Ontology of OpenMath Symbols This symbol is used to describe the minimum distance between two distinct objects, i.e. if the distance is less than this they are considered to be the same. This symbol marks an initial value for a parameter, for example this could be the point from which a newton iteration would start. This symbol marks a requirement for the absolute error ( |true-computed| ) on a computation. This symbol marks a requirement for the relative error ( |true-computed|/|computed| ) on a computation. This symbol marks an estimated upper bound for the absolute error ( |true-computed| ) on a computation. This symbol marks an estimated upper bound for the relative error ( |true-computed|/|computed| ) on a computation. This symbol represents the set of algebraic numbers. A-hypergeometric series reference: authors: "Saito, Sturmfels, Takayama" title: "Grobner Deformations of Hypergeometric Differential Equations" pages: 127 This Symbol represents the generic category of Abelian group. This symbol is the constructor for Abelian groups, that is a group such that the operation is commutative between members of the group. The Abelian_group constructor takes four arguments, the set of the Abelian group, a binary function taking two elements of the set into itself to represent the operation of the Abelian group, an element of the set to represent the identity of the Abelian group and a unary function taking the set into itself to specify inverse elements. This symbol takes one argument which should be an Abelian group. It returns the identity of the Abelian group. This symbol takes one argument which should be an Abelian group. It reurns a unary function, which should be the inverse function for the Abelian group. This symbol takes one argument which should be an Abelian group. It returns a binary function, which represents the operation of the Abelian group. This symbol takes one argument which should be an Abelian group. It returns the set of the Abelian group. This Symbol represents the generic category of Abelian monoid. This is the constructor for Abelian monoids. An Abelian monoid is a monoid, such that the operation is commutative between members of the Abelian monoid. The Abelian_monoid constructor takes three arguments, the set of the Abelian monoid, a binary function taking two elements of the set into itself to represent the operation of the Abelian monoid and an element of the set to represent the identity of the Abelian monoid. This symbol takes one argument which should be an Abelian monoid, it returns the identity of the Abelian monoid. This symbol takes one argument which should be an Abelian monoid, it returns the operation of the Abelian monoid. This symbol takes one argument which should be an Abelian monoid, it returns the set of the Abelian monoid. This Symbol represents the generic category of Abelian semigroup. This symbol is the constructor for an Abelian semigroup, that is a semigroup which has an operator which is commutative over the set of the semigroup. The Abelian semigroup constructor takes two arguments, the set of the Abelian semigroup and a binary function which represents the operation of the Abelian semigroup. This symbol takes one argument which should be an Abelian semigroup. It returns a binary function, which should represent the operation of the Abelian semigroup. This symbol takes one argument which should be an Abelian semigroup. It returns a set, which should be the set of the Abelian semigroup. This symbol is used within a limit construct to show the limit is being approached from above. It takes no arguments. A unary operator which represents the absolute value of its argument. The argument should be numerically valued. In the complex case this is often referred to as the modulus. This symbol represents the absolute zero of temperature, synonymous with the object of that temperature having zero latent heat. This symbol represents the acceleration physical dimension. It is the second derivative of distance with respect to time. This symbol represents the measure of one acre. This is the standard imperial measure for area. action of a differential operator to a function. This symbol represents a generic type for aggregates (or collections of objects. The symbol Ai defines the unary Airy Ai function; as in Abramovitz & Stegun equation 10.4.1. This is a solution to the equation: $$w^{\prime\prime}-x*w=0$$ It is linearly independent to the Airy Bi function represented by the Bi symbol in this Content Dictionary and is specifically given by: $$Ai(x)=Ai(0)~f(z)-(-Ai^\prime (0))~g(z)$$ where: $$f(z)=\sum_{k=0}^\infty 3^k{\left (\frac{1}{3}\right )}_k \frac{z^{3k}}{(3k)!}$$ and: $$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$ The symbol Ai2 takes two arguments, it represents derivatives of the Airy Ai function. The symbol Ai2(n,z) represents the n'th derivative of Ai(z). The first Airy function. This function is one of the famous two solutions of the Airy differential equation, and converges to 0 when z tends to infinity The second Airy function. This function is the another one of the famous two solutions of the Airy differential equation, and diverges when z tends to infinity This symbol represents the type of algebraic intervals. This symbol represents the error which is returned when an application raises an error due to algorithmic restrictions of the implementations. This includes operations not implemented or partially implemented, divisions by zero and other domain errors. It will have at least one argument, which is a string describing the problem. It may have a second argument which is relevant to the error. This symbol represents the measure of one amp. This is the standard metric measure for current. This symbol represents the logical and function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if all arguments are true or false otherwise. Indicates a variable that we do not want to name This symbol represents an anti-Hermitian matrix, it takes one argument. The argument should be a vector of vectors of values which determine the upper triangle of the matrix. The lower triangle of the matrix is specified by the following relation: - M^* = transpose(M), were M^* denotes the matrix consisting of all the complex conjugates of M. This rules implies that the main diagonal is zero, therefore the argument should not include it. Proposition; the type of antisymmetric binary relations. Appell's hypergeometric series F_1 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 Appell's hypergeometric series F_2 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 Appell's hypergeometric series F_3 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 Appell's hypergeometric series F_4 reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages: 14 The operation of joining one list to another This symbol is used to denote the repeated application of an n-ary function on the elements of a given list. For example when used with plus or times this can represent sums and products. The symbol takes two arguments; the first of which is the n-ary function, the second a list. This symbol is used to denote the approximate equality of its two arguments. This symbol represents the arccos function. This is the inverse of the cos function as described in Abramowitz and Stegun, section 4.4. It takes one argument. This symbol represents the arccos function. This is the multivalued inverse of the cos function. This symbol represents the arccosh function as described in Abramowitz and Stegun, section 4.6. This symbol represents the Arccosh function as described in Abramowitz and Stegun, section 4.6. This symbol represents the arccot function as described in Abramowitz and Stegun, section 4.4. This symbol represents the multi-valued arccot function as the inverse of cot This symbol represents the arccoth function as described in Abramowitz and Stegun, section 4.6. This symbol represents the Arccoth function as described in Abramowitz and Stegun, section 4.6. This symbol represents the arccsc function as described in Abramowitz and Stegun, section 4.4. This symbol represents the multivalued arccsc function as the inverse of csc. This symbol represents the arccsch function as described in Abramowitz and Stegun, section 4.6. This symbol represents the Arccsch function as described in Abramowitz and Stegun, section 4.6. This symbol represents the arcsec function as described in Abramowitz and Stegun, section 4.4. This symbol represents the multivalued arcsec function as the inverse of sec. This symbol represents the arcsech function as described in Abramowitz and Stegun, section 4.6. This symbol represents the Arcsech function as described in Abramowitz and Stegun, section 4.6. This symbol represents the arcsin function. This is the inverse of the sin function as described in Abramowitz and Stegun, section 4.4. It takes one argument. This symbol represents the arcsin function. This is the multi-valued inverse of the sin function as described in Abramowitz and Stegun, section 4.4. It takes one argument. This symbol represents the arcsinh function as described in Abramowitz and Stegun, section 4.6. This symbol represents the Arcsinh function as described in Abramowitz and Stegun, section 4.6. This symbol represents the arctan function. This is the inverse of the tan function as described in Abramowitz and Stegun, section 4.4. It takes one argument. This symbol represents the two-argument arctan function as in Fortran's ATAN2. arctan(x,y) is a value of arctan(y/x). For real x,y arctan(x,y) is positive when y is positive, negative when y is negative. If y is zero, the result is 0 if x is positive, and $\pi$ if x is negative. If x is zero, the result has absolute value $\pi/2$. This symbol represents the arctan function. This is the multi-valued inverse of the tan function. This symbol represents the arctanh function as described in Abramowitz and Stegun, section 4.6. This symbol represents the Arctanh function as described in Abramowitz and Stegun, section 4.6. This symbol is an n-ary boolean function. When applied to a_1, ..., a_n, it is true if and only if the arguments are mutually distinct (that is, a_i and a_j are equal only if i=j). This symbol represents the area physical dimension. This symbol represents the unary function which returns the argument of a complex number, viz. the angle which a straight line drawn from the number to zero makes with the Real line (measured anti-clockwise). The argument to the symbol is the complex number whos argument is being taken. The symbol is a constructor with two arguments. Its first argument should be a configuration, its second argument a statement about the configuration, called thesis. When applied to a configuration C and a thesis T, the OpenMath object assertion(C,T) expresses the assertion that T holds in C. The type of associative binary operation. This symbol represents the error which is returned when an application encounters some asynchronous error, for example if a limit in memory has been reached, or an error has occurred in some system call (I/O error, disk full, machine down). It should have one argument, which is a string describing the problem. An `attribution' object consists of pairs of keys and values. The use of the symbol `attribution' in a signature indicates that the symbol is to be used as a key. This symbol represents the number of atoms in 12 grammes of pure carbon(12). It is approximately 6.0221367*10^(23) +/- 3.6*10^(17) per mole. This symbol represents a line in a formal proof which is an instance of an axiom. The first child is the line in the proof: the second is the axiom used. This symbol represents a (p,q) banded matrix, it takes one argument. A (p,q) banded matrix should always be square. The lower non-zero subdiagonal is the first element of the argument, whilst the highest non-zero super-diagonal is given by the last element of the argument. The argument determines the band of possibly non-zero entries which are positioned around the diagonal. It should be a vector of vectors, we note that they will not all be the same length, however the length of the vectors determine p and q. The longest element specifies the diagonal of the matrix and hence the size of the matrix. Every element not in the band is zero. This symbol represents the measure of one bar. This is the standard imperial measure for pressure. This symbol represents the constructor function for integers, specifying the base. It takes two arguments, the first is a positive integer to denote the base to which the number is represented, the second argument is a string which contains an optional sign and the digits of the integer, using 0-9a-z (as a consequence of this no radix greater than 35 is supported). Base 16 and base 10 are already covered in the encodings of integers. The Bell numbers: Bell(n) is the total number of possible partitions of a set of n elements. This symbol is used within a limit construct to show the limit is being approached from below. It takes no arguments. The Bessel function. This function is one of the famous two solutions of the Bessel differential equation at z=0. The Bessel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=0. Euler's beta function The symbol Bi defines the unary Airy Bi function. This is defined in Abramivitz and Stegun 10.4.1 and is a solution to the equation: $$w^{\prime\prime}-x*w=0$$ It is linearly independant to the Airy Ai function represented by the Ai symbol in this Content Dictionary and is specifically given by: $$Bi(x)=\sqrt{3}(Bi(0)~f(z)+(-Bi^\prime (0))~g(z))$$ where: $$f(z)=\sum_{k=0}^\infty 3^k{\left (\frac{1}{3}\right )}_k \frac{z^{3k}}{(3k)!