# OpenMath Symbols

A combined list of all 1578 symbols defined in this Content Dictionary collection.

Symbol Description
setname2/
A
This symbol represents the set of algebraic numbers.
hypergeon1/
a_hypergeomeric
A-hypergeometric series reference: authors: "Saito, Sturmfels, Takayama" title: "Grobner Deformations of Hypergeometric Differential Equations" pages: 127
algebraic_cats/
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.
generic_alg_cats/
Abelian_group
This Symbol represents the generic category of Abelian group.
algebraic_cats/
Abelian_group_identity
This symbol takes one argument which should be an Abelian group. It returns the identity of the Abelian group.
algebraic_cats/
Abelian_group_inverse
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.
algebraic_cats/
Abelian_group_operation
This symbol takes one argument which should be an Abelian group. It returns a binary function, which represents the operation of the Abelian group.
algebraic_cats/
Abelian_group_set
This symbol takes one argument which should be an Abelian group. It returns the set of the Abelian group.
algebraic_cats/
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.
generic_alg_cats/
Abelian_monoid
This Symbol represents the generic category of Abelian monoid.
algebraic_cats/
Abelian_monoid_identity
This symbol takes one argument which should be an Abelian monoid, it returns the identity of the Abelian monoid.
algebraic_cats/
Abelian_monoid_operation
This symbol takes one argument which should be an Abelian monoid, it returns the operation of the Abelian monoid.
algebraic_cats/
Abelian_monoid_set
This symbol takes one argument which should be an Abelian monoid, it returns the set of the Abelian monoid.
algebraic_cats/
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.
generic_alg_cats/
Abelian_semigroup
This Symbol represents the generic category of Abelian semigroup.
algebraic_cats/
Abelian_semigroup_operation
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.
algebraic_cats/
Abelian_semigroup_set
This symbol takes one argument which should be an Abelian semigroup. It returns a set, which should be the set of the Abelian semigroup.
limit1/
above
This symbol is used within a limit construct to show the limit is being approached from above. It takes no arguments.
arith1/
abs
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.
numerical1/
absolute_error_bound
This symbol marks an estimated upper bound for the absolute error ( |true-computed| ) on a computation.
numerical1/
absolute_error_requested
This symbol marks a requirement for the absolute error ( |true-computed| ) on a computation.
physical_consts1/
absolute_zero
This symbol represents the absolute zero of temperature, synonymous with the object of that temperature having zero latent heat.
SI_DerivedQuantities1/
absorbed-dose
This symbol represents the physical quantity of absorbed dose of ionizing radiation. A variable representing an arbitrary quantity of absorbed dose is commonly represented with the italic, upper case letter, "D".
dimensions1/
acceleration
This symbol represents the acceleration physical dimension. It is the second derivative of distance with respect to time.
units_imperial1/
acre
This symbol represents the measure of one acre. This is a standard imperial measure for area.
units_us1/
acre_us_survey
This symbol represents the measure of one U.S. Survey acre.
weylalgebra1/
act
action of a differential operator to a function.
weylalgebra1/
act_of_poly
action of a polynomial as a differential operator to a function. act_of_poly is equivalent to the composition of act and diffop.
permutation1/
action
This symbols is a binary function whose first argument is a permutation (or a endomap) and whose second argument is a point. When applied to permutation or endomap p and point x, it represents the image of the point x under the permutation p.
field1/
This symbols represents a unary function, whose argument should be a field. It returns the addition map on the field. We allow for the map to be n-ary.
ring1/
This symbols represents a unary function, whose argument should be a ring. It returns the addition on the ring. We will allow for the map to be n-ary.
field1/
This symbol is a unary function, whose argument should be a field S. When applied to S its value is the monoid underlying S.
ring1/
This symbol is a unary function, whose argument should be a ring S. When applied to S its value is the monoid underlying S.
plangeo4/
affine_coordinates
This function yields the affine coordinates vector if applied to a point or line with coordinates in the affine plane.
aggregate_cats/
aggregateType
This symbol represents a generic type for aggregates (or collections of objects.
airy/
Ai
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: $$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$
airy/
Ai2
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).
hypergeo2/
airyAi
The first Airy function. This function is one of the famous two solutions of the Airy differential equation, and converges to 0 when z->\infty
hypergeo2/
airyBi
The second Airy function. This function is the another one of the famous two solutions of the Airy differential equation, and diverges when z->\infty
order1/
algebraic_integer
This is a binary function. The first argument is an order O. The second argument should be a list L of elements of the Dedekind ring R, such that O is an order over the polynomial ring of R (cf. order). The length of L should be equal to the degree n of the polynomial generating the order O. When applied to O and L, it represents the element L + L b + L b^2 + ... + L[n-1] b^(n-1) of O, where b stands for a primitive element of O.
order1/
algebraic_number
This is a binary function. The first argument is a number field F. The second argument should be a list L of elements of Q in case of an absolute number field F. Otherwise the second argument is a list L of elements of the number field whose ring of integers is the ring R over which F is defined (cf. number_field). The length of the list L should be equal to the degree n of F. When applied to F and L, it represents the element L + L b + L b^2 + ... + L[n-1] ^(b-1) of F, where b stands for a primitive element of F.
interval_types/
algebraicIntervalType
This symbol represents the type of algebraic intervals.
moreerrors/
algorithm
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.
mathmlkeys/
alternate-representation
This key specifies that the corresponding value is an alternate representation of the annotated element in some unspecified way.
group3/
alternating_group
This symbol is a function with one argument, which should be a set X. When applied to a set X it represents the group of all even permutations on X .
permgp2/
alternating_group
This symbol represents a unary function. Its argument is either a positive integer or a set. When evaluated on a set, it represents the permutation group of all even permutations of that set. When evaluated on a positive integer n, it represents the permutation group of all even permutations of the set {1,..., n}.
group3/
alternatingn
This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the group of all even permutations on the set {1,2, ...,n}.
plangeo3/
altitude
Given a point p and a line L, this defines the segment starting at p and ending in the unique point of L closest to p.
polyd1/
ambient_ring
This is a unary function, whose argument should be a DMP f. When applied to f, it represents the first argument of f, that is, ring of the form poly_ring_d(...) used to define f.
SI_BaseQuantities/
amount-of-substance
This symbol represents the SI base quantity of amount of substance. It has the short symbol form, "N".
units_metric1/
amp
This symbol represents the measure of one amp. This is the standard SI measure for current.
SI_BaseUnits1/
ampere
This symbol represents the measure of one ampere, the standard SI unit of measure for quantities of electric current. It has the short symbol form, "A", in upright roman font.
logic1/
and
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.
plangeo3/
angle
Angle of a corner, always measured in positive (anti-clockwise) direction.
SI_DerivedQuantities1/
angle
This symbol represents the quantity of a geometric planar angle. A variable representing an arbitrary quantity of angle is commonly represented with the italic, lower case greek variable, e.g., "\theta;".
polyd/
anonymous
Indicates a variable that we do not want to name
polyd1/
anonymous
Indicates a variable that we do not want to name
linalg5/
anti-Hermitian
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.
linalgsym1/
anti_Hermitian
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. These rules imply that the main diagonal is zero, therefore the argument should not include it.
relation0/
antisymmetric
Proposition; the type of antisymmetric binary relations.
hypergeon2/
appel_F1
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
hypergeon2/
appel_F2
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
hypergeon2/
appel_F3
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
hypergeon2/
appel_F4
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
list2/
append
The operation of joining one list to another
list3/
append
This symbol represents a function with two arguments, the first of which should be a list. When applied to A and b, it represents the list obtained from A by appending the element b to it.
fns2/
apply_to_list
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.
relation1/
approx
This symbol is used to denote the approximate equality of its two arguments.
relation2/
approx
This symbol is used to denote the approximate equality of its two arguments.
plangeo3/
arc
an arc of a circle M from A to B is the set of points of M that are encountered when traversing the circle clockwise from A to B.
transc1/
arccos
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.
transc3/
arccos
This symbol represents the arccos function. This is the multivalued inverse of the cos function.
