0`
0`The class of abelian groups,{} \spadignore{i.e.} additive monoids where each element has an additive inverse. \blankline
42`Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} but in general do not commute with the coefficients (which themselves may or may not be commutative). See \spadtype{FiniteAbelian
42`--MonoidRing} for the case of finite support. A useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.
130`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with an additive identity element. \blankline
175`This category represents products with coordinatewise additive operations.
235`the class of all additive (commutative) semigroups,{} \spadignore{i.e.} a set with a commutative and associative operation \spadop{+}. \blankline
282`The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as "the set of \spad{x} satisfying relation {\em r(x)}" An attribute \spadatt{finiteAggregate} is used to assert that a doma
282`--in has a finite number of elements.
320`The category of associative algebras (modules which are themselves rings). \blankline
378`Model for algebraically closed fields.
431`Model for algebraically closed function spaces.
516`An approximation to the real numbers.
560`Category for the inverse hyperbolic trigonometric functions.
619`Category for the inverse trigonometric functions.
682`An association list is a list of key entry pairs which may be viewed as a table. It is a poor mans version of a table: searching for a key is a linear operation.
764`A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.
814`\spadtype{BasicType} is the basic category for describing a collection of elements with \spadop{=} (equality).
856`A \spadtype{BiModule} is both a left and right module with respect to potentially different rings. \blankline
917`A bialgebra is a coalgebra which at the same time is an algebra such that the comultiplication is also an algebra homomorphism. MxM: Module(\spad{R}) should be replaced by a more restricted category,{} but it is not clear at this point which one.
991`A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure
1053`\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \spadfun{node} consisting of a value and a \spadfun{left} and \spadfun{right},{} both binary trees.
1116`The bit aggregate category models aggregates representing large quantities of Boolean data.
1159`A cachable set is a set whose elements keep an integer as part of their structure.
1203`This is an \spadtype{AbelianMonoid} with the cancellation property,{} \spadignore{i.e.} \spad{ a+b = a+c => b=c }. This is formalised by the partial subtraction operator,{} which satisfies the axioms listed below: \blankline
1260`The class of all domains which have canonical representaion,{} that is if mathematicaly equal elements have the same data structures.
1302`Rings of Characteristic Non Zero
1355`Rings of Characteristic Zero.
1404`A coalgebra A over a ring is an \spad{R}-module with a coassociative comultiplication from A to the tensor product of A with itself and which possesses a counit.
1478`A is coercible to \spad{B} means any element of A can automatically be converted into an element of \spad{B} by the interpreter.
1531`A is coercible to \spad{B} means any element of A can automatically be converted into an element of \spad{B} by the interpreter.
1581`A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \spadignore{e.g.} \spadfun{list} for \spadtype{List},{} \spadfun{flexibleArray} for \spadtype{FlexibleArray},{} and so on.
1629`Category for the usual combinatorial functions.
1689`CombinatorialOpsCategory is the category obtaining by adjoining summations and products to the usual combinatorial operations.
1746`The category of commutative rings with unity,{} \spadignore{i.e.} rings where \spadop{*} is commutative,{} and which have a multiplicative identity. element.
1794`The class of all commutative semigroups in multiplicative notation. In other words domains \spad{D} with \spad{"*": (D,{}D) -> D} which is commutative. Typically applied to rings.
1839`The class of set equipped with possibly unnatural linear order (needed for technical reasons).
1881`This category represents the extension of a ring by a square root of \spad{-1}.
1947`A is convertible to \spad{B} means any element of A can be converted into an element of \spad{B},{} but not automatically by the interpreter.
2002`A is convertible to \spad{B} means any element of A can be converted into an element of \spad{B},{} but not automatically by the interpreter.
2055`A dequeue is a doubly ended stack,{} that is,{} a bag where first items inserted are the first items extracted,{} at either the front or the back end of the data structure.
2109`This category is a collection of operations common to both categories \spadtype{Dictionary} and \spadtype{MultiDictionary}
2167`A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones.
2215`Differential extensions of a ring \spad{R}. Given a differentiation on \spad{R},{} extend it to a differentiation on \%.
2275`\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \spadfun{order} and \spadfun{weight} are extended from the set of derivatives of differ
2275`--ential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \spadfun{leader},{} \spadfun{initial},{} \spadfun{separant},{} \spadfun{differentialVariables},{} and \spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \spadfun{eval}. A convenient way of referencing derivati
2275`--ves is provided by the functions \spadfun{makeVariable}. \blankline To construct a domain using this constructor,{} one needs to provide a ground ring \spad{R},{} an ordered set \spad{S} of differential indeterminates,{} a ranking \spad{V} on the set of derivatives of the differential indeterminates,{} and a set \spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \blankline
2420`An ordinary differential ring,{} that is,{} a ring with an operation \spadfun{differentiate}. \blankline
2469`\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \spad{x},{}...,{}\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \spad{x},{}\spad{x'},{}\spad{x''},{}...,{} \spad{y},{}\spad{y'},{}\spad{
2469`--y''},{}... (Note: in the interpreter,{} the \spad{n}-th derivative of \spad{y} is displayed as \spad{y} with a subscript \spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \blankline A domain in this category is needed to construct a differential polynomial domain. Differential polynomials 
2469`--are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \spadfun{<}. For example,{} one may define one deriva
2469`--tive to be less than another by lexicographically comparing first the \spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For differential polynomial domains,{}
2469`-- this constructor provides a function \spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.
2543`attribute to indicate an aggregate of finite size
2628`Category of distributions formally given by moments.
2700`A division ring (sometimes called a skew field),{} \spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.
2746`A doubly-linked aggregate serves as a model for a doubly-linked list,{} that is,{} a list which can has links to both next and previous nodes and thus can be efficiently traversed in both directions.
2806`Category for the elementary functions.
2866`An eltable aggregate is one which can be viewed as a function. For example,{} the list \axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\em Dom} to an image domain {\em Im}.
2939`An eltable over domains \spad{D} and \spad{I} is a structure which can be viewed as a function from \spad{D} to \spad{I}. Examples of eltable structures range from data structures,{} \spadignore{e.g.} those of type \spadtype{List},{} to algebraic structures,{} \spadignore{e.g.} \spadtype{Polynomial}.
3003`Entire Rings (non-commutative Integral Domains),{} \spadignore{i.e.} a ring not necessarily commutative which has no zero divisors. \blankline
3047`A constructive euclidean domain,{} \spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\spadfun{euclideanSize}) than the divisor. \blankline Conditional attributes: \indented{2}{multiplicativeValuation\tab{25}\spad{Size(a*b)=Size(a)*Size(b)}} \indented{2}{additiveValuation\tab{25}\spad{Size(a*b)=Size(a)+Size(b)}}
3095`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions.
3150`An expression space is a set which is closed under certain operators.
3194`An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \spadtype{FlexibleArray} for an exception.
3258`ExtensionField {\em F} is the category of fields which extend the field \spad{F}
3309`FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.
3365`The category of commutative fields,{} \spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \spadfun{factor} operation while trivial is useful to have defined. \blankline
3402`This category provides an interface to operate on files in the computer\spad{'s} file system. The precise method of naming files is determined by the Name parameter. The type of the contents of the file is determined by \spad{S}.
3479`This category provides an interface to names in the file system.
3527`This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.
3624`FiniteAlgebraicExtensionField {\em F} is the category of fields which are finite algebraic extensions of the field {\em F}. If {\em F} is finite then any finite algebraic extension of {\em F} is finite,{} too. Let {\em K} be a finite algebraic extension of the finite field {\em F}. The exponentiation of elements of {\em K} defines a \spad{Z}-module structure on the multiplicative group of {\em K}. The additive group of {\em K} becomes
3624`-- a module over the ring of polynomials over {\em F} via the operation \spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \spad{F}) which is linear over {\em F},{} \spadignore{i.e.} for elements {\em a} from {\em K},{} {\em c,{}d} from {\em F} and {\em f,{}g} univariate polynomials over {\em F} we have \spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\em c} times \spadfun{linearAssociatedExp}(a,{}\spad{f}) plus {\em d} times \spadfun{linearAssociatedExp}(a,{}\spad{g}). Therefore \
3624`--spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\em F[X]}: \spadfun{linearAssociatedExp}(a,{}monomial(1,{}\spad{k})\spad{\$}SUP(\spad{F})) is defined to be \spadfun{Frobenius}(a,{}\spad{k}) which is {\em a^(q^k)} where {\em q=size()\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \spadfun{linearAssociatedExp}. These are the functions \spadfun{linearAssociatedOrder
3624`--} and \spadfun{linearAssociatedLog},{} respectively.
3692`This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve.
3883`FiniteFieldCategory is the category of finite fields
3936`Category of finite graphs,{} allows us to model graph theory \blankline
3993`A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \spadtype{LinearAggregate} such as \spadfun{reverse},{} \spadfun{sort},{} and so on.
4053`A FiniteRankAlgebra is an algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
4159`A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
4242`A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \spadtype{Set} for an example.
4306`The category of domains composed of a finite set of elements. We include the functions \spadfun{lookup} and \spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \blankline
4345`This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \blankline \indented{2}{1: \spadfunFrom{base}{FloatingPointSystem} of the \spadfunFrom{exponent}{FloatingPointSystem}.} \indented{9}{(actual implemena
4345`--tions are usually binary or decimal)} \indented{2}{2: \spadfunFrom{precision}{FloatingPointSystem} of the \spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \indented{2}{3: rounding error for operations} \blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\spad{"+"}) does not hold. Algorithms defined over a field need special consid
4345`--erations when the field is a floating point system.
4394`\axiomType{FortranFunctionCategory} is the category of arguments to NAG Library routines which return (sets of) function values.
4450`A category of domains which model machine arithmetic used by machines in the AXIOM-NAG link.
4507`\axiomType{FortranMatrixCategory} provides support for producing Functions and Subroutines when the input to these is an FriCAS object of type \axiomType{Matrix} or in domains involving \axiomType{FortranCode}.
4558`\axiomType{FortranMatrixFunctionCategory} provides support for producing Functions and Subroutines representing matrices of expressions.
4619`\axiomType{FortranProgramCategory} provides various models of FORTRAN subprograms. These can be transformed into actual FORTRAN code.
4675`\axiomType{FortranVectorCategory} provides support for producing Functions and Subroutines when the input to these is an FriCAS object of type \axiomType{Vector} or in domains involving \axiomType{FortranCode}.
4726`\axiomType{FortranVectorFunctionCategory} is the catagory of arguments to NAG Library routines which return the values of vectors of functions.
4787`A \spadtype{FramedAlgebra} is a \spadtype{FiniteRankAlgebra} together with a fixed \spad{R}-module basis.
4889`FramedNonAssociativeAlgebra(\spad{R}) is a \spadtype{FiniteRankNonAssociativeAlgebra} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank) over a commutative ring \spad{R} together with a fixed \spad{R}-module basis.
4968`A free abelian monoid on a set \spad{S} is the monoid of finite sums of the form \spad{reduce(+,{}[\spad{ni} * \spad{si}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are in a given abelian monoid. The operation is commutative.
5068`The category of free Lie algebras. It is used by domains of non-commutative algebra: \spadtype{LiePolynomial} and \spadtype{XPBWPolynomial}. \newline Author: Michel Petitot (petitot@lifl.\spad{fr})
5150`A domain of this category implements formal linear combinations of elements from a domain \spad{Basis} with coefficients in a domain \spad{R}. The domain \spad{Basis} needs only to belong to the category \spadtype{SetCategory} and \spad{R} to the category \spadtype{Ring}. Thus the coefficient ring may be non-commutative. See the \spadtype{XDistributedPolynomial} constructor for examples of domains built with the \spadty
5150`--pe{FreeModuleCategory} category constructor. Author: Michel Petitot (petitot@lifl.\spad{fr}) \blankline Note (Franz Lehner,{} June 2009): Since \spad{leadingTerm} makes no sense for unordered base sets,{} and at the time of this writing this domain was never used for such,{} it was changed to \spad{OrderedSet}. \spad{FreeModule} originally was not of FreeModuleCategory. Some functions (like \spad{support},{} \spad{coefficients},{} \spad{monomials},{} ...) from here could be moved to \spad{Inde
5150`--xedDirectProductCategory} but at the moment there is no need for this.
5233`This category provides a selection of evaluation operations depending on what the argument type \spad{R} provides.
5298`\spad{S} is \spadtype{FullyLinearlyExplicitRingOver R} means that \spad{S} is a \spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \spad{R} is a \spadtype{LinearlyExplicitRingOver Integer},{} then so is \spad{S}
5368`A set \spad{S} is PatternMatchable over \spad{R} if \spad{S} can lift the pattern-matching functions of \spad{S} over the integers and float to itself (necessary for matching in towers).
5430`\indented{2}{A is fully retractable to \spad{B} means that A is retractable to \spad{B},{} and,{}} \indented{2}{in addition,{} if \spad{B} is retractable to the integers or rational} \indented{2}{numbers then so is A.} \indented{2}{In particular,{} what we are asserting is that there are no integers} \indented{2}{(rationals) in A which don\spad{'t} retract into \spad{B}.} Date Created: March 1990 Date Last Updated: 9 April 1991
5489`This category is a model for the function field of a plane algebraic curve.
5658`A space of formal functions with arguments in an arbitrary ordered set.
5713`This category describes domains where \spadfun{\spad{gcd}} can be computed but where there is no guarantee of the existence of \spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \spadfun{factor} operation exist,{} factorization will be unique up to order and units.
5755`GradedAlgebra(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-algebra\spad{''}. A graded algebra is a graded module together with a degree preserving \spad{R}-linear map,{} called the {\em product}. \blankline The name ``product\spad{''} is written out in full so inner and outer products with the same mapping type can be distinguished by name.
5834`GradedModule(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-module\spad{''},{} \spadignore{i.e.} collection of \spad{R}-modules indexed by an abelian monoid \spad{E}. An element \spad{g} of \spad{G[s]} for some specific \spad{s} in \spad{E} is said to be an element of \spad{G} with {\em degree} \spad{s}. Sums are defined in each module \spad{G[s]} so two elements of \spad{G} have a sum if they have the same degree. \blankl
5834`--ine Morphisms can be defined and composed by degree to give the mathematical category of graded modules.
5912`The class of multiplicative groups,{} \spadignore{i.e.} monoids with multiplicative inverses. \blankline
5949`A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \spadatt{finiteAggregate} have a finite number of members. Of course,{} such a domain may have an infinite number of elements,{} like,{} for example \spadtype{List}. Those domains with attribute \spadatt{shallowlyMutable} allo
5949`--w an element to be modified or updated without changing its overall value.
6008`A Hopf algebra is a bialgebra with antipode.
6087`Category for the hyperbolic trigonometric functions.
6146`An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.
6223`This category represents the direct product of some set with respect to an ordered indexing set.
6309`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions. The difference between this and \spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.
6377`An \spad{IntegerNumberSystem} is a model for the integers.
6426`The category of commutative integral domains,{} \spadignore{i.e.} commutative rings with no zero divisors. \blankline
6473`\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This category implements of interval arithmetic and transcendental + functions over intervals.
6554`\axiomType{JetBundleBaseFunctionCategory} defines the category of functions (local sections) of the base space of a jet bundle,{} \spadignore{i.e.} functions depending only on the independent variables. Such a category is needed \spadignore{e.g.} for the representation of solutions.
6636`\axiomType{JetBundleCategory} provides basic data structures and procedures for jet bundles. Nearly all necessary functions are implemented already here. Only the representation and functions which direct access to it must be implemented in a domain. Two notations of derivatives are supported. Default is multi-index notation,{} where the \spad{i}-th entry of the index denotes the number of differentiations taken with respect to \axiom{x^i}. In repeated in
6636`--dex notation each entry \axiom{\spad{i}} in the index denotes a differentiation with respect to \axiom{x^i}. The choice affects,{} however,{} only in- and output. Internally,{} multi-index notation is used throughout.
6683`\axiomType{JetBundleFunctionCategory} defines the category of functions (local sections) over a jet bundle. The formal derivative is defined already here. It uses the Jacobi matrix of the functions. The columns of the matrices are enumerated by jet variables. Thus they are represented as a \axiomType{Record} of the matrix and a list of the jet variables. Several simplification routines are implemented already here.
6760`A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.
6834`LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.
6894`The category of all left algebras over an arbitrary ring.
6943`The category of left modules over an \spad{rng} (ring not necessarily with unit). This is an abelian group which supports left multiplation by elements of the \spad{rng}. \blankline
6997`This is category of left ore rings,{} that is noncommutaive rings without zero divisors where we can compute least left common multiple
7040`The category of Lie Algebras. It is used by the following domains of non-commutative algebra: \axiomType{LiePolynomial} and \axiomType{XPBWPolynomial}. \newline Author : Michel Petitot (petitot@lifl.\spad{fr}).
7101`A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \spadfun{concat} of two lists needs only to copy its first argument,{} whereas \spadfun{concat} of two arrays needs to copy both arguments. Most of the operations ex
7101`--ported here apply to infinite objects (\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \spadtype{FiniteLinearAggregate}.
7155`\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
7238`An extension ring with an explicit linear dependence test.
7302`Category for the transcendental Liouvillian functions.
7361`A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.
7413``Logic' provides the basic operations for lattices,{} \spadignore{e.g.} boolean algebra.
7450`\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by calling the function \spadfun{mi
7450`--nRowIndex}. The index of the 'first' column may be obtained by calling the function \spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.
7577`The category for general multi-variate possibly skew polynomials over a ring \spad{R},{} in variables from VarSet,{} with exponents from the \spadtype{OrderedAbelianMonoidSup}.
7689`This category specifies operations needed by \indented{1}{ModularAlgebraicGcd package.\space{2}Since we have multiple} \indented{1}{implementations we specify interface here and put} \indented{1}{implementations in separate packages.\space{2}Most operations} \indented{1}{are done using special purpose abstract representation.} \indented{1}{Apropriate types are passesd as parametes: \spad{MPT} is type} \indented{1}{of modul
7689`--ar polynomials in one variable with coefficients} \indented{1}{in some algebraic extension.\space{2}\spad{MD} is type of modulus.} \indented{1}{Final results are converted to packed representation,{}} \indented{1}{with coefficients (from prime field) stored in one} \indented{1}{array and exponents (in main variable and in auxilary} \indented{1}{variables representing generators of algebrac extension)} \indented{1}{stored in parallel array.}
7769`The category of modules over a commutative ring. \blankline
7826`\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \indented{1}{\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \indented{3}{leftIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{3}\tab{30} 1*x=x} \indented{3}{rightIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{2}\tab{30} x*1=x} Common Additional Axioms \indented{3}{unitsKnown
7826`---if "recip" says "failed",{} that PROVES input wasn\spad{'t} a unit}
7873`Monad is the class of all multiplicative monads,{} \spadignore{i.e.} sets with a binary operation.
7910`A \spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.
8015`This is the category of linear operator rings with one generator. The generator is not named by the category but can always be constructed as \spad{monomial(1,{}1)}. \blankline For convenience,{} call the generator \spad{G}. Then each value is equal to \indented{4}{\spad{sum(a(i)*G^i,{} i = 0..n)}} for some unique \spad{n} and \spad{a(i)} in \spad{R}. \blankline Note that multiplication is not necessarily commutative. In fact,{} if \spad{a} i
8015`--s in \spad{R},{} it is quite normal to have \spad{a*G \~= G*a}.
8075`\spadtype{MonoidRingCategory}(\spad{R},{}\spad{M}) defines the algebra of all maps from the monoid \spad{M} to the commutative ring \spad{R} with finite support.
8141`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with a multiplicative identity element. \blankline
8180`A multi-dictionary is a dictionary which may contain duplicates. As for any dictionary,{} its size is assumed large so that copying (non-destructive) operations are generally to be avoided.
8241`A multi-set aggregate is a set which keeps track of the multiplicity of its elements.
8418`\spadtype{MultivariateTaylorSeriesCategory} is the most general multivariate Taylor series category.
8508`NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \indented{3}{\spad{r*}(a*b) = (r*a)\spad{*b} = a*(\spad{r*b})}
8579`A NonAssociativeRing is a non associative \spad{rng} which has a unit,{} the multiplication is not necessarily commutative or associative.
8631`NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \indented{2}{\spad{x*}(\spad{y+z}) = x*y + \spad{x*z}} \indented{2}{(x+y)\spad{*z} = \spad{x*z} + \spad{y*z}} Common Additional Axioms \indented{2}{noZeroDivisors\space{2}ab = 0 \spad{=>} \spad{a=0} or \spad{b=0}}
8680`The category of normalized triangular sets. A triangular set \spad{ts} is said normalized if for every algebraic variable \spad{v} of \spad{ts} the polynomial \spad{select(ts,{}v)} is normalized \spad{w}.\spad{r}.\spad{t}. every polynomial in \spad{collectUnder(ts,{}v)}. A polynomial \spad{p} is said normalized \spad{w}.\spad{r}.\spad{t}. a non-c
8680`--onstant polynomial \spad{q} if \spad{p} is constant or \spad{degree(p,{}mdeg(q)) = 0} and \spad{init(p)} is normalized \spad{w}.\spad{r}.\spad{t}. \spad{q}. One of the important features of normalized triangular sets is that they are regular sets.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}{[2] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZ
8680`--A "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)} \indented{1}{[3] \spad{M}. MORENO MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[4] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \
8680`--spad{P}.etM. Curie,{} Paris,{} 1997.}
8838`OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.
8901`One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently grow or shrink on demand as can list
8901`-- structures (see however \spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result.
8968`\spadtype{OpenMath} provides operations for exporting an object in OpenMath format.
9005`Ordered sets which are also abelian groups,{} such that the addition preserves the ordering.
9058`This domain is an OrderedAbelianMonoid with a \spadfun{sup} operation added. The purpose of the \spadfun{sup} operator in this domain is to act as a supremum with respect to the partial order imposed by \spadop{-},{} rather than with respect to the total \spad{>} order (since that is "max"). \blankline
9114`Ordered sets which are also abelian monoids,{} such that the addition preserves the ordering.
9166`Ordered sets which are also abelian semigroups,{} such that the addition preserves the ordering. \indented{2}{\spad{ x < y => x+z < y+z}}
9221`Ordered sets which are also abelian cancellation monoids,{} such that the addition preserves the ordering.
9286`Ordered finite sets.
9332`The category of ordered commutative integral domains,{} where ordering and the arithmetic operations are compatible \blankline
9387`Ordered sets which are also monoids,{} such that multiplication preserves the ordering. \blankline
9433`to become an in order iterator
9503`Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \blankline
9548`Semigroups with compatible ordering.
9598`The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \spad{(a,{}b)} exactly one of the following relations holds \spad{a<b or a=b or b<a} and the relation is transitive,{} \spadignore{i.e.} \spad{a<b and b<c => a<c}. This order should be the natural order on given structure.
9641`This is the catefory of stream-based representations of \indented{2}{the \spad{p}-adic integers.}
9705`A partial differential ring with differentiations indexed by a parameter type \spad{S}. \blankline
9775`The class of partially ordered sets,{} that is sets equipped with transitive and reflexive relation \spad{<=}.
9820`This is the description of any package which provides partial functions on a domain belonging to TranscendentalFunctionCategory.
9891`A set \spad{R} is PatternMatchable over \spad{S} if elements of \spad{R} can be matched to patterns over \spad{S}.
9954`An object \spad{S} is Patternable over an object \spad{R} if \spad{S} can lift the conversions from \spad{R} into \spadtype{Pattern(Integer)} and \spadtype{Pattern(Float)} to itself.
10004`PermutationCategory provides a categorial environment \indented{1}{for subgroups of bijections of a set (\spadignore{i.e.} permutations)}
10071`PlottablePlaneCurveCategory is the category of curves in the plane which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the \spad{x}-coordinates and \spad{y}-coordinates of the points on the curve.
10132`PlottableSpaceCurveCategory is the category of curves in 3-space which may be plotted via the graphics facilities. Functions are provided for obtaining lists of lists of points,{} representing the branches of the curve,{} and for determining the ranges of the \spad{x-},{} \spad{y-},{} and \spad{z}-coordinates of the points on the curve.
10193`PointCategory is the category of points in space which may be plotted via the graphics facilities. Functions are provided for defining points and handling elements of points.
10245`The category for general multi-variate polynomials over a ring \spad{R},{} in variables from VarSet,{} with exponents from the \spadtype{OrderedAbelianMonoidSup}. Here variables commute with the coefficients.
10348`This is the category of domains that know "enough" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.
10412`A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \spad{R} being an integral domain,{} a set of polynomials may be viewed as a representation of the ideal it generates in the
10412`-- polynomial ring \spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.
10571`\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.
10674`Category for the functions defined by integrals.
10733`The category of constructive principal ideal domains,{} \spadignore{i.e.} where a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.
10783`A priority queue is a bag of items from an ordered set where the item extracted is always the maximum element.
10851`\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.
10921`A queue is a bag where the first item inserted is the first item extracted.
10974`QuotientField(\spad{S}) is the category of fractions of an Integral Domain \spad{S}.
11044`The \spad{RadicalCategory} is a model for the rational numbers.
11092`\axiomType{RealClosedField} provides common acces functions for all real closed fields.
11141`The category of real numeric domains,{} \spadignore{i.e.} convertible to floats.
11184`The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \spadtype{DifferentialRing} or the elementary functions (see \spadtype{TranscendentalFunctionCategory}) in the definition.
11230`\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.
11380`\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \spad{R}-modules and will be non-mutable.
11569`A recursive aggregate over a type \spad{S} is a model for a a directed graph containing values of type \spad{S}. Recursively,{} a recursive aggregate is a {\em node} consisting of a \spadfun{value} from \spad{S} and 0 or more \spadfun{children} which are recursive aggregates. A node with no children is called a \spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infinite loop.
11626`A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \spad{w}. \spad{r}. \spad{t}. to the total ordering on the elements in the ordered set,{} so that some operatio
11626`--ns usually defined for univariate polynomials make sense here.
11733`The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polynomial ring \spad{k[x1,{}...,{}xn]} where
11733`-- \spad{k} is the fraction field of \spad{R}. The triangular set \spad{[t1,{}...,{}tm]} is regular iff for every \spad{i} the initial of \spad{ti+1} is invertible in the tower of simple extensions associated with \spad{[t1,{}...,{}\spad{ti}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \spad{I} iff the radical of \spad{I} is equal to the intersection of the radical ideals generated by the saturated ideals of the \spad{[T1,{}...,{}\spad{T
11733`--i}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \spad{T} iff it is a split of Kalkbrener of the saturated ideal of \spad{T}. Let \spad{K} be an algebraic closure of \spad{k}. Assume that \spad{V} is finite with cardinality \spad{n} and let \spad{A} be the affine space \spad{K^n}. For a regular triangular set \spad{T} let denote by \spad{W(T)} the set of regular zeros of \spad{T}. A family \spad{[T1,{}...,{}Ts]} of regular tria
11733`--ngular sets is a split of Lazard of a given subset \spad{S} of \spad{A} iff the union of the \spad{W(\spad{Ti})} contains \spad{S} and is contained in the closure of \spad{S} (\spad{w}.\spad{r}.\spad{t}. Zariski topology). A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Lazard of a given triangular set \spad{T} if it is a split of Lazard of \spad{W(T)}. Note that if \spad{[T1,{}...,{}Ts]} is a split of Lazard of \spad{T} then it is also a split of Kalkbrener of \spad{T
11733`--}. The converse is \spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \newline References : \indented{1}{[1] \spad{M}. KALKBRENER "Three contributions to elimination theory"} \indented{5}{\spad{Phd} Thesis
11733`--,{} University of Linz,{} Austria,{} 1991.} \indented{1}{[2] \spad{M}. KALKBRENER "Algorithmic properties of polynomial rings"} \indented{5}{Journal of Symbol. Comp. 1998} \indented{1}{[3] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)} \indented{1}{[4] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
11889`\spad{B} is retractable from A means that some elements in A can be converted into elements of \spad{B} and any element of \spad{B} can be converted into an element of A.
11945`A is retractable to \spad{B} means that some elementsif A can be converted into elements of \spad{B} and any element of \spad{B} can be converted into an element of A.
11999`The category of right modules over an \spad{rng} (ring not necessarily with unit). This is an abelian group which supports right multiplation by elements of the \spad{rng}. \blankline
12054`The category of rings with unity,{} always associative,{} but not necessarily commutative.
12089`The category of associative rings,{} not necessarily commutative,{} and not necessarily with a 1. This is a combination of an abelian group and a semigroup,{} with multiplication distributing over addition. \blankline
12122`This category allows the manipulation of Lisp values while keeping the grunge fairly localized.
12255`PointCategory is the category of points and vectors in space. Points may be used to represent shapes in a scenegraph such as: the vertices of a cube or points along a line. Shapes such as these may be defined explicitly or may be plotted. We implement SetCategory to allow us to curry transforms from one point to another.
12302`This category provides operations on ranges,{} or {\em segments} as they are called.
12357`This category provides an interface for expanding segments to a stream of elements.
12452`the class of all multiplicative semigroups,{} \spadignore{i.e.} a set with an associative operation \spadop{*}. \blankline
12494`The category of associative semirings,{} not necessarily commutative,{} with 0 and 1.
12534`The category of associative semirings,{} not necessarily commutative,{} and not necessarily with a 1.
12572`A category for infinite sequences over a commutative ring. It is a stream with arithmetics.
12640`A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.
12699`\spadtype{SetCategory} is the basic category for describing a collection of elements with \spadop{=} (equality) and \spadfun{coerce} to output form. \blankline Conditional Attributes: \indented{3}{canonical\tab{15}data structure equality is the same as \spadop{=}}
12743`This is the category of polynomials in noncommutative variables over noncommutative rings. We do not assume that variables and elements of the base ring commute. We assume that the polynomial ring is of solvable type,{} so noncommutative version of Buchberger algorithm works.
12862`Category for the other special functions.
12918`The category of square-free and normalized triangular sets. Thus,{} up to the primitivity axiom of [1],{} these sets are Lazard triangular sets.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991}
13087`The category of square-free regular triangular sets. A regular triangular set \spad{ts} is square-free if the \spad{gcd} of any polynomial \spad{p} in \spad{ts} and \spad{differentiate(p,{}mvar(p))} \spad{w}.\spad{r}.\spad{t}. \axiomOpFrom{collectUnder}{TriangularSetCategory}(\spad{ts},{}\axiomOpFrom{mvar}{RecursivePolynomialCategory}(\sp
13087`--ad{p})) has degree zero \spad{w}.\spad{r}.\spad{t}. \spad{mvar(p)}. Thus any square-free regular set defines a tower of square-free simple extensions.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}{[2] \spad{M}. KALKBRENER "Algorithmic properties of polynomial rings"} \indented{5}{Habilitation Thesis,{} ETZH,{} Zurich,{} 1995.} \indented{1}{[3] \spad{M}. MO
13087`--RENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
13253`\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.
13426`A stack is a bag where the last item inserted is the first item extracted.
13479`A class of objects which can be 'stepped through'. Repeated applications of \spadfun{nextItem} is guaranteed never to return duplicate items and only return "failed" after exhausting all elements of the domain. This assumes that the sequence starts with \spad{init()}. For infinite domains,{} repeated application of \spadfun{nextItem} is not required to reach all possible domain elements starting from any initial element. \blankline Conditional attributes: \ind
13479`--ented{2}{infinite\tab{15}repeated \spad{nextItem}\spad{'s} are never "failed".}
13521`A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \spadtype{Stream}. From streams,{} a number of infinite structures such as power series can be built.
13575`A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.
13622`A category for string-like objects
13670`A table aggregate is a model of a table,{} \spadignore{i.e.} a discrete many-to-one mapping from keys to entries.
13963`Category of tensor powers of modules over commutative rings.
14068`Category of tensor products of modules over commutative rings.
14167`Universal property of tensor products.
14315`The category ThreeSpaceCategory is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them.
14373`Category for the transcendental elementary functions.
14437`The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \axiom{\spad{R}} be an integral domain and \axiom{\spad{V}} a finite ordered set of variables,{} say \axiom{\spad{X1} < \spad{X2} < ... < \spad{Xn}}. A set \axiom{\spad{S}} of polynomials in \axiom{\spad{R}[\spad{X1},{}\spad{X2},{}...,{}\spad{Xn}]
14437`--} is triangular if no elements of \axiom{\spad{S}} lies in \axiom{\spad{R}},{} and if two distinct elements of \axiom{\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \axiom{\spad{P}} is reduced \spad{w}.\spad{r}.\
14437`--spad{t} a non-constant polynomial \axiom{\spad{Q}} if the degree of \axiom{\spad{P}} in the main variable of \axiom{\spad{Q}} is less than the main degree of \axiom{\spad{Q}}. A polynomial \axiom{\spad{P}} is reduced \spad{w}.\spad{r}.\spad{t} a triangular set \axiom{\spad{T}} if it is reduced \spad{w}.\spad{r}.\spad{t}. every polynomial of \axiom{\spad{T}}. \newline References : \indented{1}{[1] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Tr
14437`--iangular Sets" Journal of Symbol. Comp. (to appear)}
14591`Category for the trigonometric functions.
14654`\indented{1}{TwoDimensionalArrayCategory is a general array category which} allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and columns returned as objects of type Col. The index of the 'first' row may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obt
14654`--ained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.
14782`The fundamental Type.
14817`A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \spadfun{value} of the list designating the head,{} or \spadfun{first},{} of the list,{} and the child designating the tail,{} or \spadfun{rest},{} of the list. A node with no child then designates the empty 
14817`--list. Since these aggregates are recursive aggregates,{} they may be cyclic.
14879`A constructive unique factorization domain,{} \spadignore{i.e.} where we can constructively factor members into a product of a finite number of irreducible elements.
14934`\spadtype{UnivariateLaurentSeriesCategory} is the category of Laurent series in one variable.
15008`This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair \spad{[n,{}f(x)]},{} where \spad{n} is an arbitrary integer and \spad{f(x)} is a Taylor series. This pair represents the Laurent series \spad{x^n * f(x)}.
15140`The category of univariate polynomials over a ring \spad{R}. No particular model is assumed - implementations can be either sparse or dense.
15208`\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.
15307`\spadtype{UnivariatePuiseuxSeriesCategory} is the category of Puiseux series in one variable.
15382`This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x^r)}.
15515`This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}. This category is an evolution of the types \indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \spad{M}. Watt.
15588`\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.
15661`VarCat represents a variable in Lambda and Ski domains. Since we are working in terms of functions then a variable will be a function (possibly a constant function) a variable has a name represented by a String. If the Lambda or Ski calculas is typed then the variable will also have a type.
15700`The class of all polynomials domains such that variables commute with coefficients.
15765`\spadtype{VectorCategory} represents the type of vector like objects,{} \spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.
15820`Vector Spaces (not necessarily finite dimensional) over a field.
15872`This is the category of algebras over non-commutative rings. It is used by constructors of non-commutative algebras such as: \indented{4}{\spadtype{XPolynomialRing}.} \indented{4}{\spadtype{XFreeAlgebra}} Author: Michel Petitot (petitot@lifl.\spad{fr})
15918`This category specifies opeations for polynomials and formal series with non-commutative variables.
15983`The Category of polynomial rings with non-commutative variables. The coefficient ring may be non-commutative too. However coefficients commute with variables.
16052`The class of all euclidean domains such that \spad{euclideanSize(a*b)=euclideanSize(a)+euclideanSize(b)}.
16103`Approximate numbers with arbitrarily large exponents
16154`Approximate numbers for which the user can set the precision for subsequent calculations.
16206`The class of all integral domains such that we can choose a canonical representative for each class of associate elements,{} that is \spad{associates?(a,{}b)} returns \spad{true} if and only if \spad{unitCanonical(a) = unitCanonical(b)}.
16259`The class of all integral domains such that \spad{unitCanonical(a)*unitCanonical(b) = unitCanonical(a*b)}.
16309`The class of all aggregates with a finite number of elements.
16358`The class of all domains which have have lazy representation
16407`The class of all euclidean domains such that \spad{euclideanSize(a*b)=euclideanSize(a)*euclideanSize(b)}.
16464`The class of all semirings such that \spad{x * y \~~= 0} implies both \spad{x} and \spad{y} are non-zero
16512`The class of all domains which have have immediate components that are updateable (mutable).
16562`The class of all monoids (multiplicative semigroups with a 1) such that the operation \spadfun{recip} can only return "failed" if its argument is not a unit.
16606`AlgebraGivenByStructuralConstants implements finite rank algebras over a commutative ring,{} given by the structural constants \spad{gamma} with respect to a fixed basis \spad{[a1,{}..,{}an]},{} where \spad{gamma} is an \spad{n}-vector of \spad{n} by \spad{n} matrices \spad{[(gammaijk) for k in 1..rank()]} defined by \spad{\spad{ai} * aj
16606`-- = gammaij1 * a1 + ... + gammaijn * an}. The symbols for the fixed basis have to be given as a list of symbols.
16773`Function field defined by \spad{f}(\spad{x},{} \spad{y}) = 0.
16975`Algebraic closure of the rational numbers,{} with mathematical =
17036`This domain implements anonymous functions
17087`The domain of antisymmetric polynomials.
17170`\spadtype{Any} implements a type that packages up objects and their types in objects of \spadtype{Any}. Roughly speaking that means that if \spad{s : S} then when converted to \spadtype{Any},{} the new object will include both the original object and its type. This is a way of converting arbitrary objects into a single type without losing any of the original information. Any object can be converted to one of \spadtype{Any}.
17216`A stack represented as a flexible array.
17292`\indented{1}{AssociatedJordanAlgebra takes an algebra \spad{A} and uses \spadfun{*\$A}} \indented{1}{to define the new multiplications \spad{a*b := (a *\$A b + b *\$A a)/2}} \indented{1}{(anticommutator).} \indented{1}{The usual notation \spad{{a,{}b}_+} cannot be used due to} \indented{1}{restrictions in the current language.} \indented{1}{This domain only gives a Jordan 
17292`--algebra if the} \indented{1}{Jordan-identity \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds} \indented{1}{for all \spad{a},{}\spad{b},{}\spad{c} in \spad{A}.} \indented{1}{This relation can be checked by} \indented{1}{\spadfun{jordanAdmissible?()\$A}.} \blankline If the underlying algebra is of type \spadtype{FramedNonAssociativeAlgebra(R)} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank,{} together with a fixed \spad{R}-module basis),{} 
17292`--then the same is \spad{true} for the associated Jordan algebra. Moreover,{} if the underlying algebra is of type \spadtype{FiniteRankNonAssociativeAlgebra(R)} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank),{} then the same \spad{true} for the associated Jordan algebra.
17423`AssociatedLieAlgebra takes an algebra \spad{A} and uses \spadfun{*\$A} to define the Lie bracket \spad{a*b := (a *\$A b - b *\$A a)} (commutator). Note that the notation \spad{[a,{}b]} cannot be used due to restrictions of the current compiler. This domain only gives a Lie algebra if the Jacobi-identity \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds for all \spad{a},{}\spad{b},{}\
17423`--spad{c} in \spad{A}. This relation can be checked by \spad{lieAdmissible?()\$A}. \blankline If the underlying algebra is of type \spadtype{FramedNonAssociativeAlgebra(R)} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank,{} together with a fixed \spad{R}-module basis),{} then the same is \spad{true} for the associated Lie algebra. Also,{} if the underlying algebra is of type \spadtype{FiniteRankNonAssociativeAlgebra(R)} (\spadignore{i.e.}
17423`-- a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank),{} then the same is \spad{true} for the associated Lie algebra.
17548`\spadtype{AssociationList} implements association lists. These may be viewed as lists of pairs where the first part is a key and the second is the stored value. For example,{} the key might be a string with a persons employee identification number and the value might be a record with personnel data.
17655`Automorphism \spad{R} is the multiplicative group of automorphisms of \spad{R}.
17715`\spadtype{BalancedBinaryTree(S)} is the domain of balanced binary trees (bbtree). A balanced binary tree of \spad{2^k} leaves,{} for some \spad{k > 0},{} is symmetric,{} that is,{} the left and right subtree of each interior node have identical shape. In general,{} the left and right subtree of a given node can differ by at most leaf node.
17803`Stream-based implementation of \spad{Zp:} \spad{p}-adic numbers are represented as sum(\spad{i} = 0..,{} a[\spad{i}] * p^i),{} where the a[\spad{i}] lie in -(\spad{p} - 1)\spad{/2},{}...,{}(\spad{p} - 1)\spad{/2}.
17881`Stream-based implementation of \spad{Qp:} numbers are represented as sum(\spad{i} = \spad{k}..,{} a[\spad{i}] * p^i),{} where the a[\spad{i}] lie in -(\spad{p} - 1)\spad{/2},{}...,{}(\spad{p} - 1)\spad{/2}.
17995`A basic operator is an object that can be applied to a list of arguments from a set,{} the result being a kernel over that set.
18050`This domain allows rational numbers to be presented as repeating binary expansions.
18130`This domain provides an implementation of binary files. Data is accessed one byte at a time as a small integer.
18212`BinarySearchTree(\spad{S}) is the domain of a binary trees where elements are ordered across the tree. A binary search tree is either empty or has a value which is an \spad{S},{} and a right and left which are both BinaryTree(\spad{S}) Elements are ordered across the tree.
18297`\spadtype{BinaryTournament(S)} is the domain of binary trees where elements are ordered down the tree. A binary search tree is either empty or is a node containing a \spadfun{value} of type \spad{S},{} and a \spadfun{right} and a \spadfun{left} which are both \spadtype{BinaryTree(S)}
18382`\spadtype{BinaryTree(S)} is the domain of all binary trees. A binary tree over \spad{S} is either empty or has a \spadfun{value} which is an \spad{S} and a \spadfun{right} and \spadfun{left} which are both binary trees.
18461`\spadtype{Bits} provides logical functions for Indexed Bits.
18510`\indented{1}{\spadtype{Boolean} is the elementary logic with 2 values:} \spad{true} and \spad{false}
18563`Members of the domain CardinalNumber are values indicating the cardinality of sets,{} both finite and infinite. Arithmetic operations are defined on cardinal numbers as follows. \blankline If \spad{x = \#X} and \spad{y = \#Y} then \indented{2}{\spad{x+y\space{2}= \#(X+Y)}\space{3}\tab{30}disjoint union} \indented{2}{\spad{x-y\space{2}= \#(X-Y)}\space{3}\tab{30}relative complement} \indented{2}{\spad{x*y\space{2}= \#(X*Y)}\space{3}\tab{30}cartesi
18563`--an product} \indented{2}{\spad{x^y = \#(X^Y)}\space{2}\tab{30}\spad{X^Y = \{g| g:Y->X\}}} \blankline The non-negative integers have a natural construction as cardinals \indented{2}{\spad{0 = \#\{\}},{} \spad{1 = \{0\}},{} \spad{2 = \{0,{} 1\}},{} ...,{} \spad{n = \{i| 0 <= i < n\}}.} \blankline That \spad{0} acts as a zero for the multiplication of cardinals is equivalent to the axiom of choice. \blankline The generalized continuum hypothesis asserts \center{\spad{2^Aleph i = Aleph(i+1)}} and 
18563`--is independent of the axioms of set theory [Goedel 1940]. \blankline Three commonly encountered cardinal numbers are \indented{3}{\spad{a = \#Z}\space{7}\tab{30}countable infinity} \indented{3}{\spad{c = \#R}\space{7}\tab{30}the continuum} \indented{3}{\spad{f = \#\{g| g:[0,{}1]->R\}}} \blankline In this domain,{} these values are obtained using \indented{3}{\spad{a := Aleph 0},{} \spad{c := 2^a},{} \spad{f := 2^c}.} \blankline
18620`CartesianTensor(minix,{}dim,{}\spad{R}) provides Cartesian tensors with components belonging to a commutative ring \spad{R}. These tensors can have any number of indices. Each index takes values from \spad{minix} to \spad{minix + dim - 1}.
18766`This domain allows classes of characters to be defined and manipulated efficiently.
18826`This domain provides the basic character data type.
18881`CliffordAlgebra(\spad{n},{} \spad{K},{} bLin) defines a module of dimension \spad{2^n} over \spad{K},{} given a bilinear form bLin on \spad{K^n}. \blankline Examples of Clifford Algebras are: gaussians,{} quaternions,{} exterior algebras and spin algebras.
18986`Color() specifies a domain of 27 colors provided in the \Language{} system (the colors mix additively).
19041`A type for basic commutators
19095`This is a low-level domain which implements matrices (two dimensional arrays) of complex double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
19240`This is a low-level domain which implements vectors (one dimensional arrays) of complex double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
19335`\spadtype {Complex(R)} creates the domain of elements of the form \spad{a + b * i} where \spad{a} and \spad{b} come from the ring \spad{R},{} and \spad{i} is a new element such that \spad{i^2 = -1}.
19414`\spadtype{ContinuedFraction} implements general \indented{1}{continued fractions.\space{2}This version is not restricted to simple,{}} \indented{1}{finite fractions and uses the \spadtype{Stream} as a} \indented{1}{representation.\space{2}The arithmetic functions assume that the} \indented{1}{approximants alternate below/above the convergence point.} \indented{1}{This is enforced by ensuring the partial numerators and pa
19414`--rtial} \indented{1}{denominators are greater than 0 in the Euclidean domain view of \spad{R}} \indented{1}{(\spadignore{i.e.} \spad{sizeLess?(0,{} x)}).}
19496`This domain provides some nice functions on lists
19567`\indented{1}{This domain implements a simple view of a database whose fields are} indexed by symbols
19643`The deRham complex of Euclidean space,{} that is,{} the class of differential forms of arbitary degree over a coefficient ring. See Flanders,{} Harley,{} Differential Forms,{} With Applications to the Physical Sciences,{} New York,{} Academic Press,{} 1963.
19772`This domain allows rational numbers to be presented as repeating decimal expansions.
19854`\indented{1}{A Denavit-Hartenberg Matrix is a 4x4 Matrix of the form:} \indented{1}{\spad{nx ox ax px}} \indented{1}{\spad{ny oy ay py}} \indented{1}{\spad{nz oz az pz}} \indented{2}{\spad{0\space{2}0\space{2}0\space{2}1}} (\spad{n},{} \spad{o},{} and a are the direction cosines)
19969`Linked list implementation of a Dequeue
20045`\spadtype{DifferentialSparseMultivariatePolynomial} implements an ordinary differential polynomial ring by combining a domain belonging to the category \spadtype{DifferentialVariableCategory} with the domain \spadtype{SparseMultivariatePolynomial}. \blankline
20233`This constructor provides a direct product type with a left matrix-module view.
20432`This constructor provides a direct product of \spad{R}-modules with an \spad{R}-module view.
20558`\indented{2}{This type represents the finite direct or cartesian product of an} underlying component type. This contrasts with simple vectors in that the members can be viewed as having constant length. Thus many categorical properties can by lifted from the underlying component type. Component extraction operations are provided but no updating operations. Thus new direct product elements can either be c
20558`--reated by converting vector elements using the \spadfun{directProduct} function or by taking appropriate linear combinations of basis vectors provided by the \spad{unitVector} operation.
20657`Category of directed graphs,{} allows us to model graph theory \blankline
20732`DirichletRing is the ring of arithmetical functions with Dirichlet convolution as multiplication
20795`\axiomType{DistributedJetBundlePolynomial} implements polynomials in a distributed representation. The unknows come from a finite list of jet variables. The implementation is basically a copy of the one of \axiomType{GeneralDistributedMultivariatePolynomial}.
20985`\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is lexicographic specified by the variable list parameter with the most significant variable fi
20985`--rst in the list.
21160`Domain for distributions formally given by moments. moments and different kinds of cumulants are computed and stored on demand.
21248`This is a low-level domain which implements matrices (two dimensional arrays) of double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
21362`This is a low-level domain which implements vectors (one dimensional arrays) of double precision floating point numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
21440`\indented{1}{\spadtype{DoubleFloat} is intended to make accessible} hardware floating point arithmetic in \Language{},{} either native double precision,{} or IEEE. On most machines,{} there will be hardware support for the arithmetic operations: \spadfunFrom{+}{DoubleFloat},{} \spadfunFrom{*}{DoubleFloat},{} \spadfunFrom{/}{DoubleFloat} and possibly also the \spadfunFrom{sqrt}{DoubleFloat} operation. The operations \spadfunFrom{exp}{Doub
21440`--leFloat},{} \spadfunFrom{log}{DoubleFloat},{} \spadfunFrom{sin}{DoubleFloat},{} \spadfunFrom{cos}{DoubleFloat},{} \spadfunFrom{atan}{DoubleFloat} are normally coded in software based on minimax polynomial/rational approximations. Some general comments about the accuracy of the operations: the operations \spadfunFrom{+}{DoubleFloat},{} \spadfunFrom{*}{DoubleFloat},{} \spadfunFrom{/}{DoubleFloat} and \spadfunFrom{sqrt}{DoubleFloat} are expected to be fully accurate. The operations \spadfunFrom{e
21440`--xp}{DoubleFloat},{} \spadfunFrom{log}{DoubleFloat},{} \spadfunFrom{sin}{DoubleFloat},{} \spadfunFrom{cos}{DoubleFloat} and \spadfunFrom{atan}{DoubleFloat} are not expected to be fully accurate. In particular,{} \spadfunFrom{sin}{DoubleFloat} and \spadfunFrom{cos}{DoubleFloat} will lose all precision for large arguments. \blankline The \spadtype{Float} domain provides an alternative to the \spad{DoubleFloat} domain. It provides an arbitrary precision model of floating point arithmetic. This mea
21440`--ns that accuracy problems like those above are eliminated by increasing the working precision where necessary. \spadtype{Float} provides some special functions such as \spadfunFrom{erf}{DoubleFloat},{} the error function in addition to the elementary functions. The disadvantage of \spadtype{Float} is that it is much more expensive than small floats when the latter can be used.
21505`DrawOption allows the user to specify defaults for the creation and rendering of plots.
21805`\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the corresponding functions on the Taylor series domain. We
21805`-- also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return "failed" when this is not possible.
22059`\indented{1}{This package provides elementary functions on any Laurent series} domain over a field which was constructed from a Taylor series domain. These functions are implemented by calling the correspo
22059`--nding functions on the Taylor series domain. We also provide 'partial functions' which compute transcendental functions of Laurent series when possible and return "failed" when this is not possible.
22360`\spad{Enumeration(a,b,...,c)} is a primitive type in FriCAS used to represent the object composed of the symbols \spad{a},\spad{b},..., and \spad{c}.
22408`This domain provides tables where the keys are compared using \spadfun{eq?}. Thus keys are considered equal only if they are the same instance of a structure.
22507`Equations as mathematical objects. All properties of the basis domain,{} \spadignore{e.g.} being an abelian group are carried over the equation domain,{} by performing the structural operations on the left and on the right hand side.
22557`These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \spadtype{ModularRing},{} \spadtype{ModularField}
22798`A function which does not return directly to its caller should have Exit as its return type. \blankline Note: It is convenient to have a formal \spad{coerce} into each type from type Exit. This allows,{} for example,{} errors to be raised in one half of a type-balanced \spad{if}.
22836`UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent essential singularities of functions. Objects in this domain are quotients of sums,{} where each term in the sum is a univariate Puiseux series times the exponential of a univariate Puiseux s
22836`--eries.
23071`ExponentialOfUnivariatePuiseuxSeries is a domain used to represent essential singularities of functions. An object in this domain is a function of the form \spad{exp(f(x))},{} where \spad{f(x)} is a Puiseux series with no terms of non-negative degree. Objects are ordered according to order of singularity,{} with functions which tend more rapidly to 
23071`--zero or infinity considered to be larger. Thus,{} if \spad{order(f(x)) < order(g(x))},{} \spadignore{i.e.} the first non-zero term of \spad{f(x)} has lower degree than the first non-zero term of \spad{g(x)},{} then \spad{exp(f(x)) > exp(g(x))}. If \spad{order(f(x)) = order(g(x))},{} then the ordering is essentially random. This domain is used in computing limits involving functions with essential singularities.
23226`Expressions involving symbolic functions.
23298`A domain used in the construction of the exterior algebra on a set \spad{X} over a ring \spad{R}. This domain represents the set of all ordered subsets of the set \spad{X},{} assumed to be in correspondance with {1,{}2,{}3,{} ...}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements w
23298`--hich can only be 0 or 1. \blankline The multiplicative identity element of the exterior algebra corresponds to the empty subset of \spad{X}. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.
23351`\spadtype{Factored} creates a domain whose objects are kept in factored form as long as possible. Thus certain operations like multiplication and \spad{gcd} are relatively easy to do. Others,{} like addition require somewhat more work,{} and unless the argument domain provides a factor function,{} the result may not be completely factored. Each object consists of a unit and a list of factors,{} where a factor has a member of \spad{R
23351`--} (the "base"),{} and exponent and a flag indicating what is known about the base. A flag may be one of "nil",{} "sqfr",{} "irred" or "prime",{} which respectively mean that nothing is known about the base,{} it is square-free,{} it is irreducible,{} or it is prime. The current restriction to integral domains allows simplification to be performed without worrying about multiplication order.
23421`This domain provides an interface to names in the file system.
23469`This domain provides a basic model of files to save arbitrary values. The operations provide sequential access to the contents.
23544`This domains implements finite rational divisors on a curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve.
23762`FiniteFieldCyclicGroupExtensionByPolynomial(\spad{GF},{}defpol) implements a finite extension field of the ground field {\em GF}. Its elements are represented by powers of a primitive element,{} \spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial {
23762`--\em defpol},{} which MUST be primitive (user responsibility). Zech logarithms are stored in a table of size half of the field size,{} and use \spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.
23936`FiniteFieldCyclicGroupExtension(\spad{GF},{}\spad{n}) implements a extension of degree \spad{n} over the ground field {\em GF}. Its elements are represented by powers of a primitive element,{} \spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by {\em creat
23936`--ePrimitivePoly} from \spadtype{FiniteFieldPolynomialPackage}. Zech logarithms are stored in a table of size half of the field size,{} and use \spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.
24080`FiniteFieldCyclicGroup(\spad{p},{}\spad{n}) implements a finite field extension of degee \spad{n} over the prime field with \spad{p} elements. Its elements are represented by powers of a primitive element,{} \spadignore{i.e.} a generator of the multiplicative (cyclic) group. As primitive element we choose the root of the extension polynomial,{} which is created by
24080`-- {\em createPrimitivePoly} from \spadtype{FiniteFieldPolynomialPackage}. The Zech logarithms are stored in a table of size half of the field size,{} and use \spadtype{SingleInteger} for representing field elements,{} hence,{} there are restrictions on the size of the field.
24220`FiniteFieldExtensionByPolynomial(\spad{GF},{} defpol) implements the extension of the finite field {\em GF} generated by the extension polynomial {\em defpol} which MUST be irreducible. Note: the user has the responsibility to ensure that {\em defpol} is irreducible.
24371`FiniteFieldExtensionByPolynomial(\spad{GF},{} \spad{n}) implements an extension of the finite field {\em GF} of degree \spad{n} generated by the extension polynomial constructed by \spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage} from \spadtype{FiniteFieldPolynomialPackage}.
24487`FiniteFieldNormalBasisExtensionByPolynomial(\spad{GF},{}uni) implements a finite extension of the ground field {\em GF}. The elements are represented by coordinate vectors with respect to. a normal basis,{} \spadignore{i.e.} a basis consisting of the conjugates (\spad{q}-power
24487`--s) of an element,{} in this case called normal element,{} where \spad{q} is the size of {\em GF}. The normal element is chosen as a root of the extension polynomial,{} which MUST be normal over {\em GF} (user responsibility)
24716`FiniteFieldNormalBasisExtensionByPolynomial(\spad{GF},{}\spad{n}) implements a finite extension field of degree \spad{n} over the ground field {\em GF}. The elements are represented by coordinate vectors with respect to a normal basis,{} \spadignore{i.e.} a basis consisting of the conjugates (\spad{q}-powers) of an element,{} in this case called normal element
24716`--. This is chosen as a root of the extension polynomial,{} created by {\em createNormalPoly} from \spadtype{FiniteFieldPolynomialPackage}
24860`FiniteFieldNormalBasis(\spad{p},{}\spad{n}) implements a finite extension field of degree \spad{n} over the prime field with \spad{p} elements. The elements are represented by coordinate vectors with respect to a normal basis,{} \spadignore{i.e.} a basis consisting of the conjugates (\spad{q}-powers) of an element,{} in this case called normal element. This is cho
24860`--sen as a root of the extension polynomial created by \spadfunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}.
25000`FiniteField(\spad{p},{}\spad{n}) implements finite fields with \spad{p^n} elements. This packages checks that \spad{p} is prime. For a non-checking version,{} see \spadtype{InnerFiniteField}.
25112`\indented{1}{A FlexibleArray is the notion of an array intended to allow for growth} at the end only. Hence the following efficient operations \indented{2}{\spad{concat!(a,{} x)} meaning append item \spad{x} at the end of the array \spad{a}} \indented{2}{\spad{delete!(a,{} n)} meaning delete the last item from the array \spad{a}} Flexible arrays support the other operations inherited from \spadtype{ExtensibleLinearAg
25112`--gregate}. However,{} these are not efficient. Flexible arrays combine the \spad{O(1)} access time property of arrays with growing and shrinking at the end in \spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\% larger) array. Conversely,{} when the array becomes less than 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient imple
25112`--mentation of many data structures in particular heaps,{} stacks and sets.
25252`This domain builds representations of program code segments for use with the FortranProgram domain.
25305`A domain of expressions involving functions which can be translated into standard Fortran-77,{} with some extra extensions from the NAG Fortran Library.
25460`\axiomType{FortranProgram} allows the user to build and manipulate simple models of FORTRAN subprograms. These can then be transformed into actual FORTRAN notation.
25639`Creates and manipulates objects which correspond to the basic FORTRAN data types: REAL,{} INTEGER,{} COMPLEX,{} LOGICAL and CHARACTER
25711`Code to manipulate Fortran templates
25788`Creates and manipulates objects which correspond to FORTRAN data types,{} including array dimensions.
25841`\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: 12 June 1992 Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:
25914`\indented{1}{Author: James Davenport} Date Created: 17 April 1992 Date Last Updated: Basic Functions: Related Constructors: Also See: AMS Classifications: Keywords: References: Description:
26026`\spad{Canonical} means that equal elements are in fact identical.
26108`Fractional ideals in a framed algebra.
26285`Module representation of fractional ideals.
26465`The free abelian group on a set \spad{S} is the monoid of finite sums of the form \spad{reduce(+,{}[\spad{ni} * \spad{si}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are integers. The operation is commutative.
26543`The free abelian monoid on a set \spad{S} is the monoid of finite sums of the form \spad{reduce(+,{}[\spad{ni} * \spad{si}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are non-negative integers. The operation is commutative.
26648`The free group on a set \spad{S} is the group of finite products of the form \spad{reduce(*,{}[\spad{si} ^ \spad{ni}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are integers. The multiplication is not commutative.
26711`A \spad{bi}-module is a free module over a ring with generators indexed by an ordered set. Each element can be expressed as a finite linear combination of generators. Only non-zero terms are stored. old domain \spad{FreeModule1} was merged to it in May 2009 The description of the latter: \indented{2}{This domain implements linear combinations} \indented{2}{of elements from the domain \spad{S} with coefficients} \i
26711`--ndented{2}{in the domain \spad{R} where \spad{S} is an ordered set} \indented{2}{and \spad{R} is a ring (which may be non-commutative).} \indented{2}{This domain is used by domains of non-commutative algebra such as:} \indented{6}{\spadtype{XDistributedPolynomial},{}} \indented{6}{\spadtype{XRecursivePolynomial}.} \indented{2}{Author: Michel Petitot (petitot@lifl.\spad{fr})}
26800`The free monoid on a set \spad{S} is the monoid of finite products of the form \spad{reduce(*,{}[\spad{si} ^ \spad{ni}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are nonnegative integers. The multiplication is not commutative.
26866`Generate the Free Lie Algebra over a ring \spad{R} with identity; A \spad{P}. Hall basis is generated by a package call to HallBasis.
27005`\indented{1}{Full partial fraction expansion of rational functions} Author: Manuel Bronstein Date Created: 9 December 1992 Date Last Updated: 6 October 1993 References: \spad{M}.Bronstein & \spad{B}.Salvy,{} \indented{12}{Full Partial Fraction Decomposition of Rational Functions,{}} \indented{12}{in Proceedings of \spad{ISSAC'93},{} Kiev,{} ACM Press.}
27139`This domain implements named functions
27210`allows us to model graph theory \blankline
27286`\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is specified by its third parameter. Suggested types which define term
27286`-- orderings include: \spadtype{DirectProduct},{} \spadtype{HomogeneousDirectProduct},{} \spadtype{SplitHomogeneousDirectProduct} and finally \spadtype{OrderedDirectProduct} which accepts an arbitrary user function to define a term ordering.
27486`This package \undocumented
27787`A domain for polynomial sets.
27984`\spadtype{GeneralQuaternion} implements general quaternions \indented{2}{over a commutative ring. The main constructor function is} \indented{2}{\spadfun{quatern} which takes 4 arguments: the real part,{}} \indented{2}{the \spad{i} imaginary part,{} the \spad{j} imaginary part and the \spad{k} imaginary part.}
28074`A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key.
28224`A domain constructor of the category \axiomType{TriangularSetCategory}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. Triangular sets 
28224`--are stored as sorted lists \spad{w}.\spad{r}.\spad{t}. the main variables of their members but they are displayed in reverse order.\newline References : \indented{1}{[1] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)}
28399`This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x\^r)}.
28726`AlgebraGenericElementPackage allows you to create generic elements of an algebra,{} \spadignore{i.e.} the scalars are extended to include symbolic coefficients
28922`TwoDimensionalGraph creates virtual two dimensional graphs (to be displayed on TwoDimensionalViewports).
28979`\spad{GuessOptionFunctions0} provides operations that extract the values of options for \spadtype{Guess}.
29045`GuessOption is a domain whose elements are various options used by \spadtype{Guess}.
29100`\spadtype{HtmlFormat} provides a coercion from \spadtype{OutputForm} to html.
29158`This domain supports incremental computation of hash values.
29197`This domain provides access to the underlying Lisp hash tables. By varying the hashfn parameter,{} tables suited for different purposes can be obtained.
29314`Heap implemented in a flexible array to allow for insertions
29389`This domain allows rational numbers to be presented as repeating hexadecimal expansions.
29475`\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The vectors are ordered first by the sum of their components,{} and then refined using a reverse lexicographic ordering. This type is a suitable third argument for \spadtype{GeneralDistributedMultivariatePolynomial}.
29600`\indented{2}{This type supports distributed multivariate polynomials} whose variables are from a user specified list of symbols. The coefficient ring may be non commutative,{} but the variables are assumed to commute. The term ordering is total degree ordering refined by reverse lexicographic ordering with 
29600`--respect to the position that the variables appear in the list of variables parameter.
29798`This domains implements finite rational divisors on an hyperelliptic curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve. The equation of the curve must be \spad{y^2
29798`--} = \spad{f}(\spad{x}) and \spad{f} must have odd degree.
30023`parse result includes term returned and new index
30069`\indented{1}{This domain implements a container of information} about the FriCAS library
30122`\spadtype{IndexedBits} is a domain to compactly represent large quantities of Boolean data.
30190`\indented{1}{Indexed direct products of objects over a set \spad{A}} of generators indexed by an ordered set \spad{S}. All items have finite support. If A is a monoid,{} then only non-zero terms are stored. If A has additive structure,{} it is propagated coordinatewise to the product. Similarely,{} comparisons are propagated using lexicographic ordering.
30301`converts entire exponents to OutputForm
30391`\indented{1}{Author: Michael Monagan \spad{July/87},{} modified \spad{SMW} \spad{June/91}} A FlexibleArray is the notion of an array intended to allow for growth at the end only. Hence the following efficient operations \indented{2}{\spad{concat!(a,{} x)} meaning append item \spad{x} at the end of the array \spad{a}} \indented{2}{\spad{delete!(a,{} n)} meaning delete the last item from the array \s
30391`--pad{a}} Flexible arrays support the other operations inherited from \spadtype{ExtensibleLinearAggregate}. However,{} these are not efficient. Flexible arrays combine the \spad{O(1)} access time property of arrays with growing and shrinking at the end in \spad{O(1)} (average) time. This is done by using an ordinary array which may have zero or more empty slots at the end. When the array becomes full it is copied into a new larger (50\% larger) array. Conversely,{} when the array becomes less th
30391`--an 1/2 full,{} it is copied into a smaller array. Flexible arrays provide for an efficient implementation of many data structures in particular heaps,{} stacks and sets.
30496`\axiomType{JetBundle} provides the standard implementation for a jet bundle with a given number of dependent and independent variables.
30614`\spadtype{IndexedList} is a basic implementation of the functions in \spadtype{ListAggregate},{} often using functions in the underlying LISP system. The second parameter to the constructor (\spad{mn}) is the beginning index of the list. That is,{} if \spad{l} is a list,{} then \spad{elt(l,{}mn)} is the first value. This constructor is probably best viewed as the implementation of singly-linked lists that are addressable by index rat
30614`--her than as a mere wrapper for LISP lists.
30683`An \spad{IndexedMatrix} is a matrix where the minimal row and column indices are parameters of the type. The domains Row and Col are both IndexedVectors. The index of the 'first' row may be obtained by calling the function \spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \spadfun{minColIndex}. The
30683`-- index of the first element of a 'Row' is the same as the index of the first column in a matrix and vice versa.
30845`\indented{1}{Author Micheal Monagan \spad{Aug/87}} This is the basic one dimensional array data type.
30946`This domain implements low-level strings
31011`\indented{1}{An IndexedTwoDimensionalArray is a 2-dimensional array where} the minimal row and column indices are parameters of the type. Rows and columns are returned as IndexedOneDimensionalArray\spad{'s} with minimal indices matching those of the IndexedTwoDimensionalArray. The index of the 'first' ro
31011`--w may be obtained by calling the function 'minRowIndex'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.
31212`\indented{2}{This type represents vector like objects with varying lengths} and a user-specified initial index.
31286`\indented{1}{This package implements 'infinite tuples' for the interpreter.} The representation is a stream.
31364`Algebraic closure of the rational numbers.
31431`InnerFiniteField(\spad{p},{}\spad{n}) implements finite fields with \spad{p^n} elements where \spad{p} is assumed prime but does not check. For a version which checks that \spad{p} is prime,{} see \spadtype{FiniteField}.
31554`Internal implementation of a free abelian monoid.
31680`\indented{1}{This is an internal type which provides an implementation of} 2-dimensional arrays as PrimitiveArray\spad{'s} of PrimitiveArray\spad{'s}.
31879`This domain implements \spad{Zp},{} the \spad{p}-adic completion of the integers. This is an internal domain.
31974`InnerPrimeField(\spad{p}) implements the field with \spad{p} elements. Note: argument \spad{p} MUST be a prime (this domain does not check). See \spadtype{PrimeField} for a domain that does check.
32058`InnerSparseUnivariatePowerSeries is an internal domain \indented{2}{used for creating sparse Taylor and Laurent series.}
32177`This domain is used to provide a conditional "add" domain for the implementation of \spadtype{Table}.
32327`Internal package for dense Taylor series. This is an internal Taylor series type in which Taylor series are represented by a \spadtype{Stream} of \spadtype{Ring} elements. For univariate series,{} the \spad{Stream} elements are the Taylor coefficients. For multivariate series,{} the \spad{n}th Stream element is a form of degree \spad{n} in the power series variables.
32394`Domain of parsed forms which can be passed to the interpreter. This is also the interface between algebra code and facilities in the interpreter.
32503`IntegerMod(\spad{n}) creates the ring of integers reduced modulo the integer \spad{n}.
32579`\spadtype{Integer} provides the domain of arbitrary precision integers.
32637`If a function \spad{f} has an elementary integral \spad{g},{} then \spad{g} can be written in the form \spad{g = h + c1 log(u1) + c2 log(u2) + ... + cn log(un)} where \spad{h},{} which is in the same field than \spad{f},{} is called the rational part of the integral,{} and \spad{c1 log(u1) + ... cn log(un)} is called the logarithmic part of the integral. This domain manipulates integrals represented in that form,{} by kee
32637`--ping both parts separately. The logs are not explicitly computed.
32718`\indented{1}{+ Author: Mike Dewar} + Date Created: November 1996 + Date Last Updated: + Basic Functions: + Related Constructors: + Also See: + AMS Classifications: + Keywords: + References: + Description: + This domain is an implementation of interval arithmetic and transcendental + functions over intervals.
32802`\axiomType{JetBundleExpression} defines expressions over a jet bundle based on \axiomType{Expression Integer}. It allows all kind of algebraic operations. \axiom{simplify} is implemented using Groebner bases in polynomials over kernels. Thus it might not work correctly for general expressions. This also affects \axiom{dimension}.
32903`\axiomType{JetBundleLinearFunction} implements linear functions over a jet bundle. The coefficients are functions of the independent variables only.
33050`\axiomType{JetBundlePolynomial} implements polynomial sections over a jet bundle. The order is not fixed,{} thus jet variables of any order can appear.
33161`\axiom{JetBundleSymAna} is only necessary to have a valid return type for some procedures in \axiom{SymmetryAnalysis}. It is essentially identical with \axiom{JetBundle} but computes its parameters in a more complicated way.
33309`\axiomType{JetBundleXExpression} implements arbitrary functions in a jet bundle which depend only on the independent variables \axiom{\spad{x}}. Otherwise it is identical with \axiomType{JetBundleExpression}. Such a domain is needed for \axiomType{JetLinearFunction}.
33411`\axiom{JetBundle} implements a jet bundle of arbitrary order with given names for the independent and dependent variables. It supports only repeated index notation.
33509`\axiomType{JetDifferentialEquation} provides the basic data structures and procedures for differential equations as needed in the geometric theory. Differential equation means here always a submanifold in the jet bundle. The concrete equations which define this submanifold are called system. In an object of the type \axiomType{JetDifferentialEquation} much more than only the system is stored. \axi
33509`--om{\spad{D}} denotes the class of functions allowed as equations. It is assumed that the \axiom{simplify} procedure of \axiom{\spad{D}} returns only independent equations and a system with symbol in row echelon form.
33615`\axiom{JetDifferential(\spad{JB},{} \spad{D})} implements differentials (one-forms) \indented{1}{over the jet bundle \axiom{\spad{JB}} with coefficients from \axiom{\spad{D}}.} \indented{1}{The differentials operate on \axiom{JetVectorField(\spad{JB},{} \spad{D})}.}
33731`\axiomType{JetLazyFunction} takes as argument a domain in \axiomType{JetBundleFunctionCategory} and returns another domain in the same category. This domain has basically the same properties as the argument domain,{} but there is a lazy evaluation mechanism for derivatives. This means that differentiations are not immedately performed. Instead a pointer is established t
33731`--o the function to be differentiated. Only when the exact value of the derivative is needed,{} the differentiation is executed. Special care is taken for leading derivatives and jet variables to avoid as much as possible the need to evaluate expressions. This entails that the result of \axiom{jetVariables} may contain spurious variables. Furthermore many functions in \axiomType{JetLazyFunction} destructively change their arguments. This affects,{} however,{} only their internal representation,{
33731`--} not the value obtained after full evaluation.
33865`\axiom{JetVectorField(\spad{JB},{} \spad{D})} implements vector fields over the jet bundle \axiom{\spad{JB}} with coefficients from \axiom{\spad{D}}. The fields operate on functions from \axiom{\spad{D}}.
33979`A kernel over a set \spad{S} is an operator applied to a given list of arguments from \spad{S}.
34045`This domain allows a random access file to be viewed both as a table and as a file object.
34167`parse result includes term returned and new index
34242`\indented{1}{Univariate polynomials with negative and positive exponents.} Author: Manuel Bronstein Date Created: May 1988 Date Last Updated: 26 Apr 1990
34377`This domain provides a simple way to save values in files.
34443`Management of the Lie Group associated with a free nilpotent Lie algebra. Every Lie bracket with length greater than \axiom{Order} are assumed to be null. The implementation inherits from the \spadtype{XPBWPolynomial} domain constructor: Lyndon coordinates are exponential coordinates of the second kind. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
34567`This type supports Lie polynomials in Lyndon basis see Free Lie Algebras by \spad{C}. Reutenauer (Oxford science publications). \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
34677`LieSquareMatrix(\spad{n},{}\spad{R}) implements the Lie algebra of the \spad{n} by \spad{n} matrices over the commutative ring \spad{R}. The Lie bracket (commutator) of the algebra is given by \spad{a*b := (a *\$SQMATRIX(n,{}R) b - b *\$SQMATRIX(n,{}R) a)},{} where \spadfun{*\$SQMATRIX(\spad{n},{}\spad{R})} is the usual matrix multiplication.
34827`\spad{LinearOrdinaryDifferentialOperator1} defines a ring of differential operators with coefficients in a differential ring A. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
34951`\spad{LinearOrdinaryDifferentialOperator2} defines a ring of differential operators with coefficients in a differential ring A and acting on an A-module \spad{M}. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
35115`\spad{LinearOrdinaryDifferentialOperator} defines a ring of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
35239`This internal package represents monoid (abelian or not,{} with or without inverses) as lists and provides some common operations to the various flavors of monoids.
35417`\spadtype{List} implements singly-linked lists that are addressable by indices; the index of the first element is 1. In addition to the operations provided by \spadtype{IndexedList},{} this constructor provides some LISP-like functions such as \spadfun{null} and \spadfun{cons}.
35478`LocalAlgebra produces the localization of an algebra,{} \spadignore{i.e.} fractions whose numerators come from some \spad{R} algebra.
35565`Localize(\spad{M},{}\spad{R}) produces fractions with numerators from an \spad{R} module \spad{M} and denominators being the nonzero elements of \spad{R}.
35646`This is used with graph theory code (FiniteGraph,{} DirectedGraph. FunctionGraph,{} and so on) to represent a loop as either a sequence of vertex or arrow indexes depending on context. The main benefit is that the loop is stored in a canonical way so that loops can be quickly compared using \spad{'='}.
35695`Lyndon words over arbitrary (ordered) symbols: see Free Lie Algebras by \spad{C}. Reutenauer (Oxford science publications). A Lyndon word is a word which is smaller than any of its right factors \spad{w}.\spad{r}.\spad{t}. the pure lexicographical ordering. If \axiom{a} and \axiom{\spad{b}} are two Lyndon words such that \axiom{a < \spad{b}} holds \spad{w}.\spad{r}.\spad{t} lexicographical ordering then \axiom{a*b} is a Lyndon wor
35695`--d. Parenthesized Lyndon words can be generated from symbols by using the following rule: \axiom{[[a,{}\spad{b}],{}\spad{c}]} is a Lyndon word iff \axiom{a*b < \spad{c} \spad{<=} \spad{b}} holds. Lyndon words are internally represented by binary trees using the \spadtype{Magma} domain constructor. Two ordering are provided: lexicographic and length-lexicographic. \newline Author : Michel Petitot (petitot@lifl.\spad{fr}).
35768`A domain which models the complex number representation used by machines in the AXIOM-NAG link.
35844`A domain which models the floating point representation used by machines in the AXIOM-NAG link.
35911`A domain which models the integer representation used by machines in the AXIOM-NAG link.
35985`This type is the basic representation of parenthesized words (binary trees over arbitrary symbols) useful in \spadtype{LiePolynomial}. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
36053`MakeCachableSet(\spad{S}) returns a cachable set which is equal to \spad{S} as a set.
36130`\spad{Mapping(T,S)} is a primitive type in FriCAS used to represent mappings from source type \spad{S} to target type \spad{T}. Similarly, \spad{Mapping(T,A,B)} denotes a mapping from source type \spad{(A,B)} to target type \spad{T}.
36172`\spadtype{MathMLFormat} provides a coercion from \spadtype{OutputForm} to MathML format.
36232`\spadtype{Matrix} is a matrix domain where 1-based indexing is used for both rows and columns.
36331`This package \undocumented
36822`\indented{1}{These domains are used for the factorization and gcds} of univariate polynomials over the integers in order to work modulo different primes. See \spadtype{ModularRing},{} \spadtype{EuclideanModularRing}
37015`These domains are used for the factorization and gcds of univariate polynomials over the integers in order to work modulo different primes. See \spadtype{EuclideanModularRing} ,{}\spadtype{ModularField}
37205`This package \undocumented
37370`Algebra of ADDITIVE operators on a module.
37450`\indented{1}{MoebiusTransform(\spad{F}) is the domain of fractional linear (Moebius)} transformations over \spad{F}.
37516`\spadtype{MonoidRing}(\spad{R},{}\spad{M}),{} implements the algebra of all maps from the monoid \spad{M} to the commutative ring \spad{R} with finite support. Multiplication of two maps \spad{f} and \spad{g} is defined to map an element \spad{c} of \spad{M} to the (convolution) sum over {\em f(a)g(b)} such that {\em ab = c}. Thus \spad{M} can be identified with a canonical basis and the maps can also be considered as formal li
37516`--near combinations of the elements in \spad{M}. Scalar multiples of a basis element are called monomials. A prominent example is the class of polynomials where the monoid is a direct product of the natural numbers with pointwise addition. When \spad{M} is \spadtype{FreeMonoid Symbol},{} one gets polynomials in infinitely many non-commuting variables. Another application area is representation theory of finite groups \spad{G},{} where modules over \spadtype{MonoidRing}(\spad{R},{}\spad{G}) are s
37516`--tudied.
37592`allows us to model graph theory \blankline
37674`A multiset is a set with multiplicities.
37750`\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are from a user specified list of symbols. The ordering is specified by the position of the variable in the list. The coefficient ring may be non commutative,{} but the variables are assumed to commute.
37909`A post-facto extension for \axiomType{\spad{SMP}} in order to speed up operations related to pseudo-division and \spad{gcd}. This domain is based on the \axiomType{NSUP} constructor which is itself a post-facto extension of the \axiomType{SUP} constructor.
38068`A post-facto extension for \axiomType{SUP} in order to speed up operations related to pseudo-division and \spad{gcd} for both \axiomType{SUP} and,{} consequently,{} \axiomType{NSMP}.
38169`\spadtype{NonNegativeInteger} provides functions for non \indented{2}{negative integers.}
38243`\spadtype{None} implements a type with no objects. It is mainly used in technical situations where such a thing is needed (\spadignore{e.g.} the interpreter and some of the internal \spadtype{Expression} code).
38282`Octonion implements octonions (Cayley-Dixon algebra) over a commutative ring,{} an eight-dimensional non-associative algebra,{} doubling the quaternions in the same way as doubling the complex numbers to get the quaternions the main constructor function is {\em octon} which takes 8 arguments: the real part,{} the \spad{i} imaginary part,{} the \spad{j} imaginary part,{} the \spad{k} imaginary part,{} (as with quaternions) and
38282`-- in addition the imaginary parts \spad{E},{} \spad{I},{} \spad{J},{} \spad{K}.
38360`This is the domain of 1-based one dimensional arrays
38453`Adjunction of a complex infinity to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989
38533`\spadtype{OpenMathConnection} provides low-level functions for handling connections to and from \spadtype{OpenMathDevice}\spad{s}.
38580`\spadtype{OpenMathDevice} provides support for reading and writing openMath objects to files,{} strings etc. It also provides access to low-level operations from within the interpreter.
38622`\spadtype{OpenMathEncoding} is the set of valid OpenMath encodings.
38684`\spadtype{OpenMathErrorKind} represents different kinds of OpenMath errors: specifically parse errors,{} unknown \spad{CD} or symbol errors,{} and read errors.
38748`\spadtype{OpenMathError} is the domain of OpenMath errors.
38807`Algebra of ADDITIVE operators over a ring.
38858`This constructor creates the \spadtype{MonogenicLinearOperator} domain which is ``opposite\spad{''} in the ring sense to \spad{P}. That is,{} as sets \spad{P = \$} but \spad{a * b} in \spad{\$} is equal to \spad{b * a} in \spad{P}.
38987`\indented{1}{Author : Larry Lambe} Date created : 14 August 1988 Date Last Updated : 11 March 1991 Description : A domain used in order to take the free \spad{R}-module on the Integers \spad{I}. This is actually the forgetful functor from OrderedRings to OrderedSets applied to \spad{I}
39040`Adjunction of two real infinites quantities to a set. Date Created: 4 Oct 1989 Date Last Updated: 1 Nov 1989
39119`\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The ordering on the type is determined by its third argument which represents the less than function on vectors. This type is a suitable third argument for \spadtype{GeneralDistributedMultivariatePolynomial}.
39332`\indented{3}{The free monoid on a set \spad{S} is the monoid of finite products of} the form \spad{reduce(*,{}[\spad{si} ^ \spad{ni}])} where the \spad{si}\spad{'s} are in \spad{S},{} and the \spad{ni}\spad{'s} are non-negative integers. The multiplication is not commutative. For two elements \spad{x} and \spad{y} the relation \spad{x < y} holds if either \spad{length(x) < length(y)} holds or if these lengths are equal and
39332`-- if \spad{x} is smaller than \spad{y} \spad{w}.\spad{r}.\spad{t}. the lexicographical ordering induced by \spad{S}. This domain inherits implementation from \spadtype{FreeMonoid}.
39413`This domain implements ordered variables
39505`\spadtype{OrderlyDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is orderly. This is analogous to the domain \spadtype{Polynomial}. \blankline
39706`\spadtype{OrderlyDifferentialVariable} adds a commonly used orderly ranking to the set of derivatives of an ordered list of differential indeterminates. An orderly ranking is a ranking \spadfun{<} of the derivatives with the property that for two derivatives \spad{u} and \spad{v},{} \spad{u} \spadfun{<} \spad{v} if the \spadfun{order} of \spad{u} is less than that of \spad{v}. This domain belongs to \spadtyp
39706`--e{DifferentialVariableCategory}. It defines \spadfun{weight} to be just \spadfun{order},{} and it defines an orderly ranking \spadfun{<} on derivatives \spad{u} via the lexicographic order on the pair (\spadfun{order}(\spad{u}),{} \spadfun{variable}(\spad{u})).
39802`This constructor produces an ordinary differential ring from a partial differential ring by specifying a variable.
39951`This domain represents truncated weighted polynomials over the "Polynomial" type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.
40093`This domain is used to create and manipulate mathematical expressions for output. It is intended to provide an insulating layer between the expression rendering software (\spadignore{e.g.} TeX,{} or Script) and the output coercions in the various domains.
40151`Stream-based implementation of \spad{Zp:} \spad{p}-adic numbers are represented as sum(\spad{i} = 0..,{} a[\spad{i}] * p^i),{} where the a[\spad{i}] lie in 0,{}1,{}...,{}(\spad{p} - 1).
40221`This is the category of stream-based representations of \spad{Qp}.
40358`Stream-based implementation of \spad{Qp:} numbers are represented as sum(\spad{i} = \spad{k}..,{} a[\spad{i}] * p^i) where the a[\spad{i}] lie in 0,{}1,{}...,{}(\spad{p} - 1).
40457`This domain describes four groups of color shades (palettes).
40512`ParametricPlaneCurve is used for plotting parametric plane curves in the affine plane.
40586`ParametricSpaceCurve is used for plotting parametric space curves in affine 3-space.
40660`ParametricSurface is used for plotting parametric surfaces in affine 3-space.
40730`PartialDifferentialOperator(\spad{V},{} \spad{R}) defines a ring of partial differential operators in variables from \spad{V} and with coefficients in a partial differential ring \spad{R}. Multiplication of operators corresponds to composition of operators. \blankline
40903`The domain \spadtype{PartialFraction} implements partial fractions over a euclidean domain \spad{R}. This requirement on the argument domain allows us to normalize the fractions. Of particular interest are the 2 forms for these fractions. The ``compact\spad{''} form has only one fractional term per prime in the denominator,{} while the \spad{``p}-adic\spad{''} form expands each numerator \spad{p}-adically via the prime \spad{p} in t
40903`--he denominator. For computational efficiency,{} the compact form is used,{} though the \spad{p}-adic form may be gotten by calling the function \spadfunFrom{padicFraction}{PartialFraction}. For a general euclidean domain,{} it is not known how to factor the denominator. Thus the function \spadfunFrom{partialFraction}{PartialFraction} takes as its second argument an element of \spadtype{Factored(R)}.
40974`\indented{1}{Partition is an OrderedCancellationAbelianMonoid which is used} as the basis for symmetric polynomial representation of the sums of powers in SymmetricPolynomial. Thus,{} \spad{(5 2 2 1)} will represent \spad{s5 * s2^2 * s1}.
41053`A PatternMatchListResult is an object internally returned by the pattern matcher when matching on lists. It is either a failed match,{} or a pair of PatternMatchResult,{} one for atoms (elements of the list),{} and one for lists.
41174`A PatternMatchResult is an object internally returned by the pattern matcher; It is either a failed match,{} or a list of matches of the form (var,{} expr) meaning that the variable var matches the expression expr.
41269`Patterns for use by the pattern matcher.
41338`\indented{1}{A PendantTree(\spad{S})is either a leaf? and is an \spad{S} or has} a left and a right both PendantTree(\spad{S})\spad{'s}
41428`PermutationGroup implements permutation groups acting on a set \spad{S},{} \spadignore{i.e.} all subgroups of the symmetric group of \spad{S},{} represented as a list of permutations (generators). Note that therefore the objects are not members of the \Language category \spadtype{Group}. Using the idea of base and strong generators by Sims,{} basic routines and algorithms are implemented so that the word problem for permutati
41428`--on groups can be solved.
41506`Permutation(\spad{S}) implements the group of all bijections \indented{2}{on a set \spad{S},{} which move only a finite number of points.} \indented{2}{A permutation is considered as a map from \spad{S} into \spad{S}. In particular} \indented{2}{multiplication is defined as composition of maps:} \indented{2}{{\em pi1 * pi2 = pi1 o pi2}.} \indented{2}{The internal representation of permuatations are two lists} \indented{2}{
41506`--of equal length representing preimages and images.}
41587`\indented{1}{Symbolic fractions in \%\spad{pi} with integer coefficients;} \indented{1}{The point for using \spad{Pi} as the default domain for those fractions} \indented{1}{is that \spad{Pi} is coercible to the float types,{} and not Expression.} Date Created: 21 Feb 1990 Date Last Updated: 12 Mai 1992
41630`\indented{1}{Plot a NON-SINGULAR plane algebraic curve \spad{p}(\spad{x},{}\spad{y}) = 0.} Author: Clifton \spad{J}. Williamson Date Created: Fall 1988 Date Last Updated: 27 April 1990 Keywords: algebraic curve,{} non-singular,{} plot Examples: References:
41716`Plot3D supports parametric plots defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example,{} floating point numbers and infinite continued fractions are real number systems. The facilities at this point are limited to 3-dimensional parametric plots.
41785`The Plot domain supports plotting of functions defined over a real number system. A real number system is a model for the real numbers and as such may be an approximation. For example floating point numbers and infinite continued fractions. The facilities at this point are limited to 2-dimensional plots or either a single function or a parametric function.
41850`This domain provides the internal representation of polynomials in non-commutative variables written over the Poincare-Birkhoff-Witt basis. See the \spadtype{XPBWPolynomial} domain constructor. See Free Lie Algebras by \spad{C}. Reutenauer (Oxford science publications). \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
41944`This domain implements points in coordinate space
41997`This domain represents polynomial ideals with coefficients in any field and supports the basic ideal operations,{} including intersection sum and quotient. An ideal is represented by a list of polynomials (the generators of the ideal) and a boolean that is \spad{true} if the generators are a Groebner basis. The algorithms used
41997`-- are based on Groebner basis computations. The ordering is determined by the datatype of the input polynomials. Users may use refinements of total degree orderings.
42176`This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and terms indexed by their exponents (from an arbitrary ordered abelian monoid). This type is used,{} for example,{} by the \spadtype{DistributedMultivariatePolynomial} domain where the exponent domain is a direct product of non negative integers.
42295`\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials whose variables are arbitrary symbols. The ordering is alphabetic determined by the Symbol type. The coefficient ring may be non commutative,{} but the variables are assumed to commute.
42399`\spadtype{PositiveInteger} provides functions for \indented{2}{positive integers.}
42469`PrimeField(\spad{p}) implements the field with \spad{p} elements if \spad{p} is a prime number. Error: if \spad{p} is not prime. Note: this domain does not check that argument is a prime.
42548`\indented{1}{This provides a fast array type with no bound checking on elt\spad{'s}.} Minimum index is 0 in this type,{} cannot be changed
42627`This domain implements cartesian product
42712`This domain provides modest support for quadratic forms.
42800`\spadtype{QuasiAlgebraicSet} constructs a domain representing quasi-algebraic sets,{} which is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the common zeros of a 
42800`--polynomial \spad{f} (the defining polynomial for the inequation). This domain provides simplification of a user-given representation using groebner basis computations. There are two simplification routines: the first function \spadfun{idealSimplify} uses groebner basis of ideals alone,{} while the second,{} \spadfun{simplify} uses both groebner basis and factorization. The resulting defining equations \spad{L} always form a groebner basis,{} and the resulting defining inequation \spad{f} is al
42800`--ways reduced. The function \spadfun{simplify} may be applied several times if desired. A third simplification routine \spadfun{radicalSimplify} is provided in \spadtype{QuasiAlgebraicSet2} for comparison study only,{} as it is inefficient compared to the other two,{} as well as is restricted to only certain coefficient domains. For detail analysis and a comparison of the three methods,{} please consult the reference cited. \blankline A polynomial function \spad{q} defined on the quasi-algebrai
42800`--c set is equivalent to its reduced form with respect to \spad{L}. While this may be obtained using the usual normal form algorithm,{} there is no canonical form for \spad{q}. \blankline The ordering in groebner basis computation is determined by the data type of the input polynomials. If it is possible we suggest to use refinements of total degree orderings.
42978`\spadtype{Quaternion} implements Hamilton quaternions \indented{2}{over a commutative ring.}
43051`This domain implements simple database queries
43123`Linked List implementation of a Queue
43194`Function field defined by \spad{y^n} = \spad{f}(\spad{x}).
43434`This domain allows rational numbers to be presented as repeating decimal expansions or more generally as repeating expansions in any base.
43524`This domain implements the real closure of an ordered field.
43617`\spad{Record(a:A,b:B,...,c:C)} is a primitive type in FriCAS used to represent composite objects made up of objects of type \spad{A}, \spad{B},..., \spad{C} which are indexed by "keys" (identifiers) \spad{a},\spad{b},...,\spad{c}.
43669`\spadtype{RectangularMatrix} is a matrix domain where the number of rows and the number of columns are parameters of the domain.
43859`\indented{1}{\spadtype{Reference} is for making a changeable instance} of something.
43912`A domain for regular chains (\spadignore{i.e.} regular triangular sets) over a \spad{Gcd}-Domain and with a fix list of variables. This is just a front-end for the \spadtype{RegularTriangularSet} domain constructor.
44148`This domain provides an implementation of regular chains. Moreover,{} the operation \axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\newline References : \indented{1}{[1] \spad{M}. MORENO MAZA "A new algorithm f
44148`--or computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
44337`ResidueRing is the quotient of a polynomial ring by an ideal. The ideal is given as a list of generators. The elements of the domain are equivalence classes expressed in terms of reduced elements
44533`A domain used to return the results from a call to the NAG Library. It prints as a list of names and types,{} though the user may choose to display values automatically if he or she wishes.
44601`\indented{1}{Rules for the pattern matcher} Author: Manuel Bronstein Date Created: 24 Oct 1988 Date Last Updated: 26 October 1993 Keywords: pattern,{} matching,{} rule.
44729`\axiomType{RightOpenIntervalRootCharacterization} provides work with interval root coding.
44952`\spadtype{RomanNumeral} provides functions for converting \indented{1}{integers to roman numerals.}
45018`This domain implements named rules
45086`A ruleset is a set of pattern matching rules grouped together.
45213`\indented{1}{Domain for representing points,{} vectors,{} subspaces and properties} of components in a subspace as complex numbers
45270`\indented{1}{SBoundary implements boundaries for use in scene graph} This defines the outer extent of the scene,{} or of an element in the scene,{} or of some branch in the scene. The difference between this boundary and an \spad{n}-dimensional surface,{} such as IFS,{} is that boundary must always have a well defined inside and an outside. SBoundary does not necessarily follow the exact outer contours of the shape but just constructs a simple bou
45270`--ndary shape where all points of the shape are guaranteed to be inside the boundary. Currently there are the following boundary forms: \indented{1}{Simple types:} \indented{2}{"box"::Symbol which is a \spad{n}-dimensional axis-aligned rectangle.} \indented{2}{"ellipoid"::Symbol which is a \spad{n}-dimensional axis-aligned ellipse.} \indented{1}{Compound types:} \indented{2}{This is a combination of simple and other compound types such} \indented{2}{as union or intersection.} \indented{3}{"union
45270`--"::Symbol represents union of supplied list} \indented{3}{"intersection"::Symbol represents intersection of supplied list} \indented{1}{Null types:} \indented{2}{"none" is used for a node which does not have a boundary in} \indented{6}{space,{} for example,{} when the boundary of a material node} \indented{6}{is requested.} The uses of SBoundary are: 1) So we know how much to scale a given scene to fit in a \indented{3}{given size.} 2) So that we know where to terminate arrows going in out out
45270`-- of \indented{3}{of the shape.} 3) To detect if objects intersect (collision detect),{} or if one \indented{3}{object is contained completely inside another.}
45326`an implementation of PointCategory to represent points and vectors in three dimensional coordinate space
45392`\indented{1}{number of base dimensions (excluding point at infinity and point} at origin)
45469`This domain allows the manipulation of Lisp values over arbitrary atomic types.
45646`This domain allows the manipulation of the usual Lisp values.
45745`parse result includes term returned and new index
45829`\indented{1}{Domains for transform from SPoint to SPoint} STransform transform domain can by used to represent linear transforms for instance when working with SCartesian where its Representation is a matrix of DoubleFloat Other transform domains may not necessarily be linear and will have different representations
45883`Generates various surfaces (2 dimensional subsets of 3D space). When 1 dimensional subspace (a line) needs to be represented in a 3D editor then we convert the line into a tube. Many of the functions here are from NumericTubePlot and related packages. However \spad{I} needed them to be defined over SPointCategory so \spad{I} adapted them here.
45937`Associates points or nodes in Scene structure with names (or a hierarchy of names). The aim is,{} where a point or node is used in several different constructions,{} it only needs to be specified once.
46016`\indented{1}{these are the parameters that can be set for nodes of} various types
46067`\spadtype{ScriptFormulaFormat} provides a coercion from \spadtype{OutputForm} to IBM SCRIPT/VS Mathematical Formula Format. The basic SCRIPT formula format object consists of three parts: a prologue,{} a formula part and an epilogue. The functions \spadfun{prologue},{} \spadfun{formula} and \spadfun{epilogue} extract these parts,{} respectively. The central parts of the expression go into the formula part. The other parts can be set (\s
46067`--padfun{setPrologue!},{} \spadfun{setEpilogue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain ":df." and ":edf." so that the formula section will be printed in display math mode.
46134`This domain is used to provide the function argument syntax \spad{v=a..b}. This is used,{} for example,{} by the top-level \spadfun{draw} functions.
46196`This type is used to specify a range of values from type \spad{S}.
46261`A domain for infinite sequences over a commutative ring. It is a stream with arithmetics.
46330`\spadtype{SequentialDifferentialPolynomial} implements an ordinary differential polynomial ring in arbitrary number of differential indeterminates,{} with coefficients in a ring. The ranking on the differential indeterminate is sequential. \blankline
46540`\spadtype{OrderlyDifferentialVariable} adds a commonly used sequential ranking to the set of derivatives of an ordered list of differential indeterminates. A sequential ranking is a ranking \spadfun{<} of the derivatives with the property that for any derivative \spad{v},{} there are only a finite number of derivatives \spad{u} with \spad{u} \spadfun{<} \spad{v}. This domain belongs to \spadtype{Different
46540`--ialVariableCategory}. It defines \spadfun{weight} to be just \spadfun{order},{} and it defines a sequential ranking \spadfun{<} on derivatives \spad{u} by the lexicographic order on the pair (\spadfun{variable}(\spad{u}),{} \spadfun{order}(\spad{u})).
46639`\spadtype{SetOfMIntegersInOneToN} implements the subsets of \spad{M} integers in the interval \spad{[1..n]}
46738`A set over a domain \spad{S} models the usual mathematical notion of a finite set of elements from \spad{S}. Sets are unordered collections of distinct elements (that is,{} order and duplication does not matter). The notation \spad{set [a,{}b,{}c]} can be used to create a set and the usual operations such as union and intersection are available to form new sets. If \spad{S} has OrderdSet,{} \Language{} maintains the entries in sorted order. S
46738`--pecifically,{} the parts function returns the entries as a list in ascending order and the extract operation returns the maximum entry. Given two sets \spad{s} and \spad{t} where \spad{\#s = m} and \spad{\#t = n},{} the complexity of \indented{2}{\spad{s = t} is \spad{O(min(n,{}m))}} \indented{2}{\spad{s < t} is \spad{O(max(n,{}m))}} \indented{2}{\spad{union(s,{}t)},{} \spad{intersect(s,{}t)},{} \spad{minus(s,{}t)},{} \spad{symmetricDifference(s,{}t)} is \spad{O(max(n,{}m))}} \indented{2}{\spa
46738`--d{member(x,{}t)} is \spad{O(n log n)}} \indented{2}{\spad{insert(x,{}t)} and \spad{remove(x,{}t)} is \spad{O(n)}}
46799`Domain which represents simple algebraic extensions of arbitrary rings. The first argument to the domain,{} \spad{R},{} is the underlying ring,{} the second argument is a domain of univariate polynomials over \spad{K},{} while the last argument specifies the defining minimal polynomial. The elements of the domain are canonically represented as polynomials of degree less th
46799`--an that of the minimal polynomial with coefficients in \spad{R}. The second argument is both the type of the third argument and the underlying representation used by \spadtype{SAE} itself.
46931`\axiomType{SimpleFortranProgram(\spad{f},{}type)} provides a simple model of some FORTRAN subprograms,{} making it possible to coerce objects of various domains into a FORTRAN subprogram called \axiom{\spad{f}}. These can then be translated into legal FORTRAN code.
47046`SingleInteger is intended to support machine integer arithmetic.
47112`This trivial domain lets us build Univariate Polynomials in an anonymous variable
47177`SmallOrdinal implements ordinal numbers up to \spad{epsilon_0}. \spad{+} and \spad{*} are "natural" addition and multiplication of ordinals. Avaliable separately are "ordered" operataions.
47243`\indented{1}{Domain for storing information about structure of polynomials} as vectors of exponents
47327`\axiom{SparseEchelonMatrix(\spad{C},{} \spad{D})} implements sparse matrices whose columns are enumerated by the \axiomType{OrderedSet} \axiom{\spad{C}} and whose entries belong to the \axiomType{GcdDomain} \axiom{\spad{D}}. The basic operation of this domain is the computation of an row echelon form. The used algorithm tries to maintain the sparsity and is especially adapted to matrices who are already clo
47327`--se to a row echelon form.
47579`\indented{2}{This type is the basic representation of sparse recursive multivariate} polynomials. It is parameterized by the coefficient ring and the variable set which may be infinite. The variable ordering is determined by the variable set parameter. The coefficient ring may be non-commutative,{} but the variables are assumed to commute.
47715`SparseMultivariateSkewPolynomial(\spad{R},{} Var,{} sigma,{} delta) defines a mutivariate Ore ring over \spad{R} in variables from \spad{V}. \spad{sigma(v)} gives automorphism of \spad{R} corresponding to variable \spad{v} and \spad{delta(v)} gives corresponding derivative. \blankline
47922`This domain provides multivariate Taylor series with variables from an arbitrary ordered set. A Taylor series is represented by a stream of polynomials from the polynomial domain \spad{SMP}. The \spad{n}th element of the stream is a form of degree \spad{n}. SMTS is an internal domain.
48121`A sparse table has a default entry,{} which is returned if no other value has been explicitly stored for a key.
48222`Sparse Laurent series in one variable \indented{2}{\spadtype{SparseUnivariateLaurentSeries} is a domain representing Laurent} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented
48222`--{2}{variable,{} and the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{SparseUnivariateLaurentSeries(Integer,{}x,{}3)} represents Laurent} \indented{2}{series in \spad{(x - 3)} with integer coefficients.}
48419`This domain represents univariate polynomials over arbitrary (not necessarily commutative) coefficient rings. The variable is unspecified so that the variable displays as \spad{?} on output. If it is necessary to specify the variable name,{} use type \spadtype{UnivariatePolynomial}. The representation is sparse in the sense that only non-zero terms are represented.
48516`Sparse Puiseux series in one variable \indented{2}{\spadtype{SparseUnivariatePuiseuxSeries} is a domain representing Puiseux} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indent
48516`--ed{2}{variable,{} and the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{SparseUnivariatePuiseuxSeries(Integer,{}x,{}3)} represents Puiseux} \indented{2}{series in \spad{(x - 3)} with \spadtype{Integer} coefficients.}
48715`This is the domain of sparse univariate skew polynomials over an Ore coefficient field. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}.
48860`Sparse Taylor series in one variable \indented{2}{\spadtype{SparseUnivariateTaylorSeries} is a domain representing Taylor} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented{2}{variable,{} and the center of the power series expansion.\s
48860`--pace{2}For example,{}} \indented{2}{\spadtype{SparseUnivariateTaylorSeries}(Integer,{}\spad{x},{}3) represents Taylor} \indented{2}{series in \spad{(x - 3)} with \spadtype{Integer} coefficients.}
48997`\indented{2}{This type represents the finite direct or cartesian product of an} underlying ordered component type. The vectors are ordered as if they were split into two blocks. The \spad{dim1} parameter specifies the length of the first block. The ordering is lexicographic between the blocks but acts like \spadtype{HomogeneousDirectProduct} w
48997`--ithin each block. This type is a suitable third argument for \spadtype{GeneralDistributedMultivariatePolynomial}.
49159`This domain exports a modest implementation for the vertices of splitting trees. These vertices are called here splitting nodes. Every of these nodes store 3 informations. The first one is its value,{} that is the current expression to evaluate. The second one is its condition,{} that is the hypothesis under which the value has to be evaluated. The last one is its status,{} that is a boolean f
49159`--lag which is \spad{true} iff the value is the result of its evaluation under its condition. Two splitting vertices are equal iff they have the sane values and the same conditions (so their status do not matter).
49270`This domain exports a modest implementation of splitting trees. Spliiting trees are needed when the evaluation of some quantity under some hypothesis requires to split the hypothesis into sub-cases. For instance by adding some new hypothesis on one hand and its negation on another hand. The computations are terminated is a splitting tree \axiom{a} when \axiom{status(
49270`--value(a))} is \axiom{\spad{true}}. Thus,{} if for the splitting tree \axiom{a} the flag \axiom{status(value(a))} is \axiom{\spad{true}},{} then \axiom{status(value(\spad{d}))} is \axiom{\spad{true}} for any subtree \axiom{\spad{d}} of \axiom{a}. This property of splitting trees is called the termination condition. If no vertex in a splitting tree \axiom{a} is equal to another,{} \axiom{a} is said to satisfy the no-duplicates condition. The splitting tree \axiom{a} will satisfy this condition i
49270`--f nodes are added to \axiom{a} by mean of \axiom{splitNodeOf!} and if \axiom{construct} is only used to create the root of \axiom{a} with no children.
49408`This domain provides an implementation of square-free regular chains. Moreover,{} the operation \axiomOpFrom{zeroSetSplit}{SquareFreeRegularTriangularSetCategory} is an implementation of a new algorithm for solving polynomial systems by means of regular chains.\newline References : \indented{1}{[
49408`--1] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.} \indented{2}{Version: 2}
49618`\spadtype{SquareMatrix} is a matrix domain of square matrices,{} where the number of rows (= number of columns) is a parameter of the type.
49786`Linked List implementation of a Stack
49857`A stream is an implementation of a possibly infinite sequence using a list of terms that have been computed and a function closure to compute additional terms when needed.
49928`This domain provides tables where the keys are strings. A specialized hash function for strings is used.
50065`This domain implements some global properties of subspaces.
50139`This domain \undocumented
50223`This domain implements "such that" forms
50309`This domain builds representations of boolean expressions for use with the \axiomType{FortranCode} domain.
50374`Create and manipulate a symbol table for generated FORTRAN code
50444`Basic and scripted symbols.
50496`This domain implements symmetric polynomial
50595`This is the general purpose table type. The keys are hashed to look up the entries. This creates a \spadtype{HashTable} if equal for the Key domain is consistent with Lisp EQUAL otherwise an \spadtype{AssociationList}
50693`\indented{1}{The tableau domain is for printing Young tableaux,{} and} coercions to and from List List \spad{S} where \spad{S} is a set.
51864`\spadtype{TaylorSeries} is a general multivariate Taylor series domain over the ring Coef and with variables of type Symbol.
51963`Tensor powers of a free module over a commutative ring. It is represented as a free module over the cartesian power of the basis.
52121`Tensor product of free modules over a commutative ring. It is represented as a free module over the direct product of the respective bases. The factor domains must provide operations \spad{listOfTerms},{} whose result is assumed to be stored in reverse order.
52316`\spadtype{TexFormat} provides a coercion from \spadtype{OutputForm} to \TeX{} format. The particular dialect of \TeX{} used is \LaTeX{}. The basic object consists of three parts: a prologue,{} a tex part and an epilogue. The functions \spadfun{prologue},{} \spadfun{tex} and \spadfun{epilogue} extract these parts,{} respectively. The main guts of the expression go into the tex part. The other parts can be set (\spadfun{setPrologue!},{} \spadfun{setEpi
52316`--logue!}) so that contain the appropriate tags for printing. For example,{} the prologue and epilogue might simply contain \spad{``}\verb+\spad{\[}+\spad{''} and \spad{``}\verb+\spad{\]}+\spad{''},{} respectively,{} so that the TeX section will be printed in LaTeX display math mode.
52369`\spadtype{TexmacsFormat} provides a coercion from \spadtype{OutputForm} to TeXmacs format.
52429`This domain provides an implementation of text files. Text is stored in these files using the native character set of the computer.
52504`Creates and manipulates one global symbol table for FORTRAN code generation,{} containing details of types,{} dimensions,{} and argument lists.
52575`This domain represents three dimensional matrices over a general object type
52733`The domain ThreeSpace is used for creating three dimensional objects using functions for defining points,{} curves,{} polygons,{} constructs and the subspaces containing them.
52797`\spadtype{Tree(S)} is a basic domains of tree structures. Each tree is either empty or else is a {\it node} consisting of a value and a list of (sub)trees.
52870`\indented{2}{Package for constructing tubes around 3-dimensional parametric curves.} Domain of tubes around 3-dimensional parametric curves.
52939`\indented{1}{This domain is used to interface with the interpreter\spad{'s} notion} of comma-delimited sequences of values.
53016`\indented{1}{A TwoDimensionalArray is a two dimensional array with} 1-based indexing for both rows and columns.
53144`TwoDimensionalViewport creates viewports to display graphs.
53213`\indented{1}{name of variable} and type of variable
53259`This is a low-level domain which implements matrices (two dimensional arrays) of 16-bit integers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
53347`This is a low-level domain which implements vectors (one dimensional arrays) of unsigned 16-bit numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
53419`This is a low-level domain which implements matrices (two dimensional arrays) of 32-bit integers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
53507`This is a low-level domain which implements vectors (one dimensional arrays) of unsigned 32-bit numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
53579`This is a low-level domain which implements matrices (two dimensional arrays) of 8-bit integers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
53663`This is a low-level domain which implements vectors (one dimensional arrays) of unsigned 8-bit numbers. Indexing is 0 based,{} there is no bound checking (unless provided by lower level).
53733`allows us to model graph theory \blankline
53812`\spad{Union(A,B,...,C)} is a primitive type in FriCAS used to represent objects of type \spad{A} or of type \spad{B} or...or of type \spad{C}.
53958`This package enables one to construct a univariate Laurent series domain from a univariate Taylor series domain. Univariate Laurent series are represented by a pair \spad{[n,{}f(x)]},{} where \spad{n} is an arbitrary integer and \spad{f(x)} is a Taylor series. This pair represents the Laurent series \spad{x^n * f(x)}.
54131`Dense Laurent series in one variable \indented{2}{\spadtype{UnivariateLaurentSeries} is a domain representing Laurent} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented{2}{variable,{} and 
54131`--the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{UnivariateLaurentSeries(Integer,{}x,{}3)} represents Laurent series in} \indented{2}{\spad{(x - 3)} with integer coefficients.}
54315`This domain represents univariate polynomials in some symbol over arbitrary (not necessarily commutative) coefficient rings. The representation is sparse in the sense that only non-zero terms are represented.
54416`This package enables one to construct a univariate Puiseux series domain from a univariate Laurent series domain. Univariate Puiseux series are represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x^r)}.
54591`UnivariatePuiseuxSeriesWithExponentialSingularity is a domain used to represent functions with essential singularities. Objects in this domain are sums,{} where each term in the sum is a univariate Puiseux series times 
54591`--the exponential of a univariate Puiseux series. Thus,{} the elements of this domain are sums of expressions of the form \spad{g(x) * exp(f(x))},{} where \spad{g}(\spad{x}) is a univariate Puiseux series and \spad{f}(\spad{x}) is a univariate Puiseux series with no terms of non-negative degree.
54879`Dense Puiseux series in one variable \indented{2}{\spadtype{UnivariatePuiseuxSeries} is a domain representing Puiseux} \indented{2}{series in one variable with coefficients in an arbitrary ring.\space{2}The} \indented{2}{parameters of the type specify the coefficient ring,{} the power series} \indented{2}{variable,{} an
54879`--d the center of the power series expansion.\space{2}For example,{}} \indented{2}{\spad{UnivariatePuiseuxSeries(Integer,{}x,{}3)} represents Puiseux series in} \indented{2}{\spad{(x - 3)} with \spadtype{Integer} coefficients.}
55065`This is the domain of univariate skew polynomials over an Ore coefficient field in a named variable. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}.
55212`Dense Taylor series in one variable \spadtype{UnivariateTaylorSeries} is a domain representing Taylor series in one variable with coefficients in an arbitrary ring. The parameters of the type specify the coefficient ring,{} the power series variable,{} and the center of the power series expansion. For example,{} \spadtype{UnivariateTaylorSeries}(Integer,{}\spad{x},{}3) repre
55212`--sents Taylor series in \spad{(x - 3)} with \spadtype{Integer} coefficients.
55342`This domain provides segments which may be half open. That is,{} ranges of the form \spad{a..} or \spad{a..b}.
55419`\indented{1}{At the moment an untyped variable is represented by a string} representing the name of the variable. So why wrap it in a Record structue. The main reason is because \spad{I} am planning to expand this as explained in the text part of this pamphlet. Also it seems like a good idea to keep a similar structure to typed variables. Also \spad{I} could not get it to work unwrapped. \spad{I} vaguely recall that \spad{I} couldn\spad{'t} get the cons
55419`--tructor to return an unwrapped value,{} but it may have been some other issue and that\spad{'s} not the main reason.
55469`This domain implements variables
55537`This domain supports modular methods based on \indented{1}{evaluation and rational reconstruction.\space{2}Each evaluation} \indented{1}{is done modulo machine sized prime \spad{p}.\space{2}Both Chinese} \indented{1}{remaindering and (linear) Hensel lift are supported.} \indented{1}{Once enough evaluations are known rational reconstruction} \indented{1}{produces vector of rational numbers or integers.}
55593`This domain supports modular methods based on \indented{1}{evaluation and rational reconstruction.\space{2}All computation} \indented{1}{are done on polynomials modulo machine sized prime \spad{p} \spad{
55593`--} \spad{p} must} \indented{1}{be choosen small enough to avoid overflow in intermediate} \indented{1}{calculations. Each evaluation is supposed to produce vector of} \indented{1}{values. Once enough evaluations are known rational reconstruction} 
55593`--\indented{1}{produces vector of rational functions or multivariate polynomials.}
55649`The domain \spad{VectorSpaceBasis} implements subspaces of vector spaces. A subspace is represented by a basis which is a list of vectors. Elements of basis are linearly independent so the list list describes the space with only the null vector. The scalar ring must be a field in order to avoid lattice (diofantine) questions.
55703`This type represents vector like objects with varying lengths and indexed by a finite segment of integers starting at 1.
55769`This type is used when no value is needed,{} \spadignore{e.g.} in the \spad{then} part of a one armed \spad{if}. All values can be coerced to type Void. Once a value has been coerced to Void,{} it cannot be recovered.
55800`allows us to model weighted graphs This graph assigns a weight to each arrow. This is used when calculating say,{} lowest 'cost' routes. An arrow with a higher weight is more 'costly' in some way and therefore we try to choose the minimum weight. For arrows in series the overall weight is the sum of the individual weights. For arrows in parallel the overall weight is the minimum of the individual weight
55800`--s. \spad{I} may change this in future to allow different cost metrics to be plugged in. \blankline
55901`This domain represents truncated weighted polynomials over a general (not necessarily commutative) polynomial type. The variables must be specified,{} as must the weights. The representation is sparse in the sense that only non-zero terms are represented.
56124`A domain constructor of the category \axiomType{GeneralTriangularSet}. The only requirement for a list of polynomials to be a member of such a domain is the following: no polynomial is constant and two distinct polynomials have distinct main variables. Such a triangular set may not be auto-reduced or consistent. The 
56124`--\axiomOpFrom{construct}{WuWenTsunTriangularSet} operation does not check the previous requirement. Triangular sets are stored as sorted lists \spad{w}.\spad{r}.\spad{t}. the main variables of their members. Furthermore,{} this domain exports operations dealing with the characteristic set method of Wu Wen Tsun and some optimizations mainly proposed by Dong Ming Wang.\newline References : \indented{1}{[1] \spad{W}. \spad{T}. WU "A Zero Structure Theorem for polynomial equations solving"} \indent
56124`--ed{6}{\spad{MM} Research Preprints,{} 1987.} \indented{1}{[2] \spad{D}. \spad{M}. WANG "An implementation of the characteristic set method in Maple"} \indented{6}{Proc. \spad{DISCO'92}. Bath,{} England.}
56313`\indented{2}{This type supports distributed multivariate polynomials} whose variables do not commute. The coefficient ring may be non-commutative too. However,{} coefficients and variables commute.
56437`An implementation of a hash table that uses equality of the key domain to decide upon equality of keys.
56543`This domain constructor implements polynomials in non-commutative variables written in the Poincare-Birkhoff-Witt basis from the Lyndon basis. These polynomials can be used to compute Baker-Campbell-Hausdorff relations. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
56658`This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring),{} and words belonging to an arbitrary \spadtype{OrderedMonoid}. This type is used,{} for instance,{} by the \spadtype{XDistributedPolynomial} domain constructor where the Monoid is free.
56735`\indented{2}{This type supports multivariate polynomials} whose set of variables is \spadtype{Symbol}. The representation is recursive. The coefficient ring may be non-commutative and the variables do not commute. However,{} coefficients and variables commute.
56813`\indented{2}{This type supports multivariate polynomials} whose variables do not commute. The representation is recursive. The coefficient ring may be non-commutative. Coefficients and variables commute.
56921`This package provides support for \spad{XML} files \spad{XML} attributes are inside the opening tag: <tagname attributes> and usually have the form: name=value
56961`This package provides support for \spad{XML} files an \spad{XML} element is something which either has the form: <tagname attributes> content </tagname> or <tagname attributes/> where content may be a sequence of sub-elements or unstructured text
57390`\spad{\# p} returns the number of terms in \spad{p}.
57452`\spad{\#((a1,{}...,{}an))} returns \spad{n}.
57524`\spad{\# u} returns the number of items in \spad{u}.
57598`\spad{r * a} returns the power series scalar multiplication of \spad{r} by \spad{a}: \spad{r * [a0,{}a1,{}...] = [r * a0,{}r * a1,{}...]}
57733`\spad{c*x} returns the product of \spad{c} and the series \spad{x}.
57787`\axiom{d*exp} is provided mainly for internal use,{} as basically all calculations should be performed within \axiomType{JetLazyFunction}.
57836`\spad{s * p} returns a point whose coordinates are the scalar multiple of the point \spad{p} by the scalar \spad{s},{} preserving the color,{} or fourth coordinate,{} of \spad{p}.
57919`*(\spad{s},{}\spad{x}) gives scalar multiplication Although,{} scalar multiplication is strictly only a valid operation for vectors and not points,{} we allow it here because there are some useful short cuts,{} for instance,{} to find the central point between two points we can add them together and scalar multiply by 0.5.
58939`\axiom{L*A} implements left multiplication with a usual matrix.
58999`\axiom{L*A} implements left multiplication with a usual matrix over the quotient field of \axiom{\spad{D}}.
59756`\spad{p*x} \undocumented
60951`\axiom{\spad{r} * \spad{v}} computes the product of \axiom{\spad{r}} and \axiom{\spad{v}}
62423`\spad{s * x} returns the product of \spad{x} by \spad{s} on the left.
62463`\spad{s * x} returns the product of \spad{x} by \spad{s} on the left.
62504`\spad{s * x} returns the product of \spad{x} by \spad{s} on the left.
62552`\spad{smp*ts} multiplies a TaylorSeries by a monomial \spad{SMP}.
62626`\spad{a * r} returns the power series scalar multiplication of \spad{a} by \spad{r:} \spad{[a0,{}a1,{}...] * r = [a0 * r,{}a1 * r,{}...]}
62699`\spad{a * b} returns the power series (Cauchy) product of \spad{a} and \spad{b:} \spad{[a0,{}a1,{}...] * [b0,{}b1,{}...] = [c0,{}c1,{}...]} where \spad{ck = sum(i + j = k,{}\spad{ai} * bk)}.
62780`\spad{a * b} returns the power series (Cauchy) product of \spad{a} and \spad{b:} \spad{[a0,{}a1,{}...] * [b0,{}b1,{}...] = [c0,{}c1,{}...]} where \spad{ck = sum(i + j = k,{}binomial(k,{}i) * \spad{ai} * bj)}.
62878`\spad{x*y} \undocumented{} See \axiomFunFrom{*}{SemiGroup}
62965`\spad{x*c} returns the product of \spad{c} and the series \spad{x}.
63019`\spad{x*i} returns the product of integer \spad{i} and the series \spad{x}.
63076`\spad{p*r} returns the product of \spad{p} by \spad{r}.
63124`\spad{x * s} returns the product of \spad{x} by \spad{s} on the right.
63164`\spad{x * s} returns the product of \spad{x} by \spad{s} on the right.
63205`\spad{x * s} returns the product of \spad{x} by \spad{s} on the right.
63253`\spad{"*"}(a,{}\spad{b}) returns a tensor product : the tensor product \spad{G*H} of graphs \spad{G} and \spad{H} is a graph such that the vertex set of \spad{G*H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}); and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) are adjacent in \spad{G} \times \spad{H} if and only if u' is adjacent with \spad{v'} and \spad{u} is adjacent with \spad{v}.
63323`tensor product : the tensor product \spad{G*H} of graphs \spad{G} and \spad{H} is a graph such that the vertex set of \spad{G*H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}); and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) are adjacent in \spad{G} \times \spad{H} if and only if u' is adjacent with \spad{v'} and \spad{u} is adjacent with \spad{v}. Cartesian product does apply to function graph 
63323`--produces two arrows out of every node
63393`tensor product : the tensor product \spad{G*H} of graphs \spad{G} and \spad{H} is a graph such that the vertex set of \spad{G*H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}); and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) are adjacent in \spad{G} \times \spad{H} if and only if u' is adjacent with \spad{v'} and \spad{u} is adjacent with \spad{v}.
63473`tensor product : the tensor product \spad{G*H} of graphs \spad{G} and \spad{H} is a graph such that the vertex set of \spad{G*H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}); and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) are adjacent in \spad{G} \times \spad{H} if and only if u' is adjacent with \spad{v'} and \spad{u} is adjacent with \spad{v}.
63547`tensor product : the tensor product \spad{G*H} of graphs \spad{G} and \spad{H} is a graph such that the vertex set of \spad{G*H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}); and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) are adjacent in \spad{G} \times \spad{H} if and only if u' is adjacent with \spad{v'} and \spad{u} is adjacent with \spad{v}.
63621`\spad{a * b} returns the input form corresponding to \spad{a * b}.
63659`\axiom{x*y} returns the tree \axiom{[\spad{x},{}\spad{y}]}.
63701`\spad{f * g} creates the equivalent infix form.
63740`\spad{a * b} returns the pattern \spad{a * b}.
63779`\spad{b1*b2} computes a basis of the intersection of two of subspaces.
63827`\spad{f*g} is the function \spad{h} \indented{1}{such that \spad{h x= f(g x)}.}
63890`\spad{s * c},{} returns the color \spad{c},{} whose weighted shade has been scaled by \spad{s}.
63933`\spad{e * s} returns \spad{e} times \spad{s}.
63990`\spad{n*x} is the product of \spad{x} by the integer \spad{n}.
64036`\spad{n * x} is an integer multiple.
64114`\spad{n * y} multiplies each component of the vector \spad{y} by the integer \spad{n}.
64184`\spad{n * x} is left-multiplication by a non negative integer
64242`\spad{n*x} computes the left-multiplication of \spad{x} by the positive integer \spad{n}. This is equivalent to adding \spad{x} to itself \spad{n} times.
64300`\spad{s * c},{} returns the color \spad{c},{} whose weighted shade has been scaled by \spad{s}.
64347`\spad{r*b} returns the product of \spad{r} by \spad{b}.
64397`\spad{r * y} multiplies the element \spad{r} times each component of the vector \spad{y}.
64466`\spad{r*g} is left module multiplication.
64511`\spad{r*x} returns the left multiplication of the module element \spad{x} by the ring element \spad{r}.
64552`\spad{r*x} is the left scalar multiple of the scalar \spad{r} and the matrix \spad{x}.
64605`\spad{r * y} multiplies the element \spad{r} times each component of the vector \spad{y}.
64663`\spad{r * x} is the product of the row vector \spad{r} and the matrix \spad{x}. Error: if the dimensions are incompatible.
64719`\spad{r * x} is the product of the row vector \spad{r} and the matrix \spad{x}. Error: if the dimensions are incompatible.
64786`\spad{s*r} returns the product \spad{r*s} used by \spadtype{XRecursivePolynomial}
64836`\spad{x*eqn} produces a new equation by multiplying both sides of equation eqn by \spad{x}.
64891`\spad{x * c} is the product of the matrix \spad{x} and the column vector \spad{c}. Error: if the dimensions are incompatible.
64947`\spad{x * c} is the product of the matrix \spad{x} and the column vector \spad{c}. Error: if the dimensions are incompatible.
65014`\spad{t*p} applies the dhmatrix \spad{t} to point \spad{p}
65081`\spad{y * r} multiplies each component of the vector \spad{y} by the element \spad{r}.
65150`\spad{g*r} is right module multiplication.
65195`\spad{x * r} is the right scalar multiple of the scalar \spad{r} and the matrix \spad{x}.
65248`\spad{x*r} returns the right multiplication of the module element \spad{x} by the ring element \spad{r}.
65290`\spad{y * r} multiplies each component of the vector \spad{y} by the element \spad{r}.
65348`\spad{x * r} returns the product of \spad{x} by \spad{r}. Usefull if \spad{R} is a non-commutative Ring.
65394`\spad{eqn*x} produces a new equation by multiplying both sides of equation eqn by \spad{x}.
65449`\spad{x * y} is the product of the matrices \spad{x} and \spad{y}. Error: if the dimensions are incompatible.
65503`\spad{a*b} is the product of \spad{a} and \spad{b} in a set with a binary operation.
65537`\spad{x*y} returns the product of \spad{x} and \spad{y}.
65575`\spad{s*t} is the inner product of the tensors \spad{s} and \spad{t} which contracts the last index of \spad{s} with the first index of \spad{t},{} \spadignore{i.e.} \indented{4}{\spad{t*s = contract(t,{}rank t,{} s,{} 1)}} \indented{4}{\spad{t*s = sum(k=1..N,{} t[i1,{}..,{}iN,{}k]*s[k,{}j1,{}..,{}jM])}} This is compatible with the use of \spad{M*v} to denote the matrix-vector inner product.
65632`\spad{I*J} computes the product of the ideal \spad{I} and \spad{J}.
65698`\spad{v * x} returns the product of a variable \spad{x} by \spad{x}.
65745`\spad{p + q} computes and returns a point whose coordinates are the sums of the coordinates of the two points \spad{p} and \spad{q},{} using the color,{} or fourth coordinate,{} of the first point \spad{p} as the color also of the point \spad{q}.
66055`\spad{a + b} returns the power series sum of \spad{a} and \spad{b}: \spad{[a0,{}a1,{}..] + [b0,{}b1,{}..] = [a0 + b0,{}a1 + b1,{}..]}
66136`sum : disjoint union of nodes with arrows from appropriate input
66179`+(\spad{x},{}\spad{y}) gives addition add the corresponding elements of the vector or point. The type (vector or point) depends on the following table: vector + vector = vector point + vector = point point + point = invalid (but interpret as vector) Although,{} adding two points is not strictly a valid operation,{} we allow it here by interpreting the point as a vector relative to the origin. We do that because there are some useful short cuts,{} for instance,
66179`--{} to find the central point between two points we can add them together and scalar multiply by 0.5.
66222`\spad{a + b} returns the input form corresponding to \spad{a + b}.
66260`\spad{f + g} creates the equivalent infix form.
66299`\spad{a + b} returns the pattern \spad{a + b}.
66338`\spad{B1+B2} computes a basis of the sum of two subspaces.
66386`\spad{s + x} returns the sum of \spad{s} and \spad{x}.
66444`\spad{x+eqn} produces a new equation by adding \spad{x} to both sides of equation eqn.
66506`\spad{p + x} returns the sum of \spad{p} and \spad{x}
66569`\spad{eqn+x} produces a new equation by adding \spad{x} to both sides of equation eqn.
66631`\spad{x+y} computes the sum of \spad{x} and \spad{y}.
66676`\spad{g+h} is the sum of \spad{g} and \spad{h} in the module of elements of the same degree as \spad{g} and \spad{h}. Error: if \spad{g} and \spad{h} have different degrees.
66722`\spad{x + y} is the sum of the matrices \spad{x} and \spad{y}. Error: if the dimensions are incompatible.
66776`\spad{x + y} returns the component-wise sum of the vectors \spad{x} and \spad{y}. Error: if \spad{x} and \spad{y} are not of the same length.
66845`\spad{c1 + c2} additively mixes the two colors \spad{c1} and \spad{c2}.
66879`\spad{db1+db2} returns the merge of databases \spad{db1} and \spad{db2}
66919`\spad{I+J} computes the ideal generated by the union of \spad{I} and \spad{J}.
67326`\spad{- a} returns the power series negative of \spad{a}: \spad{- [a0,{}a1,{}...] = [- a0,{}- a1,{}...]}
67397`\spad{- f} creates the equivalent prefix form.
67433`\spad{-x} is the additive inverse of \spad{x}.
67471`\spad{-g} is the additive inverse of \spad{g} in the module of elements of the same grade as \spad{g}.
67514`\spad{-x} returns the negative of the matrix \spad{x}.
67584`\spad{-x} negates all components of the vector \spad{x}.
67646`\spad{p - q} computes and returns a point whose coordinates are the differences of the coordinates of two points \spad{p} and \spad{q},{} using the color,{} or fourth coordinate,{} of the first point \spad{p} as the color also of the point \spad{q}.
68093`\spad{a - b} returns the power series difference of \spad{a} and \spad{b}: \spad{[a0,{}a1,{}..] - [b0,{}b1,{}..] = [a0 - b0,{}a1 - b1,{}..]}
68174`\spad{-(x,{}y)} gives subtraction subtract the corresponding elements of the vector or point. The type (vector or point) depends on the following table: vector - vector = vector point - vector = point vector - point = invalid (but interpret as point) point - point = invalid (but interpret as vector) Although,{} subtracting two points is not strictly a valid operation,{} we allow it here by interpreting the result as the vector distance between the two points.
68217`\spad{f - g} creates the equivalent infix form.
68256`\spad{x-eqn} produces a new equation by subtracting both sides of equation eqn from \spad{x}.
68314`\spad{eqn-x} produces a new equation by subtracting \spad{x} from both sides of equation eqn.
68372`\spad{x - y} returns an element \spad{z} such that \spad{z+y=x} or "failed" if no such element exists.
68431`\spad{x-y} is the difference of \spad{x} and \spad{y} \spadignore{i.e.} \spad{x + (-y)}.
68472`\spad{g-h} is the difference of \spad{g} and \spad{h} in the module of elements of the same degree as \spad{g} and \spad{h}. Error: if \spad{g} and \spad{h} have different degrees.
68518`\spad{x - y} is the difference of the matrices \spad{x} and \spad{y}. Error: if the dimensions are incompatible.
68591`\spad{x - y} returns the component-wise difference of the vectors \spad{x} and \spad{y}. Error: if \spad{x} and \spad{y} are not of the same length.
68656`\spad{db1-db2} returns the difference of databases \spad{db1} and \spad{db2} \spadignore{i.e.} consisting of elements in \spad{db1} but not in \spad{db2}
68734`\spadignore { /\ }returns the logical `meet',{} \spadignore{e.g.} `and'.
68769`Implement exterior grassmann product operator need to check precidence when used as an infix operator
68824`\spad{n} \spad{/\} \spad{m} returns the bit-by-bit logical {\em and} of the single integers \spad{n} and \spad{m}.
68867`\spad{a / d} divides the element \spad{a} by \spad{d}.
68949`\spad{m / d} divides the element \spad{m} by \spad{d}.
69301`\spad{a / b} returns the power series quotient of \spad{a} by \spad{b}. An error message is returned if \spad{b} is not invertible. This function is used in fixed point computations.
69382`\spad{x/y} \undocumented{} See \axiomFunFrom{/}{Field}
69469`\spad{x / d} divides the element \spad{x} by \spad{d}.
69514`\spad{x / d} divides the element \spad{x} by \spad{d}.
69555`\spad{p/r} returns \spad{p*(1/r)}.
69615`\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \spad{R} is a Field)
69716`\spad{x/y} division (only works if minimum weight of divisor is zero,{} and if \spad{R} is a Field)
69820`\spad{a / b} returns the input form corresponding to \spad{a / b}.
69858`\spad{f / g} creates the equivalent infix form.
69897`\spad{a / b} returns the pattern \spad{a / b}.
69936`\spad{d1 / d2} returns the fraction \spad{d1} divided by \spad{d2}.
69987`\spad{p1/p2} returns the quotient of \spad{p1} and \spad{p2} as an element of \%.
70133`\spad{x / i} computes the division from \spad{x} by an integer \spad{i}.
70178`\spad{x / i} computes the division from \spad{x} by an integer \spad{i}.
70217`\spad{p/c} divides \spad{p} by the coefficient \spad{c}.
70279`\axiom{\spad{x/r}} returns the division of \axiom{\spad{x}} by \axiom{\spad{r}}.
70332`\spad{m/r} divides the elements of \spad{m} by \spad{r}. Error: if \spad{r = 0}.
70397`\spad{m/r} divides the elements of \spad{m} by \spad{r}. Error: if \spad{r = 0}.
70477`\spad{x/y} divides the vector \spad{x} by the scalar \spad{y}.
70519`\spad{x/y} divides the element \spad{x} by the element \spad{y}. Error: if \spad{y} is 0.
70553`\spad{x/y} is the same as \spad{x} times the inverse of \spad{y}.
70587`\spad{e1/e2} produces a new equation by dividing the left and right hand sides of equations \spad{e1} and \spad{e2}.
71261`\spad{f <= g} creates the equivalent infix form.
71301`\spad{x <= y} is a less than or equal test.
71348`\spad{gp1 <= gp2} returns \spad{true} if and only if {\em gp1} is a subgroup of {\em gp2}. Note: because of a bug in the parser you have to call this function explicitly by {\em gp1 <=\$(PERMGRP S) gp2}.
71866`\spad{f < g} creates the equivalent infix form.
71905`\spad{x < y} is a less than test.
71951`\spad{p < q} is an order relation on permutations. Note: this order is only total if and only if \spad{S} is totally ordered or \spad{S} is finite.
72007`\spad{gp1 < gp2} returns \spad{true} if and only if {\em gp1} is a proper subgroup of {\em gp2}.
72982`return \spad{true} if equal,{} that is names are equal and if there is a type then tey mast also be equal
73022`\spad{x=y} tests for equality
73073`returns \spad{true} (boolean \spad{true}) if intuitionisticLogic values are the same. Translates from Intuitionistic Logic to Boolean Logic
73113`return \spad{true} if equal (deep search) that is: all terms at all levels in tree must be alpha-equivalent to return \spad{true} That is the names,{} but not the deBruijn index,{} of the bound variables can be different. beta-equivalenty is not implemented because it is not decidable.
73157`\spad{a=b} tests if \spad{a} and \spad{b} are equal.
73203`return \spad{true} if equal (deep search) that is: all terms at all levels in tree must be equal to return \spad{true} all terms must be exactly equal,{} not just equivalent,{} that is SKK=I will return \spad{false} even though 'SKK' and 'I' have the same effect
73255`\spad{B1=B2} tests if the two subspaces are equals.
73308`\spad{f = g} creates the equivalent infix form.
73347`\spad{a=b} creates an equation.
73385`\spad{x=y} tests if \spad{x} and \spad{y} are equal.
73514`\spad{f >= g} creates the equivalent infix form.
73554`\spad{x >= y} is a greater than or equal test.
73733`\axiom{\spad{jv1} > \spad{jv2}} checks whether \axiom{\spad{jv1}} is greater than \axiom{\spad{jv2}} in the internal ordering.
73784`\spad{f > g} creates the equivalent infix form.
73823`\spad{x > y} is a greater than test.
73869`\spad{AND(x,{}y)} returns the \axiomType{Switch} expression representing \spad{x and y}.
74078`\spad{Aleph(n)} provides the named (infinite) cardinal number.
74138`\spad{An(x)} \undocumented
74187`\spad{And(n,{}m)} returns the bit-by-bit logical {\em And} of \spad{n} and \spad{m}.
74233`\spad{And(n,{}m)} returns the bit-by-bit logical {\em and} of the single integers \spad{n} and \spad{m}.
74277`\spad{B1solve(s)} solves the system (\spad{s}.mat) \spad{z} = \spad{s}.vec for the variables given by the column indices of \spad{s}.cols in terms of the other variables and the right hand side \spad{s}.vec b
74277`--y assuming that the rank is \spad{s}.rank,{} that the system is consistent,{} with the linearly independent equations indexed by the given row indices \spad{s}.rows; the coefficients in \spad{s}.mat involving parameters are treated as polynomials. B1solve(\spad{s}) returns a particular solution to the system and a basis of the homogeneous system (\spad{s}.mat) \spad{z} = 0.
74576`\spad{s by n} creates a new segment in which only every \spad{n}\spad{-}th element is used.
74629`\spad{BasicMethod(n)} returns the factorization of integer \spad{n} by trial division
74703`\spad{Beta(x,{}y)} returns the beta function applied to \spad{x} and \spad{y}
74762`\spad{Beta(n,{}m)} \undocumented
74857`\spad{Beta(x,{} y)} is the Euler beta function,{} \spad{B(x,{}y)},{} defined by \indented{2}{\spad{Beta(x,{}y) = integrate(t^(x-1)*(1-t)^(y-1),{} t=0..1)}.} This is related to \spad{Gamma(x)} by \indented{2}{\spad{Beta(x,{}y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}
74970`\spad{Beta(x,{} y)} is the Euler Beta function.
75059`\spad{Beta(x,{} y)} is the Euler beta function,{} \spad{B(x,{}y)},{} defined by \indented{2}{\spad{Beta(x,{}y) = integrate(t^(x-1)*(1-t)^(y-1),{} t=0..1)}.} This is related to \spad{Gamma(x)} by \indented{2}{\spad{Beta(x,{}y) = Gamma(x)*Gamma(y) / Gamma(x + y)}.}
75145`\spad{Beta(x,{} y)} is the Euler Beta function.
75207`\spad{Beta(x,{}y)} is \spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.
75262`\spad{BumInSepFFE(f)} is a local function,{} exported only because it has multiple conditional definitions.
75464`\spad{Chi(f)} denotes the hyperbolic cosine integral
75514`\spad{Chi(f)} denotes the hyperbolic cosine integral
75590`\spad{Chi(f)} denotes the hyperbolic cosine integral
75648`\spad{Chi(x)} returns the hyperbolic cosine integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{cosh(x) / x dx}.
75703`\spad{\spad{Ci}(f)} denotes the cosine integral
75752`\spad{\spad{Ci}(f)} denotes the cosine integral
75827`\spad{\spad{Ci}(f)} denotes the cosine integral
75884`\spad{\spad{Ci}(x)} returns the cosine integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{cos(x) / x dx}.
76004`\spad{D()} provides the operator corresponding to a derivation in the ring \spad{A}.
76158`\spad{D(v)} returns the operator corresponding to derivative with with respect to \spad{v} in \spad{R}.
76219`\spad{D(v)} returns operator corresponding to derivative with with respect to \spad{v} in \spad{R}.
76297`\spad{D(x)} returns the derivative of \spad{x}. This function is a simple differential operator where no variable needs to be specified.
76339`\spad{D(f)} returns the derivative of \spad{f}.
76793`\axiom{\spad{D}(\spad{U},{} [\spad{X1},{} ...,{} \spad{Xq}])} generates a derivative. \axiom{\spad{U}} must be in \axiom{DVar}; all other symbols in \axiom{IVar}.
76856`\axiom{\spad{D}(\spad{u},{} der)} generates a derivative.
76926`\spad{D(x,{} deriv)} differentiates \spad{x} extending the derivation deriv on \spad{R}.
76983`\spad{D(x,{}[s1,{}...sn])} computes successive partial derivatives,{} \spadignore{i.e.} \spad{D(...D(x,{} s1)...,{} sn)}.
77043`\spad{D(x,{} n)} returns the \spad{n}-th derivative of \spad{x}.
77104`\spad{D(f,{} n)} returns the \spad{n}-th derivative of \spad{f}.
77183`\spad{D(x,{}v)} computes the partial derivative of \spad{x} with respect to \spad{v}.
77615`\spad{D(x,{} deriv,{} n)} differentiate \spad{x} \spad{n} times using a derivation which extends \spad{deriv} on \spad{R}.
77691`\spad{D(x,{} [s1,{}...,{}sn],{} [n1,{}...,{}nn])} computes multiple partial derivatives,{} \spadignore{i.e.} \spad{D(...D(x,{} s1,{} n1)...,{} sn,{} nn)}.
77776`\spad{D(x,{} s,{} n)} computes multiple partial derivatives,{} \spadignore{i.e.} \spad{n}-th derivative of \spad{x} with respect to \spad{s}.
77849`\spad{Delta(s)} returns operator corresponding to derivative with with respect to \spad{s} in \spad{R}.
77974`\spad{DiffAction(k,{} l,{} g)} gives the coefficient of \spad{x^k} in \spad{z^l} \spad{g}(\spad{x}),{} where \spad{z*}(a+b*x+c*x^2+d*x^3+...) = (a*x+b*x^2+c*x^3+...),{} \spadignore{i.e.} multiplication with \spad{x}.
78074`\spad{DiffC} gives the coefficients \spad{c_}{\spad{k},{}\spad{k}} in the expansion <x^k> \spad{z} \spad{g}(\spad{x}) = sum_{\spad{i=0}}\spad{^k} \spad{c_}{\spad{k},{}\spad{i}} <x^i> \spad{g}(\spad{x}),{} where \spad{z} acts on \spad{g}(\spad{x}) by shifting. In fact,{} the result is [0,{}0,{}0,{}...]
78154`\spad{EQ(x,{}y)} returns the \axiomType{Switch} expression representing \spad{x = y}.
78520`\spad{\spad{Ei}(f)} denotes the exponential integral
78569`\spad{\spad{Ei}(f)} denotes the exponential integral
78644`\spad{\spad{Ei}(f)} denotes the exponential integral
78701`\spad{\spad{Ei}(x)} returns the exponential integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{exp(x)/x dx}.
78913`\spad{F2FG(a + sqrt(-1) b)} returns \spad{a + i b}.
78980`\spad{FG2F(a + i b)} returns \spad{a + sqrt(-1) b}.
79047`\spad{F(n,{}m)} \undocumented
79139`\spad{FormatArabic(n)} forms an Arabic numeral string from an integer \spad{n}.
79206`\spad{FormatRoman(n)} forms a Roman numeral string from an integer \spad{n}.
79272`\spad{Frobenius(x)} \undocumented
79395`\spad{Frobenius(a)} returns \spad{a ^ q} where \spad{q} is the \spad{size()\$F}.
79525`\spad{Frobenius(a,{}s)} returns \spad{a^(q^s)} where \spad{q} is the size()\$\spad{F}.
79608`\spad{GE(x,{}y)} returns the \axiomType{Switch} expression representing \spad{x>=y}.
79816`\spad{GF2FG(a + i b)} returns \spad{a + i b} viewed as a function with the \spad{i} pushed down into the coefficient domain.
79893`\spad{GT(x,{}y)} returns the \axiomType{Switch} expression representing \spad{x>y}.
80101`\spad{Gamma(f)} returns the formal Gamma function applied to \spad{f}
80159`\spad{Gamma(x)} is the Euler gamma function,{} \spad{Gamma(x)},{} defined by \indented{2}{\spad{Gamma(x) = integrate(t^(x-1)*exp(-t),{} t=0..\%infinity)}.}
80252`\spad{Gamma(x)} is the Euler Gamma function.
80327`\spad{Gamma(x)} is the Euler gamma function,{} \spad{Gamma(x)},{} defined by \indented{2}{\spad{Gamma(x) = integrate(t^(x-1)*exp(-t),{} t=0..\%infinity)}.}
80402`\spad{Gamma(x)} is the Euler Gamma function.
80459`\spad{Gamma(x)} is the Euler Gamma function.
80512`\spad{Gamma(a,{}x)} returns the incomplete Gamma function applied to a and \spad{x}
80572`\spad{Gamma(a,{}x)} is the incomplete Gamma function.
80628`\spad{GospersMethod(b,{} n,{} new)} returns a rational function \spad{rf(n)} such that \spad{a(n) * rf(n)} is the indefinite sum of \spad{a(n)} with respect to upward difference on \spad{n},{} \spadignore{i.e.} \spad{a(n+1) * rf(n+1) - a(n) * rf(n) = a(n)},{} where \spad{b(n) = a(n)/a(n-1)} is a rational function. Returns "failed" if no such rational function \spad{rf(n)} exists. Note: \spad{new} is a nullary fu
80628`--nction returning a new \spad{V} every time. The condition on \spad{a(n)} is that \spad{a(n)/a(n-1)} is a rational function of \spad{n}.
80720`Fake solver over algebraic numbers.
81045`implements HP_solve over integers.
81356`implements HP_solve over \$\spad{Z_p}\$ with machine-sized \$\spad{p}\$.
81665`Fake solver over polynomials having algebraic coefficients.
82044`implements HP_solve over integer polynomials.
82398`\spad{HP_solve(l,{} degs,{} kind,{} sigma,{} gen check)} finds a \spad{sigma} basis with degree bounds given by \spad{degs}. It returns \spad{false} if there is no interpolating vector that is correct up to (
82398`--and including) \spad{z^}(\spad{sigma}-1) with \spad{sigma} \spad{>=} |degs+[1,{}1,{}...,{}1]| or if check rejects all solutions. It returns \spad{true} if it can not handle given problem.
82697`\spad{HankelMatrix([a_0,{} a_1,{} a_2,{} ...,{} a_{2n}])} builds the Hankel matrix [ \spad{a0},{} \spad{a1},{} ...,{} an ] [ \spad{a1},{} \spad{a2},{} ... ] [ ... ] [ an,{} ...,{} a2n]
82762`\axiom{Hausdorff(a,{}\spad{b},{}\spad{n})} returns log(exp(a)*exp(\spad{b})) truncated at order \axiom{\spad{n}}.
82862`\spad{HenselLift(pol,{}lfacts,{}prime,{}bound)} lifts \spad{lfacts},{} that are the factors of \spad{pol} mod \spad{prime},{} to factors of \spad{pol} mod prime^k > \spad{bound}. No recombining is done .
82982`\spad{HermiteIntegrate(f,{} D)} returns \spad{[g,{} h,{} s,{} p]} such that \spad{f = Dg + h + s + p},{} \spad{h} has a squarefree denominator normal \spad{w}.\spad{r}.\spad{t}. \spad{D},{} and all the squarefree factors of the denominator of \spad{s} are special \spad{w}.\spad{r}.\spad{t}. \spad{D}. Furthermore,{} \spad{h} and
82982`-- \spad{s} have no polynomial parts. \spad{D} is the derivation to use on \spadtype{UP}.
83160`\spad{HermiteIntegrate(f,{} ')} returns \spad{[g,{}h]} such that \spad{f = g' + h} and \spad{h} has a only simple finite normal poles.
83273`Constructs a \spad{I} combinator
83315`\spad{Is([e1,{}...,{}en],{} pat)} matches the pattern pat on the list of expressions \spad{[e1,{}...,{}en]} and returns the result.
83435`\spad{Is(expr,{} pat)} matches the pattern pat on the expression \spad{expr} and returns a list of matches \spad{[v1 = e1,{}...,{}vn = en]}; returns an empty list if either \spad{expr} is exactly equal to pat or if pat does not match \spad{expr}.
83596`\spad{Is(expr,{} pat)} matches the pattern pat on the expression \spad{expr} and returns a list of matches \spad{[v1 = e1,{}...,{}vn = en]}; returns an empty list if either \spad{expr} is exactly equal to pat or if pat does not match \spad{expr}.
83717`\spad{Is(expr,{} pat)} matches the pattern pat on the expression \spad{expr} and returns a match of the form \spad{[v1 = e1,{}...,{}vn = en]}; returns an empty match if \spad{expr} is exactly equal to pat. returns a \spadfun{failed} match if pat does not match \spad{expr}.
83880`\spad{JContinuedFraction(d,{} z)} returns the Cauchy transform as a continued fraction at \spad{z}.
84109`\spad{JContinuedFraction(d,{} z)} returns the Cauchy transform as a continued fraction at \spad{z}.
84286`Constructs a \spad{K} combinator
84544`\spad{LE(x,{}y)} returns the \axiomType{Switch} expression representing \spad{x<=y}.
84752`evaluation routine
84878`evaluation routine
85044`modular reduction
85191`\spad{LODO2FUN(op)} returns the function to pass to the series ODE solver in order to solve \spad{op y = 0}.
85260`\spad{LT(x,{}y)} returns the \axiomType{Switch} expression representing \spad{x<y}.
85468`\axiom{LUDecomp(A)} computes a LU decomposition of \axiom{A} using the algorithm of Crout. \axiom{LU} contains both triangular matrices; \axiom{Perm} is the permutation used for partial pivoting and \axiom{Pivots} yields the used pivots.
85580`\axiom{LUInverse(A)} computes the inverse of \axiom{A} using a LU decomposition.
85673`\axiom{LUSolve(LU,{} Perm,{} \spad{B})} uses a previously computed LU decomposition to solve a linear system with right hand side \axiom{\spad{B}}. \axiom{LU} and \axiom{Perm} are as given by \axiom{LUDecomp}.
85763`\spad{LagrangeInterpolation(l1,{}l2)} \undocumented
85859`\axiom{\spad{Lazard2}(\spad{F},{} \spad{x},{} \spad{y},{} \spad{n})} computes \axiom{(x/y)^(\spad{n}-1) * \spad{F}}
85949`\axiom{\spad{LazardQuotient2}(\spad{p},{}a,{}\spad{b},{}\spad{n})} returns \axiom{(a^(\spad{n}-1) * \spad{p}) exquo \spad{b^}(\spad{n}-1)} assuming that this quotient does not fail.
86069`\axiom{LazardQuotient(a,{}\spad{b},{}\spad{n})} returns \axiom{a^n exquo \spad{b^}(\spad{n}-1)} assuming that this quotient does not fail.
86185`\axiom{Lazard(\spad{x},{} \spad{y},{} \spad{n})} computes \axiom{x^n/y^(\spad{n}-1)}
86266`\axiom{LiePolyIfCan(\spad{p})} returns \axiom{\spad{p}} in Lyndon basis if \axiom{\spad{p}} is a Lie polynomial,{} otherwise \axiom{"failed"} is returned.
86372`\axiom{LiePolyIfCan(\spad{p})} return \axiom{\spad{p}} if \axiom{\spad{p}} is a Lie polynomial.
86470`\axiom{LiePoly(\spad{l})} returns the bracketed form of \axiom{\spad{l}} as a Lie polynomial.
86542`\spad{LowTriBddDenomInv(B,{}d)} returns \spad{M},{} where \spad{B} is a non-singular lower triangular matrix and \spad{d} is an element of \spad{R} such that \spad{M = d * inv(B)} has entries in \spad{R}.
86623`\axiom{LyndonBasis(\spad{lv})} returns the Lyndon basis of the nilpotent free Lie algebra.
86727`\axiom{LyndonCoordinates(\spad{g})} returns the exponential coordinates of \axiom{\spad{g}}.
86836`\axiom{\spad{LyndonWordsList1}(\spad{vl},{} \spad{n})} returns an array of lists of Lyndon words over the alphabet \axiom{\spad{vl}},{} up to order \axiom{\spad{n}}.
86948`\axiom{LyndonWordsList(\spad{vl},{} \spad{n})} returns the list of Lyndon words over the alphabet \axiom{\spad{vl}},{} up to order \axiom{\spad{n}}.
87038`\spad{MPtoMPT(p,{} s,{} ls,{} m)} coverts \spad{p} to packed represntation.
87148`\spad{NOT(x)} returns the \axiomType{Switch} expression representing \spad{\~~x}.
87268`\spad{NOT(x)} returns the \axiomType{Switch} expression representing \spad{\~~x}.
87302`\spad{Not(n)} returns the bit-by-bit logical {\em Not} of \spad{n}.
87345`\spad{Not(n)} returns the bit-by-bit logical {\em not} of the single integer \spad{n}.
87386`\spad{Nul()} gives the basis element 1 for the algebra generated by \spad{n} generators.
87441`\spad{OMParseError?(u)} tests whether \spad{u} is an OpenMath parsing error.
87501`\spad{OMReadError?(u)} tests whether \spad{u} is an OpenMath read error.
87560`\spad{OMUnknownCD?(u)} tests whether \spad{u} is an OpenMath unknown \spad{CD} error.
87619`\spad{OMUnknownSymbol?(u)} tests whether \spad{u} is an OpenMath unknown symbol error.
87682`\spad{OMbindTCP}
87753`\spad{OMcloseConn}
87809`\spad{OMclose(dev)} closes \axiom{\spad{dev}},{} flushing output if necessary.
87857`\spad{OMconnInDevice:}
87926`\spad{OMconnOutDevice:}
87996`\spad{OMconnectTCP}
88077`\spad{OMencodingBinary()} is the constant for the OpenMath binary encoding.
88132`\spad{OMencodingSGML()} is the constant for the deprecated OpenMath SGML encoding.
88185`\spad{OMencodingUnknown()} is the constant for unknown encoding types. If this is used on an input device,{} the encoding will be autodetected. It is invalid to use it on an output device.
88241`\spad{OMencodingXML()} is the constant for the OpenMath \spad{XML} encoding.
88293`\spad{OMgetApp(dev)} reads a begin application token from \axiom{\spad{dev}}.
88342`\spad{OMgetAtp(dev)} reads a begin attribute pair token from \axiom{\spad{dev}}.
88391`\spad{OMgetAttr(dev)} reads a begin attribute token from \axiom{\spad{dev}}.
88441`\spad{OMgetBVar(dev)} reads a begin bound variable list token from \axiom{\spad{dev}}.
88491`\spad{OMgetBind(dev)} reads a begin binder token from \axiom{\spad{dev}}.
88541`\spad{OMgetEndApp(dev)} reads an end application token from \axiom{\spad{dev}}.
88593`\spad{OMgetEndAtp(dev)} reads an end attribute pair token from \axiom{\spad{dev}}.
88645`\spad{OMgetEndAttr(dev)} reads an end attribute token from \axiom{\spad{dev}}.
88698`\spad{OMgetEndBVar(dev)} reads an end bound variable list token from \axiom{\spad{dev}}.
88751`\spad{OMgetEndBind(dev)} reads an end binder token from \axiom{\spad{dev}}.
88804`\spad{OMgetEndError(dev)} reads an end error token from \axiom{\spad{dev}}.
88858`\spad{OMgetEndObject(dev)} reads an end object token from \axiom{\spad{dev}}.
88913`\spad{OMgetError(dev)} reads a begin error token from \axiom{\spad{dev}}.
88964`\spad{OMgetFloat(dev)} reads a float from \axiom{\spad{dev}}.
89022`\spad{OMgetInteger(dev)} reads an integer from \axiom{\spad{dev}}.
89078`\spad{OMgetObject(dev)} reads a begin object token from \axiom{\spad{dev}}.
89130`\spad{OMgetString(dev)} reads a string from \axiom{\spad{dev}}.
89184`\spad{OMgetSymbol(dev)} reads a symbol from \axiom{\spad{dev}}.
89261`\spad{OMgetType(dev)} returns the type of the next object on \axiom{\spad{dev}}.
89313`\spad{OMgetVariable(dev)} reads a variable from \axiom{\spad{dev}}.
89369`\spad{OMlistCDs()} lists all the \spad{CDs} supported by AXIOM.
89426`\spad{OMlistSymbols(cd)} lists all the symbols in \axiom{\spad{cd}}.
89493`\spad{OMmakeConn}
89557`\spad{OMopenFile(f,{}mode,{}enc)} opens file \axiom{\spad{f}} for reading or writing OpenMath objects (depending on \axiom{\spad{mode}} which can be \spad{"r"},{} \spad{"w"} or "a" for read,{} write and append respectively),{} in the encoding \axiom{\spad{enc}}.
89634`\spad{OMopenString(s,{}mode)} opens the string \axiom{\spad{s}} for reading or writing OpenMath objects in encoding \axiom{enc}.
89706`\spad{OMputApp(dev)} writes a begin application token to \axiom{\spad{dev}}.
89755`\spad{OMputAtp(dev)} writes a begin attribute pair token to \axiom{\spad{dev}}.
89804`\spad{OMputAttr(dev)} writes a begin attribute token to \axiom{\spad{dev}}.
89854`\spad{OMputBVar(dev)} writes a begin bound variable list token to \axiom{\spad{dev}}.
89904`\spad{OMputBind(dev)} writes a begin binder token to \axiom{\spad{dev}}.
89954`\spad{OMputEndApp(dev)} writes an end application token to \axiom{\spad{dev}}.
90006`\spad{OMputEndAtp(dev)} writes an end attribute pair token to \axiom{\spad{dev}}.
90058`\spad{OMputEndAttr(dev)} writes an end attribute token to \axiom{\spad{dev}}.
90111`\spad{OMputEndBVar(dev)} writes an end bound variable list token to \axiom{\spad{dev}}.
90164`\spad{OMputEndBind(dev)} writes an end binder token to \axiom{\spad{dev}}.
90217`\spad{OMputEndError(dev)} writes an end error token to \axiom{\spad{dev}}.
90271`\spad{OMputEndObject(dev)} writes an end object token to \axiom{\spad{dev}}.
90326`\spad{OMputError(dev)} writes a begin error token to \axiom{\spad{dev}}.
90377`\spad{OMputFloat(dev,{}i)} writes the float \axiom{\spad{i}} to \axiom{\spad{dev}}.
90440`\spad{OMputInteger(dev,{}i)} writes the integer \axiom{\spad{i}} to \axiom{\spad{dev}}.
90501`\spad{OMputObject(dev)} writes a begin object token to \axiom{\spad{dev}}.
90553`\spad{OMputString(dev,{}i)} writes the string \axiom{\spad{i}} to \axiom{\spad{dev}}.
90612`\spad{OMputSymbol(dev,{}cd,{}s)} writes the symbol \axiom{\spad{s}} from \spad{CD} \axiom{\spad{cd}} to \axiom{\spad{dev}}.
90678`\spad{OMputVariable(dev,{}i)} writes the variable \axiom{\spad{i}} to \axiom{\spad{dev}}.
90739`\spad{OMreadFile(f)} reads an OpenMath object from \axiom{\spad{f}} and passes it to AXIOM.
90794`\spad{OMreadStr(f)} reads an OpenMath object from \axiom{\spad{f}} and passes it to AXIOM.
90848`\spad{OMread(dev)} reads an OpenMath object from \axiom{\spad{dev}} and passes it to AXIOM.
90907`\spad{OMreceive(c)} reads an OpenMath object from connection \axiom{\spad{c}} and returns the appropriate AXIOM object.
90979`\spad{OMsend(c,{}u)} attempts to output \axiom{\spad{u}} on \aciom{\spad{c}} in OpenMath.
91053`\spad{OMserve(portnum,{}timeout)} puts AXIOM into server mode on port number \axiom{\spad{portnum}}. The parameter \axiom{\spad{timeout}} specifies the \spad{timeout} period for the connection.
91133`\spad{OMsetEncoding(dev,{}enc)} sets the encoding used for reading or writing OpenMath objects to or from \axiom{\spad{dev}} to \axiom{\spad{enc}}.
91204`\spad{OMsupportsCD?(cd)} returns \spad{true} if AXIOM supports \axiom{\spad{cd}},{} \spad{false} otherwise.
91266`\spad{OMsupportsSymbol?(s,{}cd)} returns \spad{true} if AXIOM supports symbol \axiom{\spad{s}} from \spad{CD} \axiom{\spad{cd}},{} \spad{false} otherwise.
91339`\spad{OMunhandledSymbol(s,{}cd)} raises an error if AXIOM reads a symbol which it is unable to handle. Note that this is different from an unexpected symbol.
91474`\spad{OMwrite(u)} returns the OpenMath \spad{XML} encoding of \axiom{\spad{u}} as a complete OpenMath object.
91669`\spad{OMwrite(dev,{} u)} writes the OpenMath form of \axiom{\spad{u}} to the OpenMath device \axiom{\spad{dev}} as a complete OpenMath object.
91726`\spad{OMwrite(u,{} true)} returns the OpenMath \spad{XML} encoding of \axiom{\spad{u}} as a complete OpenMath object; OMwrite(\spad{u},{} \spad{false}) returns the OpenMath \spad{XML} encoding of \axiom{\spad{u}} as an OpenMath fragment.
91864`\spad{OMwrite(dev,{} u,{} true)} writes the OpenMath form of \axiom{\spad{u}} to the OpenMath device \axiom{\spad{dev}} as a complete OpenMath object; OMwrite(\spad{dev},{} \spad{u},{} \spad{false}) writes the object as an OpenMath fragment.
91929`\spad{OR(x,{}y)} returns the \axiomType{Switch} expression representing \spad{x or y}.
92137`\spad{Or(n,{}m)} returns the bit-by-bit logical {\em Or} of \spad{n} and \spad{m}.
92182`\spad{Or(n,{}m)} returns the bit-by-bit logical {\em or} of the single integers \spad{n} and \spad{m}.
92440`\axiom{\spad{P}(ind)} generates the derivative of the only dependent variable \spad{wrt} the index \axiom{ind}.
92774`\axiom{\spad{P}(\spad{i})} generates the \axiom{\spad{i}}\spad{-}th derivative of the only dependent variable \spad{wrt} the only independent variable.
93162`\axiom{\spad{P}(\spad{i},{} ind)} generates the derivative of the \axiom{\spad{i}}\spad{-}th dependent variable \spad{wrt} the index \axiom{ind}. Whether \axiom{ind} is interpreted as multi-index or as repeated index depends on the chosen notation.
93576`\axiom{\spad{P}(\spad{i},{} \spad{j})} generates the \axiom{\spad{j}}\spad{-}th derivative of the \axiom{\spad{i}}\spad{-}th independent variable \spad{wrt} the only independent variable.
93733`\spad{ParCondList(c,{}r)} computes a list of subdeterminants of each rank \spad{>=} \spad{r} of the matrix \spad{c} and returns a groebner basis for the ideal they generate
93949`\spad{ParCond(m,{}k)} returns the list of all \spad{k} by \spad{k} subdeterminants in the matrix \spad{m}
94159`\spad{Pfaffian(m)} returns the Pfaffian of the matrix \spad{m}. Error: if the matrix is not antisymmetric.
94238`\spad{Pfaffian(m)} returns the Pfaffian of the matrix \spad{m}. Error: if the matrix is not antisymmetric.
94328`\axiom{\spad{Pm}(\spad{i},{} ind)} is like \axiom{\spad{P}(\spad{i},{} ind)} but \axiom{ind} is always a multi-index.
94410`\spad{Pnan?(p)} returns \spad{true} if any of its elements are not valid DoubleFloat values. Examples of invalid floating point numbers are when we have divided a given element by zero or taken sqrt of negative number at some stage? Note: we can still represent a point at infinity and we can represent points using complex numbers,{} but we do this using projective or conformal space or by using the appropriate instance of SPointCategory,{} not by using i
94410`--nvalid floating point numbers.
94459`\spad{PollardSmallFactor(n)} returns a factor of \spad{n} or "failed" if no one is found
94623`\axiom{\spad{Pr}(\spad{i},{} ind)} is like \axiom{\spad{P}(\spad{i},{} ind)} but \axiom{ind} is always a repeated index.
94702`\spad{R1_to_R2_coercion should} be local but conditional
94773`\spad{RF2UTS(f)} converts \spad{f} to a Taylor series.
94859`\spad{ReduceOrder(op,{} s)} returns \spad{op1} such that for any solution \spad{z} of \spad{op1 z = 0},{} \spad{y = s \int z} is a solution of \spad{op y = 0}. \spad{s} must satisfy \spad{op s = 0}.
94916`\spad{ReduceOrder(op,{} [f1,{}...,{}fk])} returns \spad{[op1,{}[g1,{}...,{}gk]]} such that for any solution \spad{z} of \spad{op1 z = 0},{} \spad{y = gk \int(g_{k-1} \int(... \int(g1 \int z)...)} is a solution of \spad{op y = 0}. Each \spad{\spad{fi}} must satisfy \spad{op \spad{fi} = 0}.
95073`\spad{RemainderList(p)} returns the regular part of \spad{p} as a list of terms.
95258`\axiom{RittWuCompare(a,{}\spad{b})} returns \axiom{"failed"} if \axiom{a} and \axiom{\spad{b}} have same rank \spad{w}.\spad{r}.\spad{t}. Ritt and Wu Wen Tsun ordering using the refinement of Lazard,{} otherwise returns \axiom{infRittWu?(a,{}\spad{b})}.
95354`\spad{SEGMENT(x)} creates the prefix form: \spad{x..}.
95396`\spad{l..} produces a half open segment,{} that is,{} one with no upper bound.
95446`\spad{SEGMENT(x,{}y)} creates the infix form: \spad{x..y}.
95491`\spad{l..h} creates a segment with \spad{l} and \spad{h} as the endpoints.
95542`\spad{SFunction(\spad{li})} is the \spad{S}-function of the partition \spad{\spad{li}} \indented{1}{expressed in terms of power sum symmetric functions.}
95638`\spad{STransform(x)} returns the Taylor expansion the \spad{S}-transform \spad{S(z)} of the distribution \spad{x} if the mean is nonzero.
95724`\spad{STransform(x)} returns the Puiseux expansion of the \spad{S}-transform \spad{S(z)} if the mean is zero.
95810`\spad{STransform(x)} returns the Puiseux expansion of the \spad{S}-transform \spad{S(z)}.
95895`Constructs a \spad{S} combinator
95937`\spad{ScanArabic(s)} forms an integer from an Arabic numeral string \spad{s}.
96002`\spad{ScanFloatIgnoreSpacesIfCan(s)} tries to form a floating point number from the string \spad{s} ignoring any spaces.
96089`\spad{ScanFloatIgnoreSpaces(s)} forms a floating point number from the string \spad{s} ignoring any spaces. Error is generated if the string is not recognised as a floating point number.
96155`\spad{ScanRoman(s)} forms an integer from a Roman numeral string \spad{s}.
96219`\spad{Shi(f)} denotes the hyperbolic sine integral
96269`\spad{Shi(f)} denotes the hyperbolic sine integral
96345`\spad{Shi(f)} denotes the hyperbolic sine integral
96403`\spad{Shi(x)} returns the hyperbolic sine integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{sinh(x) / x dx}.
96458`\spad{ShiftAction(k,{} l,{} g)} gives the coefficient of \spad{x^k} in \spad{z^l} \spad{g}(\spad{x}),{} where \spad{z*(a+b*x+c*x^2+d*x^3+...) = (b*x+2*c*x^2+3*d*x^3+...)}. In terms of sequences,{} z*u(\spad{n})=n*u(\spad{n}).
96559`\spad{ShiftC} gives the coefficients \spad{c_}{\spad{k},{}\spad{k}} in the expansion <x^k> \spad{z} \spad{g}(\spad{x}) = sum_{\spad{i=0}}\spad{^k} \spad{c_}{\spad{k},{}\spad{i}} <x^i> \spad{g}(\spad{x}),{} where \spad{z} acts on \spad{g}(\spad{x}) by shifting. In fact,{} the result is [0,{}1,{}2,{}...]
96640`\spad{\spad{Si}(f)} denotes the sine integral
96689`\spad{\spad{Si}(f)} denotes the sine integral
96764`\spad{\spad{Si}(f)} denotes the sine integral
96821`\spad{\spad{Si}(x)} returns the sine integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{sin(x) / x dx}.
96875`\spad{Somos returns} whether we allow only Somos-like operators,{} default being \spad{false}
96969`\spad{Somos(d)} specifies whether we want that the total degree of the differential operators is constant,{} and equal to \spad{d},{} or maxDerivative if \spad{true}. If \spad{true},{} maxDerivative must be set,{} too.
97038`\spad{SturmHabichtCoefficients(p1,{}p2)} computes the principal Sturm-Habicht coefficients of \spad{p1} and \spad{p2}
97165`\spad{SturmHabichtMultiple(p1,{}p2)} computes \spad{c_}{+}\spad{-c_}{-} where \spad{c_}{+} is the number of real roots of \spad{p1} with \spad{p2>0} and \spad{c_}{-} is the number of real roots of \spad{p1} with \spad{p2<0}. If \spad{p2=1} what you get is the number of real roots of \spad{p1}.
97304`\spad{SturmHabichtSequence(p1,{}p2)} computes the Sturm-Habicht sequence of \spad{p1} and \spad{p2}
97451`\spad{SturmHabicht(p1,{}p2)} computes \spad{c_}{+}\spad{-c_}{-} where \spad{c_}{+} is the number of real roots of \spad{p1} with \spad{p2>0} and \spad{c_}{-} is the number of real roots of \spad{p1} with \spad{p2<0}. If \spad{p2=1} what you get is the number of real roots of \spad{p1}.
97566`\spad{UP2UTS(p)} converts \spad{p} to a Taylor series.
97621`\spad{UP2ifCan(x)} should be local but conditional.
97823`\spad{UTS2UP(s,{} n)} converts the first \spad{n} terms of \spad{s} to a univariate polynomial.
98040`\axiom{\spad{U}()} generates the only dependent variable.
98317`\axiom{\spad{U}(\spad{i})} generates the \axiom{\spad{i}}\spad{-}th dependent variable.
98436`\spad{UnVectorise(v)} \undocumented
98494`\spad{UpTriBddDenomInv(B,{}d)} returns \spad{M},{} where \spad{B} is a non-singular upper triangular matrix and \spad{d} is an element of \spad{R} such that \spad{M = d * inv(B)} has entries in \spad{R}.
98574`evaluation routine
98693`evaluation routine
98852`modular reduction
99003`\spad{Vectorise(x)} \undocumented
99148`\axiom{\spad{X}()} generates the only independent variable.
99356`\axiom{\spad{X}(\spad{i})} generates the \axiom{\spad{i}}\spad{-}th independent variable.
99475`\spad{Y(f)} computes a fixed point of the function \spad{f}.
99563`\spad{Y(g,{}n)} computes a fixed point of the function \spad{g},{} where \spad{g} takes a list of \spad{n} streams and returns a list of \spad{n} streams.
