!}\opLeftPren{}u\opRightPren{}} returns \smath{u} with +Thus both \smath{\mbox{\bf sort}\opLeftPren{}u\opRightPren{}} and +\smath{\mbox{\bf sort!}\opLeftPren{}u\opRightPren{}} returns \smath{u} with its elements in ascending order. \newitem -Also: \smath{\mbox{\bf sort}\opLeftPren{}lp\opRightPren{}} sorts a list of permutations {\it lp} +Also: \smath{\mbox{\bf sort}\opLeftPren{}lp\opRightPren{}} +sorts a list of permutations {\it lp} according to cycle structure, first according to the length of cycles, -second, if \smath{S} has \spadtype{Finite} or \smath{S} has \spadtype{OrderedSet}, +second, if \smath{S} has \spadtype{Finite} or \smath{S} has +\spadtype{OrderedSet}, according to lexicographical order of entries in cycles of equal length. } @@ -79688,70 +83616,86 @@ according to lexicographical order of entries in cycles of equal length. }% }% {1}{(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf spherical}\opLeftPren{}pt\opRightPren{}} transforms point \smath{pt} from spherical coordinates to +{\smath{\mbox{\bf spherical}\opLeftPren{}pt\opRightPren{}} +transforms point \smath{pt} from spherical coordinates to Cartesian coordinates, mapping \smath{(r, \theta, \phi)} to \smath{x = r \sin(\phi) \cos(\theta)}, \smath{y = r \sin(\phi) \sin(\theta)}, \smath{z = r \cos(\phi)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{split}}\opLeftPren{}{\it element}, \allowbreak{}{\it binarySearchTree}\opRightPren{}% +\opdata{{\mbox{\axiomFun{split}}\opLeftPren{}{\it element}, \allowbreak{} +{\it binarySearchTree}\opRightPren{}% }% }% {2}{(\$, Integer)->\$}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf split}\opLeftPren{}x, \allowbreak{} t\opRightPren{}} splits binary search tree \smath{t} into two components, returning a +{\smath{\mbox{\bf split}\opLeftPren{}x, \allowbreak{} t\opRightPren{}} +splits binary search tree \smath{t} into two components, returning a record of two components: \smath{less}, a binary search tree whose components are all less than x; and, -\smath{greater}, a binary search tree with all the rest of the components of \spad{t}. +\smath{greater}, a binary search tree with all the rest of the +components of \spad{t}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{split!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{split!}}\opLeftPren{}{\it aggregate}, +\allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, Integer)->\$}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf split!}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} splits \smath{u} into two aggregates: +{\smath{\mbox{\bf split!}\opLeftPren{}u, +\allowbreak{} n\opRightPren{}} splits \smath{u} into two aggregates: the first consisting of \smath{v}, the first \smath{n} elements of \smath{u}, and \smath{w} consisting of all the rest. The value of \smath{w} is returned. Thus \smath{v = {\bf first}(u, n)} and \smath{w := {\bf rest}(u, n)}. -Note: afterwards \smath{\mbox{\bf rest}\opLeftPren{}u, \allowbreak{} n\opRightPren{}} returns \smath{\mbox{\bf empty}\opLeftPren{}\opRightPren{}}. +Note: afterwards \smath{\mbox{\bf rest}\opLeftPren{}u, +\allowbreak{} n\opRightPren{}} returns +\smath{\mbox{\bf empty}\opLeftPren{}\opRightPren{}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{splitDenominator}}\opLeftPren{}{\it listOfFractions}\opRightPren{}% +\opdata{{\mbox{\axiomFun{splitDenominator}}\opLeftPren{} +{\it listOfFractions}\opRightPren{}% }% }% {1}{(A)->Record(num:A, den:R)}{CommonDenominator} {\smath{\mbox{\bf splitDenominator}\opLeftPren{}u\opRightPren{}}, where \smath{u} is a list of fractions \smath{[q_1, \ldots, q_n]}, returns \smath{[[p_1, \ldots, p_n], d]} such that -\smath{q_i = p_i/d} and \smath{d} is a common denominator for the \smath{q_i}'s. +\smath{q_i = p_i/d} and \smath{d} is a common denominator +for the \smath{q_i}'s. Similarly, the function is defined for a matrix (respectively, a polynomial) -\smath{u} in which case the \smath{q_i} are the elements of (respectively, the coefficients of) +\smath{u} in which case the \smath{q_i} are the elements of +(respectively, the coefficients of) \smath{u}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sqfrFactor}}\opLeftPren{}{\it element}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sqfrFactor}}\opLeftPren{} +{\it element}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(R, Integer)->\$}{Factored} -{\smath{\mbox{\bf sqfrFactor}\opLeftPren{}base, \allowbreak{} exponent\opRightPren{}} creates a factored object with +{\smath{\mbox{\bf sqfrFactor}\opLeftPren{}base, +\allowbreak{} exponent\opRightPren{}} creates a factored object with a single factor whose \smath{base} is asserted to be square-free (flag = {\tt "sqfr"}). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sqrt}}\opLeftPren{}{\it expression\opt{, option}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sqrt}}\opLeftPren{} +{\it expression\opt{, option}}\opRightPren{}% }% }% {1}{(\$)->\$}{RadicalCategory} -{\smath{\mbox{\bf sqrt}\opLeftPren{}x\opRightPren{}} returns the square root of \smath{x}. +{\smath{\mbox{\bf sqrt}\opLeftPren{}x\opRightPren{}} +returns the square root of \smath{x}. \newitem -\smath{\mbox{\bf sqrt}\opLeftPren{}x, \allowbreak{} y\opRightPren{}}, where \smath{x} and +\smath{\mbox{\bf sqrt}\opLeftPren{}x, \allowbreak{} y\opRightPren{}}, +where \smath{x} and \smath{y} are \smath{p}-adic integers, returns a square root of \smath{x} where argument \smath{y} is a square root of \smath{x \mod p}. See also \spadtype{PAdicIntegerCategory}. @@ -79762,37 +83706,46 @@ See also \spadtype{PAdicIntegerCategory}. }% }% {1}{(\$)->Boolean}{MatrixCategory} -{\smath{\mbox{\bf square?}\opLeftPren{}m\opRightPren{}} tests if \smath{m} is a square matrix. +{\smath{\mbox{\bf square?}\opLeftPren{}m\opRightPren{}} +tests if \smath{m} is a square matrix. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{squareFree}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{squareFree}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->Factored(\$)}{UniqueFactorizationDomain} -{\smath{\mbox{\bf squareFree}\opLeftPren{}x\opRightPren{}} returns the square-free factorization of \smath{x}, -that is, such that the factors are pairwise relatively prime and each has multiple prime factors. +{\smath{\mbox{\bf squareFree}\opLeftPren{}x\opRightPren{}} +returns the square-free factorization of \smath{x}, +that is, such that the factors are pairwise relatively +prime and each has multiple prime factors. Argument \smath{x} can be a member of any domain of category \spadtype{UniqueFactorizationDomain} such as a polynomial or integer. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{squareFreePart}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{squareFreePart}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->\$}{PolynomialCategory} -{\smath{\mbox{\bf squareFreePart}\opLeftPren{}p\opRightPren{}} returns product of all the prime factors of +{\smath{\mbox{\bf squareFreePart}\opLeftPren{}p\opRightPren{}} +returns product of all the prime factors of \smath{p} each taken with multiplicity one. Argument \smath{p} can be a member of any domain of category \spadtype{UniqueFactorizationDomain} such as a polynomial or integer. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{squareFreePolynomial}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{squareFreePolynomial}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% -{1}{(SparseUnivariatePolynomial(\$))->Factored(SparseUnivariatePolynomial(\$))}{PolynomialFactorizationExplicit} -{\smath{\mbox{\bf squareFreePolynomial}\opLeftPren{}p\opRightPren{}} returns the square-free factorization of the univariate polynomial \smath{p}. +{1}{(SparseUnivariatePolynomial(\$))-> +Factored(SparseUnivariatePolynomial(\$))}{PolynomialFactorizationExplicit} +{\smath{\mbox{\bf squareFreePolynomial}\opLeftPren{}p\opRightPren{}} +returns the square-free factorization of the univariate polynomial \smath{p}. } % ---------------------------------------------------------------------- @@ -79800,7 +83753,8 @@ category \spadtype{UniqueFactorizationDomain} such as a polynomial or integer. }% }% {1}{(\$)->\$}{MatrixCategory} -{\smath{\mbox{\bf squareTop}\opLeftPren{}A\opRightPren{}} returns an \smath{n}-by-\smath{n} matrix +{\smath{\mbox{\bf squareTop}\opLeftPren{}A\opRightPren{}} +returns an \smath{n}-by-\smath{n} matrix consisting of the first \smath{n} rows of the \smath{m}-by-\smath{n} matrix \smath{A}. The operation calls \spadfun{error} if \smath{m < n}. @@ -79811,17 +83765,20 @@ The operation calls \spadfun{error} if \smath{m < n}. }% }% {1}{(List(S))->\$}{StackAggregate} -{\smath{\mbox{\bf stack}\opLeftPren{}[x, \allowbreak{} y, \allowbreak{} \ldots, z]\opRightPren{}} creates a stack with first (top) +{\smath{\mbox{\bf stack}\opLeftPren{}[x, \allowbreak{} y, +\allowbreak{} \ldots, z]\opRightPren{}} creates a stack with first (top) element \smath{x}, second element \smath{y}, \ldots, and last element \smath{z}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{standardBasisOfCyclicSubmodule}}\opLeftPren{}{\it listOfMatrices}, \allowbreak{}{\it vector}\opRightPren{}% +\opdata{{\mbox{\axiomFun{standardBasisOfCyclicSubmodule}}\opLeftPren{} +{\it listOfMatrices}, \allowbreak{}{\it vector}\opRightPren{}% }% }% {2}{(List(Matrix(R)), Vector(R))->Matrix(R)}{RepresentationPackage2} -{\smath{\mbox{\bf standardBasisOfCyclicSubmodule}\opLeftPren{}lm, \allowbreak{} v\opRightPren{}} returns a matrix +{\smath{\mbox{\bf standardBasisOfCyclicSubmodule} +\opLeftPren{}lm, \allowbreak{} v\opRightPren{}} returns a matrix representation of cyclic submodule over a ring \smath{R}, where \smath{lm} is a list of matrices and \smath{v} is a vector, such that the non-zero column vectors are an \smath{R}-basis @@ -79830,14 +83787,18 @@ for \smath{A v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{stirling1}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% - \opand \mbox{\axiomFun{stirling2}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{stirling1}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% + \opand \mbox{\axiomFun{stirling2}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(I, I)->I}{IntegerCombinatoricFunctions} -{\smath{\mbox{\bf stirling1}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} returns the Stirling number of the first kind. +{\smath{\mbox{\bf stirling1}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} +returns the Stirling number of the first kind. \newitem -\smath{\mbox{\bf stirling2}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} returns the Stirling number of the second kind. +\smath{\mbox{\bf stirling2}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} +returns the Stirling number of the second kind. } % ---------------------------------------------------------------------- @@ -79846,33 +83807,41 @@ for \smath{A v}. }% }% {1}{(\$)->Boolean}{SExpressionCategory} -{\smath{\mbox{\bf string?}\opLeftPren{}s\opRightPren{}} tests if \spadtype{SExpression} object \smath{s} is a string. +{\smath{\mbox{\bf string?}\opLeftPren{}s\opRightPren{}} +tests if \spadtype{SExpression} object \smath{s} is a string. \newitem -\smath{\mbox{\bf string}\opLeftPren{}s\opRightPren{}} converts the symbol \smath{s} to a string. +\smath{\mbox{\bf string}\opLeftPren{}s\opRightPren{}} +converts the symbol \smath{s} to a string. An \spadfun{error} is called if the symbol is subscripted. \newitem -\smath{\mbox{\bf string}\opLeftPren{}s\opRightPren{}} returns \spadtype{SExpression} object \smath{s} as +\smath{\mbox{\bf string}\opLeftPren{}s\opRightPren{}} +returns \spadtype{SExpression} object \smath{s} as an element of \spadtype{String} if possible, and otherwise calls \spadfun{error}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{strongGenerators}}\opLeftPren{}{\it listOfPermutations}\opRightPren{}% +\opdata{{\mbox{\axiomFun{strongGenerators}}\opLeftPren{} +{\it listOfPermutations}\opRightPren{}% }% }% {1}{(\$)->List(Permutation(S))}{PermutationGroup} -{\smath{\mbox{\bf strongGenerators}\opLeftPren{}gp\opRightPren{}} returns strong generators for the permutation +{\smath{\mbox{\bf strongGenerators}\opLeftPren{}gp\opRightPren{}} +returns strong generators for the permutation group {\it gp}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{structuralConstants}}\opLeftPren{}{\it basis}\opRightPren{}% +\opdata{{\mbox{\axiomFun{structuralConstants}}\opLeftPren{} +{\it basis}\opRightPren{}% }% }% {0}{()->Vector(Matrix(R))}{FramedNonAssociativeAlgebra} -{\smath{\mbox{\bf structuralConstants}\opLeftPren{}basis\opRightPren{}} calculates the structural constants +{\smath{\mbox{\bf structuralConstants}\opLeftPren{}basis\opRightPren{}} +calculates the structural constants \smath{[(\gamma_{i, j, k}) {\ \tt for\ } k {\ \tt in\ } 1..rank()\$R]} -of a domain \smath{R} of category \spadtype{FramedNonAssociativeAlgebra} over a ring \smath{R}, +of a domain \smath{R} of category +\spadtype{FramedNonAssociativeAlgebra} over a ring \smath{R}, defined by: \smath{v_i v_j = \gamma_{i, j, 1} v_1 + \cdots + \gamma_{i, j, n} v_n}, where \smath{v_1}, \ldots, \smath{v_n} is the fixed \smath{R}-module basis. @@ -79883,13 +83852,15 @@ where \smath{v_1}, \ldots, \smath{v_n} is the fixed \smath{R}-module basis. }% }% {1}{(String)->\$}{DrawOption} -{\smath{\mbox{\bf style}\opLeftPren{}s\opRightPren{}} specifies the drawing style in which the graph +{\smath{\mbox{\bf style}\opLeftPren{}s\opRightPren{}} +specifies the drawing style in which the graph will be plotted by the indicated string \smath{s}. This option is expressed in the form \code{style == s}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sub}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sub}}\opLeftPren{}{\it outputForm}, \allowbreak{} +{\it outputForm}\opRightPren{}% }% }% {}{}{} @@ -79900,157 +83871,212 @@ creates an output form for \smath{o_1} subscripted by \smath{o_2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{subMatrix}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{subMatrix}}\opLeftPren{}{\it matrix}, \allowbreak{} +{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}, +\allowbreak{}{\it integer}\opRightPren{}% }% }% {5}{(\$, Integer, Integer, Integer, Integer)->\$}{MatrixCategory} -{\smath{\mbox{\bf subMatrix}\opLeftPren{}m, \allowbreak{} i_1, \allowbreak{} i_2, \allowbreak{} j_1, \allowbreak{} j_2\opRightPren{}} extracts the submatrix \smath{[m(i, j)]} where the index +{\smath{\mbox{\bf subMatrix}\opLeftPren{}m, \allowbreak{} i_1, +\allowbreak{} i_2, \allowbreak{} j_1, \allowbreak{} j_2\opRightPren{}} +extracts the submatrix \smath{[m(i, j)]} where the index \smath{i} ranges from \smath{i_1} to \smath{i_2} and the index \smath{j} ranges from \smath{j_1} to \smath{j_2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{submod}}\opLeftPren{}{\it integerNumber}, \allowbreak{}{\it integerNumber}, \allowbreak{}{\it integerNumber}\opRightPren{}% +\opdata{{\mbox{\axiomFun{submod}}\opLeftPren{}{\it integerNumber}, +\allowbreak{}{\it integerNumber}, \allowbreak{} +{\it integerNumber}\opRightPren{}% }% }% {3}{(\$, \$, \$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf submod}\opLeftPren{}a, \allowbreak{} b, \allowbreak{} p\opRightPren{}}, where \smath{0 \leq a < b < p > 1}, returns \smath{a-b\mod p}, +{\smath{\mbox{\bf submod}\opLeftPren{}a, \allowbreak{} b, +\allowbreak{} p\opRightPren{}}, where \smath{0 \leq a < b < p > 1}, returns +\smath{a-b\mod p}, for integer numbers \smath{a}, \smath{b} and \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{subResultantGcd}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{subResultantGcd}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it polynomial}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf subResultantGcd}\opLeftPren{}p, \allowbreak{} q\opRightPren{}} computes the \smath{gcd} of the polynomials \smath{p} and \smath{q} using the SubResultant \smath{GCD} algorithm. +{\smath{\mbox{\bf subResultantGcd}\opLeftPren{}p, +\allowbreak{} q\opRightPren{}} computes the \smath{gcd} of the polynomials +\smath{p} and \smath{q} using the SubResultant \smath{GCD} algorithm. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{subscript}}\opLeftPren{}{\it symbol}, \allowbreak{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{subscript}}\opLeftPren{}{\it symbol}, +\allowbreak{}{\it listOfOutputForms}\opRightPren{}% }% }% {2}{(\$, List(OutputForm))->\$}{Symbol} -{\smath{\mbox{\bf subscript}\opLeftPren{}s, \allowbreak{} [a1, \allowbreak{} \ldots, an]\opRightPren{}} returns symbol \smath{s} subscripted by output forms +{\smath{\mbox{\bf subscript}\opLeftPren{}s, \allowbreak{} +[a1, \allowbreak{} \ldots, an]\opRightPren{}} returns symbol \smath{s} +subscripted by output forms \smath{a_1, \ldots, a_n} as a symbol. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{subset}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{subset}}\opLeftPren{}{\it integer}, +\allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% -{3}{(Integer, Integer, Integer)->List(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf subSet}\opLeftPren{}n, \allowbreak{} m, \allowbreak{} k\opRightPren{}} calculates the \eth{\smath{k}} \smath{m}-subset of the set -\smath{0, 1, \ldots, (n-1)} in the lexicographic order considered as a decreasing map from +{3}{(Integer, Integer, Integer)->List(Integer)} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf subSet}\opLeftPren{}n, \allowbreak{} m, +\allowbreak{} k\opRightPren{}} calculates the \eth{\smath{k}} \smath{m}-subset +of the set +\smath{0, 1, \ldots, (n-1)} in the lexicographic order +considered as a decreasing map from \smath{0, \ldots, (m-1)} into \smath{0, \ldots, (n-1)}. See \spadtype{SymmetricGroupCombinatoricFunctions}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{subset?