}$$ and: $$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$ The symbol Bi2 takes two arguments, it represents derivatives of the Airy Bi function. The symbol Bi2(n,z) represents the n'th derivative of Bi(z). The bigfloat constructor takes three arguments, a mantissa, a base and the exponent. The bigfloat "with precision specified in (another) radix" constructor. Takes 3 arguments, the first argument is a floating point number constructed with the bigfloat constructor, the second is the new radix, whilst the third specifies how many digits are significant. An `OMBIND' object has three parts: a "binder" such as "lambda" or "for all", a (list of) bound variables, and an expression. The use of `binder' in a signature indicates that we are describing something which can only be used as the first child of an OMBIND construct. The binomial coefficients. binomial(n, m) is the number of ways of choosing m objects from a collection of n distinct objects without regard to the order. A constant which describes the relationship between temperature and kinetic energy for molecules in an ideal gas. It is approximately 1.380658*10^(-23) +/- 1.2*10^(-28) Joules per Kelvin. This symbol represents the set of Booleans. That is the truth values, true and false. This symbol is used within a limit construct to show the limit is being approached from both sides. It takes no arguments. The type of byte arrays This symbol represents the set of complex numbers. This symbol represents an n-ary construction function for constructing the Cartesian product of multisets. It takes n multiset arguments in order to construct their Cartesian product. This symbol represents an n-ary construction function for constructing the Cartesian product of sets. It takes n set arguments in order to construct their Cartesian product. the cartesian product of n copies of the first argument. Binary function. This symbol is intended to denote the type of a category. The top level element for the Content Dictionary. It just acts as a container for the elements described below. This symbol is used to represent the element of a content dictionary which explains some aspect of that content dictionary. It should have one string argument which makes that explanation. This symbol is used to represent the element of a CDGroup which explains some aspect of the corresponding content dictionary. It should have one string argument which makes that explanation. An element which contains a date as a string in the ISO-8601 YYYY-MM-DD format. This gives the date at which the Content Dictionary was last edited. This symbol is used to represent the element which contains the definition of each symbol in a content dictionary. That is: it must contain a 'Name' element and a 'Description' element, and it may contain an arbitrary number of 'Example', 'FMP' or 'CMP' elements. This symbol represents the outermost element of a CDGroup. It has an arbitrary number of arguments which may be elements of type corresponding to the other symbols defined in this file. This symbol represents the element of a CDGroup which describes the CDGroupDescription element. It has one string argument, this should be the contents of the CDGroupDescription element intended to describe the mathematical area of the CDGroup. This symbol represents the element of a CDGroup which describes each CDGroupMember element. It has one string argument, this should be the contents of the intended CDGroupMember element of the CDGroup. This should be used to identify each member of the CDGroup. This symbol represents the element of a CDGroup which describes the name of that CDGroup, it has one argument that should be a string corresponding to the name. The syntactical requirements are given in the OpenMath standard. This symbol represents the element of a CDGroup which describes the CDGroupURL element. It has one string argument which should describe the URL for that CDGroup, not necessarily for the member Content Dictionaries, The syntactical requirements are given in the OpenMath standard. An element which contains the string corresponding to the name of the CD. The string must match the syntax for CD names given in the OpenMath Standard. Here and elsewhere white space occurring at the beginning or end of the string will be ignored. This symbol represents the element of a CDGroup which describes each CDName element. It has one string argument, this should be the string corresponding to the name of a content dictionary which is in this CDGroup. An element which contains a date as a string in the ISO-8601 YYYY-MM-DD format. This gives the date at which the Content Dictionary is next scheduled for review. It should be expected to be stable until at least this date. An element which contains a revision number (or minor version number) This should be a non-negative integer starting from zero for each new version. Additional examples would be typical changes to a CD requiring a new revision number. This symbol is used to represent the element of a signature file which explains some aspect of that signature file. It should have one string argument which makes that explanation. This symbol is used to represent the outermost element of the Signature File which is characterized by two required attributes that identify the type system and the Content Dictionary whose signatures are defined. The value of the XML attribute 'type' is the name of the Content Dictionary or of the CDGroup that represents the type system. The value of the XML attribute 'cd' is the name of the Content Dictionary whose symbols are assigned signatures in this Signature File. It has an arbitrary number of arguments which may be elements of type corresponding to the other symbols defined in this file. This symbol is used to represent the element of a signature file which specifies the earliest possible revision date of the signature file. It should have one string argument which specifies that date. The date should be in the format YYYY-MM-DD, e.g. 2000-02-29. This symbol is used to represent the element of a signature file which specifies the status of that signature file. It should have one string argument, which should be one of 'official' (approved by the OpenMath Society according to the procedure outlined in the OpenMath standard), 'experimental' (currently being tested), 'private' (used by a private group of OpenMath users) or 'obsolete' (an obsolete signature file, kept only for archival purposes). An element giving information on the status of the CD. The content of the element