99746`\spadignore{ \/ } returns the logical `join',{} \spadignore{e.g.} `or'.
99781`Implement regressive inner,{}meet product operator need to check precidence when used as an infix operator
99836`\spad{n} \spad{\/} \spad{m} returns the bit-by-bit logical {\em or} of the single integers \spad{n} and \spad{m}.
99879`\spad{a ^ b} is the formal exponential a^b.
99931`\spad{x ^ q} is \spad{x} raised to the rational power \spad{q}.
100024`\spad{x ^ n} computes the \spad{n}-th power of a square matrix. The power \spad{n} is assumed greater than 1.
100665`\spad{s ^ n} returns the product of \spad{s} by itself \spad{n} times.
101250`\spad{s ^ n} returns the product of \spad{s} by itself \spad{n} times.
101307`\spad{s ^ n} returns the product of \spad{s} by itself \spad{n} times.
102066`\spad{st1 ^ st2} computes the power of a power series \spad{st1} by another power series \spad{st2}.
102160`\spad{st1 ^ st2} computes the power of a power series \spad{st1} by another power series \spad{st2}.
102268`\spad{s ^ r} raises a Laurent series \spad{s} to a rational power \spad{r}
102385`\spad{z ^ r} raises a Puiseaux series \spad{z} to a rational power \spad{r}
102511`\spad{x^n} \undocumented{} See \axiomFunFrom{^}{DivisionRing}
102595`\spad{a ^ b} returns the input form corresponding to \spad{a ^ b}.
102678`\spad{a ^ b} returns the input form corresponding to \spad{a ^ b}.
102732`\spad{a ^ n} returns the pattern \spad{a ^ n}.
102787`\spad{x^k} constructs the distribution of the \spad{k}th power of the random variable with distribution \spad{X} by picking every \spad{k}-th moment.
102852`\spad{f ^ g} creates the equivalent infix form.
102891`\spad{a ^ b} returns the pattern \spad{a ^ b}.
102930`\spad{o1^o2} returns \spad{o1} to power \spad{o2},{} where power is inductively defined using succesive natural multiplication from the left
102971`\spad{f^n} is the function which is the \spad{n}-fold application \indented{1}{of \spad{f}.}
103042`\spad{f(x) ^ a} computes a power of a power series. When the coefficient ring is a field,{} we may raise a series to an exponent from the coefficient ring provided that the constant coefficient of the series is 1.
103124`\spad{x ^ y} is the rational exponentiation of \spad{x} by the power \spad{y}.
103183`\spad{x^n} returns \spad{x} raised to the integer power \spad{n}.
103229`\spad{x^n} returns \spad{x} raised to the integer power \spad{n}.
103268`\spad{m^n} computes an integral power of the matrix \spad{m}. Error: if matrix is not square or if the matrix is square but not invertible.
103339`\spad{m^n} computes an integral power of the matrix \spad{m}. Error: if the matrix is not invertible.
103421`\spad{op^n} \undocumented
103474`\spad{x ^ n} computes a non-negative integral power of the matrix \spad{x}. Error: if the matrix is not square.
103557`\spad{a^n} returns the \spad{n}\spad{-}th power of \spad{a},{} defined by repeated squaring.
103615`\spad{x^n} returns the repeated product of \spad{x} \spad{n} times,{} \spadignore{i.e.} exponentiation.
103666`\spad{I^n} computes the \spad{n}th power of the ideal \spad{I}.
103748`\spad{a^n} returns the \spad{n}\spad{-}th power of \spad{a},{} defined by repeated squaring.
103795`\spad{x^n} returns the repeated product of \spad{x} \spad{n} times,{} \spadignore{i.e.} exponentiation.
103846`\spad{x^y} returns \spad{x} to the power \spad{y}.
103901`\spad{x^y} returns \spad{\#(X^Y)} where \spad{X^Y} is defined \indented{1}{as \spad{\{g| g:Y->X\}}.}
103944`\spad{aCubic(f,{}g,{}h,{}j)} \undocumented
104010`\spad{aLinear(f,{}g)} \undocumented
104073`\spad{aQuadratic(f,{}g,{}h)} \undocumented
104141`\spad{aQuartic(f,{}g,{}h,{}i,{}k)} \undocumented
104211`\spad{abelianGroup([n1,{}...,{}nk])} constructs the abelian group that is the direct product of cyclic groups with order {\em \spad{ni}}.
104314`\spad{abs(f)} returns the absolute value operator applied to \spad{f}
104536`\spad{abs(x)} represents the Fortran intrinsic function ABS
104616`\spad{abs(o)} computes the absolute value of an octonion,{} equal to the square root of the \spadfunFrom{norm}{Octonion}.
104685`\spad{abs(q)} computes the absolute value of quaternion \spad{q} (sqrt of norm).
104756`\spad{abs(x)} returns the absolute value of \spad{x} = sqrt(norm(\spad{x})).
104825`\spad{abs(x)} returns the absolute value of \spad{x}.
104864`\spad{abs x} returns the absolute value of \spad{x}.
104908`\spad{abs(x)} returns the absolute value of \spad{x}.
104959`\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?
105044`\spad{absolutelyIrreducible?()} tests if the curve absolutely irreducible?
105126`\spad{acosIfCan(z)} returns acos(\spad{z}) if possible,{} and "failed" otherwise.
105207`\spad{acos(x)} applies the inverse cosine operator to \spad{x}
105417`\spad{acos(st)} computes arccosine of a power series \spad{st}.
105501`\spad{acos(st)} computes arccosine of a power series \spad{st}.
105599`\spad{acos(z)} returns the arc-cosine of Laurent series \spad{z}.
105686`\spad{acos(z)} returns the arc-cosine of a Puiseux series \spad{z}.
105782`\spad{acos(x)} represents the Fortran intrinsic function ACOS
105863`\spad{acos(x)} returns the arc-cosine of \spad{x}. When evaluated into some subset of the complex numbers,{} one branch cut for acos lies along the negative real axis to the left of \spad{-1} (inclusive),{} continuous with the upper half plane,{} the other along the positive real axis to the right of 1 (inclusive),{} continuous with the lower half plane.
105924`\spad{acoshIfCan(z)} returns acosh(\spad{z}) if possible,{} and "failed" otherwise.
106006`\spad{acosh(x)} applies the inverse hyperbolic cosine operator to \spad{x}
106220`\spad{acosh(st)} computes the inverse hyperbolic cosine of a power series \spad{st}.
106305`\spad{acosh(st)} computes the inverse hyperbolic cosine of a power series \spad{st}.
106404`\spad{acosh(z)} returns the inverse hyperbolic cosine of Laurent series \spad{z}.
106492`\spad{acosh(z)} returns the inverse hyperbolic cosine of a Puiseux series \spad{z}.
106589`\spad{acosh(x)} returns the hyperbolic arc-cosine of \spad{x}.
106648`\spad{acotIfCan(z)} returns acot(\spad{z}) if possible,{} and "failed" otherwise.
106729`\spad{acot(x)} applies the inverse cotangent operator to \spad{x}
106896`\spad{acot(st)} computes arccotangent of a power series \spad{st}.
106980`\spad{acot(st)} computes arccotangent of a power series \spad{st}.
107078`\spad{acot(z)} returns the arc-cotangent of Laurent series \spad{z}.
107165`\spad{acot(z)} returns the arc-cotangent of a Puiseux series \spad{z}.
107261`\spad{acot(x)} returns the arc-cotangent of \spad{x}.
107322`\spad{acothIfCan(z)} returns acoth(\spad{z}) if possible,{} and "failed" otherwise.
107404`\spad{acoth(x)} applies the inverse hyperbolic cotangent operator to \spad{x}
107574`\spad{acoth(st)} computes the inverse hyperbolic cotangent of a power series \spad{st}.
107659`\spad{acoth(st)} computes the inverse hyperbolic cotangent of a power series \spad{st}.
107758`\spad{acoth(z)} returns the inverse hyperbolic cotangent of Laurent series \spad{z}.
107846`\spad{acoth(z)} returns the inverse hyperbolic cotangent of a Puiseux series \spad{z}.
107943`\spad{acoth(x)} returns the hyperbolic arc-cotangent of \spad{x}.
108002`\spad{acscIfCan(z)} returns acsc(\spad{z}) if possible,{} and "failed" otherwise.
108083`\spad{acsc(x)} applies the inverse cosecant operator to \spad{x}
108252`\spad{acsc(st)} computes arccosecant of a power series \spad{st}.
108336`\spad{acsc(st)} computes arccosecant of a power series \spad{st}.
108434`\spad{acsc(z)} returns the arc-cosecant of Laurent series \spad{z}.
108521`\spad{acsc(z)} returns the arc-cosecant of a Puiseux series \spad{z}.
108617`\spad{acsc(x)} returns the arc-cosecant of \spad{x}.
108678`\spad{acschIfCan(z)} returns acsch(\spad{z}) if possible,{} and "failed" otherwise.
108760`\spad{acsch(x)} applies the inverse hyperbolic cosecant operator to \spad{x}
108930`\spad{acsch(st)} computes the inverse hyperbolic cosecant of a power series \spad{st}.
109015`\spad{acsch(st)} computes the inverse hyperbolic cosecant of a power series \spad{st}.
109114`\spad{acsch(z)} returns the inverse hyperbolic cosecant of Laurent series \spad{z}.
109202`\spad{acsch(z)} returns the inverse hyperbolic cosecant of a Puiseux series \spad{z}.
109299`\spad{acsch(x)} returns the hyperbolic arc-cosecant of \spad{x}.
109358`\spad{adaptive3D?()} determines whether plotting be done adaptively.
109403`\spad{adaptive?()} determines whether plotting be done adaptively
109444`\spad{adaptive()} determines whether plotting will be done adaptively.
109496`\spad{adaptive(true)} turns adaptive plotting on; \spad{adaptive(false)} turns adaptive plotting off.
109555`\spad{adaptive(b)} turns adaptive 2D plotting on if \spad{b} is \spad{true},{} or off if \spad{b} is \spad{false}. This option is expressed in the form \spad{adaptive == b}.
109603`\spad{adaptive(l,{}b)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{adaptive}. If the option does not exist the value,{} \spad{b} is returned.
109827`\spad{addArrow!(s,{} nm,{} n1,{} n2)} adds an arrow to the graph \spad{s},{} where: \spad{nm} is the name of the arrow \spad{n1} is the index of the start object \spad{n2} is the index of the end object
109920`\spad{addArrow!(s,{} nm,{} o1,{} o2)} adds an arrow to the graph \spad{s},{} where: \spad{nm} is the name of the arrow \spad{o1} is the start object \spad{o2} is the end object
110093`\spad{addArrow!(s,{}nm,{}n1,{}n2,{}mp)} adds an arrow to the graph \spad{s},{} where: \spad{nm} is the name of the arrow \spad{n1} is the index of the start object \spad{n2} is the index of the end object \spad{mp} is a map represented by this arrow
110211`\spad{addArrows2Din2D(n,{}ptFun,{}uSeg,{}vSeg,{}numPts)} is a convenience function which combines createArrows2Din2D with addChild!
110330`\spad{addBadValue(p,{} v)} adds \spad{v} to the list of "bad values" for \spad{p}; \spad{p} is not allowed to match any of its "bad values".
110406`\spad{addBadValue(p,{} v)} adds \spad{v} to the list of "bad values" for \spad{p}. Note: \spad{p} is not allowed to match any of its "bad values".
110456`\spad{addChild!(n,{}c)} adds the specified child node \spad{'c'} below node \spad{'n'}
110504`\spad{addMatchRestricted(var,{} expr,{} r,{} val)} adds the match (\spad{var},{} \spad{expr}) in \spad{r},{} provided that \spad{expr} satisfies the predicates attached to \spad{var},{} that \spad{var} is not matched to another expression already,{} and that either \spad{var} is an optional pattern variable or that \spad{expr} is not equal to val (usually an identity).
110585`\spad{addMatch(var,{} expr,{} r)} adds the match (\spad{var},{} \spad{expr}) in \spad{r},{} provided that \spad{expr} satisfies the predicates attached to \spad{var},{} and that \spad{var} is not matched to another expression already.
110654`\spad{addNode!(n,{}ptName,{}sc)} adds a scene node.
110724`\spad{addObject!(s,{} n)} adds object with coordinates \spad{n} to the graph \spad{s}.
110837`\spad{addObject!(s,{} n)} adds object \spad{n} to the graph \spad{s}. Use this version if you don\spad{'t} intend to create diagrams and therefore don\spad{'t} care about \spad{x},{}\spad{y} coordinates.
110888`\spad{addPlot1Din2D(n,{}f,{}tRange,{}numPts)} is a convenience function which combines createPlot1Din2D with addChild!
111002`\spad{addPlot1Din2D(n,{}f,{}tRange,{}numPts)} is a convenience function which combines createPlot1Din2D with addChild!
111107`\spad{addPlot1Din2Dparametric(n,{}ppc,{}tRange,{}numPts)} is a convenience function which combines createPlot1Din2Dparametric with addChild!
111253`\spad{addPlot1Din3Dparametric(n,{}pcfun,{}tRange,{}numPts)} is a convenience function which combines createPlot1Din3Dparametric with addChild!
111368`\spad{addPlot1Din3Dparametric(n,{}psc,{}tRange,{}numPts)} is a convenience function which combines createPlot1Din3Dparametric with addChild!
111514`\spad{addPlot2Din3D(n,{}f,{}uSeg,{}vSeg,{}numPts)} is a convenience function which combines createPlot2Din3D with addChild!
111661`\spad{addPlot2Din3Dparametric(n,{}f,{}uSeg,{}vSeg,{}numPts)} is a convenience function which combines createPlot2Din3Dparametric with addChild!
111809`\spad{addPlot2Din3Dparametric(n,{}f,{}uSeg,{}vSeg,{}numPts)} is a convenience function which combines createPlot2Din3Dparametric with addChild!
111985`\spad{addPoint2(s,{}p)} adds the 4 dimensional point,{} \spad{p},{} to the 3 dimensional subspace,{} \spad{s}. The subspace \spad{s} is returned with the additional point.
112041`\spad{addPointLast(s,{}s2,{}\spad{li},{}p)} adds the 4 dimensional point,{} \spad{p},{} to the 3 dimensional subspace,{} \spad{s}. \spad{s2} point to the end of the subspace \spad{s}. \spad{n} is the path in the \spad{s2} component. The subspace \spad{s} is returned with the additional point.
112122`\spad{addPoint(s,{}p)} adds the point,{} \spad{p},{} to the 3 dimensional subspace,{} \spad{s},{} and returns the new total number of points in \spad{s}.
112193`\spad{addPoint(s,{}\spad{li},{}i)} adds the 4 dimensional point indicated by the index location,{} \spad{i},{} to the 3 dimensional subspace,{} \spad{s}. The list of non negative integers,{} \spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the point is to be added. It\spad{'s} length should range from 0 to \spad{n - 1} where \spad{n} is the dimension o
112193`--f the subspace. If the length is \spad{n - 1},{} then a specific lowest level component is being referenced. If it is less than \spad{n - 1},{} then some higher level component (0 indicates top level component) is being referenced and a component of that level with the desired point is created. The subspace \spad{s} is returned with the additional point.
112283`\spad{addPoint(s,{}\spad{li},{}p)} adds the 4 dimensional point,{} \spad{p},{} to the 3 dimensional subspace,{} \spad{s}. The list of non negative integers,{} \spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the point is to be added. It\spad{'s} length should range from 0 to \spad{n - 1} where \spad{n} is the dimension of the subspace. If the length is \spad{n 
112283`--- 1},{} then a specific lowest level component is being referenced. If it is less than \spad{n - 1},{} then some higher level component (0 indicates top level component) is being referenced and a component of that level with the desired point is created. The subspace \spad{s} is returned with the additional point.
112363`\spad{addPoints!(n,{}ptName,{}pts)} adds a points node.
112428`\spad{addSceneArrow(n,{}st,{}en,{}offset,{}mode,{}size)} is a convenience function which combines createSceneArrow with addChild!
112511`\spad{addSceneArrow(n,{}st,{}en,{}offset,{}mode,{}size,{}fontScale)} is a convenience function which combines createSceneArrow with addChild!
112598`\spad{addSceneArrows(n,{}line,{}mode,{}size)} is a convenience function which combines createSceneArrows with addChild!
112680`\spad{addSceneBox(n,{}size)} is a convenience function which combines createSceneBox with addChild!
112737`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
112807`\spad{addSceneClip(n,{}bb)} is a convenience function which combines createSceneClip with addChild!
112867`\spad{addSceneDef(n,{}nam,{}nde)} is a convenience function which combines createSceneDef with addChild!
112922`\spad{addSceneGraph(n,{}g,{}x,{}y,{}dispArrowName)} is a convenience function which combines createSceneGraph with addChild!
113037`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
113107`\spad{addSceneGrid(n,{}bb)} is a convenience function which combines createSceneGrid with addChild!
113167`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
113249`\spad{addSceneGrid(n,{}stepSize,{}bb)} is a convenience function which combines createSceneGrid with addChild!
113321`\spad{addSceneGroup(n)} is a convenience function which combines createSceneGroup with addChild!
113368`\spad{addSceneIFS(n,{}in1)} is a convenience function which combines createSceneIFS with addChild!
113426`\spad{addSceneIFS(n,{}inx,{}pts)} is a convenience function which combines createSceneIFS with addChild!
113511`\spad{addSceneLine(n,{}line)} is a convenience function which combines createSceneLine with addChild!
113566`\spad{addSceneLine(n,{}st,{}en,{}fontScale)} is a convenience function which combines createSceneLine with addChild!
113630`\spad{addSceneLines(n,{}line)} is a convenience function which combines createSceneLines with addChild!
113692`\spad{addSceneMaterial(n,{}mat)} is a convenience function which combines createSceneMaterial with addChild!
113825`\spad{addSceneMaterial(n,{}lineW,{}lineC,{}fillC)} is a convenience function which combines createSceneMaterial with addChild!
113901`\spad{addSceneNamedPoints(n,{}np)} is a convenience function which combines createSceneNamedPoints with addChild!
113975`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
114074`\spad{addScenePattern(n,{}ptype,{}step,{}bb)} is a convenience function which combines createScenePattern with addChild!
114163`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
114244`\spad{addSceneRuler(n,{}ptype,{}offset,{}bb)} is a convenience function which combines createSceneRuler with addChild!
114315`\spad{addSceneShape(n,{}shape)} is a convenience function which combines createSceneShape with addChild!
114416`\spad{addSceneShape(n,{}shape)} is a convenience function which combines createSceneShape with addChild!
114477`\spad{addSceneText(n,{}text)} is a convenience function which combines createSceneText with addChild!
114588`\spad{addSceneText(n,{}str,{}sz,{}pz)} is a convenience function which combines createSceneText with addChild!
114669`\spad{addSceneText(n,{}str,{}sz,{}pz)} is a convenience function which combines createSceneText with addChild!
114744`\spad{addSceneTransform(n,{}tran)} is a convenience function which combines createSceneTransform with addChild!
114810`\spad{addSceneUse(n,{}nam)} is a convenience function which combines createSceneUse with addChild!
114862`\spad{addWArrow!(s,{} name,{} n1,{} n2,{} w)} adds a weighted arrow to the graph \spad{s},{} where \spad{name} is \spad{name} of the arrow,{} \spad{w} is weight of the arrow,{} \spad{n1} is starting point of the arrow,{} \spad{n2} is endpoint of the arrow.
114962`\spad{addWarning s} is an internal function that adds \spad{s},{} together with an indication of the current testcase and the current testsuite to the list of warnings.
115020`extend reconstructor adding zeros at specified positions.
115096`\spad{addiag(x)} performs diagonal addition of a stream of streams. if \spad{x} = \spad{[[a<0,{}0>,{}a<0,{}1>,{}..],{}[a<1,{}0>,{}a<1,{}1>,{}..],{}[a<2,{}0>,{}a<2,{}1>,{}..],{}..]} and \spad{addiag(x) = [b<0,{}b<1>,{}...],{} then b<k> = sum(i+j=k,{}a<i,{}j>)}.
115180`\spad{additive?(a,{} n)} returns \spad{true} if the first \spad{n} coefficients of a are additive
115254`\spad{addmod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a+b mod p}.
115385`\spad{adjacencyMatrix(s)} returns an \spad{n} by \spad{n} matrix A,{} where \spad{n} is the number of vertices in the graph. If there is an edge from a vertex \spad{x} to a vertex \spad{y},{} then the element ax,{}\spad{y} is 1 (or in general the number of xy edges),{} otherwise it is 0. In computing,{} this matrix makes it easy to find subgraphs,{} and to reverse a directed graph.
115596`\spad{adjoint(p)} returns the adjoint of operator \spad{p}.
115662`\spad{adjoint(m)} returns the ajoint matrix of \spad{m} (\spadignore{i.e.} the matrix \spad{n} such that \spad{m*n} = determinant(\spad{m})*id) and the detrminant of \spad{m}.
115778`\spad{adjoint(a)} returns the adjoint operator of a.
115855`\spad{adjoint(op)} returns the adjoint of the operator \spad{op}.
115991`\spad{adjoint(op1,{} op2)} sets the adjoint of \spad{op1} to be \spad{op2}. \spad{op1} must be a basic operator
116067`\spad{airyAiPrime(x)} returns the derivative of Airy \spad{Ai} function applied to \spad{x}
116131`\spad{airyAiPrime(x)} is the derivative of the Airy function \spad{\spad{Ai}(x)}.
116190`\spad{airyAi(x)} returns the Airy \spad{Ai} function applied to \spad{x}
116249`\spad{airyAi(x)} is the Airy function \spad{\spad{Ai}(x)}. This function satisfies the differential equation: \indented{2}{\spad{\spad{Ai}''(x) - x * \spad{Ai}(x) = 0}.}
116343`\spad{airyAi(x)} is the Airy function \spad{\spad{Ai}(x)}. This function satisfies the differential equation: \indented{2}{\spad{\spad{Ai}''(x) - x * \spad{Ai}(x) = 0}.}
116419`\spad{airyAi(x)} is the Airy function \spad{\spad{Ai}(x)}.
116473`\spad{airyBiPrime(x)} returns the derivative of Airy \spad{Bi} function applied to \spad{x}
116537`airyBiPrime is the derivative of the Airy function \spad{\spad{Bi}(x)}.
116596`\spad{airyBi(x)} returns the Airy \spad{Bi} function applied to \spad{x}
116655`\spad{airyBi(x)} is the Airy function \spad{\spad{Bi}(x)}. This function satisfies the differential equation: \indented{2}{\spad{\spad{Bi}''(x) - x * \spad{Bi}(x) = 0}.}
116749`\spad{airyBi(x)} is the Airy function \spad{\spad{Bi}(x)}. This function satisfies the differential equation: \indented{2}{\spad{\spad{Bi}''(x) - x * \spad{Bi}(x) = 0}.}
116825`\spad{airyBi(x)} is the Airy function \spad{\spad{Bi}(x)}.
117968`\spad{algDepHP(list,{} options)} returns a specification for Hermite-Pade approximation looking for 
117968`--algebraic dependencies
123225`\spad{algDsolve(op,{} g)} returns \spad{["failed",{} []]} if the equation \spad{op y = g} has no solution in \spad{R}. Otherwise,{} it returns \spad{[f,{} [y1,{}...,{}ym]]} where \spad{f} is a particular rational solution and the \spad{y_i's} form a basis for the solutions in \spad{R} of the homogeneous equation.
123488`\spad{algSplitSimple(f,{} D)} returns \spad{[h,{}d,{}d',{}g]} such that \spad{f=h/d},{} \spad{h} is integral at all the normal places \spad{w}.\spad{r}.\spad{t}. \spad{D},{} \spad{d' = Dd},{} \spad{g = gcd(d,{} discriminant())} and \spad{D} is the derivation to use. \spad{f} must have at most simple finite poles.
123605`\spad{alg_reduce0(x,{} m,{} lv,{} z)} performs single reduction step.
123797`\spad{alg_reduce(x,{} lm,{} lv,{} lz)} reduces \spad{x} modulo elements of \spad{lm}.
124000`\spad{alg_trial_division(x,{} y,{} lm,{} lv,{} lz)} checks if \spad{x} is divisible by \spad{y} in algebraic extension defined by \spad{lm}. \spad{lz} is list of algebraic variables,{} \spad{lv} is list of independent (polynomial) variables. Other variables serve as parameters.
124342`\spad{algebraic?(a)} tests whether an element \spad{a} is algebraic with respect to the ground field \spad{F}.
124468`\axiom{algebraic?(\spad{v},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{v}} is the main variable of some polynomial in \axiom{\spad{ts}}.
124628`\spad{algebraicCoefficients?(p,{}ts)} returns \spad{true} iff every variable of \spad{p} which is not the main one of \spad{p} is algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}.
125019`\spad{algebraicGcd(x,{} y,{} lm,{} lp,{} v,{} la)} computes \spad{gcd} of \spad{x} and \spad{y} modulo polynomials in \spad{lm}. la is list of algebraic parameters,{} \spad{lp} is a list of transcendental parameters,{} \spad{v} is main variable.
125199`\axiom{algebraicOf(char)} is the external number
125337`\axiom{algebraicSort(\spad{lts})} sorts \axiom{\spad{lts}} \spad{w}.\spad{r}.\spad{t} \axiomOpFrom{supDimElseRittWu?}{QuasiComponentPackage}.
125420`\axiom{algebraicSort(\spad{lts})} sorts \axiom{\spad{lts}} \spad{w}.\spad{r}.\spad{t} \axiomOpFrom{supDimElseRittWu}{QuasiComponentPackage}.
125588`\axiom{algebraicVariables(\spad{ts})} returns the decreasingly sorted list of the main variables of the polynomials of \axiom{\spad{ts}}.
125666`\spad{algint(f,{} x,{} y,{} d)} returns the integral of \spad{f(x,{}y)dx} where \spad{y} is an algebraic function of \spad{x}; \spad{d} is the derivation to use on \spad{k[x]}.
125822`\spad{algintegrate(f,{} d)} integrates \spad{f} with respect to the derivation \spad{d}.
125919`\spad{allDegrees returns} whether all possibilities of the degree vector should be tried,{} the default being \spad{false}.
125995`\spad{allDegrees(d)} specifies whether all possibilities of the degree vector - taking into account maxDegree - should be tried. This is mainly interesting for rational interpolation. This option is expressed in the form \spad{allDegrees == d}.
126046`\axiom{allIndices(A)} returns all indices used for enumerating the columns of the matrix \axiom{A}.
126110`\axiom{allRepeated(ind)} returns a list of all possible realizations of a given multi-index as repeated index.
126210`\axiom{allRepeated(ind)} returns a list of all possible realizations of a given multi-index as repeated index.
126796`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} naming each uniquely
126879`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} naming each uniquely
126952`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} naming each uniquely
127020`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} naming each uniquely
127119`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} naming each uniquely
127208`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} naming each uniquely
127292`\axiom{allRootsOf(pol)} creates all the roots of \axiom{pol} in the Real Closure,{} assumed in order.
127388`\axiom{alphaHilbert(\spad{hp})} computes the Cartan characters for the Hilbert polynomial \axiom{\spad{hp}}.
127511`\axiom{alpha(\spad{q},{} beta)} computes the Cartan characters for a differential equation of order \axiom{\spad{q}} and with characters \axiom{beta}.
127625`\spad{alphabetic?(c)} tests if \spad{c} is a letter,{} \spadignore{i.e.} one of a..\spad{z} or A..\spad{Z}.
127675`\spad{alphabetic()} returns the class of all characters for which \spadfunFrom{alphabetic?}{Character} is \spad{true}.
127722`\spad{alphanumeric?(c)} tests if \spad{c} is either a letter or number,{} \spadignore{i.e.} one of 0..9,{} a..\spad{z} or A..\spad{Z}.
127774`\spad{alphanumeric()} returns the class of all characters for which \spadfunFrom{alphanumeric?}{Character} is \spad{true}.
127823`\spad{alternatingGroup(\spad{li})} constructs the alternating group acting on the integers in the list {\em \spad{li}},{} generators are in general the {\em n-2}-cycle {\em (\spad{li}.3,{}...,{}\spad{li}.n)} and the 3-cycle {\em (\spad{li}.1,{}\spad{li}.2,{}\spad{li}.3)},{} if \spad{n} is odd and product of the 2-cycle {\em (\spad{li}.1,{}\spad{li}.2)} with {\em n-2}-cycle {\em (\spad{li}.3,{}...,{}\spad{
127823`--li}.n)} and the 3-cycle {\em (\spad{li}.1,{}\spad{li}.2,{}\spad{li}.3)},{} if \spad{n} is even. Note: duplicates in the list will be removed.
127922`\spad{alternatingGroup(n)} constructs the alternating group {\em An} acting on the integers 1,{}...,{}\spad{n},{} generators are in general the {\em n-2}-cycle {\em (3,{}...,{}n)} and the 3-cycle {\em (1,{}2,{}3)} if \spad{n} is odd and the product of the 2-cycle {\em (1,{}2)} with {\em n-2}-cycle {\em (3,{}...,{}n)} and the 3-cycle {\em (1,{}2,{}3)} if \spad{n} is even.
128023`\spad{alternating n} is the cycle index of the \indented{1}{alternating group of degree \spad{n}.}
128115`\spad{alternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0 = 2*associator(a,{}b,{}b)} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
128192`\spad{alternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0 = 2*associator(a,{}b,{}b)} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
128266`\axiom{analyseSymbol(symb)} computes the multiplicative variables of the symbol \axiom{symb}.
128446`\spad{f and g} creates the equivalent infix form.
128487`\spad{a and b} returns the logical {\em and} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
128530`\spad{a and b} returns the logical {\em and} of Boolean \spad{a} and \spad{b}.
128568`\spad{anfactor(p)} tries to factor \spad{p} over algebraic numbers,{} returning "failed" if it cannot
128752`\spad{angerJ(v,{} z)} is the Anger \spad{J} function
128813`\spad{angerJ(v,{} z)} is the Anger \spad{J} function.
128870`\axiom{ansatz()} returns a general ansatz for a symmetry generator.
128969`\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \spadignore{i.e.} \spad{(a*b)*c + a*(b*c) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra.
129050`\spad{antiAssociative?()} tests if multiplication in algebra is anti-associative,{} \spadignore{i.e.} \spad{(a*b)*c + a*(b*c) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra.
129128`\spad{antiCommutative?()} tests if \spad{a*a = 0} for all \spad{a} in the algebra. Note: this implies \spad{a*b + b*a = 0} for all \spad{a} and \spad{b}.
129209`\spad{antiCommutative?()} tests if \spad{a*a = 0} for all \spad{a} in the algebra. Note: this implies \spad{a*b + b*a = 0} for all \spad{a} and \spad{b}.
129342`\spad{antiCommutator(a,{}b)} returns \spad{a*b+b*a}.
129401`\spad{anticoord }\undocumented
129587`\spad{antipode(x)} computes the antipode of an element \spad{x}.
129785`\spad{antisymmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and antisymmetric (\spadignore{i.e.} \spad{m[i,{} j] = -m[j,{} i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
129873`\spad{antisymmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and antisymmetric (\spadignore{i.e.} \spad{m[i,{}j] = -m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
129957`\spad{antisymmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and antisymmetric (\spadignore{i.e.} \spad{m[i,{}j] = -m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
130060`\spad{antisymmetricTensors(la,{}n)} applies to each \spad{m}-by-\spad{m} square matrix in the list {\em la} the irreducible,{} polynomial representation of the general linear group {\em GLm} which corresponds to the partition {\em (1,{}1,{}...,{}1,{}0,{}0,{}...,{}0)} of \spad{n}. Error: if \spad{n} is greater than \spad{m}. Note: this corresponds to the symmetrization of
130060`-- the representation with the sign representation of the symmetric group {\em Sn}. The carrier spaces of the representation are the antisymmetric tensors of the \spad{n}-fold tensor product.
130194`\spad{antisymmetricTensors(a,{}n)} applies to the square matrix {\em a} the irreducible,{} polynomial representation of the general linear group {\em GLm},{} where \spad{m} is the number of rows of {\em a},{} which corresponds to the partition {\em (1,{}1,{}...,{}1,{}0,{}0,{}...,{}0)} of \spad{n}. Error: if \spad{n} is greater than \spad{m}. Note: this corresponds to the symmetrizat
130194`--ion of the representation with the sign representation of the symmetric group {\em Sn}. The carrier spaces of the representation are the antisymmetric tensors of the \spad{n}-fold tensor product.
130774`\spad{any?(p,{}u)} tests if \axiom{\spad{p}(\spad{x})} is \spad{true} for any element \spad{x} of \spad{u}. Note: for collections,{} \axiom{any?(\spad{p},{}\spad{u}) = reduce(or,{}map(\spad{f},{}\spad{u}),{}\spad{false},{}\spad{true})}. However,{} \axiom{any?(\spad{p},{}\spad{u})} returns as soon as it finds an element for which \axiom{\spad{p}} gives \spad{true}.
130867`\spad{any(type,{}object)} is a technical function for creating an \spad{object} of \spadtype{Any}. Arugment \spad{type} is a \spadgloss{LISP} form for the \spad{type} of \spad{object}.
130912`\spad{appendPoint(\spad{gi},{}pt)} appends the point \spad{pt} to the end of the list of points component for the graph,{} \spad{\spad{gi}},{} which is of the domain \spadtype{GraphImage}.
130979`\axiom{appendRow!(A,{} \spad{r})} appends the row \axiom{\spad{r}} at the end of the matrix \axiom{A}.
131080`\spad{append(u1,{}u2)} appends the elements of list \spad{u1} onto the front of list \spad{u2}. This new list and \spad{u2} will share some structure.
131235`\spad{applyQuote(foo,{} [x1,{}...,{}xn])} returns \spad{'foo(x1,{}...,{}xn)}.
131299`\spad{applyQuote(foo,{} x)} returns \spad{'foo(x)}.
131413`\spad{applyQuote(foo,{} x,{} y)} returns \spad{'foo(x,{}y)}.
131532`\spad{applyQuote(foo,{} x,{} y,{} z)} returns \spad{'foo(x,{}y,{}z)}.
131656`\spad{applyQuote(foo,{} x,{} y,{} z,{} t)} returns \spad{'foo(x,{}y,{}z,{}t)}.
131723`\spad{applyRules([r1,{}...,{}rn],{} expr)} applies the rules \spad{r1},{}...,{}\spad{rn} to \spad{f} an unlimited number of times,{} \spadignore{i.e.} until none of \spad{r1},{}...,{}\spad{rn} is applicable to the expression.
131804`\spad{applyRules([r1,{}...,{}rn],{} expr,{} n)} applies the rules \spad{r1},{}...,{}\spad{rn} to \spad{f} a most \spad{n} times.
131901`\spad{applyTaylor(g,{} f)} applies \spad{g} to \spad{f}
132155`\spad{apply '}function' represented by this graph to vertex index 'a'
132236`\spad{elt(mm,{} \spad{pi})} returns the product of the entries indexed by the integer partition \spad{pi} (as in partitionend moments)
132294`\spad{apply(m,{}a)} defines a left operation of \spad{n} by \spad{n} matrices where \spad{n} is the rank of the algebra in terms of matrix-vector multiplication,{} this is a substitute for a left module structure. Error: if shape of matrix doesn\spad{'t} fit.
132364`\spad{apply(p,{} d)} computes the distribution of the random variable \spad{p(X)}\$ where \spad{X} has distribution \spad{d}.
132462`\spad{apply(p,{} f,{} m)} returns \spad{p(m)} where the action is given by \spad{x m = f(m)}. \spad{f} must be an \spad{R}-pseudo linear map on \spad{M}.
132535`\spad{apply '}function' represented by this graph to vertex index 'a'
132640`\spad{apply(p,{} c,{} m)} returns \spad{p(m)} where the action is given by \spad{x m = c sigma(m) + delta(m)}.
132708`\spad{apply(p,{} c,{} m,{} sigma,{} delta)} returns \spad{p(m)} where the action is given by \spad{x m = c sigma(m) + delta(m)}.
132803`\spad{approxRoot(n,{}r)} returns an approximation \spad{x} to \spad{n^(1/r)} such that \spad{-1 < x - n^(1/r) < 1}
132873`\spad{approxSqrt(n)} returns an approximation \spad{x} to \spad{sqrt(n)} such that \spad{-1 < x - sqrt(n) < 1}. Compute an approximation \spad{s} to \spad{sqrt(n)} such that \indented{10}{\spad{-1 < s - sqrt(n) < 1}} A variable precision Newton iteration is used. The running time is \spad{O( log(n)^2 )}.
132921`\spad{approximants(x)} returns the stream of approximants of the continued fraction \spadvar{\spad{x}}. If the continued fraction is finite,{} then the stream will be infinite and periodic with period 1.
133143`\spad{approximate(x,{}n)} returns a rational number \spad{y} such that \spad{y = x (mod p^n)}.
133235`\spad{approximate(f)} returns a truncated power series with the series variable viewed as an element of the coefficient domain.
133405`\spad{approximate(x,{}n)} returns an integer \spad{y} such that \spad{y = x (mod p^n)} when \spad{n} is positive,{} and 0 otherwise.
133477`\axiom{approximate(\spad{n},{}\spad{p})} gives an approximation of \axiom{\spad{n}} that has precision \axiom{\spad{p}}
133546`\axiom{approximate(term,{}root,{}prec)} gives an approximation of \axiom{term} over \axiom{root} with precision \axiom{prec}
133655`\axiom{arbFunctions(\spad{q},{} \spad{j},{} \spad{cc})} uses the Cartan characters \axiom{\spad{cc}} to compute the number of arbitrary functions of differentiation order \axiom{\spad{j}} in the general solution of a differential equation of order \axiom{\spad{q}}.
133773`\spad{arcsineDistribution(a)} produces the arcsine distribution of variance \spad{a}.
133884`\spad{areEquivalent?(aG0,{}aG1)} calls {\em areEquivalent?(aG0,{}aG1,{}true,{}25)}. Note: the choice of 25 was rather arbitrary.
133996`\spad{areEquivalent?(aG0,{}aG1,{}numberOfTries)} calls {\em areEquivalent?(aG0,{}aG1,{}true,{}25)}. Note: the choice of 25 was rather arbitrary.
134116`\spad{areEquivalent?(aG0,{}aG1,{}randomelements,{}numberOfTries)} tests whether the two lists of matrices,{} all assumed of same square shape,{} can be simultaneously conjugated by a non-singular matrix. If these matrices represent the same group generators,{} the representations are equivalent. The algorithm tries {\em numberOfTries} times to create elements in the generated 
134116`--algebras in the same fashion. If their ranks differ,{} they are not equivalent. If an isomorphism is assumed,{} then the kernel of an element of the first algebra is mapped to the kernel of the corresponding element in the second algebra. Now consider the one-dimensional ones. If they generate the whole space (\spadignore{e.g.} irreducibility !) we use {\em standardBasisOfCyclicSubmodule} to create the only possible transition matrix. The method checks whether the matrix conjugates all corresp
134116`--onding matrices from {\em aGi}. The way to choose the singular matrices is as in {\em meatAxe}. If the two representations are equivalent,{} this routine returns the transformation matrix {\em TM} with {\em aG0.i * TM = TM * aG1.i} for all \spad{i}. If the representations are not equivalent,{} a small 0-matrix is returned. Note: the case with different sets of group generators cannot be handled.
134244`\spad{arg1(a,{}c)} selects its first argument.
134306`\spad{arg2(a,{}c)} selects its second argument.
134368`\spad{argscript(s,{} [a1,{}...,{}an])} returns \spad{s} arg-scripted by \spad{[a1,{}...,{}an]}.
134425`\spad{argumentList!(l)} declares that the argument list for the current subprogram in the global symbol table is \spad{l}.
134489`\spad{argumentList!(f,{}l)} declares that the argument list for subprogram \spad{f} in the global symbol table is \spad{l}.
134560`\spad{argumentList!(f,{}l,{}tab)} declares that the argument list for subprogram \spad{f} in symbol table \spad{tab} is \spad{l}.
134634`\spad{argumentListOf(f,{}tab)} returns the argument list of \spad{f}
134751`\spad{argument(x)} returns the argument of a given sin/cos expressions
134802`\spad{argument(op(a1,{}...,{}an))} returns \spad{[a1,{}...,{}an]}.
134849`\spad{argument(x)} returns the angle made by (0,{}1) and (0,{}\spad{x}).
134936`\spad{arity(op)} returns \spad{n} if \spad{op} is \spad{n}-ary,{} and "failed" if \spad{op} has arbitrary arity.
135011`\spad{aromberg(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax,{}nint)} uses the adaptive romberg method to numerically integrate function \spad{fn} over the closed interval from \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary fr
135011`--om \spad{nmin} to \spad{nmax},{} and where \spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
135185`\spad{arrayStack([x,{}y,{}...,{}z])} creates an array stack with first (top) element \spad{x},{} second element \spad{y},{}...,{}and last element \spad{z}.
135325`\spad{arrowName(s,{} a,{} b)} retrieves the name of arrow a-\spad{>b} if it does not exist then return \spad{"?"}
135507`index of all arrows leading to a given arrow
135693`\spad{arrowsFromNode(s,{} a)} gives list of all arrows leading to a given node
135877`arrowsToArrow:(\spad{s:}\%,{}a:NNI) returns index of all arrows leading from a given arrow
136059`\spad{arrowsToNode(s,{} a)} gives list of all arrows leading from a given node
136151`\spad{asecIfCan(z)} returns asec(\spad{z}) if possible,{} and "failed" otherwise.
136232`\spad{asec(x)} applies the inverse secant operator to \spad{x}
136401`\spad{asec(st)} computes arcsecant of a power series \spad{st}.
136485`\spad{asec(st)} computes arcsecant of a power series \spad{st}.
136583`\spad{asec(z)} returns the arc-secant of Laurent series \spad{z}.
136670`\spad{asec(z)} returns the arc-secant of a Puiseux series \spad{z}.
136766`\spad{asec(x)} returns the arc-secant of \spad{x}.
136827`\spad{asechIfCan(z)} returns asech(\spad{z}) if possible,{} and "failed" otherwise.
136909`\spad{asech(x)} applies the inverse hyperbolic secant operator to \spad{x}
137079`\spad{asech(st)} computes the inverse hyperbolic secant of a power series \spad{st}.
137164`\spad{asech(st)} computes the inverse hyperbolic secant of a power series \spad{st}.
137263`\spad{asech(z)} returns the inverse hyperbolic secant of Laurent series \spad{z}.
137351`\spad{asech(z)} returns the inverse hyperbolic secant of a Puiseux series \spad{z}.
137448`\spad{asech(x)} returns the hyperbolic arc-secant of \spad{x}.
137507`\spad{asimpson(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax,{}nint)} uses the adaptive simpson method to numerically integrate function \spad{fn} over the closed interval from \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary fr
137507`--om \spad{nmin} to \spad{nmax},{} and where \spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
137681`\spad{asinIfCan(z)} returns asin(\spad{z}) if possible,{} and "failed" otherwise.
137762`\spad{asin(x)} applies the inverse sine operator to \spad{x}
137972`\spad{asin(st)} computes arcsine of a power series \spad{st}.
138056`\spad{asin(st)} computes arcsine of a power series \spad{st}.
138154`\spad{asin(z)} returns the arc-sine of Laurent series \spad{z}.
138241`\spad{asin(z)} returns the arc-sine of a Puiseux series \spad{z}.
138337`\spad{asin(x)} represents the Fortran intrinsic function ASIN
138418`\spad{asin(x)} returns the arc-sine of \spad{x}. When evaluated into some subset of the complex numbers,{} one branch cut for asin lies along the negative real axis to the left of \spad{-1} (inclusive),{} continuous with the upper half plane,{} the other along the positive real axis to the right of 1 (inclusive),{} continuous with the lower half plane.
138479`\spad{asinhIfCan(z)} returns asinh(\spad{z}) if possible,{} and "failed" otherwise.
138561`\spad{asinh(x)} applies the inverse hyperbolic sine operator to \spad{x}
138775`\spad{asinh(st)} computes the inverse hyperbolic sine of a power series \spad{st}.
138860`\spad{asinh(st)} computes the inverse hyperbolic sine of a power series \spad{st}.
138959`\spad{asinh(z)} returns the inverse hyperbolic sine of Laurent series \spad{z}.
139047`\spad{asinh(z)} returns the inverse hyperbolic sine of a Puiseux series \spad{z}.
139144`\spad{asinh(x)} returns the hyperbolic arc-sine of \spad{x}.
139203`\spad{assert(op,{} s)} attaches property \spad{s} to \spad{op}. Argument \spad{op} is modified "in place",{} \spadignore{i.e.} no copy is made.
139254`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139327`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139391`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139457`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139530`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139601`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139674`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139755`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139827`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139901`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
139982`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140061`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140142`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140211`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140278`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140347`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140400`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140481`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140553`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140627`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140708`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140787`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140868`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
140937`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
141004`\spad{assign(x,{}y)} creates a representation of the FORTRAN expression x=y.
141073`\spad{assign(f,{}g)} creates a form for the assignment \spad{f := g}.
141117`\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \spad{y} to the \spad{l}\spad{'}th element of array \spad{x} (\spad{l} is a list of indices).
141216`\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \spad{y} to the \spad{l}\spad{'}th element of array \spad{x} (\spad{l} is a list of indices).
141306`\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \spad{y} to the \spad{l}\spad{'}th element of array \spad{x} (\spad{l} is a list of indices).
141398`\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \spad{y} to the \spad{l}\spad{'}th element of array \spad{x} (\spad{l} is a list of indices).
141497`\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \spad{y} to the \spad{l}\spad{'}th element of array \spad{x} (\spad{l} is a list of indices).
141594`\spad{assign(x,{}l,{}y)} creates a representation of the assignment of \spad{y} to the \spad{l}\spad{'}th element of array \spad{x} (\spad{l} is a list of indices).
141693`\spad{assoc(k,{}u)} returns the element \spad{x} in association list \spad{u} stored with key \spad{k},{} or "failed" if \spad{u} has no key \spad{k}.
141803`\spad{associatedEquations(op,{} m)} returns \spad{[w,{} eq,{} lw,{} lop]} such that \spad{eq(w) = 0} where \spad{w} is the given minor,{} and \spad{lw_i = lop_i(w)} for all the other minors.
141983`\spad{associatedSystem(op,{} m)} returns \spad{[M,{}w]} such that the \spad{m}-th associated equation system to \spad{L} is \spad{w' = M w}.
142213`\spad{associates?(x,{}y)} tests whether \spad{x} and \spad{y} are associates,{} \spadignore{i.e.} differ by a unit factor.
142267`\spad{associative?()} tests if multiplication in algebra is associative.
142344`\spad{associative?()} tests if multiplication in algebra is associative.
142418`\spad{associatorDependence()} looks for the associator identities,{} \spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \spad{associator(a,{}b,{}c)} which yield 0,{} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. The order of the permutations is \spad{123 231 312 132 321 213}.
142511`\spad{associatorDependence()} looks for the associator identities,{} \spadignore{i.e.} finds a basis of the solutions of the linear combinations of the six permutations of \spad{associator(a,{}b,{}c)} which yield 0,{} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. The order of the permutations is \spad{123 231 312 132 321 213}.
142675`\spad{associator(a,{}b,{}c)} returns \spad{(a*b)*c-a*(b*c)}.
142733`\spad{atType(f)} returns the input form corresponding to \spad{f@R}
142906`\spad{atanIfCan(z)} returns atan(\spad{z}) if possible,{} and "failed" otherwise.
142987`\spad{atan(x)} applies the inverse tangent operator to \spad{x}
143141`\spad{atan(st)} computes arctangent of a power series \spad{st}.
143225`\spad{atan(st)} computes arctangent of a power series \spad{st}.
143323`\spad{atan(z)} returns the arc-tangent of Laurent series \spad{z}.
143410`\spad{atan(z)} returns the arc-tangent of a Puiseux series \spad{z}.
143506`\spad{atan(x)} represents the Fortran intrinsic function ATAN
143587`\spad{atan(x)} returns the arc-tangent of \spad{x}. When evaluated into some subset of the complex numbers,{} one branch cut for atan lies along the positive imaginary axis above \%\spad{i} (exclusive),{} continuous with the left half plane,{} the other along the negative imaginary axis below -\%\spad{i} (exclusive) continuous with the right half plane. The domain does not contain \%\spad{i} and -\%\spad{i}.
143648`\spad{atan(x,{}y)} computes the arc tangent from \spad{x} with phase \spad{y}.
143691`\spad{atan(x,{}y)} computes the arc tangent from \spad{x} with phase \spad{y}.
143728`\spad{atanhIfCan(z)} returns atanh(\spad{z}) if possible,{} and "failed" otherwise.
143810`\spad{atanh(x)} applies the inverse hyperbolic tangent operator to \spad{x}
144024`\spad{atanh(st)} computes the inverse hyperbolic tangent of a power series \spad{st}.
144109`\spad{atanh(st)} computes the inverse hyperbolic tangent of a power series \spad{st}.
144208`\spad{atanh(z)} returns the inverse hyperbolic tangent of Laurent series \spad{z}.
144296`\spad{atanh(z)} returns the inverse hyperbolic tangent of a Puiseux series \spad{z}.
144393`\spad{atanh(x)} returns the hyperbolic arc-tangent of \spad{x}.
144452`returns \spad{true} if this is an atom,{} that is a leaf node otherwise return \spad{false} if this is a compound term
144493`returns \spad{true} if this is an atom,{} that is free or bound variable otherwise return \spad{false} if this is a compound or lambda definition
144538`returns \spad{true} if this is an atom,{} that is a leaf node otherwise return \spad{false} if this is a compound term
144591`\spad{atom?(s)} is \spad{true} if \spad{s} is a Lisp atom.
144667`\spad{atoms(r)} returns the list of matches that match atoms (elements of the lists).
144747`\spad{atrapezoidal(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax,{}nint)} uses the adaptive trapezoidal method to numerically integrate function \spad{fn} over the closed interval from \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence chec
144747`--king vary from \spad{nmin} to \spad{nmax},{} and where \spad{nint} is the number of independent intervals to apply the integrator. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
145241`\spad{augment(lp,{}lts)} returns the same as \spad{concat([augment(lp,{}ts) for ts in lts])}
145330`\spad{augment(lp,{}ts)} returns \spad{ts} if \spad{empty? lp},{} \spad{augment(p,{}ts)} if \spad{lp = [p]},{} otherwise \spad{augment(first lp,{} augment(rest lp,{} ts))}
145413`\spad{augment(p,{}lts)} returns the same as \spad{concat([augment(p,{}ts) for ts in lts])}
145496`\spad{augment(p,{}ts)} assumes that \spad{p} is a non-constant polynomial whose main variable is greater than any variable of \spad{ts}. This operation assumes also that if \spad{p} is added to \spad{ts} the resulting set,{} say \spad{ts+p},{} is a regular triangular set. Then it returns a split of Kalkbrener of \spad{ts+p}. This may not be \spad{ts+p} itself,{} if for instance \spad{ts+p} is required to be square-free.
145645`\axiom{autoReduce(\spad{sys})} tries to simplify a system by solving each equation for its leading term and substituting it into the other equations.
145811`\axiom{autoReduced?(\spad{ts},{}redOp?)} returns \spad{true} iff every element of \axiom{\spad{ts}} is reduced \spad{w}.\spad{r}.\spad{t} to every other in the sense of \axiom{redOp?}
145904`\spad{axesColorDefault()} returns the default color of the axes in a 2D viewport.
145967`\spad{axesColorDefault(p)} sets the default color of the axes in a 2D viewport to the palette \spad{p}.
146037`\spad{axes(v,{}s)} displays the axes of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} if \spad{s} is "on",{} or does not display the axes if \spad{s} is "off".
146099`\spad{axes(v,{}n,{}c)} displays the axes of the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} with the axes color set to the given palette color \spad{c}.
146176`\spad{axes(v,{}n,{}s)} displays the axes of the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} if \spad{s} is "on",{} or does not display the axes if \spad{s} is "off".
146252`\spad{backOldPos(genPos)} takes the result produced by \spadfunFrom{generalPosition}{PolynomialIdeal} and performs the inverse transformation,{} returning the original ideal \spad{backOldPos(generalPosition(I,{}listvar))} = \spad{I}.
146374`\spad{back(q)} returns the element at the back of the queue. The queue \spad{q} is unchanged by this operation. Error: if \spad{q} is empty.
146419`\spad{badNum(p)} \undocumented
146515`\spad{badNum(u)} \undocumented
146586`\spad{badValues(p)} returns the list of "bad values" for \spad{p}; \spad{p} is not allowed to match any of its "bad values".
146700`\spad{bag([x,{}y,{}...,{}z])} creates a bag with elements \spad{x},{}\spad{y},{}...,{}\spad{z}.
146748`\spad{balancedBinaryTree(n,{} s)} creates a balanced binary tree with \spad{n} nodes each with value \spad{s}.
146830`\spad{balancedFactorisation(a,{} [b1,{}...,{}bn])} returns a factorisation \spad{a = p1^e1 ... pm^em} such that each \spad{pi} is balanced with respect to \spad{[b1,{}...,{}bm]}.
146922`\spad{balancedFactorisation(a,{} b)} returns a factorisation \spad{a = p1^e1 ... pm^em} such that each \spad{\spad{pi}} is balanced with respect to \spad{b}.
147008`\spad{bandedHessian(v,{}xlist,{}k)} computes the hessian,{} the matrix of second partial derivatives,{} of the scalar field \spad{v},{} \spad{v} a function of the variables listed in \spad{xlist},{} \spad{k} is the semi-bandwidth,{} the number of nonzero subdiagonals,{} 2*k+1 being actual bandwidth. Stores the nonzero band in lower triangle in a matrix,{} dimensions \spad{k+1} by \#xlist,{} 
147008`--whose rows are the vectors formed by diagonal,{} subdiagonal,{} etc. of the real,{} full-matrix,{} hessian. (The notation conforms to LAPACK/NAG-\spad{F07} conventions.)
147121`\spad{bandedJacobian(vf,{}xlist,{}kl,{}ku)} computes the jacobian,{} the matrix of first partial derivatives,{} of the vector field \spad{vf},{} \spad{vf} a vector function of the variables listed in \spad{xlist},{} \spad{kl} is the number of nonzero subdiagonals,{} \spad{ku} is the number of nonzero superdiagonals,{} \spad{kl+ku+1} being actual bandwidth. Stores the n
147121`--onzero band in a matrix,{} dimensions \spad{kl+ku+1} by \#xlist. The upper triangle is in the top \spad{ku} rows,{} the diagonal is in row \spad{ku+1},{} the lower triangle in the last \spad{kl} rows. Entries in a column in the band store correspond to entries in same column of full store. (The notation conforms to LAPACK/NAG-\spad{F07} conventions.)
147257`\spad{baseRDE(f,{} g)} returns a \spad{[y,{} b]} such that \spad{y' + fy = g} if \spad{b = true},{} \spad{y} is a partial solution otherwise (no solution in that case). \spad{D} is the derivation to use.
147375`\spad{baseRDEsys(f,{} g1,{} g2)} returns fractions \spad{y_1.y_2} such that \spad{(y1',{} y2') + ((0,{} -f),{} (f,{} 0)) (y1,{}y2) = (g1,{}g2)} if \spad{y_1,{}y_2} exist,{} "failed" otherwise.
147511`\spad{base()} returns the base of the model
147563`\spad{base()} returns the base of the \spadfunFrom{exponent}{FloatingPointSystem}.
147622`\spad{base()} returns the base for the operations of \spad{IntegerNumberSystem}.
147668`\spad{base(b)} sets the base of the model to \spad{b}
147735`\spad{base(gp)} returns a base for the group {\em gp}.
147908`\axiom{basicSet(\spad{ps},{}redOp?)} returns \axiom{[\spad{bs},{}\spad{ts}]} where \axiom{concat(\spad{bs},{}\spad{ts})} is \axiom{\spad{ps}} and \axiom{\spad{bs}} is a basic set in Wu Wen Tsun sense of \axiom{\spad{ps}} \spad{w}.\spad{r}.\spad{t} the reduction-test \axiom{redOp?},{} if no non-zero constant polynomial lie in \axiom{\spad{ps}},{} otherwise \axiom{"failed"} is return
147908`--ed.
148164`\axiom{basicSet(\spad{ps},{}pred?,{}redOp?)} returns the same as \axiom{basicSet(\spad{qs},{}redOp?)} where \axiom{\spad{qs}} consists of the polynomials of \axiom{\spad{ps}} satisfying property \axiom{pred?}.
148300`\spad{basisOfCenter()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{commutator(x,{}a) = 0} and \spad{associator(x,{}a,{}b) = associator(a,{}x,{}b) = associator(a,{}b,{}x) = 0} for all \spad{a},{}\spad{b} in \spad{A}.
148360`\spad{basisOfCentroid()} returns a basis of the centroid,{} \spadignore{i.e.} the endomorphism ring of \spad{A} considered as \spad{(A,{}A)}-bimodule.
148430`\spad{basisOfCommutingElements()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = commutator(x,{}a)} for all \spad{a} in \spad{A}.
148501`\spad{basisOfLeftAnnihilator(a)} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = x*a}.
148571`\spad{basisOfLeftNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = associator(x,{}a,{}b)} for all \spad{a},{}\spad{b} in \spad{A}.
148636`\spad{basisOfLeftNucloid()} returns a basis of the space of endomorphisms of \spad{A} as right module. Note: left nucloid coincides with left nucleus if \spad{A} has a unit.
148709`\spad{basisOfMiddleNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = associator(a,{}x,{}b)} for all \spad{a},{}\spad{b} in \spad{A}.
148776`\spad{basisOfNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{associator(x,{}a,{}b) = associator(a,{}x,{}b) = associator(a,{}b,{}x) = 0} for all \spad{a},{}\spad{b} in \spad{A}.
148837`\spad{basisOfRightAnnihilator(a)} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = a*x}.
148908`\spad{basisOfRightNucleus()} returns a basis of the space of all \spad{x} of \spad{A} satisfying \spad{0 = associator(a,{}b,{}x)} for all \spad{a},{}\spad{b} in \spad{A}.
148974`\spad{basisOfRightNucloid()} returns a basis of the space of endomorphisms of \spad{A} as left module. Note: right nucloid coincides with right nucleus if \spad{A} has a unit.
149103`\spad{basis()} returns a fixed basis of \% as \spad{F}-vector space.
149171`\spad{basis()} returns the fixed \spad{R}-module basis.
149226`\spad{basis()} returns the fixed \spad{R}-module basis.
149292`\spad{basis LV} extracts a basis of the subspace spanned by the \spad{LV} list of vectors.
149354`\spad{basis(n)} \undocumented{} See \axiomFunFrom{basis}{FiniteAlgebraicExtensionField}
149447`\spad{basis((f1,{}...,{}fn))} returns the vector \spad{[f1,{}...,{}fn]}.
149509`\spad{basis((f1,{}...,{}fn))} = the vector \spad{[f1,{}...,{}fn]}.
149575`\spad{basis(n)} returns a fixed basis of a subfield of \% as \spad{F}-vector space.
149658`\spad{basis(va)} selects a basis from the elements of \spad{va}.
149743`\spad{bat1(llp)} unbumps a tableau \spad{llp}. Operation \spad{bat1} is the inverse of \spad{tab1}.
149818`\spad{bat(ls)} unbumps a tableau \spad{ls}
149889`\spad{beauzamyBound(p)} returns a bound on the larger coefficient of any factor of \spad{p}.
149973`\spad{belong?(op)} is \spad{true} if \spad{op} is an algebraic operator,{} that is,{} an \spad{n}th root or implicit algebraic operator.
150043`\spad{belong?(op)} is \spad{true} if \spad{op} is a combinatorial operator.
150117`\spad{belong?(p)} returns \spad{true} if operator \spad{p} is elementary
150188`\spad{belong?(op)} is \spad{true} if \spad{op} is a special function operator.
150266`\spad{belong?(op)} checks if \spad{op} is Liouvillian
150338`\spad{belong?(op)} tests if \% accepts \spad{op} as applicable to its elements.
150467`\spad{belong?(op)} tests if \% accepts \spad{op} as applicable to its elements.
150530`\spad{bernoulliB(n,{}r)} returns value of the \spad{n}th Bernoulli polynomial at \spad{r}.
150652`\spad{bernoulliDistribution01(a)} produces the distribution of a projection of trace a
150734`\spad{bernoulli(n)} returns the \spad{n}th Bernoulli polynomial \spad{B[n](x)}. Note: Bernoulli polynomials denoted \spad{B(n,{}x)} computed by solving the differential equation \spad{differentiate(B(n,{}x),{}x) = n B(n-1,{}x)} where \spad{B(0,{}x) = 1} and initial condition comes from \spad{B(n) = B(n,{}0)}.
150847`\spad{bernoulli(n)} returns the \spad{n}th Bernoulli number. this is \spad{B(n,{}0)},{} where \spad{B(n,{}x)} is the \spad{n}th Bernoulli polynomial.
150929`\spad{besselI(x,{}y)} returns the besseli function applied to \spad{x} and \spad{y}
150991`\spad{besselI(v,{}x)} is the modified Bessel function of the first kind,{} \spad{I(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}
151107`\spad{besselI(v,{}x)} is the modified Bessel function of the first kind,{} \spad{I(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.}
151196`\spad{besselI(v,{}z)} is the modified Bessel function of the first kind.
151254`\spad{besselJ(x,{}y)} returns the besselj function applied to \spad{x} and \spad{y}
151316`\spad{besselJ(v,{}x)} is the Bessel function of the first kind,{} \spad{J(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}
151432`\spad{besselJ(v,{}x)} is the Bessel function of the first kind,{} \spad{J(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.}
151521`\spad{besselJ(v,{}z)} is the Bessel function of the first kind.
151579`\spad{besselK(x,{}y)} returns the besselk function applied to \spad{x} and \spad{y}
151641`\spad{besselK(v,{}x)} is the modified Bessel function of the second kind,{} \spad{K(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implementation uses the relation \indented{2}{\spad{K(v,{}x) = \%pi/2*(I(-v,{}x) - I(v,{}x))/sin(v*\%\spad{pi})}} so is not valid for integer values of \spad{v}.
151757`\spad{besselK(v,{}x)} is the modified Bessel function of the second kind,{} \spad{K(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) - (x^2+v^2)w(x) = 0}.} Note: The default implementation uses the relation \indented{2}{\spad{K(v,{}x) = \%pi/2*(I(-v,{}x) - I(v,{}x))/sin(v*\%\spad{pi})}.} so is not valid for integer values of \spad{v}.
151846`\spad{besselK(v,{}z)} is the modified Bessel function of the second kind.
151904`\spad{besselY(x,{}y)} returns the bessely function applied to \spad{x} and \spad{y}
151966`\spad{besselY(v,{}x)} is the Bessel function of the second kind,{} \spad{Y(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implementation uses the relation \indented{2}{\spad{Y(v,{}x) = (J(v,{}x) cos(v*\%\spad{pi}) - J(-v,{}x))/sin(v*\%\spad{pi})}} so is not valid for integer values of \spad{v}.
152082`\spad{besselY(v,{}x)} is the Bessel function of the second kind,{} \spad{Y(v,{}x)}. This function satisfies the differential equation: \indented{2}{\spad{x^2 w''(x) + x w'(x) + (x^2-v^2)w(x) = 0}.} Note: The default implementation uses the relation \indented{2}{\spad{Y(v,{}x) = (J(v,{}x) cos(v*\%\spad{pi}) - J(-v,{}x))/sin(v*\%\spad{pi})}} so is not valid for integer values of \spad{v}.
152171`\spad{besselY(v,{}z)} is the Bessel function of the second kind.
152229`\spad{bezoutDiscriminant(p)} computes the discriminant of a polynomial \spad{p} by computing the determinant of a Bezout matrix.
152321`\spad{bezoutMatrix(p,{}q)} returns the Bezout matrix for the two polynomials \spad{p} and \spad{q}.
152388`\spad{bezoutResultant(p,{}q)} computes the resultant of the two polynomials \spad{p} and \spad{q} by computing the determinant of a Bezout matrix.
152480`\spad{biRank(x)} determines the number of linearly independent elements in \spad{x},{} \spad{x*bi},{} \spad{bi*x},{} \spad{bi*x*bj},{} \spad{i,{}j=1,{}...,{}n},{} where \spad{b=[b1,{}...,{}bn]} is a basis. Note: if \spad{A} has a unit,{} then \spadfunFrom{doubleRank}{AlgebraPackage},{} \spadfunFrom{weakBiRank}{AlgebraPackage} and \spadfunFrom{biRank}{AlgebraPackage} coincide.
152545`\spad{binaryFunction(s)} is a local function
152633`\spad{binarySearchTree(l)} \undocumented
152698`\spad{binaryTournament(ls)} creates a binary tournament with the elements of \spad{ls} as values at the nodes.
152763`\spad{binaryTree(v)} is an non-empty binary tree with value \spad{v},{} and left and right empty.
152810`\spad{binaryTree(l,{}v,{}r)} creates a binary tree with value \spad{v} with left subtree \spad{l} and right subtree \spad{r}.
152863`\spad{binary(r)} converts a rational number to a binary expansion.
152924`\spad{binary(op,{} [a1,{}...,{}an])} returns the input form corresponding to \spad{a1 op a2 op ... op an}.
152973`if this is a lambda term then replace string name in sub-nodes with De Bruijn index
153094`\spad{binomThmExpt(p,{}q,{}n)} returns \spad{(x+y)^n} by means of the binomial theorem trick.
153204`\spad{binomial(n,{} r)} returns the number of subsets of \spad{r} objects taken among \spad{n} objects,{} \spadignore{i.e.} \spad{n!/}(\spad{r!} * (\spad{n}-\spad{r})!).
153263`\spad{binomial(n,{}f)} \undocumented
153405`\spad{binomial(n,{}m)} creates a form for the binomial coefficient of \spad{n} and \spad{m}.
153451`\spad{binomial(n,{}r)} returns the binomial coefficient \spad{C(n,{}r) = n!/(r! (n-r)!)},{} where \spad{n >= r >= 0}. This is the number of combinations of \spad{n} objects taken \spad{r} at a time.
153515`\spad{binomial(n,{}r)} returns the \spad{(n,{}r)} binomial coefficient (often denoted in the literature by \spad{C(n,{}r)}). Note: \spad{C(n,{}r) = n!/(r!(n-r)!)} where \spad{n >= r >= 0}.
153580`\spad{bipolarCylindrical(a)} transforms from bipolar cylindrical coordinates to Cartesian coordinates: \spad{bipolarCylindrical(a)} is a function which will map the point \spad{(u,{}v,{}z)} to \spad{x = a*sinh(v)/(cosh(v)-cos(u))},{} \spad{y = a*sin(u)/(cosh(v)-cos(u))},{} \spad{z}.
153660`\spad{bipolar(a)} transforms from bipolar coordinates to Cartesian coordinates: \spad{bipolar(a)} is a function which will map the point \spad{(u,{}v)} to \spad{x = a*sinh(v)/(cosh(v)-cos(u))},{} \spad{y = a*sin(u)/(cosh(v)-cos(u))}.
153729`\spad{birth(x)} \undocumented
153825`\spad{bit?(n,{}i)} returns \spad{true} if and only if \spad{i}-th bit of \spad{n} is a 1.
153881`\spad{bitCoef(n,{}m)} returns the coefficient of 2^m in abs(\spad{n})
153942`\spad{bitLength(n)} returns the number of bits to represent abs(\spad{n})
153997`\spad{bitTruth(n,{}m)} returns \spad{true} if coefficient of 2^m in abs(\spad{n}) is 1
154059`\spad{bits()} returns ceiling\spad{'s} precision in bits.
154118`\spad{bits(n)} set the \spadfunFrom{precision}{FloatingPointSystem} to \spad{n} bits.
154218`\spad{bits(n,{}b)} creates bits with \spad{n} values of \spad{b}
154275`\axiom{bivariate?(\spad{p})} returns \spad{true} iff \axiom{\spad{p}} involves two and only two variables.
154352`\axiom{bivariatePolynomials(\spad{lp})} returns \axiom{\spad{bps},{}nbps} where \axiom{\spad{bps}} is a list of the bivariate polynomials,{} and \axiom{nbps} are the other ones.
154478`\spad{bivariateSLPEBR(lp,{}p,{}v)} implements the bivariate case of \spadfunFrom{solveLinearPolynomialEquationByRecursion}{PolynomialFactorizationByRecursionUnivariate}; its implementation depends on \spad{R}
154685`\spad{blankSeparate(l)} creates the form separating the elements of \spad{l} by blanks.
154739`\spad{block(l)} creates a representation of the statements in \spad{l} as a block.
154786`\spad{blue()} returns the position of the blue hue from total hues.
154818`\spad{bombieriNorm(p)} returns quadratic Bombieri\spad{'s} norm of \spad{p}.
154895`\spad{bombieriNorm(p,{}n)} returns the \spad{n}th Bombieri\spad{'s} norm of \spad{p}.
154988`\spad{booleanConvolution(x,{} y)} returns the boolean convolution of the distributions \spad{x} and \spad{y}
155057`\spad{booleanCumulantFromJacobi(n,{} aa,{} bb)} computes the \spad{n}th Boolean cumulant from the given Jacobiparameters \spad{aa} and \spad{bb}.
155150`\spad{booleanCumulant(x,{} n)} returns the \spad{n}-th boolean cumulant of the distribution \spad{x}
155228`\spad{booleanCumulants(x)} returns the sequence of boolean cumulants of the distribution \spad{x}.
155301`\spad{booleancumulant2moment(cc)} computes the sequence of moments from the sequence of boolean cumulants \spad{cc}
155382`\spad{bottom!(d)} returns the element at the bottom (back) of the dequeue.
155432`\spad{bottom(x)} returns the bottom entry of each column.
155502`\axiom{boundOfCauchy(\spad{p})} bounds the roots of \spad{p}
155624`\axiom{bound(\spad{n},{} \spad{m},{} \spad{q})} computes an upper bound for the number of prolongations needed to make the symbol of an equation of order \axiom{\spad{q}} with \axiom{\spad{n}} independent and \axiom{\spad{m}} dependent variables involutive.
155745`\spad{boundary(n,{}fontScale)} returns the boundary of a given node,{} so if this is called on the root node,{} it will return the boundary of the whole scene. There is a difficulty involving text nodes,{} that is they have a fixed font size. But we want to calculate the boundary in local coordinates,{} to do this conversion we need to know the total boundary size but that is what we are trying to calculate! To get round this contradictio
155745`--n the parameter \spad{fontScale} is required since fonts are rendered at a fixed size we need a scaling factor so that we can give the boundary in our coordinate system. \spad{fontScale} should be set to the full width of the diagram (not just the width of this node). If this is not known then choose an approximate value. This only affects text nodes,{} if this node,{} or its subnodes,{} do not contain a text node then its value is not significant.
155810`\spad{boxBoundary(c1,{}c2)} constructs a box boundary (\spad{n}-dimensional axis-aligned rectangle) with given parameters c1:PT holds minimum coordinates c2:PT holds maximum coordinates
155948`\spad{box(f)} encloses \spad{f} in a box.
155986`\spad{box([f1,{}...,{}fn])} returns \spad{(f1,{}...,{}fn)} with a 'box' around them that prevents the \spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \spad{atan(box [x,{} 2])} returns the formal kernel \spad{atan(x,{} 2)}.
156035`\spad{box(f)} returns \spad{f} with a 'box' around it that prevents \spad{f} from being evaluated when operators are applied to it. For example,{} \spad{log(1)} returns 0,{} but \spad{log(box 1)} returns the formal kernel log(1).
156124`\spad{brace()}\$\spad{D} (otherwise written {}\$\spad{D}) creates an empty set aggregate of type \spad{D}. This form is considered obsolete. Use \axiomFun{set} instead.
156220`\spad{brace(lf)} creates the form separating the elements of \spad{lf} by commas and encloses the result in curly brackets.
156266`\spad{brace(f)} creates the form enclosing \spad{f} in braces (curly brackets).
156306`\spad{brace([x,{}y,{}...,{}z])} creates a set aggregate containing items \spad{x},{}\spad{y},{}...,{}\spad{z}. This form is considered obsolete. Use \axiomFun{set} instead.
156356`\spad{bracket(lf)} creates the form separating the elements of \spad{lf} by commas and encloses the result in square brackets.
156404`\spad{bracket(f)} creates the form enclosing \spad{f} in square brackets.
156446`\axiom{branchIfCan(leq,{}\spad{ts},{}lineq,{}\spad{b1},{}\spad{b2},{}\spad{b3},{}\spad{b4},{}\spad{b5})} is an internal subroutine,{} exported only for developement.
156625`\axiom{branchIfCan(leq,{}\spad{ts},{}lineq,{}\spad{b1},{}\spad{b2},{}\spad{b3},{}\spad{b4},{}\spad{b5})} is an internal subroutine,{} exported only for developement.
156814`\spad{branchPoint?(a)} tests whether \spad{x = a} is a branch point.
156887`\spad{branchPoint?(p)} tests whether \spad{p(x) = 0} is a branch point.
156961`\spad{branchPointAtInfinity?()} tests if there is a branch point at infinity.
157043`\spad{bright(c)} sets the shade of a hue,{} \spad{c},{} above dim,{} but below pastel.
157084`\spad{bright(l)} sets the font property of a list of strings,{} \spad{l},{} to bold-face type.
157149`\spad{bright(s)} sets the font property of the string \spad{s} to bold-face type.
157208`\spad{brillhartIrreducible?(p)} returns \spad{true} if \spad{p} can be shown to be irreducible by a remark of Brillhart,{} \spad{false} is inconclusive.
157277`\spad{brillhartIrreducible?(p,{}noLinears)} returns \spad{true} if \spad{p} can be shown to be irreducible by a remark of Brillhart,{} \spad{false} else. If \spad{noLinears} is \spad{true},{} we are being told \spad{p} has no linear factors \spad{false} does not mean that \spad{p} is reducible.
157354`\spad{brillhartTrials()} returns the number of tests in \spadfun{brillhartIrreducible?}.
157426`\spad{brillhartTrials(n)} sets to \spad{n} the number of tests in \spadfun{brillhartIrreducible?} and returns the previous value.
157516`\spad{bringDown(f)} \undocumented
157588`\spad{bringDown(f,{}k)} \undocumented
157698`\spad{bsolve(c,{} w,{} r,{} s,{} m)} returns a list of regimes and solutions of the system \spad{c} \spad{z} = \spad{w} for ranks at least \spad{r}; depending on the mode \
157698`--spad{m} chosen,{} it writes the output to a file given by the string \spad{s}.
158033`\spad{btwFact(p,{}sqf,{}pd,{}r)} returns the factorization of \spad{p},{} the result is a Record such that \spad{contp=}content \spad{p},{} \spad{factors=}List of irreducible factors of \spad{p} with exponent. If \spad{sqf=true} the polynomial is assumed to be square free (\spadignore{i.e.} without repeated factors). \spad{pd} is the \sp
158033`--adtype{Set} of possible degrees. \spad{r} is a lower bound for the number of factors of \spad{p}. Please do not use this function in your code because its design may change.
158201`\spad{bubbleSort!(a)} \undocumented
158268`\spad{bubbleSort!(a,{}f)} \undocumented
158333`\spad{build(r,{}i,{}e)} \undocumented
158406`\spad{bumprow(cf,{}pr,{}r)} is an auxiliary function which bumps a row \spad{r} with a pair \spad{pr} using comparison function \spad{cf},{} and returns a record
158534`\spad{bumptab1(pr,{}t)} bumps a tableau \spad{t} with a pair \spad{pr} using comparison function \spadfun{<},{} returning a new tableau
158627`\spad{bumptab(cf,{}pr,{}t)} bumps a tableau \spad{t} with a pair \spad{pr} using comparison function \spad{cf},{} returning a new tableau
158734`\spad{cAcos(f)} computes the arccosine of the power series \spad{f}. For use when the coefficient ring is commutative.
158838`\spad{cAcosh(f)} computes the inverse hyperbolic cosine of the power series \spad{f}. For use when the coefficient ring is commutative.
158943`\spad{cAcot(f)} computes the arccotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
159047`\spad{cAcoth(f)} computes the inverse hyperbolic cotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
159152`\spad{cAcsc(f)} computes the arccosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
159256`\spad{cAcsch(f)} computes the inverse hyperbolic cosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
159361`\spad{cAsec(f)} computes the arcsecant of the power series \spad{f}. For use when the coefficient ring is commutative.
159465`\spad{cAsech(f)} computes the inverse hyperbolic secant of the power series \spad{f}. For use when the coefficient ring is commutative.
159570`\spad{cAsin(f)} computes the arcsine of the power series \spad{f}. For use when the coefficient ring is commutative.
159674`\spad{cAsinh(f)} computes the inverse hyperbolic sine of the power series \spad{f}. For use when the coefficient ring is commutative.
159779`\spad{cAtan(f)} computes the arctangent of the power series \spad{f}. For use when the coefficient ring is commutative.
159883`\spad{cAtanh(f)} computes the inverse hyperbolic tangent of the power series \spad{f}. For use when the coefficient ring is commutative.
159988`\spad{cCos(f)} computes the cosine of the power series \spad{f}. For use when the coefficient ring is commutative.
160091`\spad{cCosh(f)} computes the hyperbolic cosine of the power series \spad{f}. For use when the coefficient ring is commutative.
160195`\spad{cCot(f)} computes the cotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
160298`\spad{cCoth(f)} computes the hyperbolic cotangent of the power series \spad{f}. For use when the coefficient ring is commutative.
160402`\spad{cCsc(f)} computes the cosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
160505`\spad{cCsch(f)} computes the hyperbolic cosecant of the power series \spad{f}. For use when the coefficient ring is commutative.
160609`\spad{cExp(f)} computes the exponential of the power series \spad{f}. For use when the coefficient ring is commutative.
160712`\spad{cLog(f)} computes the logarithm of the power series \spad{f}. For use when the coefficient ring is commutative.
160815`\spad{cPower(f,{}r)} computes \spad{f^r},{} where \spad{f} has constant coefficient 1. For use when the coefficient ring is commutative.
160925`\spad{cRationalPower(f,{}r)} computes \spad{f^r}. For use when the coefficient ring is commutative.
161056`\spad{cSec(f)} computes the secant of the power series \spad{f}. For use when the coefficient ring is commutative.
161159`\spad{cSech(f)} computes the hyperbolic secant of the power series \spad{f}. For use when the coefficient ring is commutative.
161263`\spad{cSin(f)} computes the sine of the power series \spad{f}. For use when the coefficient ring is commutative.
161366`\spad{cSinh(f)} computes the hyperbolic sine of the power series \spad{f}. For use when the coefficient ring is commutative.
161470`\spad{cTan(f)} computes the tangent of the power series \spad{f}. For use when the coefficient ring is commutative.
161573`\spad{cTanh(f)} computes the hyperbolic tangent of the power series \spad{f}. For use when the coefficient ring is commutative.
161677`\spad{calcRanges(l)} \undocumented
161773`\spad{call(s)} creates a representation of a FORTRAN CALL statement
161817`\spad{canonicalBasis n} build the canononical basis of the vector space with \spad{n} coordinates.
161891`\spad{canonicalIfCan(x,{} m)} tries to divide \spad{x} by its leading coefficient modulo \spad{m}.
161988`\spad{cap(s1,{}s2)},{} introduced by Redfield,{} \indented{1}{is the scalar product of two cycle indices.}
162121`\spad{car((a1,{}...,{}an))} returns \spad{a1}.
162260`\spad{cardinality(u)} returns the number of elements of \spad{u}. Note: \axiom{cardinality(\spad{u}) = \#u}.
162333`\spad{carmichaelLambda(n)} returns exponent of the multiplicative group of integers modulo \spad{n},{} that is smallest positive integer \spad{k} such that \spad{i^k rem n = 1} for all \spad{i} relatively prime to \spad{n}.
162412`\spad{cartesian(pt)} returns the Cartesian coordinates of point \spad{pt}.
162478`\spad{cartesian(a,{}b)} returns a Cartesian product: the vertex set of \spad{G} \spad{o} \spad{H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}) and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) \indented{1}{are adjacent in \spad{G} \spad{o} \spad{H} if and only if either} \spad{u} = \spad{v} and u' is adjacent with \spad{v'} in \spad{H},{} or u' = \spad{v'} and \spad{u} is adjacent with \
162478`--spad{v} in \spad{G}.
162556`Cartesian product doubles the size of next list in each object,{} that is it produces two arrows out of every node
162644`Cartesian product: the vertex set of \spad{G} \spad{o} \spad{H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}) and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) \indented{1}{are adjacent in \spad{G} \spad{o} \spad{H} if and only if either} \spad{u} = \spad{v} and u' is adjacent with \spad{v'} in \spad{H},{} or u' = \spad{v'} and \spad{u} is adjacent with \spad{v} in \spad{G}.
162726`Cartesian product: the vertex set of \spad{G} \spad{o} \spad{H} is the Cartesian product \spad{V}(\spad{G}) \times \spad{V}(\spad{H}) and any two vertices (\spad{u},{}u') and (\spad{v},{}\spad{v'}) \indented{1}{are adjacent in \spad{G} \spad{o} \spad{H} if and only if either} \spad{u} = \spad{v} and u' is adjacent with \spad{v'} in \spad{H},{} or u' = \spad{v'} and \spad{u} is adjacent with \spad{v} in \spad{G}.
162808`\spad{catalan(n)} returns the \spad{n}-th Catalan number
162869`\spad{cdr((a1,{}...,{}an))} returns \spad{(a2,{}...,{}an)}.
162983`\spad{ceiling(x)} returns the smallest integral element above \spad{x}.
163064`\spad{ceiling x} returns the small integer \spad{>= x}.
163112`\spad{center(f)} centers form \spad{f} in total space.
163153`\spad{center(f)} returns the point about which the series \spad{f} is expanded.
163227`\spad{center(f,{}n)} centers form \spad{f} within space of width \spad{n}.
163276`\spad{center(l,{}i,{}s)} takes a list of strings \spad{l},{} and centers them within a list of strings which is \spad{i} characters long,{} in which the remaining spaces are filled with strings composed of as many repetitions as possible of the last string parameter \spad{s}.
163356`\spad{center(s,{}i,{}s)} takes the first string \spad{s},{} and centers it within a string of length \spad{i},{} in which the other elements of the string are composed of as many replications as possible of the second indicated string,{} \spad{s} which must have a length greater than that of an empty string.
163424`\axiom{certainlySubVariety?(newlp,{}\spad{lp})} returns \spad{true} iff for every \axiom{\spad{p}} in \axiom{\spad{lp}} the remainder of \axiom{\spad{p}} by \axiom{newlp} using the division algorithm of Groebner techniques is zero.
163525`\axiom{chainSubResultants(\spad{P},{} \spad{Q})} computes the list of non zero subresultants of \axiom{\spad{P}} and \axiom{\spad{Q}}.
163614`\spad{changeBase(exp,{}man,{}base)} \undocumented{}
163690`\spad{changeBase(M,{} A,{} sig,{} der)}: computes the new matrix of a pseudo-linear transform given by the matrix \spad{M} under the change of base A
163797`\spad{changeVar(+/[\spad{ai} D^i],{} a)} returns the operator \spad{+/[\spad{ai} (D+a)\spad{^i}]}.
163870`\spad{changeVar(+/[\spad{ai} D^i],{} a)} returns the operator \spad{+/[\spad{ai} (D+a)\spad{^i}]}.
163933`\spad{changeWeightLevel(n)} This changes the weight level to the new value given: \spad{NB:} previously calculated terms are not affected
164037`\spad{changeWeightLevel(n)} changes the weight level to the new value given: \spad{NB:} previously calculated terms are not affected
164144`\spad{charClass(l)} creates a character class which contains exactly the characters given in the list \spad{l}.
164205`\spad{charClass(s)} creates a character class which contains exactly the characters given in the string \spad{s}.
164257`\spad{char(i)} provides a character corresponding to the integer code \spad{i}. It is always \spad{true} that \spad{ord char i = i}.
164300`\spad{char(s)} provides a character from a string \spad{s} of length one.
164342`\spad{character?(t)} tests whether \spad{t} is equivalent to the FORTRAN type CHARACTER.
164399`\spad{characteristicPolynomial(e)} returns the characteristic polynomial of \spad{e} using resultants
164727`\spad{characteristicPolynomial(m)} returns the characteristic polynomial of the matrix \spad{m} expressed as polynomial over complex rationals with a new symbol as variable.
164875`\spad{characteristicPolynomial(m)} returns the characteristic polynomial of the matrix \spad{m} expressed as polynomial over \spad{RN} with a new symbol as variable.
165002`\spad{characteristicPolynomial(m)} returns the characteristicPolynomial of the matrix \spad{m} using a new generated symbol symbol as the main variable.
165106`\spad{characteristicPolynomial(a)} returns the characteristic polynomial of the regular representation of \spad{a} with respect to any basis.
165178`\spad{characteristicPolynomial(m,{}x)} returns the characteristic polynomial of the matrix \spad{m} expressed as polynomial over Complex Rationals with variable \spad{x}.
165333`\spad{characteristicPolynomial(m,{}x)} returns the characteristic polynomial of the matrix \spad{m} expressed as polynomial over \spad{RN} with variable \spad{x}. Fraction \spad{P} \spad{RN}.
165467`\spad{characteristicPolynomial(m,{}var)} returns the characteristicPolynomial of the matrix \spad{m} using the symbol \spad{var} as the main variable.
165578`\spad{characteristicPolynomial(m,{}r)} computes the characteristic polynomial of the matrix \spad{m} evaluated at the point \spad{r}. In particular,{} if \spad{r} is the polynomial \spad{'x},{} then it returns the characteristic polynomial expressed as a polynomial in \spad{'x}.
165669`\axiom{characteristicSerie(\spad{ps})} returns the same as \axiom{characteristicSerie(\spad{ps},{}initiallyReduced?,{}initiallyReduce)}.
165755`\axiom{characteristicSerie(\spad{ps},{}redOp?,{}redOp)} returns a list \axiom{\spad{lts}} of triangular sets such that the zero set of \axiom{\spad{ps}} is the union of the regular zero sets of the members of \axiom{\spad{lts}}. This is made by the Ritt and Wu Wen Tsun process applying the operation \axiom{characteristicSet(\spad{ps},{}redOp?,{}redOp)} to compute characteristic sets in Wu Wen T
165755`--sun sense.
165865`\axiom{characteristicSet(\spad{ps})} returns the same as \axiom{characteristicSet(\spad{ps},{}initiallyReduced?,{}initiallyReduce)}.
165959`\axiom{characteristicSet(\spad{ps},{}redOp?,{}redOp)} returns a non-contradictory characteristic set of \axiom{\spad{ps}} in Wu Wen Tsun sense \spad{w}.\spad{r}.\spad{t} the reduction-test \axiom{redOp?} (using \axiom{redOp} to reduce polynomials \spad{w}.\spad{r}.\spad{t} a \axiom{redOp?} basic set),{} if no non-zero constant polynomial appear during those reductions,{} else \axiom{"fa
165959`--iled"} is returned. The operations \axiom{redOp} and \axiom{redOp?} must satisfy the following conditions: \axiom{redOp?(redOp(\spad{p},{}\spad{q}),{}\spad{q})} holds for every polynomials \axiom{\spad{p},{}\spad{q}} and there exists an integer \axiom{\spad{e}} and a polynomial \axiom{\spad{f}} such that we have \axiom{init(\spad{q})^e*p = \spad{f*q} + redOp(\spad{p},{}\spad{q})}.
166795`\spad{characteristic()} returns the characteristic of the ring.
166866`\spad{characteristic()} returns the characteristic of the ring this is the smallest positive integer \spad{n} such that \spad{n*x=0} for all \spad{x} in the ring,{} or zero if no such \spad{n} exists.
166923`\spad{charpol(m)} computes the characteristic polynomial of a matrix \spad{m} with entries in \spad{K}. This function returns a polynomial over \spad{K},{} while the general one (that is in EiegenPackage) returns Fraction \spad{P} \spad{K}
167438`\spad{charthRoot(x)} returns the \spad{p}th root of \spad{x} where \spad{p} is the characteristic of the ring.
167510`\spad{charthRoot(r)} returns the \spad{p}\spad{-}th root of \spad{r},{} or "failed" if none exists in the domain.
167621`\spad{charthRoot(a)} takes the characteristic'th root of {\em a}. Note: such a root is alway defined in finite fields.
167675`\spad{chebyshevT(n)} returns the \spad{n}th Chebyshev polynomial \spad{T[n](x)}. Note: Chebyshev polynomials of the first kind,{} denoted \spad{T[n](x)},{} computed from the two term recurrence. The generating function \spad{(1-t*x)/(1-2*t*x+t^2) = sum(T[n](x)*t^n,{} n=0..infinity)}.
167779`\spad{chebyshevT(n,{}x)} is the \spad{n}-th Chebyshev polynomial of the first kind,{} \spad{T[n](x)}. These are defined by \spad{(1-t*x)/(1-2*t*x+t^2) = sum(T[n](x) *t^n,{} n = 0..)}.
167863`\spad{chebyshevU(n)} returns the \spad{n}th Chebyshev polynomial \spad{U[n](x)}. Note: Chebyshev polynomials of the second kind,{} denoted \spad{U[n](x)},{} computed from the two term recurrence. The generating function \spad{1/(1-2*t*x+t^2) = sum(T[n](x)*t^n,{} n=0..infinity)}.
167967`\spad{chebyshevU(n,{}x)} is the \spad{n}-th Chebyshev polynomial of the second kind,{} \spad{U[n](x)}. These are defined by \spad{1/(1-2*t*x+t^2) = sum(T[n](x) *t^n,{} n = 0..)}.
168051`\spad{checkExtraValues(d)} specifies whether we want to check the solution beyond the order given by the degree bounds. The default is \spad{true}.
168133`\spad{checkExtraValues(d)} specifies whether we want to check the solution beyond the order given by the degree bounds. This option is expressed in the form \spad{checkExtraValues == d}
168190`\spad{checkForZero(p,{} a,{} b,{} incl?)} is \spad{true} if \spad{p} has a zero between a and \spad{b},{} \spad{false} otherwise,{} "failed" if this cannot be determined. Check for a and \spad{b} inclusive if incl? is \spad{true},{} exclusive otherwise.
168354`\spad{checkForZero(p,{} x,{} a,{} b,{} incl?)} is \spad{true} if \spad{p} has a zero for \spad{x} between a and \spad{b},{} \spad{false} otherwise,{} "failed" if this cannot be determined. Check for a and \spad{b} inclusive if incl? is \spad{true},{} exclusive otherwise.
168509`\spad{checkOptions checks} whether the given options are consistent,{} and yields an error otherwise
168584`\spad{checkRur(ts,{}lus)} returns \spad{true} if \spad{lus} is a rational univariate representation of \spad{ts}.
168690`one round of modular Monte Carlo check for solutions.
168859`\spad{check(d)} specifies how we want to check the solution. If the value is "skip",{} we return the solutions found by the interpolation routine without checking. If the value is "MonteCarlo",{} we use a probabilistic check. The default is "deterministic".
168965`\spad{check(d)} specifies how we want to check the solution. If the value is "skip",{} we return the solutions found by the interpolation routine without checking. If the value is "MonteCarlo",{} we use a probabilistic check. This option is expressed in the form \spad{check == d}
169046`\spad{check(s)} returns lllpt,{} list of lists of lists of point information about the \spadtype{ThreeSpace} \spad{s}.
169097`\spad{check should} be local
169297`\spad{chiSquare1(n)} \undocumented
169375`\spad{chiSquare(n)} \undocumented
169569`\spad{child?(u,{}v)} tests if node \spad{u} is a child of node \spad{v}.
169647`\spad{child(x,{}n)} \undocumented
169889`\spad{children(x)} \undocumented
169941`\spad{children(u)} returns a list of the children of aggregate \spad{u}.
170001`\spad{chineseRemainder(llv,{}lm)} returns a list of values,{} each of which corresponds to the Chinese remainder of the associated element of \axiom{\spad{llv}} and axiom{\spad{lm}}. This is more efficient than applying chineseRemainder several times.
170079`\spad{chineseRemainder(lv,{}lm)} returns a value \axiom{\spad{v}} such that,{} if \spad{x} is \axiom{\spad{lv}.\spad{i}} modulo \axiom{\spad{lm}.\spad{i}} for all \axiom{\spad{i}},{} then \spad{x} is \axiom{\spad{v}} modulo \axiom{\spad{lm}(1)\spad{*lm}(2)*...\spad{*lm}(\spad{n})}.
170145`\spad{chineseRemainder(lu,{}lr,{}n)} \undocumented
170367`\spad{chineseRemainder(x1,{}m1,{}x2,{}m2)} returns \spad{w},{} where \spad{w} is such that \spad{w = x1 mod m1} and \spad{w = x2 mod m2}. Note: \spad{m1} and \spad{m2} must be relatively prime.
170470`\spad{chinese_update(v,{} p,{} r)} informs \spad{r} about evaluation at \spad{p}
170555`\spad{chinese_update(v,{} pt,{} r)} informs \spad{r} that evaluation at \spad{pt} gave vector of values \spad{v}
170640`chkLibraryError is an internal function that steps the number of tests.
170693`\spad{choosemon }\undocumented
170871`\spad{chvar(f(x,{}y),{} p(x,{}y))} returns \spad{[g(z,{}t),{} q(z,{}t),{} c1(x),{} c2(x),{} n]} such that under the change of variable \spad{z = c1(x)},{} \spad{t = y * c2(x)},{} one gets \spad{f(x,{}y) = c1'(x)g(c1(x),{}c2(x)y)} The algebraic relation between \spad{x} and \spad{y} is \spad{p(x,{} y) = 0}. The algebraic relation between \spad{z} and \spad{t} is
170871`-- \spad{q(z,{} t) = 0}.
171015`\axiom{class(ind)} yields the class of the multi-index \axiom{ind} (Position for first non-vanishing entry).
171100`\axiom{class(ind)} yields the class of the multi-index \axiom{ind} (Position for first non-vanishing entry).
171322`\axiom{class(\spad{jv})} yields the class of the jet variable \axiom{\spad{jv}} (Class of multi-index for derivative,{} 0 else).
171385`\axiom{class(\spad{f})} is defined as the highest class of the jet variables effectively occurring in \axiom{\spad{f}}.
171460`\spad{classicalConvolution(x,{} y)} returns the classical convolution of the distributions \spad{x} and \spad{y}
171531`\spad{classicalCumulant(x,{} n)} returns the \spad{n}-th classical cumulant of the distribution \spad{x}
171611`\spad{classicalCumulants(x)} returns sequence of classical cumulants of the distribution \spad{x}
171686`\spad{clearCache()} empties the cache.
171735`\spad{clearDenominator([q1,{}...,{}qn])} returns \spad{[p1,{}...,{}pn]} such that \spad{\spad{qi} = pi/d} where \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
171805`\spad{clearDenominator([q1,{}...,{}qn])} returns \spad{[p1,{}...,{}pn]} such that \spad{\spad{qi} = pi/d} where \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
171868`\spad{clearDenominator(q)} returns \spad{p} such that \spad{q = p/d} where \spad{d} is a common denominator for the elements of \spad{q}.
171951`\spad{clearDenominator(q)} returns \spad{p} such that \spad{q = p/d} where \spad{d} is a common denominator for the coefficients of \spad{q}.
172037`\spad{clearFortranOutputStack()} clears the Fortran output stack
172119`\axiom{clearTable!()} clears the hash-table and assumes that it will no longer be used.
172193`\spad{clearTheSymbolTable()} clears the current symbol table.
172251`\spad{clearTheSymbolTable(x)} removes the symbol \spad{x} from the table
172315`\spad{clipBoolean(l,{}b)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{clipBoolean}. If the option does not exist the value,{} \spad{b} is returned.
172398`\spad{clipParametric(p)} performs two-dimensional clipping on a plot,{} \spad{p},{} from the domain \spadtype{Plot} for the parametric curve \spad{x = f(t)},{} \spad{y = g(t)}; the default parameters \spad{1/2} for the fraction and \spad{5/1} for the scale are used in the \fakeAxiomFun{iClipParametric} subroutine,{} which is called by thi
172398`--s function.
172565`\spad{clipParametric(p,{}frac,{}sc)} performs two-dimensional clipping on a plot,{} \spad{p},{} from the domain \spadtype{Plot} for the parametric curve \spad{x = f(t)},{} \spad{y = g(t)}; the fraction parameter is specified by \spad{frac} and the scale parameter is specified by \spad{sc} for use in the
172565`-- \fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.
172768`\spad{clipPointsDefault()} determines whether or not automatic clipping is to be done.
172829`\spad{clipPointsDefault(true)} turns on automatic clipping; \spad{clipPointsDefault(false)} turns off automatic clipping. The default setting is \spad{true}.
172897`\spad{clipSurface(v,{}s)} displays the graph with the specified clipping region removed if \spad{s} is "on",{} or displays the graph without clipping implemented if \spad{s} is "off",{} for the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport}.
172966`\spad{clipWithRanges(pointLists,{}xMin,{}xMax,{}yMin,{}yMax)} performs clipping on a list of lists of points,{} \spad{pointLists}. Clipping is done within the specified ranges of \spad{xMin},{} \spad{xMax} and \spad{yMin},{} \spad{yMax}. This function is used intern
172966`--ally by the \fakeAxiomFun{iClipParametric} subroutine in this package.
173207`\spad{clip(ll)} performs two-dimensional clipping on a list of lists of points,{} \spad{ll}; the default parameters \spad{1/2} for the fraction and \spad{5/1} for the scale are used in the \fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.
173390`\spad{clip(l)} performs two-dimensional clipping on a curve \spad{l},{} which is a list of points; the default parameters \spad{1/2} for the fraction and \spad{5/1} for the scale are used in the \fakeAxiomFun{iClipParametric} subroutine,{} which is called by this function.
173567`\spad{clip(p)} performs two-dimensional clipping on a plot,{} \spad{p},{} from the domain \spadtype{Plot} for the graph of one variable,{} \spad{y = f(x)}; the default parameters \spad{1/4} for the fraction and \spad{5/1} for the scale are used in the \spadfun{clip} function.
173724`\spad{clip(b)} turns 2D clipping on if \spad{b} is \spad{true},{} or off if \spad{b} is \spad{false}. This option is expressed in the form \spad{clip == b}.
173768`\spad{clip([l])} provides ranges for user-defined clipping as specified in the list \spad{l}. This option is expressed in the form \spad{clip == [l]}.
173825`\spad{clip(p,{}frac,{}sc)} performs two-dimensional clipping on a plot,{} \spad{p},{} from the domain \spadtype{Plot} for the graph of one variable \spad{y = f(x)}; the fraction parameter is specified by \spad{frac} and the scale parameter is specified by \spad{sc} for use in the \spadfun{clip} function.
174018`\spad{close!(f)} returns the file \spad{f} closed to input and output.
174069`\spad{close!(f)} returns the library \spad{f} closed to input and output.
174107`\spad{closeComponent(s,{}\spad{li},{}b)} sets the property of the component in the 3 dimensional subspace,{} \spad{s},{} to be closed if \spad{b} is \spad{true},{} or open if \spad{b} is \spad{false}. The list of non negative integers,{} \spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component whose closed property is to be set. The subspace,{} \spad{s},{} is returned with 
174107`--the component property modification.
174192`\spad{close(v)} closes the viewport window of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} and terminates the corresponding process ID.
174246`\spad{close(v)} closes the viewport window of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} and terminates the corresponding process ID.
174302`\spad{close(x,{}b)} \undocumented
174370`\spad{closed?(x)} \undocumented
174432`\spad{closed?(t)} tests whether the given tube plot \spad{t} is closed.
174484`\spad{closedCartesian(a,{} b,{} f)} builds Cartesian product of a and \spad{b} and then maps it back to \% using \spad{f}.
174552`Cartesian product doubles the size of next list in each object,{} that is it produces two arrows out of every node
174625`as Cartesian product but returns \%.
174695`as Cartesian product but returns \%.
174765`\spad{closedCurve?(s)} returns \spad{true} if the \spadtype{ThreeSpace} \spad{s} contains a single closed curve component,{} \spadignore{i.e.} the first element of the curve is also the last element,{} or \spad{false} otherwise.
174828`\spad{closedCurve(lp)} sets a list of points defined by the first element of \spad{lp} through the last element of \spad{lp} and back to the first elelment again and returns a \spadtype{ThreeSpace} whose component is the closed curve defined by \spad{lp}.
174897`\spad{closedCurve(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a single closed curve component defined by a list of points in which the first point is also the last point,{} all of which are from the domain \spad{PointDomain(m,{}R)} and if so,{} returns the list of points. An error is signaled otherwise.
174966`\spad{closedCurve(s,{}[[lr0],{}[lr1],{}...,{}[lrn],{}[lr0]])} adds a closed curve component defined by a list of points \spad{lr0} through \spad{lrn},{} which are lists of elements from the domain \spad{PointDomain(m,{}R)},{} where \spad{R} is the \spadtype{Ring} over which the point elements are defined and \spad{m} is the dimension of the points,{} in which the last element of the list of points contains a copy of the first elemen
174966`--t list,{} \spad{lr0}. The closed curve is added to the \spadtype{ThreeSpace},{} \spad{s}.
175037`\spad{closedCurve(s,{}[p0,{}p1,{}...,{}pn,{}p0])} adds a closed curve component which is a list of points defined by the first element \spad{p0} through the last element \spad{pn} and back to the first element \spad{p0} again,{} to the \spadtype{ThreeSpace} \spad{s}.
175109`\spad{closedTensor(a,{} b,{} f)} builds tensor product of a and \spad{b} and then maps it back to \% using \spad{f}.
175174`as tensor product but returns \%. Cartesian product does apply to function graph produces two arrows out of every node
175239`as tensor product but returns \%.
175309`as tensor product but returns \%.
175376`as tensor product but returns \%.
175443`\spad{cmult(x)} should be local but conditional
175594`given a mapping from this graph this function tries to calculate a unique reverse mapping back to this graph
175707`given a mapping from this graph this function tries to calculate a unique reverse mapping back to this graph
175901`\axiom{coHeight(\spad{ts})} returns \axiom{size()\spad{\$}\spad{V}} minus \axiom{\spad{\#}\spad{ts}}.
175993`\spad{code(f)} returns the internal representation of the object represented by \spad{f}.
176723`\spad{coefChoose should} be local
176806`\spad{coef(p,{}e)} extracts the coefficient of the monomial \spad{e}. Returns zero if \spad{e} is not present.
176856`\axiom{coef(\spad{x},{}\spad{y})} returns the scalar product of \axiom{\spad{x}} by \axiom{\spad{y}},{} the set of words being regarded as an orthogonal basis.
176939`\spad{coef(x,{}w)} returns the coefficient of the word \spad{w} in \spad{x}.
177007`\spad{coef(x,{}y)} returns scalar product of \spad{x} by \spad{y},{} the set of words being regarded as an orthogonal basis.
177219`\spad{coefficient(x,{}n)} \undocumented
177285`\axiom{coefficient(om,{} \spad{jb})} returns the coefficient of \axiom{om} for the differential of \axiom{\spad{jb}}.
177344`\axiom{coefficient(\spad{v},{} \spad{jb})} returns the coefficient of \axiom{\spad{v}} in direction \axiom{\spad{jb}}.
177402`\spad{coefficient(s,{} n)} gives the terms of total degree \spad{n}.
177502`\spad{coefficient(df,{}u)},{} where \spad{df} is a differential form,{} returns the coefficient of \spad{df} containing the basis term \spad{u} if such a term exists,{} and 0 otherwise.
177593`\spad{coefficient(p,{}u)} returns the coefficient of the term in \spad{p} containing the basis term \spad{u} if such a term exists,{} and 0 otherwise. Error: if the second argument \spad{u} is not a basis element.
177647`\spad{coefficient(s,{} e1 a1 + ... + en an)} returns \spad{ei} such that \spad{ai} = \spad{s},{} or 0 if \spad{s} is not one of the \spad{ai}\spad{'s}.
177714`\spad{coefficient(p,{}e)} extracts the coefficient of the monomial with exponent \spad{e} from polynomial \spad{p},{} or returns zero if exponent is not present.
177773`\spad{coefficient(x,{}[i1,{}i2,{}...,{}iN])} extracts the coefficient of \spad{e(i1)*e(i2)*...*e(iN)} in \spad{x}.
177855`\spad{coefficient(f,{}m)} extracts the coefficient of \spad{m} in \spad{f} with respect to the canonical basis \spad{M}.
177915`\spad{coefficient(s,{} n)} gives the terms of total degree \spad{n}.
178002`\spad{coefficient(l,{}k)} is \spad{a(k)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
178082`\spad{coefficient(l,{}k)} is \spad{a(k)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
178171`\spad{coefficient(x,{}s)} returns the coefficient of the basis element \spad{s}
178625`\spad{coefficient(p,{} lv,{} ln)} views the polynomial \spad{p} as a polynomial in the variables of \spad{lv} and returns the coefficient of the term \spad{lv^ln},{} \spadignore{i.e.} \spad{prod(lv_i ^ ln_i)}.
178738`\spad{coefficient(p,{}v,{}n)} views the polynomial \spad{p} as a univariate polynomial in \spad{v} and returns the coefficient of the \spad{v^n} term.
178839`\spad{coefficient(f,{}[x1,{}x2,{}...,{}xk],{}[n1,{}n2,{}...,{}nk])} returns the coefficient of \spad{x1^n1 * ... * xk^nk} in \spad{f}.
178952`\spad{coefficient(f,{}x,{}n)} returns the coefficient of \spad{x^n} in \spad{f}.
179194`\axiom{coefficients(om)} yields the coefficients of \axiom{om}.
179257`\axiom{coefficients(\spad{v})} yields the coefficients of \axiom{\spad{v}}.
179319`\spad{coefficients(x)} returns a stream of ring elements. \indented{1}{When \spad{x} is a univariate series,{} this is a stream of Taylor} \indented{1}{coefficients. When \spad{x} is a multivariate series,{} the} \indented{1}{\spad{n}th element of the stream is a form of} \indented{1}{degree \spad{n} in the power series variables.}
179389`\spad{coefficients(p)} gives the list of non-zero coefficients of polynomial \spad{p}.
179459`\spad{coefficients(x)} returns the list of coefficients of \spad{x}.
179524`\spad{coefficients(f)} lists all non-zero coefficients.
179589`\spad{coefficients(l)} returns the list of all the nonzero coefficients of \spad{l}.
179666`\spad{coefficients(a0 + a1 x + a2 x^2 + ...)} returns a stream of coefficients: \spad{[a0,{}a1,{}a2,{}...]}. The entries of the stream may be zero.
179749`\spad{coeffs0(x,{} lv,{} lp)} is used by \spad{coeffs1}
179942`\spad{coeffs1(x,{} lv)} computes list of coefficients of \spad{x} with respect to variables in \spad{lv}. Variables in \spad{lv} must be decreasing and bigger than all other variables of \spad{x}.
180084`\spad{coerceImages(ls)} coerces the list {\em ls} to a permutation whose image is given by {\em ls} and the preimage is fixed to be {\em [1,{}...,{}n]}. Note: {coerceImages(\spad{ls})=coercePreimagesImages([1,{}...,{}\spad{n}],{}\spad{ls})}. We assume that both preimage and image do not contain repetitions.
180190`\spad{coerceImages(ls)} coerces the list {\em ls} to a permutation whose image is given by {\em ls} and the preimage is fixed to be {\em [1,{}...,{}n]}. Note: {coerceImages(\spad{ls})=coercePreimagesImages([1,{}...,{}\spad{n}],{}\spad{ls})}. We assume that both preimage and image do not contain repetitions.
180272`\spad{coerceL(o)} changes \spad{o} in the standard output format to html format and displays result as one long string.
180326`coerceS(\spad{o}) changes \spad{o} in the standard output format to MathML format and displays result as one long string.
180382`coerceS(\spad{o}) changes \spad{o} in the standard output format to Texmacs format and displays result as one long string.
180439`\spad{coerceListOfPairs(lls)} coerces a list of pairs {\em lls} to a permutation. Error: if not consistent,{} \spadignore{i.e.} the set of the first elements coincides with the set of second elements. coerce(\spad{p}) generates output of the permutation \spad{p} with domain OutputForm.
180506`\spad{coerceP(v)} coerces a vector \spad{v} with entries in \spadtype{Matrix R} as vector over \spadtype{Matrix Polynomial R}
180608`\spad{coercePreimagesImages(lls)} coerces the representation {\em lls} of a permutation as a list of preimages and images to a permutation. We assume that both preimage and image do not contain repetitions.
180679`\spad{coerceS(o)} changes \spad{o} in the standard output format to html format and displays formatted result.
180733`\spad{coerceS(o)} changes \spad{o} in the standard output format to MathML format and displays formatted result.
180789`\spad{coerceToType(f)} returns the input form corresponding to f::R
181652`\spad{coerce(r)} converts a ring element \spad{r} to a stream with one element.
181942`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
181999`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
182058`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
182122`\spad{coerce(u)} transforms \spad{u} into a MachineComplex
182242`\axiom{coerce(\spad{d})} coerces an element of \axiom{\spad{D}} into the new domain. This includes the calculation of its leading derivative and its jet variables.
182293`\spad{coerce(e)} returns \spad{1*e}
182343`\spad{coerce(eq)} \undocumented{}
182457`\spad{coerce(eq)} \undocumented{}
182562`\spad{coerce(eq)} \undocumented{}
182669`\spad{coerce(eq)} \undocumented{}
182783`\spad{coerce(eq)} \undocumented{}
182895`\spad{coerce(eq)} \undocumented{}
183009`\spad{coerce(e)} \undocumented{}
183113`\spad{coerce(e)} \undocumented{}
183208`\spad{coerce(x)} returns \spad{x} with coefficients in the domain
183282`\spad{coerce(e)} \undocumented{}
183512`\spad{coerce(e)} \undocumented{}
183616`\spad{coerce(e)} \undocumented{}
183718`\spad{coerce(e)} \undocumented{}
183822`\spad{coerce(f)} produces the equivalence class of \spad{f} in the residue ring
183893`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
183956`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
184017`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
184086`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
184147`\spad{coerce(e)} takes an object from \spadtype{FortranCode} and \indented{1}{uses it as the body of an ASP.}
184216`\spad{coerce(fc)} \undocumented{}
184305`\spad{coerce(t)} creates an element from a scalar type
185592`\spad{coerce(\spad{gi})} converts the indicated \spadtype{GraphImage},{} \spad{gi},{} into the \spadtype{TwoDimensionalViewport} form.
186733`\axiom{coerce(\spad{jv})} coerces the jet variable \axiom{\spad{jv}} into a local section.
186844`\spad{coerce Lambda} term to SKI combinators. this process is known as abstraction elimination. it is done by applying the following rules until all lambda terms have been eliminated. rule LS1: Lam[\spad{x}] \spad{=>} \spad{x} rule LS2: Lam[(\spad{E1} \spad{E2})] \spad{=>} (Lam[\spad{E1}] Lam[\spad{E2}]) rule LS3: Lam[\spad{\x}.\spad{E}] \spad{=>} (\spad{K} Lam[\spad{E}]) (if \spad{x} does not occur free in \spad{E}) rule LS4: Lam[\spad
186844`--{\x}.\spad{x}] \spad{=>} \spad{I} rule LS5: Lam[\spad{\x}.\y.\spad{E}] \spad{=>} Lam[\spad{\x}.Lam[\y.\spad{E}]] (if \spad{x} occurs free in \spad{E}) rule LS6: Lam[\spad{\x}.(\spad{E1} \spad{E2})] \spad{=>} (\spad{S} Lam[\spad{\x}.\spad{E1}] Lam[\spad{\x}.\spad{E2}])
186911`\axiom{coerce(\spad{p})} returns \axiom{\spad{p}}.
186987`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
187056`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
187123`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
187198`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
187265`\spad{coerce(e)} takes an object from \spadtype{List FortranCode} and \indented{1}{uses it as the body of an ASP.}
187340`\spad{coerce(lfc)} \undocumented{}
187435`\spad{coerce(l)} converts a list of 0\spad{'s} and 1\spad{'s} into a basis element,{} where 1 (respectively 0) designates that the variable of the corresponding index of \spad{l} is (respectively,{} is not) present. Error: if an element of \spad{l} is not 0 or 1.
187488`\spad{coerce(llp)} component(\spad{gi},{}\spad{pt}) creates and returns a graph of the domain \spadtype{GraphImage} which is composed of the list of list of points given by \spad{llp},{} and whose point colors,{} line colors and point sizes are determined by the default functions \spadfun{pointColorDefault},{} \spadfun{lineColorDefault},{} and \spadfun{pointSizeDefault}. The graph data is then sent to the viewport manager where it wai
187488`--ts to be included in a two-dimensional viewport window.
187557`\spad{coerce List} to graph
187693`\spad{coerce(u)} transforms a MachineInteger into a MachineFloat
187748`\spad{coerce(v)} produces an ASP which returns the value of \spad{v}.
187818`\spad{coerce(o)} changes \spad{o} in the standard output format to html format.
187871`\spad{coerce(p)} coerces \spad{p} into Weighted form,{} applying weights and ignoring terms
187948`\spad{coerce PermutationGroup} to graph
188012`\spad{coerce PermutationGroup} to graph which represents the generators of the group
188201`\spad{coerce(p)} coerces a Polynomial(\spad{R}) into Weighted form,{} applying weights and ignoring terms
188287`\spad{coerce(p)} moves from the representation type (PrimitiveArray PrimitiveArray PrimitiveArray \spad{R}) to the domain
188390`\spad{coerce(a)} makes a tuple from primitive array a
189069`\spad{coerce(x)} \undocumented
189148`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
189255`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
189360`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
189473`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
189578`\spad{coerce(e)} takes the component of \spad{e} from \spadtype{List FortranCode} and uses it as the body of the ASP,{} making the declarations in the \spadtype{SymbolTable} component.
189691`\spad{coerce(r)} \undocumented{}
189824`\spad{coerce(x)} \undocumented
191980`\spad{coerce(a)} transforms a into an element of \%.
192026`\spad{coerce(s)} returns \spad{s} viewed as an element of \%.
192074`\spad{coerce combinators} to intuitionistic logic this is known as the Curry-Howard isomorphism it uses the following rules: rule SI1: Ski[Kab] \spad{=>} a \spad{->} (\spad{b} \spad{->} a),{} rule SI2: Ski[Sabc] \spad{=>} (a \spad{->} (\spad{b} \spad{->} \spad{c})) \spad{->} ((a \spad{->} \spad{b}) \spad{->} (a \spad{->} \spad{c})),{} rule SI3: Ski[a a-\spad{>b}] \spad{=>} \spad{b} the last rule is function application (modus ponens)
192137`\spad{coerce SKI} combinators to Lambda term. this conversion is done by applying the following rules rule SL1: Ski[\spad{I}] = \spad{\x}.0 rule SL2: Ski[\spad{K}] = \spad{\x}.\y.1 rule SL3: Ski[\spad{S}] = \spad{\x}.\y.\spad{\z}.(2 0 (1 0)) rule SL4: Ski[(\spad{E1} \spad{E2})] = (Ski[\spad{E1}] Ski[\spad{E2}])
192204`\spad{coerce(poly)} regroups the terms by total degree and forms a series.
192358`coercion to matrices over ground domain.
192546`\spad{coerce(p)} returns \spad{p} viewed as an algebraic number.
192876`\spad{coerce(x)} creation of elements
192933`\spad{coerce(s)} transforms the string \spad{s} into an element of FortranScalarType provided \spad{s} is one of "real",{} "double precision",{} "complex",{} "logical",{} "integer",{} "character",{} "REAL",{} "COMPLEX",{} "LOGICAL",{} "INTEGER",{} "CHARACTER",{} "DOUBLE PRECISION"
193089`\spad{coerce(s)} transforms the symbol \spad{s} into an element of FortranScalarType provided \spad{s} is one of real,{} complex,{}double precision,{} logical,{} integer,{} character,{} REAL,{} COMPLEX,{} LOGICAL,{} INTEGER,{} CHARACTER,{} DOUBLE PRECISION
193141`\axiom{coerce(sy)} allows for the input of jet variables as symbols. Yields an error,{} if the symbol is not contained in \axiom{IVar} or \axiom{DVar}.
193196`\axiom{coerce(\spad{s})} allows for the input of jet variables as symbols.
193258`\spad{coerce(s)} \undocumented{}
193385`\spad{coerce(p)} converts a univariate polynomial \spad{p} in the variable \spad{var} to a univariate Taylor series in \spad{var}.
193486`\spad{coerce(p)} converts a univariate polynomial \spad{p} in the variable \spad{var} to a univariate Taylor series in \spad{var}.
193581`\spad{coerce(f)} converts a \spadtype{UnivariatePuiseuxSeries} to an \spadtype{ExponentialExpansion}.
193806`\spad{coerce(var)} converts a variable to a Taylor series
193882`\spad{coerce(v)} return \spad{v}
194025`\spad{coerce(var)} converts the series variable \spad{var} into a Laurent series.
194110`\spad{coerce(var)} converts the series variable \spad{var} into a Puiseux series.
194195`\spad{coerce(var)} converts the series variable \spad{var} into a \indented{1}{Taylor series.}
194279`\spad{coerce(var)} converts the series variable \spad{var} into a Laurent series.
194358`\spad{coerce(var)} converts the series variable \spad{var} into a Puiseux series.
194437`\spad{coerce(var)} converts the series variable \spad{var} into a \indented{1}{Taylor series.}
194515`\spad{coerce(x)} returns \spad{x} as a skew-polynomial.
194596`\spad{coerce(v)} assumes that it is called with a vector of length equal to the dimension of the algebra,{} then a linear combination with the basis element is formed
194698`\spad{coerce(v)} produces an ASP which returns the value of \spad{v}.
194768`\spad{coerce(v)} coerces a vector \spad{v} with entries in \spadtype{Matrix R} as vector over \spadtype{Matrix Fraction Polynomial R}
194996`\spad{coerce(u)} transforms \spad{u} into a COmplex Float
195053`\axiom{coerce(exp)} retracts an element to the base domain \axiom{\spad{D}}. This looses all informations about its leading derivative and its jet variables and requires complete evaluation of the expression.
195237`\spad{coerce(x)} \undocumented{}
195331`\spad{coerce(u)} transforms a MachineFloat to a standard Float
195377`\spad{coerce(p)} coerces a partition into a list of integers
195428`\spad{coerce(x)} outputs \spad{x} as a list of strings. Each element in the list represents a new line. This is an alternative to creating a long single string with 'newline' embedded in it.
195479`\spad{coerce b} retracts a basis to this list of vectors.
195542`\axiom{coerce(\spad{x})} returns the element of \axiomType{Magma}(VarSet) corresponding to \axiom{\spad{x}}.
195602`\axiom{coerce(A)} yields the matrix \axiom{A} in the usual matrix type.
195664`\spad{coerce(m)} converts a matrix of type \spadtype{RectangularMatrix} to a matrix of type \spad{Matrix}.
195726`\spad{coerce(m)} converts a matrix of type \spadtype{SquareMatrix} to a matrix of type \spadtype{Matrix}.
195784`\axiom{coerce(\spad{x})} returns the element of \axiomType{OrderedFreeMonoid}(VarSet) corresponding to \axiom{\spad{x}}.
195856`\axiom{coerce(\spad{x})} returns the element of \axiomType{OrderedFreeMonoid}(VarSet) corresponding to \axiom{\spad{x}} by removing parentheses.
195923`\spad{coerce([l1]*[l2]*...[ln])} returns the word \spad{l1*l2*...*ln},{} where \spad{[l_i]} is the backeted form of the Lyndon word \spad{l_i}.
196016`output
196069`output
196114`\spad{coerce(f)} returns an object of type OutputForm.
196164`\spad{coerce(x)} provides a printable form for \spad{x}
196214`\spad{coerce(\spad{gi})} returns the indicated graph,{} \spad{\spad{gi}},{} of domain \spadtype{GraphImage} as output of the domain \spadtype{OutputForm}.
196263`output
196308`\axiom{coerce(de)} transforms the differential equation \axiom{de} to \axiomType{OutputForm}.
196376`output
196425`\spad{coerce(bd)} outputs description of \spad{bd} as formatted text
196477`output
196534`\spad{coerce(tr)} output
196587`output a full scenegraph could contain a lot of data and so we have to be careful to restrict the output to what is reasonable on the command line. Therefore we do not display all subnodes or the full information from nodes such as IFS.
196635`\spad{coerce(me)} list this instance to the command line
196686`\spad{coerce(v)} returns the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport} as output of the domain \spadtype{OutputForm}.
196747`\spad{coerce b} prepares the display of a basis.
196805`convert back into a \spad{"P"},{} ignoring weights
196882`\spad{coerce(p)} converts back into a Polynomial(\spad{R}),{} ignoring weights
196968`\spad{coerce(x)} moves from the domain to the representation type
197071`\spad{coerce(x)} \undocumented
197155`\spad{coerce(x)} \undocumented
197229`\spad{coerce(x)} \undocumented
197302`\spad{coerce(x)} \undocumented
197381`\spad{coerce(x)} returns the \spad{s}-expression associated with \spad{x}
197438`\spad{coerce(a)} transforms a into an element of \spad{S}.
197541`\spad{coerce(x)} outputs \spad{x} as a string
197588`\spad{coerce(x)} returns the symbol associated with \spad{x}
197640`\spad{coerce(x)} returns the symbol
197688`\spad{coerce(x)} returns a table view of \spad{x}
197753`\axiom{coerce(\spad{g})} returns the internal representation of \axiom{\spad{g}}.
197845`\axiom{coerce(\spad{p})} returns \axiom{\spad{p}} as a distributed polynomial.
197930`\axiom{coerce(\spad{g})} returns the internal representation of \axiom{\spad{g}}.
198014`\axiom{coerce(\spad{p})} returns \axiom{\spad{p}} as a recursive polynomial.
198097`\spad{coerce A} changes its argument into a \indented{1}{nullary function.}
198148`\spad{coerce(a)} coerces the element \spad{a} of the algebra \spad{A} to an element of the Jordan algebra \spadtype{AssociatedJordanAlgebra}(\spad{R},{}A).
198206`\spad{coerce(a)} coerces the element \spad{a} of the algebra \spad{A} to an element of the Lie algebra \spadtype{AssociatedLieAlgebra}(\spad{R},{}A).
198261`\spad{coerce(c)} returns \spad{c} as a string \spad{s} with the character \spad{c}.
198314`\spad{coerce(col)} converts the column \spad{col} to a column matrix.
198371`\spad{coerce(c)} sets the average shade for the palette to that of the indicated color \spad{c}.
198412`\spad{coerce(v)} views a vector as a rank 1 tensor.
198489`\spad{coerce(x)} coerces a distribution from the ring \spad{R1} to \spad{R2}.
198577`\spad{coerce(e)} is never really evaluated. This coercion is used for formal type correctness when a function will not return directly to its caller.
198636`\spad{coerce(x)} is the homomorphic image of \spad{x} from {\em F1} in {\em F2}. Thus {\em coerce} is a field homomorphism between the fields extensions {\em F1} and {\em F2} both over ground field {\em GF} (the second argument to the package). Error: if the extension degree of {\em F1} doesn\spad{'t} divide the extension degree of {\em F2}. Note that the other coercion function in the \spadtype{FiniteFieldHomomorphisms} is a left inverse
198636`--.
198701`\spad{coerce(x)} is the homomorphic image of \spad{x} from {\em F2} in {\em F1},{} where {\em coerce} is a field homomorphism between the fields extensions {\em F2} and {\em F1} both over ground field {\em GF} (the second argument to the package). Error: if the extension degree of {\em F2} doesn\spad{'t} divide the extension degree of {\em F1}. Note that the other coercion function in the \spadtype{FiniteFieldHomomorphisms} is a left inve
198701`--rse.
198766`\spad{coerce(c)} converts sin/cos terms into Fourier Series
198832`\spad{coerce(f)} takes a fraction with numerator and denominator in factored form and creates a partial fraction. It is necessary for the parts to be factored because it is not known in general how to factor elements of \spad{R} and this is needed to decompose into partial fractions.
198900`\spad{coerce(rf)} converts \spad{rf},{} a fraction of polynomial \spad{p} with algebraic number coefficients to \spadtype{Expression Integer}.
199006`\spad{coerce(f)} returns \spad{f} as an element of \%.
199105`\spad{coerce(q)} returns \spad{q} as an element of \%.
199182`\spad{coerce(n)} coerces the integer \spad{n} to an element of the ring.
199236`\spad{coerce(i)} converts the integer \spad{i} to a member of the given domain.
199276`\spad{coerce(i)} returns the element corresponding to \spad{i}
199322`\spad{coerce(polyList)} converts the list of polynomials \spad{polyList} to an ideal.
199399`\spad{coerce(lls)} coerces a list of cycles {\em lls} to a permutation,{} each cycle being a list with no repetitions,{} is coerced to the permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.
199455`\spad{coerce(ls)} coerces a list of permutations {\em ls} to the group generated by this list.
199523`\spad{coerce([r_1,{}...,{}r_dim])} allows tensors to be constructed using lists.
199587`\spad{coerce(lt)} converts a list of terms and coefficients to a member of the domain.
199667`\spad{coerce(l)} creates a datalist from \spad{l}
199714`\spad{coerce(l)} makes a database out of a list
199761`\spad{coerce(ls)} coerces a cycle {\em ls},{} \spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur.
199811`\spad{coerce(l)} converts a list \spad{l} to a stream.
199856`\spad{coerce([t_1,{}...,{}t_dim])} allows tensors to be constructed using lists.
199921`coerceS(\spad{o}) changes \spad{o} in the standard output format to MathML format.
199976`coerceS(\spad{o}) changes \spad{o} in the standard output format to Texmacs format.
200032`\spad{coerce(o)} changes \spad{o} in the standard output format to SCRIPT formula format.
200090`\spad{coerce(o)} changes \spad{o} in the standard output format to TeX format.
200138`\spad{coerce(f)} returns \spad{f} as an Expression(\spad{R}).
200194`\spad{coerce(p)} converts the polynomial \spad{p} with algebraic number coefficients to \spadtype{Expression Integer}.
200290`\spad{coerce(s)} regroups terms of \spad{s} by total degree \indented{1}{and forms a series.}
200353`\spad{coerce(p)} returns \spad{p} as an element of \%.
200442`\spad{coerce(r)} returns \spad{r} viewed as a rational function over \spad{R}.
200512`\spad{coerce(r)} maps the ring element \spad{r} to a member of the algebra.
200552`\spad{coerce(r)} returns \spad{r} * 1 where 1 is the identity of the left algebra.
200596`\spad{coerce(r)} equals \spad{r*1}.
200637`\spad{coerce(r)} converts coefficients into Fourier Series
200685`\spad{coerce(r)} views \spad{r} as a value in the ordinary differential ring.
200754`\spad{coerce(s)} creates an object of \spadtype{Any} from the object \spad{s} of type \spad{S}.
200801`\spad{coerce(x)} changes \spad{x} into an object of type \spadtype{None}.
200850`\spad{coerce(s)} provides a direct coercion from an expression \spad{s} of domain \spad{S} to SCRIPT formula format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to SCRIPT formula format.
200920`\spad{coerce(s)} provides a direct coercion from a domain \spad{S} to TeX format. This allows the user to skip the step of first manually coercing the object to standard output format before it is coerced to TeX format.
200970`\spad{coerce(s)} throws all information about \spad{s} away. This coercion allows values of any type to appear in contexts where they will not be used. For example,{} it allows the resolution of different types in the \spad{then} and \spad{else} branches when an \spad{if} is in a context where the resulting value is not used.
201029`\spad{coerce(s)} returns \spad{s},{} viewed as the zero-th order derivative of \spad{s}.
201090`\spad{coerce(a)} transforms a into an element of \%.
201136`\spad{coerce(x)} allows \spadtype{Segment} values to be used as \%.
201194`\spad{coerce(x = a..b)} returns \spad{x = c..d} where \spad{c} and \spad{d} are the numerical values of \spad{a} and \spad{b}.
201289`\spad{coerce(p)} returns \spad{p} viewed as an algebraic number.
201381`\spad{coerce(p)} returns \spad{p} as an element of \%.
201479`\spad{coerce(m)} views a matrix as a rank 2 tensor.
201555`\spad{coerce(s)} converts a string to a file name according to operating system-dependent conventions.
201606`\spad{coerce(s)} converts \axiom{\spad{s}} into an \axiom{IndexCard}. Warning: if \axiom{\spad{s}} is not of the right format then an error will occur when using it.
201650`\spad{coerce(s)} converts the string \spad{s} to a symbol.
201691`\spad{coerce(u)} creates an OpenMath error object of an appropriate type if \axiom{\spad{u}} is one of \axiom{OMParseError},{} \axiom{OMReadError},{} \axiom{OMUnknownCD} or \axiom{OMUnknownSymbol},{} otherwise it raises a runtime error.
201743`\spad{coerce(s)} converts a variable to a Taylor series
201796`\spad{coerce(f(x))} converts the Laurent series \spad{f(x)} to a Puiseux series.
201880`\spad{coerce(f(x))} converts the Taylor series \spad{f(x)} to a Laurent series.
201964`\spad{coerce(f)} converts a Puiseux series to a general power series.
202072`\axiom{coerce(\spad{x})} returns \axiom{\spad{x}} as a Lie polynomial.
202131`\spad{coerce(var)} converts the series variable \spad{var} into a Puiseux series.
202215`\spad{coerce(x)} converts the variable \spad{x} to a univariate polynomial.
202280`\spad{coerce(v)} converts a vector to a member of the algebra by forming a linear combination with the basis element. Note: the vector is assumed to have length equal to the dimension of the algebra.
202365`\spad{coerce(b)} converts a binary expansion to a rational number.
202426`\spad{coerce(d)} converts a decimal expansion to a rational number.
202488`\spad{coerce(h)} converts a hexadecimal expansion to a rational number.
202554`\spad{coerce(rx)} converts a radix expansion to a rational number.
202618`\spad{coerce(p)} sums up the components of the partial fraction and returns a single fraction.
202676`\spad{coerce(gp)} returns the generators of the group {\em gp}.
202744`\spad{coerce(x)} returns the list of elements in \spad{x}
202791`\spad{coerce(s)} returns the \spadtype{ThreeSpace} \spad{s} to Output format.
202851`\spad{coerce(v)} coerces void object to outputForm.
202894`\spad{coerce(p)} views \spad{p} as a valie in the partial differential ring.
202963`\spad{coerce(d)} converts a decimal expansion to a radix expansion with base 10.
203026`\spad{coerce(h)} converts a hexadecimal expansion to a radix expansion with base 16.
203093`\spad{coerce(b)} converts a binary expansion to a radix expansion with base 2.
203154`\spad{coerce(a)} transforms a into an element of \spad{S}.
203198`\spad{coerce(fn)} produces a string for a file name according to operating system-dependent conventions.
203249`\spad{coerce(x)} \undocumented
203299`\axiom{coerce(\spad{x})} returns \axiom{\spad{x}} as distributed polynomial.
203384`\axiom{coerce(\spad{x})} returns \axiom{\spad{x}} as a recursive polynomial.
203467`\spad{coerce(v)} returns \spad{v}.
203516`\spad{coleman(alpha,{}beta,{}\spad{pi})}: there is a bijection from the set of matrices having nonnegative entries and row sums {\em alpha},{} column sums {\em beta} to the set of {\em Salpha - Sbeta} double cosets of the symmetric group {\em Sn}. ({\em Salpha} is the Young subgroup corresponding to the improper partition {\em alpha}). For a representing element {\em \spad{pi}} of such
203516`-- a double coset,{} coleman(\spad{alpha},{}\spad{beta},{}\spad{pi}) generates the Coleman-matrix corresponding to {\em alpha,{} beta,{} \spad{pi}}. Note: The permutation {\em \spad{pi}} of {\em {1,{}2,{}...,{}n}} has to be given in list form. Note: the inverse of this map is {\em inverseColeman} (if {\em \spad{pi}} is the lexicographical smallest permutation in the coset). For details see James/Kerber.
203694`\spad{colinearity(x,{}y)} returns a measure of how parallel two vectors are independent of the length of the vectors: 1.0 is completely parallel 0.0 is completely perpendicular returned value will be between these
203824`\axiom{collectQuasiMonic(\spad{ts})} returns the subset of \axiom{\spad{ts}} consisting of the polynomials with initial in \axiom{\spad{R}}.
204036`\axiom{collectUnder(\spad{ps},{}\spad{v})} returns the set consisting of the polynomials of \axiom{\spad{ps}} with main variable less than \axiom{\spad{v}}.
204255`\axiom{collectUpper(\spad{ps},{}\spad{v})} returns the set consisting of the polynomials of \axiom{\spad{ps}} with main variable greater than \axiom{\spad{v}}.
204334`\axiom{collect(exp)} "collects" former lazy terms which have been meanwhile evaluated.
204457`\axiom{collect(\spad{ps},{}\spad{v})} returns the set consisting of the polynomials of \axiom{\spad{ps}} with \axiom{\spad{v}} as main variable.
204531`\spad{colorDef(v,{}c1,{}c2)} sets the range of colors along the colormap so that the lower end of the colormap is defined by \spad{c1} and the top end of the colormap is defined by \spad{c2},{} for the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport}.
204602`\spad{colorFunction(f(z))} specifies the color based upon the \spad{z}-component of three dimensional plots. This option is expressed in the form \spad{colorFunction == f(z)}.
204674`\spad{colorFunction(f(u,{}v))} specifies the color for three dimensional plots as a function based upon the two parametric variables. This option is expressed in the form \spad{colorFunction == f(u,{}v)}.
204758`\spad{colorFunction(f(x,{}y,{}z))} specifies the color for three dimensional plots as a function of \spad{x},{} \spad{y},{} and \spad{z} coordinates. This option is expressed in the form \spad{colorFunction == f(x,{}y,{}z)}.
204854`\spad{color(pt)} returns the fourth element of the point,{} \spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} color to express a fourth dimension.
204904`\spad{color(i)} returns a color of the indicated hue \spad{i}.
204944`\spad{columnSpace M} builds a basis of the kernel of the linear transformation defined by the matrix \spad{M}. This function calls the \spad{columnSpace} in the \spad{Matrix} domain. Only the signature changes.
205071`\spad{columnSpace(m)} returns a sublist of columns of the matrix \spad{m} forming a basis of its column space
205161`\spad{columnSpace(m)} returns a sublist of columns of the matrix \spad{m} forming a basis of its column space.
205341`\spad{column(m,{}j)} returns the \spad{j}th column of the matrix \spad{m}. Error: if the index outside the proper range.
205421`\spad{column(m,{}j)} returns the \spad{j}th column of \spad{m} error check to determine if index is in proper ranges
205499`\spad{commaSeparate(l)} creates the form separating the elements of \spad{l} by commas.
205553`\spad{comment(s)} creates a representation of the Strings \spad{s} as a multi-line FORTRAN comment.
205606`\spad{comment(s)} creates a representation of the String \spad{s} as a single FORTRAN comment.
205653`\spad{commonDenominator([q1,{}...,{}qn])} returns a common denominator \spad{d} for \spad{q1},{}...,{}\spad{qn}.
205724`\spad{commonDenominator([q1,{}...,{}qn])} returns a common denominator \spad{d} for \spad{q1},{}...,{}\spad{qn}.
205788`\spad{commonDenominator(q)} returns a common denominator \spad{d} for the elements of \spad{q}.
205864`\spad{commonDenominator(q)} returns a common denominator \spad{d} for the coefficients of \spad{q}.
205950`\spad{common(name,{}contents)} creates a representation a named common block.
206009`\spad{commutative?()} tests if multiplication in the algebra is commutative.
206086`\spad{commutative?()} tests if multiplication in the algebra is commutative.
206160`\spad{commutativeEquality(x,{}y)} returns \spad{true} if \spad{x} and \spad{y} are equal assuming commutativity
206323`\axiom{commutator(\spad{v},{} \spad{w})} calculates the commutator of two vector fields.
206381`\spad{commutator(p,{}q)} computes \spad{inv(p) * inv(q) * p * q}.
206424`\spad{commutator(a,{}b)} returns \spad{a*b-b*a}.
206479`\spad{compBound(p,{}lp)} computes a bound for the coefficients of the solution polynomials. Given a polynomial right hand side \spad{p},{} and a list \spad{lp} of left hand side polynomials. Exported because it depends on the valuation.
206555`\spad{comp(f,{}g,{}x)} is \spad{f(g x)}.
206631`\spad{compactFraction(p)} normalizes the partial fraction \spad{p} to the compact representation. In this form,{} the partial fraction has only one fractional term per prime in the denominator.
206689`\spad{companionBlocks(m,{} v)} returns \spad{[[C_1,{} g_1],{}...,{}[C_k,{} g_k]]} such that each \spad{C_i} is a companion block and \spad{m = diagonal(C_1,{}...,{}C_k)}.
206806`\spad{comparison(op,{} foo?)} attaches foo? as the "\%less?" property to \spad{op}. If \spad{op1} and \spad{op2} have the same name,{} and one of them has a "\%less?" property \spad{f},{} then \spad{f(op1,{} op2)} is called to decide whether \spad{op1 < op2}.
206871`\spad{compdegd should} be local
207006`\spad{compile(f,{} [t1,{}...,{}tn])} forces the interpreter to compile the function \spad{f} with signature \spad{(t1,{}...,{}tn) -> ?}. returns the symbol \spad{f} if successful. Error: if \spad{f} was not defined beforehand in the interpreter,{} or if the \spad{ti}\spad{'s} are not valid types,{} or if the compiler fails.
207064`\spad{compiledFunction(expr,{} x)} returns a function \spad{f: D -> I} defined by \spad{f(x) == expr}. Function \spad{f} is compiled and directly applicable to objects of type \spad{D}.
207147`\spad{compiledFunction(expr,{}x,{}y)} returns a function \spad{f: (D1,{} D2) -> I} defined by \spad{f(x,{} y) == expr}. Function \spad{f} is compiled and directly applicable to objects of type \spad{(D1,{} D2)}
207246`\spad{complementSpace Lv} build a complement space of the subspace spaned by the list of vectors \spad{LV}. This function fails with the empty basis.
207318`\spad{complementSpace B} build a complement space of the basis \spad{B} with some vectors of the canonical basis. This function fails with the empty basis.
207377`\spad{complementSpace(Lv,{}n)} build a complement space of the subspace spaned by the list of vectors \spad{LV}. The integer \spad{n} gives the length of the vectors if the basis is empty.
207468`\spad{complementSpace(B,{}n)} build a complement space of the basis \spad{B} with some vectors of the canonical basis \spad{B}. The integer \spad{n} gives the length of the vectors if the basis is empty.
207598`\spad{complement(u)} returns the complement of the set \spad{u},{} \spadignore{i.e.} the set of all values not in \spad{u}.
207754`\spad{complementaryBasis(b1,{}...,{}bn)} returns the complementary basis \spad{(b1',{}...,{}bn')} of \spad{(b1,{}...,{}bn)}.
207845`\axiom{\spad{complete2}(de)} is like \axiomFun{complete} but returns the involutive equation \axiom{IDe},{} a basis \axiom{ISys} for the involutive system without prolongations,{} its order \axiom{Order},{} the number \axiom{NumProj} of needed projections and the C
207845`--artan characters \axiom{CarChar}.
208087`\spad{completeEchelonBasis(lv)} completes the basis {\em lv} assumed to be in echelon form of a subspace of {\em R^n} (\spad{n} the length of all the vectors in {\em lv}) with unit vectors to a basis of {\em R^n}. It is assumed that the argument is not an empty vector and that it is not the basis of the 0-subspace. Note: the rows of the result correspond to the vectors of the basis.
208179`\spad{completeEval(upoly,{} lvar,{} lval)} evaluates the polynomial \spad{upoly} with each variable in \spad{lvar} replaced by the corresponding value in lval. Substitutions are done for all variables in \spad{upoly} producing a univariate polynomial over \spad{R}.
208315`\spad{completeHensel(pol,{}lfact,{}prime,{}bound)} lifts \spad{lfact},{} the factorization mod \spad{prime} of \spad{pol},{} to the factorization mod prime^k>bound. Factors are recombined on the way.
208415`\spad{completeHermite} returns a record that contains the Hermite normal form \spad{H} of the matrix and the equivalence matrix \spad{U} such that U*m = \spad{H}
208505`\spad{completeSmith} returns a record that contains the Smith normal form \spad{H} of the matrix and the left and right equivalence matrices \spad{U} and \spad{V} such that U*m*v = \spad{H}
208659`\axiom{complete(de)} completes \axiom{de} to an involutive equation. No result is returned; the display depends of the setting of the output flags with \axiomFun{setOutput}.
208742`\spad{complete n} is the \spad{n} th complete homogeneous \indented{1}{symmetric function expressed in terms of power sums.} \indented{1}{Alternatively it is the cycle index of the symmetric} \indented{1}{group of degree \spad{n}.}
208831`\spad{complete(st)} causes all entries of 'st' to be computed. this function should only be called on streams which are known to be finite.
208886`\spad{complete(x)} forces the computation of all digits.
208942`\spad{complete(f)} causes all terms of \spad{f} to be computed. Note: this results in an infinite loop if \spad{f} has infinitely many terms.
209010`\spad{complete(x)} causes all entries in \spadvar{\spad{x}} to be computed. Normally entries are only computed as needed. If \spadvar{\spad{x}} is an infinite continued fraction,{} a user-initiated interrupt is necessary to stop the computation.
209063`\spad{complex?(t)} tests whether \spad{t} is equivalent to the FORTRAN type COMPLEX.
209118`\spad{complexEigenvalues(m,{}eps)} computes the eigenvalues of the matrix \spad{m} to precision \spad{eps}. The eigenvalues are expressed as complex floats or complex rational numbers depending on the type of \spad{eps} (float or rational).
209244`\spad{complexEigenvectors(m,{}eps)} returns a list of records each one containing a complex eigenvalue,{} its algebraic multiplicity,{} and a list of associated eigenvectors. All these results are computed to precision \spad{eps} and are expressed as complex floats or complex rational numbers depending on the typ
209244`--e of \spad{eps} (float or rational).
209437`\spad{complexElementary(f)} rewrites \spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \spad{log,{} exp}.
209515`\spad{complexElementary(f)} rewrites \spad{f} in terms of the 2 fundamental complex transcendental elementary functions: \spad{log,{} exp}.
209586`\spad{complexElementary(f,{} x)} rewrites the kernels of \spad{f} involving \spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \spad{log,{} exp}.
209671`\spad{complexElementary(f,{} x)} rewrites the kernels of \spad{f} involving \spad{x} in terms of the 2 fundamental complex transcendental elementary functions: \spad{log,{} exp}.
209749`\spad{complexExpand(i)} returns the expanded complex function corresponding to \spad{i}.
209836`\spad{complexExpand(i)} returns the expanded complex function corresponding to \spad{i}.
209957`\spad{complexForm(f)} returns \spad{[real f,{} imag f]}.
210050`\spad{complexForm(f)} returns \spad{[real f,{} imag f]}.
210124`\spad{complexIntegrate(f,{} x)} returns the integral of \spad{f(x)dx} where \spad{x} is viewed as a complex variable.
210206`\spad{complexIntegrate(f,{} x)} returns the integral of \spad{f(x)dx} where \spad{x} is viewed as a complex variable.
210343`\spad{complexLimit(f(x),{}x = a)} computes the complex limit \spad{lim(x -> a,{}f(x))}.
210478`\spad{complexLimit(f(x),{}x = a)} computes the complex limit of \spad{f} as its argument \spad{x} approaches \spad{a}.
210642`\spad{complexLimit(f(x),{}x = a)} computes the complex limit of \spad{f} as its argument \spad{x} approaches \spad{a}.
210816`\spad{complexNormalize(f)} rewrites \spad{f} using the least possible number of complex independent kernels.
210893`\spad{complexNormalize(f)} rewrites \spad{f} using the least possible number of complex independent kernels.
210963`\spad{complexNormalize(f,{} x)} rewrites \spad{f} using the least possible number of complex independent kernels involving \spad{x}.
211047`\spad{complexNormalize(f,{} x)} rewrites \spad{f} using the least possible number of complex independent kernels involving \spad{x}.
211124`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
211270`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
211407`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
211540`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
211664`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not constant.
211788`\spad{complexNumericIfCan(x)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
211892`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
212054`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
212207`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
212356`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
212496`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
212636`\spad{complexNumericIfCan(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
212756`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
212847`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
212972`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
213088`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
213200`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
213303`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
213406`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
213489`\spad{complexNumeric(x)} returns a complex approximation of \spad{x}.
213549`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
213656`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
213797`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
213929`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
214057`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x}
214176`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
214295`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
214394`\spad{complexNumeric(x,{} n)} returns a complex approximation of \spad{x} up to \spad{n} decimal places.
214470`\spad{complexRoots(rf,{} eps)} finds all the complex solutions of a univariate rational function with rational number coefficients. The solutions are computed to precision eps.
214590`\spad{complexRoots(lrf,{} lv,{} eps)} finds all the complex solutions of a list of rational functions with rational number coefficients with respect the the variables appearing in \spad{lv}. Each solution is computed to precision eps and returned as list corresponding to the order of variables in \spad{lv}.
214735`\spad{complexSolve(eq,{}eps)} finds all the complex solutions of the equation \spad{eq} of rational functions with rational rational coefficients with respect to all the variables appearing in \spad{eq},{} with precision \spad{eps}.
214887`\spad{complexSolve(p,{}eps)} find all the complex solutions of the rational function \spad{p} with complex rational coefficients with respect to all the variables appearing in \spad{p},{} with precision \spad{eps}.
215029`\spad{complexSolve(leq,{}eps)} finds all the complex solutions to precision \spad{eps} of the system \spad{leq} of equations of rational functions over complex rationals with respect to all the variables appearing in \spad{lp}.
215193`\spad{complexSolve(lp,{}eps)} finds all the complex solutions to precision \spad{eps} of the system \spad{lp} of rational functions over the complex rationals with respect to all the variables appearing in \spad{lp}.
215347`\spad{complexZeros(p)} tries to determine all complex zeros of the polynomial \spad{p} with accuracy given by the package constant {\em globalEps} which you may change by {\em setErrorBound}.
215429`\spad{complexZeros(p,{} eps)} tries to determine all complex zeros of the polynomial \spad{p} with accuracy given by {\em eps}.
215513`\spad{complexZeros(poly,{} eps)} finds the complex zeros of the univariate polynomial \spad{poly} to precision eps with solutions returned as complex floats or rationals depending on the type of eps.
215596`\spad{complex(x,{}y)} constructs \spad{x} + \%i*y.
215647`\spad{component(\spad{gi},{}pt)} modifies the graph \spad{\spad{gi}} of the domain \spadtype{GraphImage} to contain one point component,{} \spad{pt} whose point color,{} line color and point size are determined by the default functions \spadfun{pointColorDefault},{} \spadfun{lineColorDefault},{} and \spadfun{pointSizeDefault}.
215712`\spad{component(\spad{gi},{}lp,{}pal1,{}pal2,{}p)} sets the components of the graph,{} \spad{\spad{gi}} of the domain \spadtype{GraphImage},{} to the values given. The point list for \spad{\spad{gi}} is set to the list \spad{lp},{} the color of the points in \spad{lp} is set to the palette color \spad{pal1},{} the color of the lines which connect the points \spad{lp} is set to the palette color \spad{
215712`--pal2},{} and the size of the points in \spad{lp} is given by the integer \spad{p}.
215815`\spad{component(\spad{gi},{}pt,{}pal1,{}pal2,{}ps)} modifies the graph \spad{\spad{gi}} of the domain \spadtype{GraphImage} to contain one point component,{} \spad{pt} whose point color is set to the palette color \spad{pal1},{} line color is set to the palette color \spad{pal2},{} and point size is set to the positive integer \spad{ps}.
215912`\spad{components(s)} takes the \spadtype{ThreeSpace} \spad{s},{} and creates a list containing a unique \spadtype{ThreeSpace} for each single component of \spad{s}. If \spad{s} has no components defined,{} the list returned is empty.
215974`\spad{compose(a,{}b)} composes the power series \spad{a} with the power series \spad{b}.
216061`\spad{compose(p,{}q)} \undocumented
216123`\spad{composite([s1,{}s2,{}...,{}sn])} will create a new \spadtype{ThreeSpace} that is a union of all the components from each \spadtype{ThreeSpace} in the parameter list,{} grouped as a composite.
216362`\spad{composite(f,{} q)} returns \spad{h} if \spad{f} = \spad{h}(\spad{q}),{} and "failed" is no such \spad{h} exists.
216487`\spad{composite(p,{} q)} returns \spad{h} if \spad{p = h(q)},{} and "failed" no such \spad{h} exists.
216592`\spad{composites(s)} takes the \spadtype{ThreeSpace} \spad{s},{} and creates a list containing a unique \spadtype{ThreeSpace} for each single composite of \spad{s}. If \spad{s} has no composites defined (composites need to be explicitly created),{} the list returned is empty. Note that not all the components need to be part of a composite.
216654`\spad{compound(tr,{}inpt)} combine two transforms
216704`computeBasis is a helper function which computes a basis of a linear space which contains a lexicographic Groebner basis. It expects a basis (ordered by total degree,{} reverse lexicographic) of a zero dimensional ideal.
216873`\spad{computeCycleEntry(x,{}cycElt)},{} where \spad{cycElt} is a pointer to a node in the cyclic part of the cyclic stream \spad{x},{} returns a pointer to the first node in the cycle
216941`\spad{computeCycleLength(s)} returns the length of the cycle of a cyclic stream \spad{t},{} where \spad{s} is a pointer to a node in the cyclic part of \spad{t}.
217023`\spad{computeInt(x,{} g,{} a,{} b,{} eval?)} returns the integral of \spad{f} for \spad{x} between a and \spad{b},{} assuming that \spad{g} is an indefinite integral of \spad{f} and \spad{f} has no pole between a and \spad{b}. If \spad{eval?} is \spad{true},{} then \spad{g} can be evaluated safely at \spad{a} and \spad{b},{} provided that they are 
217023`--finite values. Otherwise,{} limits must be computed.
217180`\spad{computePowers()} \undocumented
217459`\spad{concat!(u,{}x)} destructively adds element \spad{x} to the end of \spad{u}.
217521`\spad{concat!(u,{}x)} destructively adds element \spad{x} to the end of \spad{u}. Note: \axiom{concat!(a,{}\spad{x}) = setlast!(a,{}[\spad{x}])}.
217605`\spad{concat!(u,{}v)} destructively concatenates doubly-linked aggregate \spad{v} to the end of doubly-linked aggregate \spad{u}.
217688`\spad{concat!(u,{}v)} destructively appends \spad{v} to the end of \spad{u}. \spad{v} is unchanged
217751`\spad{concat!(u,{}v)} destructively concatenates \spad{v} to the end of \spad{u}. Note: \axiom{concat!(\spad{u},{}\spad{v}) = setlast!(\spad{u},{}\spad{v})}.
218063`\spad{concat(u)},{} where \spad{u} is a lists of aggregates \axiom{[a,{}\spad{b},{}...,{}\spad{c}]},{} returns a single aggregate consisting of the elements of \axiom{a} followed by those of \spad{b} followed ... by the elements of \spad{c}. Note: \axiom{concat(a,{}\spad{b},{}...,{}\spad{c}) = concat(a,{}concat(\spad{b},{}...,{}\spad{c}))}.
218118`\spad{concat(u)} returns the left-to-right concatentation of the streams in \spad{u}. Note: \spad{concat(u) = reduce(concat,{}u)}.
218933`\spad{concat(x,{}u)} returns aggregate \spad{u} with additional element at the front. Note: for lists: \axiom{concat(\spad{x},{}\spad{u}) \spad{==} concat([\spad{x}],{}\spad{u})}.
218984`\spad{concat(x,{}u)} returns aggregate consisting of \spad{x} followed by the elements of \spad{u}. Note: if \axiom{\spad{v} = concat(\spad{x},{}\spad{u})} then \axiom{\spad{x} = first \spad{v}} and \axiom{\spad{u} = rest \spad{v}}.
219043`\spad{concat(u,{}x)} returns aggregate \spad{u} with additional element \spad{x} at the end. Note: for lists,{} \axiom{concat(\spad{u},{}\spad{x}) \spad{==} concat(\spad{u},{}[\spad{x}])}
219094`\spad{concat(u,{}v)} returns an aggregate consisting of the elements of \spad{u} followed by the elements of \spad{v}. Note: if \axiom{\spad{w} = concat(\spad{u},{}\spad{v})} then \axiom{\spad{w}.\spad{i} = \spad{u}.\spad{i} for \spad{i} in indices \spad{u}} and \axiom{\spad{w}.(\spad{j} + maxIndex \spad{u}) = \spad{v}.\spad{j} for \spad{j} in indices \spad{v}}.
219146`\spad{concat(u,{}v)} returns an aggregate \spad{w} consisting of the elements of \spad{u} followed by the elements of \spad{v}. Note: \axiom{\spad{v} = rest(\spad{w},{}\#a)}.
219206`\spad{cond(s,{}e)} creates a representation of the FORTRAN expression IF (\spad{s}) THEN \spad{e}.
219253`\spad{cond(s,{}e,{}f)} creates a representation of the FORTRAN expression IF (\spad{s}) THEN \spad{e} ELSE \spad{f}.
219479`\spad{conditionP(mat)},{} given a matrix representing a homogeneous system of equations,{} returns a vector whose characteristic'th powers is a non-trivial solution,{} or "failed" if no such vector exists.
219565`\spad{conditionP(m)} returns a vector of elements,{} not all zero,{} whose \spad{p}\spad{-}th powers (\spad{p} is the characteristic of the domain) are a solution of the homogenous linear system represented by \spad{m},{} or "failed" is there is no such vector.
219692`\axiom{condition(\spad{n})} returns the condition of the node \spad{n}.
219743`\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \spad{R}-module basis
219870`\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \spad{R}-module basis.
219967`\spad{conditionsForIdempotents()} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the fixed \spad{R}-module basis.
220162`\spad{conditionsForIdempotents([v1,{}...,{}vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}
220299`\spad{conditionsForIdempotents([v1,{}...,{}vn])} determines a complete list of polynomial equations for the coefficients of idempotents with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}.
220407`\axiom{conditions(a)} returns the list of the conditions of the leaves of a
220465`\spad{conical(a,{}b)} transforms from conical coordinates to Cartesian coordinates: \spad{conical(a,{}b)} is a function which will map the point \spad{(lambda,{}mu,{}nu)} to \spad{x = lambda*mu*nu/(a*b)},{} \spad{y = lambda/a*sqrt((mu^2-a^2)*(nu^2-a^2)/(a^2-b^2))},{} \spad{z = lambda/b*sqrt((mu^2-b^2)*(nu^2-b^2)/(b^2-a^2))}.
220536`implements Clifford conjugate for a multivector by involution and reverse of each term separately using: grade: 0 1 2 3... multi: 1 \spad{-1} \spad{-1} 1...
220647`\spad{conjug(x)}should be local but conditional
220865`\spad{conjugate(p)} returns the conjugate partition of a partition \spad{p}
220908`\spad{conjugate(pt)} is the conjugate of the partition \spad{pt}.
220989`\spad{conjugate(x + \%i y)} returns \spad{x} - \%\spad{i} \spad{y}.
221041`\spad{conjugate(o)} negates the imaginary parts \spad{i},{}\spad{j},{}\spad{k},{}\spad{E},{}\spad{I},{}\spad{J},{}\spad{K} of octonian \spad{o}.
221094`\spad{conjugate(q)} negates the imaginary parts of quaternion \spad{q}.
221187`\spad{conjugate(p,{}q)} computes \spad{inv(q) * p * q}; this is 'right action by conjugation'.
221229`\spad{conjugates(lp)} is the stream of conjugates of a stream \indented{1}{of partitions \spad{lp}.}
221327`\spad{connect(v,{}n,{}s)} displays the lines connecting the graph points in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} if \spad{s} is "on",{} or does not display the lines if \spad{s} is "off".
221406`\axiom{consRow!(A,{} \spad{r})} inserts the row \axiom{\spad{r}} at the top of the matrix \axiom{A}.
221505`\spad{cons(r,{} s)} prepends \spad{r} to the stream \spad{s}
221555`\spad{cons(element,{}u)} appends \spad{element} onto the front of list \spad{u} and returns the new list. This new list and the old one will share some structure.
221593`\spad{cons(a,{}s)} returns a stream whose \spad{first} is \spad{a} and whose \spad{rest} is \spad{s}. Note: \spad{cons(a,{}s) = concat(a,{}s)}.
221633`\spad{consnewpol should} be local
221903`\axiom{const?(\spad{f})} checks whether \axiom{\spad{f}} depends of jet variables.
221968`\spad{constDsolve(op,{} g,{} x)} returns \spad{[f,{} [y1,{}...,{}ym]]} where \spad{f} is a particular solution of the equation \spad{op y = g},{} and the \spad{\spad{yi}}\spad{'s} form a basis for the solutions of \spad{op y = 0}.
222063`\axiom{const(\spad{p})} coerces a polynomial into an element of the coefficient ring,{} if it is constant. Otherwise an error occurs.
222134`\spad{const c} is a function which produces \spad{c} when \indented{1}{applied to its argument.}
222187`\spad{constant?(p)} tests if \spad{p} contains no matching variables.
222236`\spad{constant?(p)} tests whether the polynomial \spad{p} belongs to the coefficient ring.
222295`\spad{constant?(x)} returns \spad{true} if \spad{x} is constant.
222352`\spad{constantCoefficientRicDE(op,{} ric)} returns \spad{[[a1,{} L1],{} [a2,{} L2],{} ... ,{} [ak,{} Lk]]} such that any rational solution with no polynomial part of the associated Riccati equation of \spad{op y = 0} must be one of the \spad{ai}\spad{'s} in which case the equation for \spad{z = y e^{-int \spad{ai}}} is \spad{\spad{Li} z = 0}. \spad{ric} is a Riccati equation solver over 
222352`--\spad{F},{} whose input is the associated linear equation.
222469`\spad{constantIfCan(k)} \undocumented
222550`\spad{constantKernel(r)} \undocumented
222616`\spad{constantLeft(f)} is the function \spad{g} \indented{1}{such that \spad{g (a,{}b)= f b}.}
222685`\spad{constantOpIfCan(op)} returns \spad{a} if \spad{op} is the constant nullary operator always returning \spad{a},{} "failed" otherwise.
222777`\spad{constantOperator(a)} returns a nullary operator op such that \spad{op()} always evaluate to \spad{a}.
222854`\spad{constantRight(f)} is the function \spad{g} \indented{1}{such that \spad{g (a,{}b)= f a}.}
222924`\spad{constantToUnaryFunction(s)} is a local function which takes the value of \spad{s},{} which may be a function of a constant,{} and returns a function which always returns the value \spadtype{DoubleFloat} \spad{s}.
223023`\spad{constant(p)} return the constant term of \spad{p}.
223075`\spad{vu(f)} is the function \spad{g} \indented{1}{such that \spad{g a= f ()}.}
223135`\spad{constant(x)} tells the pattern matcher that \spad{x} should match only the symbol \spad{'x} and no other quantity. Error: if \spad{x} is not a symbol.
223194`\spad{constant(x)} tells the pattern matcher that \spad{x} should match only the symbol \spad{'x} and no other quantity.
223270`\spad{constant(x)} returns the constant term of \spad{x}.
223320`\spad{constructOrdered(l)} takes a list of terms and creates the object with these components. The list is assumed to be sorted (in reverse order) with respect to the ordering of \spad{S} and no checking is performed. Caution: This should only be used in cases where this condition is assured. If in doubt use construct
223413`\spad{construct(lll)} creates a 3-\spad{D} matrix from a List List List \spad{R} \spad{lll}
223555`\spad{construct([v1,{}e1],{}...,{}[vn,{}en])} returns the match result containing the matches (\spad{v1},{}\spad{e1}),{}...,{}(\spad{vn},{}en).
223642`\axiom{construct(\spad{lvt})} returns the same as \axiom{[construct(\spad{vt}.val,{}\spad{vt}.tower) for \spad{vt} in \spad{lvt}]}
223966`\axiom{construct(\spad{vt})} returns the same as \axiom{construct(\spad{vt}.val,{}\spad{vt}.tower)}
224037`\axiom{construct(\spad{s})} creates a splitting tree with value (\spadignore{i.e.} root vertex) given by \axiom{\spad{s}} and no children. Thus,{} if the status of \axiom{\spad{s}} is \spad{false},{} \axiom{[\spad{s}]} represents the starting point of the evaluation \axiom{value(\spad{s})} under the hypothesis \axiom{condition(\spad{s})}.
224105`\spad{construct(l)} is the inverse of fracPart.
224229`\spad{construct(l)} takes a list of terms and creates the object with these components.
224315`\axiom{construct(\spad{x},{}\spad{y},{}...,{}\spad{z})} returns the collection of elements \axiom{\spad{x},{}\spad{y},{}...,{}\spad{z}} ordered as given. Equivalently written as \axiom{[\spad{x},{}\spad{y},{}...,{}\spad{z}]\$\spad{D}},{} where \spad{D} is the domain. \spad{D} may be omitted for those of type List.
224367`\spad{construct(t)} converts an infinite tuple to a stream.
224424`\spad{construct(a,{}b)} creates element of the product with components a and \spad{b}.
224471`\spad{construct(i,{}e)} \undocumented
224530`\axiom{construct(\spad{x},{}\spad{y})} returns the Lie bracket \axiom{[\spad{x},{}\spad{y}]}.
224622`\axiom{construct(\spad{x},{}\spad{y})} returns the Lie bracket \axiom{[\spad{x},{}\spad{y}]}.
224698`\spad{construct(s,{}t)} makes a form \spad{s:t}
224750`\axiom{construct(\spad{v},{}\spad{t})} returns the same as \axiom{construct(\spad{v},{}\spad{t},{}\spad{false})}
224803`\axiom{construct(\spad{v},{}\spad{lt})} returns the same as \axiom{[construct(\spad{v},{}\spad{t}) for \spad{t} in \spad{lt}]}
224868`\axiom{construct(\spad{x},{}\spad{y})} returns the Lie bracket \axiom{[\spad{x},{}\spad{y}]}.
224944`\axiom{construct(\spad{x},{}\spad{y})} returns the Lie bracket of \axiom{\spad{x}} and \axiom{\spad{y}}.
224994`\spad{construct(type,{}dims)} creates an element of FortranType
225111`\spad{construct(type,{}dims)} creates an element of FortranType
225215`\axiom{construct(\spad{v},{}\spad{t},{}\spad{b})} returns the non-empty node with value \spad{v},{} condition \spad{t} and flag \spad{b}
225276`\axiom{construct(\spad{v},{}\spad{t},{}\spad{ls})} creates a splitting tree with value (\spadignore{i.e.} root vertex) given by \axiom{[\spad{v},{}\spad{t}]\$\spad{S}} and with children list given by \axiom{[[\spad{s}]\$\% for \spad{s} in \spad{ls}]}.
225354`\axiom{construct(\spad{v},{}\spad{t},{}la)} creates a splitting tree with value (\spadignore{i.e.} root vertex) given by \axiom{[\spad{v},{}\spad{t}]\$\spad{S}} and with \axiom{la} as children list.
225416`\axiom{construct(\spad{v1},{}\spad{t},{}\spad{v2},{}\spad{lt})} creates a splitting tree with value (\spadignore{i.e.} root vertex) given by \axiom{[\spad{v},{}\spad{t}]\$\spad{S}} and with children list given by \axiom{[[[\spad{v},{}\spad{t}]\$\spad{S}]\$\% for \spad{s} in \spad{ls}]}.
225479`\spad{construct(mom,{} ccum,{} fcum,{} bcum)} constructs a distribution with moments \spad{mom},{} classical cumulants \spad{ccum},{} free cumulants \spad{fcum} and boolean cumulants \spad{bcum}. The user must make sure that these are consistent,{} otherwise the results are unpredictable!
225573`\spad{contains?(i,{}f)} returns \spad{true} if \axiom{\spad{f}} is contained within the interval \axiom{\spad{i}},{} \spad{false} otherwise.
225633`\spad{containsPoint?(n,{}p)} returns \spad{true} if the point is inside \indented{2}{the boundary.} \spad{n} is the boundary \spad{p} is the point
225693`\spad{content([f1,{}..,{}fk])} = content of a list of univariate polynonials
225988`\spad{content(p)} gives the \spad{gcd} of the coefficients of polynomial \spad{p}.
226063`\spad{content(l)} returns the \spad{gcd} of all the coefficients of \spad{l}.
226291`\spad{content(p,{}v)} is the \spad{gcd} of the coefficients of the polynomial \spad{p} when \spad{p} is viewed as a univariate polynomial with respect to the variable \spad{v}. Thus,{} for polynomial 7*x^2*y + 14*x*y^2,{} the \spad{gcd} of the coefficients with respect to \spad{x} is 7*y.
226376`\spad{continue(l)} creates a representation of a FORTRAN CONTINUE labelled with \spad{l}
226613`\spad{continuedFraction(x)} converts the \spad{p}-adic rational number \spad{x} to a continued fraction.
226722`\spad{continuedFraction(f)} converts the floating point number \spad{f} to a reduced continued fraction.
226814`\spad{continuedFraction(r)} converts the fraction \spadvar{\spad{r}} with components of type \spad{R} to a continued fraction over \spad{R}.
226885`\spad{continuedFraction(b0,{}a,{}b)} constructs a continued fraction in the following way: if \spad{a = [a1,{}a2,{}...]} and \spad{b = [b1,{}b2,{}...]} then the result is the continued fraction \spad{b0 + a1/(b1 + a2/(b2 + ...))}.
226966`given a mapping from this graph this function tries to calculate a unique reverse mapping back to this graph
227083`given a mapping from this graph this function tries to calculate a unique reverse mapping back to this graph
227205`\spad{contractSolve(eq,{}x)} finds the solutions expressed in terms of radicals of the equation of rational functions \spad{eq} with respect to the symbol \spad{x}. The result contains new symbols for common subexpressions in order to reduce the size of the output.
227360`\spad{contractSolve(rf,{}x)} finds the solutions expressed in terms of radicals of the equation \spad{rf} = 0 with respect to the symbol \spad{x},{} where \spad{rf} is a rational function. The result contains new symbols for common subexpressions in order to reduce the size of the output.
227505`\axiom{contract(\spad{v},{} om)} computes the interior derivative of \axiom{om} with respect to \axiom{\spad{v}}.
227579`\spad{contract(I,{}lvar)} contracts the ideal \spad{I} to the polynomial ring \spad{F[lvar]}.
227975`\spad{contract(t,{}i,{}j)} is the contraction of tensor \spad{t} which sums along the \spad{i}\spad{-}th and \spad{j}\spad{-}th indices. For example,{} if \spad{r = contract(t,{}1,{}3)} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank 2 \spad{(= 4 - 2)} tensor given by \indented{4}{\spad{r(i,{}j) = sum(h=1..dim,{}t(h,{}i,{}h,{}j))}.}
228052`\spad{contract(t,{}i,{}s,{}j)} is the inner product of tenors \spad{s} and \spad{t} which sums along the \spad{k1}\spad{-}th index of \spad{t} and the \spad{k2}\spad{-}th index of \spad{s}. For example,{} if \spad{r = contract(s,{}2,{}t,{}1)} for rank 3 tensors rank 3 tensors \spad{s} and \spad{t},{} then \spad{r} is the rank 4 \spad{(= 3 + 3 - 2)} tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = sum(h=1..dim,{}s(i,{}h
228052`--,{}j)*t(h,{}k,{}l))}.}
228132`\spad{controlPanel(v,{}s)} displays the control panel of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} if \spad{s} is "on",{} or hides the control panel if \spad{s} is "off".
228200`\spad{controlPanel(v,{}s)} displays the control panel of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} if \spad{s} is "on",{} or hides the control panel if \spad{s} is "off".
228270`\spad{convergents(x)} returns the stream of the convergents of the continued fraction \spadvar{\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.
228702`\spad{convert(cs)} converts the complex expression \spad{cs} to a pattern
228818`\axiom{convert(\spad{p})} converts a polynomial \axiom{\spad{p}} in recursive representation into a polynomial in distributive representation.