transc1/
arccosh
This symbol represents the arccosh function as described in Abramowitz and Stegun, section 4.6.
transc3/
arccosh
This symbol represents the Arccosh function as described in Abramowitz and Stegun, section 4.6.
transc1/
arccot
This symbol represents the arccot function as described in Abramowitz and Stegun, section 4.4.
transc3/
arccot
This symbol represents the multi-valued arccot function as the inverse of cot
transc1/
arccoth
This symbol represents the arccoth function as described in Abramowitz and Stegun, section 4.6.
transc3/
arccoth
This symbol represents the Arccoth function as described in Abramowitz and Stegun, section 4.6.
transc1/
arccsc
This symbol represents the arccsc function as described in Abramowitz and Stegun, section 4.4.
transc3/
arccsc
This symbol represents the multivalued arccsc function as the inverse of csc.
transc1/
arccsch
This symbol represents the arccsch function as described in Abramowitz and Stegun, section 4.6.
transc3/
arccsch
This symbol represents the Arccsch function as described in Abramowitz and Stegun, section 4.6.
transc1/
arcsec
This symbol represents the arcsec function as described in Abramowitz and Stegun, section 4.4.
transc3/
arcsec
This symbol represents the multivalued arcsec function as the inverse of sec.
transc1/
arcsech
This symbol represents the arcsech function as described in Abramowitz and Stegun, section 4.6.
transc3/
arcsech
This symbol represents the Arcsech function as described in Abramowitz and Stegun, section 4.6.
transc1/
arcsin
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.
transc3/
arcsin
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.
transc1/
arcsinh
This symbol represents the arcsinh function as described in Abramowitz and Stegun, section 4.6.
transc3/
arcsinh
This symbol represents the Arcsinh function as described in Abramowitz and Stegun, section 4.6.
transc1/
arctan
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.
transc2/
arctan
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 $\pi/2$.
transc3/
arctan
This symbol represents the arctan function. This is the multi-valued inverse of the tan function.
transc1/
arctanh
This symbol represents the arctanh function as described in Abramowitz and Stegun, section 4.6.
transc3/
arctanh
This symbol represents the Arctanh function as described in Abramowitz and Stegun, section 4.6.
group4/
are_conjugate
This symbol represents a boolean ternary function whose first argument is a group G and whose second and third arguments are elements x and y of G. Its value on G, x, and y is true if and only if x and y are conjugate in G.
permutation1/
are_distinct
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).
permutation1/
are_distinct
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).
plangeo3/
are_on_circle
The statement that a set of points is on one circle.
plangeo6/
are_on_conic
The symbol is a boolean n-ary function. Its arguments should be points. When applied to a sequence of points, its evaluated to true if and only if there is a conic on which all arguments lie.
plangeo1/
are_on_line
The statement that a set of points is collinear.
ThreeDgeo2/
are_on_line
The symbol is a boolean n-ary function. Its arguments should be points. When applied to a sequence of points in 3-dimensional Euclidean space, its evaluated to true if and only if there is a line on which all arguments lie.
ThreeDgeo2/
are_on_plane
The symbol is a boolean n-ary function. Its arguments should be points. When applied to a sequence of points in 3-dimensional Euclidean space, its evaluated to true if and only if there is a plane on which all arguments lie.
dimensions1/
area
This symbol represents the area physical dimension.
SI_DerivedQuantities1/
area
This symbol represents the physical quantity of area.
complex1/
argument
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.
graph1/
arrowset
This symbol represents the set of arrows of a directed graph. It takes one argument, the directed graph.
plangeo1/
assertion
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.
ThreeDgeo3/
assertion
The symbol is a constructor with two arguments. Its first argument is a 3-dimensional Euclidean geometry 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.
fns4/
assign_to
This symbol denotes a function with a even and positive number of arguments. When applied to a_1,b_1,....a_n,b_n, it represents the assignment of b_i to a_i for each index i=1,...,n. It can be used to specify the behaviour of a function f, by giving the images b_i of a_i under f.
prog1/
assignment
This symbol is used to assign values to variables. The syntax is assignment(variable, value), where variable is the encoding of an OpenMath variable (OMV) and value is an OpenMath object.
semigroup/
associative
The type of associative binary operation.
SIUsed_OffSystemMeasuredUnits1/
astronomical-unit
This symbol represents the measure of one astronomical unit of distance. It has the short symbol form, "ua". It is the mean distance between the sun and the earth. Its measured value is 1 ua = 1.49597870691(6) * 10^11 m [DRAFT INTERNATIONAL STANDARD ISO/DIS 80000-1, International Organization for Standardization, 2008]
asymp1/
asymptotic
The asymptotic symbol represents a binary relation between two functions of type reals to reals. The asymptotic relation between two functions returns true if the two functions have the same rate of growth and more precisely there ratio approaches 1 as the variable approaches infinity. Formally we say that f(x) is asymptotic to g(x) if and only if the limit as x tends to infinity of f(x)/g(x) = 1.
moreerrors/
asynchronousError
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.
units_siprefix1/
atto
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^-18$
sts/
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.
field4/
automorphism_group
This is a function with a single argument which must be a field. It refers to the automorphism group of its argument.
graph2/
automorphism_group
This symbol is a unary function whose argument is an undirected graph. When applied to an undirected graph G, it represents the automorphism group of G. The resulting automorphism group is represented as a permutation group on the vertices of the graph G.
group3/
automorphism_group
This is a function with a single argument which must be a group. It refers to the automorphism group of its argument.
magma3/
automorphism_group
This is a function with a single argument which must be a magma. It refers to the automorphism group of its argument.
monoid3/
automorphism_group
This is a function with a single argument which must be a monoid. It refers to the automorphism group of its argument.
ring5/
automorphism_group
This is a function with a single argument which must be a ring. It refers to the automorphism group of its argument.
semigroup3/
automorphism_group
This is a function with a single argument which must be a semigroup. It refers to the automorphism group of its argument.
semigroup4/
automorphism_group
This is a function with a single argument which must be a semigroup. It refers to the automorphism group of its argument.
physical_consts1/
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).
logic3/
axiom_instance
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.
linalg4mat/
banded
This symbol represents a square (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.
linalg5/
banded
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.
matrix1/
banded
This symbol is a constructor for banded matrices. It takes at least 2 arguments, the first of which being the number of upper bands and the second being the number of lower bands. Amongst the further arguments you can specify AT MOST one matrix1.diagonal object. You MUST exactly as many matrix1.upper_band objects as you specified upper bands, and you MUST specify as many matrix1.lower_band objects as you specified lower bands. This symbol facilitates the use of blas based systems which expect to know the bands structure upfront.
units_imperial1/
bar
This symbol represents the measure of one bar. This is the standard imperial measure for pressure.
permgp1/
base
This is a function with one argument, which should be a permutation group. When evaluated with argument G it returns a list of points permuted by G such that the stabilizer of all elements of the list in G is trivial. Besides, the list is minimal with respect to the latter property (in the sense that the stabilizer in G of the elements of no proper subset is trivial).
nums1/
based_float
This symbol represents the constructor function for floating point numbers, 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 number, using 0-9a-z and optionally a "." (as a consequence of this no radix greater than 36 is supported).
nums1/
based_integer
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 36 is supported). Base 16 and base 10 are already covered in the encodings of integers.
tensor1/
basis_selector
This symbol takes 2 arguments, a tuple of basis elements and a covar_index or a contra_index, and returns the basis element indicated by the index value.
SI_NamedDerivedUnits1/
becquerel
This symbol represents an SI unit of radio nuclide activity, or radioactivity. A becquerel of activity represents one nuclear decay event per second. It has the short symbol form, "Bq".