}}\opLeftPren{}{\it set}, \allowbreak{}{\it set}\opRightPren{}% +\opdata{{\mbox{\axiomFun{subset?}}\opLeftPren{}{\it set}, +\allowbreak{}{\it set}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{SetAggregate} -{\smath{\mbox{\bf subset?}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} tests if set \smath{u} is a subset of set \smath{v}. +{\smath{\mbox{\bf subset?}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} tests if set \smath{u} +is a subset of set \smath{v}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{subspace}}\opLeftPren{}{\it threeSpace}\opRightPren{}% +\opdata{{\mbox{\axiomFun{subspace}}\opLeftPren{} +{\it threeSpace}\opRightPren{}% }% }% {1}{(\$)->SubSpace(3, R)}{ThreeSpace} -{\smath{\mbox{\bf subspace}\opLeftPren{}s\opRightPren{}} returns the space component which holds all the point +{\smath{\mbox{\bf subspace}\opLeftPren{}s\opRightPren{}} +returns the space component which holds all the point information in the \spadtype{ThreeSpace} object \smath{s}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{substring?}}\opLeftPren{}{\it string}, \allowbreak{}{\it string}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{substring?}}\opLeftPren{}{\it string}, +\allowbreak{}{\it string}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {3}{(\$, \$, Integer)->Boolean}{StringAggregate} -{\smath{\mbox{\bf substring?}\opLeftPren{}s, \allowbreak{} t, \allowbreak{} i\opRightPren{}} tests if \smath{s} is a substring of \smath{t} +{\smath{\mbox{\bf substring?}\opLeftPren{}s, \allowbreak{} t, +\allowbreak{} i\opRightPren{}} tests if \smath{s} is a substring of \smath{t} beginning at index \smath{i}. Note: \code{substring?(s, t, 0) = prefix?(s, t)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{subst}}\opLeftPren{}{\it expression}, \allowbreak{}{\it equations}\opRightPren{}% +\opdata{{\mbox{\axiomFun{subst}}\opLeftPren{}{\it expression}, +\allowbreak{}{\it equations}\opRightPren{}% }% }% {2}{(\$, Equation(\$))->\$}{ExpressionSpace} -{\smath{\mbox{\bf subst}\opLeftPren{}f, \allowbreak{} k = g\opRightPren{}} formally replaces the kernel \smath{k} by \smath{g} in \smath{f}. +{\smath{\mbox{\bf subst}\opLeftPren{}f, \allowbreak{} k = g\opRightPren{}} +formally replaces the kernel \smath{k} by \smath{g} in \smath{f}. \newitem -\smath{\mbox{\bf subst}\opLeftPren{}f, \allowbreak{} [k_1 = g_1, \allowbreak{} \ldots, k_n = g_n]\opRightPren{}} formally replaces the kernels -\smath{k_1}, \ldots, \smath{k_n} by \smath{g_1}, \ldots, \smath{g_n} in \smath{f}. +\smath{\mbox{\bf subst}\opLeftPren{}f, \allowbreak{} +[k_1 = g_1, \allowbreak{} \ldots, k_n = g_n]\opRightPren{}} +formally replaces the kernels +\smath{k_1}, \ldots, \smath{k_n} by \smath{g_1}, \ldots, +\smath{g_n} in \smath{f}. \newitem -\smath{\mbox{\bf subst}\opLeftPren{}f, \allowbreak{} [k_1, \allowbreak{} \ldots, k_n], [g_1, \ldots, g_n]\opRightPren{}} formally replaces kernels +\smath{\mbox{\bf subst}\opLeftPren{}f, \allowbreak{} [k_1, +\allowbreak{} \ldots, k_n], [g_1, \ldots, g_n]\opRightPren{}} +formally replaces kernels \smath{k_i} by \smath{g_i} in \smath{f}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{suchThat}}\opLeftPren{}{\it symbol}, \allowbreak{}{\it predicates}\opRightPren{}% +\opdata{{\mbox{\axiomFun{suchThat}}\opLeftPren{}{\it symbol}, +\allowbreak{}{\it predicates}\opRightPren{}% }% }% {2}{(Symbol, List((D)->Boolean))->Expression(Integer)}{AttachPredicates} -{\smath{\mbox{\bf suchThat}\opLeftPren{}sy, \allowbreak{} pred\opRightPren{}} attaches the predicate \smath{pred} to symbol \smath{sy}. +{\smath{\mbox{\bf suchThat}\opLeftPren{}sy, +\allowbreak{} pred\opRightPren{}} attaches the +predicate \smath{pred} to symbol \smath{sy}. Argument \smath{pred} may also be a list \smath{[p_1, \ldots, p_n]} of predicates \smath{p_i}. In this case, the predicate \smath{pred} attached to \smath{sy} is \smath{p_1 \and \ldots \and p_n}. \newitem -\smath{\mbox{\bf suchThat}\opLeftPren{}r, \allowbreak{} [a_1, \allowbreak{} \ldots, a_n], f\opRightPren{}} returns the rewrite rule \smath{r} with the +\smath{\mbox{\bf suchThat}\opLeftPren{}r, \allowbreak{} [a_1, +\allowbreak{} \ldots, a_n], f\opRightPren{}} returns the rewrite rule +\smath{r} with the predicate \smath{f(a1, \ldots, an)} attached to it. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{suffix?}}\opLeftPren{}{\it string}, \allowbreak{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{suffix?}}\opLeftPren{}{\it string}, +\allowbreak{}{\it string}\opRightPren{}% }% }% {2}{(\$, \$)->Boolean}{StringAggregate} -{\smath{\mbox{\bf suffix?}\opLeftPren{}s, \allowbreak{} t\opRightPren{}} tests if the string \smath{s} is the final substring of \smath{t}. +{\smath{\mbox{\bf suffix?}\opLeftPren{}s, +\allowbreak{} t\opRightPren{}} tests if the string \smath{s} +is the final substring of \smath{t}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sum}}\opLeftPren{}{\it rationalFunction}, \allowbreak{}{\it symbolOrSegmentBinding}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sum}}\opLeftPren{} +{\it rationalFunction}, \allowbreak{} +{\it symbolOrSegmentBinding}\opRightPren{}% }% }% -{2}{(Fraction(Polynomial(R)), Symbol)->Union(Fraction(Polynomial(R)), Expression(R))}{RationalFunctionSum} +{2}{(Fraction(Polynomial(R)), +Symbol)->Union(Fraction(Polynomial(R)), Expression(R))}{RationalFunctionSum} {\smath{\mbox{\bf sum}\opLeftPren{}a(n), \allowbreak{} n\opRightPren{}}, where \smath{a(n)} is an rational function or expression involving the symbol \smath{n}, returns the indefinite sum \smath{A} of \smath{a} with respect -to upward difference on \smath{n}, that is, \smath{A(n+1) - A(n) = a(n)}. +to upward difference on \smath{n}, that is, +\smath{A(n+1) - A(n) = a(n)}. \newitem -\smath{\mbox{\bf sum}\opLeftPren{}f(n), \allowbreak{} n = a..b\opRightPren{}}, where +\smath{\mbox{\bf sum}\opLeftPren{}f(n), +\allowbreak{} n = a..b\opRightPren{}}, where \smath{f(n)}, \smath{a}, and \smath{b} are rational functions (or polynomials), -computes and returns the sum \smath{f(a) + f(a+1) + \cdots + f(b)} as a rational function +computes and returns the sum \smath{f(a) + f(a+1) + \cdots + f(b)} +as a rational function (or polynomial). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{summation}}\opLeftPren{}{\it expression}, \allowbreak{}{\it segmentBinding}\opRightPren{}% +\opdata{{\mbox{\axiomFun{summation}}\opLeftPren{} +{\it expression}, \allowbreak{}{\it segmentBinding}\opRightPren{}% }% }% {2}{(\$, SegmentBinding(\$))->\$}{CombinatorialOpsCategory} -{\smath{\mbox{\bf summation}\opLeftPren{}f, \allowbreak{} n = a..b\opRightPren{}} returns the formal sum \smath{\sum_{n=a}^b f(n)}. +{\smath{\mbox{\bf summation}\opLeftPren{}f, +\allowbreak{} n = a..b\opRightPren{}} returns the formal sum +\smath{\sum_{n=a}^b f(n)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sumOfDivisors}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sumOfDivisors}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% {1}{(Integer)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf sumOfDivisors}\opLeftPren{}n\opRightPren{}} returns the sum of the integers between 1 and +{\smath{\mbox{\bf sumOfDivisors}\opLeftPren{}n\opRightPren{}} +returns the sum of the integers between 1 and integer \smath{n} (inclusive) which divide \smath{n}. This sum is often denoted in the literature by \smath{\sigma(n)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sumOfKthPowerDivisors}}\opLeftPren{}{\it integer}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sumOfKthPowerDivisors}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(Integer, NonNegativeInteger)->Integer}{IntegerNumberTheoryFunctions} -{\smath{\mbox{\bf sumOfKthPowerDivisors}\opLeftPren{}n, \allowbreak{} k\opRightPren{}} returns the sum of the -\eth{\smath{k}} powers of the integers between 1 and \smath{n} (inclusive) which divide \smath{n}. +{\smath{\mbox{\bf sumOfKthPowerDivisors}\opLeftPren{}n, +\allowbreak{} k\opRightPren{}} returns the sum of the +\eth{\smath{k}} powers of the integers between 1 and \smath{n} (inclusive) +which divide \smath{n}. This sum is often denoted in the literature by \smath{\sigma_k(n)}. } @@ -80059,7 +84085,8 @@ This sum is often denoted in the literature by \smath{\sigma_k(n)}. }% }% {1}{(Integer)->List(Integer)}{GaussianFactorizationPackage} -{\smath{\mbox{\bf sumSquares}\opLeftPren{}p\opRightPren{}} returns the list \smath{[a, b]} such that +{\smath{\mbox{\bf sumSquares}\opLeftPren{}p\opRightPren{}} returns the +list \smath{[a, b]} such that \smath{a^2+b^2} is equal to the integer prime \smath{p}, and calls \spadfun{error} if this is not possible. It will succeed if \smath{p} is 2 or congruent to \smath{1 \mod 4}. @@ -80068,37 +84095,46 @@ It will succeed if \smath{p} is 2 or congruent to \smath{1 \mod 4}. % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{sup}}\opLeftPren{}{\it element}, \allowbreak{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{sup}}\opLeftPren{}{\it element}, \allowbreak{} +{\it element}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{OrderedAbelianMonoidSup} -{\smath{\mbox{\bf sup}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} returns the least element from which both \smath{x} and \smath{y} can be subtracted. +{\smath{\mbox{\bf sup}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +returns the least element from which both \smath{x} and \smath{y} can +be subtracted. The purpose of \spadfun{sup} is to act as a supremum with respect to the partial order imposed by the \smath{-} operation on the domain. See \spadtype{OrderedAbelianMonoidSup} for details. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{super}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{super}}\opLeftPren{}{\it outputForm}, +\allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf super}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, -where \smath{o_1} and \smath{o_2} are objects of type \spadtype{OutputForm} (normally unexposed), +where \smath{o_1} and \smath{o_2} are objects of type +\spadtype{OutputForm} (normally unexposed), creates an output form for \smath{o_1} superscripted by \smath{o_2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{superscript}}\opLeftPren{}{\it symbol}, \allowbreak{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{superscript}}\opLeftPren{}{\it symbol}, +\allowbreak{}{\it listOfOutputForms}\opRightPren{}% }% }% {2}{(\$, List(OutputForm))->\$}{Symbol} -{\smath{\mbox{\bf superscript}\opLeftPren{}s, \allowbreak{} [a_1, \allowbreak{} \ldots, a_n]\opRightPren{}} returns symbol \smath{s} superscripted by +{\smath{\mbox{\bf superscript}\opLeftPren{}s, \allowbreak{} [a_1, +\allowbreak{} \ldots, a_n]\opRightPren{}} returns symbol \smath{s} +superscripted by output forms \smath{[a_1, \ldots, a_n]}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{supersub}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it listOfOutputForms}\opRightPren{}% +\opdata{{\mbox{\axiomFun{supersub}}\opLeftPren{}{\it outputForm}, +\allowbreak{}{\it listOfOutputForms}\opRightPren{}% }% }% {}{}{} @@ -80111,37 +84147,50 @@ creates an output form with each subscript aligned under each superscript. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{surface}}\opLeftPren{}{\it function}, \allowbreak{}{\it function}, \allowbreak{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{surface}}\opLeftPren{}{\it function}, +\allowbreak{}{\it function}, \allowbreak{}{\it function}\opRightPren{}% }% }% -{3}{(ComponentFunction, ComponentFunction, ComponentFunction)->\$}{ParametricSurface} -{\smath{\mbox{\bf surface}\opLeftPren{}c_1, \allowbreak{} c_2, \allowbreak{} c_3\opRightPren{}} creates a surface from three parametric component functions +{3}{(ComponentFunction, ComponentFunction, ComponentFunction)->\$} +{ParametricSurface} +{\smath{\mbox{\bf surface}\opLeftPren{}c_1, \allowbreak{} c_2, +\allowbreak{} c_3\opRightPren{}} creates a surface from three +parametric component functions \smath{c_1}, \smath{c_2}, and \smath{c_3}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{swap!}}\opLeftPren{}{\it aggregate}, \allowbreak{}{\it index}, \allowbreak{}{\it index}\opRightPren{}% +\opdata{{\mbox{\axiomFun{swap!}}\opLeftPren{}{\it aggregate}, +\allowbreak{}{\it index}, \allowbreak{}{\it index}\opRightPren{}% }% }% {3}{(\$, Index, Index)->Void}{IndexedAggregate} -{\smath{\mbox{\bf swap!}\opLeftPren{}u, \allowbreak{} i, \allowbreak{} j\opRightPren{}} interchanges elements \smath{i} and \smath{j} of aggregate \smath{u}. No meaningful value is returned. +{\smath{\mbox{\bf swap!}\opLeftPren{}u, \allowbreak{} i, +\allowbreak{} j\opRightPren{}} interchanges elements \smath{i} and \smath{j} +of aggregate \smath{u}. No meaningful value is returned. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{swapColumns!}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{swapColumns!}}\opLeftPren{}{\it matrix}, +\allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {3}{(\$, Integer, Integer)->\$}{MatrixCategory} -{\smath{\mbox{\bf swapColumns!}\opLeftPren{}m, \allowbreak{} i, \allowbreak{} j\opRightPren{}} interchanges the \eth{\smath{i}} and \eth{\smath{j}} columns of \smath{m} +{\smath{\mbox{\bf swapColumns!}\opLeftPren{}m, \allowbreak{} i, +\allowbreak{} j\opRightPren{}} interchanges the \eth{\smath{i}} and +\eth{\smath{j}} columns of \smath{m} returning \smath{m} which is destructively altered. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{swapRows!}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{swapRows!}}\opLeftPren{}{\it matrix}, +\allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {3}{(\$, Integer, Integer)->\$}{MatrixCategory} -{\smath{\mbox{\bf swapRows!}\opLeftPren{}m, \allowbreak{} i, \allowbreak{} j\opRightPren{}} interchanges the \eth{\smath{i}} and \eth{\smath{j}} rows of \smath{m}, +{\smath{\mbox{\bf swapRows!}\opLeftPren{}m, \allowbreak{} i, +\allowbreak{} j\opRightPren{}} interchanges the \eth{\smath{i}} and +\eth{\smath{j}} rows of \smath{m}, returning \smath{m} which is destructively altered. } @@ -80150,7 +84199,8 @@ returning \smath{m} which is destructively altered. }% }% {1}{(\$)->Boolean}{SExpressionCategory} -{\smath{\mbox{\bf symbol?}\opLeftPren{}s\opRightPren{}} tests if \spadtype{SExpression} object \smath{s} +{\smath{\mbox{\bf symbol?}\opLeftPren{}s\opRightPren{}} +tests if \spadtype{SExpression} object \smath{s} is a symbol. } @@ -80159,7 +84209,8 @@ is a symbol. }% }% {1}{(\$)->Sym}{SExpressionCategory} -{\smath{\mbox{\bf symbol}\opLeftPren{}s\opRightPren{}} returns \smath{s} as an element of type \spadtype{Symbol}, +{\smath{\mbox{\bf symbol}\opLeftPren{}s\opRightPren{}} +returns \smath{s} as an element of type \spadtype{Symbol}, or calls \spadfun{error} if this is not possible. } @@ -80168,35 +84219,43 @@ or calls \spadfun{error} if this is not possible. }% }% {1}{(\$)->Boolean}{MatrixCategory} -{\smath{\mbox{\bf symmetric?}\opLeftPren{}m\opRightPren{}} tests if the matrix \smath{m} is square and symmetric, +{\smath{\mbox{\bf symmetric?}\opLeftPren{}m\opRightPren{}} tests +if the matrix \smath{m} is square and symmetric, that is, if each \smath{m(i, j) = m(j, i)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{symmetricDifference}}\opLeftPren{}{\it set}, \allowbreak{}{\it set}\opRightPren{}% +\opdata{{\mbox{\axiomFun{symmetricDifference}}\opLeftPren{}{\it set}, +\allowbreak{}{\it set}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{SetAggregate} -{\smath{\mbox{\bf symmetricDifference}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} returns the set aggregate of +{\smath{\mbox{\bf symmetricDifference}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} returns the set aggregate of elements \smath{x} which are members of set aggregate \smath{u} or set aggregate \smath{v} but not both. If \smath{u} and \smath{v} have no elements in common, -\smath{\mbox{\bf symmetricDifference}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} returns a copy of \smath{u}. +\smath{\mbox{\bf symmetricDifference}\opLeftPren{}u, +\allowbreak{} v\opRightPren{}} returns a copy of \smath{u}. Note: \smath{symmetricDifference(u, v) = {\bf union}({\bf difference}(u, v), {\bf difference}(v, u))} } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{symmetricGroup}}\opLeftPren{}{\it integers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{symmetricGroup}}\opLeftPren{} +{\it integers}\opRightPren{}% }% }% {1}{(List(Integer))->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf symmetricGroup}\opLeftPren{}n\opRightPren{}} constructs the symmetric group \smath{S_n} acting on the +{\smath{\mbox{\bf symmetricGroup}\opLeftPren{}n\opRightPren{}} +constructs the symmetric group \smath{S_n} acting on the integers 1, \ldots, \smath{n}. -The generators are the \smath{n}-cycle \smath{(1, \ldots, n)} and the 2-cycle \smath{(1, 2)}. +The generators are the \smath{n}-cycle \smath{(1, \ldots, n)} +and the 2-cycle \smath{(1, 2)}. \newitem -\smath{\mbox{\bf symmetricGroup}\opLeftPren{}li\opRightPren{}}, where \smath{li} is a list of integers, +\smath{\mbox{\bf symmetricGroup}\opLeftPren{}li\opRightPren{}}, +where \smath{li} is a list of integers, constructs the symmetric group acting on the integers in the list \smath{li}. The generators are the cycle given by \smath{li} and @@ -80205,20 +84264,25 @@ Duplicates in the list will be removed. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{symmetricRemainder}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{symmetricRemainder}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{IntegerNumberSystem} -{\smath{\mbox{\bf symmetricRemainder}\opLeftPren{}a, \allowbreak{} b\opRightPren{}}, where \smath{b > 1}, yields +{\smath{\mbox{\bf symmetricRemainder}\opLeftPren{}a, +\allowbreak{} b\opRightPren{}}, where \smath{b > 1}, yields \smath{r} where \smath{ -b/2 \leq r < b/2}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{symmetricTensors}}\opLeftPren{}{\it matrices}, \allowbreak{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{symmetricTensors}}\opLeftPren{} +{\it matrices}, \allowbreak{}{\it positiveInteger}\opRightPren{}% }% }% -{2}{(List(Matrix(R)), PositiveInteger)->List(Matrix(R))}{RepresentationPackage1} -{\smath{\mbox{\bf symmetricTensors}\opLeftPren{}la, \allowbreak{} n\opRightPren{}}, where \smath{la} is a list +{2}{(List(Matrix(R)), PositiveInteger)->List(Matrix(R))} +{RepresentationPackage1} +{\smath{\mbox{\bf symmetricTensors}\opLeftPren{}la, +\allowbreak{} n\opRightPren{}}, where \smath{la} is a list \smath{[a_1, \ldots, a_k]} of \smath{m}-by-\smath{m} square matrices, applies to each matrix \smath{a_i}, the irreducible, polynomial representation of the general linear group \smath{GL_m} @@ -80227,49 +84291,59 @@ corresponding to the partition \smath{(n, 0, \ldots, 0)} of } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{systemCommand}}\opLeftPren{}{\it string}\opRightPren{}% +\opdata{{\mbox{\axiomFun{systemCommand}}\opLeftPren{} +{\it string}\opRightPren{}% }% }% {1}{(String)->Void}{MoreSystemCommands} -{\smath{\mbox{\bf systemCommand}\opLeftPren{}cmd\opRightPren{}} takes the string \smath{cmd} and +{\smath{\mbox{\bf systemCommand}\opLeftPren{}cmd\opRightPren{}} +takes the string \smath{cmd} and passes it to the runtime environment for execution as a system command. Although various things may be printed, no usable value is returned. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tableau}}\opLeftPren{}{\it listOfListOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tableau}}\opLeftPren{} +{\it listOfListOfElements}\opRightPren{}% }% }% {1}{(List(List(S)))->\$}{Tableau} -{\smath{\mbox{\bf tableau}\opLeftPren{}ll\opRightPren{}} converts a list of lists \smath{ll} to an object +{\smath{\mbox{\bf tableau}\opLeftPren{}ll\opRightPren{}} converts a +list of lists \smath{ll} to an object of type \spadtype{Tableau}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tableForDiscreteLogarithm}}\opLeftPren{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tableForDiscreteLogarithm}}\opLeftPren{} +{\it integer}\opRightPren{}% }% }% {1}{(Integer)->Table(PositiveInteger, NonNegativeInteger)}{FiniteFieldCategory} -{\smath{\mbox{\bf tableForDiscreteLogarithm}\opLeftPren{}n\opRightPren{}} returns a table of the +{\smath{\mbox{\bf tableForDiscreteLogarithm}\opLeftPren{}n\opRightPren{}} +returns a table of the discrete logarithms of \smath{a^0} up to \smath{a^{n-1}} which, -when called with the key \smath{\mbox{\bf lookup}\opLeftPren{}a^i\opRightPren{}}, returns \smath{i} for +when called with the key +\smath{\mbox{\bf lookup}\opLeftPren{}a^i\opRightPren{}}, returns \smath{i} for \smath{i} in \smath{0..n-1} for a finite field. This operation calls \spadfun{error} if not called for prime divisors of order of multiplicative group.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{table}}\opLeftPren{}{\it \opt{listOfRecords}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{table}}\opLeftPren{} +{\it \opt{listOfRecords}}\opRightPren{}% }% }% {1}{(List(Record(key:Key, entry:Entry)))->\$}{TableAggregate} -{\smath{\mbox{\bf table}\opLeftPren{}[p_1, \allowbreak{} p_2, \allowbreak{} \ldots, p_n]\opRightPren{}} creates a table with keys of +{\smath{\mbox{\bf table}\opLeftPren{}[p_1, \allowbreak{} p_2, +\allowbreak{} \ldots, p_n]\opRightPren{}} creates a table with keys of type \smath{Key} and entries of type \smath{Entry}. Each pair \smath{p_i} is a record with selectors \smath{key} and \smath{entry} with values from the corresponding domains \smath{Key} and \smath{Entry}. \newitem -\smath{\mbox{\bf table}\opLeftPren{}\opRightPren{}\$T} creates a empty table of domain \smath{T} +\smath{\mbox{\bf table}\opLeftPren{}\opRightPren{}\$T} +creates a empty table of domain \smath{T} of category \spadtype{TableAggregate}. } @@ -80278,7 +84352,8 @@ of category \spadtype{TableAggregate}. }% }% {1}{(\$)->\$}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf tail}\opLeftPren{}a\opRightPren{}} returns the last node of recursive aggregate \smath{a}. +{\smath{\mbox{\bf tail}\opLeftPren{}a\opRightPren{}} +returns the last node of recursive aggregate \smath{a}. } % ---------------------------------------------------------------------- @@ -80291,25 +84366,34 @@ of category \spadtype{TableAggregate}. \spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } \newitem -\smath{\mbox{\bf tan}\opLeftPren{}x\opRightPren{}} returns the tangent of \smath{x}. +\smath{\mbox{\bf tan}\opLeftPren{}x\opRightPren{}} +returns the tangent of \smath{x}. \newitem -\smath{\mbox{\bf tanIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf tan}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf tanIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf tan}\opLeftPren{}x\opRightPren{}} if possible, +and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tan2cot}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tan2cot}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf tan2cot}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf tan}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\rm cot}(u)}. +{\smath{\mbox{\bf tan2cot}\opLeftPren{}f\opRightPren{}} converts +every \smath{\mbox{\bf tan}\opLeftPren{}u\opRightPren{}} appearing in +\smath{f} into \smath{1/{\rm cot}(u)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tan2trig}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tan2trig}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf tan2trig}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf tan}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{\mbox{\bf sin}\opLeftPren{}u)/{\rm cos}(u)}. +{\smath{\mbox{\bf tan2trig}\opLeftPren{}f\opRightPren{}} converts +every \smath{\mbox{\bf tan}\opLeftPren{}u\opRightPren{}} appearing in +\smath{f} into \smath{\mbox{\bf sin}\opLeftPren{}u)/{\rm cos}(u)}. } % ---------------------------------------------------------------------- @@ -80318,93 +84402,123 @@ of category \spadtype{TableAggregate}. }% }% {1}{(K)->Union(K, "failed")}{PartialTranscendentalFunctions} -{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, \spadtype{DoubleFloat}, or +{\opkey{Argument x can be a \spadtype{Complex}, \spadtype{Float}, +\spadtype{DoubleFloat}, or \spadtype{Expression} value or a series. } \newitem -\smath{\mbox{\bf tanh}\opLeftPren{}x\opRightPren{}} returns the hyperbolic tangent of \smath{x}. +\smath{\mbox{\bf tanh}\opLeftPren{}x\opRightPren{}} +returns the hyperbolic tangent of \smath{x}. \newitem -\smath{\mbox{\bf tanhIfCan}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf tanh}\opLeftPren{}x\opRightPren{}} if possible, and \mbox{\tt "failed"} otherwise. +\smath{\mbox{\bf tanhIfCan}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf tanh}\opLeftPren{}x\opRightPren{}} if possible, +and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tanh2coth}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tanh2coth}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf tanh2coth}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf tanh}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{1/{\rm coth}(u)}. +{\smath{\mbox{\bf tanh2coth}\opLeftPren{}f\opRightPren{}} +converts every \smath{\mbox{\bf tanh}\opLeftPren{}u\opRightPren{}} +appearing in \smath{f} into \smath{1/{\rm coth}(u)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tanh2trigh}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tanh2trigh}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(F)->F}{TranscendentalManipulations} -{\smath{\mbox{\bf tanh2trigh}\opLeftPren{}f\opRightPren{}} converts every \smath{\mbox{\bf tanh}\opLeftPren{}u\opRightPren{}} appearing in \smath{f} into \smath{\mbox{\bf sinh}\opLeftPren{}u)/{\rm cosh}(u)}. +{\smath{\mbox{\bf tanh2trigh}\opLeftPren{}f\opRightPren{}} converts +every \smath{\mbox{\bf tanh}\opLeftPren{}u\opRightPren{}} appearing in +\smath{f} into \smath{\mbox{\bf sinh}\opLeftPren{}u)/{\rm cosh}(u)}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{taylor}}\opLeftPren{}{\it various}, \allowbreak{}{\it ..}\opRightPren{}% +\opdata{{\mbox{\axiomFun{taylor}}\opLeftPren{}{\it various}, +\allowbreak{}{\it ..}\opRightPren{}% }% }% {1}{(\$)->UTS}{UnivariateLaurentSeriesConstructorCategory} -{\smath{\mbox{\bf taylor}\opLeftPren{}u\opRightPren{}} converts the Laurent series \smath{u(x)} to a Taylor series if possible, +{\smath{\mbox{\bf taylor}\opLeftPren{}u\opRightPren{}} +converts the Laurent series \smath{u(x)} to a Taylor series if possible, and if not, calls \spadfun{error}. \newitem -\smath{\mbox{\bf taylor}\opLeftPren{}f\opRightPren{}} converts the expression \smath{f} into a +\smath{\mbox{\bf taylor}\opLeftPren{}f\opRightPren{}} +converts the expression \smath{f} into a Taylor expansion of the expression \smath{f}. Note: \smath{f} must have only one variable. \newitem -\smath{\mbox{\bf taylor}\opLeftPren{}sy\opRightPren{}}, where \smath{sy} is a symbol, returns \smath{sy} as a Taylor series. +\smath{\mbox{\bf taylor}\opLeftPren{}sy\opRightPren{}}, +where \smath{sy} is a symbol, returns \smath{sy} as a Taylor series. \newitem -\smath{\mbox{\bf taylor}\opLeftPren{}n +-> a(n), \allowbreak{} x = a\opRightPren{}} returns \smath{\sum_{n = 0 \ldots} +\smath{\mbox{\bf taylor}\opLeftPren{}n +-> a(n), +\allowbreak{} x = a\opRightPren{}} returns \smath{\sum_{n = 0 \ldots} a(n)(x-a)^n)}. \newitem -\smath{\mbox{\bf taylor}\opLeftPren{}f, \allowbreak{} x = a\optinner{, n}\opRightPren{}} expands the expression \smath{f} as a +\smath{\mbox{\bf taylor}\opLeftPren{}f, +\allowbreak{} x = a\optinner{, n}\opRightPren{}} +expands the expression \smath{f} as a series in powers of \smath{(x - a)} with \smath{n} terms. If \smath{n} is missing, the number of terms is governed by the value set by the system command \spadsyscom{)set streams calculate}. \newitem -\smath{\mbox{\bf taylor}\opLeftPren{}i +-> a(i), \allowbreak{} x = a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} creates the +\smath{\mbox{\bf taylor}\opLeftPren{}i +-> a(i), +\allowbreak{} x = a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} creates the Taylor series \smath{\sum\nolimits_{i = m..n {\ \tt by\ } k}{a(i) (x-a)^i}}. Here \smath{m}, \smath{n} and \smath{k} are integers. -Upper-limit \smath{n} and stepsize \smath{k} are optional and have default values +Upper-limit \smath{n} and stepsize \smath{k} are optional +and have default values \smath{n = \infty} and \smath{k = 1}. \newitem -\smath{\mbox{\bf taylor}\opLeftPren{}a(i), \allowbreak{} i, \allowbreak{} x=a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} returns +\smath{\mbox{\bf taylor}\opLeftPren{}a(i), \allowbreak{} i, +\allowbreak{} x=a, \allowbreak{} m..\optinner{n, k}\opRightPren{}} returns \smath{\sum\nolimits_{i = m..n {\bf by} k}{a(n) (x - a)^n}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{taylorIfCan}}\opLeftPren{}{\it laurentSeries}\opRightPren{}% +\opdata{{\mbox{\axiomFun{taylorIfCan}}\opLeftPren{} +{\it laurentSeries}\opRightPren{}% }% }% {1}{(\$)->Union(UTS, "failed")}{UnivariateLaurentSeriesConstructorCategory} -{\smath{\mbox{\bf taylorIfCan}\opLeftPren{}f(x)\opRightPren{}} converts the Laurent series +{\smath{\mbox{\bf taylorIfCan}\opLeftPren{}f(x)\opRightPren{}} +converts the Laurent series \smath{f(x)} to a Taylor series if possible, and returns \mbox{\tt "failed"} if this is not possible. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{taylorRep}}\opLeftPren{}{\it laurentSeries}\opRightPren{}% +\opdata{{\mbox{\axiomFun{taylorRep}}\opLeftPren{} +{\it laurentSeries}\opRightPren{}% }% }% {1}{(\$)->UTS}{UnivariateLaurentSeriesConstructorCategory} -{\smath{\mbox{\bf taylorRep}\opLeftPren{}f(x)\opRightPren{}} returns \smath{g(x)}, where \smath{f = +{\smath{\mbox{\bf taylorRep}\opLeftPren{}f(x)\opRightPren{}} +returns \smath{g(x)}, where \smath{f = x^n g(x)} is represented by \smath{[n, g(x)]}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tensorProduct}}\opLeftPren{}{\it listOfMatrices\opt{, listOfMatrices}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tensorProduct}}\opLeftPren{} +{\it listOfMatrices\opt{, listOfMatrices}}\opRightPren{}% }% }% -{2}{(List(Matrix(R)), List(Matrix(R)))->List(Matrix(R))}{RepresentationPackage1} -{\smath{\mbox{\bf tensorProduct}\opLeftPren{}[a_1, \allowbreak{} \ldots, a_k]\optinner{, [b_1, \ldots, b_k]}\opRightPren{}} +{2}{(List(Matrix(R)), List(Matrix(R)))->List(Matrix(R))} +{RepresentationPackage1} +{\smath{\mbox{\bf tensorProduct}\opLeftPren{}[a_1, +\allowbreak{} \ldots, a_k]\optinner{, [b_1, \ldots, b_k]}\opRightPren{}} calculates the list of Kronecker products of the matrices \smath{a_i} and \smath{b_i} for \smath{1 \leq i \leq k}. -If a second argument is missing, the \smath{b_i} is defined as the corresponding \smath{a_i}. -Also, \smath{\mbox{\bf tensorProduct}\opLeftPren{}m\opRightPren{}}, where \smath{m} is a matrix, -is defined as \smath{\mbox{\bf tensorProduct}\opLeftPren{}[m], \allowbreak{} [m]\opRightPren{}}. +If a second argument is missing, the \smath{b_i} is defined as the +corresponding \smath{a_i}. +Also, \smath{\mbox{\bf tensorProduct}\opLeftPren{}m\opRightPren{}}, +where \smath{m} is a matrix, +is defined as \smath{\mbox{\bf tensorProduct}\opLeftPren{}[m], +\allowbreak{} [m]\opRightPren{}}. Note: If each list of matrices corresponds to a group representation (representation of generators) of one group, then these matrices correspond to the tensor product of the two representations. @@ -80415,7 +84529,8 @@ these matrices correspond to the tensor product of the two representations. }% }% {1}{(\$)->List(Record(gen:S, exp:E))}{FreeAbelianMonoidCategory} -{\smath{\mbox{\bf terms}\opLeftPren{}s\opRightPren{}} returns a stream of the non-zero terms +{\smath{\mbox{\bf terms}\opLeftPren{}s\opRightPren{}} +returns a stream of the non-zero terms of series \smath{s}. Each term is returned as a record with selectors \smath{k} and \smath{c}, which correspond to the @@ -80430,11 +84545,13 @@ See \spadtype{FreeAbelianMonoidCategory}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tex}}\opLeftPren{}{\it formattedObject}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tex}}\opLeftPren{} +{\it formattedObject}\opRightPren{}% }% }% {1}{(\$)->List(String)}{TexFormat} -{\smath{\mbox{\bf tex}\opLeftPren{}t\opRightPren{}} extracts the TeX section of a TeX formatted object \smath{t}. +{\smath{\mbox{\bf tex}\opLeftPren{}t\opRightPren{}} +extracts the TeX section of a TeX formatted object \smath{t}. } % ---------------------------------------------------------------------- @@ -80442,7 +84559,8 @@ See \spadtype{FreeAbelianMonoidCategory}. }% }% {1}{(\$)->S}{UnaryRecursiveAggregate} -{\smath{\mbox{\bf third}\opLeftPren{}u\opRightPren{}} returns the third element of a recursive aggregate \smath{u}. +{\smath{\mbox{\bf third}\opLeftPren{}u\opRightPren{}} +returns the third element of a recursive aggregate \smath{u}. Note: \smath{\mbox{\bf third}\opLeftPren{}u) = first(rest(rest(u)))}. } @@ -80451,7 +84569,8 @@ Note: \smath{\mbox{\bf third}\opLeftPren{}u) = first(rest(rest(u)))}. }% }% {1}{(String)->\$}{DrawOption} -{\smath{\mbox{\bf title}\opLeftPren{}s\opRightPren{}} specifies string \smath{s} as the title for a plot. +{\smath{\mbox{\bf title}\opLeftPren{}s\opRightPren{}} +specifies string \smath{s} as the title for a plot. This option is expressed as a option to the \spadfun{draw} command in the form \code{title == s}. } @@ -80462,9 +84581,11 @@ form \code{title == s}. }% }% {1}{(\$)->S}{StackAggregate} -{\smath{\mbox{\bf top}\opLeftPren{}s\opRightPren{}} returns the top element \smath{x} from \smath{s}. +{\smath{\mbox{\bf top}\opLeftPren{}s\opRightPren{}} +returns the top element \smath{x} from \smath{s}. \newitem -\smath{\mbox{\bf top!}\opLeftPren{}d\opRightPren{}} returns the element at the top (front) of the dequeue. +\smath{\mbox{\bf top!