228915`\axiom{convert(\spad{lp})} returns the polynomial set whose members are the polynomials of \axiom{\spad{lp}}.
228986`\spad{convert([a1,{}...,{}an])} returns the pattern \spad{[a1,{}...,{}an]}.
231333`\spad{convert(a)} transforms a into element of \%.
231382`\spad{convert(s)} makes \spad{s} into an input form.
231790`\axiom{convert(\spad{p})} converts a polynomial \axiom{\spad{p}} in distributive representation into a polynomial in recursive representation.
231939`\spad{convert(x)} converts a \spadtype{DoubleFloat} \spad{x} to a \spadtype{Float}.
231985`\spad{convert(x)} returns the Lisp atom \spad{x}.
232060`\spad{convert(f1\^e1 ... fm\^em)} returns \spad{(f1)\^e1 ... (fm)\^em} as an element of \%,{} using formal kernels created using a \spadfunFrom{paren}{ExpressionSpace}.
232139`\spad{convert(x)} returns the Lisp atom \spad{x}.
232213`\spad{convert(x)} returns the Lisp atom \spad{x}.
232287`\spad{convert(l)} takes a list of elements,{} \spad{l},{} from the domain Ring and returns the form of point category.
232340`\spad{convert([a1,{}...,{}an])} returns the \spad{S}-expression \spad{(a1,{}...,{}an)}.
232419`\spad{convert(q)} converts \spad{q}.
232601`\spad{convert(q)} converts \spad{q}.
232761`\axiom{convert(\spad{p})} returns the same as \axiom{retract(\spad{p})}.
232921`\axiom{convert(\spad{p})} returns the same as \axiom{retract(\spad{p})}.
233071`\axiom{convert(\spad{p})} returns the same as \axiom{retract(\spad{p})}.
233250`\spad{convert(p)} converts \spad{p}.
233364`\axiom{convert(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}},{} otherwise an error is produced.
233677`\spad{convert(i)} creates the segment \spad{i..i}.
233726`\spad{convert(u)} converts \spad{u}.
233872`\spad{convert(st)} returns the members of \spad{st}.
234163`\spad{convert(x)} returns the Lisp atom \spad{x}.
234237`\spad{convert(x)} returns the Lisp atom \spad{x}.
234311`\spad{convert(n)} creates a roman numeral for symbol \spad{n}.
234359`\spad{convert(up)} converts the univariate polynomial \spad{up} to an algebra element,{} reducing by the \spad{definingPolynomial()} if necessary.
234413`\spad{convert([a1,{}..,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
234471`\spad{convert([a1,{}...,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed \spad{R}-module basis.
234540`\spad{convert(a)} transforms a into an element of \spad{S}.
234587`\spad{convert(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
234645`\spad{convert(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
234714`\spad{convert(o,{}step)} changes \spad{o} in standard output format to SCRIPT formula format and also adds the given \spad{step} number. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \spad{step} numbers.
234781`\spad{convert(o,{}step)} changes \spad{o} in standard output format to TeX format and also adds the given \spad{step} number. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \spad{step} numbers.
234838`\spad{convert(o,{}step,{}type)} changes \spad{o} in standard output format to TeX format and also adds the given \spad{step} number and \spad{type}. This is useful if you want to create equations with given numbers or have the equation numbers correspond to the interpreter \spad{step} numbers.
234906`\spad{coord(p)} specifies a change of coordinates of point \spad{p}. This option is expressed in the form \spad{coord == p}.
234984`\spad{coord }\undocumented
235150`\spad{coord(l,{}p)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{coord}. If the option does not exist the value,{} \spad{p} is returned.
235293`\spad{coordinate(c,{}i)} returns a coordinate function for \spad{c} using 1-based indexing according to \spad{i}. This indicates what the function for the coordinate component \spad{i} of the plane curve is.
235401`\spad{coordinate(c,{}i)} returns a coordinate function of \spad{c} using 1-based indexing according to \spad{i}. This indicates what the function for the coordinate component,{} \spad{i},{} of the space curve is.
235509`\spad{coordinate(s,{}i)} returns a coordinate function of \spad{s} using 1-based indexing according to \spad{i}. This indicates what the function for the coordinate component,{} \spad{i},{} of the surface is.
235614`\spad{coordinatesIfCan(v,{}B)} either computes the coordinates of the vector \spad{v} in the basis \spad{B} either returns \spad{"failed"} if vector \spad{v} isn\spad{'t} in the subspace described by \spad{B}.
236257`\spad{coordinates(p)} specifies a change of coordinate systems of point \spad{p}. This option is expressed in the form \spad{coordinates == p}.
236341`\spad{coordinates([v1,{}...,{}vm])} returns the coordinates of the \spad{vi}\spad{'s} with to the fixed basis. The coordinates of \spad{vi} are contained in the \spad{i}th row of the matrix returned by this function.
236424`\spad{coordinates([v1,{}...,{}vm])} returns the coordinates of the \spad{vi}\spad{'s} with to the fixed basis. The coordinates of \spad{vi} are contained in the \spad{i}th row of the matrix returned by this function.
236494`\spad{coordinates([a1,{}...,{}am])} returns a matrix whose \spad{i}-th row is formed by the coordinates of \spad{\spad{ai}} with respect to the fixed \spad{R}-module basis.
236575`\spad{coordinates(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{F}-vector space basis.
236650`\spad{coordinates(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
236712`\spad{coordinates(a)} returns the coordinates of \spad{a} with respect to the fixed \spad{R}-module basis.
237159`\spad{coordinates(v,{}B)} computes the coordinates of the vector \spad{v} in the basis \spad{B} either returns \spad{"failed"},{} and produces an error if vector \spad{v} isn\spad{'t} in the subspace described
237645`\spad{coordinates(a,{}[v1,{}...,{}vn])} returns the coordinates of \spad{a} with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}.
237740`\spad{coordinates([v1,{}...,{}vm],{} basis)} returns the coordinates of the \spad{vi}\spad{'s} with to the basis \spad{basis}. The coordinates of \spad{vi} are contained in the \spad{i}th row of the matrix returned by this function.
237825`\spad{coordinates([a1,{}...,{}am],{}[v1,{}...,{}vn])} returns a matrix whose \spad{i}-th row is formed by the coordinates of \spad{\spad{ai}} with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}.
237921`\spad{coordinates(a,{}basis)} returns the coordinates of \spad{a} with respect to the \spad{basis} \spad{basis}.
237998`\spad{coordinates(a,{}[v1,{}...,{}vn])} returns the coordinates of \spad{a} with respect to the \spad{R}-module basis \spad{v1},{}...,{}\spad{vn}.
238086`\spad{copies(i,{}s)} will take a string \spad{s} and create a new string composed of \spad{i} copies of \spad{s}.
238147`\spad{coproduct(x)} computes the coproduct of an element \spad{x}
238198`\spad{copy!(c,{}a)} copies the matrix \spad{a} into the matrix \spad{c}. Error: if \spad{a} and \spad{c} do not have the same dimensions.
238414`\spad{copyInto!(u,{}v,{}i)} returns aggregate \spad{u} containing a copy of \spad{v} inserted at element \spad{i}.
238507`\spad{copy_first(v1,{} v2,{} n)} copies first \spad{n} elements of \spad{v2} into \spad{n} first positions in \spad{v1}.
238598`copy_first(\spad{v1},{} \spad{v2},{} \spad{m},{} \spad{n}) copies the slice of \spad{v2} starting at \spad{m} elements and having \spad{n} elements into corresponding positions in \spad{v1}.
238996`\axiom{copy(om)} returns a copy of the differential \axiom{om}.
239046`\axiom{copy(De)} returns a copy of the equation \axiom{De}.
239104`\axiom{copy(\spad{v})} returns a copy of the vector field \axiom{\spad{v}}.
239153`\spad{copy(p)} returns a recursive copy of \spad{p}.
239192`\axiom{copy(A)} returns a copy of the matrix \axiom{A}.
239245`\axiom{copy(\spad{n})} returns a copy of \spad{n}.
239292`\spad{copy(x)} \undocumented
239346`\spad{copy(u)} returns a top-level (non-recursive) copy of \spad{u}. Note: for collections,{} \axiom{copy(\spad{u}) \spad{==} [\spad{x} for \spad{x} in \spad{u}]}.
239384`\spad{copy(n)} gives a copy of \spad{n}.
239432`\spad{copy(s)} returns a new \spadtype{ThreeSpace} that is an exact copy of \spad{s}.
239482`\spad{copy(op)} returns a copy of \spad{op}.
239524`\spad{corrPoly(u,{}lv,{}lr,{}ln,{}lu,{}t,{}r)} \undocumented
239786`\spad{cos2sec(f)} converts every \spad{cos(u)} appearing in \spad{f} into \spad{1/sec(u)}.
239848`\spad{cosIfCan(z)} returns cos(\spad{z}) if possible,{} and "failed" otherwise.
239928`\spad{cosSinInfo(n)} returns the list of lists of values for \spad{n},{} in the form: \spad{[[cos(n - 1) a,{}sin(n - 1) a],{}...,{}[cos 2 a,{}sin 2 a],{}[cos a,{}sin a]]} where \spad{a = 2 pi/n}. Note: \spad{n} should be greater than 2.
240002`\spad{cos(x)} makes a cos kernel for use in Fourier series
240048`\spad{cos(x)} applies the cosine operator to \spad{x}
240199`\spad{cos(st)} computes cosine of a power series \spad{st}.
240282`\spad{cos(st)} computes cosine of a power series \spad{st}.
240379`\spad{cos(z)} returns the cosine of Laurent series \spad{z}.
240465`\spad{cos(z)} returns the cosine of a Puiseux series \spad{z}.
240560`\spad{cos(x)} represents the Fortran intrinsic function COS
240640`\spad{cos(x)} returns the cosine of \spad{x}.
240697`\spad{cosh2sech(f)} converts every \spad{cosh(u)} appearing in \spad{f} into \spad{1/sech(u)}.
240761`\spad{coshIfCan(z)} returns cosh(\spad{z}) if possible,{} and "failed" otherwise.
240842`\spad{cosh(x)} applies the hyperbolic cosine operator to \spad{x}
241048`\spad{cosh(st)} computes the hyperbolic cosine of a power series \spad{st}.
241132`\spad{cosh(st)} computes the hyperbolic cosine of a power series \spad{st}.
241230`\spad{cosh(z)} returns the hyperbolic cosine of Laurent series \spad{z}.
241317`\spad{cosh(z)} returns the hyperbolic cosine of a Puiseux series \spad{z}.
241413`\spad{cosh(x)} represents the Fortran intrinsic function COSH
241494`\spad{cosh(x)} returns the hyperbolic cosine of \spad{x}.
241549`\spad{cot2tan(f)} converts every \spad{cot(u)} appearing in \spad{f} into \spad{1/tan(u)}.
241611`\spad{cot2trig(f)} converts every \spad{cot(u)} appearing in \spad{f} into \spad{cos(u)/sin(u)}.
241674`\spad{cotIfCan(z)} returns cot(\spad{z}) if possible,{} and "failed" otherwise.
241754`\spad{cot(x)} applies the cotangent operator to \spad{x}
241917`\spad{cot(st)} computes cotangent of a power series \spad{st}.
242000`\spad{cot(st)} computes cotangent of a power series \spad{st}.
242097`\spad{cot(z)} returns the cotangent of Laurent series \spad{z}.
242183`\spad{cot(z)} returns the cotangent of a Puiseux series \spad{z}.
242278`\spad{cot(x)} returns the cotangent of \spad{x}.
242335`\spad{coth2tanh(f)} converts every \spad{coth(u)} appearing in \spad{f} into \spad{1/tanh(u)}.
242399`\spad{coth2trigh(f)} converts every \spad{coth(u)} appearing in \spad{f} into \spad{cosh(u)/sinh(u)}.
242464`\spad{cothIfCan(z)} returns coth(\spad{z}) if possible,{} and "failed" otherwise.
242545`\spad{coth(x)} applies the hyperbolic cotangent operator to \spad{x}
242708`\spad{coth(st)} computes the hyperbolic cotangent of a power series \spad{st}.
242792`\spad{coth(st)} computes the hyperbolic cotangent of a power series \spad{st}.
242890`\spad{coth(z)} returns the hyperbolic cotangent of Laurent series \spad{z}.
242977`\spad{coth(z)} returns the hyperbolic cotangent of a Puiseux series \spad{z}.
243073`\spad{coth(x)} returns the hyperbolic cotangent of \spad{x}.
243128`\spad{counit(x)} evaluates the counit at an element \spad{x}
243174`\spad{countRealRootsMultiple(p)} says how many real roots \spad{p} has,{} counted with multiplicity
243289`\spad{countRealRoots(p)} says how many real roots \spad{p} has
244372`\spad{count(p,{}u)} returns the number of elements \spad{x} in \spad{u} such that \axiom{\spad{p}(\spad{x})} is \spad{true}. For collections,{} \axiom{count(\spad{p},{}\spad{u}) = reduce(+,{}[1 for \spad{x} in \spad{u} | \spad{p}(\spad{x})],{}0)}.
244477`\spad{count(x,{}u)} returns the number of occurrences of \spad{x} in \spad{u}. For collections,{} \axiom{count(\spad{x},{}\spad{u}) = reduce(+,{}[x=y for \spad{y} in \spad{u}],{}0)}.
244593`\spad{countable?(\spad{a})} determines whether \spad{a} is a countable cardinal,{} \spadignore{i.e.} an integer or \spad{Aleph 0}.
244647`\spad{create3Space()} creates a \spadtype{ThreeSpace} object capable of holding point,{} curve,{} mesh components and any combination.
244703`\spad{create3Space(s)} creates a \spadtype{ThreeSpace} object containing objects pre-defined within some \spadtype{SubSpace} \spad{s}.
244772`\spad{createArrows2Din2D(ptFun,{}uSeg,{}vSeg,{}numPts)} creates arrows to represent output for each input point in other words this represents a two dimensional force field with the arrows showing the direction of the field
244891`\spad{createGenericMatrix(m)} creates a square matrix of dimension \spad{k} whose entry at the \spad{i}-th row and \spad{j}-th column is the indeterminate {\em x[i,{}j]} (double subscripted).
244995`\spad{createIrreduciblePoly(n)}\$FFPOLY(\spad{GF}) generates a monic irreducible univariate polynomial of degree \spad{n} over the finite field {\em GF}.
245114`\spad{createLowComplexityNormalBasis(n)} tries to find a a low complexity normal basis of degree {\em n} over {\em GF} and returns its multiplication matrix If no low complexity basis is found it calls \axiomFunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}(\spad{n}) to produce a normal polynomial of degree {\em n} over {
245114`--\em GF}
245292`\spad{createLowComplexityTable(n)} tries to find a low complexity normal basis of degree {\em n} over {\em GF} and returns its multiplication matrix Fails,{} if it does not find a low complexity basis
245442`\spad{createMultiplicationMatrix(m)} forms the multiplication table {\em m} into a matrix over the ground field.
245573`\spad{createMultiplicationTable(f)} generates a multiplication table for the normal basis of the field extension determined by {\em f}. This is needed to perform multiplications between elements represented as coordinate vectors to this basis. See \spadtype{FFNBP},{} \spadtype{FFNBX}.
245794`\spad{createNormalElement()} computes a normal element over the ground field \spad{F},{} that is,{} \spad{a^(q^i),{} 0 <= i < extensionDegree()} is an \spad{F}-basis,{} where \spad{q = size()\$F}. Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35.
245881`\spad{createNormalPoly(n)}\$FFPOLY(\spad{GF}) generates a normal polynomial of degree \spad{n} over the finite field {\em GF}.
245995`\spad{createNormalPrimitivePoly(n)}\$FFPOLY(\spad{GF}) generates a normal and primitive polynomial of degree \spad{n} over the field {\em GF}. Note: this function is equivalent to createPrimitiveNormalPoly(\spad{n})
246118`\spad{createPlot1Din2D(f,{}tRange,{}numPts)} creates two dimensional function plot. Draws the graph of \spad{y} = \spad{f}(\spad{x}) as \spad{x} ranges from min(a,{}\spad{b}) to max(a,{}\spad{b}). \spad{tRange} is a..\spad{b}
246232`\spad{createPlot1Din2D(f,{}tRange,{}numPts)} creates a two dimensional function plot draws the graph from a mapping of float to point as \spad{x} ranges from min(a,{}\spad{b}) to max(a,{}\spad{b}). \spad{tRange} is a..\spad{b}
246337`\spad{createPlot1Din2Dparametric(ppc,{}tRange,{}numPts)} creates a node from plot using Parametric Plane Curve. This represents 1 dimension (line - possibly curved) in 2 dimensions (plane) In theory a line has no width but in that case we would not see it so we give it a width given by the material node that is applicable in this part of the scene graph \spad
246337`--{PPC} is ParametricPlaneCurve(\spad{DF} \spad{->} \spad{DF}) which is created with curve(\spad{f1},{}\spad{f2}) where \spad{f1} and \spad{f2} are functions of type ComponentFunction,{} in this case \spad{DF} \spad{->} \spad{DF}
246483`\spad{createPlot1Din3Dparametric(pcfun,{}tRange,{}numPts)} create a line (1D subspace) in 3D space. This represents 1 dimension (line - possibly curved) in 3 dimensions In theory a line has no width but in that case we would not see it so we give it a width given by the material node that is applicable in this part of the scene graph PCFUN is a function from float to point: \spad{DF} \spad
246483`--{->} \spad{PT}
246598`\spad{createPlot1Din3Dparametric(psc,{}tRange,{}numPts)} create a line (1D subspace) in 3D space. This represents 1 dimension (line - possibly curved) in 3 dimensions In theory a line has no width but in that case we would not see it so we give it a width given by the material node that is applicable in this part of the scene graph \spad{PSC} ParametricSpaceC
246598`--urve(\spad{DF} \spad{->} \spad{DF}) is created with curve(\spad{f1},{}\spad{f2},{}\spad{f3}) where \spad{f1},{}\spad{f2} and \spad{f3} are functions of type ComponentFunction,{} in this case \spad{DF} \spad{->} \spad{DF}
246744`\spad{createPlot2Din3D(f,{}a..b,{}c..d)} returns a scene node which contains the graph of \spad{z} = \spad{f}(\spad{x},{}\spad{y}) as \spad{x} ranges from min(a,{}\spad{b}) to max(a,{}\spad{b}) and \spad{y} ranges from min(\spad{c},{}\spad{d}) to max(\spad{c},{}\spad{d}). create a surface (2D subspace) in 3D space The surface is approximated by polygons whic
246744`--h are represented by in indexed face set (IFS) node
246891`\spad{createPlot2Din3D(ptFun,{}uSeg,{}vSeg,{}numPts)} create a surface (2D subspace) in 3D space. The surface is approximated by polygons which are represented by in indexed face set (IFS) node
247029`\spad{createPlot2Din3Dparametric(f,{}a..b,{}c..d,{}l)} returns a scene node which contains the graph of the parametric surface \spad{f}(\spad{u},{}\spad{v}) as \spad{u} ranges from min(a,{}\spad{b}) to max(a,{}\spad{b}) and \spad{v} ranges from min(\spad{c},{}\spad{d}) to max(\spad{c},{}\spad{d}). This creates a surface (2D subspace) in 3D space The surface
247029`-- is approximated by polygons which are represented by in indexed face set (IFS) node
247177`\spad{createPlot2Din3Dparametric(surface(f,{}g,{}h),{}a..b,{}c..d,{}l)} returns a scene node which contains the graph of the parametric surface \spad{x} = \spad{f}(\spad{u},{}\spad{v}),{} \spad{y} = \spad{g}(\spad{u},{}\spad{v}),{} \spad{z} = \spad{h}(\spad{u},{}\spad{v}) as \spad{u} ranges from min(a,{}\spad{b}) to max(a,{}\spad
247177`--{b}) and \spad{v} ranges from min(\spad{c},{}\spad{d}) to max(\spad{c},{}\spad{d}). This creates a surface (2D subspace) in 3D space The surface is approximated by polygons which are represented by in indexed face set (IFS) node
247415`\spad{createPrimitiveElement()} computes a generator of the (cyclic) multiplicative group of the field.
247479`\spad{createPrimitiveNormalPoly(n)}\$FFPOLY(\spad{GF}) generates a normal and primitive polynomial of degree \spad{n} over the field {\em GF}. polynomial of degree \spad{n} over the field {\em GF}.
247602`\spad{createPrimitivePoly(n)}\$FFPOLY(\spad{GF}) generates a primitive polynomial of degree \spad{n} over the finite field {\em GF}.
247719`\spad{createRandomElement(aG,{}x)} creates a random element of the group algebra generated by {\em aG}.
247818`\spad{createSceneArrow(st,{}en,{}offset,{}mode,{}size)} constructs an arrow node,{} in \spad{n}-dimensional space,{} going from \spad{st} to \spad{en}. 'st' and 'en' are determined from a list of points from named points node. \spad{st} is the start of the arrow \spad{en} is the end of the arrow 'mode' and 'size' parameters determine the \spad{size} of the arrowhead. \spad{mode} can have the following values: \indented{2
247818`--}{"fixed"::Symbol \spad{
247818`--} fixed \spad{size} line width given by 'size' parameter} \indented{2}{"proportional"::Symbol \spad{
247818`--} \spad{size} as a proportion of the overall bounds} \indented{2}{"variable"::Symbol \spad{
247818`--} \spad{size} as a proportion of the arrow length} So "proportional" would typically be used when drawing a graph (in graph theory) where it looks better if each arrow head is the same. "variable" would typically be used when drawing a force field where a bigger arrow head mig
247818`--ht indicate a stronger force.
247901`\spad{createSceneArrow(st,{}en,{}offset,{}mode,{}size,{}fontScale)} constructs an arrow node,{} in \spad{n}-dimensional space,{} going from \spad{st} to \spad{en}. 'st' and 'en' are nodes in the scene and the arrows will be drawn upto their boundary. \spad{st} is the node at the start of the arrow \spad{en} is the node at the end of the arrow 'mode' and 'size' parameters determine the \spad{size} of the arrowhead. \s
247901`--pad{mode} can have the following values: \indented{2}{"fixed"::Symbol \spad{
247901`--} fixed \spad{size} line width given by 'size' parameter} \indented{2}{"proportional"::Symbol \spad{
247901`--} \spad{size} as a proportion of the overall bounds} \indented{2}{"variable"::Symbol \spad{
247901`--} \spad{size} as a proportion of the arrow length} So "proportional" would typically be used when drawing a graph (in graph theory) where it looks better if each arrow head is the same. "variable" would typically be used when
247901`-- drawing a force field where a bigger arrow head might indicate a stronger force. \spad{fontScale} should be set to the full width of the diagram (not just the width of this node). If this is not known then choose an approximate value. This only affects text nodes,{} if \spad{st} or \spad{en},{} or its subnodes,{} do not contain a text node then its value is not significant.
247988`\spad{createSceneArrows(line,{}mode,{}size)} constructs an arrow node,{} this contains arrows (represented by a list of points) in \spad{n}-dimensional space. The space is implicit in the \spad{PT} and \spad{TR} parameters supplied when this Scene was created. 'mode' and 'size' parameters determine the \spad{size} of the arrowhead. \spad{mode} can have the following values: \indented{2}{"fixed"::Symbol \spad{
247988`--} fixed \sp
247988`--ad{size} \spad{line} width given by 'size' parameter} \indented{2}{"proportional"::Symbol \spad{
247988`--} \spad{size} as a proportion of the overall bounds} \indented{2}{"variable"::Symbol \spad{
247988`--} \spad{size} as a proportion of the arrow length} So "proportional" would typically be used when drawing a graph (in graph theory) where it looks better if each arrow head is the same. "variable" would typically be used when drawing a force field where a bigger arrow head might indicate a stronger force.
248070`\spad{createSceneBox(size)} constructs an indexed face set node which is a 3D box of a given \spad{size}
248127`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
248197`\spad{createSceneClip(bb)} constructs a clip node,{} clips its sub nodes in the coordinate system in force at the clip node.
248257`\spad{createSceneDef(nam,{}nde)} defines a point in the scenegraph so that it can be used elsewhere.
248312`\spad{createSceneGraph(g,{}x,{}y,{}dispArrowName)} creates a graph theory diagram
248427`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
248497`\spad{createSceneGrid(bb)} constructs a grid with narrow blue lines every 20 units wide blue lines every 100 units wide red lines every 200 units
248557`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
248639`\spad{createSceneGrid(stepSize,{}bb)} constructs a grid,{} consists of a group,{} under which are vertical and horizontal lines. The spacing between lines is given by step.
248711`\spad{createSceneGroup()} constructs a group node,{} this node does not do anything itself but contains other nodes
248759`\spad{createSceneIFS(in1)} constructs an indexed face set node,{} this defines a surface represented by a set of polygons in \spad{n}-dimensional space.
248817`\spad{createSceneIFS(inx,{}pts)} constructs an indexed face set node,{} this defines a surface represented by a set of polygons in \spad{n}-dimensional space.
248902`\spad{createSceneLine(line)} constructs a \spad{line} node,{} this contains a \spad{line} (represented by a list of points) in \spad{n}-dimensional space. The space is implicit in the \spad{PT} and \spad{TR} parameters supplied when this Scene was created.
248957`\spad{createSceneLine(st,{}en,{}fontScale)} constructs a line node,{} this contains a line,{} in \spad{n}-dimensions,{} from \spad{st} to \spad{en}. 'st' and 'en' are nodes in the scene and the line will be drawn upto their boundary. \spad{st} is the node at the start of the line \spad{en} is the node at the end of the line \spad{fontScale} should be set to the full width of the diagram (not just the width of this node). If this is not kno
248957`--wn then choose an approximate value. This only affects text nodes,{} if \spad{st} or \spad{en},{} or its subnodes,{} do not contain a text node then its value is not significant.
249021`\spad{createSceneLines(line)} constructs a \spad{line} node,{} this contains lines (represented by a list of points) in \spad{n}-dimensional space. The space is implicit in the \spad{PT} and \spad{TR} parameters supplied when this Scene was created.
249083`\spad{createSceneMaterial(mat)} constructs a material node This sets the lineWidth,{}lineCol and fillCol for all nodes under this,{} unless overridden by another material node. That is the material parameters that apply to a given node are those of the closest material node above it in the hierarchy line width is expressed relative to the width of current boundary
249216`\spad{createSceneMaterial(lineW,{}lineC,{}fillC)} constructs a material node This sets the lineWidth,{}lineCol and fillCol for all nodes under this,{} unless overridden by another material node. That is the material parameters that apply to a given node are those of the closest material node above it in the hierarchy
249292`\spad{createSceneNamedPoints(np)} constructs a named points node,{} this allows us to define a set of points which can be used multiple times in the scenegraph.
249366`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
249465`\spad{createScenePattern(ptype,{}step,{}bb)} creates a test pattern suitable for showing the effect of transforms. The pattern depends on \spad{ptype} parameter as follows: ptype="GRID"::Symbol : construct a set of horizontal and vertical lines in the current clip boundary and current material with a spacing between lines given by the \spad{step} parameter. ptype="SIERPINSKI"::Symbol: constructs a Sierpinski fracta
249465`--l. \spad{step} parameter gives the level of subdivision. ptype="HOUSE"::Symbol: constructs a house shape.
249554`\spad{createSceneRoot()} constructs the root node,{} all other nodes are contained in a tree structure under this node
249601`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
249671`\spad{createSceneRoot(bb)} constructs the root node,{} all other nodes are contained in a tree structure under this node
249731`\spad{createSceneRoot(minx,{}miny,{}maxx,{}maxy)} constructs the root node,{} all other nodes are contained in a tree structure under this node
249809`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
249890`\spad{createSceneRuler(ptype,{}offset,{}bb)} creates a scale that can be used to provide numeric values for an axis: ptype= "HORIZONTAL"::Symbol : horizontal axis. ptype= "VERTICAL"::Symbol: vertical axis. ptype= "DEBTH"::Symbol: depth axis.
249961`\spad{createSceneShape(shape)} constructs a \spad{shape} node,{} this contains a \spad{shape} such as a rectangle or an ellipse.
250062`\spad{createSceneShape(shape)} constructs a \spad{shape} node from a boundary,{} this contains a \spad{shape} such as a rectangle or an ellipse.
250123`\spad{createSceneText(text)} constructs a \spad{text} node,{} \spad{text} can be used for labelling anything such as graphs,{} axes and so on.
250234`\spad{createSceneText(str,{}sz,{}pz)} constructs a text node using information from named points node.
250315`\spad{createSceneText(str,{}sz,{}pz)} constructs a text node,{} text can be used for labelling anything such as graphs,{} axes and so on.
250390`\spad{createSceneTransform(tran)} constructs a transform node This transforms the points and vectors below this node If a given node has more than one transform node above it in the hierarchy then the transforms are compound
250456`\spad{createSceneUse(nam)} uses another point in the scenegraph.
250508`\spad{createThreeSpace()} creates a \spadtype{ThreeSpace(DoubleFloat)} object capable of holding point,{} curve,{} mesh components and any combination.
250586`\spad{createWidth(x)} can be used by domains which extend graph to help in creating coordinates for objects in a graph
250668`\spad{createWidth(x)} can be used by domains which extend graph to help in creating coordinates for objects in a graph
250753`\spad{createX(x,{}n)} can be used by domains which extend graph to help in creating the \spad{x} coordinate for objects in a graph
250850`\spad{createX(x,{}n)} can be used by domains which extend graph to help in creating the \spad{x} coordinate for objects in a graph
250950`\spad{createY(x,{}n)} can be used by domains which extend graph to help in creating the \spad{y} coordinate for objects in a graph
251047`\spad{createY(x,{}n)} can be used by domains which extend graph to help in creating the \spad{y} coordinate for objects in a graph
251147`\spad{createZechTable(f)} generates a Zech logarithm table for the cyclic group representation of a extension of the ground field by the primitive polynomial {\em f(x)},{} \spadignore{i.e.} \spad{Z(i)},{} defined by {\em x^Z(i) = 1+x^i} is stored at index \spad{i}. This is needed in particular to perform addition of field elements in finite fields represented in this way. See \spadtype
251147`--{FFCGP},{} \spadtype{FFCGX}.
251311`\spad{credPol }\undocumented
251404`\spad{critB }\undocumented
251506`\spad{critBonD }\undocumented
251730`\spad{critMTonD1 }\undocumented
251951`\spad{critM }\undocumented
252041`\spad{critMonD1 }\undocumented
252267`\spad{critT }\undocumented
252412`\spad{critpOrder }\undocumented
252629`\spad{cross(p,{}q)} computes the cross product of the two points \spad{p} and \spad{q} using only the first three coordinates,{} and keeping the color of the first point \spad{p}. The result is returned as a point.
252768`\spad{cross(p,{}q)} computes the cross product of the two points \spad{p} and \spad{q}. Error if the \spad{p} and \spad{q} are not 3 dimensional
252817`\spad{cross(u,{}v)} constructs the cross product of \spad{u} and \spad{v}. Error: if \spad{u} and \spad{v} are not of length 3.
252878`\axiom{crushedSet(\spad{lp})} returns \axiom{\spad{lq}} such that \axiom{\spad{lp}} and and \axiom{\spad{lq}} generate the same ideal and no rough basic sets reduce (in the sense of Groebner bases) the other polynomials in \axiom{\spad{lq}}.
252961`\spad{csc2sin(f)} converts every \spad{csc(u)} appearing in \spad{f} into \spad{1/sin(u)}.
253023`\spad{cscIfCan(z)} returns \spad{csc}(\spad{z}) if possible,{} and "failed" otherwise.
253103`\spad{csc(x)} applies the cosecant operator to \spad{x}
253266`\spad{csc(st)} computes cosecant of a power series \spad{st}.
253349`\spad{csc(st)} computes cosecant of a power series \spad{st}.
253446`\spad{csc(z)} returns the cosecant of Laurent series \spad{z}.
253532`\spad{csc(z)} returns the cosecant of a Puiseux series \spad{z}.
253627`\spad{csc(x)} returns the cosecant of \spad{x}.
253684`\spad{csch2sinh(f)} converts every \spad{csch(u)} appearing in \spad{f} into \spad{1/sinh(u)}.
253748`\spad{cschIfCan(z)} returns csch(\spad{z}) if possible,{} and "failed" otherwise.
253829`\spad{csch(x)} applies the hyperbolic cosecant operator to \spad{x}
253992`\spad{csch(st)} computes the hyperbolic cosecant of a power series \spad{st}.
254076`\spad{csch(st)} computes the hyperbolic cosecant of a power series \spad{st}.
254174`\spad{csch(z)} returns the hyperbolic cosecant of Laurent series \spad{z}.
254261`\spad{csch(z)} returns the hyperbolic cosecant of a Puiseux series \spad{z}.
254357`\spad{csch(x)} returns the hyperbolic cosecant of \spad{x}.
254412`\spad{csubst(a,{}b)} is for internal use only
254528`\spad{cubic(u)} \undocumented
254594`\spad{cubic(f,{}g,{}h,{}i)} \undocumented
254665`\spad{cumulant2moment(cc)} computes the sequence of moments from the sequence of classical cumulants \spad{cc}
254739`\spad{cup(s1,{}s2)},{} introduced by Redfield,{} \indented{1}{is the scalar product of two cycle indices,{} in which the} \indented{1}{power sums are retained to produce a cycle index.}
254893`\spad{currentSubProgram()} returns the name of the current subprogram being processed
254951`\spad{curryLeft(f,{}a)} is the function \spad{g} \indented{1}{such that \spad{g b = f(a,{}b)}.}
255019`\spad{curryRight(f,{}b)} is the function \spad{g} such that \indented{1}{\spad{g a = f(a,{}b)}.}
255088`\spad{cu(f,{}a)} is the function \spad{g} \indented{1}{such that \spad{g ()= f a}.}
255147`\spad{curve?(s)} queries whether the \spadtype{ThreeSpace},{} \spad{s},{} is a curve,{} \spadignore{i.e.} has one component,{} a list of list of points,{} and returns \spad{true} if it is,{} or \spad{false} otherwise.
255204`\spad{curveColorPalette(l,{}p)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{curveColorPalette}. If the option does not exist the value,{} \spad{p} is returned.
255293`\spad{curveColor(v)} specifies a color,{} \spad{v},{} for 2D graph curves. This option is expressed in the form \spad{curveColor == v}.
255341`\spad{curveColor(p)} specifies a color index for 2D graph curves from the spadcolors palette \spad{p}. This option is expressed in the form \spad{curveColor ==p}.
255391`\spad{curveLoops(pts1,{}r,{}nn)} generates ifs from line given by \spad{pts1}
255476`\spad{curve([p0,{}p1,{}p2,{}...,{}pn])} creates a space curve defined by the list of points \spad{p0} through \spad{pn},{} and returns the \spadtype{ThreeSpace} whose component is the curve.
255539`\spad{curve(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a single curve defined by a list of points and if so,{} returns the curve,{} \spadignore{i.e.} list of points. An error is signaled otherwise.
255602`\spad{curve(c1,{}c2)} creates a plane curve from 2 component functions \spad{c1} and \spad{c2}.
255704`\spad{curve(s,{}[[p0],{}[p1],{}...,{}[pn]])} adds a space curve which is a list of points \spad{p0} through \spad{pn} defined by lists of elements from the domain \spad{PointDomain(m,{}R)},{} where \spad{R} is the \spadtype{Ring} over which the point elements are defined and \spad{m} is the dimension of the points,{} to the \spadtype{ThreeSpace} \spad{s}.
255769`\spad{curve(s,{}[p0,{}p1,{}...,{}pn])} adds a space curve component defined by a list of points \spad{p0} through \spad{pn},{} to the \spadtype{ThreeSpace} \spad{s}.
255835`\spad{curve(c1,{}c2,{}c3)} creates a space curve from 3 component functions \spad{c1},{} \spad{c2},{} and \spad{c3}.
255955`cycleClosed:(objs:List \spad{S},{}arrowName:String) constructs a graph with vertices (from objs) connected in a cycle. arrowName is a prefix for all arrow names,{} this will be followed by a number starting at 1 and incremented for each arrow
256017`\spad{cycleElt(s)} returns a pointer to a node in the cycle if the stream \spad{s} is cyclic and returns "failed" if \spad{s} is not cyclic
256199`\spad{cycleEntry(u)} returns the head of a top-level cycle contained in aggregate \spad{u},{} or \axiom{empty()} if none exists.
256406`\spad{cycleLength(u)} returns the length of a top-level cycle contained in aggregate \spad{u},{} or 0 is \spad{u} has no such cycle.
256484`\spad{cycleOpen(objs,{}arrowName)} constructs a graph with vertices (from \spad{objs}) connected in a cycle but with one gap. The last vertex in the sequence loops back to itself so all vertices have one outgoing arrow. \spad{arrowName} is a prefix for all arrow names,{} this will be followed by a number starting at 1 and incremented for each arrow
256544`\spad{cyclePartition(p)} returns the cycle structure of a permutation \spad{p} including cycles of length 1 only if \spad{S} is finite.
256604`\spad{cycleRagits(rx)} returns the cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \spad{x = 3/28 = 0.10 714285 714285 ...},{} then \spad{cycleRagits(x) = [7,{}1,{}4,{}2,{}8,{}5]}.
256727`\spad{cycleSplit!(u)} splits the aggregate by dropping off the cycle. The value returned is the cycle entry,{} or nil if none exists. For example,{} if \axiom{\spad{w} = concat(\spad{u},{}\spad{v})} is the cyclic list where \spad{v} is the head of the cycle,{} \axiom{cycleSplit!(\spad{w})} will drop \spad{v} off \spad{w} thus destructively changing \spad{w} to \spad{u},{} and returning \spad{v}.
256921`\spad{cycleTail(u)} returns the last node in the cycle,{} or empty if none exists.
256981`\spad{cycle(ls)} coerces a cycle {\em ls},{} \spadignore{i.e.} a list with not repetitions to a permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list. Error: if repetitions occur.
257038`\spad{cycles(lls)} coerces a list list of cycles {\em lls} to a permutation,{} each cycle being a list with not repetitions,{} is coerced to the permutation,{} which maps {\em ls.i} to {\em ls.i+1},{} indices modulo the length of the list,{} then these permutations are mutiplied. Error: if repetitions occur in one cycle.
257279`\spad{cyclic?(u)} tests if \spad{u} has a cycle.
257337`\spad{cyclic?(t)} tests if \spad{t} is a cyclic tree.
257381`\spad{cyclicCopy(l)} makes a copy of a (possibly) cyclic tree \spad{l}.
257423`\spad{cyclicEntries(t)} returns a list of top-level cycles in tree \spad{t}.
257474`\spad{cyclicEqual?(t1,{} t2)} tests of two cyclic trees have the same structure.
257526`\spad{cyclicGroup([i1,{}...,{}ik])} constructs the cyclic group of order \spad{k} acting on the integers {\em i1},{}...,{}{\em ik}. Note: duplicates in the list will be removed.
257620`\spad{cyclicGroup(n)} constructs the cyclic group of order \spad{n} acting on the integers 1,{}...,{}\spad{n}.
257716`\spad{cyclicParents(t)} returns a list of cycles that are parents of \spad{t}.
257767`\spad{cyclicSubmodule(lm,{}v)} generates a basis as follows. It is assumed that the size \spad{n} of the vector equals the number of rows and columns of the matrices. Then the matrices generate a subalgebra,{} say \spad{A},{} of the algebra of all square matrices of dimension \spad{n}. {\em V R} is an \spad{A}-module in the natural way. cyclicSubmodule(\spad{lm},{}\spad{v}) gener
257767`--ates the \spad{R}-Basis of {\em Av} as described in section 6 of \spad{R}. A. Parker\spad{'s} "The Meat-Axe". Note: in contrast to the description in "The Meat-Axe" and to {\em standardBasisOfCyclicSubmodule} the result is in echelon form.
257892`\spad{cyclic n} is the cycle index of the \indented{1}{cyclic group of degree \spad{n}.}
257979`\spad{cyclotomicDecomposition(n)} returns prime factors of \spad{x^n - 1} over integers.
258098`\spad{cyclotomicFactorization(n)} returns prime factors of \spad{x^n - 1} over integers.
258221`\spad{cyclotomic(n)} returns \spad{n}-th cyclotomic polynomial,{} that is minimal polynomial over rationals of primitive \spad{n}-th root of 1.
258321`\spad{cyclotomic(n)} returns the \spad{n}th cyclotomic polynomial \spad{phi[n](x)}. Note: \spad{phi[n](x)} is the factor of \spad{x^n - 1} whose roots are the primitive \spad{n}th roots of unity.
258425`\spad{cyclotomic(n,{}r)} returns value of the \spad{n}th cyclotomic polynomial at \spad{r}.
258514`\spad{cylindrical(pt)} transforms \spad{pt} from polar coordinates to Cartesian coordinates: the function produced will map the point \spad{(r,{}theta,{}z)} to \spad{x = r * cos(theta)},{} \spad{y = r * sin(theta)},{} \spad{z}.
258582`\spad{dAndcExp(v,{}n,{}k)} computes \spad{v^e} interpreting \spad{v} as an element of normal basis field. A divide and conquer algorithm similar to the one from \spad{D}.\spad{R}.Stinson,{} "Some observations on parallel Algorithms for fast exponentiation in \spad{GF}(2^n)",{} Siam \spad{J}. Computation,{} Vol.19,{} No.4,{} \spad{pp}.711-717,{} August 1990 is used. Argument \spad{k} is a 
258582`--parameter of this algorithm.
258698`\axiom{\spad{dP}(\spad{i},{} mu)} returns the differential of \axiom{\spad{P}(\spad{i},{} mu)}.
258845`\axiom{dSubst(\spad{f},{} \spad{jv},{} exp)} is like \axiom{subst(\spad{f},{} \spad{jv},{} exp)}. But additionally for all derivatives of \axiom{\spad{jv}} the corresponding substitutions are performed.
258911`\axiom{dU(\spad{i})} returns the differential of \axiom{\spad{U}(\spad{i})}.
258972`\axiom{\spad{dX}(\spad{i})} returns the differential of \axiom{\spad{X}(\spad{i})}.
259033`\axiom{\spad{d}(\spad{f})} computes the differential of \axiom{\spad{f}}.
259079`\axiom{\spad{d}(\spad{jb})} returns the differential of \axiom{\spad{jb}}.
259126`\spad{dark(c)} sets the shade of the indicated hue of \spad{c} to it\spad{'s} lowest value.
259165`\spad{datalist(l)} creates a datalist from \spad{l}
259214`\spad{ddFact(f,{}p)} computes a distinct degree factorization of the polynomial \spad{f} modulo the prime \spad{p},{} \spadignore{i.e.} such that each factor is a product of irreducibles of the same degrees. The input polynomial \spad{f} is assumed to be square-free modulo \spad{p}.
259321`\spad{debug3D(true)} turns debug mode on; debug3D(\spad{false}) turns debug mode off.
259369`\spad{debug(true)} turns debug mode on \spad{debug(false)} turns debug mode off
259413`\spad{debug returns} whether we want additional output on the progress,{} default being \spad{false}
259484`\spad{debug(d)} specifies whether we want additional output on the progress. This option is expressed in the form \spad{debug == d}.
259530`decFatal is an internal function that declares that the preceding test did not fail with a fatal error.
259576`like decFatal,{} but for expected failures.
259624`\spad{decXfPass()} is an internal function that declares that the preceding test did not pass unexpectedly. This function should only be used for testing library errors
259671`\spad{dec(x)} returns \spad{x - 1}.
259718`\spad{decimal(r)} converts a rational number to a decimal expansion.
259781`\spad{declare!(u,{}t)} declares the parameter \spad{u} to have type \spad{t} in the current level of the symbol table.
259853`\spad{declare!(l,{}t,{}tab)} creates new entrys in \spad{tab},{} declaring each of \spad{l} to be of type \spad{t}
259931`\spad{declare!(u,{}t,{}asp)} declares the parameter \spad{u} to have type \spad{t} in \spad{asp}.
260010`\spad{declare!(u,{}t,{}tab)} creates a new entry in \spad{tab},{} declaring \spad{u} to be of type \spad{t}
260082`\spad{declare!(u,{}t,{}asp,{}tab)} declares the parameters \spad{u} of subprogram \spad{asp} to have type \spad{t} in symbol table \spad{tab}.
260170`\spad{declare!(u,{}t,{}asp,{}tab)} declares the parameter \spad{u} of subprogram \spad{asp} to have type \spad{t} in symbol table \spad{tab}.
260252`\spad{declare(t)} returns a name \spad{f} such that \spad{f} has been declared to the interpreter to be of type \spad{t},{} but has not been assigned a value yet. Note: \spad{t} should be created as \spad{devaluate(T)\$Lisp} where \spad{T} is the actual type of \spad{f} (this hack is required for the case where \spad{T} is a mapping type).
260303`\spad{decomposeFunc(func1,{} func2,{} newvar)} returns a function \spad{func3} where \spad{func1} = \spad{func3}(\spad{func2}) and expresses it in the new variable newvar. If there is no solution then \spad{func1} will be returned.
260570`\spad{decompose(up)} \undocumented
260639`\spad{decompose(d)} returns \spad{[id,{} f]} where \spad{d = (id) + div(f)}.
260771`\spad{decompose(f,{} D)} returns \spad{[p,{}n,{}s]} such that \spad{f = p+n+s},{} all the squarefree factors of \spad{denom(n)} are normal \spad{w}.\spad{r}.\spad{t}. \spad{D},{} \spad{denom(s)} is special \spad{w}.\spad{r}.\spad{t}. \spad{D},{} and \spad{n} and \spad{s} are proper fractions (no pole at infinity). \spad{D} is the derivation to use.
260906`\spad{decompose(up,{}m,{}n)} \undocumented
261541`\spad{decreasePrecision(n)} decreases the current \spadfunFrom{precision}{FloatingPointSystem} precision by \spad{n} decimal digits.
261646`assumes \spad{ln} contains a list of factors which must be \spad{true} for the whole to be \spad{true} (such as the list produced by factor). From this deductions attemps to produce a list of other proposition that must also be \spad{true} by using modus ponens. This is used to determine the returned type when converting ILogic to types by using the Curry-Howard isomorphism.
261699`\spad{deepCopy(x)} \undocumented
261877`\spad{deepExpand(x)} \undocumented{}
262016`\axiom{deepestInitial(\spad{p})} returns an error if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns the last term of \axiom{iteratedInitials(\spad{p})}.
262155`\axiom{deepestTail(\spad{p})} returns \axiom{0} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns tail(\spad{p}),{} if \axiom{tail(\spad{p})} belongs to \axiom{\spad{R}} or \axiom{mvar(tail(\spad{p})) < mvar(\spad{p})},{} otherwise returns \axiom{deepestTail(tail(\spad{p}))}.
262225`\spad{defineProperty(s,{}\spad{li},{}p)} defines the component property in the 3 dimensional subspace,{} \spad{s},{} to be that of \spad{p},{} where \spad{p} is of the domain \spadtype{SubSpaceComponentProperty}. The list of non negative integers,{} \spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component whose property is being defined. The subspace,{} \s
262225`--pad{s},{} is returned with the component property definition.
262328`\spad{definingEquations(s)} returns a list of defining polynomials for equations,{} that is,{} for the Zariski closed part of \spad{s}.
262415`\spad{definingInequation(s)} returns a single defining polynomial for the inequation,{} that is,{} the Zariski open part of \spad{s}.
262581`\spad{definingPolynomial()} returns the polynomial used to define the field extension.
262681`\spad{definingPolynomial()} returns the minimal polynomial which \spad{generator()} satisfies.
262744`\spad{definingPolynomial(f)} returns the defining polynomial of \spad{f} as an element of \spad{F}. Error: if \spad{f} is not a kernel.
262836`\axiom{definingPolynomial(aRoot)} gives a polynomial such that \axiom{definingPolynomial(aRoot).aRoot = 0}
262934`\spad{definingPolynomial(x)} returns an expression \spad{p} such that \spad{p(x) = 0}.
263004`\spad{degreePartition(f)} returns the degree partition (\spadignore{i.e.} the multiset of the degrees of the irreducible factors) of the polynomial \spad{f}.
263116`\spad{degreePartition(ddfactorization)} returns the degree partition of the polynomial \spad{f} modulo \spad{p} where \spad{ddfactorization} is the distinct degree factorization of \spad{f} computed by \spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \spad{p}.
263243`\axiom{indiceSubResultant(\spad{P},{} \spad{Q},{} \spad{i})} returns a subresultant \axiom{\spad{S}} of degree \axiom{\spad{d}} and carries out the equality \axiom{coef1*P + coef2*Q = S_i}.
263397`\axiom{degreeSubResultant(\spad{P},{} \spad{Q},{} \spad{d})} computes a subresultant of degree \axiom{\spad{d}}.
263565`\spad{degree(x)} gives degree of \spad{x}.
264026`\spad{degree(v)} is degree of \spad{v} treated as polynomial
264098`\spad{degree(p)} returns the maximum of the exponents of the terms of \spad{p}.
264178`\spad{degree(df)} returns the homogeneous degree of differential form \spad{df}.
264248`\spad{degree(x)} \undocumented
264307`\spad{degree(p)} returns the homogeneous degree of \spad{p}.
264370`\spad{degree(x)} gives the numbers of 1\spad{'s} in \spad{x},{} \spadignore{i.e.} the number of non-zero exponents in the basis element that \spad{x} represents.
264428`\spad{degree(p)} returns the maximum of the exponents of the terms of \spad{p}.
264480`\spad{degree(g)} names the degree of \spad{g}. The set of all elements of a given degree form an \spad{R}-module.
264527`\spad{degree(f)} returns the exponent of the lowest order term of \spad{f}.
264596`\spad{degree(f(x))} returns the degree of the leading term of the Puiseux series \spad{f(x)},{} which may have zero as a coefficient.
264694`\spad{degree(f(x))} returns the degree of the lowest order term of \spad{f(x)},{} which may have zero as a coefficient.
264782`\axiom{degree(\spad{x})} returns the greatest length of a word in the support of \axiom{\spad{x}}.
264853`\spad{degree(l)} is \spad{n} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
264926`\axiom{degree(\spad{ts})} returns the product of main degrees of the members of \axiom{\spad{ts}}.
265003`\spad{degree(l)} is \spad{n} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
265085`\spad{degree(p)} returns the degree of \spad{p}. \indented{1}{Note that the degree of a word is its length.}
265153`\spad{degree(p)} retuns the number of points moved by the permutation \spad{p}.
265214`\spad{degree(gp)} returns the number of points moved by all permutations of the group {\em gp}.
265280`\spad{degree(a)} returns the degree of minimal polynomial of an element \spad{a} if \spad{a} is algebraic with respect to the ground field \spad{F},{} and \spad{infinity} otherwise.
265361`\spad{degree(a)} returns the degree of the minimal polynomial of an element \spad{a} over the ground field \spad{F}.
265910`\spad{degree(upoly,{} lvar)} returns a list containing the maximum degree for each variable in lvar.
266027`\spad{degree(p,{}lv)} gives the list of degrees of polynomial \spad{p} with respect to each of the variables in the list \spad{lv}.
266132`\spad{degree(p,{}v)} gives the degree of polynomial \spad{p} with respect to the variable \spad{v}.
266225`\spad{degree(p,{} s)} returns the maximum degree of the differential polynomial \spad{p} viewed as a differential polynomial in the differential indeterminate \spad{s} alone.
266313`\spad{delay(f)} creates a stream with a lazy evaluation defined by function \spad{f}. Caution: This function can only be called in compiled code.
266478`\spad{delete!(u,{}i)} destructively deletes the \axiom{\spad{i}}th element of \spad{u}.
266546`\spad{delete!(u,{}i..j)} destructively deletes elements \spad{u}.\spad{i} through \spad{u}.\spad{j}.
266632`\spad{deleteProperty!(op,{} s)} unattaches property \spad{s} from \spad{op}. Argument \spad{op} is modified "in place",{} \spadignore{i.e.} no copy is made.
266692`\axiom{deleteRow(A,{} \spad{i})} deletes the \axiom{\spad{i}}\spad{-}th row of the matrix \axiom{A}.
267187`\spad{delete(u,{}i)} returns a copy of \spad{u} with the \axiom{\spad{i}}th element deleted. Note: for lists,{} \axiom{delete(a,{}\spad{i}) \spad{==} concat(a(0..\spad{i} - 1),{}a(\spad{i} + 1,{}..))}.
267244`\spad{delete(u,{}i..j)} returns a copy of \spad{u} with the \axiom{\spad{i}}th through \axiom{\spad{j}}th element deleted. Note: \axiom{delete(a,{}\spad{i}..\spad{j}) = concat(a(0..\spad{i}-1),{}a(\spad{j+1}..))}.
267319`\spad{delta(S,{}k,{}p)} returns the number of elements of \spad{S} which are strictly between \spad{p} and the \spad{k^}{th} element of \spad{S}.
267424`\spad{denomLODE(op,{} g)} returns a polynomial \spad{d} such that any rational solution of \spad{op y = g} is of the form \spad{p/d} for some polynomial \spad{p},{} and "failed",{} if the equation has no rational solution.
267511`\spad{denomLODE(op,{} [g1,{}...,{}gm])} returns a polynomial \spad{d} such that any rational solution of \spad{op y = c1 g1 + ... + cm gm} is of the form \spad{p/d} for some polynomial \spad{p}.
267588`\spad{denomRicDE(op)} returns a polynomial \spad{d} such that any rational solution of the associated Riccati equation of \spad{op y = 0} is of the form \spad{p/d + q'/q + r} for some polynomials \spad{p} and \spad{q} and a reduced \spad{r}. Also,{} \spad{deg(p) < deg(d)} and {\spad{gcd}(\spad{d},{}\spad{q}) = 1}.
267650`\spad{denom(1/d * (f1,{}...,{}fn))} returns \spad{d}.
267704`\spad{denom x} returns the denominator of \spad{x}.
267750`\spad{denom x} returns the denominator of \spad{x}.
267792`\spad{denom(f)} returns the denominator of \spad{f} viewed as a polynomial in the kernels over \spad{Z}.
267888`\spad{denom(x)} returns the denominator of the fraction \spad{x}.
267941`\spad{denom(f)} returns the denominator of \spad{f} viewed as a polynomial in the kernels over \spad{Z}.
268032`\spad{denom(f)} returns the denominator of \spad{f} viewed as a polynomial in the kernels over \spad{R}.
268243`\axiom{denominator(\spad{f})} yields the denominator of \axiom{\spad{f}}.
268308`\spad{denominator(f)} returns the denominator of \spad{f} converted to \%.
268381`\spad{denominator(x)} is the denominator of the fraction \spad{x} converted to \%.
268441`\spad{denominators(x)} returns the stream of denominators of the approximants of the continued fraction \spadvar{\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.
268505`\spad{depth(p)} returns the nesting level of \spad{p}.
268561`\spad{depth(s)} returns the number of elements of stack \spad{s}. Note: \axiom{depth(\spad{s}) = \spad{\#s}}.
268624`\spad{dequeue! s} destructively extracts the first (top) element from queue \spad{q}. The element previously second in the queue becomes the first element. Error: if \spad{q} is empty.
268673`\spad{dequeue()}\$\spad{D} creates an empty dequeue of type \spad{D}.
268722`\spad{dequeue([x,{}y,{}...,{}z])} creates a dequeue with first (top or front) element \spad{x},{} second element \spad{y},{}...,{}and last (bottom or back) element \spad{z}.
268778`\spad{dequeue([x,{}y,{}...,{}z])} creates a dequeue with first (top or front) element \spad{x},{} second element \spad{y},{}...,{}and last (bottom or back) element \spad{z}.
268825`\spad{deref(n)} is equivalent to \spad{elt(n)}.
268866`\spad{deriv(a)} returns the derivative of the power series with respect to the power series variable. Thus \spad{deriv([a0,{}a1,{}a2,{}...])} returns \spad{[a1,{}2 a2,{}3 a3,{}...]}.
268941`\spad{deriv(f)} is the derivative,{} which simply coincides with left shift
269117`\spad{derivationCoordinates(b,{} ')} returns \spad{M} such that \spad{b' = M b}.
269296`\axiom{derivativeOf?(\spad{jv1},{} \spad{jv2})} checks whether \axiom{\spad{jv1}} is a derivative of \axiom{\spad{jv2}}. In this case,{} the difference of their multi-indices is returned. Otherwise,{} an empty list is returned.
269376`\spad{derivative(op)} returns the value of the "\%diff" property of \spad{op} if it has one,{} and "failed" otherwise.
269480`\spad{derivative(op,{} foo)} attaches foo as the "\%diff" property of \spad{op}. If \spad{op} has an "\%diff" property \spad{f},{} then applying a derivation \spad{D} to \spad{op}(a) returns \spad{f(a) * D(a)}. Argument \spad{op} must be unary.
269570`\spad{derivative(op,{} [foo1,{}...,{}foon])} attaches [\spad{foo1},{}...,{}foon] as the "\%diff" property of \spad{op}. If \spad{op} has an "\%diff" property \spad{[f1,{}...,{}fn]} then applying a derivation \spad{D} to \spad{op(a1,{}...,{}an)} returns \spad{f1(a1,{}...,{}an) * D(a1) + ... + fn(a1,{}...,{}an) * D(an)}.
269672`\spad{destruct(r)} returns the list of matches (var,{} expr) in \spad{r}. Error: if \spad{r} is a failed match.
269758`\spad{destruct((a1,{}...,{}an))} returns the list [\spad{a1},{}...,{}an].
269838`\axiom{detSysNS(\spad{sys},{} \spad{vf})} computes the determining system for symmetry generators without assuming solvability of the equations for some derivatives.
269979`\axiom{detSysNS(\spad{sys},{} \spad{vf})} computes the determining system for symmetry generators without assuming solvability of the equations for some derivatives.
270141`\axiom{detSys(\spad{sys})} computes the determining system for symmetry generators of the system \axiom{\spad{sys}}. It is assumed that each equation can be solved for its leading derivative.
270280`\axiom{detSys(\spad{sys},{} \spad{vf})} computes the determining system for symmetry generators of the system \axiom{\spad{sys}} with a given ansatz for the generators. It is assumed that each equation can be solved for its leading derivative.
270440`\axiom{detSys(\spad{sys},{} \spad{sjb},{} \spad{vf})} computes the determining system for symmetry generators of the system \axiom{\spad{sys}} with a given ansatz for the generators. \axiom{\spad{sjb}} contains jet variables for which the individual equations can be solved.
270610`\spad{determinant(m)} returns the determinant of the matrix \spad{m}. an error message is returned if the matrix is not square.
270690`\spad{determinant(m)} returns the determinant of the matrix \spad{m}. an error message is returned if the matrix is not square.
270765`\spad{determinant(m)} returns the determinant of the matrix \spad{m}. Error: if the matrix is not square.
270847`\spad{determinant(m)} returns the determinant of the matrix \spad{m}.
270940`\spad{diag(f)} is the function \spad{g} \indented{1}{such that \spad{g a = f(a,{}a)}.}
271136`\spad{diagonal?(m)} returns \spad{true} if the matrix \spad{m} is square and diagonal (\spadignore{i.e.} all entries of \spad{m} not on the diagonal are zero) and \spad{false} otherwise.
271200`\spad{diagonal?(m)} returns \spad{true} if the matrix \spad{m} is square and diagonal (\spadignore{i.e.} all entries of \spad{m} not on the diagonal are zero) and \spad{false} otherwise.
271411`\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \spad{l} on the diagonal.
271480`\spad{diagonalMatrix(l)} returns a diagonal matrix with the elements of \spad{l} on the diagonal.
271560`\spad{diagonalMatrix([m1,{}...,{}mk])} creates a block diagonal matrix \spad{M} with block matrices {\em m1},{}...,{}{\em mk} down the diagonal,{} with 0 block matrices elsewhere. More precisly: if \spad{\spad{ri} := nrows \spad{mi}},{} \spad{\spad{ci} := ncols \spad{mi}},{} then \spad{m} is an (\spad{r1+}..\spad{+rk}) by (\spad{c1+}..\spad{+ck}) - matrix with entries \spad{m.i.j = ml.(i-r1-..-r(l-1)).(j-n1-..-n(l-1))},{} if \spad{(r
271560`--1+..+r(l-1)) < i <= r1+..+rl} and \spad{(c1+..+c(l-1)) < i <= c1+..+cl},{} \spad{m.i.j} = 0 otherwise.
271630`\spad{diagonalMatrix(v)} returns a diagonal matrix where the elements of \spad{v} appear on the diagonal.
271685`\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \spad{m}
271829`\spad{diagonalProduct(m)} returns the product of the elements on the diagonal of the matrix \spad{m}.
271971`\spad{diagonal(m)} returns a row consisting of the elements on the diagonal of the matrix \spad{m}.
272041`\spad{diagonals(v,{}s)} displays the diagonals of the polygon outline showing a triangularized surface instead of a quadrilateral surface outline,{} for the given three-dimensional viewport \spad{v} which is of domain \spadtype{ThreeDimensionalViewport},{} if \spad{s} is "on",{} or does not display the diagonals if \spad{s} is "off".
272173`\spad{diagramHeight(s)} returns the height of the diagram that will be generated by diagramSvg. This is the maximum posY of all vertices in graph \spad{s}
272304`\spad{diagramSvg(fileName,{} n,{} dispArrowName)} creates an \spad{SVG} diagram fileName:String is the name of the \spad{SVG} file that will be created \spad{n:}\% is the graph that will be written dispArrowName:Boolean is \spad{true} to include the name of each arrow
272434`\spad{diagramWidth(s)} returns the width of the diagram that will be generated by diagramSvg. This is the maximum posX of all vertices in graph \spad{s}
272597`\spad{dictionary()}\$\spad{D} creates an empty dictionary of type \spad{D}.
272763`\spad{dictionary([x,{}y,{}...,{}z])} creates a dictionary consisting of entries \axiom{\spad{x},{}\spad{y},{}...,{}\spad{z}}.
274996`\spad{diffHP options} returns a specification for Hermite-Pade approximation with the differential operator
280107`\spad{diffHP options} returns a specificatio
280107`--n for Hermite-Pade approximation with the \$\spad{q}\$-dilation operator
285646`\axiom{diffP(\spad{i},{} mu)} returns the base vector field in direction \axiom{\spad{P}(\spad{i},{} mu)}.
285734`\axiom{diffU(\spad{i})} returns the base vector field in direction \axiom{\spad{U}(\spad{i})}.
285797`\axiom{diffX(\spad{i})} returns the base vector field in direction \axiom{\spad{X}(\spad{i})}.
285933`\axiom{diff(\spad{jb})} returns the base vector field in direction \axiom{\spad{jb}}.
285982`\spad{diff(x)} returns the derivation with respect to \spad{x}.
286242`\spad{difference(u,{}x)} returns the set aggregate \spad{u} with element \spad{x} removed. If \spad{u} does not contain \spad{x},{} a copy of \spad{u} is returned. Note: \axiom{difference(\spad{s},{} \spad{x}) = difference(\spad{s},{} {\spad{x}})}.
286294`\spad{difference(u,{}v)} returns the set aggregate \spad{w} consisting of elements in set aggregate \spad{u} but not in set aggregate \spad{v}. If \spad{u} and \spad{v} have no elements in common,{} \axiom{difference(\spad{u},{}\spad{v})} returns a copy of \spad{u}. Note: equivalent to the notation (not currently supported) \axiom{{\spad{x} for \spad{x} in \spad{u} | not member?(\spad{x},{}\spad{v})}}.
286434`\spad{differentialVariables(p)} returns a list of differential indeterminates occurring in a differential polynomial \spad{p}.
286524`\axiom{directions(om)} yields the differentials where \axiom{om} has non-vanishing coefficients.
287428`\spad{differentiate(x)} returns the derivative of \spad{x}. This function is a simple differential operator where no variable needs to be specified.
287482`\spad{differentiate(v)} returns the derivative of \spad{v}.
287551`\spad{differentiate(f)} returns the derivative of \spad{f}.
291605`Polynomial differentiation.
291694`\spad{differentiate(ir,{}D)} differentiates \spad{ir} with respect to the derivation \spad{D}.
291758`\axiom{differentiate(\spad{f},{} \spad{jv})} differentiates the function \axiom{\spad{f}} \spad{wrt} the jet variable \axiom{\spad{jv}}.
291828`\spad{differentiate(f,{}n)} creates a form for the \spad{n}th derivative of \spad{f},{} \spadignore{e.g.} \spad{f'},{} \spad{f''},{} \spad{f'''},{} \spad{"f} super \spad{iv}".
291895`\axiom{differentiate(\spad{jv},{} \spad{i})} differentiates \axiom{\spad{jv}} \spad{wrt} the \axiom{\spad{i}}\spad{-}th independent variable.
291977`\spad{differentiate(ir,{}x)} differentiates \spad{ir} with respect to \spad{x}
292158`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
292253`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
292348`\spad{differentiate(f(x),{}x)} computes the derivative of \spad{f(x)} with respect to \spad{x}.
292442`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
292531`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
292620`\spad{differentiate(f(x),{}x)} computes the derivative of \spad{f(x)} with respect to \spad{x}.
292708`\spad{differentiate(x,{} deriv)} differentiates \spad{x} extending the derivation deriv on \spad{R}.
292777`\spad{differentiate(x,{} d)} extends the derivation \spad{d} from UP to \$ and applies it to \spad{x}.
292856`\spad{differentiate(x,{}[s1,{}...sn])} computes successive partial derivatives,{} \spadignore{i.e.} \spad{differentiate(...differentiate(x,{} s1)...,{} sn)}.
292928`\spad{differentiate(x,{} n)} returns the \spad{n}-th derivative of \spad{x}.
293001`\spad{differentiate(v,{} n)} returns the \spad{n}-th derivative of \spad{v}.
293089`\spad{differentiate(f,{} n)} returns the \spad{n}-th derivative of \spad{f}.
293180`\spad{differentiate(x,{}v)} computes the partial derivative of \spad{x} with respect to \spad{v}.
293246`\spad{differentiate(f(x),{}x)} returns the derivative of \spad{f(x)} with respect to \spad{x}.
296943`Polynomial differentiation.
297051`\spad{differentiate(x,{} deriv,{} n)} differentiate \spad{x} \spad{n} times using a derivation which extends \spad{deriv} on \spad{R}.
297139`\spad{differentiate(p,{} d,{} x')} extends the \spad{R}-derivation \spad{d} to an extension \spad{D} in \spad{R[x]} where \spad{Dx} is given by \spad{x'},{} and returns \spad{Dp}.
297218`\spad{differentiate(x,{} [s1,{}...,{}sn],{} [n1,{}...,{}nn])} computes multiple partial derivatives,{} \spadignore{i.e.}
297315`\spad{differentiate(x,{} s,{} n)} computes multiple partial derivatives,{} \spadignore{i.e.} \spad{n}-th derivative of \spad{x} with respect to \spad{s}.
297400`\spad{digamma(x)} returns the digamma function applied to \spad{x}
297460`\spad{digamma(x)} is the function,{} \spad{psi(x)},{} defined by \indented{2}{\spad{psi(x) = Gamma'(x)/Gamma(x)}.}
297555`\spad{digamma(z)} is the logarithmic derivative of \spad{Gamma(x)}.
297632`\spad{digamma(x)} is the function,{} \spad{psi(x)},{} defined by \indented{2}{\spad{psi(x) = Gamma'(x)/Gamma(x)}.}
297709`\spad{digamma(z)} is the logarithmic derivative of \spad{Gamma(x)}.
297768`\spad{digamma(x)} is the logarithmic derivative of \spad{Gamma(x)} (often written \spad{psi(x)} in the literature).
297823`\spad{digit?(c)} tests if \spad{c} is a digit character,{} \spadignore{i.e.} one of 0..9.
297868`\spad{digit()} returns the class of all characters for which \spadfunFrom{digit?}{Character} is \spad{true}.
297910`\spad{digits()} returns ceiling\spad{'s} precision in decimal digits.
297975`\spad{digits()} returns ceiling\spad{'s} precision in decimal digits.
298036`\spad{digits(d)} set the \spadfunFrom{precision}{FloatingPointSystem} to \spad{d} digits.
298116`\spad{digits(d)} set the \spadfunFrom{precision}{FloatingPointSystem} to \spad{d} digits.
298218`\spad{digits(x)} returns a stream of \spad{p}-adic digits of \spad{x}.
298285`\spad{dihedralGroup([i1,{}...,{}ik])} constructs the dihedral group of order 2k acting on the integers out of {\em i1},{}...,{}{\em ik}. Note: duplicates in the list will be removed.
298381`\spad{dihedralGroup(n)} constructs the dihedral group of order 2n acting on integers 1,{}...,{}\spad{N}.
298479`\spad{dihedral n} is the cycle index of the \indented{1}{dihedral group of degree \spad{n}.}
298568`\spad{dilate(a,{} x)} returns the sequence \$a^n \spad{x_n}\$ (starting at \$\spad{n=1}\$)
298620`\spad{dilog(f)} denotes the dilogarithm
298672`\spad{dilog(z)} is the dilogaritm
298747`\spad{dilog(x)} returns the dilogarithm of \spad{x},{} \spadignore{i.e.} the integral of \spad{log(x) / (1 - x) dx}.
298804`\axiom{dimJ(\spad{q})} computes the (fibre) dimension of the \axiom{\spad{q}}\spad{-}th order jet bundle.
298882`\axiom{dimJ(\spad{q})} computes the (fibre) dimension of the \axiom{\spad{q}}\spad{-}th order jet bundle.
298964`\axiom{dimS(\spad{q})} computes dimension of \spad{SqT} \spad{x} VE (= number of derivatives of order \axiom{\spad{q}}).
299042`\axiom{dimS(\spad{q})} computes dimension of \spad{SqT} \spad{x} VE (= number of derivatives of order \axiom{\spad{q}}).
299124`\spad{dim(c)} sets the shade of a hue,{} \spad{c},{} above dark,{} but below bright.
299162`\spad{dimensionOfIrreducibleRepresentation(lambda)} is the dimension of the ordinary irreducible representation of the symmetric group corresponding to {\em lambda}. Note: the Robinson-Thrall hook formula is implemented.
299413`\spad{dimension()} returns the dimensionality of the vector space.
299471`\spad{dimension()} is the rank of this Lie algebra
299546`\spad{dimension(p)} returns the number of dimensions that make up the point category \spad{p}.
299607`\spad{dimension(I)} gives the dimension of the ideal \spad{I}. in the ring \spad{F[lvar]},{} where lvar are the variables appearing in \spad{I}
299683`\spad{dimension(s)} returns the dimension of the point category \spad{s}.
299746`\axiom{dimension(de,{} \spad{q})} computes the dimension of the differential equation \axiom{de} as a submanifold of the \axiom{\spad{q}}\spad{-}th order jet bundle. The result is correct only,{} if \axiom{de} is simplified.
299844`\spad{dimension(I,{}lvar)} gives the dimension of the ideal \spad{I},{} in the ring \spad{F[lvar]}
300060`\axiom{dimension(\spad{sys},{} \spad{jm},{} \spad{q})} computes the dimension of the manifold described by the system \axiom{\spad{sys}} with Jacobi matrix \axiom{\spad{jm}} in the jet bundle of order \axiom{\spad{q}}.
300191`\spad{dimensionsOf(t)} returns the dimensions of \spad{t}
300262`\spad{dimensions(v,{}x,{}y,{}width,{}height)} sets the position of the upper left-hand corner of the two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} to the window coordinate \spad{x},{} \spad{y},{} and sets the dimensions of the window to that of \spad{width},{} \spad{height}. The new dimensions are not displayed until the funct
300262`--ion \spadfun{makeViewport2D} is executed again for \spad{v}.
300391`\spad{dimensions(v,{}x,{}y,{}width,{}height)} sets the position of the upper left-hand corner of the three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} to the window coordinate \spad{x},{} \spad{y},{} and sets the dimensions of the window to that of \spad{width},{} \spad{height}. The new dimensions are not displayed until the
300391`-- function \spadfun{makeViewport3D} is executed again for \spad{v}.
300522`\spad{dioSolve(u)} computes a basis of all minimal solutions for linear homogeneous Diophantine equation \spad{u},{} then all minimal solutions of inhomogeneous equation
300727`\spad{diophantineSystem(A,{}B)} returns a particular integer solution and an integer basis of the equation \spad{AX = B}.
300852`\spad{directProduct(v)} converts the vector \spad{v} to become a direct product. Error: if the length of \spad{v} is different from dim.
300925`\spad{directSum(a,{}b)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the sums of a solution of \spad{a} by a solution of \spad{b}.
301019`\spad{directSum(a,{}b,{}D)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the sums of a solution of \spad{a} by a solution of \spad{b}. \spad{D} is the derivation to use.
301103`\spad{directedGraph(perms)} constructs graph from a list of permutations: \spad{perms}.
301175`\spad{directedGraph(ob)} is a constructor for graph with given objects \spad{ob},{} more objects and arrows can be added later if required.
301296`\spad{directedGraph(ob)} is a constructor for graph with given list of object names and no arrows. Use this version of the constructor if you don\spad{'t} want to create specific \spad{x},{}\spad{y} coordinates. more objects and arrows can be added later if required.
301355`\spad{directedGraph(ob,{} ar)} construcs graph with objects \spad{ob} and arrows ar,{} more objects and arrows can be added later if required.
301640`\spad{directedGraph(ob,{} am)} construcs graph with objects \spad{ob} and adjacency matrix am.
301730`\spad{direction(s)} \undocumented
301788`\axiom{directions(\spad{v})} yields the directions of the base vectors where \axiom{\spad{v}} has non-vanishing coefficients.
301849`\spad{directory(f)} returns the directory part of the file name.
301972`\spad{discreteLog(a)} computes the discrete logarithm of \spad{a} with respect to \spad{primitiveElement()} of the field.
302130`\spad{discreteLog(b,{}a)} computes \spad{s} with \spad{b^s = a} if such an \spad{s} exists.
302227`\axiom{discriminantEuclidean(\spad{P})} carries out the equality \axiom{\spad{coef1} * \spad{P} + \spad{coef2} * \spad{D}(\spad{P}) = discriminant(\spad{P})}.
302348`\spad{discriminant()} returns the discriminant of the integral closure of \spad{Z} in the quotient field of the framed algebra \spad{F}.
302468`\spad{discriminant()} = determinant(traceMatrix()).
302524`\spad{discriminant()} = determinant(traceMatrix()).
302760`\axiom{discriminant(\spad{P},{} \spad{Q})} returns the discriminant of \axiom{\spad{P}} and \axiom{\spad{Q}}.
302829`\spad{discriminant([v1,{}..,{}vn])} returns \spad{determinant(traceMatrix([v1,{}..,{}vn]))}.
302896`\spad{discriminant(p)} returns the discriminant of the polynomial \spad{p}.
303055`\spad{discriminant(p,{}v)} returns the disriminant of the polynomial \spad{p} with respect to the variable \spad{v}.
303151`\spad{displayKind(d)} specifies kind of the result: generating function,{} recurrence or equation. This option should not be set by the user,{} but rather by the \spad{HP}-specification.
303227`\spad{displayKind(d)} specifies kind of the result: generating function,{} recurrence or equation. This option should not be set by the user,{} but rather by the \spad{HP}-specification.
303278`\spad{display(o)} prints the string returned by coerce.
303326`\axiom{display(de)} prints all information stored about the differential equation \axiom{de}. This comprises the system ordered by the order of the equations,{} the Jacobi matrices separately for each order and the index of the independent variable with respect to which the equation was lastly differentiated (1 for not prolonged equations).
303389`prints the string returned by coerce,{} adding <math ...> tags.
303439`prints the string returned by coerce,{} adding <math ...> tags.
303490`\spad{display(op)} returns the "\%display" property of \spad{op} if it has one attached,{} and "failed" otherwise.
303579`\spad{display(db)} prints a summary line for each entry in \axiom{\spad{db}}.
303624`\spad{display(ic)} prints a summary of the information contained in \axiom{\spad{ic}}.
303667`\spad{display(t)} outputs the formatted code \spad{t} so that each line has length less than or equal to the value set by the system command \spadsyscom{set output length}.
303720`\spad{display(t)} outputs the TeX formatted code \spad{t} so that each line has length less than or equal to the value set by the system command \spadsyscom{set output length}.
303763`\spad{display(op,{} foo)} attaches foo as the "\%display" property of \spad{op}. If \spad{op} has a "\%display" property \spad{f},{} then \spad{op(a1,{}...,{}an)} gets converted to OutputForm as \spad{f(a1,{}...,{}an)}.
303839`\spad{display(op,{} foo)} attaches foo as the "\%display" property of \spad{op}. If \spad{op} has a "\%display" property \spad{f},{} then \spad{op(a)} gets converted to OutputForm as \spad{f(a)}. Argument \spad{op} must be unary.
303909`\spad{display(t,{}width)} outputs the formatted code \spad{t} so that each line has length less than or equal to \spadvar{\spad{width}}.
303970`\spad{display(t,{}width)} outputs the TeX formatted code \spad{t} so that each line has length less than or equal to \spadvar{\spad{width}}.
304021`\spad{distFact(contm,{}unilist,{}plead,{}vl,{}lvar,{}lval)},{} where \spad{contm} is the content of the evaluated polynomial,{} \spad{unilist} is the list of factors of the evaluated polynomial,{} \spad{plead} is the complete factori
304021`--zation of the leading coefficient,{} \spad{vl} is the list of factors of the leading coefficient evaluated,{} \spad{lvar} is the list of variables,{} \spad{lval} is the list of values,{} returns a record giving the list of leading coefficients to impose on the univariate factors,{}
304358`\spad{distanceMatrix(s)} gives matrix of distances between vertices. Element a_{\spad{i},{} \spad{j}} is the distance from \spad{i} to \spad{j}. Distance matrices are related to adjacency matrices,{} with the differences that: a) the latter only provides the information which vertices are connected \indented{3}{but does not tell about costs or distances between the vertices} \spad{b}) adjacency matrix only tells us about directly connect
304358`--ed \indented{3}{vertices while distance matrix also considers indirect connections.}
304424`\spad{distanceSquared(p1,{}p2)} returns the distance between the points \spad{p1} and \spad{p2}. There are 2 metrics associated with any SPointCategory the underlying space and the space that it is embedded in this is the distance in the underlying space.
304490`the shortest distance between 'a' and \spad{'b'} in terms of weight.
304719`\spad{distance(p1,{}p2)} returns the distance between the points \spad{p1} is the first point. \spad{p2} is the second point. There are 2 metrics associated with any SPointCategory the underlying space and the space that it is embedded in this is the distance in the underlying space.
304778`\spad{distance(u,{}v)} returns the path length (an integer) from node \spad{u} to \spad{v}.
304927`\spad{distance(s,{} a,{} b)} gives the shortest distance between nodes 'a' and \spad{'b'} as a number of hops. 0 if 'a' = \spad{'b'},{} \spad{-1} if it is not possible to go from 'a' to \spad{'b'}
305017`\spad{distdfact(p,{}sqfrflag)} produces the complete factorization of the polynomial \spad{p} returning an internal data structure. If argument \spad{sqfrflag} is \spad{true},{} the polynomial is assumed square free.
305188`\spad{distribute(f)} expands all the kernels in \spad{f} that are formally enclosed by a \spadfunFrom{box}{ExpressionSpace} or \spadfunFrom{paren}{ExpressionSpace} expression.
305287`\spad{distribute(f,{} g)} expands all the kernels in \spad{f} that contain \spad{g} in their arguments and that are formally enclosed by a \spadfunFrom{box}{ExpressionSpace} or a \spadfunFrom{paren}{ExpressionSpace} expression.
305340`\spad{distributionByBooleanCumulants(bb)} initiates a distribution with given Boolean cumulants \spad{bb}.
305419`\spad{distributionByBooleanCumulants(bb)} initiates a distribution with given Boolean cumulants \spad{bb}.
305496`\spad{distributionByEvenMoments(kk)} initiates a distribution with given classical cumulants \spad{kk}.
305577`\spad{distributionByEvenMoments(kk)} initiates a distribution with given classical cumulants \spad{kk}.
305656`\spad{distributionByEvenMoments(mm)} initiates a distribution with given even moments \spad{mm} and odd moments zero.
305730`\spad{distributionByEvenMoments(mm)} initiates a distribution with given even moments \spad{mm} and odd moments zero.
305802`\spad{distributionByFreeCumulants(cc)} initiates a distribution with given free cumulants \spad{cc}.
305878`\spad{distributionByFreeCumulants(cc)} initiates a distribution with given free cumulants \spad{cc}.
305952`\spad{distributionByJacobiParameters(aa,{} bb)} initiates a distribution with given Jacobi parameters \spad{[aa,{} bb]}.
306043`\spad{distributionByJacobiParameters(aa,{} bb)} initiates a distribution with given Jacobi parameters \spad{[aa,{} bb]}.
306130`\spad{distributionByMoments(mm)} initiates a distribution with given moments \spad{mm}.
306200`\spad{distributionByMoments(mm)} initiates a distribution with given moments \spad{mm}.
306268`\spad{distributionBySTransform(series)} initiates a distribution with given \spad{S}-transform \spad{series}.
306439`of the distributions \spad{x} if the mean is zero \spad{distributionBySTransform(x)} returns the distribution with \spad{S}-transform \spad{x}.
306538`\spad{distributionBySTransform(series)} initiates a distribution with given \spad{S}-transform \spad{series}.
306680`\spad{divergence(vf,{}xlist)} computes the divergence of the vector field \spad{vf},{} \spad{vf} a vector function of the variables listed in \spad{xlist}.
306766`Polynomial division.
306864`\spad{divideExponents(p,{}n)} returns a new polynomial resulting from dividing all exponents of the polynomial \spad{p} by the non negative integer \spad{n},{} or "failed" if some exponent is not exactly divisible by \spad{n}.
306970`\spad{divideIfCan!(matrix,{}matrixOut,{}prime,{}n)} attempts to divide the entries of \spad{matrix} by \spad{prime} and store the result in \spad{matrixOut}. If it is successful,{} 1 is returned and if not,{} \spad{prime} is returned. Here both \spad{matrix} and \spad{matrixOut} are \spad{n}-by-\spad{n} upper triangular matrices.
307059`\spad{divideIfCan(f,{}g)} returns quotient and remainder of the division of \spad{f} by \spad{g} or "failed" if it has not succeeded.
307414`\spad{divide(x,{} y)} returns the left and right exact quotients of \spad{x} by \spad{y},{} \spadignore{i.e.} \spad{[l,{} r]} such that \spad{x = l * y * r},{} "failed" if \spad{x} is not of the form \spad{l * y * r}.
307494`\spad{divide(x,{} y)} returns the left and right exact quotients of \spad{x} by \spad{y},{} that is \spad{[l,{} r]} such that \spad{x = l * y * r}. "failed" is returned iff \spad{x} is not of the form \spad{l * y * r}.
307581`\axiom{divide(\spad{F},{}\spad{G})} computes quotient and rest of the exact euclidean division of \axiom{\spad{F}} by \axiom{\spad{G}}.
307684`\spad{divide(x,{}y)} divides \spad{x} by \spad{y} producing a record containing a \spad{quotient} and \spad{remainder},{} where the remainder is smaller (see \spadfunFrom{sizeLess?}{EuclideanDomain}) than the divisor \spad{y}.
307763`\spad{divide(a,{}b)} returns a record containing both remainder and quotient.
307845`\spad{divisorCascade(p,{}tp)} assumes that degree of polynomial {\em tp} is smaller than degree of polynomial \spad{p},{} both monic. A sequence of divisions is calculated using the remainder,{} made monic,{} as divisor for the the next division. The result contains also the error of the factorizations,{} \spadignore{i.e.} the norm of the remainder polynomial.
307954`\spad{divisorCascade(p,{}tp)} assumes that degree of polynomial {\em tp} is smaller than degree of polynomial \spad{p},{} both monic. A sequence of divisions are calculated using the remainder,{} made monic,{} as divisor for the the next division. The result contains also the error of the factorizations,{} \spadignore{i.e.} the norm of the remainder polynomial. If {\em info} is {\em true
307954`--},{} then information messages are issued.
308071`\spad{divisor(I)} makes a divisor \spad{D} from an ideal \spad{I}.
308174`\spad{divisor(g)} returns the divisor of the function \spad{g}.
308239`\spad{divisor(a,{} b)} makes the divisor \spad{P:} \spad{(x = a,{} y = b)}. Error: if \spad{P} is singular.
308306`\spad{divisor(a,{} b,{} n)} makes the divisor \spad{nP} where \spad{P:} \spad{(x = a,{} y = b)}. \spad{P} is allowed to be singular if \spad{n} is a multiple of the rank.
308381`\spad{divisor(h,{} d,{} d',{} g,{} r)} returns the sum of all the finite points where \spad{h/d} has residue \spad{r}. \spad{h} must be integral. \spad{d} must be squarefree. \spad{d'} is some derivative of \spad{d} (not necessarily dd/dx). \spad{g = gcd(d,{}discriminant)} contains the ramified zeros of \spad{d}
308457`\spad{divisors(n)} returns a list of the divisors of \spad{n}.
308534`\spad{dmp2rfi(p)} converts \spad{p} to target domain
308628`\spad{dmp2rfi(l)} converts \spad{l} to target domain
308734`\spad{dmp2rfi(m)} converts \spad{m} to target domain
308844`\spad{dmpToHdmp(p)} converts \spad{p} from a \spadtype{DMP} to a \spadtype{HDMP}.
308977`\spad{dmpToP(p)} converts \spad{p} from a \spadtype{DMP} to a \spadtype{POLY}.
309070`\spad{doFactor(p,{} n,{} ext?)} returns the factorisation of polynomial \spad{p},{} \spad{p} is assumed to be primitive and squarefree,{} \spad{n} is degree in auxilary variable,{} ext? if \spad{false} inhibits use of extension field.
310643`\spad{dom(a)} returns a \spadgloss{LISP} form of the type of the original object that was converted to \spadtype{Any}.
310683`\spad{domainOf(a)} returns a printable form of the type of the original object that was converted to \spadtype{Any}.
310727`\spad{dominantTerm(f(var))} returns the term that dominates the limiting behavior of \spad{f(var)} as \spad{var -> cen+} together with a \spadtype{String} which briefly describes that behavior. The value of the \
310727`--spadtype{String} will be \spad{"zero"} (resp. \spad{"infinity"}) if the term tends to zero (resp. infinity) exponentially and will \spad{"series"} if the term is a Puiseux series.
311022`\spad{dot(f)} creates the form with a one dot overhead.
311060`\spad{dot(p,{}q)} computes the dot product of the two points \spad{p} and \spad{q} using only the first three coordinates,{} and returns the resulting \spadtype{DoubleFloat}.
311188`\spad{dot(f,{}n)} creates the form \spad{f} with \spad{n} dots overhead.
311245`\spad{dot(x,{}y)} computes the inner product of the vectors \spad{x} and \spad{y}.
311314`\spad{dot(x,{}y)} computes the inner product of the two vectors \spad{x} and \spad{y}. Error: if \spad{x} and \spad{y} are not of the same length.
311372`\spad{double?(t)} tests whether \spad{t} is equivalent to the FORTRAN type DOUBLE PRECISION
311426`\spad{doubleComplex?(t)} tests whether \spad{t} is equivalent to the (non-standard) FORTRAN type DOUBLE COMPLEX.
311487`\spad{doubleDisc(u)} \undocumented
311562`change the output format for doublefloats using lisp format strings
311623`\spad{doubleRank(x)} determines the number of linearly independent elements in \spad{b1*x},{}...,{}\spad{x*bn},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
311692`\spad{doubleResultant(f,{} ')} returns \spad{p}(\spad{x}) whose roots are rational multiples of the residues of \spad{f} at all its finite poles. Argument ' is the derivation to use.
311775`\spad{double(i,{} r)} multiplies \spad{r} by \spad{i} using repeated doubling.
311839`\spad{doublyTransitive?(p)} is \spad{true} if \spad{p} is irreducible over over the field \spad{K} generated by its coefficients,{} and if \spad{p(X) / (X - a)} is irreducible over \spad{K(a)} where \spad{p(a) = 0}.
311899`\spad{drawComplexVectorField(f,{}rRange,{}iRange)} draws a complex vector field using arrows on the \spad{x
311899`--y} plane. These vector fields should be viewed from the top by pressing the "XY" translate button on the 3-\spad{d} viewport control panel.\newline Sample call: \indented{3}{\spad{f z == sin z}} \indented{3}{\spad{drawComplexVectorFie
311899`--ld(f,{} -2..2,{} -2..2)}} Parameter descriptions: \indented{2}{\spad{f} : the function to draw} \indented{2}{\spad{rRange} : the range of the real values} \indented{2}{\spad{iRange} : the range of the imaginary values} Call the functions \axiomFunFrom{setRealSteps}{DrawComplex} and \axiomFunFrom{setImagSteps}{DrawComplex} to change the number of steps used in each direction.
312063`\spad{drawComplex(f,{}rRange,{}iRange,{}arrows?)} draws a complex function as a height field. It uses the complex norm as the height and the complex argument as the color. It will optionally draw arrows on the surface indicating the direction of the complex value.\newline Sample call: \indented{2}{\spad{f z == exp(1/z)}} \indented{2}{\spad{draw
312063`--Complex(f,{} 0.3..3,{} 0..2*\%\spad{pi},{} false)}} Parameter descriptions: \indented{2}{\spad{f:}\space{2}the function to draw} \indented{2}{\spad{rRange} : the range of the real values} \indented{2}{\spad{iRange} : the range of imaginary values} \indented{2}{\spad{arrows?} : a flag indicating whether to draw the phase arrows for \spad{f}} Call the functions \axiomFunFrom{setRealSteps}{DrawComplex} and \axiomFunFrom{setImagSteps}{DrawComplex} to change the number of steps used in each directi
312063`--on.
312224`\spad{drawCurves([[p0],{}[p1],{}...,{}[pn]],{}[options])} creates a \spadtype{TwoDimensionalViewport} from the list of lists of points,{} \spad{p0} throught \spad{pn},{} using the options specified in the list \spad{options}.
312339`\spad{drawCurves([[p0],{}[p1],{}...,{}[pn]],{}ptColor,{}lineColor,{}ptSize,{}[options])} creates a \spadtype{TwoDimensionalViewport} from the list of lists of points,{} \spad{p0} throught \spad{pn},{} using the options specified in the list \spad{options}. The point color is specified by \spad{ptColor},{} the line color is specified by \spad{lineColor},{} an
312339`--d the point size is specified by \spad{ptSize}.
312486`\spad{drawStyle(v,{}s)} displays the surface for the given three-dimensional viewport \spad{v} which is of domain \spadtype{ThreeDimensionalViewport} in the style of drawing indicated by \spad{s}. If \spad{s} is not a valid drawing style the style is wireframe by default. Possible styles are \spad{"shade"},{} \spad{"solid"} or \spad{"opaque"},{} \spad{"smooth"},{} and \spad{"wireMesh"}.
312553`\spad{drawToScale()} determines whether or not plots are to be drawn to scale.
312608`\spad{drawToScale(true)} causes plots to be drawn to scale. \spad{drawToScale(false)} causes plots to be drawn so that they fill up the viewport window. The default setting is \spad{false}.
312670`\spad{draw(lp)} plots the curve constructed from the list of points \spad{lp}.
312771`\spad{draw(f,{}a..b)} draws the graph of \spad{y = f(x)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}.
312900`\spad{draw(f,{}a..b,{}l)} draws the graph of the parametric curve \spad{f} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}.
313038`\spad{draw(f(x),{}x = a..b)} draws the graph of \spad{y = f(x)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{f(x)} appears in the title bar.
313134`\spad{draw(lx,{}ly)} plots the curve constructed of points (\spad{x},{}\spad{y}) for \spad{x} in \spad{lx} for \spad{y} in \spad{ly}.
313246`\spad{draw(lp,{}l)} plots the curve constructed from the list of points \spad{lp}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
313364`\spad{draw(curve(f,{}g),{}a..b)} draws the graph of the parametric curve \spad{x = f(t),{} y = g(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}.
313515`\spad{draw(curve(f(t),{}g(t)),{}t = a..b)} draws the graph of the parametric curve \spad{x = f(t),{} y = g(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{(f(t),{}g(t))} appears in the title bar.
313633`\spad{draw(curve(f,{}g,{}h),{}a..b,{}l)} draws the graph of the parametric curve \spad{x = f(t),{} y = g(t),{} z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}.
313786`\spad{draw(curve(f(t),{}g(t),{}h(t)),{}t = a..b)} draws the graph of the parametric curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{h(t)} is the default title.
313906`\spad{draw(f,{}a..b,{}l)} draws the graph of \spad{y = f(x)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
314052`\spad{draw(f,{}a..b,{}l)} draws the graph of the parametric curve \spad{f} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
314207`\spad{draw(f,{}a..b,{}c..d)} draws the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}.
314365`\spad{draw(f,{}a..b,{}c..d)} draws the graph of the parametric surface \spad{f(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)} The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
314530`\spad{draw(f(x),{}x = a..b,{}l)} draws the graph of \spad{y = f(x)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{f(x)} is the default title,{} and the options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
314643`\spad{draw(f(x,{}y),{}x = a..b,{}y = c..d)} draws the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; \spad{f(x,{}y)} appears in the title bar.
314763`\spad{draw(lx,{}ly,{}lz)} draws the surface constructed by projecting the values in the \axiom{\spad{lz}} list onto the rectangular grid formed by the \axiom{\spad{lx} \spad{X} \spad{ly}}.
314895`\spad{draw(lx,{}ly,{}l)} plots the curve constructed of points (\spad{x},{}\spad{y}) for \spad{x} in \spad{lx} for \spad{y} in \spad{ly}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
315024`\spad{draw(curve(f,{}g),{}a..b,{}l)} draws the graph of the parametric curve \spad{x = f(t),{} y = g(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
315192`\spad{draw(curve(f(t),{}g(t)),{}t = a..b,{}l)} draws the graph of the parametric curve \spad{x = f(t),{} y = g(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{(f(t),{}g(t))} is the default title,{} and the options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
315327`\spad{draw(curve(f,{}g,{}h),{}a..b,{}l)} draws the graph of the parametric curve \spad{x = f(t),{} y = g(t),{} z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
315497`\spad{draw(curve(f(t),{}g(t),{}h(t)),{}t = a..b,{}l)} draws the graph of the parametric curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{h(t)} is the default title,{} and the options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
315634`\spad{draw(surface(f,{}g,{}h),{}a..b,{}c..d)} draws the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}.
315811`\spad{draw(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d)} draws the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; \spad{h(t)} is the default title.
315950`\spad{draw(f,{}a..b,{}c..d,{}l)} draws the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}. and the options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
316125`\spad{draw(f,{}a..b,{}c..d)} draws the graph of the parametric surface \spad{f(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
316307`\spad{draw(f(x,{}y) = g(x,{}y),{}x,{}y,{}l)} draws the graph of a polynomial equation. The list \spad{l} of draw options must specify a region in the plane in which the curve is to sketched.
316442`\spad{draw(f(x,{}y),{}x = a..b,{}y = c..d,{}l)} draws the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; \spad{f(x,{}y)} is the default title,{} and the options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
316579`\spad{draw(lx,{}ly,{}lz,{}l)} draws the surface constructed by projecting the values in the \axiom{\spad{lz}} list onto the rectangular grid formed by the The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
316728`\spad{draw(surface(f,{}g,{}h),{}a..b,{}c..d)} draws the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; The options contained in the li
316728`--st \spad{l} of the domain \spad{DrawOption} are applied.
316922`\spad{draw(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d,{}l)} draws the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; \spad{h(t)} is the default title,{} 
316922`--and the options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
317078`\spad{droot(l)} should be a non-exported function.
317143`\spad{duplicates?(d)} tests if dictionary \spad{d} has duplicate entries.
317206`\spad{duplicates(d)} returns a list of values which have duplicates in \spad{d}
317303`\spad{eFromBinaryMap(n)} sets the appropriate Grassmann basis,{} for example: eFromBinaryMap(0) = 1 (scalar) eFromBinaryMap(1) = \spad{e1} eFromBinaryMap(2) = \spad{e2} eFromBinaryMap(3) = e1/\e2
317383`unit pseudoscalar
317444`\spad{e(n)} produces phi(e_i) where e_i is \spad{i}-th basis vector in \spad{K^n} and phi is canonical embedding of \spad{K^n} into Clifford algebra.
317508`to allow entries like: ee[1,{}2]
317579`\spad{eigenMatrix(m)} returns the matrix \spad{b} such that \spad{b*m*(inverse b)} is diagonal,{} or "failed" if no such \spad{b} exists.
317710`\spad{eigenvalues(m)} returns the eigenvalues of the matrix \spad{m} which are expressible as rational functions over the rational numbers.
317855`\spad{eigenvector(eigval,{}m)} returns the eigenvectors belonging to the eigenvalue \spad{eigval} for the matrix \spad{m}.
318032`\spad{eigenvectors(m)} returns the eigenvalues and eigenvectors for the matrix \spad{m}. The rational eigenvalues and the correspondent eigenvectors are explicitely computed,{} while the non rational ones are given via their minimal polynomial and the corresponding eigenvec
318032`--tors are expressed in terms of a "generic" root of such a polynomial.
318265`\spad{eisensteinIrreducible?(p)} returns \spad{true} if \spad{p} can be shown to be irreducible by Eisenstein\spad{'s} criterion,{} \spad{false} is inconclusive.
318342`\spad{elColumn2!(m,{}a,{}i,{}j)} adds to column \spad{i} a*column(\spad{m},{}\spad{j}) : elementary operation of second kind. (\spad{i} \spad{~=j})
318434`\spad{elRow1!(m,{}i,{}j)} swaps rows \spad{i} and \spad{j} of matrix \spad{m} : elementary operation of first kind
318521`\spad{elRow2!(m,{}a,{}i,{}j)} adds to row \spad{i} a*row(\spad{m},{}\spad{j}) : elementary operation of second kind. (\spad{i} \spad{~=j})
318610`\spad{elem?(ir)} tests if an integration result is elementary over \spad{F?}
318665`\spad{element?(f,{}I)} tests whether the polynomial \spad{f} belongs to the ideal \spad{I}.
318746`\spad{elementary n} is the \spad{n} th elementary symmetric \indented{1}{function expressed in terms of power sums.}
318837`\spad{elements(S)} returns the list of the elements of \spad{S} in increasing order.
318916`\axiom{elimZeroCols!(A)} removes columns which contain only zeros. This effects basically only the value of \axiom{allIndices(A)}.
318980`\spad{ellipseBoundary(c1,{}c2)} constructs an ellipse boundary \spad{n}-dimensional axis-aligned ellipoid with given parameters c1:PT holds centre point c2:PT holds extent (radius for each coordinate)
319036`\spad{ellipticCylindrical(a)} transforms from elliptic cylindrical coordinates to Cartesian coordinates: \spad{ellipticCylindrical(a)} is a function which will map the point \spad{(u,{}v,{}z)} to \spad{x = a*cosh(u)*cos(v)},{} \spad{y = a*sinh(u)*sin(v)},{} \spad{z}.
319117`\spad{ellipticE(m)} is the complete elliptic integral of the second kind: \spad{ellipticE(m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..1)}
319179`\spad{ellipticE(m)} is the complete elliptic integral of the second kind
319277`\spad{ellipticE(m)} is the complete elliptic integral of the second kind.
319357`\spad{ellipticE(m)} is the complete elliptic integral of the second kind
319437`\spad{ellipticE(m)} is the complete elliptic integral of the second kind.
319499`\spad{ellipticE(m)} is the complete elliptic integral of the second kind: \spad{ellipticE(m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..1)}.
319556`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind: \spad{ellipticE(z,{} m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..z)}
319620`\spad{ellipticE(z,{} m)} expands the elliptic integral \spad{E} as a Taylor series in \spad{z}.
319746`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind.
319865`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind.
319960`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind.
320052`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind.
320120`\spad{ellipticE(z,{} m)} is the incomplete elliptic integral of the second kind: \spad{ellipticE(z,{} m) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),{} t = 0..z)}.
320180`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of the first kind : \spad{ellipticF(z,{} m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..z)}
320244`\spad{ellipticF(z,{} m)} expands the elliptic integral \spad{F} as a Taylor series in \spad{z}.
320370`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of the first kind.
320489`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of first kind.
320584`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of the first kind.
320676`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of first kind.
320744`\spad{ellipticF(z,{} m)} is the incomplete elliptic integral of the first kind : \spad{ellipticF(z,{} m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..z)}.
320804`\spad{ellipticK(m)} is the complete elliptic integral of the first kind: \spad{ellipticK(m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..1)}
320866`\spad{ellipticK(z,{} m)} is the incomplete elliptic integral of the first kind.
320964`\spad{ellipticK(m)} is the complete elliptic integral of the first kind.
321044`\spad{ellipticK(z,{} m)} is the complete elliptic integral of the first kind.
321124`\spad{ellipticK(m)} is the complete elliptic integral of the first kind.
321186`\spad{ellipticK(m)} is the complete elliptic integral of the first kind: \spad{ellipticK(m) = integrate(1/sqrt((1-t^2)*(1-m*t^2)),{} t = 0..1)}.
321243`\spad{ellipticPi(z,{} n,{} m)} is the incomplete elliptic integral of the third kind: \spad{ellipticPi(z,{} n,{} m) = integrate(1/((1-n*t^2)*sqrt((1-t^2)*(1-m*t^2))),{} t = 0..z)}
321310`\spad{ellipticPi(z,{} n,{} m)} expands the elliptic integral \spad{Pi} as a Taylor series in \spad{z}.
321442`\spad{ellipticPi(z,{} n,{} m)} is the incomplete elliptic integral of the third kind.
321583`\spad{ellipticPi(z,{} n,{} m)} is the incomplete elliptic integral of the third kind.
321688`\spad{ellipticPi(z,{} n,{} m)} is the incomplete elliptic integral of the third kind: \spad{ellipticPi(z,{} n,{} m) = integrate(1/((1-n*t^2)*sqrt((1-t^2)*(1-m*t^2))),{} t = 0..z)}.
321752`\spad{ellipticRC(x,{} y)} computes integral from 0 to infinity of \spad{(1/2)*(t+x)^(-1/2)*(t+y)^(-1)dt}.
321872`\spad{ellipticRC(x,{} y)} computes integral from 0 to infinity of \spad{(1/2)*(t+x)^(-1/2)*(t+y)^(-1)dt}.
321965`\spad{ellipticRD(x,{} y,{} z)} computes integral from 0 to infinity of \spad{(3/2)*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+Z)^(-3/2)dt}.
322106`\spad{ellipticRD(x,{} y,{} z)} computes integral from 0 to infinity of \spad{(3/2)*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+Z)^(-3/2)dt}.
322211`\spad{ellipticRF(x,{} y,{} z)} computes integral from 0 to infinity of \spad{(1/2)*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+Z)^(-1/2)dt}.
322352`\spad{ellipticRF(x,{} y,{} z)} computes integral from 0 to infinity of \spad{(1/2)*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+Z)^(-1/2)dt}.
322457`ellipticRF(\spad{x},{} \spad{y},{} \spad{z},{} \spad{p}) computes integral from 0 to infinity of \spad{(3/2)*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+Z)^(-1/2)*(t+p)^(-1)dt}.
322619`ellipticRF(\spad{x},{} \spad{y},{} \spad{z},{} \spad{p}) computes integral from 0 to infinity of \spad{(3/2)*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+Z)^(-1/2)*(t+p)^(-1)dt}.
322736`\spad{elliptic()} returns \spad{p(x)} if the curve is the elliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
322818`\spad{elliptic()} returns \spad{p(x)} if the curve is the elliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
322897`\spad{elliptic(a)} transforms from elliptic coordinates to Cartesian coordinates: \spad{elliptic(a)} is a function which will map the point \spad{(u,{}v)} to \spad{x = a*cosh(u)*cos(v)},{} \spad{y = a*sinh(u)*sin(v)}.
323109`\spad{elt(op,{} x,{} y,{} z,{} t,{} s,{} r,{} q,{} p,{} o)} applies the 9-ary operator \spad{op} to \spad{x},{} \spad{y},{} \spad{z},{} \spad{t},{} \spad{s},{} \spad{r},{} \spad{q},{} \spad{p} and \spad{o}
323191`\spad{elt(n)} returns the object \spad{n}.
323230`\spad{elt(t)} gives the component of a rank 0 tensor.
324664`\spad{elt(op,{} [a1,{}...,{}an])} returns \spad{op(a1,{}...,{}an)}.
325505`\spad{elt(mm,{} n)} returns the \spad{n}th element of a sequence.
325559`\spad{elt(op,{}l)} creates a form for application of \spad{op} to list of arguments \spad{l}.
325606`\spad{elt(x,{}r)} or \spad{x}.\spad{r} \undocumented
325689`\spad{elt(op,{}[x1,{}...,{}xn])} or \spad{op}([\spad{x1},{}...,{}\spad{xn}]) applies the \spad{n}-ary operator \spad{op} to \spad{x1},{}...,{}\spad{xn}.
325752`\spad{elt(op,{}x)} or \spad{op}(\spad{x}) applies the unary operator \spad{op} to \spad{x}.
325809`\spad{elt(a,{}b)} evaluates the fraction of univariate polynomials \spad{a} with the distinguished variable replaced by \spad{b}.
325922`\spad{elt(a,{}r)} evaluates the fraction of univariate polynomials \spad{a} with the distinguished variable replaced by the constant \spad{r}.
326004`\axiom{\spad{l}."count"} returns the number of elements in \axiom{\spad{l}}.
326067`\spad{elt(u,{}"first")} (also written: \axiom{\spad{u} . first}) is equivalent to first \spad{u}.
326128`\spad{elt(u,{}"last")} (also written: \axiom{\spad{u} . last}) is equivalent to last \spad{u}.
326188`\spad{elt(u,{}"left")} (also written: \axiom{a . left}) is equivalent to \axiom{left(a)}.
326250`\spad{elt(\%,{}"rest")} (also written: \axiom{\spad{u}.rest}) is equivalent to \axiom{rest \spad{u}}.
326311`\spad{elt(a,{}"right")} (also written: \axiom{a . right}) is equivalent to \axiom{right(a)}.
326374`\axiom{\spad{l}.sort} returns \axiom{\spad{l}} with elements sorted. Note: \axiom{\spad{l}.sort = sort(\spad{l})}
326420`\axiom{\spad{l}.unique} returns \axiom{\spad{l}} with duplicates removed. Note: \axiom{\spad{l}.unique = removeDuplicates(\spad{l})}.
326468`\spad{elt(u,{}"value")} (also written: \axiom{a. value}) is equivalent to \axiom{value(a)}.
326524`\spad{elt(qf,{}v)} evaluates the quadratic form \spad{qf} on the vector \spad{v},{} producing a scalar.
326588`\spad{elt(f(x),{}r)} returns the coefficient of the term of degree \spad{r} in \spad{f(x)}. This is the same as the function \spadfun{coefficient}.
326665`\spad{elt(a,{}i)} returns the \spad{i}-th coefficient of \spad{a} with respect to the fixed \spad{R}-module basis.
326730`\spad{elt(t,{}i)} gives a component of a rank 1 tensor.
326793`\spad{elt((a1,{}...,{}an),{} i)} returns \spad{\spad{ai}}.
326870`\spad{elt(t,{}[i1,{}...,{}iN])} gives a component of a rank \spad{N} tensor.
326939`\spad{elt((a1,{}...,{}an),{} [i1,{}...,{}im])} returns \spad{(a_i1,{}...,{}a_im)}.
327022`\spad{elt(s,{}[a1,{}...,{}an])} or \spad{s}([\spad{a1},{}...,{}an]) returns \spad{s} subscripted by \spad{[a1,{}...,{}an]}.
327073`\spad{elt(gp,{}i)} returns the \spad{i}-th generator of the group {\em gp}.
327151`\spad{elt(db,{}q)} returns all elements of \axiom{\spad{db}} which satisfy \axiom{\spad{q}}.
327204`\spad{elt(u,{}i)} (also written: \spad{u} . \spad{i}) returns the element of \spad{u} indexed by \spad{i}. Error: if \spad{i} is not an index of \spad{u}.
327253`\spad{elt(p,{} el)} returns the image of {\em el} under the permutation \spad{p}.
327304`\spad{elt(lib,{}k)} or \spad{lib}.\spad{k} extracts the value corresponding to the key \spad{k} from the library \spad{lib}.
327347`\spad{elt(db,{}s)} returns the \axiom{\spad{s}} field of each element of \axiom{\spad{db}}.
327407`\spad{elt(ic,{}s)} selects a particular field from \axiom{\spad{ic}}. Valid fields are \axiom{name,{} nargs,{} exposed,{} type,{} abbreviation,{} kind,{} origin,{} params,{} condition,{} doc}.
327455`\spad{elt(u,{}i..j)} (also written: \axiom{a(\spad{i}..\spad{j})}) returns the aggregate of elements \axiom{\spad{u}} for \spad{k} from \spad{i} to \spad{j} in that order. Note: in general,{} \axiom{a.\spad{s} = [a.\spad{k} for \spad{i} in \spad{s}]}.
327527`\spad{elt(s,{}t)} returns the concatenation of \spad{s} and \spad{t}. It is provided to allow juxtaposition of strings to work as concatenation. For example,{} \axiom{"smoo" "shed"} returns \axiom{"smooshed"}.
328336`\axiom{elt(A,{} \spad{i},{} \spad{c})} returns the entry of the matrix \axiom{A} in row \axiom{\spad{i}} and in the column with index \axiom{\spad{c}}.
328397`\spad{elt(op,{}x,{}y)} or \spad{op}(\spad{x},{} \spad{y}) applies the binary operator \spad{op} to \spad{x} and \spad{y}.
328457`\spad{elt(u,{} x,{} y)} applies \spad{u} to \spad{x} if \spad{x} is in the domain of \spad{u},{} and returns \spad{y} otherwise. For example,{} if \spad{u} is a polynomial in \axiom{\spad{x}} over the rationals,{} \axiom{elt(\spad{u},{}\spad{n},{}0)} may define the coefficient of \axiom{\spad{x}} to the power \spad{n},{} returning 0 when \spad{n} is out of range.
328516`\spad{elt(f,{}a,{}b)} or \spad{f}(a,{} \spad{b}) returns the value of \spad{f} at the point \spad{(x = a,{} y = b)} if it is not singular.
328579`\spad{elt(r,{}f,{}n)} or \spad{r}(\spad{f},{} \spad{n}) applies the rule \spad{r} to \spad{f} at most \spad{n} times.
328645`\spad{elt(r,{}f,{}n)} or \spad{r}(\spad{f},{} \spad{n}) applies all the rules of \spad{r} to \spad{f} at most \spad{n} times.
328707`\spad{elt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the matrix \spad{m}. Error: if indices are outside the proper ranges.
328790`\spad{elt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the array \spad{m} error check to determine if indices are in proper ranges
328871`\spad{elt(t,{}i,{}j)} gives a component of a rank 2 tensor.
328942`\spad{elt(x,{}rowList,{}colList)} returns an \spad{m}-by-\spad{n} matrix consisting of elements of \spad{x},{} where \spad{m = \# rowList} and \spad{n = \# colList}. If \spad{rowList = [i<1>,{}i<2>,{}...,{}i<m>]} and \spad{colList = [j<1>,{}j<2>,{}...,{}j<n>]},{} then the \spad{(k,{}l)}th entry of \spad{elt(x,{}rowList,{}colList)} is \spad{x(i<k>,{}j<l>)}.
329286`\spad{elt(x,{}i,{}j,{}k)} extract an element from the matrix \spad{x}
329395`\spad{elt(op,{}x,{}y,{}z)} or \spad{op}(\spad{x},{} \spad{y},{} \spad{z}) applies the ternary operator \spad{op} to \spad{x},{} \spad{y} and \spad{z}.
329458`\spad{elt(t,{}i,{}j,{}k)} gives a component of a rank 3 tensor.
329537`\spad{elt(m,{}i,{}j,{}r)} returns the element in the \spad{i}th row and \spad{j}th column of the matrix \spad{m},{} if \spad{m} has an \spad{i}th row and a \spad{j}th column,{} and returns \spad{r} otherwise.
329622`\spad{elt(m,{}i,{}j,{}r)} returns the element in the \spad{i}th row and \spad{j}th column of the array \spad{m},{} if \spad{m} has an \spad{i}th row and a \spad{j}th column,{} and returns \spad{r} otherwise
329825`\spad{elt(op,{}x,{}y,{}z,{}t)} or \spad{op}(\spad{x},{} \spad{y},{} \spad{z},{} \spad{t}) applies the 4-ary operator \spad{op} to \spad{x},{} \spad{y},{} \spad{z} and \spad{t}.
329891`\spad{elt(t,{}i,{}j,{}k,{}l)} gives a component of a rank 4 tensor.
330102`\spad{elt(op,{} x,{} y,{} z,{} t,{} s)} applies the 5-ary operator \spad{op} to \spad{x},{} \spad{y},{} \spad{z},{} \spad{t} and \spad{s}
330299`\spad{elt(op,{} x,{} y,{} z,{} t,{} s,{} r)} applies the 6-ary operator \spad{op} to \spad{x},{} \spad{y},{} \spad{z},{} \spad{t},{} \spad{s} and \spad{r}
330503`\spad{elt(op,{} x,{} y,{} z,{} t,{} s,{} r,{} q)} applies the 7-ary operator \spad{op} to \spad{x},{} \spad{y},{} \spad{z},{} \spad{t},{} \spad{s},{} \spad{r} and \spad{q}
330714`\spad{elt(op,{} x,{} y,{} z,{} t,{} s,{} r,{} q,{} p)} applies the 8-ary operator \spad{op} to \spad{x},{} \spad{y},{} \spad{z},{} \spad{t},{} \spad{s},{} \spad{r},{} \spad{q} and \spad{p}
330792`\spad{eltable? r} returns whether the operator determines (most likely) the sequence completely.
330895`\spad{empty?(s)} returns \spad{true} if the quasialgebraic set \spad{s} has no points,{} and \spad{false} otherwise.
330967`\axiom{empty?(\spad{n})} returns \spad{true} iff the node \spad{n} is \axiom{empty()\$\%}.
331021`\spad{empty?(el)} returns \spad{true} is this element does not contain anything
331067`\spad{empty?(u)} tests if \spad{u} has 0 elements.
331112`\spad{empty()} creates an empty form.
331150`\spad{empty()} returns the empty quasi-algebraic set
331214`\axiom{empty()} returns the same as \axiom{[empty()\$\spad{V},{}empty()\$\spad{C},{}\spad{false}]\$\%}
331260`\spad{empty()} returns a new,{} empty symbol table
331299`\spad{empty()} creates a new,{} empty symbol table.
331341`\spad{empty()}\$\spad{D} creates an aggregate of type \spad{D} with 0 elements. Note: The {\em \$D} can be dropped if understood by context,{} \spadignore{e.g.} \axiom{u: \spad{D} \spad{:=} empty()}.
331378`\spad{empty(n)} produces reconstructor with \spad{n} slots
331439`\spad{empty(n,{} p)} initializes reconstructor with \spad{n} slots working modulo \spad{p}
331508`\spad{endOfFile?(f)} tests whether the file \spad{f} is positioned after the end of all text. If the file is open for output,{} then this test is always \spad{true}.
331556`\spad{endSubProgram()} asserts that we are no longer processing the current subprogram.
331610`\spad{enqueue!(x,{}q)} inserts \spad{x} into the queue \spad{q} at the back end.
331661`\spad{enterInCache(x,{} f)} enters \spad{x} in the cache,{} calling \spad{f(y)} to determine whether \spad{x} is equal to \spad{y}. It returns \spad{x} with an integer associated with it.
331723`\spad{enterInCache(x,{} f)} enters \spad{x} in the cache,{} calling \spad{f(x,{} y)} to determine whether \spad{x < y (f(x,{}y) < 0),{} x = y (f(x,{}y) = 0)},{} or \spad{x > y (f(x,{}y) > 0)}. It returns \spad{x} with an integer associated with it.
331787`\spad{enterPointData(s,{}[p0,{}p1,{}...,{}pn])} adds a list of points from \spad{p0} through \spad{pn} to the \spadtype{ThreeSpace},{} \spad{s},{} and returns the index,{} to the starting point of the list.
332064`\spad{entries(lp)} returns list of indexes that make this loop
332122`\spad{entries(u)} returns a list of all the entries of aggregate \spad{u} in no assumed order.
332260`\spad{entry?(x,{}u)} tests if \spad{x} equals \axiom{\spad{u} . \spad{i}} for some index \spad{i}.
332424`\spad{enumerate()} returns a vector of all the sets of \spad{M} integers in \spad{1..n}.
332491`\spad{enumerate()} returns list of elements of the set.
332535`\spad{epilogue(t)} extracts the epilogue section of a formatted object \spad{t}.
332597`\spad{epilogue(t)} extracts the epilogue section of a TeX form \spad{t}.
332691`\spad{eq?(u,{}v)} tests if \spad{u} and \spad{v} are same objects.
332736`\axiom{eqRep?(\spad{x},{} \spad{y})} compares the representations of \axiom{\spad{x}} and \axiom{\spad{y}} without any evaluation. Thus it is much weaker than \axiom{=} and cannot decide equality of the evaluated expressions.
332796`\spad{eq(s,{} t)} is \spad{true} if EQ(\spad{s},{}\spad{t}) is \spad{true} in Lisp.
332872`\spad{equality(op,{} foo?)} attaches foo? as the "\%equal?" property to \spad{op}. If \spad{op1} and \spad{op2} have the same name,{} and one of them has an "\%equal?" property \spad{f},{} then \spad{f(op1,{} op2)} is called to decide whether \spad{op1} and \spad{op2} should be considered equal.
332935`\spad{equation(a,{}b)} creates an equation.
332980`\spad{equation(v,{}a..b)} creates a segment binding value with variable \spad{v} and segment \spad{a..b}. Note that the interpreter parses \spad{v=a..b} to this form.
333045`\spad{equation(s,{}"a")} creates a new equation.
333102`\spad{erf(f)} denotes the error function
333152`\spad{erf(f)} denotes the error function
333228`\spad{erf(f)} denotes the error function
333286`\spad{erf(x)} returns the error function of \spad{x},{} \spadignore{i.e.} \spad{2 / sqrt(\%\spad{pi})} times the integral of \spad{exp(-x^2) dx}.
333341`\spad{errorInfo(u)} returns information about the error \spad{u}.
333398`\spad{errorKind(u)} returns the type of error which \spad{u} represents.
333460`\spad{error(lmsg)} displays error message \spad{lmsg} and terminates.
333516`\spad{error(msg)} displays error message \spad{msg} and terminates.
333566`\spad{error(nam,{}lmsg)} displays error messages \spad{lmsg} preceded by a message containing the name \spad{nam} of the function in which the error is contained.
333629`\spad{error(nam,{}msg)} displays error message \spad{msg} preceded by a message containing the name \spad{nam} of the function in which the error is contained.
333686`\spad{escape()} provides the escape character,{} \spad{_},{} which is used to allow quotes and other characters {\em within} strings.
333724`\spad{euclideanGroebner(lp)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \spad{lp}.
333834`\spad{euclideanGroebner(lp,{} infoflag)} computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials \spad{lp}. During computation,{} additional information is printed out if infoflag is given as either "info" (for summary information) or "redcrit" (for reduced critical pairs)
333951`\spad{euclideanGroebner(lp,{} "info",{} "redcrit")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp}. If the second argument is \spad{"info"},{} a summary is given of the critical pairs. If the third argument is "redcrit",{} critical pairs are printed.
334075`\spad{euclideanNormalForm(poly,{}gb)} reduces the polynomial \spad{poly} modulo the precomputed groebner basis \spad{gb} giving a canonical representative of the residue class.
334465`\spad{euclideanSize(x)} returns the euclidean size of the element \spad{x}. Error: if \spad{x} is zero.
334534`\spad{eulerE(n,{}r)} returns value of the \spad{n}th Euler polynomial at \spad{r}.
334652`\spad{eulerPhi(n)} returns the number of integers between 1 and \spad{n} (including 1) which are relatively prime to \spad{n}. This is the Euler phi function \spad{\phi(n)} is also called the totient function.
334723`\spad{euler(n)} returns the \spad{n}th Euler polynomial \spad{E[n](x)}. Note: Euler polynomials denoted \spad{E(n,{}x)} are computed by solving the differential equation \spad{differentiate(E(n,{}x),{}x) = n E(n-1,{}x)} where \spad{E(0,{}x) = 1} and initial condition comes from \spad{E(n) = 2^n E(n,{}1/2)}.
334832`\spad{euler(n)} returns the \spad{n}th Euler number. This is \spad{2^n E(n,{}1/2)},{} where \spad{E(n,{}x)} is the \spad{n}th Euler polynomial.
334900`\axiom{\spad{eval1}(exp)} explicitly evaluates the next term in \axiom{exp}. \axiom{exp} is destructively altered.
335584`\spad{eval_at(v,{} deg,{} pt,{} p)} treats \spad{v} as coefficients of polynomial of degree \spad{deg} and evaluates the polynomial at point \spad{pt} modulo \spad{p}
335722`\axiom{eval(exp)} explicitly evaluates all terms in \axiom{exp}. \axiom{exp} is destructively altered.
335772`\spad{eval s} is the sum of the coefficients of a cycle index.
335867`\spad{eval(f)} unquotes all the quoted operators in \spad{f}.
335943`\spad{eval(f,{}s)} evaluates the cycle index \spad{s} by applying \indented{1}{the function \spad{f} to each integer in a monomial partition,{}} \indented{1}{forms their product and sums the results over all monomials.}
337160`\spad{eval(a,{}r)} returns a stream of partial sums of the power series \spad{a} evaluated at the power series variable equal to \spad{r}.
337236`\axiom{eval(\spad{v},{} \spad{f})} applies the vector field \axiom{\spad{v}} to the function \axiom{\spad{f}}.
337286`\spad{eval(m,{}x)} returns \spad{(a*x + b)/(c*x + d)} where \spad{m = moebius(a,{}b,{}c,{}d)} (see \spadfunFrom{moebius}{MoebiusTransform}).
337335`\axiom{eval(om,{} \spad{v})} applies the differential \axiom{om} to the vector field \axiom{\spad{v}}.
337405`\spad{eval(m,{}x)} returns \spad{(a*x + b)/(c*x + d)} where \spad{m = moebius(a,{}b,{}c,{}d)} (see \spadfunFrom{moebius}{MoebiusTransform}).
337494`\spad{eval(d,{} p)} evaluates the distribution \spad{d} as a linear functional on the polynomial \spad{p}. Same as \spad{integrate(p,{} d)}.
337577`\spad{eval(f,{} v = g)} returns \spad{f} with \spad{v} replaced by \spad{g}. Error: if \spad{v} is not a symbol.
337701`\spad{eval(f,{} [v1 = g1,{}...,{}vn = gn])} returns \spad{f} with each \spad{vi} replaced by \spad{gi} in parallel,{} \spadignore{i.e.} \spad{vi}\spad{'s} appearing inside the \spad{gi}\spad{'s} are not replaced. Error: if any \spad{vi} is not a symbol.
337831`\spad{eval(x,{}f)} evaluates the bivariate function \spad{f} linearly on the tensor product.
337903`\spad{eval(f,{}a)} evaluates a power series at a value in the ground ring by returning a stream of partial sums.
338027`\spad{eval(f,{}x = v)} replaces \spad{x} by \spad{v} in \spad{f}.
338079`\spad{eval(f,{} [x1 = v1,{}...,{}xn = vn])} replaces \spad{xi} by \spad{vi} in \spad{f}.
338137`\spad{eval(f,{} [foo1,{}...,{}foon])} unquotes all the \spad{fooi}\spad{'s} in \spad{f}.
338226`\spad{eval(eqn,{} [x1=v1,{} ... xn=vn])} replaces \spad{xi} by \spad{vi} in equation \spad{eqn}.
338317`\spad{eval(p,{} el)} returns the image of {\em el} under the permutation \spad{p}.
338369`\spad{eval(f,{} foo)} unquotes all the foo\spad{'s} in \spad{f}.
338452`\spad{eval(eqn,{} x=f)} replaces \spad{x} by \spad{f} in equation \spad{eqn}.
342227`\spad{eval(p(x,{}y),{} f(x),{} g(x))} returns \spad{p(f(x),{} y * g(x))}.
342313`\spad{eval(f,{} [v1,{}...,{}vn],{} [g1,{}...,{}gn])} returns \spad{f} with each \spad{vi} replaced by \spad{gi} in parallel,{} \spadignore{i.e.} \spad{vi}\spad{'s} appearing inside the \spad{gi}\spad{'s} are not replaced.
342446`\spad{eval(f,{} v,{} g)} returns \spad{f} with \spad{v} replaced by \spad{g}.
342567`\spad{eval(f,{} x,{} v)} replaces \spad{x} by \spad{v} in \spad{f}.
342618`\spad{eval(x,{} s,{} f)} replaces every \spad{s(a1,{}..,{}am)} in \spad{x} by \spad{f(a1,{}..,{}am)} for any \spad{a1},{}...,{}\spad{am}.
342691`\spad{eval(x,{} s,{} f)} replaces every \spad{s(a)} in \spad{x} by \spad{f(a)} for any \spad{a}.
342758`\spad{eval(f,{} [x1,{}...,{}xn],{} [v1,{}...,{}vn])} replaces \spad{xi} by \spad{vi} in \spad{f}.
342821`\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \spad{\spad{si}(a1,{}...,{}an)} in \spad{x} by \spad{\spad{fi}(a1,{}...,{}an)} for any \spad{a1},{}...,{}\spad{an}.
342906`\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \spad{\spad{si}(a)} in \spad{x} by \spad{\spad{fi}(a)} for any \spad{a}.
342985`\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \spad{\spad{si}(a1,{}...,{}an)} in \spad{x} by \spad{\spad{fi}(a1,{}...,{}an)} for any \spad{a1},{}...,{}\spad{an}.
343063`\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm])} replaces every \spad{\spad{si}(a)} in \spad{x} by \spad{\spad{fi}(a)} for any \spad{a}.
343135`\axiom{eval(\spad{p},{} [\spad{x1},{}...,{}\spad{xn}],{} [\spad{v1},{}...,{}\spad{vn}])} replaces \axiom{\spad{xi}} by \axiom{\spad{vi}} in \axiom{\spad{p}}.
343210`\spad{eval(x,{} s,{} f)} replaces every \spad{s(a1,{}..,{}am)} in \spad{x} by \spad{f(a1,{}..,{}am)} for any \spad{a1},{}...,{}\spad{am}.
343276`\spad{eval(x,{} s,{} f)} replaces every \spad{s(a)} in \spad{x} by \spad{f(a)} for any \spad{a}.
343336`\axiom{eval(\spad{p},{} \spad{x},{} \spad{v})} replaces \axiom{\spad{x}} by \axiom{\spad{v}} in \axiom{\spad{p}}.
343883`\spad{eval(x,{} s,{} f,{} y)} replaces every \spad{s(a)} in \spad{x} by \spad{f(y)} with \spad{y} replaced by \spad{a} for any \spad{a}.
343983`\spad{eval(x,{} [s1,{}...,{}sm],{} [f1,{}...,{}fm],{} y)} replaces every \spad{\spad{si}(a)} in \spad{x} by \spad{\spad{fi}(y)} with \spad{y} replaced by \spad{a} for any \spad{a}.
344095`\spad{eval(x,{} [s1,{}...,{}sm],{} [n1,{}...,{}nm],{} [f1,{}...,{}fm])} replaces every \spad{\spad{si}(a1,{}...,{}an)^ni} in \spad{x} by \spad{\spad{fi}(a1,{}...,{}an)} for any \spad{a1},{}...,{}am.
344210`\spad{eval(x,{} [s1,{}...,{}sm],{} [n1,{}...,{}nm],{} [f1,{}...,{}fm])} replaces every \spad{\spad{si}(a)^ni} in \spad{x} by \spad{\spad{fi}(a)} for any \spad{a}.
344319`\spad{eval(x,{} s,{} n,{} f)} replaces every \spad{s(a1,{}...,{}am)^n} in \spad{x} by \spad{f(a1,{}...,{}am)} for any \spad{a1},{}...,{}am.
344416`\spad{eval(x,{} s,{} n,{} f)} replaces every \spad{s(a)^n} in \spad{x} by \spad{f(a)} for any \spad{a}.
344560`\spad{evaluateInverse(x,{}f)} \undocumented
344626`\spad{evaluate(op)} returns the value of the "\%eval" property of \spad{op} if it has one,{} and "failed" otherwise.
344768`\spad{evaluate(op,{} foo)} attaches foo as the "\%eval" property of \spad{op}. If \spad{op} has an "\%eval" property \spad{f},{} then applying \spad{op} to a returns the result of \spad{f(a)}. Argument \spad{op} must be unary.
344856`\spad{evaluate(op,{} foo)} attaches foo as the "\%eval" property of \spad{op}. If \spad{op} has an "\%eval" property \spad{f},{} then applying \spad{op} to \spad{(a1,{}...,{}an)} returns the result of \spad{f(a1,{}...,{}an)}.
344950`\spad{evaluate(op,{} [a1,{}...,{}an])} checks if \spad{op} has an "\%eval" property \spad{f}. If it has,{} then \spad{f(a1,{}...,{}an)} is returned,{} and "failed" otherwise.
345043`\spad{evaluate(f,{} u +-> g u)} attaches the map \spad{g} to \spad{f}. \spad{f} must be a basic operator \spad{g} MUST be additive,{} \spadignore{i.e.} \spad{g(a + b) = g(a) + g(b)} for any \spad{a},{} \spad{b} in \spad{M}. This implies that \spad{g(n a) = n g(a)} for any \spad{a} in \spad{M} and integer \spad{n > 0}.
345202`\spad{even? x} is \spad{true} if \spad{x} is an even integer.
345282`\spad{even?(n)} returns \spad{true} if and only if \spad{n} is even.
345336`\spad{even?(p)} returns \spad{true} if and only if \spad{p} is an even permutation,{} \spadignore{i.e.} {\em sign(p)} is 1.
345385`\spad{evenInfiniteProduct(f(x))} computes \spad{product(n=2,{}4,{}6...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
345476`\spad{evenInfiniteProduct(f(x))} computes \spad{product(n=2,{}4,{}6...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
345565`\spad{evenInfiniteProduct(f(x))} computes \spad{product(n=2,{}4,{}6...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
345652`\spad{evenInfiniteProduct(f(x))} computes \spad{product(n=2,{}4,{}6...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
345733`\spad{evenlambert(st)} computes \spad{f(x^2) + f(x^4) + f(x^6) + ...} if \spad{st} is a stream representing \spad{f(x)}. This function is used for computing infinite products. If \spad{f(x)} is a power series with constant coefficient 1,{} then \spad{prod(f(x^(2*n)),{}n=1..infinity) = exp(evenlambert(log(f(x))))}.
345878`\spad{evenlambert(f(x))} returns \spad{f(x^2) + f(x^4) + f(x^6) + ...}. \indented{1}{\spad{f(x)} should have a zero constant coefficient.} \indented{1}{This function is used for computing infinite products.} \indented{1}{If \spad{f(x)} is a Taylor series with constant term 1,{} then} \indented{1}{\spad{product(n=1..infinity,{}f(x^(2*n))) = exp(log(evenlambert(f(x))))}.}
346420`\spad{every?(f,{}u)} tests if \spad{p}(\spad{x}) is \spad{true} for all elements \spad{x} of \spad{u}. Note: for collections,{} \axiom{every?(\spad{p},{}\spad{u}) = reduce(and,{}map(\spad{f},{}\spad{u}),{}\spad{true},{}\spad{false})}. However,{} \axiom{every?(\spad{p},{}\spad{u})} returns as soon as it finds an element for which \axiom{\spad{p}} gives \spad{false}.
346515`\spad{exQuo(x,{}y)} \undocumented
346618`\spad{exQuo(x,{}y)} \undocumented
346711`\spad{exQuo(x,{}y)} \undocumented
346945`\axiom{exactQuotient!(\spad{p},{}\spad{r})} replaces \axiom{\spad{p}} by \axiom{exactQuotient(\spad{p},{}\spad{r})}.
347041`\axiom{exactQuotient!(a,{}\spad{b})} replaces \axiom{a} by \axiom{exactQuotient(a,{}\spad{b})}
347278`\axiom{exactQuotient(\spad{p},{}\spad{r})} computes the exact quotient of \axiom{\spad{p}} by \axiom{\spad{r}},{} which is assumed to be a divisor of \axiom{\spad{p}}. No error is returned if this exact quotient fails!
347373`\axiom{exactQuotient(a,{}\spad{b})} computes the exact quotient of \axiom{a} by \axiom{\spad{b}},{} which is assumed to be a divisor of \axiom{a}. No error is returned if this exact quotient fails!
347469`\spad{exists?(f)} tests if the file exists in the file system.
347522`\spad{exp0(f)} returns the exponential of the power series represented by cons(0,{}\spad{f}),{} \spadignore{i.e.} assuming zero constant term and therefore transcendentality is not involved.
347610`\spad{exp1()} returns the natural log base \spad{2.718281828...}.
347648`\spad{exp1()} returns exp 1: \spad{2.7182818284...}.
347680`\spad{expIfCan(z)} returns exp(\spad{z}) if possible,{} and "failed" otherwise.
347760`\spad{expPot(v,{}e,{}d)} returns the sum from \spad{i = 0} to \spad{e - 1} of \spad{v^(q^i*d)},{} interpreting \spad{v} as an element of a normal basis field and where \spad{q} is the size of the ground field. Note: for a description of the algorithm,{} see \spad{T}.Itoh and \spad{S}.Tsujii,{} "A fast algorithm for computing multiplicative inverses in \spad{GF}(2^m) using normal bases",{} Inform
347760`--ation and Computation 78,{} \spad{pp}.171-177,{} 1988.
347869`\spad{exp(x)} applies the exponential operator to \spad{x}
347918`\axiom{exp(\spad{p})} returns the exponential of \axiom{\spad{p}}.
347998`\spad{exp([i1,{}...in])} returns \spad{u_1\^{i_1} ... u_n\^{i_n}}
348250`\spad{exp(st)} computes the exponential of a power series \spad{st}.
348333`\spad{exp(st)} computes the exponential of a power series \spad{st}.
348430`\spad{exp(z)} returns the exponential of Laurent series \spad{z}.
348516`\spad{exp(z)} returns the exponential of a Puiseux series \spad{z}.
348611`\spad{exp(x)} represents the Fortran intrinsic function EXP
348691`\spad{exp(x)} returns \%\spad{e} to the power \spad{x}.
348745`\axiom{exp(\spad{p},{} \spad{n})} returns the exponential of \axiom{\spad{p}} truncated at order \axiom{\spad{n}}.
348833`\axiom{exp(\spad{p},{}\spad{n})} returns the exponential of \axiom{\spad{p}} (truncated up to order \axiom{\spad{n}}).
348936`\spad{expandLog(f)} converts every \spad{log(a/b)} appearing in \spad{f} into \spad{log(a) - log(b)},{} and every \spad{log(a*b)} into \spad{log(a) + log(b)}..
349000`\spad{expandPower(f)} converts every power \spad{(a/b)^c} appearing in \spad{f} into \spad{a^c * b^(-c)}.
349066`\spad{expandTrigProducts(e)} replaces \axiom{sin(\spad{x})*sin(\spad{y})} by \spad{(cos(x-y)-cos(x+y))/2},{} \axiom{cos(\spad{x})*cos(\spad{y})} by \spad{(cos(x-y)+cos(x+y))/2},{} and \axiom{sin(\spad{x})*cos(\spad{y})} by \spad{(sin(x-y)+sin(x+y))/2}. Note that this operation uses the pattern matcher and so 
349066`--is relatively expensive. To avoid getting into an infinite loop the transformations are applied at most ten times.
349333`\spad{expand(p)} returns \spad{p} in distributed form.
349424`\spad{expand(f)} performs the following expansions on \spad{f:}\begin{items} \item 1. logs of products are expanded into sums of logs,{} \item 2. trigonometric and hyperbolic trigonometric functions of sums are expanded into sums of products of trigonometric and hyperbolic trigonometric functions. \item 3. formal powers of the form \spad{(a/b)^c} are expanded into \spad{a^c * b^(-c)}. \end{items}
349485`\spad{expand(i)} returns the list of possible real functions corresponding to \spad{i}.
349571`\spad{expand(i)} returns the list of possible real functions corresponding to \spad{i}.
349691`\spad{expand(l)} creates a new value of type \spad{L} in which each segment \spad{l..h by k} is replaced with \spad{l,{} l+k,{} ... lN},{} where \spad{lN <= h < lN+k}. For example,{} \spad{expand [1..4,{} 7..9] = [1,{}2,{}3,{}4,{}7,{}8,{}9]}.
349756`\spad{expand(l..h by k)} creates value of type \spad{L} with elements \spad{l,{} l+k,{} ... lN} where \spad{lN <= h < lN+k}. For example,{} \spad{expand(1..5 by 2) = [1,{}3,{}5]}.
349815`\spad{expand(f)} multiplies the unit and factors together,{} yielding an "unfactored" object. Note: this is purposely not called \spadfun{coerce} which would cause the interpreter to do this automatically.
349856`\spad{expand(f,{}n)} \undocumented{}
349962`\spad{expextendedint(f,{} ',{} foo,{} g)} returns either \spad{[v,{} c]} such that \spad{f = v' + c g} and \spad{c' = 0},{} or \spad{[v,{} a]} such that \spad{f = g' + a},{} and \spad{a = 0} or \spad{a} has no integral in \spad{F}. Returns "failed" if neither cas
349962`--e can hold. Argument \spad{foo} is a Risch differential equation function on \spad{F}.
350206`\spad{expint(f,{} x)} returns e^{the integral of \spad{f} with respect to \spad{x}}.
350261`\spad{expintegrate(f,{} ',{} foo)} returns \spad{[g,{} a]} such that \spad{f = g' + a},{} and \spad{a = 0} or \spad{a} has no integral in \spad{F}; Argument foo is a Risch differential equation solver on \spad{F}.
350545`\spad{expintfldpoly(p,{} foo)} returns \spad{q} such that \spad{p' = q} or "failed" if no such \spad{q} exists. Argument foo is a Risch differential equation function on \spad{F}.
350720`\spad{explicitEntries?(s)} returns \spad{true} if the stream \spad{s} has explicitly computed entries,{} and \spad{false} otherwise.
350788`\spad{explicitlyEmpty?(s)} returns \spad{true} if the stream is an (explicitly) empty stream. Note: this is a null test which will not cause lazy evaluation.
350984`\spad{explicitlyFinite?(s)} tests if the stream has a finite number of elements,{} and \spad{false} otherwise. Note: for many datatypes,{} \axiom{explicitlyFinite?(\spad{s}) = not possiblyInfinite?(\spad{s})}.
351049`\spad{explimitedint(f,{} ',{} foo,{} [u1,{}...,{}un])} returns \spad{[v,{} [c1,{}...,{}cn],{} a]} such that \spad{ci' = 0},{} \spad{f = v' + a + reduce(+,{}[\spad{ci} * ui'/ui])},{} and \spad{a = 0} or \spad{a} has no integr
351049`--al in \spad{F}. Returns "failed" if no such \spad{v},{} \spad{ci},{} a exist. Argument \spad{foo} is a Risch differential equation function on \spad{F}.
351332`\spad{explogs2trigs(f)} rewrites all the complex logs and exponentials appearing in \spad{f} in terms of trigonometric functions.
351417`\spad{exponent(x)} \undocumented
351472`\spad{exponent(u)} returns the exponent of \spad{u}
351522`\spad{exponent(exp(f(x)))} returns \spad{f(x)}
351636`\spad{exponent(x)} returns the \spadfunFrom{exponent}{FloatingPointSystem} part of \spad{x}.
351693`\spad{exponent(u)} returns the exponent of the first factor of \spadvar{\spad{u}},{} or 0 if the factored form consists solely of a unit.
351742`\spad{exponential1()} \undocumented
351804`\spad{exponentialOrder(exp(c * x ^(-n) + ...))} returns \spad{-n}. exponentialOrder(0) returns \spad{0}.
351908`\spad{exponential(f)} \undocumented
351978`\spad{exponential(f(x))} returns \spad{exp(f(x))}. Note: the function does NOT check that \spad{f(x)} has no non-negative terms.
352095`\spad{exponents(x)} converts a domain element into a list of zeros and ones corresponding to the exponents in the basis element that \spad{x} represents.
352151`\spad{exprToGenUPS(fcn,{}posCheck?,{}atanFlag)} converts the expression \spad{fcn} to a generalized power series. If \spad{posCheck?} is \spad{true},{} log\spad{'s} of negative numbers are not allowed nor are \spad{n}th roots of negative numbers with \
352151`--spad{n} even. If \spad{posCheck?} is \spad{false},{} these are allowed. \spad{atanFlag} determines how the case \spad{atan(f(x))},{} where \spad{f(x)} has a pole,{} will be treated. The possible values of \spad{atanFlag} are \spad{"complex"},{} \spad{"real: two sides"},{} \spad{"real: left side"},{} \spad{"real: right side"},{} and \spad{"just do it"}. If \spad{atanFlag} is \spad{"complex"},{} then no series expansion will be computed because,{} viewed as a function of a complex variable,{} \s
352151`--pad{atan(f(x))} has an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \spad{f(x)} determines the constant coefficient in the series expansion of \spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \spad{atanFlag} is \spad{"just do it"}. When the leading term in the series expansion of \spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficient in the s
352151`--eries expansion of \spad{atan(f(x))} for values to the left differs from that for values to the right. If \spad{atanFlag} is \spad{"real: two sides"},{} no series expansion will be computed. If \spad{atanFlag} is \spad{"real: left side"} the constant coefficient for values to the left will be used and if \spad{atanFlag} \spad{"real: right side"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} we return a recor
352151`--d containing the name of the function that caused the problem and a brief description of the problem. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \spad{x -> x + a} before calling this function.
352406`\spad{exprToGenUPS(fcn,{}posCheck?,{}atanFlag)} converts the expression \spad{fcn} to a generalized power series. If \spad{posCheck?} is \spad{true},{} log\spad{'s} of negative numbers are not allowed nor are \spad{n}th roots of negative num
352406`--bers with \spad{n} even. If \spad{posCheck?} is \spad{false},{} these are allowed. \spad{atanFlag} determines how the case \spad{atan(f(x))},{} where \spad{f(x)} has a pole,{} will be treated. The possible values of \spad{atanFlag} are \spad{"complex"},{} \spad{"real: two sides"},{} \spad{"real: left side"},{} \spad{"real: right side"},{} and \spad{"just do it"}. If \spad{atanFlag} is \spad{"complex"},{} then no series expansion will be computed because,{} viewed as a function of a complex var
352406`--iable,{} \spad{atan(f(x))} has an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \spad{f(x)} determines the constant coefficient in the series expansion of \spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \spad{atanFlag} is \spad{"just do it"}. When the leading term in the series expansion of \spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficie
352406`--nt in the series expansion of \spad{atan(f(x))} for values to the left differs from that for values to the right. If \spad{atanFlag} is \spad{"real: two sides"},{} no series expansion will be computed. If \spad{atanFlag} is \spad{"real: left side"} the constant coefficient for values to the left will be used and if \spad{atanFlag} \spad{"real: right side"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} we ret
352406`--urn a record containing the name of the function that caused the problem and a brief description of the problem. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \spad{x -> x + a} before calling this function.
352672`\spad{exprToPS(fcn,{} posCheck?,{} atanFlag,{} log_flag,{} log_val)} is like exprToUPS,{} but if \spad{log_flag} is \spad{true} it replaces logarithms of expansion variable by \spad{log_val}
352934`\spad{exprToPS(fcn,{} posCheck?,{} atanFlag,{} log_flag,{} log_val)} is like exprToUPS,{} but if \spad{log_flag} is \spad{true} it replaces logarithms of expansion variable by \spad{log_val}
353207`exprToPS(\spad{fcn},{} posCheck?,{} atanFlag,{} log_flag,{} log_val,{} coef_chk,{} inv_chk) is like exprToPS(\spad{fcn},{} posCheck?,{} atanFlag,{} log_flag,{} log_val),{} but fails if \spad{coef_chk} is \spad{false} 
353207`--for some coefficient of \spad{fcn} or if expansion process needs to invert element of \spad{FE} for which \spad{inv_chk} is \spad{false}
353497`exprToPS(\spad{fcn},{} posCheck?,{} atanFlag,{} log_flag,{} log_val,{} coef_chk,{} inv_chk) is like exprToPS(\spad{fcn},{} posCheck?,{} atanFlag,{} log_flag,{} log_val),{} but fails if \spad{coef_chk} is \s
353497`--pad{false} for some coefficient of \spad{fcn} or if expansion process needs to invert element of \spad{FE} for which \spad{inv_chk} is \spad{false}
353798`\spad{exprToUPS(fcn,{}posCheck?,{}atanFlag)} converts the expression \spad{fcn} to a power series. If \spad{posCheck?} is \spad{true},{} log\spad{'s} of negative numbers are not allowed nor are \spad{n}th roots of negative numbers with \spad{n} even. If \
353798`--spad{posCheck?} is \spad{false},{} these are allowed. \spad{atanFlag} determines how the case \spad{atan(f(x))},{} where \spad{f(x)} has a pole,{} will be treated. The possible values of \spad{atanFlag} are \spad{"complex"},{} \spad{"real: two sides"},{} \spad{"real: left side"},{} \spad{"real: right side"},{} and \spad{"just do it"}. If \spad{atanFlag} is \spad{"complex"},{} then no series expansion will be computed because,{} viewed as a function of a complex variable,{} \spad{atan(f(x))} ha
353798`--s an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \spad{f(x)} determines the constant coefficient in the series expansion of \spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \spad{atanFlag} is \spad{"just do it"}. When the leading term in the series expansion of \spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficient in the series expansion of
353798`-- \spad{atan(f(x))} for values to the left differs from that for values to the right. If \spad{atanFlag} is \spad{"real: two sides"},{} no series expansion will be computed. If \spad{atanFlag} is \spad{"real: left side"} the constant coefficient for values to the left will be used and if \spad{atanFlag} \spad{"real: right side"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} a record containing the name of the
353798`-- function that caused the problem and a brief description of the problem is returned. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \spad{x -> x + a} before calling this function.
354050`\spad{exprToUPS(fcn,{}posCheck?,{}atanFlag)} converts the expression \spad{fcn} to a power series. If \spad{posCheck?} is \spad{true},{} log\spad{'s} of negative numbers are not allowed nor are \spad{n}th roots of negative numbers with \spad{n}
354050`-- even. If \spad{posCheck?} is \spad{false},{} these are allowed. \spad{atanFlag} determines how the case \spad{atan(f(x))},{} where \spad{f(x)} has a pole,{} will be treated. The possible values of \spad{atanFlag} are \spad{"complex"},{} \spad{"real: two sides"},{} \spad{"real: left side"},{} \spad{"real: right side"},{} and \spad{"just do it"}. If \spad{atanFlag} is \spad{"complex"},{} then no series expansion will be computed because,{} viewed as a function of a complex variable,{} \spad{ata
354050`--n(f(x))} has an essential singularity. Otherwise,{} the sign of the leading coefficient of the series expansion of \spad{f(x)} determines the constant coefficient in the series expansion of \spad{atan(f(x))}. If this sign cannot be determined,{} a series expansion is computed only when \spad{atanFlag} is \spad{"just do it"}. When the leading term in the series expansion of \spad{f(x)} is of odd degree (or is a rational degree with odd numerator),{} then the constant coefficient in the series e
354050`--xpansion of \spad{atan(f(x))} for values to the left differs from that for values to the right. If \spad{atanFlag} is \spad{"real: two sides"},{} no series expansion will be computed. If \spad{atanFlag} is \spad{"real: left side"} the constant coefficient for values to the left will be used and if \spad{atanFlag} \spad{"real: right side"} the constant coefficient for values to the right will be used. If there is a problem in converting the function to a power series,{} a record containing the 
354050`--name of the function that caused the problem and a brief description of the problem is returned. When expanding the expression into a series it is assumed that the series is centered at 0. For a series centered at a,{} the user should perform the substitution \spad{x -> x + a} before calling this function.
354313`\spad{exprToXXP(fcn,{}posCheck?)} converts the expression \spad{fcn} to an exponential expansion. If \spad{posCheck?} is \spad{true},{} log\spad{'s} of negative numbers are not allowed nor are \spad{n}th roots of negative numbers with \spad{n} even. If \spad{posCheck?} is \spad{false},{} these are allowed.
354491`Compute degree and leading coefficinet of series expansion
354802`\spad{expr(s)} returns \spad{s} as an element of Expr; Error: if \spad{s} is not an atom that also belongs to Expr.
354959`\spad{expressIdealMember([f1,{}...,{}fn],{}h)} returns a representation of \spad{h} as a linear combination of the \spad{fi} or "failed" if \spad{h} is not in the ideal generated by the \spad{fi}.
355053`\spad{exprex(o)} coverts \spadtype{OutputForm} to \spadtype{String}
355106`converts \spadtype{OutputForm} to \spadtype{String} with the structure preserved with braces. Actually this is not quite accurate. The function \spadfun{precondition} is first applied to the \spadtype{OutputForm} expression before \spadfun{exprex}. The raw \spadtype{OutputForm} and the nature of the \spadfun{precondition} function is still obscure to me at the time of this writing (2007-02-14).
355161`\spad{exptMod(u,{}k,{}v)} raises the polynomial \spad{u} to the \spad{k}th power modulo the polynomial \spad{v}.
355244`\spad{exptMod(f,{}n,{}g,{}p)} raises the univariate polynomial \spad{f} to the \spad{n}th power modulo the polynomial \spad{g} and the prime \spad{p}.
355326`\spad{expt(r,{} i)} computes r^i by repeated squaring
355914`\spad{exquo(a,{}b)} returns the power series quotient of \spad{a} by \spad{b},{} if the quotient exists,{} and "failed" otherwise
356015`\axiom{\spad{v} exquo \spad{r}} computes the exact quotient of \axiom{\spad{v}} by \axiom{\spad{r}}
356098`\spad{exquo(f,{}g)} creates the equivalent infix form.
356141`\spad{exquo(x,{} r)} returns the exact quotient of \spad{x} by \spad{r},{} or "failed" if \spad{r} does not divide \spad{x} exactly.
356228`\spad{exquo(p,{}r)} returns the exact quotient of polynomial \spad{p} by \spad{r},{} or "failed" if none exists.
356325`\spad{exquo(m,{}r)} computes the exact quotient of the elements of \spad{m} by \spad{r},{} returning \axiom{"failed"} if this is not possible.
356419`\spad{exquo(m,{}r)} computes the exact quotient of the elements of \spad{m} by \spad{r},{} returning \axiom{"failed"} if this is not possible.
356528`\spad{exquo(l,{} a)} returns the exact quotient of \spad{l} by a,{} returning \axiom{"failed"} if this is not possible.
356632`\spad{exquo(a,{}b)} either returns an element \spad{c} such that \spad{c*b=a} or "failed" if no such element can be found.
356691`\spad{exquo(a,{}b)} returns the quotient of \spad{a} and \spad{b},{} or "failed" if \spad{b} is zero or \spad{a} rem \spad{b} is zero.
356758`\axiom{extendIfCan(\spad{ts},{}\spad{p})} returns a triangular set which encodes the simple extension by \axiom{\spad{p}} of the extension of the base field defined by \axiom{\spad{ts}},{} according to the properties of triangular sets of the current domain. If the required properties do not hold then "failed" is returned. This operation encodes in some sense the properties of the triangular sets of the current category
356758`--. Is is used to implement the \axiom{construct} operation to guarantee that every triangular set build from a list of polynomials has the required properties.
356842`\spad{extendToPoint(n,{}p)} creates a boundary that is the union \indented{2}{of a boundary and a point.} \spad{n} is the boundary \spad{p} is the point
357390`\spad{extend(lp,{}lts)} returns the same as \spad{concat([extend(lp,{}ts) for ts in lts])|}
357478`\spad{extend(lp,{}ts)} returns \spad{ts} if \spad{empty? lp} \spad{extend(p,{}ts)} if \spad{lp = [p]} else \spad{extend(first lp,{} extend(rest lp,{} ts))}
357560`\spad{extend(p,{}lts)} returns the same as \spad{concat([extend(p,{}ts) for ts in lts])|}
357642`\spad{extend(p,{}ts)} assumes that \spad{p} is a non-constant polynomial whose main variable is greater than any variable of \spad{ts}. Then it returns a split of Kalkbrener of \spad{ts+p}. This may not be \spad{ts+p} itself,{} if for instance \spad{ts+p} is not a regular triangular set.
357718`\spad{extend(f,{}n)} causes all terms of \spad{f} of degree \spad{<=} \spad{n} to be computed.
357796`\spad{extend(st,{}n)} causes entries to be computed,{} if necessary,{} so that 'st' will have at least \spad{'n'} explicit entries or so that all entries of 'st' will be computed if 'st' is finite with length \spad{<=} \spad{n}.
357857`\spad{extend(x,{}n)} forces the computation of digits up to order \spad{n}.
357919`\spad{extend(x,{}n)} causes the first \spadvar{\spad{n}} entries in the continued fraction \spadvar{\spad{x}} to be computed. Normally entries are only computed as needed.
357978`\spad{extend(x,{}l,{}r)} \undocumented
358033`\spad{extend(f,{}n)} causes all terms of \spad{f} of degree \spad{<= n} to be computed.
358125`\axiom{extend(\spad{ts},{}\spad{p})} returns a triangular set which encodes the simple extension by \axiom{\spad{p}} of the extension of the base field defined by \axiom{\spad{ts}},{} according to the properties of triangular sets of the current category If the required properties do not hold an error is returned.
358188`\spad{extendedCoords(pt)} returns extended coordinates as a list.
358346`\spad{extendedEuclidean(x,{}y)} returns a record rec where \spad{rec.coef1*x+rec.coef2*y = rec.generator} and rec.generator is a \spad{gcd} of \spad{x} and \spad{y}. The \spad{gcd} is unique only up to associates if \spadatt{canonicalUnitNormal} is not asserted. \spadfun{principalIdeal} provides a version of this operation which accepts an arbitrary length list of arguments.
358538`\spad{extendedEuclidean(x,{}y,{}z)} either returns a record rec where \spad{rec.coef1*x+rec.coef2*y=z} or returns "failed" if \spad{z} cannot be expressed as a linear combination of \spad{x} and \spad{y}.
358640`\spad{extendedIntegrate(f,{} x,{} g)} returns fractions \spad{[h,{} c]} such that \spad{dc/dx = 0} and \spad{dh/dx = f - cg},{} if \spad{(h,{} c)} exist,{} "failed" otherwise.
358847`\axiom{extendedResultant(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca,{}\spad{cb}]} such that \axiom{\spad{r}} is the resultant of \axiom{a} and \axiom{\spad{b}} and \axiom{\spad{r} = ca * a + \spad{cb} * \spad{b}}
358980`\axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{} \spad{cb}]} such that \axiom{\spad{g}} is a \spad{gcd} of \axiom{a} and \axiom{\spad{b}} in \axiom{\spad{R^}(\spad{-1}) \spad{P}} and \axiom{\spad{g} = ca * a + \spad{cb} * \spad{b}}
359114`\axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[ca,{}\spad{cb},{}\spad{r}]} such that \axiom{\spad{r}} is \axiom{subResultantGcd(a,{}\spad{b})} and we have \axiom{ca * a + \spad{cb} * \spad{cb} = \spad{r}} .
359250`\spad{extended_gcd(v1,{} v2,{} p)} gives [\spad{g},{} \spad{c1},{} \spad{c2}] such that \spad{g} is \spad{gcd(v1,{} v2,{} p)},{} \spad{g = c1*v1 + c2*v2} and degree(\spad{c1}) < max(degree(\spad{v2}) - degree(\spad{g}),{} 0) and degree(\spad{c2}) < max(degree(\spad{v1}) - degree(\spad{g}),{} 1)
359354`\spad{extendedint(f,{} g)} returns fractions \spad{[h,{} c]} such that \spad{c' = 0} and \spad{h' = f - cg},{} if \spad{(h,{} c)} exist,{} "failed" otherwise.
359600`\spad{extensionDegree()} returns the degree of field extension.
359686`\spad{extensionDegree()} returns the degree of the field extension if the extension is algebraic,{} and \spad{infinity} if it is not.
359774`\spad{extensionDegree()} returns the degree of field extension.
359857`\spad{extension(f)} returns the type part of the file name.
359911`\spad{exteriorDifferential(df)} returns the exterior derivative (gradient,{} curl,{} divergence,{} ...) of the differential form \spad{df}.
359990`\spad{external?(u)} returns \spad{true} if \spad{u} is declared to be EXTERNAL
360040`\spad{externalList(tab)} returns a list of all the external symbols in \spad{tab}
360178`\spad{extract!(u)} destructively removes a (random) item from bag \spad{u}.
360225`\spad{extractBottom!(d)} destructively extracts the bottom (back) element from the dequeue \spad{d}. Error: if \spad{d} is empty.
360282`\spad{extractClosed(s)} returns the \spadtype{Boolean} value of the closed property for the indicated 3 dimensional subspace \spad{s}. If the property is closed,{} \spad{True} is returned,{} otherwise \spad{False} is returned.
360338`\axiom{extractIfCan(\spad{x})} searches the item whose key is \axiom{\spad{x}}.
360433`\spad{extractIndex(s)} returns a non negative integer which is the current index of the 3 dimensional subspace \spad{s}.
360499`\spad{extractPoint(s)} returns the point which is given by the current index location into the point data field of the 3 dimensional subspace \spad{s}.
360555`\spad{extractProperty(s)} returns the property of domain \spadtype{SubSpaceComponentProperty} of the indicated 3 dimensional subspace \spad{s}.
360631`\axiom{extractSplittingLeaf(a)} returns the left most leaf (as a tree) whose status is \spad{false} if any,{} else "failed" is returned.
360821`\axiom{extractSymbol(\spad{jm})} extracts the highest order part of the Jacobi matrix.
360936`\axiom{extractSymbol(de,{} solved?)} computes the symbol of the differential equation \axiom{de}. If \axiom{solved?} is \spad{true},{} the row echelon form of the symbol is computed at once.
361034`\spad{extractTop!(d)} destructively extracts the top (front) element from the dequeue \spad{d}. Error: if \spad{d} is empty.
361088`\axiom{extract(A,{} \spad{i1},{} \spad{i2})} extracts the rows \axiom{\spad{i1}} to \axiom{\spad{i2}} and returns them as a new matrix.
361160`\spad{eyeDistance(v,{}d)} sets the distance of the observer from the center of the graph to \spad{d},{} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}.
361228`\spad{factor1(a)} returns the factorisation of a,{} assuming that a has no first-order right factor.
361441`\spad{factorAndSplit(eq)} make the right hand side 0 and factors the new left hand side. Each factor is equated to 0 and put into the resulting list without repetitions.
361518`\spad{factorByRecursion(p)} factors polynomial \spad{p}. This function performs the recursion step for factorPolynomial,{} as defined in \spadfun{PolynomialFactorizationExplicit} category (see \spadfun{factorPolynomial})
361672`\spad{factorByRecursion(p)} factors polynomial \spad{p}. This function performs the recursion step for factorPolynomial,{} as defined in \spadfun{PolynomialFactorizationExplicit} category (see \spadfun{factorPolynomial})
361827`\spad{factorFraction(r)} factors the numerator and the denominator of the polynomial fraction \spad{r}.
361947`\spad{factorGroebnerBasis(basis)} checks whether the \spad{basis} contains reducible polynomials and uses these to split the \spad{basis}.
362064`\spad{factorGroebnerBasis(basis,{}info)} checks whether the \spad{basis} contains reducible polynomials and uses these to split the \spad{basis}. If argument {\em info} is \spad{true},{} information is printed about partial results.
362189`\spad{factorList(u)} returns the list of factors with flags (for use by factoring code).
362306`\spad{factorList(k,{}n,{}m,{}j)} \undocumented
362475`\spad{factorOfDegree(d,{}f)} returns a factor of degree \spad{d} of the factored polynomial \spad{f}. Such a factor shall exist.
362576`\spad{factorOfDegree(d,{}p)} returns a factor of \spad{p} of degree \spad{d}.
362672`\spad{factorOfDegree(d,{}p,{}listOfDegrees)} returns a factor of \spad{p} of degree \spad{d} knowing that \spad{p} has for possible splitting of its degree \spad{listOfDegrees}.
362793`\spad{factorOfDegree(d,{}p,{}r)} returns a factor of \spad{p} of degree \spad{d} knowing that \spad{p} has at least \spad{r} factors.
362908`\spad{factorOfDegree(d,{}p,{}listOfDegrees,{}r)} returns a factor of \spad{p} of degree \spad{d} knowing that \spad{p} has for possible splitting of its degree \spad{listOfDegrees},{} and that \spad{p} has at least \spad{r} factors.
363048`\spad{factorOfDegree(d,{}p,{}listOfDegrees,{}r,{}sqf)} returns a factor of \spad{p} of degree \spad{d} knowing that \spad{p} has for possible splitting of its degree \spad{listOfDegrees},{} and that \spad{p} has at least \spad{r} factors. If \spad{sqf=true} the polynomial is assumed to be square free (\spadignore{i.e.} without repeated factors).
363584`\spad{factorPolynomial(p)} returns the factorization into irreducibles of the univariate polynomial \spad{p}.
363722`\spad{factorPolynomial(p)} \undocumented{}
363885`\spad{factorSFBRlcUnit(p)} returns the square free factorization of polynomial \spad{p} (see \spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate}) in the case where the leading coefficient of \spad{p} is a unit.
364039`\spad{factorSFBRlcUnit(p)} returns the square free factorization of polynomial \spad{p} (see \spadfun{factorSquareFreeByRecursion}{PolynomialFactorizationByRecursionUnivariate}) in the case where the leading coefficient of \spad{p} is a unit.
364205`\spad{factorSquareFreeByRecursion(p)} returns the square free factorization of \spad{p}. This functions performs the recursion step for factorSquareFreePolynomial,{} as defined in \spadfun{PolynomialFactorizationExplicit} category (see \spadfun{factorSquareFreePolynomial}).
364369`\spad{factorSquareFreeByRecursion(p)} returns the square free factorization of \spad{p}. This functions performs the recursion step for factorSquareFreePolynomial,{} as defined in \spadfun{PolynomialFactorizationExplicit} category (see \spadfun{factorSquareFreePolynomial}).
364952`\spad{factorSquareFreePolynomial(p)} factors the univariate polynomial \spad{p} into irreducibles where \spad{p} is known to be square free and primitive with respect to its main variable.
365100`\spad{factorSquareFree(p)} factors an extended squareFree polynomial \spad{p} over the rational numbers.
365172`\spad{factorSquareFree(p)} returns the factorization of \spad{p} which is supposed not having any repeated factor (this is not checked).
365248`\spad{factorSquareFree(m)} returns the factorization of \spad{m} square free polynomial
365322`\spad{factorSquareFree(p)} produces the complete factorization of the square free polynomial \spad{p}.
365402`\spad{factorSquareFree(p,{}listOfDegrees)} factorizes the polynomial \spad{p} using the single factor bound algorithm and knowing that \spad{p} has for possible splitting of its degree \spad{listOfDegrees}. \spad{f} is supposed not having any repeated factor (this is not checked).
365503`\spad{factorSquareFree(p,{}r)} factorizes the polynomial \spad{p} using the single factor bound algorithm and knowing that \spad{p} has at least \spad{r} factors. \spad{f} is supposed not having any repeated factor (this is not checked).
365598`\spad{factorSquareFree(p,{}listOfDegrees,{}r)} factorizes the polynomial \spad{p} using the single factor bound algorithm,{} knowing that \spad{p} has for possible splitting of its degree \spad{listOfDegrees} and that \spad{p} has at least \spad{r} factors. \spad{f} is supposed not having any repeated factor (this is not checked).
365718`\spad{factorSquareFree(p,{}d,{}r)} factorizes the polynomial \spad{p} using the single factor bound algorithm,{} knowing that \spad{d} divides the degree of all factors of \spad{p} and that \spad{p} has at least \spad{r} factors. \spad{f} is supposed not having any repeated factor (this is not checked).
365832`\spad{factor(n)} returns the full factorization of integer \spad{n}
365901`\spad{factor(a)} returns the factorisation of a.
366113`\axiom{factor(\spad{x})} returns the decreasing factorization into Lyndon words.
366191`\spad{factor(p)} factors the multivariate polynomial \spad{p} over its coefficient domain
366274`\spad{factor(p)} factors the multivariate polynomial \spad{p} with coefficients which are fractions of elements of \spad{R}.
366341`\spad{factor(p)} factors a polynomial with polynomial coefficients.
366418`\spad{factor(p)} factors an extended polynomial \spad{p} over the rational numbers.
366770`\spad{factor(p)} should be local but conditional.
366938`\spad{factor(p)} factors the univariate polynomial \spad{p} with coefficients which are fractions of polynomials over \spad{R}.
367084`\spad{factor(p)} returns the factorisation of \spad{p}
367195`\spad{factor(p)} tries to factor \spad{p} into linear factors with error atmost {\em globalEps},{} the internal error bound,{} which can be set by {\em setErrorBound}. An overall error bound {\em eps0} is determined and iterated tree-like calls to {\em pleskenSplit} are used to get the factorization.
367267`\spad{factor(p)} returns the factorization of \spad{p} over the integers.
367333`\spad{factor(m)} returns the factorization of \spad{m}
367397`splits \spad{n} into a list of factors which must be \spad{true} for the whole to be \spad{true}. This assumes that the top level is already a set of factors seperated by \spad{/\} otherwise the result will just be a list with one entry: \spad{'n'}. This is used when converting ILogic to types by using the Curry-Howard isomorphism.
367440`\spad{factor(\spad{zi})} produces the complete factorization of the complex integer \spad{zi}.
367537`\spad{factor(p)} produces the complete factorization of the polynomial \spad{p}.
367607`\spad{factor(p)} factors the multivariate polynomial \spad{p} over its coefficient domain
367677`\spad{factor(p)} produces the complete factorization of the multivariate polynomial \spad{p} over a finite field.
367748`\spad{factor(p)} factorizes the polynomial \spad{p} with complex coefficients.
367816`\spad{factor(prf)} factors a polynomial with rational function coefficients.
367905`\spad{factor(p)} factors the multivariate polynomial \spad{p} over its coefficient domain where \spad{p} is represented as a univariate polynomial with multivariate coefficients
368031`\spad{factor(p)} produces the complete factorization of the multivariate polynomial \spad{p} over a finite field. \spad{p} is represented as a univariate polynomial with multivariate coefficients over a finite field.
368158`\spad{factor(p)} returns a prime factorisation of \spad{p} over the field generated by its coefficients.
368212`\spad{factor(p)} returns a prime factorisation of \spad{p}.
368279`\spad{factor(p)} returns a prime factorisation of \spad{p}.
368362`\spad{factor(p)} returns a prime factorisation of \spad{p}.
368450`\spad{factor(x)} returns the factorization of \spad{x} into irreducibles.
368516`\spad{factor(p,{} f)} returns a prime factorisation of \spad{p}; \spad{f} is a factorisation map for elements of UP.
368614`\spad{factor(a,{} zeros)} returns the factorisation of a. \spad{zeros} is a zero finder in \spad{UP}.
368809`\spad{factor(p,{}ufact)} factors the multivariate polynomial \spad{p} by specializing variables and calling the univariate factorizer \spad{ufact}.
368944`\spad{factor(p,{}ufact)} factors the multivariate polynomial \spad{p} by specializing variables and calling the univariate factorizer \spad{ufact}. \spad{p} is represented as a univariate polynomial with multivariate coefficients.
369135`\spad{factor(p,{}listOfDegrees)} factorizes the polynomial \spad{p} using the single factor bound algorithm and knowing that \spad{p} has for possible splitting of its degree \spad{listOfDegrees}.
369226`\spad{factor(p,{}r)} factorizes the polynomial \spad{p} using the single factor bound algorithm and knowing that \spad{p} has at least \spad{r} factors.