SIUsed_OffSystemUnits1/
bel
This symbol represents the dimensionless measure of one bel. It has the short symbol form, "B".
combinat1/
Bell
The Bell numbers: Bell(n) is the total number of possible partitions of a set of n elements.
limit1/
below
This symbol is used within a limit construct to show the limit is being approached from below. It takes no arguments.
hypergeo2/
besselJ
The Bessel function. This function is one of the famous two solutions of the Bessel differential equation at z=0.
hypergeo2/
besselY
The Bessel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=0.
hypergeo0/
beta
Euler's beta function
airy/
Bi
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: $$g(z)=\sum_{k=0}^\infty 3^k{\left (\frac{2}{3}\right )}_k \frac{z^{3k+1}}{(3k+1)!}$$
airy/
Bi2
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).
set3/
big_intersect
This symbol is a unary function whose argument should be a collection C of subsets of a given set. When applied to C, it represents the intersection over all members of C.
set3/
big_union
This symbol is a unary function whose argument should be a collection C of subsets of a given set. When applied to C, it represents the union over all members of C.
bigfloat1/
bigfloat
The bigfloat constructor takes three arguments, a mantissa, a base and the exponent.
bigfloat1/
bigfloatprec
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.
sts/
binder
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.
combinat1/
binomial
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.
matrix1/
block
This symbol is like the matrix constructor as described above, but intended for use inside matrix to form submatrices''. The symbol takes at least two arguments: a column_dimension and a row_dimension object which denote the total extent of the block.
prog1/
block
This symbol is meant to represent an arbitray block of code. A block of code can be empty. The syntax is block(obj1, obj2,...,objN), where obji is the OpenMath encoding of the ith sentence (or action) inside the body.
FundamentalPhysicalConstants1/
Boltzmann-constant
The Boltzmann constant relates energy at the particle level with temperature observed at the bulk level via the ideal gas law, pV = NkT. By measurement it is found to be approximately equal to 1.3806504(24)*10^(-23) joule per kelvin. It is commonly represented with the short, italic symbol, "k".
physical_consts1/
Boltzmann_constant
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.
setname2/
Boolean
This symbol represents the set of Booleans. That is the truth values, true and false.
limit1/
both_sides
This symbol is used within a limit construct to show the limit is being approached from both sides. It takes no arguments.
omtypes/
bytearray
The type of byte arrays
fieldname1/
C
This is a symbol representing the field of complex numbers.
setname1/
C
This symbol represents the set of complex numbers.
units_time1/
calendar_month
This symbol represents the measure of one month of (calendar) time.
units_time1/
calendar_year
This symbol represents the measure of one year of (calendar) time.
prog1/
call_arguments
This symbol can be used to encode the arguments that will be passed to a function or procedure.
scscp1/
call_id
Uniquely identifies a procedure call. Used in subsequent communication, so the parties know which call they are talking about.
SI_BaseUnits1/
candela
This symbol represents the measure of one candela, the standard SI unit measure for quantities of luminous intensity. It has the short symbol form, "cd", in upright roman font.
SI_DerivedQuantities1/
capacitance
This symbol represents the physical quantity of electric capacitance. A variable representing an arbitrary quantity of capacitance is commonly represented with the italic, upper case letter, "C".
field1/
carrier
This symbol represents a unary function, whose argument should be a field S (for instance constructed by field). When applied to S, its value should be the set of elements of S.
group1/
carrier
This symbol represents a unary function, whose argument should be a group G (for instance constructed by group). When applied to G, its value should be the set of elements of G.
magma1/
carrier
This symbol represents a unary function, whose argument should be a magma G (for instance constructed by magma). When applied to G, its value should be the set of elements of a magma.
monoid1/
carrier
This symbol represents a unary function, whose argument should be a monoid M (for instance constructed by monoid). When applied to M, its value should be the set of elements of a monoid.
ring1/
carrier
This symbol represents a unary function, whose argument should be a ring S (for instance constructed by ring). When applied to S, its value should be the set of elements of S.
semigroup1/
carrier
This symbol represents a unary function, whose argument should be a semigroup S (for instance constructed by semigroup). When applied to S, its value should be the set of elements of S.
tensor1/
Cartesian
This symbol takes one argument, a natural number, and returns the Cartesian coordinate, of a right handed Cartesian coordinate frame, corresponding to the value of the argument. These coordinates are commonly named X, Y, and Z in three dimensions, though X, Y, and Z are non-exclusively used for this and other purposes.
set3/
cartesian_power
This symbol is a binary function whose first argument should be a set A and whose second argument should be a natural number k. When applied to A and k, it represents the Cartesian product of k copies of A.
multiset1/
cartesian_product
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.
set1/
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.
hypergeon0/
cartesian_product_n
the cartesian product of n copies of the first argument. Binary function.
set1p/
cartesian_product_n
the cartesian product of n copies of the first argument. Binary function.
SI_DerivedQuantities1/
catalytic-activity
This symbol represents the physical quantity of catalytic activity, an amount of catalyst that effects a rate of catalytic conversion of an amount of substance.
meta_cats/
Category
This symbol is intended to denote the type of a category.
meta/
CD
The top level element for the Content Dictionary. It just acts as a container for the elements described below.
meta/
CDBase
An optional element. If it is used it contains a string representing the URI to be used as the base for generated canonical URI references for symbols in the CD.
meta/
CDComment
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.
metagrp/
CDComment
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.
meta/
CDDate
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.
meta/
CDDefinition
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.
metagrp/
CDGroup
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.
metagrp/
CDGroupDescription
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.
metagrp/
CDGroupMember
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.
metagrp/
CDGroupName
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.
metagrp/
CDGroupURL
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.
metagrp/
CDGroupVersion
meta/
CDName
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.
metagrp/
CDName
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.
meta/
CDReviewDate
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.
meta/
CDRevision
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.
metasig/
CDSComment
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.
metasig/
CDSignatures
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.
metasig/
CDSReviewDate
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.
metasig/
CDSStatus
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).
meta/
CDStatus
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).
meta/
CDURL
An optional element. If it is used it contains a string representing the URL where the canonical reference copy of this CD is stored.
metagrp/
CDURL
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.
meta/
CDUses
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 may be present here.
meta/
CDVersion
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.
metagrp/
CDVersion
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.
rounding1/
ceiling
The round up (to +infinity) operation.
SI_DerivedQuantities1/
Celsius-temperature
This symbol represents the physical quantity of Celsius temperature. A variable representing an arbitrary quantity of temperature is commonly represented with the italic, upper case letter, "T".
group3/
center
This symbols represents a unary function whose argument should be a group G. Its value is the biggest subgroup of G all of whose elements commute with all elements of G.
plangeo3/
center
Defines the center of a circle.
plangeo3/
center_of
Gives the center of the circle
plangeo3/
center_of_gravity
Center of gravity of a number of points.
units_siprefix1/
centi
This symbol represents the fact that the subsequent unit has been effectively multiplied by $0.01$
group3/
centralizer
This symbols represents a binary function whose first argument should be a group G and whose second argument should be an element g or a list of elements L of the group G. Its value is the subgroup of G of all elements commuting with g or, if the second argument is a list, all elements of L.
gp1/
character_table
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.
gp1/
character_table_of_group
Refers to the character table of its argument which must be a group.
linalg4/
characteristic_eqn
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.
linalgpoly1/
characteristic_poly
This symbol represents a binary function. This first argument should be a square matrix A defined over a field F, the second argument a variable X. When applied to A and X, it represents the characteristic polynomial of A in the variable X over the field F. (The output should be semantically equivalent to an object obtained by the poly_ring_d_named constructor of the CD polyd1.)
dimensions1/
charge
This symbol represents the charge physical dimension.
SI_DerivedQuantities1/
charge
This symbol represents the physical quantity of electric charge. A variable representing an arbitrary quantity of charge is commonly represented with the italic, upper case letter, "Q".
intpath1/
circle
The symbol circle(c,r) is the circle in the Riemann sphere of which center is c and the radius is r. The direction of the circle is the counter clockwise. When the center is intpath1.infty, the radius should be given in the standard coordinate t=1/z at the infinity.
plangeo3/
circle
The symbol represents a circle. The circle may be subject to constraints.