}\opLeftPren{}d\opRightPren{}} +returns the element at the top (front) of the dequeue. } % ---------------------------------------------------------------------- @@ -80472,8 +84593,10 @@ form \code{title == s}. }% }% {1}{(R)->(Point(R))->Point(R)}{CoordinateSystems} -{\smath{\mbox{\bf toroidal}\opLeftPren{}element\opRightPren{}} transforms from toroidal coordinates to Cartesian coordinates: -\smath{\mbox{\bf toroidal}\opLeftPren{}a\opRightPren{}} is a function that maps the point \smath{(u, v, \phi)} to +{\smath{\mbox{\bf toroidal}\opLeftPren{}element\opRightPren{}} +transforms from toroidal coordinates to Cartesian coordinates: +\smath{\mbox{\bf toroidal}\opLeftPren{}a\opRightPren{}} +is a function that maps the point \smath{(u, v, \phi)} to \smath{x = a{\sinh}(v){\cos}(\phi)/({\cosh}(v)-{\cos}(u))}, \smath{y = a{\sinh}(v){\sin}(\phi)/({\cosh}(v)-{\cos}(u))}, \smath{z = a{\sin}(u)/({\cosh}(v)-{\cos}(u))}. @@ -80484,18 +84607,21 @@ form \code{title == s}. }% }% {1}{(Boolean)->\$}{DrawOption} -{\smath{\mbox{\bf toScale}\opLeftPren{}b\opRightPren{}} specifies whether or not a plot is to be drawn +{\smath{\mbox{\bf toScale}\opLeftPren{}b\opRightPren{}} +specifies whether or not a plot is to be drawn to scale. This command may be expressed as an option to the \spadfun{draw} command in the form \smath{toScale == b}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{totalDegree}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it listOfVariables}\opRightPren{}% +\opdata{{\mbox{\axiomFun{totalDegree}}\opLeftPren{} +{\it polynomial}, \allowbreak{}{\it listOfVariables}\opRightPren{}% }% }% {2}{(\$, List(VarSet))->NonNegativeInteger}{PolynomialCategory} -{\smath{\mbox{\bf totalDegree}\opLeftPren{}p\optinner{, lv}\opRightPren{}} returns the maximum sum +{\smath{\mbox{\bf totalDegree}\opLeftPren{}p\optinner{, lv}\opRightPren{}} +returns the maximum sum (over all monomials of polynomial \smath{p}) of the variables in the list \smath{lv}. If a second argument is missing, \smath{lv} is defined @@ -80503,22 +84629,27 @@ to be all the variables appearing in \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{totalfract}}\opLeftPren{}{\it polynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{totalfract}}\opLeftPren{} +{\it polynomial}\opRightPren{}% }% }% -{1}{(PRF)->Record(sup:Polynomial(R), inf:Polynomial(R))}{MPolyCatRationalFunctionFactorizer} -{\smath{\mbox{\bf totalfract}\opLeftPren{}prf\opRightPren{}} takes a polynomial whose coefficients are +{1}{(PRF)->Record(sup:Polynomial(R), inf:Polynomial(R))} +{MPolyCatRationalFunctionFactorizer} +{\smath{\mbox{\bf totalfract}\opLeftPren{}prf\opRightPren{}} +takes a polynomial whose coefficients are themselves fractions of polynomials and returns a record containing the numerator and denominator resulting from putting \smath{prf} over a common denominator. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{totalGroebner}}\opLeftPren{}{\it listOfPolynomials}, \allowbreak{}{\it listOfVariables}\opRightPren{}% +\opdata{{\mbox{\axiomFun{totalGroebner}}\opLeftPren{} +{\it listOfPolynomials}, \allowbreak{}{\it listOfVariables}\opRightPren{}% }% }% {2}{(List(Polynomial(F)), List(Symbol))->List(Polynomial(F))}{PolyGroebner} -{\smath{\mbox{\bf totalGroebner}\opLeftPren{}lp, \allowbreak{} lv\opRightPren{}} computes the Gr\"obner basis for the +{\smath{\mbox{\bf totalGroebner}\opLeftPren{}lp, +\allowbreak{} lv\opRightPren{}} computes the Gr\"obner basis for the \index{Groebner basis@{Gr\protect\"{o}bner basis}} list of polynomials \smath{lp} with the terms ordered first by \index{basis!Groebner@{Gr\protect\"{o}bner}} @@ -80532,16 +84663,20 @@ The variables are ordered by their position in the list }% }% {1}{(\$)->List(Kernel(\$))}{ExpressionSpace} -{\smath{\mbox{\bf tower}\opLeftPren{}f\opRightPren{}} returns all the kernels appearing in \smath{f}, regardless of level.} +{\smath{\mbox{\bf tower}\opLeftPren{}f\opRightPren{}} +returns all the kernels appearing in \smath{f}, regardless of level.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{trace}}\opLeftPren{}{\it various}, \allowbreak{}{\it ..}\opRightPren{}% +\opdata{{\mbox{\axiomFun{trace}}\opLeftPren{} +{\it various}, \allowbreak{}{\it ..}\opRightPren{}% }% }% {1}{(\$)->R}{SquareMatrixCategory} -{\smath{\mbox{\bf trace}\opLeftPren{}m\opRightPren{}} returns the trace of the matrix \smath{m}, that is, the +{\smath{\mbox{\bf trace}\opLeftPren{}m\opRightPren{}} +returns the trace of the matrix \smath{m}, that is, the sum of its diagonal elements. -\newitem\smath{\mbox{\bf trace}\opLeftPren{}a\opRightPren{}} returns the trace of the regular +\newitem\smath{\mbox{\bf trace}\opLeftPren{}a\opRightPren{}} +returns the trace of the regular representation of \smath{a}, an element of an algebra of finite rank. See \spadtype{FiniteRankAlgebra}. @@ -80553,15 +84688,18 @@ over the ground field of size \smath{q}. This operation calls \spadfun{error} if \smath{d} does not divide the extension degree of \smath{a}. The default value of \smath{d} is 1. -Note: \smath{\mbox{\bf trace}\opLeftPren{}a, d) = \sum_{i=0}^{n/d} a^{q^{d i}}}. +Note: +\smath{\mbox{\bf trace}\opLeftPren{}a, d) = \sum_{i=0}^{n/d} a^{q^{d i}}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{traceMatrix}}\opLeftPren{}{\it \opt{basis}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{traceMatrix}}\opLeftPren{} +{\it \opt{basis}}\opRightPren{}% }% }% {1}{(Vector(\$))->Matrix(R)}{FiniteRankAlgebra} -{\smath{\mbox{\bf traceMatrix}\opLeftPren{}[v1, \allowbreak{} .., \allowbreak{} vn]\opRightPren{}} is the \smath{n}-by-\smath{n} +{\smath{\mbox{\bf traceMatrix}\opLeftPren{}[v1, \allowbreak{} .., +\allowbreak{} vn]\opRightPren{}} is the \smath{n}-by-\smath{n} matrix whose \smath{i}, \smath{j} element is \smath{Tr(v_i v_j)}. If no argument is given, the \smath{v_i} are assumed to be elements @@ -80569,11 +84707,14 @@ of the fixed basis. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tracePowMod}}\opLeftPren{}{\it poly}, \allowbreak{}{\it nonNegativeInteger}, \allowbreak{}{\it poly}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tracePowMod}}\opLeftPren{}{\it poly}, +\allowbreak{}{\it nonNegativeInteger}, \allowbreak{}{\it poly} +\opRightPren{}% }% }% {3}{(FP, NonNegativeInteger, FP)->FP}{DistinctDegreeFactorize} -{\smath{\mbox{\bf tracePowMod}\opLeftPren{}u, \allowbreak{} k, \allowbreak{} v\opRightPren{}} returns \smath{\sum\nolimits_{i=0}^k +{\smath{\mbox{\bf tracePowMod}\opLeftPren{}u, \allowbreak{} k, +\allowbreak{} v\opRightPren{}} returns \smath{\sum\nolimits_{i=0}^k {u^{2^i}}}, all computed modulo the polynomial \smath{v}. } @@ -80582,30 +84723,36 @@ of the fixed basis. }% }% {0}{()->NonNegativeInteger}{ExtensionField} -{\smath{\mbox{\bf transcendenceDegree}\opLeftPren{}\opRightPren{}\$F} returns the transcendence degree +{\smath{\mbox{\bf transcendenceDegree}\opLeftPren{}\opRightPren{}\$F} +returns the transcendence degree of the field extension \smath{F}, or 0 if the extension is algebraic. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{transcendent?}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{transcendent?}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->Boolean}{ExtensionField} -{\smath{\mbox{\bf transcendent?}\opLeftPren{}a\opRightPren{}} tests whether an element \smath{a} of +{\smath{\mbox{\bf transcendent?}\opLeftPren{}a\opRightPren{}} +tests whether an element \smath{a} of a domain that is an extension field over a ground field \smath{F} is transcendent with respect to \smath{F}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{transpose}}\opLeftPren{}{\it matrix\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{transpose}}\opLeftPren{} +{\it matrix\opt{, options}}\opRightPren{}% }% }% {1}{(\$)->\$}{MatrixCategory} -{\smath{\mbox{\bf transpose}\opLeftPren{}m\opRightPren{}} returns the transpose of the matrix \smath{m}. +{\smath{\mbox{\bf transpose}\opLeftPren{}m\opRightPren{}} +returns the transpose of the matrix \smath{m}. \newitem -\smath{\mbox{\bf transpose}\opLeftPren{}t\optinner{, i, j}\opRightPren{}} exchanges the \eth{\smath{i}} +\smath{\mbox{\bf transpose}\opLeftPren{}t\optinner{, i, j}\opRightPren{}} +exchanges the \eth{\smath{i}} and \eth{\smath{j}} indices of \smath{t}. For example, if \smath{r = {\bf transpose}(t, 2, 3)} for a rank four tensor \smath{t}, then \smath{r} is the rank four tensor given by @@ -80615,26 +84762,37 @@ first and last index of \smath{t}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tree}}\opLeftPren{}{\it value\opt{, listOfChildren}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tree}}\opLeftPren{} +{\it value\opt{, listOfChildren}}\opRightPren{}% }% }% {1}{(\$)->\$}{MatrixCategory} -{\smath{\mbox{\bf tree}\opLeftPren{}x, \allowbreak{} ls\opRightPren{}} creates an element of \spadtype{Tree} with +{\smath{\mbox{\bf tree}\opLeftPren{}x, \allowbreak{} ls\opRightPren{}} +creates an element of \spadtype{Tree} with value \smath{x} at the root node, and immediate children \spad{ls} in left-to-right order. \newitem -\smath{\mbox{\bf tree}\opLeftPren{}x\opRightPren{}} is equivalent to \smath{\mbox{\bf tree}\opLeftPren{}x, \allowbreak{} []\$List(S)\opRightPren{}} where +\smath{\mbox{\bf tree}\opLeftPren{}x\opRightPren{}} is equivalent to +\smath{\mbox{\bf tree}\opLeftPren{}x, \allowbreak{} []\$List(S)\opRightPren{}} +where \spad{x} has type \spad{S}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{trapezoidal}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it threeIntegers}\opRightPren{}% -\optand \mbox{\axiomFun{trapezoidalClosed}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% -\opand \mbox{\axiomFun{trapezoidalOpen}}\opLeftPren{}{\it floatFunction}, \allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{trapezoidal}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it threeIntegers} +\opRightPren{}% +\optand \mbox{\axiomFun{trapezoidalClosed}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% +\opand \mbox{\axiomFun{trapezoidalOpen}}\opLeftPren{}{\it floatFunction}, +\allowbreak{}{\it fourFloats}, \allowbreak{}{\it twoIntegers}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf trapezoidal}\opLeftPren{}fn, \allowbreak{} a, \allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, \allowbreak{} nmin, \allowbreak{} nmax, \allowbreak{} nint\opRightPren{}} +{\smath{\mbox{\bf trapezoidal}\opLeftPren{}fn, \allowbreak{} a, +\allowbreak{} b, \allowbreak{} epsrel, \allowbreak{} epsabs, +\allowbreak{} nmin, +\allowbreak{} nmax, \allowbreak{} nint\opRightPren{}} uses the adaptive trapezoidal method to numerically integrate function \smath{fn} over the closed interval from \smath{a} to \smath{b}, with relative accuracy \smath{epsrel} and absolute @@ -80668,11 +84826,14 @@ it integrates function \smath{fn} over the open interval from } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{triangularSystems}}\opLeftPren{}{\it listOfFractions}, \allowbreak{}{\it listOfSymbols}\opRightPren{}% +\opdata{{\mbox{\axiomFun{triangularSystems}}\opLeftPren{} +{\it listOfFractions}, \allowbreak{}{\it listOfSymbols}\opRightPren{}% }% }% -{2}{(List(Fraction(Polynomial(R))), List(Symbol))->List(List(Polynomial(R)))}{SystemSolvePackage} -{\smath{\mbox{\bf triangularSystems}\opLeftPren{}lf, \allowbreak{} lv\opRightPren{}} solves the system of equations +{2}{(List(Fraction(Polynomial(R))), +List(Symbol))->List(List(Polynomial(R)))}{SystemSolvePackage} +{\smath{\mbox{\bf triangularSystems}\opLeftPren{}lf, +\allowbreak{} lv\opRightPren{}} solves the system of equations defined by \smath{lf} with respect to the list of symbols \smath{lv}; the system of equations is obtaining by equating to zero the list of @@ -80686,16 +84847,19 @@ as a ``reduced'' triangular system of polynomials. }% }% {1}{(F)->F}{TrigonometricManipulations} -{\smath{\mbox{\bf trigs}\opLeftPren{}f\opRightPren{}} rewrites all the complex logs and exponentials +{\smath{\mbox{\bf trigs}\opLeftPren{}f\opRightPren{}} +rewrites all the complex logs and exponentials appearing in \smath{f} in terms of trigonometric functions. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{trim}}\opLeftPren{}{\it string}, \allowbreak{}{\it characterOrCharacterClass}\opRightPren{}% +\opdata{{\mbox{\axiomFun{trim}}\opLeftPren{}{\it string}, \allowbreak{} +{\it characterOrCharacterClass}\opRightPren{}% }% }% {2}{(\$, Character)->\$}{StringAggregate} -{\smath{\mbox{\bf trim}\opLeftPren{}s, \allowbreak{} c\opRightPren{}} returns \smath{s} with all characters \smath{c} +{\smath{\mbox{\bf trim}\opLeftPren{}s, \allowbreak{} c\opRightPren{}} +returns \smath{s} with all characters \smath{c} deleted from right and left ends. For example, \code{trim(" abc ", char " ")} returns \code{"abc"}. @@ -80707,25 +84871,30 @@ For example, \code{trim("(abc)", charClass "()")} returns } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{truncate}}\opLeftPren{}{\it various\opt{, options}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{truncate}}\opLeftPren{} +{\it various\opt{, options}}\opRightPren{}% }% }% {1}{(\$)->\$}{RealNumberSystem} -{\smath{\mbox{\bf truncate}\opLeftPren{}x\opRightPren{}} returns the integer between \smath{x} and 0 +{\smath{\mbox{\bf truncate}\opLeftPren{}x\opRightPren{}} +returns the integer between \smath{x} and 0 closest to \smath{x}. \newitem -\smath{\mbox{\bf truncate}\opLeftPren{}f, \allowbreak{} m\optinner{, n}\opRightPren{}} returns a (finite) power series +\smath{\mbox{\bf truncate}\opLeftPren{}f, +\allowbreak{} m\optinner{, n}\opRightPren{}} returns a (finite) power series consisting of the sum of all terms of \smath{f} of degree \smath{d} with \smath{n \leq d \leq m}. Upper bound \smath{m} is \smath{\infty} by default. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tubePoints}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tubePoints}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(PositiveInteger)->\$}{DrawOption} -{\smath{\mbox{\bf tubePoints}\opLeftPren{}n\opRightPren{}} specifies the number of points, \smath{n}, +{\smath{\mbox{\bf tubePoints}\opLeftPren{}n\opRightPren{}} +specifies the number of points, \smath{n}, defining the circle that creates the tube around a \threedim{} curve. The default is 6. @@ -80733,15 +84902,18 @@ This option is expressed in the form \code{tubePoints == n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tubePointsDefault}}\opLeftPren{}{\it \opt{positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tubePointsDefault}}\opLeftPren{} +{\it \opt{positiveInteger}}\opRightPren{}% }% }% {1}{(PositiveInteger)->PositiveInteger}{ViewDefaultsPackage} -{\smath{\mbox{\bf tubePointsDefault}\opLeftPren{}i\opRightPren{}} sets the number of points to use +{\smath{\mbox{\bf tubePointsDefault}\opLeftPren{}i\opRightPren{}} +sets the number of points to use when creating the circle to be used in creating a \threedim{} tube plot to \smath{i}. \newitem -\smath{\mbox{\bf tubePointsDefault}\opLeftPren{}\opRightPren{}} returns the number of points to be +\smath{\mbox{\bf tubePointsDefault}\opLeftPren{}\opRightPren{}} +returns the number of points to be used when creating the circle to be used in creating a \threedim{} tube plot. } @@ -80751,21 +84923,25 @@ tube plot. }% }% {1}{(Float)->\$}{DrawOption} -{\smath{\mbox{\bf tubeRadius}\opLeftPren{}r\opRightPren{}} specifies a radius \smath{r} for a tube +{\smath{\mbox{\bf tubeRadius}\opLeftPren{}r\opRightPren{}} +specifies a radius \smath{r} for a tube plot around a \threedim{} curve. This operation may be expressed as an option to the \spadfun{draw} command in the form \code{tubeRadius == r}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{tubeRadiusDefault}}\opLeftPren{}{\it \opt{float}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{tubeRadiusDefault}}\opLeftPren{} +{\it \opt{float}}\opRightPren{}% }% }% {0}{()->DoubleFloat}{ViewDefaultsPackage} -{\smath{\mbox{\bf tubeRadiusDefault}\opLeftPren{}r\opRightPren{}} sets the default radius for a +{\smath{\mbox{\bf tubeRadiusDefault}\opLeftPren{}r\opRightPren{}} +sets the default radius for a \threedim{} tube plot to \smath{r}. \newitem -\smath{\mbox{\bf tubeRadiusDefault}\opLeftPren{}\opRightPren{}} returns the radius used for a +\smath{\mbox{\bf tubeRadiusDefault}\opLeftPren{}\opRightPren{}} +returns the radius used for a \threedim{} tube plot. } @@ -80781,25 +84957,31 @@ See \spadtype{MappingPackage} for related functions. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unary?}}\opLeftPren{}{\it basicOperator}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unary?}}\opLeftPren{} +{\it basicOperator}\opRightPren{}% }% }% {1}{(\$)->Boolean}{BasicOperator} -{\smath{\mbox{\bf unary?}\opLeftPren{}op\opRightPren{}} tests if basic operator \smath{op} is unary, +{\smath{\mbox{\bf unary?}\opLeftPren{}op\opRightPren{}} tests +if basic operator \smath{op} is unary, that is, takes exactly one argument. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{union}}\opLeftPren{}{\it set}, \allowbreak{}{\it elementOrSet}\opRightPren{}% +\opdata{{\mbox{\axiomFun{union}}\opLeftPren{}{\it set}, \allowbreak{} +{\it elementOrSet}\opRightPren{}% }% }% {2}{(\$, S)->\$}{SetAggregate} -{\smath{\mbox{\bf union}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} returns the set aggregate \smath{u} with the +{\smath{\mbox{\bf union}\opLeftPren{}u, +\allowbreak{} x\opRightPren{}} returns the set aggregate \smath{u} with the element \smath{x} added. -If \smath{u} already contains \smath{x}, \smath{\mbox{\bf union}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} +If \smath{u} already contains \smath{x}, +\smath{\mbox{\bf union}\opLeftPren{}u, \allowbreak{} x\opRightPren{}} returns a copy of \smath{x}. \newitem -\smath{\mbox{\bf union}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} returns the set aggregate of elements that are +\smath{\mbox{\bf union}\opLeftPren{}u, \allowbreak{} v\opRightPren{}} +returns the set aggregate of elements that are members of either set aggregate \smath{u} or \smath{v}. See also \axiomType{Multiset}. } @@ -80809,13 +84991,16 @@ See also \axiomType{Multiset}. }% }% {1}{(\$)->R}{Factored} -{\smath{\mbox{\bf unit}\opLeftPren{}\opRightPren{}} returns a unit of the algebra (necessarily +{\smath{\mbox{\bf unit}\opLeftPren{}\opRightPren{}} +returns a unit of the algebra (necessarily unique), or \mbox{\tt "failed"} if there is none. \newitem -\smath{\mbox{\bf unit}\opLeftPren{}u\opRightPren{}} extracts the unit part of the factored object +\smath{\mbox{\bf unit}\opLeftPren{}u\opRightPren{}} +extracts the unit part of the factored object \smath{u}. \newitem -\smath{\mbox{\bf unit}\opLeftPren{}l\opRightPren{}} marks off the units on a viewport according to +\smath{\mbox{\bf unit}\opLeftPren{}l\opRightPren{}} +marks off the units on a viewport according to the indicated list \smath{l}. This option is expressed in the draw command in the form \smath{{\tt unit ==} [f_1, f_2]}. @@ -80826,24 +85011,29 @@ This option is expressed in the draw command in the form }% }% {1}{(\$)->Boolean}{IntegralDomain} -{\smath{\mbox{\bf unit?