369311`\spad{factor(p,{} eps)} tries to factor \spad{p} into linear factors with error atmost {\em eps}. An overall error bound {\em eps0} is determined and iterated tree-like calls to {\em pleskenSplit} are used to get the factorization.
369385`\spad{factor(p,{}lan)} factors the polynomial \spad{p} over the extension generated by the algebraic numbers given by the list \spad{lan}.
369471`\spad{factor(p,{}lan)} factors the polynomial \spad{p} over the extension generated by the algebraic numbers given by the list \spad{lan}. \spad{p} is presented as a univariate polynomial with multivariate coefficients.
369613`\spad{factor(f1,{}p)} returns the list of factors of the univariate polynomial \spad{f1} modulo the integer prime \spad{p}. Error: if \spad{f1} is not square-free modulo \spad{p}.
369690`\spad{factor(p,{} [a1,{}...,{}an])} returns a prime factorisation of \spad{p} over the field generated by its coefficients and \spad{a1},{}...,{}an.
369766`\spad{factor(p,{}listOfDegrees,{}r)} factorizes the polynomial \spad{p} using the single factor bound algorithm,{} knowing that \spad{p} has for possible splitting of its degree \spad{listOfDegrees} and that \spad{p} has at least \spad{r} factors.
369876`\spad{factor(p,{}d,{}r)} factorizes the polynomial \spad{p} using the single factor bound algorithm,{} knowing that \spad{d} divides the degree of all factors of \spad{p} and that \spad{p} has at least \spad{r} factors.
369980`\spad{factor(p,{} eps,{} info)} tries to factor \spad{p} into linear factors with error atmost {\em eps}. An overall error bound {\em eps0} is determined and iterated tree-like calls to {\em pleskenSplit} are used to get the factorization. If {\em info} is {\em true},{} then information messages are given.
370062`\spad{factorial(n)} returns the factorial of \spad{n},{} \spadignore{i.e.} \spad{n!}.
370170`\spad{factorial(n)} returns \spad{n!}. this is the product of all integers between 1 and \spad{n} (inclusive). Note: \spad{0!} is defined to be 1.
370233`\spad{factorial(n)} computes the factorial of \spad{n} (denoted in the literature by \spad{n!}) Note: \spad{n! = n (n-1)! when n > 0}; also,{} \spad{0! = 1}.
370296`\spad{factorials(f)} rewrites the permutations and binomials in \spad{f} in terms of factorials.
370355`\spad{factorials(f)} rewrites the permutations and binomials in \spad{f} in terms of factorials.
370414`\spad{factorials(f,{} x)} rewrites the permutations and binomials in \spad{f} involving \spad{x} in terms of factorials.
370480`\spad{factorials(f,{} x)} rewrites the permutations and binomials in \spad{f} involving \spad{x} in terms of factorials.
370546`\spad{factorsOfCyclicGroupSize()} returns the factorization of size()\spad{-1}
370655`\spad{factorsOfDegree(d,{}f)} returns the factors of degree \spad{d} of the factored polynomial \spad{f}.
370763`\spad{factors(a1\^e1,{}...,{}an\^en)} returns \spad{[[a1,{} e1],{}...,{}[an,{} en]]}.
370836`\spad{factors(a1\^e1,{}...,{}an\^en)} returns \spad{[[a1,{} e1],{}...,{}[an,{} en]]}.
370921`\spad{factors(a1\^e1,{}...,{}an\^en)} returns \spad{[[a1,{} e1],{}...,{}[an,{} en]]}.
371013`\spad{factors(u)} returns a list of the factors in a form suitable for iteration. That is,{} it returns a list where each element is a record containing a base and exponent. The original object is the product of all the factors and the unit (which can be extracted by \axiom{unit(\spad{u})}).
371093`\spad{factorset(p)} returns the set of irreducible factors of \spad{p}.
371174`\spad{failed?(r)} tests if \spad{r} is a failed match.
371240`\spad{failed?(r)} tests if \spad{r} is a failed match.
371300`\spad{failed()} returns a failed match.
371358`\spad{failed()} returns a failed match.
371410`\spad{false} is a logical constant.
371445`\spad{ffactor(p)} tries to factor a univariate polynomial \spad{p} over \spad{F}
371534`\spad{fffg} is the general algorithm as proposed by Beckermann and Labahn. \blankline The first argument is the list of \spad{c_}{\spad{i},{}\spad{i}}. These are the only values of \spad{C} explicitely needed in \spad{fffg}. \blankline The second argument \spad{c},{} computes \spad{c_k}(\spad{M}),{} \spadignore{i.e.} \sp
371534`--ad{c_k}(.) is the dual basis of the vector space \spad{V},{} but also knows about the special multiplication rule as descibed in Equation (2). Note that the information about \spad{f} is therefore encoded in \spad{c}. \blankline The third argument is the vector of degree bounds \spad{n},{} as introduced in Definition 2.1. In particular,{} the sum of the entries is the order of the Mahler system computed.
371719`\axiom{fglmIfCan(\spad{lp})} returns the lexicographical Groebner basis of \axiom{\spad{lp}} by using the {\em FGLM} strategy,{} if \axiom{zeroDimensional?(\spad{lp})} holds .
371919`\axiom{fglmIfCan(\spad{lq1})} returns the lexicographical Groebner basis of \axiom{\spad{lq1}} by using the {\em FGLM} strategy,{} if \axiom{zeroDimensional?(\spad{lq1})} holds.
372025`\spad{fibonacci(n)} returns the \spad{n}th Fibonacci number. the Fibonacci numbers \spad{F[n]} are defined by \spad{F[0] = F[1] = 1} and \spad{F[n] = F[n-1] + F[n-2]}. The algorithm has running time \spad{O(log(n)^3)}. Reference: Knuth,{} The Art of Computer Programming Vol 2,{} Semi-Numerical Algorithms.
372097`\spad{filename(d,{}n,{}e)} creates a file name with \spad{d} as its directory,{} \spad{n} as its name and \spad{e} as its extension. This is a portable way to create file names. When \spad{d} or \spad{t} is the empty string,{} a default is used.
372337`\spad{fill!(u,{}x)} replaces each entry in aggregate \spad{u} by \spad{x}. The modified \spad{u} is returned as value.
372426`\spad{fill!(m,{}r)} fills \spad{m} with \spad{r}\spad{'s}
372496`\spad{fillPascalTriangle()} fills the stored table.
372562`\spad{filterUntil(p,{}t)} returns \spad{[x for x in t while not p(x)]}.
372627`\spad{filterUntil(p,{}s)} returns \spad{[x0,{}x1,{}...,{}x(n)]} where \spad{s = [x0,{}x1,{}x2,{}..]} and \spad{n} is the smallest index such that \spad{p(xn) = true}.
372685`\spad{filterWhile(p,{}t)} returns \spad{[x for x in t while p(x)]}.
372750`\spad{filterWhile(p,{}s)} returns \spad{[x0,{}x1,{}...,{}x(n-1)]} where \spad{s = [x0,{}x1,{}x2,{}..]} and \spad{n} is the smallest index such that \spad{p(xn) = false}.
372808`\spad{findCycle(n,{}st)} determines if \spad{st} is periodic within \spad{n}.
372942`\spad{findNode(n,{}ptName)} finds a scene node
373009`\spad{findPoint(n,{}ptName)} find the point associated with a given string.
373423`\spad{find(p,{}u)} returns the first \spad{x} in \spad{u} such that \axiom{\spad{p}(\spad{x})} is \spad{true},{} and "failed" otherwise.
373493`\spad{finite?(\spad{a})} determines whether \spad{a} is a finite cardinal,{} \spadignore{i.e.} an integer.
373544`\spad{finite?(x)} tests if \spad{x} is finite.
373602`\spad{finite?(x)} tests if \spad{x} is finite.
373659`\spad{finiteBasis(d)} returns a basis for \spad{d} as a module over {\em K[x]}.
373728`\spad{fintegrate(f,{}v,{}c)} is the integral of \spad{f()} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.} \indented{1}{The evaluation of \spad{f()} is delayed.}
373856`\spad{fintegrate(f,{}v,{}c)} is the integral of \spad{f()} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.} \indented{1}{The evaluation of \spad{f()} is delayed.}
373961`\spad{firstDenom(p)} extracts the denominator of the first fractional term. This returns 1 if there is no fractional part (use \spadfunFrom{wholePart}{PartialFraction} to get the whole part).
374023`\spad{firstNumer(p)} extracts the numerator of the first fractional term. This returns 0 if there is no fractional part (use \spadfunFrom{wholePart}{PartialFraction} to get the whole part).
374075`\spad{firstSubsetGray(n)} creates the first vector {\em ww} to start a loop using {\em nextSubsetGray(ww,{}n)}
374157`\spad{firstUncouplingMatrix(op,{} m)} returns the matrix A such that \spad{A w = (W',{}W'',{}...,{}W^N)} in the corresponding associated equations for right-factors of order \spad{m} of \spad{op}. Returns "failed" if the matrix A has not been precomputed for the particular combination \spad{degree(L),{} m}.
374499`\spad{first(x)} selects first component of the product
374540`\spad{first([l1]*[l2]*...[ln])} returns the Lyndon word \spad{l1}.
374625`\spad{first(x)} returns the first letter of \spad{x}.
374674`\axiom{first(\spad{x})} returns the first entry of the tree \axiom{\spad{x}}.
374721`\spad{first(u)} returns the first element \spad{x} of \spad{u}. Note: for collections,{} \axiom{first([\spad{x},{}\spad{y},{}...,{}\spad{z}]) = \spad{x}}. Error: if \spad{u} is empty.
374804`\spad{first(u)} returns the first element of \spad{u} (equivalently,{} the value at the current node).
374859`\axiom{first(\spad{ts})} returns the polynomial of \axiom{\spad{ts}} with greatest main variable if \axiom{\spad{ts}} is not empty,{} otherwise returns \axiom{"failed"}.
375064`\spad{first(x,{} n)} returns the sequence of the first \spad{n} entries
375132`\spad{first(u,{}n)} returns a copy of the first \spad{n} (\axiom{\spad{n} \spad{>=} 0}) elements of \spad{u}.
375207`\spad{firstn(x,{} n)} returns a list of the first \spad{n} entries
375281`\spad{fixPredicate(f)} returns \spad{g} defined by \spad{g}(a) = \spad{f}(a::B).
375365`\spad{fixedDivisor(a)} for \spad{a(x)} in \spad{Z[x]} is the largest integer \spad{f} such that \spad{f} divides \spad{a(x=k)} for all integers \spad{k}. Note: fixed divisor of \spad{a} is \spad{reduce(gcd,{}[a(x=k) for k in 0..degree(a)])}.
375471`\spad{fixedPointExquo(f,{}g)} computes the exact quotient of \spad{f} and \spad{g} using a fixed point computation.
375558`\spad{fixedPoints(p)} returns the points fixed by the permutation \spad{p}.
375625`\spad{flagFactor(base,{}exponent,{}flag)} creates a factored object with a single factor whose \spad{base} is asserted to be properly described by the information \spad{flag}.
375714`\spad{flatten(n)} takes a second order graph,{} that is a graph whose elements are themselves graphs and create a first order graph whose vertices are the vertices of the inner graphs.
375775`\spad{flatten(s)} returns an input form corresponding to \spad{s} with all the nested operations flattened to triples using new local variables. If \spad{s} is a piece of code,{} this speeds up the compilation tremendously later on.
375816`\spad{flexible?()} tests if \spad{2*associator(a,{}b,{}a) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
375890`\spad{flexible?()} tests if \spad{2*associator(a,{}b,{}a) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
375961`\spad{flexibleArray(l)} creates a flexible array from the list of elements \spad{l}
376084`\spad{float?(s)} is \spad{true} if \spad{s} is an atom and belong to \spad{Flt}.
376161`\spad{float(s)} returns \spad{s} as an element of \spad{Flt}; Error: if \spad{s} is not an atom that also belongs to \spad{Flt}.
376293`\spad{float(a,{}e)} returns \spad{a * base() ^ e}.
376431`\spad{float(a,{}e,{}b)} returns \spad{a * b ^ e}.
376552`\spad{floor(x)} returns the largest integral element below \spad{x}.
376631`\spad{floor x} returns the largest integer \spad{<= x}.
376677`\spad{flush(f)} make sure that buffered data in written out to the operating system.
376729`\spad{fmecg(p1,{}e,{}r,{}p2)} finds \spad{X} : \spad{p1} - \spad{r} * X^e * \spad{p2}
376844`\axiom{fmecg(\spad{p1},{}\spad{e},{}\spad{r},{}\spad{p2})} returns \axiom{\spad{p1} - \spad{r} * X^e * \spad{p2}} where \axiom{\spad{X}} is \axiom{monomial(1,{}1)}
376930`\spad{fmecg(p1,{}e,{}r,{}p2)} finds \spad{X} : \spad{p1} - \spad{r} * X^e * \spad{p2}
377142`\spad{fmecg(p1,{}e,{}r,{}p2)} finds \spad{X} : \spad{p1} - \spad{r} * X^e * \spad{p2}
377221`\spad{forLoop(i=1..10,{}c)} creates a representation of a FORTRAN DO loop with \spad{i} ranging over the values 1 to 10.
377300`\spad{forLoop(i=1..10,{}n,{}c)} creates a representation of a FORTRAN DO loop with \spad{i} ranging over the values 1 to 10 by \spad{n}.
377548`\axiom{\spad{formalDiff2}(\spad{sys},{} \spad{i},{} \spad{jm})} is like the other \axiomFun{\spad{formalDiff2}} but for systems.
377833`\axiom{\spad{formalDiff2}(\spad{f},{} \spad{i},{} \spad{jm})} formally differentiates the function \axiom{\spad{f}} with the Jacobi matrix \axiom{\spad{jm}} \spad{wrt} the \axiom{\spad{i}}\spad{-}th independent variable. \axiom{JVars} is a list of the jet variables effectively in the result \axiom{DPhi} (might be too large).
378057`\axiom{formalDiff(\spad{sys},{} \spad{i})} formally differentiates a family \axiom{\spad{sys}} of functions \spad{wrt} the \axiom{\spad{i}}\spad{-}th independent variable.
378310`\axiom{formalDiff(\spad{f},{} mu)} formally differentiates \axiom{\spad{f}} as indicated by the multi-index \axiom{mu}.
378399`\axiom{formalDiff(\spad{f},{} \spad{i})} formally (totally) differentiates \axiom{\spad{f}} \spad{wrt} the \axiom{\spad{i}}\spad{-}th independent variable.
378479`\spad{formula(t)} extracts the formula section of a formatted object \spad{t}.
378540`\spad{fortranCarriageReturn()} produces a carriage return on the current Fortran output stream
378601`\spad{fortranCharacter()} returns CHARACTER,{} an element of FortranType
378651`\spad{fortranComplex()} returns COMPLEX,{} an element of FortranType
378699`\spad{fortranDoubleComplex()} returns DOUBLE COMPLEX,{} an element of FortranType
378753`\spad{fortranDouble()} returns DOUBLE PRECISION,{} an element of FortranType
378800`\spad{fortranInteger()} returns INTEGER,{} an element of FortranType
378848`\spad{fortranLiteralLine(s)} writes \spad{s} to the current Fortran output stream,{} followed by a carriage return
378912`\spad{fortranLiteral(s)} writes \spad{s} to the current Fortran output stream
378972`\spad{fortranLogical()} returns LOGICAL,{} an element of FortranType
379020`\spad{fortranReal()} returns REAL,{} an element of FortranType
379065`\spad{fortranTypeOf(u,{}tab)} returns the type of \spad{u} in \spad{tab}
379130`\spad{fortran(fname,{}ftype,{}body)} builds an object of type \axiomType{FortranProgramCategory}. The three arguments specify the name,{} the type and the \spad{body} of the program.
379213`\spad{fprindINFO }\undocumented
379405`\spad{fracPart(f)} returns the list of summands of the fractional part of \spad{f}.
379528`\spad{fractRadix(pre,{}cyc)} creates a fractional radix expansion from a list of prefix ragits and a list of cyclic ragits. For example,{} \spad{fractRadix([1],{}[6])} will return \spad{0.16666666...}.
379606`\spad{fractRagits(rx)} returns the ragits of the fractional part of a radix expansion.
379673`\spad{fractionFreeGauss(m)} performs the fraction free gaussian elimination on the matrix \spad{m}.
379883`\spad{fractionPart(b)} returns the fractional part of a binary expansion.
379950`\spad{fractionPart(d)} returns the fractional part of a decimal expansion.
380018`\spad{fractionPart(h)} returns the fractional part of a hexadecimal expansion.
380090`\spad{fractionPart(rx)} returns the fractional part of a radix expansion.
380160`\spad{fractionPart(x)} returns the fractional part of \spad{x}. \spad{x} = wholePart(\spad{x}) + fractionPart(\spad{x})
380243`\spad{fractionPart x} returns the fractional part of \spad{x}.
380296`if this is a lambda term then is it free,{} that is does its variable appear in its expression
380341`\spad{freeConvolution(x,{} y)} returns the free convolution of the distributions \spad{x} and \spad{y}
380407`\spad{freeCumulant(x,{} n)} returns the \spad{n}-th free cumulant of the distribution \spad{x}
380482`\spad{freeCumulants(x)} returns the sequence of free cumulants of the distribution \spad{x}.
380552`\spad{freeMultiplicativeConvolution(x,{} y)} returns the free multiplicative convolution of the distributions \spad{x} and \spad{y}.
380683`\spad{freeMultiplicativeConvolution(mu,{} nu)} computes the free multiplicative convolution of the distributions \spad{mu} and \spad{nu}.
380963`\axiom{freeOf?(fun,{} \spad{jv})} checks whether \axiom{fun} contains the jet variable \axiom{\spad{jv}}.
381032`\spad{freeOf?(x,{} s)} tests if \spad{x} does not contain any operator whose name is \spad{s}.
381091`\spad{freeOf?(x,{} y)} tests if \spad{x} does not contain any occurrence of \spad{y},{} where \spad{y} is a single kernel.
381146`\spad{freePoissonDistribution(a)} produces a free Poisson distribution of variance \spad{a}.
381228`the variable indicated by \spad{'s'} is free if it does not appear in node \spad{'n'} or any of its subnodes.
381292`\spad{freecumulant2moment(cc)} computes the sequence of moments from the sequence of free cumulants \spad{cc}
381370`\spad{fresnelC(f)} denotes the Fresnel integral \spad{C}
381425`\spad{fresnelC(f)} denotes the Fresnel integral \spad{C}
381506`\spad{fresnelC(f)} denotes the Fresnel integral \spad{C}
381569`fresnelC is the Fresnel integral \spad{C},{} defined by \spad{C(x) = integrate(cos(\%pi*t^2/2),{}t=0..x)}
381629`\spad{fresnelS(f)} denotes the Fresnel integral \spad{S}
381684`\spad{fresnelS(f)} denotes the Fresnel integral \spad{S}
381765`\spad{fresnelS(f)} denotes the Fresnel integral \spad{S}
381828`fresnelS is the Fresnel integral \spad{S},{} defined by \spad{S(x) = integrate(sin(\%pi*t^2/2),{}t=0..x)}
381888`\spad{frobenius(x)} \undocumented
381963`\spad{front(q)} returns the element at the front of the queue. The queue \spad{q} is unchanged by this operation. Error: if \spad{q} is empty.
382009`\spad{froot(f,{} n)} returns \spad{[m,{}c,{}r]} such that \spad{f^(1/n) = c * r^(1/m)}.
382150`\spad{frst(s)} returns the first element of stream \spad{s}. Caution: this function should only be called after a \spad{empty?} test has been made since there no error check.
382200`\spad{fullDisplay(db)} prints full details of each entry in \axiom{\spad{db}}.
382249`\spad{fullDisplay(ic)} prints all of the information contained in \axiom{\spad{ic}}.
382296`\spad{fullDisplay(db,{}start,{}end )} prints full details of entries in the range \axiom{\spad{start}..end} in \axiom{\spad{db}}.
382377`\spad{fullPartialFraction(f)} returns \spad{[p,{} [[j,{} Dj,{} Hj]...]]} such that \spad{f = p(x) + \sum_{[j,{}Dj,{}Hj] in l} \sum_{Dj(a)=0} Hj(a)/(x - a)\^j}.
382465`construct graph from a list of permutations.
382537`constructor for graph with given objects more objects and arrows can be added later if required.
382711`constructor for graph with given list of object names. Use this version of the constructor if you don\spad{'t} intend to create diagrams and therefore don\spad{'t} care about \spad{x},{}\spad{y} coordinates. more objects and arrows can be added later if required.
382770`constructor for graph with given objects and arrows more objects and arrows can be added later if required.
383055`\spad{functionName returns} the name of the function given by the algebraic differential equation,{} default being \spad{f}
383132`\spad{functionName(d)} specifies the name of the function given by the algebraic differential equation or recurrence. This option is expressed in the form \spad{functionName == d}.
383184`\spad{functionNames returns} the names for the function in the algebraic dependence,{} default being \%\spad{f1},{} \%\spad{f2},{} ...
383268`\spad{functionNames(d)} specifies the names for the function in algebraic dependence. This option is expressed in the form \spad{functionNames == d}.
383327`\axiom{function(\spad{f},{} arg)} generates a function with name \axiom{\spad{f}} and arguments \axiom{arg}.
383397`\spad{function(e,{} foo)} creates a function \spad{foo() == e}.
383455`\axiom{function(\spad{f},{} arg,{} show)} generates a function with name \axiom{\spad{f}} and arguments \axiom{arg}. In the output only the first \axiom{show} arguments are shown.
383543`\spad{function(code,{} [x1,{}...,{}xn],{} f)} returns the input form corresponding to \spad{f(x1,{}...,{}xn) == code}.
383605`\spad{function(e,{} foo,{} [x1,{}...,{}xn])} creates a function \spad{foo(x1,{}...,{}xn) == e}.
383676`\spad{function(e,{} foo,{} x)} creates a function \spad{foo(x) == e}.
383741`\spad{function(e,{} foo,{} x,{} y)} creates a function \spad{foo(x,{} y) = e}.
383813`\spad{gamma()} is the Euler constant
383877`\axiom{gaugeHilbert(\spad{q},{} gamma)} computes the gauge correction to the Hilbert polynomial for a system of order \axiom{\spad{q}} with \axiom{gamma} gauge functions.
384019`\axiom{gauge(\spad{q},{} \spad{j},{} gamma)} computes the gauge corrections to the number of arbitrary functions of differentiation order \axiom{\spad{j}} for a system of order \axiom{\spad{q}} with \axiom{gamma} gauge functions.
384130`\spad{gaussianDistribution(a)} produces a gaussian distribution of variance \spad{a}.
384209`\spad{gbasis }\undocumented
384318`\spad{gcdBasis(v)} returns basis part \spad{o}
384376`\spad{gcdDecomposition(v)} returns \spad{[b,{} t]} such that elements of \spad{b} are relatively prime and that \spad{v(i) = product(b(j)^(t(j,{}i)),{} j=1..n)}
384492`\spad{gcdDecomposition(v)} returns \spad{[b,{} t]} such that elements of \spad{b} are relatively prime and that \spad{v(i) = product(b(j)^(t(j,{}i)),{} j=1..n)}
384598`\spad{gcdPolynomial(p,{}q)} returns the \spad{GCD} of \spad{p} and \spad{q}
385609`\spad{gcdPolynomial(p,{}q)} returns the \spad{gcd} of the univariate polynomials \spad{p} \spad{qnd} \spad{q}.
385765`\spad{gcdPrimitive lp} computes the \spad{gcd} of the list of primitive polynomials \spad{lp}.
385836`\spad{gcdPrimitive(p,{}q)} computes the \spad{gcd} of the primitive polynomials \spad{p} and \spad{q}.
385903`\spad{gcdPrimitive(p,{}q)} computes the \spad{gcd} of the primitive polynomials \spad{p} and \spad{q}.
386054`\spad{gcd([f1,{}..,{}fk])} = \spad{gcd} of the polynomials \spad{fi}.
386098`\spad{gcd(lp)} computes the \spad{gcd} of the list of polynomials \spad{lp}.
386405`\spad{gcd(lp)} computes the \spad{gcd} of the list of polynomials \spad{lp}.
386523`\spad{gcd(l)} returns the common \spad{gcd} of the elements in the list \spad{l}.
386566`\spad{gcd(p,{}q)} computes the \spad{gcd} of the two polynomials \spad{p} and \spad{q}.
386909`\spad{gcd(p,{}q)} computes the \spad{gcd} of the two polynomials \spad{p} and \spad{q}.
387051`\axiom{\spad{gcd}(\spad{P},{} \spad{Q})} returns the \spad{gcd} of \axiom{\spad{P}} and \axiom{\spad{Q}}.
387135`\axiom{\spad{gcd}(\spad{r},{}\spad{p})} returns the \spad{gcd} of \axiom{\spad{r}} and the content of \axiom{\spad{p}}.
387214`\spad{gcd(x,{}y)} returns the greatest common divisor of \spad{x} and \spad{y}.
387254`\spad{gcd(a,{}b)} computes the greatest common divisor of two non negative integers \spad{a} and \spad{b}.
387303`\spad{gcd(a,{}b)} computes the greatest common divisor of two positive integers \spad{a} and \spad{b}.
387349`\spad{gcd(v1,{} v2,{} p)} computes monic \spad{gcd} of \spad{v1} and \spad{v2} modulo \spad{p}.
387438`\spad{gcd(f1,{}f2,{}p)} computes the \spad{gcd} of the univariate polynomials \spad{f1} and \spad{f2} modulo the integer prime \spad{p}.
387508`\spad{gcd(a,{} lo,{} \spad{hi},{} p)} computes \spad{gcd} of elements a(\spad{lo}),{} a(\spad{lo+1}),{} ...,{} a(\spad{hi}).
387619`\spad{gcdcofact([f1,{}..fk])} = \spad{gcd} and cofactors of \spad{k} univariate polynomials.
387675`\spad{gcdcofactprim([f1,{}..fk])} = \spad{gcd} and cofactors of \spad{k} primitive polynomials.
387735`\spad{gcdprim([f1,{}..,{}fk])} = \spad{gcd} of \spad{k} PRIMITIVE univariate polynomials
387783`\spad{gderiv(f,{}[a0,{}a1,{}a2,{}..])} returns \spad{[f(0)*a0,{}f(1)*a1,{}f(2)*a2,{}..]}.
387872`like genVectorStream,{} but skips every second vector.
388026`\spad{genVectorStream(sumEta,{} maxEta,{} k)} generates stream of all possible non-increasing lists \spad{eta} with maximal entry \spad{maxEta} and sum of entries at most \spad{sumEta}.
388179`\spad{gen_Monte_Carlo_check(check,{} l,{} kind,{} qvar,{} sigma,{} gen)} returns Monte Carlo \spad{check} for solutions to guessing problem. If Monte Carlo \spad{check} is unavailable re
388179`--turns \spad{check}.
388500`\spad{generalCoefficient(action,{} f,{} k,{} p)} gives the coefficient of \spad{x^k} in \spad{p}(\spad{z})\dot \spad{f}(\spad{x}),{} where the \spad{action} of \spad{z^l} on a polynomial in \spad{x} is given by \spad{action},{} \spadignore{i.e.} \spad{action}(\spad{k},{} \spad{l},{} \spad{f}) should return the coefficient of 
388500`--\spad{x^k} in \spad{z^l} \spad{f}(\spad{x}).
388680`\spad{generalInfiniteProduct(f(x),{}a,{}d)} computes \spad{product(n=a,{}a+d,{}a+2*d,{}...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
388790`\spad{generalInfiniteProduct(f(x),{}a,{}d)} computes \spad{product(n=a,{}a+d,{}a+2*d,{}...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
388898`\spad{generalInfiniteProduct(f(x),{}a,{}d)} computes \spad{product(n=a,{}a+d,{}a+2*d,{}...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
389004`\spad{generalInfiniteProduct(f(x),{}a,{}d)} computes \spad{product(n=a,{}a+d,{}a+2*d,{}...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
389104`\spad{generalInterpolation(C,{} CA,{} f,{} eta)} performs Hermite-Pade approximation using the given action \spad{CA} of polynomials on the elements of \spad{f}. The result is guaranteed to be correct up to order |eta|-1. Given that eta is a "normal" point,{} the degrees on the diagonal are given by eta. The deg
389104`--rees of column \spad{i} are in this case eta + \spad{e}.\spad{i} - [1,{}1,{}...,{}1],{} where the degree of zero is \spad{-1}. \blankline The first argument \spad{C} is the list of coefficients \spad{c_}{\spad{k},{}\spad{k}} in the expansion <x^k> \spad{z} \spad{g}(\spad{x}) = sum_{\spad{i=0}}\spad{^k} \spad{c_}{\spad{k},{}\spad{i}} <x^i> \spad{g}(\spad{x}). \blankline The second argument,{} \spad{CA}(\spad{k},{} \spad{l},{} \spad{f}),{} should return the coefficient of \spad{x^k} in \spad{z^l
389104`--} \spad{f}(\spad{x}).
389298`\spad{generalInterpolation(l,{} CA,{} f,{} eta)} performs Hermite-Pade approximation using the given action \spad{CA} of polynomials on the elements of \spad{f}. The result is guaranteed to be correct up to order |eta|-1. Given that eta is a "normal" point,{} the degrees on the diagonal are given by
389298`-- eta. The degrees of column \spad{i} are in this case eta + \spad{e}.\spad{i} - [1,{}1,{}...,{}1],{} where the degree of zero is \spad{-1}.
389505`\spad{generalLambert(f(x),{}a,{}d)} returns \spad{f(x^a) + f(x^(a + d)) + f(x^(a + 2 d)) + ...}. \spad{f(x)} should have zero constant coefficient and \spad{a} and \spad{d} should be positive.
389688`\spad{generalLambert(f(x),{}a,{}d)} returns \spad{f(x^a) + f(x^(a + d)) + \indented{1}{f(x^(a + 2 d)) + ... }. \spad{f(x)} should have zero constant} \indented{1}{coefficient and \spad{a} and \spad{d} should be positive.}
389779`\spad{generalPosition(I,{}listvar)} perform a random linear transformation on the variables in \spad{listvar} and returns the transformed ideal along with the change of basis matrix.
389919`\spad{generalSqFr(p)} returns the square-free factorisation of polynomial \spad{p},{} a sparse univariate polynomial (sup) over a sup over \spad{F}.
390090`\spad{generalTwoFactor(p)} returns the factorisation of polynomial \spad{p},{} a sparse univariate polynomial (sup) over a sup over \spad{F}.
390266`\spad{generalizedContinuumHypothesisAssumed?()} tests if the hypothesis is currently assumed.
390346`\spad{generalizedContinuumHypothesisAssumed(bool)} is used to dictate whether the hypothesis is to be assumed.
390432`\spad{generalizedEigenvector(eigen,{}m)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \spad{eigen},{} as returned by the function eigenvectors.
390707`\spad{generalizedEigenvector(alpha,{}m,{}k,{}g)} returns the generalized eigenvectors of the matrix relative to the eigenvalue \spad{alpha}. The integers \spad{k} and \spad{g} are respectively the algebraic and the geometric multiplicity of tye eigenvalue \spad{alpha}. \spad{alpha
390707`--} can be either rational or not. In the seconda case apha is the minimal polynomial of the eigenvalue.
390933`\spad{generalizedEigenvectors(m)} returns the generalized eigenvectors of the matrix \spad{m}.
391153`\spad{generalizedInverse(m)} returns the generalized (Moore
391153`--Penrose) inverse of the matrix \spad{m},{} \spadignore{i.e.} the matrix \spad{h} such that m*h*m=h,{} h*m*h=m,{} \spad{m*h} and \spad{h*m} are both symmetric matrices.
391240`\spad{generateIrredPoly(n)} generates an irreducible univariate polynomial of the given degree \spad{n} over the finite field.
391351`\spad{generate(f)} creates an infinite stream all of whose elements are equal to \spad{f()}. Note: \spad{generate(f) = [f(),{}f(),{}f(),{}...]}.
391396`\spad{generate(f,{} s0)} generates a stream recursively by applying the function \spad{f:} \spad{S} \spad{->} \spad{S} to the last computed value.
391454`\spad{generate(f,{}s)} returns \spad{[s,{}f(s),{}f(f(s)),{}...]}.
391509`\spad{generate(f,{}x)} creates an infinite stream whose first element is \spad{x} and whose \spad{n}th element (\spad{n > 1}) is \spad{f} applied to the previous element. Note: \spad{generate(f,{}x) = [x,{}f(x),{}f(f(x)),{}...]}.
391557`\spad{generate(numberOfGens,{} maximalWeight)} generates a vector of elements of the form [left,{}weight,{}right] which represents a \spad{P}. Hall basis element for the free lie algebra on \spad{numberOfGens} generators. We only generate those basis elements of weight less than or equal to maximalWeight
391704`\spad{generator()} returns a root of the defining polynomial. This element generates the field as an algebra over the ground field.
391781`\spad{generator()} returns the generator for this domain.
391835`\spad{generator(n)} returns the \spad{n}th multiplicative generator,{} a basis term.
391901`\spad{generator(n)} returns the \spad{n}th basis term for a differential form.
391985`\spad{generator(i)} is the \spad{i}th Hall Basis element
392062`\spad{generator(d)} returns \spad{f} if \spad{(f) = d},{} "failed" if \spad{d} is not principal.
392145`\spad{generators(I)} returns a list of generators for the ideal \spad{I}.
392226`\spad{generators(gp)} returns the generators of the group {\em gp}.
392298`\spad{generic?(p)} tests if \spad{p} is a single matching variable.
392346`\spad{genericLeftDiscriminant()} is the determinant of the generic left trace forms of all products of basis element,{} if the generic left trace form is associative,{} an algebra is separable if the generic left discriminant is invertible,{} if it is non-zero,{} there is some ring extension which makes the algebra separable
392476`\spad{genericLeftMinimalPolynomial(a)} substitutes the coefficients of {em a} for the generic coefficients in \spad{leftRankPolynomial()}
392641`\spad{genericLeftNorm(a)} substitutes the coefficients of \spad{a} for the generic coefficients into the coefficient of the constant term in \spadfun{leftRankPolynomial} and changes the sign if the degree of this polynomial is odd. This is a form of degree \spad{k}
392765`\spad{genericLeftTraceForm (a,{}b)} is defined to be \spad{genericLeftTrace (a*b)},{} this defines a symmetric bilinear form on the algebra
392897`\spad{genericLeftTrace(a)} substitutes the coefficients of \spad{a} for the generic coefficients into the coefficient of the second highest term in \spadfun{leftRankPolynomial} and changes the sign. \indented{1}{This is a linear form}
393022`\spad{genericPosition(lp,{}lv)} puts a radical zero dimensional ideal in general position,{} for system \spad{lp} in variables \spad{lv}.
393235`\spad{genericRightDiscriminant()} is the determinant of the generic left trace forms of all products of basis element,{} if the generic left trace form is associative,{} an algebra is separable if the generic left discriminant is invertible,{} if it is non-zero,{} there is some ring extension which makes the algebra separable
393366`\spad{genericRightMinimalPolynomial(a)} substitutes the coefficients of \spad{a} for the generic coefficients in \spadfun{rightRankPolynomial}
393532`\spad{genericRightNorm(a)} substitutes the coefficients of \spad{a} for the generic coefficients into the coefficient of the constant term in \spadfun{rightRankPolynomial} and changes the sign if the degree of this polynomial is odd
393657`\spad{genericRightTraceForm (a,{}b)} is defined to be \spadfun{genericRightTrace (a*b)},{} this defines a symmetric bilinear form on the algebra
393790`\spad{genericRightTrace(a)} substitutes the coefficients of \spad{a} for the generic coefficients into the coefficient of the second highest term in \spadfun{rightRankPolynomial} and changes the sign
393916`\spad{generic()} returns a generic element,{} \spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \spad{\%x1,{}\%x2,{}..}
393988`\spad{generic(s)} returns a generic element,{} \spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \spad{s1,{}s2,{}..}
394066`\spad{generic(vs)} returns a generic element,{} \spadignore{i.e.} the linear combination of the fixed basis with the symbolic coefficients \spad{vs}; error,{} if the vector of symbols is too short
394152`\spad{generic(ve)} returns a generic element,{} \spadignore{i.e.} the linear combination of \spad{ve} basis with the symbolic coefficients \spad{\%x1,{}\%x2,{}..}
394234`\spad{generic(s,{}v)} returns a generic element,{} \spadignore{i.e.} the linear combination of \spad{v} with the symbolic coefficients \spad{s1,{}s2,{}..}
394323`\spad{generic(vs,{}ve)} returns a generic element,{} \spadignore{i.e.} the linear combination of \spad{ve} with the symbolic coefficients \spad{vs} error,{} if the vector of symbols is shorter than the vector of elements
394420`\spad{genus()} returns the genus of one absolutely irreducible component
394499`\spad{genus()} returns the genus of one absolutely irreducible component
394575`\spad{geometric(f)} \undocumented
394762`\spad{getArrowIndex(s,{} a,{} b)} retrieves arrow index of the arrow form a to \spad{b}
394868`\spad{getArrows(s)} returns a list of all the arrows (or edges)
395077`\spad{getBadValues(p)} returns the list of "bad values" for \spad{p}. Note: \spad{p} is not allowed to match any of its "bad values".
395131`introspection: returns deBruijn index of bound variable in bound leaf node
395195`\spad{getCentre(n)} returns centre coordinate of the boundary. \spad{n:}\% is the boundary
395242`returns child nodes if this is a compound term otherwise returns []
395290`returns 2 child nodes if this is a compound term returns 1 child node if this is a lamda term otherwise returns []
395342`returns child nodes if this is a compound term otherwise returns []
395402`\spad{getCode(f)} returns a Lisp list of strings representing \spad{f} in Fortran notation. This is used by the FortranProgram domain.
395454`\spad{getCurve(t)} returns the \spadtype{PlottableSpaceCurveCategory} representing the parametric curve of the given tube plot \spad{t}.
395505`\spad{getDatabase("char")} returns a list of appropriate entries in the browser database. The legal values for \spad{"char"} are "o" (operations),{} \spad{"k"} (constructors),{} \spad{"d"} (domains),{} \spad{"c"} (categories) or \spad{"p"} (packages).
395577`\spad{getEq f} returns the defining equation,{} if \spad{f} represents the coefficient of an ADE or a recurrence.
395628`\spad{getGoodPrime n} returns the smallest prime not dividing \spad{n}
395716`\spad{getGraph(v,{}n)} returns the graph which is of the domain \spadtype{GraphImage} which is located in graph field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of the domain \spadtype{TwoDimensionalViewport}.
395795`\spad{getMatch(var,{} r)} returns the expression that \spad{var} matches in the result \spad{r},{} and "failed" if \spad{var} is not matched in \spad{r}.
395877`\spad{getMax(n)} returns maximum coordinate of the boundary. \spad{n} is the boundary
395921`\spad{getMin(n)} returns minimum coordinate of the boundary. \spad{n} is the boundary
395965`\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.
396061`\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.
396166`\spad{getMultiplicationMatrix()} returns the multiplication table in form of a matrix.
396263`\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements.
396398`\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements.
396542`\spad{getMultiplicationTable()} returns the multiplication table for the normal basis of the field. This table is used to perform multiplications between field elements.
396678`returns the name of the variable.
396720`\spad{getNames(n)} returns a list of names for this node
396783`\axiom{getNotation()} shows the currently used notation.
396965`\spad{getOrder()} returns \spad{[[b1,{}...,{}bm],{} [a1,{}...,{}an]]} such that the partial ordering on \spad{S} was given by \spad{setOrder([b1,{}...,{}bm],{}[a1,{}...,{}an])}.
397055`\spad{getPickedPoints(x)} returns a list of small floats for the points the user interactively picked on the viewport for full integration into the system,{} some design issues need to be addressed: \spadignore{e.g.} how to go through the GraphImage interface,{} how to default to graphs,{} etc.
397139`\spad{getRef(f)} returns a reference containing the order to which the terms of \spad{f} have been computed.
397243`\spad{getSimplifyDenomsFlag()} gets values of flag affecting simplification of denominators. See setSimplifyDenomsFlag.
397326`\spad{getStream(f)} returns the stream of terms representing the series \spad{f}.
397428`\spad{getType()} returns the input form corresponding to \spad{R}
397487`returns the type of the variable. This may be a compund type,{} for instance ["a",{}\spad{"b"}] represents a-\spad{>b} a function from type a to type \spad{b} untyped implementations return []
397529`\spad{getVariableOrder()} returns \spad{[[b1,{}...,{}bm],{} [a1,{}...,{}an]]} such that the ordering on the variables was given by \spad{setVariableOrder([b1,{}...,{}bm],{} [a1,{}...,{}an])}.
397635`introspection: returns value of unbound variable in unbound leaf node or bound variable in lambda term
397681`returns the variable,{} if this is not a variable then return \spad{I}
397803`\spad{getVertexIndex(s,{} o)} gives index of object \spad{o}.
397874`\spad{getVertices(s)} returns a list of all the vertices (or objects) of the graph \spad{s}.
397991`\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly.
398095`\spad{getZechTable()} returns the zech logarithm table of the field it is used to perform additions in the field quickly.
398211`\spad{getZechTable()} returns the zech logarithm table of the field. This table is used to perform additions in the field quickly.
398419`\spad{goodPoint(p,{} q)} returns an integer a such that a is neither a pole of \spad{p(x,{}y)} nor a branch point of \spad{q(x,{}y) = 0}.
398486`\spad{goto(l)} creates a representation of a FORTRAN GOTO statement
398537`\spad{x} = ((\spad{-1})^grade(\spad{x}))\spad{*x}
398602`return the max grade of multivector,{} for example 1 is grade 0 \spad{e1} is grade 1 e1/\e2 is grade 2 and so on
398673`\spad{gradient(v,{}xlist)} computes the gradient,{} the vector of first partial derivatives,{} of the scalar field \spad{v},{} \spad{v} a function of the variables listed in \spad{xlist}.
398762`\spad{graeffe p} determines \spad{q} such that \spad{q(-z^2) = p(z)*p(-z)}. Note that the roots of \spad{q} are the squares of the roots of \spad{p}.
398825`\spad{gramschmidt(lv)} converts the list of column vectors \spad{lv} into a set of orthogonal column vectors of euclidean length 1 using the Gram-Schmidt algorithm.
398942`\spad{graphCurves([[p0],{}[p1],{}...,{}[pn]])} creates a \spadtype{GraphImage} from the list of lists of points indicated by \spad{p0} through \spad{pn}.
399029`\spad{graphCurves([[p0],{}[p1],{}...,{}[pn]],{}[options])} creates a \spadtype{GraphImage} from the list of lists of points,{} \spad{p0} throught \spad{pn},{} using the options specified in the list \spad{options}.
399133`\spad{graphCurves([[p0],{}[p1],{}...,{}[pn]],{}ptColor,{}lineColor,{}ptSize,{}[options])} creates a \spadtype{GraphImage} from the list of lists of points,{} \spad{p0} throught \spad{pn},{} using the options specified in the list \spad{options}. The graph point color is specified by \spad{ptColor},{} the graph line color is specified by \spad{lineColor},{} and the size
399133`-- of the points is specified by \spad{ptSize}.
399269`\spad{graphImage()} returns an empty graph with 0 point lists of the domain \spadtype{GraphImage}. A graph image contains the graph data component of a two dimensional viewport.
399312`\spad{graphState(v,{}num,{}sX,{}sY,{}dX,{}dY,{}pts,{}lns,{}box,{}axes,{}axesC,{}un,{}unC,{}cP)} sets the state of the characteristics for the graph indicated by \spad{num} in the given two-dimensional viewport \spad{v},{} of domain \spadtype{TwoDimensionalViewport},{} to the values given as parameters. The scaling of 
399312`--the graph in the \spad{x} and \spad{y} component directions is set to be \spad{sX} and \spad{sY}; the window translation in the \spad{x} and \spad{y} component directions is set to be \spad{dX} and \spad{dY}; The graph points,{} lines,{} bounding \spad{box},{} \spad{axes},{} or units will be shown in the viewport if their given parameters \spad{pts},{} \spad{lns},{} \spad{box},{} \spad{axes} or \spad{un} are set to be \spad{1},{} but will not be shown if they are set to \spad{0}. The color of 
399312`--the \spad{axes} and the color of the units are indicated by the palette colors \spad{axesC} and \spad{unC} respectively. To display the control panel when the viewport window is displayed,{} set \spad{cP} to \spad{1},{} otherwise set it to \spad{0}.
399500`\spad{graphStates(v)} returns and shows a listing of a record containing the current state of the characteristics of each of the ten graph records in the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDi
399500`--mensionalViewport}.
399773`\spad{graphs(v)} returns a vector,{} or list,{} which is a union of all the graphs,{} of the domain \spadtype{GraphImage},{} which are allocated for the two-dimensional viewport,{} \spad{v},{} of domain \spadtype{TwoDimensionalViewport}. Those graphs which have no data are labeled "undefined",{} otherwise their contents are shown.
399861`\spad{graphs n} is the cycle index of the group induced on \indented{1}{the edges of a graph by applying the symmetric function to the} \indented{1}{\spad{n} nodes.}
399948`\spad{green()} returns the position of the green hue from total hues.
399981`\spad{groebSolve(lp,{}lv)} reduces the polynomial system \spad{lp} in variables \spad{lv} to triangular form. Algorithm based on groebner bases algorithm with linear algebra for change of ordering. Preprocessing for the general solver. The polynomials in input are of type \spadtype{DMP}.
400159`\spad{groebgen }\undocumented
400337`\spad{groebner?(I)} tests if the generators of the ideal \spad{I} are a Groebner basis.
400413`\spad{groebnerFactorize(listOfPolys)} returns a list of groebner bases. The union of their solutions is the solution of the system of equations given by {\em listOfPolys}. At each stage the polynomial \spad{p} under consideration (either from the given basis or obtained from a reduction of the next \spad{S}-polynomial) is factorized. For each irreducible factors of \spad{p},{} a new {\em c
400413`--reateGroebnerBasis} is started doing the usual updates with the factor in place of \spad{p}.
400528`\spad{groebnerFactorize(listOfPolys,{} info)} returns a list of groebner bases. The union of their solutions is the solution of the system of equations given by {\em listOfPolys}. At each stage the polynomial \spad{p} under consideration (either from the given basis or obtained from a reduction of the next \spad{S}-polynomial) is factorized. For each irreducible factors of \spad{p}
400528`--,{} a new {\em createGroebnerBasis} is started doing the usual updates with the factor in place of \spad{p}. If {\em info} is \spad{true},{} information is printed about partial results.
400651`\spad{groebnerFactorize(listOfPolys,{} nonZeroRestrictions)} returns a list of groebner basis. The union of their solutions is the solution of the system of equations given by {\em listOfPolys} under the restriction that the polynomials of {\em nonZeroRestrictions} don\spad{'t} vanish. At each stage the polynomial \spad{p} under consideration (either from the given basis or obta
400651`--ined from a reduction of the next \spad{S}-polynomial) is factorized. For each irreducible factors of \spad{p},{} a new {\em createGroebnerBasis} is started doing the usual updates with the factor in place of \spad{p}.
400777`\spad{groebnerFactorize(listOfPolys,{} nonZeroRestrictions,{} info)} returns a list of groebner basis. The union of their solutions is the solution of the system of equations given by {\em listOfPolys} under the restriction that the polynomials of {\em nonZeroRestrictions} don\spad{'t} vanish. At each stage the polynomial \spad{p} under consideration (either from the giv
400777`--en basis or obtained from a reduction of the next \spad{S}-polynomial) is factorized. For each irreducible factors of \spad{p} a new {\em createGroebnerBasis} is started doing the usual updates with the factor in place of \spad{p}. If argument {\em info} is \spad{true},{} information is printed about partial results.
400911`\spad{groebnerIdeal(polyList)} constructs the ideal generated by the list of polynomials \spad{polyList} which are assumed to be a Groebner basis. Note: this operation avoids a Groebner basis computation.
400995`\spad{groebner(lp)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp}.
401083`\axiom{groebner(\spad{lp})} returns the lexicographical Groebner basis of \axiom{\spad{lp}}. If \axiom{\spad{lp}} generates a zero-dimensional ideal then the {\em FGLM} strategy is used,{} otherwise the {\em Sugar} strategy is used.
401266`\axiom{groebner(\spad{lp})} computes a Groebner basis for the ideal generated by the polynomials \axiom{\spad{lp}}.
401334`\axiom{groebner(\spad{lq1})} returns the lexicographical Groebner basis of \axiom{\spad{lq1}}. If \axiom{\spad{lq1}} generates a zero-dimensional ideal then the {\em FGLM} strategy is used,{} otherwise the {\em Sugar} strategy is used.
401423`\axiom{groebner(\spad{lp})} computes a Groebner basis for the ideal generated by the list of polynomials \axiom{\spad{lp}}.
401526`\axiom{groebner(\spad{lp})} computes a Groebner basis for the ideal generated by \axiom{\spad{lp}} \spad{wrt} a lexicographic ordering.
401612`\spad{groebner(lp)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp}.
401699`\spad{groebner(I)} returns a set of generators of \spad{I} that are a Groebner basis for \spad{I}.
401769`\spad{groebner(lp,{} infoflag)} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp}. Argument infoflag is used to get information on the computation. If infoflag is "info",{} then summary information is displayed for each \spad{s}-polynomial generated. If infoflag is "redcrit",{} the reduced critical pairs are displayed. If infoflag is any other string,{} no informat
401769`--ion is printed during computation.
401863`\spad{groebner(lp,{} "info",{} "redcrit")} computes a groebner basis for a polynomial ideal generated by the list of polynomials \spad{lp},{} displaying both a summary of the critical pairs considered (\spad{"info"}) and the result of reducing each critical pair ("redcrit"). If the second or third arguments have any other string value,{} the indicated information is suppressed.
402076`\axiom{ground?(\spad{l})} yields \spad{true},{} if \axiom{\spad{l}} is an element of the ground domain \axiom{\spad{D}}.
402142`\axiom{ground(exp)} is \spad{true},{} if \axiom{exp} contains only fully evaluated parts.
402200`\spad{ground?(p)} tests if polynomial \spad{p} is a member of the coefficient ring.
402265`\spad{ground?(f)} tests if \spad{f} is an element of \spad{R}.
402416`\axiom{ground(\spad{l})} returns the ground part of \axiom{\spad{l}}.
402476`\spad{ground(p)} retracts polynomial \spad{p} to the coefficient ring.
402534`\spad{ground(f)} returns \spad{f} as an element of \spad{R}. An error occurs if \spad{f} is not an element of \spad{R}.
402830`\spad{guessADE l} tries to find an algebraic differential equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}.
403264`\spad{guessADE q} returns a guesser that tries to find an algebraic differential equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the given options.
404324`\spad{guessADE(l,{} options)} tries to find an algebraic differential equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the given options.
404947`\spad{guessAlgDep ll} tries to find an algebraic dependence between several power series whose first Taylor coefficients are given by members of \spad{ll},{} using the default options described in \spadtype{GuessOptionFunctions0}.
405543`\spad{guessAlgDep ll} tries to find an algebraic dependence between several power series whose first Taylor coefficients are given by members of \spad{ll},{} using the given options.
406202`\spad{guessAlg l} tries to find an algebraic equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}. It is equivalent to \spadfun{guessADE}(\spad{l},{} maxDerivative \spad{==} 0).
406744`\spad{guessAlg(l,{} options)} tries to find an algebraic equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the given options. It is equivalent to \spadfun{guessADE}(\spad{l},{} options) with \spad{maxDerivative == 0}.
407524`\spad{guessBinRat(l,{} options)} tries to find a function of the form \spad{n+}->binomial(a+b \spad{n},{} \spad{n}) \spad{r}(\spad{n}),{} where \spad{r}(\spad{n}) is a rational function,{} that fits \spad{l}.
407973`\spad{guessBinRat q} returns a guesser that tries to find a function of the form \spad{n+}->qbinomial(a+b \spad{n},{} \spad{n}) \spad{r}(\spad{n}),{} where \spad{r}(\spad{q^n}) is a \spad{q}-rational function,{} that fits \spad{l}.
409051`\spad{guessBinRat(l,{} options)} tries to find a function of the form \spad{n+}->binomial(a+b \spad{n},{} \spad{n}) \spad{r}(\spad{n}),{} where \spad{r}(\spad{n}) is a rational function,{} that fits \spad{l}.
409846`\spad{guessExpRat l} tries to find a function of the form \spad{n+}->(a+b \spad{n})\spad{^n} \spad{r}(\spad{n}),{} where \spad{r}(\spad{n}) is a rational function,{} that fits \spad{l}.
410295`\spad{guessExpRat q} returns a guesser that tries to find a function of the form \spad{n+}->(a+b \spad{q^n})\spad{^n} \spad{r}(\spad{q^n}),{} where \spad{r}(\spad{q^n}) is a \spad{q}-rational function,{} that fits \spad{l}.
411373`\spad{guessExpRat(l,{} options)} tries to find a function of the form \spad{n+}->(a+b \spad{n})\spad{^n} \spad{r}(\spad{n}),{} where \spad{r}(\spad{n}) is a rational function,{} that fits \spad{l}.
412001`\spad{guessFE l} tries to find an algebraic substitution equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}.
412537`\spad{guessFE(l,{} options)} tries to find an algebraic substitution equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the given options.
413308`\spad{guessHolo l} tries to find an ordinary linear differential equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}. It is equivalent to \spadfun{guessADE}\spad{(l,{} maxPower == 1)}.
413747`\spad{guessHolo q} returns a guesser that tries to find a linear differential equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the given options.
414813`\spad{guessHolo(l,{} options)} tries to find an ordinary linear differential equation for a generating function whose first Taylor coefficients are given by \spad{l},{} using the given options. It is equivalent to \spadfun{guessADE}\spad{(l,{} options)} with \spad{maxPower == 1}.
415594`\spad{guessPRec l} tries to find a linear recurrence with polynomial coefficients whose first values are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}. It is equivalent to \spadfun{guessRec}\spad{(l,{} maxPower == 1)}.
416033`\spad{guessPRec q} returns a guesser that tries to find a linear \spad{q}-recurrence with polynomial coefficients whose first values are given by \spad{l},{} using the given options. It is equivalent to \spadfun{guessRec}\spad{(q)} with \spad{maxPower == 1}.
417099`\spad{guessPRec(l,{} options)} tries to find a linear recurrence with polynomial coefficients whose first values are given by \spad{l},{} using the given options. It is equivalent to \spadfun{guessRec}\spad{(l,{} options)} with \spad{maxPower == 1}.
417719`\spad{guessPade(l,{} options)} tries to find a rational function whose first Taylor coefficients are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}. It is equivalent to \spadfun{guessADE}\spad{(l,{} options)} with \spad{maxDerivative == 0,{} maxPower == 1,{} allDegrees == true}.
418267`\spad{guessPade(l,{} options)} tries to find a rational function whose first Taylor coefficients are given by \spad{l},{} using the given options. It is equivalent to \spadfun{guessADE}\spad{(l,{} maxDerivative == 0,{} maxPower == 1,{} allDegrees == true)}.
419046`\spad{guessRat l} tries to find a rational function whose first values are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}. It is equivalent to \spadfun{guessRec}\spad{(l,{} maxShift == 0,{} maxPower == 1,{} allDegrees == true)}.
419480`\spad{guessRat q} returns a guesser that tries to find a \spad{q}-rational function whose first values are given by \spad{l},{} using the given options. It is equivalent to \spadfun{guessRec} with \spad{(l,{} maxShift == 0,{} maxPower == 1,{} allDegrees == true)}.
420540`\spad{guessRat(l,{} options)} tries to find a rational function whose first values are given by \spad{l},{} using the given options. It is equivalent to \spadfun{guessRec}\spad{(l,{} maxShift == 0,{} maxPower == 1,{} allDegrees == true)}.
421314`\spad{guessRec l} tries to find an ordinary difference equation whose first values are given by \spad{l},{} using the default options described in \spadtype{GuessOptionFunctions0}.
421748`\spad{guessRec q} returns a guesser that finds an ordinary \spad{q}-difference equation whose first values are given by \spad{l},{} using the given options.
422808`\spad{guessRec(l,{} options)} tries to find an ordinary difference equation whose first values are given by \spad{l},{} using the given options.
423419`\spad{guess l} applies recursively \spadfun{guessRat} to the successive differences and quotients of the list. Default options as described in \spadtype{GuessOptionFunctions0} are used.
423943`\spad{guess(l,{} options)} applies recursively \spadfun{guessRat} to the successive differences and quotients of the list. The given options are used.
424627`\spad{guess(l,{} guessers,{} ops)} applies recursively the given \spad{guessers} to the successive differences if ops contains the symbol guessSum and quotients if ops contains the symbol guessProduct to the list. Default options as described in \spadtype{GuessOptionFunctions0} are used.
425661`\spad{guess(l,{} guessers,{} ops)} applies recursively the given \spad{guessers} to the successive differences if ops contains the symbol \spad{guessSum} and quotients if ops contains the symbol \spad{guessProduct} to the list. The given options are used.
426592`\spad{hMonic }\undocumented
426673`\spad{hMonic(p)} tries to remove content from \spad{p}
426913`\axiom{\spad{halfExtendedResultant1}(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca]} such that \axiom{extendedResultant(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca,{} \spad{cb}]}
427042`\axiom{\spad{halfExtendedResultant2}(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca]} such that \axiom{extendedResultant(a,{}\spad{b})} returns \axiom{[\spad{r},{}ca,{} \spad{cb}]}
427171`\axiom{\spad{halfExtendedSubResultantGcd1}(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca]} such that \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{} \spad{cb}]}
427301`\axiom{\spad{halfExtendedSubResultantGcd1}(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca]} if \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{}\spad{cb}]} otherwise produces an error.
427433`\axiom{\spad{halfExtendedSubResultantGcd2}(a,{}\spad{b})} returns \axiom{[\spad{g},{}\spad{cb}]} such that \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{} \spad{cb}]}
427563`\axiom{\spad{halfExtendedSubResultantGcd2}(a,{}\spad{b})} returns \axiom{[\spad{g},{}\spad{cb}]} if \axiom{extendedSubResultantGcd(a,{}\spad{b})} returns \axiom{[\spad{g},{}ca,{}\spad{cb}]} otherwise produces an error.
427695`\spad{hankelDeterminant(x,{} n)} returns the \spad{n}th Hankel determinant of the sequence \spad{x}.
427780`\spad{hankelDeterminants(x)} returns the stream of hankel determinants of the distribution \spad{x}.
427853`\spad{hankelH1(v,{} z)} is first Hankel function (Bessel function of the third kind)
427916`\spad{hankelH1(v,{} z)} is first Hankel function (Bessel function of the third kind).
427975`\spad{hankelH2(v,{} z)} is the second Hankel function (Bessel function of the third kind)
428038`\spad{hankelH2(v,{} z)} is the second Hankel function (Bessel function of the third kind).
428097`\spad{harmonic(n)} returns the \spad{n}th harmonic number. This is \spad{H[n] = sum(1/k,{}k=1..n)}.
428178`\spad{has?(op,{} s)} tests if property \spad{s} is attached to \spad{op}.
428232`\spad{hasHi(s)} tests whether the segment \spad{s} has an upper bound.
428286`\spad{hasPredicate?(p)} tests if \spad{p} has predicates attached to it.
428339`\spad{hasSolution?(A,{}B)} tests if the linear system \spad{AX = B} has a solution.
428422`\spad{hasSolution?(A,{}B)} tests if the linear system \spad{AX = B} has a solution.
428510`\spad{hasTopPredicate?(p)} tests if \spad{p} has a top-level predicate.
428640`\spad{hashUpdate!(hs,{} s)} computes new HashState form old and \spad{s} used for incremental computation of hash values. hashUpdate! might destructively work on its first argument.
428804`\spad{hash(s)} calculates a hash code for \spad{s}.
428855`\spad{hasoln(g,{} l)} tests whether the quasi-algebraic set defined by \spad{p} = 0 for \spad{p} in \spad{g} and \spad{q} \spad{~=} 0 for \spad{q} in \spad{l} is empty or not and returns a simplified definition of the quasi-algebraic set
428985`\spad{hclf(x,{} y)} returns the highest common left factor of \spad{x} and \spad{y},{} \spadignore{i.e.} the largest \spad{d} such that \spad{x = d a} and \spad{y = d b}.
429030`\spad{hclf(x,{} y)} returns the highest common left factor of \spad{x} and \spad{y},{} that is the largest \spad{d} such that \spad{x = d a} and \spad{y = d b}.
429082`\spad{hconcat(u)} horizontally concatenates all forms in list \spad{u}.
429130`\spad{hconcat(f,{}g)} horizontally concatenate forms \spad{f} and \spad{g}.
429175`\spad{hcrf(x,{} y)} returns the highest common right factor of \spad{x} and \spad{y},{} \spadignore{i.e.} the largest \spad{d} such that \spad{x = a d} and \spad{y = b d}.
429220`\spad{hcrf(x,{} y)} returns the highest common right factor of \spad{x} and \spad{y},{} that is the largest \spad{d} such that \spad{x = a d} and \spad{y = b d}.
429272`\spad{hdmpToDmp(p)} converts \spad{p} from a \spadtype{HDMP} to a \spadtype{DMP}.
429405`\spad{hdmpToP(p)} converts \spad{p} from a \spadtype{HDMP} to a \spadtype{POLY}.
429640`\axiom{headReduce(\spad{p},{}\spad{ts})} returns a polynomial \axiom{\spad{r}} such that \axiom{headReduce?(\spad{r},{}\spad{ts})} holds and there exists some product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}.
429706`\axiom{headReduce(a,{}\spad{b})} returns a polynomial \axiom{\spad{r}} such that \axiom{headReduced?(\spad{r},{}\spad{b})} holds and there exists an integer \axiom{\spad{e}} such that \axiom{init(\spad{b})^e a - \spad{r}} is zero modulo \axiom{\spad{b}}.
429847`\spad{headReduced?(ts)} returns \spad{true} iff the head of every element of \axiom{\spad{ts}} is reduced \spad{w}.\spad{r}.\spad{t} to any other element of \axiom{\spad{ts}}.
430146`\axiom{headReduced?(\spad{p},{}\spad{ts})} returns \spad{true} iff the head of \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
430220`\axiom{headReduced?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{headReduced?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
430305`\axiom{headReduced?(a,{}\spad{b})} returns \spad{true} iff \axiom{degree(head(a),{}mvar(\spad{b})) < mdeg(\spad{b})}.
430473`\axiom{headRemainder(a,{}\spad{ps})} returns \axiom{[\spad{b},{}\spad{r}]} such that the leading monomial of \axiom{\spad{b}} is reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ps}} and \axiom{r*a - \spad{b}} lies in the ideal generated by \axiom{\spad{ps}}.
430645`\spad{head(l)} returns the first element of a doubly-linked aggregate \spad{l}. Error: if \spad{l} is empty.
430698`\axiom{head(\spad{p})} returns \axiom{\spad{p}} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns its leading term (monomial in the FriCAS sense),{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
430761`\spad{heapSort(f,{} agg)} sorts the aggregate agg with the ordering function \spad{f} using the heapsort algorithm.
430837`\spad{heap(ls)} creates a heap of elements consisting of the elements of \spad{ls}.
430878`\spad{height()} returns the height of the display area (an integer).
430982`\spad{height(p)} returns the maximal absolute value of the coefficients of the polynomial \spad{p}.
431053`\spad{height(f)} returns the height of form \spad{f} (an integer).
431099`\spad{height(k)} returns the nesting level of \spad{k}.
431155`\spad{height(d)} returns the number of elements in dequeue \spad{d}. Note: \axiom{height(\spad{d}) = \# \spad{d}}.
431221`\spad{height(f)} returns the highest nesting level appearing in \spad{f}. Constants have height 0. Symbols have height 1. For any operator op and expressions \spad{f1},{}...,{}\spad{fn},{} \spad{op(f1,{}...,{}fn)} has height equal to \spad{1 + max(height(f1),{}...,{}height(fn))}.
431283`\spad{henselFact(p,{}sqf)} returns the factorization of \spad{p},{} the result is a Record such that \spad{contp=}content \spad{p},{} \spad{factors=}List of irreducible factors of \spad{p} with exponent. If \spad{sqf=true} the polynomial is assumed to be square free (\spadignore{i.e.} without repeated factors).
431411`\spad{henselFact(m,{}flag)} returns the factorization of \spad{m},{} FinalFact is a Record \spad{s}.\spad{t}. FinalFact.contp=content \spad{m},{} FinalFact.factors=List of irreducible factors of \spad{m} with exponent ,{} if \spad{flag} =true the polynomial is assumed square free.
431537`one step of Hensel lifting
431621`\spad{hermiteH(n,{}x)} is the \spad{n}-th Hermite polynomial,{} \spad{H[n](x)}. These are defined by \spad{exp(2*t*x-t^2) = sum(H[n](x)*t^n/n!,{} n = 0..)}.
431703`\spad{hermite(n)} returns the \spad{n}th Hermite polynomial \spad{H[n](x)}. Note: Hermite polynomials,{} denoted \spad{H[n](x)},{} are computed from the two term recurrence. The generating function is: \spad{exp(2*t*x-t^2) = sum(H[n](x)*t^n/n!,{} n=0..infinity)}.
431804`\spad{hermite(m)} returns the Hermite normal form of the matrix \spad{m}.
431862`\spad{hessian(v,{}xlist)} computes the hessian,{} the matrix of second partial derivatives,{} of the scalar field \spad{v},{} \spad{v} a function of the variables listed in \spad{xlist}.
431950`\spad{hexDigit?(c)} tests if \spad{c} is a hexadecimal numeral,{} \spadignore{i.e.} one of 0..9,{} a..\spad{f} or A..\spad{F}.
431998`\spad{hexDigit()} returns the class of all characters for which \spadfunFrom{hexDigit?}{Character} is \spad{true}.
432043`\spad{hex(r)} converts a rational number to a hexadecimal expansion.
432106`\spad{\spad{hi}(s)} returns the second endpoint of \spad{s}. Note: \spad{\spad{hi}(l..h) = h}.
432150`\spad{highCommonTerms(e1 a1 + ... + en an,{} f1 b1 + ... + fm bm)} returns \indented{2}{\spad{reduce(+,{}[max(\spad{ei},{} \spad{fi}) \spad{ci}])}} where \spad{ci} ranges in the intersection of \spad{{a1,{}...,{}an}} and \spad{{b1,{}...,{}bm}}.
432250`\spad{high(s)} returns the second endpoint of \spad{s}. Note: \spad{high(l..h) = h}.
432296`\axiom{hilbert(\spad{cc})} computes the Hilbert polynomial to the Cartan characters \axiom{\spad{cc}}.
432414`\spad{hitherPlane(v,{}h)} sets the hither clipping plane of the graph to \spad{h},{} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}.
432482`\spad{homogeneous?(p)} tests if all of the terms of \spad{p} have the same degree.
432540`\spad{homogeneous?(df)} tests if all of the terms of differential form \spad{df} have the same degree.
432616`\spad{homogeneous returns} whether we allow only homogeneous algebraic differential equations,{} default being \spad{false}
432716`\spad{homogeneous(d)} specifies whether we allow only homogeneous algebraic differential equations. This option is expressed in the form \spad{homogeneous == d}. If \spad{true},{} then maxPower must be set,{} too,{} and ADEs with constant total degree are allowed. If a PositiveInteger is given,{} only ADE\spad{'s} with this total degree are allowed.
432850`\spad{horizConcat(x,{}y)} horizontally concatenates two matrices with an equal number of rows. The entries of \spad{y} appear to the right of the entries of \spad{x}. Error: if the matrices do not have the same number of rows.
432914`\axiom{horizJoin(A,{} \spad{B})} horizontally concats the matrices \axiom{A} and \axiom{\spad{B}}. It is assumed that all indices of \axiom{\spad{B}} are smaller than those of \axiom{A}.
432975`\axiom{horizSplit(A,{} \spad{c})} splits the matrix \axiom{A} into two at the column given by \axiom{\spad{c}}. The first column of the right matrix is enumerated by the first index less or equal to \axiom{\spad{c}}.
433058`\spad{hspace(n)} creates white space of width \spad{n}.
433104`\spad{htrigs(f)} converts all the exponentials in \spad{f} into hyperbolic sines and cosines.
433165`\spad{hue(pt)} returns the third element of the two dimensional point,{} \spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} hue to express a third dimension.
433213`\spad{hue(p)} returns the hue field of the indicated palette \spad{p}.
433251`\spad{hue(c)} returns the hue index of the indicated color \spad{c}.
433289`\spad{hyperelliptic()} returns \spad{p(x)} if the curve is the hyperelliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
433376`\spad{hyperelliptic()} returns \spad{p(x)} if the curve is the hyperelliptic defined by \spad{y^2 = p(x)},{} "failed" otherwise.
433460`\spad{hypergeometric0F1(c,{}z)} is the hypergeometric function \spad{0F1(; c; z)}.
433586`\spad{hypergeometric0F1(c,{}z)} is the hypergeometric function \spad{0F1(; c; z)}.
433685`\spad{hypergeometricF(la,{} lb,{} z)} is the generalized hypergeometric function
433798`\spad{hypergeometricF(la,{} lb,{} z)} is the generalized hypergeometric function.
433909`\spad{iAiryAiPrime(x)} should be local but conditional.
433974`\spad{iAiryAi(x)} should be local but conditional.
434034`\spad{iAiryBiPrime(x)} should be local but conditional.
434099`\spad{iAiryBi(x)} should be local but conditional.
434159`\spad{iCompose(f,{}g)} returns \spad{f(g(x))}. This is an internal function which should only be called for Taylor series \spad{f(x)} and \spad{g(x)} such that the constant coefficient of \spad{g(x)} is zero.
434233`\spad{iExquo(f,{}g,{}taylor?)} is the quotient of the power series \spad{f} and \spad{g}. If \spad{taylor?} is \spad{true},{} then we must have \spad{order(f) >= order(g)}.
434329`\spad{iLambertW(x)} should be local but conditional.
434471`\spad{id x} is \spad{x}.
434514`\spad{idealSimplify(s)} returns a different and presumably simpler representation of \spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using Buchberger\spad{'s} algorithm.
434588`\spad{ideal([f1,{}...,{}fn])} returns the ideal \spad{(f1,{}...,{}fn)}.
434650`\spad{ideal(polyList)} constructs the ideal generated by the list of polynomials \spad{polyList}.
434726`\spad{ideal(D)} returns the ideal corresponding to a divisor \spad{D}.
434827`\spad{idealiserMatrix(m1,{} m2)} returns the matrix representing the linear conditions on the Ring associatied with an ideal defined by \spad{m1} and \spad{m2}.
434917`\spad{idealiser(m1,{}m2)} computes the order of an ideal defined by \spad{m1} and \spad{m2}
435001`\spad{idealiser(m1,{}m2,{}d)} computes the order of an ideal defined by \spad{m1} and \spad{m2} where \spad{d} is the known part of the denominator
435087`\axiom{identification(\spad{g},{}\spad{h})} returns the list of equations \axiom{g_i = h_i},{} where \axiom{g_i} (resp. \axiom{h_i}) are exponential coordinates of \axiom{\spad{g}} (resp. \axiom{\spad{h}}).
435175`\spad{identityMatrix(n)} create an identity matrix we note that this must be square
435266`\spad{identitySquareMatrix(s,{}p)} \undocumented{}
435363`\spad{identity()} returns the identity element which is do nothing transform
435408`\spad{identity()} create the identity dhmatrix
435465`\spad{iexactQuo(x,{}y)} should be local but conditional
435537`\spad{iexactQuo(x,{}y)} should be local but conditional
435608`\spad{iexactQuo(x,{}y)} should be local but conditional
435698`\spad{iexactQuo(x,{}y)} should be local but conditional
435787`\spad{ignore?(s)} is \spad{true} if \spad{s} is the string that tells the integrator to assume that the function has no pole in the integration interval.
435857`\spad{iiAiryAiPrime(x)} should be local but conditional.
435923`\spad{iiAiryAi(x)} should be local but conditional.
435984`\spad{iiAiryBiPrime(x)} should be local but conditional.
436050`\spad{iiAiryBi(x)} should be local but conditional.
436111`\spad{iiBesselI(x)} should be local but conditional.
436179`\spad{iiBesselJ(x)} should be local but conditional.
436247`\spad{iiBesselK(x)} should be local but conditional.
436315`\spad{iiBesselY(x)} should be local but conditional.
436383`\spad{iiBeta(x)} should be local but conditional.
436448`\spad{iiGamma(x)} should be local but conditional.
436508`\spad{iiHypergeometricF(l)} should be local but conditional.
436584`\spad{iiPolylog(x,{} s)} should be local but conditional.
436648`\spad{iiabs(x)} should be local but conditional.
436706`\spad{iiacos(x)} should be local but conditional
436758`\spad{iiacosh(x)} should be local but conditional
436811`\spad{iiacot(x)} should be local but conditional
436863`\spad{iiacoth(x)} should be local but conditional
436916`\spad{iiacsc(x)} should be local but conditional
436968`\spad{iiacsch(x)} should be local but conditional
437021`\spad{iiasec(x)} should be local but conditional
437073`\spad{iiasech(x)} should be local but conditional
437126`\spad{iiasin(x)} should be local but conditional
437178`\spad{iiasinh(x)} should be local but conditional
437231`\spad{iiatan(x)} should be local but conditional
437283`\spad{iiatanh(x)} should be local but conditional
437336`\spad{iibinom(l)} should be local but conditional.
437398`\spad{iicos(x)} should be local but conditional
437449`\spad{iicosh(x)} should be local but conditional
437501`\spad{iicot(x)} should be local but conditional
437552`\spad{iicoth(x)} should be local but conditional
437604`\spad{iicsc(x)} should be local but conditional
437655`\spad{iicsch(x)} should be local but conditional
437707`\spad{iidigamma(x)} should be local but conditional.
437769`\spad{iidprod(l)} should be local but conditional.
437831`\spad{iidsum(l)} should be local but conditional.
437892`\spad{iiexp(x)} should be local but conditional
437943`\spad{iifact(x)} should be local but conditional.
437998`\spad{iilog(x)} should be local but conditional
438049`should be local but conditional
438121`\spad{iiperm(l)} should be local but conditional.
438182`\spad{iipolygamma(x)} should be local but conditional.
438252`\spad{iipow(l)} should be local but conditional.
438312`should be local but conditional
438412`\spad{iisec(x)} should be local but conditional
438463`\spad{iisech(x)} should be local but conditional
438515`\spad{iisin(x)} should be local but conditional
438566`\spad{iisinh(x)} should be local but conditional
438618`\spad{iisqrt2()} should be local but conditional
438670`\spad{iisqrt3()} should be local but conditional
438722`\spad{iitan(x)} should be local but conditional
438773`\spad{iitanh(x)} should be local but conditional
438825`\spad{imagE(o)} extracts the imaginary \spad{E} part of octonion \spad{o}.
438873`\spad{imagI(o)} extracts the imaginary \spad{I} part of octonion \spad{o}.
438921`\spad{imagI(q)} extracts the imaginary \spad{i} part of quaternion \spad{q}.
438971`\spad{imagJ(o)} extracts the imaginary \spad{J} part of octonion \spad{o}.
439019`\spad{imagJ(q)} extracts the imaginary \spad{j} part of quaternion \spad{q}.
439069`\spad{imagK(o)} extracts the imaginary \spad{K} part of octonion \spad{o}.
439117`\spad{imagK(q)} extracts the imaginary \spad{k} part of quaternion \spad{q}.
439167`\spad{imag(f)} returns the imaginary part of \spad{f} where \spad{f} is a complex function.
439244`\spad{imag(f)} returns the imaginary part of \spad{f} where \spad{f} is a complex function.
439302`\spad{imag(x)} returns imaginary part of \spad{x}.
439348`\spad{imagi(o)} extracts the \spad{i} part of octonion \spad{o}.
439443`\spad{imaginary()} = sqrt(\spad{-1}) = \%\spad{i}.
439493`\spad{imagj(o)} extracts the \spad{j} part of octonion \spad{o}.
439541`\spad{imagk(o)} extracts the \spad{k} part of octonion \spad{o}.
439589`\spad{implies(a,{}b)} returns the logical implication of ILogic a and \spad{b}. a is premise,{} \spad{b} is conclusion,{} result is \spad{false} (contradiction) if premise=true and conclusion=false does not mean there is a causal connection
439630`\spad{implies(a,{}b)} returns the logical implication of Boolean \spad{a} and \spad{b}.
439672`\spad{in?(I,{}J)} tests if the ideal \spad{I} is contained in the ideal \spad{J}.
439745`\spad{inBounds?(pt,{}mns,{}mxs)} returns \spad{true} if \spad{pt} is inside bounding box where bounding box is specified by min and max.
439883`\spad{inDegree(s,{} a)} gives the number of arrows leading in to node 'a' in graph \spad{'s'}
439965`\spad{inGroundField?(a)} tests whether an element \spad{a} is already in the ground field \spad{F}.
440026`\spad{inHallBasis?(numberOfGens,{} leftCandidate,{} rightCandidate,{} left)} tests to see if a new element should be added to the \spad{P}. Hall basis being constructed. The list \spad{[leftCandidate,{}wt,{}rightCandidate]} is included in the basis if in the unique factorization of \spad{rightCandidate},{} we have left factor leftOfRight,{} and leftOfRight \spad{<=} \spad{leftCandidate}
440106`\spad{inR?(p)} tests if \spad{p} is an atom (\spadignore{i.e.} an element of \spad{R}).
440150`\spad{inRadical?(f,{}I)} tests if some power of the polynomial \spad{f} belongs to the ideal \spad{I}.
440233`\spad{incFail(s,{} o)} is an internal function that steps the number of failed tests and records in- and output.
440301`\spad{incFatal s} is an internal function that steps the number of fatal tests and records input.
440353`incLibraryError is an internal function that steps the number of failed tests,{} but not the number of tests.
440429`incPass is an internal function that steps the number of passed tests.
440474`\spad{incXfFail()} is an internal function that steps the number expected failures.
440521`like incFatal,{} but for expected failures.
440575`like incLibraryError,{} but using expected failures.
440653`\spad{incXfPass(s,{} o)} is an internal function that steps the number of tests that passed but were expected to fail and records in- and output.
440723`\spad{inc(x)} returns \spad{x + 1}.
440834`\spad{incidenceMatrix(s)} represents graph \spad{s} by a matrix of size \spad{|V|} by |E| where: V=number of vertices E=number of edges entry [vertex,{} arrow] = arrow endpoint data (undirected case case: 1 - incident,{} 0 - not incident,{} directed case: \spad{-1} - start,{} 1 - end,{} 0 - not incident).
440901`inconsistant?(\spad{pl}) returns \spad{true} if the system of equations \spad{p} = 0 for \spad{p} in \spad{pl} is inconsistent. It is assumed that \spad{pl} is a groebner basis.
440991`inconsistant?(\spad{pl}) returns \spad{true} if the system of equations \spad{p} = 0 for \spad{p} in \spad{pl} is inconsistent. It is assumed that \spad{pl} is a groebner basis.
441092`\spad{incr(s)} returns \spad{n},{} where \spad{s} is a segment in which every \spad{n}\spad{-}th element is used. Note: \spad{incr(l..h by n) = n}.
441144`\spad{increasePrecision(n)} increases the current \spadfunFrom{precision}{FloatingPointSystem} by \spad{n} decimal digits.
441249`\spad{incrementBy(n)} produces a function which adds \spad{n} to whatever argument it is given. For example,{} if {\spad{f} \spad{:=} increment(\spad{n})} then \spad{f x} is \spad{x+n}.
441307`\spad{incrementKthElement(S,{}k)} increments the \spad{k^}{th} element of \spad{S},{} and returns "failed" if the result is not a set of \spad{M} integers in \spad{1..n} any more.
441410`\spad{increment()} produces a function which adds \spad{1} to whatever argument it is given. For example,{} if {\spad{f} \spad{:=} increment()} then \spad{f x} is \spad{x+1}.
441649`\spad{index?(i,{}u)} tests if \spad{i} is an index of aggregate \spad{u}.
441720`\spad{indexName returns} the name of the index variable used for the formulas,{} default being \spad{n}
441794`\spad{indexName(d)} specifies the index variable used for the formulas. This option is expressed in the form \spad{indexName == d}.
442052`\spad{index(x)} \undocumented
442105`\axiom{index(\spad{jv})} yields number of the jet variable \axiom{\spad{jv}}.
442165`\spad{index(i)} takes a positive integer \spad{i} less than or equal to \spad{size()} and returns the \spad{i}\spad{-}th element of the set. This operation establishs a bijection between the elements of the finite set and \spad{1..size()}.
442214`\spad{index(n,{}m)} is a index function for vectors of length \spad{n} over the ground field.
442315`\spad{indexes(me)} returns indexes into pointList(\spad{me})
442387`\axiom{indiceSubResultant(\spad{P},{} \spad{Q},{} \spad{i})} returns the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} and carries out the equality \axiom{coef1*P + coef2*Q = S_i(\spad{P},{}\spad{Q})}
442541`\axiom{indiceSubResultant(\spad{P},{} \spad{Q},{} \spad{i})} returns the subresultant of indice \axiom{\spad{i}}
442824`\spad{indices(u)} returns a list of indices of aggregate \spad{u} in no particular order.
442894`\spad{indicialEquationAtInfinity op} returns the indicial equation of \spad{op} at infinity.
443010`\spad{indicialEquationAtInfinity op} returns the indicial equation of \spad{op} at infinity.
443129`\spad{indicialEquation(op,{} a)} returns the indicial equation of \spad{op} at \spad{a}.
443196`\spad{indicialEquation(op,{} a)} returns the indicial equation of \spad{op} at \spad{a}.
443264`\spad{indicialEquations op} returns \spad{[[d1,{}e1],{}...,{}[dq,{}eq]]} where the \spad{d_i}\spad{'s} are the affine singularities of \spad{op},{} and the \spad{e_i}\spad{'s} are the indicial equations at each \spad{d_i}.
443363`\spad{indicialEquations op} returns \spad{[[d1,{}e1],{}...,{}[dq,{}eq]]} where the \spad{d_i}\spad{'s} are the affine singularities of \spad{op},{} and the \spad{e_i}\spad{'s} are the indicial equations at each \spad{d_i}.
443463`\spad{indicialEquations(op,{} p)} returns \spad{[[d1,{}e1],{}...,{}[dq,{}eq]]} where the \spad{d_i}\spad{'s} are the affine singularities of \spad{op} above the roots of \spad{p},{} and the \spad{e_i}\spad{'s} are the indicial equations at each \spad{d_i}.
443565`\spad{indicialEquations(op,{} p)} returns \spad{[[d1,{}e1],{}...,{}[dq,{}eq]]} where the \spad{d_i}\spad{'s} are the affine singularities of \spad{op} above the roots of \spad{p},{} and the \spad{e_i}\spad{'s} are the indicial equations at each \spad{d_i}.
443668`\axiom{infLex?(\spad{n1},{}\spad{n2},{}\spad{o1},{}\spad{o2})} returns \spad{true} iff \axiom{\spad{o1}(value(\spad{n1}),{}value(\spad{n2}))} or \axiom{value(\spad{n1}) = value(\spad{n2})} and \axiom{\spad{o2}(condition(\spad{n1}),{}condition(\spad{n2}))}.
443898`\axiom{infRittWu?(\spad{lp1},{}\spad{lp2})} is an internal subroutine,{} exported only for developement.
443984`\axiom{infRittWu?(\spad{lp1},{}\spad{lp2})} is an internal subroutine,{} exported only for developement.
444080`\axiom{infRittWu?(a,{}\spad{b})} returns \spad{true} if \axiom{a} is less than \axiom{\spad{b}} \spad{w}.\spad{r}.\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.
444157`\axiom{infRittWu?(\spad{ts1},{}\spad{ts2})} returns \spad{true} iff \axiom{\spad{ts2}} has higher rank than \axiom{\spad{ts1}} in Wu Wen Tsun sense.
444230`\spad{inf(u)} returns the infinum of \axiom{\spad{u}}.
444276`\spad{infieldIntegrate(f,{} x)} returns a fraction \spad{g} such that \spad{dg/dx = f} if \spad{g} exists,{} "failed" otherwise.
444412`\spad{infieldint(f)} returns \spad{g} such that \spad{g' = f} or "failed" if the integral of \spad{f} is not a rational function.
444508`\spad{infinite?(x)} tests if \spad{x} is infinite.
444568`\spad{infinite?(x)} tests if \spad{x} is +infinity or -infinity,{}
444627`\spad{infiniteProduct(f(x))} computes \spad{product(n=1,{}2,{}3...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
444714`\spad{infiniteProduct(f(x))} computes \spad{product(n=1,{}2,{}3...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
444799`\spad{infiniteProduct(f(x))} computes \spad{product(n=1,{}2,{}3...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
444882`\spad{infiniteProduct(f(x))} computes \spad{product(n=1,{}2,{}3...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
444959`\spad{infinityNorm(f)} returns the maximal absolute value of the coefficients of the polynomial \spad{f}.
445036`\spad{infinity()} returns infinity.
445100`\spad{infinity()} returns infinity.
445152`\spad{infix?(op)} returns \spad{true} if \spad{op} is an infix operator,{} and \spad{false} otherwise.
445198`\spad{infix(f,{}l)} creates a form depicting the \spad{n}-ary application of infix operation \spad{f} to a tuple of arguments \spad{l}.
445247`\spad{infix(op,{} a,{} b)} creates a form which prints as: a \spad{op} \spad{b}.
445293`\spad{infsum(x)} computes sum of all elements of \spad{x}. Degrees of elements of \spad{x} must be nondecreasing and tend to infinity.
445384`\axiom{initTable!()} initializes the hash-table.
445648`\spad{init()} chooses an initial object for stepping.
445745`\axiom{init(\spad{p})} returns an error if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns its leading coefficient,{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
445808`\spad{initial constructs} a graph without vertices or edges
445919`\spad{initial(p)} returns the leading coefficient when the differential polynomial \spad{p} is written as a univariate polynomial in its leader.
445990`\spad{initializeGroupForWordProblem(gp)} initializes the group {\em gp} for the word problem. Notes: it calls the other function of this name with parameters 0 and 1: {\em initializeGroupForWordProblem(gp,{}0,{}1)}. Notes: (1) be careful: invoking this routine will destroy the possibly information about your group (but will recompute it again) (2) users need not call this function normally for the soultion of the word problem.
446065`\spad{initializeGroupForWordProblem(gp,{}m,{}n)} initializes the group {\em gp} for the word problem. Notes: (1) with a small integer you get shorter words,{} but the routine takes longer than the standard routine for longer words. (2) be careful: invoking this routine will destroy the possibly stored information about your group (but will recompute it again). (3) users need not call this function normally for th
446065`--e soultion of the word problem.
446296`\axiom{initiallyReduce(\spad{p},{}\spad{ts})} returns a polynomial \axiom{\spad{r}} such that \axiom{initiallyReduced?(\spad{r},{}\spad{ts})} holds and there exists some product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}.
446367`\axiom{initiallyReduce(a,{}\spad{b})} returns a polynomial \axiom{\spad{r}} such that \axiom{initiallyReduced?(\spad{r},{}\spad{b})} holds and there exists an integer \axiom{\spad{e}} such that \axiom{init(\spad{b})^e a - \spad{r}} is zero modulo \axiom{\spad{b}}.
446518`\spad{initiallyReduced?(ts)} returns \spad{true} iff for every element \axiom{\spad{p}} of \axiom{\spad{ts}} \axiom{\spad{p}} and all its iterated initials are reduced \spad{w}.\spad{r}.\spad{t}. to the other elements of \axiom{\spad{ts}} with the same main variable.
446837`\axiom{initiallyReduced?(\spad{p},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{p}} and all its iterated initials are reduced \spad{w}.\spad{r}.\spad{t}. to the elements of \axiom{\spad{ts}} with the same main variable.
446916`\axiom{initiallyReduced?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{initiallyReduced?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
447006`\axiom{initiallyReduced?(a,{}\spad{b})} returns \spad{false} iff there exists an iterated initial of \axiom{a} which is not reduced \spad{w}.\spad{r}.\spad{t} \axiom{\spad{b}}.
447155`\axiom{initials(\spad{ts})} returns the list of the non-constant initials of the members of \axiom{\spad{ts}}.
447223`\spad{innerEigenvectors(m,{}eps,{}factor)} computes explicitly the eigenvalues and the correspondent eigenvectors of the matrix \spad{m}. The parameter \spad{eps} determines the type of the output,{} \spad{factor} is the univariate factorizer to \spad{br} used to reduce the characteristi
447223`--c polynomial into irreducible factors.
447442`\spad{innerSolve1(p,{}eps)} returns the list of the zeros of the polynomial \spad{p} with precision \spad{eps}.
447536`\spad{innerSolve1(up,{}eps)} returns the list of the zeros of the univariate polynomial \spad{up} with precision \spad{eps}.
447646`\spad{innerSolve(lnum,{}lden,{}lvar,{}eps)} returns a list of solutions of the system of polynomials \spad{lnum},{} with the side condition that none of the members of \spad{lden} vanish identically on any solution. Each solution is expressed as a list corresponding to the list of variables in \spad{lvar} and with precision specified by \spad{eps}.
447784`\spad{innerint(f,{} x,{} a,{} b,{} ignore?)} should be local but conditional
448008`\spad{input(op)} returns the "\%input" property of \spad{op} if it has one attached,{} "failed" otherwise.
448093`\spad{input(op,{} foo)} attaches foo as the "\%input" property of \spad{op}. If \spad{op} has a "\%input" property \spad{f},{} then \spad{op(a1,{}...,{}an)} gets converted to InputForm as \spad{f(a1,{}...,{}an)}.
448165`\spad{inrootof(p,{} x)} should be a non-exported function.
448248`\axiom{insert!(\spad{x},{}\spad{y})} stores the item whose key is \axiom{\spad{x}} and whose entry is \axiom{\spad{y}}.
448408`\spad{insert!(x,{}u)} inserts item \spad{x} into bag \spad{u}.
448457`\spad{insert!(x,{}b)} inserts element \spad{x} as leaves into binary search tree \spad{b}.
448510`\spad{insert!(x,{}b)} inserts element \spad{x} as leaves into binary tournament \spad{b}.
448671`\spad{insert!(x,{}u,{}i)} destructively inserts \spad{x} into \spad{u} at position \spad{i}.
448741`\spad{insert!(x,{}d,{}n)} destructively inserts \spad{n} copies of \spad{x} into dictionary \spad{d}.
448812`\spad{insert!(v,{}u,{}i)} destructively inserts aggregate \spad{v} into \spad{u} at position \spad{i}.
448883`\spad{insertBottom!(x,{}d)} destructively inserts \spad{x} into the dequeue \spad{d} at the bottom (back) of the dequeue.
448941`\spad{insertMatch(var,{} expr,{} r)} adds the match (\spad{var},{} \spad{expr}) in \spad{r},{} without checking predicates or previous matches for \spad{var}.
449013`\spad{insertRoot!(x,{}b)} inserts element \spad{x} as a root of binary search tree \spad{b}.
449070`\spad{insertTop!(x,{}d)} destructively inserts \spad{x} into the dequeue \spad{d},{} that is,{} at the top (front) of the dequeue. The element previously at the top of the dequeue becomes the second in the dequeue,{} and so on.
449125`\axiom{insert(\spad{sys},{} de)} adds the system \axiom{\spad{sys} = 0} to the differential equation \axiom{de}.
449687`\spad{insert(x,{}u,{}i)} returns a copy of \spad{u} having \spad{x} as its \axiom{\spad{i}}th element. Note: \axiom{insert(\spad{x},{}a,{}\spad{k}) = concat(concat(a(0..\spad{k}-1),{}\spad{x}),{}a(\spad{k}..))}.
449746`\spad{insert(v,{}u,{}k)} returns a copy of \spad{u} having \spad{v} inserted beginning at the \axiom{\spad{i}}th element. Note: \axiom{insert(\spad{v},{}\spad{u},{}\spad{k}) = concat( \spad{u}(0..\spad{k}-1),{} \spad{v},{} \spad{u}(\spad{k}..) )}.
449806`\spad{insertionSort! }\undocumented
449876`\spad{insertionSort!(a,{}f)} \undocumented
450023`\spad{inspect(u)} returns an (random) element from a bag.
450069`\spad{intBasis LLv} computes a basis of the intersection of list of subspaces.
450140`\spad{intBasis Lb} computes a basis of the intersection of list of subspaces.
450198`\spad{intBasis(Lv1,{}Lv2)} computes a basis of the intersection of the two subspaces spanned by the vectors of \spad{Lv1} and \spad{Lv2}
450279`\spad{intBasis(B1,{}B2)} computes a basis of the intersection of the two subspaces.
450334`\spad{intChoose should} be local
450585`\spad{intPatternMatch(f,{} x,{} int,{} pmint)} tries to integrate \spad{f} first by using the integration function \spad{int},{} and then by
450585`-- using the pattern match intetgration function \spad{pmint} on any remaining unintegrable part.
450952`\spad{int(r)} returns [\spad{r},{}\spad{r+1},{}\spad{r+2},{}...],{} where \spad{r} is a ring element.
451017`\spad{int(expr)} creates the form prefixing \spad{expr} with an integral sign.
451055`\spad{int(f,{} x)} returns the integral of \spad{f} with respect to \spad{x}.
451107`\spad{int(expr,{}lowerlimit)} creates the form prefixing \spad{expr} by an integral sign with a \spad{lowerlimit}.
451148`\spad{int(expr,{}lowerlimit,{}upperlimit)} creates the form prefixing \spad{expr} by an integral sign with both a \spad{lowerlimit} and \spad{upperlimit}.
451192`\spad{intcompBasis }\undocumented
451442`\spad{integ_df should} be local but conditional
451560`\spad{integ should} be local but conditional
451671`\spad{integer?(t)} tests whether \spad{t} is equivalent to the FORTRAN type INTEGER.
451726`\spad{integer?(x)} is \spad{true} if \spad{x} is an integer,{} \spad{false} otherwise.
451784`\spad{integer?(s)} is \spad{true} if \spad{s} is an atom and belong to Int.
451863`\spad{integerBound(p)} returns a lower bound on the negative integer roots of \spad{p},{} and 0 if \spad{p} has no negative integer roots.
451928`\spad{integerIfCan(x)} returns \spad{x} as an integer,{} "failed" if \spad{x} is not an integer.
452006`\spad{integerPart(o)} = \spad{n} when \spad{o} = \spad{l} + \spad{n} and \spad{l} is a limit ordinal
452070`\spad{integer(x)} returns \spad{x} as an integer; error if \spad{x} is not an integer.
452127`\spad{integer(s)} returns \spad{s} as an element of Int. Error: if \spad{s} is not an atom that also belongs to Int.
452201`\spad{integers(n)} returns \spad{[n,{}n+1,{}n+2,{}...]}.
452351`\spad{integral?(f)} tests if \spad{f} is integral over \spad{k[x]}.
452563`\spad{integral?(f,{} a)} tests whether \spad{f} is locally integral at \spad{x = a}.
452636`\spad{integral?(f,{} p)} tests whether \spad{f} is locally integral at \spad{p(x) = 0}.
452788`\spad{integralAtInfinity?(f)} tests if \spad{f} is locally integral at infinity.
452869`\spad{integralBasisAtInfinity()} returns the local integral basis at infinity.
452955`\spad{integralBasis()} returns a record \spad{[basis,{}basisDen,{}basisInv]} containing information regarding the integral closure of \spad{Z} in the quotient field of \spad{F},{} where \spad{F} is a framed algebra with \spad{Z}-module basis \spad{w1,{}w2,{}...,{}wn}. If \spad{basis} is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n)},{} then the \spad{i}th element of th
452955`--e integral basis is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of \spad{basis} contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix \spad{basisInv} contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if \spad{basisInv} is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1..n)}.
453090`\spad{integralBasis()} returns a record \spad{[basis,{}basisDen,{}basisInv]} containing information regarding the integral closure of \spad{R} in the quotient field of \spad{F},{} where \spad{F} is a framed algebra with \spad{R}-module basis \spad{w1,{}w2,{}...,{}wn}. If \spad{basis} is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n)},{} then the \spad{i}th element of the integral bas
453090`--is is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of \spad{basis} contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix \spad{basisInv} contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if \spad{basisInv} is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1..n)}.
453211`\spad{integralBasis()} returns a record \spad{[basis,{}basisDen,{}basisInv] } containing information regarding the integral closure of \spad{R} in the quotient field of the framed algebra \spad{F}. \spad{F} is a framed algebra with \spad{R}-module basis \spad{w1,{}w2,{}...,{}wn}. If 'basis' is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n)},{} then the \spad{i}th element o
453211`--f the integral basis is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of 'basis' contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix 'basisInv' contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if 'basisInv' is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1..n)}.
453343`\spad{integralBasis()} returns a record \spad{[basis,{}basisDen,{}basisInv]} containing information regarding the integral closure of \spad{R} in the quotient field of \spad{F},{} where \spad{F} is a framed algebra with \spad{R}-module basis \spad{w1,{}w2,{}...,{}wn}. If \spad{basis} is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n)},{} then the \spad{i}th element of the integr
453343`--al basis is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of \spad{basis} contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix \spad{basisInv} contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if \spad{basisInv} is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1..n)}.
453470`\spad{integralBasis()} returns the integral basis for the curve.
453546`\spad{integralCoordinates(f)} returns \spad{[[A1,{}...,{}An],{} D]} such that \spad{f = (A1 w1 +...+ An wn) / D} where \spad{(w1,{}...,{}wn)} is the integral basis returned by \spad{integralBasis()}.
453649`\spad{integralDerivationMatrix(d)} extends the derivation \spad{d} from UP to \$ and returns (\spad{M},{} \spad{Q}) such that the i^th row of \spad{M} divided by \spad{Q} form the coordinates of \spad{d(\spad{wi})} with respect to \spad{(w1,{}...,{}wn)} where \spad{(w1,{}...,{}wn)} is the integral basis returned by integralBasis().
453763`\axiom{integralLastSubResultant(\spad{p1},{}\spad{p2},{}\spad{ts})} is an internal subroutine,{} exported only for developement.
453884`\spad{integralMatrixAtInfinity()} returns \spad{M} such that \spad{(v1,{}...,{}vn) = M (1,{} y,{} ...,{} y^(n-1))} where \spad{(v1,{}...,{}vn)} is the local integral basis at infinity returned by \spad{infIntBasis()}.
453981`\spad{integralMatrix()} returns \spad{M} such that \spad{(w1,{}...,{}wn) = M (1,{} y,{} ...,{} y^(n-1))},{} where \spad{(w1,{}...,{}wn)} is the integral basis of \spadfunFrom{integralBasis}{FunctionFieldCategory}.
454068`\spad{integralRepresents([A1,{}...,{}An],{} D)} returns \spad{(A1 w1+...+An wn)/D} where \spad{(w1,{}...,{}wn)} is the integral basis of \spad{integralBasis()}.
454154`\spad{integral(f,{}x)} returns the formal integral of \spad{f} with respect to \spad{x}
454208`\spad{integral(f,{}x = a..b)} denotes the definite integral of \spad{f} with respect to \spad{x} from \spad{a} to \spad{b}.
454281`\spad{integral(f,{}x)} indefinite integral of \spad{f} with respect to \spad{x}.
454343`\spad{integral(f,{}x)} returns the formal integral of \spad{f} with respect to \spad{x}
454430`\spad{integral(f,{} x = a..b)} returns the formal definite integral of \spad{f} \spad{dx} for \spad{x} between \spad{a} and \spad{b}.
454507`\spad{integral(f,{} x)} returns the formal integral of \spad{f} \spad{dx}.
454657`\axiom{integrate(\spad{jv},{} \spad{i})} integrated \axiom{\spad{jv}} \spad{wrt} the \axiom{\spad{i}}\spad{-}th independent variable,{} if possible.
454745`\spad{integrate(f)} returns \spad{g} such that \spad{g' = f}.
454904`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. Warning: function does not check for a term of degree \spad{-1}.
455012`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 1. We may integrate a series when we can divide coefficients by integers.
455119`\spad{integrate(p)} integrates the univariate polynomial \spad{p} with respect to its distinguished variable.
455217`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 1. We may integrate a series when we can divide coefficients by rational numbers.
455324`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
455430`\spad{integrate(r,{}a)} returns the integral of the power series \spad{a} with respect to the power series variableintegration where \spad{r} denotes the constant of integration. Thus \spad{integrate(a,{}[a0,{}a1,{}a2,{}...]) = [a,{}a0,{}a1/2,{}a2/3,{}...]}.
455544`\spad{integrate(c,{}f)} integrates with constant term \spad{c},{} this is simply the right shift
455706`\axiom{integrate(\spad{jv},{} \spad{i})} is like \axiom{integrateIfCan(\spad{jv},{} \spad{i})} but yields an error,{} if the integration is not possible.
455884`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
456011`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
456138`\spad{integrate(f(x),{}x)} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
456264`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
456385`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
456506`\spad{integrate(f(x),{}x)} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
456626`\spad{integrate(f,{} x = a..b)} returns the integral of \spad{f(x)dx} from a to \spad{b}. Error: if \spad{f} has a pole for \spad{x} between a and \spad{b}.
456831`\spad{integrate(f,{} x)} returns the integral of \spad{f(x)dx} where \spad{x} is viewed as a real variable.
456914`\spad{integrate(f,{} x = a..b)} returns the integral of \spad{f(x)dx} from a to \spad{b}. Error: if \spad{f} has a pole for \spad{x} between a and \spad{b}.
457173`\spad{integrate(f,{} x = a..b)} returns the integral of \spad{f(x)dx} from a to \spad{b}. Error: if \spad{f} has a pole for \spad{x} between a and \spad{b}.
457442`\spad{integrate(f,{} x)} returns the integral of \spad{f(x)dx} where \spad{x} is viewed as a real variable..
457599`\spad{integrate(p,{} d)} integrates the polynomial \spad{p} against the distribution \spad{d}. Same as \spad{eval(d,{} p)}.
457687`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
457935`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
458153`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
458401`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
458619`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
458866`\spad{integrate(f(x),{}y)} returns an anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{y}.
459083`\spad{integrate(p,{}x)} computes the integral of \spad{p*dx},{} \spadignore{i.e.} integrates the polynomial \spad{p} with respect to the variable \spad{x}.
459170`\spad{integrate(f,{}x)} returns the anti-derivative of the power series \spad{f(x)} with respect to the variable \spad{x} with constant coefficient 1. We may integrate a series when we can divide coefficients by integers.
459286`\spad{integrate(f(x))} returns an anti-derivative of the power series \spad{f(x)} with constant coefficient 0. We may integrate a series when we can divide coefficients by integers.
459412`\spad{integrate(s,{}v,{}c)} is the integral of \spad{s} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.}
459535`\spad{integrate(f,{} x = a..b,{} "noPole")} returns the integral of \spad{f(x)dx} from a to \spad{b}. If it is not possible to check whether \spad{f} has a pole for \spad{x} between a and \spad{b} (because of parameters),{} then this function will assume that \spad{f} has no such pole. Error: i
459535`--f \spad{f} has a pole for \spad{x} between a and \spad{b} or if the last argument is not "noPole".
459747`\spad{integrate(f,{} x = a..b,{} "noPole")} returns the integral of \spad{f(x)dx} from a to \spad{b}. If it is not possible to check whether \spad{f} has a pole for \spad{x} between a and \spad{b} (because of parameters),{} then this functio
459747`--n will assume that \spad{f} has no such pole. Error: if \spad{f} has a pole for \spad{x} between a and \spad{b} or if the last argument is not "noPole".
460013`\spad{integrate(f,{} x = a..b,{} "noPole")} returns the integral of \spad{f(x)dx} from a to \spad{b}. If it is not possible to check whether \spad{f} has a pole for \spad{x} between a and \spad{b} (because of parameters),{} then th
460013`--is function will assume that \spad{f} has no such pole. Error: if \spad{f} has a pole for \spad{x} between a and \spad{b} or if the last argument is not "noPole".
460289`\spad{integrate(s,{}v,{}c)} is the integral of \spad{s} with respect \indented{1}{to \spad{v} and having \spad{c} as the constant of integration.}
460389`\spad{intensity(v,{}i)} sets the intensity of the light source to \spad{i},{} for the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport}.
460455`\axiom{interReduce(\spad{lp})} returns \axiom{\spad{lq}} such that \axiom{\spad{lp}} and \axiom{\spad{lq}} generate the same ideal and no polynomial in \axiom{\spad{lq}} is reducuble by the others in the sense of Groebner bases. Since no assumptions are required the result may depend on the ordering the reductions are performed.
460539`\spad{internal?(x)} \undocumented
460591`\spad{internalAugment(lp,{}ts)} returns \spad{ts} if \spad{lp} is empty otherwise returns \spad{internalAugment(rest lp,{} internalAugment(first lp,{} ts))}
460676`\spad{internalAugment(p,{}ts)} assumes that \spad{augment(p,{}ts)} returns a singleton and returns it.
460755`\axiom{internalAugment(\spad{p},{}\spad{ts},{}\spad{b1},{}\spad{b2},{}\spad{b3},{}\spad{b4},{}\spad{b5})} is an internal subroutine,{} exported only for developement.
460872`\axiom{internalAugment(\spad{p},{}\spad{ts},{}\spad{b1},{}\spad{b2},{}\spad{b3},{}\spad{b4},{}\spad{b5})} is an internal subroutine,{} exported only for developement.
461961`\axiom{internalInfRittWu?(\spad{lp1},{}\spad{lp2})} is an internal subroutine,{} exported only for developement.
462055`\axiom{internalInfRittWu?(\spad{lp1},{}\spad{lp2})} is an internal subroutine,{} exported only for developement.
462159`\spad{internalIntegrate0 should} be a local function,{} but is conditional.
462262`\spad{internalIntegrate(f,{} x)} returns the integral of \spad{f(x)dx} where \spad{x} is viewed as a complex variable.
462364`\spad{internalIntegrate(f,{} x)} returns \spad{g} such that \spad{dg/dx = f}.
462504`\axiom{internalLastSubResultant(lpwt,{}\spad{v},{}flag)} is an internal subroutine,{} exported only for developement.
462663`\axiom{internalLastSubResultant(\spad{p1},{}\spad{p2},{}\spad{ts},{}inv?,{}break?)} is an internal subroutine,{} exported only for developement.
462800`\axiom{internalSubPolSet?(\spad{lp1},{}\spad{lp2})} returns \spad{true} iff \axiom{\spad{lp1}} is a sub-set of \axiom{\spad{lp2}} assuming that these lists are sorted increasingly \spad{w}.\spad{r}.\spad{t}. \axiomOpFrom{infRittWu?}{RecursivePolynomialCategory}.
462894`\axiom{internalSubPolSet?(\spad{lp1},{}\spad{lp2})} returns \spad{true} iff \axiom{\spad{lp1}} is a sub-set of \axiom{\spad{lp2}} assuming that these lists are sorted increasingly \spad{w}.\spad{r}.\spad{t}. \axiomOpFrom{infRittWu?}{RecursivePolynomialCategory}.
462998`\axiom{internalSubQuasiComponent?(\spad{ts},{}us)} returns a boolean \spad{b} value if the fact that the regular zero set of \axiom{us} contains that of \axiom{\spad{ts}} can be decided (and in that case \axiom{\spad{b}} gives this inclusion) otherwise returns \axiom{"failed"}.
463106`\axiom{internalSubQuasiComponent?(\spad{ts},{}us)} returns a boolean \spad{b} value if the fact the regular zero set of \axiom{us} contains that of \axiom{\spad{ts}} can be decided (and in that case \axiom{\spad{b}} gives this inclusion) otherwise returns \axiom{"failed"}.
463224`\axiom{internalZeroSetSplit(\spad{lp},{}\spad{b1},{}\spad{b2},{}\spad{b3})} is an internal subroutine,{} exported only for developement.
463333`\axiom{internalZeroSetSplit(\spad{lp},{}\spad{b1},{}\spad{b2},{}\spad{b3})} is an internal subroutine,{} exported only for developement.
463452`\spad{interpolate(lf,{}lg)} \undocumented
463557`\spad{interpolate(xlist,{} ylist,{} deg)} returns the rational function with numerator degree at most \spad{deg} and denominator degree at most \spad{\#xlist-deg-1} that interpolates the given points using fraction free arithmetic. Note that rational interpolation does not guarantee that all given points are interpolated correctly: unattainable points may make this impos
463557`--sible.
463691`\spad{interpolate(xlist,{} ylist,{} deg)} returns the rational function with numerator degree \spad{deg} that interpolates the given points using fraction free arithmetic.
463845`\spad{interpolate(u,{}lf,{}lg)} \undocumented
464104`\spad{interpretString(s)} treats a string as a piece of FriCAS input,{} by parsing and interpreting it.
464166`\spad{interpret(f)} passes \spad{f} to the interpreter,{} and transforms the result into an object of type \spad{R}.
464228`\spad{interpret(f)} passes \spad{f} to the interpreter.
464272`\spad{intersect(LI)} computes the intersection of the list of ideals \spad{LI}.
464726`\spad{intersect(lp,{}lts)} returns the same as \spad{concat([intersect(lp,{}ts) for ts in lts])|}
464817`\spad{intersect(lp,{}ts)} returns \spad{lts} a split of Lazard of the intersection of the affine variety associated with \spad{lp} and the regular zero set of \spad{ts}.
464902`\spad{intersect(p,{}lts)} returns the same as \spad{intersect([p],{}lts)}
464987`\spad{intersect(p,{}ts)} returns the same as \spad{intersect([p],{}ts)}
465066`\spad{intersect(u,{}v)} returns the set aggregate \spad{w} consisting of elements common to both set aggregates \spad{u} and \spad{v}. Note: equivalent to the notation (not currently supported) {\spad{x} for \spad{x} in \spad{u} | member?(\spad{x},{}\spad{v})}.
465118`\spad{intersect(I,{}J)} computes the intersection of the ideals \spad{I} and \spad{J}.
465192`\spad{intersection(x)} constructs a compound boundary from the intersection of \spad{x} x:List \% is supplied list of boundaries to form intersection.
465248`\spad{interval(f)} creates a new interval around \spad{f}.
465315`\spad{interval(f)} creates a new interval around \spad{f}.
465366`\spad{interval(inf,{}sup)} creates a new interval,{} either \axiom{[\spad{inf},{}\spad{sup}]} if \axiom{\spad{inf} \spad{<=} \spad{sup}} or \axiom{[\spad{sup},{}in]} otherwise.
465419`\spad{intoMatrix(lp,{}lvar)} converts linear in symbols \spad{lvar} system of polynomials \spad{lp} = 0 into matrix form
465720`\spad{inv x} \undocumented{} See \axiomFunFrom{inv}{DivisionRing}
465798`\spad{inv(x)} \undocumented
465880`\spad{inv(x)} \undocumented
465951`\spad{inv x} returns the multiplicative inverse of \spad{x}. Error: if \spad{x} is 0.
465991`\spad{inv(x)} returns the inverse of \spad{x}.
466024`\spad{inv(o)} returns the inverse of \spad{o} if it exists.
466083`\spad{inv(x)} returns the multiplicative inverse of \spad{x}.
466134`\spad{inverseColeman(alpha,{}beta,{}C)}: there is a bijection from the set of matrices having nonnegative entries and row sums {\em alpha},{} column sums {\em beta} to the set of {\em Salpha - Sbeta} double cosets of the symmetric group {\em Sn}. ({\em Salpha} is the Young subgroup corresponding to the improper partition {\em alpha}). For such a matrix \spad{C},{} inverseColeman
466134`--(\spad{alpha},{}\spad{beta},{}\spad{C}) calculates the lexicographical smallest {\em \spad{pi}} in the corresponding double coset. Note: the resulting permutation {\em \spad{pi}} of {\em {1,{}2,{}...,{}n}} is given in list form. Notes: the inverse of this map is {\em coleman}. For details,{} see James/Kerber.
466260`\spad{inverseIntegralMatrixAtInfinity()} returns \spad{M} such that \spad{M (v1,{}...,{}vn) = (1,{} y,{} ...,{} y^(n-1))} where \spad{(v1,{}...,{}vn)} is the local integral basis at infinity returned by \spad{infIntBasis()}.
466364`\spad{inverseIntegralMatrix()} returns \spad{M} such that \spad{M (w1,{}...,{}wn) = (1,{} y,{} ...,{} y^(n-1))} where \spad{(w1,{}...,{}wn)} is the integral basis of \spadfunFrom{integralBasis}{FunctionFieldCategory}.
466458`\spad{inverseLaplace(f,{} s,{} t)} returns the Inverse Laplace transform of \spad{f(s)} using \spad{t} as the new variable or "failed" if unable to find a closed form. Handles only rational \spad{f(s)}.
466553`\spad{inverse(ls)} forms the inverse of a sequence \spad{ls}
466613`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
466705`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square. Note: the result will have entries in the quotient field.
466814`\spad{inverse(m)} returns the inverse of the matrix. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
466913`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
466998`\spad{inverse(m)} returns the inverse of the matrix \spad{m},{} if that matrix is invertible and returns "failed" otherwise.
467094`\spad{inverse(m)} returns the inverse of the matrix \spad{m}. If the matrix is not invertible,{} "failed" is returned. Error: if the matrix is not square.
467163`\spad{invertIfCan(m)} returns the inverse of \spad{m} over \spad{R}
467275`\spad{invertible?(p,{}ts)} returns \spad{true} iff \spad{p} is invertible in the tower associated with \spad{ts}.
467355`\spad{invertible?(p,{}ts)} returns \spad{lbwt} where \spad{lbwt.i} is the result of \spad{invertibleElseSplit?(p,{}lbwt.i.tower)} and the list of the \spad{(lqrwt.i).tower} is a split of Kalkbrener of \spad{ts}.
467462`\spad{invertibleElseSplit?(p,{}ts)} returns \spad{true} (resp. \spad{false}) if \spad{p} is invertible in the tower associated with \spad{ts} or returns a split of Kalkbrener of \spad{ts}.
467567`\spad{invertibleSet(p,{}ts)} returns a split of Kalkbrener of the quotient ideal of the ideal \axiom{\spad{I}} by \spad{p} where \spad{I} is the radical of saturated of \spad{ts}.
467699`\spad{invmod(a,{}b)},{} \spad{0<=a<b>1},{} \spad{(a,{}b)=1} means \spad{1/a mod b}.
467752`\spad{invmultisect(a,{}b,{}st)} substitutes \spad{x^((a+b)*n)} for \spad{x^n} and multiplies by \spad{x^b}.
467931`\spad{invmultisect(a,{}b,{}f(x))} substitutes \spad{x^((a+b)*n)} \indented{1}{for \spad{x^n} and multiples by \spad{x^b}.}
468020`\spad{iomode(f)} returns the status of the file \spad{f}. The input/output status of \spad{f} may be "input",{} "output" or "closed" mode.
468075`\spad{ipow(l)} should be local but conditional.
468134`\axiom{iprint(\spad{s})} prints \axiom{\spad{s}} at the current position of the cursor.
468191`\spad{irootDep(k)} is a local function with a conditional implementation.
468324`\spad{iroot(p,{} n)} should be a non-exported function.
468411`\spad{irreducible?(p)} tests whether the polynomial \spad{p} is irreducible.
468482`\spad{irreducibleFactor(base,{}exponent)} creates a factored object with a single factor whose \spad{base} is asserted to be irreducible (flag = "irred").
468542`\axiom{irreducibleFactors(\spad{lp})} returns \axiom{\spad{lf}} such that if \axiom{\spad{lp} = [\spad{p1},{}...,{}\spad{pn}]} and \axiom{\spad{lf} = [\spad{f1},{}...,{}\spad{fm}]} then \axiom{p1*p2*...\spad{*pn=0}} means \axiom{f1*f2*...\spad{*fm=0}},{} and the \axiom{\spad{fi}} are irreducible over \axiom{\spad{R}} and are pairwise distinct.
468686`\spad{irreducibleRepresentation(lambda)} is the list of the two irreducible representations corresponding to the partition {\em lambda} in Young\spad{'s} natural form for the following two generators of the symmetric group,{} whose elements permute {\em {1,{}2,{}...,{}n}},{} namely {\em (1 2)} (2-cycle) and {\em (1 2 ... n)} (\spad{n}-cycle).
468785`\spad{irreducibleRepresentation(lambda,{}listOfPerm)} is the list of the irreducible representations corresponding to {\em lambda} in Young\spad{'s} natural form for the list of permutations given by {\em listOfPerm}.
468911`\spad{irreducibleRepresentation(lambda,{}\spad{pi})} is the irreducible representation corresponding to partition {\em lambda} in Young\spad{'s} natural form of the permutation {\em \spad{pi}} in the symmetric group,{} whose elements permute {\em {1,{}2,{}...,{}n}}.
469138`\spad{is?(op(a1,{}...,{}an),{} f)} tests if op = \spad{f}.
469194`\spad{is?(op(a1,{}...,{}an),{} s)} tests if the name of op is \spad{s}.
469243`\spad{is?([e1,{}...,{}en],{} pat)} tests if the list of expressions \spad{[e1,{}...,{}en]} matches the pattern pat.
469321`\spad{is?(expr,{} pat)} tests if the expression \spad{expr} matches the pattern pat.
469393`\spad{is?(x,{} op)} tests if \spad{x} is a kernel and is its operator is op.
469455`\spad{is?(x,{} s)} tests if \spad{x} is a kernel and is the name of its operator is \spad{s}.
469510`\spad{is?(op,{} s)} tests if the name of \spad{op} is \spad{s}.
469563`\spad{isAbsolutelyIrreducible?(aG)} calls {\em isAbsolutelyIrreducible?(aG,{}25)}. Note: the choice of 25 was rather arbitrary.
469667`\spad{isAbsolutelyIrreducible?(aG,{} numberOfTries)} uses Norton\spad{'s} irreducibility test to check for absolute irreduciblity,{} assuming if a one-dimensional kernel is found. As no field extension changes create "new" elements in a one-dimensional space,{} the criterium stays \spad{true} for every extension. The method looks for one-dimensionals only by creating random elements (no finge
469667`--rprints) since a run of {\em meatAxe} would have proved absolute irreducibility anyway.
469830`\spad{isAcyclic?(s)} returns \spad{true} if there are no loops
469884`\spad{isBasis? Lv} tests if a list of vector is a basis,{} without linear relations between the vectors.
469954`introspection: returns \spad{true} if this is a bound leaf node
470006`\spad{isBox?(n)} returns \spad{true} if this is a rectangular boundary.
470055`introspection: returns \spad{true} if this is a compound term containing two nodes
470203`\spad{isDirectSuccessor?(s,{} a,{} b)} is \spad{true} if \spad{'b'} is a direct successor of 'a' that is,{} if there is a direct arrow from 'a' to \spad{'b'}
470303`isDirected? is \spad{true} iff \% is domain consisting of directed graphs,{} \spad{false} for undirected graphs.
470356`isDirected? is \spad{true} iff \% is domain consisting of directed graphs,{} \spad{false} for undirected graphs.
470412`\spad{isEllipse?(n)} returns \spad{true} if this is a ellipsoid boundary.
470465`\spad{isExpt(p)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0},{} "failed" otherwise.
470799`\spad{isExpt(p)} returns \spad{[q,{} n]} if \spad{n > 0} and \spad{p = q ^ n},{} and "failed" otherwise.
470896`\spad{isExpt(p)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0}.
471012`\spad{isExpt(p)} returns \spad{[x,{} n]} if polynomial \spad{p} has the form \spad{x^n} and \spad{n > 0}.
471346`\spad{isExpt(p,{}op)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0} and \spad{x = op(a)}.
471471`\spad{isExpt(p,{}f)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0} and \spad{x = f(a)}.
471660`\spad{isFixPoint?(s,{} a)} is \spad{true} if 'a' has an arrow to itself
471734`introspection: returns \spad{true} if this is a unbound leaf node
471839`\spad{isFunctional?(s)} returns \spad{true} if \spad{s} is a functional graph,{} that is a directed graph in which each vertex has a single outgoing arrow.
471989`isGreaterThan?((\spad{s},{} a,{} \spad{b}) is \spad{true} if we can get from vertex 'a' to \spad{'b'} through a sequence of arrows but we can\spad{'t} go in the opposite direction from \spad{'b'} to 'a'
472085`returns \spad{true} if this is an \spad{I} combinator node
472137`returns \spad{true} if this is a \spad{K} combinator node
472189`introspection: returns \spad{true} if this is a lambda definition
472238`\spad{isList(p)} returns \spad{[a1,{}...,{}an]} if \spad{p = [a1,{}...,{}an]},{} "failed" otherwise.
472393`\spad{isMult(p)} returns \spad{[n,{} x]} if \spad{p = n * x} and \spad{n <> 0}.
472512`\spad{isNodeBranch?(n)} returns \spad{true} if this is a branch.
472575`\spad{isNodeLeaf?(n)} returns \spad{true} if this is a leaf which holds Scene nodes.
472636`\spad{isNull?(n)} returns \spad{true} if this is a null boundary,{} that is no boundary
472686`\spad{isOp(p)} returns \spad{[op,{} [a1,{}...,{}an]]} if \spad{p = op(a1,{}...,{}an)},{} and "failed" otherwise.
472776`\spad{isOp(p,{} op)} returns \spad{[a1,{}...,{}an]} if \spad{p = op(a1,{}...,{}an)},{} and "failed" otherwise.
472851`\spad{isPlus(p)} returns [\spad{m1},{}...,{}\spad{mn}] if \spad{p = m1 + ... + mn} and \spad{n > 1},{} "failed" otherwise.
473092`\spad{isPlus(p)} returns \spad{[a1,{}...,{}an]} if \spad{n > 1} \indented{1}{and \spad{p = a1 + ... + an},{}} and "failed" otherwise.
473155`\spad{isPlus(p)} returns \spad{[m1,{}...,{}mn]} if \spad{p = m1 +...+ mn} and \spad{n > 1}.
473247`\spad{isPlus(p)} returns \spad{[m1,{}...,{}mn]} if polynomial \spad{p = m1 + ... + mn} and \spad{n >= 2} and each \spad{mi} is a nonzero monomial.
473330`\spad{isPoint?(p)} returns \spad{true} if this represents a point as opposed to a vector.
473382`\spad{isPointLeaf?(n)} returns \spad{true} if this is a \spad{pt} node,{} that is,{} if this node contains points
473444`\spad{isPower(p)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0},{} "failed" otherwise.
473654`\spad{isPower(p)} returns \spad{[a,{} b]} if \spad{p = a ^ b},{} and "failed" otherwise.
473736`\spad{isPower(p)} returns \spad{[x,{} n]} if \spad{p = x^n} and \spad{n <> 0}.
473840`\spad{isQuotient(p)} returns \spad{[a,{} b]} if \spad{p = a / b},{} and "failed" otherwise.
473920`returns \spad{true} if this is a \spad{S} combinator node
473972`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if \spad{p = a1 ... an} and \spad{n > 1},{} "failed" otherwise.
474216`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if \spad{n > 1} and \spad{p = a1 * ... * an},{} and "failed" otherwise.
474280`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if \spad{p = a1*...*an} and \spad{n > 1}.
474366`\spad{isTimes(p)} returns \spad{[a1,{}...,{}an]} if polynomial \spad{p = a1 ... an} and \spad{n >= 2},{} and,{} for each \spad{i},{} \spad{ai} is either a nontrivial constant in \spad{R} or else of the form \spad{x^e},{} where \spad{e > 0} is an integer and \spad{x} in a member of VarSet.
474450`\spad{isVector? returns} \spad{true} if this represents a vector as opposed to a point.
474578`\spad{isobaric?(p)} returns \spad{true} if every differential monomial appearing in the differential polynomial \spad{p} has same weight,{} and returns \spad{false} otherwise.
474656`\spad{iter(f,{}n,{}x)} applies \spad{f n} times to \spad{x}.
474817`\axiom{iteratedInitials(\spad{p})} returns \axiom{[]} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns the list of the iterated initials of \axiom{\spad{p}}.
474898`\spad{jacobi2poly(aa,{} bb)} returns the stream of orthogonal polynomials corresponding to the Jacobi parameters \spad{a_n} and \spad{b_n}.
475002`\spad{jacobiCn(z,{} m)} is the Jacobi elliptic \spad{cn} function,{} defined by \spad{jacobiCn(z,{} m)^2 + jacobiSn(z,{} m)^2 = 1} and \spad{jacobiCn(0,{} m) = 1}
475065`\spad{jacobiCn(z,{} m)} expands the elliptic function \spad{cn} as a Taylor \indented{1}{series in \spad{z}.}
475154`\spad{jacobiCn(z,{} m)} is the Jacobi \spad{cn} function
475248`\spad{jacobiCn(z,{} m)} is the Jacobi \spad{cn} function
475315`\spad{jacobiCn(z,{} m)} is the Jacobi elliptic \spad{cn} function,{} defined by \spad{jacobiCn(z,{} m)^2 + jacobiSn(z,{} m)^2 = 1} and \spad{jacobiCn(0,{} m) = 1}.
475374`\spad{jacobiDn(z,{} m)} is the Jacobi elliptic \spad{dn} function,{} defined by \spad{jacobiDn(z,{} m)^2 + m*jacobiSn(z,{} m)^2 = 1} and \spad{jacobiDn(0,{} m) = 1}
475437`\spad{jacobiDn(z,{} m)} expands the elliptic function \spad{dn} as a Taylor \indented{1}{series in \spad{z}.}
475526`\spad{jacobiDn(z,{} m)} is the Jacobi \spad{dn} function
475620`\spad{jacobiDn(z,{} m)} is the Jacobi \spad{dn} function
475687`\spad{jacobiDn(z,{} m)} is the Jacobi elliptic \spad{dn} function,{} defined by \spad{jacobiDn(z,{} m)^2 + m*jacobiSn(z,{} m)^2 = 1} and \spad{jacobiDn(0,{} m) = 1}.
475746`\spad{jacobiIdentity?()} tests if \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.
475826`\spad{jacobiIdentity?()} tests if \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra. For example,{} this holds for crossed products of 3-dimensional vectors.
475995`\axiom{jacobiMatrix(\spad{sys})} constructs the Jacobi matrix of the family \axiom{\spad{sys}} of functions.
476091`\axiom{jacobiMatrix(De)} returns a list of Jacobi matrices sorted by the order of the equations.
476293`\axiom{jacobiMatrix(\spad{sys},{} jvars)} constructs the Jacobi matrix of the family \axiom{\spad{sys}} of functions. \axiom{jvars} contains for each function the effectively occurring jet variables. The columns of the matrix are ordered.
476404`\spad{jacobiParameters(x)} returns the pair of streams of Jacobi parameters of the distribution \spad{x}.
476563`\spad{jacobiParameters(x)} returns the pair of streams of Jacobi parameters of the distribution \spad{x}.
476670`\spad{motzkinPathArray([b0,{} b1,{} ...],{} [c1,{} c2,{} ...])} computes Flajolet\spad{'s} Motzkin path array from the sequences [1,{} 1,{} ...],{} \spad{bi},{} \spad{ci}.
476759`\spad{jacobiSn(z,{} m)} is the Jacobi elliptic \spad{sn} function,{} defined by the formula \spad{jacobiSn(ellipticF(z,{} m),{} m) = z}
476822`\spad{jacobiSn(z,{} m)} expands the elliptic function \spad{sn} as a Taylor \indented{1}{series in \spad{z}.}
476911`\spad{jacobiSn(z,{} m)} is the Jacobi \spad{sn} function
477005`\spad{jacobiSn(z,{} m)} is the Jacobi \spad{sn} function
477072`\spad{jacobiSn(z,{} m)} is the Jacobi elliptic \spad{sn} function,{} defined by the formula \spad{jacobiSn(ellipticF(z,{} m),{} m) = z}.
477131`\spad{jacobiTheta(q,{} z)} is the third Jacobi Theta function
477197`\spad{jacobiTheta(q,{} z)} is the third Jacobi Theta function.
477259`\spad{jacobiZeta(z,{} m)} is the Jacobi elliptic zeta function,{} defined by \spad{D(jacobiZeta(z,{} m),{} z) = jacobiDn(z,{} m)^2 - ellipticE(m)/ellipticK(m)} and \spad{jacobiZeta(0,{} m) = 0}.
477324`\spad{jacobiZeta(z,{} m)} is the Jacobi elliptic zeta function,{} defined by \spad{D(jacobiZeta(z,{} m),{} z) = jacobiDn(z,{} m)^2 - ellipticE(m)/ellipticK(m)} and \spad{jacobiZeta(0,{} m) = 0}.
477385`\spad{jacobi(a,{}b)} returns the Jacobi symbol \spad{J(a/b)}. When \spad{b} is odd,{} \spad{J(a/b) = product(L(a/p) for p in factor b )}. Note: by convention,{} 0 is returned if \spad{gcd(a,{}b) ~= 1}. Iterative \spad{O(log(b)^2)} version coded by Michael Monagan June 1987.
477462`\spad{jacobian(vf,{}xlist)} computes the jacobian,{} the matrix of first partial derivatives,{} of the vector field \spad{vf},{} \spad{vf} a vector function of the variables listed in \spad{xlist}.
477554`\spad{janko2 constructs} the janko group acting on the integers 1,{}...,{}100.
477630`\spad{janko2(\spad{li})} constructs the janko group acting on the 100 integers given in the list {\em \spad{li}}. Note: duplicates in the list will be removed. Error: if {\em \spad{li}} has less or more than 100 different entries
477719`\axiom{jetVariables(\spad{f})} yields all jet variables effectively occurring in \axiom{\spad{f}} in an ordered list.
477791`\axiom{join(\spad{de1},{} \spad{de2})} combines \axiom{\spad{de1}} and \axiom{\spad{de2}} to a single differential equation.
477852`\axiom{join(A,{} \spad{B})} vertically concats the matrices \axiom{A} and \axiom{\spad{B}}.
477908`\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \spadignore{i.e.} satisfies the Jordan identity. The property of \spadatt{CommutativeStar} follows from by definition.
477990`\spad{jordanAdmissible?()} tests if 2 is invertible in the coefficient domain and the multiplication defined by \spad{(1/2)(a*b+b*a)} determines a Jordan algebra,{} \spadignore{i.e.} satisfies the Jordan identity. The property of \spadatt{CommutativeStar} follows from by definition.
478069`\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \spad{(a*b)*a^2 - a*(b*a^2) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Jordan algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := (a@b+b@a)/2}.
478148`\spad{jordanAlgebra?()} tests if the algebra is commutative,{} characteristic is not 2,{} and \spad{(a*b)*a^2 - a*(b*a^2) = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jordan identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Jordan algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := (a@b+b@a)/2}.
478339`\spad{karatsubaDivide(p,{}n)} returns the same as \spad{monicDivide(p,{}monomial(1,{}n))}
478459`\spad{karatsuba(a,{}b)} returns \spad{a*b} by applying Karatsuba\spad{'s} trick once. The other multiplications are performed by calling \spad{*} from \spad{U}.
478543`\spad{karatsuba(a,{}b,{}l,{}k)} returns \spad{a*b} by applying Karatsuba\spad{'s} trick provided that both \spad{a} and \spad{b} have at least \spad{l} terms and \spad{k > 0} holds and by calling \spad{noKaratsuba} otherwise. The other multiplications are performed by recursive calls with the same third argument and \spad{k-1} as fourth argument.
478661`\spad{kelvinBei(v,{} z)} is the Kelvin bei function defined by equality \spad{kelvinBei(v,{} z) = imag(besselJ(v,{} exp(3*\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
478725`\spad{kelvinBei(v,{} z)} is the Kelvin bei function defined by equality \spad{kelvinBei(v,{} z) = imag(besselJ(v,{} exp(3*\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real.
478785`\spad{kelvinBer(v,{} z)} is the Kelvin ber function defined by equality \spad{kelvinBer(v,{} z) = real(besselJ(v,{} exp(3*\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
478849`\spad{kelvinBer(v,{} z)} is the Kelvin ber function defined by equality \spad{kelvinBer(v,{} z) = real(besselJ(v,{} exp(3*\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real.
478909`\spad{kelvinKei(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKei(v,{} z) = imag(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
478973`\spad{kelvinKei(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKei(v,{} z) = imag(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real.
479033`\spad{kelvinKer(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKer(v,{} z) = real(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real
479097`\spad{kelvinKer(v,{} z)} is the Kelvin kei function defined by equality \spad{kelvinKer(v,{} z) = real(exp(-v*\%pi*\%i/2)*besselK(v,{} exp(\%pi*\%i/4)*z))} for \spad{z} and \spad{v} real.
479157`\spad{kernel(x)} returns \spad{x} viewed as a kernel.
479321`\spad{kernel(op,{} [f1,{}...,{}fn])} constructs \spad{op(f1,{}...,{}fn)} without evaluating it.
479387`\spad{kernel(op,{} x)} constructs \spad{op}(\spad{x}) without evaluating it.
479447`\spad{kernel(op,{} [a1,{}...,{}an],{} m)} returns the kernel \spad{op(a1,{}...,{}an)} of nesting level \spad{m}. Error: if \spad{op} is \spad{k}-ary for some \spad{k} not equal to \spad{n}.
479583`\spad{kernels(f)} returns the list of all the top-level kernels appearing in \spad{f},{} but not the ones appearing in the arguments of the top-level kernels.
479705`\spad{key?(k,{}t)} tests if \spad{k} is a key in table \spad{t}.
479769`\spad{key(\spad{gi})} returns the process ID of the given graph,{} \spad{\spad{gi}},{} of the domain \spadtype{GraphImage}.
479812`\spad{key(v)} returns the process ID number of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport}.
479867`\spad{key(v)} returns the process ID number of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport}.
479983`\spad{keys(t)} returns the list the keys in table \spad{t}.
480045`\spad{kgraph(objs,{} arrowName)} constructs a graph with vertices (from \spad{objs}) and fully connected arrows,{} that is,{} each object has an arrow to every other object except itself. arrowName is a prefix for all arrow names,{} this will be followed by a number starting at 1 and incremented for each arrow
480102`\spad{kmax([k1,{}...,{}kn])} returns the top-level \spad{ki} for integration.
480172`\spad{knownInfBasis(n)} \undocumented{}
480269`\spad{kovacic(a_0,{}a_1,{}a_2)} returns either "failed" or \spad{P}(\spad{u}) such that \spad{\$e^{\int(-a_1/2a_2)} e^{\int u}\$} is a solution of \indented{5}{\spad{a_2 y'' + a_1 y' + a0 y = 0}} whenever \spad{u} is a solution of \spad{P u = 0}. The equation must be already irreducible over the rational functions.
480404`\spad{kovacic(a_0,{}a_1,{}a_2,{}ezfactor)} returns either "failed" or \spad{P}(\spad{u}) such that \spad{\$e^{\int(-a_1/2a_2)} e^{\int u}\$} is a solution of \indented{5}{\spad{\$a_2 y'' + a_1 y' + a0 y = 0\$}} whenever \spad{u} is a solution of \spad{P u = 0}. The equation must be already irreducible over the rational functions. Argument \spad{ezfact
480404`--or} is a factorisation in \spad{UP},{} not necessarily into irreducibles.
480558`Undocumented.
480640`Undocumented.
480704`\spad{kroneckerDelta()} is the rank 2 tensor defined by \indented{3}{\spad{kroneckerDelta()(i,{}j)}} \indented{6}{\spad{= 1\space{2}if i = j}} \indented{6}{\spad{= 0 if\space{2}i \~= j}}
480837`\spad{kroneckerProduct([a1,{} a2,{} ...,{} an])} calculates the Kronecker product of the matrices \spad{a1},{} \spad{a2},{} ...,{} an. This corresponds to tensor product of corresponding operators.
480973`\spad{kroneckerProduct(a,{}b)} calculates the Kronecker product of the matrices a and \spad{b}. This corresponds to tensor product of corresponding operators.
481106`\spad{kroneckerSum([a1,{} a2,{} ...,{} an])} calculates the Kronecker sum of the matrices \spad{a1},{} \spad{a2},{} ...,{} an.
481234`\spad{kroneckerSum(a,{}b)} calculates the Kronecker sum of the matrices a and \spad{b}.
481463`Should be local but conditional.
481631`\spad{ksec(k,{} [k1,{}...,{}kn],{} x)} returns the second top-level \spad{ki} after \spad{k} involving \spad{x}.
481718`\spad{kummerM(a,{} b,{} z)} is the Kummer \spad{M} function
481782`\spad{kummerM(mu,{} nu,{} z)} is the Kummer \spad{M} function.
481843`\spad{kummerU(a,{} b,{} z)} is the Kummer \spad{U} function
481907`\spad{kummerU(mu,{} nu,{} z)} is the Kummer \spad{U} function.
481968`\spad{lSpaceBasis(d)} returns a basis for \spad{L(d) = {f | (f) >= -d}} as a module over \spad{K[x]}.
482037`\spad{label(n,{}f)} gives form \spad{f} an equation label \spad{n}.
482080`\spad{lagrange(g)} produces the power series for \spad{f} where \spad{f} is implicitly defined as \spad{f(z) = z*g(f(z))}.
482219`\spad{lagrange(g(x))} produces the Taylor series for \spad{f(x)} \indented{1}{where \spad{f(x)} is implicitly defined as \spad{f(x) = x*g(f(x))}.}
482288`\spad{laguerreL(n,{}x)} is the \spad{n}-th Laguerre polynomial,{} \spad{L[n](x)}. These are defined by \spad{exp(-t*x/(1-t))/(1-t) = sum(L[n](x)*t^n/n!,{} n = 0..)}.
482371`\spad{laguerreL(m,{}n,{}x)} is the associated Laguerre polynomial,{} \spad{L<m>[n](x)}. This is the \spad{m}-th derivative of \spad{L[n](x)}.
482473`\spad{laguerre(n)} returns the \spad{n}th Laguerre polynomial \spad{L[n](x)}. Note: Laguerre polynomials,{} denoted \spad{L[n](x)},{} are computed from the two term recurrence. The generating function is: \spad{exp(x*t/(t-1))/(1-t) = sum(L[n](x)*t^n/n!,{} n=0..infinity)}.
482575`Constructs a reference to a bound variable from its deBruijn index
482632`Constructs a reference to a free variable
482673`\spad{lambda(code,{} [x1,{}...,{}xn])} returns the input form corresponding to \spad{(x1,{}...,{}xn) +-> code} if \spad{n > 1},{} or to \spad{x1 +-> code} if \spad{n = 1}.
482726`Constructs lambda term and bind any variables with the name provided
482770`Constructs a node containing multiple terms
482814`\spad{lambertW0(x)} applies lambertW to \spad{x}. \spad{x} must have no term of order 0.
482897`\spad{lambertW(x)} is the Lambert \spad{W} function at \spad{x}
482958`\spad{lambertW(x)} is the Lambert \spad{W} function.
483048`\spad{lambertW(x)} is the Lambert \spad{W} function.
483126`\spad{lambertW(x)} is the Lambert \spad{W} function.
483198`\spad{lambertW(x)} is the Lambert \spad{W} function.
483258`\spad{lambertW(z)} = \spad{w} is the principial branch of the solution to the equation \spad{we^w = z}.
483314`Undocumented
483439`Undocumented
483546`Undocumented
483644`Undocumented
483724`Undocumented
483858`Undocumented
483968`Undocumented
484075`Undocumented
484158`Undocumented
484292`Undocumented
484402`Undocumented
484509`Undocumented
484592`\spad{lambert(st)} computes \spad{f(x) + f(x^2) + f(x^3) + ...} if \spad{st} is a stream representing \spad{f(x)}. This function is used for computing infinite products. If \spad{f(x)} is a power series with constant coefficient 1 then \spad{prod(f(x^n),{}n = 1..infinity) = exp(lambert(log(f(x))))}.
484729`\spad{lambert(f(x))} returns \spad{f(x) + f(x^2) + f(x^3) + ...}. \indented{1}{This function is used for computing infinite products.} \indented{1}{\spad{f(x)} should have zero constant coefficient.} \indented{1}{If \spad{f(x)} is a Taylor series with constant term 1,{} then} \indented{1}{\spad{product(n = 1..infinity,{}f(x^n)) = exp(log(lambert(f(x))))}.}
484797`Undocumented.
484902`Undocumented.
484980`Undocumented.
485091`Undocumented.
485175`Undocumented.
485264`Undocumented.
485335`\spad{laplace(f,{} t,{} s)} returns the Laplace transform of \spad{f(t)} using \spad{s} as the new variable. This is \spad{integral(exp(-s*t)*f(t),{} t = 0..\%plusInfinity)}. Returns the formal object \spad{laplace(f,{} t,{} s)} if it cannot compute the transform.
485464`\spad{laplacianMatrix(s)} returns matrix also known as "Kirchhoff matrix" or "Admittance matrix" where: entry [\spad{i},{}\spad{j}] = inDegree(\spad{vi}) if \spad{i} = \spad{j} (number of incoming links) \spad{-1} if \spad{i} not = \spad{j} and \spad{vi} is adjacent to \spad{vj} 0 otherwise Alternatively this is defined as \spad{D} - A,{} where \spad{D} is the diagonal degree matrix. It contains both adjacency information and degree inf
485464`--ormation. There are other,{} similar matrices,{} that are also called "Laplacian matrices" of a graph.
485531`\spad{laplacian(v,{}xlist)} computes the laplacian of the scalar field \spad{v},{} \spad{v} a function of the variables listed in \spad{xlist}.
485613`\spad{largest l} returns the largest element of \spad{l} where the partial ordering induced by setOrder is completed into a total one by the ordering on \spad{S}.
485695`\spad{largest(l,{} fn)} returns the largest element of \spad{l} where the partial ordering induced by setOrder is completed into a total one by \spad{fn}.
485775`\spad{lastSubResultantElseSplit(p1,{}p2,{}ts)} returns either \spad{g} a quasi-monic \spad{gcd} of \spad{p1} and \spad{p2} \spad{w}.\spad{r}.\spad{t}. the \spad{ts} or a split of Kalkbrener of \spad{ts}. This assumes that \spad{p1} and \spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \spad{ts}.
485881`\axiom{lastSubResultantEuclidean(\spad{P},{} \spad{Q})} computes the last non zero subresultant \axiom{\spad{S}} and carries out the equality \axiom{coef1*P + coef2*Q = \spad{S}}.
486014`\axiom{lastSubResultant(a,{}\spad{b})} returns \axiom{resultant(a,{}\spad{b})} if \axiom{a} and \axiom{\spad{b}} has no non-trivial \spad{gcd} in \axiom{\spad{R^}(\spad{-1}) \spad{P}} otherwise the non-zero sub-resultant with smallest index.
486111`\axiom{lastSubResultant(\spad{P},{} \spad{Q})} computes the last non zero subresultant of \axiom{\spad{P}} and \axiom{\spad{Q}}
486192`\axiom{lastSubResultant(a,{}\spad{b})} returns the last non-zero subresultant of \axiom{a} and \axiom{\spad{b}} where \axiom{a} and \axiom{\spad{b}} are assumed to have the same main variable \axiom{\spad{v}} and are viewed as univariate polynomials in \axiom{\spad{v}}.
486291`\spad{lastSubResultant(p1,{}p2,{}ts)} returns \spad{lpwt} such that \spad{lpwt.i.val} is a quasi-monic \spad{gcd} of \spad{p1} and \spad{p2} \spad{w}.\spad{r}.\spad{t}. \spad{lpwt.i.tower},{} for every \spad{i},{} and such that the list of the \spad{lpwt.i.tower} is a split of Kalkbrener of \spad{ts}. Moreover,{} if \spad{p1} and \spad{p2} do not have a non-trivial \spad{gcd} \spad{w}.\spad{r}.\s
486291`--pad{t}. \spad{lpwt.i.tower} then \spad{lpwt.i.val} is the resultant of these polynomials \spad{w}.\spad{r}.\spad{t}. \spad{lpwt.i.tower}. This assumes that \spad{p1} and \spad{p2} have the same maim variable and that this variable is greater that any variable occurring in \spad{ts}.
486568`\spad{last(l)} returns the last element of a doubly-linked aggregate \spad{l}. Error: if \spad{l} is empty.
486620`\spad{last(u)} resturn the last element of \spad{u}. Note: for lists,{} \axiom{last(\spad{u}) = \spad{u} . (maxIndex \spad{u}) = \spad{u} . (\# \spad{u} - 1)}.
486674`\axiom{last(\spad{ts})} returns the polynomial of \axiom{\spad{ts}} with smallest main variable if \axiom{\spad{ts}} is not empty,{} otherwise returns \axiom{"failed"}.
486884`\spad{last(u,{}n)} returns a copy of the last \spad{n} (\axiom{\spad{n} \spad{>=} 0}) nodes of \spad{u}. Note: \axiom{last(\spad{u},{}\spad{n})} is a list of \spad{n} elements.
487046`\spad{latex(s)} returns a LaTeX-printable output representation of \spad{s}.
487091`\spad{laurentIfCan(f(x))} converts the Puiseux series \spad{f(x)} to a Laurent series if possible. If this is not possible,{} "failed" is returned.
487197`\spad{laurentRep(f(x))} returns \spad{g(x)} where the Puiseux series \spad{f(x) = g(x^r)} is represented by \spad{[r,{}g(x)]}.
487285`\spad{laurent(f)} returns a Laurent expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
487357`\spad{laurent(x)} returns \spad{x} viewed as a Laurent series.
487433`\spad{laurent(f(x))} converts the Puiseux series \spad{f(x)} to a Laurent series if possible. Error: if this is not possible.
487518`\spad{laurent(f,{}x = a)} expands the expression \spad{f} as a Laurent series in powers of \spad{(x - a)}.
487603`\spad{laurent(f,{}n)} returns a Laurent expansion of the expression \spad{f}. Note: \spad{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 \spad{n}.
487683`\spad{laurent(n,{}f(x))} returns \spad{x^n * f(x)}.
487776`\spad{laurent(n +-> a(n),{}x = a,{}n0..)} returns \spad{sum(n = n0..,{}a(n) * (x - a)^n)}; \spad{laurent(n +-> a(n),{}x = a,{}n0..n1)} returns \spad{sum(n = n0..n1,{}a(n) * (x - a)^n)}.
487894`\spad{laurent(f,{}x = a,{}n)} expands the expression \spad{f} as a Laurent series in powers of \spad{(x - a)}; terms will be computed up to order at least \spad{n}.
487987`\spad{laurent(a(n),{}n,{}x=a,{}n0..)} returns \spad{sum(n = n0..,{}a(n) * (x - a)^n)}; \spad{laurent(a(n),{}n,{}x=a,{}n0..n1)} returns \spad{sum(n = n0..n1,{}a(n) * (x - a)^n)}.
488101`\spad{lazy?(s)} returns \spad{true} if the first node of the stream \spad{s} is a lazy evaluation mechanism which could produce an additional entry to \spad{s}.
488158`\spad{lazyEvaluate(s)} causes one lazy evaluation of stream \spad{s}. Caution: the first node must be a lazy evaluation mechanism (satisfies \spad{lazy?(s) = true}) as there is no error check. Note: a call to this function may or may not produce an explicit first entry
488217`\spad{lazyGintegrate(f,{}r,{}g)} is used for fixed point computations.
488332`\spad{lazyIntegrate(r,{}f)} is a local function used for fixed point computations.
488454`\spad{lazyIntegrate(c,{}f)} integrates with constant term \spad{c},{} this is simply the right shift
488560`\axiom{lazyIrreducibleFactors(\spad{lp})} returns \axiom{\spad{lf}} such that if \axiom{\spad{lp} = [\spad{p1},{}...,{}\spad{pn}]} and \axiom{\spad{lf} = [\spad{f1},{}...,{}\spad{fm}]} then \axiom{p1*p2*...\spad{*pn=0}} means \axiom{f1*f2*...\spad{*fm=0}},{} and the \axiom{\spad{fi}} are irreducible over \axiom{\spad{R}} and are pairwise distinct. The algor
488560`--ithm tries to avoid factorization into irreducible factors as far as possible and makes previously use of \spad{gcd} techniques over \axiom{\spad{R}}.
488773`\axiom{lazyPquo(a,{}\spad{b})} returns the polynomial \axiom{\spad{q}} such that \axiom{lazyPseudoDivide(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]}.
488910`\axiom{lazyPquo(a,{}\spad{b},{}\spad{v})} returns the polynomial \axiom{\spad{q}} such that \axiom{lazyPseudoDivide(a,{}\spad{b},{}\spad{v})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]}.
489106`\axiom{lazyPremWithDefault(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{r}]} such that \axiom{\spad{r} = lazyPrem(a,{}\spad{b})} and \axiom{(\spad{c^g})\spad{*r} = prem(a,{}\spad{b})}.
489362`\axiom{lazyPremWithDefault(a,{}\spad{b},{}\spad{v})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{r}]} such that \axiom{\spad{r} = lazyPrem(a,{}\spad{b},{}\spad{v})} and \axiom{(\spad{c^g})\spad{*r} = prem(a,{}\spad{b},{}\spad{v})}.
489559`\axiom{lazyPrem(a,{}\spad{b})} returns the polynomial \axiom{\spad{r}} reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} and such that \axiom{\spad{b}} divides \axiom{init(\spad{b})^e a - \spad{r}} where \axiom{\spad{e}} is the number of steps of this pseudo-division.
489696`\axiom{lazyPrem(a,{}\spad{b},{}\spad{v})} returns the polynomial \axiom{\spad{r}} reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} viewed as univariate polynomials in the variable \axiom{\spad{v}} such that \axiom{\spad{b}} divides \axiom{init(\spad{b})^e a - \spad{r}} where \axiom{\spad{e}} is the number of steps of this pseudo-division.
489900`\axiom{lazyPseudoDivide(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]} such that \axiom{\spad{c^n} * a = \spad{q*b} \spad{+r}} and \axiom{lazyResidueClass(a,{}\spad{b})} returns \axiom{[\spad{r},{}\spad{c},{}\spad{n}]} where \axiom{\spad{n} + \spad{g} = max(0,{} degree(\spad{b}) - degree(a) + 1)}.
490036`\axiom{lazyPseudoDivide(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]} such that \axiom{[\spad{c},{}\spad{g},{}\spad{r}] = lazyPremWithDefault(a,{}\spad{b})} and \axiom{\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.
490309`\axiom{lazyPseudoDivide(a,{}\spad{b},{}\spad{v})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]} such that \axiom{\spad{r} = lazyPrem(a,{}\spad{b},{}\spad{v})},{} \axiom{(\spad{c^g})\spad{*r} = prem(a,{}\spad{b},{}\spad{v})} and \axiom{\spad{q}} is the pseudo-quotient computed in this lazy pseudo-division.
490450`\axiom{lazyPseudoQuotient(a,{}\spad{b})} returns \axiom{\spad{q}} if \axiom{lazyPseudoDivide(a,{}\spad{b})} returns \axiom{[\spad{c},{}\spad{g},{}\spad{q},{}\spad{r}]}
490528`\axiom{lazyPseudoRemainder(a,{}\spad{b})} returns \axiom{\spad{r}} if \axiom{lazyResidueClass(a,{}\spad{b})} returns \axiom{[\spad{r},{}\spad{c},{}\spad{n}]}. This lazy pseudo-remainder is computed by means of the \axiomOpFrom{fmecg}{NewSparseUnivariatePolynomial} operation.
490729`\axiom{lazyResidueClass(a,{}\spad{b})} returns \axiom{[\spad{r},{}\spad{c},{}\spad{n}]} such that \axiom{\spad{r}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} and \axiom{\spad{b}} divides \axiom{\spad{c^n} * a - \spad{r}} where \axiom{\spad{c}} is \axiom{leadingCoefficient(\spad{b})} and \axiom{\spad{n}} is as small as possible with the previous properties.
490854`\axiom{lazyResidueClass(a,{}\spad{b})} returns \axiom{[\spad{p},{}\spad{q},{}\spad{n}]} where \axiom{\spad{p} / \spad{q^n}} represents the residue class of \axiom{a} modulo \axiom{\spad{b}} and \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} and \axiom{\spad{q}} is \axiom{init(\spad{b})}.
490982`\axiom{lazyVariations(\spad{l},{}\spad{s1},{}\spad{sn})} is the number of sign variations in the list of non null numbers [s1::l]\spad{@sn},{}
491138`left contraction inner product
491268`\spad{lcmCoef(c1,{} c2)} computes (llcm_res,{} \spad{coeff1},{} \spad{coeff2}) such that llcm_res is least left common multiple of \spad{c1} and \spad{c2} and llcm_res = \spad{coeff1*c1} = \spad{coeff2*c2}
491391`\spad{lcm(l)} returns the least common multiple of the elements of the list \spad{l}.
491470`\spad{lcm(x,{}y)} returns the least common multiple of \spad{x} and \spad{y}.
491510`\spad{lcm(a,{} lo,{} \spad{hi},{} p)} computes \spad{lcm} of elements a(\spad{lo}),{} a(\spad{lo+1}),{} ...,{} a(\spad{hi}).
491621`\spad{lcx0(x,{} lv)} computes leading coefficient of \spad{x} and corresponding product of variables (monomial with coefficient 1) with respect to variables in \spad{lv} Variables in \spad{lv} must be decreasing and bigger than all other variables of \spad{x}.
491820`\spad{lcz(x,{} z)} computes leading coefficient and degree of \spad{x} with respect to variable \spad{z}.
492036`\spad{leader(p)} returns the derivative of the highest rank appearing in the differential polynomial \spad{p} Note: an error occurs if \spad{p} is in the ground ring.
492105`\spad{leadingBasisTerm(p)} returns the leading basis term of antisymmetric polynomial \spad{p}.
492162`\spad{leadingBasisTerm(df)} returns the leading basis term of differential form \spad{df}.
492237`\spad{leadingCoefficientRicDE(op)} returns \spad{[[m1,{} p1],{} [m2,{} p2],{} ... ,{} [mk,{} pk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \spad{op y = 0} must have degree \spad{mj} for some \spad{j},{} and its leading coefficient is then a zero of \spad{pj}. In addition,{}\spad{m1>m2> ... >mk}.
492508`\spad{leadingCoefficient(df)} returns the leading coefficient of differential form \spad{df}.
492603`\spad{leadingCoefficient(p)} returns the coefficient of the highest degree term of \spad{p}.
492691`\spad{leadingCoefficient(p)} returns the leading coefficient of antisymmetric polynomial \spad{p}.
492749`\spad{leadingCoefficient(x)} \undocumented
492830`\spad{leadingCoefficient }\undocumented
492895`\spad{leadingCoefficient(z)} returns the coefficient of the leading (with respect to the ordering on the indexing set) monomial of \spad{z}. Error: if \spad{z} has no support.
492970`\spad{leadingCoefficient(f)} returns the coefficient of the lowest order term of \spad{f}
493050`\spad{leadingCoefficient(p)} returns the coefficient highest degree term of \spad{p}.
493114`\spad{leadingCoefficient(x)} returns the first coefficient which appears in \spad{listOfTerms(x)}.
493179`\spad{leadingCoefficient(l)} is \spad{a(n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
493247`\spad{leadingCoefficient(f)} gives the coefficient of \spad{f},{} whose corresponding monoid element is the greatest among all those with non-zero coefficients.
493329`\spad{leadingCoefficient(l)} is \spad{a(n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
493481`\axiom{leadingCoefficient(\spad{p},{}\spad{v})} returns the leading coefficient of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as A univariate polynomial in \axiom{\spad{v}}.
493621`\axiom{leadingDer(fun)} yields the leading derivative of \axiom{fun}. If \axiom{fun} contains no derivatives \axiom{1} is returned.
493685`\spad{leadingExponent(x)} \undocumented
493763`\spad{leadingIdeal(I)} is the ideal generated by the leading terms of the elements of the ideal \spad{I}.
493837`\spad{leadingIndex(x)} \undocumented
493990`\spad{leadingMonomial(x)} \undocumented
494090`\spad{leadingMonomial(p)} returns the monomial of \spad{p} with the highest degree.
494176`\spad{leadingMonomial(f)} gives the monomial of \spad{f} whose corresponding monoid element is the greatest among all those with non-zero coefficients.
494255`\spad{leadingMonomial(x)} returns the first element from \spad{S} which appears in \spad{listOfTerms(x)}.
494317`\spad{leadingMonomial(p)} returns the monomial of \spad{p} with the highest degree.
494379`\spad{leadingMonomial(f)} returns the monomial of \spad{f} of lowest order.
494454`\spad{leadingSupport(z)} returns the index of leading (with respect to the ordering on the indexing set) monomial of \spad{z}. Error: if \spad{z} has no support.
494525`\spad{leadingTerm(x)} returns the leading (with respect to the ordering on the indexing set) term of \spad{z}. Error: if \spad{z} has no support.
494778`\spad{leaf?(x)} \undocumented
494826`\spad{leaf?(u)} tests if \spad{u} is a terminal node.
494882`\spad{leastAffineMultiple(f)} computes the least affine polynomial which is divisible by the polynomial \spad{f} over the finite field {\em GF},{} \spadignore{i.e.} a polynomial whose exponents are 0 or a power of \spad{q},{} the size of {\em GF}.
495082`\axiom{leastMonomial(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns \axiom{1},{} otherwise,{} the monomial of \axiom{\spad{p}} with lowest degree,{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
495154`\spad{leastPower(p,{}n)} returns \spad{e},{} where \spad{e} is the smallest integer such that \spad{p ^e >= n}
495326`\spad{leaves(t)} returns the list of values in obtained by visiting the nodes of tree \axiom{\spad{t}} in left-to-right order.
495383`\spad{leftAlternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
495464`\spad{leftAlternative?()} tests if \spad{2*associator(a,{}a,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
495653`\spad{leftCharacteristicPolynomial(a)} returns the characteristic polynomial of the left regular representation of \spad{a} with respect to any basis.
495767`\spad{leftDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(leftTraceMatrix())}.
495838`\spad{leftDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(leftTraceMatrix())}.
496060`\spad{leftDiscriminant([v1,{}...,{}vn])} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj}. Note: the same as \spad{determinant(leftTraceMatrix([v1,{}...,{}vn]))}.
496142`\spad{leftDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``left division\spad{''}.
496241`\spad{leftDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``left division\spad{''}.
496356`\spad{leftDivide(a,{} b,{} sigma)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``left division\spad{''}. \spad{\sigma} is the morphism to use.
496488`\spad{leftExactQuotient(a,{}b)} computes the value \spad{q},{} if it exists,{} \indented{1}{such that \spad{a = b*q}.}
496672`\spad{leftExactQuotient(a,{}b)} computes the value \spad{q},{} if it exists,{} \indented{1}{such that \spad{a = b*q}.}
496887`\spad{leftExtendedGcd(a,{}b)} returns \spad{[c,{}d]} such that \spad{g = a * c + b * d = leftGcd(a,{} b)}.
497013`\spad{leftFactorIfCan(f,{}h)} returns the left factor (\spad{g} in \spad{f} = \spad{g} \spad{o} \spad{h}) of the functional decomposition of the polynomial \spad{f} with given \spad{h} or \spad{"failed"} if \spad{g} does not exist.
497118`\spad{leftFactor(p,{}q)} \undocumented
497201`\spad{leftGcd(a,{}b)} computes the value \spad{g} of highest degree such that \indented{3}{\spad{a = aa*g}} \indented{3}{\spad{b = bb*g}} for some values \spad{aa} and \spad{bb}. The value \spad{g} is computed using left-division.
497333`\spad{leftGcd(a,{}b)} computes the value \spad{g} of highest degree such that \indented{3}{\spad{a = g*aa}} \indented{3}{\spad{b = g*bb}} for some values \spad{aa} and \spad{bb}. The value \spad{g} is computed using left-division.
497415`\spad{leftLcm(a,{}b)} computes the value \spad{m} of lowest degree such that \spad{m = a*aa = b*bb} for some values \spad{aa} and \spad{bb}. The value \spad{m} is computed using left-division.
497547`\spad{leftLcm(a,{}b)} computes the value \spad{m} of lowest degree such that \spad{m = aa*a = bb*b} for some values \spad{aa} and \spad{bb}. The value \spad{m} is computed using right-division.
497733`\spad{leftMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of left powers of \spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.
497861`\spad{leftMult(s,{} a)} returns \spad{s * a} where \spad{*} is the monoid operation,{} which is assumed non-commutative.
497980`\spad{leftNorm(a)} returns the determinant of the left regular representation of \spad{a}.
498046`\spad{leftOne(eq)} divides by the left hand side.
498117`\spad{leftOne(eq)} divides by the left hand side.
498364`\spad{leftPower(a,{}n)} returns the \spad{n}\spad{-}th left power of \spad{a},{} \spadignore{i.e.} \spad{leftPower(a,{}n) := a * leftPower(a,{}n-1)} and \spad{leftPower(a,{}0) := 1}.
498430`\spad{leftPower(a,{}n)} returns the \spad{n}\spad{-}th left power of \spad{a},{} \spadignore{i.e.} \spad{leftPower(a,{}n) := a * leftPower(a,{}n-1)} and \spad{leftPower(a,{}1) := a}.
498485`\spad{leftQuotient(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{q} is returned.
498627`\spad{leftQuotient(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{q} is returned.
498714`\spad{leftRankPolynomial()} returns the left minimimal polynomial of the generic element
498867`\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
498980`\spad{leftRankPolynomial()} calculates the left minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
499102`\spad{leftRank(x)} determines the number of linearly independent elements in \spad{x*b1},{}...,{}\spad{x*bn},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
499249`\spad{leftRecip(a)} returns an element,{} which is a left inverse of \spad{a},{} or \spad{"failed"} if there is no unit element,{} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
499354`\spad{leftRecip(a)} returns an element,{} which is a left inverse of \spad{a},{} or \spad{"failed"} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
499501`\spad{leftRegularRepresentation(a)} returns the matrix of the linear map defined by left multiplication by \spad{a} with respect to the fixed \spad{R}-module basis.
499780`\spad{leftRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix \spad{m} of the linear map defined by left multiplication by \spad{a} with respect to the \spad{R}-module basis \spad{b = [v1,{}...,{}vn]}. That is for all \spad{x} \spad{coordinates(a*x,{} b) = m*coordinates(x,{} b)}.
499882`\spad{leftRemainder(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{r} is returned.
500026`\spad{leftRemainder(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{r} is returned.
500114`\spad{leftScalarTimes!(c,{}r,{}a)} computes the scalar product \spad{r * a} and stores the result in the matrix \spad{c}. Error: if \spad{a} and \spad{c} do not have the same dimensions.
500216`\spad{leftTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
500294`\spad{leftTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by left trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
500537`\spad{leftTraceMatrix([v1,{}...,{}vn])} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the left trace of the product \spad{vi*vj}.
500690`\spad{leftTrace(a)} returns the trace of the left regular representation of \spad{a}.
500757`\spad{leftTrim(s,{}c)} returns \spad{s} with all leading characters \spad{c} deleted. For example,{} \axiom{leftTrim(" abc ",{} char " ")} returns \axiom{"abc "}.
500815`\spad{leftTrim(s,{}cc)} returns \spad{s} with all leading characters in \spad{cc} deleted. For example,{} \axiom{leftTrim("(abc)",{} charClass "()")} returns \axiom{"abc)"}.
500952`\spad{leftUnit()} returns a left unit of the algebra (not necessarily unique),{} or \spad{"failed"} if there is none.
501162`\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \spad{"failed"} if there is none
501286`\spad{leftUnits()} returns the affine space of all left units of the algebra,{} or \spad{"failed"} if there is none.
501423`\spad{leftZero(eq)} subtracts the left hand side.
501486`\axiom{left(\spad{x})} returns left subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{LyndonWord}(\axiom{\spad{x}}) is \spad{true}.
501533`\axiom{left(\spad{x})} returns left subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true}.
501575`\spad{left(f)} left-justifies form \spad{f} in total space.
501614`\axiom{left(rootChar)} is the left bound of the isolating interval
501706`\spad{left(u)} returns the left child.
501762`\spad{left(f,{}n)} left-justifies form \spad{f} within space of width \spad{n}.
501809`\spad{legendreP(n,{}x)} is the \spad{n}-th Legendre polynomial,{} \spad{P[n](x)}. These are defined by \spad{1/sqrt(1-2*x*t+t^2) = sum(P[n](x)*t^n,{} n = 0..)}.
501925`\spad{legendreP(nu,{} mu,{} z)} is the Legendre \spad{P} function
501991`\spad{legendreP(nu,{} mu,{} z)} is the Legendre \spad{P} function.
502054`\spad{legendreQ(nu,{} mu,{} z)} is the Legendre \spad{Q} function
502120`\spad{legendreQ(nu,{} mu,{} z)} is the Legendre \spad{Q} function.
502183`\spad{legendre(n)} returns the \spad{n}th Legendre polynomial \spad{P[n](x)}. Note: Legendre polynomials,{} denoted \spad{P[n](x)},{} are computed from the two term recurrence. The generating function is: \spad{1/sqrt(1-2*t*x+t^2) = sum(P[n](x)*t^n,{} n=0..infinity)}.
502295`\spad{legendre(a,{}p)} returns the Legendre symbol \spad{L(a/p)}. \spad{L(a/p) = (-1)^((p-1)/2) mod p} (\spad{p} prime),{} which is 0 if \spad{a} is 0,{} 1 if \spad{a} is a quadratic residue \spad{mod p} and \spad{-1} otherwise. Note: because the primality test is expensive,{} if it is known that \spad{p} is prime then use \spad{jacobi(a,{}p)}.
502418`\spad{length(p)} returns the sum of the absolute values of the coefficients of the polynomial \spad{p}.
502489`\spad{length(x)} returns the length of \spad{x}.
502556`\spad{length([l1]*[l2]*...[ln])} returns the length of the word \spad{l1*l2*...*ln}.
502642`\spad{length(x)} returns the number of elements in tuple \spad{x}
502697`\axiom{length(\spad{x})} returns the number of entries in \axiom{\spad{x}}.
502759`\axiom{length(\spad{x})} returns the number of entries in \axiom{\spad{x}}.
502816`\spad{length(q)} returns the number of elements in the queue. Note: \axiom{length(\spad{q}) = \spad{\#q}}.
502880`\spad{length(v)} computes the sqrt(dot(\spad{v},{}\spad{v})),{} \spadignore{i.e.} the magnitude
502966`\spad{length(a)} length of \spad{a} in digits.
503016`\spad{lepol }\undocumented
503099`\spad{lerchPhi(z,{} s,{} a)} is the Lerch Phi function
503164`\spad{lerchPhi(z,{} s,{} a)} is the Lerch Phi function.
503526`\spad{less?(a,{} b)} compares \spad{a} and \spad{b} in the partial ordering induced by setOrder.
503607`\spad{less?(u,{}n)} tests if \spad{u} has less than \spad{n} elements.
503670`\spad{less?(a,{} b,{} fn)} compares \spad{a} and \spad{b} in the partial ordering induced by setOrder,{} and returns \spad{fn(a,{} b)} if \spad{a} and \spad{b} are not comparable in that ordering.
503750`\spad{level(s)} returns a non negative integer which is the current level field of the indicated 3 dimensional subspace \spad{s}.
503809`\spad{leviCivitaSymbol()} is the rank \spad{dim} tensor defined by \spad{leviCivitaSymbol()(i1,{}...idim) = +1/0/-1} if \spad{i1,{}...,{}idim} is an even/is nota /is an odd permutation of \spad{minix,{}...,{}minix+dim-1}.
503876`\spad{lexGroebner(lp,{}lv)} computes Groebner basis for the list of polynomials \spad{lp} in lexicographic order. The variables are ordered by their position in the list \spad{lv}.
503974`\axiom{lexTriangular(base,{} norm?)} decomposes the variety associated with \axiom{base} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \axiom{base} needs to be a lexicographical Groebner basis of a zero-dimensional ideal. If \axiom{norm?} is \axiom{\spad{true}} then th
503974`--e regular sets are normalized.
504130`\spad{lex(ls)} sorts a list of pairs to lexicographic order
504198`\axiom{lexico(\spad{x},{}\spad{y})} returns \axiom{\spad{true}} iff \axiom{\spad{x}} is smaller than \axiom{\spad{y}} \spad{w}.\spad{r}.\spad{t}. the lexicographical ordering induced by \axiom{VarSet}.