ThreeDgeo1/
circle
The symbol is used to indicate a circle in 3-dimensional Euclidean geometry by a variable. The circle may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
ThreeDgeo2/
circle_center
The statement that a circle in 3-dimensional Euclidean space has a given point as center. Takes the circle as first argument and the point as second argument.
intpath1/
circle_with_starting_point
The symbol circle_with_starting_point(c,r,z0) is the circle in the Riemann sphere of which center is c and the radius is r. The direction of the circle is the counter clockwise and the staring point is z0.
integer2/
class
This symbol represents a bivariate function, whose arguments should be integers. If a, m are integers, then class(a,m) denotes the residue class a mod m in setname2.Zm.
mathmlattr/
class
A symbol to be used within an OpenMath attribute to specify the class attribute of the object. The annotation should be an OpenMath string representing the value of the class attribute.
polynomial2/
class
This symbol represents a bivariate function, whose arguments should be polynomials. If a, m are polynomials in a polynomial ring R[X], then class(a,m) denotes the residue class a mod m in the quotient ring R[X]/ (mR[X]).
relation3/
class
This symbol represents a ternary function whose first argument is a set S, whose second argument is a relation R on S, and whose third argument is an element a of S. When applied to S, R, and a, it represents the set of all elements in S related to a by R, that is, the set {b in S | (a,b) in R}.
relation3/
classes
This symbol represents a binary function whose first argument is a set S, whose second argument is a relation R on S. When applied to S and R, it represents the set of all elements in S of the form class(S,R,a) for a in S.
intpath1/
closed_path
The symbol closed_path(start_end,points_in, points_out) is a closed path with the starting point "start_end". The direction of the path is counter clockwise. It contains the set of points "points_in" in the inside of the path. The winding number of the path for each point in the set points_in is 1. The path is zero homotope in the space (P^1 - points_in).
meta/
CMP
An optional element (which may be repeated many times) which contains a string corresponding to a property of the symbol being defined.
poly/
coefficient
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.
polynomial1/
coefficient
This symbol is a binary function whose first argument should be a polynomial f and whose second argument should be a non-negative integer n. It represents the coefficient of the i-th power of the variable in the polynomial f.
poly/
coefficient_ring
The coefficient ring.
polynomial1/
coefficient_ring
This symbol is a unary function whose argument should be a polynomial. It represents the coefficient ring of the polynomial.
polyd3/
collect
This a binary function. Its first argument should be a DMP f, its second argument a list of positive integers L. When applied to f and L, it represents the DMP with coefficients from the poly_ring_d whose variables only have indices i for i not occurring in the list L, and whose monomials are built up from the variables indexed by the entries of L.
matrix1/
column_dimension
This symbol is a unary function whose first argument must be either a non-negative OpenMath integer or nums1.infinity. When applied this creates an object that denotes the dimension of the domain of the linear mapping represented by the matrix.
linalg3/
columncount
This symbol represents the function which takes one matrix argument and returns the number of columns in that matrix.
linalg4/
columncount
This symbol represents the function which takes one matrix argument and returns the number of columns in that matrix.
logic3/
complete_pred_deduction
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).
logic3/
complete_pred_theorem
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).
logic3/
complete_prop_deduction
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).
logic3/
complete_prop_theorem
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).
polyd/
completely_reduced
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.
polygb1/
completely_reduced
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.
complex1/
complex_cartesian
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.)
mathmltypes/
complex_cartesian_type
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.
complex1/
complex_polar
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).
mathmltypes/
complex_polar_type
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.
list3/
concatenate
The operation of joining one list to another
monoid3/
concatenation
This symbol represents a binary concatenation operation on strings.
dimensions1/
concentration
This symbol represents the concentration physical dimension, it is the amount of a substance in a volume.
SI_DerivedQuantities1/
concentration
This symbol represents the physical quantity of concentration, the amount of a substance in a volume.
SI_DerivedQuantities1/
conductance
This symbol represents the physical quantity of electrical conductance, the inverse of resistance. A variable representing an arbitrary quantity of conductance is commonly represented with the italic, upper case letter, "G" or "S".
plangeo1/
configuration
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.
ThreeDgeo3/
configuration
The symbol represents a configuration in Euclidean 3-dimensional geometry consisting of a sequence of geometric objects like points, lines, etc, but also of other configurations.
plangeo6/
conic
The symbol represents a conic. The conic may be subject to constraints.
gp1/
conjugacy_class
The binary function whose value is the set of elements which are conjugate to the second argument in the first.
group4/
conjugacy_class
This symbol represents a binary function, whose first argument is a group G and whose second argument is an element x of G. Its value on G and x is the set of elements which are conjugate to x in G.
group4/
conjugacy_class_representatives
This symbol represents a unary function whose argument should be a group. Its value on a group is a set of representatives of the conjugacy classes of that group.
group4/
conjugacy_classes
This symbol represents a unary function whose argument should be a group. Its value on a group is the set of conjugacy classes of that group.
complex1/
conjugate
A unary operator representing the complex conjugate of its argument.
field2/
conjugation
This symbol is a function with two arguments, which should be a field M and a nonzero element x of M. When applied to M and x, it denotes conjugation on M by x.
group2/
conjugation
This symbol is a function with two arguments, which should be a group M and an element x of M. When applied to M and x, it denotes conjugation on M by x.
list2/
cons
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.
list2/
cons
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.
polyslp/
const_node
This constructor takes one argument, which is a value from the coefficient ring. It is intended to represent a constant node.
linalg4mat/
constant
This symbol represents a matrix which has all entries of the same value. It takes three arguments, the first is the rowcount of the matrix, the second is the column count, and the third is the constant which determines every element.
linalg4vec/
constant
This symbol represents a binary function whose first argument should be a natural number. When applied to n and c, it represents the constant (row) vector (so vector as defined in linalg2), so size (dimension) n all of whose components have the value c.
linalg5/
constant
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.
linalg7/
constant
The constant symbol represents the constant vector. It takes two parameters, the length (dimension) of the vector and the constant value, which all the elements are equal to.
mathmltypes/
constant_type
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>.
mathmlkeys/
contentequiv
This key specifies that the corresponding value is the content MathML equivalent of the annotated element.
fns3/
continuous
A predicate to indicate that a function is continuous everywhere.
function_properties/
continuous
fns3/
continuous_on
A predicate to indicate that a function is continuous over a particular region or range.
aggregate_cats/
continuousSetType
This symbol represents the type of continuous sets.
tensor1/
contra_index
This symbol takes a natural number as its argument and returns a contravariant index.
poly/
convert
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.
finfield1/
conway_polynomial
This symbol represents a binary function. Its arguments should be a prime number p and a positive integer n. Before defining which of the possible f(X) is the Conway polynomial we introduce an ordering of the (univariate) polynomials of degree n over GF(p). Here the coefficients of the polynomials are taken in {0, ..., p-1}, the indeterminate is X. Let g(X) = g_nX^n + ... + g_0 and h(X) = h_nX^n + ... + h_0. Then we define g < h if and only if there is an index k with g_i = h_i for i > k and (-1)^{n-k} g_k < (-1)^{n-k} h_k. The Conway polynomial f_{p,n}(X) for GF(p^n) is defined recursively as the smallest polynomial of degree n with respect to this ordering such that: 1) f_{p,n}(X) is monic, 2) f_{p,n}(X) is primitive, that is, it is irreducible and its zeros are generators of the (cyclic) multiplicative group of GF(p^n), 3) for each proper divisor m of n we have that f_{p,m}(X^{(p^n-1) / (p^m-1)})= 0 mod f_{p,n}(X); that is, the ((p^n-1) / (p^m-1))-th power of a zero of f_{p,n}(X) is a zero of f_{p,m}(X).
plangeo4/
coordinates
This function yields the coordinates vector if applied to a point or line with coordinates.
plangeo5/
coordinatize
This symbol is a function of one argument which must be a configuration or an assertion (as defined in plangeo1). When applied to a configuration C, it stands for the same configuration but now with coordinates attached to each object of C. The new variables are bound within an OMBIND element with head element the lambda symbol. The bound variables (placed within an OMBVAR element) are the new variables, and the last argument of OMBIND is the expression C in which each object is coordinatized. If an object already has coordinates, these are left as they are. If not, then new variables are introduced to coordinatize the object. When applied to an assertion of the form assertion(C,S), it leads to the same result except that the last argument of OMBIND is the assertion whose configuration argument is the expression C in which each object is coordinatized, and whose thesis argument is S.
plangeo2/
corner
The corner between two halflines L and M, both starting at the same point. Given three points A, B and C, the corner A, B, C is the corner of the two halflines BA and BC. Corresponding to the two cases, the symbol can have as arguments two halflines or three points.
transc1/
cos
This symbol represents the cos function as described in Abramowitz and Stegun, section 4.3. It takes one argument.
transc1/
cosh
This symbol represents the cosh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.
transc1/
cot
This symbol represents the cot function as described in Abramowitz and Stegun, section 4.3. It takes one argument.
transc1/
coth
This symbol represents the coth function as described in Abramowitz and Stegun, section 4.5. It takes one argument.