must be one of the following strings. official (approved by the OpenMath Society), experimental (currently being tested), private (used by a private group of OpenMath users), or obsolete (an obsolete CD kept only for archival purposes). An optional element. If it is used it contains a string representing the URL where the canonical reference copy of this CD is stored. This symbol represents the element of a CDGroup which describes each CDURL element. It has one string argument, this should be the string corresponding to the contents of the CDURL element for each Content Dictionary in the CDGroup. The element is optional, in case it is missing, the location of the CDGroup identified by the element CDGroupURL is assumed. An element which contains zero or more CDNames which correspond to the CDs that this CD depends on, i.e. uses in examples and FMPs. If the CD is dependent on any other CDs they must be present here. An element which contains a version number for the CD. This should be a non negative integer. Any change to the CD that affects existing OpenMath applications that support this CD should result in an increase in the version number. This symbol represents the element of a CDGroup which describes each CDVersion element. It has one integral argument, this should specify which version of the content dictionary is to be taken as member of the CDGroup. The element is optional. In case it is missing, the last version is the one included in the CDGroup. The round up (to +infinity) operation. This is the constructor for a character table. Usage: CharacterTable(centralizer_primes, centralizer_indices, classnames, power_map, irreducibles_matrix) If G has n conjugacy classes then: * centralizer_primes is of the form [p1, .., pk] i < j implies that pi < pj and the pi are precisely the primes which divide the order of some centralizer of a conjugacy class * centralizer_indices is of the form [[i11, ...,i1k] ... [in1,...ink]] so the centralizer of class 1 has order p1^i11 ... pk^i1k etc * classnames is a list of n strings which name the conjugacy classes in line with the convention used in the Atlas of Finite Groups * power_map is of the form [list1, ..., listk] where listi[j] is the name of the class where elements of class j go when raised to the power pi. * irreducibles_matrix: rows correspond to irreducible characters, columns are conjugacy classes. Entries are the value of an element of the column's conjugacy class under the character of the row. Refers to the character table of its argument which must be a group. This symbol represents the polynomial which appears in the left hand side of the characteristic equation of a matrix. It takes one argument which should be the matrix. A definition of the characteristic equation is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 2 of chapter 6, page 535. This symbol represents the charge physical dimension. An optional element (which may be repeated many times) which contains a string corresponding to a property of the symbol being defined. The coefficient with respect to a list of variables (the second argument) raised to a list of powers (the third argument). Zero if no such term is present. Not all variables need be specified. This symbol represents the function which takes one matrix argument and returns the number of columns in that matrix. This symbol is used to claim, with proof (the third child), that a statement (the first child) is a deduction of the classical predicate calculus, i.e. that it follows by applications of Modus Ponens, forall-introduction and exists-elimination, from instantiations of the axioms (which may be the common three involving applications of Modus Ponens, and generalisation from instantiations of the Axioms (which may be the common three involving 'implies', together with forall-instantiation and moving forall inside implication, but need not be), and the hypotheses (elements of the set which is the second child). This symbol is used to state, with justification, that a statement is a theorem of the classical first-order predicate calculus, i.e. that it follows by applications of Modus Ponens, and generalisation from instantiations of the Axioms (which may be the common three involving 'implies', together with forall-instantiation and moving forall inside implication, but need not be), and the hypotheses (elements of the set which is the second child). This symbol is used to claim, with proof (the third child), that a statement (the first child) is a deduction of the classical propositional calculus, i.e. that it follows by applications of Modus Ponens from instantiations of the axioms (which may be the common three involving 'implies', but need not be), and the hypotheses (elements of the set which is the second child). This symbol is used to state, with proof (the second child), that a statement (the first child) is a theorem of the classical propositional calculus, i.e. that it follows by applications of Modus Ponens from instantiations of the axioms (which may be the common three involving 'implies', but need not be). This attribute, attached to a groebnered object, says 'true' if the base is fully reduced, i.e. no monomial is divisible by the leading monomial of any other polynomial. This symbol represents a constructor function for complex numbers specified as the Cartesian coordinates of the relevant point on the complex plane. It takes two arguments, the first is a number x to denote the real part and the second a number y to denote the imaginary part of the complex number x + i y. (Where i is the square root of -1.) A symbol to be used as the argument of the type symbol to convey the type of a complex number specified in terms of its real and imaginary parts. This symbol represents a constructor function for complex numbers specified as the polar coordinates of the relevant point on the complex plane. It takes two arguments, the first is a nonnegative number r to denote the magnitude and the second a number theta (given in radians) to denote the argument of the complex number r e^(i theta). (i and e are defined as in this CD). A symbol to be used as the argument of the type symbol to convey the type of a complex number specified in terms of its modulus and argument. This symbol represents the concentration physical dimension, it is the amount of a substance in a volume. The symbol represents a configuration in Euclidean planar geometry consisting of a sequence of geometric objects like points, lines, etc, but also of other configurations. The symbol represents a configuration in Euclidean planar geometry consisting of a sequence of geometric objects like points, lines, etc, but also of other configurations. The binary function whose value is the set