}\opLeftPren{}x\opRightPren{}} tests whether \smath{x} is a unit, that is, if +{\smath{\mbox{\bf unit?}\opLeftPren{}x\opRightPren{}} +tests whether \smath{x} is a unit, that is, if \smath{x} is invertible. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unitCanonical}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unitCanonical}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(\$)->\$}{IntegralDomain} -{\smath{\mbox{\bf unitCanonical}\opLeftPren{}x\opRightPren{}} returns \smath{\mbox{\bf unitNormal}\opLeftPren{}x).{canonical}}. +{\smath{\mbox{\bf unitCanonical}\opLeftPren{}x\opRightPren{}} +returns \smath{\mbox{\bf unitNormal}\opLeftPren{}x).{canonical}}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unitNormalize}}\opLeftPren{}{\it factored}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unitNormalize}}\opLeftPren{} +{\it factored}\opRightPren{}% }% }% {1}{(\$)->\$}{Factored} -{\smath{\mbox{\bf unitNormalize}\opLeftPren{}u\opRightPren{}} normalizes the unit part of the +{\smath{\mbox{\bf unitNormalize}\opLeftPren{}u\opRightPren{}} +normalizes the unit part of the factorization. For example, when working with factored integers, this operation ensures that the bases are all positive integers. @@ -80854,44 +85044,53 @@ ensures that the bases are all positive integers. }% }% {1}{(\$)->Record(unit:\$, canonical:\$, associate:\$)}{IntegralDomain} -{\smath{\mbox{\bf unitNormal}\opLeftPren{}x\opRightPren{}} tries to choose a canonical element from +{\smath{\mbox{\bf unitNormal}\opLeftPren{}x\opRightPren{}} +tries to choose a canonical element from the associate class of \smath{x}. If successful, it returns a record with three components ``unit'', ``canonical'' and ``associate''. The attribute \spadatt{canonicalUnitNormal}, if asserted, means that the ``canonical'' element is the same across all associates of \smath{x}. -If \smath{\mbox{\bf unitNormal}\opLeftPren{}x) = [u, c, a]} then \smath{ux = c}, +If \smath{\mbox{\bf unitNormal}\opLeftPren{}x) = [u, c, a]} +then \smath{ux = c}, \smath{au = 1}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unitsColorDefault}}\opLeftPren{}{\it \opt{palette}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unitsColorDefault}}\opLeftPren{} +{\it \opt{palette}}\opRightPren{}% }% }% {0}{()->Palette}{ViewDefaultsPackage} -{\smath{\mbox{\bf unitsColorDefault}\opLeftPren{}p\opRightPren{}} sets the default color of the unit +{\smath{\mbox{\bf unitsColorDefault}\opLeftPren{}p\opRightPren{}} +sets the default color of the unit ticks in a \twodim{} viewport to the palette \smath{p}. \newitem -\smath{\mbox{\bf unitsColorDefault}\opLeftPren{}\opRightPren{}} returns the default color of the unit +\smath{\mbox{\bf unitsColorDefault}\opLeftPren{}\opRightPren{}} +returns the default color of the unit ticks in a \twodim{} viewport. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unitVector}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unitVector}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(PositiveInteger)->\$}{DirectProductCategory} -{\smath{\mbox{\bf unitVector}\opLeftPren{}n\opRightPren{}} produces a vector with 1 in position +{\smath{\mbox{\bf unitVector}\opLeftPren{}n\opRightPren{}} +produces a vector with 1 in position \smath{n} and zero elsewhere. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{univariate}}\opLeftPren{}{\it polynomial\opt{, variable}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{univariate}}\opLeftPren{} +{\it polynomial\opt{, variable}}\opRightPren{}% }% }% {1}{(\$)->SparseUnivariatePolynomial(R)}{PolynomialCategory} -{\smath{\mbox{\bf univariate}\opLeftPren{}p\optinner{, v}\opRightPren{}} converts the multivariate +{\smath{\mbox{\bf univariate}\opLeftPren{}p\optinner{, v}\opRightPren{}} +converts the multivariate polynomial \smath{p} into a univariate polynomial in \smath{v} whose coefficients are multivariate polynomials in all the other variables. @@ -80903,7 +85102,8 @@ If \smath{v} is omitted, then \smath{p} must involve exactly one variable. }% }% {0}{()->\$}{FiniteSetAggregate} -{\smath{\mbox{\bf universe}\opLeftPren{}\opRightPren{}}\$\smath{R} returns the universal set for +{\smath{\mbox{\bf universe}\opLeftPren{}\opRightPren{}}\$\smath{R} +returns the universal set for finite set aggregate \smath{R}. } @@ -80913,18 +85113,23 @@ finite set aggregate \smath{R}. }% }% { $ -> String}{}{} -{\smath{\mbox{\bf unparse}\opLeftPren{}f\opRightPren{}} returns a string \smath{s} such that the parser +{\smath{\mbox{\bf unparse}\opLeftPren{}f\opRightPren{}} +returns a string \smath{s} such that the parser would transform \smath{s} to \smath{f}, or calls \spadfun{error} if \smath{f} is not the parsed form of a string. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unrankImproperPartitions0}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unrankImproperPartitions0}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}, \allowbreak{} +{\it integer}\opRightPren{}% }% }% -{3}{(Integer, Integer, Integer)->List(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf unrankImproperPartitions0}\opLeftPren{}n, \allowbreak{} m, \allowbreak{} k\opRightPren{}} computes the +{3}{(Integer, Integer, Integer)->List(Integer)} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf unrankImproperPartitions0}\opLeftPren{}n, \allowbreak{} m, +\allowbreak{} k\opRightPren{}} computes the \eth{\smath{k}} improper partition of nonnegative \smath{n} in \smath{m} nonnegative parts in reverse lexicographical order. Example: \smath{ [0, 0, 3] < [0, 1, 2] < [0, 2, 1] < [0, 3, 0] < [1, 0, 2] < @@ -80936,11 +85141,15 @@ Note: counting of subtrees is done by } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unrankImproperPartitions1}}\opLeftPren{}{\it integer}, \allowbreak{}{\it integer}, \allowbreak{}{\it integer}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unrankImproperPartitions1}}\opLeftPren{} +{\it integer}, \allowbreak{}{\it integer}, \allowbreak{} +{\it integer}\opRightPren{}% }% }% -{3}{(Integer, Integer, Integer)->List(Integer)}{SymmetricGroupCombinatoricFunctions} -{\smath{\mbox{\bf unrankImproperPartitions1}\opLeftPren{}n, \allowbreak{} m, \allowbreak{} k\opRightPren{}} computes the +{3}{(Integer, Integer, Integer)->List(Integer)} +{SymmetricGroupCombinatoricFunctions} +{\smath{\mbox{\bf unrankImproperPartitions1}\opLeftPren{}n, \allowbreak{} m, +\allowbreak{} k\opRightPren{}} computes the \eth{\smath{k}} improper partition of nonnegative \smath{n} in at most \smath{m} nonnegative parts ordered as follows: first, in reverse lexicographical order according to their non-zero parts, then @@ -80953,11 +85162,13 @@ Note: counting of subtrees is done by } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{unravel}}\opLeftPren{}{\it listOfElement}\opRightPren{}% +\opdata{{\mbox{\axiomFun{unravel}}\opLeftPren{} +{\it listOfElement}\opRightPren{}% }% }% {1}{(List(R))->\$}{CartesianTensor} -{\smath{\mbox{\bf unravel}\opLeftPren{}t\opRightPren{}} produces a tensor from a list of components such that +{\smath{\mbox{\bf unravel}\opLeftPren{}t\opRightPren{}} +produces a tensor from a list of components such that \smath{\mbox{\bf unravel}\opLeftPren{}{\bf ravel}(t)) = t}. } @@ -80968,153 +85179,188 @@ Note: counting of subtrees is done by }% }% {1}{(\$)->\$}{StringAggregate} -{\smath{\mbox{\bf upperCase!}\opLeftPren{}s\opRightPren{}} destructively replaces the alphabetic +{\smath{\mbox{\bf upperCase!}\opLeftPren{}s\opRightPren{}} +destructively replaces the alphabetic characters in \smath{s} by upper case characters. \newitem -\smath{\mbox{\bf upperCase}\opLeftPren{}\opRightPren{}} returns the class of all characters for which +\smath{\mbox{\bf upperCase}\opLeftPren{}\opRightPren{}} +returns the class of all characters for which \spadfunFrom{upperCase?}{Character} is \smath{true}. \newitem -\smath{\mbox{\bf upperCase}\opLeftPren{}c\opRightPren{}} converts a lower case letter +\smath{\mbox{\bf upperCase}\opLeftPren{}c\opRightPren{}} +converts a lower case letter \smath{c} to the corresponding upper case letter. If \smath{c} is not a lower case letter, then it is returned unchanged. \newitem -\smath{\mbox{\bf upperCase}\opLeftPren{}s\opRightPren{}} returns the string with all characters in +\smath{\mbox{\bf upperCase}\opLeftPren{}s\opRightPren{}} +returns the string with all characters in upper case. \newitem -\smath{\mbox{\bf upperCase?}\opLeftPren{}c\opRightPren{}} tests if \smath{c} is an upper case letter, +\smath{\mbox{\bf upperCase?}\opLeftPren{}c\opRightPren{}} tests +if \smath{c} is an upper case letter, that is, one of A..\smath{Z}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{validExponential}}\opLeftPren{}{\it listOfKernels}, \allowbreak{}{\it expression}, \allowbreak{}{\it symbol}\opRightPren{}% +\opdata{{\mbox{\axiomFun{validExponential}}\opLeftPren{} +{\it listOfKernels}, \allowbreak{}{\it expression}, \allowbreak{} +{\it symbol}\opRightPren{}% }% }% -{3}{(List(Kernel(F)), F, Symbol)->Union(F, "failed")}{ElementaryFunctionStructurePackage} -{\smath{\mbox{\bf validExponential}\opLeftPren{}[k_1, \allowbreak{} \ldots, k_n], f, x\opRightPren{}} returns \smath{g} +{3}{(List(Kernel(F)), F, Symbol)->Union(F, "failed")} +{ElementaryFunctionStructurePackage} +{\smath{\mbox{\bf validExponential}\opLeftPren{}[k_1, +\allowbreak{} \ldots, k_n], f, x\opRightPren{}} returns \smath{g} if \smath{\mbox{\bf exp}\opLeftPren{}f)=g} and \smath{g} involves only \smath{k_1\ldots{}k_n}, and \mbox{\tt "failed"} otherwise. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{value}}\opLeftPren{}{\it recursiveAggregate}\opRightPren{}% +\opdata{{\mbox{\axiomFun{value}}\opLeftPren{} +{\it recursiveAggregate}\opRightPren{}% }% }% {1}{(\$)->S}{RecursiveAggregate} -{\smath{\mbox{\bf value}\opLeftPren{}a\opRightPren{}} returns the ``value'' part of a recursive +{\smath{\mbox{\bf value}\opLeftPren{}a\opRightPren{}} +returns the ``value'' part of a recursive aggregate \spad{a}, typically the root of tree. See, for example, \spadtype{BinaryTree}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{var1Steps}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{var1Steps}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(PositiveInteger)->\$}{DrawOption} -{\smath{\mbox{\bf var1Steps}\opLeftPren{}n\opRightPren{}} indicates the number of subdivisions +{\smath{\mbox{\bf var1Steps}\opLeftPren{}n\opRightPren{}} +indicates the number of subdivisions \smath{n} of the first range variable. This command may be expressed as an option to the \spadfun{draw} command in the form \smath{{\bf var1Steps}== n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{var1StepsDefault}}\opLeftPren{}{\it \opt{positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{var1StepsDefault}}\opLeftPren{} +{\it \opt{positiveInteger}}\opRightPren{}% }% }% {0}{()->PositiveInteger}{ViewDefaultsPackage} -{\smath{\mbox{\bf var1StepsDefault}\opLeftPren{}\opRightPren{}} returns the current setting for the +{\smath{\mbox{\bf var1StepsDefault}\opLeftPren{}\opRightPren{}} +returns the current setting for the number of steps to take when creating a \threedim{} mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (that is, \smath{x=0}..10)). \newitem -\smath{\mbox{\bf var1StepsDefault}\opLeftPren{}i\opRightPren{}} sets the number of steps to take when +\smath{\mbox{\bf var1StepsDefault}\opLeftPren{}i\opRightPren{}} +sets the number of steps to take when creating a \threedim{} mesh in the direction of the first defined free variable to \smath{i} (a free variable is considered defined when its range is specified (that is, \smath{x=0}..10)). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{var2Steps}}\opLeftPren{}{\it positiveInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{var2Steps}}\opLeftPren{} +{\it positiveInteger}\opRightPren{}% }% }% {1}{(PositiveInteger)->\$}{DrawOption} -{\smath{\mbox{\bf var2Steps}\opLeftPren{}n\opRightPren{}} indicates the number of subdivisions, +{\smath{\mbox{\bf var2Steps}\opLeftPren{}n\opRightPren{}} +indicates the number of subdivisions, \smath{n}, of the second range variable. This option is expressed in the form \smath{{\bf var2Steps} == n}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{var2StepsDefault}}\opLeftPren{}{\it \opt{positiveInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{var2StepsDefault}}\opLeftPren{} +{\it \opt{positiveInteger}}\opRightPren{}% }% }% {()->PositiveInteger}{ViewDefaultsPackage} -{\smath{\mbox{\bf var2StepsDefault}\opLeftPren{}\opRightPren{}} is the current setting for the number +{\smath{\mbox{\bf var2StepsDefault}\opLeftPren{}\opRightPren{}} +is the current setting for the number of steps to take when creating a \threedim{} mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (that is, \smath{x=0}..10)). \newitem -\smath{\mbox{\bf var2StepsDefault}\opLeftPren{}i\opRightPren{}} sets the number of steps to take when +\smath{\mbox{\bf var2StepsDefault}\opLeftPren{}i\opRightPren{}} +sets the number of steps to take when creating a \threedim{} mesh in the direction of the first defined free variable to \smath{i} (a free variable is considered defined when its range is specified (that is, \smath{x=0}..10)). } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{variable}}\opLeftPren{}{\it various}\opRightPren{}% +\opdata{{\mbox{\axiomFun{variable}}\opLeftPren{} +{\it various}\opRightPren{}% }% }% {1}{(\$)->Symbol}{UnivariatePowerSeriesCategory} -{\smath{\mbox{\bf variable}\opLeftPren{}f\opRightPren{}} returns the (unique) power series variable of +{\smath{\mbox{\bf variable}\opLeftPren{}f\opRightPren{}} +returns the (unique) power series variable of the power series \smath{f}. \newitem -\smath{\mbox{\bf variable}\opLeftPren{}segb\opRightPren{}} returns the variable from the left hand +\smath{\mbox{\bf variable}\opLeftPren{}segb\opRightPren{}} +returns the variable from the left hand side of the \spadtype{SegmentBinding} \smath{segb}. For example, if \smath{segb} is \smath{v=a..b}, then -\smath{\mbox{\bf variable}\opLeftPren{}segb\opRightPren{}} returns \smath{v}. +\smath{\mbox{\bf variable}\opLeftPren{}segb\opRightPren{}} +returns \smath{v}. \newitem -\smath{\mbox{\bf variable}\opLeftPren{}v\opRightPren{}} returns \smath{s} if \smath{v} is any +\smath{\mbox{\bf variable}\opLeftPren{}v\opRightPren{}} +returns \smath{s} if \smath{v} is any derivative of the differential indeterminate \smath{s}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{variables}}\opLeftPren{}{\it expression}\opRightPren{}% +\opdata{{\mbox{\axiomFun{variables}}\opLeftPren{} +{\it expression}\opRightPren{}% }% }% {1}{(\$)->List(Symbol)}{FunctionSpace} -{\smath{\mbox{\bf variables}\opLeftPren{}f\opRightPren{}} returns the list of all the variables of +{\smath{\mbox{\bf variables}\opLeftPren{}f\opRightPren{}} +returns the list of all the variables of expression, polynomial, rational function, or power series \smath{f}.} % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{vconcat}}\opLeftPren{}{\it outputForms\opt{, OutputForm} (normally unexposed)}\opRightPren{}% +\opdata{{\mbox{\axiomFun{vconcat}}\opLeftPren{} +{\it outputForms\opt{, OutputForm} (normally unexposed)}\opRightPren{}% }% }% {}{}{} -{\smath{\mbox{\bf vconcat}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, where \smath{o_1} and \smath{o_2} are +{\smath{\mbox{\bf vconcat}\opLeftPren{}o_1, +\allowbreak{} o_2\opRightPren{}}, where \smath{o_1} and \smath{o_2} are objects of type \spadtype{OutputForm} (normally unexposed), returns an output form for the vertical concatenation of forms \smath{o_1} and \smath{o_2}. \newitem -\smath{\mbox{\bf vconcat}\opLeftPren{}lo\opRightPren{}}, where \smath{lo} is a list of objects of type +\smath{\mbox{\bf vconcat}\opLeftPren{}lo\opRightPren{}}, +where \smath{lo} is a list of objects of type \spadtype{OutputForm} (normally unexposed), returns an output form for the vertical concatenation of the elements of \smath{lo}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{vector}}\opLeftPren{}{\it listOfElements}\opRightPren{}% +\opdata{{\mbox{\axiomFun{vector}}\opLeftPren{} +{\it listOfElements}\opRightPren{}% }% }% {1}{(List(R))->\$}{Vector} -{\smath{\mbox{\bf vector}\opLeftPren{}l\opRightPren{}} converts the list \smath{l} to a vector. +{\smath{\mbox{\bf vector}\opLeftPren{}l\opRightPren{}} converts the +list \smath{l} to a vector. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{vectorise}}\opLeftPren{}{\it polynomial}, \allowbreak{}{\it nonNegativeInteger}\opRightPren{}% +\opdata{{\mbox{\axiomFun{vectorise}}\opLeftPren{}{\it polynomial}, +\allowbreak{}{\it nonNegativeInteger}\opRightPren{}% }% }% {2}{(\$, NonNegativeInteger)->Vector(R)}{UnivariatePolynomialCategory} -{\smath{\mbox{\bf vectorise}\opLeftPren{}p, \allowbreak{} n\opRightPren{}} returns \smath{[a_0, \ldots, a_{n-1}]} +{\smath{\mbox{\bf vectorise}\opLeftPren{}p, \allowbreak{} n\opRightPren{}} +returns \smath{[a_0, \ldots, a_{n-1}]} where \smath{p = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1}} + higher order terms. The degree of polynomial \smath{p} can be different from @@ -81122,11 +85368,13 @@ The degree of polynomial \smath{p} can be different from } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{vertConcat}}\opLeftPren{}{\it matrix}, \allowbreak{}{\it matrix}\opRightPren{}% +\opdata{{\mbox{\axiomFun{vertConcat}}\opLeftPren{}{\it matrix}, +\allowbreak{}{\it matrix}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{MatrixCategory} -{\smath{\mbox{\bf vertConcat}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} vertically concatenates two matrices with an +{\smath{\mbox{\bf vertConcat}\opLeftPren{}x, \allowbreak{} y\opRightPren{}} +vertically concatenates two matrices with an equal number of columns. The entries of \smath{y} appear below the entries of \smath{x}. } @@ -81136,31 +85384,38 @@ The entries of \smath{y} appear below the entries of \smath{x}. }% }% {0}{()->Void}{ViewDefaultsPackage} -{\smath{\mbox{\bf viewDefaults}\opLeftPren{}\opRightPren{}} resets all the default graphics settings. +{\smath{\mbox{\bf viewDefaults}\opLeftPren{}\opRightPren{}} +resets all the default graphics settings. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{viewPosDefault}}\opLeftPren{}{\it \opt{listOfNonNegativeIntegers}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{viewPosDefault}}\opLeftPren{} +{\it \opt{listOfNonNegativeIntegers}}\opRightPren{}% }% }% {1}{(List(NonNegativeInteger))->List(NonNegativeInteger)}{ViewDefaultsPackage} -{\smath{\mbox{\bf viewPosDefault}\opLeftPren{}[x, \allowbreak{} y]\opRightPren{}} sets the default \smath{X} and +{\smath{\mbox{\bf viewPosDefault}\opLeftPren{}[x, \allowbreak{} y] +\opRightPren{}} sets the default \smath{X} and \smath{Y} position of a viewport window. Unless overridden explicitly, newly created viewports will have the \smath{X} and \smath{Y} coordinates \smath{x}, \smath{y}. \newitem -\smath{\mbox{\bf viewPosDefault}\opLeftPren{}\opRightPren{}} returns the default \smath{X} and +\smath{\mbox{\bf viewPosDefault}\opLeftPren{}\opRightPren{}} +returns the default \smath{X} and \smath{Y} position of a viewport window unless overridden explicitly, newly created viewports will have these \smath{X} and \smath{Y} coordinate. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{viewSizeDefault}}\opLeftPren{}{\it \opt{listOfPositiveIntegers}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{viewSizeDefault}}\opLeftPren{} +{\it \opt{listOfPositiveIntegers}}\opRightPren{}% }% }% {1}{(List(PositiveInteger))->List(PositiveInteger)}{ViewDefaultsPackage} -{\smath{\mbox{\bf viewSizeDefault}\opLeftPren{}[w, \allowbreak{} h]\opRightPren{}} sets the default viewport width to \smath{w} and height to \smath{h}. +{\smath{\mbox{\bf viewSizeDefault}\opLeftPren{}[w, \allowbreak{} h] +\opRightPren{}} sets the default viewport width to \smath{w} and +height to \smath{h}. } % ---------------------------------------------------------------------- @@ -81168,18 +85423,22 @@ explicitly, newly created viewports will have these \smath{X} and }% }% {0}{()->List(String)}{ViewDefaultsPackage} -{\smath{\mbox{\bf viewWriteAvailable}\opLeftPren{}\opRightPren{}} returns a list of available methods for writing, such as BITMAP, POSTSCRIPT, etc. +{\smath{\mbox{\bf viewWriteAvailable}\opLeftPren{}\opRightPren{}} returns +a list of available methods for writing, such as BITMAP, POSTSCRIPT, etc. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{viewWriteDefault}}\opLeftPren{}{\it listOfStrings}\opRightPren{}% +\opdata{{\mbox{\axiomFun{viewWriteDefault}}\opLeftPren{} +{\it listOfStrings}\opRightPren{}% }% }% {0}{()->List(String)}{ViewDefaultsPackage} -{\smath{\mbox{\bf viewWriteDefault}\opLeftPren{}\opRightPren{}} returns the list of things to write in +{\smath{\mbox{\bf viewWriteDefault}\opLeftPren{}\opRightPren{}} +returns the list of things to write in a viewport data file; a viewAlone file is always generated. \newitem -\smath{\mbox{\bf viewWriteDefault}\opLeftPren{}l\opRightPren{}} sets the default list of things to +\smath{\mbox{\bf viewWriteDefault}\opLeftPren{}l\opRightPren{}} +sets the default list of things to write in a viewport data file to the strings in \smath{l}; a viewAlone file is always generated. } @@ -81193,11 +85452,13 @@ viewAlone file is always generated. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{weakBiRank}}\opLeftPren{}{\it element}\opRightPren{}% +\opdata{{\mbox{\axiomFun{weakBiRank}}\opLeftPren{} +{\it element}\opRightPren{}% }% }% {1}{(A)->NonNegativeInteger}{AlgebraPackage} -{\smath{\mbox{\bf weakBiRank}\opLeftPren{}x\opRightPren{}} determines the number of linearly +{\smath{\mbox{\bf weakBiRank}\opLeftPren{}x\opRightPren{}} +determines the number of linearly independent elements in the \smath{b_i x b_j}, \smath{i, j=1, \ldots, n}, where \smath{b=[b_1, \ldots, b_n]} is the fixed basis of a domain of category \spadtype{FramedNonAssociativeAlgebra}.} @@ -81216,17 +85477,24 @@ differential monomials appearing in the differential polynomial the weight of the derivative \smath{u}. \item if u is a basic operator: the weight attached to \smath{u}. \end{simpleList} -\smath{\mbox{\bf weight}\opLeftPren{}p, \allowbreak{} s\opRightPren{}} returns the maximum weight of all differential monomials appearing in the differential polynomial \smath{p} when \smath{p} is viewed as a differential polynomial in the differential indeterminate \smath{s} alone. +\smath{\mbox{\bf weight}\opLeftPren{}p, \allowbreak{} s\opRightPren{}} +returns the maximum weight of all differential monomials appearing in +the differential polynomial \smath{p} when \smath{p} is viewed as a +differential polynomial in the differential indeterminate \smath{s} alone. \newitem -\smath{\mbox{\bf weight}\opLeftPren{}op, \allowbreak{} n\opRightPren{}} attaches the weight \smath{n} to \smath{op}. +\smath{\mbox{\bf weight}\opLeftPren{}op, +\allowbreak{} n\opRightPren{}} attaches the weight \smath{n} to \smath{op}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{weights}}\opLeftPren{}{\it differentialPolynomial}, \allowbreak{}{\it differentialIndeterminated}\opRightPren{}% +\opdata{{\mbox{\axiomFun{weights}}\opLeftPren{} +{\it differentialPolynomial}, \allowbreak{} +{\it differentialIndeterminated}\opRightPren{}% }% }% {2}{(\$, S)->List(NonNegativeInteger)}{DifferentialPolynomialCategory} -{\smath{\mbox{\bf weights}\opLeftPren{}p, \allowbreak{} s\opRightPren{}} returns a list of weights of differential +{\smath{\mbox{\bf weights}\opLeftPren{}p, +\allowbreak{} s\opRightPren{}} returns a list of weights of differential monomials appearing in the differential polynomial \smath{p} when \smath{p} is viewed as a differential polynomial in the differential indeterminate \smath{s} alone. @@ -81235,81 +85503,99 @@ monomials appearing in differential polynomial \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{whatInfinity}}\opLeftPren{}{\it orderedCompletion}\opRightPren{}% +\opdata{{\mbox{\axiomFun{whatInfinity}}\opLeftPren{} +{\it orderedCompletion}\opRightPren{}% }% }% {1}{(\$)->SingleInteger}{OrderedCompletion} -{\smath{\mbox{\bf whatInfinity}\opLeftPren{}x\opRightPren{}} returns 0 if \smath{x} is finite, 1 if +{\smath{\mbox{\bf whatInfinity}\opLeftPren{}x\opRightPren{}} +returns 0 if \smath{x} is finite, 1 if \smath{x} is \smath{\infty}, and \smath{-1} if \smath{x} is \smath{-\infty}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{wholePart}}\opLeftPren{}{\it various}\opRightPren{}% +\opdata{{\mbox{\axiomFun{wholePart}}\opLeftPren{} +{\it various}\opRightPren{}% }% }% {1}{(\$)->R}{ContinuedFraction} -{\smath{\mbox{\bf wholePart}\opLeftPren{}x\opRightPren{}} returns the whole part of the fraction +{\smath{\mbox{\bf wholePart}\opLeftPren{}x\opRightPren{}} +returns the whole part of the fraction \smath{x}, that is, the truncated quotient of the numerator by the denominator. \newitem -\smath{\mbox{\bf wholePart}\opLeftPren{}x\opRightPren{}} extracts the whole part of \smath{x}. +\smath{\mbox{\bf wholePart}\opLeftPren{}x\opRightPren{}} +extracts the whole part of \smath{x}. That is, -if \smath{x = {\bf continuedFraction}(b_0, [a_1, a_2, \ldots], [b_1, b_2, \ldots])}, +if \smath{x = {\bf continuedFraction} +(b_0, [a_1, a_2, \ldots], [b_1, b_2, \ldots])}, then \smath{\mbox{\bf wholePart}\opLeftPren{}x) = b_0}. \newitem -\smath{\mbox{\bf wholePart}\opLeftPren{}p\opRightPren{}} extracts the whole part of the partial +\smath{\mbox{\bf wholePart}\opLeftPren{}p\opRightPren{}} +extracts the whole part of the partial fraction \smath{p}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{wholeRadix}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{wholeRadix}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% {1}{(List(Integer))->\$}{RadixExpansion} -{\smath{\mbox{\bf wholeRadix}\opLeftPren{}l\opRightPren{}} creates an integral radix expansion from a +{\smath{\mbox{\bf wholeRadix}\opLeftPren{}l\opRightPren{}} +creates an integral radix expansion from a list of ragits. -For example, \smath{\mbox{\bf wholeRadix}\opLeftPren{}[1, \allowbreak{} 3, \allowbreak{} 4]\opRightPren{}} returns \smath{134}. +For example, \smath{\mbox{\bf wholeRadix}\opLeftPren{}[1, \allowbreak{} 3, +\allowbreak{} 4]\opRightPren{}} returns \smath{134}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{wholeRagits}}\opLeftPren{}{\it listOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{wholeRagits}}\opLeftPren{} +{\it listOfIntegers}\opRightPren{}% }% }% {1}{(\$)->List(Integer)}{RadixExpansion} -{\smath{\mbox{\bf wholeRagits}\opLeftPren{}rx\opRightPren{}} returns the ragits of the integer part of +{\smath{\mbox{\bf wholeRagits}\opLeftPren{}rx\opRightPren{}} +returns the ragits of the integer part of a radix expansion. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{wordInGenerators}}\opLeftPren{}{\it permutation}, \allowbreak{}{\it permutationGroup}\opRightPren{}% +\opdata{{\mbox{\axiomFun{wordInGenerators}}\opLeftPren{}{\it permutation}, +\allowbreak{}{\it permutationGroup}\opRightPren{}% }% }% {2}{(Permutation(S), \$)->List(NonNegativeInteger)}{PermutationGroup} -{\smath{\mbox{\bf wordInGenerators}\opLeftPren{}p, \allowbreak{} gp\opRightPren{}} returns the word for the +{\smath{\mbox{\bf wordInGenerators}\opLeftPren{}p, +\allowbreak{} gp\opRightPren{}} returns the word for the permutation \smath{p} in the original generators of the permutation group {\it gp}, represented by the indices of the list, given by \spadfun{generators}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{wordInStrongGenerators}}\opLeftPren{}{\it permutation}, \allowbreak{}{\it permutationGroup}\opRightPren{}% +\opdata{{\mbox{\axiomFun{wordInStrongGenerators}}\opLeftPren{} +{\it permutation}, \allowbreak{}{\it permutationGroup}\opRightPren{}% }% }% {2}{(Permutation(S), \$)->List(NonNegativeInteger)}{PermutationGroup} -{\smath{\mbox{\bf wordInStrongGenerators}\opLeftPren{}p, \allowbreak{} gp\opRightPren{}} returns the word for the +{\smath{\mbox{\bf wordInStrongGenerators}\opLeftPren{}p, +\allowbreak{} gp\opRightPren{}} returns the word for the permutation \smath{p} in the strong generators of the permutation group {\it gp}, represented by the indices of the list, given by \spadfun{strongGenerators}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{wordsForStrongGenerators}}\opLeftPren{}{\it listOfListsOfIntegers}\opRightPren{}% +\opdata{{\mbox{\axiomFun{wordsForStrongGenerators}}\opLeftPren{} +{\it listOfListsOfIntegers}\opRightPren{}% }% }% {1}{(\$)->List(List(NonNegativeInteger))}{PermutationGroup} -{\smath{\mbox{\bf wordsForStrongGenerators}\opLeftPren{}gp\opRightPren{}} returns the words for the +{\smath{\mbox{\bf wordsForStrongGenerators}\opLeftPren{}gp\opRightPren{}} +returns the words for the strong generators of the permutation group {\it gp} in the original generators of {\it gp}, represented by their indices in the list of nonnegative integers, @@ -81317,11 +85603,13 @@ given by \spadfun{generators}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{wreath}}\opLeftPren{}{\it symmetricPolynomial}, \allowbreak{}{\it symmetricPolynomial}\opRightPren{}% +\opdata{{\mbox{\axiomFun{wreath}}\opLeftPren{}{\it symmetricPolynomial}, +\allowbreak{}{\it symmetricPolynomial}\opRightPren{}% }% }% { (SPOL RN, SPOL RN) -> SPOL RN}{}{} -{\smath{\mbox{\bf wreath}\opLeftPren{}s_1, \allowbreak{} s_2\opRightPren{}} is the cycle index of the wreath product +{\smath{\mbox{\bf wreath}\opLeftPren{}s_1, \allowbreak{} s_2\opRightPren{}} +is the cycle index of the wreath product of the two groups whose cycle indices are \smath{s_1} and \smath{s_2}, symmetric polynomials with rational number coefficients. @@ -81332,47 +85620,58 @@ coefficients. }% }% {1}{(\$)->Boolean}{FileNameCategory} -{\smath{\mbox{\bf writable?}\opLeftPren{}f\opRightPren{}} tests if the named file can be opened for +{\smath{\mbox{\bf writable?}\opLeftPren{}f\opRightPren{}} +tests if the named file can be opened for writing. The named file need not already exist. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{write!}}\opLeftPren{}{\it file}, \allowbreak{}{\it value}\opRightPren{}% +\opdata{{\mbox{\axiomFun{write!}}\opLeftPren{}{\it file}, \allowbreak{} +{\it value}\opRightPren{}% }% }% {2}{(\$, S)->S}{FileCategory} -{\smath{\mbox{\bf write!}\opLeftPren{}f, \allowbreak{} s\opRightPren{}} puts the value \smath{s} into the file \smath{f}. +{\smath{\mbox{\bf write!}\opLeftPren{}f, \allowbreak{} s\opRightPren{}} +puts the value \smath{s} into the file \smath{f}. The state of \smath{f} is modified so that subsequent calls to {\bf write!} will append values to the end of the file. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{writeLine!}}\opLeftPren{}{\it textfile\opt{, string}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{writeLine!}}\opLeftPren{} +{\it textfile\opt{, string}}\opRightPren{}% }% }% {1}{(\$)->String}{TextFile} -{\smath{\mbox{\bf writeLine!}\opLeftPren{}f\opRightPren{}} finishes the current line in the file +{\smath{\mbox{\bf writeLine!}\opLeftPren{}f\opRightPren{}} +finishes the current line in the file \smath{f}. An empty string is returned. -The call \smath{\mbox{\bf writeLine!}\opLeftPren{}f\opRightPren{}} is equivalent to -\smath{\mbox{\bf writeLine!}\opLeftPren{}f, \allowbreak{} {\tt ""}\opRightPren{}}. +The call \smath{\mbox{\bf writeLine!}\opLeftPren{}f\opRightPren{}} +is equivalent to +\smath{\mbox{\bf writeLine!}\opLeftPren{}f, \allowbreak{} +{\tt ""}\opRightPren{}}. \newitem -\smath{\mbox{\bf writeLine!}\opLeftPren{}f, \allowbreak{} s\opRightPren{}} writes the contents of the string +\smath{\mbox{\bf writeLine!}\opLeftPren{}f, \allowbreak{} s\opRightPren{}} +writes the contents of the string \smath{s} and finishes the current line in the file \smath{f}. The value of \smath{s} is returned. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{xor}}\opLeftPren{}{\it boolean}, \allowbreak{}{\it boolean}\opRightPren{}% +\opdata{{\mbox{\axiomFun{xor}}\opLeftPren{}{\it boolean}, \allowbreak{} +{\it boolean}\opRightPren{}% }% }% {2}{(\$, \$)->\$}{BitAggregate} -{\smath{\mbox{\bf xor}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} returns the logical {\it exclusive-or} of +{\smath{\mbox{\bf xor}\opLeftPren{}a, \allowbreak{} b\opRightPren{}} +returns the logical {\it exclusive-or} of booleans or bit aggregates \smath{a} and \smath{b}. \newitem -\smath{\mbox{\bf xor}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} returns the bit-by-bit logical {\it xor} of the +\smath{\mbox{\bf xor}\opLeftPren{}n, \allowbreak{} m\opRightPren{}} +returns the bit-by-bit logical {\it xor} of the small integers \smath{n} and \smath{m}. } @@ -81381,12 +85680,14 @@ small integers \smath{n} and \smath{m}. }% }% {1}{(\$)->Segment(DoubleFloat)}{PlottablePlaneCurveCategory} -{\smath{\mbox{\bf xRange}\opLeftPren{}c\opRightPren{}} returns the range of the \smath{x}-coordinates +{\smath{\mbox{\bf xRange}\opLeftPren{}c\opRightPren{}} +returns the range of the \smath{x}-coordinates of the points on the curve \smath{c}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{yCoordinates}}\opLeftPren{}{\it function}\opRightPren{}% +\opdata{{\mbox{\axiomFun{yCoordinates}}\opLeftPren{} +{\it function}\opRightPren{}% }% }% {1}{(\$)->Record(num:Vector(UP), den:UP)}{FunctionFieldCategory} @@ -81395,7 +85696,8 @@ where \smath{f} is a function defined over a curve, returns the coordinates of \smath{f} with respect to the natural basis for the curve. Specifically, the operation returns -\smath{[ [a_1, \ldots, a_n], d]} such that \smath{f = (a_1 +\ldots+ a_n y^{n-1}) / d}. +\smath{[ [a_1, \ldots, a_n], d]} +such that \smath{f = (a_1 +\ldots+ a_n y^{n-1}) / d}. } % ---------------------------------------------------------------------- @@ -81403,7 +85705,8 @@ Specifically, the operation returns }% }% {0}{()->\$}{Color} -{\smath{\mbox{\bf yellow}\opLeftPren{}\opRightPren{}} returns the position of the yellow hue from +{\smath{\mbox{\bf yellow}\opLeftPren{}\opRightPren{}} +returns the position of the yellow hue from total hues. } @@ -81412,11 +85715,13 @@ total hues. }% }% {1}{(List(Integer))->PermutationGroup(Integer)}{PermutationGroupExamples} -{\smath{\mbox{\bf youngGroup}\opLeftPren{}[n_1, \allowbreak{} \ldots, n_k]\opRightPren{}} constructs the direct +{\smath{\mbox{\bf youngGroup}\opLeftPren{} +[n_1, \allowbreak{} \ldots, n_k]\opRightPren{}} constructs the direct product of the symmetric groups \smath{Sn_1}, \ldots, \smath{Sn_k}. \newitem -\smath{\mbox{\bf youngGroup}\opLeftPren{}lambda\opRightPren{}} constructs the direct product of the +\smath{\mbox{\bf youngGroup}\opLeftPren{}lambda\opRightPren{}} +constructs the direct product of the symmetric groups given by the parts of the partition {\it lambda}. } @@ -81425,29 +85730,35 @@ symmetric groups given by the parts of the partition {\it lambda}. }% }% {1}{(\$)->Segment(DoubleFloat)}{PlottablePlaneCurveCategory} -{\smath{\mbox{\bf yRange}\opLeftPren{}c\opRightPren{}} returns the range of the \smath{y}-coordinates +{\smath{\mbox{\bf yRange}\opLeftPren{}c\opRightPren{}} +returns the range of the \smath{y}-coordinates of the points on the curve \smath{c}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{zag}}\opLeftPren{}{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% +\opdata{{\mbox{\axiomFun{zag}}\opLeftPren{} +{\it outputForm}, \allowbreak{}{\it outputForm}\opRightPren{}% }% }% {}{}{} {\smath{\mbox{\bf zag}\opLeftPren{}o_1, \allowbreak{} o_2\opRightPren{}}, -where \smath{o_1} and \smath{o_2} are objects of type \spadtype{OutputForm} (normally unexposed), +where \smath{o_1} and \smath{o_2} are objects of +type \spadtype{OutputForm} (normally unexposed), return an output form displaying the continued fraction form for \smath{o_2} over \smath{o_1}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{zero}}\opLeftPren{}{\it nonNegativeInteger\opt{, nonNegativeInteger}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{zero}}\opLeftPren{} +{\it nonNegativeInteger\opt{, nonNegativeInteger}}\opRightPren{}% }% }% {1}{(NonNegativeInteger)->\$}{VectorCategory} -{\smath{\mbox{\bf zero}\opLeftPren{}n\opRightPren{}} creates a zero vector of length \smath{n}. +{\smath{\mbox{\bf zero}\opLeftPren{}n\opRightPren{}} +creates a zero vector of length \smath{n}. \newitem -\smath{\mbox{\bf zero}\opLeftPren{}m, \allowbreak{} n\opRightPren{}} returns an \smath{m}-by-\smath{n} zero matrix. +\smath{\mbox{\bf zero}\opLeftPren{}m, \allowbreak{} n\opRightPren{}} +returns an \smath{m}-by-\smath{n} zero matrix. } % ---------------------------------------------------------------------- @@ -81455,7 +85766,8 @@ the continued fraction form for \smath{o_2} over \smath{o_1}. }% }% {1}{(\$)->Boolean}{AbelianMonoid} -{\smath{\mbox{\bf zero?