SI_NamedDerivedUnits1/
coulomb
This symbol represents an SI unit of electric charge. It has the short symbol form, "C".
units_metric1/
Coulomb
This symbol represents the measure of one Coulomb. This is the standard SI measure for charge.
FundamentalPhysicalConstants1/
Coulomb-constant
The value of the Coulomb constant is implied by international definitions of the speed of light and the vacuum permeability. By definition, its exact value is equal to (299,792,458)^2 * 10^-7 N m^2 C^-2. It is commonly represented with the short, italic symbol, "k" subscripted with the upright letter "e".
tensor1/
covar_index
This symbol takes a natural number as its argument and returns a covariant index.
transc1/
csc
This symbol represents the csc function as described in Abramowitz and Stegun, section 4.3. It takes one argument.
transc1/
csch
This symbol represents the csch function as described in Abramowitz and Stegun, section 4.5. It takes one argument.
veccalc1/
curl
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.
dimensions1/
current
This symbol represents the current physical dimension.
SI_BaseQuantities/
current
This symbol represents the SI base quantity of electrical current. It has the short symbol form, "I".
permutation1/
cycle
This symbol is an n-ary constructor. 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.
permutation1/
cycle
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.
permutation1/
cycle_type
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.
permutation1/
cycle_type
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.
permutation1/
cycles
This symbol has one argument which should be a endomap p. It returns the list of cycles of p.
groupname1/
cyclic_group
This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the cyclic group of order n.
permgp2/
cyclic_group
This symbol represents a unary function whose argument should be a positive integer. When evaluated at the integer n, it represents the permutation group generated by the permutation (1,2,...,n).
monoid3/
cyclic_monoid
This symbol is a function of two natural numbers, the first of which should be positive. When evaluated at k and l, it denotes the cyclic monoid with a cycle of length l and a tail (including the identity element) of length k.
semigroup3/
cyclic_semigroup
This symbol denotes the cyclic semigroup with a cycle of length l and a tail of length k.
SIUsed_OffSystemMeasuredUnits1/
dalton
This symbol represents the measure of one dalton of mass. It has the short symbol form, "Da". The dalton is one-twelth the mass of an atom of Carbon-12 at rest and in its ground state. Its measured value is 1 Da = 1.660538782(83) * 10^–27 kg [CODATA 2006]
SIUsed_OffSystemUnits1/
day
This symbol represents the measure of one day of time. It has the short symbol form, "d".
units_time1/
day
This symbol represents the measure of one day of time. The definitions below ignore the possibilities of "leap seconds".
units_siprefix1/
deci
This symbol represents the fact that the subsequent unit has been effectively multiplied by $0.1$
directives1/
decide
This symbol is a function with one argument, which should be a clause. When applied to a clause, it asks whether the clause holds.
gp1/
declare_group
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.
prog1/
def_arguments
This symbol can be used to encode the arguments that a function or procedure can receive.
polynomial4/
definitely_irreducible
A symbol which denotes that a factor of the factorisation is definitely irreducible.
mathmlattr/
definitionURL
A symbol to be used within an OpenMath attribute to specify the definitionURL attribute of the object. The annotation should be an OpenMath string representing the value of the definitionURL attribute.
calculus1/
defint
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.
calculus1/
defint
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.
cauchypv/
defint
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.
poly/
degree
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.
polynomial1/
degree
This symbol represents a unary function, whose argument should be univariate polynomial. When applied to a polynomial, it represents its degree, that is the highest power of the variable occurring in a term of the polynomial. If the polynomial has no terms, it is the zero polynomial, in which case the value represented is -1.
SI_NamedDerivedUnits1/
degree-Celsius
This symbol represents an SI unit of Celsius temperature. It has the short symbol form, "ºC".
SIUsed_OffSystemUnits1/
degree-of-arc
This symbol represents the angular measure of one degree of arc. It has the short symbol form of the degree symbol, a superscript circle, Unicode: U+00B0 or HTML: °.
units_metric1/
degree_Celsius
This symbol represents the measure of one degree Celsius. This is a standard metric measure for temperature.
units_imperial1/
degree_Fahrenheit
This symbol represents the measure of one degree Fahrenheit. This is the standard imperial measure for temperature.
units_metric1/
degree_Kelvin
This symbol represents the measure of one degree Kelvin. This is a standard SI measure for temperature relative to absolute zero.
poly/
degree_wrt
The degree with respect to a variable (the second argument). We note that the degree of 0 is undefined.
units_siprefix1/
deka
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10$
equations1/
dense
A predicate to indicate that an equation or system of equations is dense.
matrix1/
dense
This symbol is an $(m \cdot n)$-ary function whose arguments specify the entries of the matrix, where $m$ is the dimension of the codomain and $n$ is the dimension of the domain. The matrix (or block) must be filled row-wise, that is the first argument denotes the entry in row 1, column 1 of the matrix (or block), the second argument denotes the entry at row 1, column 2, and so forth. The number of arguments MUST match the dimensions of either the matrix algebra or the surrounding block (see below).
dimensions1/
density
This symbol represents the density physical dimension, it is the mass per unit volume.
SI_DerivedQuantities1/
density
This symbol represents the physical quantity of volumic mass density.
polyslp/
depth
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.
gp1/
derived_subgroup
The unary function whose value is the subgroup of argument generated by all products of the form xyx^-1y^-1.
group3/
derived_subgroup
The unary function whose value is the subgroup of argument generated by all products of the form xyx^-1y^-1.
meta/
Description
An element which contains a string corresponding to the description of either the CD or the symbol (depending on which is the enclosing element).
linalg1/
determinant
This symbol denotes the unary function which returns the determinant of its argument, the argument should be a square matrix.
matrix1/
diagonal
This symbol is an $m$-ary function whose arguments specify the entries of a (generalised) matrix diagonal. The diagonal must be filled from top-left to bottom-right. That is: the first argument represents the entry in row 1, column 1 of the matrix, the second argument denotes the entry at row 2, column 2, and so forth. If used inside a sparse_entry object at location $(i, j)$, the first entry is offset accordingly. If not used inside a sparse_entry object, the number of arguments MUST match the dimensions of either the matrix algebra (the smaller of $m$ and $n$). If used inside a sparse_entry object, the number of entries MUST NOT exceed the total matrix dimensions.
linalg4mat/
diagonal_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.
linalg5/
diagonal_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.
calculus1/
diff
This symbol is used to express ordinary differentiation of a unary function. The single argument is the unary function.
calculus1/
diff
This symbol is used to express ordinary differentiation of a unary function. The single argument is the unary function.
weylalgebra1/
diff
Differentiation of a given function in one variable.
list3/
difference
This symbol takes two arguments both a list. It represents a function which returns a list made up of all the elements of the first list which are not in the second.
list4/
difference
This symbol represents a function with two arguments, both lists. When applied to two lists, it represents a list made up of all the elements of the first list which do not occur in the second, appearing in the order in which they occur in the first list.
fns3/
differentiable
A predicate to indicate that a function is differentiable over its whole domain.
function_properties/
differentiable
fns3/
differentiable_on
A predicate to indicate that a function is differentiable on a region.
weylalgebra1/
diffop
constructor of a differential operator from a polynomial or from an element of the finitely generated free algebra. The inverse of gr.
graph1/
digraph
This symbol refers to a digraph. It has two arguments. The first is the set of vertices, the second is the set of arrows. Arrows are represented by lists of length two, where a list represents the arrow from the first element to the second.
groupname1/
dihedral_group
This symbol is a function with one argument, which should be a positive integer n. When applied to n it represents the dihedral group of order 2n. This is the group of all isometries (including reflections) of the regular n-gon in the plane.
permgp2/
dihedral_group
This symbol represents a unary function whose argument should be a positive integer. When evaluated at the integer n, it represents the dihedral group of all 2n permutations of {1,2,...,n} preserving the n-gon 1,2,...,n.