of elements which are conjugate to the second argument in the first. A unary operator representing the complex conjugate of its argument. This symbol represents the cons list function. It takes 2 arguments: the second must be a list, where the elements have the same type as the type of the first. The function denotes a new list which has the first argument as its first element followed by the elements of the second argument. This constructor takes one argument, which is a value from the coefficient ring. It is intended to represent a constant node. This symbol represents a matrix which has all entries of the same value. It takes two arguments, the first is the size of the matrix, the second is the constant which determines every element. A symbol to be used as the argument of the type symbol to convey a type for the common constants, pi ~= 3.1415, e ~= 2.718, i = square root of -1, gamma ~= .5772, NaN, infinity (all in the nums cd), true and false (in the logic cd). Also for MathML variables declared to have type constant, as in <ci type="constant">x</ci>. This symbol represents the type of continuous sets. Conversion between polynomial rings. The first argument is a polynomial and the second is a polynomial ring. This represents the conversion of the given polynomial as an element of the given ring. A program that can compute the conversion is required to return a polynomial in the given ring. This symbol represents the cos function as described in Abramowitz and Stegun, section 4.3. It takes one argument. This symbol represents the cosh function as described in Abramowitz and Stegun, section 4.5. It takes one argument. This symbol represents the cot function as described in Abramowitz and Stegun, section 4.3. It takes one argument. This symbol represents the coth function as described in Abramowitz and Stegun, section 4.5. It takes one argument. This symbol represents the measure of one Coulomb. This is the standard metric measure for charge. This symbol represents the csc function as described in Abramowitz and Stegun, section 4.3. It takes one argument. This symbol represents the csch function as described in Abramowitz and Stegun, section 4.5. It takes one argument. This symbol is used to represent the curl function. It takes one argument which should be a vector of scalar valued functions, intended to represent a vector valued function and returns a vector of functions. It should satisfy the defining relation: curl(F) = i X \partial(F)/\partial(x) + j X \partial(F)/\partial(y) + j X \partial(F)/\partial(Z) where i,j,k are the unit vectors corresponding to the x,y,z axes respectively and the multiplication X is cross multiplication. This symbol represents the current physical dimension. This symbol is an n-ary function, with n at least 1. It marks a relation on the set of its arguments a_1, a_2,...,a_n consisting of the pairs (a_i,a_{i+1}) for i=1,...,n-1 and the pair (a_n,a_1). The arguments a_i should all be distinct. The number n is referred to as the length of the cycle. This symbol is a function with one argument, which is a permutation. When applied to a permutation P, it represents the multiset of lengths of cycles occurring as arguments of P. This symbol is a constructor for groups. It takes four arguments in the following order; a set to specify the elements in the group, a binary operation to specify the group operation, a unary operation to specify inverses of group elements and an element to specify the identity. Both the binary and unary operations should act on elements of the set and return an element of the set. This symbol is used to represent definite (Cauchy principal value) integration of unary functions. It takes two arguments; the first being the range (e.g. a set) of integration, and the second the function. This symbol is used to describe the variable with respect to which an integral is calculated. This symbol represents the integrand of the integral. This symbol is used to represent definite integration of unary functions. It takes two arguments; the first being the range (e.g. a set) of integration, and the second the function. The total degree of its argument. The value returned is a non-negative integer. We note that the degree of 0 is undefined. Note that this operation takes no account of any weights that have been defined: see weighted_degree in polyd. This symbol represents the measure of one degree Celsius. This is a standard metric measure for temperature. This symbol represents the measure of one degree Fahrenheit. This is the standard imperial measure for temperature. This symbol represents the measure of one degree Kelvin. This is a standard metric measure for temperature relative to absolute zero. The degree with respect to a variable (the second argument). We note that the degree of 0 is undefined. This symbol represents the density physical dimension, it is the mass per unit volume. A unary function taking an slp as argument and returning the greatest depth of any leaf node, that is the length of the longest contiguous path to any leaf node. The unary function whose value is the subgroup of argument generated by all products of the form xyx^-1y^-1. An element which contains a string corresponding to the description of either the CD or the symbol (depending on which is the enclosing element). This symbol denotes the unary function which returns the determinant of its argument, the argument should be a square matrix. This symbol denotes an n_ary function which is used to construct an (nxn) diagonal matrix, that is a matrix where every non-diagonal element is zero, the diagonal elements are equal to the n arguments. This symbol is used to express ordinary differentiation of a unary function. The single argument is the unary function. constructor of a differential operator from a polynomial or from an element of the finitely generated free algebra. This symbol represents the type of discrete sets. Function taking two arguments, it represents the discriminant of a polynomial, which is the first argument, with respect to the given variable which is the second argument. This symbol is used to represent the divergence function. It takes one argument which should be a vector of scalar valued functions, intended to represent a vector valued function and returns a scalar value. It should satisfy the defining relation: divergence(F) = \partial(F_(x_1))/\partial(x_1) + ... + \partial(F_(x_n))/\partial(x_n) This symbol represents a (binary) division function denoting the first argument right-divided by the second, i.e. divide(a,b)=a*inverse(b). It is the inverse of the multiplication function defined by the symbol times in this CD. A constant value, constructs the divide for division nodes. The constructor of DMPs. The first argument is the polynomial ring containing the polynomial and the second is a "SDMP". Should be of the form DMP(PolyRingD(...), SDMP(...)) The constructor for lists of multivariate polynomial members of the same polynomial ring. The first argument is a polynomial ring and the rest are "SDMP"s. DMPL can be attributed with the "ordering" symbol to indicate a particular ordering for monomials of all its polynomials. Should be of the form DMPL(PolyRingD(...), SDMP(...)