504255`\axiom{lexico(\spad{x},{}\spad{y})} returns \axiom{\spad{true}} iff \axiom{\spad{x}} is smaller than \axiom{\spad{y}} \spad{w}.\spad{r}.\spad{t}. the lexicographical ordering induced by \axiom{VarSet}. \spad{N}.\spad{B}. This operation does not take into account the tree structure of its arguments. Thus this is not a total ordering.
504307`\spad{lexico(x,{}y)} returns \spad{true} iff \spad{x} is smaller than \spad{y} \spad{w}.\spad{r}.\spad{t}. the pure lexicographical ordering induced by \spad{S}.
504366`\spad{lfextendedint(f,{} x,{} g)} returns functions \spad{[h,{} c]} such that \spad{dh/dx = f - cg},{} if (\spad{h},{} \spad{c}) exist,{} "failed" otherwise.
504477`\spad{lfextlimint(f,{}x,{}k,{}[k1,{}...,{}kn])} returns functions \spad{[h,{} c]} such that \spad{dh/dx = f - c dk/dx}. Value \spad{h} is looked for in a field containing \spad{f} and \spad{k1},{}...,{}\spad{kn} (the \spad{ki}\spad{'s} must be logs).
504610`\spad{lfinfieldint(f,{} x)} returns a function \spad{g} such that \spad{dg/dx = f} if \spad{g} exists,{} "failed" otherwise.
504694`\spad{lfintegrate(f,{} x)} = \spad{g} such that \spad{dg/dx = f}.
504780`\spad{lflimitedint(f,{}x,{}[g1,{}...,{}gn])} returns functions \spad{[h,{}[[\spad{ci},{} \spad{gi}]]]} such that the \spad{gi}\spad{'s} are among \spad{[g1,{}...,{}gn]},{} and \spad{d(h+sum(\spad{ci} log(\spad{gi})))/dx = f},{} if possible,{} "failed" otherwise.
504932`\spad{lfunc(d,{}n)} computes the rank of the \spad{n}th factor in the lower central series of the free \spad{d}-generated free Lie algebra; This rank is \spad{d} if \spad{n} = 1 and binom(\spad{d},{}2) if \spad{n} = 2
504989`\spad{lhs(f)} returns the left side of \spad{f}
505032`\spad{lhs(r)} returns the left hand side of the rule \spad{r}.
505080`\spad{lhs(eqn)} returns the left hand side of equation \spad{eqn}.
505118`\spad{li2(z)} is polylog(2,{} \spad{z})
505191`\spad{li_int(f,{} k,{} x)} generates logarithmic integrals.
505523`\spad{\spad{li}(f)} denotes the logarithmic integral
505572`\spad{\spad{li}(f)} denotes the logarithmic integral
505647`\spad{\spad{li}(f)} denotes the logarithmic integral
505704`\spad{\spad{li}(x)} returns the logarithmic integral of \spad{x},{} \spadignore{i.e.} the integral of \spad{dx / log(x)}.
505758`\spad{library(ln)} creates a new library file.
505803`\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.
505882`\spad{lieAdmissible?()} tests if the algebra defined by the commutators is a Lie algebra,{} \spadignore{i.e.} satisfies the Jacobi identity. The property of anticommutativity follows from definition.
505958`\spad{lieAlgebra?()} tests if the algebra is anticommutative and \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Lie algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := a@b-b@a}.
506034`\spad{lieAlgebra?()} tests if the algebra is anticommutative and \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} for all \spad{a},{}\spad{b},{}\spad{c} in the algebra (Jacobi identity). Example: for every associative algebra \spad{(A,{}+,{}@)} we can construct a Lie algebra \spad{(A,{}+,{}*)},{} where \spad{a*b := a@b-b@a}.
506107`\axiom{lie(\spad{v},{} om)} calculates the Lie derivative of \axiom{om} with respect to \axiom{\spad{v}}.
506177`\axiom{lie(\spad{v},{} \spad{w})} calculates the Lie derivative of \axiom{\spad{w}} with respect to \axiom{\spad{v}}. (This yields the commutator of the fields.)
506299`\spad{lift(x)} return the canonical representative of the equivalence class \spad{x}
506368`\spad{lift(x)} \undocumented
506413`\spad{lift(z)} returns a minimal degree univariate polynomial up such that \spad{z=reduce up}.
506464`\spad{lift(u,{}k)} \undocumented
506622`\spad{lift should} be local
506863`\spad{lifting1(u,{}lv,{}lu,{}lr,{}lp,{}lt,{}ln,{}t,{}r)} \undocumented
507185`\spad{lifting(u,{}lv,{}lu,{}lr,{}lp,{}ln,{}r)} \undocumented
507410`\spad{light(c)} sets the shade of a hue,{} \spad{c},{} to it\spad{'s} highest value.
507450`\spad{lighting(v,{}x,{}y,{}z)} sets the position of the light source to the coordinates \spad{x},{} \spad{y},{} and \spad{z} and displays the graph for the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport}.
507527`\spad{limitPart(o)} = \spad{l} when \spad{o} = \spad{l} + \spad{n} and \spad{l} is a limit ordinal and \spad{n} is a nonnegative integer
507573`\spad{limitPlus(f(var))} returns \spad{limit(var -> a+,{}f(var))}.
507676`\spad{limitPlus(f(var))} returns \spad{limit(var -> cen+,{}f(var))}.
507808`apply 'function' represented by this graph to 'a' repeatedly until we reach a loop which is returned as a sequence of vertex indexes.
507875`\spad{limit(f(x),{}x = a)} computes the real limit \spad{lim(x -> a,{}f(x))}.
508114`\spad{limit(f(x),{}x = a)} computes the real two-sided limit of \spad{f} as its argument \spad{x} approaches \spad{a}.
508441`\spad{limit(f(x),{}x = a)} computes the real two-sided limit of \spad{f} as its argument \spad{x} approaches \spad{a}.
508777`apply 'function' represented by this graph to 'a' repeatedly until we reach a loop which is returned as a sequence of vertex indexes.
508868`\spad{limit(f(x),{}x=a,{}"left")} computes the left hand real limit \spad{lim(x -> a-,{}f(x))}; \spad{limit(f(x),{}x=a,{}"right")} computes the right hand real limit \spad{lim(x -> a+,{}f(x))}.
508982`\spad{limit(f(x),{}x,{}a,{}"left")} computes the real limit of \spad{f} as its argument \spad{x} approaches \spad{a} from the left; limit(\spad{f}(\spad{x}),{}\spad{x},{}a,{}"right") computes the corresponding limit as \spad{x} approaches \spad{a} from the right.
509161`\spad{limitedIntegrate(f,{} x,{} [g1,{}...,{}gn])} returns fractions \spad{[h,{} [[\spad{ci},{}\spad{gi}]]]} such that the \spad{gi}\spad{'s} are among \spad{[g1,{}...,{}gn]},{} \spad{dci/dx = 0},{} and \spad{d(h + sum(\spad{ci} log(\spa
509161`--d{gi})))/dx = f} if possible,{} "failed" otherwise.
509431`\spad{limitedint(f,{} [g1,{}...,{}gn])} returns fractions \spad{[h,{}[[\spad{ci},{} \spad{gi}]]]} such that the \spad{gi}\spad{'s} are among \spad{[g1,{}...,{}gn]},{} \spad{ci' = 0},{} and \spad{(h+sum(\spad{ci} log(\spad{gi})))' = f},{} if possible,{} "failed" otherwise.
509628`\spad{linGenPos }\undocumented
509819`\spad{linSolve(lp,{}lvar)} finds the solutions of the linear system of polynomials \spad{lp} = 0 with respect to the list of symbols \spad{lvar}.
509989`\spad{lin_sol(eql,{} vl)} solves system of equations \spad{eql} for variables in \spad{vl}. Equations must be linear in variables from \spad{vl}.
510108`\spad{lineColorDefault()} returns the default color of lines connecting points in a 2D viewport.
510171`\spad{lineColorDefault(p)} sets the default color of lines connecting points in a 2D viewport to the palette \spad{p}.
510241`\spad{lineIntersect(n,{}p)} returns the point where a line,{} from the centre of the boundary to the point \spad{p},{} intersects with the boundary. \spad{n:}\% is the boundary p:PT is the point at the end of the line (outside boundary)
510295`\axiom{linear?(\spad{p})} returns \spad{true} iff \axiom{\spad{p}} does not lie in the base ring \axiom{\spad{R}} and has main degree \axiom{1}.
510471`\spad{linearAssociatedExp(a,{}f)} is linear over {\em F},{} \spadignore{i.e.} for elements {\em a} from {\em \$},{} {\em c,{}d} form {\em F} and {\em f,{}g} univariate polynomials over {\em F} we have \spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\em c} times \spadfun{linearAssociatedExp}(a,{}\spad{f}) plus {\em d} times \spadfun{linearAssociatedExp}(a,{}\spad{g}). Therefore \spadfu
510471`--n{linearAssociatedExp} is defined completely by its action on monomials from {\em F[X]}: \spadfun{linearAssociatedExp}(a,{}monomial(1,{}\spad{k})\spad{\$}SUP(\spad{F})) is defined to be \spadfun{Frobenius}(a,{}\spad{k}) which is {\em a^(q^k)},{} where {\em q=size()\$F}.
510690`\spad{linearAssociatedLog(a)} returns a polynomial {\em g},{} such that \spadfun{linearAssociatedExp}(normalElement(),{}\spad{g}) equals {\em a}.
510924`\spad{linearAssociatedLog(b,{}a)} returns a polynomial {\em g},{} such that the \spadfun{linearAssociatedExp}(\spad{b},{}\spad{g}) equals {\em a}. If there is no such polynomial {\em g},{} then \spadfun{linearAssociatedLog} fails.
511161`\spad{linearAssociatedOrder(a)} retruns the monic polynomial {\em g} of least degree,{} such that \spadfun{linearAssociatedExp}(a,{}\spad{g}) is 0.
511279`\spad{linearlyDependenceOverZ([v1,{}...,{}vn])} returns \spad{[c1,{}...,{}cn]} if \spad{c1*v1 + ... + cn*vn = 0} and not all the \spad{ci}\spad{'s} are 0,{} "failed" if the \spad{vi}\spad{'s} are linearly independent over the integers.
511387`\spad{linearDependence([v1,{}...,{}vn])} returns \spad{[c1,{}...,{}cn]} if \spad{c1*v1 + ... + cn*vn = 0} and not all the \spad{ci}\spad{'s} are 0,{} "failed" if the \spad{vi}\spad{'s} are linearly independent over \spad{S}.
511479`\spad{linearExtend:(f,{}x)} returns the linear extension of a multilinear map defined on the basis of \spad{M2} applied to a linear combination
511583`\spad{linearExtend:(f,{}x)} returns the linear extension of a map defined on the basis of \spad{M2} applied to a linear combination
511742`\axiom{linearPolynomials(\spad{lp})} returns \axiom{\spad{lps},{}nlps} where \axiom{\spad{lps}} is a list of the linear polynomials in \spad{lp},{} and \axiom{nlps} are the other ones.
511865`\spad{linearSearch(x,{} f)} searches \spad{x} in the cache,{} calling \spad{f(y)} to determine whether \spad{x} is equal to \spad{y}. It returns \spad{y} from cache such that \spad{f}(\spad{y}) or failed is no such \spad{y} exists.
511943`\spad{linear(u)} \undocumented
512010`\spad{linear(f,{}g)} \undocumented
512078`\axiom{linearize(\spad{sys})} tries to retract all equations in \axiom{\spad{sys}} to a linear one. If that is not possible,{} an error occurs.
512298`\spad{linearlyDependent?([v1,{}...,{}vn])} returns \spad{true} if the \spad{vi}\spad{'s} are linearly dependent over \spad{S},{} \spad{false} otherwise.
512374`\spad{linearlyDependentOverZ?([v1,{}...,{}vn])} returns \spad{true} if the \spad{vi}\spad{'s} are linearly dependent over the integers,{} \spad{false} otherwise.
512460`\spad{linears(f,{}p)} returns the product of all the linear factors of \spad{f} modulo \spad{p}. Potentially incorrect result if \spad{f} is not square-free modulo \spad{p}.
512532`\spad{linkToFortran(s,{}l,{}t,{}lv)} \undocumented{}
512638`\spad{linkToFortran(s,{}l,{}ll,{}lv)} \undocumented{}
512814`\spad{linkToFortran(s,{}l,{}ll,{}lv,{}t)} \undocumented{}
512997`\spad{link(m,{}n)} returns a line which is the shortest distance between the two boundaries. The line is represented by its two endpoints. \spad{m:}\% is the first boundary to be linked \spad{n:}\% is the second boundary to be linked
513048`\spad{lintgcd([a1,{}..,{}ak])} = \spad{gcd} of a list of integers
513106`\spad{list?(s)} is \spad{true} if \spad{s} is a Lisp list,{} possibly ().
513182`\spad{listBranches(c)} returns a list of lists of points,{} representing the branches of the curve \spad{c}.
513274`\spad{listBranches(c)} returns a list of lists of points,{} representing the branches of the curve \spad{c}.
513366`\spad{listConjugateBases(bas,{}q,{}n)} returns the list \spad{[bas,{}bas^Frob,{}bas^(Frob^2),{}...bas^(Frob^(n-1))]},{} where \spad{Frob} raises the coefficients of all polynomials appearing in the basis \spad{bas} to the \spad{q}th power.
513600`\spad{listLoops(t)} returns the list of lists of points,{} or the 'loops',{} of the given tube plot \spad{t}.
513746`\spad{listOfLists(m)} returns the rows of the matrix \spad{m} as a list of lists.
513818`\spad{listOfLists(m)} returns the rows of the matrix \spad{m} as a list of lists.
513905`\spad{listOfLists t} converts a tableau \spad{t} to a list of lists.
513962`\spad{listOfMonoms(l)} returns the list of the monomials forming \spad{l}.
514043`\spad{listOfTerms([l1]*[l2]*...[ln])} returns the list of words \spad{l1,{} l2,{} .... ln}.
514140`\axiom{listOfTerms(\spad{p})} returns the internal representation of \axiom{\spad{p}}.
514264`\spad{listOfTerms(x)} returns a list \spad{lt} of terms with type \spad{Record(k: S,{} c: R)} such that \spad{x} equals \spad{constructOrdered lt}.
514352`\spad{listRepresentation(p)} produces a representation {\em rep} of the permutation \spad{p} as a list of preimages and images,{} \spad{i}.\spad{e} \spad{p} maps {\em (rep.preimage).k} to {\em (rep.image).k} for all indices \spad{k}. Elements of \spad{S} not in {\em (rep.preimage).k} are fixed points,{} and these are the only fixed points of the permutation.
514445`\spad{listYoungTableaus(lambda)} where {\em lambda} is a proper partition generates the list of all standard tableaus of shape {\em lambda} by means of lattice permutations. The numbers of the lattice permutation are interpreted as column labels. Hence the contents of these lattice permutations are the conjugate of {\em lambda}. Notes: the functions {\em nextLatticePermutation} and {\em makeYoungT
514445`--ableau} are used. The entries are from {\em 0,{}...,{}n-1}.
514593`\spad{list(x)} returns the list of one element \spad{x}.
514637`\spad{list(sy)} takes a scripted symbol and produces a list of the name followed by the scripts.
514678`\spad{listexp }\undocumented
514751`\spad{lists(r)} returns the list of matches that match lists.
514831`\spad{lllip(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a list of components,{} which are lists of curves,{} which are lists of indices to points,{} and if so,{} returns the list of lists of lists; An error is signaled otherwise.
514916`\spad{lllp(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a list of components,{} which are lists of curves,{} which are lists of points,{} and if so,{} returns the list of lists of lists; An error is signaled otherwise.
514990`\spad{llprop(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a list of curves which are lists of the subspace component properties of the curves,{} and if so,{} returns the list of lists; An error is signaled otherwise.
515077`\spad{lo(s)} returns the first endpoint of \spad{s}. Note: \spad{lo(l..h) = l}.
515121`\spad{localAbs(fcn)} = \spad{abs(fcn)} or \spad{sqrt(fcn^2)} depending on whether or not FE has a function \spad{abs}. This should be a local function,{} but the compiler won\spad{'t} allow it.
515201`\spad{localAbs(fcn)} = \spad{abs(fcn)} or \spad{sqrt(fcn^2)} depending on whether or not FE has a function \spad{abs}. This should be a local function,{} but the compiler won\spad{'t} allow it.
515313`\spad{integralBasis(p)} returns a record \spad{[basis,{}basisDen,{}basisInv]} containing information regarding the local integral closure of \spad{Z} at the prime \spad{p} in the quotient field of \spad{F},{} where \spad{F} is a framed algebra with \spad{Z}-module basis \spad{w1,{}w2,{}...,{}wn}. If \spad{basis} is the matrix \spad{(aij,{} i = 1..n,{} j = 1.
515313`--.n)},{} then the \spad{i}th element of the integral basis is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of \spad{basis} contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix \spad{basisInv} contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if \spad{basisInv} is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,
515313`--{} j = 1..n)}.
515460`\spad{integralBasis(p)} returns a record \spad{[basis,{}basisDen,{}basisInv]} containing information regarding the local integral closure of \spad{R} at the prime \spad{p} in the quotient field of \spad{F},{} where \spad{F} is a framed algebra with \spad{R}-module basis \spad{w1,{}w2,{}...,{}wn}. If \spad{basis} is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n)},{} then the \sp
515460`--ad{i}th element of the local integral basis is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of \spad{basis} contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix \spad{basisInv} contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if \spad{basisInv} is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1..n)}.
515587`\spad{integralBasis(p)} returns a record \spad{[basis,{}basisDen,{}basisInv] } containing information regarding the local integral closure of \spad{R} at the prime \spad{p} in the quotient field of the framed algebra \spad{F}. \spad{F} is a framed algebra with \spad{R}-module basis \spad{w1,{}w2,{}...,{}wn}. If 'basis' is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n
515587`--)},{} then the \spad{i}th element of the local integral basis is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of 'basis' contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix 'basisInv' contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if 'basisInv' is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1..n
515587`--)}.
515725`\spad{integralBasis(p)} returns a record \spad{[basis,{}basisDen,{}basisInv]} containing information regarding the local integral closure of \spad{R} at the prime \spad{p} in the quotient field of \spad{F},{} where \spad{F} is a framed algebra with \spad{R}-module basis \spad{w1,{}w2,{}...,{}wn}. If \spad{basis} is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n)},{} then t
515725`--he \spad{i}th element of the local integral basis is \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of \spad{basis} contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix \spad{basisInv} contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if \spad{basisInv} is the matrix \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1
515725`--..n)}.
515858`\spad{localReal?(x)} should be local but conditional
515920`\spad{localUnquote(f,{}ls)} is a local function.
515988`\spad{log10()} returns \spad{ln 10}: \spad{2.3025809299...}.
516021`\spad{log10(x)} represents the Fortran intrinsic function \spad{LOG10}
516103`\spad{log10(x)} computes the logarithm with base 10 for \spad{x}.
516144`\spad{log10(x)} computes the logarithm for \spad{x} to base 10.
516179`\spad{log1(f)} returns the logarithm of the power series represented by cons(1,{} \spad{f}),{} \spadignore{i.e.} assuming that the constant term is 1 and therefore transcendentality is not involved.
516267`\spad{log2()} returns \spad{ln 2},{} \spadignore{i.e.} \spad{0.6931471805...}.
516299`\spad{log2(x)} computes the logarithm with base 2 for \spad{x}.
516339`\spad{log2(x)} computes the logarithm for \spad{x} to base 2.
516373`\spad{logDependenceQ([q1,{} ...,{} qn],{} q0)} finds rational constants \spad{c1,{}...cn} such that \spad{q1^c1*...*qn^cn=u*q0} where \spad{u} is a unit
516514`\spad{logGamma(x)} is the natural log of \spad{Gamma(x)}. This can often be computed even if \spad{Gamma(x)} cannot.
516610`\spad{logGamma(x)} is the natural log of \spad{Gamma(x)}.
516688`\spad{logGamma(x)} is the natural log of \spad{Gamma(x)}. This can often be computed even if \spad{Gamma(x)} cannot.
516766`\spad{logGamma(x)} is the natural log of \spad{Gamma(x)}.
516826`\spad{logIfCan(z)} returns log(\spad{z}) if possible,{} and "failed" otherwise.
516906`\spad{log(x)} applies the logarithm operator to \spad{x}
516955`\spad{log(f)} returns \spad{[(a1,{}b1),{}...,{}(am,{}bm)]} such that the logarithm of \spad{f} is equal to \spad{a1*log(b1) + ... + am*log(bm)}.
517253`\spad{log(st)} computes the log of a power series.
517336`\spad{log(st)} computes the log of a power series.
517433`\spad{log(z)} returns the logarithm of Laurent series \spad{z}.
517519`\spad{log(z)} returns the logarithm of a Puiseux series \spad{z}.
517614`\axiom{log(\spad{p})} returns the logarithm of \axiom{\spad{p}}.
517694`\spad{log(x)} represents the Fortran intrinsic function LOG
517774`\spad{log(x)} returns the natural logarithm of \spad{x}. When evaluated into some subset of the complex numbers,{} the branch cut lies along the negative real axis,{} continuous with quadrant II. The domain does not contain the origin.
517828`\axiom{log(\spad{p},{} \spad{n})} returns the logarithm of \axiom{\spad{p}} truncated at order \axiom{\spad{n}}.
517916`\axiom{log(\spad{p},{}\spad{n})} returns the logarithm of \axiom{\spad{p}} (truncated up to order \axiom{\spad{n}}).
518019`\spad{false} (contradiction) is a logical constant.
518054`\spad{true} is a logical constant.
518089`\spad{logical?(t)} tests whether \spad{t} is equivalent to the FORTRAN type LOGICAL.
518144`\spad{logpart(ir)} returns the logarithmic part of an integration result
518305`\spad{lommelS1(mu,{} nu,{} z)} is the Lommel \spad{s} function
518370`\spad{lommelS1(mu,{} nu,{} z)} is the Lommel \spad{s} function.
518432`\spad{lommelS2(mu,{} nu,{} z)} is the Lommel \spad{S} function
518497`\spad{lommelS2(mu,{} nu,{} z)} is the Lommel \spad{S} function.
518771`\spad{lookup(x)} \undocumented{} See \axiomFunFrom{lookup}{Finite}
518857`\spad{lookup(x)} returns a positive integer such that \spad{x = index lookup x}.
518907`\spad{loopPoints(p,{}n,{}b,{}r,{}lls)} creates and returns a list of points which form the loop with radius \spad{r},{} around the center point indicated by the point \spad{p},{} with the principal normal vector of the space curve at point \spad{p} given by the point(vector) \spad{n},{} and the binormal vector given by the point(vector) \spad{b}
518907`--,{} and a list of lists,{} \spad{lls},{} which is the \spadfun{cosSinInfo} of the number of points defining the loop.
519067`\spad{loop(\spad{li})} constructs loop from list of indexes \spad{li}
519177`\spad{loopsArrows(s)} returns a list of loops for this graph in this case the loop is represented by the indexes of the sequence of nodes passed through. to-do: it would be better to use a more efficient algorithm,{} currently the code calls spanningForestArrow and traverses the result for loops,{} it might be more efficient to use Floyds algorithm.
519309`\spad{loopsAtNode(s,{}a)} returns a list of loops for this graph that pass through vertex index 'a'
519440`\spad{loopsNodes(s)} returns a list of loops for this graph in this case the loop is represented by the indexes of the sequence of nodes passed through.
519551`\spad{looseEquals(x,{}y)} is \spad{true} if \spad{x} 'equals' \spad{y} this is a looser version of equality test but is not as general as isomorphism. it only requires the same number of vertices but does not require the objects themselves being equal. the arrows must be the same,{} that is it may return \spad{false} if the order of vertices is changed so this is not isomorphism test.
519609`\spad{low(s)} returns the first endpoint of \spad{s}. Note: \spad{low(l..h) = l}.
519654`\spad{lowerCase!(s)} destructively replaces the alphabetic characters in \spad{s} by lower case.
519704`\spad{lowerCase?(c)} tests if \spad{c} is an lower case letter,{} \spadignore{i.e.} one of a..\spad{z}.
519753`\spad{lowerCase()} returns the class of all characters for which \spadfunFrom{lowerCase?}{Character} is \spad{true}.
519845`\spad{lowerCase(s)} returns the string with all characters in lower case.
519894`\spad{lowerCase(c)} converts an upper case letter to the corresponding lower case letter. If \spad{c} is not an upper case letter,{} then it is returned unchanged.
519937`\spad{lowerPolynomial(upoly)} converts \spad{upoly} to be a univariate polynomial over \spad{R}. An error if the coefficients contain variables.
520059`\spad{lp(s)} returns the list of points component which the \spadtype{ThreeSpace},{} \spad{s},{} contains; these points are used by reference,{} \spadignore{i.e.} the component holds indices referring to the points rather than the points themselves. This allows for sharing of the points.
520119`\spad{lprop(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a list of subspace component properties,{} and if so,{} returns the list; An error is signaled otherwise.
520199`\spad{lquo(x,{} s)} returns the exact left quotient of \spad{x} by \spad{s}.
520266`\spad{lquo(x,{} y)} returns the exact left quotient of \spad{x} by \spad{y} \spadignore{i.e.} \spad{q} such that \spad{x = y * q},{} "failed" if \spad{x} is not of the form \spad{y * q}.
520327`\spad{lquo(x,{} y)} returns the exact left quotient of \spad{x} \indented{1}{by \spad{y} that is \spad{q} such that \spad{x = y * q},{}} "failed" if \spad{x} is not of the form \spad{y * q}.
520395`\axiom{lquo(\spad{x},{}\spad{y})} returns the left simplification of \axiom{\spad{x}} by \axiom{\spad{y}}.
520507`\spad{lquo(x,{}w)} returns the left simplification of \spad{x} by the word \spad{w}.
520576`\spad{lquo(x,{}y)} returns the left simplification of \spad{x} by \spad{y}.
520626`\spad{lquo(x,{}v)} returns the left simplification of \spad{x} by the variable \spad{v}.
520676`\axiom{lyndon?(\spad{w})} test if \axiom{\spad{w}} is a Lyndon word.
520754`\axiom{lyndonIfCan(\spad{w})} convert \axiom{\spad{w}} into a Lyndon word.
520847`\axiom{lyndon(\spad{w})} convert \axiom{\spad{w}} into a Lyndon word,{} error if \axiom{\spad{w}} is not a Lyndon word.
520919`\axiom{\spad{m2r}(ind)} transforms a multi-index into a repeated index.
521005`\axiom{\spad{m2r}(ind)} transforms a multi-index into a repeated index.
521095`\spad{m_inverse(x,{} lm,{} lv,{} p)} computes inverse of \spad{x} in algeraic extension defined by \spad{lm}.
521298`\spad{magnitude(v)} computes the sqrt(dot(\spad{v},{}\spad{v})),{} \spadignore{i.e.} the length
521387`\axiom{mainCharacterization(\spad{x})} is the main algebraic quantity of \axiom{\spad{x}} (\axiom{SEG})
521616`\axiom{mainCoefficients(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns [\spad{p}],{} otherwise returns the list of the coefficients of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
521763`\axiom{mainContent(\spad{p})} returns the content of \axiom{\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \axiom{\spad{R}}.
521849`\axiom{mainDefiningPolynomial(\spad{x})} is the defining polynomial for the main algebraic quantity of \axiom{\spad{x}}
521955`\axiom{mainForm(\spad{x})} is the main algebraic quantity name of \axiom{\spad{x}}
522169`\spad{mainKernel(f)} returns a kernel of \spad{f} with maximum nesting level,{} or if \spad{f} has no kernels (\spadignore{i.e.} \spad{f} is a constant).
522310`\axiom{mainMonomial(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns \axiom{1},{} otherwise,{} \axiom{mvar(\spad{p})} raised to the power \axiom{mdeg(\spad{p})}.
522455`\axiom{mainMonomials(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{\spad{O}},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns [1],{} otherwise returns the list of the monomials of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
522605`\axiom{mainPrimitivePart(\spad{p})} returns the primitive part of \axiom{\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \axiom{\spad{R}}.
522770`\axiom{mainSquareFreePart(\spad{p})} returns the square free part of \axiom{\spad{p}} viewed as a univariate polynomial in its main variable and with coefficients in the polynomial ring generated by its other variables over \axiom{\spad{R}}.
522863`\axiom{mainValue(\spad{x})} is the expression of \axiom{\spad{x}} in terms of \axiom{SparseUnivariatePolynomial(\%)}
523038`\axiom{mainVariable?(\spad{v},{}\spad{ps})} returns \spad{true} iff \axiom{\spad{v}} is the main variable of some polynomial in \axiom{\spad{ps}}.
523123`\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \spad{f},{} "failed" if \spad{f} has no variables.
523320`\spad{mainVariable(p)} returns the biggest variable which actually occurs in the polynomial \spad{p},{} or "failed" if no variables are present. fails precisely if polynomial satisfies ground?
523416`\spad{mainVariable(f)} returns the highest variable appearing in the numerator or the denominator of \spad{f},{} "failed" if \spad{f} has no variables.
523593`\axiom{mainVariables(\spad{ps})} returns the decreasingly sorted list of the variables which are main variables of some polynomial in \axiom{\spad{ps}}.
523676`\spad{makeCos(e,{}r)} makes a sin expression with given argument and coefficient
523727`\spad{makeCrit }\undocumented
523934`\spad{makeEq(lsol,{}lvar)} returns a list of equations formed by corresponding members of \spad{lvar} and \spad{lsol}.
524048`\spad{evalADE(f,{} x,{} n,{} eq,{} parameters,{} values)} creates an expression that stands for the coefficient of \spad{x^n} in the Taylor expansion of \spad{f}(\spad{x}),{} where \spad{f}(\spad{x}) is given by the functional equation \spad{eq}. The argument values specifies the first few Taylor coefficients.
524147`\spad{makeFR(flist)} turns the final factorization of henselFact into a \spadtype{Factored} object.
524273`\spad{makeFR(unit,{}listOfFactors)} creates a factored object (for use by factoring code).
524388`\spad{makeFloatFunction(expr,{} x)} returns a Lisp function \spad{f: \axiomType{DoubleFloat} -> \axiomType{DoubleFloat}} defined by \spad{f(x) == expr}. Function \spad{f} is compiled and directly applicable to objects of type \axiomType{DoubleFloat}.
524488`\spad{makeFloatFunction(expr,{} x,{} y)} returns a Lisp function \spad{f: (\axiomType{DoubleFloat},{} \axiomType{DoubleFloat}) -> \axiomType{DoubleFloat}} defined by \spad{f(x,{} y) == expr}. Function \spad{f} is compiled and directly applicable to objects of type \spad{(\axiomType{DoubleFloat},{} \axiomType{DoubleFloat})}.
524607`\spad{makeGraphImage(llp)} returns a graph of the domain \spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \spad{llp},{} with default point size and default point and line colours. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.
524684`\spad{makeGraphImage(\spad{gi})} takes the given graph,{} \spad{\spad{gi}} of the domain \spadtype{GraphImage},{} and sends it\spad{'s} data to the viewport manager where it waits to be included in a two-dimensional viewport window. \spad{\spad{gi}} cannot be an empty graph,{} and it\spad{'s} elements must have been created using the \spadfun{point} or \spadfun{component} functions,{} not by a previous \spadfun{makeGraphImage}.
524733`\spad{makeGraphImage(llp,{}lpal1,{}lpal2,{}lp)} returns a graph of the domain \spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \spad{llp},{} whose point colors are indicated by the list of palette colors,{} \spad{lpal1},{} and whose lines are colored according to the list of palette colors,{} \spad{lpal2}. The paramater \spad{l
524733`--p} is a list of integers which denote the size of the data points. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.
524860`\spad{makeGraphImage(llp,{}lpal1,{}lpal2,{}lp,{}lopt)} returns a graph of the domain \spadtype{GraphImage} which is composed of the points and lines from the list of lists of points,{} \spad{llp},{} whose point colors are indicated by the list of palette colors,{} \spad{lpal1},{} and whose lines are colored according to the list of palette colors,{} \spad{lpal2
524860`--}. The paramater \spad{lp} is a list of integers which denote the size of the data points,{} and \spad{lopt} is the list of draw command options. The graph data is then sent to the viewport manager where it waits to be included in a two-dimensional viewport window.
525004`\spad{makeMulti(l)} returns the element whose list of monomials is \spad{l}.
525082`\spad{makeObject(sp,{}curve(f,{}g,{}h),{}a..b)} returns the space \spad{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of the parametric curve \spad{x = f(t),{} y = g(t),{} z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}.
525225`\spad{makeObject(sp,{}curve(f,{}g,{}h),{}a..b)} returns the space \spad{sp} of the domain \spadtype{ThreeSpace} with the addition of the graph of the parametric curve \spad{x = f(t),{} y = g(t),{} z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}.
525383`\spad{makeObject(curve(f(t),{}g(t),{}h(t)),{}t = a..b)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{h(t)} is the default title.
525508`\spad{makeObject(curve(f,{}g,{}h),{}a..b,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric curve \spad{x = f(t),{} y = g(t),{} z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
525668`\spad{makeObject(f,{}a..b,{}c..d)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}.
525831`\spad{makeObject(f,{}a..b,{}c..d,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric surface \spad{f(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}.
526001`\spad{makeObject(f(x,{}y),{}x = a..b,{}y = c..d)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; \spad{f(x,{}y)} appears as the default title.
526126`\spad{makeObject(curve(f,{}g,{}h),{}a..b,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric curve \spad{x = f(t),{} y = g(t),{} z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; The options contained in the list \spad{l} of the domain \spad{DrawOption} are a
526126`--pplied.
526301`\spad{makeObject(curve(f(t),{}g(t),{}h(t)),{}t = a..b,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} as \spad{t} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)}; \spad{h(t)} is the default title,{} and the options contained in the list \spad{l} of 
526301`--the domain \spad{DrawOption} are applied.
526443`\spad{makeObject(surface(f,{}g,{}h),{}a..b,{}c..d,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,
526443`--{}d)} to \spad{max(c,{}d)}.
526625`\spad{makeObject(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spa
526625`--d{max(c,{}d)}; \spad{h(t)} is the default title.
526769`\spad{makeObject(f,{}a..b,{}c..d,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)},{} and the options contained in the list \spad{l} of the doma
526769`--in \spad{DrawOption} are applied.
526949`\spad{makeObject(f,{}a..b,{}c..d,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric surface \spad{f(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; The options contained in the list 
526949`--\spad{l} of the domain \spad{DrawOption} are applied.
527136`\spad{makeObject(f(x,{}y),{}x = a..b,{}y = c..d,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of \spad{z = f(x,{}y)} as \spad{x} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{y} ranges from \spad{min(c,{}d)} to \spad{max(c,{}d)}; \spad{f(x,{}y)} is the default title,{} and the options contained in the list \spad
527136`--{l} of the domain \spad{DrawOption} are applied.
527278`\spad{makeObject(surface(f,{}g,{}h),{}a..b,{}c..d,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges 
527278`--from \spad{min(c,{}d)} to \spad{max(c,{}d)}. The options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
527477`\spad{makeObject(surface(f(u,{}v),{}g(u,{}v),{}h(u,{}v)),{}u = a..b,{}v = c..d,{}l)} returns a space of the domain \spadtype{ThreeSpace} which contains the graph of the parametric surface \spad{x = f(u,{}v)},{} \spad{y = g(u,{}v)},{} \spad{z = h(u,{}v)} as \spad{u} ranges from \spad{min(a,{}b)} to \spad{max(a,{}b)} and \spad{v} ranges from \spa
527477`--d{min(c,{}d)} to \spad{max(c,{}d)}; \spad{h(t)} is the default title,{} and the options contained in the list \spad{l} of the domain \spad{DrawOption} are applied.
527638`\spad{makeRec(u,{} n,{} eq,{} parameters,{} values)} creates an expression that stands for \spad{u}(\spad{n}),{} where \spad{u}(\spad{n}) is given by the equation \spad{eq}. The argument \spad{parameters} has to contain all variables (or kernels) that appear eventually in the stream of values. The argument values specifies the initial values of the recurrence \spad{u}(0),{} \spad{u}(1),{}... For the moment we do
527638`--n\spad{'t} allow recursions that contain \spad{u} inside of another operator.
527730`\spad{makeRecord(a,{}b)} creates a record object with type Record(part1:S,{} part2:R),{} where \spad{part1} is \spad{a} and \spad{part2} is \spad{b}.
527802`\spad{makeResult(r1,{}r2)} makes the combined result [\spad{r1},{}\spad{r2}].
527998`\spad{makeSUP(p)} converts the polynomial \spad{p} to be of type SparseUnivariatePolynomial over the same coefficients.
528088`\spad{makeSeries(refer,{}str)} creates a power series from the reference \spad{refer} and the stream \spad{str}.
528229`\spad{makeSin(e,{}r)} makes a sin expression with given argument and coefficient
528280`\spad{makeSketch(p,{}x,{}y,{}a..b,{}c..d)} creates an ACPLOT of the curve \spad{p = 0} in the region {\em a <= x <= b,{} c <= y <= d}. More specifically,{} 'makeSketch' plots a non-singular algebraic curve \spad{p = 0} in an rectangular region {\em xMin <= x <= xMax},{} {\em yMin <= y <= yMax}. The user inputs \spad{makeSketch(p,{}x,{}y,{}xMin..xMax,{}yMin..yMax
528280`--)}. Here \spad{p} is a polynomial in the variables \spad{x} and \spad{y} with integer coefficients (\spad{p} belongs to the domain \spad{Polynomial Integer}). The case where \spad{p} is a polynomial in only one of the variables is allowed. The variables \spad{x} and \spad{y} are input to specify the the coordinate axes. The horizontal axis is the \spad{x}-axis and the vertical axis is the \spad{y}-axis. The rational numbers xMin,{}...,{}yMax specify the boundaries of the region in which the cu
528280`--rve is to be plotted.
528423`\axiom{makeSystem(\spad{sys})} creates a differential equation from a system.
528492`\spad{makeTerm(s,{} e)} returns the monomial \spad{s} exponentiated by \spad{e} (\spadignore{e.g.} s^e or \spad{e} * \spad{s}).
528547`\spad{makeUnit()} returns the unit element of the monomial.
528787`\spad{makeVariable(s)} views \spad{s} as a differential indeterminate,{} in such a way that the \spad{n}-th derivative of \spad{s} may be simply referenced as \spad{z}.\spad{n} where \spad{z} :=makeVariable(\spad{s}). Note: In the interpreter,{} \spad{z} is given as an internal map,{} which may be ignored.
528884`\spad{makeVariable(p)} views \spad{p} as an element of a differential ring,{} in such a way that the \spad{n}-th derivative of \spad{p} may be simply referenced as \spad{z}.\spad{n} where \spad{z} \spad{:=} makeVariable(\spad{p}). Note: In the interpreter,{} \spad{z} is given as an internal map,{} which may be ignored.
529005`\spad{makeVariable(s,{} n)} returns the \spad{n}-th derivative of a differential indeterminate \spad{s} as an algebraic indeterminate.
529091`\spad{makeViewport2D(v)} takes the given two-dimensional viewport,{} \spad{v},{} of the domain \spadtype{TwoDimensionalViewport} and displays a viewport window on the screen which contains the contents of \spad{v}.
529152`\spad{makeViewport2D(\spad{gi},{}lopt)} creates and displays a viewport window of the domain \spadtype{TwoDimensionalViewport} whose graph field is assigned to be the given graph,{} \spad{\spad{gi}},{} of domain \spadtype{GraphImage},{} and whose options field is set to be the list of options,{} \spad{lopt} of domain \spadtype{DrawOption}.
529238`\spad{makeViewport3D(v)} takes the given three-dimensional viewport,{} \spad{v},{} of the domain \spadtype{ThreeDimensionalViewport} and displays a viewport window on the screen which contains the contents of \spad{v}.
529301`\spad{makeViewport3D(sp,{}lopt)} takes the given space,{} \spad{sp} which is of the domain \spadtype{ThreeSpace} and displays a viewport window on the screen which contains the contents of \spad{sp},{} and whose draw options are indicated by the list \spad{lopt},{} which is a list of options from the domain \spad{DrawOption}.
529402`\spad{makeViewport3D(sp,{}s)} takes the given space,{} \spad{sp} which is of the domain \spadtype{ThreeSpace} and displays a viewport window on the screen which contains the contents of \spad{sp},{} and whose title is given by \spad{s}.
529493`\spad{makeYoungTableau(lambda,{}gitter)} computes for a given lattice permutation {\em gitter} and for an improper partition {\em lambda} the corresponding standard tableau of shape {\em lambda}. Notes: see {\em listYoungTableaus}. The entries are from {\em 0,{}...,{}n-1}.
529668`\spad{makeop should} be local but conditional
529742`\axiom{makingStats?()} returns \spad{true} iff the statisitics process is running.
529820`\spad{mantissa(u)} returns the mantissa of \spad{u}
529870`\spad{mantissa(x)} returns the mantissa part of \spad{x}.
530406`\spad{map!(f,{}a)} assign \spad{a(i,{}j)} to \spad{f(a(i,{}j))} for all \spad{i,{} j}
530480`\spad{map!(f,{}u)} destructively replaces each element \spad{x} of \spad{u} by \axiom{\spad{f}(\spad{x})}.
530563`\spad{mapBivariate(f,{}p(x,{}y))} applies the function \spad{f} to the coefficients of \spad{p(x,{}y)}.
530700`\spad{mapCoef(f,{} e1 a1 +...+ en an)} returns \spad{f(e1) a1 +...+ f(en) an}.
530769`\spad{mapContra(s,{}m,{}newOb,{}offsetX,{}offsetY)} is similar to map function but reverses the directions of the arrows
530866`\spad{mapDown!(t,{}p,{}f)} returns \spad{t} after traversing \spad{t} in "preorder" (node then left then right) fashion replacing the successive interior nodes as follows. The root value \spad{x} is replaced by \spad{q} \spad{:=} \spad{f}(\spad{p},{}\spad{x}). The mapDown!(\spad{l},{}\spad{q},{}\spad{f}) and mapDown!(\spad{r},{}\spad{q},{}\spad{f}) are evaluated for the left and right subtrees \spad{l} and \spad{r} of \spad{t}.
530931`\spad{mapDown!(t,{}p,{}f)} returns \spad{t} after traversing \spad{t} in "preorder" (node then left then right) fashion replacing the successive interior nodes as follows. Let \spad{l} and \spad{r} denote the left and right subtrees of \spad{t}. The root value \spad{x} of \spad{t} is replaced by \spad{p}. Then \spad{f}(value \spad{l},{} value \spad{r},{} \spad{p}),{} where \spad{l} and \spad{r} denote the left and right subtrees o
530931`--f \spad{t},{} is evaluated producing two values \spad{pl} and \spad{pr}. Then \spad{mapDown!(l,{}pl,{}f)} and \spad{mapDown!(l,{}pr,{}f)} are evaluated.
531004`\spad{mapExpon(f,{} a1\^e1 ... an\^en)} returns \spad{a1\^f(e1) ... an\^f(en)}.
531065`\spad{mapExpon(f,{} a1\^e1 ... an\^en)} returns \spad{a1\^f(e1) ... an\^f(en)}.
531129`\spad{mapExpon(f,{} a1\^e1 ... an\^en)} returns \spad{a1\^f(e1) ... an\^f(en)}.
531284`\spad{mapExponents(fn,{}u)} maps function \spad{fn} onto the exponents of the non-zero monomials of polynomial \spad{u}.
531356`\spad{mapGen(f,{} a1\^e1 ... an\^en)} returns \spad{f(a1)\^e1 ... f(an)\^en}.
531406`\spad{mapGen(f,{} a1\^e1 ... an\^en)} returns \spad{f(a1)\^e1 ... f(an)\^en}.
531457`\spad{mapGen(f,{} a1\^e1 ... an\^en)} returns \spad{f(a1)\^e1 ... f(an)\^en}.
531516`\spad{mapGen(f,{} e1 a1 +...+ en an)} returns \spad{e1 f(a1) +...+ en f(an)}.
531584`\spad{mapMatrixIfCan(f,{}mat)} applies the function \spad{f} to the coefficients of the entries of \spad{mat} if possible,{} and returns \spad{"failed"} otherwise.
531742`\spad{mapSolve(u,{}f)} \undocumented
531863`\spad{mapUnivariateIfCan(f,{}p(x))} applies the function \spad{f} to the coefficients of \spad{p(x)},{} if possible,{} and returns \spad{"failed"} otherwise.
532009`\spad{mapUnivariate(f,{}p(x))} applies the function \spad{f} to the coefficients of \spad{p(x)}.
532118`\spad{mapUnivariate(f,{}p(x))} applies the function \spad{f} to the coefficients of \spad{p(x)}.
532227`\spad{mapUp!(t,{}f)} traverses balanced binary tree \spad{t} in an "endorder" (left then right then node) fashion returning \spad{t} with the value at each successive interior node of \spad{t} replaced by \spad{f}(\spad{l},{}\spad{r}) where \spad{l} and \spad{r} are the values at the immediate left and right nodes.
532287`\spad{mapUp!(t,{}t1,{}f)} traverses \spad{t} in an "endorder" (left then right then node) fashion returning \spad{t} with the value at each successive interior node of \spad{t} replaced by \spad{f}(\spad{l},{}\spad{r},{}\spad{l1},{}\spad{r1}) where \spad{l} and \spad{r} are the values at the immediate left and right nodes. Values \spad{l1} and \spad{r1} are values at the corresponding nodes of a balanced binary tree \spad{t1},{} of ide
532287`--ntical shape at \spad{t}.
532487`\spad{map(f,{}df)} replaces each coefficient \spad{x} of differential form \spad{df} by \spad{f(x)}.
532594`\spad{map(f,{}p)} \undocumented{}
532943`\spad{map(f,{}p)} changes each coefficient of \spad{p} by the application of \spad{f}.
532994`\spad{map(fn,{}x)} returns \spad{Sum(fn(r_i) w_i)} if \spad{x} writes \spad{Sum(r_i w_i)}.
533049`\axiom{map(func,{} poly)} creates a new polynomial by applying func to every non-zero coefficient of the polynomial poly.
533188`\spad{map}(\spad{f},{} a) applies the map \spad{f} to each coefficient in a. It is assumed that \spad{f} maps 0 to 0
533273`\spad{map(f,{} p)} lifts \spad{f} to the domain of \spad{p} then applies it to \spad{p}.
533681`\spad{map(f,{}x)} \undocumented
533773`\spad{map(f,{}z)} returns the new element created by applying the function \spad{f} to each component of the direct product element \spad{z}.
533841`\spad{map(f,{} v)} applies the function \spad{f} to every element of the vector \spad{v} producing a new vector containing the values.
533944`\spad{map(func,{}frac)} applies the function \spad{func} to the numerator and denominator of the fraction \spad{frac}.
534020`\spad{map(f,{}[x0,{}x1,{}x2,{}...])} returns \spad{[f(x0),{}f(x1),{}f(x2),{}..]}.
534111`\spad{map(fn,{}u)} applies \spad{fn} to each element of list \spad{u} and returns a new list with the results. For example \spad{map(square,{}[1,{}2,{}3]) = [1,{}4,{}9]}.
534175`\spad{map(f,{}a)} applies function \spad{f} to each member of one-dimensional array \spad{a} resulting in a new one-dimensional array over a possibly different underlying domain.
534284`\spad{map(f,{} [(v1,{}a1),{}...,{}(vn,{}an)])} returns the matching result [(\spad{v1},{}\spad{f}(\spad{a1})),{}...,{}(\spad{vn},{}\spad{f}(an))].
534396`\spad{map(f,{}a)} applies function \spad{f} to each member of primitive array \spad{a} resulting in a new primitive array over a possibly different underlying domain.
534490`\spad{map(func,{}frac)} applies the function \spad{func} to the numerator and denominator of \spad{frac}.
534563`\spad{map(f,{}s)} returns a stream whose elements are the function \spad{f} applied to the corresponding elements of \spad{s}. Note: \spad{map(f,{}[x0,{}x1,{}x2,{}...]) = [f(x0),{}f(x1),{}f(x2),{}..]}.
534633`\spad{map(f,{} v)} applies the function \spad{f} to every element of the vector \spad{v} producing a new vector containing the values.
534703`\spad{map(f,{} v)} applies the function \spad{f} to every element of the vector \spad{v} producing a new vector containing the values or \spad{"failed"}.
534805`\spad{map(f,{}x)} \undocumented
534929`\spad{map(f,{}x)} \undocumented
535053`\spad{map(f,{}x)} \undocumented
535168`\spad{map(f,{}g(x))} applies the map \spad{f} to the coefficients of \indented{1}{the Taylor series \spad{g(x)}.}
535270`\spad{map(f,{}g(x))} applies the map \spad{f} to the coefficients of the Laurent series \spad{g(x)}.
535455`\spad{map(f,{}g(x))} applies the map \spad{f} to the coefficients of the Puiseux series \spad{g(x)}.
535640`\spad{map(f,{}ire)} \undocumented
535743`\spad{map(f,{} k)} returns \spad{g = op(f(a1),{}...,{}f(an))} where \spad{k = op(a1,{}...,{}an)}.
535814`\spad{map(f,{}ue)} \undocumented
535911`\spad{map(f,{}ufe)} \undocumented
536128`\spad{map(f,{}ure)} \undocumented
536273`\spad{map(fn,{}u)} maps function \spad{fn} onto the coefficients of the non-zero monomials of \spad{u}.
536330`\spad{map(f,{} ex)} evaluates ex,{} applying \spad{f} to values of type \spad{R} in ex.
536385`\spad{map(fn,{}u)} maps function \spad{fn} onto the coefficients of the non-zero monomials of \spad{u}.
536443`\spad{map(f,{}a)} returns \spad{b},{} where \spad{b(i,{}j) = a(i,{}j)} for all \spad{i},{} \spad{j}.
536518`\spad{map(f,{}a)} returns \spad{b},{} where \spad{b(i,{}j) = f(a(i,{}j))} for all \spad{i,{} j}
536591`\spad{map(fn,{}x)} returns \spad{Sum(fn(r_i) w_i)} if \spad{x} writes \spad{Sum(r_i w_i)}.
536644`\spad{map(fn,{}u)} maps the function \userfun{\spad{fn}} across the factors of \spadvar{\spad{u}} and creates a new factored object. Note: this clears the information flags (sets them to "nil") because the effect of \userfun{\spad{fn}} is clearly not known in general.
536690`\spad{map(f,{} a)} applies \spad{f} to all the constants in \spad{R} appearing in \spad{a}.
536755`\spad{map(f,{}u)} maps \spad{f} onto the coordinates of \spad{u} to get an element in \spad{AS} via identification of the basis of \spad{AR} as beginning part of the basis of \spad{AS}.
536838`\spad{map(f,{}u)} maps \spad{f} onto real and imaginary parts of \spad{u}.
536911`\spad{map(f,{} e)} applies \spad{f} to all the constants appearing in \spad{e}.
536993`\spad{map(fn,{}u)} is used to apply the function \userfun{\spad{fn}} to every factor of \spadvar{\spad{u}}. The new factored object will have all its information flags set to "nil". This function is used,{} for example,{} to coerce every factor base to another type.
537069`\spad{map(f,{}u)} maps \spad{f} onto the coefficients \spad{f} the element \spad{u} of the monoid ring to create an element of a monoid ring with the same monoid \spad{b}.
537157`\spad{map(f,{}u)} maps \spad{f} onto the component parts of the octonion \spad{u}.
537229`\spad{map(f,{} r)} lifts \spad{f} and applies it to \spad{r},{} assuming that \spad{f}(infinity) = infinity.
537335`\spad{map(f,{} r)} lifts \spad{f} and applies it to \spad{r},{} assuming that \spad{f}(plusInfinity) = plusInfinity and that \spad{f}(minusInfinity) = minusInfinity.
537438`\spad{map(f,{}p)} \undocumented
537515`\spad{map(f,{} p)} takes a function \spad{f} from \spad{R} to \spad{S},{} and applies it to each (non-zero) coefficient of a polynomial \spad{p} over \spad{R},{} getting a new polynomial over \spad{S}. Note: since the map is not applied to zero elements,{} it may map zero to zero.
537599`\spad{map(f,{} p)} applies \spad{f} to all the leaves of \spad{p} and returns the result as a pattern over \spad{S}.
537672`\spad{map(f,{} p)} produces a new polynomial as a result of applying the function \spad{f} to every coefficient of the polynomial \spad{p}.
537754`\spad{map(f,{}u)} maps \spad{f} onto the component parts of the quaternion \spad{u}.
537828`\spad{map(f,{}s)} expands the segment \spad{s},{} applying \spad{f} to each value. For example,{} if \spad{s = l..h by k},{} then the list \spad{[f(l),{} f(l+k),{}...,{} f(lN)]} is computed,{} where \spad{lN <= h < lN+k}.
537916`\spad{map(f,{}l..h)} returns a new segment \spad{f(l)..f(h)}.
537989`\spad{map(f,{}v=a..b)} returns the value given by \spad{v=f(a)..f(b)}.
538083`\spad{map(func,{} poly)} creates a new polynomial by applying \spad{func} to every non-zero coefficient of the polynomial poly.
538213`\spad{map(func,{} poly)} creates a new polynomial by applying \spad{func} to every non-zero coefficient of the polynomial poly.
538333`\spad{map(f,{}s)} expands the segment \spad{s},{} applying \spad{f} to each value.
538441`\spad{map(f,{}seg)} returns the new segment obtained by applying \spad{f} to the endpoints of \spad{seg}.
538541`\spad{map(f,{} p)} lifts \spad{f} to \spad{F1} and applies it to \spad{p}.
538642`\spad{map(f,{}d)} \undocumented{}
538791`\spad{map(f,{}i)} \undocumented{}
538930`\spad{map(f,{}m)} applies the function \spad{f} to the elements of the matrix \spad{m}.
539024`\spad{map(f,{}m)} applies the function \spad{f} to the elements of the matrix \spad{m}.
539133`\spad{map(f,{}p)} \undocumented
539206`\spad{map(f,{}x)} maps the function \spad{f} on the entries of \spad{x}
539288`\spad{map(f,{}m)} applies the function \spad{f} to the elements of the matrix \spad{m}.
539414`\spad{map(f,{}a)} applies function \spad{f} to each member of aggregate \spad{a} resulting in a new aggregate over a possibly different underlying domain.
539487`\spad{map(f,{}a)} applies function \spad{f} to each member of aggregate \spad{a},{} creating a new aggregate with a possibly different underlying domain.
539557`\spad{map(f,{}eq)} returns an equation where \spad{f} is applied to the sides of \spad{eq}
539633`\spad{map(f,{}l..h by k)} produces a value of type \spad{L} by applying \spad{f} to each of the succesive elements of the segment,{} that is,{} \spad{[f(l),{} f(l+k),{} ...,{} f(lN)]},{} where \spad{lN <= h < lN+k}.
539696`\spad{map(f,{}u)} returns a copy of \spad{u} with each element \spad{x} replaced by \spad{f}(\spad{x}). For collections,{} \axiom{map(\spad{f},{}\spad{u}) = [\spad{f}(\spad{x}) for \spad{x} in \spad{u}]}.
539754`\spad{map(f,{}eqn)} constructs a new equation by applying \spad{f} to both sides of \spad{eqn}.
539800`\spad{map(f,{}t)} replaces the tuple \spad{t} by \spad{[f(x) for x in t]}.
539851`\spad{map(f,{}ts)} does a componentwise conversion of the tensor \spad{ts} to a tensor with components of type \spad{T}.
539978`\spad{map(f,{} k)} returns \spad{op(f(x1),{}...,{}f(xn))} where \spad{k = op(x1,{}...,{}xn)}.
540110`\spad{map(f,{} p,{} k)} uses the property \spad{p} of the operator of \spad{k},{} in order to lift \spad{f} and apply it to \spad{k}.
540378`\spad{map(varmap,{} coefmap,{} p)} takes a \spad{varmap},{} a mapping from the variables of polynomial \spad{p} into \spad{S},{} \spad{coefmap},{} a mapping from coefficients of \spad{p} into \spad{S},{} and \spad{p},{} and produces a member of \spad{S} using the corresponding arithmetic. in \spad{S}.
540460`\spad{map(f,{}a,{}b)} \undocumented
540572`\spad{map(f,{}a,{}b)} \undocumented
540670`\spad{map(fn,{}list1,{} u2)} applies the binary function \spad{fn} to corresponding elements of lists \spad{u1} and \spad{u2} and returns a list of the results (in the same order). Thus \spad{map(/,{}[1,{}2,{}3],{}[4,{}5,{}6]) = [1/4,{}2/4,{}1/2]}. The computation terminates when the end of either list is reached. That is,{} the length of the result list is equal to the minimum of the lengths of \spad{u1} and \spad{u2}.
540746`\spad{map(f,{}a,{}b)} \undocumented
540844`\spad{map(f,{}st1,{}st2)} returns the stream whose elements are the function \spad{f} applied to the corresponding elements of \spad{st1} and \spad{st2}. Note: \spad{map(f,{}[x0,{}x1,{}x2,{}..],{}[y0,{}y1,{}y2,{}..]) = [f(x0,{}y0),{}f(x1,{}y1),{}..]}.
540928`\spad{map(fn,{}t1,{}t2)} creates a new table \spad{t} from given tables \spad{t1} and \spad{t2} with elements \spad{fn}(\spad{x},{}\spad{y}) where \spad{x} and \spad{y} are corresponding elements from \spad{t1} and \spad{t2} respectively.
541005`\spad{map(f,{} r,{} i)} lifts \spad{f} and applies it to \spad{r},{} assuming that \spad{f}(infinity) = \spad{i}.
541133`\spad{map(f,{}a,{}b)} returns \spad{c},{} where \spad{c} is such that \spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))} for all \spad{i},{} \spad{j}.
541213`\spad{map(f,{}a,{}b)} returns \spad{c},{} where \spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))} for all \spad{i,{} j}
541291`\spad{map(f,{}u,{}v)} returns a new collection \spad{w} with elements \axiom{\spad{z} = \spad{f}(\spad{x},{}\spad{y})} for corresponding elements \spad{x} and \spad{y} from \spad{u} and \spad{v}. Note: for linear aggregates,{} \axiom{\spad{w}.\spad{i} = \spad{f}(\spad{u}.\spad{i},{}\spad{v}.\spad{i})}.
541424`\spad{map(f,{} r,{} p,{} m)} lifts \spad{f} and applies it to \spad{r},{} assuming that \spad{f}(plusInfinity) = \spad{p} and that \spad{f}(minusInfinity) = \spad{m}.
541569`\spad{map(f,{}a,{}b,{}r)} returns \spad{c},{} where \spad{c(i,{}j) = f(a(i,{}j),{}b(i,{}j))} when both \spad{a(i,{}j)} and \spad{b(i,{}j)} exist; else \spad{c(i,{}j) = f(r,{} b(i,{}j))} when \spad{a(i,{}j)} does not exist; else \spad{c(i,{}j) = f(a(i,{}j),{}r)} when \spad{b(i,{}j)} does not exist; otherwise \spad{c(i,{}j) = f(r,{}r)}.
541649`\spad{map(s,{} m,{} newOb,{} offsetX,{} offsetY)} creates a new graph by mapping from this one \spad{newOb} should contain the new list of vertices. \spad{m} should contain a NNI value for each vertex,{} this is the new index into \spad{newOb}. It is allowed that \spad{newOb} may contain less objects than \spad{s} (for surjective mapping) or more objects than \spad{s} (for injective mapping)
541740`\spad{map(f,{}x)} maps the moments and cumulants from the ring \spad{R1} to \spad{R2}.
541837`\spad{mapdiv([a0,{}a1,{}..],{}[b0,{}b1,{}..])} returns \spad{[a0/b0,{}a1/b1,{}..]}.
541935`\spad{mapmult([a0,{}a1,{}..],{}[b0,{}b1,{}..])} returns \spad{[a0*b0,{}a1*b1,{}..]}.
542067`\spad{mask(n)} returns \spad{2^n-1} (an \spad{n} bit mask).
542115`\spad{match?(s,{}t,{}c)} tests if \spad{s} matches \spad{t} except perhaps for multiple and consecutive occurrences of character \spad{c}. Typically \spad{c} is the blank character.
542179`\spad{match(la,{} lb)} creates a map with no default source or target values defined by lists \spad{la} and \spad{lb} of equal length. The target of a source value \spad{x} in \spad{la} is the value \spad{y} with the same index \spad{lb}. Error: if \spad{la} and \spad{lb} are not of equal length. Note: when this map is applied,{} an error occurs when applied to a value missing from \spad{la}.
542240`\spad{match(la,{} lb,{} f)} creates a map defined by lists \spad{la} and \spad{lb} of equal length. The target of a source value \spad{x} in \spad{la} is the value \spad{y} with the same index \spad{lb}. Argument \spad{f} is used as the function to call when the given function argument is not in \spad{la}. The value returned is \spad{f} applied to that argument.
542308`\spad{match(la,{} lb,{} a)} creates a map defined by lists \spad{la} and \spad{lb} of equal length,{} where \spad{a} is used as the default source value if the given one is not in \spad{la}. The target of a source value \spad{x} in \spad{la} is the value \spad{y} with the same index \spad{lb}. Error: if \spad{la} and \spad{lb} are not of equal length.
542366`\spad{match(la,{} lb,{} b)} creates a map defined by lists \spad{la} and \spad{lb} of equal length,{} where \spad{b} is used as the default target value if the given function argument is not in \spad{la}. The target of a source value \spad{x} in \spad{la} is the value \spad{y} with the same index \spad{lb}. Error: if \spad{la} and \spad{lb} are not of equal length.
542429`\spad{match(p,{}s,{}wc)} tests if pattern \axiom{\spad{p}} matches subject \axiom{\spad{s}} where \axiom{\spad{wc}} is a wild card character. If no match occurs,{} the index \axiom{0} is returned; otheriwse,{} the value returned is the first index of the first character in the subject matching the subject (excluding that matched by an initial wild-card). For example,{} \axiom{match("*to*",{}"yorktown",{}\spad{"*"})} returns \axio
542429`--m{5} indicating a successful match starting at index \axiom{5} of \axiom{"yorktown"}.
542503`\spad{match(la,{} lb,{} a,{} f)} creates a map defined by lists \spad{la} and \spad{lb} of equal length. and applies this map to a. The target of a source value \spad{x} in \spad{la} is the value \spad{y} with the same index \spad{lb}. Argument \spad{f} is a default function to call if a is not in \spad{la}. The value returned is then obtained by applying \spad{f} to argument a.
542568`\spad{match(la,{} lb,{} a,{} b)} creates a map defined by lists \spad{la} and \spad{lb} of equal length. and applies this map to a. The target of a source value \spad{x} in \spad{la} is the value \spad{y} with the same index \spad{lb}. Argument \spad{b} is the default target value if a is not in \spad{la}. Error: if \spad{la} and \spad{lb} are not of equal length.
542628`\spad{mathieu11 constructs} the mathieu group acting on the integers 1,{}...,{}11.
542707`\spad{mathieu11(\spad{li})} constructs the mathieu group acting on the 11 integers given in the list {\em \spad{li}}. Note: duplicates in the list will be removed. error,{} if {\em \spad{li}} has less or more than 11 different entries.
542799`\spad{mathieu12 constructs} the mathieu group acting on the integers 1,{}...,{}12.
542878`\spad{mathieu12(\spad{li})} constructs the mathieu group acting on the 12 integers given in the list {\em \spad{li}}. Note: duplicates in the list will be removed Error: if {\em \spad{li}} has less or more than 12 different entries.
542970`\spad{mathieu22 constructs} the mathieu group acting on the integers 1,{}...,{}22.
543049`\spad{mathieu22(\spad{li})} constructs the mathieu group acting on the 22 integers given in the list {\em \spad{li}}. Note: duplicates in the list will be removed. Error: if {\em \spad{li}} has less or more than 22 different entries.
543141`\spad{mathieu23 constructs} the mathieu group acting on the integers 1,{}...,{}23.
543220`\spad{mathieu23(\spad{li})} constructs the mathieu group acting on the 23 integers given in the list {\em \spad{li}}. Note: duplicates in the list will be removed. Error: if {\em \spad{li}} has less or more than 23 different entries.
543312`\spad{mathieu24 constructs} the mathieu group acting on the integers 1,{}...,{}24.
543391`\spad{mathieu24(\spad{li})} constructs the mathieu group acting on the 24 integers given in the list {\em \spad{li}}. Note: duplicates in the list will be removed. Error: if {\em \spad{li}} has less or more than 24 different entries.
543483`\spad{matrixConcat3D(s,{}x,{}y)} concatenates two 3-\spad{D} matrices along a specified axis
543557`\spad{matrixDimensions(x)} returns the dimensions of a matrix
543647`\spad{matrixGcd(mat,{}sing,{}n)} is \spad{gcd(sing,{}g)} where \spad{g} is the \spad{gcd} of the entries of the \spad{n}-by-\spad{n} upper-triangular matrix \spad{mat}.
543798`\spad{matrix(llf)} makes \spad{llf} (a list of lists of forms) into a form which displays as a matrix.
543851`\spad{matrix(l)} converts the list of lists \spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.
543918`\spad{matrix(l)} converts the list of lists \spad{l} to a matrix,{} where the list of lists is viewed as a list of the rows of the matrix.
544000`\spad{matrix(qf)} creates a square matrix from the quadratic form \spad{qf}.
544064`\spad{maxColIndex(m)} returns the index of the 'last' column of the matrix \spad{m}.
544145`\spad{maxColIndex(m)} returns the index of the 'last' column of the array \spad{m}
544224`\spad{maxDegree returns} the specified maxDegree.
544329`\spad{maxDegree(d)} specifies the maximum degree of the coefficient polynomials in an algebraic differential equation or a recursion with polynomial coefficients. For rational functions with an exponential term,{} \spad{maxDegree} bounds the degree of the denominator polynomial. This option is expressed in the form \spad{maxDegree == d}.
544409`\spad{maxDerivative returns} the specified maxDerivative.
544518`\spad{maxDerivative(d)} specifies the maximum derivative in an algebraic differential equation. This option is expressed in the form \spad{maxDerivative == d}.
544774`\spad{maxIndex(u)} returns the maximum index \spad{i} of aggregate \spad{u}. Note: in general,{} \axiom{maxIndex(\spad{u}) = reduce(max,{}[\spad{i} for \spad{i} in indices \spad{u}])}; if \spad{u} is a list,{} \axiom{maxIndex(\spad{u}) = \#u}.
544860`\spad{maxLevel returns} the specified maxLevel.
544964`\spad{maxLevel(d)} specifies the maximum number of recursion levels operators guessProduct and guessSum will be applied. This option is expressed in the form spad{maxLevel \spad{==} \spad{d}}.
545043`\spad{maxMixedDegree returns} the specified maxMixedDegree.
545134`\spad{maxMixedDegree(d)} specifies the maximum \spad{q}-degree of the coefficient polynomials in a recurrence with polynomial coefficients,{} in the case of mixed shifts. Although slightly inconsistent,{} maxMixedDegree(0) specifies that no mixed shifts are allowed. This option is expressed in the form \spad{maxMixedDegree == d}.
545200`\spad{maxPoints3D()} returns the maximum number of points in a plot.
545245`\spad{maxPoints()} returns the maximum number of points in a plot
545286`\spad{maxPoints()} returns the maximum number of points in a plot.
545339`\spad{maxPoints()} sets the maximum number of points in a plot.
545399`\spad{maxPower returns} the specified maxPower.
545500`\spad{maxPower(d)} specifies the maximum degree in an algebraic differential equation. For example,{} the degree of (\spad{f}\spad{''})\spad{^3} \spad{f'} is 4. maxPower(\spad{-1}) specifies that the maximum exponent can be arbitrary. This option is expressed in the form \spad{maxPower == d}.
545576`\spad{maxRowIndex(m)} returns the index of the 'last' row of the matrix \spad{m}.
545657`\spad{maxRowIndex(m)} returns the index of the 'last' row of the array \spad{m}
545736`\spad{maxShift returns} the specified maxShift.
545840`\spad{maxShift(d)} specifies the maximum shift in a recurrence equation. This option is expressed in the form \spad{maxShift == d}.
545919`\spad{maxSubst returns} the specified maxSubst.
546020`\spad{maxSubst(d)} specifies the maximum degree of the monomial substituted into the function we are looking for. That is,{} if \spad{maxSubst == d},{} we look for polynomials such that \$\spad{p}(\spad{f}(\spad{x}),{} \spad{f}(\spad{x^2}),{} ...,{} \spad{f}(\spad{x^d}))\spad{=0}\$. equation. This option is expressed in the form \spad{maxSubst == d}.
546096`\spad{max()} returns the maximum floating point number.
546208`\spad{max()} returns the largest single integer.
546347`\spad{max(s)} returns index of the vertex which can be reached from all other vertices. Gives 0 if no such node exists or if it is not unique,{} if there is a loop for instance.
546405`\spad{max(u)} returns the largest element of aggregate \spad{u}.
546470`\spad{max(q)} returns the maximum element of priority queue \spad{q}.
546639`\spad{max(s,{}sub)} returns index of the vertex which can be reached from a given subset of the vertices. Gives 0 if no such node exists or if it is not unique,{} if there is a loop for instance.
546722`\spad{max(a,{}b)} returns a point whose value in each dimension is the maximum. So if a has the largest \spad{x} value and \spad{b} has the largest \spad{y} value then the result will combine these. This is useful for constructing boundary boxes around sets of points.
546767`\spad{max(x,{}y)} returns the maximum of \spad{x} and \spad{y} relative to \spad{"<"}.
546808`\spad{maxdeg(p)} returns the greatest word occurring in the polynomial \spad{p} with a non-zero coefficient. An error is produced if \spad{p} is zero.
546858`\spad{maxdeg(p)} returns the greatest leading word in the support of \spad{p}.
546929`\spad{maximumExponent()} returns the maximum exponent in the model
546984`\spad{maximumExponent(e)} sets the maximum exponent in the model to \spad{e}
547046`\spad{maxint()} returns the maximum integer in the model
547102`\spad{maxint(u)} sets the maximum integer in the model to \spad{u}
547173`\spad{maxrank(r)} returns the maximum rank in the list \spad{r} of regimes
547374`\spad{maxrow(a,{}b,{}c,{}d,{}e)} is an auxiliary function for \spad{mr}
547677`\axiom{mdeg(\spad{p})} returns an error if \axiom{\spad{p}} is \axiom{0},{} otherwise,{} if \axiom{\spad{p}} belongs to \axiom{\spad{R}} returns \axiom{0},{} otherwise,{} returns the degree of \axiom{\spad{p}} in its main variable.
547756`\spad{meatAxe(aG)} calls {\em meatAxe(aG,{}false,{}25,{}7)} returns a 2-list of representations as follows. All matrices of argument \spad{aG} are assumed to be square and of equal size. Then \spad{aG} generates a subalgebra,{} say \spad{A},{} of the algebra of all square matrices of dimension \spad{n}. {\em V R} is an A-module in the usual way. meatAxe(\spad{aG}) creates at most 25 r
547756`--andom elements of the algebra,{} tests them for singularity. If singular,{} it tries at most 7 elements of its kernel to generate a proper submodule. If successful a list which contains first the list of the representations of the submodule,{} then a list of the representations of the factor module is returned. Otherwise,{} if we know that all the kernel is already scanned,{} Norton\spad{'s} irreducibility test can be used either to prove irreducibility or to find the splitting. Notes: the fir
547756`--st 6 tries use Parker\spad{'s} fingerprints. Also,{} 7 covers the case of three-dimensional kernels over the field with 2 elements.
547876`\spad{meatAxe(aG,{} randomElements)} calls {\em meatAxe(aG,{}false,{}6,{}7)},{} only using Parker\spad{'s} fingerprints,{} if {\em randomElemnts} is \spad{false}. If it is \spad{true},{} it calls {\em meatAxe(aG,{}true,{}25,{}7)},{} only using random elements. Note: the choice of 25 was rather arbitrary. Also,{} 7 covers the case of three-dimensional kernels over the field wit
547876`--h 2 elements.
548004`\spad{meatAxe(aG,{} numberOfTries)} calls {\em meatAxe(aG,{}true,{}numberOfTries,{}7)}. Notes: 7 covers the case of three-dimensional kernels over the field with 2 elements.
548140`\spad{meatAxe(aG,{}randomElements,{}numberOfTries,{} maxTests)} returns a 2-list of representations as follows. All matrices of argument \spad{aG} are assumed to be square and of equal size. Then \spad{aG} generates a subalgebra,{} say \spad{A},{} of the algebra of all square matrices of dimension \spad{n}. {\em V R} is an A-module in the usual way. meatAxe(\sp
548140`--ad{aG},{}\spad{numberOfTries},{} maxTests) creates at most {\em numberOfTries} random elements of the algebra,{} tests them for singularity. If singular,{} it tries at most {\em maxTests} elements of its kernel to generate a proper submodule. If successful,{} a 2-list is returned: first,{} a list containing first the list of the representations of the submodule,{} then a list of the representations of the factor module. Otherwise,{} if we know that all the kernel is already scanned,{} Norton\s
548140`--pad{'s} irreducibility test can be used either to prove irreducibility or to find the splitting. If {\em randomElements} is {\em false},{} the first 6 tries use Parker\spad{'s} fingerprints.
548284`\axiom{medial(\spad{ps})} returns the same as \axiom{medialSet(\spad{ps},{}initiallyReduced?,{}initiallyReduce)}.
548370`\axiom{medialSet(\spad{ps},{}redOp?,{}redOp)} returns \axiom{\spad{bs}} a basic set (in Wu Wen Tsun sense \spad{w}.\spad{r}.\spad{t} the reduction-test \axiom{redOp?}) of some set generating the same ideal as \axiom{\spad{ps}} (with rank not higher than any basic set of \axiom{\spad{ps}}),{} if no non-zero constant polynomials appear during the computatioms,{} else \axiom{"failed"} is returned.
548370`-- In the former case,{} \axiom{\spad{bs}} has to be understood as a candidate for being a characteristic set of \axiom{\spad{ps}}. In the original algorithm,{} \axiom{\spad{bs}} is simply a basic set of \axiom{\spad{ps}}.
548480`\spad{meijerG(la,{} lb,{} lc,{} ld,{} z)} is the meijerG function
548601`\spad{meijerG(la,{} lb,{} lc,{} ld,{} z)} is the meijerG function.
548722`\spad{member?(p,{} s)} returns \spad{true} is \spad{p} is in \spad{s},{} \spad{false} otherwise.
549086`\spad{member?(v,{}B)} tests if the vector \spad{v} is in the subspace spaned by \spad{B}.
549152`\spad{member?(pp,{}gp)} answers the question,{} whether the permutation {\em pp} is in the group {\em gp} or not.
549223`\spad{member?(x,{}u)} tests if \spad{x} is a member of \spad{u}. For collections,{} \axiom{member?(\spad{x},{}\spad{u}) = reduce(or,{}[x=y for \spad{y} in \spad{u}],{}\spad{false})}. However,{} \axiom{member?(\spad{x},{}\spad{u})} returns as soon as it finds a member.
549387`\spad{members(u)} returns a list of the consecutive elements of \spad{u}. For collections,{} \axiom{parts([\spad{x},{}\spad{y},{}...,{}\spad{z}]) = (\spad{x},{}\spad{y},{}...,{}\spad{z})}.
549470`\spad{members(ms)} returns a list of the elements of \spad{ms} {\em without} their multiplicity. See also \spadfun{parts}.
549620`\spad{merge!(u,{}v)} destructively merges \spad{u} and \spad{v} in ascending order.
549699`\spad{merge!(q,{}q1)} destructively changes priority queue \spad{q} to include the values from priority queue \spad{q1}.
549890`\spad{merge!(p,{}u,{}v)} destructively merges \spad{u} and \spad{v} using predicate \spad{p}.
550184`\spad{mergeDifference(l1,{}l2)} returns a list of elements in \spad{l1} not present in \spad{l2}. Assumes lists are ordered and all \spad{x} in \spad{l2} are also in \spad{l1}.
550255`\spad{mergeFactors(u,{}v)} is used when the factorizations of \spadvar{\spad{u}} and \spadvar{\spad{v}} are known to be disjoint,{} \spadignore{e.g.} resulting from a content/primitive part split. Essentially,{} it creates a new factored object by multiplying the units together and appending the lists of factors.
550504`\spad{merge(ls)} a list of subspaces,{} \spad{ls},{} into one subspace.
550553`\spad{merge([s1,{}s2,{}...,{}sn])} will create a new \spadtype{ThreeSpace} that has the components of all the ones in the list; Groupings of components into composites are maintained.
550765`\spad{merge(a,{}b)} returns sum : union (not necessarily disjoint) of nodes with arrows merged in from appropriate input,{} if arrow exists from both inputs then it will be duplicated.
550812`\spad{merge(s1,{}s2)} the subspaces \spad{s1} and \spad{s2} into a single subspace.
550858`\spad{merge(u,{}v)} merges \spad{u} and \spad{v} in ascending order. Note: \axiom{merge(\spad{u},{}\spad{v}) = merge(\spad{<=},{}\spad{u},{}\spad{v})}.
550932`\spad{merge(q1,{}q2)} returns combines priority queues \spad{q1} and \spad{q2} to return a single priority queue \spad{q}.
550990`\spad{merge(s1,{}s2)} will create a new \spadtype{ThreeSpace} that has the components of \spad{s1} and \spad{s2}; Groupings of components into composites are maintained.
551244`\spad{merge(p,{}a,{}b)} returns an aggregate \spad{c} which merges \axiom{a} and \spad{b}. The result is produced by examining each element \spad{x} of \axiom{a} and \spad{y} of \spad{b} successively. If \axiom{\spad{p}(\spad{x},{}\spad{y})} is \spad{true},{} then \spad{x} is inserted into the result; otherwise \spad{y} is inserted. If \spad{x} is chosen,{} the next element of \axiom{a} is examined,{} and so on. When all the elemen
551244`--ts of one aggregate are examined,{} the remaining elements of the other are appended. For example,{} \axiom{merge(<,{}[1,{}3],{}[2,{}7,{}5])} returns \axiom{[1,{}2,{}3,{}7,{}5]}.
551316`\spad{mesh?(s)} returns \spad{true} if the \spadtype{ThreeSpace} \spad{s} is composed of one component,{} a mesh comprising a list of curves which are lists of points,{} or returns \spad{false} if otherwise
551372`\spad{meshFun2Var(f,{}g,{}s1,{}s2,{}l)} \undocumented
551632`\spad{meshPar1Var(s,{}t,{}u,{}f,{}s1,{}l)} \undocumented
551849`\spad{meshPar2Var(f,{}s1,{}s2,{}l)} \undocumented
552046`\spad{meshPar2Var(sp,{}f,{}s1,{}s2,{}l)} \undocumented
552267`\spad{meshPar2Var(f,{}g,{}h,{}j,{}s1,{}s2,{}l)} \undocumented
552605`\spad{mesh([[p0],{}[p1],{}...,{}[pn]])} creates a surface defined by a list of curves which are lists,{} \spad{p0} through \spad{pn},{} of points,{} and returns a \spadtype{ThreeSpace} whose component is the surface.
552673`\spad{mesh(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a single surface component defined by a list curves which contain lists of points,{} and if so,{} returns the list of lists of points; An error is signaled otherwise.
552741`\spad{mesh([[p0],{}[p1],{}...,{}[pn]],{} close1,{} close2)} creates a surface defined over a list of curves,{} \spad{p0} through \spad{pn},{} which are lists of points; the booleans \spad{close1} and \spad{close2} indicate how the surface is to be closed: \spad{close1} set to \spad{true} means that each individual list (a curve) is to be closed (that is,{} the last point of the list is to be connected to the first point
552741`--); \spad{close2} set to \spad{true} means that the boundary at one end of the surface is to be connected to the boundary at the other end (the boundaries are defined as the first list of points (curve) and the last list of points (curve)); the \spadtype{ThreeSpace} containing this surface is returned.
552825`\spad{mesh(s,{}[ [[r10]...,{}[r1m]],{} [[r20]...,{}[r2m]],{}...,{} [[rn0]...,{}[rnm]] ],{} close1,{} close2)} adds a surface component to the \spadtype{ThreeSpace} \spad{s},{} which is defined over a rectangular domain of size \spad{WxH} where \spad{W} is the number of lists of points from the domain \spad{PointDomain(R)} and \spad{H} is the number of elements in each of those lists; the booleans \spad{close1} and \sp
552825`--ad{close2} indicate how the surface is to be closed: if \spad{close1} is \spad{true} this means that each individual list (a curve) is to be closed (\spadignore{i.e.} the last point of the list is to be connected to the first point); if \spad{close2} is \spad{true},{} this means that the boundary at one end of the surface is to be connected to the boundary at the other end (the boundaries are defined as the first list of points (curve) and the last list of points (curve)).
552911`\spad{mesh(s,{}[ [[r10]...,{}[r1m]],{} [[r20]...,{}[r2m]],{}...,{} [[rn0]...,{}[rnm]] ],{} [props],{} prop)} adds a surface component to the \spadtype{ThreeSpace} \spad{s},{} which is defined over a rectangular domain of size \spad{WxH} where \spad{W} is the number of lists of points from the domain \spad{PointDomain(R)} and \spad{H} is the number of elements in each of those 
552911`--lists; lprops is the list of the subspace component properties for each curve list,{} and prop is the subspace component property by which the points are defined.
553039`\spad{mesh(s,{}[[p0],{}[p1],{}...,{}[pn]],{} close1,{} close2)} adds a surface component to the \spadtype{ThreeSpace},{} which is defined over a list of curves,{} in which each of these curves is a list of points. The boolean arguments \spad{close1} and \spad{close2} indicate how the surface is to be closed. Argument \spad{close1} equal \spad{true} means that each individual list (a curve) is to be closed,{} \spadign
553039`--ore{i.e.} the last point of the list is to be connected to the first point. Argument \spad{close2} equal \spad{true} means that the boundary at one end of the surface is to be connected to the boundary at the other end,{} \spadignore{i.e.} the boundaries are defined as the first list of points (curve) and the last list of points (curve).
553126`\spad{mesh(s,{}[[p0],{}[p1],{}...,{}[pn]],{}[props],{}prop)} adds a surface component,{} defined over a list curves which contains lists of points,{} to the \spadtype{ThreeSpace} \spad{s}; props is a list which contains the subspace component properties for each surface parameter,{} and \spad{prop} is the subspace component property by which the points are defined.
553255`\spad{messagePrint(s)} prints \spad{s} without string quotes. Note: \spad{messagePrint(s)} is equivalent to \spad{print message(s)}.
553308`\spad{message(s)} creates an form with no string quotes from string \spad{s}.
553354`\axiom{middle(rootChar)} is the middle of the isolating interval
553448`\spad{midpoint(int)} returns the midpoint of the interval \spad{int}.
553568`\spad{midpoints(isolist)} returns the list of midpoints for the list of intervals \spad{isolist}.
553701`\axiom{mightHaveRoots(\spad{p},{}\spad{r})} is \spad{false} if \axiom{\spad{p}.\spad{r}} is not 0
553812`\spad{minColIndex(m)} returns the index of the 'first' column of the matrix \spad{m}.
553893`\spad{minColIndex(m)} returns the index of the 'first' column of the array \spad{m}
553972`\spad{minGbasis }\undocumented
554187`\spad{minIndex(u)} returns the minimum index \spad{i} of aggregate \spad{u}. Note: in general,{} \axiom{minIndex(a) = reduce(min,{}[\spad{i} for \spad{i} in indices a])}; for lists,{} \axiom{minIndex(a) = 1}.
554273`\spad{minPoints3D()} returns the minimum number of points in a plot.
554318`\spad{minPoints()} returns the minimum number of points in a plot
554359`\spad{minPoints()} returns the minimum number of points in a plot.
554412`\spad{minPoints()} sets the minimum number of points in a plot.
554472`\spad{minPol }\undocumented
554653`\spad{minPol }\undocumented
554891`\spad{minPoly(k)} returns the defining polynomial of \spad{k}.
555008`\spad{minPoly(k)} returns \spad{p} such that \spad{p(k) = 0}.
555103`\spad{minRowIndex(m)} returns the index of the 'first' row of the matrix \spad{m}.
555184`\spad{minRowIndex(m)} returns the index of the 'first' row of the array \spad{m}
555263`\spad{min()} returns the minimum floating point number.
555375`\spad{min()} returns the smallest single integer.
555514`\spad{min(s)} returns index of the vertex which can reach to all other vertices. Gives 0 if no such node exists or if it is not unique,{} if there is a loop for instance.
555572`\spad{min(u)} returns the smallest element of aggregate \spad{u}.
555637`\spad{min(u)} returns the smallest entry in the multiset aggregate \spad{u}.
555808`\spad{min(s,{}sub)} returns index of the vertex which can reach to a given subset of the vertices. Gives 0 if no such node exists or if it is not unique,{} if there is a loop for instance.
555891`\spad{min(a,{}b)} returns a point whose value in each dimension is the minimum. So if a has the smallest \spad{x} value and \spad{b} has the smallest \spad{y} value then the result will combine these. This is useful for constructing boundary boxes around sets of points.
555936`\spad{min(x,{}y)} returns the minimum of \spad{x} and \spad{y} relative to \spad{"<"}.
555977`\spad{mindegTerm(x)} returns the term whose word is \spad{mindeg(x)}.
556063`\spad{mindeg(p)} returns the smallest word occurring in the polynomial \spad{p} with a non-zero coefficient. An error is produced if \spad{p} is zero.
556113`\spad{mindeg(x)} returns the little word which appears in \spad{x}. Error if \spad{x=0}.
556421`\spad{minimalPolynomial(x)} \undocumented{} See \axiomFunFrom{minimalPolynomial}{FiniteAlgebraicExtensionField}
556533`\spad{minimalPolynomial(a)} returns the minimal polynomial of an element \spad{a} over the ground field \spad{F}.
556634`\spad{minimalPolynomial(a)} returns the minimal polynomial of \spad{a}.
556825`\spad{minimalPolynomial(x,{}n)} computes the minimal polynomial of \spad{x} over the field of extension degree \spad{n} over the ground field \spad{F}.
556956`\spad{minimize(I)} returns a reduced set of generators for \spad{I}.
557096`\spad{minimumDegree(p)} gives the least exponent of a non-zero term of polynomial \spad{p}. Error: if applied to 0.
557161`\spad{minimumDegree(l)} is the smallest \spad{k} such that \spad{a(k) \~= 0} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
557241`\spad{minimumDegree(l)} is the smallest \spad{k} such that \spad{a(k) ~= 0} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
557550`\spad{minimumDegree(p,{} lv)} gives the list of minimum degrees of the polynomial \spad{p} with respect to each of the variables in the list \spad{lv}
557653`\spad{minimumDegree(p,{}v)} gives the minimum degree of polynomial \spad{p} with respect to \spad{v},{} \spadignore{i.e.} viewed a univariate polynomial in \spad{v}
557744`\spad{minimumExponent()} returns the minimum exponent in the model
557799`\spad{minimumExponent(e)} sets the minimum exponent in the model to \spad{e}
557861`\spad{minordet(m)} computes the determinant of the matrix \spad{m} using minors. Error: if the matrix is not square.
557933`\spad{minordet(m)} computes the determinant of the matrix \spad{m} using minors. Error: if the matrix is not square.
558012`\spad{minordet(m)} computes the determinant of the matrix \spad{m} using minors.
558102`\spad{minrank(r)} returns the minimum rank in the list \spad{r} of regimes
558303`\spad{minset(sl)} returns the sublist of \spad{sl} consisting of the minimal lists (with respect to inclusion) in the list \spad{sl} of lists
558399`\spad{minus!(c,{}a)} computes \spad{-a} and stores the result in the matrix \spad{c}. Error: if a and \spad{c} do not have the same dimensions.
558489`\spad{!minus!(c,{}a,{}b)} computes the matrix difference \spad{a - b} and stores the result in the matrix \spad{c}. Error: if \spad{a},{} \spad{b},{} and \spad{c} do not have the same dimensions.
558589`\spad{minusInfinity()} returns minusInfinity.
558657`\spad{minusInfinity()} returns -infinity.
558713`\axiom{mirror(\spad{g})} is the mirror of the internal representation of \axiom{\spad{g}}.
558775`\axiom{mirror(\spad{x})} returns the reversed word of \axiom{\spad{x}}. That is \axiom{\spad{x}} itself if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true} and \axiom{mirror(\spad{z}) * mirror(\spad{y})} if \axiom{\spad{x}} is \axiom{\spad{y*z}}.
558819`\spad{mirror(x)} returns the reversed word of \spad{x}.
558870`\axiom{mirror(\spad{x})} returns \axiom{Sum(r_i mirror(w_i))} if \axiom{\spad{x}} is \axiom{Sum(r_i w_i)}.
558925`\spad{mirror(x)} returns \spad{Sum(r_i mirror(w_i))} if \spad{x} writes \spad{Sum(r_i w_i)}.
558974`\spad{mix(l)} \undocumented
559078`\spad{mkAnswer(r,{}l,{}ne)} creates an integration result from a rational part \spad{r},{} a logarithmic part \spad{l},{} and a non-elementary part \spad{ne}.
559277`\spad{mkIntegral(p(x,{}y))} returns \spad{[c(x),{} q(x,{}z)]} such that \spad{z = c * y} is integral. The algebraic relation between \spad{x} and \spad{y} is \spad{p(x,{} y) = 0}. The algebraic relation between \spad{x} and \spad{z} is \spad{q(x,{} z) = 0}.
559374`\spad{mkPrim(f,{} x)} makes the logs in \spad{f} which are linear in \spad{x} primitive with respect to \spad{x}.
559486`\spad{mkcomm(i)} \undocumented{}
559532`\spad{mkcomm(i,{}j)} \undocumented{}
559576`\spad{modTree(r,{}l)} \undocumented{}
559633`\spad{modifyPointData(v,{}ind,{}pt)} takes the viewport,{} \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport},{} and places the data point,{} \spad{pt} into the list of points database of \spad{v} at the index location given by \spad{ind}.
559737`\spad{modifyPointData(s,{}i,{}p)} changes the point at the indexed location \spad{i} in the \spadtype{ThreeSpace},{} \spad{s},{} to that of point \spad{p}. This is useful for making changes to a point which has been transformed.
559826`\spad{modifyPoint(s,{}\spad{li},{}i)} replaces an existing point in the 3 dimensional subspace,{} \spad{s},{} with the 4 dimensional point indicated by the index location,{} \spad{i}. The list of non negative integers,{} \spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the existing point is to be modified. An error message occurs if \spad{s} is empt
559826`--y,{} otherwise the subspace \spad{s} is returned with the point modification.
559919`\spad{modifyPoint(s,{}\spad{li},{}p)} replaces an existing point in the 3 dimensional subspace,{} \spad{s},{} with the 4 dimensional point,{} \spad{p}. The list of non negative integers,{} \spad{li},{} dictates the path to follow,{} or,{} to look at it another way,{} points to the component in which the existing point is to be modified. An error message occurs if \spad{s} is empty,{} otherwise the subspace \spad{s} is re
559919`--turned with the point modification.
560002`\spad{modifyPoint(s,{}ind,{}p)} modifies the point referenced by the index location,{} \spad{ind},{} by replacing it with the point,{} \spad{p} in the 3 dimensional subspace,{} \spad{s}. An error message occurs if \spad{s} is empty,{} otherwise the subspace \spad{s} is returned with the point modification.
560285`\spad{modularFactor(f)} chooses a "good" prime and returns the factorization of \spad{f} modulo this prime in a form that may be used by \spadfunFrom{completeHensel}{GeneralHenselPackage}. If prime is zero it means that \spad{f} has been proved to be irreducible over the integers or that \spad{f} is a unit (\spadignore{i.e.} 1 or \spad{-1}). \spad{f} shall be primitive (\spadignore{i.e.} content(\spad{p})
560285`--\spad{=1}) and square free (\spadignore{i.e.} without repeated factors).
560384`\spad{modularGcdPrimitive(listf)} computes the \spad{gcd} of the list of primitive polynomials \spad{listf} by modular methods.
560468`\spad{modularGcd(listf)} computes the \spad{gcd} of the list of polynomials \spad{listf} by modular methods.
560543`\spad{modularInvariantJ(tau)} computes modular invariant \spad{j},{} that is \spad{1728*g2^3/(g2^3 - 27*g3^2)} where \spad{g2},{} \spad{g3} are invariants corresponding to half periods \spad{w1},{} \spad{w2} such that \spad{tau = w1/w2}.
560631`\spad{moduleSum(m1,{}m2)} returns the sum of two modules in the framed algebra \spad{F}. Each module \spad{\spad{mi}} is represented as follows: \spad{F} is a framed algebra with \spad{R}-module basis \spad{w1,{}w2,{}...,{}wn} and \spad{\spad{mi}} is a record \spad{[basis,{}basisDen,{}basis
560631`--Inv]}. If \spad{basis} is the matrix \spad{(aij,{} i = 1..n,{} j = 1..n)},{} then a basis \spad{v1,{}...,{}vn} for \spad{\spad{mi}} is given by \spad{\spad{vi} = (1/basisDen) * sum(aij * wj,{} j = 1..n)},{} \spadignore{i.e.} the \spad{i}th row of 'basis' contains the coordinates of the \spad{i}th basis vector. Similarly,{} the \spad{i}th row of the matrix \spad{basisInv} contains the coordinates of \spad{\spad{wi}} with respect to the basis \spad{v1,{}...,{}vn}: if \spad{basisInv} is the matri
560631`--x \spad{(bij,{} i = 1..n,{} j = 1..n)},{} then \spad{\spad{wi} = sum(bij * vj,{} j = 1..n)}.
560847`\spad{module(I)} returns \spad{I} viewed has a module over \spad{R}.
560953`\spad{module([f1,{}...,{}fn])} = the module generated by \spad{(f1,{}...,{}fn)} over \spad{R}.
561020`\spad{modulo(x)} returns a,{} where \spad{x = a + b p}.
561080`\spad{modulus()} \undocumented
561126`\spad{modulus()} returns the value of \spad{p}.
561184`\spad{modulus(x)} \undocumented
561271`\spad{modulus(x)} \undocumented
561348`\spad{modulus(x)} \undocumented
561424`\spad{moebiusMu(n)} returns the Moebius function \spad{mu(n)}. \spad{mu(n)} is either \spad{-1},{}0 or 1 as follows: \spad{mu(n) = 0} if \spad{n} is divisible by a square > 1,{} \spad{mu(n) = (-1)^k} if \spad{n} is square-free and has \spad{k} distinct prime divisors.
561496`\spad{moebius(a,{}b,{}c,{}d)} returns \spad{matrix [[a,{}b],{}[c,{}d]]}.
561552`\spad{moment2Stransform(x)} returns the Puiseux and Laurent order and coefficients of the \spad{S} transform of \spad{x}
561726`\spad{moment2booleancumulant(mm)} computes the sequence of boolean cumulants the sequence of moments \spad{mm}
561807`\spad{moment2cumulant(mm)} computes the sequence of classical cumulants the sequence of moments \spad{mm}
561881`\spad{moment2freecumulant(mm)} computes the sequence of free cumulants the sequence of moments \spad{mm}
561959`\spad{moment2jacobi2(mm)} computes the Jacobi parameters as stream of pairs \$(an,{} \spad{bn})\$.
562058`\spad{moment2jacobi(mm)} computes the Jacobi parameters as pair of streams.
562164`\spad{moment2monotonecumulant(x)} returns the sequence of monotone cumulants of the moment sequence \spad{x}
562279`\spad{moment2nthjacobi(mm)} computes the list of Jacobi parameters from the list of moments \spad{mm} as far as possible.
562380`\spad{moment(x,{} n)} returns the \spad{n}-th moment of the distribution \spad{x}
562452`\spad{moments(x)} returns the sequence of moments of the distribution \spad{x}
562583`\spad{monic?(p)} tests if \spad{p} is monic (\spadignore{i.e.} leading coefficient equal to 1).
562655`\axiom{monic?(\spad{p})} returns \spad{false} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns \spad{true} iff \axiom{\spad{p}} is monic as a univariate polynomial in its main variable.
562725`\spad{monicCompleteDecompose(f)} returns a list of factors of \spad{f} for the functional decomposition ([ \spad{f1},{} ...,{} \spad{fn} ] means \spad{f} = \spad{f1} \spad{o} ... \spad{o} \spad{fn}).
562824`\spad{monicDecomposeIfCan(f)} returns a functional decomposition of the monic polynomial \spad{f} of "failed" if it has not found any.
562952`\spad{monicDivide(p,{}q)} divide the polynomial \spad{p} by the monic polynomial \spad{q},{} returning the pair \spad{[quotient,{} remainder]}. Error: if \spad{q} isn\spad{'t} monic.
563152`\spad{monicDivide(a,{}b,{}v)} divides the polynomial a by the polynomial \spad{b},{} with each viewed as a univariate polynomial in \spad{v} returning both the quotient and remainder. Error: if \spad{b} is not monic with respect to \spad{v}.
563258`\spad{monicLeftDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. \spad{b} must be monic. This process is called ``left division\spad{''}.
563387`\spad{monicLeftDivide(a,{} b,{} sigma)} returns the pair \spad{[q,{}r]} such that \spad{a = b*q + r} and the degree of \spad{r} is less than the degree of \spad{b}. \spad{b} must be monic. This process is called ``left division\spad{''}. \spad{\sigma} is the morphism to use.
563601`\axiom{monicModulo(a,{}\spad{b})} returns \axiom{\spad{r}} such that \axiom{\spad{r}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{b}} and \axiom{\spad{b}} divides \axiom{a \spad{-r}} where \axiom{\spad{b}} is monic.
563672`\axiom{monicModulo(a,{}\spad{b})} computes \axiom{a mod \spad{b}},{} if \axiom{\spad{b}} is monic as univariate polynomial in its main variable.
563745`\spad{monicRightDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. \spad{b} must be monic. This process is called ``right division\spad{''}.
563875`\spad{monicRightDivide(a,{} b,{} sigma)} returns the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. \spad{b} must be monic. This process is called ``right division\spad{''}. \spad{\sigma} is the morphism to use.
564022`\spad{monicRightFactorIfCan(f,{}d)} returns a candidate to be the monic right factor (\spad{h} in \spad{f} = \spad{g} \spad{o} \spad{h}) of degree \spad{d} of a functional decomposition of the polynomial \spad{f} or \spad{"failed"} if no such candidate.
564149`\spad{monomRDE(f,{}g,{}D)} returns \spad{[A,{} B,{} C,{} T]} such that \spad{y' + f y = g} has a solution if and only if \spad{y = Q / T},{} where \spad{Q} satisfies \spad{A Q' + B Q = C} and has no normal pole. A and \spad{T} are polynomials and \spad{B} and \spad{C} have no normal poles. \spad{D} is the derivation to use.
564302`\spad{monomRDEsys(f,{}g1,{}g2,{}D)} returns \spad{[A,{} B,{} H,{} C1,{} C2,{} T]} such that \spad{(y1',{} y2') + ((0,{} -f),{} (f,{} 0)) (y1,{}y2) = (g1,{}g2)} has a solution if and only if \spad{y1 = Q1 / T,{} y2 = Q2 / T},{} where \spad{B,{}C1,{}C2,{}Q1,{}Q2} have no normal poles and satisfy A \spad{(Q1',{}
564302`-- Q2') + ((H,{} -B),{} (B,{} H)) (Q1,{}Q2) = (C1,{}C2)} \spad{D} is the derivation to use.
564499`\spad{monom(deg,{}coef)} is a monomial of degree \spad{deg} with coefficient \spad{coef}.
564574`\spad{monom(w,{}r)} returns the product of the word \spad{w} by the coefficient \spad{r}.
564643`\spad{monom(s,{}r)} returns the product of the basis element \spad{s} by the coefficient \spad{r}.
564755`\spad{monomial?(f)} tests if \spad{f} is a single monomial.
564832`\spad{monomial?(x)} \undocumented
564894`\spad{monomial?(p)} tests if \spad{p} is a single monomial.
564955`\spad{monomial?(x)} returns \spad{true} if \spad{x} contains a single monomial.
565017`\spad{monomial?(f)} tests if \spad{f} is a single monomial.
565079`\spad{monomial?(x)} returns \spad{true} if \spad{x} is a monomial
565136`\spad{monomialIntPoly(p,{} ')} returns [\spad{q},{} \spad{r}] such that \spad{p = q' + r} and \spad{degree(r) < degree(t')}. Error if \spad{degree(t') < 2}.
565243`\spad{monomialIntegrate(f,{} ')} returns \spad{[ir,{} s,{} p]} such that \spad{f = ir' + s + p} and all the squarefree factors of the denominator of \spad{s} are special \spad{w}.\spad{r}.\spad{t} the derivation '.
565558`\spad{monomial(r,{}e)} makes a term from a coefficient \spad{r} and an exponent \spad{e}.
565642`\spad{monomial(x,{}n)} \undocumented
565705`\spad{monomial(r,{}x)} \undocumented
565879`\spad{monomial(a,{}s)} constructs a direct product element with the \spad{s} component set to \spad{a}
565946`\spad{monomial(c,{}[i1,{}i2,{}...,{}iN])} produces the value given by \spad{c*e(i1)*e(i2)*...*e(iN)}.
566025`\spad{monomial(r,{}e)} makes a term from a coefficient \spad{r} and an exponent \spad{e}.
566081`\spad{monomial(r,{}m)} creates a scalar multiple of the basis element \spad{m}.
566138`\spad{monomial(c,{}k)} produces \spad{c} times the \spad{k}-th power of the generating operator,{} \spad{monomial(1,{}1)}.
566215`\spad{monomial(c,{}k)} produces \spad{c} times the \spad{k}-th power of the generating operator,{} \spad{monomial(1,{}1)}.
566923`\spad{monomial(a,{}[v1..vn],{}[e1..en])} returns \spad{a*prod(vi^ei)}.
567033`\spad{monomial(a,{}x,{}n)} creates the monomial \spad{a*x^n} where \spad{a} is a polynomial,{} \spad{x} is a variable and \spad{n} is a nonnegative integer.
567131`\spad{monomial(a,{}[x1,{}..,{}xk],{}[n1,{}..,{}nk])} computes \spad{a * x1^n1 * .. * xk^nk}.
567221`\spad{monomial(a,{}[x1,{}x2,{}...,{}xk],{}[n1,{}n2,{}...,{}nk])} returns \spad{a * x1^n1 * ... * xk^nk}.
567331`\spad{monomial(a,{}x,{}n)} computes \spad{a*x^n}.
567409`\spad{monomial(a,{}x,{}n)} returns \spad{a*x^n}.
567582`\spad{monomials(p)} returns the list of non-zero monomials of polynomial \spad{p},{} \spadignore{i.e.} \spad{monomials(sum(a_(i) X^(i))) = [a_(1) X^(1),{}...,{}a_(n) X^(n)]}.
567661`\spad{monomials(x)} returns the list of \spad{r_i*b_i} whose sum is \spad{x}.
567724`\spad{monomials(f)} gives the list of all monomials whose sum is \spad{f}.
567787`\spad{monotoneConvolution(x,{} y)} returns the monotone convolution of the distributions \spad{x} and \spad{y}
567857`\spad{monotoneCumulants(x)} returns the sequence of monotone cumulants of the distribution \spad{x}.
568256`\spad{more?(a,{} b)} compares \spad{a} and \spad{b} in the partial ordering induced by setOrder,{} and uses the ordering on \spad{S} if \spad{a} and \spad{b} are not comparable in the partial ordering.
568338`\spad{more?(u,{}n)} tests if \spad{u} has greater than \spad{n} elements.
568401`\axiom{moreAlgebraic?(\spad{ts},{}us)} returns \spad{false} iff \axiom{\spad{ts}} and \axiom{us} are both empty,{} or \axiom{\spad{ts}} has less elements than \axiom{us},{} or some variable is algebraic \spad{w}.\spad{r}.\spad{t}. \axiom{us} and is not \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
568481`\axiom{moreAlgebraic?(\spad{ts},{}us)} returns \spad{false} iff \axiom{\spad{ts}} and \axiom{us} are both empty,{} or \axiom{\spad{ts}} has less elements than \axiom{us},{} or some variable is algebraic \spad{w}.\spad{r}.\spad{t}. \axiom{us} and is not \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
568571`\spad{morphism(f)} returns the non-invertible morphism given by \spad{f}.
568623`\spad{morphism(f)} returns the morphism given by \spad{f^n(x) = f(x,{}n)}.
568683`\spad{morphism(f,{} g)} returns the invertible morphism given by \spad{f},{} where \spad{g} is the inverse of \spad{f}..
568742`\spad{motzkinPathArray([a0,{} a1,{} ...],{} [b0,{} b1,{} ...],{} [c1,{} c2,{} ...])} computes Flajolet\spad{'s} Motzkin path array from the sequences \spad{ai},{} \spad{bi},{} \spad{ci}.
568842`\spad{move(v,{}x,{}y)} displays the two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} with the upper left-hand corner of the viewport window at the screen coordinate position \spad{x},{} \spad{y}.
568933`\spad{move(v,{}x,{}y)} displays the three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} with the upper left-hand corner of the viewport window at the screen coordinate position \spad{x},{} \spad{y}.
569026`\spad{movedPoints(p)} returns the set of points moved by the permutation \spad{p}.
569080`\spad{movedPoints(gp)} returns the points moved by the group {\em gp}.
569139`\spad{mpsode(r,{}f)} solves the system of differential equations \spad{dy[i]/dx =f[i] [x,{}y[1],{}y[2],{}...,{}y[n]]},{} \spad{y[i](a) = r[i]} for \spad{i} in 1..\spad{n}.
569249`\spad{mr(t)} is an auxiliary function which finds the position of the maximum element of a tableau \spad{t} which is in the lowest row,{} producing a record of results
569390`Compare kernels
569697`normalize and compute limit.
569823`Compute limit(\spad{f}(\spad{x}),{} x=a) for a finite or infinite limit point a
570137`Compute limit(\spad{f}(\spad{x}),{} x=a) for a strictly finite limit point a This function computes one-sided limits from the left or right
570292`Compute limit.
570557`\spad{mrv_normalize(f,{} x)} transform \spad{f} to the form acceptable by core variant of mrv_limit.
570791`\spad{mrv_rewrite0}
570957`rewrite
571304`Compute \spad{MRV} set
571830`\spad{mul_by_binomial(v,{} pt,{} p)} treats \spad{v} a polynomial and multiplies in place this polynomial by binomial (\spad{x} + \spad{pt}). Highest coefficient of product is ignored.
571924`\spad{mul_by_binomial(v,{} deg,{} pt,{} p)} treats \spad{v} as coefficients of polynomial of degree \spad{deg} and multiplies in place this polynomial by binomial (\spad{x} + \spad{pt}). Highest coefficient of product is ignored.
572026`\spad{mul_by_scalar(v,{} deg,{} c,{} p)} treats \spad{v} as coefficients of polynomial of degree \spad{deg} and multiplies in place this polynomial by scalar \spad{c}
572126`Polynomial multiplication.
572215`\spad{mulmod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a*b mod p}.
572271`\spad{multMonom(r,{}e,{}x)} \undocumented
572348`\spad{multiEuclideanTree(l,{}r)} \undocumented{}
572497`\spad{multiEuclidean([f1,{}...,{}fn],{}z)} returns a list of coefficients \spad{[a1,{} ...,{} an]} such that \spad{ z / prod \spad{fi} = sum aj/fj}. If no such list of coefficients exists,{} "failed" is returned.
572582`\axiom{multiIndex(\spad{jv})} returns the multi-index of the jet variable \axiom{\spad{jv}}.
572656`construct graph from a list of permutations.
572738`constructor for graph with given objects more objects and arrows can be added later if required.
572934`constructor for graph with given list of object names. Use this version of the constructor if you don\spad{'t} intend to create diagrams and therefore don\spad{'t} care about \spad{x},{}\spad{y} coordinates. more objects and arrows can be added later if required.
573003`constructor for graph with given objects and arrows more objects and arrows can be added later if required.
573298`constructor for graph with given objects and adjacency matrix.
573398`\spad{multinomial(n,{}[m1,{}m2,{}...,{}mk])} returns the multinomial coefficient \spad{n!/(m1! m2! ... mk!)}.
573471`\spad{multiple?(p)} tests if \spad{p} is a single matching variable allowing list matching or multiple term matching in a sum or product.
573520`\spad{multiple(x)} tells the pattern matcher that \spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\spad{x}) tells the pattern matcher that \spad{x} should match a list instead of an element of a list. Error: if \spad{x} is not a symbol.
573579`\spad{multiple(x)} tells the pattern matcher that \spad{x} should preferably match a multi-term quantity in a sum or product. For matching on lists,{} multiple(\spad{x}) tells the pattern matcher that \spad{x} should match a list instead of an element of a list.
573655`\spad{multiplicative?(a,{} n)} returns \spad{true} if the first \spad{n} coefficients of a are multiplicative
573735`\spad{multiplyCoefficients(fn,{}f)} returns the series \spad{sum(fn(n) * an * x^n,{}n = n0..)},{} where \spad{f} is the series \spad{sum(an * x^n,{}n = n0..)}.
573834`\spad{multiplyCoefficients(f,{}sum(n = n0..infinity,{}a[n] * x^n)) = sum(n = 0..infinity,{}f(n) * a[n] * x^n)}. This function is used when Puiseux series are represented by a Laurent series and an exponent.
573932`\spad{multiplyCoefficients(f,{}sum(n = 0..infinity,{}a[n] * x^n))} returns \spad{sum(n = 0..infinity,{}f(n) * a[n] * x^n)}. This function is used when Laurent series are represented by a Taylor series and an order.
574029`\spad{multiplyExponents(f,{}r)} multiplies all exponents of the power series \spad{f} by the positive rational number \spad{r}.
574126`\spad{multiplyExponents(p,{}n)} returns a new polynomial resulting from multiplying all exponents of the polynomial \spad{p} by the non negative integer \spad{n}.
574218`\spad{multiplyExponents(f,{}n)} multiplies all exponents of the power series \spad{f} by the positive integer \spad{n}.
574317`\spad{multisect(a,{}b,{}st)} selects the coefficients of \spad{x^((a+b)*n+a)},{} and changes them to \spad{x^n}.
574490`\spad{multisect(a,{}b,{}f(x))} selects the coefficients of \indented{1}{\spad{x^((a+b)*n+a)},{} and changes this monomial to \spad{x^n}.}
574576`\spad{multiset()}\$\spad{D} creates an empty multiset of domain \spad{D}.
574618`\spad{multiset(ls)} creates a multiset with elements from \spad{ls}.
574667`\spad{multiset(s)} creates a multiset with singleton \spad{s}.
574710`\spad{multivariate(f,{} v)} applies both the numerator and denominator of \spad{f} to \spad{v}.
574831`\spad{multivariate(f,{} v)} applies both the numerator and denominator of \spad{f} to \spad{v}.
574974`\spad{multivariate(sup,{}v)} converts an anonymous univariable polynomial \spad{sup} to a polynomial in the variable \spad{v}.
575075`\spad{multivariate(sup,{}v)} converts an anonymous univariable polynomial \spad{sup} to a polynomial in the variable \spad{v}.
575177`\spad{multivariate(u,{}k,{}f)} \undocumented
575317`to allow entries like: 1+2*e1+3*e2+4*e1e2 = multivector[1,{}2,{}3,{}4]
575383`\spad{musserTrials()} returns the number of primes that are tried in \spadfun{modularFactor}.
575456`\spad{musserTrials(n)} sets to \spad{n} the number of primes to be tried in \spadfun{modularFactor} and returns the previous value.
575658`\axiom{mvar(\spad{p})} returns an error if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns its main variable \spad{w}. \spad{r}. \spad{t}. to the total ordering on the elements in \axiom{\spad{V}}.
575720`\axiom{mvar(\spad{ps})} returns the main variable of the non constant polynomial with the greatest main variable,{} if any,{} else an error is returned.
575788`\spad{myDegree should} be local
575930`main solver routine
576640`naiveBeckermannLabahn(\spad{f},{} degs,{} sigma,{} \spad{p}) solves modulo \$\spad{p}\$ multipoint Hermite-Pade interpolation problem of order \$sigma\$. \$degs\$ specifies degree bounds for solution. The first element of result contains sigma-basis. The second gives defects of elements of sigma-basis. The third one contains residuals \spa
576640`--d{
576640`--} we stop updating solution if this would violate degree bounds,{} so residual is zero for elements of sigma-basis which are within bounds,{} but non-zero otherwise.
576806`naiveBeckermannLabahn(\spad{f},{} degs,{} sigma,{} \spad{p}) solves modulo \$\spad{p}\$ multipoint Hermite-Pade interpolation problem of order \$sigma\$. \$degs\$ specifies degree bounds for solution. The first element of result contains sigma-basis. The second gives defects of elements of sigma-basis. The third one contains residuals \spad{
576806`--} we stop updating sol
576806`--ution if this would violate degree bounds,{} so residual is zero for elements of sigma-basis which are within bounds,{} but non-zero otherwise.
576946`\spad{naiveBeckermannLabahn(f,{} degs,{} sigma,{} p)} solves modulo \$\spad{p}\$ Hermite-Pade interpolation problem of order \$\spad{sigma}\$. \$\spad{degs}\$ specifies degree bounds for solution. The first element of result contains \spad{sigma}-basis. The second gives defects of elements of \spad{sigma}-basis. The third one contains resid
576946`--uals \spad{
576946`--} we stop updating solution if this would violate degree bounds,{} so residual is zero for elements of \spad{sigma}-basis which are within bounds,{} but non-zero otherwise.
577111`\spad{naiveBeckermannLabahn(f,{} degs,{} sigma,{} p)} solves modulo \$\spad{p}\$ Hermite-Pade interpolation problem of order \$\spad{sigma}\$. \$\spad{degs}\$ specifies degree bounds for solution. The first element of result contains \spad{sigma}-basis. The second gives defects of elements of \spad{sigma}-basis. The third one contains residuals \spad{
577111`--} we stop upd
577111`--ating solution if this would violate degree bounds,{} so residual is zero for elements of \spad{sigma}-basis which are within bounds,{} but non-zero otherwise.
577250`\axiom{name(\spad{jv})} yields the name of the jet variable \axiom{\spad{jv}}.
577300`\spad{name(x)} returns the symbol
577350`\spad{name(op(a1,{}...,{}an))} returns the name of op.
577392`\spad{name(x)} returns the symbol
577438`\spad{name(f)} returns the external name of the file \spad{f}.
577489`\spad{name(f)} returns the name part of the file name.
577538`\spad{name(op)} returns the name of \spad{op}.
577584`\spad{name(s)} returns \spad{s} without its scripts.
577619`\spad{namedBranch(chin,{}nmin)} constructs a branch in the tree structure
577694`\spad{namedPoints(ptin,{}nmin)} constructs a list of named points.
577769`\spad{namedPoints(scin,{}nmin)} constructs a list of named nodes.
577891`\spad{nand(a,{}b)} returns the logical {\em nand} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
577935`\spad{nand(a,{}b)} returns the logical negation of \spad{a} and \spad{b}.
577974`\spad{nary?(op)} tests if \spad{op} has arbitrary arity.
578022`\axiom{ncDetSys(\spad{sys})} computes the determining system for generators of conditional symmetries of \axiom{\spad{sys}}. It makes the same assumptions as \axiom{detSys}.
578163`\axiom{ncDetSys(\spad{sys},{} \spad{vf})} computes the determining system for generators of conditional symmetries of \axiom{\spad{sys}} with a given ansatz for the vector fields.
578325`\axiom{ncDetSys(\spad{sys},{} \spad{sjb},{} \spad{vf})} computes the determining system for generators of conditional symmetries of \axiom{\spad{sys}}. The meaning of the further arguments is the same as in \axiom{detSys}.
578580`\axiom{ncols(A)} returns the number of columns of the matrix \axiom{A}.
578650`\spad{ncols(m)} returns the number of columns in the matrix \spad{m}.
578736`\spad{ncols(m)} returns the number of columns in the array \spad{m}
578868`\spad{negative?(u)} returns \axiom{\spad{true}} if every element of \spad{u} is negative,{} \axiom{\spad{false}} otherwise.
578926`\spad{negative?(x)} tests whether \spad{x} is strictly less than 0.
579070`\axiom{negative?(pol,{}aRoot)} answers if \axiom{pol} interpreted as \axiom{aRoot} is negative
579167`\spad{newLine()} sends a new line command to output.
579215`\spad{newReduc()} \undocumented
579272`\spad{newSubProgram(f)} asserts that from now on type declarations are part of subprogram \spad{f}.
579330`\spad{newTypeLists(x)} \undocumented
579387`\spad{new()} return a new HashState.
579422`\spad{new()} returns a new empty match result.
579477`\spad{new()} returns a new empty match result.
579526`\spad{new()} \undocumented
579565`\spad{new()} \undocumented
579616`\spad{new()} create a new,{} empty object. Use \spadfun{setPrologue!},{} \spadfun{setFormula!} and \spadfun{setEpilogue!} to set the various components of this object.
579661`\spad{new()} returns a new symbol whose name starts with \%.
579693`\spad{new()} create a new,{} empty object. Use \spadfun{setPrologue!},{} \spadfun{setTex!} and \spadfun{setEpilogue!} to set the various components of this object.
579728`\spad{new(s)} returns a new symbol whose name starts with \%\spad{s}.
579762`\axiom{new(inds,{} nrows)} generates a new matrix with \axiom{nrows} rows and columns enumerated by the indices \axiom{inds}. The matrix is empty,{} \spadignore{i.e.} the zero matrix.
579886`\spad{new(n,{}x)} returns \axiom{fill!(new \spad{n},{}\spad{x})}.
579950`\spad{new(m,{}n,{}r)} is an \spad{m}-by-\spad{n} array all of whose entries are \spad{r}
580053`\spad{new(d,{}pref,{}e)} constructs the name of a new writable file with \spad{d} as its directory,{} \spad{pref} as a prefix of its name and \spad{e} as its extension. When \spad{d} or \spad{t} is the empty string,{} a default is used. An error occurs if a new file cannot be written in the given directory.
580115`\spad{newline()} provides the newline character.
580154`\spad{newton}(\spad{l}) returns the interpolating polynomial for the values \spad{l},{} where the \spad{x}-coordinates are assumed to be [1,{}2,{}3,{}...,{}\spad{n}] and the coefficients of the interpolating polynomial are known to be in the domain \spad{F}. \spad{I}.\spad{e}.,{} it is a very streamlined version for a special case of interpolation.
580239`\spad{nextColeman(alpha,{}beta,{}C)} generates the next Coleman matrix of column sums {\em alpha} and row sums {\em beta} according to the lexicographical order from bottom-to-top. The first Coleman matrix is achieved by {\em C=new(1,{}1,{}0)}. Also,{} {\em new(1,{}1,{}0)} indicates that \spad{C} is the last Coleman matrix.
580364`\spad{nextIrreduciblePoly(f)} yields the next monic irreducible polynomial over a finite field {\em GF} of the same degree as \spad{f} in the following order,{} or "failed" if there are no greater ones. Error: if \spad{f} has degree 0. Note: the input polynomial \spad{f} is made monic. Also,{} \spad{f < g} if the number of monomials of \spad{f} is less than
580364`-- this number for \spad{g}. If \spad{f} and \spad{g} have the same number of monomials,{} the lists of exponents are compared lexicographically. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\em GF} given by {\em lookup}.
580787`\spad{nextItem(x)} returns the next item,{} or "failed" if domain is exhausted.
580847`\spad{nextLatticePermutation(lambda,{}lattP,{}constructNotFirst)} generates the lattice permutation according to the proper partition {\em lambda} succeeding the lattice permutation {\em lattP} in lexicographical order as long as {\em constructNotFirst} is \spad{true}. If {\em constructNotFirst} is \spad{false},{} the first lattice permutation is returned. The result {\em nil} i
580847`--ndicates that {\em lattP} has no successor.
580973`\spad{nextNormalPoly(f)} yields the next normal polynomial over a finite field {\em GF} of the same degree as \spad{f} in the following order,{} or "failed" if there are no greater ones. Error: if \spad{f} has degree 0. Note: the input polynomial \spad{f} is made monic. Also,{} \spad{f < g} if the {\em lookup} of the coefficient of the term of degree {\em n-1} o
580973`--f \spad{f} is less than that for \spad{g}. In case these numbers are equal,{} \spad{f < g} if if the number of monomials of \spad{f} is less that for \spad{g} or if the list of exponents of \spad{f} are lexicographically less than the corresponding list for \spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\em GF} given by {\em lookup}.
581116`\spad{nextNormalPrimitivePoly(f)} yields the next normal primitive polynomial over a finite field {\em GF} of the same degree as \spad{f} in the following order,{} or "failed" if there are no greater ones. Error: if \spad{f} has degree 0. Note: the input polynomial \spad{f} is made monic. Also,{} \spad{f < g} if the {\em lookup} of the constant term of 
581116`--\spad{f} is less than this number for \spad{g} or if {\em lookup} of the coefficient of the term of degree {\em n-1} of \spad{f} is less than this number for \spad{g}. Otherwise,{} \spad{f < g} if the number of monomials of \spad{f} is less than that for \spad{g} or if the lists of exponents for \spad{f} are lexicographically less than those for \spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of t
581116`--he elements of {\em GF} given by {\em lookup}. This operation is equivalent to nextPrimitiveNormalPoly(\spad{f}).
581268`\spad{nextPartition(gamma,{}part,{}number)} generates the partition of {\em number} which follows {\em part} according to the right-to-left lexicographical order. The partition has the property that its components do not exceed the corresponding components of {\em gamma}. the first partition is achieved by {\em part=[]}. Also,{} {\em []} indicates that {\em part} is the last partitio
581268`--n.
581389`\spad{nextPartition(gamma,{}part,{}number)} generates the partition of {\em number} which follows {\em part} according to the right-to-left lexicographical order. The partition has the property that its components do not exceed the corresponding components of {\em gamma}. The first partition is achieved by {\em part=[]}. Also,{} {\em []} indicates that {\em part} is the last partit
581389`--ion.
581512`\spad{nextPrime(n)} returns the smallest prime strictly larger than \spad{n}
581567`\spad{nextPrimitiveNormalPoly(f)} yields the next primitive normal polynomial over a finite field {\em GF} of the same degree as \spad{f} in the following order,{} or "failed" if there are no greater ones. Error: if \spad{f} has degree 0. Note: the input polynomial \spad{f} is made monic. Also,{} \spad{f < g} if the {\em lookup} of the constant term of 
581567`--\spad{f} is less than this number for \spad{g} or,{} in case these numbers are equal,{} if the {\em lookup} of the coefficient of the term of degree {\em n-1} of \spad{f} is less than this number for \spad{g}. If these numbers are equals,{} \spad{f < g} if the number of monomials of \spad{f} is less than that for \spad{g},{} or if the lists of exponents for \spad{f} are lexicographically less than those for \spad{g}. If these lists are also equal,{} the lists of coefficients are coefficients a
581567`--ccording to the lexicographic ordering induced by the ordering of the elements of {\em GF} given by {\em lookup}. This operation is equivalent to nextNormalPrimitivePoly(\spad{f}).
581719`\spad{nextPrimitivePoly(f)} yields the next primitive polynomial over a finite field {\em GF} of the same degree as \spad{f} in the following order,{} or "failed" if there are no greater ones. Error: if \spad{f} has degree 0. Note: the input polynomial \spad{f} is made monic. Also,{} \spad{f < g} if the {\em lookup} of the constant term of \spad{f} is less th
581719`--an this number for \spad{g}. If these values are equal,{} then \spad{f < g} if if the number of monomials of \spad{f} is less than that for \spad{g} or if the lists of exponents of \spad{f} are lexicographically less than the corresponding list for \spad{g}. If these lists are also equal,{} the lists of coefficients are compared according to the lexicographic ordering induced by the ordering of the elements of {\em GF} given by {\em lookup}.
581865`\spad{nextSublist(n,{}k)} returns a list of \spad{k}-subsets of {1,{} ...,{} \spad{n}}.
581972`\spad{nextSubsetGray(ww,{}n)} returns a vector {\em vv} whose components have the following meanings:\begin{items} \item {\em vv.1}: a vector of length \spad{n} whose entries are 0 or 1. This \indented{3}{can be interpreted as a code for a subset of the set 1,{}...,{}\spad{n};} \indented{3}{{\em vv.1} differs from {\em ww.1} by exactly one entry;} \item {\em vv.2.1} is the number of the entry of {\e
581972`--m vv.1} which \indented{3}{will be changed next time;} \item {\em vv.2.1 = n+1} means that {\em vv.1} is the last subset; \indented{3}{trying to compute nextSubsetGray(\spad{vv}) if {\em vv.2.1 = n+1}} \indented{3}{will produce an error!} \end{items} The other components of {\em vv.2} are needed to compute nextSubsetGray efficiently. Note: this is an implementation of [Williamson,{} Topic II,{} 3.54,{} \spad{p}. 112] for the special case {\em r1 = r2 = ... = rn = 2}; Note: nextSubsetGray produ
581972`--ces a side-effect,{} \spadignore{i.e.} {\em nextSubsetGray(vv)} and {\em vv := nextSubsetGray(vv)} will have the same effect.
582077`\axiom{\spad{nextsousResultant2}(\spad{P},{} \spad{Q},{} \spad{Z},{} \spad{s})} returns the subresultant \axiom{\spad{S_}{\spad{e}-1}} where \axiom{\spad{P} ~ \spad{S_d},{} \spad{Q} = \spad{S_}{\spad{d}-1},{} \spad{Z} = S_e,{} \spad{s} = \spad{lc}(\spad{S_d})}
582168`\axiom{\spad{nextsubResultant2}(\spad{p},{}\spad{q},{}\spad{z},{}\spad{s})} is the multivariate version of the operation \axiomOpFrom{\spad{next_sousResultant2}}{PseudoRemainderSequence} from the \axiomType{PseudoRemainderSequence} constructor.
582275`\spad{next(l)} returns the doubly-linked aggregate beginning with its next element. Error: if \spad{l} has no next element. Note: \axiom{next(\spad{l}) = rest(\spad{l})} and \axiom{previous(next(\spad{l})) = \spad{l}}.
582328`\spad{nilFactor(base,{}exponent)} creates a factored object with a single factor with no information about the kind of \spad{base} (flag = "nil").
582380`\spad{nil()} returns the empty list.
582413`\spad{nlde(u)} solves a first order non-linear differential equation described by \spad{u} of the form \spad{[[b<0,{}0>,{}b<0,{}1>,{}...],{}[b<1,{}0>,{}b<1,{}1>,{}.],{}...]}. the differential equation has the form \spad{y' = sum(i=0 to infinity,{}j=0 to infinity,{}b<i,{}j>*(x^i)*(y^j))}.
582528`\spad{noKaratsuba(a,{}b)} returns \spad{a*b} without using Karatsuba\spad{'s} trick at all.
582610`\spad{noLinearFactor?(p)} returns \spad{true} if \spad{p} can be shown to have no linear factor by a theorem of Lehmer,{} \spad{false} else. \spad{I} insist on the fact that \spad{false} does not mean that \spad{p} has a linear factor.
582850`\spad{node?(u,{}v)} tests if node \spad{u} is contained in node \spad{v} (either as a child,{} a child of a child,{} etc.).
583017`\spad{nodeFromArrow(s,{}a)} returns index of all nodes with a direct arrow leading in to arrow 'a' in graph \spad{'s'}
583199`\spad{nodeFromNode(s,{} a)} gives list of all nodes with a direct arrow leading in to node 'a' in graph \spad{'s'}
583291`\axiom{nodeOf?(\spad{s},{}a)} returns \spad{true} iff some node of \axiom{a} is equal to \axiom{\spad{s}}
583453`\spad{nodeToArrow(s,{}a)} returns index of all nodes with a direct arrow leading out of arrow 'a' in graph \spad{'s'}
583631`\spad{nodeToNode(s,{} a)} gives list of all nodes with a direct arrow leading out of node 'a' in graph \spad{'s'}
583721`\spad{node(left,{}v,{}right)} creates a binary tree with value \spad{v},{} a binary tree \spad{left},{} and a binary tree \spad{right}.
583947`\spad{nodes(u)} returns a list of all of the nodes of aggregate \spad{u}.
584004`\spad{nonQsign(r)} \undocumented
584072`\spad{nonSingularModel(u)} returns the equations in \spad{u1},{}...,{}un of an affine non-singular model for the curve.
584169`\spad{nonSingularModel(u)} returns the equations in \spad{u1},{}...,{}un of an affine non-singular model for the curve.
584275`\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.
584368`\spad{noncommutativeJordanAlgebra?()} tests if the algebra is flexible and Jordan admissible.
584497`\spad{nor(a,{}b)} returns the logical {\em nor} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
584540`\spad{nor(a,{}b)} returns the logical negation of \spad{a} or \spad{b}.
584578`\spad{normDeriv2 should} be local
584707`\spad{normFactors(x)} \undocumented
584791`\axiom{normInvertible?(\spad{p},{}\spad{ts})} is an internal subroutine,{} exported only for developement.
585176`\spad{norm(p)} determines sum of absolute values of coefficients Note: this function depends on \spadfunFrom{abs}{Complex}.
585235`\spad{norm(I)} returns the norm of the ideal \spad{I}.
585288`\spad{norm(f)} returns the norm of the module \spad{f}.
585345`\spad{norm q} returns the norm of \spad{q},{} \spadignore{i.e.} the product of all the conjugates of \spad{q}.
585415`\spad{norm(a)} computes the norm of \spad{a} with respect to the field considered as an algebra with 1 over the ground field \spad{F}.
585475`\spad{norm(x)} returns \spad{x} * conjugate(\spad{x})
585521`\spad{norm(a)} returns the determinant of the regular representation of \spad{a} with respect to any basis.
585572`\spad{norm(o)} returns the norm of an octonion,{} equal to the sum of the squares of its coefficients.
585619`\spad{norm(q)} computes the norm of \spad{q} (the sum of the squares of the components).
585668`\spad{norm x} returns the same as absolute value.
585786`\spad{norm(p,{}k)} computes the norm of the polynomial \spad{p} with respect to the extension generated by kernel \spad{k}
585902`\spad{norm(p,{}l)} computes the norm of the polynomial \spad{p} with respect to the extension generated by kernels \spad{l}
586024`\spad{norm(f,{}p)} returns the \spad{lp} norm of the polynomial \spad{f}.
586109`\spad{norm(x,{}n)} \undocumented{} See \axiomFunFrom{norm}{FiniteAlgebraicExtensionField}
586204`\spad{norm(f,{}k)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernel \spad{k}
586264`\spad{norm(f,{}l)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernels \spad{l}
586330`\spad{norm(p,{}k)} computes the norm of the polynomial \spad{p} with respect to the extension generated by kernel \spad{k}
586441`\spad{norm(p,{}l)} computes the norm of the polynomial \spad{p} with respect to the extension generated by kernels \spad{l}
586558`\spad{norm(f,{}k)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernel \spad{k}
586613`\spad{norm(f,{}l)} computes the norm of the algebraic number \spad{f} with respect to the extension generated by kernels \spad{l}
586674`\spad{norm(a,{}d)} computes the norm of \spad{a} with respect to the field of extension degree \spad{d} over the ground field of size. Error: if \spad{d} does not divide the extension degree of \spad{a}. Note: norm(a,{}\spad{d}) = reduce(*,{}[a^(\spad{q^}(d*i)) for \spad{i} in 0..\spad{n/d}])
586764`\spad{normal01()} \undocumented
586888`\spad{normal?(f)} tests whether the polynomial \spad{f} over a finite field is normal,{} \spadignore{i.e.} its roots are linearly independent over the field.
586985`\spad{normal?(x)} \undocumented{} See \axiomFunFrom{normal?}{FiniteAlgebraicExtensionField}
587064`\spad{normal?(a)} tests whether the element \spad{a} is normal over the ground field \spad{F},{} \spadignore{i.e.} \spad{a^(q^i),{} 0 <= i <= extensionDegree()-1} is an \spad{F}-basis,{} where \spad{q = size()\$F}. Implementation according to Lidl/Niederreiter: Theorem 2.39.
587146`\spad{normalDenom(f,{} D)} returns the product of all the normal factors of \spad{denom(f)}. \spad{D} is the derivation to use.
587229`\spad{normalDeriv(poly,{}i)} computes the \spad{i}th derivative of \spad{poly} divided by i!.
587355`\spad{normalElement()} returns a element,{} normal over the ground field \spad{F},{} \spadignore{i.e.} \spad{a^(q^i),{} 0 <= i < extensionDegree()} is an \spad{F}-basis,{} where \spad{q = size()\$F}. At the first call,{} the element is computed by \spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} then cached in a global variable. On subsequent calls,{} the element is retrieved by referencing the global varia
587355`--ble.
587436`\spad{normalElement(n)} \undocumented{} See \axiomFunFrom{normalElement}{FiniteAlgebraicExtensionField}
587529`\spad{normalForm(poly,{}gb)} reduces the polynomial \spad{poly} modulo the precomputed groebner basis \spad{gb} giving a canonical representative of the residue class.
587631`\spad{normalForm(M,{} sig,{} der)} returns \spad{[R,{} A,{} A^{-1}]} such that the pseudo-linear operator whose matrix in the basis \spad{y} is \spad{M} had matrix \spad{R} in the basis \spad{z = A y}. \spad{der} is a \spad{sig}-derivation.
587765`\spad{normal(f,{}g)} \undocumented
587836`normalalised point has scalar value of one,{} if this is not the case then scale values
587888`\spad{normalise(v)} returns the column vector \spad{v} divided by its euclidean norm; when possible,{} the vector \spad{v} is expressed in terms of radicals.
588079`\spad{normalizeAtInfinity(v)} makes \spad{v} normal at infinity.
588171`\axiom{normalizeIfCan(\spad{ts})} returns \axiom{\spad{ts}} in an normalized shape if \axiom{\spad{ts}} is zero-dimensional.
588248`\spad{normalize(f)} rewrites \spad{f} using the least possible number of real algebraically independent kernels.
588319`\spad{normalize(x)} normalizes \spad{x} at current precision.
588358`\spad{normalize(f,{} x)} rewrites \spad{f} using the least possible number of real algebraically independent kernels involving \spad{x}.
588436`\axiom{normalize(\spad{p},{}\spad{ts})} normalizes \axiom{\spad{p}} \spad{w}.\spad{r}.\spad{t} \spad{ts}.
588598`\axiom{normalized?(\spad{ts})} returns \spad{true} iff for every axiom{\spad{p}} in axiom{\spad{ts}} we have \axiom{normalized?(\spad{p},{}us)} where \axiom{us} is \axiom{collectUnder(\spad{ts},{}mvar(\spad{p}))}.
588893`\axiom{normalized?(\spad{p},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{p}} and all its iterated initials have degree zero \spad{w}.\spad{r}.\spad{t}. the main variables of the polynomials of \axiom{\spad{ts}}
588966`\axiom{normalized?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{normalized?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
589050`\axiom{normalized?(a,{}\spad{b})} returns \spad{true} iff \axiom{a} and its iterated initials have degree zero \spad{w}.\spad{r}.\spad{t}. the main variable of \axiom{\spad{b}}
589128`\axiom{normalizedAssociate(\spad{p},{}\spad{ts})} returns a normalized polynomial \axiom{\spad{n}} \spad{w}.\spad{r}.\spad{t}. \spad{ts} such that \axiom{\spad{n}} and \axiom{\spad{p}} are associates \spad{w}.\spad{r}.\spad{t} \spad{ts} and assuming that \axiom{\spad{p}} is invertible \spad{w}.\spad{r}.\spad{t} \spad{ts}.
589205`\spad{normalizedDivide(n,{}d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \spadignore{e.g.} positive remainders
589306`\spad{normalizedDivide(n,{}d)} returns a normalized quotient and remainder such that consistently unique representatives for the residue class are chosen,{} \spadignore{e.g.} positive remainders
589476`\spad{not f} creates the equivalent prefix form.
589514`\spad{not(b)} returns the logical {\em not} of bit aggregate \axiom{\spad{b}}.
589554`\spad{not n} returns the negation of \spad{n}.
589589`\spad{not(n)} returns the bit-by-bit logical {\em not} of the single integer \spad{n}.
589630`\spad{notelem(ir)} returns the non-elementary part of an integration result
589714`\spad{npcoef }\undocumented
590026`\axiom{nrows(A)} returns the number of rows of the matrix \axiom{A}.
590096`\spad{nrows(m)} returns the number of rows in the matrix \spad{m}.
590182`\spad{nrows(m)} returns the number of rows in the array \spad{m}
590266`\spad{nsqfree should} be local
590471`\spad{nthCoef(x,{} n)} returns the coefficient of the n^th term of \spad{x}.
590540`\spad{nthExpon(l,{} n)} returns the exponent of the n^th monomial of \spad{l}.
590601`\spad{nthExpon(x,{} n)} returns the exponent of the n^th monomial of \spad{x}.
590659`\spad{nthExpon(x,{} n)} returns the exponent of the n^th monomial of \spad{x}.
590729`\spad{nthExpon(x,{} n)} returns the exponent of the \spad{n-th} monomial of \spad{x}.
590806`\spad{nthExponent(u,{}n)} returns the exponent of the \spad{n}th factor of \spadvar{\spad{u}}. If \spadvar{\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 0 is returned.
590866`\spad{nthFactor(x,{} n)} returns the factor of the n^th monomial of \spad{x}.
590919`\spad{nthFactor(x,{} n)} returns the factor of the n^th monomial of \spad{x}.
590973`\spad{nthFactor(l,{} n)} returns the factor of the n^th monomial of \spad{l}.
591035`\spad{nthFactor(x,{} n)} returns the factor of the \spad{n-th} monomial of \spad{x}.
591096`\spad{nthFactor(u,{}n)} returns the base of the \spad{n}th factor of \spadvar{\spad{u}}. If \spadvar{\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} 1 is returned. If \spadvar{\spad{u}} consists only of a unit,{} the unit is returned.
591148`\spad{nthFactor(x,{} n)} returns the factor of the n^th term of \spad{x}.
591219`\spad{nthFlag(u,{}n)} returns the information flag of the \spad{n}th factor of \spadvar{\spad{u}}. If \spadvar{\spad{n}} is not a valid index for a factor (for example,{} less than 1 or too big),{} "nil" is returned.
591303`\spad{nthFractionalTerm(p,{}n)} extracts the \spad{n}th fractional term from the partial fraction \spad{p}. This returns 0 if the index \spad{n} is out of range.
591371`\spad{nthRootIfCan(z,{}n)} returns the \spad{n}th root of \spad{z} if possible,{} and "failed" otherwise.
591474`\spad{nthRoot(f,{} n)} returns \spad{(p,{} r,{} [r1,{}...,{}rm])} such that the \spad{n}th-root of \spad{f} is equal to \spad{r * \spad{p}th-root(r1 * ... * rm)},{} where \spad{r1},{}...,{}\spad{rm} are distinct factors of \spad{f},{} each of which has an exponent smaller than \spad{p} in \spad{f}.
591715`\spad{nthRoot(x,{}n)} returns the \spad{n}th root of \spad{x}.
591770`\spad{nthr(p,{}n)} should be local but conditional
591900`\spad{null?(s)} is \spad{true} if \spad{s} is the \spad{S}-expression ().
591976`\spad{nullBoundary()} constructs a null boundary,{} for example,{} when the boundary of a material node is requested.
592024`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
592135`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
592263`\spad{nullSpace M} builds a basis of the kernel of the linear transformation defined by the matrix \spad{M}. This function calls the \spad{nullSpace} in the \spad{Matrix} domain. Only the signature changes.
592323`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
592425`\spad{nullSpace(m)} returns a basis for the null space of the matrix \spad{m}.
592512`\spad{nullSpace(m)}+ returns a basis for the null space of the matrix \spad{m}.
592614`\spad{null(u)} tests if list \spad{u} is the empty list.
592655`\spad{nullary?(op)} tests if \spad{op} is nullary.
592706`\spad{nullary A} changes its argument into a \indented{1}{nullary function.}
592758`\spad{nullity(m)} returns the mullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
592851`\spad{nullity(m)} returns the mullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
592960`\spad{nullity(m)} returns the nullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
593054`\spad{nullity(m)} returns the nullity of the matrix \spad{m}. This is the dimension of the null space of the matrix \spad{m}.
593163`\axiom{numDepVar} returns the number of dependent variables.
593366`\spad{numFunEvals3D()} returns the number of points computed.
593413`\spad{numFunEvals()} returns the number of points computed
593456`\axiom{numIndVar} returns the number of independent variables.
593659`\spad{number?(f)} tests if \spad{f} is rational
593730`\spad{numberOfChildren(x)} \undocumented
593800`\spad{numberOfComponents()} returns the number of absolutely irreducible components.
593892`\spad{numberOfComponents()} returns the number of absolutely irreducible components.
593981`\spad{numberOfComponents(s)} returns the number of distinct object components in the indicated \spadtype{ThreeSpace},{} \spad{s},{} such as points,{} curves,{} polygons,{} and constructs.
594061`\spad{numberOfComposites(s)} returns the number of supercomponents,{} or composites,{} in the \spadtype{ThreeSpace},{} \spad{s}; Composites are arbitrary groupings of otherwise distinct and unrelated components; A \spadtype{ThreeSpace} need not have any composites defined at all and,{} outside of the requirement that no component can belong to more than one composite at a time,{} the definition and interpretation of composi
594061`--tes are unrestricted.
594141`\spad{numberOfComputedEntries(st)} returns the number of explicitly computed entries of stream \spad{st} which exist immediately prior to the time this function is called.
594227`\spad{numberOfCycles(p)} returns the number of non-trivial cycles of the permutation \spad{p}.
594296`\spad{numberOfDivisors(n)} returns the number of integers between 1 and \spad{n} (inclusive) which divide \spad{n}. The number of divisors of \spad{n} is often denoted by \spad{tau(n)}.
594375`\spad{numberOfFactors(ddfactorization)} returns the number of factors of the polynomial \spad{f} modulo \spad{p} where \spad{ddfactorization} is the distinct degree factorization of \spad{f} computed by \spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} for some prime \spad{p}.
594492`\spad{numberOfFactors(u)} returns the number of factors in \spadvar{\spad{u}}.
594559`\spad{numberOfFractionalTerms(p)} computes the number of fractional terms in \spad{p}. This returns 0 if there is no fractional part.
594630`\spad{numberOfHues()} returns the number of total hues,{} set in totalHues.
594683`\spad{numberOfImproperPartitions(n,{}m)} computes the number of partitions of the nonnegative integer \spad{n} in \spad{m} nonnegative parts with regarding the order (improper partitions). Example: {\em numberOfImproperPartitions (3,{}3)} is 10,{} since {\em [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,{}
594683`--0]} are the possibilities. Note: this operation has a recursive implementation.
594787`\spad{numberOfIrreduciblePoly(n)}\$FFPOLY(\spad{GF}) yields the number of monic irreducible univariate polynomials of degree \spad{n} over the finite field {\em GF}.
594893`\spad{numberOfMonomials(p)} gives the number of non-zero monomials in polynomial \spad{p}.
594979`\spad{numberOfMonomials(x)} returns the number of monomials of \spad{x}.
595060`\spad{numberOfMonomials(f)} is the number of non-zero coefficients with respect to the canonical basis.
595141`\spad{numberOfNormalPoly(n)}\$FFPOLY(\spad{GF}) yields the number of normal polynomials of degree \spad{n} over the finite field {\em GF}.
595242`\spad{numberOfPrimitivePoly(n)}\$FFPOLY(\spad{GF}) yields the number of primitive polynomials of degree \spad{n} over the finite field {\em GF}.
595574`\axiom{numerJP(\spad{f})} writes \axiom{\spad{f}} as polynomial over \axiom{\spad{JB}}.
595679`\spad{numer x} returns the numerator of \spad{x}.
595725`\spad{numer x} returns the numerator of \spad{x}.
595767`\spad{numer(f)} returns the numerator of \spad{f} viewed as a polynomial in the kernels over \spad{Z}.
595863`\spad{numer(1/d * (f1,{}...,{}fn))} = the vector \spad{[f1,{}...,{}fn]}.
595925`\spad{numer(x)} returns the numerator of the fraction \spad{x}.
595978`\spad{numer(f)} returns the numerator of \spad{f} viewed as a polynomial in the kernels over \spad{Z}.
596069`\spad{numer(f)} returns the numerator of \spad{f} viewed as a polynomial in the kernels over \spad{R} if \spad{R} is an integral domain. If not,{} then numer(\spad{f}) = \spad{f} viewed as a polynomial in the kernels over \spad{R}.
596266`\axiom{numerator(\spad{f})} yields the numerator of \axiom{\spad{f}}.
596329`\spad{numerator(f)} returns the numerator of \spad{f} converted to \%.
596390`\spad{numerator(x)} is the numerator of the fraction \spad{x} converted to \%.
596448`\spad{numerators(x)} returns the stream of numerators of the approximants of the continued fraction \spadvar{\spad{x}}. If the continued fraction is finite,{} then the stream will be finite.
596510`\spad{numericIfCan(x)} returns a real approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
596631`\spad{numericIfCan(x)} returns a real approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
596739`\spad{numericIfCan(x)} returns a real approximation of \spad{x},{} or "failed" if \axiom{\spad{x}} is not a constant.
596827`\spad{numericIfCan(x,{} n)} returns a real approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
596964`\spad{numericIfCan(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
597088`\spad{numericIfCan(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places,{} or "failed" if \axiom{\spad{x}} is not a constant.
597192`\spad{numeric(x)} returns a real approximation of \spad{x}.
597292`\spad{numeric(x)} returns a real approximation of \spad{x}.
597379`\spad{numeric(x)} returns a real approximation of \spad{x}.
597446`\spad{numeric(x)} returns a real approximation of \spad{x}.
597490`\spad{numeric(x,{} n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
597606`\spad{numeric(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
597709`\spad{numeric(x,{}n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
597792`\spad{numeric(x,{} n)} returns a real approximation of \spad{x} up to \spad{n} decimal places.
597852`\spad{obj(a)} essentially returns the original object that was converted to \spadtype{Any} except that the type is forced to be \spadtype{None}.
597885`\spad{objectOf(a)} returns a printable form of the original object that was converted to \spadtype{Any}.
597929`\spad{objects(s)} returns the \spadtype{ThreeSpace},{} \spad{s},{} in the form of a 3D object record containing information on the number of points,{} curves,{} polygons and constructs comprising the \spadtype{ThreeSpace}..
598097`\spad{oblateSpheroidal(a)} transforms from oblate spheroidal coordinates to Cartesian coordinates: \spad{oblateSpheroidal(a)} is a function which will map the point \spad{(\spad{xi},{}eta,{}phi)} to \spad{x = a*sinh(\spad{xi})*sin(eta)*cos(phi)},{} \spad{y = a*sinh(\spad{xi})*sin(eta)*sin(phi)},{} \spad{z = a*cosh(\spad{xi})*cos(eta)}.
598175`\spad{octon(qe,{}qE)} constructs an octonion from two quaternions using the relation {\em O = Q + QE}.
598241`\spad{octon(re,{}\spad{ri},{}rj,{}rk,{}rE,{}rI,{}rJ,{}rK)} constructs an octonion from scalars.
598350`\spad{odd? x} is \spad{true} if \spad{x} is an odd integer.
598429`\spad{odd?(n)} returns \spad{true} if and only if \spad{n} is odd.
598482`\spad{odd?(p)} returns \spad{true} if and only if \spad{p} is an odd permutation \spadignore{i.e.} {\em sign(p)} is {\em -1}.
598530`\spad{oddInfiniteProduct(f(x))} computes \spad{product(n=1,{}3,{}5...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
598620`\spad{oddInfiniteProduct(f(x))} computes \spad{product(n=1,{}3,{}5...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
598708`\spad{oddInfiniteProduct(f(x))} computes \spad{product(n=1,{}3,{}5...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
598794`\spad{oddInfiniteProduct(f(x))} computes \spad{product(n=1,{}3,{}5...,{}f(x^n))}. The series \spad{f(x)} should have constant coefficient 1.
598874`\spad{oddintegers(n)} returns \spad{[n,{}n+2,{}n+4,{}...]}.
598959`\spad{oddlambert(st)} computes \spad{f(x) + f(x^3) + f(x^5) + ...} if \spad{st} is a stream representing \spad{f(x)}. This function is used for computing infinite products. If \spad{f}(\spad{x}) is a power series with constant coefficient 1 then \spad{prod(f(x^(2*n-1)),{}n=1..infinity) = exp(oddlambert(log(f(x))))}.
599102`\spad{oddlambert(f(x))} returns \spad{f(x) + f(x^3) + f(x^5) + ...}. \indented{1}{\spad{f(x)} should have a zero constant coefficient.} \indented{1}{This function is used for computing infinite products.} \indented{1}{If \spad{f(x)} is a Taylor series with constant term 1,{} then} \indented{1}{\spad{product(n=1..infinity,{}f(x^(2*n-1)))=exp(log(oddlambert(f(x))))}.}
599173`\spad{ode1(f,{}c)} is the solution to \spad{y' = f(y)} such that \spad{y(a) = c}.
599257`\spad{ode2(f,{}c0,{}c1)} is the solution to \spad{y'' = f(y,{}y')} such that \spad{y(a) = c0} and \spad{y'(a) = c1}.
599350`\spad{ode(f,{}cl)} is the solution to \spad{y<n>=f(y,{}y',{}..,{}y<n-1>)} such that \spad{y<i>(a) = cl.(i+1)} for \spad{i} in 0..(\spad{n} - 1).
599445`\spad{omError(k,{}l)} creates an instance of OpenMathError.
599518`\spad{omega()} is the first infinite ordinal
599558`\spad{omegapower(p)} returns omega^p
599789`\axiom{one?(\spad{jv})} checks whether the jet variables \axiom{\spad{jv}} is the special variable 1.
599840`\spad{one?(a)} tests whether \spad{a} is the unit 1.
599887`\spad{one?(x)} tests if \spad{x} is equal to 1.
599927`\spad{one?(I)} tests whether the ideal \spad{I} is the unit ideal,{} \spadignore{i.e.} contains 1.
599998`\spad{oneDimensionalArray(l)} creates an array from a list of elements \spad{l}
600069`\spad{oneDimensionalArray(n,{}s)} creates an array from \spad{n} copies of element \spad{s}
600153`\spad{one returns} whether we need only one solution,{} default being \spad{true}.
600222`\spad{one(d)} specifies whether we are happy with one solution. This option is expressed in the form \spad{one == d}.
600266`if this is a compound op then opType returns the type of that op: "IMPLY"::Symbol =implies "AND"::Symbol=/\ "OR"::Symbol=\spad{\/} "NOT"::Symbol=~ "OTHER"::Symbol=not compound op
600307`\spad{op(p)} creates a value in \% equal to \spad{p} in \spad{P}.
600369`\spad{open?(t)} tests whether the given tube plot \spad{t} is open.
600419`\spad{open(s)} returns the file \spad{s} open for input.
600470`\spad{open(s,{}mode)} returns a file \spad{s} open for operation in the indicated mode: "input" or "output".
600528`\spad{operation(f)} returns the name of the operation represented by \spad{f}.
600859`\spad{operator(op)} returns a copy of \spad{op} with the domain-dependent properties appropriate for \spad{F}. Error: if \spad{op} is not an algebraic operator,{} that is,{} an \spad{n}th root or implicit algebraic operator.
600936`\spad{operator(op)} returns a copy of \spad{op} with the domain-dependent properties appropriate for \spad{F}; error if \spad{op} is not a combinatorial operator.
601017`\spad{operator(p)} returns an elementary operator with the same symbol as \spad{p}
601095`\spad{operator(op)} returns a copy of \spad{op} with the domain-dependent properties appropriate for \spad{F}; error if \spad{op} is not a special function operator
601180`\spad{operator(op)} returns the Liouvillian operator based on \spad{op}
601259`\spad{operator(op)} returns a copy of \spad{op} with the domain-dependent properties appropriate for \%.
601402`\spad{operator(s)} returns an operator with name \spad{s},{} with the appropriate semantics if \spad{s} is known. If \spad{s} is not known,{} the result has no semantics.
601465`\spad{operator(op(a1,{}...,{}an))} returns the operator op.
601518`\spad{operator(op)} returns a copy of \spad{op} with the domain-dependent properties appropriate for \%.
601588`\spad{operator(f)} makes \spad{f} into an operator with arbitrary arity.
601638`\spad{operator(f,{} n)} makes \spad{f} into an \spad{n}-ary operator.
601771`\spad{operators(f)} returns all the basic operators appearing in \spad{f},{} no matter what their levels are.
601886`\spad{opeval should} be local but conditional
601948`\spad{option?()} is not to be used at the top level; option? internally returns \spad{true} for drawing options which are indicated in a draw command,{} or \spad{false} for those which are not.
602008`\spad{option(l,{}s)} determines whether the indicated drawing option,{} \spad{s},{} is contained in the list of drawing options,{} \spad{l},{} which is defined by the draw command.
602098`\spad{option()} is not to be used at the top level; option determines internally which drawing options are indicated in a draw command.
602169`\spad{option(l,{} option)} returns which options are given.
602241`\spad{optional?(p)} tests if \spad{p} is a single matching variable which can match an identity.
602290`\spad{optional(x)} tells the pattern matcher that \spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation). Error: if \spad{x} is not a symbol.
602349`\spad{optional(x)} tells the pattern matcher that \spad{x} can match an identity (0 in a sum,{} 1 in a product or exponentiation)..
602425`\spad{options(v)} takes the given two-dimensional viewport,{} \spad{v},{} of the domain \spadtype{TwoDimensionalViewport} and returns a list containing the draw options from the domain \spadtype{DrawOption} for \spad{v}.
602493`\spad{options(v)} takes the viewport,{} \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport} and returns a list of all the draw options from the domain \spad{DrawOption} which are being used by \spad{v}.
602563`\spad{options(v,{}lopt)} takes the given two-dimensional viewport,{} \spad{v},{} of the domain \spadtype{TwoDimensionalViewport} and returns \spad{v} with it\spad{'s} draw options modified to be those which are indicated in the given list,{} \spad{lopt} of domain \spadtype{DrawOption}.
602634`\spad{options(v,{}lopt)} takes the viewport,{} \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport} and sets the draw options being used by \spad{v} to those indicated in the list,{} \spad{lopt},{} which is a list of options from the domain \spad{DrawOption}.
602707`\spad{optpair(l)} returns \spad{l} has the form \spad{[a,{} b]} and a is optional,{} and "failed" otherwise.
602777`\spad{f or g} creates the equivalent infix form.
602817`\spad{a or b} returns the logical {\em or} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
602859`\spad{a or b} returns the logical inclusive {\em or} of Boolean \spad{a} and \spad{b}.
602896`\spad{orbit(gp,{}ls)} returns the orbit of the ordered list {\em ls} under the group {\em gp}. Note: return type is \spad{L} \spad{L} \spad{S} temporarily because FSET \spad{L} \spad{S} has an error.
602963`\spad{orbit(p,{} el)} returns the orbit of {\em el} under the permutation \spad{p},{} \spadignore{i.e.} the set which is given by applications of the powers of \spad{p} to {\em el}.
603021`\spad{orbit(gp,{}el)} returns the orbit of the element {\em el} under the group {\em gp},{} \spadignore{i.e.} the set of all points gained by applying each group element to {\em el}.
603076`\spad{orbit(gp,{}els)} returns the orbit of the unordered set {\em els} under the group {\em gp}.
603141`\spad{orbits(gp)} returns the orbits of the group {\em gp},{} \spadignore{i.e.} it partitions the (finite) of all moved points.
603200`\spad{ord(c)} provides an integral code corresponding to the character \spad{c}. It is always \spad{true} that \spad{char ord c = c}.
603368`\axiom{orderDim(\spad{sys},{} \spad{jm},{} \spad{q})} computes the dimension of the manifold described by the system \axiom{\spad{sys}} with Jacobi matrix \axiom{\spad{jm}} in the jet bundle of order \axiom{\spad{q}} over the jet bundle of order \axiom{\spad{q}-1}.
603580`\spad{order(x)} \undocumented
603678`\spad{order(f)} \undocumented
603799`\spad{order(f)} \undocumented
604212`\spad{order(x)} \undocumented
604270`\axiom{order(\spad{jv})} yields the order of the jet variable \axiom{\spad{jv}} (Order as derivative).
604333`\axiom{order(\spad{f})} gives highest order of the jet variables effectively occurring in \axiom{\spad{f}}.
604408`\spad{order(x)} returns the order of a power series \spad{x},{} \indented{1}{\spadignore{i.e.} the degree of the first non-zero term of the series.}
604477`\axiom{order(de)} yields the order of the differential equation \axiom{de}.
604552`\spad{order(f)} is the degree of the lowest order non-zero term in \spad{f}. This will result in an infinite loop if \spad{f} has no non-zero terms.
604626`\spad{order x} is the order of magnitude of \spad{x}. Note: \spad{base ^ order x <= |x| < base ^ (1 + order x)}.
604680`\spad{order(p)} returns the order of the differential polynomial \spad{p},{} which is the maximum number of differentiations of a differential indeterminate,{} among all those appearing in \spad{p}.
604765`\spad{order(v)} returns \spad{n} if \spad{v} is the \spad{n}-th derivative of any differential indeterminate.
604842`\spad{order(x)} returns the exponent of the highest power of \spad{p} dividing \spad{x}.
604911`\spad{order(p)} returns the order of a permutation \spad{p} as a group element.
604971`\spad{order(gp)} returns the order of the group {\em gp}.
605036`\spad{order(a)} computes the order of an element in the multiplicative group of the field. Error: if \spad{a} is 0.
605125`\spad{order(b)} computes the order of an element \spad{b} in the multiplicative group of the field. Error: if \spad{b} equals 0.
605347`\spad{order(x,{}n)} returns the minimum of \spad{n} and the order of \spad{x}.
605435`\spad{order(f,{}n) = min(m,{}n)},{} where \spad{m} is the degree of the lowest order non-zero term in \spad{f}.
605515`\spad{order(p,{}s)} returns the order of the differential polynomial \spad{p} in differential indeterminate \spad{s}.
605602`\spad{order(f,{}x)} returns the order of \spad{f} viewed as a series in \spad{x} may result in an infinite loop if \spad{f} has no non-zero terms.
605694`\spad{order(p,{} q)} returns the largest \spad{n} such that \spad{q^n} divides polynomial \spad{p} \spadignore{i.e.} the order of \spad{p(x)} at \spad{q(x)=0}.
605795`\spad{order(f,{}u,{}g)} \undocumented
605911`\spad{order(f,{}x,{}n)} returns \spad{min(n,{}order(f,{}x))}.
606022`\spad{ordinalAdd(o1,{} o2)} returns sum of \spad{o1} and \spad{o2} as ordered sets
606072`\spad{ordinalMul(o1,{} o2)} returns product of \spad{o1} and \spad{o2} as ordered sets
606122`\spad{ordinalPower(o1,{} o2)} returns \spad{o1} to power \spad{o2},{} where power is inductively defined using succesive ordinal multiplication from the left
606174`\spad{orthogonalPolynomials(x)} returns the stream of orthogonal polynomials.
606332`\spad{orthogonalPolynomials(x)} returns the stream of orthogonal polynomials.
606448`\spad{orthonormalBasis(m)} returns the orthogonal matrix \spad{b} such that \spad{b*m*(inverse b)} is diagonal. Error: if \spad{m} is not a symmetric matrix.
606654`\spad{outDegree(s,{} a)} gives the number of arrows leading out of node 'a' in graph \spad{'s'}
606797`\spad{outerProduct(u,{}v)} constructs the matrix whose (\spad{i},{}\spad{j})\spad{'}th element is \spad{u}(\spad{i})\spad{*v}(\spad{j}).
606872`\spad{outlineRender(v,{}s)} displays the polygon outline showing either triangularized surface or a quadrilateral surface outline depending on the whether the \spadfun{diagonals} function has been set,{} for the given three-dimensional viewport \spad{v} which is of domain \spadtype{ThreeDimensionalViewport},{} if \spad{s} is "on",{} or does not display the polygon outline if \spad{s} is "off".
606943`\axiom{outputArgs(\spad{s1},{}\spad{s2},{}\spad{p},{}\spad{ts})} is an internal subroutine,{} exported only for developement.
607028`\spad{outputAsFortran(fn)} \undocumented{}
607090`\axiom{outputAsFortran(\spad{u})} translates \axiom{\spad{u}} into a legal FORTRAN subprogram.
607154`\spad{outputAsFortran(l)} sends (for each expression in the list \spad{l}) output in FORTRAN format to the destination defined by \spadsyscom{set output fortran}.
607230`\spad{outputAsFortran(o)} sends output \spad{o} in FORTRAN format.
607300`\spad{outputAsFortran(v,{}o)} sends output \spad{v} = \spad{o} in FORTRAN format to the destination defined by \spadsyscom{set output fortran}.
607377`\spad{outputAsScript(l)} sends (for each expression in the list \spad{l}) output in Script Formula Formatter format to the destination defined. by \spadsyscom{set output forumula}.
607452`\spad{outputAsScript(o)} sends output \spad{o} in Script Formula Formatter format to the destination defined by \spadsyscom{set output formula}.
607521`\spad{outputAsTex(l)} sends (for each expression in the list \spad{l}) output in Tex format to the destination as defined by \spadsyscom{set output tex}.
607593`\spad{outputAsTex(o)} sends output \spad{o} in Tex format to the destination defined by \spadsyscom{set output tex}.
607659`\spad{outputFixed()} sets the output mode to fixed point notation; the output will contain a decimal point.
607700`\spad{outputFixed(n)} sets the output mode to fixed point notation,{} with \spad{n} digits displayed after the decimal point.
607759`\spad{outputFloating()} sets the output mode to floating (scientific) notation,{} \spadignore{i.e.} \spad{mantissa * 10 exponent} is displayed as \spad{0.mantissa E exponent}.
607803`\spad{outputFloating(n)} sets the output mode to floating (scientific) notation with \spad{n} significant digits displayed after the decimal point.
607865`\spad{outputForm(sf)} creates an form for small float \spad{sf}.
607919`\spad{outputForm(n)} creates an form for integer \spad{n}.
607969`\spad{outputForm(s)} creates an form for string \spad{s}.
608018`\spad{outputForm(s)} creates an form for symbol \spad{s}.
608067`\spad{outputForm(p,{}var)} converts the SparseUnivariatePolynomial \spad{p} to an output form (see \spadtype{OutputForm}) printed as a polynomial in the output form variable.
608150`\spad{outputForm(p,{} x)} returns the output form of \spad{p} using \spad{x} for the otherwise anonymous variable.
608249`\spad{outputForm(l,{} fop,{} fexp,{} unit)} converts the monoid element represented by \spad{l} to an \spadtype{OutputForm}. Argument unit is the output form for the \spadignore{unit} of the monoid (\spadignore{e.g.} 0 or 1),{} \spad{fop(a,{} b)} is the output form for the monoid operation applied to \spad{a} and \spad{b} (\spadignore{e.g.} \spad{a + b},{} \spa
608249`--d{a * b},{} \spad{ab}),{} and \spad{fexp(a,{} n)} is the output form for the exponentiation operation applied to \spad{a} and \spad{n} (\spadignore{e.g.} \spad{n a},{} \spad{n * a},{} \spad{a ^ n},{} \spad{a\^n}).
608393`\spad{outputGeneral()} sets the output mode (default mode) to general notation; numbers will be displayed in either fixed or floating (scientific) notation depending on the magnitude.
608436`\spad{outputGeneral(n)} sets the output mode to general notation with \spad{n} significant digits displayed.
608497`\spad{outputList(l)} displays the concatenated components of the list \spad{l} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}; quotes are stripped from strings.
608554`\spad{outputSpacing(n)} inserts a space after \spad{n} (default 10) digits on output; outputSpacing(0) means no spaces are inserted.
608615`\spad{outputVRML(rp,{}f1)} writes the attribute as \spad{VRML2}. \spad{VRML2} is not \spad{xml} but it has the same semantics and node names as \spad{X3D} but a different syntax so it makes sense to create an \spad{XML} structure and then format the output differently.
608673`\spad{outputVRML(rp,{}f1)} writes the element as \spad{VRML2}. \spad{VRML2} is not \spad{xml} but it has the same semantics and node names as \spad{X3D} but a different syntax so it makes sense to create an \spad{XML} structure and then format the output differently.
608729`\spad{output(x)} displays the output form \spad{x} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}.
608783`\spad{output(s)} displays the string \spad{s} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}.
608833`\spad{output(n,{}st)} computes and displays the first \spad{n} entries of \spad{st}.
608901`\spad{output(s,{}x)} displays the string \spad{s} followed by the form \spad{x} on the ``algebra output\spad{''} stream,{} as defined by \spadsyscom{set output algebra}.
608962`\spad{over(f,{}g)} creates a form for the vertical fraction of \spad{f} over \spad{g}.
609004`\spad{overbar(f)} creates the form \spad{f} with an overbar.
609046`\spad{overlabel(x,{}f)} creates the form \spad{f} with \spad{"x} overbar" over the top.
609093`\spad{overlap(x,{} y)} returns \spad{[l,{} m,{} r]} such that \spad{x = l * m},{} \spad{y = m * r} and \spad{l} and \spad{r} have no overlap,{} \spadignore{i.e.} \spad{overlap(l,{} r) = [l,{} 1,{} r]}.
609164`\spad{overlap(x,{} y)} returns \spad{[l,{} m,{} r]} such that \spad{x = l * m} and \spad{y = m * r} hold and such that \spad{l} and \spad{r} have no overlap,{} that is \spad{overlap(l,{} r) = [l,{} 1,{} r]}.
609242`\spad{overset?(s,{}sl)} returns \spad{true} if \spad{s} properly a sublist of a member of \spad{sl}; otherwise it returns \spad{false}
609342`\spad{pToDmp(p)} converts \spad{p} from a \spadtype{POLY} to a \spadtype{DMP}.
609435`\spad{pToHdmp(p)} converts \spad{p} from a \spadtype{POLY} to a \spadtype{HDMP}.
609540`\spad{pack!(f)} reorganizes the file \spad{f} on disk to recover unused space.
609592`\spad{pack!(f)} reorganizes the file \spad{f} on disk to recover unused space.
609629`\spad{pack_exps0(exps,{} sizes,{} ns,{} start)} is used by pack_exps.
609739`\spad{pack_exps(d,{} s,{} m)} produces vector of exponents up to degree \spad{d}. \spad{s} is size (degree) of algebraic extension. Use together with \spad{repack1}.
609844`\spad{pack_modulus(lp,{} ls,{} p)} converts \spad{lp},{} \spad{ls} and prime \spad{p} which together describe algebraic extension to packed representation.
609978`\spad{packageCall(f)} returns the input form corresponding to \spad{f}\$\spad{R}.
610047`\spad{packageCall(f,{} l)} returns the input form corresponding to \spad{f}(\spad{l}.1,{} \spad{l}.2,{}...)\$\spad{R}.
610132`\spad{pade(nd,{}dd,{}s)} computes the quotient of polynomials (if it exists) with numerator degree at most \spad{nd} and denominator degree at most \spad{dd} which matches the series \spad{s} to order \spad{nd + dd}.
610306`\spad{pade(nd,{}dd,{}ns,{}ds)} computes the approximant as a quotient of polynomials (if it exists) for arguments \spad{nd} (numerator degree of approximant),{} \spad{dd} (denominator degree of approximant),{} \spad{ns} (numerator series of function),{} and \spad{ds} (denominator series of function).
610427`\spad{pade(nd,{}dd,{}ns,{}ds)} computes the approximant as a quotient of polynomials (if it exists) for arguments \spad{nd} (numerator degree of approximant),{} \spad{dd} (denominator degree of approximant),{} \spad{ns} (numerator series of function),{} and \spad{ds} (denominator series of function).
610632`\spad{padecf(nd,{}dd,{}ns,{}ds)} computes the approximant as a continued fraction of polynomials (if it exists) for arguments \spad{nd} (numerator degree of approximant),{} \spad{dd} (denominator degree of approximant),{} \spad{ns} (numerator series of function),{} and \spad{ds} (denominator series of function).