SI_functions1/
dim
The symbol to represent the function that returns the physical dimension of its argument in terms of products of powers of SI base quantities. The dim operation may be meaningfully applied to SI quantities, SI units, and numbers without physical dimension.
group3/
direct_power
This is a binary function whose first argument should be a group G and whose second argument should be a natural number n. It refers to the direct product of n copies of G.
monoid3/
direct_power
This is a binary function whose first argument should be a monoid M and whose second argument should be a natural number n. It refers to the direct product of n copies of M.
ring3/
direct_power
This is a symbol with two arguments. The first argument should be a ring S and the second argument a positive integer n. It denotes the direct product of n copies of S.
semigroup3/
direct_power
This is a binary function whose first argument should be a semigroup M and whose second argument should be a natural number n. It refers to the direct product of n copies of M.
group3/
direct_product
This is an n-ary function whose arguments must be groups. It refers to the direct product of its arguments.
magma3/
direct_product
This is an n-ary function whose arguments must be magmas. It refers to the direct product of its arguments.
monoid3/
direct_product
This is an n-ary function whose arguments must be monoids. It refers to the direct product of its arguments.
ring3/
direct_product
This is a symbol with two or more arguments, all of which are rings. It denotes the ring that is the direct product of its arguments.
semigroup3/
direct_product
This is an n-ary function whose arguments must be semigroups. It refers to the direct product of its arguments.
ThreeDgeo3/
discovery
The symbol is used to describe the task of finding necessary conditions for some properties to hold in a geometric configuration in 3-dimensional Euclidean geometry. The symbol takes a configuration as the first argument and the properties for which necessary conditions are to be sought as further arguments.
finfield1/
discrete_log
This symbol represents a binary function. The first argument is the base b, a primitive element of a finite field F. The second argument is a nonzero element x in F. It returns the smallest nonnegative integer i such that x=b^i.
aggregate_cats/
discreteSetType
This symbol represents the type of discrete sets.
poly/
discriminant
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.
numerical2/
discrimination_criteria
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.
dimensions1/
displacement
This symbol represents the spatial difference between two points. The direction of the displacement is taken into account as well as the distance between the points.
directives1/
disprove
This symbol is a function with one argument, which should be a clause. When applied to a clause C, it asks for a proof of that C does not hold.
plangeo3/
distance
The distance between two affine points is the Euclidean distance. The distance between two geometric objects O and O' is the infimum of the distances between two affine points, one on O and one on O'.
ThreeDgeo3/
distance
The distance between two affine points in 3-dimensional Euclidean space is the Euclidean distance. The distance between two geometric objects O and O' in 3-dimensional Euclidean space is the infimum of the distances between two affine points, one on O and one on O'.
veccalc1/
divergence
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)
arith1/
divide
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.
opnode/
divide
A constant value, constructs the divide for division nodes.
polynomial4/
divide
This symbol is a binary function whose arguments are polynomials f and g which must be defined over the same ground domain. When applied to f and g, it represents the quotient arising from dividing f by g and the remainder h such that h is congruent f modulo g. The result is gathered in a polynomial4.quotient_remainder cell. Hint: We consider named polynomial rings, i.e. the indeterminate is explicitly specified by a named variable, different once the variable names differ. That is, a polynomial in Z[X] cannot be divided by a polynomial in Z[Y] a priori. However, we leave it up to the implementor to handle this differently, though we strongly encourage implementors to return a polynomial in an anonymous indeterminate (using e.g. polyd1.poly_ring_d rather than polyd1.poly_ring_d_named).
integer2/
divides
This symbol represents a bivariate Boolean function, whose arguments should be integers. When applied to integers a and b, it denotes the property that a divides b.
polynomial2/
divides
This symbol represents a bivariate Boolean function, whose arguments should be polynomials in the same polynomial ring. When applied to a and b, it denotes the property that a divides b.
monoid1/
divisor_of
This symbol is a ternary function. Its first argument should be a monoid M and the second and third arguments should be elements of M. When applied to M, a, and b, it denotes the fact that a is a divisor of b in M. This means that there are u,v in carrier(M) such that uav=b.
polyd/
DMP
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(...))
polyd1/
DMP
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(poly_ring_d(...), SDMP(...))
polyd/
DMPL
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(...)+)
polyd1/
DMPL
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(poly_ring_d(...), SDMP(...)+)
fns1/
domain
This symbol denotes the domain of a given function, which is the set of values it is defined over.
meta_cats/
domain
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).
permutation1/
domain
This symbol is a function with one argument which is a endomap. When applied to a endomap whose arguments are a_1,...,a_n, it represents the set {1,...,n}.
fns1/
domainofapplication
Deprecated. This symbol was intended to model MathML domainofapplication but as defined it is a synonym for domain. In MathML3, MathML compatibility is defined to use the new restriction symbol.
linalgrank1/
dual_kernel_matrix
This symbol represents a unary function whose argument should be a matrix. When applied to a matrix, it represents a list of column vectors spanning the kernel of the matrix acting on the left.
expint/
E
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.
nums1/
e
This symbol represents the base of the natural logarithm, approximately 2.718. See Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1.
graph1/
edgeset
This symbol represents the set of edges of an undirected graph. It takes one argument, the undirected graph.
expint/
Ei
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
linalgeig1/
eigenspace
This symbol represents a binary function, whose arguments should be a square matrix A over a field F and an element lambda of F. When applied to A and lambda, it returns a matrix whose rows are a basis of the eigenspace of A with respect to lambda.
linalg4/
eigenvalue
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.
linalgeig2/
eigenvalue
This symbol represents a binary function. The first argument should be a square matrix A defined over the field of complex numbers, the second should be an index i to specify the eigenvalue. When applied to A and i it represents the i-th eigenvalue of A (counted without multiplicities). 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.
linalgeig1/
eigenvalues
This symbol represents a binary function, whose arguments should be a square matrix A and a field F over which the matrix A is defined. When applied to A and F, it returns a vector whose entries are the eigenvalues of A contained in F, with multiplicities.
linalg4/
eigenvector
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.
linalgeig2/
eigenvector
This symbol represents a binary function. Its first argument should be a square matrix A defined over the complex numbers, the second should be an index i to specify which eigenvalue this eigenvector should be paired with, with the ordering specified in the definition of eigenvalue in this CD. When applied to A and i, it represents an eigenvector for A with respect to the i-th eigenvalue.
SIUsed_OffSystemMeasuredUnits1/
electronvolt
This symbol represents the measure of one electronvolt of energy. It has the short symbol form, "eV". It is the kinetic energy acquired by an electron in passing through a potential difference of 1 volt in a vacuum. Its measured value is 1 eV = 1.602176487(40) * 10^–19 J [CODATA 2006]
gp1/
element_set
The unary function which returns the set of elements of a group.
polyd/
elimination
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.
polyd2/
elimination
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.
multiset1/
emptyset
This symbol is used to represent the empty multiset, that is the multiset which contains no members. It takes no parameters.
set1/
emptyset
This symbol is used to represent the empty set, that is the set which contains no members. It takes no parameters.
monoid3/
emptyword
This symbol represents a constant. It represents the empty string.
moreerrors/
encodingError
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.
permutation1/
endomap
This symbol is an n-ary constructor. Its arguments should be positive integers. When applied to arguments a_1,...,a_n, the resulting value is the map sending i to a_i for i=1,...,n.
permutation1/
endomap_left_compose
This symbol is a binary function. Its arguments should be endomaps with identical domain D. When applied to arguments P1 and P2, the resulting value is the endomap which maps x in D to P1(P2(x)).
permutation1/
endomap_right_compose
This symbol is a binary function. Its arguments should be endomaps with identical domain D. When applied to arguments P1 and P2, the resulting value is the endomap which maps x in D to P2(P1(x)).
plangeo2/
endpoint
The endpoint of a halfline.
plangeo2/
endpoints
The two endpoints of a segment.