+) These objects have categories as there types and specific implementations of their functions. *** details to be worked out *** *** for now *** The first argument is a Category, the remaining arguments are the functions (e.g. lambda bindings or unapplied functions). This symbol denotes the domain of a given function, which is the set of values it is defined over. The domainofapplication element denotes the domain over which a given function is being applied. It is intended in MathML to be a more general alternative to specification of this domain using such quantifier elements as bvar, lowlimit or condition. This symbol represents the base of the natural logarithm, approximately 2.718. See Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1. The symbol E defines the generalised exponential integral as in Abramovitz & Stegun equation 5.1.4. This is an ordinary integral: $$E_n(z)=\int_1^{-\infty}\frac{e^{-zt}}{t^n} dt\qquad(\Re z>0)$$ which is then extended by analytic continuation (this latter is not currently represented in the FMPs) to the complex plane slit along the negative real axis. Note that OpenMath's definition is curried, i.e. E(n) is a function. The symbol Ei defines the basic exponential integral as in Abramovitz & Stegun equation 5.1.2. This is a Cauchy principal value integral: $$Ei(x)=\int_{-\infty}^x\frac{e^t}t dt\qquad(x>0)$$ which is then extended by analytic continuation (this latter is not currently represented in the FMPs) to the complex plane slit along the negative real axis This symbol represents the eigenvalue of a matrix. It takes two arguments the first should be the matrix, the second should be an index to specify the eigenvalue. The ordering imposed on the eigenvalues is first on the modulus of the value, and second on the argument of the value. A definition of eigenvalue is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1 of chapter 6, page 533. This symbol represents the eigenvector of a matrix. It takes two arguments the first should be the matrix, the second should be an index to specify which eigenvalue this eigenvector should be paired with. The ordering is as given in the eigenvalue symbol. A definition of eigenvector is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1 of chapter 6, page 533. The unary function which returns the set of elements of a group. This is an ordering, which is partially in terms of one ordering, and partially in terms of another. First argument is a number of variables. Second is ordering to apply on the first so many variables. Third is an ordering on the rest, to be used to break ties. This symbol is used to represent the empty multiset, that is the multiset which contains no members. It takes no parameters. This symbol is used to represent the empty set, that is the set which contains no members. It takes no parameters. This symbol represents the error which is returned when an application detects a lexical or syntactic error. It should have one argument which is a string, which should explain the error that occurred. This symbol represents the energy physical dimension. This symbol represents the binary equality function. Proposition; the type of equivalence relations, namely relations that are reflexive, symmetric and transitive. This symbol is used to show that two boolean expressions are logically equivalent, that is have the same boolean value for any inputs. The error symbol is the 'return type' of error symbols in the error signature file. This Symbol represents the generic category of Euclidean domain. This symbol is the constructor for Euclidean domains. A Euclidean domain is a ring on which there is no zero divisors together with an integer norm function. The Euclidean_domain constructor takes six arguments: The set of the Euclidean domain. A binary function into itself to represent the multiplication operation, *. A binary function into itself to represent the addition operation, +. A member of the set of the Euclidean domain to specify the additive identity, 0. A unary function taking the set of the Euclidean domain into itself to represent the additive inverses (i.e. inverses under +, or negatives). And a unary function taking elements of the set into the positive integers, to represent the integer norm function. This symbol takes one argument which should be a Euclidean domain. It returns a unary function, which is the absolute value function of the Euclidean domain. This symbol takes one argument which should be a Euclidean domain. It returns a unary function, which represents additive inverses of the Euclidean domain. This symbol takes one argument which should be a Euclidean domain. It returns a binary function, which represents the additive operation of the Euclidean domain. This symbol takes one argument which should be a Euclidean domain. It returns the set of the Euclidean domain. This symbol takes one argument which should be a Euclidean domain. It returns a binary function, which represents the multiplicative operation of the Euclidean domain. This symbol takes one argument which should be a Euclidean domain. It returns the additive identity of the Euclidean domain. An element which contains an arbitrary number of children, each of which is either a string or an OpenMath Object. These children give examples in natural language, or in OpenMath, of the enclosing symbol definition. This symbol represents the existential ("there exists") quantifier which takes two arguments. It must be placed within an OMBIND element. The first argument is the bound variables (placed within an OMBVAR element), and the second is an expression. This symbol represents the exponentiation function as described in Abramowitz and Stegun, section 4.2. It takes one argument. Converts a factored or squarefreed form into the expanded polynomial over the same ring, so that factored(recursive) becomes recursive, etc. The decomposition of its argument into irreducible factors. A program that can compute the factorization is required to return a "factored" object - see above. It is currently an open question whether powers of 1 can be omitted. The constructor for a factorization. Its arguments are formal powers (see previous operator), where the polynomials are supposed to be irreducible (except possibly for a content from the ground ring). Note that "factored" is not a call to factorise something, rather a statement that we know a factorisation. The symbol to represent a unary factorial function on non-negative integers. This is the binary OpenMath operator that is used to indicate the mathematical relationship a "is a factor of" b, where a is the first argument and b is the second. This relationship is true if and only if b mod a = 0. falling_multi_factorial is a product of falling pochhammer symbols. 