610764`\spad{padicFraction(q)} expands the fraction \spad{p}-adically in the primes \spad{p} in the denominator of \spad{q}. For example,{} \spad{padicFraction(3/(2^2)) = 1/2 + 1/(2^2)}. Use \spadfunFrom{compactFraction}{PartialFraction} to return to compact form.
610820`\spad{padicallyExpand(p,{}x)} is a utility function that expands the second argument \spad{x} \spad{``p}-adically\spad{''} in the first.
610906`\spad{pair?(s)} is \spad{true} if \spad{s} has is a non-null Lisp list.
610982`\spad{palgLODE0(op,{} g,{} x,{} y,{} d,{} p)} returns the solution of \spad{op f = g}. Argument \spad{y} is an algebraic function of \spad{x} satisfying \spad{d(x)\^2y(x)\^2 = P(x)}.
611196`\spad{palgLODE0(op,{}g,{}x,{}y,{}z,{}t,{}c)} returns the solution of \spad{op f = g} Argument \spad{y} is an algebraic function of \spad{x} satisfying \spad{f(x,{}y)dx = c f(t,{}y) dy}; \spad{c} and \spad{t} are rational functions of \spad{y}.
611430`\spad{palgLODE(op,{} g,{} kx,{} y,{} x)} returns the solution of \spad{op f = g}. \spad{y} is an algebraic function of \spad{x}.
611622`\spad{palgRDE0(f,{} g,{} x,{} y,{} foo,{} d,{} p)} returns a function \spad{z(x,{}y)} such that \spad{dz/dx + n * df/dx z(x,{}y) = g(x,{}y)} if such a \spad{z} exists,{} and "failed" otherwise. Argument \spad{y} is an algebraic function of \spad{x} satisfying \spad{d(x)\^2y(x)\^2 = P(x)}. Argument \spad{foo},{} called by \spad{foo(a,{} b,{} x)},
611622`--{} is a function that solves \spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \spad{u(x)} not involving \spad{y}.
611782`\spad{palgRDE0(f,{} g,{} x,{} y,{} foo,{} t,{} c)} returns a function \spad{z(x,{}y)} such that \spad{dz/dx + n * df/dx z(x,{}y) = g(x,{}y)} if such a \spad{z} exists,{} and "failed" otherwise. Argument \spad{y} is an algebraic function of \spad{x} satisfying \spad{f(x,{}y)dx = c f(t,{}y) dy}; \spad{c} and \spad{t} are ration
611782`--al functions of \spad{y}. Argument \spad{foo},{} called by \spad{foo(a,{} b,{} x)},{} is a function that solves \spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \spad{u(x)} not involving \spad{y}.
611962`\spad{palgRDE(nfp,{} f,{} g,{} x,{} y,{} foo)} returns a function \spad{z(x,{}y)} such that \spad{dz/dx + n * df/dx z(x,{}y) = g(x,{}y)} if such a \spad{z} exists,{} "failed" otherwise; \spad{y} is an algebraic function of \spad{x}; \spad{foo(a,{} b,{} x)} is a function that solves \spad{du/dx + n * da/dx u(x) = u(x)} for an unknown \spad{u(x)} not involving \spad{y}. \sp
611962`--ad{nfp} is \spad{n * df/dx}.
612095`\spad{palgextint0(f,{} x,{} y,{} g,{} d,{} p)} returns functions \spad{[h,{} c]} such that \spad{dh/dx = f(x,{}y) - c g},{} where \spad{y} is an algebraic function of \spad{x} satisfying \spad{d(x)\^2 y(x)\^2 = P(x)},{} or "failed" if no such functions exist.
612250`\spad{palgextint0(f,{} x,{} y,{} g,{} z,{} t,{} c)} returns functions \spad{[h,{} d]} such that \spad{dh/dx = f(x,{}y) - d g},{} where \spad{y} is an algebraic function of \spad{x} satisfying \spad{f(x,{}y)dx = c f(t,{}y) dy},{} and \spad{c} and \spad{t} are rational functions of \spad{y}. Argument \spad{z} is a dummy variable not
612250`-- appearing in \spad{f(x,{}y)}. The operation returns "failed" if no such functions exist.
612425`\spad{palgextint(f,{} x,{} y,{} g)} returns functions \spad{[h,{} c]} such that \spad{dh/dx = f(x,{}y) - c g},{} where \spad{y} is an algebraic function of \spad{x}; returns "failed" if no such functions exist.
612551`\spad{palgextintegrate(f,{} d,{} g,{} c1,{} c2)} returns functions \spad{[h,{} c]} such that using \spad{c1,{} c2} as a change of variables we have \spad{(d(h))(c1(x),{} c2(y)) = d(c1(x))f(c1(x),{}c2(y)) - c g(x,{}y)},{} where \spad{y} is an algebraic function of \spad{x}; returns "failed" if no such functions exist.
612704`\spad{palginfieldint(f,{} d)} returns an algebraic function \spad{g} such that \spad{dg = f} if such a \spad{g} exists,{} "failed" otherwise. Argument \spad{f} must be a pure algebraic function.
612800`\spad{palgint0(f,{} x,{} y,{} d,{} p)} returns the integral of \spad{f(x,{}y)dx} where \spad{y} is an algebraic function of \spad{x} satisfying \spad{d(x)\^2 y(x)\^2 = P(x)}.
612929`\spad{palgint0(f,{} x,{} y,{} z,{} t,{} c)} returns the integral of \spad{f(x,{}y)dx} where \spad{y} is an algebraic function of \spad{x} satisfying \spad{f(x,{}y)dx = c f(t,{}y) dy}; \spad{c} and \spad{t} are rational functions of \spad{y}. Argument \spad{z} is a dummy variable not appearing in \spad{f(x,{}y)}.
613078`\spad{palgint(f,{} x,{} y)} returns the integral of \spad{f(x,{}y)dx} where \spad{y} is an algebraic function of \spad{x}.
613178`\spad{palgintegrate(f,{} d)} integrates \spad{f} with respect to the derivation \spad{d}. Argument \spad{f} must be a pure algebraic function.
613276`\spad{palglimint0(f,{} x,{} y,{} [u1,{}...,{}un],{} d,{} p)} returns functions \spad{[h,{}[[\spad{ci},{} \spad{ui}]]]} such that the \spad{ui}\spad{'s} are among \spad{[u1,{}...,{}un]} and \spad{d(h + sum(\spad{ci} log(\spad{ui})))/dx = f(x,{}y)} if such functions exist,{} and "failed" otherwise. Argument \sp
613276`--ad{y} is an algebraic function of \spad{x} satisfying \spad{d(x)\^2y(x)\^2 = P(x)}.
613473`\spad{palglimint0(f,{} x,{} y,{} [u1,{}...,{}un],{} z,{} t,{} c)} returns functions \spad{[h,{}[[\spad{ci},{} \spad{ui}]]]} such that the \spad{ui}\spad{'s} are among \spad{[u1,{}...,{}un]} and \spad{d(h + sum(\spad{ci} log(\spad{ui})))/dx = f(x,{}y)} if such functions exist,{} and "failed
613473`--" otherwise. Argument \spad{y} is an algebraic function of \spad{x} satisfying \spad{f(x,{}y)dx = c f(t,{}y) dy}; \spad{c} and \spad{t} are rational functions of \spad{y}.
613690`\spad{palglimint(f,{} x,{} y,{} [u1,{}...,{}un])} returns functions \spad{[h,{}[[\spad{ci},{} \spad{ui}]]]} such that the \spad{ui}\spad{'s} are among \spad{[u1,{}...,{}un]} and \spad{d(h + sum(\spad{ci} log(\spad{ui})))/dx = f(x,{}y)} if such functions exist,{} "failed" otherwise; \spad{y} is an algebraic function of \spad{x}.
613858`\spad{parabolicCylindrical(pt)} transforms \spad{pt} from parabolic cylindrical coordinates to Cartesian coordinates: the function produced will map the point \spad{(u,{}v,{}z)} to \spad{x = 1/2*(u^2 - v^2)},{} \spad{y = u*v},{} \spad{z}.
613935`\spad{parabolic(pt)} transforms \spad{pt} from parabolic coordinates to Cartesian coordinates: the function produced will map the point \spad{(u,{}v)} to \spad{x = 1/2*(u^2 - v^2)},{} \spad{y = u*v}.
614001`\spad{paraboloidal(pt)} transforms \spad{pt} from paraboloidal coordinates to Cartesian coordinates: the function produced will map the point \spad{(u,{}v,{}phi)} to \spad{x = u*v*cos(phi)},{} \spad{y = u*v*sin(phi)},{} \spad{z = 1/2 * (u^2 - v^2)}.
614070`\spad{parallel(x,{}y)} returns the length of the parallel component of two vectors. In the case of Euclidean space this is the dot product of two vectors. We use this function to test the colinearity
614129`\spad{parametersOf(tab)} returns a list of all the symbols declared in \spad{tab}
614187`\spad{parametric? determines} whether it is a parametric plot?
614280`\spad{paren(lf)} creates the form separating the elements of \spad{lf} by commas and encloses the result in parentheses.
614368`\spad{paren(f)} creates the form enclosing \spad{f} in parentheses.
614408`\spad{paren([f1,{}...,{}fn])} returns \spad{(f1,{}...,{}fn)}. This prevents the \spad{fi} from being evaluated when operators are applied to them,{} and makes them applicable to a unary operator. For example,{} \spad{atan(paren [x,{} 2])} returns the formal kernel \spad{atan((x,{} 2))}.
614459`\spad{paren(f)} returns (\spad{f}). This prevents \spad{f} from being evaluated when operators are applied to it. For example,{} \spad{log(1)} returns 0,{} but \spad{log(paren 1)} returns the formal kernel log((1)).
614504`\spad{parent(s)} returns the subspace which is the parent of the indicated 3 dimensional subspace \spad{s}. If \spad{s} is the top level subspace an error message is returned.
614548`Constructs intuitionistic logic terms from a string notation assumes format like this: <term2> \spad{::=} var | "("<term>")" <term> \spad{::=} var | <term>/\spad{\<}term> | <term>\spad{\/}<term> | \indented{11}{<term>-><term> | "("<term>")"}
614646`parseTerm is used by parseIL. It would rarely be called externally but it is here to allow it to call parseIL that is to allow circular calls
614747`Constructs intuitionistic logic terms from a string notation assumes format like this: <term> \spad{::=} var | <term>/\spad{\<}term> | <term>\spad{\/}<term> | \indented{11}{<term>-><term> | "("<term>")"}
614789`Constructs nested lambda terms from a string notation assumes format like this: <term> \spad{::=} "\spad{\"} var "."<term> | \spad{n} | <term><term> | "("<term>")" where: \\ = lambda (\spad{I} would like to use unicode lambda symbol but \indented{12}{\spad{I} would also like to keep maximum compatibility} \indented{12}{with non-unicode versions of Lisp)} \spad{n} = De Bruijn index which is a integer where,{} 1=inside inner \indented{12}{lambda term,{} 2
614789`--= next outer lambda term,{} 3= next} \indented{12}{outer and so on.} brackets can be used around whole terms.
614839`Constructs combinators from a string
614894`parseTerm is used by parseLambda. It would rarely be called externally but it is here to allow it to call parseLambda that is to allow circular calls
614997`parseTerm is used by parseSki. It would rarely be called externally but it is here to allow it to call parseSki that is to allow circular calls
615108`construct a variable by parsing a string pin is index to string at start of parse pout is index to string at end of parse
615210`construct a variable by parsing a string
615312`\spad{parse(s)} is the inverse of unparse. It parses a string to InputForm
615355`\spad{partialDenominators(x)} extracts the denominators in \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{partialDenominators(x) = [b1,{}b2,{}b3,{}...]}.
615426`\spad{partialFraction(rf,{} var)} returns the partial fraction decomposition of the rational function \spad{rf} with respect to the variable var.
615520`\spad{partialFraction(numer,{}denom)} is the main function for constructing partial fractions. The second argument is the denominator and should be factored.
615589`\spad{partialFraction(num,{} facdenom,{} var)} returns the partial fraction decomposition of the rational function whose numerator is \spad{num} and whose factored denominator is \spad{facdenom} with respect to the variable var.
615697`\spad{partialNumerators(x)} extracts the numerators in \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{partialNumerators(x) = [a1,{}a2,{}a3,{}...]}.
615766`\spad{partialQuotients(x)} extracts the partial quotients in \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{partialQuotients(x) = [b0,{}b1,{}b2,{}b3,{}...]}.
615834`\spad{particularSolution(u)} \undocumented
615907`\spad{particularSolution(A,{}B)} finds a particular solution of the linear system \spad{AX = B}.
616008`\spad{particularSolution(A,{}B)} finds a particular solution of the linear system \spad{AX = B}.
616120`\spad{particularSolution(op,{} g,{} [f1,{}...,{}fm],{} I)} returns a particular solution \spad{h} of the equation \spad{op y = g} where \spad{[f1,{}...,{}fm]} are linearly independent and \spad{op(\spad{fi})=0}. The value "failed" is returned if no particular solution is found. Note: the method of variations of parameters is used.
616213`\spad{partition(\spad{li})} converts a list of integers \spad{li} to a partition
616267`\spad{partition(n)} returns the number of partitions of the integer \spad{n}. This is the number of distinct ways that \spad{n} can be written as a sum of positive integers.
616330`\spad{partitions(n)} is the stream of all partitions of \spad{n}.
616414`\spad{partitions(p,{}l)} is the stream of all \indented{1}{partitions whose number of} \indented{1}{parts and largest part are no greater than \spad{p} and \spad{l}.}
616506`\spad{partitions(p,{}l,{}n)} is the stream of partitions \indented{1}{of \spad{n} whose number of parts is no greater than \spad{p}} \indented{1}{and whose largest part is no greater than \spad{l}.}
616936`\spad{parts(m)} returns a list of the elements of \spad{m} in row major order
617009`\spad{parts(u)} returns a list of the consecutive elements of \spad{u}. For collections,{} \axiom{parts([\spad{x},{}\spad{y},{}...,{}\spad{z}]) = (\spad{x},{}\spad{y},{}...,{}\spad{z})}.
617090`\spad{pascalTriangle(n,{}r)} returns the binomial coefficient \spad{C(n,{}r)=n!/(r! (n-r)!)} and stores it in a table to prevent recomputation.
617175`\spad{pastel(c)} sets the shade of a hue,{} \spad{c},{} above bright,{} but below light.
617216`\spad{patternMatchTimes(lsubj,{} lpat,{} res,{} match)} matches the product of patterns \spad{reduce(*,{}lpat)} to the product of subjects \spad{reduce(*,{}lsubj)}; \spad{r} contains the previous matches and match is a pattern-matching function on \spad{P}.
617671`\spad{patternMatch(expr,{} pat,{} res)} matches the pattern \spad{pat} to the expression \spad{expr}; res contains the variables of \spad{pat} which are already matched and their matches. Note: this function handles type towers by changing the predicates and calling the matching function provided by \spad{A}.
617790`\spad{patternMatch(cexpr,{} pat,{} res)} matches the pattern \spad{pat} to the complex expression \spad{cexpr}. res contains the variables of \spad{pat} which are already matched and their matches.
617950`\spad{patternMatch(expr,{} pat,{} res)} matches the pattern \spad{pat} to the expression \spad{expr}; res contains the variables of \spad{pat} which are already matched and their matches.
618074`\spad{patternMatch(n,{} pat,{} res)} matches the pattern \spad{pat} to the integer \spad{n}; res contains the variables of \spad{pat} which are already matched and their matches.
618218`\spad{patternMatch(f(e1,{}...,{}en),{} pat,{} res)} matches the pattern \spad{pat} to \spad{f(e1,{}...,{}en)}; res contains the variables of \spad{pat} which are already matched and their matches.
618341`\spad{patternMatch(l,{} pat,{} res)} matches the pattern \spad{pat} to the list \spad{l}; res contains the variables of \spad{pat} which are already matched and their matches.
618477`\spad{patternMatch(p,{} pat,{} res)} matches the pattern \spad{pat} to the polynomial \spad{p}; res contains the variables of \spad{pat} which are already matched and their matches.
618636`\spad{patternMatch(a/b,{} pat,{} res)} matches the pattern \spad{pat} to the quotient \spad{a/b}; res contains the variables of \spad{pat} which are already matched and their matches.
619531`\spad{patternMatch(expr,{} pat,{} res)} matches the pattern \spad{pat} to the expression \spad{expr}; res contains the variables of \spad{pat} which are already matched and their matches (necessary for recursion).
619659`\spad{patternMatch(expr,{} pat,{} res)} matches the pattern \spad{pat} to the expression \spad{expr}. res contains the variables of \spad{pat} which are already matched and their matches (necessary for recursion). Initially,{} res is just the result of \spadfun{new} which is an empty list of matches.
619773`\spad{patternMatch(p,{} pat,{} res,{} vmatch)} matches the pattern \spad{pat} to the polynomial \spad{p}. \spad{res} contains the variables of \spad{pat} which are already matched and their matches; vmatch is the matching function to use on the variables.
619970`\spad{patternMatch(lsubj,{} lpat,{} op,{} res,{} match)} matches the list of patterns \spad{lpat} to the list of subjects \spad{lsubj},{} allowing for commutativity; \spad{op} is the operator such that \spad{op}(\spad{lpat}) should match \spad{op}(\spad{lsubj}) at the end,{} \spad{r} contains the prev
619970`--ious matches,{} and match is a pattern-matching function on \spad{P}.
620175`\spad{patternVariable(x,{} c?,{} o?,{} m?)} creates a pattern variable \spad{x},{} which is constant if \spad{c? = true},{} optional if \spad{o? = true},{} and multiple if \spad{m? = true}.
620253`\spad{pattern(r)} returns the pattern corresponding to the left hand side of the rule \spad{r}.
620317`\spad{pdct(a1^n1 a2^n2 ...)} returns \spad{n1! * a1^n1 * n2! * a2^n2 * ...}. This function is used in the package \spadtype{CycleIndicators}.
620360`\spad{perfectNthPower?(n,{}r)} returns \spad{true} if \spad{n} is an \spad{r}th power and \spad{false} otherwise
620439`\spad{perfectNthRoot(n)} returns \spad{[x,{}r]},{} where \spad{n = x\^r} and \spad{r} is the largest integer such that \spad{n} is a perfect \spad{r}th power
620532`\spad{perfectNthRoot(n,{}r)} returns the \spad{r}th root of \spad{n} if \spad{n} is an \spad{r}th power and returns "failed" otherwise
620619`\spad{perfectSqrt(n)} returns the square root of \spad{n} if \spad{n} is a perfect square and returns "failed" otherwise
620684`\spad{perfectSquare?(n)} returns \spad{true} if \spad{n} is a perfect square and \spad{false} otherwise
620742`\spad{perm_to_vec(supp,{} p,{} degree)} should be local but conditional
620860`\spad{permanent(x)} computes the permanent of a square matrix \spad{x}. The {\em permanent} is equivalent to the \spadfun{determinant} except that coefficients have no change of sign. This function is much more difficult to compute than the {\em determinant}. The formula used is by \spad{H}.\spad{J}. Ryser,{} improved by [Nijenhuis and Wilf,{} \spad{Ch}. 19]. Note: permanent(\spad{x}) choose one of three algorithms,{} depending on the underl
620860`--ying ring \spad{R} and on \spad{n},{} the number of rows (and columns) of \spad{x:}\begin{items} \item 1. if 2 has an inverse in \spad{R} we can use the algorithm of \indented{3}{[Nijenhuis and Wilf,{} \spad{ch}.19,{}\spad{p}.158]; if 2 has no inverse,{}} \indented{3}{some modifications are necessary:} \item 2. if {\em n > 6} and \spad{R} is an integral domain with characteristic \indented{3}{different from 2 (the algorithm works if and only 2 is not a} \indented{3}{zero-divisor of \spad{R} an
620860`--d {\em characteristic()\$R ~= 2},{}} \indented{3}{but how to check that for any given \spad{R} ?),{}} \indented{3}{the local function {\em permanent2} is called;} \item 3. else,{} the local function {\em permanent3} is called \indented{3}{(works for all commutative rings \spad{R}).} \end{items}
620922`\spad{permutationGroup(ls)} coerces a list of permutations {\em ls} to the group generated by this list.
621000`\spad{permutationRepresentation(\spad{pi},{}n)} returns the matrix {\em (deltai,{}\spad{pi}(i))} (Kronecker delta) if the permutation {\em \spad{pi}} is in list notation and permutes {\em {1,{}2,{}...,{}n}}.
621099`\spad{permutationRepresentation([pi1,{}...,{}pik],{}n)} returns the list of matrices {\em [(deltai,{}pi1(i)),{}...,{}(deltai,{}pik(i))]} if the permutations {\em pi1},{}...,{}{\em pik} are in list notation and are permuting {\em {1,{}2,{}...,{}n}}.
621210`\spad{permutationRepresentation([pi1,{}...,{}pik],{}n)} returns the list of matrices {\em [(deltai,{}pi1(i)),{}...,{}(deltai,{}pik(i))]} (Kronecker delta) for the permutations {\em pi1,{}...,{}pik} of {\em {1,{}2,{}...,{}n}}.
621336`\spad{permutationRepresentation(\spad{pi},{}n)} returns the matrix {\em (deltai,{}\spad{pi}(i))} (Kronecker delta) for a permutation {\em \spad{pi}} of {\em {1,{}2,{}...,{}n}}.
621450`\spad{permutation(n,{} r)} returns the number of permutations of \spad{n} objects taken \spad{r} at a time,{} \spadignore{i.e.} \spad{n!/}(\spad{n}-\spad{r})!.
621566`\spad{permutation(n)} returns \spad{!P(n,{}r) = n!/(n-r)!}. This is the number of permutations of \spad{n} objects taken \spad{r} at a time.
621633`\spad{permutation(n,{} m)} returns the number of permutations of \spad{n} objects taken \spad{m} at a time. Note: \spad{permutation(n,{}m) = n!/(n-m)!}.
621701`\spad{permutations(n)} is the stream of permutations \indented{1}{formed from \spad{1,{}2,{}3,{}...,{}n}.}
621787`\spad{perpendicular(x,{}y)} returns a vector perpendicular to the two vectors in the case of 3D Euclidean space this is the cross the product of two vectors. We use this function to compute orthogonal
621842`\spad{perspective(v,{}s)} displays the graph in perspective if \spad{s} is "on",{} or does not display perspective if \spad{s} is "off" for the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport}.
621911`\spad{phiCoord(pt)} returns the third element of the point,{} \spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical coordinate system.
621964`\spad{physicalLength!(x,{}n)} changes the physical length of \spad{x} to be \spad{n} and returns the new array.
622097`\spad{physicalLength(x)} returns the number of elements \spad{x} can accomodate before growing
622244`\spad{\spad{pi}()} returns the \spad{pi} operator
622384`\spad{\spad{pi}(x)} represents the NAG Library function X01AAF which returns \indented{1}{an approximation to the value of \spad{pi}}
622461`\spad{\spad{pi}()} returns the constant \spad{pi}.
622516`\spad{\spad{pi}()} returns the symbolic \%\spad{pi}.
622543`\spad{pile(l)} creates the form consisting of the elements of \spad{l} which displays as a pile,{} \spadignore{i.e.} the elements begin on a new line and are indented right to the same margin.
622588`\axiom{pivot(A,{} \spad{i})} returns the leading entry of the \axiom{\spad{i}}\spad{-}th row of the matrix \axiom{A} together with its index.
622671`\axiom{pivots(A)} returns all leading entries of the matrix \axiom{A} together with their indices.
622836`\spad{plenaryPower(a,{}n)} is recursively defined to be \spad{plenaryPower(a,{}n-1)*plenaryPower(a,{}n-1)} for \spad{n>1} and \spad{a} for \spad{n=1}.
622913`\spad{pleskenSplit(poly,{} eps)} determines a start polynomial {\em start}\\ by using "startPolynomial then it increases the exponent \spad{n} of {\em start ^ n mod poly} to get an approximate factor of {\em poly},{} in general of degree "degree \spad{poly} \spad{-1"}. Then a divisor cascade is calculated and the best splitting is chosen,{} as soon as the error is small enough.
622993`\spad{pleskenSplit(poly,{}eps,{}info)} determines a start polynomial {\em start} by using "startPolynomial then it increases the exponent \spad{n} of {\em start ^ n mod poly} to get an approximate factor of {\em poly},{} in general of degree "degree \spad{poly} \spad{-1"}. Then a divisor cascade is calculated and the best splitting is chosen,{} as soon as the error is small enough. If {\em info} is {\em true},{} the
622993`--n information messages are issued.
623081`\spad{plotPolar(f)} plots the polar curve \spad{r = f(theta)} as theta ranges over the interval \spad{[0,{}2*\%\spad{pi}]}; this is the same as the parametric curve \spad{x = f(t) * cos(t)},{} \spad{y = f(t) * sin(t)}.
623143`\spad{plotPolar(f,{}a..b)} plots the polar curve \spad{r = f(theta)} as theta ranges over the interval \spad{[a,{}b]}; this is the same as the parametric curve \spad{x = f(t) * cos(t)},{} \spad{y = f(t) * sin(t)}.
623226`\spad{plotPolar(f,{}theta)} plots the graph of \spad{r = f(theta)} as \spad{theta} ranges from 0 to 2 \spad{pi}
623285`\spad{plotPolar(f,{}theta,{}seg)} plots the graph of \spad{r = f(theta)} as \spad{theta} ranges over an interval
623365`\spad{plot(f,{}a..b)} plots the function \spad{f(x)} on the interval \spad{[a,{}b]}.
623443`\spad{plot([f1,{}...,{}fm],{}a..b)} plots the functions \spad{y = f1(x)},{}...,{} \spad{y = fm(x)} on the interval \spad{a..b}.
623527`\spad{plot(x,{}r)} \undocumented
623583`\spad{plot(x,{}r)} \undocumented
623637`\spad{plot(f,{}g,{}a..b)} plots the parametric curve \spad{x = f(t)},{} \spad{y = g(t)} as \spad{t} ranges over the interval \spad{[a,{}b]}.
623742`\spad{plot(f,{}a..b,{}c..d)} plots the function \spad{f(x)} on the interval \spad{[a,{}b]}; \spad{y}-range of \spad{[c,{}d]} is noted in Plot object.
623841`\spad{plot([f1,{}...,{}fm],{}a..b,{}c..d)} plots the functions \spad{y = f1(x)},{}...,{} \spad{y = fm(x)} on the interval \spad{a..b}; \spad{y}-range of \spad{[c,{}d]} is noted in Plot object.
623946`\spad{plot(fcn,{}x,{}seg)} plots the graph of \spad{y = f(x)} on a interval
624021`\spad{plot(f,{}g,{}t,{}seg)} plots the graph of \spad{x = f(t)},{} \spad{y = g(t)} as \spad{t} ranges over an interval.
624098`\spad{plot(f,{}g,{}h,{}a..b)} plots {/emx = \spad{f}(\spad{t}),{} \spad{y} = \spad{g}(\spad{t}),{} \spad{z} = \spad{h}(\spad{t})} as \spad{t} ranges over {/em[a,{}\spad{b}]}.
624259`\spad{plot(f,{}g,{}a..b,{}c..d,{}e..f)} plots the parametric curve \spad{x = f(t)},{} \spad{y = g(t)} as \spad{t} ranges over the interval \spad{[a,{}b]}; \spad{x}-range of \spad{[c,{}d]} and \spad{y}-range of \spad{[e,{}f]} are noted in Plot object.
624406`\spad{plot(f1,{}f2,{}f3,{}f4,{}x,{}y,{}z,{}w)} \undocumented
624630`\spad{plus!(c,{}a,{}b)} computes the matrix sum \spad{a + b} and stores the result in the matrix \spad{c}. Error: if \spad{a},{} \spad{b},{} and \spad{c} do not have the same dimensions.
624729`\spad{plusInfinity()} returns plusIinfinity.
624796`\spad{plusInfinity()} returns +infinity.
624851`\spad{plus(x,{} y)} returns \spad{x + y} where \spad{+} is the monoid operation,{} which is assumed commutative.
624904`\spad{plus(x,{}y)} adds two matrices,{} term by term we note that they must be the same size
624972`\spad{plus(s,{} e,{} x)} returns \spad{e * s + x} where \spad{+} is the monoid operation,{} which is assumed commutative.
625026`\spad{pmComplexintegrate(f,{} x)} returns either "failed" or \spad{[g,{}h]} such that \spad{integrate(f,{}x) = g + integrate(h,{}x)}. It only looks for special complex integrals that pmintegrate does not return.
625257`\spad{pmintegrate(f,{} x)} returns either "failed" or \spad{[g,{}h]} such that \spad{integrate(f,{}x) = g + integrate(h,{}x)}.
625481`\spad{pmintegrate(f,{} x = a..b)} returns the integral of \spad{f(x)dx} from a to \spad{b} if it can be found by the built-in pattern matching rules.
625715`\spad{po(q)} creates a value in \spad{P} equal to \spad{q} in \%.
625777`\spad{point?(s)} queries whether the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a single component which is a point and returns the boolean result.
625834`\spad{pointColorDefault()} returns the default color of points in a 2D viewport.
625898`\spad{pointColorDefault(p)} sets the default color of points in a 2D viewport to the palette \spad{p}.
625969`\spad{pointColorPalette(l,{}p)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{pointColorPalette}. If the option does not exist the value,{} \spad{p} is returned.
626058`\spad{pointColor(v)} specifies a color,{} \spad{v},{} for 2D graph points. This option is expressed in the form \spad{pointColor == v}.
626106`\spad{pointColor(p)} specifies a color index for 2D graph points from the spadcolors palette \spad{p}. This option is expressed in the form \spad{pointColor == p}.
626156`\spad{pointData(s)} returns the list of points from the point data field of the 3 dimensional subspace \spad{s}.
626215`\spad{pointList(me)} returns points at vertices of this mesh.
626267`\spad{pointList(gp)} should be local but conditional
626325`\spad{pointLists(\spad{gi})} returns the list of lists of points which compose the given graph,{} \spad{\spad{gi}},{} of the domain \spadtype{GraphImage}.
626398`\spad{pointPlot(f,{}g,{}h,{}a..b)} plots {/emx = \spad{f}(\spad{t}),{} \spad{y} = \spad{g}(\spad{t}),{} \spad{z} = \spad{h}(\spad{t})} as \spad{t} ranges over {/em[a,{}\spad{b}]}.
626490`\spad{pointPlot(t +-> (f(t),{}g(t)),{}a..b)} plots the parametric curve \spad{x = f(t)},{} \spad{y = g(t)} as \spad{t} ranges over the interval \spad{[a,{}b]}.
626580`\spad{pointPlot(t +-> (f(t),{}g(t)),{}a..b,{}c..d,{}e..f)} plots the parametric curve \spad{x = f(t)},{} \spad{y = g(t)} as \spad{t} ranges over the interval \spad{[a,{}b]}; \spad{x}-range of \spad{[c,{}d]} and \spad{y}-range of \spad{[e,{}f]} are noted in Plot object.
626712`\spad{pointPlot(f,{}x,{}y,{}z,{}w)} \undocumented
626867`\spad{pointSizeDefault()} returns the default size of the points in a 2D viewport.
626938`\spad{pointSizeDefault(i)} sets the default size of the points in a 2D viewport to \spad{i}.
627024`\spad{point(l)} returns a point category defined by a list \spad{l} of elements from the domain \spad{R}.
627075`\spad{point(p)} returns a \spadtype{ThreeSpace} object which is composed of one component,{} the point \spad{p}.
627132`\spad{point(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of only a single point and if so,{} returns the point. An error is signaled otherwise.
627189`\spad{point(s,{}[x,{}y,{}z])} adds a point component defined by a list of elements which are from the \spad{PointDomain(R)} to the \spadtype{ThreeSpace},{} \spad{s},{} where \spad{R} is the \spadtype{Ring} over which the point elements are defined.
627248`\spad{point(s,{}i)} adds a point component which is placed into a component list of the \spadtype{ThreeSpace},{} \spad{s},{} at the index given by \spad{i}.
627318`\spad{point(s,{}p)} adds a point component defined by the point,{} \spad{p},{} specified as a list from \spad{List(R)},{} to the \spadtype{ThreeSpace},{} \spad{s},{} where \spad{R} is the \spadtype{Ring} over which the point is defined.
627378`\spad{point(\spad{gi},{}pt,{}pal)} modifies the graph \spad{\spad{gi}} of the domain \spadtype{GraphImage} to contain one point component,{} \spad{pt} whose point color is set to be the palette color \spad{pal},{} and whose line color and point size are determined by the default functions \spadfun{lineColorDefault} and \spadfun{pointSizeDefault}.
627447`\spad{point(x1,{}x2,{}x3,{}c)} creates and returns a point from the three specified coordinates \spad{x1},{} \spad{x2},{} \spad{x3},{} and also a fourth coordinate,{} \spad{c},{} which is generally used to specify the color of the point.
627551`\spad{points(v,{}n,{}s)} displays the points of the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} if \spad{s} is "on",{} or does not display the points if \spad{s} is "off".
627629`\spad{poissonDistribution(a)} produces a Poisson distribution of variance \spad{a}.
627707`\spad{poisson(f)} \undocumented
627789`\spad{polCase(contprod,{} numFacts,{} evallcs)},{} where \spad{contprod} is the product of the content of the leading coefficient of the polynomial to be factored with the content of the evaluated polynomial,{} \spad{numFacts} is the number of factors of the leadingCoefficient,{} and evallcs is the list of the evaluated factors of the leadingCoefficient,{} returns \spad{true} if the factors of the leading C
627789`--oefficient can be distributed with this valuation.
627886`\spad{pol(v)} turns the vector \spad{[v0,{}...,{}vn]} into the polynomial \spad{v0+v1*x+ ... + vn*x^n}.
628055`\spad{polarCoordinates(x)} returns (\spad{r},{} phi) such that \spad{x} = \spad{r} * exp(\%\spad{i} * phi).
628195`\spad{polar(pt)} transforms \spad{pt} from polar coordinates to Cartesian coordinates: the function produced will map the point \spad{(r,{}theta)} to \spad{x = r * cos(theta)} ,{} \spad{y = r * sin(theta)}.
628257`\spad{pole?(x)} tests if the series \spad{x} has a pole. \indented{1}{Note: this is \spad{false} when \spad{x} is a Taylor series.}
628315`\spad{pole?(f)} determines if the power series \spad{f} has a pole.
628385`\spad{polyPart(f)} returns the polynomial part of \spad{f}.
628452`\spad{polyRDE(a,{} B,{} C,{} n,{} D)} returns either: 1. \spad{[Q,{} b]} such that \spad{degree(Q) <= n} and \indented{3}{\spad{a Q'+ B Q = C} if \spad{b = true},{} \spad{Q} is a partial solution} \indented{3}{otherwise.} 2. \spad{[B1,{} C1,{} m,{} \alpha,{} \beta]} such that any polynomial solution \indented{3}{of degree at most \spad{n}
628452`-- of \spad{A Q' + BQ = C} must be of the form} \indented{3}{\spad{Q = \alpha H + \beta} where \spad{degree(H) <= m} and} \indented{3}{\spad{H} satisfies \spad{H' + B1 H = C1}.} \spad{D} is the derivation to use.
628619`\spad{polyRicDE(op,{} zeros)} returns \spad{[[p1,{} L1],{} [p2,{} L2],{} ... ,{} [pk,{} Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \spad{op y=0} must be one of the \spad{pi}\spad{'s} (up to the constant coefficient),{} in which case the equation for \spad{z=y e^{-int p}} is \spad{\spad{Li} z =0}. \spad{zeros} is a zero finder in \spad{UP}.
628718`\spad{polyRicDE(op,{} zeros)} returns \spad{[[p1,{} L1],{} [p2,{} L2],{} ... ,{} [pk,{}Lk]]} such that the polynomial part of any rational solution of the associated Riccati equation of \spad{op y = 0} must be one of the \spad{pi}\spad{'s} (up to the constant coefficient),{} in which case the equation for \spad{z 
628718`--= y e^{-int p}} is \spad{\spad{Li} z = 0}. \spad{zeros} is a zero finder in \spad{UP}.
628910`\spad{poly_int(f,{} k,{} x)} generates polylogarithms.
629244`\spad{polygamma(x,{}y)} returns the polygamma function applied to \spad{x} and \spad{y}
629308`\spad{polygamma(n,{} x)} is the \spad{n}-th derivative of \spad{digamma(x)}.
629424`\spad{polygamma(n,{} x)} is the \spad{n}-th derivative of \spad{digamma(x)}.
629522`\spad{polygamma(k,{}x)} is the \spad{k-th} derivative of \spad{digamma(x)},{} (often written \spad{psi(k,{}x)} in the literature).
629582`\spad{polygon?(s)} returns \spad{true} if the \spadtype{ThreeSpace} \spad{s} contains a single polygon component,{} or \spad{false} otherwise.
629641`\spad{polygon([p0,{}p1,{}...,{}pn])} creates a polygon defined by a list of points,{} \spad{p0} through \spad{pn},{} and returns a \spadtype{ThreeSpace} whose component is the polygon.
629706`\spad{polygon(s)} checks to see if the \spadtype{ThreeSpace},{} \spad{s},{} is composed of a single polygon component defined by a list of points,{} and if so,{} returns the list of points; An error is signaled otherwise.
629771`\spad{polygon(s,{}[[r0],{}[r1],{}...,{}[rn]])} adds a polygon component defined by a list of points \spad{r0} through \spad{rn},{} which are lists of elements from the domain \spad{PointDomain(m,{}R)} to the \spadtype{ThreeSpace} \spad{s},{} where \spad{m} is the dimension of the points and \spad{R} is the \spadtype{Ring} over which the points are defined.
629838`\spad{polygon(s,{}[p0,{}p1,{}...,{}pn])} adds a polygon component defined by a list of points,{} \spad{p0} throught \spad{pn},{} to the \spadtype{ThreeSpace} \spad{s}.
629906`\spad{polylog(s,{} x)} is the polylogarithm of order \spad{s} at \spad{x}
629968`\spad{polylog(s,{} x)} is the polylogarithm of order \spad{s} at \spad{x}.
630026`\spad{polynomial(f,{}k)} returns a polynomial consisting of the sum of all terms of \spad{f} of degree \spad{<= k}.
630136`\spad{polynomial(f,{}k)} returns a polynomial consisting of the sum of all terms of \spad{f} of degree \spad{<= k}.
630240`\spad{polynomial(f,{}k1,{}k2)} returns a polynomial consisting of the sum of all terms of \spad{f} of degree \spad{d} with \spad{k1 <= d <= k2}.
630369`\spad{polynomial(f,{}k1,{}k2)} returns a polynomial consisting of the sum of all terms of \spad{f} of degree \spad{d} with \spad{k1 <= d <= k2}.
630492`\spad{polyred(u)} \undocumented
630623`\spad{pomopo!(p1,{}r,{}e,{}p2)} returns \spad{p1 + monomial(e,{}r) * p2} and may use \spad{p1} as workspace. The constant \spad{r} is assumed to be nonzero.
630690`\spad{pop!(s)} returns the top element \spad{x},{} destructively removing \spad{x} from \spad{s}. Note: Use \axiom{top(\spad{s})} to obtain \spad{x} without removing it from \spad{s}. Error: if \spad{s} is empty.
630735`\spad{popFortranOutputStack()} pops the Fortran output stack
630806`\spad{position!(f,{} i)} sets the current byte-position to \spad{i}.
630875`\spad{position(x)} returns the integer \spad{n} associated to \spad{x}.
630935`\spad{position(f)} returns the current byte-position in the file \spad{f}.
631386`\spad{position(p,{}a)} returns the index \spad{i} of the first \spad{x} in \axiom{a} such that \axiom{\spad{p}(\spad{x})} is \spad{true},{} and \axiom{minIndex(a) - 1} if there is no such \spad{x}.
631461`\spad{position(x,{}a)} returns the index \spad{i} of the first occurrence of \spad{x} in a,{} and \axiom{minIndex(a) - 1} if there is no such \spad{x}.
631747`\spad{position(cc,{}t,{}i)} returns the position \axiom{\spad{j} \spad{>=} \spad{i}} in \spad{t} of the first character belonging to \spad{cc}.
631823`\spad{position(x,{}a,{}n)} returns the index \spad{i} of the first occurrence of \spad{x} in \axiom{a} where \axiom{\spad{i} \spad{>=} \spad{n}},{} and \axiom{minIndex(a) - 1} if no such \spad{x} is found.
631911`\spad{position(s,{}t,{}i)} returns the position \spad{j} of the substring \spad{s} in string \spad{t},{} where \axiom{\spad{j} \spad{>=} \spad{i}} is required.
632079`\spad{positive?(u)} returns \axiom{\spad{true}} if every element of \spad{u} is positive,{} \axiom{\spad{false}} otherwise.
632137`\spad{positive?(x)} tests whether \spad{x} is strictly greater than 0.
632281`\axiom{positive?(pol,{}aRoot)} answers if \axiom{pol} interpreted as \axiom{aRoot} is positive
632445`\spad{positivePower(x,{} n)} computes a positive integral power of the matrix \spad{x}. Error: if the matrix is not square.
632516`\spad{positiveRemainder(a,{}b)} (where \spad{b > 1}) yields \spad{r} where \spad{0 <= r < b} and \spad{r == a rem b}.
632580`\spad{positiveSolve(lp)} returns the same as \spad{positiveSolve(lp,{}false,{}false)}.
632706`\spad{positiveSolve(ts)} returns the points of the regular set of \spad{ts} with (real) strictly positive coordinates.
632831`\spad{positiveSolve(lp)} returns the same as \spad{positiveSolve(lp,{}info?,{}false)}.
632965`\spad{positiveSolve(lp,{}info?,{}lextri?)} returns the set of the points in the variety associated with \spad{lp} whose coordinates are (real) strictly positive. Moreover,{} if \spad{info?} is \spad{true} then some information is displayed during decomposition into regular chains. If \spad{lextri?} is \spad{true} then the lexTriangular algorithm is called from th
632965`--e \spadtype{LexTriangularPackage} constructor (see \axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\spad{lp},{}\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \axiomOpFrom{zeroSetSplit}{RegularChain} from \spadtype{RegularChain}. WARNING: For each set of coordinates given by \spad{positiveSolve(lp,{}info?,{}lextri?)} the ordering of the indeterminates is reversed \spad{w}.\spad{r}.\spad{t}. \spad{ls}.
633343`\spad{possiblyInfinite?(s)} tests if the stream \spad{s} could possibly have an infinite number of elements. Note: for many datatypes,{} \axiom{possiblyInfinite?(\spad{s}) = not explictlyFinite?(\spad{s})}.
633408`\axiom{possiblyNewVariety?(newlp,{}\spad{llp})} returns \spad{true} iff for every \axiom{\spad{lp}} in \axiom{\spad{llp}} certainlySubVariety?(newlp,{}\spad{lp}) does not hold.
633514`\spad{postfix(op,{} a)} creates a form which prints as: a \spad{op}.
633559`\spad{powToUPS should} be local but conditional
633971`\spad{pow()} \undocumented
634028`\spad{pow(u,{} n,{} d,{} p)} returns u^n truncated after degree \spad{d},{} except if \spad{n=1},{} in which case \spad{u} itself is returned
634142`\spad{power!(a,{}b,{}c,{}m,{}n)} computes \spad{m} ^ \spad{n} and stores the result in \spad{a}. The matrices \spad{b} and \spad{c} are used to store intermediate results. Error: if \spad{a},{} \spad{b},{} \spad{c},{} and \spad{m} are not square and of the same dimensions.
634271`\spad{powerAssociative?()} tests if all subalgebras generated by a single element are associative.
634350`\spad{powerSum n} is the \spad{n} th power sum symmetric \indented{1}{function.}
634439`\spad{power(a,{}f)} returns the power series \spad{f} raised to the power \spad{a}.
634528`\spad{powern(r,{}f)} raises power series \spad{f} to the power \spad{r}.
634655`\spad{powers(\spad{li})} returns a list of 2-element lists. For each 2-element list,{} the first element is an entry of \spad{li} and the second element is the multiplicity with which the first element occurs in \spad{li}. There is a 2-element list for each value occurring in \spad{l}.
634774`\spad{powmod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a^b mod p}.
634891`\axiom{pquo(a,{}\spad{b})} computes the pseudo-quotient of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in the main variable of \axiom{\spad{b}}.
635020`\axiom{pquo(a,{}\spad{b},{}\spad{v})} computes the pseudo-quotient of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in \axiom{\spad{v}}.
635088`\spad{pr2dmp(p)} converts \spad{p} to target domain
635171`\axiom{pre_process(\spad{lp},{}\spad{b1},{}\spad{b2})} is an internal subroutine,{} exported only for developement.
635290`\axiom{pre_process(\spad{lp},{}\spad{b1},{}\spad{b2})} is an internal subroutine,{} exported only for developement.
635419`\spad{precision()} returns the number of digits in the model
635476`\spad{precision()} returns the precision in digits base.
635540`\spad{precision(p)} sets the number of digits in the model to \spad{p}
635612`\spad{precision(n)} set the precision in the base to \spad{n} decimal digits.
635717`\spad{predicate(p)} returns the predicate attached to \spad{p},{} the constant function \spad{true} if \spad{p} has no predicates attached to it.
635791`\spad{predicates(p)} returns \spad{[p1,{}...,{}pn]} such that the predicate attached to \spad{p} is \spad{p1} and ... and \spad{pn}.
635895`\spad{prefix?(s,{}t)} tests if the string \spad{s} is the initial substring of \spad{t}. Note: \axiom{prefix?(\spad{s},{}\spad{t}) \spad{==} reduce(and,{}[\spad{s}.\spad{i} = \spad{t}.\spad{i} for \spad{i} in 0..maxIndex \spad{s}])}.
635950`\spad{prefixRagits(rx)} returns the non-cyclic part of the ragits of the fractional part of a radix expansion. For example,{} if \spad{x = 3/28 = 0.10 714285 714285 ...},{} then \spad{prefixRagits(x)=[1,{}0]}.
636016`\spad{prefix(f,{}l)} creates a form depicting the \spad{n}-ary prefix application of \spad{f} to a tuple of arguments given by list \spad{l}.
636127`\axiom{prem(a,{}\spad{b})} computes the pseudo-remainder of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in the main variable of \axiom{\spad{b}}.
636256`\axiom{prem(a,{}\spad{b},{}\spad{v})} computes the pseudo-remainder of \axiom{a} by \axiom{\spad{b}},{} both viewed as univariate polynomials in \axiom{\spad{v}}.
636324`\axiom{prepareDecompose(\spad{lp},{}\spad{lts},{}\spad{b1},{}\spad{b2})} is an internal subroutine,{} exported only for developement.
636472`\axiom{prepareDecompose(\spad{lp},{}\spad{lts},{}\spad{b1},{}\spad{b2})} is an internal subroutine,{} exported only for developement.
636630`\axiom{prepareSubResAlgo(\spad{p1},{}\spad{p2},{}\spad{ts})} is an internal subroutine,{} exported only for developement.
636750`\spad{presub(f,{}n)} creates a form for \spad{f} presubscripted by \spad{n}.
636794`\spad{presuper(f,{}n)} creates a form for \spad{f} presuperscripted by \spad{n}.
636840`\spad{pretendOfType(f)} returns the input form corresponding to \spad{f pretend R}
636914`\spad{prevPrime(n)} returns the largest prime strictly smaller than \spad{n}
636969`\spad{previous(l)} returns the doubly-link list beginning with its previous element. Error: if \spad{l} has no previous element. Note: \axiom{next(previous(\spad{l})) = \spad{l}}.
637107`\axiom{primPartElseUnitCanonical!(\spad{p})} replaces \axiom{\spad{p}} by \axiom{primPartElseUnitCanonical(\spad{p})}.
637293`\axiom{primPartElseUnitCanonical(\spad{p})} returns \axiom{primitivePart(\spad{p})} if \axiom{\spad{R}} is a \spad{gcd}-domain,{} otherwise \axiom{unitCanonical(\spad{p})}.
637398`\spad{primaryDecomp(I)} returns a list of primary ideals such that their intersection is the ideal \spad{I}.
637977`\spad{prime?(\spad{zi})} tests if the complex integer \spad{zi} is prime.
638055`\spad{prime?(n)} returns \spad{true} if \spad{n} is prime and \spad{false} if not. Note that we ignore sign of \spad{n},{} so \spad{-5} is considered prime. The algorithm used is Rabin\spad{'s} probabilistic primality test (reference: Knuth Volume 2 Semi Numerical Algorithms). If \spad{prime? n} returns \spad{false},{} \spad{n} is proven composite. If \spad{prime? n} returns \spad{true},{} prime? may be in error however,{} the probability of err
638055`--or is very low. and is zero below 25*10^9 (due to a result of Pomerance et al),{} below 10^12 and 10^13 due to results of Pinch,{} and below 341550071728321 due to a result of Jaeschke. Specifically,{} this implementation does at least 10 pseudo prime tests and so the probability of error is \spad{< 4^(-10)}. The running time of this method is cubic in the length of the input \spad{n},{} that is \spad{O( (log n)^3 )},{} for \spad{n<10^20}. beyond that,{} the algorithm is quartic,{} \spad{O( (l
638055`--og n)^4 )}. Two improvements due to Davenport have been incorporated which catches some trivial strong pseudo-primes,{} such as [Jaeschke,{} 1991] 1377161253229053 * 413148375987157,{} which the original algorithm regards as prime
638113`\spad{prime?(I)} tests if the ideal \spad{I} is prime.
638330`\spad{prime?(x)} tests if \spad{x} can never be written as the product of two non-units of the ring,{} \spadignore{i.e.} \spad{x} is an irreducible element.
638391`\spad{primeFactor(base,{}exponent)} creates a factored object with a single factor whose \spad{base} is asserted to be prime (flag = "prime").
638507`\spad{primeFrobenius(a)} returns \spad{a ^ p} where \spad{p} is the characteristic.
638653`\spad{primeFrobenius(a,{}s)} returns \spad{a^(p^s)} where \spad{p} is the characteristic.
638737`\spad{prime(f)} creates the form \spad{f} followed by a suffix prime (single quote).
638777`\spad{prime(f,{}n)} creates the form \spad{f} followed by \spad{n} primes.
638836`\spad{primes(a,{}b)} returns a list of all primes \spad{p} with \spad{a <= p <= b}
638896`\spad{primextendedint(f,{} ',{} foo,{} g)} returns either \spad{[v,{} c]} such that \spad{f = v' + c g} and \spad{c' = 0},{} or \spad{[v,{} a]} such that \spad{f = g' + a},{} and \spad{a = 0} or \spad{a} has no integral in UP. Returns "failed" if neither case can
638896`-- hold. Argument \spad{foo} is an extended integration function on \spad{F}.
639140`\spad{primextintfrac(f,{} ',{} g)} returns \spad{[v,{} c]} such that \spad{f = v' + c g} and \spad{c' = 0}. Error: if \spad{degree numer f >= degree denom f} or if \spad{degree numer g >= degree denom g} or if \spad{denom g} is not squarefree.
639303`\spad{primintegrate(f,{} ',{} foo)} returns \spad{[g,{} a]} such that \spad{f = g' + a},{} and \spad{a = 0} or \spad{a} has no integral in UP. Argument foo is an extended integration function on \spad{F}.
639587`\spad{primintfldpoly(p,{} ',{} t')} returns \spad{q} such that \spad{p' = q} or "failed" if no such \spad{q} exists. Argument \spad{t'} is the derivative of the primitive generating the extension.
639779`\spad{primitive?(f)} tests whether the polynomial \spad{f} over a finite field is primitive,{} \spadignore{i.e.} all its roots are primitive.
639879`\spad{primitive?(b)} tests whether the element \spad{b} is a generator of the (cyclic) multiplicative group of the field,{} \spadignore{i.e.} is a primitive element. Implementation Note: see \spad{ch}.IX.1.3,{} th.2 in \spad{D}. Lipson.
639938`\spad{primitiveElement()} returns a primitive element stored in a global variable in the domain. At first call,{} the primitive element is computed by calling \spadfun{createPrimitiveElement}.
639996`\spad{primitiveElement([a1,{}...,{}an])} returns \spad{[a,{} [q1,{}...,{}qn],{} q]} such that then \spad{k(a1,{}...,{}an) = k(a)},{} \spad{\spad{ai} = \spad{qi}(a)},{} and \spad{q(a) = 0}. This operation uses the technique of \spadglossSee{groebner bases}{Groebner basis}.
640167`\spad{primitiveElement([p1,{}...,{}pn],{} [a1,{}...,{}an])} returns \spad{[[c1,{}...,{}cn],{} [q1,{}...,{}qn],{} q]} such that then \spad{k(a1,{}...,{}an) = k(a)},{} where \spad{a = a1 c1 + ... + an cn},{} \spad{\spad{ai} = \spad{qi}(a)},{} and \spad{q(a) = 0}. The \spad{pi}\spad{'s} are the defining polynomials for
640167`-- the \spad{ai}\spad{'s}. This operation uses the technique of \spadglossSee{groebner bases}{Groebner basis}.
640357`\spad{primitiveElement(a1,{} a2)} returns \spad{[a,{} q1,{} q2,{} q]} such that \spad{k(a1,{} a2) = k(a)},{} \spad{\spad{ai} = \spad{qi}(a)},{} and \spad{q(a) = 0}. The minimal polynomial for \spad{a2} may involve \spad{a1},{} but the minimal polynomial for \spad{a1} may not invo
640357`--lve \spad{a2}; This operations uses \spadfun{resultant}.
640584`\spad{primitiveElement([p1,{}...,{}pn],{} [a1,{}...,{}an],{} a)} returns \spad{[[c1,{}...,{}cn],{} [q1,{}...,{}qn],{} q]} such that then \spad{k(a1,{}...,{}an) = k(a)},{} where \spad{a = a1 c1 + ... + an cn},{} \spad{\spad{ai} = \spad{qi}(a)},{} and \spad{q(a) = 0}. The \spad{pi}\spad{'s} are the defining pol
640584`--ynomials for the \spad{ai}\spad{'s}. This operation uses the technique of \spadglossSee{groebner bases}{Groebner basis}.
640781`\spad{primitiveElement(p1,{} a1,{} p2,{} a2)} returns \spad{[c1,{} c2,{} q]} such that \spad{k(a1,{} a2) = k(a)} where \spad{a = c1 a1 + c2 a2,{} and q(a) = 0}. The \spad{pi}\spad{'s} are the defining polynomials for the \spad{ai}\spad{'s}. The \spad{p2} may involve \spad{a1},{} but \spad{p1} must not involve \spad{a2}. This operation use
640781`--s \spadfun{resultant}.
641032`\spad{primitiveMonomials(p)} gives the list of monomials of the polynomial \spad{p} with their coefficients removed. Note: \spad{primitiveMonomials(sum(a_(i) X^(i))) = [X^(1),{}...,{}X^(n)]}.
641189`\axiom{primitivePart!(\spad{p})} replaces \axiom{\spad{p}} by its primitive part.
641539`\spad{primitivePart(p)} returns the unit normalized form of polynomial \spad{p} divided by the content of \spad{p}.
641621`\spad{primitivePart(f)} removes the content of the denominator and the common content of the numerator of \spad{f}.
641691`\spad{primitivePart(p)} returns the unitCanonical associate of the polynomial \spad{p} with its content divided out.
641775`\spad{primitivePart(l)} returns \spad{l0} such that \spad{l = a * l0} for some a in \spad{R},{} and \spad{content(l0) = 1}.
641935`\spad{primitivePart(p,{} q)} reduces the coefficient of \spad{p} modulo \spad{q},{} takes the primitive part of the result,{} and ensures that the leading coefficient of that result is monic.
642022`\spad{primitivePart(p,{}v)} returns the unitCanonical associate of the polynomial \spad{p} with its content with respect to the variable \spad{v} divided out.
642113`\axiom{primitiveRowEchelon(A)} computes a row echelon form for the matrix \axiom{A}. The algorithm used is fraction-free elimination. Every row is made primitive by division by the \spad{gcd}. The algorithm is especially adapted to matrices already close to row echelon form. The transformation matrix,{} the used pivots and the rank of the matrix a
642113`--re also returned.
642271`\spad{primlimintfrac(f,{} ',{} [u1,{}...,{}un])} returns \spad{[v,{} [c1,{}...,{}cn]]} such that \spad{ci' = 0} and \spad{f = v' + +/[\spad{ci} * ui'/ui]}. Error: if \spad{degree numer f >= degree denom f}.
642487`\spad{primlimitedint(f,{} ',{} foo,{} [u1,{}...,{}un])} returns \spad{[v,{} [c1,{}...,{}cn],{} a]} such that \spad{ci' = 0},{} \spad{f = v' + a + reduce(+,{}[\spad{ci} * ui'/ui])},{} and \spad{a = 0} or \spad{a} has no integ
642487`--ral in UP. Returns "failed" if no such \spad{v},{} \spad{ci},{} a exist. Argument \spad{foo} is an extended integration function on \spad{F}.
642770`\spad{prinb }\undocumented
642924`\spad{principal?(D)} tests if the argument is the divisor of a function.
643086`\spad{principalIdeal([f1,{}...,{}fn])} returns a record whose generator component is a generator of the ideal generated by \spad{[f1,{}...,{}fn]} whose coef component satisfies \spad{generator = sum (input.i * coef.i)}
643183`\spad{prindINFO }\undocumented
643366`\spad{prinpolINFO }\undocumented
643458`\spad{prinshINFO }\undocumented
643543`\spad{printCode(f)} prints out \spad{f} in FORTRAN notation.
643590`\spad{printHeader()} produces the FORTRAN header for the current subprogram in the global symbol table on the current FORTRAN output stream.
643640`\spad{printHeader(f)} produces the FORTRAN header for subprogram \spad{f} in the global symbol table on the current FORTRAN output stream.
643696`\spad{printHeader(f,{}tab)} produces the FORTRAN header for subprogram \spad{f} in symbol table \spad{tab} on the current FORTRAN output stream.
643755`\axiom{printInfo!(\spad{x},{}\spad{y})} initializes the mesages to be printed when manipulating items from the hash-table. If a key is retrieved then \axiom{\spad{x}} is displayed. If an item is stored then \axiom{\spad{y}} is displayed.
644099`\spad{printStatement(l)} creates a representation of a PRINT statement.
644163`\axiom{printStats!()} prints the statistics.
644237`\axiom{printSys(\spad{sys})} writes a list of functions as a vector of equations (with right hand side 0) and coerces the result to \axiomType{OutputForm}.
644312`\spad{printTypes(tab)} produces FORTRAN type declarations from \spad{tab},{} on the current FORTRAN output stream
644367`\spad{printTypes(tab)} produces FORTRAN type declarations from \spad{tab},{} on the current FORTRAN output stream
644415`\spad{print(u)} prints the form \spad{u}.
644457`\spad{print(o)} writes the output form \spad{o} on standard output using the two-dimensional formatter.
644509`\axiom{printingInfo?()} returns \spad{true} iff messages are printed when manipulating items from the hash-table.
644588`\axiom{probablyZeroDim?(\spad{lp})} returns \spad{true} iff the number of polynomials in \axiom{\spad{lp}} is not smaller than the number of variables occurring in these polynomials.
644677`\spad{processTemplate(tp)} processes the template \spad{tp},{} writing the result to the current FORTRAN output stream.
644744`\spad{processTemplate(tp,{}fn)} processes the template \spad{tp},{} writing the result out to \spad{fn}.
644820`\spad{prod(expr)} creates the form prefixing \spad{expr} by a capital \spad{pi}.
644859`\spad{prod(expr,{}lowerlimit)} creates the form prefixing \spad{expr} by a capital \spad{pi} with a \spad{lowerlimit}.
644901`\spad{prod(expr,{}lowerlimit,{}upperlimit)} creates the form prefixing \spad{expr} by a capital \spad{pi} with both a \spad{lowerlimit} and \spad{upperlimit}.
644946`\spad{product(f(n),{} n = a..b)} returns \spad{f}(a) * ... * \spad{f}(\spad{b}) as a formal product.
645020`\spad{product(f(n),{} n)} returns the formal product \spad{P}(\spad{n}) which verifies \spad{P}(\spad{n+1})\spad{/P}(\spad{n}) = \spad{f}(\spad{n}).
645083`\spad{product(f(n),{} n = a..b)} returns \spad{f}(a) * ... * \spad{f}(\spad{b}) as a formal product.
645158`\spad{product(f(n),{} n)} returns the formal product \spad{P}(\spad{n}) which verifies \spad{P}(\spad{n+1})\spad{/P}(\spad{n}) = \spad{f}(\spad{n}).
645221`\spad{product(a,{}b)} is the degree-preserving \spad{R}-linear product: \blankline \indented{2}{\spad{degree product(a,{}b) = degree a + degree b}} \indented{2}{\spad{product(a1+a2,{}b) = product(a1,{}b) + product(a2,{}b)}} \indented{2}{\spad{product(a,{}b1+b2) = product(a,{}b1) + product(a,{}b2)}} \indented{2}{\spad{product(r*a,{}b) = product(a,{}r*b) = r*product(a,{}b)}} \indented{2}{\spad{product(a,{}product(b,{}c)) = product(product(a,{}b),{}c)}}
645274`\spad{product(s,{}t)} is the outer product of the tensors \spad{s} and \spad{t}. For example,{} if \spad{r = product(s,{}t)} for rank 2 tensors \spad{s} and \spad{t},{} then \spad{r} is a rank 4 tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = s(i,{}j)*t(k,{}l)}.}
645337`\axiom{product(a,{}\spad{b},{}\spad{n})} returns \axiom{a*b} (truncated up to order \axiom{\spad{n}}).
645415`\axiom{project(de,{} \spad{q})} projects the differential equation \axiom{de} of order higher than \axiom{\spad{q}} into the \axiom{\spad{q}}\spad{-}th order jet bundle.
645495`\spad{prolateSpheroidal(a)} transforms from prolate spheroidal coordinates to Cartesian coordinates: \spad{prolateSpheroidal(a)} is a function which will map the point \spad{(\spad{xi},{}eta,{}phi)} to \spad{x = a*sinh(\spad{xi})*sin(eta)*cos(phi)},{} \spad{y = a*sinh(\spad{xi})*sin(eta)*sin(phi)},{} \spad{z = a*cosh(\spad{xi})*cos(eta)}.
645574`\spad{prologue(t)} extracts the prologue section of a formatted object \spad{t}.
645636`\spad{prologue(t)} extracts the prologue section of a TeX form \spad{t}.
645688`\axiom{prolongMV(\spad{mv})} calculates the number of multiplicative variables for the prolongation of an involutive symbol.
645931`\axiom{prolongSymbol(symb)} prolongs directly the symbol \axiom{symb}.
646044`\axiom{prolong(de)} prolongs the differential equation \axiom{de}. Additionally the arising integrability conditions are returned.
646128`\axiom{prolong(de,{} \spad{q})} is like \axiom{prolong(de)}. However,{} only equations of lower order than \axiom{\spad{q}} are prolonged.
646231`\axiom{prolong(\spad{v},{} \spad{q})} prolongs a vector field \axiom{\spad{v}} defined on the base space into the jet bundle of order \axiom{\spad{q}}.
646302`\spad{properties(op)} returns the list of all the properties currently attached to \spad{op}.
646376`\spad{property(op,{} s)} returns the value of property \spad{s} if it is attached to \spad{op},{} and "failed" otherwise.
646447`Constructs a proposition
646693`\axiom{pseudoDivide(\spad{P},{}\spad{Q})} computes the pseudoDivide of \axiom{\spad{P}} by \axiom{\spad{Q}}.
646809`\spad{pseudoDivide(p,{}q)} returns \spad{[c,{} q,{} r]},{} when \spad{p' := p*lc(q)^(deg p - deg q + 1) = c * p} is pseudo right-divided by \spad{q},{} \spadignore{i.e.} \spad{p' = s q + r}.
646938`\axiom{pseudoDivide(a,{}\spad{b})} computes \axiom{[pquo(a,{}\spad{b}),{}prem(a,{}\spad{b})]},{} both polynomials viewed as univariate polynomials in the main variable of \axiom{\spad{b}},{} if \axiom{\spad{b}} is not a constant polynomial.
647110`\spad{pseudoQuotient(p,{}q)} returns \spad{r},{} the quotient when \spad{p' := p*lc(q)^(deg p - deg q + 1)} is pseudo right-divided by \spad{q},{} \spadignore{i.e.} \spad{p' = s q + r}.
647204`\spad{pseudoRem(x,{} y,{} m)} computes pseudoremainder of \spad{x} by \spad{y} modulo \spad{m}.
647284`\spad{pseudoRemainder(p,{}q)} = \spad{r},{} for polynomials \spad{p} and \spad{q},{} returns the remainder when \spad{p' := p*lc(q)^(deg p - deg q + 1)} is pseudo right-divided by \spad{q},{} \spadignore{i.e.} \spad{p' = s q + r}.
647358`\spad{psolve(c)} solves the homogeneous linear system \spad{c} \spad{z} = 0 for all possible ranks of the matrix \spad{c}
647604`\spad{psolve(c,{}w)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks of the matrix \spad{c} and given right hand side vector \spad{w}
647859`\spad{psolve(c,{}w)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks of the matrix \spad{c} and indeterminate right hand side \spad{w}
648118`\spad{psolve(c)} solves the homogeneous linear system \spad{c} \spad{z} = 0 for all possible ranks \spad{>=} \spad{k} of the matrix \spad{c}
648380`\spad{psolve(c,{}s)} solves \spad{c} \spad{z} = 0 for all possible ranks of the matrix \spad{c} and given right hand side vector \spad{w},{} writes the results to a file named \spad{s},{} and returns the number of regimes
648472`\spad{psolve(c,{}w,{}k)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks \spad{>=} \spad{k} of the matrix \spad{c} and given right hand side vector \spad{w}
648743`\spad{psolve(c,{}w,{}s)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks of the matrix \spad{c} and given right hand side vector \spad{w},{} writes the results to a file named \spad{s},{} and returns the number of regimes
648844`\spad{psolve(c,{}w,{}k)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks \spad{>=} \spad{k} of the matrix \spad{c} and indeterminate right hand side \spad{w}
649119`\spad{psolve(c,{}w,{}s)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks of the matrix \spad{c} and indeterminate right hand side \spad{w},{} writes the results to a file named \spad{s},{} and returns the number of regimes
649224`\spad{psolve(c,{}k,{}s)} solves \spad{c} \spad{z} = 0 for all possible ranks \spad{>=} \spad{k} of the matrix \spad{c},{} writes the results to a file named \spad{s},{} and returns the number of regimes
649332`\spad{psolve(c,{}w,{}k,{}s)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks \spad{>=} \spad{k} of the matrix \spad{c} and given right hand side \spad{w},{} writes the results to a file named \spad{s},{} and returns the number of regimes
649449`\spad{psolve(c,{}w,{}k,{}s)} solves \spad{c} \spad{z} = \spad{w} for all possible ranks \spad{>=} \spad{k} of the matrix \spad{c} and indeterminate right hand side \spad{w},{} writes the results to a file named \spad{s},{} and returns the number of regimes
649570`\spad{ptFunc(a,{}b,{}c,{}d)} is an internal function exported in order to compile packages.
649847`\spad{ptree(s)} is a leaf? pendant tree
649890`\spad{ptree(x,{}y)} \undocumented
649937`\spad{puiseux(f)} returns a Puiseux expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
650009`\spad{puiseux(x)} returns \spad{x} viewed as a Puiseux series.
650085`\spad{puiseux(f,{}x = a)} expands the expression \spad{f} as a Puiseux series in powers of \spad{(x - a)}.
650170`\spad{puiseux(f,{}n)} returns a Puiseux expansion of the expression \spad{f}. Note: \spad{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 \spad{n}.
650260`\spad{puiseux(r,{}f(x))} returns \spad{f(x^r)}.
650363`\spad{puiseux(f,{}x = a,{}n)} expands the expression \spad{f} as a Puiseux series in powers of \spad{(x - a)}; terms will be computed up to order at least \spad{n}.
650466`\spad{puiseux(n +-> a(n),{}x = a,{}r0..,{}r)} returns \spad{sum(n = r0,{}r0 + r,{}r0 + 2*r...,{} a(n) * (x - a)^n)}; \spad{puiseux(n +-> a(n),{}x = a,{}r0..r1,{}r)} returns \spad{sum(n = r0 + k*r while n <= r1,{} a(n) * (x - a)^n)}.
650622`\spad{puiseux(a(n),{}n,{}x = a,{}r0..,{}r)} returns \spad{sum(n = r0,{}r0 + r,{}r0 + 2*r...,{} a(n) * (x - a)^n)}; \spad{puiseux(a(n),{}n,{}x = a,{}r0..r1,{}r)} returns \spad{sum(n = r0 + k*r while n <= r1,{} a(n) * (x - a)^n)}.
650764`\spad{pureLex(v1,{}v2)} return \spad{true} if the vector \spad{v1} is less than the vector \spad{v2} in the lexicographic ordering.
650922`\spad{purelyAlgebraic?(ts)} returns \spad{true} iff for every algebraic variable \spad{v} of \spad{ts} we have \spad{algebraicCoefficients?(t_v,{}ts_v_-)} where \spad{ts_v} is \axiomOpFrom{select}{TriangularSetCategory}(\spad{ts},{}\spad{v}) and \spad{ts_v_-} is \axiomOpFrom{collectUnder}{TriangularSetCategory}(\spad{ts},{}\spad{v}).
651087`\spad{purelyAlgebraic?(p,{}ts)} returns \spad{true} iff every variable of \spad{p} is algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}.
651269`\spad{purelyAlgebraicLeadingMonomial?(p,{}ts)} returns \spad{true} iff the main variable of any non-constant iterarted initial of \spad{p} is algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}.
651456`\spad{purelyTranscendental?(p,{}ts)} returns \spad{true} iff every variable of \spad{p} is not algebraic \spad{w}.\spad{r}.\spad{t}. \spad{ts}
651546`\axiom{purge!(A,{} crit)} eliminates all columns belonging to an index \axiom{\spad{c}} such that \axiom{crit(\spad{c})} yields \axiom{\spad{true}}.
651616`\spad{push!(x,{}s)} pushes \spad{x} onto stack \spad{s},{} \spadignore{i.e.} destructively changing \spad{s} so as to have a new first (top) element \spad{x}. Afterwards,{} pop!(\spad{s}) produces \spad{x} and pop!(\spad{s}) produces the original \spad{s}.
651664`\spad{pushFortranOutputStack(f)} pushes \spad{f} onto the Fortran output stack
651744`\spad{pushFortranOutputStack(f)} pushes \spad{f} onto the Fortran output stack
651822`\spad{pushdown(p,{}lv)} \undocumented{}
651891`\spad{pushdown(p,{}v)} \undocumented{}
651954`\spad{pushdown(prf,{}var)} pushes all top level occurences of the variable \spad{var} into the coefficient domain for the polynomial \spad{prf}.
652038`\spad{pushdterm(monom,{}var)} pushes all top level occurences of the variable \spad{var} into the coefficient domain for the monomial \spad{monom}.
652151`\spad{pushucoef(upoly,{}var)} converts the anonymous univariate polynomial \spad{upoly} to a polynomial in \spad{var} over rational functions.
652274`\spad{pushuconst(r,{}var)} takes a rational function and raises all occurances of the variable \spad{var} to the polynomial level.
652380`\spad{pushup(p,{}lv)} \undocumented{}
652447`\spad{pushup(p,{}v)} \undocumented{}
652508`\spad{pushup(prf,{}var)} raises all occurences of the variable \spad{var} in the coefficients of the polynomial \spad{prf} back to the polynomial level.
652590`\spad{putColorInfo(llp,{}lpal)} takes a list of list of points,{} \spad{llp},{} and returns the points with their hue and shade components set according to the list of palette colors,{} \spad{lpal}.
652707`\spad{putGraph(v,{}\spad{gi},{}n)} sets the graph field indicated by \spad{n},{} of the indicated two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} to be the graph,{} \spad{\spad{gi}} of domain \spadtype{GraphImage}. The contents of viewport,{} \spad{v},{} will contain \spad{\spad{gi}} when the function \spadfun{makeViewport2D} is called to create the an updated viewport \sp
652707`--ad{v}.
652791`\spad{qPot(v,{}e)} computes \spad{v^(q^e)},{} interpreting \spad{v} as an element of normal basis field,{} \spad{q} the size of the ground field. This is done by a cyclic \spad{e}-shift of the vector \spad{v}.
652878`\spad{qShiftAction(q,{} k,{} l,{} g)} gives the coefficient of \spad{x^k} in \spad{z^l} \spad{g}(\spad{x}),{} where \spad{z*}(a+b*x+c*x^2+d*x^3+...) = (a+q*b*x+q^2*c*x^2+q^3*d*x^3+...). In terms of sequences,{} z*u(\spad{n})=q^n*u(\spad{n}).
652982`\spad{qShiftC} gives the coefficients \spad{c_}{\spad{k},{}\spad{k}} in the expansion <x^k> \spad{z} \spad{g}(\spad{x}) = sum_{\spad{i=0}}\spad{^k} \spad{c_}{\spad{k},{}\spad{i}} <x^i> \spad{g}(\spad{x}),{} where \spad{z} acts on \spad{g}(\spad{x}) by shifting. In fact,{} the result is [1,{}\spad{q},{}\spad{q^2},{}...]
653066`\spad{qcoerce(n)} coerces \spad{n} to \spad{\%} trusting that \spad{n} is nonnegative
653121`\spad{qconvert(x)} converts \spad{x} to \% trusting that \spad{x} is in correct range.
653172`\spad{qcumulant2nthmoment(q,{} cc,{} n)} computes the \spad{n}th moment from the sequence of reduced \spad{q}-cumulants \spad{cc}
653315`\spad{qelt(u,{} x)} applies \axiom{\spad{u}} to \axiom{\spad{x}} without checking whether \axiom{\spad{x}} is in the domain of \axiom{\spad{u}}. If \axiom{\spad{x}} is not in the domain of \axiom{\spad{u}} a memory-access violation may occur. If a check on whether \axiom{\spad{x}} is in the domain of \axiom{\spad{u}} is required,{} use the function \axiom{elt}.
653372`\spad{qelt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the matrix \spad{m}. Note: there is NO error check to determine if indices are in the proper ranges.
653456`\spad{qelt(m,{}i,{}j)} returns the element in the \spad{i}th row and \spad{j}th column of the array \spad{m} NO error check to determine if indices are in proper ranges
653538`\spad{qfactor(p)} tries to factor \spad{p} over fractions of integers,{} returning "failed" if it cannot
653686`\spad{qinterval(inf,{}sup)} creates a new interval \axiom{[\spad{inf},{}\spad{sup}]},{} without checking the ordering on the elements.
653740`\spad{qlog(x)} computes natural logarithm of \spad{x}. It assumes that \spad{x > 0}.
653780`\spad{qnew(n)} creates a new uninitialized vector of length \spad{n}.
653838`\spad{qnew(n)} creates a new uninitialized vector of length \spad{n}.
653889`\spad{qnew(n,{} m)} creates a new uninitialized \spad{n} by \spad{m} matrix.
653955`\spad{qnew(n,{} m)} creates a new uninitialized \spad{n} by \spad{m} matrix.
654014`\spad{qnew(n,{} m)} creates a new uninitialized \spad{n} by \spad{m} matrix.
654065`\spad{qnew(n,{} m)} creates a new uninitialized \spad{n} by \spad{m} matrix.
654116`\spad{qnew(n,{} m)} creates a new uninitialized \spad{n} by \spad{m} matrix.
654166`\spad{qroot(f,{} n)} returns \spad{[m,{}c,{}r]} such that \spad{f^(1/n) = c * r^(1/m)}.
654368`\spad{qsetelt!(u,{}x,{}y)} sets the image of \axiom{\spad{x}} to be \axiom{\spad{y}} under \axiom{\spad{u}},{} without checking that \axiom{\spad{x}} is in the domain of \axiom{\spad{u}}. If such a check is required use the function \axiom{setelt}.
654456`\spad{qsetelt!(m,{}i,{}j,{}r)} sets the element in the \spad{i}th row and \spad{j}th column of \spad{m} to \spad{r} NO error check to determine if indices are in proper ranges
654544`\spad{qsqrt(x)} computes square root of \spad{x}. It assumes that \spad{x >= 0}.
654585`\spad{quadraticForm(m)} creates a quadratic form from a symmetric,{} square matrix \spad{m}.
654656`\spad{quadraticNorm(f)} returns the \spad{l2} norm of the polynomial \spad{f}.
654734`\spad{quadratic(u)} \undocumented
654804`\spad{quadratic(f,{}g,{}h)} \undocumented
654877`\spad{quartic(u)} \undocumented
654945`\spad{quartic(f,{}g,{}h,{}i,{}j)} \undocumented
655020`\spad{quasiAlgebraicSet(pl,{}q)} returns the quasi-algebraic set with defining equations \spad{p} = 0 for \spad{p} belonging to the list \spad{pl},{} and defining inequation \spad{q} \spad{~=} 0.
655211`\axiom{quasiComponent(\spad{ts})} returns \axiom{[\spad{lp},{}\spad{lq}]} where \axiom{\spad{lp}} is the list of the members of \axiom{\spad{ts}} and \axiom{\spad{lq}}is \axiom{initials(\spad{ts})}.
655384`\axiom{quasiMonic?(\spad{p})} returns \spad{false} if \axiom{\spad{p}} belongs to \axiom{\spad{R}},{} otherwise returns \spad{true} iff the initial of \axiom{\spad{p}} lies in the base ring \axiom{\spad{R}}.
655459`\axiom{quasiMonicPolynomials(\spad{lp})} returns \axiom{qmps,{}nqmps} where \axiom{qmps} is a list of the quasi-monic polynomials in \axiom{\spad{lp}} and \axiom{nqmps} are the other ones.
655586`\spad{quasiRegular?(x)} return \spad{true} if \spad{constant(p)} is zero.
655649`\spad{quasiRegular?(x)} return \spad{true} if \spad{constant(x)} is zero.
655710`\spad{quasiRegular(x)} return \spad{x} minus its constant term.
655767`\spad{quasiRegular(x)} return \spad{x} minus its constant term.
655822`\spad{quatern(r,{}i,{}j,{}k)} constructs a quaternion from scalars.
655880`\spad{queue([x,{}y,{}...,{}z])} creates a queue with first (top) element \spad{x},{} second element \spad{y},{}...,{}and last (bottom) element \spad{z}.
655923`\spad{quickSort(f,{} agg)} sorts the aggregate agg with the ordering function \spad{f} using the quicksort algorithm.
656000`\spad{quoByVar(a0 + a1 x + a2 x^2 + ...)} returns \spad{a1 + a2 x + a3 x^2 + ...} Thus,{} this function substracts the constant term and divides by the series variable. This function is used when Laurent series are represented by a Taylor series and an order.
656155`\spad{f quo g} creates the equivalent infix form.
656196`\spad{x quo y} is the same as \spad{divide(x,{}y).quotient}. See \spadfunFrom{divide}{EuclideanDomain}.
656242`\spad{a quo b} returns the quotient of \spad{a} and \spad{b},{} forgetting the remainder.
656291`\spad{quote()} provides the string quote character,{} \spad{"}.
656328`\spad{quote(f)} creates the form \spad{f} with a prefix quote.
656368`\spad{quoted?(p)} tests if \spad{p} is of the form \spad{'s} for a symbol \spad{s}.
656415`\spad{quotedOperators(r)} returns the list of operators on the right hand side of \spad{r} that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.
656486`\spad{quotientByP(x)} returns \spad{b},{} where \spad{x = a + b p}.
656545`\spad{quotient(I,{}f)} computes the quotient of the ideal \spad{I} by the principal ideal generated by the polynomial \spad{f},{} \spad{(I:(f))}.
656621`\spad{quotient(I,{}J)} computes the quotient of the ideals \spad{I} and \spad{J},{} \spad{(I:J)}.
656694`\axiom{\spad{r2m}(ind)} transforms a repeated index into a multi-index.
656780`\axiom{\spad{r2m}(ind)} transforms a repeated index into a multi-index.
656870`\spad{rCoord(pt)} returns the first element of the point,{} \spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical or a cylindrical coordinate system.
656921`Undocumented
656998`Undocumented.
657064`Undocumented
657129`Undocumented
657197`Undocumented.
657254`Undocumented
657310`\spad{radPoly(p(x,{} y))} returns \spad{[c(x),{} n]} if \spad{p} is of the form \spad{y^n - c(x)},{} "failed" otherwise.
657437`\spad{radicalEigenvalues(m)} computes the eigenvalues of the matrix \spad{m}; when possible,{} the eigenvalues are expressed in terms of radicals.
657557`\spad{radicalEigenvector(c,{}m)} computes the eigenvector(\spad{s}) of the matrix \spad{m} corresponding to the eigenvalue \spad{c}; when possible,{} values are expressed in terms of radicals.
657705`\spad{radicalEigenvectors(m)} computes the eigenvalues and the corresponding eigenvectors of the matrix \spad{m}; when possible,{} values are expressed in terms of radicals.
657899`\spad{radicalOfLeftTraceForm()} returns basis for null space of \spad{leftTraceMatrix()},{} if the algebra is associative,{} alternative or a Jordan algebra,{} then this space equals the radical (maximal nil ideal) of the algebra.
657968`\spad{radicalRoots(rf,{}x)} finds the roots expressed in terms of radicals of the rational function \spad{rf} with respect to the symbol \spad{x}.
658072`\spad{radicalRoots(lrf,{}lvar)} finds the roots expressed in terms of radicals of the list of rational functions \spad{lrf} with respect to the list of symbols \spad{lvar}.
658194`\spad{radicalSimplify(s)} returns a different and presumably simpler representation of \spad{s} with the defining polynomials for the equ
658194`--ations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using using groebner basis of radical ideals
658564`\spad{radicalSolve(eq)} finds the solutions expressed in terms of radicals of the equation of rational functions \spad{eq} with respect to the unique symbol \spad{x} appearing in \spad{eq}.
658681`\spad{radicalSolve(rf)} finds the solutions expressed in terms of radicals of the equation \spad{rf} = 0,{} where \spad{rf} is a univariate rational function.
658788`\spad{radicalSolve(leq)} finds the solutions expressed in terms of radicals of the system of equations of rational functions \spad{leq} with respect to the unique symbol \spad{x} appearing in \spad{leq}.
658917`\spad{radicalSolve(lrf)} finds the solutions expressed in terms of radicals of the system of equations \spad{lrf} = 0,{} where \spad{lrf} is a system of univariate rational functions.
659036`\spad{radicalSolve(eq,{}x)} finds the solutions expressed in terms of radicals of the equation of rational functions \spad{eq} with respect to the symbol \spad{x}.
659160`\spad{radicalSolve(rf,{}x)} finds the solutions expressed in terms of radicals of the equation \spad{rf} = 0 with respect to the symbol \spad{x},{} where \spad{rf} is a rational function.
659274`\spad{radicalSolve(leq,{}lvar)} finds the solutions expressed in terms of radicals of the system of equations of rational functions \spad{leq} with respect to the list of symbols \spad{lvar}.
659416`\spad{radicalSolve(lrf,{}lvar)} finds the solutions expressed in terms of radicals of the system of equations \spad{lrf} = 0 with respect to the list of symbols \spad{lvar},{} where \spad{lrf} is a list of rational functions.
659548`\spad{radical(I)} returns the radical of the ideal \spad{I}.
659913`\spad{radix(x,{}b)} converts \spad{x} to a radix expansion in base \spad{b}.
659981`\spad{raisePolynomial(rpoly)} converts \spad{rpoly} from a univariate polynomial over \spad{r} to be a univariate polynomial with polynomial coefficients.
660103`\spad{ramified?(a)} tests whether \spad{x = a} is ramified.
660173`\spad{ramified?(p)} tests whether \spad{p(x) = 0} is ramified.
660244`\spad{ramifiedAtInfinity?()} tests if infinity is ramified.
660323`\spad{ran(k)} computes a random integer between \spad{-k} and \spad{k} as a member of \spad{R}.
660383`\spad{randnum()} is a random number between 0 and size().
660436`\spad{randnum(n)} is a random number between 0 and \spad{n}.
660496`\spad{randomLC(n,{}x)} should be local but conditional.
660579`\spad{randomR()} should be local but conditional
660645`\spad{randomR produces} a random element of \spad{R}
660722`\spad{randomR()} produces a random element of \spad{R}
660997`\spad{random()} returns a random element from the set.
661032`\spad{random()} creates a random element. This function is deprecated.
661080`\spad{random()} returns a random fraction. This function is deprecated.
661159`\spad{random(n)}\$FFPOLY(\spad{GF}) generates a random monic polynomial of degree \spad{n} over the finite field {\em GF}.
661263`\spad{random(n)} creates a vector over the ground field with random entries.
661349`\spad{random(gp)} returns a random product of maximal 20 generators of the group {\em gp}. Note: {\em random(gp)=random(gp,{}20)}.
661411`\spad{random(a)} creates a random element from 0 to \spad{n-1}.
661461`\spad{random(n)} returns a random integer from 0 to \spad{n-1}.
661499`\spad{random(n)} returns a random integer from 0 to \spad{n-1}.
661548`\spad{random(m,{}n)}\$FFPOLY(\spad{GF}) generates a random monic polynomial of degree \spad{d} over the finite field {\em GF},{} \spad{d} between \spad{m} and \spad{n}.
661668`\spad{random(gp,{}i)} returns a random product of maximal \spad{i} generators of the group {\em gp}.
661738`\spad{rangePascalTriangle()} returns the maximal number of lines stored.
661819`\spad{rangePascalTriangle(n)} sets the maximal number of lines which are stored and returns the previous value.
661918`\spad{range([l])} provides a user-specified range \spad{l}. This option is expressed in the form \spad{range == [l]}.
661976`\spad{range([i])} provides a user-specified range \spad{i}. This option is expressed in the form \spad{range == [i]}.
662046`\spad{ranges(\spad{gi})} returns the list of ranges of the point components from the indicated graph,{} \spad{\spad{gi}},{} of the domain \spadtype{GraphImage}.
662105`\spad{ranges(l)} provides a list of user-specified ranges \spad{l}. This option is expressed in the form \spad{ranges == l}.
662164`\spad{ranges(l,{}r)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{ranges}. If the option does not exist the value,{} \spad{r} is returned.
662268`\spad{ranges(\spad{gi},{}lr)} modifies the list of ranges for the given graph,{} \spad{\spad{gi}} of the domain \spadtype{GraphImage},{} to be that of the list of range segments,{} \spad{lr},{} and returns the new range list for \spad{\spad{gi}}.
662404`\spad{rank()} returns the rank of the algebra.
662467`\spad{rank()} returns the rank of the algebra as \spad{R}-module.
662541`\spad{rank Lv} computes the rank of a list of vectors.
662618`\spad{rank(m)} returns the rank of the matrix \spad{m}.
662708`\spad{rank b} computes the rank of a basis,{} it\spad{'s} the number of vectors in the basis.
662772`\spad{rank(m)} returns the rank of the matrix \spad{m}.
662878`\spad{rank(m)} returns the rank of the matrix \spad{m}.
662969`\spad{rank(m)} returns the rank of the matrix \spad{m}.
663075`\spad{rank(t)} returns the tensorial rank of \spad{t} (that is,{} the number of indices). This is the same as the graded module degree.
663148`\spad{rank(A,{}B)} computes the rank of the complete matrix \spad{(A|B)} of the linear system \spad{AX = B}.
663234`\spad{rank(A,{}B)} computes the rank of the complete matrix \spad{(A|B)} of the linear system \spad{AX = B}.
663325`\spad{rarrow(f,{}g)} creates a form for the mapping \spad{f -> g}.
663369`\spad{ratDenom(f)} rationalizes the denominators appearing in \spad{f} by moving all the algebraic quantities into the numerators.
663427`\spad{ratDenom(f,{} a)} removes \spad{a} from the denominators in \spad{f} if \spad{a} is an algebraic kernel.
663487`\spad{ratDenom(f,{} [a1,{}...,{}an])} removes the \spad{ai}\spad{'s} which are algebraic kernels from the denominators in \spad{f}.
663553`\spad{ratDenom(f,{} [a1,{}...,{}an])} removes the \spad{ai}\spad{'s} which are algebraic from the denominators in \spad{f}.
663627`\spad{ratDsolve(op,{} g)} returns \spad{["failed",{} []]} if the equation \spad{op y = g} has no rational solution. Otherwise,{} it returns \spad{[f,{} [y1,{}...,{}ym]]} where \spad{f} is a particular rational solution and the \spad{yi}\spad{'s} form a basis for the rational solutions of the homogeneous equation.
663809`\spad{ratDsolve(op,{} [g1,{}...,{}gm])} returns \spad{[[h1,{}...,{}hq],{} M]} such that any rational solution of \spad{op y = c1 g1 + ... + cm gm} is of the form \spad{d1 h1 + ... + dq hq} where \spad{M [d1,{}...,{}dq,{}c1,{}...,{}cm] = 0}.
663971`\spad{ratDsolve(op,{} g)} returns \spad{["failed",{} []]} if the equation \spad{op y = g} has no rational solution. Otherwise,{} it returns \spad{[f,{} [y1,{}...,{}ym]]} where \spad{f} is a particular rational solution and the \spad{yi}\spad{'s} form a basis for the rational solutions of the homogeneous equation.
664156`\spad{ratDsolve(op,{} [g1,{}...,{}gm])} returns \spad{[[h1,{}...,{}hq],{} M]} such that any rational solution of \spad{op y = c1 g1 + ... + cm gm} is of the form \spad{d1 h1 + ... + dq hq} where \spad{M [d1,{}...,{}dq,{}c1,{}...,{}cm] = 0}.
664321`\spad{ratPoly(f)} returns a polynomial \spad{p} such that \spad{p} has no algebraic coefficients,{} and \spad{p(f) = 0}.
664628`\spad{rational?(x)} returns \spad{true} if \spad{x} is a rational number,{} \spad{false} otherwise.
664688`\spad{rational?(x)} tests if \spad{x} is a rational number.
664771`\spad{rational?(n)} tests if \spad{n} is a rational number (see \spadtype{Fraction Integer}).
664829`\spad{rational?(o)} tests if \spad{o} is rational,{} \spadignore{i.e.} that all seven imaginary parts are 0.
664913`\spad{rational?(q)} returns {\it \spad{true}} if all the imaginary parts of \spad{q} are zero and the real part can be converted into a rational number,{} and {\it \spad{false}} otherwise.
664999`\spad{rational?(u)} tests if \spadvar{\spad{u}} is actually a rational number (see \spadtype{Fraction Integer}).
665075`\spad{rational?(x)} tests if \spad{x} is a finite rational number.
665161`\spad{rational?(x)} tests if \spad{x} is a finite rational number.
665246`\spad{rationalApproximation(f,{} n)} computes a rational approximation \spad{r} to \spad{f} with relative error \spad{< 10^(-n)}.
665337`\spad{rationalApproximation(f,{} n)} computes a rational approximation \spad{r} to \spad{f} with relative error \spad{< 10^(-n)}.
665422`\spad{rationalApproximation(f,{} n,{} b)} computes a rational approximation \spad{r} to \spad{f} with relative error \spad{< b^(-n)} (that is,{} \spad{|(r-f)/f| < b^(-n)}).
665532`\spad{rationalApproximation(f,{} n,{} b)} computes a rational approximation \spad{r} to \spad{f} with relative error \spad{< b^(-n)},{} that is \spad{|(r-f)/f| < b^(-n)}.
665636`\spad{rationalFunction(f,{}k)} returns a rational function consisting of the sum of all terms of \spad{f} of degree \spad{<=} \spad{k}.
665770`\spad{rationalFunction(f,{}k1,{}k2)} returns a rational function consisting of the sum of all terms of \spad{f} of degree \spad{d} with \spad{k1 <= d <= k2}.
666254`\spad{rationalIfCan(x)} returns \spad{x} as a rational number,{} "failed" if \spad{x} is not a rational number.
666344`\spad{rationalIfCan(x)} returns \spad{x} as a rational number,{} or "failed" if \spad{x} is not a rational number.
666457`\spad{rationalIfCan(n)} creates a rational number,{} or returns "failed" if this is not possible.
666545`\spad{rationalIfCan(o)} returns the real part if all seven imaginary parts are 0,{} and "failed" otherwise.
666659`\spad{rationalIfCan(q)} returns \spad{q} as a rational number,{} or "failed" if this is not possible. Note: if \spad{rational?(q)} is \spad{true},{} the conversion can be done and the rational number will be returned.
666775`\spad{rationalIfCan(u)} returns a rational number if \spad{u} really is one,{} and "failed" otherwise.
666881`\spad{rationalIfCan(x)} returns \spad{x} as a finite rational number if it is one,{} "failed" otherwise.
666997`\spad{rationalIfCan(x)} returns \spad{x} as a finite rational number if it is one and "failed" otherwise.
667112`\spad{rationalPoint?(a,{} b)} tests if \spad{(x=a,{}y=b)} is on the curve.
667192`\spad{rationalPoint?(a,{} b)} tests if \spad{(x=a,{}y=b)} is on the curve.
667269`\spad{rationalPoints()} returns the list of all the affine rational points.
667352`\spad{rationalPoints()} returns the list of all the affine rational points.
667445`\spad{rationalPower(f(x))} returns \spad{r} where the Puiseux series \spad{f(x) = g(x^r)}.
667702`reconstruct vector of rational functions based on information stored in reconstructor.
668006`reconstruct rational function
668424`\spad{rational(x)} returns \spad{x} as a rational number; error if \spad{x} is not a rational number.
668493`\spad{rational(x)} returns \spad{x} as a rational number. Error: if \spad{x} is not a rational number.
668585`\spad{rational(n)} creates a rational number (see \spadtype{Fraction Integer})..
668652`\spad{rational(o)} returns the real part if all seven imaginary parts are 0. Error: if \spad{o} is not rational.
668745`\spad{rational(q)} tries to convert \spad{q} into a rational number. Error: if this is not possible. If \spad{rational?(q)} is \spad{true},{} the conversion will be done and the rational number returned.
668840`\spad{rational(u)} assumes spadvar{\spad{u}} is actually a rational number and does the conversion to rational number (see \spadtype{Fraction Integer}).
668925`\spad{rational(x)} returns \spad{x} as a finite rational number. Error: if \spad{x} is not a rational number.
669020`\spad{rational(x)} returns \spad{x} as a finite rational number. Error: if \spad{x} cannot be so converted.
669114`\spad{ratpart(ir)} returns the rational part of an integration result
669165`\spad{ravel(t)} produces a list of components from a tensor such that \indented{2}{\spad{unravel(ravel(t)) = t}.}
669228`right contraction inner product
669283`\spad{rdHack1(v,{}u,{}n)} \undocumented
669371`\spad{rdregime(s)} reads in a list from a file with name \spad{s}
669615`\spad{read!(f)} extracts a value from file \spad{f}. The state of \spad{f} is modified so a subsequent call to \spadfun{read!} will return the next element.
669664`\spad{readIfCan!(f)} returns a value from the file \spad{f},{} if possible. If \spad{f} is not open for reading,{} or if \spad{f} is at the end of file then \spad{"failed"} is the result.
669721`\spad{readIfCan!(f)} returns a value from the file \spad{f},{} if possible. If \spad{f} is not open for reading,{} or if \spad{f} is at the end of file then \spad{"failed"} is the result.
669793`\spad{readIfCan!(f)} returns a string of the contents of a line from file \spad{f},{} if possible. If \spad{f} is not readable or if it is positioned at the end of file,{} then \spad{"failed"} is returned.
669856`\spad{readLine!(f)} returns a string of the contents of a line from the file \spad{f}.
669902`\spad{readLineIfCan!(f)} returns a string of the contents of a line from file \spad{f},{} if possible. If \spad{f} is not readable or if it is positioned at the end of file,{} then \spad{"failed"} is returned.
669969`\spad{readable?(f)} tests if the named file exist and can it be opened for reading.
670024`\spad{real?(t)} tests whether \spad{t} is equivalent to the FORTRAN type REAL.
670076`\spad{real?(f)} returns \spad{true} if \spad{f = real f}.
670148`\spad{real?(f)} returns \spad{true} if \spad{f = real f}.
670213`\spad{realEigenvalues(m,{}eps)} computes the eigenvalues of the matrix \spad{m} to precision \spad{eps}. The eigenvalues are expressed as floats or rational numbers depending on the type of \spad{eps} (float or rational).
670315`\spad{realEigenvectors(m,{}eps)} returns a list of records each one containing a real eigenvalue,{} its algebraic multiplicity,{} and a list of associated eigenvectors. All these results are computed to precision \spad{eps} as floats or rational numbers depending on the type of \spad{eps} .
670475`\spad{realElementary(f)} rewrites \spad{f} in terms of the 4 fundamental real transcendental elementary functions: \spad{log,{} exp,{} tan,{} atan}.
670551`\spad{realElementary(f,{}x)} rewrites the kernels of \spad{f} involving \spad{x} in terms of the 4 fundamental real transcendental elementary functions: \spad{log,{} exp,{} tan,{} atan}.
670634`\spad{realLiouvillian(f)} rewrites \spad{f} elementary kernels of \spad{f} in terms 4 fundamental real elementary functions: \spad{log,{} exp,{} tan,{} atan}. Additionally,{} it rewrites Liouvillian functions as indefinite integrals to support better normalization.
670711`\spad{realLiouvillian(f,{} x)} rewrites \spad{f} elementary kernels of \spad{f} in terms 4 fundamental real elementary functions: \spad{log,{} exp,{} tan,{} atan}. Additionally,{} it rewrites Liouvillian functions of \spad{x} as indefinite integrals to support better normalization.
670795`\spad{realRoots(rf,{} eps)} finds the real zeros of a univariate rational function with precision given by eps.
670891`\spad{realRoots(lp,{}lv,{}eps)} computes the list of the real solutions of the list \spad{lp} of rational functions with rational coefficients with respect to the variables in \spad{lv},{} with precision \spad{eps}. Each solution is expressed as a list of numbers in order corresponding to the variables in \spad{lv}.
671012`\spad{realSolve(lp)} returns the same as \spad{realSolve(ts,{}false,{}false,{}false)}
671134`\spad{realSolve(ts)} returns the set of the points in the regular zero set of \spad{ts} whose coordinates are all real. WARNING: For each set of coordinates given by \spad{realSolve(ts)} the ordering of the indeterminates is reversed \spad{w}.\spad{r}.\spad{t}. \spad{ls}.
671255`\spad{realSolve(ts,{}info?)} returns the same as \spad{realSolve(ts,{}info?,{}false,{}false)}.
671385`\spad{realSolve(lp,{}lv,{}eps)} = compute the list of the real solutions of the list \spad{lp} of polynomials with integer coefficients with respect to the variables in \spad{lv},{} with precision \spad{eps}.
671492`\spad{realSolve(ts,{}info?,{}check?)} returns the same as \spad{realSolve(ts,{}info?,{}check?,{}false)}.
671630`\spad{realSolve(ts,{}info?,{}check?,{}lextri?)} returns the set of the points in the variety associated with \spad{lp} whose coordinates are all real. Moreover,{} if \spad{info?} is \spad{true} then some information is displayed during decomposition into regular chains. If \spad{check?} is \spad{true} then the result is checked. If \spad{lextri?} is \spad{tru
671630`--e} then the lexTriangular algorithm is called from the \spadtype{LexTriangularPackage} constructor (see \axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\spad{lp},{}\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \axiomOpFrom{zeroSetSplit}{RegularChain} from \spadtype{RegularChain}. WARNING: For each set of coordinates given by \spad{realSolve(ts,{}info?,{}check?,{}lextri?)} the ordering of the indeterminates is reversed \spad{w
671630`--}.\spad{r}.\spad{t}. \spad{ls}.
671776`\spad{realZeros(pol)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \spad{pol}.
671889`\spad{realZeros(pol)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \spad{pol}.
672003`\spad{realZeros(pol,{} eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \spad{pol}.
672134`\spad{realZeros(pol,{} eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \spad{pol}.
672266`\spad{realZeros(pol,{} range)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \spad{pol} which lie in the interval expressed by the record range.
672434`\spad{realZeros(pol,{} range)} returns a list of isolating intervals for all the real zeros of the univariate polynomial \spad{pol} which lie in the interval expressed by the record range.
672603`\spad{realZeros(pol,{} int,{} eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \spad{pol} which lie in the interval expressed by the record \spad{int}.
672789`\spad{realZeros(pol,{} int,{} eps)} returns a list of intervals of length less than the rational number eps for all the real roots of the polynomial \spad{pol} which lie in the interval expressed by the record \spad{int}.
672976`\spad{real(f)} returns the real part of \spad{f} where \spad{f} is a complex function.
673053`\spad{real(f)} returns the real part of \spad{f} where \spad{f} is a complex function.
673111`\spad{real(x)} returns real part of \spad{x}.
673157`\spad{real(o)} extracts real part of octonion \spad{o}.
673204`\spad{real(q)} extracts the real part of quaternion \spad{q}.
673253`\spad{recip()} returns \spad{matrix [[0,{}1],{}[1,{}0]]} representing the map \spad{x -> 1 / x}.
673650`\spad{recip(a)} returns the power series reciprocal of \spad{a},{} or "failed" if not possible.
673741`\spad{recip(x)} \undocumented
673841`\spad{recip(x)} \undocumented
673930`\spad{recip(m)} = recip() * \spad{m}
673979`\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \spad{a},{} or \spad{"failed"} if there is no unit element,{} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
674080`\spad{recip(a)} returns an element,{} which is both a left and a right inverse of \spad{a},{} or \spad{"failed"} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
674139`\spad{recip(x)} tries to compute the multiplicative inverse for \spad{x} or "failed" if it cannot find the inverse (see unitsKnown).
674191`\spad{recip(x)} computes the multiplicative inverse of \spad{x} or "failed" if \spad{x} is not invertible.
674368`\axiom{recip(\spad{p},{}\spad{ts})} returns the inverse of \axiom{\spad{p}} \spad{w}.\spad{r}.\spad{t} \spad{ts} assuming that \axiom{\spad{p}} is invertible \spad{w}.\spad{r}.\spad{t} \spad{ts}.
674449`\axiom{recip(pol,{}aRoot)} tries to inverse \axiom{pol} interpreted as \axiom{aRoot}
674558`\spad{reciprocalPolynomial(p)} calulates a polynomial which has exactly the inverses of the non-zero roots of \spad{p} as roots,{} and the same number of 0-roots.
674634`\spad{recolor()},{} uninteresting to top level user; exported in order to compile package.
674844`combines rational reconstruction with removal of common denominators in blocks.
674959`\spad{reconstruct polynomials} in \spad{n} + 1 variables
675193`\spad{reconstruct polynomials} in \spad{n} + 1 variables from result of rational reconstruction
675468`\spad{rectangularMatrix(m)} converts a matrix of type \spadtype{Matrix} to a matrix of type \spad{RectangularMatrix}.
675541`\spad{recur(g)} is the function \spad{h} such that \indented{1}{\spad{h(n,{}x)= g(n,{}g(n-1,{}..g(1,{}x)..))}.}
675635`\spad{recur(n,{}g,{}x)} is \spad{g(n,{}g(n-1,{}..g(1,{}x)..))}.
675739`\spad{redPo }\undocumented
675852`\spad{redPol }\undocumented
675944`normalForm(poly,{}\spad{gb}) reduces the polynomial poly modulo the precomputed groebner basis \spad{gb} giving up to a constant factor a canonical representative of the residue class.
676029`\spad{red()} returns the position of the red hue from total hues.
676060`\spad{redmat(m,{}g)} returns a matrix whose entries are those of \spad{m} modulo the ideal generated by the groebner basis \spad{g}
676157`\spad{redpps(s,{}g)} returns the simplified form of \spad{s} after reducing modulo a groebner basis \spad{g}
676722`\spad{reduceBasisAtInfinity(b1,{}...,{}bn)} returns \spad{(x^i * bj)} for all \spad{i},{}\spad{j} such that \spad{x^i*bj} is locally integral at infinity.
676816`Skips elements of sigma basis which does not satisfy degree bounds and puts sigma-basis into canonical form.
677094`\axiom{reduceByQuasiMonic(\spad{p},{}\spad{ts})} returns the same as \axiom{remainder(\spad{p},{}collectQuasiMonic(\spad{ts})).polnum}.
677168`\spad{reduceLODE(op,{} g)} returns \spad{[m,{} v]} such that any solution in \spad{A} of \spad{op z = g} is of the form \spad{z = (z_1,{}...,{}z_m) . (b_1,{}...,{}b_m)} where the \spad{b_i's} are the basis of \spad{A} over \spad{F} returned by \spadfun{basis}() from \spad{A},{} and the \spad{z_i's} satisfy the differential system \spad{M.z = v}.
677340`\axiom{reduceMod(\spad{sys1},{} \spad{sys2})} reduces the system \axiom{\spad{sys1}} modulo the system \axiom{\spad{sys2}}.
677424`\spad{reduce(f)} produces the equivalence class of \spad{f} in the residue ring
677670`\spad{reduce(x)} \undocumented
677790`\spad{reduce(p)} \undocumented{}
677988`\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \spad{f} by applying their defining relations.
678039`\axiom{reduce(\spad{x})} reduces algebraics in \axiom{\spad{x}}.
678093`\axiom{reduce(\spad{x})} reduces the algebraics in \axiom{\spad{x}}.
678148`\spad{reduce(frac)} converts the fraction \spad{frac} to an algebra element.
678239`\spad{reduce(up)} converts the univariate polynomial \spad{up} to an algebra element,{} reducing by the \spad{definingPolynomial()} if necessary.
678292`\spad{reduce(D)} converts \spad{D} to some reduced form (the reduced forms can be differents in different implementations).
678357`\spad{reduce(f)} simplifies all the unreduced algebraic numbers present in \spad{f} by applying their defining relations.
678403`\spad{reduce(f)} simplifies all the unreduced algebraic quantities present in \spad{f} by applying their defining relations.
678702`\spad{reduce(r,{}m)} \undocumented
678790`\spad{reduce(r,{}m)} \undocumented
678868`\spad{reduce(r,{}m)} \undocumented
678945`\spad{reduce(f,{}u)} reduces the binary operation \spad{f} across \spad{u}. For example,{} if \spad{u} is \axiom{[\spad{x},{}\spad{y},{}...,{}\spad{z}]} then \axiom{reduce(\spad{f},{}\spad{u})} returns \axiom{\spad{f}(..\spad{f}(\spad{f}(\spad{x},{}\spad{y}),{}...),{}\spad{z})}. Note: if \spad{u} has one element \spad{x},{} \axiom{reduce(\spad{f},{}\spad{u})} returns \spad{x}. Error: if \spad{u} is empty.
679262`\spad{reduce(func,{}vec,{}ident)} combines the elements in \spad{vec} using the binary function \spad{func}. Argument \spad{ident} is returned if the vector is empty.
679353`\spad{reduce(fn,{}u,{}ident)} successively uses the binary function \spad{fn} on the elements of list \spad{u} and the result of previous applications. \spad{ident} is returned if the \spad{u} is empty. Note the order of application in the following examples: \spad{reduce(fn,{}[1,{}2,{}3],{}0) = fn(3,{}fn(2,{}fn(1,{}0)))} and \spad{reduce(*,{}[2,{}3],{}1) = 3 * (2 * 1)}.
679418`\spad{reduce(f,{}a,{}r)} applies function \spad{f} to each successive element of the one-dimensional array \spad{a} and an accumulant initialized to \spad{r}. For example,{} \spad{reduce(_+\$Integer,{}[1,{}2,{}3],{}0)} does \spad{3+(2+(1+0))}. Note: third argument \spad{r} may be regarded as the identity element for the function \spad{f}.
679513`\spad{reduce(f,{}a,{}r)} applies function \spad{f} to each successive element of the primitive array \spad{a} and an accumulant initialized to \spad{r}. For example,{} \spad{reduce(_+\$Integer,{}[1,{}2,{}3],{}0)} does \spad{3+(2+(1+0))}. Note: third argument \spad{r} may be regarded as the identity element for the function \spad{f}.
679598`\spad{reduce(func,{}vec,{}ident)} combines the elements in \spad{vec} using the binary function \spad{func}. Argument \spad{ident} is returned if \spad{vec} is empty.
679667`\spad{reduce(f,{}m,{}r)} returns a matrix \spad{n} where \spad{n[i,{}j] = f(m[i,{}j],{}r)} for all indices \spad{i} and \spad{j}.
679770`\spad{reduce(f,{}m,{}r)} returns a matrix \spad{n} where \spad{n[i,{}j] = f(m[i,{}j],{}r)} for all indices spad{\spad{i}} and \spad{j}.
679888`\spad{reduce(f,{}a,{}r)} applies function \spad{f} to each successive element of the aggregate \spad{a} and an accumulant initialized to \spad{r}. For example,{} \spad{reduce(_+\$Integer,{}[1,{}2,{}3],{}0)} does \spad{3+(2+(1+0))}. Note: third argument \spad{r} may be regarded as the identity element for the function \spad{f}.
679968`\spad{reduce(f,{}a,{}r)} applies function \spad{f} to each successive element of the aggregate \spad{a} and an accumulant initialised to \spad{r}. For example,{} \spad{reduce(_+\$Integer,{}[1,{}2,{}3],{}0)} does a \spad{3+(2+(1+0))}. Note: third argument \spad{r} may be regarded as an identity element for the function.
680045`\spad{reduce(f,{}u,{}x)} reduces the binary operation \spad{f} across \spad{u},{} where \spad{x} is the identity operation of \spad{f}. Same as \axiom{reduce(\spad{f},{}\spad{u})} if \spad{u} has 2 or more elements. Returns \axiom{\spad{f}(\spad{x},{}\spad{y})} if \spad{u} has one element \spad{y},{} \spad{x} if \spad{u} is empty. For example,{} \axiom{reduce(+,{}\spad{u},{}0)} returns the sum of the elements of \spad{u}.
680122`\spad{reduce(b,{}f,{}u)},{} where \spad{u} is a finite stream \spad{[x0,{}x1,{}...,{}xn]},{} returns the value \spad{r(n)} computed as follows: \spad{r0 = f(x0,{}b),{} r1 = f(x1,{}r0),{}...,{} r(n) = f(xn,{}r(n-1))}.
680524`\spad{reduce(f,{}u,{}x,{}z)} reduces the binary operation \spad{f} across \spad{u},{} stopping when an "absorbing element" \spad{z} is encountered. As for \axiom{reduce(\spad{f},{}\spad{u},{}\spad{x})},{} \spad{x} is the identity operation of \spad{f}. Same as \axiom{reduce(\spad{f},{}\spad{u},{}\spad{x})} when \spad{u} contains no element \spad{z}. Thus the third argument \spad{x} is returned when \spa
680524`--d{u} is empty.
680625`\axiom{reduce(\spad{p},{}\spad{ts},{}redOp,{}redOp?)} returns a polynomial \axiom{\spad{r}} such that \axiom{redOp?(\spad{r},{}\spad{p})} holds for every \axiom{\spad{p}} of \axiom{\spad{ts}} and there exists some product \axiom{\spad{h}} of the initials of the members of \axiom{\spad{ts}} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}. The operation \axiom{redOp} must satisfy
680625`-- the following conditions. For every \axiom{\spad{p}} and \axiom{\spad{q}} we have \axiom{redOp?(redOp(\spad{p},{}\spad{q}),{}\spad{q})} and there exists an integer \axiom{\spad{e}} and a polynomial \axiom{\spad{f}} such that \axiom{init(\spad{q})^e*p = \spad{f*q} + redOp(\spad{p},{}\spad{q})}.
680859`\axiom{reduced?(\spad{q},{}\spad{lp})} returns \spad{true} iff \axiom{reduced?(\spad{q},{}\spad{p})} holds for every \axiom{\spad{p}} in \axiom{\spad{lp}}.
680940`\axiom{reduced?(a,{}\spad{b})} returns \spad{true} iff \axiom{degree(a,{}mvar(\spad{b})) < mdeg(\spad{b})}.
681097`\axiom{reduced?(\spad{p},{}\spad{ts},{}redOp?)} returns \spad{true} iff \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. in the sense of the operation \axiom{redOp?},{} that is if for every \axiom{\spad{t}} in \axiom{\spad{ts}} \axiom{redOp?(\spad{p},{}\spad{t})} holds.
681182`\spad{reducedContinuedFraction(b0,{}b)} constructs a continued fraction in the following way: if \spad{b = [b1,{}b2,{}...]} then the result is the continued fraction \spad{b0 + 1/(b1 + 1/(b2 + ...))}. That is,{} the result is the same as \spad{continuedFraction(b0,{}[1,{}1,{}1,{}...],{}[b1,{}b2,{}b3,{}...])}.
681260`\spad{reducedDiscriminant(up)} \undocumented
681348`\spad{reducedForm(x)} puts the continued fraction \spadvar{\spad{x}} in reduced form,{} \spadignore{i.e.} the function returns an equivalent continued fraction of the form \spad{continuedFraction(b0,{}[1,{}1,{}1,{}...],{}[b1,{}b2,{}b3,{}...])}.
681404`\spad{reducedQPowers(f)} generates \spad{[x,{}x^q,{}x^(q^2),{}...,{}x^(q^(n-1))]} reduced modulo \spad{f} where \spad{q = size()\$GF} and \spad{n = degree f}.
682519`\spad{reducedSystem(A)} returns a matrix \spad{B} such that \spad{A x = 0} and \spad{B x = 0} have the same solutions in \spad{R}.
684039`\spad{reducedSystem(A,{} v)} returns a matrix \spad{B} and a vector \spad{w} such that \spad{A x = v} and \spad{B x = w} have the same solutions in \spad{R}.
684156`\spad{reduction(p,{}prime)} reduces the polynomial \spad{p} modulo \spad{prime} of \spad{R}. Note: this function is exported only because it\spad{'s} conditional.
684209`\spad{reduction(f,{}p)} reduces the coefficients of the polynomial \spad{f} modulo the prime \spad{p}.
684279`\spad{reduction(u,{}pol)} computes the symmetric reduction of \spad{u} mod \spad{pol}
684476`\spad{reductum(u)} returns \spad{u} minus its leading monomial returns zero if handed the zero element.
684555`\spad{reductum(p)},{} where \spad{p} is an antisymmetric polynomial,{} returns \spad{p} minus the leading term of \spad{p} if \spad{p} has at least two terms,{} and 0 otherwise.
684604`\spad{reductum(df)},{} where \spad{df} is a differential form,{} returns \spad{df} minus the leading term of \spad{df} if \spad{df} has two or more terms,{} and 0 otherwise.
684671`\spad{reductum(x)} \undocumented
684743`\spad{reductum(x)} \undocumented
684799`\spad{reductum(p)} returns \spad{p} minus its leading term. An error is produced if \spad{p} is zero.
684852`\spad{reductum(u)} returns \spad{u} minus its leading monomial returns zero if handed the zero element.
684907`\spad{reductum(z)} returns a new element created by removing the leading coefficient/support pair from the element \spad{z}. Error: if \spad{z} has no support.
684973`\spad{reductum(l)} is \spad{l - monomial(a(n),{}n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
685032`\spad{reductum(f)} is \spad{f} minus its leading monomial.
685105`\spad{reductum(l)} is \spad{l - monomial(a(n),{}n)} if \indented{2}{\spad{l = sum(monomial(a(i),{}i),{} i = 0..n)}.}
685238`\axiom{reductum(\spad{p},{}\spad{v})} returns the reductum of \axiom{\spad{p}},{} where \axiom{\spad{p}} is viewed as a univariate polynomial in \axiom{\spad{v}}.
685307`attempt to simplify theory apply recursively to subnodes normally this should not be necessary since logic values are interpreted when constructed
685343`beta reduction - apply beta reduction recusivly to all subnodes
685383`weak reduction - apply this combinator to rearange its subnodes then apply recusivly to its subnodes
685431`\spad{ref(n)} creates a pointer (reference) to the object \spad{n}.
685470`\spad{refine(x)} \undocumented
685507`\spad{refine(p)} performs a refinement on the plot \spad{p}
685542`\axiom{refine(rootChar)} shrinks isolating interval around \axiom{rootChar}
685630`\spad{refine(u,{}fn)} is used to apply the function \userfun{\spad{fn}} to each factor of \spadvar{\spad{u}} and then build a new factored object from the results. For example,{} if \spadvar{\spad{u}} were created by calling \spad{nilFactor(10,{}2)} then \spad{refine(u,{}factor)} would create a factored object equal to that created by \spad{factor(100)} or \spad{primeFactor(2,{}2) * primeFactor(5,{}2)}.
685724`\spad{refine(p,{}x)} \undocumented{}
685790`\spad{refine(x,{}r)} \undocumented
685848`\spad{refine(x,{}r)} \undocumented
685904`\spad{refine(pol,{} int,{} eps)} refines the interval \spad{int} containing exactly one root of the univariate polynomial \spad{pol} to size less than the rational number eps.
686081`\spad{refine(pol,{} int,{} eps)} refines the interval \spad{int} containing exactly one root of the univariate polynomial \spad{pol} to size less than the rational number eps.
686259`\spad{refine(pol,{} int,{} range)} takes a univariate polynomial \spad{pol} and and isolating interval \spad{int} containing exactly one real root of \spad{pol}; the operation returns an isolating interval which is contained within range,{} or "failed" if no such isolating int
686259`--erval exists.
686489`\spad{refine(pol,{} int,{} range)} takes a univariate polynomial \spad{pol} and and isolating interval \spad{int} which must contain exactly one real root of \spad{pol},{} and returns an isolating interval which is contained within range,{} or "failed" if no such isolating in
686489`--terval exists.
686720`\spad{regime(y,{}c,{} w,{} p,{} r,{} rm,{} m)} returns a regime,{} a list of polynomials specifying the consistency conditi
686720`--ons,{} a particular solution and basis representing the general solution of the parametric linear system \spad{c} \spad{z} = \spad{w} on that regime. The regime returned depends on the subdeterminant \spad{y}.det and the row and column indices. The solutions are simplified using the assumption that the system has rank \spad{r} and maximum rank \spad{rm}. The list \spad{p} represents a list of list of factors of polynomials in a groebner basis of the ideal generated by higher order subdetermina
686720`--nts,{} and ius used for the simplification. The mode \spad{m} distinguishes the cases when the system is homogeneous,{} or the right hand side is arbitrary,{} or when there is no new right hand side variables.
687104`\spad{region(v,{}n,{}s)} displays the bounding box of the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} if \spad{s} is "on",{} or does not display the bounding box if \spad{s} is "off".
687251`\spad{regularRepresentation(a)} returns the matrix \spad{m} of the linear map defined by left multiplication by \spad{a} with respect to the fixed basis. That is for all \spad{x} we have \spad{coordinates(a*x) = m*coordinates(x)}.
687485`\spad{regularRepresentation(a,{} basis)} returns the matrix \spad{m} of the linear map defined by left multiplication by \spad{a} with respect to the basis \spad{basis}. That is for all \spad{x} we have \spad{coordinates(a*x,{} basis) = m*coordinates(x,{} basis)}.
687572`\spad{reindex(t,{}[i1,{}...,{}idim])} permutes the indices of \spad{t}. For example,{} if \spad{r = reindex(t,{} [4,{}1,{}2,{}3])} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank for tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = t(l,{}i,{}j,{}k)}.}
687646`\spad{relationsIdeal(polyList)} returns the ideal of relations among the polynomials in \spad{polyList}.
687821`\axiom{relativeApprox(\spad{n},{}\spad{p})} gives a relative approximation of \axiom{\spad{n}} that has precision \axiom{\spad{p}}
687899`\axiom{relativeApprox(exp,{}\spad{c},{}\spad{p}) = a} is relatively close to exp as a polynomial in \spad{c} ip to precision \spad{p}
688020`\axiom{approximate(term,{}root,{}prec)} gives an approximation of \axiom{term} over \axiom{root} with precision \axiom{prec}
688132`\spad{relerror(x,{}y)} computes the absolute value of \spad{x - y} divided by \spad{y},{} when \spad{y \~= 0}.
688264`\spad{f rem g} creates the equivalent infix form.
688305`\spad{x rem y} is the same as \spad{divide(x,{}y).remainder}. See \spadfunFrom{divide}{EuclideanDomain}.
688351`\spad{a rem b} returns the remainder of \spad{a} and \spad{b}.
688400`Polynomial remainder
688586`\axiom{remainder(a,{}\spad{ps})} returns \axiom{[\spad{c},{}\spad{b},{}\spad{r}]} such that \axiom{\spad{b}} is fully reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ps}},{} \axiom{r*a - \spad{c*b}} lies in the ideal generated by \axiom{\spad{ps}}. Furthermore,{} if \axiom{\spad{R}} is a \spad{gcd}-domain,{} \axiom{\spad{b}} is primitive.
688705`\spad{remap_variables(p,{} lv1,{} lv2)} replaces corresponding variables from \spad{l1} by variables from \spad{lv2}. Assumes that \spad{lv1} and \spad{lv2} have the same length and that \spad{p} has variables in \spad{lv1}.
689474`\axiom{remove!(\spad{s},{}a)} replaces a by remove(\spad{s},{}a)
689543`\spad{remove!(p,{}d)} destructively changes dictionary \spad{d} by removeing all entries \spad{x} such that \axiom{\spad{p}(\spad{x})} is \spad{true}.
689634`\spad{remove!(p,{}u)} destructively removes all elements \spad{x} of \spad{u} such that \axiom{\spad{p}(\spad{x})} is \spad{true}.
689707`\spad{remove!(k,{}t)} searches the table \spad{t} for the key \spad{k} removing (and return) the entry if there. If \spad{t} has no such key,{} \axiom{remove!(\spad{k},{}\spad{t})} returns "failed".
689788`\spad{remove!(x,{}d)} destructively changes dictionary \spad{d} by removing all entries \spad{y} such that \axiom{\spad{y} = \spad{x}}.
689890`\spad{remove!(x,{}u)} destructively removes all values \spad{x} from \spad{u}.
689970`\spad{remove!(p,{}ms,{}number)} removes destructively at most \spad{number} copies of elements \spad{x} such that \spad{p(x)} is \spadfun{\spad{true}} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
690034`\spad{remove!(x,{}ms,{}number)} removes destructively at most \spad{number} copies of element \spad{x} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
690087`\spad{removeChild!(n,{}c)} removes the specified child node \spad{'c'} below node \spad{'n'}
690138`\spad{removeConstantTerm(f,{} x)} returns \spad{f} minus any additive constant with respect to \spad{x}.
690228`\spad{removeCosSq(f)} converts every \spad{cos(u)^2} appearing in \spad{f} into \spad{1 - sin(x)^2},{} and also reduces higher powers of \spad{cos(u)} with that formula.
690294`\spad{removeCoshSq(f)} converts every \spad{cosh(u)^2} appearing in \spad{f} into \spad{1 - sinh(x)^2},{} and also reduces higher powers of \spad{cosh(u)} with that formula.
690415`\spad{removeDuplicates!(u)} destructively removes duplicates from \spad{u}.
690503`\spad{removeDuplicates!(d)} destructively removes any duplicate values in dictionary \spad{d}.
690678`\spad{removeDuplicates(u)} returns a copy of \spad{u} with all duplicates removed.
690777`\axiom{removeIrreducibleRedundantFactors(\spad{lp},{}\spad{lq})} returns the same as \axiom{irreducibleFactors(concat(\spad{lp},{}\spad{lq}))} assuming that \axiom{irreducibleFactors(\spad{lp})} returns \axiom{\spad{lp}} up to replacing some polynomial \axiom{\spad{pj}} in \axiom{\spad{lp}} by some polynomial \axiom{\spad{qj}} associated 
690777`--to \axiom{\spad{pj}}.
690944`\axiom{removeRedundantFactorsInContents(\spad{lp},{}\spad{lf})} returns \axiom{newlp} where \axiom{newlp} is obtained from \axiom{\spad{lp}} by removing in the content of every polynomial of \axiom{\spad{lp}} any non trivial factor of any polynomial \axiom{\spad{f}} in \axiom{\spad{lf}}. Moreover,{} squares over \axiom{\spad{R}} are first removed in the content of every polyn
690944`--omial of \axiom{\spad{lp}}.
691073`\axiom{removeRedundantFactorsInPols(\spad{lp},{}\spad{lf})} returns \axiom{newlp} where \axiom{newlp} is obtained from \axiom{\spad{lp}} by removing in every polynomial \axiom{\spad{p}} of \axiom{\spad{lp}} any non trivial factor of any polynomial \axiom{\spad{f}} in \axiom{\spad{lf}}. Moreover,{} squares over \axiom{\spad{R}} are first removed in every polynomial \axiom{\spad{lp
691073`--}}.
691198`\axiom{removeRedundantFactors(\spad{lp})} returns \axiom{\spad{lq}} such that if \axiom{\spad{lp} = [\spad{p1},{}...,{}\spad{pn}]} and \axiom{\spad{lq} = [\spad{q1},{}...,{}\spad{qm}]} then the product \axiom{p1*p2*...\spad{*pn}} vanishes iff the product \axiom{q1*q2*...\spad{*qm}} vanishes,{} and the product of degrees of the \axiom{\spad{qi}} is not greater than the one of the \axiom{\spad{pj}},{} and no po
691198`--lynomial in \axiom{\spad{lq}} divides another polynomial in \axiom{\spad{lq}}. In particular,{} polynomials lying in the base ring \axiom{\spad{R}} are removed. Moreover,{} \axiom{\spad{lq}} is sorted \spad{w}.\spad{r}.\spad{t} \axiom{infRittWu?}. Furthermore,{} if \spad{R} is \spad{gcd}-domain,{} the polynomials in \axiom{\spad{lq}} are pairwise without common non trivial factor.
691293`\axiom{removeRedundantFactors(\spad{lp},{}\spad{lq})} returns the same as \axiom{removeRedundantFactors(concat(\spad{lp},{}\spad{lq}))} assuming that \axiom{removeRedundantFactors(\spad{lp})} returns \axiom{\spad{lp}} up to replacing some polynomial \axiom{\spad{pj}} in \axiom{\spad{lp}} by some polynomial \axiom{\spad{qj}} associated to \axiom{\spad{pj}}.
691396`\axiom{removeRedundantFactors(\spad{lp},{}\spad{q})} returns the same as \axiom{removeRedundantFactors(cons(\spad{q},{}\spad{lp}))} assuming that \axiom{removeRedundantFactors(\spad{lp})} returns \axiom{\spad{lp}} up to replacing some polynomial \axiom{\spad{pj}} in \axiom{\spad{lp}} by some some polynomial \axiom{\spad{qj}} associated to \axiom{\spad{pj}}.
691493`\axiom{removeRedundantFactors(\spad{p},{}\spad{q})} returns the same as \axiom{removeRedundantFactors([\spad{p},{}\spad{q}])}
691584`\axiom{removeRedundantFactors(\spad{lp},{}\spad{lq},{}remOp)} returns the same as \axiom{concat(remOp(removeRoughlyRedundantFactorsInPols(\spad{lp},{}\spad{lq})),{}\spad{lq})} assuming that \axiom{remOp(\spad{lq})} returns \axiom{\spad{lq}} up to similarity.
691706`\spad{removeRepeats!(u)} destructively replaces runs of consequtive equal elements of \spad{u} by single elements.
691821`\axiom{removeRoughlyRedundantFactorsInContents(\spad{lp},{}\spad{lf})} returns \axiom{newlp}where \axiom{newlp} is obtained from \axiom{\spad{lp}} by removing in the content of every polynomial of \axiom{\spad{lp}} any occurence of a polynomial \axiom{\spad{f}} in \axiom{\spad{lf}}. Moreover,{} squares over \axiom{\spad{R}} are first removed in the content of every pol
691821`--ynomial of \axiom{\spad{lp}}.
691957`\axiom{removeRoughlyRedundantFactorsInPol(\spad{p},{}\spad{lf})} returns the same as removeRoughlyRedundantFactorsInPols([\spad{p}],{}\spad{lf},{}\spad{true})
692060`\axiom{removeRoughlyRedundantFactorsInPols(\spad{lp},{}\spad{lf})} returns \axiom{newlp}where \axiom{newlp} is obtained from \axiom{\spad{lp}} by removing in every polynomial \axiom{\spad{p}} of \axiom{\spad{lp}} any occurence of a polynomial \axiom{\spad{f}} in \axiom{\spad{lf}}. This may involve a lot of exact-quotients computations.
692176`\axiom{removeRoughlyRedundantFactorsInPols(\spad{lp},{}\spad{lf},{}opt)} returns the same as \axiom{removeRoughlyRedundantFactorsInPols(\spad{lp},{}\spad{lf})} if \axiom{opt} is \axiom{\spad{false}} and if the previous operation does not return any non null and constant polynomial,{} else return \axiom{[1]}.
692300`\spad{removeSinSq(f)} converts every \spad{sin(u)^2} appearing in \spad{f} into \spad{1 - cos(x)^2},{} and also reduces higher powers of \spad{sin(u)} with that formula.
692366`\spad{removeSinhSq(f)} converts every \spad{sinh(u)^2} appearing in \spad{f} into \spad{1 - cosh(x)^2},{} and also reduces higher powers of \spad{sinh(u)} with that formula.
692433`\axiom{removeSquaresIfCan(\spad{lp})} returns \axiom{removeDuplicates [squareFreePart(\spad{p})\$\spad{P} for \spad{p} in \spad{lp}]} if \axiom{\spad{R}} is \spad{gcd}-domain else returns \axiom{\spad{lp}}.
692524`\axiom{removeSuperfluousCases(llpwt)} is an internal subroutine,{} exported only for developement.
692668`\axiom{removeSuperfluousCases(llpwt)} is an internal subroutine,{} exported only for developement.
692822`\axiom{removeSuperfluousQuasiComponents(\spad{lts})} removes from \axiom{\spad{lts}} any \spad{ts} such that \axiom{subQuasiComponent?(\spad{ts},{}us)} holds for another \spad{us} in \axiom{\spad{lts}}.
692924`\axiom{removeSuperfluousQuasiComponents(\spad{lts})} removes from \axiom{\spad{lts}} any \spad{ts} such that \axiom{subQuasiComponent?(\spad{ts},{}us)} holds for another \spad{us} in \axiom{\spad{lts}}.
693099`\axiom{removeZero(\spad{p},{}\spad{ts})} returns \axiom{0} if \axiom{\spad{p}} reduces to \axiom{0} by pseudo-division \spad{w}.\spad{r}.\spad{t} \axiom{\spad{ts}} otherwise returns a polynomial \axiom{\spad{q}} computed from \axiom{\spad{p}} by removing any coefficient in \axiom{\spad{p}} reducing to \axiom{0}.
693269`\spad{removeZeroes(x)} removes leading zeroes from the representation of the \spad{p}-adic rational \spad{x}. A \spad{p}-adic rational is represented by (1) an exponent and (2) a \spad{p}-adic integer which may have leading zero digits. When the \spad{p}-adic integer has a leading zero digit,{} a 'leading zero' is removed from the \spad{p}-adic rational as follows: the number is rewritten by increasing the exponent by 1 and dividing 
693269`--the \spad{p}-adic integer by \spad{p}. Note: \spad{removeZeroes(f)} removes all leading zeroes from \spad{f}.
693339`\spad{removeZeroes(f(x))} removes leading zeroes from the representation of the Laurent series \spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and dividing t
693339`--he Taylor series by its variable. Note: \spad{removeZeroes(f)} removes all leading zeroes from \spad{f}
693548`\spad{removeZeroes(n,{}x)} removes up to \spad{n} leading zeroes from the \spad{p}-adic rational \spad{x}.
693626`\spad{removeZeroes(n,{}f(x))} removes up to \spad{n} leading zeroes from the Laurent series \spad{f(x)}. A Laurent series is represented by (1) an exponent and (2) a Taylor series which may have leading zero coefficients. When the Taylor series has a leading zero coefficient,{} the 'leading zero' is removed from the Laurent series as follows: the series is rewritten by increasing the exponent by 1 and divid
693626`--ing the Taylor series by its variable.
693812`remove common denominators in blocks
693958`\spad{remove_denoms(offsets,{} nums,{} denoms,{} p)} removes common denominator from vectors of rational functions. Several vectors of rational functions are packed into \spad{nums} (storing numerators) and dens (storing denominators) Vector \spad{i} starts at position \spad{offsets}(\spad{i}). Computations are done modulo \spad{p}.
694338`\axiom{remove(\spad{s},{}a)} returns the splitting tree obtained from a by removing every sub-tree \axiom{\spad{b}} such that \axiom{value(\spad{b})} and \axiom{\spad{s}} have the same value,{} condition and status.
694406`\spad{remove(p,{}u)} returns a copy of \spad{u} removing all elements \spad{x} such that \axiom{\spad{p}(\spad{x})} is \spad{true}. Note: \axiom{remove(\spad{p},{}\spad{u}) \spad{==} [\spad{x} for \spad{x} in \spad{u} | not \spad{p}(\spad{x})]}.
694486`\spad{remove(f,{}st)} returns a stream consisting of those elements of stream \spad{st} which do not satisfy the predicate \spad{f}. Note: \spad{remove(f,{}st) = [x for x in st | not f(x)]}.
694552`\spad{remove(x,{}u)} returns a copy of \spad{u} with all elements \axiom{\spad{y} = \spad{x}} removed. Note: \axiom{remove(\spad{y},{}\spad{c}) \spad{==} [\spad{x} for \spad{x} in \spad{c} | \spad{x} \spad{~=} \spad{y}]}.
694643`\spad{remove(p,{}ms,{}number)} removes at most \spad{number} copies of elements \spad{x} such that \spad{p(x)} is \spadfun{\spad{true}} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
694706`\spad{remove(x,{}ms,{}number)} removes at most \spad{number} copies of element \spad{x} if \spad{number} is positive,{} all of them if \spad{number} equals zero,{} and all but at most \spad{-number} if \spad{number} is negative.
694758`\axiom{rename!(\spad{x},{}name)} changes the way \axiom{\spad{x}} is printed
694816`\axiom{rename(\spad{x},{}name)} gives a new number that prints as name
694873`\spad{reopen!(f,{}mode)} returns a file \spad{f} reopened for operation in the indicated mode: "input" or "output". \spad{reopen!(f,{}"input")} will reopen the file \spad{f} for input.
694932`\spad{reorder(p,{} perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial
695017`\spad{reorder(p,{} perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial
695111`\spad{reorder(p,{} perm)} applies the permutation perm to the variables in a polynomial and returns the new correctly ordered polynomial
695207`\spad{repSq(v,{}e)} computes \spad{v^e} by repeated squaring,{} interpreting \spad{v} as an element of a normal basis field.
695306`\spad{repack1(x,{} a,{} d,{} m)} stores coefficients of \spad{x} in a. \spad{d} is degree of \spad{x}. Corresponding exponents are given by pack_exps.
695399`convert polynomials represented as parallel vector of exponents in \spad{k} variables and vector of univariate polynomials to parallel vector of exponents in \spad{k+1} variables and coefficients
695625`\spad{repeatUntilLoop(s,{}c)} creates a repeat ... until loop in FORTRAN.
695755`\axiom{repeatedIndex(\spad{jv})} returns the multi-index of the jet variable \axiom{\spad{jv}} in repeated index notation.
695829`\spad{repeating?(l,{}s)} returns \spad{true} if a stream \spad{s} is periodic with period \spad{l},{} and \spad{false} otherwise.
695904`\spad{repeating(l)} is a repeating stream whose period is the list \spad{l}.
696036`\spad{replaceKthElement(S,{}k,{}p)} replaces the \spad{k^}{th} element of \spad{S} by \spad{p},{} and returns "failed" if the result is not a set of \spad{M} integers in \spad{1..n} any more.
696153`\spad{replace(s,{}i..j,{}t)} replaces the substring \axiom{\spad{s}(\spad{i}..\spad{j})} of \spad{s} by string \spad{t}.
696229`\spad{representationType()} returns the type of the representation,{} one of: \spad{prime},{} \spad{polynomial},{} \spad{normal},{} or \spad{cyclic}.
696612`\spad{represents([a1,{}..,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed basis.
696686`\spad{represents([a1,{}..,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
696747`\spad{represents([a1,{}...,{}an])} returns \spad{a1*v1 + ... + an*vn},{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed \spad{R}-module basis.
697288`\spad{represents([a1,{}..,{}an],{}[v1,{}..,{}vn])} returns \spad{a1*v1 + ... + an*vn}.
697364`\spad{represents([a1,{}...,{}am],{}[v1,{}...,{}vm])} returns the linear combination \spad{a1*vm + ... + an*vm}.
697451`\spad{represents([A0,{}...,{}A(n-1)],{}D)} returns \spad{(A0 + A1 y +...+ A(n-1)*y^(n-1))/D}.
697529`\spad{reseed(n)} restarts the random number generator at \spad{n}.
697585`\spad{resetBadValues(p)} initializes the list of "bad values" for \spad{p} to \spad{[]}. Note: \spad{p} is not allowed to match any of its "bad values".
697634`\spad{resetNew()} resets the internals counters that new() and new(\spad{s}) use to return distinct symbols every time.
697673`\spad{resetVariableOrder()} cancels any previous use of setVariableOrder and returns to the default system ordering.
697743`\spad{reset(v)} sets the current state of the graph characteristics of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} back to their initial settings.
697797`\spad{reset(v)} sets the current state of the graph characteristics of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} back to their initial settings.
697853`\spad{reshape(l,{}p)} \undocumented
697935`\spad{reshape(lt,{}ts)} organizes the list of components \spad{lt} into a tensor with the same shape as \spad{ts}.
698067`\spad{resize(v,{}w,{}h)} displays the two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} with a width of \spad{w} and a height of \spad{h},{} keeping the upper left-hand corner position unchanged.
698154`\spad{resize(v,{}w,{}h)} displays the three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} with a width of \spad{w} and a height of \spad{h},{} keeping the upper left-hand corner position unchanged.
698412`\axiom{rest(\spad{x})} return \axiom{\spad{x}} without the first entry or error if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true}.
698454`\spad{rest(x)} returns \spad{x} except the first letter.
698503`\spad{rest([l1]*[l2]*...[ln])} returns the list \spad{l2,{} .... ln}.
698571`\axiom{rest(\spad{ts})} returns the polynomials of \axiom{\spad{ts}} with smaller main variable than \axiom{mvar(\spad{ts})} if \axiom{\spad{ts}} is not empty,{} otherwise returns "failed"
698646`\spad{rest(u)} returns an aggregate consisting of all but the first element of \spad{u} (equivalently,{} the next node of \spad{u}).
698837`\spad{rest(u,{}n)} returns the \axiom{\spad{n}}th (\spad{n} \spad{>=} 0) node of \spad{u}. Note: \axiom{rest(\spad{u},{}0) = \spad{u}}.
698911`\axiom{result(a)} where \axiom{\spad{ls}} is the leaves list of \axiom{a} returns \axiom{[[value(\spad{s}),{}condition(\spad{s})]\$\spad{VT} for \spad{s} in \spad{ls}]} if the computations are terminated in \axiom{a} else an error is produced.
698985`\axiom{resultantEuclidean_naif(\spad{P},{}\spad{Q})} returns the extended resultant of \axiom{\spad{P}} and \axiom{\spad{Q}} computed by means of the naive algorithm.
699110`\axiom{resultantEuclidean(\spad{P},{}\spad{Q})} carries out the equality \axiom{coef1*P + coef2*Q = resultant(\spad{P},{}\spad{Q})}
699230`\axiom{resultantReduitEuclidean(\spad{P},{}\spad{Q})} returns the "reduce resultant" and carries out the equality \axiom{coef1*P + coef2*Q = resultantReduit(\spad{P},{}\spad{Q})}.
699378`\axiom{resultantReduit(\spad{P},{}\spad{Q})} returns the "reduce resultant" of \axiom{\spad{P}} and \axiom{\spad{Q}}.
699471`\axiom{resultantEuclidean_naif(\spad{P},{}\spad{Q})} returns the resultant of \axiom{\spad{P}} and \axiom{\spad{Q}} computed by means of the naive algorithm.
699547`\axiom{resultant(\spad{P},{} \spad{Q})} returns the resultant of \axiom{\spad{P}} and \axiom{\spad{Q}}
699618`\spad{resultant(p,{}q)} returns the resultant of the polynomials \spad{p} and \spad{q}.
699707`\axiom{resultant(a,{}\spad{b})} computes the resultant of \axiom{a} and \axiom{\spad{b}} where \axiom{a} and \axiom{\spad{b}} are assumed to have the same main variable \axiom{\spad{v}} and are viewed as univariate polynomials in \axiom{\spad{v}}.
699868`\spad{resultant(v1,{} v2,{} p)} computes resultant of \spad{v1} and \spad{v2} modulo \spad{p}.
699961`\spad{resultant(p,{}q,{}v)} returns the resultant of the polynomials \spad{p} and \spad{q} with respect to the variable \spad{v}.
701150`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
701241`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
701388`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
701482`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
701634`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
701751`\spad{retractIfCan(x)} \undocumented
701900`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
702002`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
702160`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
702264`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
702426`\axiom{retractIfCan(\spad{p})} tries to write a general expression as a linear function.
702603`\axiom{retractIfCan(\spad{p})} checks whether \axiom{\spad{p}} depends only on the independent variables. If yes,{} it is coerced.
702787`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
702893`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
703001`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
703117`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
703235`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
703341`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
703449`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
703541`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
703901`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
703995`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
708055`\spad{retractIfCan(a)} transforms a into an element of \% if possible. Returns "failed" if a cannot be made into an element of \%.
708126`\spad{retractIfCan(e)} takes \spad{e} and tries to transform it into a FortranExpression \indented{1}{checking that it is one of the given basic symbols} \indented{1}{or subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
708236`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
708342`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
708450`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
708566`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
708684`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
708790`\spad{retractIfCan(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
708898`\spad{retractIfCan(a)} tries change \spad{a} into an object of type \spad{S}. If it can,{} then such an object is returned. Otherwise,{} "failed" is returned.
708968`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
709150`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
709322`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
709523`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
709732`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
709959`\axiom{retractIfCan(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if all its variables belong to \axiom{\spad{V}}.
710159`\spad{retractIfCan(a)} transforms a into an element of \spad{S} if possible. Returns "failed" if a cannot be made into an element of \spad{S}.
710735`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
710806`\spad{retract(e)} takes \spad{e} and transforms it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
710933`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
711006`\spad{retract(e)} takes \spad{e} and transforms it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
711137`\spad{retract(e)} takes \spad{e} and transforms it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
711233`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
711314`\spad{retract(e)} takes \spad{e} and transforms it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
711451`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
711534`\spad{retract(e)} takes \spad{e} and transforms it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
711675`\axiom{retract(\spad{p})} is like \axiom{retractIfCan(\spad{p})} put yields a hard error,{} if \axiom{\spad{p}} contains further jet variables.
711831`\axiom{retract(\spad{p})} is like \axiom{retractIfCan(\spad{p})} put yields a hard error,{} if \axiom{\spad{p}} contains further jet variables.
711909`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
711994`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
712081`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
712176`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
712273`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
712358`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
712445`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
712516`\spad{retract(e)} takes \spad{e} and transforms it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
712813`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
712886`\spad{retract(e)} takes \spad{e} and transforms it into a \indented{1}{FortranExpression checking that it contains no non-Fortran} \indented{1}{functions,{} and that it only contains the given basic symbols} \indented{1}{and subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
715736`\spad{retract(a)} transforms a into an element of \% if possible. Error: if a cannot be made into an element of \%.
715786`\spad{retract(e)} takes \spad{e} and transforms it into a FortranExpression \indented{1}{checking that it is one of the given basic symbols} \indented{1}{or subscripted symbols which correspond to scalar and array} \indented{1}{parameters respectively.}
715875`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
715960`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
716047`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
716142`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
716239`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
716324`\spad{retract(e)} tries to convert \spad{e} into an ASP,{} checking that \indented{1}{legal Fortran-77 is produced.}
716411`\axiom{retract(de)} returns the system defining the differential equation \axiom{de}.
716478`\spad{retract(a)} tries to convert \spad{a} into an object of type \spad{S}. If possible,{} it returns the object. Error: if no such retraction is possible.
716527`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
716688`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
716839`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
717019`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
717207`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
717413`\axiom{retract(\spad{p})} returns \axiom{\spad{p}} as an element of the current domain if \axiom{retractIfCan(\spad{p})} does not return "failed",{} otherwise an error is produced.
717592`\spad{retract(a)} transforms a into an element of \spad{S} if possible. Error: if a cannot be made into an element of \spad{S}.
717640`\spad{retractable?(p)} tests if \spad{p} is a 0-form,{} \spadignore{i.e.} if degree(\spad{p}) = 0.
717699`\spad{retractable?(df)} tests if differential form \spad{df} is a 0-form,{} \spadignore{i.e.} if degree(\spad{df}) = 0.
717776`\axiom{retractable?(\spad{x})} tests if \axiom{\spad{x}} is a tree with only one entry.
717837`\axiom{retractable?(\spad{x})} tests if \axiom{\spad{x}} is a tree with only one entry.
717893`\spad{retractable?([l1]*[l2]*...[ln])} returns \spad{true} iff \spad{n} equals \spad{1}.
717975`\spad{retractable?(a)} tests if \spad{a} can be converted into an object of type \spad{S}.
718035`\spad{returnType!(t)} declares that the return type of he current subprogram in the global symbol table is \spad{t}.
718126`\spad{returnType!(f,{}t)} declares that the return type of subprogram \spad{f} in the global symbol table is \spad{t}.
718224`\spad{returnType!(f,{}t,{}tab)} declares that the return type of subprogram \spad{f} in symbol table \spad{tab} is \spad{t}.
718325`\spad{returnTypeOf(f,{}tab)} returns the type of the object returned by \spad{f}
718422`\spad{returns()} creates a representation of a FORTRAN RETURN statement.
718464`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
718532`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
718591`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
718652`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
718720`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
718786`\spad{returns(e)} creates a representation of a FORTRAN RETURN statement with a returned value.
718959`\spad{reverse!(l)} reverses the list of monomials forming \spad{l},{} destroying the element \spad{l}.
719014`\spad{reverse!(d)} destructively replaces \spad{d} by its reverse dequeue,{} \spadignore{i.e.} the top (front) element is now the bottom (back) element,{} and so on.
719067`\spad{reverse!(u)} returns \spad{u} with its elements in reverse order.
719149`\spad{reverseLex(v1,{}v2)} return \spad{true} if the vector \spad{v1} is less than the vector \spad{v2} in the ordering which is total degree refined by the reverse lexicographic ordering.
719283`\spad{reverse(p)} returns the reverse polynomial of \spad{p}.
719352`\spad{reverse(l)} reverses the list of monomials forming \spad{l}. This has some effect if the monoid is non-abelian,{} \spadignore{i.e.} \spad{reverse(a1\^e1 ... an\^en) = an\^en ... a1\^e1} which is different.
719406`\spad{reverse(a)} returns a copy of \axiom{a} with elements in reverse order.
719463`implements reverse for a single term by using: grade: 0 1 2 3... multi: 1 1 \spad{-1} \spad{-1}...
719521`\spad{revert(a)} computes the inverse of a power series \spad{a} with respect to composition. the series should have constant coefficient 0 and invertible first order coefficient.
719657`\spad{revert(f(x))} returns a Taylor series \spad{g(x)} such that \spad{f(g(x)) = g(f(x)) = x}. Series \spad{f(x)} should have constant coefficient 0 and invertible 1st order coefficient.
719824`\axiom{rewriteIdealWithHeadRemainder(\spad{lp},{}\spad{cs})} returns \axiom{\spad{lr}} such that the leading monomial of every polynomial in \axiom{\spad{lr}} is reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{cs}} and \axiom{(\spad{lp},{}\spad{cs})} and \axiom{(\spad{lr},{}\spad{cs})} generate the same ideal in \axiom{(\spad{R})^(\spad{-1}) \spad{P}
719824`--}.
719948`\axiom{rewriteIdealWithQuasiMonicGenerators(\spad{lp},{}redOp?,{}redOp)} returns \axiom{\spad{lq}} where \axiom{\spad{lq}} and \axiom{\spad{lp}} generate the same ideal in \axiom{\spad{R^}(\spad{-1}) \spad{P}} and \axiom{\spad{lq}} has rank not higher than the one of \axiom{\spad{lp}}. Moreover,{} \axiom{\spad{lq}} is computed by reducing \axiom{\spad{lp}} \spad{w}.\spad{
719948`--r}.\spad{t}. some basic set of the ideal generated by the quasi-monic polynomials in \axiom{\spad{lp}}.
720177`\axiom{rewriteIdealWithRemainder(\spad{lp},{}\spad{cs})} returns \axiom{\spad{lr}} such that every polynomial in \axiom{\spad{lr}} is fully reduced in the sense of Groebner bases \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{cs}} and \axiom{(\spad{lp},{}\spad{cs})} and \axiom{(\spad{lr},{}\spad{cs})} generate the same ideal in \axiom{(\spad{R})^(\spad{-1}) \spad{P}}.
720297`\axiom{rewriteSetByReducingWithParticularGenerators(\spad{lp},{}pred?,{}redOp?,{}redOp)} returns \axiom{\spad{lq}} where \axiom{\spad{lq}} is computed by the following algorithm. Chose a basic set \spad{w}.\spad{r}.\spad{t}. the reduction-test \axiom{redOp?} among the polynomials satisfying property \axiom{pred?},{} if it is empty then leave,{} else r
720297`--educe the other polynomials by this basic set \spad{w}.\spad{r}.\spad{t}. the reduction-operation \axiom{redOp}. Repeat while another basic set with smaller rank can be computed. See code. If \axiom{pred?} is \axiom{quasiMonic?} the ideal is unchanged.
720564`\axiom{rewriteSetWithReduction(\spad{lp},{}\spad{ts},{}redOp,{}redOp?)} returns a list \axiom{\spad{lq}} of polynomials such that \axiom{[reduce(\spad{p},{}\spad{ts},{}redOp,{}redOp?) for \spad{p} in \spad{lp}]} and \axiom{\spad{lp}} have the same zeros inside the regular zero set of \axiom{\spad{ts}}. Moreover,{} for every polynomial \axiom{\spad{q}} in \axiom{\spad{lq}} and every polynom
720564`--ial \axiom{\spad{t}} in \axiom{\spad{ts}} \axiom{redOp?(\spad{q},{}\spad{t})} holds and there exists a polynomial \axiom{\spad{p}} in the ideal generated by \axiom{\spad{lp}} and a product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}. The operation \axiom{redOp} must satisfy the following conditions. For every \axiom{\spad{p}} and \axiom{\spad{q}} we have \axiom{redOp?(redOp(\spad{p},{}\spad{q}),{}\spad
720564`--{q})} and there exists an integer \axiom{\spad{e}} and a polynomial \axiom{\spad{f}} such that \axiom{init(\spad{q})^e*p = \spad{f*q} + redOp(\spad{p},{}\spad{q})}.
720680`\spad{rhs(f)} returns the right side of \spad{f}
720724`\spad{rhs(r)} returns the right hand side of the rule \spad{r}.
720773`\spad{rhs(eqn)} returns the right hand side of equation \spad{eqn}.
720812`\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}.
720960`\spad{ricDsolve(op)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}.
721111`\spad{ricDsolve(op,{} ezfactor)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}. Argument \spad{ezfactor} is a factorisation in \spad{UP},{} not necessarily into irreducibles.
721278`\spad{ricDsolve(op,{} zeros)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}. \spad{zeros} is a zero finder in \spad{UP}.
721409`\spad{ricDsolve(op,{} ezfactor)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}. Argument \spad{ezfactor} is a factorisation in \spad{UP},{} not necessarily into irreducibles.
721579`\spad{ricDsolve(op,{} zeros)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}. \spad{zeros} is a zero finder in \spad{UP}.
721713`\spad{ricDsolve(op,{} zeros,{} ezfactor)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}. \spad{zeros} is a zero finder in \spad{UP}. Argument \spad{ezfactor} is a factorisation in \spad{UP},{} not necessarily into irreducibles.
721863`\spad{ricDsolve(op,{} zeros,{} ezfactor)} returns the rational solutions of the associated Riccati equation of \spad{op y = 0}. \spad{zeros} is a zero finder in \spad{UP}. Argument \spad{ezfactor} is a factorisation in \spad{UP},{} not necessarily into irreducibles.
722016`\spad{ridHack1(i,{}j,{}k,{}l)} \undocumented
722110`\spad{riemannZeta(z)} is the Riemann Zeta function
722175`\spad{riemannZeta(z)} is the Riemann Zeta function.
722235`\spad{rightAlternative?()} tests if \spad{2*associator(a,{}b,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
722318`\spad{rightAlternative?()} tests if \spad{2*associator(a,{}b,{}b) = 0} for all \spad{a},{} \spad{b} in the algebra. Note: we only can test this; in general we don\spad{'t} know whether \spad{2*a=0} implies \spad{a=0}.
722510`\spad{rightCharacteristicPolynomial(a)} returns the characteristic polynomial of the right regular representation of \spad{a} with respect to any basis.
722626`\spad{rightDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(rightTraceMatrix())}.
722699`\spad{rightDiscriminant()} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis. Note: the same as \spad{determinant(rightTraceMatrix())}.
722925`\spad{rightDiscriminant([v1,{}...,{}vn])} returns the determinant of the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj}. Note: the same as \spad{determinant(rightTraceMatrix([v1,{}...,{}vn]))}.
723009`\spad{rightDivide(a,{}b)} returns the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``right division\spad{''}.
723126`\spad{rightDivide(a,{} b,{} sigma)} returns the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. This process is called ``right division\spad{''}. \spad{\sigma} is the morphism to use.
723352`\spad{rightExactQuotient(a,{}b)} computes the value \spad{q},{} if it exists such that \spad{a = q*b}.
723570`\spad{rightExtendedGcd(a,{}b)} returns \spad{[c,{}d]} such that \spad{g = c * a + d * b = rightGcd(a,{} b)}.
723698`\spad{rightFactorCandidate(p,{}n)} \undocumented
723792`\spad{rightFactorIfCan(f,{}d,{}c)} returns a candidate to be the right factor (\spad{h} in \spad{f} = \spad{g} \spad{o} \spad{h}) of degree \spad{d} with leading coefficient \spad{c} of a functional decomposition of the polynomial \spad{f} or \spad{"failed"} if no such candidate.
723983`\spad{rightGcd(a,{}b)} computes the value \spad{g} of highest degree such that \indented{3}{\spad{a = aa*g}} \indented{3}{\spad{b = bb*g}} for some values \spad{aa} and \spad{bb}. The value \spad{g} is computed using right-division.
724133`\spad{rightLcm(a,{}b)} computes the value \spad{m} of lowest degree such that \spad{m = a*aa = b*bb} for some values \spad{aa} and \spad{bb}. The value \spad{m} is computed using left-division.
724322`\spad{rightMinimalPolynomial(a)} returns the polynomial determined by the smallest non-trivial linear combination of right powers of \spad{a}. Note: the polynomial never has a constant term as in general the algebra has no unit.
724452`\spad{rightMult(a,{} s)} returns \spad{a * s} where \spad{*} is the monoid operation,{} which is assumed non-commutative.
724574`\spad{rightNorm(a)} returns the determinant of the right regular representation of \spad{a}.
724642`\spad{rightOne(eq)} divides by the right hand side.
724715`\spad{rightOne(eq)} divides by the right hand side.
724967`\spad{rightPower(a,{}n)} returns the \spad{n}\spad{-}th right power of \spad{a},{} \spadignore{i.e.} \spad{rightPower(a,{}n) := rightPower(a,{}n-1) * a} and \spad{rightPower(a,{}0) := 1}.
725035`\spad{rightPower(a,{}n)} returns the \spad{n}\spad{-}th right power of \spad{a},{} \spadignore{i.e.} \spad{rightPower(a,{}n) := rightPower(a,{}n-1) * a} and \spad{rightPower(a,{}1) := a}.
725163`\spad{rightQuotient(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{q} is returned.
725252`\spad{rightRankPolynomial()} returns the right minimimal polynomial of the generic element
725407`\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
725522`\spad{rightRankPolynomial()} calculates the right minimal polynomial of the generic element in the algebra,{} defined by the same structural constants over the polynomial ring in symbolic coefficients with respect to the fixed basis.
725646`\spad{rightRank(x)} determines the number of linearly independent elements in \spad{b1*x},{}...,{}\spad{bn*x},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
725796`\spad{rightRecip(a)} returns an element,{} which is a right inverse of \spad{a},{} or \spad{"failed"} if there is no unit element,{} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
725903`\spad{rightRecip(a)} returns an element,{} which is a right inverse of \spad{a},{} or \spad{"failed"} if such an element doesn\spad{'t} exist or cannot be determined (see unitsKnown).
726053`\spad{rightRegularRepresentation(a)} returns the matrix of the linear map defined by right multiplication by \spad{a} with respect to the fixed \spad{R}-module basis.
726336`\spad{rightRegularRepresentation(a,{}[v1,{}...,{}vn])} returns the matrix \spad{m} of the linear map defined by right multiplication by \spad{a} with respect to the \spad{R}-module basis \spad{b = [v1,{}...,{}vn]}. That is for all \spad{x} \spad{coordinates(x*a,{} b) = m*coordinates(x,{} b)}.
726512`\spad{rightRemainder(a,{}b)} computes the pair \spad{[q,{}r]} such that \spad{a = q*b + r} and the degree of \spad{r} is less than the degree of \spad{b}. The value \spad{r} is returned.
726602`\spad{rightScalarTimes!(c,{}a,{}r)} computes the scalar product \spad{a * r} and stores the result in the matrix \spad{c}. Error: if \spad{a} and \spad{c} do not have the same dimensions.
726706`\spad{rightTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
726786`\spad{rightTraceMatrix()} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj},{} where \spad{v1},{}...,{}\spad{vn} are the elements of the fixed \spad{R}-module basis.
727033`\spad{rightTraceMatrix([v1,{}...,{}vn])} is the \spad{n}-by-\spad{n} matrix whose element at the \spad{i}\spad{-}th row and \spad{j}\spad{-}th column is given by the right trace of the product \spad{vi*vj}.
727189`\spad{rightTrace(a)} returns the trace of the right regular representation of \spad{a}.
727258`\spad{rightTrim(s,{}c)} returns \spad{s} with all trailing occurrences of \spad{c} deleted. For example,{} \axiom{rightTrim(" abc ",{} char " ")} returns \axiom{" abc"}.
727318`\spad{rightTrim(s,{}cc)} returns \spad{s} with all trailing occurences of characters in \spad{cc} deleted. For example,{} \axiom{rightTrim("(abc)",{} charClass "()")} returns \axiom{"(abc"}.
727458`\spad{rightUnit()} returns a right unit of the algebra (not necessarily unique),{} or \spad{"failed"} if there is none.
727671`\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \spad{"failed"} if there is none
727797`\spad{rightUnits()} returns the affine space of all right units of the algebra,{} or \spad{"failed"} if there is none.
727936`\spad{rightZero(eq)} subtracts the right hand side.
728001`\axiom{right(\spad{x})} returns right subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{LyndonWord}(\axiom{\spad{x}}) is \spad{true}.
728050`\axiom{right(\spad{x})} returns right subtree of \axiom{\spad{x}} or error if \axiomOpFrom{retractable?}{Magma}(\axiom{\spad{x}}) is \spad{true}.
728094`\spad{right(f)} right-justifies form \spad{f} in total space.
728135`\axiom{right(rootChar)} is the right bound of the isolating interval
728229`\spad{right(a)} returns the right child.
728287`\spad{right(f,{}n)} right-justifies form \spad{f} within space of width \spad{n}.
728336`\spad{rischDE(n,{} f,{} g,{} x,{} lim,{} ext)} returns \spad{[y,{} h,{} b]} such that \spad{dy/dx + n df/dx y = h} and \spad{b := h = g}. The equation \spad{dy/dx + n df/dx y = g} has no solution if \spad{h \~~= g} (\spad{y} is a partial solution in that case). Not
728336`--es: \spad{lim} is a limited integration function,{} and ext is an extended integration function.
728579`\spad{rischDEsys(n,{} f,{} g_1,{} g_2,{} x,{}lim,{}ext)} returns \spad{y_1.y_2} such that \spad{(dy1/dx,{}dy2/dx) + ((0,{} - n df/dx),{}(n df/dx,{}0)) (y1,{}y2) = (g1,{}g2)} if \spad{y_1,{}y_2} exist,{} "failed" otherwise. \spad{lim} is a limited integration functi
728579`--on,{} \spad{ext} is an extended integration function.
728822`\spad{rischNormalize(f,{} lx)} returns \spad{[g,{} [k1,{}...,{}kn],{} [h1,{}...,{}hn]]} such that \spad{g = normalize(f,{} lx)} and each \spad{\spad{ki}} was rewritten as \spad{\spad{hi}} during the normalization.
728959`\spad{rischNormalize(f,{} x)} returns \spad{[g,{} [k1,{}...,{}kn],{} [h1,{}...,{}hn]]} such that \spad{g = normalize(f,{} x)} and each \spad{\spad{ki}} was rewritten as \spad{\spad{hi}} during the normalization.
729090`\spad{rk4(y,{}n,{}x1,{}h,{}derivs)} uses a 4-th order Runge-Kutta method to numerically integrate the ordinary differential equation {\em dy/dx = f(y,{}x)} of \spad{n} variables,{} where \spad{y} is an \spad{n}-vector. Argument \spad{y} is a vector of initial conditions of length \spad{n} which upon exit contains the solution at \spad{x1 + h},{} \spad{n} is the nu
729090`--mber of dependent variables,{} \spad{x1} is the initial point,{} \spad{h} is the step size,{} and \spad{derivs} is a function which computes the right hand side of the ordinary differential equation. For details,{} see \spadtype{NumericalOrdinaryDifferentialEquations}.
729232`\spad{rk4(y,{}n,{}x1,{}h,{}derivs,{}t1,{}t2,{}t3,{}t4)} is the same as \spad{rk4(y,{}n,{}x1,{}h,{}derivs)} except that you must provide 4 scratch arrays \spad{t1}-\spad{t4} of size \spad{n}. For details,{} see \con{NumericalOrdinaryDifferentialEquations}.
729430`\spad{rk4a(y,{}n,{}x1,{}x2,{}eps,{}h,{}ns,{}derivs)} is a driver function for the numerical integration of an ordinary differential equation {\em dy/dx = f(y,{}x)} of \spad{n} variables,{} where \spad{y} is an \spad{n}-vector using a 4-th order Runge-Kutta method. For details,{} see \con{NumericalOrdinaryDifferentialEquations}.
729593`\spad{rk4f(y,{}n,{}x1,{}x2,{}ns,{}derivs)} uses a 4-th order Runge-Kutta method to numerically integrate the ordinary differential equation {\em dy/dx = f(y,{}x)} of \spad{n} variables,{} where \spad{y} is an \spad{n}-vector. Starting with \spad{y} at \spad{x1},{} this function uses \spad{ns} fixed steps of a 4-th order Runge-Kutta integrator to advance t
729593`--he solution vector to \spad{x2} and return the values in \spad{y}. For details,{} see \con{NumericalOrdinaryDifferentialEquations}.
729744`\spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs,{}t1,{}t2,{}t3,{}t4,{}t5,{}t6,{}t7)} is a subfunction for the numerical integration of an ordinary differential equation {\em dy/dx = f(y,{}x)} of \spad{n} 
729744`--variables,{} where \spad{y} is an \spad{n}-vector using a 4-th order Runge-Kutta method. This function takes a 5-th order Runge-Kutta \spad{step} with monitoring of local truncation to ensure accuracy and adjust stepsize. For details,{} see \con{NumericalOrdinaryDifferentialEquations}.
730043`\spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs)} is a subfunction for the numerical integration of an ordinary differential equation {\em dy/dx = f(y,{}x)} of \spad{n} variables,{} where \spad{y} is an \spad{n}-vector using a 4-th order Runge-Kutta method. This function takes a 5-th order Runge-Kutta 
730043`--\spad{step} with monitoring of local truncation to ensure accuracy and adjust stepsize. For details,{} see \con{NumericalOrdinaryDifferentialEquations}.
730243`\spad{rmap(f,{} e)} rewrites \spad{e} replacing each kernel \spad{k} in \spad{e} by \spad{f}(\spad{k})
730325`\spad{roman(n)} creates a roman numeral for \spad{n}.
730373`\spad{roman(n)} creates a roman numeral for symbol \spad{n}.
730420`\spad{romberg(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax)} uses the romberg method to numerically integrate function \spadvar{\spad{fn}} over the closed interval \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary from \spad{nmin} to \sp
730420`--ad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
730586`\spad{rombergo(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax)} uses the romberg method to numerically integrate function \spad{fn} over the open interval from \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary from \spad{nmin} to \spad{nm
730586`--ax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
730753`\spad{root?(x)} \undocumented
730802`\spad{rootBound(p)} returns a bound on the largest norm of the complex roots of \spad{p}.
730883`\spad{rootFactor(f)} transforms every radical of the form \spad{(a1*...*am)^(1/n)} appearing in \spad{f} into \spad{a^(1/n)*...*am^(1/n)}. This transformation is not in general valid for all complex numbers \spad{a} and \spad{b}.
731070`\spad{rootKerSimp(op,{}f,{}n)} should be local but conditional.
731258`\spad{rootNormalize(f,{} k)} returns \spad{f} rewriting either \spad{k} which must be an \spad{n}th-root in terms of radicals already in \spad{f},{} or some radicals in \spad{f} in terms of \spad{k}.
731344`\spad{rootOfIrreduciblePoly(f)} computes one root of the monic,{} irreducible polynomial \spad{f},{} which degree must divide the extension degree of {\em F} over {\em GF},{} \spadignore{i.e.} \spad{f} splits into linear factors over {\em F}.
731823`\spad{rootOf(p)} returns \spad{y} such that \spad{p(y) = 0}. Error: if \spad{p} has more than one variable \spad{y}.
731891`\spad{rootOf(p)} returns \spad{y} such that \spad{p(y) = 0}.
731975`\spad{rootOf(p)} returns \spad{y} such that \spad{p(y) = 0}. Error: if \spad{p} has more than one variable \spad{y}.
732111`\spad{rootOf(p,{} y)} returns \spad{y} such that \spad{p(y) = 0}. The object returned displays as \spad{'y}.
732600`\axiom{rootOf(pol,{}\spad{n})} creates the \spad{n}th root for the order of \axiom{pol} and gives it unique name
732707`\spad{rootOf(p,{} y)} returns \spad{y} such that \spad{p(y) = 0}. The object returned displays as \spad{'y}.
732798`\axiom{rootOf(pol,{}\spad{n})} gives the \spad{n}th root for the order of the Real Closure
732917`\spad{rootOf(p,{}y)} returns \spad{y} such that \spad{p(y) = 0}. The object returned displays as \spad{'y}.
733105`\axiom{rootOf(pol,{}\spad{n},{}name)} creates the \spad{n}th root for the order of \axiom{pol} and names it \axiom{name}
733223`\spad{rootPoly(g,{} n)} returns \spad{[m,{} c,{} P]} such that \spad{c * g ^ (1/n) = P ^ (1/m)} thus if \spad{y^n = g},{} then \spad{z^m = P} where \spad{z = c * y}.
733376`\spad{rootPower(f)} transforms every radical power of the form \spad{(a^(1/n))^m} into a simpler form if \spad{m} and \spad{n} have a common factor.
733529`\spad{rootProduct(f)} combines every product of the form \spad{(a^(1/n))^m * (a^(1/s))^t} into a single power of a root of \spad{a},{} and transforms every radical power of the form \spad{(a^(1/n))^m} into a simpler form.
733684`\spad{rootRadius(p)} calculates the root radius of \spad{p} with a maximal error quotient of {\em 1+globalEps},{} where {\em globalEps} is the internal error bound,{} which can be set by {\em setErrorBound}.
733750`\spad{rootRadius(p,{}errQuot)} calculates the root radius of \spad{p} with a maximal error quotient of {\em errQuot}.