ThreeDgeo1/
endpoints
The symbol is used to indicate the set of the two endpoints of a segment in 3-dimensional Euclidean geometry by a variable. The symbol takes the variable as the first argument and the segment as second argument.
dimensions1/
energy
This symbol represents the energy physical dimension.
SI_DerivedQuantities1/
energy
This symbol represents the physical quantity of energy. A variable representing an arbitrary quantity of energy is commonly represented with the italic, upper case letter, "E".
SI_DerivedQuantities1/
entropy
This symbol represents the physical quantity of entropy, a measure of the disorder of a system.
list3/
entry
This symbol represents a binary function whose first argument should be a list L and whose second argument should be a positive integer i such that the absolute value of i is in the interval [1..n], where n is the length of L. If i is positive, it returns the i-th entry L[i] of L, if i is negative it returns the (n+1-i)-th entry of L.
list4/
entry
This symbol represents a binary function whose first argument should be a list L and whose second argument should be a positive integer i such that the absolute value of i is in the interval [1..n], where n is the length of L. If i is positive, it returns the i-th entry L[i] of L, if i is negative it returns the (n+1-i)-th entry of L.
matrix1/
entry_domain
This symbol is a unary function, whose argument should be a ring r. When applied to r, it represents the matrix-algebra ground domain (MAD).
relation1/
eq
This symbol represents the binary equality function.
integer2/
eqmod
This symbol represents a Boolean valued trivariate function, whose arguments should be integers. When applied to integers a, b, m, it denotes the Boolean evalue of the assertion that a and b are equal modulo m.
polynomial2/
eqmod
This symbol represents a Boolean valued trivariate function, whose arguments should be polynomials. When applied to polynomials a, b, m, it denotes the Boolean evalue of the assertion that a and b are equal modulo m.
relation4/
eqs
This symbol is used to denote the n-ary version of equality. When applied to n arguments a1, ..., an, it represents the boolean expression that a1, a2, ,,, and an are equal.
relation0/
equivalence
Proposition; the type of equivalence relations, namely relations that are reflexive, symmetric and transitive.
relation3/
equivalence_closure
This symbol represents a binary function whose first argument is a set S, whose second argument is a relation R on S. When applied to S and R, it represents the smallest equivalence relation (with respect to inclusion) on S containing R.
logic1/
equivalent
This symbol is used to show that two boolean expressions are logically equivalent, that is have the same boolean value for any inputs.
SI_DerivedQuantities1/
equivalent-dose
This symbol represents the physical quantity of equivalent dose of ionizing radiation. Equivalent dose is similar to absorbed dose but is weighted to reflect differing biological effects and different radiation types. A variable representing an arbitrary quantity of equivalent dose is commonly represented with the italic, upper case letter, "H".
sts/
error
The error symbol is the 'return type' of error symbols in the error signature file.
scscp1/
error_memory
A description of the error that caused a procedure call to be terminated. This symbol is used with a procedure_terminated, when the system exceeded the amount of memory specified in the option_max_memory option given in the corresponding procedure call. It carries one argument: An OMSTR, which may be empty.
scscp1/
error_runtime
A description of the error that caused a procedure call to be terminated. This symbol is used with a procedure_terminated, when the system exceeded the runtime specified in the option_runtime option given in the corresponding procedure call. It carries one argument: An OMSTR, which may be empty. Note that this symbol is not intended to be used when a different runtime error occurred. In those cases, one should use error_system_specific.
scscp1/
error_system_specific
A description of the error that caused a procedure call to be terminated. This symbol is used with a procedure_terminated, when the error is specific to the system that carried out the calculation. This error must carry exactly one argument, and it must be an OMSTR describing the error that occurred.
algebraic_cats/
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.
generic_alg_cats/
Euclidean_domain
This Symbol represents the generic category of Euclidean domain.
algebraic_cats/
Euclidean_domain_abs
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.
algebraic_cats/
Euclidean_domain_negative
This symbol takes one argument which should be a Euclidean domain. It returns a unary function, which represents additive inverses of the Euclidean domain.
algebraic_cats/
Euclidean_domain_plus
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.
algebraic_cats/
Euclidean_domain_set
This symbol takes one argument which should be a Euclidean domain. It returns the set of the Euclidean domain.
algebraic_cats/
Euclidean_domain_times
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.
algebraic_cats/
Euclidean_domain_zero
This symbol takes one argument which should be a Euclidean domain. It returns the additive identity of the Euclidean domain.
norm1/
Euclidean_norm
This symbol signifies the Euclidean ($L_2$) norm.
integer2/
euler
This symbol denotes the univariate Euler totient function. If m is an integer, then euler(m) denotes the order of the multiplicative group of invertible elements in the residue class ring Z/mZ.
directives1/
evaluate
This symbol is a function with one argument, which should be a mathematical expression. When applied to a mathematical expression, it asks for an evaluation or simplification of the expression. The evaluation or simplification to be carried out by a service is a simpler mathematical expression (in some definition of complexity of objects) which is equal to the argument.
poly/
evaluate
Evaluation of a polynomial at a value or vector of values.
directives1/
evaluate_to_type
This symbol is a function with two arguments, which should be a mathematical expression and a type. When applied to a mathematical expression E and a type T, it asks for an evaluation or simplification of E to a mathematical expression of type T.
units_siprefix1/
exa
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^18$
meta/
Example
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.
units_binaryprefix1/
exbi
This symbol represents the fact that the subsequent unit has been effectively multiplied by $2^60$. The full technical name is exabinary.
quant1/
exists
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.
transc1/
exp
This symbol represents the exponentiation function as described in Abramowitz and Stegun, section 4.2. It takes one argument.
poly/
expand
Converts a factored or squarefreed form into the expanded polynomial over the same ring, so that factored(recursive) -> recursive, etc.
polynomial1/
expand
Expands a polynomial.
polyoperators1/
expand
Expands a polynomial. Acts as expand(expresion).
directives1/
explore
This symbol is a unary function whose argument should be a mathematical assertion. When applied to an assertion A, it asks for conditions under which the assertion holds.
field1/
expression
This symbol is a function with two arguments. Its first argument should be a field. The second should be an arithmetic expression A, whose operators are times, plus, minus, unary_minus, and power, and whose leaves are members of the carrier of G. When applied to G and A, it denotes the element (of G) that is the element obtained from the leaves of A by applying the operations of G instead of those from the CD arith1 according to A. Here multiplication, addition, subtraction, minus, and power take over the roles of times, plus, minus, unary_minus, and power, respectively. Also, an integer m occurring in A will be interpreted as a member of G by interpreting it as the sum of m copies of the identity element, the symbol alg1.one will be interpreted as the identity, and the symbol alg1.zero will be interpreted as the zero of G.
group1/
expression
This symbol is a function with two arguments. Its first argument should be a group. The second should be an arithmetic expression A, whose operators are times and power, and whose leaves are members of the carrier of G. When applied to G and A, it denotes the element (of G) that is obtained from the leaves of A by applying the multiplication and the power map of G instead of the times and power from the CD arith1 appearing in A. The symbol alg1.one occurring in A will be interpreted as the identity of G.
monoid1/
expression
This symbol is a function with two arguments. Its first argument should be a monoid. The second should be an arithmetic expression A, whose operators are times and power, and whose leaves are members of the carrier of G. The second argument of power should be nonnegative. When applied to G and A, it denotes the element (of G) that is obtained from the leaves of A by applying the multiplication and the power map of G instead of the times and power from the CD arith1 appearing in A. The symbol alg1.one occurring in A will be interpreted as the identity of G.