2-ary function. reference: authors: "Saito, Sturmfels, Takayama" title: "Grobner Deformations of Hypergeometric Differential Equations" pages: 127 This symbol represents the boolean value false. This symbol represents the electric charge carried by one mole of electrons. it is approximately 96485.309 +/- 0.029 Coulombs per mole. The Fibonacci numbers, defined by the linear recurrence: Fibonacci(0) = 0, Fibonacci(1) = 1, and Fibonacci(n + 1) = Fibonacci(n) + Fibonacci(n - 1). Note that some authors define Fibonacci(0) = 1. This Symbol represents the generic category of field. This symbol is the constructor for fields. A field is an Abelian group under +, the set of the field complement {0} with * is an Abelian group, a field has a further rule which associates the two operations, that is left and right distributivity. The field constructor takes seven arguments: The set of the field. A binary function into itself to represent the multiplication operation, *. A binary function into itself to represent the addition operation, +. A member of the set of the field to specify the multiplicative identity, 1. A member of the set of the field to specify the additive identity, 0. A unary function taking the set of the field into itself to represent the multiplicative inverses (i.e. inverses under *). A unary function taking the set of the field into itself to represent the additive inverses (i.e. inverses under +, or negatives). This symbol takes one argument which should be a field. It returns a unary function, which is the additive inverse function of the field. This symbol takes one argument which should be a field. It returns the multiplicative identity of the field. This symbol takes one argument which should be a field. It returns a binary function, to represent the additive operation of the field. This symbol takes one argument which should be a field. It returns a unary function, which is the multiplicative inverse function of the field. This symbol takes one argument which should be a field. It returns the set of the field. This symbol takes one argument which should be a field. It returns a binary function, to represent the multiplicative operation of the field. This symbol takes one argument which should be a field. It returns the additive identity of the field. This symbol represents the type of finite sets. This symbol represents a function which returns the first elements of its argument, which should be a list. The type of floating point numbers The round down (to -infinity) operation. An optional element which contains an OpenMath Object. This corresponds to a property of the symbol being defined. A symbol to be used as the argument of the type symbol to convey the type for a function name. This symbol represents the universal ("for all") quantifier which takes two arguments. It must be placed within an OMBIND element. The first argument is the bound variables (placed within an OMBVAR element), and the second is an expression. This symbol represents the force physical dimension. Euler's gamma function A symbol to convey the notion of the gamma constant as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 6.1.3. It is the limit of 1 + 1/2 + 1/3 + ... + 1/m - ln m as m tends to infinity, this is approximately 0.5772 15664. This symbol represents the constant which is equal to the ratio of the pressure times the volume and the temperature of an ideal gas. It is approximately 8.31451 +/- 7.0*10^(-05) Joules per mole per Kelvin. The symbol to represent the n-ary function to return the gcd (greatest common divisor) of its arguments. The n-ary greatest common divisor of its polynomial arguments. This is unique up to units. This symbol represents the generation of a line of a proof by application of Generalisation. The first argument is the new well-formed formula (forall x.B) and the second is the line number in the proof for B. This symbol represents the binary greater than or equal to function which returns true if the first argument is greater than or equal to the second, it returns false otherwise. This symbol represents the finite field of integers modulo p, where p is a prime. This symbol represents the finite field with p^n elements, where p is a prime. the symbol polynomial of a given differential operator. This symbol is used to represent the grad function. It takes one argument which should be a scalar valued function and returns a vector of functions. It should satisfy the defining relation: grad(F) = (\partial(F)/\partial(x_1), ... ,\partial(F)/partial(x_n)) Total degree order, graded with the lexicographic ordering. Note that, if a poly_ring_d_named is used, lexigographic refers to the order of the variables in the poly_ring_d_named, not to their order as strings. Total degree order, graded with the reverse lexicographic ordering. Note that, if a poly_ring_d_named is used, lexigographic refers to the order of the variables in the poly_ring_d_named, not to their order as strings. This symbol represents the measure of one gramme. This is the standard metric measure for mass. This symbol represents the constant of proportionality in Newtons law of universal gravitation which states; Two bodies attract each other with equal and opposite forces; the magnitude of this force is proportional to the product of the two masses and is also proportional to the inverse square of the distance between the centers of mass of the two bodies. It is approximately equal to: 6.672*10^(-11) Newton square metres per kilogramme squared. The groebner basis (lt-reduced, minimal) of a set of polynomials, with respect to a given ordering. First argument is an ordering, the second is a list of polynomials. A program that can compute the basis is required to return a "groebnered" object. The constructor for a Groebner basis (reduced, minimal). The first argument is an ordering, the second is the Groebner Basis itself (with respect to the ordering) that should be represented as a DMPL. This Symbol represents the generic category of group. This symbol is the constructor for groups, that is a monoid for which every element is invertible. The group constructor takes four arguments, the set of the group, a binary function taking two elements of the set