}\opLeftPren{}x\opRightPren{}} tests if \smath{x} is equal to 0. +{\smath{\mbox{\bf zero?}\opLeftPren{}x\opRightPren{}} +tests if \smath{x} is equal to 0. } % ---------------------------------------------------------------------- @@ -81463,34 +85775,47 @@ the continued fraction form for \smath{o_2} over \smath{o_1}. }% }% {1}{(\$)->Boolean}{PolynomialIdeals} -{\smath{\mbox{\bf zeroDim?}\opLeftPren{}I\opRightPren{}} tests if the ideal \smath{I} is zero +{\smath{\mbox{\bf zeroDim?}\opLeftPren{}I\opRightPren{}} +tests if the ideal \smath{I} is zero dimensional, that is, all its associated primes are maximal. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{zeroDimPrimary?}}\opLeftPren{}{\it ideal}\opRightPren{}% +\opdata{{\mbox{\axiomFun{zeroDimPrimary?}}\opLeftPren{} +{\it ideal}\opRightPren{}% }% }% -{1}{(PolynomialIdeals(Fraction(Integer), DirectProduct(nv, NonNegativeInteger), vl, DistributedMultivariatePolynomial(vl, Fraction(Integer))))->Boolean}{IdealDecompositionPackage} -{\smath{\mbox{\bf zeroDimPrimary?}\opLeftPren{}I\opRightPren{}} tests if the ideal \smath{I} is +{1}{(PolynomialIdeals(Fraction(Integer), +DirectProduct(nv, NonNegativeInteger), vl, +DistributedMultivariatePolynomial(vl, Fraction(Integer))))->Boolean} +{IdealDecompositionPackage} +{\smath{\mbox{\bf zeroDimPrimary?}\opLeftPren{}I\opRightPren{}} +tests if the ideal \smath{I} is 0-dimensional primary. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{zeroDimPrime?}}\opLeftPren{}{\it ideal}\opRightPren{}% +\opdata{{\mbox{\axiomFun{zeroDimPrime?}}\opLeftPren{} +{\it ideal}\opRightPren{}% }% }% -{1}{(PolynomialIdeals(Fraction(Integer), DirectProduct(nv, NonNegativeInteger), vl, DistributedMultivariatePolynomial(vl, Fraction(Integer))))->Boolean}{IdealDecompositionPackage} -{\smath{\mbox{\bf zeroDimPrime?}\opLeftPren{}I\opRightPren{}} tests if the ideal \smath{I} is a +{1}{(PolynomialIdeals(Fraction(Integer), +DirectProduct(nv, NonNegativeInteger), vl, +DistributedMultivariatePolynomial(vl, Fraction(Integer))))->Boolean} +{IdealDecompositionPackage} +{\smath{\mbox{\bf zeroDimPrime?}\opLeftPren{}I\opRightPren{}} +tests if the ideal \smath{I} is a 0-dimensional prime. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{zeroOf}}\opLeftPren{}{\it polynomial\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{zeroOf}}\opLeftPren{} +{\it polynomial\opt{, symbol}}\opRightPren{}% }% }% {2}{(SparseUnivariatePolynomial(\$), Symbol)->\$}{AlgebraicallyClosedField} -{\smath{\mbox{\bf zeroOf}\opLeftPren{}p\optinner{, y}\opRightPren{}} returns \smath{y} such that +{\smath{\mbox{\bf zeroOf}\opLeftPren{}p\optinner{, y}\opRightPren{}} +returns \smath{y} such that \smath{p(y) = 0}. If possible, \smath{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity that displays as \smath{'y}. @@ -81499,11 +85824,13 @@ variable \smath{y}. } % ---------------------------------------------------------------------- -\opdata{{\mbox{\axiomFun{zerosOf}}\opLeftPren{}{\it polynomial\opt{, symbol}}\opRightPren{}% +\opdata{{\mbox{\axiomFun{zerosOf}}\opLeftPren{} +{\it polynomial\opt{, symbol}}\opRightPren{}% }% }% {1}{(Polynomial(\$))->List(\$)}{AlgebraicallyClosedField} -{\smath{\mbox{\bf zerosOf}\opLeftPren{}p, \allowbreak{} y\opRightPren{}} returns \smath{[y_1, \ldots, y_n]} such +{\smath{\mbox{\bf zerosOf}\opLeftPren{}p, \allowbreak{} y\opRightPren{}} +returns \smath{[y_1, \ldots, y_n]} such that \smath{p(y_i) = 0}. The \smath{y_i}'s are expressed in radicals if possible. Otherwise they are implicit algebraic quantities that display as @@ -81519,7 +85846,8 @@ variable \smath{y}. }% }% {1}{(\$)->Segment(DoubleFloat)}{PlottableSpaceCurveCategory} -{\smath{\mbox{\bf zRange}\opLeftPren{}c\opRightPren{}} returns the range of the \smath{z}-coordinates +{\smath{\mbox{\bf zRange}\opLeftPren{}c\opRightPren{}} +returns the range of the \smath{z}-coordinates of the points on the curve \smath{c}. } }\onecolumn @@ -81527,7 +85855,7 @@ of the points on the curve \smath{c}. %Original Page 691 -\chapter{Programs for AXIOM Images} +\chapter{Programs for Axiom Images} \label{ugAppGraphics} % @@ -81573,10 +85901,10 @@ method. The poles in these images represent bad starting values. The flat areas are the regions of convergence to the three roots. \begin{verbatim} -)read newton Read the programs from -)read vectors Chapter 10 -f := newtonStep(x**3 - 2) Create a Newton's iteration - function for $x^3 = 2$ +)read newton Read the programs from +)read vectors Chapter 10 +f := newtonStep(x**3 - 2) Create a Newton's iteration + function for $x^3 = 2$ \end{verbatim} The function $f^n$ computes $n$ steps of Newton's method. @@ -81670,9 +85998,9 @@ twist2(u, t) == Create a twisting circle p := point [sin u, cos(u)] rotateBy(p, theta) -cf(u,v) == sin(21*u) Color function with $21$ stripes +cf(u,v) == sin(21*u) Color function with $21$ stripes -ntubeDrawOpt(bcircle, twist2, 0..2*\%pi, 0..2*\%pi, Striped Torus +ntubeDrawOpt(bcircle, twist2, 0..2*\%pi, 0..2*\%pi, Striped Torus colorFunction == cf, var1Steps == 168, var2Steps == 126) \end{verbatim} @@ -81681,10 +86009,12 @@ ntubeDrawOpt(bcircle, twist2, 0..2*\%pi, 0..2*\%pi, Striped Torus \label{ugFimagesSix} \begin{verbatim} -gam(x,y) == The height and color are the - g := Gamma complex(x,y) real and argument parts - point [x,y,max(min(real g, 4), -4), argument g] of the Gamma function, - respectively. +-- The height and coloar are the real and argument parts +-- of the Gamma function, respectively. +gam(x,y) == + g := Gamma complex(x,y) + point [x,y,max(min(real g, 4), -4), argument g] + draw(gam, -\%pi..\%pi, -\%pi..\%pi, The Gamma Function title == "Gamma(x + \%i*y)", _ @@ -81712,16 +86042,21 @@ First we look at the conformal \index{conformal map} map $z \mapsto z + 1/z$. \begin{verbatim} -)read conformal Read program for drawing - conformal maps +)read conformal +-- Read program for drawing conformal maps + +-- The coordinate grid for the complex plane +f z == z -f z == z The coordinate grid for the - complex plane -conformalDraw(f, -2..2, -2..2, 9, 9, "cartesian") Mapping 1: Source +-- Mapping 1: Source +conformalDraw(f, -2..2, -2..2, 9, 9, "cartesian") -f z == z + 1/z The map $z \mapsto z + 1/z$ +-- The map z mapsto z + 1/z +f z == z + 1/z + +-- Mapping 1: Target +conformalDraw(f, -2..2, -2..2, 9, 9, "cartesian") -conformalDraw(f, -2..2, -2..2, 9, 9, "cartesian") Mapping 1: Target \end{verbatim} The map $z \mapsto -(z+1)/(z-1)$ maps @@ -81730,15 +86065,20 @@ the unit disk to the right half-plane, as shown on the Riemann sphere. \begin{verbatim} -f z == z The unit disk +-- The unit disk +f z == z -riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar") Mapping 2: Source +-- Mapping 2: Source +riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar") -f z == -(z+1)/(z-1) The map $x \mapsto -(z+1)/(z-1)$ +-- The map x mapsto -(z+1)/(z-1) +f z == -(z+1)/(z-1) -riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar") Mapping 2: Target +-- Mapping 2: Target +riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar") -riemannSphereDraw(-4..4, -4..4, 7, 7, "cartesian") Riemann Sphere Mapping +-- Riemann Sphere Mapping +riemannSphereDraw(-4..4, -4..4, 7, 7, "cartesian") \end{verbatim} \section{images8.input} @@ -81747,18 +86087,18 @@ riemannSphereDraw(-4..4, -4..4, 7, 7, "cartesian") Riemann Sphere Mapping \begin{verbatim} )read dhtri )read tetra -drawPyramid 4 Sierpinsky's Tetrahedron +drawPyramid 4 Sierpinsky's Tetrahedron Sierpinsky's Tetrahedron )read antoine -drawRings 2 Antoine's Necklace +drawRings 2 Antoine's Necklace Aintoine's Necklace )read scherk -drawScherk(3,3) Scherk's Minimal Surface +drawScherk(3,3) Scherk's Minimal Surface )read ribbonsnew -drawRibbons([x**i for i in 1..5], x=-1..1, y=0..2) Ribbon Plot +drawRibbons([x**i for i in 1..5], x=-1..1, y=0..2) Ribbon Plot \end{verbatim} \index{Scherk's minimal surface} @@ -81775,9 +86115,9 @@ plane and on the Riemann sphere. %-- Compile, don't interpret functions. %\xmpLine{)set fun comp on}{} \begin{verbatim} -C := Complex DoubleFloat Complex Numbers -S := Segment DoubleFloat Draw ranges -R3 := Point DFLOAT Points in 3-space +C := Complex DoubleFloat Complex Numbers +S := Segment DoubleFloat Draw ranges +R3 := Point DFLOAT Points in 3-space \end{verbatim} {\bf conformalDraw}{\it (f, rRange, tRange, rSteps, tSteps, coord)} @@ -81794,13 +86134,18 @@ $\theta$ directions; and \begin{verbatim} conformalDraw: (C -> C, S, S, PI, PI, String) -> VIEW3D conformalDraw(f,rRange,tRange,rSteps,tSteps,coord) == - transformC := Function for changing an $(x,y)$ - coord = "polar" => polar2Complex pair into a complex number + -- Function for changing an (x,y) + transformC := + -- pair into a complex number + coord = "polar" => polar2Complex cartesian2Complex cm := makeConformalMap(f, transformC) - sp := createThreeSpace() Create a fresh space - adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) Plot the coordinate lines - makeViewport3D(sp, "Conformal Map") Draw the image + -- Create a fresh space + sp := createThreeSpace() + -- Plot the coordinate lines + adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) + -- Draw the image + makeViewport3D(sp, "Conformal Map") \end{verbatim} {\bf riemannConformalDraw}{\it (f, rRange, tRange, rSteps, tSteps, coord)} @@ -81812,40 +86157,52 @@ Its arguments are the same as those for {\bf conformalDraw}. riemannConformalDraw:(C->C,S,S,PI,PI,String)->VIEW3D riemannConformalDraw(f, rRange, tRange, rSteps, tSteps, coord) == - transformC := Function for changing an $(x,y)$ - coord = "polar" => polar2Complex pair into a complex number + -- Function for changing an $(x,y)$ + transformC := + -- pair into a complex number + coord = "polar" => polar2Complex cartesian2Complex - sp := createThreeSpace() Create a fresh space + -- Create a fresh space + sp := createThreeSpace() cm := makeRiemannConformalMap(f, transformC) - adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) Plot the coordinate lines + -- Plot the coordinate lines + adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) curve(sp,[point [0,0,2.0@DFLOAT,0],point [0,0,2.0@DFLOAT,0] ]) - Add an invisible point at - makeViewport3D(sp,"Map on the Riemann Sphere") the north pole for scaling - -adaptGrid(sp, f, uRange, vRange, uSteps, vSteps) == Plot the coordinate grid - delU := (hi(uRange) - lo(uRange))/uSteps using adaptive plotting for - delV := (hi(vRange) - lo(vRange))/vSteps coordinate lines, and draw - uSteps := uSteps + 1; vSteps := vSteps + 1 tubes around the lines + -- Add an invisible point at the north pole for scaling + makeViewport3D(sp,"Map on the Riemann Sphere") + +-- Plot the coordinate grid +adaptGrid(sp, f, uRange, vRange, uSteps, vSteps) == + -- using adaptive plotting for coordinate lines, and draw + -- tubes around the lines + delU := (hi(uRange) - lo(uRange))/uSteps + delV := (hi(vRange) - lo(vRange))/vSteps + uSteps := uSteps + 1; vSteps := vSteps + 1 u := lo uRange - for i in 1..uSteps repeat Draw coordinate lines in the $v$ - c := curryLeft(f,u) direction; curve $c$ fixes the - cf := (t:DFLOAT):DFLOAT +-> 0 current value of $u$ + -- Draw coordinate lines in the v direction; curve c fixes the + -- current value of u + for i in 1..uSteps repeat + c := curryLeft(f,u) + cf := (t:DFLOAT):DFLOAT +-> 0 makeObject(c,vRange::SEG Float,colorFunction==cf, - Draw the $v$ coordinate line + -- Draw the v coordinate line space == sp, tubeRadius == .02, tubePoints == 6) u := u + delU v := lo vRange - for i in 1..vSteps repeat Draw coodinate lines in the $u$ - c := curryRight(f,v) direction; curve $c$ fixes the - cf := (t:DFLOAT):DFLOAT +-> 1 current value of $v$ + -- Draw coodinate lines in the u direction; curve c fixes the + -- current value of v + for i in 1..vSteps repeat + c := curryRight(f,v) + cf := (t:DFLOAT):DFLOAT +-> 1 makeObject(c,uRange::SEG Float,colorFunction==cf, - Draw the $u$ coordinate line + -- Draw the u coordinate line space == sp, tubeRadius == .02, tubePoints == 6) v := v + delV void() -riemannTransform(z) == Map a point in the complex - r := sqrt norm z plane to the Riemann sphere +-- Map a point in the complex plane to the Riemann sphere +riemannTransform(z) == + r := sqrt norm z cosTheta := (real z)/r sinTheta := (imag z)/r cp := 4*r/(4+r**2) @@ -81853,69 +86210,83 @@ riemannTransform(z) == Map a point in the complex if r>2 then sp := -sp point [cosTheta*cp, sinTheta*cp, -sp + 1] -cartesian2Complex(r:DFLOAT, i:DFLOAT):C == Convert Cartesian coordinates to - complex(r, i) complex Cartesian form +-- Convert Cartesian coordinates to complex Cartesian form +cartesian2Complex(r:DFLOAT, i:DFLOAT):C == + complex(r, i) -polar2Complex(r:DFLOAT, th:DFLOAT):C == Convert polar coordinates to - complex(r*cos(th), r*sin(th)) complex Cartesian form +-- Convert polar coordinates to complex Cartesian form +polar2Complex(r:DFLOAT, th:DFLOAT):C == + complex(r*cos(th), r*sin(th)) -makeConformalMap(f, transformC) == Convert complex function $f$ - (u:DFLOAT,v:DFLOAT):R3 +-> to a mapping: - (DFLOAT,DFLOAT) $mapsto$ R3 - z := f transformC(u, v) in the complex plane +-- Convert complex function f to a mapping: (DFLOAT,DFLOAT) maps to R3 +-- in the complex plane +makeConformalMap(f, transformC) == + (u:DFLOAT,v:DFLOAT):R3 +-> + z := f transformC(u, v) point [real z, imag z, 0.0@DFLOAT] -makeRiemannConformalMap(f, transformC) == Convert a complex function $f$ - (u:DFLOAT, v:DFLOAT):R3 +-> to a mapping: - (DFLOAT,DFLOAT) $\mapsto$ R3 - riemannTransform f transformC(u, v) on the Riemann sphere +-- Convert a complex function f to a mapping: (DFLOAT,DFLOAT) maps to R3 +-- on the Riemann sphere +makeRiemannConformalMap(f, transformC) == + (u:DFLOAT, v:DFLOAT):R3 +-> + riemannTransform f transformC(u, v) +-- Draw a picture of the mapping of the complex plane to +-- the Riemann sphere riemannSphereDraw: (S, S, PI, PI, String) -> VIEW3D - Draw a picture of the mapping - of the complex plane to - the Riemann sphere riemannSphereDraw(rRange,tRange,rSteps,tSteps,coord) == transformC := coord = "polar" => polar2Complex cartesian2Complex - grid := (u:DFLOAT, v:DFLOAT): R3 +-> Coordinate grid function + -- Coordinate grid function + grid := (u:DFLOAT, v:DFLOAT): R3 +-> z1 := transformC(u, v) point [real z1, imag z1, 0] - sp := createThreeSpace() Create a fresh space - adaptGrid(sp, grid, rRange, tRange, rSteps, tSteps) Draw the flat grid + -- Create a fresh space + sp := createThreeSpace() + -- Draw the flat grid + adaptGrid(sp, grid, rRange, tRange, rSteps, tSteps) connectingLines(sp,grid,rRange,tRange,rSteps,tSteps) - makeObject(riemannSphere,0..2*\%pi,0..\%pi,space==sp) Draw the sphere + -- Draw the sphere + makeObject(riemannSphere,0..2*\%pi,0..