733818`\spad{rootSimp(f)} transforms every radical of the form \spad{(a * b^(q*n+r))^(1/n)} appearing in \spad{f} into \spad{b^q * (a * b^r)^(1/n)}. This transformation is not in general valid for all complex numbers \spad{b}.
733970`\spad{rootSplit(f)} transforms every radical of the form \spad{(a/b)^(1/n)} appearing in \spad{f} into \spad{a^(1/n) / b^(1/n)}. This transformation is not in general valid for all complex numbers \spad{a} and \spad{b}.
734030`\spad{root(f)} creates a form for the square root of form \spad{f}.
734070`\spad{root(f,{}n)} creates a form for the \spad{n}th root of form \spad{f}.
734113`\spad{root(f,{}a)} returns a root of the polynomial \spad{f}. Argument \spad{a} must be a root of \spad{f} \spad{(mod p)}.
734612`\spad{rootsOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable \spad{y}.
734687`\spad{rootsOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values.
734778`\spad{rootsOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}; Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable \spad{y}.
735126`\spad{rootsOf(p,{} z)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}; The returned roots contain new symbols \spad{'\%z0},{} \spad{'\%z1} ...; Note: the new symbols are bound in the interpreter to the respective values.
735224`\spad{rootsOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}; The returned roots contain new symbols \spad{'\%z0},{} \spad{'\%z1} ...; Note: the new symbols are bound in the interpreter to the respective values.
735305`\spad{rotate! q} rotates queue \spad{q} so that the element at the front of the queue goes to the back of the queue. Note: rotate! \spad{q} is equivalent to enqueue!(dequeue!(\spad{q})).
735355`\spad{rotate(v,{}th,{}phi)} rotates the graph to the longitudinal view angle \spad{th} radians and the latitudinal view angle \spad{phi} radians for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}.
735425`\spad{rotate(v,{}th,{}phi)} rotates the graph to the longitudinal view angle \spad{th} degrees and the latitudinal view angle \spad{phi} degrees for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}. The new rotation position is not displayed until the function \spadfun{makeViewport3D} is executed again for \spad{v}.
735499`\spad{rotatex(r)} returns a dhmatrix for rotation about axis \spad{X} for \spad{r} degrees
735557`\spad{rotatey(r)} returns a dhmatrix for rotation about axis \spad{Y} for \spad{r} degrees
735615`\spad{rotatez(r)} returns a dhmatrix for rotation about axis \spad{Z} for \spad{r} degrees
735745`\axiom{roughBase?(\spad{ps})} returns \spad{true} iff for every pair \axiom{{\spad{p},{}\spad{q}}} of polynomials in \axiom{\spad{ps}} their leading monomials are relatively prime.
735842`\axiom{roughBasicSet(\spad{lp})} returns the smallest (with Ritt-Wu ordering) triangular set contained in \axiom{\spad{lp}}.
736072`\axiom{roughEqualIdeals?(\spad{ps1},{}\spad{ps2})} returns \spad{true} iff it can proved that \axiom{\spad{ps1}} and \axiom{\spad{ps2}} generate the same ideal in \axiom{(\spad{R})^(\spad{-1}) \spad{P}} without computing Groebner bases.
736257`\axiom{roughSubIdeal?(\spad{ps1},{}\spad{ps2})} returns \spad{true} iff it can proved that all polynomials in \axiom{\spad{ps1}} lie in the ideal generated by \axiom{\spad{ps2}} in \axiom{\axiom{(\spad{R})^(\spad{-1}) \spad{P}}} without computing Groebner bases.
736438`\axiom{roughUnitIdeal?(\spad{ps})} returns \spad{true} iff \axiom{\spad{ps}} contains some non null element lying in the base ring \axiom{\spad{R}}.
736583`\spad{round x} computes the integer closest to \spad{x}.
736630`the cumulative weight of a given route. The route is given in terms of a sequence of arrow indexes.
736821`\spad{routeArrows(s,{} a,{} b)} gives the shortest route between nodes 'a' and \spad{'b'} as a sequence of arrow indexes. [] if 'a' = \spad{'b'} [0] if it is not possible to go from 'a' to \spad{'b'}
736932`the cumulative weight of a given route. The route is given in terms of a sequence of node indexes.
737121`\spad{routeNodes(s,{} a,{} b)} gives the shortest route between nodes 'a' and \spad{'b'} as a sequence of node indexes. [a] if 'a' = \spad{'b'} [] if it is not possible to go from 'a' to \spad{'b'}
737231`\spad{rowEchLocal(m,{}p)} computes a modular row-echelon form of \spad{m},{} finding an appropriate modulus over a local ring where \spad{p} is the only prime.
737315`\spad{rowEch(m)} computes a modular row-echelon form of \spad{m},{} finding an appropriate modulus.
737392`\spad{rowEchelonLocal(m,{} d,{} p)} computes the row-echelon form of \spad{m} concatenated with \spad{d} times the identity matrix over a local ring where \spad{p} is the only prime.
737482`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}. the result will have entries in the quotient field.
737579`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
737659`\axiom{primitiveRowEchelon(A)} computes a row echelon form for the matrix \axiom{A}. The algorithm used is fraction-free elimination. It is especially adapted to matrices already close to row echelon form. The transformation matrix,{} the used pivots and the rank of the matrix are also returned.
737783`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
737880`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
737963`\spad{rowEchelon(m)} returns the row echelon form of the matrix \spad{m}.
738061`\spad{rowEchelon(m,{} d)} computes a modular row-echelon form mod \spad{d} of \indented{3}{[\spad{d}\space{5}]} \indented{3}{[\space{2}\spad{d}\space{3}]} \indented{3}{[\space{4}. ]} \indented{3}{[\space{5}\spad{d}]} \indented{3}{[\space{3}\spad{M}\space{2}]} where \spad{M = m mod d}.
738216`\axiom{row(A,{} \spad{i})} returns the \axiom{\spad{i}}\spad{-}th row of the matrix \axiom{A}.
738314`\spad{row(m,{}i)} returns the \spad{i}th row of the matrix \spad{m}. Error: if the index is outside the proper range.
738392`\spad{row(m,{}i)} returns the \spad{i}th row of \spad{m} error check to determine if index is in proper ranges
738468`\spad{rquo(x,{} s)} returns the exact right quotient of \spad{x} by \spad{s}.
738536`\spad{rquo(x,{} y)} returns the exact right quotient of \spad{x} by \spad{y} \spadignore{i.e.} \spad{q} such that \spad{x = q * y},{} "failed" if \spad{x} is not of the form \spad{q * y}.
738598`\spad{rquo(x,{} y)} returns the exact right quotient of \spad{x} by \spad{y} that is \spad{q} such that \spad{x = q * y},{} "failed" if \spad{x} is not of the form \spad{q * y}.
738667`\axiom{rquo(\spad{x},{}\spad{y})} returns the right simplification of \axiom{\spad{x}} by \axiom{\spad{y}}.
738780`\spad{rquo(x,{}w)} returns the right simplification of \spad{x} by \spad{w}.
738850`\spad{rquo(x,{}y)} returns the right simplification of \spad{x} by \spad{y}.
738901`\spad{rquo(x,{}v)} returns the right simplification of \spad{x} by the variable \spad{v}.
738952`\spad{rroot(f,{} n)} returns \spad{[m,{}c,{}r]} such that \spad{f^(1/n) = c * r^(1/m)}.
739078`\spad{rspace(n,{}m)} creates rectangular white space,{} \spad{n} wide by \spad{m} high.
739133`\spad{rst(s)} returns a pointer to the next node of stream \spad{s}. Caution: this function should only be called after a \spad{empty?} test has been made since there no error check.
739184`\spad{rubiksGroup constructs} the permutation group representing Rubic\spad{'s} Cube acting on integers {\em 10*i+j} for {\em 1 <= i <= 6},{} {\em 1 <= j <= 8}. The faces of Rubik\spad{'s} Cube are labelled in the obvious way Front,{} Right,{} Up,{} Down,{} Left,{} Back and numbered from 1 to 6 in this given ordering,{} the pieces on each face (except the unmoveable center piece) are clockwise numbered from 1 to 8 starting
739184`-- 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 {\em ij} where \spad{i} is the numer of theface (1 to 6) and \spad{j} is the number of the piece on this face. The remaining ambiguities are resolved by looking at the 6 generators,{} which represent a 90 degree turns of the faces,{} or from the following pictorial description. Permutation group representing Rubic\spad{'s} Cube acting on integers
739184`-- 10*i+j for 1 \spad{<=} \spad{i} \spad{<=} 6,{} 1 \spad{<=} \spad{j} \spad{<=8}. \blankline\begin{verbatim}Rubik's Cube:   +
739184`--
739184`---+ +
739184`-- B   where: marks Side # :               / U   /|/              /     / |         F(ront)    <->    1      L 
739184`-->  +
739184`--
739184`---+ R|         R(ight)    <->    2             |     |  +         U(p)       <->    3             |  F  | /          D(own)     <->    4             |     |/           L(eft)     <->    5             +
739184`--
739184`---+            B(ack)     <->    
739184`--6                ^                |                DThe Cube's surface:                               The pieces on each side            +
739184`---+              (except the unmoveable center            |567|              piece) are clockwise numbered            |4U8|              from 1 to 8 starting with the            |321|              piece in the upper left        +
739184`---+
739184`---+
739184`---+          corner (see figure on the        |781|123|345|          left).  The moves of the cube        |
739184`--6L2|8F4|2R6|          are represented as        |543|765|187|          permutations on these pieces.        +
739184`---+
739184`---+
739184`---+          Each of the pieces is            |123|              represented as a two digit            |8D4|              integer ij where i is the            |765|              # of the side ( 1 to 6 for            +
739184`---+              F to B (see table above ))            |567|              and j is the # of the piece.            |4B8|            |321|            +
739184`--
739184`---+\end{verbatim}
739266`\spad{rule(f,{} g)} creates the rewrite rule: \spad{f == eval(g,{} g is f)},{} with left-hand side \spad{f} and right-hand side \spad{g}.
739318`\spad{rule(f,{} g,{} [f1,{}...,{}fn])} creates the rewrite rule \spad{f == eval(eval(g,{} g is f),{} [f1,{}...,{}fn])},{} that is a rule with left-hand side \spad{f} and right-hand side \spad{g}; The symbols \spad{f1},{}...,{}\spad{fn} are the operators that are considered quoted,{} that is they are not evaluated during any rewrite,{} but just applied formally to their arguments.
739383`\spad{rules(r)} returns the rules contained in \spad{r}.
739456`\spad{ruleset([r1,{}...,{}rn])} creates the rule set \spad{{r1,{}...,{}rn}}.
739531`\spad{rur(lp)} returns the same as \spad{rur(lp,{}true)}
739708`\spad{rur(lp,{}univ?)} returns a rational univariate representation of \spad{lp}. This assumes that \spad{lp} defines a regular triangular \spad{ts} whose associated variety is zero-dimensional over \spad{R}. \spad{rur(lp,{}univ?)} returns a list of items \spad{[u,{}lc]} where \spad{u} is an irreducible univariate polynom
739708`--ial and each \spad{c} in \spad{lc} involves two variables: one from \spad{ls},{} called the coordinate of \spad{c},{} and an extra variable which represents any root of \spad{u}. Every root of \spad{u} leads to a tuple of values for the coordinates of \spad{lc}. Moreover,{} a point \spad{x} belongs to the variety associated with \spad{lp} iff there exists an item \spad{[u,{}lc]} in \spad{rur(lp,{}univ?)} and a root \spad{r} of \spad{u} such that \spad{x} is given by the tuple of values for the
739708`-- coordinates of \spad{lc} evaluated at \spad{r}. If \spad{univ?} is \spad{true} then each polynomial \spad{c} will have a constant leading coefficient \spad{w}.\spad{r}.\spad{t}. its coordinate. See the example which illustrates the \spadtype{ZeroDimensionalSolvePackage} package constructor.
739893`\spad{rur(ts,{}univ?)} returns a rational univariate representation of \spad{ts}. This assumes that the lowest polynomial in \spad{ts} is a variable \spad{v} which does not occur in the other polynomials of \spad{ts}. This variable will be used to define the simple algebraic extension over which these other polynomials will be rewritten as univariate polynomials with degree one. If \spad{univ?} is \spad
739893`--{true} then these polynomials will have a constant initial.
739995`\spad{rur(lp,{}univ?,{}check?)} returns the same as \spad{rur(lp,{}true)}. Moreover,{} if \spad{check?} is \spad{true} then the result is checked.
740188`\spad{sPol }\undocumented
740330`sPol
740465`\spad{safeCeiling(x)} returns the integer which is greater than any integer with the same floating point number representation.
740555`\spad{safeFloor(x)} returns the integer which is lower or equal to the largest integer which has the same floating point number representation.
740643`\spad{safetyMargin()} returns the number of low weight digits we do not trust in the floating point representation (used by \spadfun{safeCeiling}).
740744`\spad{safetyMargin(n)} sets to \spad{n} the number of low weight digits we do not trust in the floating point representation and returns the previous value (for use by \spadfun{safeCeiling}).
740863`\spad{safety returns} the specified safety or 1 as default.
740947`\spad{safety(d)} specifies the number of values reserved for testing any solutions found. This option is expressed in the form \spad{safety == d}.
741115`\spad{sample yields} a value of type \%
741158`\spad{sample yields} a value of type \%
741197`\spad{sample yields} a value of type \%
741233`\spad{sample()} returns an object of type \%.
741291`\spad{sample()} returns a sample of \%
741327`\spad{satisfy?(v,{} p)} returns \spad{f}(\spad{v}) where \spad{f} is the predicate attached to \spad{p}.
741398`\spad{satisfy?([v1,{}...,{}vn],{} p)} returns \spad{f(v1,{}...,{}vn)} where \spad{f} is the top-level predicate attached to \spad{p}.
741475`\spad{satisfy?(r,{} p)} returns \spad{true} if the matches satisfy the top-level predicate of \spad{p},{} \spad{false} if they don\spad{'t},{} and "failed" if not enough variables of \spad{p} are matched in \spad{r} to decide.
741564`\spad{saturate(I,{}f)} is the saturation of the ideal \spad{I} with respect to the multiplicative set generated by the polynomial \spad{f}.
741641`\spad{saturate(I,{}f,{}lvar)} is the saturation with respect to the prime principal ideal which is generated by \spad{f} in the polynomial ring \spad{F[lvar]}.
741731`\spad{save()} creates a representation of a SAVE statement.
741770`\spad{sayLength(l)} returns the length of a list of strings \spad{l} as an integer.
741834`\spad{sayLength(s)} returns the length of a string \spad{s} as an integer.
741892`\spad{say(l)} sends a list of strings \spad{l} to output.
741947`\spad{say(s)} sends a string \spad{s} to output.
741996`\spad{scalarMatrix(r)} returns an \spad{n}-by-\spad{n} matrix with \spad{r}\spad{'s} on the diagonal and zeroes elsewhere.
742146`\spad{scalarMatrix(n,{}r)} returns an \spad{n}-by-\spad{n} matrix with \spad{r}\spad{'s} on the diagonal and zeroes elsewhere.
742227`\spad{scalarTypeOf(t)} returns the FORTRAN data type of \spad{t}
742314`\spad{scaleRoots(p,{}c)} returns the polynomial which has \spad{c} times the roots of \spad{p}.
742388`\spad{scale(k)} returns \spad{matrix [[k,{}0],{}[0,{}1]]} representing the map \spad{x -> k * x}.
742437`\spad{scale(m,{}h)} returns \spad{scale(h) * m} (see \spadfunFrom{shift}{MoebiusTransform}).
742489`\spad{scale(sx,{}sy,{}sz)} returns a dhmatrix for scaling in the \spad{X},{} \spad{Y} and \spad{Z} directions
742549`\spad{scale(v,{}n,{}sx,{}sy)} displays the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} scaled by the factor \spad{sx} in the \spad{x}-coordinate direction and by the factor \spad{sy} in the \spad{y}-coordinate direction.
742632`\spad{scanOneDimSubspaces(basis,{}n)} gives a canonical representative of the {\em n}\spad{-}th one-dimensional subspace of the vector space generated by the elements of {\em basis},{} all from {\em R^n}. The coefficients of the representative are of shape {\em (0,{}...,{}0,{}1,{}*,{}...,{}*)},{} {\em *} in \spad{R}. If the size of \spad{R} is \spad{q},{} then there are {\em (
742632`--q^n-1)/(q-1)} of them. We first reduce \spad{n} modulo this number,{} then find the largest \spad{i} such that {\em +/[q^i for i in 0..i-1] <= n}. Subtracting this sum of powers from \spad{n} results in an \spad{i}-digit number to \spad{basis} \spad{q}. This fills the positions of the stars.
742761`\spad{scan(func,{}vec,{}ident)} creates a new vector whose elements are the result of applying reduce to the binary function \spad{func},{} increasing initial subsequences of the vector \spad{vec},{} and the element \spad{ident}.
742870`\spad{scan(fn,{}u,{}ident)} successively uses the binary function \spad{fn} to reduce more and more of list \spad{u}. \spad{ident} is returned if the \spad{u} is empty. The result is a list of the reductions at each step. See \spadfun{reduce} for more information. Examples: \spad{scan(fn,{}[1,{}2],{}0) = [fn(2,{}fn(1,{}0)),{}fn(1,{}0)]} and \spad{scan(*,{}[2,{}3],{}1) = [2 * 1,{} 3 * (2 * 1)]}.
742940`\spad{scan(f,{}a,{}r)} successively applies \spad{reduce(f,{}x,{}r)} to more and more leading sub-arrays \spad{x} of one-dimensional array \spad{a}. More precisely,{} if \spad{a} is \spad{[a1,{}a2,{}...]},{} then \spad{scan(f,{}a,{}r)} returns \spad{[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.
743055`\spad{scan(f,{}a,{}r)} successively applies \spad{reduce(f,{}x,{}r)} to more and more leading sub-arrays \spad{x} of primitive array \spad{a}. More precisely,{} if \spad{a} is \spad{[a1,{}a2,{}...]},{} then \spad{scan(f,{}a,{}r)} returns \spad{[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.
743155`\spad{scan(func,{}vec,{}ident)} creates a new vector whose elements are the result of applying reduce to the binary function \spad{func},{} increasing initial subsequences of the vector \spad{vec},{} and the element \spad{ident}.
743231`\spad{scan(f,{}a,{}r)} successively applies \spad{reduce(f,{}x,{}r)} to more and more leading sub-aggregates \spad{x} of aggregrate \spad{a}. More precisely,{} if \spad{a} is \spad{[a1,{}a2,{}...]},{} then \spad{scan(f,{}a,{}r)} returns \spad{[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.
743310`\spad{scan(f,{}a,{}r)} successively applies \spad{reduce(f,{}x,{}r)} to more and more leading sub-aggregates \spad{x} of aggregate \spad{a}. More precisely,{} if \spad{a} is \spad{[a1,{}a2,{}...]},{} then \spad{scan(f,{}a,{}r)} returns \spad {[reduce(f,{}[a1],{}r),{}reduce(f,{}[a1,{}a2],{}r),{}...]}.
743386`\spad{scan(b,{}h,{}[x0,{}x1,{}x2,{}...])} returns \spad{[y0,{}y1,{}y2,{}...]},{} where \spad{y0 = h(x0,{}b)},{} \spad{y1 = h(x1,{}y0)},{}\spad{...} \spad{yn = h(xn,{}y(n-1))}.
743462`\axiom{schema(\spad{P},{}\spad{Q})} returns the list of degrees of non zero subresultants of \axiom{\spad{P}} and \axiom{\spad{Q}}.
743554`\spad{schwerpunkt(p)} determines the 'Schwerpunkt' of the roots of the polynomial \spad{p} of degree \spad{n},{} \spadignore{i.e.} the center of gravity,{} which is {\em coefficient of \spad{x^(n-1)}} divided by {\em n times coefficient of \spad{x^n}}.
743630`\spad{screenCoordX(pt)} returns the \spad{'x'}(left to right) screen coordinate.
743691`\spad{screenCoordY(pt)} returns the \spad{'y'}(bottom to top) screen coordinate.
743752`\spad{screenCoordZ(pt)} returns the \spad{'z'}(out from screen) screen coordinate.
743813`\spad{screenCoords(pt)} returns screen coordinates as a list.
743880`\spad{screenResolution3D()} returns the screen resolution for a 3d graph.
743933`\spad{screenResolution()} returns the screen resolution
743982`\spad{screenResolution()} returns the screen resolution \spad{n}.
744043`\spad{screenResolution(n)} sets the screen resolution to \spad{n}.
744111`\spad{script(s,{} [a,{}b,{}c,{}d,{}e])} returns \spad{s} with subscripts a,{} superscripts \spad{b},{} pre-superscripts \spad{c},{} pre-subscripts \spad{d},{} and argument-scripts \spad{e}. Omitted components are taken to be empty. For example,{} \spad{script(s,{} [a,{}b,{}c])} is equivalent to \spad{script(s,{}[a,{}b,{}c,{}[],{}[]])}.
744172`\spad{script(s,{} [a,{}b,{}c,{}d,{}e])} returns \spad{s} with subscripts a,{} superscripts \spad{b},{} pre-superscripts \spad{c},{} pre-subscripts \spad{d},{} and argument-scripts \spad{e}.
744330`\spad{scripted?(s)} is \spad{true} if \spad{s} has been given any scripts.
744376`\spad{scripts(s)} returns all the scripts of \spad{s}.
744532`\spad{scripts(f,{} [sub,{} super,{} presuper,{} presub])} \indented{1}{creates a form for \spad{f} with scripts on all 4 corners.}
744584`\spad{se2rfi(l)} converts \spad{l} to target domain
744694`\spad{search(k,{}t)} searches the table \spad{t} for the key \spad{k},{} returning the entry stored in \spad{t} for key \spad{k}. If \spad{t} has no such key,{} \axiom{search(\spad{k},{}\spad{t})} returns "failed".
744775`\spad{sec2cos(f)} converts every \spad{sec(u)} appearing in \spad{f} into \spad{1/cos(u)}.
744838`\spad{secIfCan(z)} returns sec(\spad{z}) if possible,{} and "failed" otherwise.
744919`\spad{sec(x)} applies the secant operator to \spad{x}
745083`\spad{sec(st)} computes secant of a power series \spad{st}.
745167`\spad{sec(st)} computes secant of a power series \spad{st}.
745265`\spad{sec(z)} returns the secant of Laurent series \spad{z}.
745352`\spad{sec(z)} returns the secant of a Puiseux series \spad{z}.
745448`\spad{sec(x)} returns the secant of \spad{x}.
745506`\spad{sech2cosh(f)} converts every \spad{sech(u)} appearing in \spad{f} into \spad{1/cosh(u)}.
745571`\spad{sechIfCan(z)} returns sech(\spad{z}) if possible,{} and "failed" otherwise.
745653`\spad{sech(x)} applies the hyperbolic secant operator to \spad{x}
745817`\spad{sech(st)} computes the hyperbolic secant of a power series \spad{st}.
745902`\spad{sech(st)} computes the hyperbolic secant of a power series \spad{st}.
746001`\spad{sech(z)} returns the hyperbolic secant of Laurent series \spad{z}.
746089`\spad{sech(z)} returns the hyperbolic secant of a Puiseux series \spad{z}.
746186`\spad{sech(x)} returns the hyperbolic secant of \spad{x}.
746295`\spad{second(x)} selects second component of the product
746338`\spad{second(u)} returns the second element of \spad{u}. Note: \axiom{second(\spad{u}) = first(rest(\spad{u}))}.
746395`\spad{seed()} returns the current seed value.
746446`\spad{segment(l)} is an alternate way to construct the segment \spad{l..}.
746497`\spad{segment(segb)} returns the segment from the right hand side of the \spadtype{SegmentBinding}. For example,{} if \spad{segb} is \spad{v=a..b},{} then \spad{segment(segb)} returns \spad{a..b}.
746555`\spad{segment(i,{}j)} is an alternate way to create the segment \spad{i..j}.
746718`\spad{select!(p,{}d)} destructively changes dictionary \spad{d} by removing all entries \spad{x} such that \axiom{\spad{p}(\spad{x})} is not \spad{true}.
746810`\spad{select!(p,{}u)} destructively changes \spad{u} by keeping only values \spad{x} such that \axiom{\spad{p}(\spad{x})}.
746884`\axiom{selectAndPolynomials(lpred?,{}\spad{ps})} returns \axiom{\spad{gps},{}\spad{bps}} where \axiom{\spad{gps}} is a list of the polynomial \axiom{\spad{p}} in \axiom{\spad{ps}} such that \axiom{pred?(\spad{p})} holds for every \axiom{pred?} in \axiom{lpred?} and \axiom{\spad{bps}} are the other ones.
747030`\axiom{selectOrPolynomials(lpred?,{}\spad{ps})} returns \axiom{\spad{gps},{}\spad{bps}} where \axiom{\spad{gps}} is a list of the polynomial \axiom{\spad{p}} in \axiom{\spad{ps}} such that \axiom{pred?(\spad{p})} holds for some \axiom{pred?} in \axiom{lpred?} and \axiom{\spad{bps}} are the other ones.
747175`\axiom{selectPolynomials(pred?,{}\spad{ps})} returns \axiom{\spad{gps},{}\spad{bps}} where \axiom{\spad{gps}} is a list of the polynomial \axiom{\spad{p}} in \axiom{\spad{ps}} such that \axiom{pred?(\spad{p})} holds and \axiom{\spad{bps}} are the other ones.
747578`\spad{select(x,{}n)} returns the \spad{n}-th element of tuple \spad{x}. tuples are 0-based
747636`\spad{select(p,{}u)} returns a copy of \spad{u} containing only those elements such \axiom{\spad{p}(\spad{x})} is \spad{true}. Note: \axiom{select(\spad{p},{}\spad{u}) \spad{==} [\spad{x} for \spad{x} in \spad{u} | \spad{p}(\spad{x})]}.
747717`\spad{select(f,{}st)} returns a stream consisting of those elements of stream \spad{st} satisfying the predicate \spad{f}. Note: \spad{select(f,{}st) = [x for x in st | f(x)]}.
747784`\spad{select(p,{}t)} returns \spad{[x for x in t | p(x)]}.
747845`\axiom{select(\spad{ts},{}\spad{v})} returns the polynomial of \axiom{\spad{ts}} with \axiom{\spad{v}} as main variable,{} if any.
747924`\axiom{indiceSubResultant(\spad{P},{} \spad{Q},{} \spad{i})} returns a subresultant \axiom{\spad{S}} of degree \axiom{\spad{d}} and carries out the equality \axiom{...\spad{P} + coef2*Q = S_i}. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
748072`\axiom{discriminantEuclidean(\spad{P})} carries out the equality \axiom{...\spad{P} + \spad{coef2} * \spad{D}(\spad{P}) = discriminant(\spad{P})}. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
748187`\axiom{semiIndiceSubResultantEuclidean(\spad{P},{} \spad{Q},{} \spad{i})} returns the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} and carries out the equality \axiom{...\spad{P} + coef2*Q = S_i(\spad{P},{}\spad{Q})} Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
748335`\axiom{semiLastSubResultantEuclidean(\spad{P},{} \spad{Q})} computes the last non zero subresultant \axiom{\spad{S}} and carries out the equality \axiom{...\spad{P} + coef2*Q = \spad{S}}. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
748462`\axiom{\spad{semiResultantEuclidean1}(\spad{P},{}\spad{Q})} carries out the equality \axiom{\spad{coef1}.\spad{P} + ? \spad{Q} = resultant(\spad{P},{}\spad{Q})}.
748577`\axiom{\spad{semiResultantEuclidean2}(\spad{P},{}\spad{Q})} carries out the equality \axiom{...\spad{P} + coef2*Q = resultant(\spad{P},{}\spad{Q})}. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
748692`\axiom{resultantEuclidean_naif(\spad{P},{}\spad{Q})} returns the semi-extended resultant of \axiom{\spad{P}} and \axiom{\spad{Q}} computed by means of the naive algorithm.
748811`\axiom{semiResultantReduitEuclidean(\spad{P},{}\spad{Q})} returns the "reduce resultant" and carries out the equality \axiom{...\spad{P} + coef2*Q = resultantReduit(\spad{P},{}\spad{Q})}.
748953`\axiom{\spad{semiSubResultantGcdEuclidean1}(\spad{P},{}\spad{Q})} carries out the equality \axiom{coef1*P + ? \spad{Q} = \spad{+/-} S_i(\spad{P},{}\spad{Q})} where the degree (not the indice) of the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} is the smaller as possible.
749071`\axiom{\spad{semiSubResultantGcdEuclidean2}(\spad{P},{}\spad{Q})} carries out the equality \axiom{...\spad{P} + coef2*Q = \spad{+/-} S_i(\spad{P},{}\spad{Q})} where the degree (not the indice) of the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} is the smaller as possible. Warning: \axiom{degree(\spad{P}) \spad{>=} degree(\spad{Q})}.
749189`\spad{semicolonSeparate(l)} creates the form separating the elements of \spad{l} by semicolons.
749316`\spad{separant(p)} returns the partial derivative of the differential polynomial \spad{p} with respect to its leader.
749389`\spad{separateDegrees(p)} splits the square free polynomial \spad{p} into factors each of which is a product of irreducibles of the same degree.
749501`\spad{separateFactors(lfact)} takes the list produced by \spadfunFrom{separateDegrees}{DistinctDegreeFactorization} and produces the complete list of factors.
749619`\spad{separateFactors(ddl,{} p)} refines the distinct degree factorization produced by \spadfunFrom{ddFact}{ModularDistinctDegreeFactorizer} to give a complete list of factors.
749742`\spad{separate(x)} \undocumented
749860`\spad{separate(s)} makes each of the components of the \spadtype{SubSpace},{} \spad{s},{} into a list of separate and distinct subspaces and returns the list.
750006`\spad{separate(p,{} q)} returns \spad{[a,{} b]} such that polynomial \spad{p = a b} and \spad{a} is relatively prime to \spad{q}.
750122`\spad{sequence(x)} turns the stream \spad{x} into a sequence
750182`\spad{sequences([l0,{}l1,{}l2,{}..,{}ln])} is the set of \indented{1}{all sequences formed from} \spad{l0} 0\spad{'s},{}\spad{l1} 1\spad{'s},{}\spad{l2} 2\spad{'s},{}...,{}\spad{ln} \spad{n}\spad{'s}.
750272`\spad{sequences(l1,{}l2)} is the stream of all sequences that \indented{1}{can be composed from the multiset defined from} \indented{1}{two lists of integers \spad{l1} and \spad{l2}.} \indented{1}{For example,{}the pair \spad{([1,{}2,{}4],{}[2,{}3,{}5])} represents} \indented{1}{multi-set with 1 \spad{2},{} 2 \spad{3}\spad{'s},{} and 4 \spad{5}\spad{'s}.}
750554`\spad{seriesSolve(eq,{}y,{}x=a,{} y a = b)} returns a Taylor series solution of \spad{eq} around \spad{x} = a with initial condition \spad{y(a) = b}. Note: \spad{eq} must be of the form \spad{f(x,{} y x) y'(x) + g(x,{} y x) = h(x,{} y x)}.
750668`\spad{seriesSolve(eq,{}y,{} x=a,{} b)} is equivalent to \spad{seriesSolve(eq,{} y,{} x=a,{} y a = b)}.
750772`\spad{seriesSolve(eq,{}y,{}x=a,{}[b0,{}...,{}b(n-1)])} returns a Taylor series solution of \spad{eq} around \spad{x = a} with initial conditions \spad{y(a) = b0},{} \spad{y'(a) = b1},{} \spad{y''(a) = b2},{} ...,{}\spad{y(n-1)(a) = b(n-1)} \spad{eq} must be of the form \spad{f(x,{} y x,{} y'(x),{}...,{} y(n-1)(x)) y(n)(x) + g(x,{}y x,{}y'(x),{}...,{}y(n-1)(x)) = h(x,{}y x,{} y'(x),{}...,{} y(n-1
750772`--)(x))}.
750882`\spad{seriesSolve(eq,{} y,{} x = a,{} y a = b)} is equivalent to \spad{seriesSolve(eq=0,{} y,{} x=a,{} y a = b)}.
750986`\spad{seriesSolve(eq,{} y,{} x = a,{} b)} is equivalent to \spad{seriesSolve(eq = 0,{} y,{} x = a,{} y a = b)}.
751080`\spad{seriesSolve(eq,{} y,{} x = a,{} [b0,{}...,{}bn])} is equivalent to \spad{seriesSolve(eq = 0,{} y,{} x = a,{} [b0,{}...,{}b(n-1)])}.
751180`\spad{seriesSolve([eq1,{}...,{}eqn],{}[y1,{}...,{}yn],{}x = a,{}[y1 a = b1,{}...,{}yn a = bn])} returns a taylor series solution of \spad{[eq1,{}...,{}eqn]} around \spad{x = a} with initial conditions \spad{\spad{yi}(a) = \spad{bi}}. Note: eqi must be of the form \spad{\spad{fi}(x,{} y1 x,{} y2 x,{}...,{} yn x) y1'(x) + \spad{gi}(x,{} y1 x,{} y2 x,{}...,{} yn x) = h(x,{} y1
751180`-- x,{} y2 x,{}...,{} yn x)}.
751312`\spad{seriesSolve([eq1,{}...,{}eqn],{} [y1,{}...,{}yn],{} x=a,{} [b1,{}...,{}bn])} is equivalent to \spad{seriesSolve([eq1,{}...,{}eqn],{} [y1,{}...,{}yn],{} x = a,{} [y1 a = b1,{}...,{} yn a = bn])}.
751434`\spad{seriesSolve([eq1,{}...,{}eqn],{} [y1,{}...,{}yn],{} x = a,{}[y1 a = b1,{}...,{} yn a = bn])} is equivalent to \spad{seriesSolve([eq1=0,{}...,{}eqn=0],{} [y1,{}...,{}yn],{} x = a,{} [y1 a = b1,{}...,{} yn a = bn])}.
751556`\spad{seriesSolve([eq1,{}...,{}eqn],{} [y1,{}...,{}yn],{} x=a,{} [b1,{}...,{}bn])} is equivalent to \spad{seriesSolve([eq1=0,{}...,{}eqn=0],{} [y1,{}...,{}yn],{} x=a,{} [b1,{}...,{}bn])}.
751668`\spad{seriesToOutputForm(st,{}refer,{}var,{}cen,{}r)} prints the series \spad{f((var - cen)^r)}.
751856`\spad{series(s)} creates a power series from a stream of \indented{1}{ring elements.} \indented{1}{For univariate series types,{} the stream \spad{s} should be a stream} \indented{1}{of Taylor coefficients. For multivariate series types,{} the} \indented{1}{stream \spad{s} should be a stream of forms the \spad{n}th element} \indented{1}{of which is a} \indented{1}{form of degree \spad{n} in the power series variables.}
751921`\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.
752021`\spad{series(f)} returns a series expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
752093`\spad{series([a0,{}a1,{}a2,{}...])} is the Taylor series \spad{a0 + a1 x + a2 x^2 + ...}.
752171`\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.
752270`\spad{series(st)} creates a series from a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents.
752379`\spad{series(x)} returns \spad{x} viewed as a series.
752455`\spad{series(n +-> a(n),{}x = a)} returns \spad{sum(n = 0..,{}a(n)*(x-a)^n)}.
752547`\spad{series(f,{}x = a)} expands the expression \spad{f} as a series in powers of (\spad{x} - a).
752632`\spad{series(f,{}n)} returns a series expansion of the expression \spad{f}. Note: \spad{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 \spad{n}.
752722`\spad{series(n,{}st)} creates a series from a common denomiator and a stream of non-zero terms,{} where a term is an exponent-coefficient pair. The terms in the stream should be ordered by increasing order of exponents and \spad{n} should be a common denominator for the exponents in the stream of terms.
752850`\spad{series(n +-> a(n),{}x = a,{}n0..)} returns \spad{sum(n = n0..,{}a(n) * (x - a)^n)}; \spad{series(n +-> a(n),{}x = a,{}n0..n1)} returns \spad{sum(n = n0..n1,{}a(n) * (x - a)^n)}.
752968`\spad{series(f,{}x = a,{}n)} expands the expression \spad{f} as a series in powers of (\spad{x} - a); terms will be computed up to order at least \spad{n}.
753071`\spad{series(a(n),{}n,{}x = a)} returns \spad{sum(n = 0..,{}a(n)*(x-a)^n)}.
753159`\spad{series(n +-> a(n),{}x = a,{}r0..,{}r)} returns \spad{sum(n = r0,{}r0 + r,{}r0 + 2*r...,{} a(n) * (x - a)^n)}; \spad{series(n +-> a(n),{}x = a,{}r0..r1,{}r)} returns \spad{sum(n = r0 + k*r while n <= r1,{} a(n) * (x - a)^n)}.
753315`\spad{series(a(n),{}n,{}x=a,{}n0..)} returns \spad{sum(n = n0..,{}a(n) * (x - a)^n)}; \spad{series(a(n),{}n,{}x=a,{}n0..n1)} returns \spad{sum(n = n0..n1,{}a(n) * (x - a)^n)}.
753429`\spad{series(a(n),{}n,{}x = a,{}r0..,{}r)} returns \spad{sum(n = r0,{}r0 + r,{}r0 + 2*r...,{} a(n) * (x - a)^n)}; \spad{series(a(n),{}n,{}x = a,{}r0..r1,{}r)} returns \spad{sum(n = r0 + k*r while n <= r1,{} a(n) * (x - a)^n)}.
753571`\spad{setAdaptive3D(true)} turns adaptive plotting on; setAdaptive3D(\spad{false}) turns adaptive plotting off.
753626`\spad{setAdaptive(true)} turns adaptive plotting on \spad{setAdaptive(false)} turns adaptive plotting off
753677`\spad{setClipValue(x)} sets to \spad{x} the maximum value to plot when drawing complex functions. Returns \spad{x}.
753744`\spad{setClosed(t,{}b)} declares the given tube plot \spad{t} to be closed if \spad{b} is \spad{true},{} or if \spad{b} is \spad{false},{} \spad{t} is set to be open.
753888`\spad{setColumn!(m,{}j,{}v)} sets to \spad{j}th column of \spad{m} to \spad{v}
753974`\axiom{setCondition!(\spad{n},{}\spad{t})} returns \spad{n} whose condition has been replaced by \spad{t} if it is not empty,{} else an error is produced.
754033`\spad{setDifference(u1,{}u2)} returns a list of the elements of \spad{u1} that are not also in \spad{u2}. The order of elements in the resulting list is unspecified.
754098`\axiom{setEmpty!(\spad{n})} replaces \spad{n} by \axiom{empty()\$\%}.
754151`\spad{setEpilogue!(t,{}strings)} sets the epilogue section of a formatted object \spad{t} to \spad{strings}.
754231`\spad{setEpilogue!(t,{}strings)} sets the epilogue section of a TeX form \spad{t} to \spad{strings}.
754301`\spad{setErrorBound(eps)} changes the internal error bound,{} by default being {\em 10 ^ (-3)} to \spad{eps},{} if \spad{R} is a member in the category \spadtype{QuotientFieldCategory Integer}. The internal {\em globalDigits} is set to {\em ceiling(1/r)^2*10} being {\em 10^7} by default.
754369`\spad{setFieldInfo(m,{}p)} initializes the field arithmetic,{} where \spad{m} is the multiplication table and \spad{p} is the respective normal element of the ground field \spad{GF}.
754494`\spad{setFormula!(t,{}strings)} sets the formula section of a formatted object \spad{t} to \spad{strings}.
754573`\axiom{setGcdMode(\spad{s})} sets a new value for the flag deciding on the method used to compute gcd`s for lists. Possible values for \axiom{\spad{s}} are \axiom{iterated} and \axiom{random}.
754657`\spad{setImagSteps(i)} sets to \spad{i} the number of steps to use in the imaginary direction when drawing complex functions. Returns \spad{i}.
754716`\spad{setIntersection(u1,{}u2)} returns a list of the elements that lists \spad{u1} and \spad{u2} have in common. The order of elements in the resulting list is unspecified.
754783`\spad{setLabelValue(i)} resets the counter which produces labels to \spad{i}
754855`\spad{setLegalFortranSourceExtensions(l)} \undocumented{}
754946`\spad{setMaxPoints3D(i)} sets the maximum number of points in a plot to \spad{i}.
755002`\spad{setMaxPoints(i)} sets the maximum number of points in a plot to \spad{i}
755054`\spad{setMinPoints3D(i)} sets the minimum number of points in a plot to \spad{i}.
755110`\spad{setMinPoints(i)} sets the minimum number of points in a plot to \spad{i}
755162`allows override of parameters such as orthogonal used for debugging
755237`\axiom{setNotation(\spad{s})} chooses the notation used for derivatives. Returns the old value.
755434`\spad{setOfMinN([a_1,{}...,{}a_m])} returns the set {\spad{a_1},{}...,{}a_m}. Error if {\spad{a_1},{}...,{}a_m} is not a set of \spad{M} integers in \spad{1..n}.
755515`\spad{setOrder([a1,{}...,{}an])} defines a partial ordering on \spad{S} given \spad{by:} \indented{3}{(1)\space{2}\spad{a1 < a2 < ... < an}.} \indented{3}{(2)\space{2}\spad{b < \spad{ai}\space{3}for i = 1..n} and \spad{b} not among the \spad{ai}\spad{'s}.} \indented{3}{(3)\space{2}undefined on \spad{(b,{} c)} if neither is among the \spad{ai}\spad{'s}.}
755585`\spad{setOrder([b1,{}...,{}bm],{} [a1,{}...,{}an])} defines a partial ordering on \spad{S} given \spad{by:} \indented{3}{(1)\space{2}\spad{b1 < b2 < ... < bm < a1 < a2 < ... < an}.} \indented{3}{(2)\space{2}\spad{bj < c < \spad{ai}}\space{2}for \spad{c} not among the \spad{ai}\spad{'s} and \spad{bj}\spad{'s}.} \indented{3}{(3)\space{2}undefined on \spad{(c,{}d)} if neither is among the \spad{ai}\spad{'s},{}\spad{bj}\spad{'s}.}
755663`\axiom{setOutput(\spad{i})} controls amount of generated output during the completion algorithm: \axiom{\spad{i} = 0} 
755663`--> no display,{} \axiom{\spad{i} = 1} 
755663`--> result is displayed,{} \axiom{\spad{i} = 2} 
755663`--> Cartan characters are displayed,{} \axiom{\spad{i} = 3} 
755663`--> integrability conditions are traced,{} \axiom{\spad{i} = 4} 
755663`--> intermediate dimensions are traced,{} \axiom{\spad{i} = 5} 
755663`--> all intermediate system
755663`--s are traced,{} \axiom{\spad{i} = 6} 
755663`--> all intermediate systems and symbols are traced,{} if \axiom{\spad{i} > 10} then TeX output is produced. Default is 0. The old value is returned.
755752`\spad{setPoly(x)} \undocumented
755802`\spad{setPosition(x,{} n)} associates the integer \spad{n} to \spad{x}.
755871`\spad{setPredicates(p,{} [p1,{}...,{}pn])} attaches the predicate \spad{p1} and ... and \spad{pn} to \spad{p}.
755930`\spad{setPrologue!(t,{}strings)} sets the prologue section of a formatted object \spad{t} to \spad{strings}.
756010`\spad{setPrologue!(t,{}strings)} sets the prologue section of a TeX form \spad{t} to \spad{strings}.
756080`\spad{setProperties(op,{} l)} sets the property list of \spad{op} to \spad{l}. Argument \spad{op} is modified "in place",{} \spadignore{i.e.} no copy is made.
756161`\spad{setProperty(op,{} s,{} v)} attaches property \spad{s} to \spad{op},{} and sets its value to \spad{v}. Argument \spad{op} is modified "in place",{} \spadignore{i.e.} no copy is made.
756223`\spad{setRealSteps(i)} sets to \spad{i} the number of steps to use in the real direction when drawing complex functions. Returns \spad{i}.
756282`\axiom{setRedMode(\spad{i})} sets the flag for the reduction mode. Returns old value. Current values are: \axiom{\spad{i} = 0} 
756282`--> No reduction of integrability conditions etc. \axiom{\spad{i} = 1} 
756282`--> Autoreduction of complete system and reduction \indented{16}{of all integrability conditions.} Default is 0.
756449`\axiom{setRow!(A,{} \spad{i},{} ind,{} ent)} sets the \axiom{\spad{i}}\spad{-}th row of the matrix \axiom{A} to the value \axiom{\spad{r}}.
756556`\spad{setRow!(m,{}i,{}v)} sets to \spad{i}th row of \spad{m} to \spad{v}
756639`\axiom{setRow!(A,{} \spad{i},{} ind,{} ent)} sets the \axiom{\spad{i}}\spad{-}th row of the matrix \axiom{A}. Its indices are \axiom{ind}; the entries \axiom{ent}.
756722`\spad{setScreenResolution3D(i)} sets the screen resolution for a 3d graph to \spad{i}.
756785`\spad{setScreenResolution(i)} sets the screen resolution to \spad{i}
756844`\axiom{setSimpMode(\spad{i})} sets the flag controlling the used simplifications and returns the old value. Current values are: \axiom{\spad{i} = 0} \spad{->} No simplification modulo lower order equations. \axiom{\spad{i} = 1} \spad{->} Simplification modulo lower order equations. Default is 0.
756942`\axiom{setSimpMode(\spad{i})} sets the simplification mode used in \axiom{JetDifferentialEquation}. Returns old value.
757032`\spad{setSimplifyDenomsFlag(x)} sets flag affecting simplification of denominators. If \spad{true} irrational algebraics are removed from denominators. If \spad{false} they are kept.
757123`\axiom{setStatus!(\spad{n},{}\spad{b})} returns \spad{n} whose status has been replaced by \spad{b} if it is not empty,{} else an error is produced.
757185`\spad{setStatus(s,{}t)} returns the same representation for \spad{s},{} but asserts the following: if \spad{t} is \spad{true},{} then \spad{s} is empty,{} if \spad{t} is \spad{false},{} then \spad{s} is non-empty,{} and if \spad{t} = "failed",{} then no assertion is made (that is,{} "don\spad{'t} know"). Note: for internal use only,{} with care.
757280`\spad{setTex!(t,{}strings)} sets the TeX section of a TeX form \spad{t} to \spad{strings}.
757345`\spad{setTopPredicate(x,{} [a1,{}...,{}an],{} f)} returns \spad{x} with the top-level predicate set to \spad{f(a1,{}...,{}an)}.
757413`\spad{setTransform!(n,{}tran)} changes the transform without altering the scene hierarchy
757478`\spad{setUnion(u1,{}u2)} appends the two lists \spad{u1} and \spad{u2},{} then removes all duplicates. The order of elements in the resulting list is unspecified.
757538`\axiom{setValue!(\spad{n},{}\spad{v})} returns \spad{n} whose value has been replaced by \spad{v} if it is not empty,{} else an error is produced.
757593`\spad{setVariableOrder([a1,{}...,{}an])} defines an ordering on the variables given by \spad{a1 > a2 > ... > an > other variables}.
757674`\spad{setVariableOrder([b1,{}...,{}bm],{} [a1,{}...,{}an])} defines an ordering on the variables given by \spad{b1 > b2 > ... > bm >} other variables \spad{> a1 > a2 > ... > an}.
757812`\spad{set()}\$\spad{D} creates an empty set aggregate of type \spad{D}.
757905`\spad{set([x,{}y,{}...,{}z])} creates a set aggregate containing items \spad{x},{}\spad{y},{}...,{}\spad{z}.
758021`\spad{setchildren!(u,{}v)} replaces the current children of node \spad{u} with the members of \spad{v} in left-to-right order.
758113`\axiom{setelt!(A,{} \spad{i},{} \spad{c},{} \spad{d})} sets the entry of the matrix \axiom{A} in row \axiom{\spad{i}} and in the column with index \axiom{\spad{c}} to the value \axiom{\spad{d}}.
758184`\spad{setelt!(x,{}i,{}j,{}k,{}s)} (or \spad{x}.\spad{i}.\spad{j}.k:=s) sets a specific element of the array to some value of type \spad{R}
758300`\spad{setelt(n,{}m)} changes the value of the object \spad{n} to \spad{m}.
759344`\spad{setelt(u,{}"first",{}x)} (also written: \axiom{\spad{u}.first \spad{:=} \spad{x}}) is equivalent to \axiom{setfirst!(\spad{u},{}\spad{x})}.
759435`\spad{setelt(u,{}"last",{}x)} (also written: \axiom{\spad{u}.last \spad{:=} \spad{b}}) is equivalent to \axiom{setlast!(\spad{u},{}\spad{v})}.
759525`\spad{setelt(a,{}"left",{}b)} (also written \axiom{a . left \spad{:=} \spad{b}}) is equivalent to \axiom{setleft!(a,{}\spad{b})}.
759618`\spad{setelt(u,{}"rest",{}v)} (also written: \axiom{\spad{u}.rest \spad{:=} \spad{v}}) is equivalent to \axiom{setrest!(\spad{u},{}\spad{v})}.
759710`\spad{setelt(a,{}"right",{}b)} (also written \axiom{\spad{b} . right \spad{:=} \spad{b}}) is equivalent to \axiom{setright!(a,{}\spad{b})}.
759804`\spad{setelt(a,{}"value",{}x)} (also written \axiom{a . value \spad{:=} \spad{x}}) is equivalent to \axiom{setvalue!(a,{}\spad{x})}
759890`\spad{setelt(u,{}x,{}y)} sets the image of \spad{x} to be \spad{y} under \spad{u},{} assuming \spad{x} is in the domain of \spad{u}. Error: if \spad{x} is not in the domain of \spad{u}.
759977`\spad{setelt(t,{}k,{}e)} (also written \axiom{\spad{t}.\spad{k} \spad{:=} \spad{e}}) is equivalent to \axiom{(insert([\spad{k},{}\spad{e}],{}\spad{t}); \spad{e})}.
760047`\spad{lib.k := v} saves the value \spad{v} in the library \spad{lib}. It can later be extracted using the key \spad{k}.
760098`\spad{setelt(u,{}i..j,{}x)} (also written: \axiom{\spad{u}(\spad{i}..\spad{j}) \spad{:=} \spad{x}}) destructively replaces each element in the segment \axiom{\spad{u}(\spad{i}..\spad{j})} by \spad{x}. The value \spad{x} is returned. Note: \spad{u} is destructively change so that \axiom{\spad{u}.\spad{k} \spad{:=} \spad{x} for \spad{k} in \spad{i}..\spad{j}}; its length remains unchanged.
760351`\spad{setelt(m,{}i,{}j,{}r)} sets the element in the \spad{i}th row and \spad{j}th column of \spad{m} to \spad{r} error check to determine if indices are in proper ranges
760438`\spad{setelt(x,{}rowList,{}colList,{}y)} destructively alters the matrix \spad{x}. If \spad{y} is \spad{m}-by-\spad{n},{} \spad{rowList = [i<1>,{}i<2>,{}...,{}i<m>]} and \spad{colList = [j<1>,{}j<2>,{}...,{}j<n>]},{} then \spad{x(i<k>,{}j<l>)} is set to \spad{y(k,{}l)} for \spad{k = 1,{}...,{}m} and \spad{l = 1,{}...,{}n}.
760526`\spad{setfirst!(u,{}x)} destructively changes the first element of a to \spad{x}.
760669`\spad{setlast!(u,{}x)} destructively changes the last element of \spad{u} to \spad{x}.
760754`\spad{setleaves!(t,{} ls)} sets the leaves of \spad{t} in left-to-right order to the elements of \spad{ls}.
760819`\spad{setleft!(a,{}b)} sets the left child of \axiom{a} to be \spad{b}.
760907`\spad{setnext!(u,{}v)} destructively sets the next node of doubly-linked aggregate \spad{u} to \spad{v},{} returning \spad{v}.
760992`\spad{setprevious!(u,{}v)} destructively sets the previous node of doubly-linked aggregate \spad{u} to \spad{v},{} returning \spad{v}.
761081`\spad{setref(n,{}m)} same as \spad{setelt(n,{}m)}.
761126`\spad{setrest!(u,{}v)} destructively changes the rest of \spad{u} to \spad{v}.
761213`\spad{setrest!(x,{}n,{}y)} sets rest(\spad{x},{}\spad{n}) to \spad{y}. The function will expand cycles if necessary.
761267`\spad{setright!(a,{}x)} sets the right child of \spad{t} to be \spad{x}.
761433`\spad{setsubMatrix(x,{}i1,{}j1,{}y)} destructively alters the matrix \spad{x}. Here \spad{x(i,{}j)} is set to \spad{y(i-i1+1,{}j-j1+1)} for \spad{i = i1,{}...,{}i1-1+nrows y} and \spad{j = j1,{}...,{}j1-1+ncols y}.
761574`\spad{setvalue!(u,{}x)} sets the value of node \spad{u} to \spad{x}.
761655`\spad{sh(x,{}n)} returns the shuffle power of \spad{x} to the \spad{n}.
761742`\spad{sh(x,{}y)} returns the shuffle-product of \spad{x} by \spad{y}. This multiplication is associative and commutative.
761813`\spad{shade(pt)} returns the fourth element of the two dimensional point,{} \spad{pt},{} although no assumptions are made with regards as to how the components of higher dimensional points are interpreted. This function is defined for the convenience of the user using specifically,{} shade to express a fourth dimension.
761864`\spad{shade(p)} returns the shade index of the indicated palette \spad{p}.
761907`\spad{shallowCopy(x)} \undocumented
761957`\spad{shallowExpand(x)} \undocumented{}
762031`\spad{shanksDiscLogAlgorithm(b,{}a,{}p)} computes \spad{s} with \spad{b^s = a} for assuming that \spad{a} and \spad{b} are elements in a 'small' cyclic group of order \spad{p} by Shank\spad{'s} algorithm. Note: this is a subroutine of the function \spadfun{discreteLog}.
762158`\spad{shellSort(f,{} agg)} sorts the aggregate agg with the ordering function \spad{f} using the shellSort algorithm.
764408`\spad{shiftHP options} returns a specification for Hermite-Pade approximation with the shift operator
769525`\spad{shiftHP options} returns a specificat
769525`--ion for Hermite-Pade approximation with the \$\spad{q}\$-shift operator,{} or,{} if \spad{maxMixedDegree > 0} for mixed shifts
775150`\spad{shiftLeft(p,{}n)} returns \spad{p * monomial(1,{}n)}
775316`\spad{shiftRight(p,{}n)} returns \spad{monicDivide(p,{}monomial(1,{}n)).quotient}
775402`\spad{shiftRoots(p,{}c)} returns the polynomial which has for roots \spad{c} added to the roots of \spad{p}.
775476`\spad{shift(k)} returns \spad{matrix [[1,{}k],{}[0,{}1]]} representing the map \spad{x -> x + k}.
775525`\spad{shift(m,{}h)} returns \spad{shift(h) * m} (see \spadfunFrom{shift}{MoebiusTransform}).
775577`\spad{shift(x,{}n)} adds \spad{n} to the exponent of float \spad{x}.
775621`\spad{shift(a,{}i)} shift \spad{a} by \spad{i} bits.
775678`\spad{shift(a,{}i)} shift \spad{a} by \spad{i} digits.
775731`\spad{showAll?()} returns \spad{true} if all computed entries of streams will be displayed.
775795`\spad{showAllElements(s)} creates an output form which displays all computed elements.
775871`\spad{showArrayValues(true)} forces the values of array components to be \indented{1}{displayed rather than just their types.}
775928`\spad{showClipRegion(v,{}s)} displays the clipping region of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} if \spad{s} is "on",{} or does not display the region if \spad{s} is "off".
776001`\spad{showFortranOutputStack()} returns the Fortran output stack
776083`\spad{showRegion(v,{}s)} displays the bounding box of the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} if \spad{s} is "on",{} or does not display the box if \spad{s} is "off".
776152`\spad{showScalarValues(true)} forces the values of scalar components to be \indented{1}{displayed rather than just their types.}
776210`\spad{showTheSymbolTable()} returns the current symbol table.
776266`\spad{showTypeInOutput(bool)} affects the way objects of \spadtype{Any} are displayed. If \spad{bool} is \spad{true} then the type of the original object that was converted to \spadtype{Any} will be printed. If \spad{bool} is \spad{false},{} it will not be printed.
776320`\spad{show(v,{}n,{}s)} displays the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} if \spad{s} is "on",{} or does not display the graph if \spad{s} is "off".
776397`\spad{shrinkable(b)} sets the shrinkable attribute of flexible arrays to \spad{b} and returns the previous value
776525`\spad{shuffle(l1,{}l2)} forms the stream of all shuffles of \spad{l1} \indented{1}{and \spad{l2},{} \spadignore{i.e.} all sequences that can be formed from} \indented{1}{merging \spad{l1} and \spad{l2}.}
776627`\spad{shufflein(l,{}st)} maps shuffle(\spad{l},{}\spad{u}) on to all \indented{1}{members \spad{u} of \spad{st},{} concatenating the results.}
776739`\spad{sierpinskiDivide(me,{}level)} for each edge of each face calculate a new point in the centre of the edge,{} then build a new set of faces which include the new points.
776812`\spad{signAround(u,{}i,{}f)} \undocumented
776925`\spad{signAround(u,{}r,{}f)} \undocumented
777032`\spad{signAround(u,{}r,{}i,{}f)} \undocumented
777147`\spad{sign(r)} \undocumented
777255`\spad{sign(f)} returns the sign of \spad{f} if it is constant everywhere.
777332`\spad{sign f} returns the sign of \spad{f} if it is constant everywhere.
777427`\spad{sign(x)} is 1 if \spad{x} is positive,{} \spad{-1} if \spad{x} is negative,{} 0 if \spad{x} equals 0.
777473`\spad{sign(p)} returns the signum of the permutation \spad{p},{} \spad{+1} or \spad{-1}.
777522`\axiom{sign(pol,{}aRoot)} gives the sign of \axiom{pol} interpreted as \axiom{aRoot}
777615`\spad{sign(f,{} x,{} a)} returns the sign of \spad{f} as \spad{x} nears \spad{a},{} from both sides if \spad{a} is finite.
777720`\spad{sign(f,{} x,{} a)} returns the sign of \spad{f} as \spad{x} approaches \spad{a},{} from both sides if \spad{a} is finite.
777865`\spad{sign(f,{} x,{} a,{} s)} returns the sign of \spad{f} as \spad{x} nears \spad{a} from below if \spad{s} is "left",{} or above if \spad{s} is "right".
777958`\spad{sign(f,{} x,{} a,{} s)} returns the sign of \spad{f} as \spad{x} nears \spad{a} from the left (below) if \spad{s} is the string \spad{"left"},{} or from the right (above) if \spad{s} is the string \spad{"right"}.
778168`\axiom{simpMod(\spad{sys1},{} \spad{sys2})} simplifies the system \axiom{\spad{sys1}} modulo the system \axiom{\spad{sys2}}.
778449`\axiom{simpMod(\spad{sys1},{} \spad{sys2})} simplifies the system \axiom{\spad{sys1}} modulo the system \axiom{\spad{sys2}}. Returns the same information as \axiom{simplify}.
778712`\axiom{simpOne(\spad{f})} removes unnecessary coefficients and exponents,{} denominators etc.
778774`\spad{simplifyCoeffs(d,{} la)} should be local but conditional
778968`\spad{simplifyExp(f)} converts every product \spad{exp(a)*exp(b)} appearing in \spad{f} into \spad{exp(a+b)}.
779035`\spad{simplifyLog(f)} converts every \spad{log(a) - log(b)} appearing in \spad{f} into \spad{log(a/b)},{} every \spad{log(a) + log(b)} into \spad{log(a*b)} and every \spad{n*log(a)} into \spad{log(a^n)}.
779102`\spad{simplifyPower(f,{}n)} \undocumented{}
779183`\axiom{simplify(de)} simplifies the equations of each order separately using the procedure \axiom{simplify} from \axiom{\spad{D}}. Found integrability conditions are also returned separately.
779269`\spad{simplify(s)} returns a different and presumably simpler representation of \spad{s} with the defining polynomials for the equations forming a groebner basis,{} and the defining polynomial for the inequation reduced with respect to the basis,{} using a heuristic algorithm based on factoring.
779392`\spad{simplify(an)} applies simplifications to \spad{an}
779493`\spad{simplify(f)} performs the following simplifications on \spad{f:}\begin{items} \item 1. rewrites trigs and hyperbolic trigs in terms of \spad{sin} ,{}\spad{cos},{} \spad{sinh},{} \spad{cosh}. \item 2. rewrites \spad{sin^2} and \spad{sinh^2} in terms of \spad{cos} and \spad{cosh},{} \item 3. rewrites \spad{exp(a)*exp(b)} as \spad{exp(a+b)}. \item 4. rewrites \spad{(a^(1/n))^m * (a^(1/s))^t} as a single power of a single radical of \spad
779493`--{a}. \end{items}
779748`\axiom{simplify(\spad{sys},{} \spad{jm})} simplifies a system with given Jacobi matrix. The Jacobi matrix of the simplified system is returned,{} too. \axiom{Depend} contains for each equation of the simplified system the numbers of the equations of the original system out of which it is build,{} if it is pos
779748`--sible to obtain this information. If one can generate equations of lower order by purely algebraic operations,{} then \axiom{simplify} should do this.
779946`\spad{simpson(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax)} uses the simpson method to numerically integrate function \spad{fn} over the closed interval \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary from \spad{nmin} to \spad{nmax}. 
779946`--The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
780112`\spad{simpsono(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax)} uses the simpson method to numerically integrate function \spad{fn} over the open interval from \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary from \spad{nmin} to \spad{nm
780112`--ax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
780279`\spad{sin2csc(f)} converts every \spad{sin(u)} appearing in \spad{f} into \spad{1/csc(u)}.
780342`\spad{sin?(x)} returns \spad{true} if term is a sin,{} otherwise \spad{false}
780396`\spad{sinIfCan(z)} returns sin(\spad{z}) if possible,{} and "failed" otherwise.
780477`\spad{sin(x)} makes a sin kernel for use in Fourier series
780524`\spad{sin(x)} applies the sine operator to \spad{x}
780676`\spad{sin(st)} computes sine of a power series \spad{st}.
780760`\spad{sin(st)} computes sine of a power series \spad{st}.
780858`\spad{sin(z)} returns the sine of Laurent series \spad{z}.
780945`\spad{sin(z)} returns the sine of a Puiseux series \spad{z}.
781041`\spad{sin(x)} represents the Fortran intrinsic function SIN
781122`\spad{sin(x)} returns the sine of \spad{x}.
781180`\spad{sincos(st)} returns a record containing the sine and cosine of a power series \spad{st}.
781296`\spad{singRicDE(op,{} ezfactor)} returns \spad{[[f1,{}L1],{} [f2,{}L2],{}...,{} [fk,{}Lk]]} such that the singular \spad{++} part of any rational solution of the associated Riccati equation of \spad{op y = 0} must be one of the \spad{fi}\spad{'s} (up to the constant coefficient),{} in which case the
781296`-- equation for \spad{z = y e^{-int \spad{ai}}} is \spad{\spad{Li} z = 0}. Argument \spad{ezfactor} is a factorisation in \spad{UP},{} not necessarily into irreducibles.
781504`\spad{singRicDE(op,{} zeros,{} ezfactor)} returns \spad{[[f1,{} L1],{} [f2,{} L2],{} ... ,{} [fk,{} Lk]]} such that the singular part of any rational solution of the associated Riccati equation of \spad{op y=0} must be one of the \spad{fi}\spad{'s} (up to the constant coefficient),{} in which case the equation for \spad{z=y e^{-int p}} is \spad{
781504`--\spad{Li} z=0}. \spad{zeros(C(x),{}H(x,{}y))} returns all the \spad{P_i(x)}\spad{'s} such that \spad{H(x,{}P_i(x)) = 0 modulo C(x)}. Argument \spad{ezfactor} is a factorisation in \spad{UP},{} not necessarily into irreducibles.
781665`\spad{singleFace(ptin)} constructs a ifs representation of a face from an array of points
781719`\spad{singleFactorBound(p,{}r)} returns a bound on the infinite norm of the factor of \spad{p} with smallest Bombieri\spad{'s} norm. \spad{p} shall be of degree higher or equal to 2.
781808`\spad{singleFactorBound(p,{}r)} returns a bound on the infinite norm of the factor of \spad{p} with smallest Bombieri\spad{'s} norm. \spad{r} is a lower bound for the number of factors of \spad{p}. \spad{p} shall be of degree higher or equal to 2.
781916`\spad{singular?(a)} tests whether \spad{x = a} is singular.
781987`\spad{singular?(p)} tests whether \spad{p(x) = 0} is singular.
782059`\spad{singularAtInfinity?()} tests if there is a singularity at infinity.
782139`\spad{sinh2csch(f)} converts every \spad{sinh(u)} appearing in \spad{f} into \spad{1/csch(u)}.
782204`\spad{sinhIfCan(z)} returns sinh(\spad{z}) if possible,{} and "failed" otherwise.
782286`\spad{sinh(x)} applies the hyperbolic sine operator to \spad{x}
782493`\spad{sinh(st)} computes the hyperbolic sine of a power series \spad{st}.
782578`\spad{sinh(st)} computes the hyperbolic sine of a power series \spad{st}.
782677`\spad{sinh(z)} returns the hyperbolic sine of Laurent series \spad{z}.
782765`\spad{sinh(z)} returns the hyperbolic sine of a Puiseux series \spad{z}.
782862`\spad{sinh(x)} represents the Fortran intrinsic function SINH
782944`\spad{sinh(x)} returns the hyperbolic sine of \spad{x}.
783000`\spad{sinhcosh(st)} returns a record containing the hyperbolic sine and cosine of a power series \spad{st}.
783175`\spad{sipnt(a,{}b)} constructs a 2D point defined by a and \spad{b} of type Integer
783296`\spad{sipnt(a,{}b,{}c)} constructs a 3D point defined by a,{}\spad{b} and \spad{c} of type Integer
783417`\spad{sivec(a,{}b)} constructs a 2D vector defined by a and \spad{b} of type Integer
783538`\spad{sivec(a,{}b,{}c)} constructs a 3D vector defined by a,{}\spad{b} and \spad{c} of type Integer
783904`\spad{size?(u,{}n)} tests if \spad{u} has exactly \spad{n} elements.
784022`\spad{sizeLess?(x,{}y)} tests whether \spad{x} is strictly smaller than \spad{y} with respect to the \spadfunFrom{euclideanSize}{EuclideanDomain}.
784080`\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.
784180`\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: the time of multiplication of field elements depends on this size.
784289`\spad{sizeMultiplication()} returns the number of entries in the multiplication table of the field. Note: The time of multiplication of field elements depends on this size.
784379`\spad{sizeMultiplication(m)} returns the number of entries of the multiplication table {\em m}.
784511`\spad{sizePascalTriangle()} returns the number of entries currently stored in the table.
784940`\spad{size()} is the base of the random number generator
784991`\spad{size()} returns the number of elements in the set.
785041`\spad{size(x)} returns the number of monomials in \spad{x}.
785099`\spad{size(x)} returns the number of monomials in \spad{x}.
785158`\spad{size(l)} returns the number of monomials forming \spad{l}.
785225`\spad{size(x)} returns the number of monomials in \spad{x}.
785291`\spad{size(x)} returns the number of terms in \spad{x}. mapGen(\spad{f},{} \spad{a1}\spad{\^}\spad{e1} ... an\spad{\^}en) returns \spad{f(a1)\^e1 ... f(an)\^en}.
785367`The size of the isolating interval
785460`\spad{skewSFunction(li1,{}li2)} is the \spad{S}-function \indented{1}{of the partition difference \spad{li1 - li2}} \indented{1}{expressed in terms of power sum symmetric functions.}
785575`Constructs variable combinator
785622`Constructs a node combinator over combinators
785672`\spad{slash(f,{}g)} creates a form for the horizontal fraction of \spad{f} over \spad{g}.
785716`\spad{slex(ls)} sorts the argument sequence \spad{ls},{} then zips (see \spadfunFrom{map}{\spad{ListFunctions3}}) the original argument sequence with the sorted result to a list of pairs
786071`\spad{smaller?(x,{} y)} is a strict total ordering on the elements of the set.
786123`\spad{smesh(ptin,{}clos)} constructs a ifs representation of a mesh from a 2 dimensional array of points
786186`\spad{smith(m)} returns the Smith Normal form of the matrix \spad{m}.
786243`Undocumented.
786339`Undocumented.
786408`\spad{sncndn(s,{}c)} is used internally.
786520`\spad{solid?(x)} \undocumented
786582`\spad{solid(x,{}b)} \undocumented
786651`\spad{solve1(pol,{} eps)} finds the roots of the univariate polynomial polynomial \spad{pol} to precision eps. If \spad{K} is \spad{Fraction Integer} then only the real roots are returned,{} if \spad{K} is \spad{Complex Fraction Integer} then all roots are found.
786752`\axiom{solveFor(fun,{} \spad{jv})} tries to solve \axiom{fun} for the jet variable \axiom{\spad{jv}}.
786834`\spad{solveInField(lp)} finds the solution of the list \spad{lp} of rational functions with respect to all the symbols appearing in \spad{lp}.
786956`\spad{solveInField(lp,{}lv)} finds the solutions of the list \spad{lp} of rational functions with respect to the list of symbols \spad{lv}.
787091`\spad{solveInField(m,{} v,{} solve)} returns \spad{[[v_1,{}...,{}v_m],{} v_p]} such that the solutions in \spad{F} of the system \spad{m x = v} are \spad{v_p + c_1 v_1 + ... + c_m v_m} where the \spad{c_i's} are constants,{} and the \spad{v_i's} form a basis for the solutions of \spad{m x = 0}. Argument \sp
787091`--ad{solve} is a function for solving a single linear ordinary differential equation in \spad{F}.
787291`\spad{solveLinearPolynomialEquationByFractions([f1,{} ...,{} fn],{} g)} (where the \spad{fi} are relatively prime to each other) returns a list of \spad{ai} such that \spad{g/prod \spad{fi} = sum ai/fi} or returns "failed" if no such exists.
787507`\spad{solveLinearPolynomialEquationByRecursion([p1,{}...,{}pn],{}p)} returns the list of polynomials \spad{[q1,{}...,{}qn]} such that \spad{sum qi/pi = p / prod \spad{pi}},{} a recursion step for solveLinearPolynomialEquation as defined in \spadfun{PolynomialFactorizationExplicit} 
787507`--category (see \spadfun{solveLinearPolynomialEquation}). If no such list of \spad{qi} exists,{} then "failed" is returned.
787733`\spad{solveLinearPolynomialEquationByRecursion([p1,{}...,{}pn],{}p)} returns the list of polynomials \spad{[q1,{}...,{}qn]} such that \spad{sum qi/pi = p / prod \spad{pi}},{} a recursion step for solveLinearPolynomialEquation as defined in \spadfun{PolynomialFactorizationExplicit}
787733`-- category (see \spadfun{solveLinearPolynomialEquation}). If no such list of \spad{qi} exists,{} then "failed" is returned.
787960`\spad{solveLinearPolynomialEquation([f1,{} ...,{} fn],{} g)} (where the \spad{fi} are relatively prime to each other) returns a list of \spad{ai} such that \spad{g/prod \spad{fi} = sum ai/fi} or returns "failed" if no such list of \spad{ai}\spad{'s} exists.
788099`\spad{solveLinearPolynomialEquation([f1,{} ...,{} fn],{} g)} where (\spad{fi} relatively prime to each other) returns a list of \spad{ai} such that \spad{g} = sum \spad{ai} prod \spad{fj} (\spad{j} \spad{\=} \spad{i}) or equivalently g/prod \spad{fj} = sum (ai/fi) or returns "failed" if n
788099`--o such list exists
788318`\spad{solveLinearPolynomialEquation([f1,{} ...,{} fn],{} g)} (where the \spad{fi} are relatively prime to each other) returns a list of \spad{ai} such that \spad{g/prod \spad{fi} = sum ai/fi} or returns "failed" if no such list of \spad{ai}\spad{'s} exists.
789306`\spad{solveLinearPolynomialEquation([f1,{} ...,{} fn],{} g)} (where the \spad{fi} are relatively prime to each other) returns a list of \spad{ai} such that \spad{g/prod \spad{fi} = sum ai/fi} or returns "failed" if no such list of \spad{ai}\spad{'s} exists.
789507`\spad{solveLinear([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such \spad{ci}\spad{'s} exist in the quotient field of \spad{S}.
789632`\spad{solveLinear([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such \spad{ci}\spad{'s} exist in \spad{S}.
789742`\spad{solveLinear([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such \spad{ci}\spad{'s} exist in the quotient field of \spad{S}.
789859`\spad{solveLinear([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such \spad{ci}\spad{'s} exist in \spad{S}.
789961`\spad{solveLinearlyOverQ([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such rational numbers \spad{ci}\spad{'s} exist.
790087`\spad{solveLinearlyOverQ([v1,{}...,{}vn],{} u)} returns \spad{[c1,{}...,{}cn]} such that \spad{c1*v1 + ... + cn*vn = u},{} "failed" if no such rational numbers \spad{ci}\spad{'s} exist.
790205`\spad{solveRetract(lp,{}lv)} finds the solutions of the list \spad{lp} of rational functions with respect to the list of symbols \spad{lv}. The function tries to retract all the coefficients of the equations to \spad{Q} before solving if possible.
790340`\spad{solve(lp)} finds the solution in the algebraic closure of \spad{R} of the list \spad{lp} of rational functions with respect to all the symbols appearing in \spad{lp}.
790455`\spad{solve(u)} \undocumented
790522`\spad{solve(eq)} finds the solutions of the equation \spad{eq} where \spad{eq} is an equation of functions of type Expression(\spad{R}) with respect to the unique symbol \spad{x} appearing in \spad{eq}.
790621`\spad{solve(eq)} finds the solutions of the equation \spad{eq} with respect to the unique variable appearing in \spad{eq}.
790741`\spad{solve(expr)} finds the solutions of the equation \spad{expr} = 0 where \spad{expr} is a function of type Expression(\spad{R}) with respect to the unique symbol \spad{x} appearing in eq.
790830`\spad{solve(p)} finds the solution of a rational function \spad{p} = 0 with respect to the unique variable appearing in \spad{p}.
790940`\spad{solve(le)} finds the solutions of the list \spad{le} of equations of rational functions with respect to all symbols appearing in \spad{le}.
791072`\spad{solve(lp)} finds the solutions of the list \spad{lp} of rational functions with respect to all symbols appearing in \spad{lp}.
791194`\spad{solve(lp,{}lv)} finds the solutions in the algebraic closure of \spad{R} of the list \spad{lp} of rational functions with respect to the list of symbols \spad{lv}.
791322`\spad{solve(p,{}eps)} finds the real zeroes of a univariate rational polynomial \spad{p} with precision \spad{eps}.
791411`\spad{solve(p,{}eps)} finds the real zeroes of a univariate integer polynomial \spad{p} with precision \spad{eps}.
791490`\spad{solve(eq,{}x)} finds the solutions of the equation \spad{eq} where \spad{eq} is an equation of functions of type Expression(\spad{R}) with respect to the symbol \spad{x}.
791596`\spad{solve(eq,{}eps)} finds all of the real solutions of the univariate equation \spad{eq} of rational functions with respect to the unique variables appearing in \spad{eq},{} with precision \spad{eps}.
791721`\spad{solve(eq,{}v)} finds the solutions of the equation \spad{eq} with respect to the variable \spad{v}.
791848`\spad{solve(expr,{}x)} finds the solutions of the equation \spad{expr} = 0 with respect to the symbol \spad{x} where \spad{expr} is a function of type Expression(\spad{R}).
791944`\spad{solve(p,{}eps)} finds all of the real solutions of the univariate rational function \spad{p} with rational coefficients with respect to the unique variable appearing in \spad{p},{} with precision \spad{eps}.
792059`\spad{solve(p,{}v)} solves the equation \spad{p=0},{} where \spad{p} is a rational function with respect to the variable \spad{v}.
792176`\spad{solve(leqs,{} lvar)} returns a list of solutions to the list of equations \spad{leqs} with respect to the list of symbols lvar.
792300`\spad{solve(leq,{}eps)} finds all of the real solutions of the system \spad{leq} of equationas of rational functions with respect to all the variables appearing in \spad{lp},{} with precision \spad{eps}.
792437`\spad{solve(le,{}lv)} finds the solutions of the list \spad{le} of equations of rational functions with respect to the list of symbols \spad{lv}.
792582`\spad{solve(lp,{}eps)} finds all of the real solutions of the system \spad{lp} of rational functions over the rational numbers with respect to all the variables appearing in \spad{lp},{} with precision \spad{eps}.
792709`\spad{solve(lp,{}lv)} finds the solutions of the list \spad{lp} of rational functions with respect to the list of symbols \spad{lv}.
792844`\spad{solve(A,{}LB)} finds a particular soln of the systems \spad{AX = B} and a basis of the associated homogeneous systems \spad{AX = 0} where \spad{B} varies in the list of column vectors \spad{LB}.
793001`\spad{solve(A,{}B)} finds a particular solution of the system \spad{AX = B} and a basis of the associated homogeneous system \spad{AX = 0}.
793146`\spad{solve(A,{}B)} finds a particular solution of the system \spad{AX = B} and a basis of the associated homogeneous system \spad{AX = 0}.
793270`\spad{solve(A,{}LB)} finds a particular soln of the systems \spad{AX = B} and a basis of the associated homogeneous systems \spad{AX = 0} where \spad{B} varies in the list of column vectors \spad{LB}.
793406`\spad{solve(A,{}LB)} finds a particular soln of the systems \spad{AX = B} and a basis of the associated homogeneous systems \spad{AX = 0} where \spad{B} varies in the list of column vectors \spad{LB}.
793559`\spad{solve(m,{} x)} returns a basis for the solutions of \spad{D y = m y}. \spad{x} is the dependent variable.
793665`\spad{solve(A,{}B)} finds a particular solution of the system \spad{AX = B} and a basis of the associated homogeneous system \spad{AX = 0}.
793806`\spad{solve(m,{} v,{} solve)} returns \spad{[[v_1,{}...,{}v_m],{} v_p]} such that the solutions in \spad{F} of the system \spad{D x = m x + v} are \spad{v_p + c_1 v_1 + ... + c_m v_m} where the \spad{c_i's} are constants,{} and the \spad{v_i's} form a basis for the solutions of \spad{D x = m x}. Argument \spad{solve} is 
793806`--a function for solving a single linear ordinary differential equation in \spad{F}.
793992`\spad{solve(eq,{} y,{} x)} returns either a solution of the ordinary differential equation \spad{eq} or "failed" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \spad{[h,{} [b1,{}...,{}bm]]} where \spad{h} is a particular solution and \spad{[b1,{}...bm]} are linearly independent solutions of the associated 
793992`--homogenuous equation \spad{f(x,{}y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \spad{f}(\spad{x},{}\spad{y})},{} a solution is of the form \spad{h(x,{}y)} where \spad{h(x,{}y) = c} is a first integral of the equation for any constant \spad{c}; error if the equation is not one of those 2 forms.
794135`\spad{solve(eq,{} y,{} x)} returns either a solution of the ordinary differential equation \spad{eq} or "failed" if no non-trivial solution can be found; If the equation is linear ordinary,{} a solution is of the form \spad{[h,{} [b1,{}...,{}bm]]} where \spad{h} is a particular solution and and \spad{[b1,{}...bm]} are linearly independent solutions of the associated homoge
794135`--nuous equation \spad{f(x,{}y) = 0}; A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; If the equation is of the form {dy/dx = \spad{f}(\spad{x},{}\spad{y})},{} a solution is of the form \spad{h(x,{}y)} where \spad{h(x,{}y) = c} is a first integral of the equation for any constant \spad{c}.
794268`\spad{solve(op,{} g,{} x)} returns either a solution of the ordinary differential equation \spad{op y = g} or "failed" if no non-trivial solution can be found; When found,{} the solution is returned in the form \spad{[h,{} [b1,{}...,{}bm]]} where \spad{h} is a particular solution and and \spad{[b1,{}...bm]} are linearly independent solutions of the associated homogenuous equation \sp
794268`--ad{op y = 0}. A full basis for the solutions of the homogenuous equation is not always returned,{} only the solutions which were found; \spad{x} is the dependent variable.
794390`\spad{solve([eq_1,{}...,{}eq_n],{} [y_1,{}...,{}y_n],{} x)} returns either "failed" or,{} if the equations form a fist order linear system,{} a solution of the form \spad{[y_p,{} [b_1,{}...,{}b_n]]} where \spad{h_p} is a particular solution and \spad{[b_1,{}...b_m]} are linearly independent solutions of the associated homogenuous system.
794390`-- error if the equations do not form a first order linear system
794559`\spad{solve([eq_1,{}...,{}eq_n],{} [y_1,{}...,{}y_n],{} x)} returns either "failed" or,{} if the equations form a fist order linear system,{} a solution of the form \spad{[y_p,{} [b_1,{}...,{}b_n]]} where \spad{h_p} is a particular solution and \spad{[b_1,{}...b_m]} are linearly independent solutions of the associated homogenuous system. error if 
794559`--the equations do not form a first order linear system
794718`\spad{solve(m,{} v,{} x)} returns \spad{[v_p,{} [v_1,{}...,{}v_m]]} such that the solutions of the system \spad{D y = m y + v} are \spad{v_p + c_1 v_1 + ... + c_m v_m} where the \spad{c_i's} are constants,{} and the \spad{v_i's} form a basis for the solutions of \spad{D y = m y}. \spad{x} is the dependent variable.
794869`\spad{solve(M(x,{}y),{} N(x,{}y),{} y,{} x)} returns \spad{F(x,{}y)} such that \spad{F(x,{}y) = c} for a constant \spad{c} is a first integral of the equation \spad{M(x,{}y) dx + N(x,{}y) dy = 0},{} or "failed" if no first-integral can be found.
794970`\spad{solve(eq,{} y,{} x = a,{} [y0,{}...,{}ym])} returns either the solution of the initial value problem \spad{eq,{} y(a) = y0,{} y'(a) = y1,{}...} or "failed" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \spad{dy/dx = f(x,{}y)}.
795091`\spad{solve(eq,{} y,{} x = a,{} [y0,{}...,{}ym])} returns either the solution of the initial value problem \spad{eq,{} y(a) = y0,{} y'(a) = y1,{}...} or "failed" if the solution cannot be found; error if the equation is not one linear ordinary or of the form \spad{dy/dx = f(x,{}y)}.
795202`\spad{solve(op,{} g,{} x,{} a,{} [y0,{}...,{}ym])} returns either the solution of the initial value problem \spad{op y = g,{} y(a) = y0,{} y'(a) = y1,{}...} or "failed" if the solution cannot be found; \spad{x} is the dependent variable.
795301`\spad{solveid(h,{} prime,{} table)} computes the coefficients of the extended euclidean algorithm for a list of polynomials whose tablePow is table and with right side \spad{h}.
795392`\spad{someBasis()} returns some \spad{R}-module basis.
795616`\spad{sort!(u)} returns \spad{u} with its elements in ascending order.
795912`\spad{sort!(p,{}u)} returns \spad{u} with its elements ordered by \spad{p}.
796074`\axiom{sortLD(\spad{sys})} sorts the functions in \axiom{\spad{sys}} according to their leading derivatives.
796196`\spad{sort(lp)} sorts a list of permutations {\em lp} according to cycle structure first according to length of cycles,{} second,{} if \spad{S} has \spadtype{Finite} or \spad{S} has \spadtype{OrderedSet} according to lexicographical order of entries in cycles of equal length.
796252`\spad{sort(u)} returns an \spad{u} with elements in ascending order. Note: \axiom{sort(\spad{u}) = sort(\spad{<=},{}\spad{u})}.
796484`\spad{sort(p,{}a)} returns a copy of \axiom{a} sorted using total ordering predicate \spad{p}.
796553`\axiom{sort(\spad{v},{}\spad{ps})} returns \axiom{us,{}\spad{vs},{}\spad{ws}} such that \axiom{us} is \axiom{collectUnder(\spad{ps},{}\spad{v})},{} \axiom{\spad{vs}} is \axiom{collect(\spad{ps},{}\spad{v})} and \axiom{\spad{ws}} is \axiom{collectUpper(\spad{ps},{}\spad{v})}.
796830`\spad{sorted?(u)} tests if the elements of \spad{u} are in ascending order.
797127`\spad{sorted?(p,{}a)} tests if \axiom{a} is sorted according to predicate \spad{p}.
797204`\axiom{sortedPurge!(A,{} crit)} is like \axiom{purge},{} however,{} with the additional assumption that \axiom{crit} respects the ordering of the indices.
797281`\spad{space()} provides the blank character.
797319`\spad{space(l)} takes a list of draw options,{} \spad{l},{} and checks to see if it contains the option \spad{space}. If the the option doesn\spad{'t} exist,{} then an empty space is returned.
797405`\spad{space specifies} the space into which we will draw. If none is given then a new space is created.
797539`\spad{spanningForestArrow(s)} constructs a spanning tree for every arrow.
797686`\spad{spanningForestNode(s)} constructs a spanning tree for every vertex.
797844`\spad{spanningTreeArrow(s,{}i)} constructs a spanning tree for graph \spad{'s'} rooted at the arrow indexed by 'i'. The tree will expand out from 'i' only stopping when reaching a arrow that has already been visited (that is: loop detected). Elements in the tree are Integer,{} a positive Integer represents a arrow and a negative Integer represents a repeated arrow. note: it is possible that nodes may be visited many t
797844`--imes,{} only arrows must not be re-visited.
798013`\spad{spanningTreeNode(s,{} i)} constructs a spanning tree for graph \spad{'s'} rooted at the node indexed by 'i'. The tree will expand out from 'i' only stopping when reaching a vertex that has already been visited (that is: loop detected). Elements in the tree are Integer,{} a positive Integer represents a vertex and a negative Integer represents a repeated vertex.
798099`\spad{specialTrigs(x,{}l)} should be local but conditional
798208`\spad{spherical(pt)} transforms \spad{pt} from spherical coordinates to Cartesian coordinates: the function produced will map the point \spad{(r,{}theta,{}phi)} to \spad{x = r*sin(phi)*cos(theta)},{} \spad{y = r*sin(phi)*sin(theta)},{} \spad{z = r*cos(phi)}.
798336`\spad{split!(u,{}n)} splits \spad{u} into two aggregates: \axiom{\spad{v} = rest(\spad{u},{}\spad{n})} and \axiom{\spad{w} = first(\spad{u},{}\spad{n})},{} returning \axiom{\spad{v}}. Note: afterwards \axiom{rest(\spad{u},{}\spad{n})} returns \axiom{empty()}.
798426`\spad{splitConstant(f,{} x)} returns \spad{[c,{} g]} such that \spad{f = c * g} and \spad{c} does not involve \spad{t}.
798521`\spad{splitDenominator([q1,{}...,{}qn])} returns \spad{[[p1,{}...,{}pn],{} d]} such that \spad{\spad{qi} = pi/d} and \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
798610`\spad{splitDenominator([q1,{}...,{}qn])} returns \spad{[[p1,{}...,{}pn],{} d]} such that \spad{\spad{qi} = pi/d} and \spad{d} is a common denominator for the \spad{qi}\spad{'s}.
798692`\spad{splitDenominator(q)} returns \spad{[p,{} d]} such that \spad{q = p/d} and \spad{d} is a common denominator for the elements of \spad{q}.
798794`\spad{splitDenominator(q)} returns \spad{[p,{} d]} such that \spad{q = p/d} and \spad{d} is a common denominator for the coefficients of \spad{q}.
798899`\spad{splitDenominator(op,{} [g1,{}...,{}gm])} returns \spad{op0,{} [h1,{}...,{}hm]} such that the equations \spad{op y = c1 g1 + ... + cm gm} and \spad{op0 y = c1 h1 + ... + cm hm} have the same solutions.
799017`\axiom{splitNodeOf!(\spad{l},{}a,{}\spad{ls})} returns \axiom{a} where the children list of \axiom{\spad{l}} has been set to \axiom{[[\spad{s}]\$\% for \spad{s} in \spad{ls} | not nodeOf?(\spad{s},{}a)]}. Thus,{} if \axiom{\spad{l}} is not a node of \axiom{a},{} this latter splitting tree is unchanged.
799101`\axiom{splitNodeOf!(\spad{l},{}a,{}\spad{ls},{}sub?)} returns \axiom{a} where the children list of \axiom{\spad{l}} has been set to \axiom{[[\spad{s}]\$\% for \spad{s} in \spad{ls} | not subNodeOf?(\spad{s},{}a,{}sub?)]}. Thus,{} if \axiom{\spad{l}} is not a node of \axiom{a},{} this latter splitting tree is unchanged.
799200`\spad{splitSquarefree(p,{} D)} returns \spad{[n_1 n_2\^2 ... n_m\^m,{} s_1 s_2\^2 ... s_q\^q]} such that \spad{p = n_1 n_2\^2 ... n_m\^m s_1 s_2\^2 ... s_q\^q},{} each \spad{n_i} is normal \spad{w}.\spad{r}.\spad{t}. \spad{D} and each \spad{s_i} is special \spad{w}.\spad{r}.\spad{t} \spad{D}. \spad{D} is the derivation to use.
799324`\spad{split(u(x) + sum_{P(a)=0} Q(a,{}x))} returns \spad{u(x) + sum_{P1(a)=0} Q(a,{}x) + ... + sum_{Pn(a)=0} Q(a,{}x)} where \spad{P1},{}...,{}\spad{Pn} are the factors of \spad{P}.
799423`\spad{split(u(x) + sum_{P(a)=0} Q(a,{}x))} returns \spad{u(x) + sum_{P1(a)=0} Q(a,{}x) + ... + sum_{Pn(a)=0} Q(a,{}x)} where \spad{P1},{}...,{}\spad{Pn} are the factors of \spad{P}.
799566`\spad{split(p)} returns a prime factorisation of \spad{p} over its splitting field.
799620`\spad{split(p,{} D)} returns \spad{[n,{}s]} such that \spad{p = n s},{} all the squarefree factors of \spad{n} are normal \spad{w}.\spad{r}.\spad{t}. \spad{D},{} and \spad{s} is special \spad{w}.\spad{r}.\spad{t}. \spad{D}. \spad{D} is the derivation to use.
799714`\spad{split(aG,{} vector)} returns a subalgebra \spad{A} of all square matrix of dimension \spad{n} as a list of list of matrices,{} generated by the list of matrices \spad{aG},{} where \spad{n} denotes both the size of vector as well as the dimension of each of the square matrices. {\em V R} is an A-module in the natural way. split(\spad{aG},{} vector) then checks whether the cyclic submodule g
799714`--enerated by {\em vector} is a proper submodule of {\em V R}. If successful,{} it returns a two-element list,{} which contains first the list of the representations of the submodule,{} then the list of the representations of the factor module. If the vector generates the whole module,{} a one-element list of the old representation is given. Note: a later version this should call the other split.
799824`\spad{split(aG,{}submodule)} uses a proper \spad{submodule} of {\em R^n} to create the representations of the \spad{submodule} and of the factor module.
799942`\spad{split(x,{}b)} splits binary tree \spad{b} into two trees,{} one with elements greater than \spad{x},{} the other with elements less than \spad{x}.
800018`\spad{split(s,{}c)} returns a list of substrings delimited by character \spad{c}.
800080`\spad{split(s,{}cc)} returns a list of substrings delimited by characters in \spad{cc}.
800147`\spad{spnt(a,{}b)} constructs a 2D point defined by a and \spad{b} of type \spad{R}
800212`\spad{spnt(a,{}b,{}c)} constructs a 3D point defined by a,{}\spad{b} and \spad{c} of type \spad{R}
800289`\spad{sqfrFactor(base,{}exponent)} creates a factored object with a single factor whose \spad{base} is asserted to be square-free (flag = "sqfr").
800343`\spad{sqfree(p)} returns the product of square free factors of \spad{p}
800602`\spad{sqrt(x)} represents the Fortran intrinsic function SQRT
800684`\axiom{sqrt(\spad{x})} is \axiom{\spad{x} ^ (1/2)}
800744`\axiom{sqrt(\spad{x})} is \axiom{\spad{x} ^ (1/2)}
800794`\spad{sqrt(x)} returns the square root of \spad{x}. The branch cut lies along the negative real axis,{} continuous with quadrant II.
800839`\axiom{sqrt(\spad{x})} is \axiom{\spad{x} ^ (1/2)}
800941`\spad{sqrt(b,{}a)} returns a square root of \spad{b}. Argument \spad{a} is a square root of \spad{b} \spad{(mod p)}.
801002`\axiom{sqrt(\spad{x},{}\spad{n})} is \axiom{\spad{x} ^ (1/n)}
801196`\spad{square?(m)} returns \spad{true} if \spad{m} is a square matrix (\spadignore{i.e.} if \spad{m} has the same number of rows as columns) and \spad{false} otherwise.
801259`\spad{square?(m)} returns \spad{true} if \spad{m} is a square matrix (\spadignore{i.e.} if \spad{m} has the same number of rows as columns) and \spad{false} otherwise.
801337`\axiom{squareFreeFactors(\spad{p})} returns the square-free factors of \axiom{\spad{p}} over \axiom{\spad{R}}
801438`\axiom{squareFreeLexTriangular(base,{} norm?)} decomposes the variety associated with \axiom{base} into square-free regular chains. Thus a point belongs to this variety iff it is a regular zero of a re
801438`--gular set in in the output. Note that \axiom{base} needs to be a lexicographical Groebner basis of a zero-dimensional ideal. If \axiom{norm?} is \axiom{\spad{true}} then the regular sets are normalized.
801745`\spad{squareFreePart(p)} returns a polynomial which has the same irreducible factors as the univariate polynomial \spad{p},{} but each factor has multiplicity one.
802011`\spad{squareFreePart(p)} returns product of all the irreducible factors of polynomial \spad{p} each taken with multiplicity one.
802097`\spad{squareFreePart(x)} returns a product of prime factors of \spad{x} each taken with multiplicity one.
802162`\spad{squareFreePart(p,{}ts)} returns \spad{lpwt} such that \spad{lpwt.i.val} is a square-free polynomial \spad{w}.\spad{r}.\spad{t}. \spad{lpwt.i.tower},{} this polynomial being associated with \spad{p} modulo \spad{lpwt.i.tower},{} for every \spad{i}. Moreover,{} the list of the \spad{lpwt.i.tower} is a split of Kalkbrener of \spad{ts}. WARNING: This assumes that \spad{p} is a non-constant polynomi
802162`--al such that if \spad{p} is added to \spad{ts},{} then the resulting set is a regular triangular set.
802405`\spad{squareFreePolynomial(p)} returns the square-free factorization of the univariate polynomial \spad{p}.
802548`\spad{squareFreePolynomial(p)} \undocumented{}
802716`\spad{squareFreePrim(p)} compute the square free decomposition of a primitive multivariate polynomial \spad{p}.
802796`\spad{squareFree(n)} returns the square free factorization of integer \spad{n}
802870`\spad{squareFree(p)} computes the square free decomposition of a multivariate polynomial \spad{p}.
802946`\spad{squareFree(p)} returns the square-free factorization of the polynomial \spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime.
803025`\spad{squareFree(p)} computes the square-free factorization of the univariate polynomial \spad{p}. Each factor has no repeated roots,{} and the factors are pairwise relatively prime.
803415`\spad{squareFree(p)} returns the square-free factorization of the univariate polynomial \spad{p} with coefficients which are fractions of polynomials over \spad{R}. Each factor has no repeated roots and the factors are pairwise relatively prime.
803566`\spad{squareFree(p)} computes the square free decomposition of a multivariate polynomial \spad{p} presented as a univariate polynomial with multivariate coefficients.
803698`\spad{squareFree(ts)} returns the square-free factorization of \spad{ts}. Moreover,{} each factor is a Lazard triangular set and the decomposition is a Kalkbrener split of \spad{ts},{} which is enough here for the matter of solving zero-dimensional algeb
803698`--raic systems. WARNING: \spad{ts} is not checked to be zero-dimensional.
803952`\spad{squareFree(p)} returns the square free factorization of the polynomial \spad{p}.
804044`\spad{squareFree(x)} returns the square-free factorization of \spad{x} \spadignore{i.e.} such that the factors are pairwise relatively prime and each has multiple prime factors.
804115`\spad{squareMatrix(m)} converts a matrix of type \spadtype{Matrix} to a matrix of type \spadtype{SquareMatrix}.
804235`\spad{squareTop(m)} returns an \spad{n}-by-\spad{n} matrix consisting of the first \spad{n} rows of the \spad{m}-by-\spad{n} matrix \spad{m}. Error: if \spad{m < n}.
804295`\spad{stFunc1(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user.
804403`\spad{stFunc2(f)} is a local function exported due to compiler problem. This function is of no interest to the top-level user.
804528`\spad{stFuncN(f)} is a local function xported due to compiler problem. This function is of no interest to the top-level user.
804648`\spad{stack([x,{}y,{}...,{}z])} creates a stack with first (top) element \spad{x},{} second element \spad{y},{}...,{}and last element \spad{z}.
804692`\spad{standardBasisOfCyclicSubmodule(lm,{}v)} returns a matrix as follows. It is assumed that the size \spad{n} of the vector equals the number of rows and columns of the matrices. Then the matrices generate a subalgebra,{} say \spad{A},{} of the algebra of all square matrices of dimension \spad{n}. {\em V R} is an \spad{A}-module in the natural way. standardBasisOfCyclicS
804692`--ubmodule(\spad{lm},{}\spad{v}) calculates a matrix whose non-zero column vectors are the \spad{R}-Basis of {\em Av} achieved in the way as described in section 6 of \spad{R}. A. Parker\spad{'s} "The Meat-Axe". Note: in contrast to {\em cyclicSubmodule},{} the result is not in echelon form.
804825`\spad{startPolynomial(p)} uses the ideas of Schoenhage\spad{'s} variant of Graeffe\spad{'s} method to construct circles which separate roots to get a good start polynomial,{} \spadignore{i.e.} one whose image under the Chinese Remainder Isomorphism has both entries of norm smaller and greater or equal to 1. In case the roots are found during internal calculations. The corresponding factors are in {
804825`--\em factors} which are otherwise 1.
804932`\axiom{startStats!(\spad{x})} initializes the statisitics process and sets the comments to display when statistics are printed
805013`\axiom{startTableGcd!(\spad{s1},{}\spad{s2},{}\spad{s3})} is an internal subroutine,{} exported only for developement.
805103`\axiom{startTableGcd!(\spad{s1},{}\spad{s2},{}\spad{s3})} is an internal subroutine,{} exported only for developement.
805203`\axiom{startTableGcd!(\spad{s1},{}\spad{s2},{}\spad{s3})} is an internal subroutine,{} exported only for developement.
805411`\axiom{startTableInvSet!(\spad{s1},{}\spad{s2},{}\spad{s3})} is an internal subroutine,{} exported only for developement.
805625`\axiom{statistics()} prints a statistic on the use of the lazy evaluation mechanism. It displays the number of lazy differentiations performed and how many of them had to be executed explicitly later on.
805682`\spad{statistics()} prints out a summary of the outcome of the testcases so far. Use \axiom{clear completely} to reset the statistics.
805731`\axiom{status(\spad{n})} returns the status of the node \spad{n}.
805786`\spad{status(s)} returns \spad{true} if the quasi-algebraic set is empty,{} \spad{false} if it is not,{} and "failed" if not yet known
805875`\spad{stirling1(n,{}m)} returns the Stirling number of the first kind denoted \spad{S[n,{}m]}.
805941`\spad{stirling2(n,{}m)} returns the Stirling number of the second kind denoted \spad{SS[n,{}m]}.
806007`\axiom{stirling(\spad{i},{} \spad{k},{} \spad{q})} computes the corresponding modified Stirling number.
806132`\spad{stopMusserTrials()} returns the bound on the number of factors for which \spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \spad{2^stopMusserTrials()} trials.
806210`\spad{stopMusserTrials(n)} sets to \spad{n} the bound on the number of factors for which \spadfun{modularFactor} stops to look for an other prime. You will have to remember that the step of recombining the extraneous factors may take up to \spad{2^n} trials. Returns the previous value.
806303`\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.
806372`\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.
806451`\axiom{stopTableGcd!()} is an internal subroutine,{} exported only for developement.
806617`\axiom{stopTableInvSet!()} is an internal subroutine,{} exported only for developement.
806789`\spad{stop()} creates a representation of a STOP statement.
808424`\spad{stransform(cpx)} construct transform as function of complex variable can only be used when \spad{PT} is SArgand so this can be converted to \spad{PT} \spad{->} \spad{PT}
808516`\spad{stransform(gen)} construct transform in general form as a mapping from \spad{PT} to \spad{PT}
808572`\spad{stransform(m)} construct transorm using a multivector
808637`\spad{stransform(m)} constructs transform with given matrix elements
808708`\spad{stranslate(offsetx,{}offsety,{}offsetz,{}scalex,{}scaley,{}scalez)} constructs a transform which represents pure translation we can also combine with scale which,{} for instance,{} is useful when writing to \spad{SVG} file because the \spad{y} dimension is inverted
808827`\spad{stream(x)} returns stream of entries
808885`\spad{string?(s)} is \spad{true} if \spad{s} is an atom and belong to \spad{Str}.
808964`\spad{string(f)} creates \spad{f} with string quotes.
809006`\spad{string(i)} returns the decimal representation of \spad{i} in a string
809057`\spad{string(s)} returns \spad{s} as an element of \spad{Str}. Error: if \spad{s} is not an atom that also belongs to \spad{Str}.
809131`\spad{string(s)} converts the symbol \spad{s} to a string. Error: if the symbol is subscripted.
809173`\spad{stripCommentsAndBlanks(s)} treats \spad{s} as a piece of FriCAS input,{} and removes comments,{} and leading and trailing blanks.
809246`\spad{strongGenerators(gp)} returns strong generators for the group {\em gp}.
809392`\axiom{stronglyReduce(\spad{p},{}\spad{ts})} returns a polynomial \axiom{\spad{r}} such that \axiom{stronglyReduced?(\spad{r},{}\spad{ts})} holds and there exists some product \axiom{\spad{h}} of \axiom{initials(\spad{ts})} such that \axiom{\spad{h*p} - \spad{r}} lies in the ideal generated by \axiom{\spad{ts}}.
809536`\axiom{stronglyReduced?(\spad{ts})} returns \spad{true} iff every element of \axiom{\spad{ts}} is reduced \spad{w}.\spad{r}.\spad{t} to any other element of \axiom{\spad{ts}}.
809688`\axiom{stronglyReduced?(\spad{p},{}\spad{ts})} returns \spad{true} iff \axiom{\spad{p}} is reduced \spad{w}.\spad{r}.\spad{t}. \axiom{\spad{ts}}.
809767`\spad{structuralConstants()} calculates the structural constants \spad{[(gammaijk) for k in 1..rank()]} defined by \spad{\spad{vi} * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \spad{v1},{}...,{}\spad{vn} is the fixed \spad{R}-module basis.
809858`\spad{structuralConstants()} calculates the structural constants \spad{[(gammaijk) for k in 1..rank()]} defined by \spad{\spad{vi} * vj = gammaij1 * v1 + ... + gammaijn * vn},{} where \spad{v1},{}...,{}\spad{vn} is the fixed \spad{R}-module basis.
810138`\spad{structuralConstants(basis)} takes the \spad{basis} of a matrix algebra,{} \spadignore{e.g.} the result of \spadfun{basisOfCentroid} and calculates the structural constants. Note,{} that the it is not checked,{} whether \spad{basis} really is a \spad{basis} of a matrix algebra.
810240`\spad{structuralConstants([v1,{}v2,{}...,{}vm])} calculates the structural constants \spad{[(gammaijk) for k in 1..m]} defined by \spad{\spad{vi} * vj = gammaij1 * v1 + ... + gammaijm * vm},{} where \spad{[v1,{}...,{}vm]} is an \spad{R}-module basis of a subalgebra.
810342`\spad{structuralConstants(ls,{}mt)} determines the structural constants of an algebra with generators \spad{ls} and multiplication table \spad{mt},{} the entries of which must be given as linear polynomials in the indeterminates given by \spad{ls}. The result is in particular useful \indented{1}{as fourth argument for \spadtype{AlgebraGivenByStructuralC
810342`--onstants}} \indented{1}{and \spadtype{GenericNonAssociativeAlgebra}.}
810495`\spad{structuralConstants(ls,{}mt)} determines the structural constants of an algebra with generators \spad{ls} and multiplication table \spad{mt},{} the entries of which must be given as linear polynomials in the indeterminates given by \spad{ls}. The result is in particular useful \indented{1}{as fourth argument for \spadtype{AlgebraGivenByStructuralConstants}} \indented
810495`--{1}{and \spadtype{GenericNonAssociativeAlgebra}.}
810628`\spad{struveH(v,{} z)} is the Struve \spad{H} function
810691`\spad{struveH(v,{} z)} is the Struve \spad{H} function.
810750`\spad{struveL(v,{} z)} is the Struve \spad{L} function defined by the formula \spad{struveL(v,{} z) = -\%i^exp(-v*\%pi*\%i/2)*struveH(v,{} \%i*z)}
810813`\spad{struveL(v,{} z)} is the Struve \spad{L} function defined by the formula \spad{struveL(v,{} z) = -\%i^exp(-v*\%pi*\%i/2)*struveH(v,{} \%i*z)}.
810872`\spad{stube(functions:List Clipd,{}r,{}n)} creates a tube of radius \spad{r} around the curve functions.
811045`\axiom{sturmSequence(\spad{p}) = sylvesterSequence(\spad{p},{}\spad{p'})}
811148`\axiom{sturmVariationsOf(\spad{l})} is the number of sign variations in the list of numbers \spad{l},{} note that the first term counts as a sign
811292`\spad{style(s)} specifies the drawing style in which the graph will be plotted by the indicated string \spad{s}. This option is expressed in the form \spad{style == s}.
811337`\spad{style(l,{}s)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{style}. If the option does not exist the value,{} \spad{s} is returned.
811413`\axiom{subCase?(\spad{lpwt1},{}\spad{lpwt2})} is an internal subroutine,{} exported only for developement.
811540`\axiom{subCase?(\spad{lpwt1},{}\spad{lpwt2})} is an internal subroutine,{} exported only for developement.
811677`\spad{subHeight(f)} returns the height of form \spad{f} below the base line.
811814`\spad{subMatrix(x,{}i1,{}i2,{}j1,{}j2)} extracts the submatrix \spad{[x(i,{}j)]} where the index \spad{i} ranges from \spad{i1} to \spad{i2} and the index \spad{j} ranges from \spad{j1} to \spad{j2}.
811906`\axiom{subNode?(\spad{n1},{}\spad{n2},{}\spad{o2})} returns \spad{true} iff \axiom{value(\spad{n1}) = value(\spad{n2})} and \axiom{\spad{o2}(condition(\spad{n1}),{}condition(\spad{n2}))}
811981`\axiom{subNodeOf?(\spad{s},{}a,{}sub?)} returns \spad{true} iff for some node \axiom{\spad{n}} in \axiom{a} we have \axiom{\spad{s} = \spad{n}} or \axiom{status(\spad{n})} and \axiom{subNode?(\spad{s},{}\spad{n},{}sub?)}.
812074`\axiom{subPolSet?(\spad{lp1},{}\spad{lp2})} returns \spad{true} iff \axiom{\spad{lp1}} is a sub-set of \axiom{\spad{lp2}}.
812161`\axiom{subPolSet?(\spad{lp1},{}\spad{lp2})} returns \spad{true} iff \axiom{\spad{lp1}} is a sub-set of \axiom{\spad{lp2}}.
812258`\axiom{subQuasiComponent?(\spad{ts},{}lus)} returns \spad{true} iff \axiom{subQuasiComponent?(\spad{ts},{}us)} holds for one \spad{us} in \spad{lus}.
812349`\axiom{subQuasiComponent?(\spad{ts},{}lus)} returns \spad{true} iff \axiom{subQuasiComponent?(\spad{ts},{}us)} holds for one \spad{us} in \spad{lus}.
812450`\axiom{subQuasiComponent?(\spad{ts},{}us)} returns \spad{true} iff \axiomOpFrom{internalSubQuasiComponent?}{QuasiComponentPackage} returs \spad{true}.
812535`\axiom{subQuasiComponent?(\spad{ts},{}us)} returns \spad{true} iff \axiomOpFrom{internalSubQuasiComponent?(\spad{ts},{}us)}{QuasiComponentPackage} returs \spad{true}.
812630`\axiom{subResultantChain(a,{}\spad{b})},{} where \axiom{a} and \axiom{\spad{b}} are not contant polynomials with the same main variable,{} returns the subresultant chain of \axiom{a} and \axiom{\spad{b}}.
812737`\axiom{subResultantGcdEuclidean(\spad{P},{}\spad{Q})} carries out the equality \axiom{coef1*P + coef2*Q = \spad{+/-} S_i(\spad{P},{}\spad{Q})} where the degree (not the indice) of the subresultant \axiom{S_i(\spad{P},{}\spad{Q})} is the smaller as possible.
812861`\axiom{subResultantGcd(\spad{P},{} \spad{Q})} returns the \spad{gcd} of two primitive polynomials \axiom{\spad{P}} and \axiom{\spad{Q}}.
812942`\axiom{subResultantGcd(a,{}\spad{b})} computes a \spad{gcd} of \axiom{a} and \axiom{\spad{b}} where \axiom{a} and \axiom{\spad{b}} are assumed to have the same main variable \axiom{\spad{v}} and are viewed as univariate polynomials in \axiom{\spad{v}} with coefficients in the fraction field of the polynomial ring generated by their other variables over \axiom{\spad{R}}.
813041`\spad{subResultantGcd(p,{}q)} computes the \spad{gcd} of the polynomials \spad{p} and \spad{q} using the SubResultant \spad{GCD} algorithm.
813137`\axiom{subResultantsChain(a,{}\spad{b})} returns the list of the non-zero sub-resultants of \axiom{a} and \axiom{\spad{b}} sorted by increasing degree.
813243`\spad{subSet(n,{}m,{}k)} calculates the {\em k}\spad{-}th {\em m}-subset of the set {\em 0,{}1,{}...,{}(n-1)} in the lexicographic order considered as a decreasing map from {\em 0,{}...,{}(m-1)} into {\em 0,{}...,{}(n-1)}. See \spad{S}.\spad{G}. Williamson: Theorem 1.60. Error: if not {\em (0 <= m <= n and 0 < = k < (n choose m))}.
813342`\axiom{subTriSet?(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} is a sub-set of \axiom{us}.
813419`\axiom{subTriSet?(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} is a sub-set of \axiom{us}.
813506`\spad{sub(f,{}n)} creates a form for \spad{f} subscripted by \spad{n}.
813636`\spad{subdiagramSvg(sc,{}n,{}dispArrowName,{}deep)} creates a branch of an \spad{SVG} diagram diagram under an already existing scene node \spad{sc}. \spad{n:}\% is the graph that will be written dispArrowName:Boolean is \spad{true} to include the name of each arrow
813728`\spad{subdivide(me)} for each face calculate a new point in the centre of the face,{} then build a new set of faces which include the new point.
813775`\spad{submod(a,{}b,{}p)},{} \spad{0<=a,{}b<p>1},{} means \spad{a-b mod p}.
813832`\spad{subresultantSequence(p1,{}p2)} computes the (standard) subresultant sequence of \spad{p1} and \spad{p2}
813980`\spad{subresultantVector(p,{} q)} returns \spad{[p0,{}...,{}pn]} where \spad{pi} is the \spad{i}-th subresultant of \spad{p} and \spad{q}. In particular,{} \spad{p0 = resultant(p,{} q)}.
814068`\spad{subscript(s,{} [a1,{}...,{}an])} returns \spad{s} subscripted by \spad{[a1,{}...,{}an]}.
814183`\spad{subset?(u,{}v)} tests if \spad{u} is a subset of \spad{v}. Note: equivalent to \axiom{reduce(and,{}{member?(\spad{x},{}\spad{v}) for \spad{x} in \spad{u}},{}\spad{true},{}\spad{false})}.
814239`\spad{subspace?(b1,{}b2)} tests if the first space is include in the second.
814301`\spad{subspace()} \undocumented
814346`\spad{subspace(s)} returns the \spadtype{SubSpace} which holds all the point information in the \spadtype{ThreeSpace},{} \spad{s}.
814412`\spad{subspace(v)} returns the contents of the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport},{} as a subspace of the domain \spad{ThreeSpace}.
814491`\spad{subspace(v,{}sp)} places the contents of the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport},{} in the subspace \spad{sp},{} which is of the domain \spad{ThreeSpace}.
815633`\spad{substHP options} returns a specification for Hermite-Pade approximation with the substitution operator
820978`\spad{subst(f,{} k = g)} replaces the kernel \spad{k} by \spad{g} formally in \spad{f}.
821037`\spad{subst(f,{} [k1 = g1,{}...,{}kn = gn])} replaces the kernels \spad{k1},{}...,{}\spad{kn} by \spad{g1},{}...,{}\spad{gn} formally in \spad{f}.
821102`\spad{subst(eq1,{}eq2)} substitutes \spad{eq2} into both sides of \spad{eq1} the \spad{lhs} of \spad{eq2} should be a kernel
821301`\axiom{subst(\spad{f},{} \spad{jv},{} exp)} substitutes \axiom{exp} for the jet variable \axiom{\spad{jv}} in the function \axiom{\spad{f}}.
821367`substitution of 'a' for \spad{'b'} in \spad{'n'}
821414`\spad{subst(f,{} [k1...,{}kn],{} [g1,{}...,{}gn])} replaces the kernels \spad{k1},{}...,{}\spad{kn} by \spad{g1},{}...,{}\spad{gn} formally in \spad{f}.
821486`\spad{substitute(x,{}y,{}d)} replace \spad{x}\spad{'s} with \spad{y}\spad{'s} in dictionary \spad{d}.
821548`\spad{substring?(s,{}t,{}i)} tests if \spad{s} is a substring of \spad{t} beginning at index \spad{i}. Note: \axiom{substring?(\spad{s},{}\spad{t},{}0) = prefix?(\spad{s},{}\spad{t})}.
821680`\spad{subtractIfCan(x,{} y)} returns an element \spad{z} such that \spad{z+y=x} or "failed" if no such element exists.
821763`\spad{suchThat(p,{} f)} makes a copy of \spad{p} and adds the predicate \spad{f} to the copy,{} which is returned.
821848`\spad{suchThat(p,{} [f1,{}...,{}fn])} makes a copy of \spad{p} and adds the predicate \spad{f1} and ... and \spad{fn} to the copy,{} which is returned.
821939`\spad{suchThat(x,{} foo)} attaches the predicate foo to \spad{x}; error if \spad{x} is not a symbol.
822020`\spad{suchThat(x,{} [f1,{} f2,{} ...,{} fn])} attaches the predicate \spad{f1} and \spad{f2} and ... and \spad{fn} to \spad{x}. Error: if \spad{x} is not a symbol.
822107`\spad{suchThat(x,{} foo)} attaches the predicate foo to \spad{x}.
822194`\spad{suchThat(x,{} [f1,{} f2,{} ...,{} fn])} attaches the predicate \spad{f1} and \spad{f2} and ... and \spad{fn} to \spad{x}.
822287`\spad{suchThat(p,{} [a1,{}...,{}an],{} f)} returns a copy of \spad{p} with the top-level predicate set to \spad{f(a1,{}...,{}an)}.
822391`\spad{suchThat(r,{} [a1,{}...,{}an],{} f)} returns the rewrite rule \spad{r} with the predicate \spad{f(a1,{}...,{}an)} attached to it.
822478`\spad{suffix?(s,{}t)} tests if the string \spad{s} is the final substring of \spad{t}. Note: \axiom{suffix?(\spad{s},{}\spad{t}) \spad{==} reduce(and,{}[\spad{s}.\spad{i} = \spad{t}.(\spad{n} - \spad{m} + \spad{i}) for \spad{i} in 0..maxIndex \spad{s}])} where \spad{m} and \spad{n} denote the maxIndex of \spad{s} and \spad{t} respectively.
822534`\spad{sumBasis LLv} computes a basis of the sum of list of subspace. This function is slightly more efficient than \spad{reduce(sumBasis,{}LLv)}
822606`\spad{sumBasis Lb} computes a basis of the sum of list of subspaces. This function is slightly more efficient than \spad{reduce(sumBasis,{}Lb)}
822665`\spad{sumBasis(Lv1,{}Lv2)} computes a basis of the sum of the two subspaces spanned by the vectors of \spad{Lv1} and \spad{Lv2}
822747`\spad{sumBasis(B1,{}B2)} computes a basis of the sum of the two subspaces.
822803`\spad{sumOfDivisors(n)} returns the sum of the integers between 1 and \spad{n} (inclusive) which divide \spad{n}. The sum of the divisors of \spad{n} is often denoted by \spad{sigma(n)}.
822880`\spad{sumOfKthPowerDivisors(n,{}k)} returns the sum of the \spad{k}th powers of the integers between 1 and \spad{n} (inclusive) which divide \spad{n}. the sum of the \spad{k}th powers of the divisors of \spad{n} is often denoted by \spad{sigma_k(n)}.
822984`\spad{sumSquares(p)} construct \spad{a} and \spad{b} such that \spad{a^2+b^2} is equal to the integer prime \spad{p},{} and otherwise returns an error. It will succeed if the prime number \spad{p} is 2 or congruent to 1 mod 4.
823064`\spad{sum(expr)} creates the form prefixing \spad{expr} by a capital sigma.
823103`\spad{sum(p(n),{} n)} returns \spad{P(n)},{} the indefinite sum of \spad{p(n)} with respect to upward difference on \spad{n},{} \spadignore{i.e.} \spad{P(n+1) - P(n) = a(n)}.
823177`\spad{sum(expr,{}lowerlimit)} creates the form prefixing \spad{expr} by a capital sigma with a \spad{lowerlimit}.
823219`\spad{sum(f(n),{} n = a..b)} returns \spad{f}(a) + \spad{f}(\spad{a+1}) + ... + \spad{f}(\spad{b}).
823285`\spad{sum(a(n),{} n)} returns A(\spad{n}) such that A(\spad{n+1}) - A(\spad{n}) = a(\spad{n}).
823340`\spad{sum(f(n),{} n = a..b)} returns \spad{f(a) + f(a+1) + ... f(b)}.
823494`\spad{sum(a(n),{} n)} returns \spad{A} which is the indefinite sum of \spad{a} with respect to upward difference on \spad{n},{} \spadignore{i.e.} \spad{A(n+1) - A(n) = a(n)}.
823615`\spad{sum(f(n),{} n = a..b)} returns \spad{f(a) + f(a+1) + ... f(b)}.
823728`\spad{sum(a(n),{} n)} returns \spad{A} which is the indefinite sum of \spad{a} with respect to upward difference on \spad{n},{} \spadignore{i.e.} \spad{A(n+1) - A(n) = a(n)}.
823818`\spad{sum(p(n),{} n = a..b)} returns \spad{p(a) + p(a+1) + ... + p(b)}.
823903`\spad{sum(expr,{}lowerlimit,{}upperlimit)} creates the form prefixing \spad{expr} by a capital sigma with both a \spad{lowerlimit} and \spad{upperlimit}.
823948`\spad{summation(f(n),{} n = a..b)} returns \spad{f}(a) + ... + \spad{f}(\spad{b}) as a formal sum.
824025`\spad{summation(f(n),{} n)} returns the formal sum \spad{S}(\spad{n}) which verifies \spad{S}(\spad{n+1}) - \spad{S}(\spad{n}) = \spad{f}(\spad{n}).
824091`\spad{summation(f(n),{} n = a..b)} returns \spad{f}(a) + ... + \spad{f}(\spad{b}) as a formal sum.
824169`\spad{summation(f(n),{} n)} returns the formal sum \spad{S}(\spad{n}) which verifies \spad{S}(\spad{n+1}) - \spad{S}(\spad{n}) = \spad{f}(\spad{n}).
824235`\spad{sunion(x,{}y)} gives the union of two boundaries as a box,{} \indented{2}{that is it gives a box that is external to both.} \spad{x:}\% is first boundary to be combined \spad{y:}\% is second boundary to be combined
824283`\axiom{supDimElseRittWu(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} has less elements than \axiom{us} otherwise if \axiom{\spad{ts}} has higher rank than \axiom{us} \spad{w}.\spad{r}.\spad{t}. Riit and Wu ordering.
824367`\axiom{supDimElseRittWu(\spad{ts},{}us)} returns \spad{true} iff \axiom{\spad{ts}} has less elements than \axiom{us} otherwise if \axiom{\spad{ts}} has higher rank than \axiom{us} \spad{w}.\spad{r}.\spad{t}. Ritt and Wu ordering.
824534`\axiom{supRittWu?(a,{}\spad{b})} returns \spad{true} if \axiom{a} is greater than \axiom{\spad{b}} \spad{w}.\spad{r}.\spad{t}. the Ritt and Wu Wen Tsun ordering using the refinement of Lazard.
824612`\spad{sup(u)} returns the supremum of \axiom{\spad{u}}.
824659`\spad{sup(x,{}y)} returns the least element from which both \spad{x} and \spad{y} can be subtracted.
824714`\spad{superHeight(f)} returns the height of form \spad{f} above the base line.
824766`\spad{super(f,{}n)} creates a form for \spad{f} superscripted by \spad{n}.
824810`\spad{superscript(s,{} [a1,{}...,{}an])} returns \spad{s} superscripted by \spad{[a1,{}...,{}an]}.
824870`\spad{supersub(a,{}[sub1,{}super1,{}sub2,{}super2,{}...])} creates a form with each subscript aligned under each superscript.
824923`\spad{support(x)} returns the list of basis elements with nonzero coefficients.
824984`\spad{surface(c1,{}c2,{}c3)} creates a surface from 3 parametric component functions \spad{c1},{} \spad{c2},{} and \spad{c3}.
825104`\spad{svec(a,{}b)} constructs a 2D vector defined by a and \spad{b} of type \spad{R}
825169`\spad{svec(a,{}b,{}c)} constructs a 3D vector defined by a,{}\spad{b} and \spad{c} of type \spad{R}
825316`\spad{swap!(u,{}i,{}j)} interchanges elements \spad{i} and \spad{j} of aggregate \spad{u}. No meaningful value is returned.
825488`\spad{swapColumns!(m,{}i,{}j)} interchanges the \spad{i}th and \spad{j}th columns of \spad{m}. This destructively alters the matrix.
825638`\spad{swapRows!(m,{}i,{}j)} interchanges the \spad{i}th and \spad{j}th rows of \spad{m}. This destructively alters the matrix.
825714`\spad{swap(p(x,{}y))} returns \spad{p}(\spad{y},{}\spad{x}).
825794`\spad{swap(eq)} interchanges left and right hand side of equation \spad{eq}.
825835`\spad{sylvesterMatrix(p,{}q)} returns the Sylvester matrix for the two polynomials \spad{p} and \spad{q}.
825906`\axiom{sylvesterSequence(\spad{p},{}\spad{q})} is the negated remainder sequence of \spad{p} and \spad{q} divided by the last computed term
826021`\spad{symFunc([r1,{}...,{}rn])} returns the vector of the elementary symmetric functions in the \spad{\spad{ri}'s}: \spad{[r1 + ... + rn,{} r1 r2 + ... + r(n-1) rn,{} ...,{} r1 r2 ... rn]}.
826087`\spad{symFunc(r,{} n)} returns the vector of the elementary symmetric functions in \spad{[r,{}r,{}...,{}r]} \spad{n} times.
826163`\spad{symbol?(p)} tests if \spad{p} is a symbol.
826211`\spad{symbol?(s)} is \spad{true} if \spad{s} is an atom and belong to \spad{Sym}.
826290`\spad{symbolIfCan(k)} returns \spad{k} viewed as a symbol if \spad{k} is a symbol,{} and "failed" otherwise.
826356`\spad{symbolTableOf(f,{}tab)} returns the symbol table of \spad{f}
826425`\spad{symbolTable(l)} creates a symbol table from the elements of \spad{l}.
826599`\axiom{symbol(\spad{sys})} computes directly the symbol of the family \axiom{\spad{sys}} of functions.
826690`\spad{symbol(s)} returns \spad{s} as an element of \spad{Sym}. Error: if \spad{s} is not an atom that also belongs to \spad{Sym}.
826903`\spad{symmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and symmetric (\spadignore{i.e.} \spad{m[i,{}j] = m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
826969`\spad{symmetric?(m)} returns \spad{true} if the matrix \spad{m} is square and symmetric (\spadignore{i.e.} \spad{m[i,{}j] = m[j,{}i]} for all \spad{i} and \spad{j}) and \spad{false} otherwise.
827170`\spad{symmetricDifference(u,{}v)} returns the set aggregate of elements \spad{x} which are members of set aggregate \spad{u} or set aggregate \spad{v} but not both. If \spad{u} and \spad{v} have no elements in common,{} \axiom{symmetricDifference(\spad{u},{}\spad{v})} returns a copy of \spad{u}. Note: \axiom{symmetricDifference(\spad{u},{}\spad{v}) = union(difference(\spad{u},{}\spad{v}),{}difference(\spad{v},{}\spad{u}))}
827233`\spad{symmetricGroup(\spad{li})} constructs the symmetric group acting on the integers in the list {\em \spad{li}},{} generators are the cycle given by {\em \spad{li}} and the 2-cycle {\em (\spad{li}.1,{}\spad{li}.2)}. Note: duplicates in the list will be removed.
827331`\spad{symmetricGroup(n)} constructs the symmetric group {\em Sn} acting on the integers 1,{}...,{}\spad{n},{} generators are the {\em n}-cycle {\em (1,{}...,{}n)} and the 2-cycle {\em (1,{}2)}.
827431`\spad{symmetricPower(a,{}n)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the products of \spad{n} solutions of \spad{a}.
827547`\spad{symmetricPower(a,{}n,{}D)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the products of \spad{n} solutions of \spad{a}. \spad{D} is the derivation to use.
827654`\spad{symmetricProduct(a,{}b)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the products of a solution of \spad{a} by a solution of \spad{b}.
827756`\spad{symmetricProduct(a,{}b,{}D)} computes an operator \spad{c} of minimal order such that the nullspace of \spad{c} is generated by all the products of a solution of \spad{a} by a solution of \spad{b}. \spad{D} is the derivation to use.
827909`\spad{symmetricRemainder(a,{}b)} (where \spad{b > 1}) yields \spad{r} where \spad{ -b/2 <= r < b/2 }.
828056`\spad{symmetricSquare(a)} computes \spad{symmetricProduct(a,{}a)} using a more efficient method.
828154`\spad{symmetricTensors(la,{}n)} applies to each \spad{m}-by-\spad{m} square matrix in the list {\em la} the irreducible,{} polynomial representation of the general linear group {\em GLm} which corresponds to the partition {\em (n,{}0,{}...,{}0)} of \spad{n}. Error: if the matrices in {\em la} are not square matrices. Note: this corresponds to the symmetrization of the representation with the triv
828154`--ial representation of the symmetric group {\em Sn}. The carrier spaces of the representation are the symmetric tensors of the \spad{n}-fold tensor product.
828263`\spad{symmetricTensors(a,{}n)} applies to the \spad{m}-by-\spad{m} square matrix {\em a} the irreducible,{} polynomial representation of the general linear group {\em GLm} which corresponds to the partition {\em (n,{}0,{}...,{}0)} of \spad{n}. Error: if {\em a} is not a square matrix. Note: this corresponds to the symmetrization of the representation with the trivial representation of the symmetric group {\e
828263`--m Sn}. The carrier spaces of the representation are the symmetric tensors of the \spad{n}-fold tensor product.
828360`\spad{systemCommand(cmd)} takes the string \spadvar{\spad{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.
828423`\spad{tRange(p)} returns the range of the parameter in a parametric plot \spad{p}.
828479`\spad{tRange(p)} returns the range of the parameter in a parametric plot \spad{p}
828533`\spad{tValues(p)} returns a list of lists of the values of the parameter for which a point is computed,{} one list for each curve in the plot \spad{p}.
828593`\spad{t(n)} \undocumented
828667`\spad{tab1(lp)} creates a tableau from a list of pairs \spad{lp}
828743`\spad{tab(ls)} creates a tableau from \spad{ls} by first creating a list of pairs using \spadfunFrom{slex}{TableauBumpers},{} then creating a tableau using \spadfunFrom{\spad{tab1}}{TableauBumpers}.
828809`\spad{tableForDiscreteLogarithm(a,{}n)} returns a table of the discrete logarithms of \spad{a^0} up to \spad{a^(n-1)} which,{} called with key \spad{lookup(a^i)} returns \spad{i} for \spad{i} in \spad{0..n-1}. Error: if not called for prime divisors of order of \indented{7}{multiplicative group.}
828923`\spad{tablePow(maxdeg,{}prime,{}lpol)} constructs the table with the coefficients of the Extended Euclidean Algorithm for \spad{lpol}. Here the right side is \spad{x^k},{} for \spad{k} less or equal to \spad{maxdeg}. The operation returns "failed" when the elements are not coprime modulo \spad{prime}.
829081`\spad{table()}\$\spad{T} creates an empty table of type \spad{T}.
829135`\spad{table(h)} creates an empty hash table that uses \spad{h} instead of hash\$Key. Note that \spad{h} should be a mathematical function in the sense that from \spad{k1=k2} follows \spad{h}(\spad{k1})\spad{=h}(\spad{k2}). If that is not the case,{} \spad{k1} and \spad{k2} will internally be considered as being different keys.
829288`\axiom{table(\spad{lv})} computes the commutator table for a given list of vector fields.
829366`\spad{table([x,{}y,{}...,{}z])} creates a table consisting of entries \axiom{\spad{x},{}\spad{y},{}...,{}\spad{z}}.
829453`\spad{tableau(ll)} converts a list of lists \spad{ll} to a tableau.
829507`\axiom{tableau(symb,{} chi)} computes the tableau parametrized by a given one-form.
829637`\axiom{tableau(symb,{} lchi)} computes the extended tableau parametrized by a given list of one-forms.
829930`\spad{tail(l)} returns the doubly-linked aggregate \spad{l} starting at its second element. Error: if \spad{l} is empty.
829984`\axiom{tail(\spad{p})} returns its reductum,{} where \axiom{\spad{p}} is viewed as a univariate polynomial in its main variable.
830048`\spad{tail(u)} returns the last node of \spad{u}. Note: if \spad{u} is \axiom{shallowlyMutable},{} \axiom{setrest(tail(\spad{u}),{}\spad{v}) = concat(\spad{u},{}\spad{v})}.
830104`\spad{tan2cot(f)} converts every \spad{tan(u)} appearing in \spad{f} into \spad{1/cot(u)}.
830167`\spad{tan2trig(f)} converts every \spad{tan(u)} appearing in \spad{f} into \spad{sin(u)/cos(u)}.
830231`\spad{tanAn(a,{} n)} returns \spad{P(x)} such that if \spad{a = tan(u)} then \spad{P(tan(u/n)) = 0}.
830324`\spad{tanIfCan(z)} returns tan(\spad{z}) if possible,{} and "failed" otherwise.
830405`\spad{tanNa(a,{} n)} returns \spad{f(a)} such that if \spad{a = tan(u)} then \spad{f(a) = tan(n * u)}.
830462`\spad{tanQ(q,{}a)} is a local function with a conditional implementation.
830547`\spad{tanSum([a1,{}...,{}an])} returns \spad{f(a1,{}...,{}an)} such that if \spad{\spad{ai} = tan(\spad{ui})} then \spad{f(a1,{}...,{}an) = tan(u1 + ... + un)}.
830603`\spad{tan(x)} applies the tangent operator to \spad{x}
830809`\spad{tan(st)} computes tangent of a power series \spad{st}.
830893`\spad{tan(st)} computes tangent of a power series \spad{st}.
830991`\spad{tan(z)} returns the tangent of Laurent series \spad{z}.
831078`\spad{tan(z)} returns the tangent of a Puiseux series \spad{z}.
831174`\spad{tan(x)} represents the Fortran intrinsic function TAN
831255`\spad{tan(x)} returns the tangent of \spad{x}.
831313`\spad{tanh2coth(f)} converts every \spad{tanh(u)} appearing in \spad{f} into \spad{1/coth(u)}.
831378`\spad{tanh2trigh(f)} converts every \spad{tanh(u)} appearing in \spad{f} into \spad{sinh(u)/cosh(u)}.
831444`\spad{tanhIfCan(z)} returns tanh(\spad{z}) if possible,{} and "failed" otherwise.
831526`\spad{tanh(x)} applies the hyperbolic tangent operator to \spad{x}
831733`\spad{tanh(st)} computes the hyperbolic tangent of a power series \spad{st}.
831818`\spad{tanh(st)} computes the hyperbolic tangent of a power series \spad{st}.
831917`\spad{tanh(z)} returns the hyperbolic tangent of Laurent series \spad{z}.
832005`\spad{tanh(z)} returns the hyperbolic tangent of a Puiseux series \spad{z}.
832102`\spad{tanh(x)} represents the Fortran intrinsic function TANH
832184`\spad{tanh(x)} returns the hyperbolic tangent of \spad{x}.
832240`\spad{tanintegrate(f,{} ',{} foo)} returns \spad{[g,{} a]} such that \spad{f = g' + a},{} and \spad{a = 0} or \spad{a} has no integral in \spad{F}; Argument foo is a Risch differential system solver on \spad{F}.
832416`\spad{taylorIfCan(f(x))} converts the Laurent series \spad{f(x)} to a Taylor series,{} if possible. If this is not possible,{} "failed" is returned.
832522`\spad{taylorQuoByVar(a0 + a1 x + a2 x^2 + ...)} returns \spad{a1 + a2 x + a3 x^2 + ...}
832600`\spad{taylorRep(f(x))} returns \spad{g(x)},{} where \spad{f = x^n * g(x)} is represented by \spad{[n,{}g(x)]}.
832688`\spad{taylor_via_deriv(f,{} [x1,{}...,{}xn],{} [D1,{}...,{}Dn])} expands \spad{f} into Taylor series at (\spad{x1},{}...,{}\spad{xn}) using \spad{D1},{}...,{}\spad{Dn} as partial derivatives
832804`\spad{taylor_via_deriv(f,{} [x1,{}...,{}xn],{} [D1,{}...,{}Dn])} expands \spad{f} into Taylor series at (\spad{x1},{}...,{}\spad{xn}) using \spad{D1},{}...,{}\spad{Dn} as partial derivatives
832923`\spad{taylor_via_deriv(f,{} [x1,{}...,{}xn],{} [D1,{}...,{}Dn])} expands \spad{f} into Taylor series at (\spad{x1},{}...,{}\spad{xn}) using \spad{D1},{}...,{}\spad{Dn} as partial derivatives
833037`\spad{taylor_via_deriv(f,{} [x1,{}...,{}xn],{} [D1,{}...,{}Dn])} expands \spad{f} into Taylor series at (\spad{x1},{}...,{}\spad{xn}) using \spad{D1},{}...,{}\spad{Dn} as partial derivatives
833154`\spad{taylor_via_lode([a0,{}...,{}ak],{} z,{} [c0,{} ...,{} ck])} expands into Taylor series solution of the linear ordinary differential equation
833260`\spad{taylor(f)} returns a Taylor expansion of the expression \spad{f}. Note: \spad{f} should have only one variable; the series will be expanded in powers of that variable.
833332`\spad{taylor(x)} returns \spad{x} viewed as a Taylor series.
833408`\spad{taylor(f(x))} converts the Laurent series \spad{f}(\spad{x}) to a Taylor series,{} if possible. Error: if this is not possible.
833493`\spad{taylor(n +-> a(n),{}x = a)} returns \spad{sum(n = 0..,{}a(n)*(x-a)^n)}.
833585`\spad{taylor(f,{}x = a)} expands the expression \spad{f} as a Taylor series in powers of \spad{(x - a)}.
833670`\spad{taylor(f,{}n)} returns a Taylor expansion of the expression \spad{f}. Note: \spad{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 \spad{n}.
833761`\spad{taylor(n +-> a(n),{}x = a,{}n0..)} returns \spad{sum(n=n0..,{}a(n)*(x-a)^n)}; \spad{taylor(n +-> a(n),{}x = a,{}n0..n1)} returns \spad{sum(n = n0..,{}a(n)*(x-a)^n)}.
833890`\spad{taylor(f,{}x = a)} expands the expression \spad{f} as a Taylor series in powers of \spad{(x - a)}; terms will be computed up to order at least \spad{n}.
833994`\spad{taylor(a(n),{}n,{}x = a)} returns \spad{sum(n = 0..,{}a(n)*(x-a)^n)}.
834082`\spad{taylor(a(n),{}n,{}x = a,{}n0..)} returns \spad{sum(n = n0..,{}a(n)*(x-a)^n)}; \spad{taylor(a(n),{}n,{}x = a,{}n0..n1)} returns \spad{sum(n = n0..,{}a(n)*(x-a)^n)}.
834207`\spad{tensorMap([s1,{} s2,{} ...],{} f)} returns the stream consisting of all elements of \spad{f}(\spad{s1}) followed by all elements of \spad{f}(\spad{s2}) and so on.
834284`\spad{tensorProduct([a1,{}...ak])} calculates the list of Kronecker products of each matrix {\em \spad{ai}} with itself for {1 \spad{<=} \spad{i} \spad{<=} \spad{k}}. Note: If the list of matrices corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the representation with itself.
834374`\spad{tensorProduct(a)} calculates the Kronecker product of the matrix {\em a} with itself.
834452`\spad{tensorProduct([a1,{}...,{}ak],{}[b1,{}...,{}bk])} calculates the list of Kronecker products of the matrices {\em \spad{ai}} and {\em \spad{bi}} for {1 \spad{<=} \spad{i} \spad{<=} \spad{k}}. Note: If each list of matrices corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the two representations.
834558`\spad{tensorProduct(a,{}b)} calculates the Kronecker product of the matrices {\em a} and \spad{b}. Note: if each matrix corresponds to a group representation (repr. of generators) of one group,{} then these matrices correspond to the tensor product of the two representations.
834756`\spad{tensor([x1,{}x2,{}...,{}xn])} constructs the tensor product of \spad{x1,{}x2,{}...,{}xn}.
834874`\spad{tensor(a,{}b)} creates a form for a tensor \spad{b}
834919`\spad{tensor(x,{}y)} constructs the tensor product of the elements \spad{x} and \spad{y}.
834980`\spad{terminal(a)} constructs a graph over a with a single vertex and a single loop
835027`\spad{terms(f)} gives the list of non-zero coefficients combined with their corresponding basis element as records. This is the internal representation.
835107`\spad{terms(e1 a1 + ... + en an)} returns \spad{[[a1,{} e1],{}...,{}[an,{} en]]}.
835191`\spad{terms(f(x))} returns a stream of non-zero terms,{} where a a term is an exponent-coefficient pair. The terms in the stream are ordered by increasing order of exponents.
835291`\spad{testAbsolutePrecision(eps)} returns the current absolute precision used for floating point comparison,{} and then sets it to \spad{eps}. The default value is 1.0e-15.
835373`testComplexEquals(inp \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be approximately equal as complex numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
835484`\spad{testComplexEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be approximately equal as complex numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
835548`\spad{testDim(lp,{}lv)} tests if the polynomial system \spad{lp} in variables \spad{lv} is zero dimensional.
835756`\spad{testEqualsAuxCmp should} be local but is conditional
835818`testEquals(inp,{} \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be equal. To sidestep the possibility that the equality function of the domain \spad{R} performs some simplifications,{} we convert \spad{ex1} and \spad{ex2} to \axiom{InputForm},{} if possible.
835881`\spad{testEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be equal. To sidestep the possibility that the equality function of the domain \spad{R} performs some simplifications,{} we convert \spad{ex1} and \spad{ex2} to \axiom{InputForm},{} if possible
835938`testLibraryError(inp,{} ex) states that ex should throw an error. Such a test will never count as a fatal error.
836005`\spad{testLibraryError ex} states that ex should throw an error. Such a test will never count as a fatal error.
836061`\spad{testModulus(p,{}lp)} returns \spad{true} if the the prime \spad{p} is valid for the list of polynomials \spad{lp},{} \spadignore{i.e.} preserves the degree and they remain relatively prime.
836128`testNotEquals(inp,{} \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be different.
836194`\spad{testNotEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be different.
836254`testRealEquals(inp,{} \spad{ex1},{} \spad{ex2}) states that \spad{ex1} and \spad{ex2} should be approximately equal as real numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
836344`\spad{testRealEquals(ex1,{} ex2)} states that \spad{ex1} and \spad{ex2} should be approximately equal as real numbers,{} taking into acount \spadfun{testAbsolutePrecision} and \spadfun{testRelativePrecision}.
836405`\spad{testRelativePrecision(eps)} returns the current absolute precision used for floating point comparison,{} and then sets it to \spad{eps}. The default value is 1.0e-15.
836487`\spad{testTrueAux(inp,{} ex)} states that ex should be \spad{true}.
836546`\spad{testTrue ex} states that ex should be \spad{true}.
836594`\spad{test(b)} returns \spad{b} and is provided for compatibility with the new compiler.
836636`\spad{testcaseNoClear s},{} starts a new testcase with \spad{s} as title,{} without calling \axiom{clear all}. A testcase is composed of several tests.
836696`\spad{testcase s} starts a new testcase with \spad{s} as title. It also calls \axiom{clear all}. A testcase is composed of several tests.
836749`\spad{testsuiteNoClear s} starts a new testsuite with \spad{s} as title,{} without calling \axiom{clear all}. A testsuite is composed of several testcases.
836810`\spad{testsuite s} starts a new testsuite with \spad{s} as title. It also calls \axiom{clear all}. A testsuite is composed of several testcases.
836864`\spad{tex(t)} extracts the TeX section of a TeX form \spad{t}.
836912`\spad{thetaCoord(pt)} returns the second element of the point,{} \spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a spherical or a cylindrical coordinate system.
837020`\spad{third(u)} returns the third element of \spad{u}. Note: \axiom{third(\spad{u}) = first(rest(rest(\spad{u})))}.
837076`\spad{times!(c,{}a,{}b)} computes the matrix product \spad{a * b} and stores the result in the matrix \spad{c}. Error: if \spad{a},{} \spad{b},{} and \spad{c} do not have compatible dimensions.
837177`\spad{times(p,{} q,{} sigma,{} delta)} returns \spad{p * q}. \spad{\sigma} and \spad{\delta} are the maps to use.
837271`\spad{title(s)} specifies a title for a plot by the indicated string \spad{s}. This option is expressed in the form \spad{title == s}.
837316`\spad{title(l,{}s)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{title}. If the option does not exist the value,{} \spad{s} is returned.
837392`\spad{title(v,{}s)} changes the title which is shown in the two-dimensional viewport window,{} \spad{v} of domain \spadtype{TwoDimensionalViewport}.
837454`\spad{title(v,{}s)} changes the title which is shown in the three-dimensional viewport window,{} \spad{v} of domain \spadtype{ThreeDimensionalViewport}.
837518`convert PermutationGroup to a Cayley graph
837619`convert permutation generators to a Cayley graph permList should contain generator permutations and should not contain identity permutation. if permutationNames then names generated represent permutation
837729`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
837902`\spad{toObj(n,{}ptLst,{}indexLst,{}indexNxt,{}tran,{}bb)} creates an .OBJ (Wavefront) file from scenegraph tree structure called recursively for each node,{} so when called on root node in scenegraph all other nodes in the scenegraph will get called. \spad{n} is the scene to be output to \spad{ptLst} and \spad{indexLst} \spad{ptLst} is the poi
837902`--nts. This will be set to ref([]) at the start \indented{6}{and will be built up as this function is called recursively} \spad{indexLst} is list of faces,{} where faces are list of points indexed \indented{9}{by \spad{ptLst} This will be set to ref(nil()) at the start} \indented{6}{and will be built up as this function is called recursively} \spad{tran} is the default transform which may be overridden by subnodes \spad{bb} is the default boundary which may be overridden by subnodes
838065`generates a permutation group from this graph assumes this graph represents a valid group
838159`\spad{toPoint(p)} returns a Point with the same coordinates as \spad{p} If \spad{p} was originally a vector then that will be treated as the distance from the origin.
838206`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
838425`\spad{toSVG(n,{}mat,{}tran,{}bb,{}scale,{}clipEn,{}useInteger,{}npt)} creates an XmlElement containing a 'SVG' representation of node \spad{'n'} and the nodes below it. \spad{n} is the scene to be output to \spad{XML} elements \spad{mat} is the default material which may be overridden by subnodes \
838425`--spad{tran} is the default transform which may be overridden by subnodes \spad{bb} is the default boundary which may be overridden by subnodes \spad{scale} is the \spad{scale} used to \spad{scale} into canvas,{} it would be better \indented{6}{to merge this into trans.} \spad{clipEn} if \spad{true} this clips the output to the boundary \spad{useInteger} if \spad{true} quantise to integer values. \spad{npt} names points can be provided. If not required then set \indented{4}{to namedBranch([],{}[
838425`--])}
838634`\spad{toScale(b)} specifies whether or not a plot is to be drawn to scale; if \spad{b} is \spad{true} it is drawn to scale,{} if \spad{b} is \spad{false} it is not. This option is expressed in the form \spad{toScale == b}.
838682`\spad{toScale(l,{}b)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{toScale}. If the option does not exist the value,{} \spad{b} is returned.
838762`return string representation using conventional notation,{} that is deBruijn index is replaced by name using String value for bound variables. notation assumes association to the left,{} in the absence of brackets,{} the term to the left binds more tightly than the one on the right.
838829`similar to 'toString' but does not put outer compound terms in brackets
838882`returns the name and the type in string form.
838926`creates a string representaion of this term and its sub-terms
838970`return string representation using deBruijn index for bound variables. notation assumes association to the left,{} in the absence of brackets,{} the term to the left binds more tightly than the one on the right.
839018`output
839074`\spad{toString(n)} returns string representation for debug purposes.
839205`\spad{toString(x,{}n)} returns a string representation of \spad{x} truncated to \spad{n} decimal digits.
839281`displays table of unary function such as inverse,{} reverse,{} complement,{} or dual basis could have returned type 'List List \%' but matrix displays better
839353`displays multiplication table for binary operation which is represented as a function with two parameters. row number represents first operand in binary order column number represents second operand in binary order could have returned type 'List List \%' but matrix displays better
839428`\spad{toVector returns} a Vector with the same coordinates as \spad{p} If \spad{p} was originally a point then the vector will be the distance from the origin.
839476`deprecated: this has been superseded by version with SBoundary(\spad{PT}) do not use,{} may be removed in the future.
839646`\spad{toX3D(n,{}mat,{}tran,{}bb)} creates an XmlElement containing a 'X3D' representation of node \spad{'n'} and the nodes below it. \spad{n} is the scene to be output to \spad{XML} elements \spad{mat} is the default material which may be overridden by subnodes \spad{tran} is the default transform which may be overridden by subnodes \spad{bb} is 
839646`--the default boundary which may be overridden by subnodes
839806`\spad{to_mod_pa(s,{} p)} reduces coefficients of polynomial \spad{s} modulo prime \spad{p} and converts the result to vector
839918`\spad{top!(d)} returns the element at the top (front) of the dequeue.
839966`\spad{topFortranOutputStack()} returns the top element of the Fortran output stack
840040`\spad{topPredicate(x)} returns \spad{[[a1,{}...,{}an],{} f]} where the top-level predicate of \spad{x} is \spad{f(a1,{}...,{}an)}. Note: \spad{n} is 0 if \spad{x} has no top-level predicate.
840119`\spad{top(s)} returns the top element \spad{x} from \spad{s}; \spad{s} remains unchanged. Note: Use \axiom{pop!(\spad{s})} to obtain \spad{x} and remove it from \spad{s}.
840164`\spad{toroidal(a)} transforms from toroidal coordinates to Cartesian coordinates: \spad{toroidal(a)} is a function which will map the point \spad{(u,{}v,{}phi)} to \spad{x = a*sinh(v)*cos(phi)/(cosh(v)-cos(u))},{} \spad{y = a*sinh(v)*sin(phi)/(cosh(v)-cos(u))},{} \spad{z = a*sin(u)/(cosh(v)-cos(u))}.
840235`\spad{torsion?(f)} \undocumented
840333`\spad{torsion?(f)} \undocumented
840450`\spad{torsionIfCan(f)}\\ undocumented
840604`\spad{torsionIfCan(f)} \undocumented
840777`\axiom{toseInvertible?(\spad{p1},{}\spad{p2},{}\spad{ts})} has the same specifications as \axiomOpFrom{invertible?}{RegularTriangularSetCategory}.
840867`\axiom{toseInvertible?(\spad{p1},{}\spad{p2},{}\spad{ts})} has the same specifications as \axiomOpFrom{invertible?}{RegularTriangularSetCategory}.
840984`\axiom{toseInvertibleSet(\spad{p1},{}\spad{p2},{}\spad{ts})} has the same specifications as \axiomOpFrom{invertibleSet}{RegularTriangularSetCategory}.
841077`\axiom{toseLastSubResultant(\spad{p1},{}\spad{p2},{}\spad{ts})} has the same specifications as \axiomOpFrom{lastSubResultant}{RegularTriangularSetCategory}.
841195`\axiom{toseSquareFreePart(\spad{p},{}\spad{ts})} has the same specifications as \axiomOpFrom{squareFreePart}{RegularTriangularSetCategory}.
841407`\spad{totalDegreeSorted(p,{} lv)} returns the maximum sum (over all monomials of polynomial \spad{p}) of the degree in variables in the list \spad{lv}. \spad{lv} is assumed to be sorted in decreasing order.
841677`\spad{totalDegree(p)} returns the largest sum over all monomials of all exponents of a monomial.
841969`\spad{totalDegree(p,{} lv)} returns the maximum sum (over all monomials of polynomial \spad{p}) of the variables in the list \spad{lv}.
842074`\spad{totalDifferential(x)} returns the total differential (gradient) form for element \spad{x}.
842169`\spad{totalGroebner(lp,{}lv)} computes Groebner basis for the list of polynomials \spad{lp} with the terms ordered first by total degree and then refined by reverse lexicographic ordering. The variables are ordered by their position in the list \spad{lv}.
842270`\spad{totalLex(v1,{}v2)} return \spad{true} if the vector \spad{v1} is less than the vector \spad{v2} in the ordering which is total degree refined by lexicographic ordering.
842350`\spad{totalfract(prf)} takes a polynomial whose coefficients are themselves fractions of polynomials and returns a record containing the numerator and denominator resulting from putting \spad{prf} over a common denominator.
842474`\spad{totolex }\undocumented
842684`\spad{tower(f)} returns all the kernels appearing in \spad{f},{} no matter what their levels are.
842744`\spad{trace2PowMod(u,{}k,{}v)} produces the sum of \spad{u^(2^i)} for \spad{i} running from 1 to \spad{k} all computed modulo the polynomial \spad{v}.
842833`\spad{traceMatrix()} is the \spad{n}-by-\spad{n} matrix ( \spad{Tr(\spad{vi} * vj)} ),{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
842897`\spad{traceMatrix()} is the \spad{n}-by-\spad{n} matrix ( \spad{Tr(\spad{vi} * vj)} ),{} where \spad{v1},{} ...,{} \spad{vn} are the elements of the fixed basis.
843096`\spad{traceMatrix([v1,{}..,{}vn])} is the \spad{n}-by-\spad{n} matrix ( \spad{Tr}(\spad{vi} * \spad{vj}) )
843171`\spad{tracePowMod(u,{}k,{}v)} produces the sum of \spad{u^(q^i)} for \spad{i} running and \spad{q=} size \spad{F}
843423`\spad{trace(a)} computes the trace of \spad{a} with respect to the field considered as an algebra with 1 over the ground field \spad{F}.
843485`\spad{trace(a)} returns the trace of the regular representation of \spad{a} with respect to any basis.
843538`\spad{trace(m)} returns the trace of the matrix \spad{m}. this is the sum of the elements on the diagonal of the matrix \spad{m}.
843678`\spad{trace(x,{}n)} \undocumented{} See \axiomFunFrom{trace}{FiniteAlgebraicExtensionField}
843775`\spad{trace(a,{}d)} computes the trace of \spad{a} with respect to the field of extension degree \spad{d} over the ground field of size \spad{q}. Error: if \spad{d} does not divide the extension degree of \spad{a}. Note: \spad{trace(a,{}d) = reduce(+,{}[a^(q^(d*i)) for i in 0..n/d])}.
843867`\spad{trailingCoefficient }\undocumented
844022`\spad{transcendenceDegree()} returns the transcendence degree of the field extension,{} 0 if the extension is algebraic.
844227`\spad{transcendent?(a)} tests whether an element \spad{a} is transcendent with respect to the ground field \spad{F}.
844930`\spad{transform }\undocumented
845074`\axiom{transform(\spad{jv})} transforms a jet variable \axiom{\spad{jv}} into a function of the new coordinates.
845172`\axiom{transform(\spad{f})} rewrites a function of the old coordinates in the new ones.
845291`\axiom{transform(\spad{f})} transforms expressions between the different function types involved.
845422`\spad{translate(X,{}Y,{}Z)} returns a dhmatrix for translation by \spad{X},{} \spad{Y},{} and \spad{Z}
845486`\spad{translate(v,{}dx,{}dy)} sets the horizontal viewport offset to \spad{dx} and the vertical viewport offset to \spad{dy},{} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}.
845559`\spad{translate(v,{}n,{}dx,{}dy)} displays the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} translated by \spad{dx} in the \spad{x}-coordinate direction from the center of the viewport,{} and by \spad{dy} in the \spad{y}-coordinate direction from the center. Setting \spad{dx} and \spad{dy} to \spad{0} places the center of the graph
845559`-- at the center of the viewport.
845758`\spad{transpose(m)} returns the transpose of the matrix \spad{m}.
845813`\spad{transpose(r)} converts the row \spad{r} to a row matrix.
845874`\spad{transpose(m)} returns the transpose of the matrix \spad{m}.
845934`\spad{transpose(t)} exchanges the first and last indices of \spad{t}. For example,{} if \spad{r = transpose(t)} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank 4 tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = t(l,{}j,{}k,{}i)}.}
845997`\spad{transpose(t,{}i,{}j)} exchanges the \spad{i}\spad{-}th and \spad{j}\spad{-}th indices of \spad{t}. For example,{} if \spad{r = transpose(t,{}2,{}3)} for a rank 4 tensor \spad{t},{} then \spad{r} is the rank 4 tensor given by \indented{4}{\spad{r(i,{}j,{}k,{}l) = t(i,{}k,{}j,{}l)}.}
846076`\spad{trapezoidal(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax)} uses the trapezoidal method to numerically integrate function \spadvar{\spad{fn}} over the closed interval \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary from \spad{
846076`--nmin} to \spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
846246`\spad{trapezoidalo(fn,{}a,{}b,{}epsrel,{}epsabs,{}nmin,{}nmax)} uses the trapezoidal method to numerically integrate function \spad{fn} over the open interval from \spad{a} to \spad{b},{} with relative accuracy \spad{epsrel} and absolute accuracy \spad{epsabs},{} with the refinement levels for convergence checking vary from \spad{nmin}
846246`-- to \spad{nmax}. The value returned is a record containing the value of the integral,{} the estimate of the error in the computation,{} the total number of function evaluations,{} and either a boolean value which is \spad{true} if the integral was computed within the user specified error criterion. See \spadtype{NumericalQuadrature} for details.
846417`\spad{traverse(s,{}\spad{li})} follows the branch list of the 3 dimensional subspace,{} \spad{s},{} along the path dictated by the list of non negative integers,{} \spad{li},{} which points to the component which has been traversed to. The subspace,{} \spad{s},{} is returned,{} where \spad{s} is now the subspace pointed to by \spad{li}.
846489`\spad{tree(ls)} creates a tree from a list of elements of \spad{s}.
846531`\spad{tree(nd)} creates a tree with value \spad{nd},{} and no children
846567`\spad{tree(nd,{}ls)} creates a tree with value \spad{nd},{} and children \spad{ls}.
846612`\spad{triangSolve(lp)} returns the same as \spad{triangSolve(lp,{}false,{}false)}
846725`\spad{triangSolve(lp,{}info?)} returns the same as \spad{triangSolve(lp,{}false)}
846846`\spad{triangSolve(lp,{}info?,{}lextri?)} decomposes the variety associated with \axiom{\spad{lp}} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \axiom{\spad{lp}} needs to generate a zero-dimensional ideal. If \axiom{\spad{lp}} is not zero-dimensional then the result is only a decomposition of its
846846`-- zero-set in the sense of the closure (\spad{w}.\spad{r}.\spad{t}. Zarisky topology). Moreover,{} if \spad{info?} is \spad{true} then some information is displayed during the computations. See \axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory}(\spad{lp},{}\spad{true},{}\spad{info?}). If \spad{lextri?} is \spad{true} then the lexTriangular algorithm is called from the \spadtype{LexTriangularPackage} constructor (see \axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\spad{lp},{}\spad{false}
846846`--)). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \axiomOpFrom{zeroSetSplit}{RegularChain} from \spadtype{RegularChain}.
847048`\axiom{triangular?(\spad{ps})} returns \spad{true} iff \axiom{\spad{ps}} is a triangular set,{} \spadignore{i.e.} two distinct polynomials have distinct main variables and no constant lies in \axiom{\spad{ps}}.
847146`\spad{triangularSystems(lf,{}lv)} solves the system of equations defined by \spad{lf} with respect to the list of symbols \spad{lv}; the system of equations is obtaining by equating to zero the list of rational functions \spad{lf}. The output is a list of solutions where each solution is expressed as a "reduced" triangular system of polynomials.
847273`\spad{triangulate(M,{}v)} returns \spad{A,{}[[C_1,{}g_1,{}L_1,{}h_1],{}...,{}[C_k,{}g_k,{}L_k,{}h_k]]} such that under the change of variable \spad{y = A z},{} the first order linear system \spad{D y = M y + v} is uncoupled as \spad{D z_i = C_i z_i + g_i} and each \spad{C_i} is a companion matrix corresponding to the scalar equation \spad{L_i z_j = h_i}.
847418`\spad{triangulate(m,{} v)} returns \spad{[m_0,{} v_0]} such that \spad{m_0} is upper triangular and the system \spad{m_0 x = v_0} is equivalent to \spad{m x = v}.
847528`\spad{trigs2explogs(f,{} [k1,{}...,{}kn])} rewrites all the trigonometric functions appearing in \spad{f} in terms of complex logarithms and exponentials. A kernel of the form \spad{tan(u)} is expressed using \spad{exp(u)^2} if it is one of the \spad{\spad{ki}'s},{} in terms of \spad{exp(2*u)} otherwise.
847623`\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \spad{f} in terms of trigonometric functions.
847690`\spad{trigs(f)} rewrites all the complex logs and exponentials appearing in \spad{f} in terms of trigonometric functions.
847857`\spad{trim(s,{}c)} returns \spad{s} with all characters \spad{c} deleted from right and left ends. For example,{} \axiom{trim(" abc ",{} char " ")} returns \axiom{"abc"}.
847912`\spad{trim(s,{}cc)} returns \spad{s} with all characters in \spad{cc} deleted from right and left ends. For example,{} \axiom{trim("(abc)",{} charClass "()")} returns \axiom{"abc"}.
848047`\axiom{trivialIdeal?(\spad{ps})} returns \spad{true} iff \axiom{\spad{ps}} does not contain non-zero elements.
848126`\spad{trueEqual(x,{}y)} tries to determine if the two numbers are equal
848189`\spad{true} is a logical constant.
848224`\axiom{trunc(\spad{p},{}\spad{n})} returns the polynomial \axiom{\spad{p}} truncated at order \axiom{\spad{n}}.
848298`\spad{trunc(p,{}n)} returns the polynomial \spad{p} truncated at order \spad{n}.
848415`\spad{truncate x} returns the integer between \spad{x} and 0 closest to \spad{x}.
848465`\spad{truncate(f,{}k)} returns a (finite) power series consisting of the sum of all terms of \spad{f} of degree \spad{<= k}.
848546`\spad{truncate(f,{}k1,{}k2)} returns a (finite) power series consisting of the sum of all terms of \spad{f} of degree \spad{d} with \spad{k1 <= d <= k2}.
848633`\spad{truncated_mul_add(x,{} y,{} z,{} d,{} p)} adds to \spad{z} the produce x*y truncated after degree \spad{d}
848750`\spad{truncated_multiplication(x,{} y,{} d,{} p)} computes x*y truncated after degree \spad{d}
848869`\spad{tryFunctionalDecomposition?()} returns \spad{true} if factorizers try functional decomposition of polynomials before factoring them.
848950`\spad{tryFunctionalDecomposition(b)} chooses whether factorizers have to look for functional decomposition of polynomials (\spad{true}) or not (\spad{false}). Returns the previous value.
849037`\spad{tryTwoFactor(p)} returns the factorisation of polynomial \spad{p},{} if it does not require using field extensions,{} otherwise returns \spad{p} unfactored (nil factorization).
849210`\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n)} puts a tube of radius \spad{r}(\spad{t}) with \spad{n} points on each circle about the curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} for \spad{t} in \spad{[a,{}b]}. The tube is considered to be open.
849416`\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n)} puts a tube of radius \spad{r} with \spad{n} points on each circle about the curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} for \spad{t} in \spad{[a,{}b]}. The tube is considered to be open.
849607`\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n,{}s)} puts a tube of radius \spad{r(t)} with \spad{n} points on each circle about the curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} for \spad{t} in \spad{[a,{}b]}. If \spad{s} = "closed",{} the tube is considered to be closed; if 
849607`--\spad{s} = "open",{} the tube is considered to be open.
849820`\spad{tubePlot(f,{}g,{}h,{}colorFcn,{}a..b,{}r,{}n,{}s)} puts a tube of radius \spad{r} with \spad{n} points on each circle about the curve \spad{x = f(t)},{} \spad{y = g(t)},{} \spad{z = h(t)} for \spad{t} in \spad{[a,{}b]}. If \spad{s} = "closed",{} the tube is considered to be closed; if \spad{s} = "open",
849820`--{} the tube is considered to be open.
850018`\spad{tubePointsDefault()} returns the number of points to be used when creating the circle to be used in creating a 3D tube plot.
850091`\spad{tubePointsDefault(i)} sets the number of points to use when creating the circle to be used in creating a 3D tube plot to \spad{i}.
850179`\spad{tubePoints(n)} specifies the number of points,{} \spad{n},{} defining the circle which creates the tube around a 3D curve,{} the default is 6. This option is expressed in the form \spad{tubePoints == n}.
850238`\spad{tubePoints(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{tubePoints}. If the option does not exist the value,{} \spad{n} is returned.
850337`\spad{tubeRadiusDefault()} returns the radius used for a 3D tube plot.
850406`\spad{tubeRadiusDefault(r)} sets the default radius for a 3D tube plot to \spad{r}.
850480`\spad{tubeRadius(r)} specifies a radius,{} \spad{r},{} for a tube plot around a 3D curve; is expressed in the form \spad{tubeRadius == 4}.
850529`\spad{tubeRadius(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{tubeRadius}. If the option does not exist the value,{} \spad{n} is returned.
850608`\spad{tube(c,{}r,{}n)} creates a tube of radius \spad{r} around the curve \spad{c}.
850696`\spad{tube(c,{}ll,{}b)} creates a tube of the domain \spadtype{TubePlot} from a space curve \spad{c} of the category \spadtype{PlottableSpaceCurveCategory},{} a list of lists of points (loops) \spad{ll} and a boolean \spad{b} which if \spad{true} indicates a closed tube,{} or if \spad{false} an open tube.
850783`\spad{twist(f)} is the function \spad{g} \indented{1}{such that \spad{g (a,{}b)= f(b,{}a)}.}
850848`\spad{twoFactor(p,{}n)} returns the factorisation of polynomial \spad{p},{} a sparse univariate polynomial (sup) over a sup over \spad{F}. Also,{} \spad{p} is assumed primitive and square-free and \spad{n} is the degree of the inner variable of \spad{p} (maximum of the degrees of the coefficients of \spad{p}).
851026`\spad{typeList(t,{}tab)} returns a list of all the objects of type \spad{t} in \spad{tab}
851162`\spad{typeLists(tab)} returns a list of lists of types of objects in \spad{tab}
851287`\axiom{type(\spad{jv})} yields the type (\axiom{Const,{} \spad{Indep},{} \spad{Dep},{} Deriv}) of the jet variable \axiom{\spad{jv}}.
851338`\spad{ucodeToString(n)} converts Unicode code point to one-character string. If UTF-8 is in use the string may contain multiple octets.
851388`\spad{unary?(op)} tests if \spad{op} is unary.
851438`\spad{unaryFunction(a)} is a local function
851517`if this is a lambda term then replace De Bruijn index in sub-nodes with string name
851559`\spad{uncouplingMatrices(M)} returns \spad{[A_1,{}...,{}A_n]} such that if \spad{y = [y_1,{}...,{}y_n]} is a solution of \spad{y' = M y},{} then \spad{[\$y_j',{}y_j'',{}...,{}y_j^{(n)}\$] = \$A_j y\$} for all \spad{j}\spad{'s}.
851649`constructor for graph with given objects more objects and arrows can be added later if required.
851775`constructor for graph with given list of object names. Use this version of the constructor if you don\spad{'t} intend to create diagrams and therefore don\spad{'t} care about \spad{x},{}\spad{y} coordinates. more objects and arrows can be added later if required.
851839`constructor for graph with given objects and arrows more objects and arrows can be added later if required.
852129`constructor for graph with given objects and adjacency matrix.
852296`\spad{unexpand(p)} returns \spad{p} in recursive form.
852390`\spad{uniform01()} \undocumented
852450`\spad{uniform(s)} \undocumented
852532`\spad{uniform(s)} \undocumented
852594`\spad{uniform(f,{}g)} \undocumented
852667`\spad{union(x)} constructs a compound boundary from the union of \spad{x} x:List \% is supplied list of boundaries to form union
852901`\spad{union(l1,{} l2)} returns set-theoretic union of \spad{l1} and \spad{l2}.
853087`\spad{union(a,{} b)} makes the set-union of two match results.
853144`\spad{union(x,{}u)} returns the set aggregate \spad{u} with the element \spad{x} added. If \spad{u} already contains \spad{x},{} \axiom{union(\spad{x},{}\spad{u})} returns a copy of \spad{u}.
853192`\spad{union(u,{}x)} returns the set aggregate \spad{u} with the element \spad{x} added. If \spad{u} already contains \spad{x},{} \axiom{union(\spad{u},{}\spad{x})} returns a copy of \spad{u}.
853240`\spad{union(u,{}v)} returns the set aggregate of elements which are members of either set aggregate \spad{u} or \spad{v}.
853332`\spad{unit?(x)} tests whether \spad{x} is a unit,{} \spadignore{i.e.} is invertible.
853463`\spad{unitCanonical(x)} returns \spad{unitNormal(x).canonical}.
853751`\spad{unitNormal(x)} tries to choose a canonical element from the associate class of \spad{x}. The attribute canonicalUnitNormal,{} if asserted,{} means that the "canonical" element is the same across all associates of \spad{x} if \spad{unitNormal(x) = [u,{}c,{}a]} then \spad{u*c = x},{} \spad{a*u = 1}.
853836`\spad{unitNormalize(u)} normalizes the unit part of the factorization. For example,{} when working with factored integers,{} this operation will ensure that the bases are all positive integers.
853886`\spad{unitVector(x)} \undocumented
853961`\spad{unitVector(p)} creates the unit vector of the point \spad{p} and returns the result as a point. Note: \spad{unitVector(p) = p/|p|}.
854042`\spad{unitVector(p)} returns a vector with the same direction as \spad{p} but with unit length. We scale until length is 1
854092`\spad{unitVector(n)} produces a vector with 1 in position \spad{n} and zero elsewhere.
854250`\spad{unit()} returns a unit of the algebra (necessarily unique),{} or \spad{"failed"} if there is none.
854349`\spad{unit(lf)} will mark off the units according to the indicated list \spad{lf}. This option is expressed in the form \spad{unit == [f1,{}f2]}.
854398`\spad{unit(u)} extracts the unit part of the factorization.