into itself to represent the operation of the group, an element of the set to represent the identity of the group and a unary function taking the set into itself to specify inverse elements of the group. The n-ary function Group. The group generated by its arguments. The arguments must have a natural group operation associated with them. This symbol takes one argument which should be a group. It returns the identity of the group. This symbol takes one argument which should be a group. It returns a unary function, which is the inverse mapping for the group. This symbol takes one argument which should be a group. It returns a binary function, which represents the operation of the group. This symbol takes one argument which should be a group. It returns a set, which should be the set of the group. This Symbol represents the generic category of groupoid. This symbol is the constructor for groupoids, that is an algebraic structure on a set, with a binary operation. The operator of the groupoid must be closed over the set of the groupoid. The groupoid constructor takes two arguments, the set of the groupoid and a binary function which represents the operation of the groupoid. This symbol takes one argument which should be a groupoid. It returns a binary function which should represent the operation of the groupoid. This symbol takes one argument which should be a groupoid. It returns the set of the groupoid. This symbol represents the binary greater than function which returns true if the first argument is greater than the second, it returns false otherwise. This symbol represents the set of quaternions. The first Hankel function. This function is one of the famous two solutions of the Bessel differential equation at z=\infty. The second Hankel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=\infty. This symbol represents the notion of category inclusion. It takes two arguments, which should both be categories. It implies that axioms of the second argument apply to the first, and that function signatures in the second category are also in the first. This symbol represents a Hermitian matrix, it takes one argument. The argument should be a vector of vectors of values which determine the upper triangle of the matrix. The lower triangle of the matrix is specified by the following relation: M^* = transpose(M), were M^* denotes the matrix consisting of all the complex conjugates of M. Generalized hypergeometric function. This function has a branch cut on [1,+infinity). Hypergeometric function {}_0 F_1. Kummer's confluent hypergeometric function. The Gauss hypergeometric function. This function has a branch cut on [1,+infinity). This symbol represents that a wellformed formula is a hypothesis of a deduction of the propositional or predicate calculus. This symbol represents the square root of -1. The identity function, it takes one argument and returns the same value. This symbol denotes a unary function which is used to construct an (nxn) identity matrix where n is the single positive integral argument. This symbol denotes the image of a given function, which is the set of values the domain of the given function maps to. This represents the imaginary part of a complex number This symbol represents the logical implies function which takes two boolean expressions as arguments. It evaluates to false if the first argument is true and the second argument is false, otherwise it evaluates to true. This symbol has two arguments, an element and a multiset. It is used to denote that the element is in the given multiset. This symbol has two arguments, an element and a set. It is used to denote that the element is in the given set. The symbol represents the logical incidence function which is a binary function taking arguments representing geometric objects like points and lines and returning a boolean value. It is true if and only if the first argument is incident to the second. The symbol represents the logical incidence function which is a binary function taking arguments representing geometric objects like points and lines and returning a boolean value. It is true if and only if the first argument is incident to the second. index returns the index of a given indexed variable. indexed_variable(x,i) returns the variable x_i Attribution tag to denote the type of inductively defined natural numbers. It is also denoted as setname1:N. Constructor for Inductive Types. Takes arguments the constructor functions for the inhabitants of the type and their signatures. This symbol represents the type of infinite sets. A symbol to represent the notion of infinity. This constructor takes one argument, which is a variable. The return value is intended to represent an input node. This symbol is used to represent indefinite integration of unary functions. The argument is the unary function. The function that converts an integer to a float. The type of integers A symbol to denote a discrete 1 dimensional interval from the first argument to the second (inclusive), where the discretisation occurs at unit intervals. The arguments are the start and the end points of the interval in that order. A symbol to be used as the argument of the type symbol to convey the type of an integer. This symbol represents the type of integer intervals. This Symbol represents the generic category of integral domain. This symbol is the constructor for integral domains. An integral domain is a ring which is commutative under *, it has a multiplicative identity (under *), and has no zero divisors. The integral_domain constructor takes six arguments. The set of the integral domain, a binary function from the set into itself to represent the * operation, a binary function from the set into itself to represent the + operation, an element of the set of the ring to represent the multiplicative identity 1, an element of the set of the ring to represent the additive identity 0, and a unary function from the set into itself to represent additive inverses (i.e. inverses under +, or negatives). This symbol takes one argument which should be an integral domain. It returns a unary function which represents the additive inverse function of the integral domain. This symbol takes one argument which should be an integral domain. It returns the multiplicative identity of the integral domain. This symbol takes one argument which should be an integral domain. It returns a binary function which represents the additive operation of the integral domain. This symbol takes one argument which should be an integral domain. It returns the set of the integral domain. This symbol takes one argument which should be an integral domain. It returns a binary function which represents the multiplicative operation of the integral domain.