\%pi,space==sp) f := (z:C):C +-> z cm := makeRiemannConformalMap(f, transformC) - adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) Draw the sphere grid + -- Draw the sphere grid + adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) makeViewport3D(sp, "Riemann Sphere") +-- Draw the lines that connect the points in the complex +-- plane to the north pole of the Riemann sphere connectingLines(sp,f,uRange,vRange,uSteps,vSteps) == - Draw the lines that connect - delU := (hi(uRange) - lo(uRange))/uSteps the points in the complex - delV := (hi(vRange) - lo(vRange))/vSteps plane to the north pole - uSteps := uSteps + 1; vSteps := vSteps + 1 of the Riemann sphere + delU := (hi(uRange) - lo(uRange))/uSteps + delV := (hi(vRange) - lo(vRange))/vSteps + uSteps := uSteps + 1; vSteps := vSteps + 1 u := lo uRange - for i in 1..uSteps repeat For each u + for i in 1..uSteps repeat v := lo vRange - for j in 1..vSteps repeat For each v + for j in 1..vSteps repeat p1 := f(u,v) - p2 := riemannTransform complex(p1.1, p1.2) Project p1 onto the sphere - fun := lineFromTo(p1,p2) Create a line function + -- Project p1 onto the sphere + p2 := riemannTransform complex(p1.1, p1.2) + -- Create a line function + fun := lineFromTo(p1,p2) cf := (t:DFLOAT):DFLOAT +-> 3 - makeObject(fun, 0..1,space==sp,tubePoints==4, Draw the connecting line + -- Draw the connecting line + makeObject(fun, 0..1,space==sp,tubePoints==4, tubeRadius==0.01,colorFunction==cf) v := v + delV u := u + delU void() -riemannSphere(u,v) == A sphere sitting on the - sv := sin(v) complex plane, with radius 1 +-- A sphere sitting on the complex plane, with radius 1 +riemannSphere(u,v) == + sv := sin(v) 0.99@DFLOAT*(point [cos(u)*sv,sin(u)*sv,cos(v),0.0@DFLOAT])+ point [0.0@DFLOAT, 0.0@DFLOAT, 1.0@DFLOAT, 4.0@DFLOAT] -lineFromTo(p1, p2) == Create a line function - d := p2 - p1 that goes from p1 to p2 +-- Create a line function that goes from p1 to p2 +lineFromTo(p1, p2) == + d := p2 - p1 (t:DFLOAT):Point DFLOAT +-> p1 + t*d \end{verbatim} @@ -81931,13 +86302,15 @@ The formula was derived by Larry Lambe. )read ntube torusKnot: (DFLOAT, DFLOAT, DFLOAT, PI, PI) -> VIEW3D torusKnot(p, q ,r, uSteps, tSteps) == - knot := (t:DFLOAT):Point DFLOAT +-> Function for the torus knot + -- Function for the torus knot + knot := (t:DFLOAT):Point DFLOAT +-> fac := 4/(2.2@DFLOAT-sin(q*t)) fac * point [cos(p*t), sin(p*t), cos(q*t)] - circle := (u:DFLOAT, t:DFLOAT): Point DFLOAT +-> The cross section + -- The cross section + circle := (u:DFLOAT, t:DFLOAT): Point DFLOAT +-> r * point [cos u, sin u] - ntubeDrawOpt(knot, circle, 0..2*\%pi, 0..2*\%pi, - Draw the circle around the knot + -- Draw the circle around the knot + ntubeDrawOpt(knot, circle, 0..2*\%pi, 0..2*\%pi, var1Steps == uSteps, var2Steps == tSteps) \end{verbatim} @@ -81952,19 +86325,19 @@ These functions draw a 2-d curve in the normal planes around a 3-d curve. \begin{verbatim} -R3 := Point DFLOAT Points in 3-Space -R2 := Point DFLOAT Points in 2-Space -S := Segment Float Draw ranges - Introduce types for functions for: -ThreeCurve := DFLOAT -> R3 --the space curve function -TwoCurve := (DFLOAT, DFLOAT) -> R2 --the plane curve function -Surface := (DFLOAT, DFLOAT) -> R3 --the surface function - Frenet frames define a -FrenetFrame := coordinate system around a +R3 := Point DFLOAT Points in 3-Space +R2 := Point DFLOAT Points in 2-Space +S := Segment Float Draw ranges + Introduce types for functions: +ThreeCurve := DFLOAT -> R3 --the space curve function +TwoCurve := (DFLOAT, DFLOAT) -> R2 --the plane curve function +Surface := (DFLOAT, DFLOAT) -> R3 --the surface function + Frenet frames define a +FrenetFrame := coordinate system around a Record(value:R3,tangent:R3,normal:R3,binormal:R3) - point on a space curve -frame: FrenetFrame The current Frenet frame - for a point on a curve + point on a space curve +frame: FrenetFrame The current Frenet frame + for a point on a curve \end{verbatim} {\bf ntubeDraw}{\it (spaceCurve, planeCurve,} @@ -81985,13 +86358,12 @@ ntubeDraw(spaceCurve,planeCurve,uRange,tRange) == ntubeDrawOpt(spaceCurve, planeCurve, uRange, _ tRange, []$List DROPT) -ntubeDrawOpt: (ThreeCurve,TwoCurve,S,S,List DROPT) - -> VIEW3D +ntubeDrawOpt: (ThreeCurve,TwoCurve,S,S,List DROPT) -> VIEW3D +-- This function is similar to ntubeDraw, but takes +-- optional parameters that it passes to the draw command ntubeDrawOpt(spaceCurve,planeCurve,uRange,tRange,l) == - This function is similar - to ntubeDraw, but takes - delT:DFLOAT := (hi(tRange) - lo(tRange))/10000 optional parameters that it - oldT:DFLOAT := lo(tRange) - 1 passes to the draw command + delT:DFLOAT := (hi(tRange) - lo(tRange))/10000 + oldT:DFLOAT := lo(tRange) - 1 fun := ngeneralTube(spaceCurve,planeCurve,delT,oldT) draw(fun, uRange, tRange, l) @@ -82029,14 +86401,17 @@ to ensure that frames are only computed once for each value of {\it t}. \begin{verbatim} ngeneralTube: (ThreeCurve, TwoCurve, DFLOAT, DFLOAT) -> Surface ngeneralTube(spaceCurve, planeCurve, delT, oldT) == - free frame Indicate that $frame$ is global + -- Indicate that frame is global + free frame (v:DFLOAT, t: DFLOAT): R3 +-> - if (t $\sim$= oldT) then If not already computed - frame := nfrenetFrame(spaceCurve, t, delT) compute new frame + -- If not already computed compute new frame + if (t $\sim$= oldT) then + frame := nfrenetFrame(spaceCurve, t, delT) oldT := t p := planeCurve(v, t) + -- Project $p$ into the normal plane frame.value + p.1*frame.normal + p.2*frame.binormal - Project $p$ into the normal plane + \end{verbatim} %\input{gallery/dhtri.htex} @@ -82048,40 +86423,46 @@ a given triangle into another. \begin{verbatim} tri2tri: (List Point DFLOAT, List Point DFLOAT) -> DHMATRIX(DFLOAT) - Compute a DHMATRIX that -tri2tri(t1, t2) == transforms $t1$ to $t2,$ where - n1 := triangleNormal(t1) $t1$ and $t2$ are the vertices - n2 := triangleNormal(t2) of two triangles in 3-space +-- Compute a DHMATRIX that transforms t1 to t2, where +-- t1 and t2 are the vertices of two triangles in 3-space +tri2tri(t1, t2) == + n1 := triangleNormal(t1) + n2 := triangleNormal(t2) tet2tet(concat(t1, n1), concat(t2, n2)) tet2tet: (List Point DFLOAT, List Point DFLOAT) -> DHMATRIX(DFLOAT) - Compute a DHMATRIX that -tet2tet(t1, t2) == transforms $t1$ to $t2,$ - m1 := makeColumnMatrix t1 where $t1$ and $t2$ are the - m2 := makeColumnMatrix t2 vertices of two tetrahedrons - m2 * inverse(m1) in 3-space - -makeColumnMatrix(t) == Put the vertices of a tetra- - m := new(4,4,0)$DHMATRIX(DFLOAT) hedron into matrix form +-- Compute a DHMATRIX that transforms t1 to t2, where t1 and t2 +-- are the vertices of two tetrahedrons in 3-space +tet2tet(t1, t2) == + m1 := makeColumnMatrix t1 + m2 := makeColumnMatrix t2 + m2 * inverse(m1) + +-- Put the vertices of a tetrahedron into matrix form +makeColumnMatrix(t) == + m := new(4,4,0)$DHMATRIX(DFLOAT) for x in t for i in 1..repeat for j in 1..3 repeat m(j,i) := x.j m(4,i) := 1 m -triangleNormal(t) == Compute a vector normal to - a := triangleArea t the given triangle, whose - p1 := t.2 - t.1 length is the square root - p2 := t.3 - t.2 of the area of the triangle +-- Compute a vector normal to the given triangle, whose +-- length is the square root of the area of the triangle +triangleNormal(t) == + a := triangleArea t + p1 := t.2 - t.1 + p2 := t.3 - t.2 c := cross(p1, p2) len := length(c) len = 0 => error "degenerate triangle!" c := (1/len)*c t.1 + sqrt(a) * c -triangleArea t == Compute the area of a - a := length(t.2 - t.1) triangle using Heron's - b := length(t.3 - t.2) formula +-- Compute the area of a triangle using Heron's formula +triangleArea t == + a := length(t.2 - t.1) + b := length(t.3 - t.2) c := length(t.1 - t.3) s := (a+b+c)/2 sqrt(s*(s-a)*(s-b)*(s-c)) @@ -82094,48 +86475,60 @@ triangleArea t == Compute the area of a %\outdent{Sierpinsky's Tetrahedron} \begin{verbatim} -)set expose add con DenavitHartenbergMatrix Bring DH matrices into the - environment -x1:DFLOAT := sqrt(2.0@DFLOAT/3.0@DFLOAT) Set up the coordinates of the -x2:DFLOAT := sqrt(3.0@DFLOAT)/6 corners of the tetrahedron. +-- Bring DH matrices into the environment +)set expose add con DenavitHartenbergMatrix + +-- Set up the coordinates of the corners of the tetrahedron. +x1:DFLOAT := sqrt(2.0@DFLOAT/3.0@DFLOAT) +x2:DFLOAT := sqrt(3.0@DFLOAT)/6 -p1 := point [-0.5@DFLOAT, -x2, 0.0@DFLOAT] Some needed points +p1 := point [-0.5@DFLOAT, -x2, 0.0@DFLOAT] p2 := point [0.5@DFLOAT, -x2, 0.0@DFLOAT] p3 := point [0.0@DFLOAT, 2*x2, 0.0@DFLOAT] p4 := point [0.0@DFLOAT, 0.0@DFLOAT, x1] -baseTriangle := [p2, p1, p3] The base of the tetrahedron +-- The base of the tetrahedron +baseTriangle := [p2, p1, p3] +-- The middle triangle inscribed in the base of the tetrahedron +-- The bases of the triangles of the subdivided tetrahedron mt := [0.5@DFLOAT*(p2+p1), 0.5@DFLOAT*(p1+p3), 0.5@DFLOAT*(p3+p2)] - The middle triangle inscribed - in the base of the tetrahedron -bt1 := [mt.1, p1, mt.2] The bases of the triangles of -bt2 := [p2, mt.1, mt.3] the subdivided tetrahedron + +bt1 := [mt.1, p1, mt.2] +bt2 := [p2, mt.1, mt.3] bt3 := [mt.2, p3, mt.3] bt4 := [0.5@DFLOAT*(p2+p4), 0.5@DFLOAT*(p1+p4), 0.5@DFLOAT*(p3+p4)] -tt1 := tri2tri(baseTriangle, bt1) Create the transformations -tt2 := tri2tri(baseTriangle, bt2) that bring the base of the -tt3 := tri2tri(baseTriangle, bt3) tetrahedron to the bases of -tt4 := tri2tri(baseTriangle, bt4) the subdivided tetrahedron - -drawPyramid(n) == Draw a Sierpinsky tetrahedron - s := createThreeSpace() with $n$ levels of recursive - dh := rotatex(0.0@DFLOAT) subdivision +-- Create the transformations that bring the base of the +-- tetrahedron to the bases of the subdivided tetrahedron +tt1 := tri2tri(baseTriangle, bt1) +tt2 := tri2tri(baseTriangle, bt2) +tt3 := tri2tri(baseTriangle, bt3) +tt4 := tri2tri(baseTriangle, bt4) + +-- Draw a Sierpinsky tetrahedron with n levels of recursive +-- subdivision +drawPyramid(n) == + s := createThreeSpace() + dh := rotatex(0.0@DFLOAT) drawPyramidInner(s, n, dh) makeViewport3D(s, "Sierpinsky Tetrahedron") -drawPyramidInner(s, n, dh) == Recursively draw a Sierpinsky - n = 0 => makeTetrahedron(s, dh, n) tetrahedron - drawPyramidInner(s, n-1, dh * tt1) Draw the 4 recursive pyramids +-- Recursively draw a Sierpinsky tetrahedron +-- Draw the 4 recursive pyramids +drawPyramidInner(s, n, dh) == + n = 0 => makeTetrahedron(s, dh, n) + drawPyramidInner(s, n-1, dh * tt1) drawPyramidInner(s, n-1, dh * tt2) drawPyramidInner(s, n-1, dh * tt3) drawPyramidInner(s, n-1, dh * tt4) -makeTetrahedron(sp, dh, color) == Draw a tetrahedron into the - w1 := dh*p1 given space with the given - w2 := dh*p2 color, transforming it by - w3 := dh*p3 the given DH matrix +-- Draw a tetrahedron into the given space with the given +-- color, transforming it by the given DH matrix +makeTetrahedron(sp, dh, color) == + w1 := dh*p1 + w2 := dh*p2 + w3 := dh*p3 w4 := dh*p4 polygon(sp, [w1, w2, w4]) polygon(sp, [w1, w3, w4]) @@ -82154,15 +86547,18 @@ Draw Antoine's Necklace. Thank you to Matthew Grayson at IBM's T.J Watson Research Center for the idea. \begin{verbatim} -)set expose add con DenavitHartenbergMatrix Bring DH matrices into - the environment -torusRot: DHMATRIX(DFLOAT) The transformation for - drawing a sub ring - -drawRings(n) == Draw Antoine's Necklace with $n$ - s := createThreeSpace() levels of recursive subdivision - dh:DHMATRIX(DFLOAT) := identity() The number of subrings is $10^n$ - drawRingsInner(s, n, dh) Do the real work +-- Bring DH matrices into the environment +)set expose add con DenavitHartenbergMatrix + +-- The transformation for drawing a sub ring +torusRot: DHMATRIX(DFLOAT) + +-- Draw Antoine's Necklace with n levels of recursive subdivision +-- The number of subrings is 10^n. Do the real work +drawRings(n) == + s := createThreeSpace() + dh:DHMATRIX(DFLOAT) := identity() + drawRingsInner(s, n, dh) makeViewport3D(s, "Antoine's Necklace") \end{verbatim} @@ -82177,36 +86573,39 @@ rings. Notice how the {\bf DHMATRIX} operations are used to build up the proper matrix composing all these transformations. \begin{verbatim} -drawRingsInner(s, n, dh) == Recursively draw Antoine's - n = 0 => Necklace +-- Recursively draw Antoine's Necklace +drawRingsInner(s, n, dh) == + n = 0 => drawRing(s, dh) void() - t := 0.0@DFLOAT Angle around ring - p := 0.0@DFLOAT Angle of subring from plane - tr := 1.0@DFLOAT Amount to translate subring - inc := 0.1@DFLOAT The translation increment - for i in 1..10 repeat Subdivide into 10 linked rings + t := 0.0@DFLOAT Angle around ring + p := 0.0@DFLOAT Angle of subring from plane + tr := 1.0@DFLOAT Amount to translate subring + inc := 0.1@DFLOAT The translation increment + for i in 1..10 repeat Subdivide into 10 linked rings tr := tr + inc inc := -inc + -- Transform ring in center to a link dh' := dh*rotatez(t)*translate(tr,0.0@DFLOAT,0.0@DFLOAT)* - Transform ring in center - to a link rotatey(p)*scale(0.35@DFLOAT, 0.48@DFLOAT, 0.4@DFLOAT) drawRingsInner(s, n-1, dh') t := t + 36.0@DFLOAT p := p + 90.0@DFLOAT void() -drawRing(s, dh) == Draw a single ring into - free torusRot the given subspace, - torusRot := dh transformed by the given - DHMATRIX +-- Draw a single ring into the given subspace, +-- transformed by the given DHMATRIX +drawRing(s, dh) == + free torusRot + torusRot := dh makeObject(torus, 0..2*\%pi, 0..2*\%pi, var1Steps == 6, space == s, var2Steps == 15) -torus(u ,v) == Parameterization of a torus, - cu := cos(u)/6 transformed by the - DHMATRIX in $torusRot.$ +-- Parameterization of a torus, transformed by the +-- DHMATRIX in torusRot. +torus(u ,v) == + cu := cos(u)/6 + torusRot*point [(1+cu)*cos(v),(1+cu)*sin(v),(sin u)/6] \end{verbatim} @@ -82222,40 +86621,47 @@ See: {\it A Comprehensive Introduction to Differential Geometry,} Vol. 3, by Michael Spivak, Publish Or Perish, Berkeley, 1979, pp. 249-252. \begin{verbatim} -(xOffset, yOffset):DFLOAT Offsets for a single piece - of Scherk's minimal surface +-- Offsets for a single piece of Scherk's minimal surface +(xOffset, yOffset):DFLOAT -drawScherk(m,n) == Draw Scherk's minimal surface - free xOffset, yOffset on an $m$ by $n$ patch +-- Draw Scherk's minimal surface on an m by n patch +drawScherk(m,n) == + free xOffset, yOffset space := createThreeSpace() for i in 0..m-1 repeat xOffset := i*\%pi for j in 0 .. n-1 repeat - rem(i+j, 2) = 0 => 'iter Draw only odd patches + -- Draw only odd patches + rem(i+j, 2) = 0 => 'iter yOffset := j*\%pi - drawOneScherk(space) Draw a patch + -- Draw a patch + drawOneScherk(space) makeViewport3D(space, "Scherk's Minimal Surface") -scherk1(u,v) == The first patch that makes - x := cos(u)/exp(v) up a single piece of - point [xOffset + acos(x), yOffset + u, v, abs(v)] Scherk's minimal surface +-- The first patch that makes up a single piece of +-- Scherk's minimal surface +scherk1(u,v) == + x := cos(u)/exp(v) + point [xOffset + acos(x), yOffset + u, v, abs(v)] -scherk2(u,v) == The second patch +-- The second patch +scherk2(u,v) == x := cos(u)/exp(v) point [xOffset - acos(x), yOffset + u, v, abs(v)] -scherk3(u,v) == The third patch +-- The third patch +scherk3(u,v) == x := exp(v) * cos(u) point [xOffset + u, yOffset + acos(x), v, abs(v)] -scherk4(u,v) == The fourth patch +-- The fourth patch +scherk4(u,v) == x := exp(v) * cos(u) point [xOffset + u, yOffset - acos(x), v, abs(v)] -drawOneScherk(s) == Draw the surface by - breaking it into four - patches and then drawing - the patches +-- Draw the surface by breaking it into four +-- patches and then drawing the patches +drawOneScherk(s) == makeObject(scherk1,-\%pi/2..\%pi/2,0..\%pi/2,space==s, var1Steps == 28, var2Steps == 28) makeObject(scherk2,-\%pi/2..\%pi/2,0..\%pi/2,space==s, @@ -82267,11 +86673,10 @@ drawOneScherk(s) == Draw the surface by void() \end{verbatim} +%Original Page 703 { -%Original Page 703 - \setcounter{chapter}{6} % Appendix G \chapter{Glossary} \label{ugGlossary} @@ -82429,7 +86834,8 @@ $11.0$ in order to assign a floating-point value to $x$. \ourGloss{\glossaryTerm{attribute}}{% -a name or functional form denoting {\it any} useful computational or mathematical +a name or functional form denoting {\it any} +useful computational or mathematical property. For example, {\bf commutative($"*"$)} asserts that $*$ is commutative. Also, {\bf finiteAggregate} is used to assert that an aggregate has @@ -83524,8 +87930,8 @@ an argument of an {\it operator} (regarding an operator as a \ourGloss{\glossaryTerm{operation}}{% an abstraction of a {\it function}, described by a -{\it signature}. For example, $fact: -NonNegativeInteger -> NonNegativeInteger$ describes an operation for +{\it signature}. For example,\\ +$fact: NonNegativeInteger -> NonNegativeInteger$ describes an operation for ``the factorial of a (non-negative) integer.'' } @@ -83589,7 +87995,8 @@ function. } \ourGloss{\glossaryTerm{parent}}{% -(of a domain or category) a category which is explicitly declared in the source code +(of a domain or category) a category which is +explicitly declared in the source code definition for the domain either to the left of the {\tt with} or as an {\it export} of the domain. See {\it category extension}. @@ -83684,8 +88091,6 @@ Thus the change to $V$ made by the third statement does not affect $y$. } -\newpage - \ourGloss{\glossaryTerm{polymorphic}}{% a {\it function} (for example, one implementing an {\it algorithm}) defined with categorical types so as to be @@ -84259,7 +88664,7 @@ Addison-Wesley Publishing Company, Inc., \bibitem{3} Jenks, Richard D. and Sutor, Robert S.,\\ {\it Axiom, The Scientific Computation System} \\ Springer-Verlag, New York, NY 1992 ISBN 0-387-97855-0 -\bibitem{4} Daly, Timothy, "The Axiom Literate Documentation"\\ +\bibitem{4} Daly, Timothy, ``The Axiom Literate Documentation''\\ {\bf http://axiom.axiom-developer.org/axiom-website/documentation.html} \end{thebibliography} \printindex diff --git a/changelog b/changelog index 8ecb2a0..95c21fc 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20130418 jzc src/axiom-website/patches.html 20130418.01.jzc.patch +20130418 jzc books/bookvol0 fix Jenks book format issues 20130414 tpd src/axiom-website/patches.html 20130414.01.tpd.patch 20130414 tpd src/axiom-website/documentation.html add quote 20130411 tpd src/axiom-website/patches.html 20130411.01.tpd.patch diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 50fb8d5..e91ed76 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4137,5 +4137,7 @@ books/bookvol10.4 fix OUT.spad warning (bug 329) src/interp/Makefile, books/bookvol5 remove bc-matrix.lisp 20130414.01.tpd.patch src/axiom-website/documentation.html add quote +20130418.01.jzc.patch +books/bookvol0 fix Jenks book format issues