854438`\spad{unit(objs,{} arrowName)} constructs a graph with vertices (from \spad{objs}) and arrows from each object to itself. arrowName is a prefix for all arrow names,{} this will be followed by a number starting at 1 and incremented for each arrow
854494`\spad{unitsColorDefault()} returns the default color of the unit ticks in a 2D viewport.
854559`\spad{unitsColorDefault(p)} sets the default color of the unit ticks in a 2D viewport to the palette \spad{p}.
854631`\spad{units(\spad{gi})} returns the list of unit increments for the \spad{x} and \spad{y} axes of the indicated graph,{} \spad{\spad{gi}},{} of the domain \spadtype{GraphImage}.
854681`\spad{units(l,{}u)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{unit}. If the option does not exist the value,{} \spad{u} is returned.
854767`\spad{units(\spad{gi},{}lu)} modifies the list of unit increments for the \spad{x} and \spad{y} axes of the given graph,{} \spad{\spad{gi}} of the domain \spadtype{GraphImage},{} to be that of the list of unit increments,{} \spad{lu},{} and returns the new list of units for \spad{\spad{gi}}.
854829`\spad{units(v,{}n,{}c)} displays the units of the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} with the units color set to the given palette color \spad{c}.
854908`\spad{units(v,{}n,{}s)} displays the units of the graph in field \spad{n} of the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} if \spad{s} is "on",{} or does not display the units if \spad{s} is "off".
854986`\axiom{univariate?(\spad{p})} returns \spad{true} iff \axiom{\spad{p}} involves one and only one variable.
855184`\spad{univariatePolynomial(f,{}k)} returns a univariate polynomial \indented{1}{consisting of the sum of all terms of \spad{f} of degree \spad{<= k}.}
855319`\spad{univariatePolynomial(f,{}k)} returns a univariate polynomial \indented{1}{consisting of the sum of all terms of \spad{f} of degree \spad{<= k}.}
855448`\axiom{univariatePolynomialsGcds(\spad{lp})} returns \axiom{\spad{lg}} where \axiom{\spad{lg}} is a list of the gcds of every pair in \axiom{\spad{lp}} of univariate polynomials in the same main variable.
855563`\axiom{univariatePolynomialsGcds(\spad{lp},{}opt)} returns the same as \axiom{univariatePolynomialsGcds(\spad{lp})} if \axiom{opt} is \axiom{\spad{false}} and if the previous operation does not return any non null and constant polynomial,{} else return \axiom{[1]}.
855686`\axiom{univariatePolynomials(\spad{lp})} returns \axiom{ups,{}nups} where \axiom{ups} is a list of the univariate polynomials,{} and \axiom{nups} are the other ones.
855814`\spad{univariateSolve(lp)} returns the same as \spad{univariateSolve(lp,{}false,{}false,{}false)}.
855995`\spad{univariateSolve(ts)} returns a univariate representation of \spad{ts}. See \axiomOpFrom{rur}{RationalUnivariateRepresentationPackage}(\spad{lp},{}\spad{true}).
856175`\spad{univariateSolve(lp,{}info?)} returns the same as \spad{univariateSolve(lp,{}info?,{}false,{}false)}.
856364`\spad{univariateSolve(lp,{}info?,{}check?)} returns the same as \spad{univariateSolve(lp,{}info?,{}check?,{}false)}.
856561`\spad{univariateSolve(lp,{}info?,{}check?,{}lextri?)} returns a univariate representation of the variety associated with \spad{lp}. Moreover,{} if \spad{info?} is \spad{true} then some information is displayed during the decomposition into regular chains. If \spad{check?} is \spad{true} then the result
856561`-- is checked. See \axiomOpFrom{rur}{RationalUnivariateRepresentationPackage}(\spad{lp},{}\spad{true}). If \spad{lextri?} is \spad{true} then the lexTriangular algorithm is called from the \spadtype{LexTriangularPackage} constructor (see \axiomOpFrom{zeroSetSplit}{LexTriangularPackage}(\spad{lp},{}\spad{false})). Otherwise,{} the triangular decomposition is computed directly from the input system by using the \axiomOpFrom{zeroSetSplit}{RegularChain} from \spadtype{RegularChain}.
856766`\spad{univariate(p)} converts the multivariate polynomial \spad{p},{} which should actually involve only one variable,{} into a univariate polynomial in that variable,{} whose coefficients are in the ground ring. Error: if polynomial is genuinely multivariate
856859`\spad{univariate(f,{} v)} returns \spad{f} viewed as a univariate rational function in \spad{v}.
857074`\spad{univariate(f,{} v)} returns \spad{f} viewed as a univariate rational function in \spad{v}.
857216`\spad{univariate(p,{} x)} converts the polynomial \spad{p} to a one of type \spad{UnivariatePolynomial(x,{}Polynomial(R))},{} ie. as a member of \spad{R[...][x]}.
857347`\spad{univariate(f,{} k)} returns \spad{f} viewed as a univariate fraction in \spad{k}.
857469`\spad{univariate(p,{}v)} converts the multivariate polynomial \spad{p} into a univariate polynomial in \spad{v},{} whose coefficients are still multivariate polynomials (in all the other variables).
857570`\spad{univariate(f,{} x,{} p)} returns \spad{f} viewed as a univariate polynomial in \spad{x},{} using the side-condition \spad{p(x) = 0}.
857710`\spad{univariate(f,{}k,{}k,{}p)} \undocumented
857887`\spad{univcase should} be local
858013`\spad{universe()}\$\spad{D} returns the universal set for finite set aggregate \spad{D}.
858168`\spad{unmakeSUP(sup)} converts \spad{sup} of type \spadtype{SparseUnivariatePolynomial(R)} to be a member of the given type. Note: converse of makeSUP.
858261`\spad{unpack_poly(vars,{} exps,{} coeffs,{} lo,{} \spad{hi})} converts polynomial from packed to sparse representation. Coefficients of packed polynomial are stored in \spad{coeffs}(\spad{l0}) up to \spad{coeffs}(\spad{hi}) Exponents are stored in corresponding elements of \spad{exps}.
858432`\spad{unparse(f)} returns a string \spad{s} such that the parser would transform \spad{s} to \spad{f}. Error: if \spad{f} is not the parsed form of a string.
858478`\axiom{unprotectedRemoveRedundantFactors(\spad{p},{}\spad{q})} returns the same as \axiom{removeRedundantFactors(\spad{p},{}\spad{q})} but does assume that neither \axiom{\spad{p}} nor \axiom{\spad{q}} lie in the base ring \axiom{\spad{R}} and assumes that \axiom{infRittWu?(\spad{p},{}\spad{q})} holds. Moreover,{} if \axiom{\spad{R}} is \spad{gcd}-domain,{} then \axiom{\spad{p}} and \axiom{\spad{q}} ar
858478`--e assumed to be square free.
858581`\spad{unrankImproperPartitions0(n,{}m,{}k)} computes the {\em k}\spad{-}th improper partition of nonnegative \spad{n} in \spad{m} nonnegative parts in reverse lexicographical order. Example: {\em [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]}. Error: if \spad{k} is negative or too big. Note: count
858581`--ing of subtrees is done by \spadfunFrom{numberOfImproperPartitions}{SymmetricGroupCombinatoricFunctions}.
858699`\spad{unrankImproperPartitions1(n,{}m,{}k)} computes the {\em k}\spad{-}th improper partition of nonnegative \spad{n} in at most \spad{m} nonnegative parts ordered as follows: first,{} in reverse lexicographically according to their non-zero parts,{} then according to their positions (\spadignore{i.e.} lexicographical order using {\em subSet}: {\em [3,{}0,{}0] < [0,{}3,{}0] < [0,{}0,{}3]
858699`-- < [2,{}1,{}0] < [2,{}0,{}1] < [0,{}2,{}1] < [1,{}2,{}0] < [1,{}0,{}2] < [0,{}1,{}2] < [1,{}1,{}1]}). Note: counting of subtrees is done by {\em numberOfImproperPartitionsInternal}.
858817`\spad{unravel(t)} produces a tensor from a list of components such that \indented{2}{\spad{unravel(ravel(t)) = t}.}
858883`\spad{untab(lp,{}llp)} is an auxiliary function which unbumps a tableau \spad{llp},{} using \spad{lp} to accumulate pairs
858974`\spad{unvectorise(v)} returns the polynomial which has for coefficients the entries of \spad{v} in the increasing order.
859054`\spad{unvectorise(vect,{} var,{} n)} returns \spad{vect(1) + vect(2)*var + ... + vect(n+1)*var^(n)} where \spad{vect} is the vector of the coefficients of the polynomail ,{} \spad{var} the new variable and \spad{n} the degree.
859732`\spad{updatD }\undocumented
860023`\spad{updatF }\undocumented
860217`\spad{update!(hs,{} x)} computes new values of HashState from \spad{hs} and \spad{x} and might destructively operate on its first argument.
860273`\axiom{updateStatus!(a)} returns a where the status of the vertices are updated to satisfy the "termination condition".
860330`\spad{update(v,{}gr,{}n)} drops the graph \spad{gr} in slot \spad{n} of viewport \spad{v}. The graph \spad{gr} must have been transmitted already and acquired an integer key.
860413`\spad{upperCase!(s)} destructively replaces the alphabetic characters in \spad{s} by upper case characters.
860464`\spad{upperCase?(c)} tests if \spad{c} is an upper case letter,{} \spadignore{i.e.} one of A..\spad{Z}.
860514`\spad{upperCase()} returns the class of all characters for which \spadfunFrom{upperCase?}{Character} is \spad{true}.
860607`\spad{upperCase(s)} returns the string with all characters in upper case.
860657`\spad{upperCase(c)} converts a lower case letter to the corresponding upper case letter. If \spad{c} is not a lower case letter,{} then it is returned unchanged.
860701`\spad{useEisensteinCriterion?()} returns \spad{true} if factorizers check Eisenstein\spad{'s} criterion before factoring.
860778`\spad{useEisensteinCriterion(b)} chooses whether factorizers check Eisenstein\spad{'s} criterion before factoring: \spad{true} for using it,{} \spad{false} else. Returns the previous value.
860861`\spad{useNagFunctions()} indicates whether NAG functions are being used \indented{1}{for mathematical and machine constants.}
860957`\spad{useNagFunctions(v)} sets the flag which controls whether NAG functions \indented{1}{are being used for mathematical and machine constants.\space{2}The previous} \indented{1}{value is returned.}
861060`\spad{useSingleFactorBound?()} returns \spad{true} if algorithm with single factor bound is used for factorization,{} \spad{false} for algorithm with overall bound.
861135`\spad{useSingleFactorBound(b)} chooses the algorithm to be used by the factorizers: \spad{true} for algorithm with single factor bound,{} \spad{false} for algorithm with overall bound. Returns the previous value.
861216`\spad{userOrdered?()} tests if the partial ordering induced by \spadfunFrom{setOrder}{UserDefinedPartialOrdering} is not empty.
861286`\axiom{usingTable?()} returns \spad{true} iff the hash-table is used
861364`\spad{validExponential([k1,{}...,{}kn],{}f,{}x)} returns \spad{g} if \spad{exp(f)=g} and \spad{g} involves only \spad{k1...kn},{} and "failed" otherwise.
861582`\spad{value(x)} returns a SingleInteger value corresponding to \spad{x}.
861633`returns: \indented{2}{"T"::Symbol = \spad{T}} \indented{2}{"F"::Symbol = \spad{_|}\space{4}\spad{++}\space{3}"E"::Symbol = error} \indented{2}{"P"::Symbol = proposition} \indented{2}{"C"::Symbol = compound} Constructs lambda term and bind any variables with the name provided
861674`\axiom{value(\spad{n})} returns the value of the node \spad{n}.
861722`\spad{value(x)} returns the integer associated with \spad{x}
861768`\spad{value(u)} returns the value of the node \spad{u}.
861819`\spad{value(q)} returns the value (\spadignore{i.e.} right hand side) of \axiom{\spad{q}}.
861867`\spad{values()} returns the number of initial values to display at most.
861936`\spad{values N} sets the number of initial values to display at most.
862023`\spad{var1StepsDefault()} is the current setting for the number of steps to take when creating a 3D mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (\spadignore{e.g.} \spad{x=0}..10)).
862095`\spad{var1StepsDefault(i)} sets the number of steps to take when creating a 3D mesh in the direction of the first defined free variable to \spad{i} (a free variable is considered defined when its range is specified (\spadignore{e.g.} \spad{x=0}..10)).
862182`\spad{var1Steps(n)} indicates the number of subdivisions,{} \spad{n},{} of the first range variable. This option is expressed in the form \spad{var1Steps == n}.
862240`\spad{var1Steps(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{var1Steps}. If the option does not exist the value,{} \spad{n} is returned.
862338`\spad{var2StepsDefault()} is the current setting for the number of steps to take when creating a 3D mesh in the direction of the first defined free variable (a free variable is considered defined when its range is specified (\spadignore{e.g.} \spad{x=0}..10)).
862410`\spad{var2StepsDefault(i)} sets the number of steps to take when creating a 3D mesh in the direction of the first defined free variable to \spad{i} (a free variable is considered defined when its range is specified (\spadignore{e.g.} \spad{x=0}..10)).
862497`\spad{var2Steps(n)} indicates the number of subdivisions,{} \spad{n},{} of the second range variable. This option is expressed in the form \spad{var2Steps == n}.
862555`\spad{var2Steps(l,{}n)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{var2Steps}. If the option does not exist the value,{} \spad{n} is returned.
862653`\spad{varList(x)} returns the list of variables of \spad{x}.
862711`\axiom{varList(\spad{g})} returns the list of variables of \axiom{\spad{g}}.
862785`\axiom{varList(\spad{x})} returns the list of distinct entries of \axiom{\spad{x}}.
862846`\axiom{varList(\spad{x})} returns the list of distinct entries of \axiom{\spad{x}}.
862902`\spad{varList([l1]*[l2]*...[ln])} returns the list of variables in the word \spad{l1*l2*...*ln}.
862984`\axiom{varList(\spad{x})} returns the list of distinct entries of \axiom{\spad{x}}.
863051`\spad{varList(x)} returns the list of variables which appear in \spad{x}.
863108`constructs variable with a name but no type.
863147`constructs variable with a name and also a type.
863193`returns \spad{true} if this is an variable
863251`\spad{variableName returns} the name of the variable used in by the algebraic differential equation,{} default being \spad{x}
863329`\spad{variableName(d)} specifies the variable used in by the algebraic differential equation. This option is expressed in the form \spad{variableName == d}.
863382`\spad{variable()} returns the symbol
863431`\spad{variable(s)} returns a member of the variable set or failed
863518`\spad{variable(v)} returns \spad{s} if \spad{v} is any derivative of the differential indeterminate \spad{s}.
863582`\spad{variable(f)} returns the (unique) power series variable of the power series \spad{f}.
863661`\spad{variable(q)} returns the variable (\spadignore{i.e.} left hand side) of \axiom{\spad{q}}.
863712`\spad{variable(segb)} returns the variable from the left hand side of the \spadtype{SegmentBinding}. For example,{} if \spad{segb} is \spad{v=a..b},{} then \spad{variable(segb)} returns \spad{v}.
863767`\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \spad{f}.
863923`\axiom{variables(\spad{q})} computes the list of all jet variables up to order \axiom{\spad{q}}.
864314`\spad{variables(upoly)} returns the list of variables for the coefficients of \spad{upoly}.
864410`\spad{variables(e)} return a list of all the variables in \spad{e}.
864507`\spad{variables(p)} returns the list of those variables actually appearing in the polynomial \spad{p}.
864591`\spad{variables(p)} returns the list of matching variables appearing in \spad{p}.
864642`\spad{variables(f)} returns the list of variables appearing in the numerator or the denominator of \spad{f}.
864727`\spad{variables(f)} returns the list of all the variables of \spad{f}.
864788`\spad{variables(f)} returns a list of the variables occurring in the power series \spad{f}.
864865`\axiom{variables(\spad{ps})} returns the decreasingly sorted list of the variables which are variables of some polynomial in \axiom{\spad{ps}}.
865032`\axiom{variables(\spad{q},{} \spad{c})} computes all jet variables of order \axiom{\spad{q}} whose class is greater than or equal to \axiom{\spad{c}}.
865122`\spad{variationOfParameters(op,{} g,{} [f1,{}...,{}fm])} returns \spad{[u1,{}...,{}um]} such that a particular solution of the equation \spad{op y = g} is \spad{f1 int(u1) + ... + fm int(um)} where \spad{[f1,{}...,{}fm]} are linearly independent and \spad{op(\spad{fi})=0}. The value "failed" is returned if \spad{m < n} and no particular solution is found.
865220`\spad{vark([f1,{}...,{}fn],{}x)} returns the set-theoretic union of \spad{(varselect(f1,{}x),{}...,{}varselect(fn,{}x))}.
865296`\spad{varselect([k1,{}...,{}kn],{} x)} returns the \spad{ki} which involve \spad{x}.
865385`\spad{vconcat(u)} vertically concatenates all forms in list \spad{u}.
865434`\spad{vconcat(f,{}g)} vertically concatenates forms \spad{f} and \spad{g}.
865480`\spad{vector_add_mul(v1,{} v2,{} m,{} n,{} c,{} p)} sets \spad{v1}(\spad{m}),{} ...,{} \spad{v1}(\spad{n}) to corresponding extries in \spad{v1} + \spad{c*v2} modulo \spad{p}.
865600`\spad{vector_combination(v1,{} c1,{} v2,{} c2,{} n,{} delta,{} p)} replaces first \spad{n} + 1 entires of \spad{v1} by corresponding entries of \spad{c1*v1+c2*x^delta*v2} mod \spad{p}.
865732`\spad{vector(l)} converts the list \spad{l} to a vector.
865812`\spad{vector(l)} converts the list \spad{l} to a vector.
865946`\spad{vectorise(p,{} n)} returns \spad{[a0,{}...,{}a(n-1)]} where \spad{p = a0 + a1*x + ... + a(n-1)*x^(n-1)} + higher order terms. The degree of polynomial \spad{p} can be different from \spad{n-1}.
866096`\spad{vertConcat(x,{}y)} vertically concatenates two matrices with an equal number of columns. The entries of \spad{y} appear below of the entries of \spad{x}. Error: if the matrices do not have the same number of columns.
866160`\spad{viewDefaults()} resets all the default graphics settings.
866217`\spad{viewDeltaXDefault()} returns the current default horizontal offset from the center of the viewport window.
866285`\spad{viewDeltaXDefault(dx)} sets the current default horizontal offset from the center of the viewport window to be \spad{dx} and returns \spad{dx}.
866358`\spad{viewDeltaYDefault()} returns the current default vertical offset from the center of the viewport window.
866426`\spad{viewDeltaYDefault(dy)} sets the current default vertical offset from the center of the viewport window to be \spad{dy} and returns \spad{dy}.
866499`\spad{viewPhiDefault()} returns the current default latitudinal view angle in radians.
866564`\spad{viewPhiDefault(p)} sets the current default latitudinal view angle in radians to the value \spad{p} and returns \spad{p}.
866634`\spad{viewPosDefault()} returns the default \spad{X} and \spad{Y} position of a viewport window unless overriden explicityly,{} newly created viewports will have this \spad{X} and \spad{Y} coordinate.
866713`\spad{viewPosDefault([x,{}y])} sets the default \spad{X} and \spad{Y} position of a viewport window unless overriden explicityly,{} newly created viewports will have th \spad{X} and \spad{Y} coordinates \spad{x},{} \spad{y}.
866816`\spad{viewSizeDefault()} returns the default viewport width and height.
866893`\spad{viewSizeDefault([w,{}h])} sets the default viewport width to \spad{w} and height to \spad{h}.
866991`\spad{viewThetaDefault()} returns the current default longitudinal view angle in radians.
867058`\spad{viewThetaDefault(t)} sets the current default longitudinal view angle in radians to the value \spad{t} and returns \spad{t}.
867130`\spad{viewWriteAvailable()} returns a list of available methods for writing,{} such as BITMAP,{} POSTSCRIPT,{} etc.
867201`\spad{viewWriteDefault()} returns the list of things to write in a viewport data file; a viewAlone file is always generated.
867270`\spad{viewWriteDefault(l)} sets the default list of things to write in a viewport data file to the strings in \spad{l}; a viewAlone file is always genereated.
867351`\spad{viewZoomDefault()} returns the current default graph scaling value.
867417`\spad{viewZoomDefault(s)} sets the current default graph scaling value to \spad{s} and returns \spad{s}.
867488`\spad{viewpoint(vp)} creates a viewpoint data structure corresponding to the list of values. The values are interpreted as [theta,{} phi,{} scale,{} scaleX,{} scaleY,{} scaleZ,{} deltaX,{} deltaY]. This option is expressed in the form \spad{viewpoint == ls}.
867685`\spad{viewpoint(v)} returns the current viewpoint setting of the given viewport,{} \spad{v}. This function is useful in the situation where the user has created a viewport,{} proceeded to interact with it via the control panel and desires to save the values of the viewpoint as the default setting
867685`--s for another viewport to be created using the system.
867896`\spad{viewpoint(l,{}ls)} takes the list of draw options,{} \spad{l},{} and checks the list to see if it contains the option \spad{vi
867896`--ewpoint}. IF the option does not exist,{} the value \spad{ls} is returned.
868272`\spad{viewpoint(v,{}viewpt)} sets the viewpoint for the viewport. The viewport record consists of the latitudal and longitudal angles,{} the zoom factor,{} the \spad{X},{} \spad{Y},{} and \spad{Z} scales,{} and the \spad{X} and \spad{Y} displacements.
868488`\spad{viewpoint(v,{}th,{}phi)} sets the longitudinal view angle to \spad{th} radians and the latitudinal view angle to \spad{phi} radians for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until the function \spadfun{makeViewport3D} is executed again for \spad{v}.
868561`\spad{viewpoint(v,{}rotx,{}roty,{}rotz)} sets the rotation about the \spad{x}-axis to be \spad{rotx} radians,{} sets the rotation about the \spad{y}-axis to be \spad{roty} radians,{} and sets the rotation about the \spad{z}-axis to be \spad{rotz} radians,{} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport} and displays \spad{v} with the new view position.
868640`\spad{viewpoint(v,{}th,{}phi,{}s,{}dx,{}dy)} sets the longitudinal view angle to \spad{th} radians,{} the latitudinal view angle to \spad{phi} radians,{} the scale factor to \spad{s},{} the horizontal viewport offset to \spad{dx},{} and the vertical viewport offset to \spad{dy} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed until
868640`-- the function \spadfun{makeViewport3D} is executed again for \spad{v}.
868731`\spad{viewpoint(v,{}th,{}phi,{}s,{}dx,{}dy)} sets the longitudinal view angle to \spad{th} degrees,{} the latitudinal view angle to \spad{phi} degrees,{} the scale factor to \spad{s},{} the horizontal viewport offset to \spad{dx},{} and the vertical viewport offset to \spad{dy} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}. The new viewpoint position is not displayed u
868731`--ntil the function \spadfun{makeViewport3D} is executed again for \spad{v}.
868826`\spad{viewport2D()} returns an undefined two-dimensional viewport of the domain \spadtype{TwoDimensionalViewport} whose contents are empty.
868882`\spad{viewport3D()} returns an undefined three-dimensional viewport of the domain \spadtype{ThreeDimensionalViewport} whose contents are empty.
868940`\spad{virtualDegree }\undocumented
869043`virtualDegree
869139`\spad{void()} produces a void object.
869171`\spad{vspace(n)} creates white space of height \spad{n}.
869218`\spad{weakBiRank(x)} determines the number of linearly independent elements in the \spad{bi*x*bj},{} \spad{i,{}j=1,{}...,{}n},{} where \spad{b=[b1,{}...,{}bn]} is a basis.
869288`\spad{weberE(v,{} z)} is the Weber \spad{E} function
869350`\spad{weberE(v,{} z)} is the Weber \spad{E} function.
869408`\spad{weierstrassHalfPeriods(g2,{} g3)} computes half periods of Weierstrass elliptic functions from invariants \spad{g2},{} \spad{g3}.
869523`\spad{weierstrassInvariants(w1,{} w2)} computes invariants \spad{g2},{} \spad{g3} of Weierstrass elliptic functions from half periods \spad{w1},{} \spad{w2}.
869637`\spad{weierstrassP0(g2,{} g3,{} x)} computes finite part of weierstrassP(\spad{g2},{} \spad{g3},{} \spad{x}). \spad{x} must have no term of order 0.
869735`\spad{weierstrassPPrime0(g2,{} g3,{} x)} computes finite part of weierstrassPPrime(\spad{g2},{} \spad{g3},{} \spad{x}). \spad{x} must have no term of order 0.
869838`\spad{weierstrassPPrime(g2,{} g3,{} x)}
869913`\spad{weierstrassPPrime(g2,{} g3,{} x)} is the derivative of the Weierstrass \spad{P} function
870032`\spad{weierstrassPPrime(g2,{} g3,{} x)} is the derivative of the Weierstrass \spad{P} function
870115`\spad{weierstrassPPrime(g2,{} g3,{} z)} is the derivative of Weierstrass \spad{P} function.
870187`\spad{weierstrassP(g2,{} g3,{} x)}
870257`\spad{weierstrassP(g2,{} g3,{} x)} is the Weierstrass \spad{P} function
870371`\spad{weierstrassP(g2,{} g3,{} x)} is the Weierstrass \spad{P} function
870449`\spad{weierstrassP(g2,{} g3,{} z)} is the Weierstrass \spad{P} function.
870516`\spad{weierstrassSigma0(g2,{} g3,{} x)} computes weierstrassSigma(\spad{g2},{} \spad{g3},{} \spad{x}). \spad{x} must have no term of order 0.
870618`\spad{weierstrassSigma(g2,{} g3,{} x)}
870692`\spad{weierstrassSigma(g2,{} g3,{} z)} is the Weierstrass Sigma function.
870763`\spad{weierstrassZeta0(g2,{} g3,{} x)} computes finite part of weierstrassZeta(\spad{g2},{} \spad{g3},{} \spad{x}). \spad{x} must have no term of order 0.
870864`\spad{weierstrassZeta(g2,{} g3,{} x)}
870937`\spad{weierstrassZeta(g2,{} g3,{} z)} is the Weierstrass Zeta function.
871007`\spad{weierstrass(v,{}ts)} where \spad{v} is a variable and \spad{ts} is \indented{1}{a TaylorSeries,{} implements the Weierstrass Preparation} \indented{1}{Theorem. The result is a list of TaylorSeries that} \indented{1}{are the coefficients of the equivalent series.}
871328`\axiom{weight(\spad{jv})} assigns each jet variable a unique integer reflecting its position in the internal ordering. The variable with the greater weight is also greater in this ordering.
871393`\spad{weight(p)} returns the maximum weight of all differential monomials appearing in the differential polynomial \spad{p}.
871480`\spad{weight(v)} returns the weight of the derivative \spad{v}.
871559`\spad{weight(op)} returns the weight attached to \spad{op}.
871705`\spad{weight(op,{} n)} attaches the weight \spad{n} to \spad{op}.
871769`\spad{weight(p,{} s)} returns the maximum weight of all differential monomials appearing in the differential polynomial \spad{p} when \spad{p} is viewed as a differential polynomial in the differential indeterminate \spad{s} alone.
871858`The elements of the returned array will represent the distance weights between vertices. if there is no connection between the vertices then the element will be set to "disjoint"
871962`constructor for graph with given objects more objects and arrows can be added later if required.
872095`constructor for graph with given list of object names. Use this version of the constructor if you don\spad{'t} intend to create diagrams and therefore don\spad{'t} care about \spad{x},{}\spad{y} coordinates. more objects and arrows can be added later if required.
872157`constructor for graph with given objects and arrows more objects and arrows can be added later if required.
872445`\spad{weightedGraph(ob,{} ar)} constructs graph with objects on and arrows ar. More objects and arrows can be added later if required.
872724`constructor for graph with given objects and adjacency matrix.
872817`\spad{weighted(l)} \undocumented
873000`\spad{weights(p)} returns a list of weights of differential monomials appearing in differential polynomial \spad{p}.
873186`\spad{weights(p,{} s)} returns a list of weights of differential monomials appearing in the differential polynomial \spad{p} when \spad{p} is viewed as a differential polynomial in the differential indeterminate \spad{s} alone.
873282`\spad{whatInfinity(x)} returns 0 if \spad{x} is finite,{} 1 if \spad{x} is +infinity,{} and \spad{-1} if \spad{x} is -infinity.
873351`\spad{whileLoop(s,{}c)} creates a while loop in FORTRAN.
873404`\spad{whittakerM(k,{} m,{} z)} is the Whittaker \spad{M} function
873472`\spad{whittakerM(k,{} m,{} z)} is the Whittaker \spad{M} function.
873537`\spad{whittakerW(k,{} m,{} z)} is the Whittaker \spad{W} function
873605`\spad{whittakerW(k,{} m,{} z)} is the Whittaker \spad{W} function.
873670`\spad{wholePart x} returns the integer part of \spad{x}.
873726`\spad{wholePart(x)} extracts the whole part of \spadvar{\spad{x}}. That is,{} if \spad{x = continuedFraction(b0,{} [a1,{}a2,{}a3,{}...],{} [b1,{}b2,{}b3,{}...])},{} then \spad{wholePart(x) = b0}.
873780`\spad{wholePart(p)} extracts the whole part of the partial fraction \spad{p}.
873832`\spad{wholePart(x)} returns the whole part of the fraction \spad{x} \spadignore{i.e.} the truncated quotient of the numerator by the denominator.
873912`\spad{wholeRadix(l)} creates an integral radix expansion from a list of ragits. For example,{} \spad{wholeRadix([1,{}3,{}4])} will return \spad{134}.
873977`\spad{wholeRagits(rx)} returns the ragits of the integer part of a radix expansion.
874043`\spad{width()} returns the width of the display area (an integer).
874087`\spad{width(f)} returns the width of form \spad{f} (an integer).
874133`\spad{width(u)} returns \axiom{sup(\spad{u}) - inf(\spad{u})}.
874182`\spad{wignerDistribution(a)} produces a Wigner distribution of variance \spad{a}.
874260`\spad{withPredicates(p,{} [p1,{}...,{}pn])} makes a copy of \spad{p} and attaches the predicate \spad{p1} and ... and \spad{pn} to the copy,{} which is returned.
874320`\spad{wordInGenerators(p,{}gp)} returns the word for the permutation \spad{p} in the original generators of the group {\em gp},{} represented by the indices of the list,{} given by {\em generators}.
874418`\spad{wordInStrongGenerators(p,{}gp)} returns the word for the permutation \spad{p} in the strong generators of the group {\em gp},{} represented by the indices of the list,{} given by {\em strongGenerators}.
874522`\spad{wordsForStrongGenerators(gp)} returns the words for the strong generators of the group {\em gp} in the original generators of {\em gp},{} represented by their indices in the list,{} given by {\em generators}.
874619`\spad{wreath(s1,{}s2)} is the cycle index of the wreath product \indented{1}{of the two groups whose cycle indices are \spad{s1} and} \indented{1}{\spad{s2}.}
874777`\spad{writable?(f)} tests if the named file be opened for writing. The named file need not already exist.
874833`\spad{write!(f,{}s)} puts the value \spad{s} into the file \spad{f}. The state of \spad{f} is modified so subsequents call to \spad{write!} will append one after another.
874886`writes a category to a file.
874966`\spad{writeLine!(f)} finishes the current line in the file \spad{f}. An empty string is returned. The call \spad{writeLine!(f)} is equivalent to \spad{writeLine!(f,{}"")}.
875014`\spad{writeLine!(f,{}s)} writes the contents of the string \spad{s} and finishes the current line in the file \spad{f}. The value of \spad{s} is returned.
875069`writes 3D SubSpace to a file in Wavefront (.OBJ) format
875141`\spad{writeObj(n,{}filename)} writes an 'OBJ' (Wavefront) representation of node \spad{'n'} to the \spad{filename} supplied.
875193`writes a package to a file.
875286`\spad{writeSvgQuantised(n,{}filename)} writes an 'SVG' representation of node \spad{'n'} to the \spad{filename} supplied.
875347`\spad{writeSvg(n,{}filename)} writes an 'SVG' representation of node \spad{'n'} to the \spad{filename} supplied.
875399`\spad{writeVRML(content,{} filename)} writes the element as \spad{VRML2}. \spad{VRML2} is not \spad{xml} but it has the same semantics and node names as \spad{X3D} but a different syntax so it makes sense to create an \spad{XML} structure and then format the output differently.
875460`\spad{writeVRML(n,{}filename)} writes an 'VRML' representation of node \spad{'n'} to the \spad{filename} supplied.
875513`writeX3d:(\spad{n},{}filename) writes an 'X3D' representation of node \spad{'n'} to the filename supplied.
875565`\spad{writeXml(content,{}filename)} writes an \spad{xml} element and all the elements below it to an \spad{xml} file. This assumes that all elements,{} in a file,{} have a single root which is usually the case.
875625`\spad{write(v,{}s)} takes the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \spad{s},{} which contains the graph data files for \spad{v}.
875689`\spad{write(v,{}s)} takes the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \spad{s},{} which contains the graph data file for \spad{v}.
875755`\spad{write(v,{}s,{}lf)} takes the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \spad{s},{} which contains the graph data files for \spad{v} and the optional file types indicated by the list \spad{lf}.
875832`\spad{write(v,{}s,{}f)} takes the given two-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{TwoDimensionalViewport},{} and creates a directory indicated by \spad{s},{} which contains the graph data files for \spad{v} and an optional file type \spad{f}.
875903`\spad{write(v,{}s,{}lf)} takes the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \spad{s},{} which contains the graph data file for \spad{v} and the optional file types indicated by the list \spad{lf}.
875982`\spad{write(v,{}s,{}f)} takes the given three-dimensional viewport,{} \spad{v},{} which is of domain \spadtype{ThreeDimensionalViewport},{} and creates a directory indicated by \spad{s},{} which contains the graph data file for \spad{v} and an optional file type \spad{f}.
876055`\spad{wronskianMatrix([f1,{}...,{}fn])} returns the \spad{n x n} matrix \spad{m} whose i^th row is \spad{[f1^(i-1),{}...,{}fn^(i-1)]}.
876124`\spad{wronskianMatrix([f1,{}...,{}fn],{} q,{} D)} returns the \spad{q x n} matrix \spad{m} whose i^th row is \spad{[f1^(i-1),{}...,{}fn^(i-1)]}.
876212`\spad{wrregime(l,{}s)} writes a list of regimes to a file named \spad{s} and returns the number of regimes written
876465`\spad{xCoord(pt)} returns the first element of the point,{} \spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian coordinate system.
876517`\spad{xRange(c)} returns the range of the \spad{x}-coordinates of the points on the curve \spad{c}.
876594`\spad{xRange(c)} returns the range of the \spad{x}-coordinates of the points on the curve \spad{c}.
876671`\spad{xform(tr,{}inpt)} apply transform to a point producing another point
876719`like testEquals,{} but expects failure.
876832`like testComplexEquals,{} but expects failure.
876898`like testEquals,{} but expects failure.
876963`like testEquals,{} but expects failure.
877022`like testEquals,{} but expects failure.
877091`like testLibraryError,{} but expects failure.
877149`like testEquals,{} but expects failure.
877217`like testNotEquals,{} but expects failure.
877279`like testEquals,{} but expects failure.
877371`like testRealEquals,{} but expects failure.
877434`like testEquals,{} but expects failure.
877495`like testTrue,{} but expects failure.
877545`xmlAttribute:(name:String,{}values:List String) constructs a xmlAttribute from its name and list of values
877612`\spad{xmlAttribute(name,{}value)} construct a xmlAttribute from its \spad{name} and \spad{value}
877673`\spad{xmlElement(name,{}elements,{}attributes)} construct an element with structured \spad{elements}
877751`\spad{xmlElement(name,{}txt,{}attributes)} construct an element with unstructured text
877827`\spad{xn(n)} returns the polynomial \spad{x^n-1}.
877933`\spad{xor(a,{}b)} returns the logical {\em exclusive-or} of bit aggregates \axiom{a} and \axiom{\spad{b}}.
877977`\spad{xor(a,{}b)} returns the logical exclusive {\em or} of Boolean \spad{a} and \spad{b}.
878016`\spad{xor(n,{}m)} returns the bit-by-bit logical {\em xor} of the single integers \spad{n} and \spad{m}.
878061`\spad{yCoord(pt)} returns the second element of the point,{} \spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian coordinate system.
878206`\spad{yCoordinates(f)} returns \spad{[[A1,{}...,{}An],{} D]} such that \spad{f = (A1 + A2 y +...+ An y^(n-1)) / D}.
878303`\spad{yRange(c)} returns the range of the \spad{y}-coordinates of the points on the curve \spad{c}.
878380`\spad{yRange(c)} returns the range of the \spad{y}-coordinates of the points on the curve \spad{c}.
878457`\spad{yellow()} returns the position of the yellow hue from total hues.
878492`\spad{youngGroup([n1,{}...,{}nk])} constructs the direct product of the symmetric groups {\em Sn1},{}...,{}{\em Snk}.
878586`\spad{youngGroup(lambda)} constructs the direct product of the symmetric groups given by the parts of the partition {\em lambda}.
878676`\spad{zCoord(pt)} returns the third element of the point,{} \spad{pt},{} although no assumptions are made as to the coordinate system being used. This function is defined for the convenience of the user dealing with a Cartesian or a cylindrical coordinate system.
878728`\spad{zRange(c)} returns the range of the \spad{z}-coordinates of the points on the curve \spad{c}.
878805`\spad{zag(f,{}g)} creates a form for the continued fraction form for \spad{f} over \spad{g}.
878847`\spad{zero?(x)} checks if \spad{x} is zero.
879307`\spad{zero?(x)} tests if \spad{x} is equal to 0.
879356`\spad{zero?(I)} tests whether the ideal \spad{I} is the zero ideal
879519`\axiom{zero?(pol,{}aRoot)} answers if \axiom{pol} interpreted as \axiom{aRoot} is \axiom{0}
879613`\spad{zeroDim?(I)} tests if the ideal \spad{I} is zero dimensional,{} \spadignore{i.e.} all its associated primes are maximal,{} in the ring \spad{F[lvar]},{} where lvar are the variables appearing in \spad{I}
879689`\spad{zeroDim?(I,{}lvar)} tests if the ideal \spad{I} is zero dimensional,{} \spadignore{i.e.} all its associated primes are maximal,{} in the ring \spad{F[lvar]}
879778`\spad{zeroDimPrimary?(I)} tests if the ideal \spad{I} is 0-dimensional primary.
880005`\spad{zeroDimPrime?(I)} tests if the ideal \spad{I} is a 0-dimensional prime.
880230`\axiom{zeroDimensional?(\spad{lp})} returns \spad{true} iff \axiom{\spad{lp}} generates a zero-dimensional ideal \spad{w}.\spad{r}.\spad{t}. the variables involved in \axiom{\spad{lp}}.
880365`\axiom{zeroDimensional?(\spad{lq1})} returns \spad{true} iff \axiom{\spad{lq1}} generates a zero-dimensional ideal \spad{w}.\spad{r}.\spad{t}. the variables of \axiom{\spad{ls}}.
880451`\spad{zeroMatrix(i,{}j,{}k)} create a matrix with all zero terms
880577`\spad{zeroMatrix(s,{}p,{}q)} uses loop variables in the Fortran,{} \spad{I1} and \spad{I2}
880685`\spad{zeroMatrix(s,{}b,{}d)} in this version gives the user control over names of Fortran variables used in loops.
881195`\spad{zeroOf(p)} returns \spad{y} such that \spad{p(y) = 0}. If possible,{} \spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity. Error: if \spad{p} has more than one variable \spad{y}.
881263`\spad{zeroOf(p)} returns \spad{y} such that \spad{p(y) = 0}; if possible,{} \spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity.
881347`\spad{zeroOf(p)} returns \spad{y} such that \spad{p(y) = 0}. The value \spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity. Error: if \spad{p} has more than one variable.
881667`\spad{zeroOf(p,{} y)} returns \spad{y} such that \spad{p(y) = 0}; if possible,{} \spad{y} is expressed in terms of radicals. Otherwise it is an implicit algebraic quantity which displays as \spad{'y}.
881758`\spad{zeroOf(p,{} y)} returns \spad{y} such that \spad{p(y) = 0}. The value \spad{y} is expressed in terms of radicals if possible,{}and otherwise as an implicit algebraic quantity which displays as \spad{'y}.
881832`\axiom{zeroSetSplitIntoTriangularSystems(\spad{lp})} returns a list of triangular systems \axiom{[[\spad{ts1},{}\spad{qs1}],{}...,{}[\spad{tsn},{}\spad{qsn}]]} such that the zero set of \axiom{\spad{lp}} is the union of the closures of the \axiom{W_i} where \axiom{W_i} consists of the zeros of \axiom{\spad{ts}} which do not cancel any polynomial in \axiom{qsi}.
881959`\axiom{zeroSetSplit(\spad{lp})} returns a list \axiom{\spad{lts}} of triangular sets such that the zero set of \axiom{\spad{lp}} is the union of the closures of the regular zero sets of the members of \axiom{\spad{lts}}.
882038`\axiom{zeroSetSplit(\spad{lp},{} norm?)} decomposes the variety associated with \axiom{\spad{lp}} into regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in in the output. Note that \axiom{\spad{lp}} needs to generate a zero-dimensional ideal. If \axiom{norm?} is \axiom{\spad{true}} then the regular sets ar
882038`--e normalized.
882194`\axiom{zeroSetSplit(\spad{lp},{} norm?)} decomposes the variety associated with \axiom{\spad{lp}} into square-free regular chains. Thus a point belongs to this variety iff it is a regular zero of a regular set in
882194`-- in the output. Note that \axiom{\spad{lp}} needs to generate a zero-dimensional ideal. If \axiom{norm?} is \axiom{\spad{true}} then the regular sets are normalized.
882490`\axiom{zeroSetSplit(\spad{lp},{}clos?)} has the same specifications as \axiomOpFrom{zeroSetSplit(\spad{lp},{}clos?)}{RegularTriangularSetCategory}.
882585`\spad{zeroSetSplit(lp,{}clos?)} returns \spad{lts} a split of Kalkbrener of the radical ideal associated with \spad{lp}. If \spad{clos?} is \spad{false},{} it is also a decomposition of the variety associated with \spad{lp} into the regular zero set of the \spad{ts} in \spad{lts} (or,{} in other words,{} a split of Lazard of this variety). See the example illustrating the \spadtype{RegularTriangularSet} constru
882585`--ctor for more explanations about decompositions by means of regular triangular sets.
882679`\spad{zeroSetSplit(lp,{}clos?,{}info?)} returns a list \spad{lts} of regular chains such that the union of the closures of their regular zero sets equals the affine variety associated with \spad{lp}. Moreover,{} if \spad{clos?} is \spad{false} then the union of the regular zero set of the \spad{ts} (for \spad{ts} in \spad{lts}) equals this variety. If \spad{info?} i
882679`--s \spad{true} then some information is displayed during the computations. See \axiomOpFrom{zeroSetSplit}{RegularTriangularSet}.
882819`\axiom{zeroSetSplit(\spad{lp},{}clos?,{}info?)} has the same specifications as \axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory}. Moreover,{} if \axiom{clos?} then solves in the sense of the Zariski closure else solves in the sense of the regular zeros. If \axiom{info?} then do print messages during the computations.
882913`\axiom{zeroSetSplit(\spad{lp},{}clos?,{}info?)} has the same specifications as \axiomOpFrom{zeroSetSplit}{RegularTriangularSetCategory} from \spadtype{RegularTriangularSetCategory} Moreover,{} if \axiom{clos?} then solves in the sense of the Zariski closure else solves in the sense of the regular zeros. If \axiom{info?} then do print messages during the computations.
883017`\axiom{zeroSetSplit(\spad{lp},{}\spad{b1},{}\spad{b2}.\spad{b3},{}\spad{b4})} is an internal subroutine,{} exported only for developement.
883127`\axiom{zeroSetSplit(\spad{lp},{}\spad{b1},{}\spad{b2}.\spad{b3},{}\spad{b4})} is an internal subroutine,{} exported only for developement.
883247`\spad{zeroSquareMatrix(s,{}p)} \undocumented{}
883341`\spad{zeroVector(s,{}p)} \undocumented{}
883488`\spad{zero(n)} creates a zero vector of length \spad{n}.
883657`\spad{zero(m,{}n)} returns an \spad{m}-by-\spad{n} zero matrix.
884152`\spad{zerosOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible. Otherwise they are implicit algebraic quantities containing new symbols. The new symbols are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable \spad{y}.
884227`\spad{zerosOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible. Otherwise they are implicit algebraic quantities containing new symbols. The new symbols are bound in the interpreter to the respective values.
884318`\spad{zerosOf(p)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible. Note: the returned values \spad{y1},{}...,{}\spad{yn} contain new symbols which are bound in the interpreter to the respective values. Error: if \spad{p} has more than one variable.
884666`\spad{zerosOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities containing new symbols which display as \spad{'\%z0},{} \spad{'\%z1},{} ...; The new symbols are bound in the interpreter to respective values.
884764`\spad{zerosOf(p,{} y)} returns \spad{[y1,{}...,{}yn]} such that \spad{p(\spad{yi}) = 0}. The \spad{yi}\spad{'s} are expressed in radicals if possible,{} and otherwise as implicit algebraic quantities containing new symbols which display as \spad{'\%z0},{} \spad{'\%z1},{} ...; The new symbols are bound in the interpreter to the respective values.
884845`\spad{zeta()} returns the function which is constantly one
884892`\spad{zoom(x,{}r)} \undocumented
884947`\spad{zoom(v,{}s)} sets the graph scaling factor to \spad{s},{} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}.
885009`\spad{zoom(x,{}r,{}s)} \undocumented
885085`\spad{zoom(x,{}r,{}s,{}t)} \undocumented
885184`\spad{zoom(v,{}sx,{}sy,{}sz)} sets the graph scaling factors for the \spad{x}-coordinate axis to \spad{sx},{} the \spad{y}-coordinate axis to \spad{sy} and the \spad{z}-coordinate axis to \spad{sz} for the viewport \spad{v},{} which is of the domain \spadtype{ThreeDimensionalViewport}.
885344`\spad{f ~= g} creates the equivalent infix form.
885385`\spad{x~=y} tests if \spad{x} and \spad{y} are not equal.
885465`The complement or inverse of a graph is a graph on the same vertices such that there is an arrow if and only if there is not an arrow in its compliment. That is,{} it is the compliment of the arrows but is not the set complement. for more information see: http://en.wikipedia.org/wiki/Complement_graph
885508`The complement or inverse of a graph is a graph on the same vertices such that there is an arrow if and only if there is not an arrow in its compliment. That is,{} it is the compliment of the arrows but is not the set complement. for more information see: http://en.wikipedia.org/wiki/Complement_graph
885556`\spad{~(x)} returns the logical complement of \spad{x}.
885588`reverse,{} complement,{}canonical dual basis
885640`\spad{~ n} returns the bit-by-bit logical {\em not } of the single integer \spad{n}.
885680`Factorization of univariate polynomials with coefficients in \spadtype{AlgebraicNumber}.
885769`AlgebraPackage assembles a variety of useful functions for general algebras.
885867`This package provides algebraic functions over an integral domain.
885956`algebraic Hermite redution.
886153`This package provides functions for integrating a function on an algebraic curve.
886395`This package provides functions for the integration of algebraic integrands over transcendental functions.
886508`AlgebraicManipulations provides functions to simplify and expand expressions involving algebraic operators.
886601`This package factors multivariate polynomials over the domain of \spadtype{AlgebraicNumber} by allowing the user to specify a list of algebraic numbers generating the particular extension to factor over.
886740`\spadtype{AnyFunctions1} implements several utility functions for working with \spadtype{Any}. These functions are used to go back and forth between objects of \spadtype{Any} and objects of other types.
886788`This package apply rewrite rules to expressions,{} calling the pattern matcher.
886901`\spad{ApplyUnivariateSkewPolynomial} (internal) allows univariate skew polynomials to be applied to appropriate modules.
887029`\spadtype{AssociatedEquations} provides functions to compute the associated equations needed for factoring operators
887148`Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
887201`This package provides balanced factorisations of polynomials.
887315`This package exports functions to set some commonly used properties of operators,{} including properties which contain functions.
887380`\spadtype{BezoutMatrix} contains functions for computing resultants and discriminants using Bezout matrices.
887561`\spadtype{BoundIntegerRoots} provides functions to find lower bounds on the integer roots of a polynomial.
887668`\indented{1}{Author: Frederic Lehobey,{} James \spad{H}. Davenport} Date Created: 28 June 1994 Date Last Updated: 11 July 1997 Basic Operations: brillhartIrreducible? Related Domains: Also See: AMS Classifications: Keywords: factorization Examples: References: [1] John Brillhart,{} Note on Irreducibility Testing,{} Mathematics of Computation,{} vol. 35,{} num. 35,{} Oct. 1980,{} 1379-1381 [2] James Davenport,{} On Brillh
887668`--art Irreducibility. To appear. [3] John Brillhart,{} On the Euler and Bernoulli polynomials,{} \spad{J}. Reine Angew. Math.,{} \spad{v}. 234,{} (1969),{} \spad{pp}. 45-64
887752`This package implements asymptotically efficient Chinese reconstruction
887811`\axiomType{CartanKuranishi} is a package for the completion of a given differential equation to an involutive equation. Procedures for Cartan characters and Hilbert polynomial are also provided. Based on the Cartan-Kuranishi theorem as it is used in formal theory.
887911`This package provides functions to enable conversion of tensors given conversion of the components.
888048`Tools to send a point to infinity on an algebraic curve.
888214`This package implements characteristicPolynomials for monogenic algebras using resultants
888378`This package provides a characteristicPolynomial function for any matrix over a commutative ring.
888458`\indented{1}{Author: Clifton Williamson} Date Created: 9 August 1993 Date Last Updated: 3 December 1993 Basic Operations: chineseRemainder,{} factorList Related Domains: PAdicWildFunctionFieldIntegralBasis(\spad{K},{}\spad{R},{}UP,{}\spad{F}) Also See: WildFunctionFieldIntegralBasis,{} FunctionFieldIntegralBasis AMS Classifications: Keyword
888458`--s: function field,{} finite field,{} integral basis Examples: References: Description:
888624`CoerceVectorMatrixPackage: an unexposed,{} technical package for data conversions
888695`Provides combinatorial functions over an integral domain.
888791`CommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.
888913`This package exports the elementary operators,{} with some semantics already attached to them. The semantics that is attached here is not dependent on the set in which the operators will be applied.
888960`A package for swapping the order of two variables in a tower of two UnivariatePolynomialCategory extensions.
889116`\indented{1}{Author:} Date Created: Date Last Updated: Basic Functions: Related Constructors: Complex,{} UnivariatePolynomial Also See: AMS Classifications: Keywords: complex,{} polynomial factorization,{} factor References:
889238`This package extends maps from underlying rings to maps between complex over those rings.
889325`This package provides the generalized euclidean algorithm which is needed as the basic step for factoring polynomials.
889449`This package supports matching patterns involving complex expressions
889574`This package supports converting complex expressions to patterns
889701`\spadtype{ComplexRootFindingPackage} provides functions to find all roots of a polynomial \spad{p} over the complex number by using Plesken\spad{'s} idea to calculate in the polynomial ring modulo \spad{f} and employing the Chinese Remainder Theorem. In this first version,{} the precision (see \spadfunFrom{digits}{Float}) is not increased when this is necessary to avoid rounding errors
889701`--. Hence it is the user\spad{'s} responsibility to increase the precision if necessary. Note also,{} if this package is called with \spadignore{e.g.} \spadtype{Fraction Integer},{} the precise calculations could require a lot of time. Also note that evaluating the zeros is not necessarily a good check whether the result is correct: already evaluation can cause rounding errors.
889938`\spadtype{ComplexTrigonometricManipulations} provides function that compute the real and imaginary parts of complex functions.
890071`Solution of linear ordinary differential equations,{} constant coefficient case.
890240`CoordinateSystems provides coordinate transformation functions for plotting. Functions in this package return conversion functions which take points expressed in other coordinate systems and return points with the corresponding Cartesian coordinates.
890307`Enumeration by cycle indices.
890352`This package provides tools for working with cyclic streams.
890439`This package \undocumented{}
890498`\spadtype{DefiniteIntegrationTools} provides common tools used by the definite integration of both rational and elementary functions.
890623`This package \undocumented{}
890710`This package generates logarithmic integrals and polylogs corresponding to denominator of the integrand.
890784`any solution of a homogeneous linear Diophantine equation can be represented as a sum of minimal solutions,{} which form a "basis" (a minimal solution cannot be represented as a nontrivial sum of solutions) in the case of an inhomogeneous linear Diophantine equation,{} each solution is the sum of a inhomogeneous solution and any number of homogeneous solutions therefore,{} it suffices to compute two sets: \indented{3}{1. all minimal inhomogeneous so
890784`--lutions} \indented{3}{2. all minimal homogeneous solutions} the algorithm implemented is a completion procedure,{} which enumerates all solutions in a recursive depth-first-search it can be seen as finding monotone paths in a graph for more details see Reference
890839`\indented{2}{This package provides operations which all take as arguments} direct products of elements of some type \spad{A} and functions from \spad{A} to another type \spad{B}. The operations all iterate over their vector argument and either return a value of type \spad{B} or a direct product over \spad{B}.
890935`DiscreteLogarithmPackage implements help functions for discrete logarithms in monoids using small cyclic groups.
891005`DisplayPackage allows one to print strings in a nice manner,{} including highlighting substrings.
891050`Package for the factorization of a univariate polynomial with coefficients in a finite field. The algorithm used is the "distinct degree" algorithm of Cantor-Zassenhaus,{} modified to use trace instead of the norm and a table for computing Frobenius as suggested by Naudin and Quitte .
891165`A package to apply polynomials transformations to distributions.
891261`A package to map distributions between different rings.
891355`Various popular distributions.
891422`A package to apply polynomials transformations to distributions and distributions to polynomials.
891572`\spad{DoubleFloatEllipticIntegrals} implements machine A package for computing machine precision real and complex elliptic integrals,{} using algorithms given by Calrson. Note: Complex versions may misbehave for very large/small arguments and close to branch cuts.
891631`This package provides special functions for double precision real and complex floating point.
891688`This package provides functions for computing the residues of a function on an algebraic curve.
891914`Hack for the draw interface. DrawNumericHack provides a "coercion" from something of the form \spad{x = a..b} where \spad{a} and \spad{b} are formal expressions to a binding of the form \spad{x = c..d} where \spad{c} and \spad{d} are the numerical values of \spad{a} and \spad{b}. This "coercion" fails if \spad{a} and \spad{b} contains symbolic variables,{} but is meant for expressions involving \%\spad{pi}.
891984`This package \undocumented{}
892034`This package \undocumented{}
892091`This is a package for the exact computation of eigenvalues and eigenvectors. This package can be made to work for matrices with coefficients which are rational functions over a ring where we can factor polynomials. Rational eigenvalues are always explicitly computed while the non-rational ones are expressed in terms of their minimal polynomial.
892141`\spadtype{ElementaryFunctionDefiniteIntegration} provides functions to compute definite integrals of elementary functions.
892285`\spad{ElementaryFunctionLODESolver} provides the top-level functions for finding closed form solutions of linear ordinary differential equations and initial value problems.
892468`\spad{ElementaryFunctionODESolver} provides the top-level functions for finding closed form solutions of ordinary differential equations and initial value problems.
892599`This package provides functions to determine the sign of an elementary function around a point or infinity.
892719`ElementaryFunctionStructurePackage provides functions to test the algebraic independence of various elementary functions,{} using the Risch structure theorem (real and complex versions). It also provides transformations on elementary functions which are not considered simplifications.
892852`Provides elementary functions over an integral domain.
892952`This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for elemntary functions.
893065`\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 12 August 1992 Date Last Updated: 17 August 1992 Keywords: elementary,{} function,{} integration.
893187`\indented{1}{Risch differential equation,{} elementary case.} Author: Manuel Bronstein Date Created: 1 February 1988 Date Last Updated: 2 November 1995 Keywords: elementary,{} function,{} integration.
893302`The elliptic functions \spad{sn},{} \spad{sc} and \spad{dn} are expanded as Taylor series.
893432`This package provides operations for mapping the sides of equations.
893493`ErrorFunctions implements error functions callable from the system interpreter. Typically,{} these functions would be called in user functions. The simple forms of the functions take one argument which is either a string (an error message) or a list of strings which all together make up a message. The list can contain formatting codes (see below). The more sophisticated versions takes two arguments where the first argument is the name of the function from which
893493`-- the error was invoked and the second argument is either a string or a list of strings,{} as above. When you use the one argument version in an interpreter function,{} the system will automatically insert the name of the function as the new first argument. Thus in the user interpreter function \indented{2}{\spad{f x == if x < 0 then error "negative argument" else x}} the call to error will actually be of the form \indented{2}{\spad{error("f",{}"negative argument")}} because the interpreter wil
893493`--l have created a new first argument. \blankline Formatting codes: error messages may contain the following formatting codes (they should either start or end a string or else have blanks around them): \indented{3}{\spad{\%l}\space{6}start a new line} \indented{3}{\spad{\%b}\space{6}start printing in a bold font (where available)} \indented{3}{\spad{\%d}\space{6}stop\space{2}printing in a bold font (where available)} \indented{3}{\spad{ \%ceon}\space{2}start centering message lines} \indented{3}
893493`--{\spad{\%ceoff}\space{2}stop\space{2}centering message lines} \indented{3}{\spad{\%rjon}\space{3}start displaying lines "ragged left"} \indented{3}{\spad{\%rjoff}\space{2}stop\space{2}displaying lines "ragged left"} \indented{3}{\spad{\%i}\space{6}indent\space{3}following lines 3 additional spaces} \indented{3}{\spad{\%u}\space{6}unindent following lines 3 additional spaces} \indented{3}{\spad{\%xN}\space{5}insert \spad{N} blanks (eg,{} \spad{\%x10} inserts 10 blanks)} \blankline
893536`\spadtype{EuclideanGroebnerBasisPackage} computes groebner bases for polynomial ideals over euclidean domains. The basic computation provides a distinguished set of generators for these ideals. This basis allows an easy test for membership: the operation \spadfun{euclideanNormalForm} returns zero on ideal members.
893536`-- The string "info" and "redcrit" can be given as additional args to provide incremental information during the computation. If "info" is given,{} \indented{1}{a computational summary is given for each \spad{s}-polynomial. If "redcrit"} is given,{} the reduced critical pairs are printed. The term ordering is determined by the polynomial type used. Suggested types include \spadtype{DistributedMultivariatePolynomial},{} \spadtype{HomogeneousDistributedMultivariatePolynomial},{} \spadtype{GeneralD
893536`--istributedMultivariatePolynomial}.
893729`This package is to be used in conjuction with \indented{12}{the CycleIndicators package. It provides an evaluation} \indented{12}{function for SymmetricPolynomials.}
893843`This package provides support for \spad{XML} files
893879`Lifting of maps to Expressions. Date Created: 16 Jan 1989 Date Last Updated: 22 Jan 1990
893956`Solver for linear systems represented as list \indented{1}{of expressions.\space{2}More efficient than using solve because} \indented{1}{it does not check that system really is linear.}
894248`This package allows a map from any expression space into any object to be lifted to a kernel over the expression set,{} using a given property of the operator of the kernel.
894327`This package allows a mapping \spad{E} \spad{->} \spad{F} to be lifted to a kernel over \spad{E}; This lifting can fail if the operator of the kernel cannot be applied in \spad{F}; Do not use this package with \spad{E} = \spad{F},{} since this may drop some properties of the operators.
894417`Taylor series solutions of explicit ODE\spad{'s}.
894518`\spadtype{ExpressionToOpenMath} provides support for converting objects of type \spadtype{Expression} into OpenMath.
894589`This package provides functions to convert functional expressions to power series.
894719`\indented{1}{Author: Clifton \spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.
894769`This is just an interface between several packages and domains. The goal is to compute lexicographical Groebner bases of sets of polynomial with type \spadtype{Polynomial R} by the {\em FGLM} algorithm if this is possible (\spadignore{i.e.} if the input system generates a zero-dimensional ideal).
894845`\spadtype{FactoredFunctionUtilities} implements some utility functions for manipulating factored objects.
894917`\spadtype{FactoredFunctions2} contains functions that involve factored objects whose underlying domains may not be the same. For example,{} \spadfun{map} might be used to coerce an object of type \spadtype{Factored(Integer)} to \spadtype{Factored(Complex(Integer))}.
894998`computes various functions on factored arguments.
895064`This package provides utilities used by the factorizers which operate on polynomials represented as univariate polynomials with multivariate coefficients.
895198`\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 11 Jul 1990
895392`This package provides a mapping function for \spadtype{FiniteAbelianMonoidRing}
895573`\indented{1}{Lift a map to finite divisors.} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 19 May 1993
895927`FiniteFieldFunctions(\spad{GF}) is a package with functions concerning finite extension fields of the finite ground field {\em GF},{} \spadignore{e.g.} Zech logarithms.
895997`FiniteFieldHomomorphisms(\spad{F1},{}\spad{GF},{}\spad{F2}) exports coercion functions of elements between the fields {\em F1} and {\em F2},{} which both must be finite simple algebraic extensions of the finite ground field {\em GF}.
896153`\spad{FiniteFieldPolynomialPackage2}(\spad{F},{}\spad{GF}) exports some functions concerning finite fields,{} which depend on a finite field {\em GF} and an algebraic extension \spad{F} of {\em GF},{} \spadignore{e.g.} a zero of a polynomial over {\em GF} in \spad{F}.
896278`This package provides a number of functions for generating,{} counting and testing irreducible,{} normal,{} primitive,{} random polynomials over finite fields.
896359`This package solves linear diophantine equations for Bivariate polynomials over finite fields
896532`\spad{FiniteLinearAggregateFunctions2} provides functions involving two FiniteLinearAggregates where the underlying domains might be different. An example of this might be creating a list of rational numbers by mapping a function across a list of integers where the function divides each integer by 1000.
896669`This package exports 3 sorting algorithms which work over FiniteLinearAggregates.
896773`\spad{FiniteSetAggregateFunctions2} provides functions involving two finite set aggregates where the underlying domains might be different. An example of this is to create a set of rational numbers by mapping a function across a set of integers,{} where the function divides each integer by 1000.
896915`This package implements arbitrary precision numerical elliptic functions. The method is based on descending Landen transform.
897061`\indented{3}{This is a package for the approximation of complex solutions for} systems of equations of rational functions with complex rational coefficients. The results are expressed as either complex rational numbers or complex floats depending on the type of the precision parameter which can be either a rational number or a floating point number.
897134`\indented{3}{This is a package for the approximation of real solutions for} systems of polynomial equations over the rational numbers. The results are expressed as either rational numbers or floats depending on the type of the precision parameter which can be either a rational number or a floating point number.
897210`\spadtype{FortranCodePackage1} provides some utilities for producing useful objects in FortranCode domain. The Package may be used with the FortranCode domain and its \spad{printCode} or possibly via an outputAsFortran. (The package provides items of use in connection with ASPs in the AXIOM-NAG link and,{} where appropriate,{} naming accords with that in IRENA.) The easy-to-use functions use Fortran loop variables \spad{I1},{} \spad{I2},{} and it is users
897210`--' responsibility to check that this is sensible. The advanced functions use SegmentBinding to allow users control over Fortran loop variable names.
897259`Code to manipulate Fortran Output Stack
897311`provides an interface to the boot code for calling Fortran
897353`This package lifts the interpolation functions from \spadtype{FractionFreeFastGaussian} to fractions.
897556`This package implements the interpolation algorithm proposed in Beckermann,{} Bernhard and Labahn,{} George,{} Fraction-free computation of matrix rational interpolants and matrix GCDs,{} SIAM Journal on Matrix Analysis and Applications 22.
897681`This package extends a map between integral domains to a map between Fractions over those domains by applying the map to the numerators and denominators.
897764`\indented{1}{Lifting of morphisms to fractional ideals.} Author: Manuel Bronstein Date Created: 1 Feb 1989 Date Last Updated: 27 Feb 1990 Keywords: ideal,{} algebra,{} module.
898081`\spad{FramedNonAssociativeAlgebraFunctions2} implements functions between two framed non associative algebra domains defined over different rings. The function map is used to coerce between algebras over different domains having the same structural constants.
898260`linear extensions of maps on the basis
898388`Lifts a map from rings to function fields over them.
898791`In this package \spad{R} is a Euclidean domain and \spad{F} is a framed algebra over \spad{R}. The package provides functions to compute the integral closure of \spad{R} in the quotient field of \spad{F}. It is assumed that \spad{char(R/P) = char(R)} for any prime \spad{P} of \spad{R}. A typical instance of this is when \spad{R = K[x]} and \spad{F} is a func
898791`--tion field over \spad{R}.
898939`Attaching assertions to symbols for pattern matching; Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
899029`Attaching predicates to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
899135`\spadtype{FunctionSpaceComplexIntegration} provides functions for the indefinite integration of complex-valued functions.
899271`This package allows a mapping \spad{R} \spad{->} \spad{S} to be lifted to a mapping from a function space over \spad{R} to a function space over \spad{S}.
899401`\spadtype{FunctionSpaceIntegration} provides functions for the indefinite integration of real-valued functions.
899529`FunctionsSpacePrimitiveElement provides functions to compute primitive elements in functions spaces.
899640`This package provides function which replaces transcendental kernels in a function space by random integers. The correspondence between the kernels and the integers is fixed between calls to new().
899734`computes sums of top-level expressions.
899839`This package converts expressions in some function space to exponential expansions.
899993`This package converts expressions in some function space to power series in a variable \spad
899993`--{x} with coefficients in that function space. The function \spadfun{exprToUPS} converts expressions to power series whose coefficients do not contain the variable \spad{x}. The function \spadfun{exprToGenUPS} converts functional expressions to power series whose coefficients may involve functions of \spad{log(x)}.
900409`This package converts expressions in some function space to power series in a variable \spad{x} with coefficients in
900409`-- that function space. The function \spadfun{exprToUPS} converts expressions to power series whose coefficients do not contain the variable \spad{x}. The function \spadfun{exprToGenUPS} converts functional expressions to power series whose coefficients may involve functions of \spad{log(x)}.
900801`\indented{1}{Used internally by IR2F} Author: Manuel Bronstein Date Created: 12 May 1988 Date Last Updated: 22 September 1993 Keywords: function,{} space,{} polynomial,{} factoring
900960`Provides some special functions over an integral domain.
901061`\spadtype{GaloisGroupFactorizationUtilities} provides functions that will be used by the factorizer.
901207`\spadtype{GaloisGroupFactorizer} provides functions to factor resolvents.
901300`\spadtype{GaloisGroupPolynomialUtilities} provides useful functions for univariate polynomials which should be added to \spadtype{UnivariatePolynomialCategory} or to \spadtype{Factored} (July 1994).
901409`\spadtype{GaloisGroupUtilities} provides several useful functions.
901467`Package for the factorization of complex or gaussian integers.
901527`\spad{Gcd} basis provides functions to find structure of multiplicative group (and semigroup) generated by elements of \spad{Gcd} domain.
901587`\indented{1}{Author : \spad{P}.Gianni.} January 1990 The equation \spad{Af+Bg=h} and its generalization to \spad{n} polynomials is solved for solutions over the \spad{R},{} euclidean domain. A table containing the solutions of \spad{Af+Bg=x^k} is used. The operations are performed modulus a prime which are in principle big enough,{} but the solutions are tested and,{} in case of failure,{} a hensel lifting
901587`-- process is used to get to the right solutions. It will be used in the factorization of multivariate polynomials over finite field,{} with \spad{R=F[x]}.
901686`\indented{1}{Description} This package provides operations for the factorization of univariate polynomials with integer coefficients. The factorization is done by "lifting" the finite "berlekamp's" factorization
901749`\indented{1}{Author : \spad{P}.Gianni} General Hensel Lifting Used for Factorization of bivariate polynomials over a finite field.
901861`This package provides operations for \spad{GCD} computations on polynomials
902031`\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \spadtype{Integer} or \spadtype{Fraction Integer}.
902217`\spadtype{GenerateUnivariatePowerSeries} provides functions that create power series from explicit formulas for their \spad{n}th coefficient.
902346`This internal package rationalises integrands on curves of the form: \indented{2}{\spad{y\^2 = a x\^2 + b x + c}} \indented{2}{\spad{y\^2 = (a x + b) / (c x + d)}} \indented{2}{\spad{f(x,{} y) = 0} where \spad{f} has degree 1 in \spad{x}} The rationalization is done for integration,{} limited integration,{} extended integration and the risch differential equation.
902466`Gosper\spad{'s} summation algorithm.
902656`TwoDimensionalPlotSettings sets global flags and constants for 2-dimensional plotting.
902701`GrayCode provides a function for efficiently running through all subsets of a finite set,{} only changing one element by another one.
902737`\spadtype{GroebnerFactorizationPackage} provides the function groebnerFactor" which uses the factorization routines of \Language{} to factor each polynomial under consideration while doing the groebner basis algorithm. Then it writes the ideal as an intersection of ideals determined by the irreducible factors.
902737`-- Note that the whole ring may occur as well as other redundancies. We also use the fact,{} that from the second factor on we can assume that the preceding factors are not equal to 0 and we divide all polynomials under considerations by the elements of this list of "nonZeroRestrictions". The result is a list of groebner bases,{} whose union of solutions of the corresponding systems of equations is the solution of the system of equation corresponding to the input list. The term ordering is deter
902737`--mined by the polynomial type used. Suggested types include \spadtype{DistributedMultivariatePolynomial},{} \spadtype{HomogeneousDistributedMultivariatePolynomial},{} \spadtype{GeneralDistributedMultivariatePolynomial}.
902934`\indented{1}{Author:} Date Created: Date Last Updated: Keywords: Description This package provides low level tools for Groebner basis computations
903115`\spadtype{GroebnerPackage} computes groebner bases for polynomial ideals. The basic computation provides a distinguished set of generators for polynomial ideals over fields. This basis allows an easy test for membership: the operation \spadfun{normalForm} returns zero on ideal members. When the provided coefficient domain,{} Dom,{} is not 
903115`--a field,{} the result is equivalent to considering the extended ideal with \spadtype{Fraction(Dom)} as coefficients,{} but considerably more efficient since all calculations are performed in Dom. Additional argument "info" and "redcrit" can be given to provide incremental information during computation. Argument "info" produces a computational summary for each \spad{s}-polynomial. Argument "redcrit" prints out the reduced critical pairs. The term ordering is determined by the polynomial type u
903115`--sed. Suggested types include \spadtype{DistributedMultivariatePolynomial},{} \spadtype{HomogeneousDistributedMultivariatePolynomial},{} \spadtype{GeneralDistributedMultivariatePolynomial}.
903282`\indented{1}{Author : \spad{P}.Gianni,{} Summer \spad{'88},{} revised November \spad{'89}} Solve systems of polynomial equations using Groebner bases Total order Groebner bases are computed and then converted to lex ones This package is mostly intended for internal use.
903369`This package exports guessing of sequences of rational functions
903420`This package exports coercion and retraction from EXPR INT to finite fields
903503`This package exports guessing of sequences of numbers in a finite field
903576`This package exports guessing of sequences of rational numbers
903620`This package exports coercion and retraction from EXPR INT to polynomials
903682`This package exports guessing of sequences of rational functions over the integers
903735`This package exports guessing of sequences of rational functions
903787`This package implements guessing of sequences. Packages for the most common cases are provided as \spadtype{GuessInteger},{} \spadtype{GuessPolynomial},{} etc.
903927`\indented{1}{Author : Larry Lambe} Date Created : August 1988 Date Last Updated : March 9 1990 Related Constructors: OrderedSetInts,{} Commutator,{} FreeNilpotentLie AMS Classification: Primary 17B05,{} 17B30; Secondary 17A50 Keywords: free Lie algebra,{} Hall basis,{} basic commutators Description : Generate a basis for the free Lie algebra on \spad{n} generators over a ring \spad{R} with identity up to basic commutators of length \spad{c} using the algorithm of \spad
903927`--{P}. Hall as given in Serre\spad{'s} book Lie Groups \spad{
903927`--} Lie Algebras
903962`Package to generate Hankel matrices.
904011`This package provides the functions for the heuristic integer \spad{gcd}. Geddes\spad{'s} algorithm,{}for univariate polynomials with integer coefficients
904088`\indented{2}{This package provides functions for the primary decomposition of} polynomial ideals over the rational numbers. The ideals are members of the \spadtype{PolynomialIdeal} domain,{} and the polynomial generators are required to be from the \spadtype{DistributedMultivariatePolynomial} domain.
904160`This package provides operations to create incrementing functions.
904227`This package computes infinite products of univariate Taylor series over an integral domain of characteristic 0.
904372`This package computes infinite products of univariate Taylor series over an arbitrary finite field.
904566`This package computes infinite products of univariate Taylor series over a field of prime order.
904694`Functions defined on streams with entries in two sets.
904763`Functions defined on streams with entries in two sets.
904841`Default infinity signatures for the interpreter; Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989
904881`Factorization of univariate polynomials with coefficients in an algebraic extension of a field over which we can factor UP\spad{'s}.
905051`InnerCommonDenominator provides functions to compute the common denominator of a finite linear aggregate of elements of the quotient field of an integral domain.
905208`\spadtype{InnerMatrixLinearAlgebraFunctions} is an internal package which provides standard linear algebra functions on domains in \spad{MatrixCategory}
905378`\spadtype{InnerMatrixQuotientFieldFunctions} provides functions on matrices over an integral domain which involve the quotient field of that integral domain. The functions rowEchelon and inver
905378`--se return matrices with entries in the quotient field.
905881`\indented{2}{This is an inner package for factoring multivariate polynomials} over various coefficient domains in characteristic 0. The univariate factor operation is passed as a parameter. Multivariate hensel lifting is used to lift the univariate factorization
906032`InnerNormalBasisFieldFunctions(\spad{GF}) (unexposed): This package has functions used by every normal basis finite field extension domain.
906114`This package is the inner package to be used by NumericRealEigenPackage and NumericComplexEigenPackage for the computation of numeric eigenvalues and eigenvectors.
906208`This is an internal package for computing approximate solutions to systems of polynomial equations. The parameter \spad{K} specifies the coefficient field of the input polynomials and must be either \spad{Fraction(Integer)} or \spad{Complex(Fraction Integer)}. The parameter \spad{F} specifies where the solutions must lie and can be one of the following: \spad{Float},{} \spad{Fraction(Integer)},{} \spa
906208`--d{Complex(Float)},{} \spad{Complex(Fraction Integer)}. The last parameter specifies the type of the precision operand and must be either \spad{Fraction(Integer)} or \spad{Float}.
906312`Find the sign of a polynomial around a point or infinity.
906403`tools for the summation packages.
906536`This package provides transformations from trigonometric functions to exponentials and logarithms,{} and back. \spad{F} and \spad{FG} should be the same type of function space.
906701`Tools for manipulating input forms.
906758`This package provides functions to lookup bits in integers
906875`\indented{1}{This Package contains basic methods for integer factorization.} The factor operation employs trial division up to 10,{}000. It then tests to see if \spad{n} is a perfect power before using Pollards rho method. Because Pollards method may fail,{} the result of factor may contain composite factors. We should also employ Lenstra\spad{'s} eliptic curve method.
906955`Test for linear dependence over the integers.
907045`This package provides various number theoretic functions on the integers.
907105`The \spadtype{IntegerPrimesPackage} implements a modification of Rabin\spad{'s} probabilistic primality test and the utility functions \spadfun{nextPrime},{} \spadfun{prevPrime} and \spadfun{primes}.
907177`Provides integer testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991
907250`The \spadtype{IntegerRoots} package computes square roots and \indented{2}{\spad{n}th roots of integers efficiently.}
907313`This package provides the implementation for the \spadfun{solveLinearPolynomialEquation} operation over the integers. It uses a lifting technique from the package GenExEuclid
907380`IntegralBasisPolynomialTools provides functions for \indented{1}{mapping functions on the coefficients of univariate and bivariate} \indented{1}{polynomials.}
907532`This package contains functions used in the packages FunctionFieldIntegralBasis and NumberFieldIntegralBasis.
907671`\indented{1}{Internally used by the integration packages} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 12 August 1992 Keywords: integration.
907743`\indented{2}{This package allows a sum of logs over the roots of a polynomial} \indented{2}{to be expressed as explicit logarithms and arc tangents,{} provided} \indented{2}{that the indexing polynomial can be factored into quadratics.} Date Created: 21 August 1988 Date Last Updated: 4 October 1993
907824`This package allows a sum of logs over the roots of a polynomial to be expressed as explicit logarithms and arc tangents,{} provided that the indexing polynomial can be factored into quadratics.
907953`\indented{1}{Tools for the integrator} Author: Manuel Bronstein Date Created: 25 April 1990 Date Last Updated: 9 June 1993 Keywords: elementary,{} function,{} integration.
908037`A package to print strings without line-feed nor carriage-return.
908088`\indented{1}{An internal package for computing the rational univariate representation} \indented{1}{of a zero-dimensional algebraic variety given by a square-free} \indented{1}{triangular set.} \indented{1}{The main operation is \axiomOpFrom{rur}{InternalRationalUn
908088`--ivariateRepresentationPackage}.} \indented{1}{It is based on the {\em generic} algorithm description in [1]. \newline References:} [1] \spad{D}. LAZARD "Solving Zero-dimensional Algebraic Systems" \indented{4}{Journal of Symbolic Computation,{} 1992,{} 13,{} 117-131}
908331`This package computes the inverse Laplace Transform.
908461`IrrRepSymNatPackage contains functions for computing the ordinary irreducible representations of symmetric groups on \spad{n} letters {\em {1,{}2,{}...,{}n}} in Young\spad{'s} natural form and their dimensions. These representations can be labelled by number partitions of \spad{n},{} \spadignore{i.e.} a weakly decreasing sequence of integers summing up to \spad{n},{} \spadignore{e.g.} {\em [3,{}3,{}3,{}1]} labels an irreducible representation for \spad{n} e
908461`--quals 10. Note: whenever a \spadtype{List Integer} appears in a signature,{} a partition required.
908508`This package exports the function generateIrredPoly that computes a monic irreducible polynomial of degree \spad{n} over a finite field.
908587`\axiomType{JetCoordinateTransformation} implements changes of local coordinates. Given are the changes of the coordinates of the base space,{} \spadignore{i.e.} the independent and dependent variables. The transformations of the derivatives are computed via the chain rule. \axiom{\spad{Y}} (\axiom{\spad{W}}) contains expressions for t
908587`--he old variables in terms of the new ones.
908759`\axiomType{JetGroebner} provides a procedure to compute Groebner bases for arbitrary domains of jet polynomials. Two internal procedures transform to and from \axiomType{DistributedJetBundlePolynomial} where the actual computation is done. The argument \axiom{\spad{LJV}} contains all jet variables effectively occurring
908759`-- in the polynomials. The ordering is determined by the ordering in \axiom{\spad{P}}.
908947`This package exports some auxiliary functions on kernels
909022`\spadtype{Kovacic} provides a modified Kovacic\spad{'s} algorithm for solving explicitely irreducible 2nd order linear ordinary differential equations.
909131`\axiomType{LUDecomposition} contains procedures to solve linear systems of equations or to compute inverses using a LU decomposition.
909181`This package computes the forward Laplace Transform.
909303`A package for solving polynomial systems by means of Lazard triangular sets [1]. This package provides two operations. One for solving in the sense of the regular zeros,{} and the other for solving in the sense of the Zariski closure. Both produce square-free reg
909303`--ular sets. Moreover,{} the decompositions do not contain any redundant component. However,{} only zero-dimensional regular sets are normalized,{} since normalization may be time consumming in positive dimension. The decomposition process is that of [2].\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}{[2] \spad{M}. MORENO MAZA "A new algorithm for computing t
909303`--riangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
909548`Package for leading coefficient determination in the lifting step. Package working for every \spad{R} euclidean with property \spad{"F"}. Uses algorithm given in section 3 of \spad{P}. \spad{S}. Wang "An improved multivariate polynomial factoring algorithm",{} Math. of Comput. 32,{} 1978,{} \spad{p}. 1215-\spad{-1231},{} generalized to arbitrary euclidea
909548`--n domain.
909700`A package for solving polynomial systems with finitely many solutions. The decompositions are given by means of regular triangular sets. The computations use lexicographical Groebner bases. The main operations are \axiomOpFrom{lexTriangular}{LexTriangularPackage} and \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}. The second one provide decompositions by means of square-free regular triangular sets. Both are 
909700`--based on the {\em lexTriangular} method described in [1]. They differ from the algorithm described in [2] by the fact that multiciplities of the roots are not kept. With the \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} operation all multiciplities are removed. With the other operation some multiciplities may remain. Both operations admit an optional argument to produce normalized triangular sets. \newline
909783`\indented{1}{Given a Groebner basis \spad{B} with respect to the total degree ordering for a} zero-dimensional ideal \spad{I},{} compute a Groebner basis with respect to the lexicographical ordering by using linear algebra. This is an internal package,{} operations in this package may crash the system on invalid input.
909861`Test for linear dependence.
909958`\spadtype{LinearOrdinaryDifferentialOperatorFactorizer} provides a factorizer for linear ordinary differential operators whose coefficients are rational functions.
910089`\spad{LinearOrdinaryDifferentialOperatorsOps} provides symmetric products and sums for linear ordinary differential operators.
910218`Given a PolynomialFactorizationExplicit ring,{} this package provides a defaulting rule for the \spad{solveLinearPolynomialEquation} operation,{} by moving into the field of fractions,{} and solving it there via the \spad{multiEuclidean} operation.
910318`This package solves linear system in the matrix form \spad{AX = B}. It is essentially a particular instantiation of the package \spadtype{LinearSystemMatrixPackage} for Matrix and Vector. This package\spad{'s} existence makes it easier to use \spadfun{solve} in the FriCAS interpreter.
910381`This package solves linear system in the matrix form \spad{AX = B}.
910560`this package finds the solutions of linear systems presented as a list of polynomials.
910712`This package provides liouvillian functions over an integral domain.
910813`\spadtype{ListFunctions2} implements utility functions that operate on two kinds of lists,{} each with a possibly different type of element.
910872`\spadtype{ListFunctions3} implements utility functions that operate on three kinds of lists,{} each with a possibly different type of element.
910940`\spadtype{ListToMap} allows mappings to be described by a pair of lists of equal lengths. The image of an element \spad{x},{} which appears in position \spad{n} in the first list,{} is then the \spad{n}th element of the second list. A default value or default function can be specified to be used when \spad{x} does not appear in the first list. In the absence of defaults,{} an error will occur in that case.
911004`\indented{1}{Utilities for MPolyCat} Author: Manuel Bronstein Date Created: 1987 Date Last Updated: 28 March 1990 (\spad{PG})
911229`This package \undocumented
911468`\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are polynomials over some ring \spad{R} over which we can factor. It is used internally by packages such as the solve package which need to work with polynomials in a specific set of variables with coefficients which are polynom
911468`--ials in all the other variables.
911641`\indented{3}{This package exports a factor operation for multivariate polynomials} with coefficients which are rational functions over some ring \spad{R} over which we can factor. It is used internally by packages such as primary decomposition which need to work with polynomials with rational function coefficient
911641`--s,{} \spadignore{i.e.} themselves fractions of polynomials.
911835`\indented{1}{MRationalFactorize contains the factor function for multivariate} polynomials over the quotient field of a ring \spad{R} such that the package MultivariateFactorize can factor multivariate polynomials over \spad{R}.
912000`transforms top-level objects into compiled functions.
912116`MakeFloatCompiledFunction transforms top-level objects into compiled Lisp functions whose arguments are Lisp floats. This by-passes the \Language{} compiler and interpreter,{} thereby gaining several orders of magnitude.
912199`transforms top-level objects into interpreter functions.
912268`MakeRecord is used internally by the interpreter to create record types which are used for doing parallel iterations on streams.
912326`transforms top-level objects into compiled functions.
912428`various Currying operations.
912488`various Currying operations.
912564`various Currying operations.
912656`various Currying operations.
912730`various Currying operations.
912820`various Currying operations.
912926`\spadtype{MatrixCategoryFunctions2} provides functions between two matrix domains. The functions provided are \spadfun{map} and \spadfun{reduce}.
913243`MatrixCommonDenominator provides functions to compute the common denominator of a matrix of elements of the quotient field of an integral domain.
913343`\spadtype{MatrixLinearAlgebraFunctions} provides functions to compute inverses and canonical forms.
913517`This package exports tools for merging lists
913570`\indented{1}{<description of package>} Author: Jim Wen Date Created: \spad{??} Date Last Updated: October 1991 by Jon Steinbach Keywords: Examples: References:
913636`This package computes \spad{gcd} over field of algebraic \indented{1}{functions over \spad{Q} using modular method based on \spad{M}. Monagan and} \indented{1}{van Hoej paper.} \indented{1}{Core modular operations are passed as parameter (MO) to this} \indented{1}{package.}
913749`This package supports factorization and gcds of univariate polynomials over the integers modulo different primes. The inputs are given as polynomials over the integers with the prime passed explicitly as an extra argument.
913851`This package provides generic interface to modular \indented{1}{Hermite-Pade solver.}
913984`\indented{1}{Modular hermitian row reduction.} Author: Manuel Bronstein Date Created: 22 February 1989 Date Last Updated: 24 November 1993 Keywords: matrix,{} reduction.
914062`An auxiliary package for various moment and cumulant transformations used in \spad{Distribution}.
914123`\spad{MonoidRingFunctions2} implements functions between two monoid rings defined with the same monoid over different rings.
914198`Tools for handling monomial extensions.
914300`\spadtype{MoreSystemCommands} implements an interface with the system command facility. These are the commands that are issued from source files or the system interpreter and they start with a close parenthesis,{} \spadignore{e.g.} \spadsyscom{what} commands.
914349`Compute limits using Gruntz-Gonnet \spad{MRV} algorithm
914394`Package for factorization of multivariate polynomials over finite fields.
914547`\indented{1}{\spadtype{MultiVariableCalculusFunctions} Package provides several} \indented{1}{functions for multivariable calculus.} These include gradient,{} hessian and jacobian,{} divergence and laplacian. Various forms for banded and sparse storage of matrices are included.
914725`Lifting of a map through 2 levels of polynomials.
915048`\indented{2}{This is the top level package for doing multivariate factorization} over basic domains like \spadtype{Integer} or \spadtype{Fraction Integer}.
915207`This package provides the functions for the multivariate "lifting",{} using an algorithm of Paul Wang. This package will work for every euclidean domain \spad{R} which has property \spad{F},{} \spadignore{i.e.} there exists a factor operation in \spad{R[x]}.
915351`Author : \spad{P}.Gianni This package provides the functions for the computation of the square free decomposition of a multivariate polynomial. It uses the package GenExEuclid for the resolution of the equation \spad{Af + Bg = h} and its generalization to \spad{n} polynomials over an integral domain and the package \spad{MultivariateLifting} for the "multi
915351`--variate" lifting.
915501`This is package computes noncommutative Groebner basis. Based on commutative version.
915687`Package for the determination of the coefficients in the lifting process. Used by \spadtype{MultivariateLifting}. This package will work for every euclidean domain \spad{R} which has property \spad{F},{} \spadignore{i.e.} there exists a factor operation in \spad{R[x]}.
915904`This package lifts a mapping from coefficient rings \spad{R} to \spad{S} to a mapping from sparse univariate polynomial over \spad{R} to a sparse univariate polynomial over \spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.
915988`This package exports Newton interpolation for the special case where the result is known to be in the original integral domain
916054`This package provides a division and related operations for \spadtype{MonogenicLinearOperator}\spad{s} over a \spadtype{Field}. Since the multiplication is in general non-commutative,{} these operations all have left- and right-hand versions. This package provides the operations based on left-division.
916151`NonLinearFirstOrderODESolver provides a function for finding closed form first integrals of nonlinear ordinary differential equations of order 1.
916283`NonLinearSolvePackage is an interface to \spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving. The solutions are given in the algebraic closure of \spad{R} whenever possible.
916352`\spadtype{NoneFunctions1} implements functions on \spadtype{None}. It particular it includes a particulary dangerous coercion from any other type to \spadtype{None}.
916402`This package implements the norm of a polynomial with coefficients in a monogenic algebra (using resultants)
916581`This package \undocumented
916806`A package for computing normalized assocites of univariate polynomials with coefficients in a tower of simple extensions of a field.\newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{1}
916806`--{[2] \spad{M}. MORENO MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[3] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.}
916996`In this package \spad{F} is a framed algebra over the integers (typically \spad{F = Z[a]} for some algebraic integer a). The package provides functions to compute the integral closure of \spad{Z} in the quotient quotient field of \spad{F}.
917124`NumberFormats provides function to format and read arabic and roman numbers,{} to convert numbers to strings and to read floating-point numbers.
917167`This package provides polynomials as functions on a ring.
917250`This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the complex rational numbers. The results are expressed either as complex floating numbers or as complex rational numbers depending on the type of the precision parameter.
917324`\spadtype{NumericContinuedFraction} provides functions \indented{2}{for converting floating point numbers to continued fractions.}
917402`This package computes explicitly eigenvalues and eigenvectors of matrices with entries over the Rational Numbers. The results are expressed as floating numbers or as rational numbers depending on the type of the parameter Par.
917473`\indented{1}{Author: Clifton \spad{J}. Williamson} Date Created: Bastille Day 1989 Date Last Updated: 5 June 1990 Keywords: Examples: Package for constructing tubes around 3-dimensional parametric curves.
917553`Numeric provides real and complex numerical evaluation functions for various symbolic types.
917614`This package is a suite of functions for the numerical integration of an ordinary differential equation of \spad{n} variables: \blankline \indented{8}{\center{dy/dx = \spad{f}(\spad{y},{}\spad{x})\space{5}\spad{y} is an \spad{n}-vector}} \blankline \par All the routines are based on a 4-th order Runge-Kutta kernel. These routines generally have as arguments: \spad{n},{} the number of dependent variables; \spad{x1},{} the initial point; 
917614`--\spad{h},{} the step size; \spad{y},{} a vector of initial conditions of length \spad{n} which upon exit contains the solution at \spad{x1 + h}; \spad{derivs},{} a function which computes the right hand side of the ordinary differential equation: \spad{derivs(dydx,{}y,{}x)} computes \spad{dydx},{} a vector which contains the derivative information. \blankline \par In order of increasing complexity:\begin{items} \blankline \item \spad{rk4(y,{}n,{}x1,{}h,{}derivs)} advances the solution vector t
917614`--o \spad{x1 + h} and return the values in \spad{y}. \blankline \item \spad{rk4(y,{}n,{}x1,{}h,{}derivs,{}t1,{}t2,{}t3,{}t4)} is the same as \spad{rk4(y,{}n,{}x1,{}h,{}derivs)} except that you must provide 4 scratch arrays \spad{t1}-\spad{t4} of size \spad{n}. \blankline \item Starting with \spad{y} at \spad{x1},{} \spad{rk4f(y,{}n,{}x1,{}x2,{}ns,{}derivs)} uses \spad{ns} fixed steps of a 4-th order Runge-Kutta integrator to advance the solution vector to \spad{x2} and return the values in \spad
917614`--{y}. Argument \spad{x2},{} is the final point,{} and \spad{ns},{} the number of steps to take. \blankline \item \spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs)} takes a 5-th order Runge-Kutta step with monitoring of local truncation to ensure accuracy and adjust stepsize. The function takes two half steps and one full step and scales the difference in solutions at the final point. If the error is within \spad{eps},{} the step is taken and the result is returned. If the error is not withi
917614`--n \spad{eps},{} the stepsize if decreased and the procedure is tried again until the desired accuracy is reached. Upon input,{} an trial step size must be given and upon return,{} an estimate of the next step size to use is returned as well as the step size which produced the desired accuracy. The scaled error is computed as \center{\spad{error = MAX(ABS((y2steps(i) - y1step(i))/yscal(i)))}} and this is compared against \spad{eps}. If this is greater than \spad{eps},{} the step size is reduced
917614`-- accordingly to \center{\spad{hnew = 0.9 * hdid * (error/eps)^(-1/4)}} If the error criterion is satisfied,{} then we check if the step size was too fine and return a more efficient one. If \spad{error > \spad{eps} * (6.0E-04)} then the next step size should be \center{\spad{hnext = 0.9 * hdid * (error/\spad{eps})^(\spad{-1/5})}} Otherwise \spad{hnext = 4.0 * hdid} is returned. A more detailed discussion of this and related topics can be found in the book "Numerical Recipies" by \spad{W}.Press
917614`--,{} \spad{B}.\spad{P}. Flannery,{} \spad{S}.A. Teukolsky,{} \spad{W}.\spad{T}. Vetterling published by Cambridge University Press. Argument \spad{step} is a record of 3 floating point numbers \spad{(to_try ,{} did ,{} next)},{} \spad{eps} is the required accuracy,{} \spad{yscal} is the scaling vector for the difference in solutions. On input,{} \spad{step.to_try} should be the guess at a step size to achieve the accuracy. On output,{} \spad{step.did} contains the step size which achieved the a
917614`--ccuracy and \spad{step.next} is the next step size to use. \blankline \item \spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs,{}t1,{}t2,{}t3,{}t4,{}t5,{}t6,{}t7)} is the same as \spad{rk4qc(y,{}n,{}x1,{}step,{}eps,{}yscal,{}derivs)} except that the user must provide the 7 scratch arrays \spad{t1-t7} of size \spad{n}. \blankline \item \spad{rk4a(y,{}n,{}x1,{}x2,{}eps,{}h,{}ns,{}derivs)} is a driver program which uses \spad{rk4qc} to integrate \spad{n} ordinary differential equations starting
917614`-- at \spad{x1} to \spad{x2},{} keeping the local truncation error to within \spad{eps} by changing the local step size. The scaling vector is defined as \center{\spad{yscal(i) = abs(y(i)) + abs(h*dydx(i)) + tiny}} where \spad{y(i)} is the solution at location \spad{x},{} \spad{dydx} is the ordinary differential equation\spad{'s} right hand side,{} \spad{h} is the current step size and \spad{tiny} is 10 times the smallest positive number representable. The user must supply an estimate for a tria
917614`--l step size and the maximum number of calls to \spad{rk4qc} to use. Argument \spad{x2} is the final point,{} \spad{eps} is local truncation,{} \spad{ns} is the maximum number of call to \spad{rk4qc} to use. \end{items}
917682`This suite of routines performs numerical quadrature using algorithms derived from the basic trapezoidal rule. Because the error term of this rule contains only even powers of the step size (for open and closed versions),{} fast convergence can be obtained if the integrand is sufficiently smooth. \blankline Each routine returns a Record of type TrapAns,{} which contains\indent{3} \newline value (\spadtype{Float}):\tab{20} estimate of the integral \newlin
917682`--e error (\spadtype{Float}):\tab{20} estimate of the error in the computation \newline totalpts (\spadtype{Integer}):\tab{20} total number of function evaluations \newline success (\spadtype{Boolean}):\tab{20} if the integral was computed within the user specified error criterion \indent{0}\indent{0} To produce this estimate,{} each routine generates an internal sequence of sub-estimates,{} denoted by {\em S(i)},{} depending on the routine,{} to which the various convergence criteria are applie
917682`--d. The user must supply a relative accuracy,{} \spad{eps_r},{} and an absolute accuracy,{} \spad{eps_a}. Convergence is obtained when either \center{\spad{ABS(S(i) - S(i-1)) < eps_r * ABS(S(i-1))}} \center{or \spad{ABS(S(i) - S(i-1)) < eps_a}} are \spad{true} statements. \blankline The routines come in three families and three flavors: \newline\tab{3} closed:\tab{20}romberg,{}\tab{30}simpson,{}\tab{42}trapezoidal \newline\tab{3} open: \tab{20}rombergo,{}\tab{30}simpsono,{}\tab{42}trapezoidalo 
917682`--\newline\tab{3} adaptive closed:\tab{20}aromberg,{}\tab{30}asimpson,{}\tab{42}atrapezoidal \par The {\em S(i)} for the trapezoidal family is the value of the integral using an equally spaced absicca trapezoidal rule for that level of refinement. \par The {\em S(i)} for the simpson family is the value of the integral using an equally spaced absicca simpson rule for that level of refinement. \par The {\em S(i)} for the romberg family is the estimate of the integral using an equally spaced absicc
917682`--a romberg method. For the \spad{i}\spad{-}th level,{} this is an appropriate combination of all the previous trapezodial estimates so that the error term starts with the \spad{2*(i+1)} power only. \par The three families come in a closed version,{} where the formulas include the endpoints,{} an open version where the formulas do not include the endpoints and an adaptive version,{} where the user is required to input the number of subintervals over which the appropriate closed family integrator
917682`-- will apply with the usual convergence parmeters for each subinterval. This is useful where a large number of points are needed only in a small fraction of the entire domain. \par Each routine takes as arguments: \newline \spad{f}\tab{10} integrand \newline a\tab{10} starting point \newline \spad{b}\tab{10} ending point \newline \spad{eps_r}\tab{10} relative error \newline \spad{eps_a}\tab{10} absolute error \newline \spad{nmin} \tab{10} refinement level when to start checking for convergence 
917682`--(> 1) \newline \spad{nmax} \tab{10} maximum level of refinement \par The adaptive routines take as an additional parameter \newline \spad{nint}\tab{10} the number of independent intervals to apply a closed \indented{1}{family integrator of the same name.} \par Notes: \newline Closed family level \spad{i} uses \spad{1 + 2^i} points. \newline Open family level \spad{i} uses \spad{1 + 3^i} points.
917732`\spadtype{ODEIntegration} provides an interface to the integrator. This package is intended for use by the differential equations solver but not at top-level.
917851`\spad{ODETools} provides tools for the linear ODE solver.
917957`\spad{OctonionCategoryFunctions2} implements functions between two octonion domains defined over different rings. The function map is used to coerce between octonion types.
918102`\indented{1}{This package provides tools for operating on one-dimensional arrays} with unary and binary functions involving different underlying types
918178`Lifting of maps to one-point completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989
918268`\spadtype{OpenMathPackage} provides some simple utilities to make reading OpenMath objects easier.
918312`\spadtype{OpenMathServerPackage} provides the necessary operations to run AXIOM as an OpenMath server,{} reading/writing objects to/from a port. Please note the facilities available here are very basic. The idea is that a user calls \spadignore{e.g.} \axiom{Omserve(4000,{}60)} and then another process sends OpenMath objects to port 4000 and reads the result.
918365`This package exports tools to create FriCAS Library information databases.
918411`Lifting of maps to ordered completions. Date Created: 4 Oct 1989 Date Last Updated: 4 Oct 1989
918500`\indented{3}{This package provides ordering functions on vectors which} are suitable parameters for OrderedDirectProduct.
918596`This package provides orthogonal polynomials as functions on a ring.
918674`OutPackage allows pretty-printing from programs.
918714`In this package \spad{K} is a finite field,{} \spad{R} is a ring of univariate polynomials over \spad{K},{} and \spad{F} is a monogenic algebra over \spad{R}. We require that \spad{F} is monogenic,{} \spadignore{i.e.} that \spad{F = K[x,{}y]/(f(x,{}y))},{} because the integral basis algorithm used will factor the pol
918714`--ynomial \spad{f(x,{}y)}. The package provides a function to compute the integral closure of \spad{R} in the quotient field of \spad{F} as well as a function to compute a "local integral basis" at a specific prime.
918904`\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Trager,{}Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \indented{2}{"Pade Approximants,{} Part I: Basic Theory",{} Baker & Graves-Morris.}
918986`\indented{1}{This package computes reliable Pad&ea. approximants using} a generalized Viskovatov continued fraction algorithm. Authors: Burge,{} Hassner & Watt. Date Created: April 1987 Date Last Updated: 12 April 1990 Keywords: Pade,{} series Examples: References: \indented{2}{"Pade Approximants,{} Part I: Basic Theory",{} Baker & Graves-Morris.}
919118`This package implements fixed-point computations on streams.
919188`Author: William Sit,{} spring 89
919375`This package \undocumented
919457`This package \undocumented
919539`This package \undocumented
919725`The package \spadtype{PartialFractionPackage} gives an easier to use interfact the domain \spadtype{PartialFraction}. The user gives a fraction of polynomials,{} and a variable and the package converts it to the proper datatype for the \spadtype{PartialFraction} domain.
919805`PartitionsAndPermutations contains functions for generating streams of integer partitions,{} and streams of sequences of integers composed from a multi-set.
919862`A domain for infinite tridiagonal matrices over a ring.
919916`Tools for patterns.
919988`Lifts maps to patterns.
920067`Attaching assertions to symbols for pattern matching. Date Created: 21 Mar 1989 Date Last Updated: 23 May 1990
920118`This package provides pattern matching functions on function spaces.
920247`This package provides pattern matching functions on integers.
920329`\spadtype{PatternMatchIntegration} provides functions that use the pattern matcher to find some indefinite and definite integrals involving special functions and found in the literature.
920445`This package provides pattern matching functions on kernels.
920536`This package provides pattern matching functions on lists.
920653`This package provides pattern matching functions on polynomials.
920832`This packages provides tools for matching recursively in type towers.
920948`This package provides pattern matching functions on quotients.
921086`Lifts maps to pattern matching results.
921191`This package provides pattern matching functions on symbols.
921252`This package provides tools for the pattern matcher.
921360`This package provides the top-level pattern macthing functions.
921502`Permanent implements the functions {\em permanent},{} the permanent for square matrices.
921578`PermutationGroupExamples provides permutation groups for some classes of groups: symmetric,{} alternating,{} dihedral,{} cyclic,{} direct products of cyclic,{} which are in fact the finite abelian groups of symmetric groups called Young subgroups. Furthermore,{} Rubik\spad{'s} group as permutation group of 48 integers and a list of sporadic simple groups derived from the atlas of finite groups.
921629`\indented{1}{Provides a coercion from the symbolic fractions in \%\spad{pi} with} integer coefficients to any Expression type. Date Created: 21 Feb 1990 Date Last Updated: 21 Feb 1990
921695`\spad{PlotFunctions1} provides facilities for plotting curves where functions \spad{SF} \spad{->} \spad{SF} are specified by giving an expression
921765`This package exports plotting tools
921806`This package \undocumented
921871`This package \undocumented
921920`This package provides function for testing whether a divisor on a curve is a torsion divisor.
922135`\indented{1}{Utilities for PFOQ and PFO} Author: Manuel Bronstein Date Created: 25 Aug 1988 Date Last Updated: 11 Jul 1990
922298`This package provides function for testing whether a divisor on a curve is a torsion divisor.
922521`Package with the conversion functions among different kind of polynomials
922586`Groebner functions for \spad{P} \spad{F} \indented{2}{This package is an interface package to the groebner basis} package which allows you to compute groebner bases for polynomials in either lexicographic ordering or total degree ordering refined by reverse lex. The input is the ordinary polynomial type which is internally converted to a type with the required ordering. The resulting grobner basis is converted back to ordinary polynomials. The orderi
922586`--ng among the variables is controlled by an explicit list of variables which is passed as a second argument. The coefficient domain is allowed to be any \spad{gcd} domain,{} but the groebner basis is computed as if the polynomials were over a field.
922640`This package provides a coerce from polynomials over algebraic numbers to \spadtype{Expression AlgebraicNumber}.
922695`This package provides a very general map function,{} which given a set \spad{S} and polynomials over \spad{R} with maps from the variables into \spad{S} and the coefficients into \spad{S},{} maps polynomials into \spad{S}. \spad{S} is assumed to support \spad{+},{} \spad{*} and \spad{^}.
922869`This package transforms multivariate polynomials or fractions into univariate polynomials or fractions,{} and back.
923038`This package \undocumented
923130`This package \undocumented
923227`\indented{1}{PolynomialFactorizationByRecursionUnivariate} \spad{R} is a \spadfun{PolynomialFactorizationExplicit} domain,{} \spad{S} is univariate polynomials over \spad{R} We are interested in handling SparseUnivariatePolynomials over \spad{S},{} is a variable we shall call \spad{z}
923372`PolynomialFactorizationByRecursion(\spad{R},{}\spad{E},{}\spad{VarSet},{}\spad{S}) is used for factorization of sparse univariate polynomials over a domain \spad{S} of multivariate polynomials over \spad{R}.
923558`\indented{2}{This package takes a mapping between coefficient rings,{} and lifts} it to a mapping between polynomials over those rings.
923761`This package exports interpolation algorithms
923872`This package exports interpolation algorithms
923946`This package provides various polynomial number theoretic functions over the integers.
924009`computes \spad{n}-th roots of quotients of multivariate polynomials
924168`This package provides modest routines for polynomial system solving. The aim of many of the operations of this package is to remove certain factors in some polynomials in order to avoid unnecessary computations in algorithms involving splitting techniques by partial factorization.
924328`This package factors the formulas out of the general solve code,{} allowing their recursive use over different domains. Care is taken to introduce few radicals so that radical extension domains can more easily simplify the results.
924440`This package computes square-free decomposition of multivariate polynomials over a coefficient ring which is an arbitrary \spad{gcd} domain. The requirement on the coefficient domain guarantees that the \spadfun{content} can be removed so that factors will be primitive as well as square-free. Over an infinite ring of finite characteristic,{}it may not 
924440`--be possible to guarantee that the factors are square-free.
924594`This package is primarily to help the interpreter do coercions. It allows you to view a polynomial as a univariate polynomial in one of its variables with coefficients which are again a polynomial in all the other variables.
924675`PowerSeriesLimitPackage implements limits of expressions in one or more variables as one of the variables approaches a limiting value. Included are two-sided limits,{} left- and right- hand limits,{} and limits at plus or minus infinity.
924794`\spadtype{PrecomputedAssociatedEquations} stores some generic precomputations which speed up the computations of the associated equations needed for factoring operators.
924925`This is unfinished package for computing primitive \indented{1}{\spad{gcd} over algebraic extensions.\space{2}Algebraic extension is defined} \indented{1}{by list of polynomial forming triangular system.} \indented{1}{Currently implemented is only trial division.}
924963`\indented{1}{This package provides tools for operating on primitive arrays} with unary and binary functions involving different underlying types
925035`PrimitiveElement provides functions to compute primitive elements in algebraic extensions.
925100`\spad{PrimitiveRatDE} provides functions for in-field solutions of linear \indented{1}{ordinary differential equations,{} in the transcendental case.} \indented{1}{The derivation to use is given by the parameter \spad{L}.}
925317`In-field solution of Riccati equations,{} primitive case.
925535`PrintPackage provides a print function for output forms.
925576`PseudoLinearNormalForm provides a function for computing a block-companion form for pseudo-linear operators.
925638`This package contains some functions: \axiomOpFrom{discriminant}{PseudoRemainderSequence},{} \axiomOpFrom{resultant}{PseudoRemainderSequence},{} \axiomOpFrom{subResultantGcd}{PseudoRemainderSequence},{} \axiomOpFrom{chainSubResultants}{PseudoRemainderSequence},{} \axiomOpFrom{degreeSubResultant}{PseudoRemainderSequence},{} \axiomOpFrom{lastSubResultant}{PseudoRemainderSequence},{} \axiomOpFrom{
925638`--resultantEuclidean}{PseudoRemainderSequence},{} \axiomOpFrom{subResultantGcdEuclidean}{PseudoRemainderSequence},{} \axiomOpFrom{\spad{semiSubResultantGcdEuclidean1}}{PseudoRemainderSequence},{} \axiomOpFrom{\spad{semiSubResultantGcdEuclidean2}}{PseudoRemainderSequence},{} etc. This procedures are coming from improvements of the subresultants algorithm. \indented{2}{Version : 7} \indented{2}{References : Lionel Ducos "Optimizations of the subresultant algorithm"} \indented{2}{to appear in the J
925638`--ournal of Pure and Applied Algebra.} \indented{2}{Author : Ducos Lionel \axiom{Lionel.Ducos@mathlabo.univ-poitiers.\spad{fr}}}
925749`This package provides functions for integration,{} limited integration,{} extended integration and the risch differential equation for pure algebraic integrands.
925874`In-field solution of an linear ordinary differential equation,{} pure algebraic case.
926069`This package \undocumented{}
926224`\spadtype{QuasiAlgebraicSet2} adds a function \spadfun{radicalSimplify} which uses \spadtype{IdealDecompositionPackage} to simplify the representation of a quasi-algebraic set. A quasi-algebraic set is the intersection of a Zariski closed set,{} defined as the common zeros of a given list of polynomials (the defining polynomials for equations),{} and a principal Zariski open set,{} defined as the complement of the common zeros of a polyno
926224`--mial \spad{f} (the defining polynomial for the inequation). Quasi-algebraic sets are implemented in the domain \spadtype{QuasiAlgebraicSet},{} where two simplification routines are provided: \spadfun{idealSimplify} and \spadfun{simplify}. The function \spadfun{radicalSimplify} is added for comparison study only. Because the domain \spadtype{IdealDecompositionPackage} provides facilities for computing with radical ideals,{} it is necessary to restrict the ground ring to the domain \spadtype{Fra
926224`--ction Integer},{} and the polynomial ring to be of type \spadtype{DistributedMultivariatePolynomial}. The routine \spadfun{radicalSimplify} uses these to compute groebner basis of radical ideals and is inefficient and restricted when compared to the two in \spadtype{QuasiAlgebraicSet}.
926290`A package for removing redundant quasi-components and redundant branches when decomposing a variety by means of quasi-components of regular triangular sets. \newline References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic systems of} \indented{5}{positive dimension" Discr. App. Math. 33:1
926290`--47-160,{}1991} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
926482`\spadtype{QuaternionCategoryFunctions2} implements functions between two quaternion domains. The function \spadfun{map} is used by the system interpreter to coerce between quaternion types.
926634`This package extends a function between integral domains to a mapping between their quotient fields.
926791`Package for the computation of eigenvalues and eigenvectors. This package works for matrices with coefficients which are rational functions over the integers. (see \spadtype{Fraction Polynomial Integer}). The eigenvalues and eigenvectors are expressed in terms of radicals.
926837`This package tries to find solutions expressed in terms of radicals for systems of equations of rational functions with coefficients in an integral domain \spad{R}.
926916`This package provides tools for creating radix expansions.
926961`This package exports random distributions
927023`This package exports random floating-point distributions
927127`Random number generators \indented{2}{All random numbers used in the system should originate from} \indented{2}{the same generator.\space{2}This package is intended to be the source.}
927269`\spadtype{RationalFunctionDefiniteIntegration} provides functions to compute definite integrals of rational functions.
927364`Factorization of univariate polynomials with coefficients which are rational functions with integer coefficients.
927479`\spadtype{RationalFunctionFactorizer} contains the factor function (called factorFraction) which factors fractions of polynomials by factoring the numerator and denominator. Since any non zero fraction is a unit the usual factor operation will just return the original fraction.
927555`This package provides functions for the integration of rational functions.
927638`Computation of limits for rational functions.
927709`Find the sign of a rational function around a point or infinity.
927771`Computes sums of rational functions.
927846`Utilities that provide the same top-level manipulations on fractions than on polynomials.
927905`This package provides functions for the base case of the Risch algorithm.
928012`This package exports rational interpolation algorithms
928084`\spad{RationalLODE} provides functions for in-field solutions of linear \indented{1}{ordinary differential equations,{} in the rational case.}
928184`rational number testing and retraction functions. Date Created: March 1990 Date Last Updated: 9 April 1991
928268`In-field solution of Riccati equations,{} rational case.
928371`\indented{1}{A package for computing the rational univariate representation} \indented{1}{of a zero-dimensional algebraic variety given by a regular} \indented{1}{triangular set. This package is essentially an interface for the} \spadtype{InternalRationalUnivariateRepresentationPackage} constructor. It is used in the \spadtype{ZeroDimensionalSolvePackage} for solving polynomial systems with fi
928371`--nitely many solutions.
928483`\axiomType{RealPolynomialUtilitiesPackage} provides common functions used by interval coding.
928623`\indented{1}{This package provides numerical solutions of systems of polynomial} equations for use in ACPLOT.
928671`\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the rational numbers to arbitrary user-specified precision. The results are returned as a list of isolating intervals,{} expressed as records with "left" and "right" rational number components.
928769`\indented{2}{This package provides functions for finding the real zeros} of univariate polynomials over the integers to arbitrary user-specified precision. The results are returned as a list of isolating intervals which are expressed as records with "left" and "right" rational number components.
928855`\spadtype{RectangularMatrixCategoryFunctions2} provides functions between two matrix domains. Th
928855`--e functions provided are \spadfun{map} and \spadfun{reduce}.
929267`This package provides an operator for the \spad{n}-th term of a recurrence and an operator for the coefficient of \spad{x^n} in a function specified by a functional equation.
929370`Elimination of an algebraic from the coefficentss of a linear ordinary differential equation.
929587`\indented{1}{Finds the order of a divisor over a finite field} Author: Manuel Bronstein Date Created: 1988 Date Last Updated: 8 November 1994
929794`\spadtype{ReductionOfOrder} provides functions for reducing the order of linear ordinary differential equations once some solutions are known.
929902`A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are proposed: in the sense of Zariski closure (like in Kalkbrener\spad{'s} algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard methods). This algorithm is valid for n
929902`--ay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \spad{TS} and the packages \axiomType{QCMPACK}(\spad{R},{}\spad{E},{}\spad{V},{}\spad{P},{}\spad{TS}) and \axiomType{RSETGCD}(\spad{R},{}\spad{E},{}\spad{V},{}\spad{P},{}\sp
929902`--ad{TS}). The same way it does not care about the way univariate polynomial \spad{gcd} (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these \spad{gcd} need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \axiom{\spad{TS}}. Thus,{} the operations of this package are not documented.\newline Refe
929902`--rences : \indented{1}{[1] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
930103`An internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field.\newline References : \indented{1}{[1] \spad{M}. MORENO MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions
930103`--" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
930304`Implements multiplication by repeated addition
930373`Implements exponentiation by repeated squaring
930442`\spad{RepresentationPackage1} provides functions for representation theory for finite groups and algebras. The package creates permutation representations and uses tensor products and its symmetric and antisymmetric components to create new representations of larger degree from given ones. Note: instead of having parameters from \spadtype{Permutation} this package allows list notation of permutations as well: \spadignore{e.g.} \spad{[1,{}4,{}3,{}2
930442`--]} denotes permutes 2 and 4 and fixes 1 and 3.
930499`\spad{RepresentationPackage2} provides functions for working with modular representations of finite groups and algebra. The routines in this package are created,{} using ideas of \spad{R}. Parker,{} (the meat-Axe) to get smaller representations from bigger ones,{} \spadignore{i.e.} finding sub- and factormodules,{} or to show,{} that such the representations are irreducible. Note: most functions are randomized functions of Las Vegas type \spadigno
930499`--re{i.e.} every answer is correct,{} but with small probability the algorithm fails to get an answer.
930556`This package provides coercions for the special types \spadtype{Exit} and \spadtype{Void}.
930618`RetractSolvePackage is an interface to \spadtype{SystemSolvePackage} that attempts to retract the coefficients of the equations before solving.
930715`Factorization of univariate polynomials with coefficients in an algebraic extension of \spadtype{Fraction Polynomial Integer}.
930906`A package to computes Taylor series of \spad{S}-transforms.
931192`\spadtype{ScriptFormulaFormat1} provides a utility coercion for changing to SCRIPT formula format anything that has a coercion to the standard output format.
931258`This package provides operations for mapping functions onto \spadtype{SegmentBinding}\spad{s}.
931330`This package provides operations for mapping functions onto segments.
931391`A domain for mappings between sequences.
931479`Factorization of univariate polynomials with coefficients in an algebraic extension of the rational numbers (\spadtype{Fraction Integer}).
931663`\indented{1}{Package to allow simplify to be called on AlgebraicNumbers} by converting to EXPR(INT)
931730`\spadtype{SmithNormalForm} is a package which provides some standard canonical forms for matrices.
931890`This package exports sorting algorithnms
931983`\indented{1}{Cache of elements in a set} Author: Manuel Bronstein Date Created: 31 Oct 1988 Date Last Updated: 14 May 1991 \indented{2}{A sorted cache of a cachable set \spad{S} is a dynamic structure that} \indented{2}{keeps the elements of \spad{S} sorted and assigns an integer to each} \indented{2}{element of \spad{S} once it is in the cache. This way,{} equality and ordering} \indented{2}{on \spad{S} are tested directly on the integers associate
931983`--d with the elements} \indented{2}{of \spad{S},{} once they have been entered in the cache.}
932155`This package lifts a mapping from coefficient rings \spad{R} to \spad{S} to a mapping from sparse univariate polynomial over \spad{R} to a sparse univariate polynomial over \spad{S}. Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.
932357`SpecialOutputPackage allows FORTRAN,{} Tex and \indented{2}{Script Formula Formatter output from programs.}
932408`\indented{2}{A internal package for removing redundant quasi-components and redundant} \indented{2}{branches when decomposing a variety by means of quasi-components} \indented{2}{of regular triangular sets. \newline} References : \indented{1}{[1] \spad{D}. LAZARD "A new method for solving algebraic system
932408`--s of} \indented{5}{positive dimension" Discr. App. Math. 33:147-160,{}1991} \indented{5}{Tech. Report (PoSSo project)} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. 
932408`--Rep. 4/98.}
932610`A package providing a new algorithm for solving polynomial systems by means of regular chains. Two ways of solving are provided: in the sense of Zariski closure (like in Kalkbrener\spad{'s} algorithm) or in the sense of the regular zeros (like in Wu,{} Wang or Lazard- Moreno methods). T
932610`--his algorithm is valid for nay type of regular set. It does not care about the way a polynomial is added in an regular set,{} or how two quasi-components are compared (by an inclusion-test),{} or how the invertibility test is made in the tower of simple extensions associated with a regular set. These operations are realized respectively by the domain \spad{TS} and the packages \spad{QCMPPK(R,{}E,{}V,{}P,{}TS)} and \spad{RSETGCD(R,{}E,{}V,{}P,{}TS)}. The same way it does not care about the way 
932610`--univariate polynomial gcds (with coefficients in the tower of simple extensions associated with a regular set) are computed. The only requirement is that these gcds need to have invertible initials (normalized or not). WARNING. There is no need for a user to call diectly any operation of this package since they can be accessed by the domain \axiomType{\spad{TS}}. Thus,{} the operations of this package are not documented.\newline References : \indented{1}{[1] \spad{M}. MORENO MAZA "A new algori
932610`--thm for computing triangular} \indented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
932831`A internal package for computing gcds and resultants of univariate polynomials with coefficients in a tower of simple extensions of a field. There is no need to use directly this package since its main operations are available from \spad{TS}. \newline References : \indented{1}{[1] \spad{M}. MORENO
932831`-- MAZA and \spad{R}. RIOBOO "Computations of \spad{gcd} over} \indented{5}{algebraic towers of simple extensions" In proceedings of \spad{AAECC11}} \indented{5}{Paris,{} 1995.} \indented{1}{[2] \spad{M}. MORENO MAZA "Calculs de pgcd au-dessus des tours} \indented{5}{d'extensions simples et resolution des systemes d'equations} \indented{5}{algebriques" These,{} Universite \spad{P}.etM. Curie,{} Paris,{} 1997.} \indented{1}{[3] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \ind
932831`--ented{5}{decomposition of algebraic varieties" NAG Tech. Rep. 4/98.}
933041`This package provides standard arithmetic operations on matrices. The functions in this package store the results of computations in existing matrices,{} rather than creating new matrices. This package works only for matrices of type Matrix and uses the internal representation of this type.
933111`StreamExponentialSeriesOperations implements arithmetic of exponential power series,{} where a power series is represented by a stream of its coefficients.
933195`\indented{1}{StreamExponentialSeriesTranscendentalFunctions implements} \indented{1}{exp and log of exponential power series,{}} where a power series is represented by a stream of its coefficients.
933314`Functions defined on streams with entries in one set.
933368`Functions defined on streams with entries in two sets.
933431`Functions defined on streams with entries in three sets.
933503`This package computes infinite products of Taylor series over an integral domain of characteristic 0. Here Taylor series are represented by streams of Taylor coefficients.
933586`StreamTaylorSeriesOperations implements Taylor series arithmetic,{} where a Taylor series is represented by a stream of its coefficients.
933695`StreamTranscendentalFunctionsNonCommutative implements transcendental functions on Taylor series over a non-commutative ring,{} where a Taylor series is represented by a stream of its coefficients.
933800`StreamTranscendentalFunctions implements transcendental functions on Taylor series,{} where a Taylor series is represented by a stream of its coefficients.
933889`StructuralConstantsPackage provides functions creating structural constants from a multiplication tables or a basis of a matrix algebra and other useful functions in this context.
933952`This package produces functions for counting etc. real roots of univariate polynomials in \spad{x} over \spad{R},{} which must be an OrderedIntegralDomain
934033`This package computes the subresultants of two polynomials which is needed for the `Lazard Rioboo' enhancement to Tragers integrations formula For efficiency reasons this has been rewritten to call Lionel Ducos package which is currently the best one. \blankline
934140`\indented{1}{SupFractionFactorize} contains the factor function for univariate polynomials over the quotient field of a ring \spad{S} such that the package MultivariateFactorize works for \spad{S}
934283`Computes all the symmetric functions in \spad{n} variables.
934339`SymmetricGroupCombinatoricFunctions contains combinatoric functions concerning symmetric groups and representation theory: list young tableaus,{} improper partitions,{} subsets bijection of Coleman.
934402`\axiom{SymmetryAnalysis(\spad{JB},{} \spad{xi},{} eta)} provides procedures for the symmetry analysis of differential equations over a given jet bundle. Currently there exist only some procedures to set up the determining system for the symmetry generators of Lie point symmetries.
934538`SystemODESolver provides tools for triangulating and solving some systems of linear ordinary differential equations.
934705`TableauBumpers implements the Schenstead-Knuth correspondence between sequences and pairs of Young tableaux. The 2 Young tableaux are represented as a single tableau with pairs as components.
934765`\axiom{TabulatedComputationPackage(Key ,{}Entry)} provides some modest support for dealing with operations with type \axiom{Key \spad{->} Entry}. The result of such operations can be stored and retrieved with this package by using a hash-table. The user does not need to worry about the management of this hash-table. However,{} onnly one hash-table is built by calling \axiom{TabulatedComputationPackage(Key ,
934765`--{}Entry)}.
934863`Expands tangents of sums and scalar products.
935098`This package provides functions for template manipulation
935145`linear extensions of maps on the basis
935299`\spadtype{TexFormat1} provides a utility coercion for changing to TeX format anything that has a coercion to the standard output format.
935351`Tools for the sign finding utilities.
935402`TopLevelDrawFunctionsForAlgebraicCurves provides top level functions for drawing non-singular algebraic curves.
935525`TopLevelDrawFunctionsForCompiledFunctions provides top level functions for drawing graphics of expressions.
935598`TopLevelDrawFunctionsForPoints provides top level functions for drawing curves and surfaces described by sets of points.
935658`TopLevelDrawFunctions provides top level functions for drawing graphics of expressions.
935745`This package exports a function for making a \spadtype{ThreeSpace}
935792`This package finds the function \spad{func3} where \spad{func1} and \spad{func2} \indented{1}{are given and\space{2}\spad{func1} = \spad{func3}(\spad{func2}) .\space{2}If there is no solution then} \indented{1}{function \spad{func1} will be returned.} \indented{1}{An example would be\space{2}\spad{func1:= 8*X^3+32*X^2-14*X ::EXPR INT} and} \indented{1}{\spad{func2:=2*X ::EXPR INT} convert them via univariate} \indented{1}{to 
935792`--FRAC SUP EXPR INT and then the solution is \spad{func3:=X^3+X^2-X}} \indented{1}{of type FRAC SUP EXPR INT}
935871`This package tries to find solutions of equations of type Expression(\spad{R}). This means expressions involving transcendental,{} exponential,{} logarithmic and nthRoot functions. After trying to transform different kernels to one kernel by applying several rules,{} it calls zerosOf for the SparseUnivariatePolynomial in the remaining kernel. For example the expression \spad{sin(x)*cos(x)-2} will be transformed to \indented{3}{\spad
935871`--{-2 tan(x/2)^4 -2 tan(x/2)^3 -4 tan(x/2)^2 +2 tan(x/2) -2}} by using the function normalize and then to \indented{3}{\spad{-2 tan(x)^2 + tan(x) -2}} with help of subsTan. This function tries to express the given function in terms of \spad{tan(x/2)} to express in terms of \spad{tan(x)} . Other examples are the expressions \spad{sqrt(x+1)+sqrt(x+7)+1} or \indented{1}{\spad{sqrt(sin(x))+1} .}
935943`Hermite integration,{} transcendental case.
936055`This package provides functions for the transcendental case of the Risch algorithm.
936157`TranscendentalManipulations provides functions to simplify and expand expressions involving transcendental operators.
936271`\indented{1}{Risch differential equation system,{} transcendental case.} Author: Manuel Bronstein Date Created: 17 August 1992 Date Last Updated: 3 February 1994
936386`\indented{1}{Risch differential equation,{} transcendental case.} Author: Manuel Bronstein Date Created: Jan 1988 Date Last Updated: 2 November 1995
936494`This package provides functions that compute "fraction-free" inverses of upper and lower triangular matrices over a integral domain. By "fraction-free inverses" we mean the following: given a matrix \spad{B} with entries in \spad{R} and an element \spad{d} of \spad{R} such that \spad{d} * inv(\spad{B}) also has entries in \spad{R},{} w
936494`--e return \spad{d} * inv(\spad{B}). Thus,{} it is not necessary to pass to the quotient field in any of our computations.
936665`\spadtype{TrigonometricManipulations} provides transformations from trigonometric functions to complex exponentials and logarithms,{} and back.
936786`Tools for constructing tubes around 3-dimensional parametric curves.
936831`\indented{1}{The purpose of this package is to provide reasonable plots of} functions with singularities.
936885`A basic package for the factorization of bivariate polynomials over a finite field. The functions here represent the base step for the multivariate factorizer.
936950`This is a low-level package which implements operations \indented{1}{on vectors treated as univariate modular polynomials.\space{2}Most} \indented{1}{operations takes modulus as parameter.\space{2}Modulus is machine} \indented{1}{sized prime which should be small enough to avoid overflow} \indented{1}{in intermediate calculations.}
937010`\spad{RUTSodetools} provides tools to interface with the series \indented{1}{ODE solver when presented with linear ODEs.}
937193`UnittestAux is an auxiliary package to Unittest,{} that performs \indented{13}{checking of tests,{} dealing also with fatal errors.}
937247`UnittestCount is a package that keeps track of statistics \indented{13}{collected during testing.\space{2}It is separate from} \indented{13}{\spadtype{UnittestAux} because the latter takes an argument,{}} \indented{13}{and we want the state to be common for all arguments.}
937291`Unittest is a package that provides functions for regression \indented{13}{testing.\space{2}Note that it calls the interpreter,{} thus,{} all} \indented{13}{functions require that there arguments be passed as strings.}
937331`Package for the factorization of univariate polynomials with integer coefficients. The factorization is done by "lifting" (HENSEL) the factorization over a finite field.
937491`Mapping package for univariate Laurent series \indented{2}{This package allows one to apply a function to the coefficients of} \indented{2}{a univariate Laurent series.}
937652`Mapping from polynomials over \spad{R} to polynomials over \spad{S} given a map from \spad{R} to \spad{S} assumed to send zero to zero.
937815`UnivariatePolynomialCommonDenominator provides functions to compute the common denominator of the coefficients of univariate polynomials over the quotient field of a \spad{gcd} domain.
937968`UnivariatePolynomialDecompositionPackage implements functional decomposition of univariate polynomial with coefficients in an \spad{IntegralDomain} of \spad{CharacteristicZero}.
938107`UnivariatePolynomialDivisionPackage provides a division for non monic univarite polynomials with coefficients in an \spad{IntegralDomain}.
938229`This package lifts a mapping from coefficient rings \spad{R} to \spad{S} to a mapping from \spadtype{UnivariatePolynomial}(\spad{x},{}\spad{R}) to \spadtype{UnivariatePolynomial}(\spad{y},{}\spad{S}). Note that the mapping is assumed to send zero to zero,{} since it will only be applied to the non-zero coefficients of the polynomial.
938324`This package implements Karatsuba\spad{'s} trick for multiplying (large) univariate polynomials. It could be improved with a version doing the work on place and also with a special case for squares. We've done this in Basicmath,{} but we believe that this out of the scope of FriCAS.
938438`This package provides for square-free decomposition of univariate polynomials over arbitrary rings,{} \spadignore{i.e.} a partial factorization such that each factor is a product of irreducibles with multiplicity one and the factors are pairwise relatively prime. If the ring has characteristic zero,{} the result is guaranteed to satisfy this condition. If the ring is an infin
938438`--ite ring of finite characteristic,{} then it may not be possible to decide when polynomials contain factors which are \spad{p}th powers. In this case,{} the flag associated with that polynomial is set to "nil" (meaning that that polynomials are not guaranteed to be square-free).
938568`Mapping package for univariate Puiseux series. This package allows one to apply a function to the coefficients of a univariate Puiseux series.
938730`\spad{UnivariateSkewPolynomialCategoryOps} provides products and \indented{1}{divisions of univariate skew polynomials.}
938845`Mapping package for univariate Taylor series. \indented{2}{This package allows one to apply a function to the coefficients of} \indented{2}{a univariate Taylor series.}
939035`\indented{1}{This package provides Taylor series solutions to regular} linear or non-linear ordinary differential equations of arbitrary order.
939178`This package provides operations for mapping functions onto segments.
939251`Provides functions to force a partial ordering on any set.
939319`This packages provides functions to allow the user to select the ordering on the variables and operators for displaying polynomials,{} fractions and expressions. The ordering affects the display only and not the computations.
939374`\indented{2}{This package provides operations which all take as arguments} vectors of elements of some type \spad{A} and functions from \spad{A} to another of type \spad{B}. The operations all iterate over their vector argument and either return a value of type \spad{B} or a vector over \spad{B}.
939437`ViewportDefaultsPackage describes default and user definable values for graphics
939487`ViewportPackage provides functions for creating GraphImages and TwoDimensionalViewports from lists of lists of points.
939530`This package implements the Weierstrass preparation theorem \spad{f} for multivariate power series. weierstrass(\spad{v},{}\spad{p}) where \spad{v} is a variable,{} and \spad{p} is a TaylorSeries(\spad{R}) in which there is term of form c*v^s where \spad{c} is a constant. Let \spad{s} be minimal as above. The result is a list of TaylorSeries coefficients A[\spad{i}] of the equivalent polynomial A = A[0] + A[1]\spad{*v} + A[2]\spad{*v^2} + ... + 
939530`--A[\spad{s}-1]\spad{*v^}(\spad{s}-1) + \spad{v^s} such that p=A*B ,{} \spad{B} being a TaylorSeries of minimum degree 0
939589`In this package \spad{K} is a finite field,{} \spad{R} is a ring of univariate polynomials over \spad{K},{} and \spad{F} is a framed algebra over \spad{R}. The package provides a function to compute the integral closure of \spad{R} in the quotient field of \spad{F} as well as a function to compute a "local integral basis" at
939589`-- a specific prime.
939771`This package provides computations of logarithms and exponentials for polynomials in non-commutative variables. \newline Author: Michel Petitot (petitot@lifl.\spad{fr}).
939903`A package for computing symbolically the complex and real roots of zero-dimensional algebraic systems over the integer or rational numbers. Complex roots are given by means of univariate representations of irreducible regular chains. Real roots are given by means of tuples of coordinates lying in the \spadtype{RealClosure} of the coefficient ring. This constructor takes three argumen
939903`--ts. The first one \spad{R} is the coefficient ring. The second one \spad{ls} is the list of variables involved in the systems to solve. The third one must be \spad{concat(ls,{}s)} where \spad{s} is an additional symbol used for the univariate representations. WARNING: The third argument is not checked. All operations are based on triangular decompositions. The default is to compute these decompositions directly from the input system by using the \spadtype{RegularChain} domain constructor. The 
939903`--lexTriangular algorithm can also be used for computing these decompositions (see the \spadtype{LexTriangularPackage} package constructor). For that purpose,{} the operations \axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage},{} \axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage} and \axiomOpFrom{positiveSolve}{ZeroDimensionalSolvePackage} admit an optional argument. \newline Author: Marc Moreno Maza.
940025`Maps abstract computational structures to real-world FriCAS code \blankline This allows FriCAS domains to be created from an instance of Lambda it also allows FriCAS categories to be created from an instance of intuitionisticLogic. \blankline So by the Curry-Howard isomorphism we can coerce to the intuitionisticLogic from Lambda. This gives an isomorphism where theorems in intuitionistic logic correspond to type signatures in combinatory logic and programs in intu
940025`--itionistic logic correspond to the proofs of those theorems in combinatory logic. \blankline As an example of this in Haskell see "Djinn,{} a theorem prover in Haskell,{} for Haskell" here: http://lambda-the-ultimate.\spad{org/node/1178} http://www.augustsson.net/Darcs/Djinn/ \blankline see also: Philip Wadler - Theorems for free! http://homepages.inf.ed.ac.uk/wadler/topics/parametricity.html
940065`The compUtil package provides utilities to convert between the computational domains: Lambda,{} Ski and ILogic. Both Lambda are Ski are Turing complete and can be coerced to each other. Lambda and Ski are not equal and they are only isomorphic up to beta-equivalence and beta-equivalence is undecidable so there is not a direct correspondence between the nodes in their trees. Also the names of bound variables and other such constructions may be lost in Lam
940065`--bda \spad{->} Ski \spad{->} Lambda round trip. An element of ILogic cannot be coerced to the other types. However ILogic can be used to produce a theory which can be concerted to/from the other domains using Curry-Howard isomorphism.
940115`The class of abelian groups,{} \spadignore{i.e.} additive monoids where each element has an additive inverse. \blankline
940175`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with an additive identity element. \blankline
940237`Abelian monoid ring elements (not necessarily of finite support) of this ring are of the form formal SUM (r_i * e_i) where the r_i are coefficents and the e_i,{} elements of the ordered abelian monoid,{} are thought of as exponents or monomials. The monomials commute with each other,{} but in general do not commute with the coefficients (which themselves may or may not be commutative). S
940237`--ee \spadtype{FiniteAbelianMonoidRing} for the case of finite support. A useful common model for polynomials and power series. Conceptually at least,{} only the non-zero terms are ever operated on.
940355`the class of all additive (commutative) semigroups,{} \spadignore{i.e.} a set with a commutative and associative operation \spadop{+}. \blankline
940422`The notion of aggregate serves to model any data structure aggregate,{} designating any collection of objects,{} with heterogenous or homogeneous members,{} with a finite or infinite number of members,{} explicitly or implicitly represented. An aggregate can in principle represent everything from a string of characters to abstract sets such as "the set of \spad{x} satisfying relation {\em r(x)}" An attribute \spadatt{finiteAggregate} is used to assert th
940422`--at a domain has a finite number of elements.
940472`The category of associative algebras (modules which are themselves rings). \blankline
940545`Model for algebraically closed fields.
940625`Model for algebraically closed function spaces.
940748`Category for the inverse trigonometric functions.
940846`A bag aggregate is an aggregate for which one can insert and extract objects,{} and where the order in which objects are inserted determines the order of extraction. Examples of bags are stacks,{} queues,{} and dequeues.
940916`\spadtype{BasicType} is the basic category for describing a collection of elements with \spadop{=} (equality).
940970`A binary-recursive aggregate has 0,{} 1 or 2 children and serves as a model for a binary tree or a doubly-linked aggregate structure
941064`\spadtype{BinaryTreeCategory(S)} is the category of binary trees: a tree which is either empty or else is a \spadfun{node} consisting of a value and a \spadfun{left} and \spadfun{right},{} both binary trees.
941153`The bit aggregate category models aggregates representing large quantities of Boolean data.
941211`A collection is a homogeneous aggregate which can built from list of members. The operation used to build the aggregate is generically named \spadfun{construct}. However,{} each collection provides its own special function with the same name as the data type,{} except with an initial lower case letter,{} \spadignore{e.g.} \spadfun{list} for \spadtype{List},{} \spadfun{flexibleArray} for \spadtype{FlexibleArray},{} and so on.
941277`This category represents the extension of a ring by a square root of \spad{-1}.
941366`A dictionary is an aggregate in which entries can be inserted,{} searched for and removed. Duplicates are thrown away on insertion. This category models the usual notion of dictionary which involves large amounts of data where copying is impractical. Principal operations are thus destructive (non-copying) ones.
941432`This category is a collection of operations common to both categories \spadtype{Dictionary} and \spadtype{MultiDictionary}
941518`Differential extensions of a ring \spad{R}. Given a differentiation on \spad{R},{} extend it to a differentiation on \%.
941607`\spadtype{DifferentialPolynomialCategory} is a category constructor specifying basic functions in an ordinary differential polynomial ring with a given ordered set of differential indeterminates. In addition,{} it implements defaults for the basic functions. The functions \spadfun{order} and \spadfun{weight} are e
941607`--xtended from the set of derivatives of differential indeterminates to the set of differential polynomials. Other operations provided on differential polynomials are \spadfun{leader},{} \spadfun{initial},{} \spadfun{separant},{} \spadfun{differentialVariables},{} and \spadfun{isobaric?}. Furthermore,{} if the ground ring is a differential ring,{} then evaluation (substitution of differential indeterminates by elements of the ground ring or by differential polynomials) is provided by \spadfun{ev
941607`--al}. A convenient way of referencing derivatives is provided by the functions \spadfun{makeVariable}. \blankline To construct a domain using this constructor,{} one needs to provide a ground ring \spad{R},{} an ordered set \spad{S} of differential indeterminates,{} a ranking \spad{V} on the set of derivatives of the differential indeterminates,{} and a set \spad{E} of exponents in bijection with the set of differential monomials in the given differential indeterminates. \blankline
941800`An ordinary differential ring,{} that is,{} a ring with an operation \spadfun{differentiate}. \blankline
941868`\spadtype{DifferentialVariableCategory} constructs the set of derivatives of a given set of (ordinary) differential indeterminates. If \spad{x},{}...,{}\spad{y} is an ordered set of differential indeterminates,{} and the prime notation is used for differentiation,{} then the set of derivatives (including zero-th order) of the differential indeterminates is \spad{x},{}\spad{x'},{}\spad{x''},{}...
941868`--,{} \spad{y},{}\spad{y'},{}\spad{y''},{}... (Note: in the interpreter,{} the \spad{n}-th derivative of \spad{y} is displayed as \spad{y} with a subscript \spad{n}.) This set is viewed as a set of algebraic indeterminates,{} totally ordered in a way compatible with differentiation and the given order on the differential indeterminates. Such a total order is called a ranking of the differential indeterminates. \blankline A domain in this category is needed to construct a differential polynomial 
941868`--domain. Differential polynomials are ordered by a ranking on the derivatives,{} and by an order (extending the ranking) on on the set of differential monomials. One may thus associate a domain in this category with a ranking of the differential indeterminates,{} just as one associates a domain in the category \spadtype{OrderedAbelianMonoidSup} with an ordering of the set of monomials in a set of algebraic indeterminates. The ranking is specified through the binary relation \spadfun{<}. For exa
941868`--mple,{} one may define one derivative to be less than another by lexicographically comparing first the \spadfun{order},{} then the given order of the differential indeterminates appearing in the derivatives. This is the default implementation. \blankline The notion of weight generalizes that of degree. A polynomial domain may be made into a graded ring if a weight function is given on the set of indeterminates,{} Very often,{} a grading is the first step in ordering the set of monomials. For d
941868`--ifferential polynomial domains,{} this constructor provides a function \spadfun{weight},{} which allows the assignment of a non-negative number to each derivative of a differential indeterminate. For example,{} one may define the weight of a derivative to be simply its \spadfun{order} (this is the default assignment). This weight function can then be extended to the set of all differential polynomials,{} providing a graded ring structure.
941978`attribute to indicate an aggregate of finite size
942095`A division ring (sometimes called a skew field),{} \spadignore{i.e.} a not necessarily commutative ring where all non-zero elements have multiplicative inverses.
942155`Category for the elementary functions.
942243`An eltable aggregate is one which can be viewed as a function. For example,{} the list \axiom{[1,{}7,{}4]} can applied to 0,{}1,{} and 2 respectively will return the integers 1,{}7,{} and 4; thus this list may be viewed as mapping 0 to 1,{} 1 to 7 and 2 to 4. In general,{} an aggregate can map members of a domain {\em Dom} to an image domain {\em Im}.
942349`Entire Rings (non-commutative Integral Domains),{} \spadignore{i.e.} a ring not necessarily commutative which has no zero divisors. \blankline
942405`A constructive euclidean domain,{} \spadignore{i.e.} one can divide producing a quotient and a remainder where the remainder is either zero or is smaller (\spadfun{euclideanSize}) than the divisor. \blankline Conditional attributes: \indented{2}{multiplicativeValuation\tab{25}\spad{Size(a*b)=Size(a)*Size(b)}} \indented{2}{additiveValuation\tab{25}\spad{Size(a*b)=Size(a)+Size(b)}}
942470`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions.
942540`An expression space is a set which is closed under certain operators.
942601`An extensible aggregate is one which allows insertion and deletion of entries. These aggregates are models of lists and streams which are represented by linked structures so as to make insertion,{} deletion,{} and concatenation efficient. However,{} access to elements of these extensible aggregates is generally slow since access is made from the end. See \spadtype{FlexibleArray} for an exception.
942697`ExtensionField {\em F} is the category of fields which extend the field \spad{F}
942767`The category of commutative fields,{} \spadignore{i.e.} commutative rings where all non-zero elements have multiplicative inverses. The \spadfun{factor} operation while trivial is useful to have defined. \blankline
942811`FieldOfPrimeCharacteristic is the category of fields of prime characteristic,{} \spadignore{e.g.} finite fields,{} algebraic closures of fields of prime characteristic,{} transcendental extensions of of fields of prime characteristic.
942895`The category of domains composed of a finite set of elements. We include the functions \spadfun{lookup} and \spadfun{index} to give a bijection between the finite set and an initial segment of positive integers. \blankline
942942`This category is similar to AbelianMonoidRing,{} except that the sum is assumed to be finite. It is a useful model for polynomials,{} but is somewhat more general.
943073`FiniteAlgebraicExtensionField {\em F} is the category of fields which are finite algebraic extensions of the field {\em F}. If {\em F} is finite then any finite algebraic extension of {\em F} is finite,{} too. Let {\em K} be a finite algebraic extension of the finite field {\em F}. The exponentiation of elements of {\em K} defines a \spad{Z}-module structure on the multiplicative group of {\em K}. The
943073`-- additive group of {\em K} becomes a module over the ring of polynomials over {\em F} via the operation \spadfun{linearAssociatedExp}(a:K,{}f:SparseUnivariatePolynomial \spad{F}) which is linear over {\em F},{} \spadignore{i.e.} for elements {\em a} from {\em K},{} {\em c,{}d} from {\em F} and {\em f,{}g} univariate polynomials over {\em F} we have \spadfun{linearAssociatedExp}(a,{}cf+dg) equals {\em c} times \spadfun{linearAssociatedExp}(a,{}\spad{f}) plus {\em d} times \spadfun{linearAssocia
943073`--tedExp}(a,{}\spad{g}). Therefore \spadfun{linearAssociatedExp} is defined completely by its action on monomials from {\em F[X]}: \spadfun{linearAssociatedExp}(a,{}monomial(1,{}\spad{k})\spad{\$}SUP(\spad{F})) is defined to be \spadfun{Frobenius}(a,{}\spad{k}) which is {\em a^(q^k)} where {\em q=size()\$F}. The operations order and discreteLog associated with the multiplicative exponentiation have additive analogues associated to the operation \spadfun{linearAssociatedExp}. These are the functi
943073`--ons \spadfun{linearAssociatedOrder} and \spadfun{linearAssociatedLog},{} respectively.
943177`This category describes finite rational divisors on a curve,{} that is finite formal sums SUM(\spad{n} * \spad{P}) where the \spad{n}\spad{'s} are integers and the \spad{P}\spad{'s} are finite rational points on the curve.
943406`FiniteFieldCategory is the category of finite fields
943480`Category of finite graphs,{} allows us to model graph theory \blankline
943555`A finite linear aggregate is a linear aggregate of finite length. The finite property of the aggregate adds several exports to the list of exports from \spadtype{LinearAggregate} such as \spadfun{reverse},{} \spadfun{sort},{} and so on.
943643`A FiniteRankAlgebra is an algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
943777`A FiniteRankNonAssociativeAlgebra is a non associative algebra over a commutative ring \spad{R} which is a free \spad{R}-module of finite rank.
943898`A finite-set aggregate models the notion of a finite set,{} that is,{} a collection of elements characterized by membership,{} but not by order or multiplicity. See \spadtype{Set} for an example.
943987`This category is intended as a model for floating point systems. A floating point system is a model for the real numbers. In fact,{} it is an approximation in the sense that not all real numbers are exactly representable by floating point numbers. A floating point system is characterized by the following: \blankline \indented{2}{1: \spadfunFrom{base}{FloatingPointSystem} of the \spadfunFrom{exponent}{FloatingPointSystem}.} \indented{9
943987`--}{(actual implemenations are usually binary or decimal)} \indented{2}{2: \spadfunFrom{precision}{FloatingPointSystem} of the \spadfunFrom{mantissa}{FloatingPointSystem} (arbitrary or fixed)} \indented{2}{3: rounding error for operations} \blankline Because a Float is an approximation to the real numbers,{} even though it is defined to be a join of a Field and OrderedRing,{} some of the attributes do not hold. In particular associative(\spad{"+"}) does not hold. Algorithms defined over a field 
943987`--need special considerations when the field is a floating point system.
944057`A \spadtype{FramedAlgebra} is a \spadtype{FiniteRankAlgebra} together with a fixed \spad{R}-module basis.
944183`FramedNonAssociativeAlgebra(\spad{R}) is a \spadtype{FiniteRankNonAssociativeAlgebra} (\spadignore{i.e.} a non associative algebra over \spad{R} which is a free \spad{R}-module of finite rank) over a commutative ring \spad{R} together with a fixed \spad{R}-module basis.
944296`A domain of this category implements formal linear combinations of elements from a domain \spad{Basis} with coefficients in a domain \spad{R}. The domain \spad{Basis} needs only to belong to the category \spadtype{SetCategory} and \spad{R} to the category \spadtype{Ring}. Thus the coefficient ring may be non-commutative. See the \spadtype{XDistributedPolynomial} constructor for examples of domai
944296`--ns built with the \spadtype{FreeModuleCategory} category constructor. Author: Michel Petitot (petitot@lifl.\spad{fr}) \blankline Note (Franz Lehner,{} June 2009): Since \spad{leadingTerm} makes no sense for unordered base sets,{} and at the time of this writing this domain was never used for such,{} it was changed to \spad{OrderedSet}. \spad{FreeModule} originally was not of FreeModuleCategory. Some functions (like \spad{support},{} \spad{coefficients},{} \spad{monomials},{} ...) from here cou
944296`--ld be moved to \spad{IndexedDirectProductCategory} but at the moment there is no need for this.
944406`This category provides a selection of evaluation operations depending on what the argument type \spad{R} provides.
944495`\spad{S} is \spadtype{FullyLinearlyExplicitRingOver R} means that \spad{S} is a \spadtype{LinearlyExplicitRingOver R} and,{} in addition,{} if \spad{R} is a \spadtype{LinearlyExplicitRingOver Integer},{} then so is \spad{S}
944601`\indented{2}{A is fully retractable to \spad{B} means that A is retractable to \spad{B},{} and,{}} \indented{2}{in addition,{} if \spad{B} is retractable to the integers or rational} \indented{2}{numbers then so is A.} \indented{2}{In particular,{} what we are asserting is that there are no integers} \indented{2}{(rationals) in A which don\spad{'t} retract into \spad{B}.} Date Created: March 1990 Date Last Updated: 9 Apr
944601`--il 1991
944685`This category is a model for the function field of a plane algebraic curve.
944893`A space of formal functions with arguments in an arbitrary ordered set.
944968`This category describes domains where \spadfun{\spad{gcd}} can be computed but where there is no guarantee of the existence of \spadfun{factor} operation for factorisation into irreducibles. However,{} if such a \spadfun{factor} operation exist,{} factorization will be unique up to order and units.
945021`GradedAlgebra(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-algebra\spad{''}. A graded algebra is a graded module together with a degree preserving \spad{R}-linear map,{} called the {\em product}. \blankline The name ``product\spad{''} is written out in full so inner and outer products with the same mapping type can be distinguished by name.
945124`GradedModule(\spad{R},{}\spad{E}) denotes ``E-graded \spad{R}-module\spad{''},{} \spadignore{i.e.} collection of \spad{R}-modules indexed by an abelian monoid \spad{E}. An element \spad{g} of \spad{G[s]} for some specific \spad{s} in \spad{E} is said to be an element of \spad{G} with {\em degree} \spad{s}. Sums are defined in each module \spad{G[s]} so two elements of \spad{G} have a sum if they have the s
945124`--ame degree. \blankline Morphisms can be defined and composed by degree to give the mathematical category of graded modules.
945223`The class of multiplicative groups,{} \spadignore{i.e.} monoids with multiplicative inverses. \blankline
945267`A homogeneous aggregate is an aggregate of elements all of the same type. In the current system,{} all aggregates are homogeneous. Two attributes characterize classes of aggregates. Aggregates from domains with attribute \spadatt{finiteAggregate} have a finite number of members. Of course,{} such a domain may have an infinite number of elements,{} like,{} for example \spadtype{List}. Those domains with attribute \spada
945267`--tt{shallowlyMutable} allow an element to be modified or updated without changing its overall value.
945353`Category for the hyperbolic trigonometric functions.
945440`An indexed aggregate is a many-to-one mapping of indices to entries. For example,{} a one-dimensional-array is an indexed aggregate where the index is an integer. Also,{} a table is an indexed aggregate where the indices and entries may have any type.
945560`This category provides \spadfun{eval} operations. A domain may belong to this category if it is possible to make ``evaluation\spad{''} substitutions. The difference between this and \spadtype{Evalable} is that the operations in this category specify the substitution as a pair of arguments rather than as an equation.
945652`An \spad{IntegerNumberSystem} is a model for the integers.
945722`\axiomType{JetBundleBaseFunctionCategory} defines the category of functions (local sections) of the base space of a jet bundle,{} \spadignore{i.e.} functions depending only on the independent variables. Such a category is needed \spadignore{e.g.} for the representation of solutions.
945839`\axiomType{JetBundleCategory} provides basic data structures and procedures for jet bundles. Nearly all necessary functions are implemented already here. Only the representation and functions which direct access to it must be implemented in a domain. Two notations of derivatives are supported. Default is multi-index notation,{} where the \spad{i}-th entry of the index denotes the number of differentiations taken with respect to \axiom{x^i
945839`--}. In repeated index notation each entry \axiom{\spad{i}} in the index denotes a differentiation with respect to \axiom{x^i}. The choice affects,{} however,{} only in- and output. Internally,{} multi-index notation is used throughout.
945905`\axiomType{JetBundleFunctionCategory} defines the category of functions (local sections) over a jet bundle. The formal derivative is defined already here. It uses the Jacobi matrix of the functions. The columns of the matrices are enumerated by jet variables. Thus they are represented as a \axiomType{Record} of the matrix and a list of the jet variables. Several simplification routines are impl
945905`--emented already here.
946016`A keyed dictionary is a dictionary of key-entry pairs for which there is a unique entry for each key.
946128`LazyStreamAggregate is the category of streams with lazy evaluation. It is understood that the function 'empty?' will cause lazy evaluation if necessary to determine if there are entries. Functions which call 'empty?',{} \spadignore{e.g.} 'first' and 'rest',{} will also cause lazy evaluation if necessary.
946214`The category of all left algebras over an arbitrary ring.
946281`The category of Lie Algebras. It is used by the following domains of non-commutative algebra: \axiomType{LiePolynomial} and \axiomType{XPBWPolynomial}. \newline Author : Michel Petitot (petitot@lifl.\spad{fr}).
946359`A linear aggregate is an aggregate whose elements are indexed by integers. Examples of linear aggregates are strings,{} lists,{} and arrays. Most of the exported operations for linear aggregates are non-destructive but are not always efficient for a particular aggregate. For example,{} \spadfun{concat} of two lists needs only to copy its first argument,{} whereas \spadfun{concat} of two arrays needs to copy both arguments. Most 
946359`--of the operations exported here apply to infinite objects (\spadignore{e.g.} streams) as well to finite ones. For finite linear aggregates,{} see \spadtype{FiniteLinearAggregate}.
946435`\spad{LinearOrdinaryDifferentialOperatorCategory} is the category of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: \indented{4}{\spad{(L1 * L2).(f) = L1 L2 f}}
946565`A list aggregate is a model for a linked list data structure. A linked list is a versatile data structure. Insertion and deletion are efficient and searching is a linear operation.
946637``Logic' provides the basic operations for lattices,{} \spadignore{e.g.} boolean algebra.
946681`\spadtype{MatrixCategory} is a general matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col. A domain belonging to this category will be shallowly mutable. The index of the 'first' row may be obtained by
946681`-- calling the function \spadfun{minRowIndex}. The index of the 'first' column may be obtained by calling the function \spadfun{minColIndex}. The index of the first element of a Row is the same as the index of the first column in a matrix and vice versa.
946843`The category for general multi-variate possibly skew polynomials over a ring \spad{R},{} in variables from VarSet,{} with exponents from the \spadtype{OrderedAbelianMonoidSup}.
947007`The category of modules over a commutative ring. \blankline
947077`Monad is the class of all multiplicative monads,{} \spadignore{i.e.} sets with a binary operation.
947121`\indented{1}{MonadWithUnit is the class of multiplicative monads with unit,{}} \indented{1}{\spadignore{i.e.} sets with a binary operation and a unit element.} Axioms \indented{3}{leftIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{3}\tab{30} 1*x=x} \indented{3}{rightIdentity("*":(\%,{}\%)\spad{->}\%,{}1)\space{2}\tab{30} x*1=x} Common Additional Axioms \indented{3}{unitsKnown
947121`---if "recip" says "failed",{} that PROVES input wasn\spad{'t} a unit}
947183`A \spadtype{MonogenicAlgebra} is an algebra of finite rank which can be generated by a single element.
947315`The class of multiplicative monoids,{} \spadignore{i.e.} semigroups with a multiplicative identity element. \blankline
947362`\spadtype{MultivariateTaylorSeriesCategory} is the most general multivariate Taylor series category.
947505`NonAssociativeAlgebra is the category of non associative algebras (modules which are themselves non associative rngs). Axioms \indented{3}{\spad{r*}(a*b) = (r*a)\spad{*b} = a*(\spad{r*b})}
947602`A NonAssociativeRing is a non associative \spad{rng} which has a unit,{} the multiplication is not necessarily commutative or associative.
947674`NonAssociativeRng is a basic ring-type structure,{} not necessarily commutative or associative,{} and not necessarily with unit. Axioms \indented{2}{\spad{x*}(\spad{y+z}) = x*y + \spad{x*z}} \indented{2}{(x+y)\spad{*z} = \spad{x*z} + \spad{y*z}} Common Additional Axioms \indented{2}{noZeroDivisors\space{2}ab = 0 \spad{=>} \spad{a=0} or \spad{b=0}}
947742`OctonionCategory gives the categorial frame for the octonions,{} and eight-dimensional non-associative algebra,{} doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.
947828`One-dimensional-array aggregates serves as models for one-dimensional arrays. Categorically,{} these aggregates are finite linear aggregates with the \spadatt{shallowlyMutable} property,{} that is,{} any component of the array may be changed without affecting the identity of the overall array. Array data structures are typically represented by a fixed area in storage and therefore cannot efficiently gro
947828`--w or shrink on demand as can list structures (see however \spadtype{FlexibleArray} for a data structure which is a cross between a list and an array). Iteration over,{} and access to,{} elements of arrays is extremely fast (and often can be optimized to open-code). Insertion and deletion however is generally slow since an entirely new data structure must be created for the result.
947930`Ordered sets which are also rings,{} that is,{} domains where the ring operations are compatible with the ordering. \blankline
947988`The class of totally ordered sets,{} that is,{} sets such that for each pair of elements \spad{(a,{}b)} exactly one of the following relations holds \spad{a<b or a=b or b<a} and the relation is transitive,{} \spadignore{i.e.} \spad{a<b and b<c => a<c}. This order should be the natural order on given structure.
948043`A partial differential ring with differentiations indexed by a parameter type \spad{S}. \blankline
948143`The class of partially ordered sets,{} that is sets equipped with transitive and reflexive relation \spad{<=}.
948202`The category for general multi-variate polynomials over a ring \spad{R},{} in variables from VarSet,{} with exponents from the \spadtype{OrderedAbelianMonoidSup}. Here variables commute with the coefficients.
948348`This is the category of domains that know "enough" about themselves in order to factor univariate polynomials over themselves. This will be used in future releases for supporting factorization over finitely generated coefficient fields,{} it is not yet available in the current release of axiom.
948445`A category for finite subsets of a polynomial ring. Such a set is only regarded as a set of polynomials and not identified to the ideal it generates. So two distinct sets may generate the same the ideal. Furthermore,{} for \spad{R} being an integral domain,{} a set of polynomials may be viewed as a representa
948445`--tion of the ideal it generates in the polynomial ring \spad{(R)^(-1) P},{} or the set of its zeros (described for instance by the radical of the previous ideal,{} or a split of the associated affine variety) and so on. So this category provides operations about those different notions.
948643`\spadtype{PowerSeriesCategory} is the most general power series category with exponents in an ordered abelian monoid.
948798`\spadtype{QuaternionCategory} describes the category of quaternions and implements functions that are not representation specific.
948893`QuotientField(\spad{S}) is the category of fractions of an Integral Domain \spad{S}.
948991`The \spad{RadicalCategory} is a model for the rational numbers.
949056`\axiomType{RealClosedField} provides common acces functions for all real closed fields.
949122`The real number system category is intended as a model for the real numbers. The real numbers form an ordered normed field. Note that we have purposely not included \spadtype{DifferentialRing} or the elementary functions (see \spadtype{TranscendentalFunctionCategory}) in the definition.
949186`\axiomType{RealRootCharacterizationCategory} provides common acces functions for all real root codings.
949396`\spadtype{RectangularMatrixCategory} is a category of matrices of fixed dimensions. The dimensions of the matrix will be parameters of the domain. Domains in this category will be \spad{R}-modules and will be non-mutable.
949635`A recursive aggregate over a type \spad{S} is a model for a a directed graph containing values of type \spad{S}. Recursively,{} a recursive aggregate is a {\em node} consisting of a \spadfun{value} from \spad{S} and 0 or more \spadfun{children} which are recursive aggregates. A node with no children is called a \spadfun{leaf} node. A recursive aggregate may be cyclic for which some operations as noted may go into an infini
949635`--te loop.
949717`A category for general multi-variate polynomials with coefficients in a ring,{} variables in an ordered set,{} and exponents from an ordered abelian monoid,{} with a \axiomOp{sup} operation. When not constant,{} such a polynomial is viewed as a univariate polynomial in its main variable \spad{w}. \spad{r}. \spad{t}. to the total ordering on the elements in 
949717`--the ordered set,{} so that some operations usually defined for univariate polynomials make sense here.
949866`The category of regular triangular sets,{} introduced under the name regular chains in [1] (and other papers). In [3] it is proved that regular triangular sets and towers of simple extensions of a field are equivalent notions. In the following definitions,{} all polynomials and ideals are taken from the polyno
949866`--mial ring \spad{k[x1,{}...,{}xn]} where \spad{k} is the fraction field of \spad{R}. The triangular set \spad{[t1,{}...,{}tm]} is regular iff for every \spad{i} the initial of \spad{ti+1} is invertible in the tower of simple extensions associated with \spad{[t1,{}...,{}\spad{ti}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given ideal \spad{I} iff the radical of \spad{I} is equal to the intersection of the radical ideals generated by the saturated 
949866`--ideals of the \spad{[T1,{}...,{}\spad{Ti}]}. A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Kalkbrener of a given triangular set \spad{T} iff it is a split of Kalkbrener of the saturated ideal of \spad{T}. Let \spad{K} be an algebraic closure of \spad{k}. Assume that \spad{V} is finite with cardinality \spad{n} and let \spad{A} be the affine space \spad{K^n}. For a regular triangular set \spad{T} let denote by \spad{W(T)} the set of regular zeros of \spad{T}. A family
949866`-- \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Lazard of a given subset \spad{S} of \spad{A} iff the union of the \spad{W(\spad{Ti})} contains \spad{S} and is contained in the closure of \spad{S} (\spad{w}.\spad{r}.\spad{t}. Zariski topology). A family \spad{[T1,{}...,{}Ts]} of regular triangular sets is a split of Lazard of a given triangular set \spad{T} if it is a split of Lazard of \spad{W(T)}. Note that if \spad{[T1,{}...,{}Ts]} is a split of Lazard of \spad{T} then it i
949866`--s also a split of Kalkbrener of \spad{T}. The converse is \spad{false}. This category provides operations related to both kinds of splits,{} the former being related to ideals decomposition whereas the latter deals with varieties decomposition. See the example illustrating the \spadtype{RegularTriangularSet} constructor for more explanations about decompositions by means of regular triangular sets. \newline References : \indented{1}{[1] \spad{M}. KALKBRENER "Three contributions to elimination 
949866`--theory"} \indented{5}{\spad{Phd} Thesis,{} University of Linz,{} Austria,{} 1991.} \indented{1}{[2] \spad{M}. KALKBRENER "Algorithmic properties of polynomial rings"} \indented{5}{Journal of Symbol. Comp. 1998} \indented{1}{[3] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)} \indented{1}{[4] \spad{M}. MORENO MAZA "A new algorithm for computing triangular} \indented{5}{decomposition of algebra
949866`--ic varieties" NAG Tech. Rep. 4/98.}
950063`\spad{B} is retractable from A means that some elements in A can be converted into elements of \spad{B} and any element of \spad{B} can be converted into an element of A.
950141`A is retractable to \spad{B} means that some elementsif A can be converted into elements of \spad{B} and any element of \spad{B} can be converted into an element of A.
950215`The category of rings with unity,{} always associative,{} but not necessarily commutative.
950256`PointCategory is the category of points and vectors in space. Points may be used to represent shapes in a scenegraph such as: the vertices of a cube or points along a line. Shapes such as these may be defined explicitly or may be plotted. We implement SetCategory to allow us to curry transforms from one point to another.
950319`the class of all multiplicative semigroups,{} \spadignore{i.e.} a set with an associative operation \spadop{*}. \blankline
950372`A set category lists a collection of set-theoretic operations useful for both finite sets and multisets. Note however that finite sets are distinct from multisets. Although the operations defined for set categories are common to both,{} the relationship between the two cannot be described by inclusion or inheritance.
950450`\spadtype{SetCategory} is the basic category for describing a collection of elements with \spadop{=} (equality) and \spadfun{coerce} to output form. \blankline Conditional Attributes: \indented{3}{canonical\tab{15}data structure equality is the same as \spadop{=}}
950507`\spadtype{SquareMatrixCategory} is a general square matrix category which allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and colums returned as objects of type Col.
950726`A stream aggregate is a linear aggregate which possibly has an infinite number of elements. A basic domain constructor which builds stream aggregates is \spadtype{Stream}. From streams,{} a number of infinite structures such as power series can be built.
950802`A string aggregate is a category for strings,{} that is,{} one dimensional arrays of characters.
950866`A table aggregate is a model of a table,{} \spadignore{i.e.} a discrete many-to-one mapping from keys to entries.
951247`Category of tensor powers of modules over commutative rings.
951382`Category for the transcendental elementary functions.
951478`The category of triangular sets of multivariate polynomials with coefficients in an integral domain. Let \axiom{\spad{R}} be an integral domain and \axiom{\spad{V}} a finite ordered set of variables,{} say \axiom{\spad{X1} < \spad{X2} < ... < \spad{Xn}}. A set \axiom{\spad{S}} of polynomials in \axiom{\spad{R}[\spad{X1
951478`--},{}\spad{X2},{}...,{}\spad{Xn}]} is triangular if no elements of \axiom{\spad{S}} lies in \axiom{\spad{R}},{} and if two distinct elements of \axiom{\spad{S}} have distinct main variables. Note that the empty set is a triangular set. A triangular set is not necessarily a (lexicographical) Groebner basis and the notion of reduction related to triangular sets is based on the recursive view of polynomials. We recall this notion here and refer to [1] for more details. A polynomial \axiom{\spad{P}
951478`--} is reduced \spad{w}.\spad{r}.\spad{t} a non-constant polynomial \axiom{\spad{Q}} if the degree of \axiom{\spad{P}} in the main variable of \axiom{\spad{Q}} is less than the main degree of \axiom{\spad{Q}}. A polynomial \axiom{\spad{P}} is reduced \spad{w}.\spad{r}.\spad{t} a triangular set \axiom{\spad{T}} if it is reduced \spad{w}.\spad{r}.\spad{t}. every polynomial of \axiom{\spad{T}}. \newline References : \indented{1}{[1] \spad{P}. AUBRY,{} \spad{D}. LAZARD and \spad{M}. MORENO MAZA "On 
951478`--the Theories} \indented{5}{of Triangular Sets" Journal of Symbol. Comp. (to appear)}
951666`Category for the trigonometric functions.
951760`\indented{1}{TwoDimensionalArrayCategory is a general array category which} allows different representations and indexing schemes. Rows and columns may be extracted with rows returned as objects of type Row and columns returned as objects of type Col. The index of the 'first' row may be obtained by calling the function 'minRowInde
951760`--x'. The index of the 'first' column may be obtained by calling the function 'minColIndex'. The index of the first element of a 'Row' is the same as the index of the first column in an array and vice versa.
951936`A unary-recursive aggregate is a one where nodes may have either 0 or 1 children. This aggregate models,{} though not precisely,{} a linked list possibly with a single cycle. A node with one children models a non-empty list,{} with the \spadfun{value} of the list designating the head,{} or \spadfun{first},{} of the list,{} and the child designating the tail,{} or \spadfun{rest},{} of the list. A node with no chil
951936`--d then designates the empty list. Since these aggregates are recursive aggregates,{} they may be cyclic.
952028`A constructive unique factorization domain,{} \spadignore{i.e.} where we can constructively factor members into a product of a finite number of irreducible elements.
952110`This is a category of univariate Laurent series constructed from univariate Taylor series. A Laurent series is represented by a pair \spad{[n,{}f(x)]},{} where \spad{n} is an arbitrary integer and \spad{f(x)} is a Taylor series. This pair represents the Laurent series \spad{x^n * f(x)}.
952300`The category of univariate polynomials over a ring \spad{R}. No particular model is assumed - implementations can be either sparse or dense.
952403`\spadtype{UnivariatePowerSeriesCategory} is the most general univariate power series category with exponents in an ordered abelian monoid. Note: this category exports a substitution function if it is possible to multiply exponents. Note: this category exports a derivative operation if it is possible to multiply coefficients by exponents.
952556`This is a category of univariate Puiseux series constructed from univariate Laurent series. A Puiseux series is represented by a pair \spad{[r,{}f(x)]},{} where \spad{r} is a positive rational number and \spad{f(x)} is a Laurent series. This pair represents the Puiseux series \spad{f(x^r)}.
952747`This is the category of univariate skew polynomials over an Ore coefficient ring. The multiplication is given by \spad{x a = \sigma(a) x + \delta a}. This category is an evolution of the types \indented{2}{MonogenicLinearOperator,{} OppositeMonogenicLinearOperator,{} and} \indented{2}{NonCommutativeOperatorDivision} developped by Jean Della Dora and Stephen \spad{M}. Watt.
952859`\spadtype{UnivariateTaylorSeriesCategory} is the category of Taylor series in one variable.
952975`\spadtype{VectorCategory} represents the type of vector like objects,{} \spadignore{i.e.} finite sequences indexed by some finite segment of the integers. The operations available on vectors depend on the structure of the underlying components. Many operations from the component domain are defined for vectors componentwise. It can by assumed that extraction or updating components can be done in constant time.
953051`Vector Spaces (not necessarily finite dimensional) over a field.