ring1/
expression
This symbol is a function with two arguments. Its first argument should be a ring. The second should be an arithmetic expression A, whose operators are times, plus, minus, unary_minus, and power, and whose leaves are members of the carrier of G. (Here an integer m will be interpreted as a member of G by interpreting it as the sum of m copies of the identity element, the symbol alg1.one will be interpreted as the identity, and the symbol alg1.zero will be interpreted as the zero of G.) When applied to G and A, it denotes the element (of G) that is the element obtained from the leaves by applying the arithmetic operations of G instead of those from the CD arith1.
semigroup1/
expression
This symbol is a function with two arguments. Its first argument should be a semigroup G. The second should be an arithmetic expression A, whose operators are times and power, and whose leaves are members of the carrier of G. The second argument of power should be positive. When applied to G and A, it denotes the element (of G) that is obtained from the leaves of A by applying the multiplication and the power map of G instead of the times and power of the CD arith1 appearing in A.
arith3/
extended_gcd
The symbol represents the n-ary function, a_1,...,a_n to return a list consisting of the gcd (greatest common divisor) of its arguments, together with n elements x_1,...,x_n such that gcd(a_1,...,a_n)=x_1 a_1+...+x_n a_n
polygb2/
extended_in
This symbol is a function of at least 3 arguments. The first argument is a list of variables. The second and third argument are lists of polynomials in the variables from the first argument, C and T respectively. When applied to its arguments, it represents the boolean value of the assertion that all elements t in T can be written as t = f_1*c_1 + ... + f_n*c_n (c_i in C). If the optional 4th argument is 1, those f_i are returned.
poly/
factor
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.
polynomial4/
factor
A symbol which represents one factor of a factorisation, it takes at least 2 arguments, the first of which being the factor polynomial, e.g. a polyd1.DMP, and the second being its multiplicity specified as an integer >= 1. Optionally, the third argument is one of polynomial4.definitely_irreducible, polynomial4.possibly_reducible to indicate whether or not the given factor is guaranteed to be irreducible. Furthermore, this symbol may contain polynomial4.ground_ring_injected to indicate that the ground ring is considered to be embedded in the polynomial algebra and hence the factor is actually the factorisation of a polynomial coefficient.
polyoperators1/
factor
The action of factoring a polynomial into irreducible factors (I know this is field dependent but lets keep it simple by now).
semigroup1/
factor_of
This symbol is a ternary function. Its first argument should be a semigroup S and the second and third arguments should be elements of S. When applied to S, a, and b, it denotes the fact that a is a divisor of b in S. This means that there are u,v in carrier(S) such that uav=b.
poly/
factored
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.
integer1/
factorial
The symbol to represent a unary factorial function on non-negative integers.
polynomial4/
factorisations
This symbol may be used in the reply of polynomial4.factorise and takes at least 1 argument. The first argument is one of polynomial4.factorisations_complete to indicate that the following list of polynomial4.factors cells covers all possible factorisations. The counterpart would be polynomial4.factorisations_possibly_incomplete to indicate that the following list of factorisations are some of possibly many more factorisations. Note: If the polynomial algebra is a UFD (unique factorisation domain) the uniqueness can be underpinned by giving exactly one polynomial4.factors cell and using the symbol polynomial4.factorisations_complete here. The rest of the arguments are polynomial4.factors cells, each of which being a possible factorisation. Using the call of polynomial4.factorise above we might obtain:
polynomial4/
factorisations_complete
A symbol to indicate that a given list of factorisations of a polynomial covers in fact all possible factorisations.
polynomial4/
factorisations_incomplete
A symbol to indicate that a given list of factorisations is an assortment of all possible factorisations.
polynomial4/
factorise
This symbol is a unary function, whose argument should be a polynomial f. When applied to f, it represents a list of factors of f. Cf. polynomial4.factorisations for a description of the expected reply.
integer1/
factorof
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.
polynomial3/
factors
This symbol is a unary function, whose argument should be a polynomial f. When applied to f, it represents a complete list of irreducible factors of f.
polynomial4/
factors
This symbol is used in the reply of polynomial4.factorise and takes at least 2 arguments. Note this symbol may also be used in a polynomial4.factorisations cell. The first argument is one of polynomial4.definitely_irreducible or polynomial4.possibly_reducible and specifies whether the computed factorisation is known to be irreducible or if the irreducibility of some of the factors is not guaranteed. Note: This symbol is mandatory even if the factors themselves (see polynomial4.factor) can carry that information, this is simply to connive at computer algebra systems that cannot figure out which of the factors is the possibly reducible one. Generally this slot must be polynomial4.possibly_reducible if at least one of the factors is possibly reducible. The second argument contains a polyd1.poly_ring_d or polyd1.poly_ring_d_named cell, as specified in e.g. polyd or polyd1 to indicate the underlying polynomial algebra. The third argument is a symbol polynomial4.common_coefficient and denotes the common coefficient of the factorisation. Note: In case the ground ring itself is regarded as being injected into the polynomial algebra, or the factorisation is normalised, this field may be used to specify the unit giving the normalisation. Furthermore, the cell comprises polynomial4.factor cells which in turn represent the factors of the polynomial in a factorisation along with their multiplicities. Using the call of polynomial4.factorise above we might obtain:
polyoperators1/
factors
The action of returning a list composed of the irreducible factors of a polynomial. (I know this is field dependent but lets keep it simple by now).
hypergeon1/
falling_factorial
falling_factorial(n,i) is equal to n*(n-1)* ... *(n-i+1).
hypergeon1/
falling_multi_factorial
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
logic1/
false
This symbol represents the boolean value false.
SI_NamedDerivedUnits1/
This symbol represents an SI unit of electric capacitance. It has the short symbol form, "F".
physical_consts1/
This symbol represents the electric charge carried by one mole of electrons. It is approximately 96485.309 +/- 0.029 Coulombs per mole.
units_siprefix1/
femto
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^-15$
combinat1/
Fibonacci
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.
algebraic_cats/
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).
field1/
field
This symbol is a constructor for fields. It takes seven arguments R, a, o, n, m, e, i: which are, respectively, a set R to specify the elements in the field, a binary operation a on R, an element o of R, and a unary operation n on R such that [R,a,o,n] is a commutative group, a binary operation m on R, an element e of R, and a map from R - {o} to itself such that [R,m,e] is a monoid and such that [R - {o},m',e,i] is a group, where m' is the restriction of m to R -{o}.
generic_alg_cats/
field
This Symbol represents the generic category of field.
finfield1/
field_by_conway
This symbol represents a binary function. The first argument should be a prime number p, the second argument a positive integer n. This symbol returns the field GF(q)[X]/ (C(X)), where q = p^n, X is an indeterminate, C(X) is the Conway polynomial f_{n,p}(X), and (C(X)) is the ideal in the polynomial ring GF(q)[X] generated by C(X).
finfield1/
field_by_conway
This symbol has two arguments. Its first argument should be a prime number p and its second argument a positive integer n. When applied to p and n, the result is the field defined as the quotient ring GF(p)[X]/(c(X)), where c(X)=conway_polynomial(p,n).
field3/
field_by_poly
This symbol is a binary function whose first argument is a univariate polynomial ring R over a field, and whose second argument is an irreducible polynomial f in this polynomial ring R. So, when applied to R and f, the function has value the quotient ring R/(f).
field4/
field_by_poly_map
Same as quotient_by_poly_map in CD ring5, except that R and the quotient ring R[X]/(f) are fields (so f is irreducible in R[X]).
field4/
field_by_poly_vector
This symbol is a binary function. Its first argument should be a field_by_poly(R,f). Its second argument should be a list L of elements of F, the coefficient field of the univariate polynomial ring R = F[X]. The length of the list L should be equal to the degree d of f. When applied to R and L, it represents the element L + L x + L x^2 + ... + L[d-1] ^(d-1) of R/(f), where x stands for the image of x under the natural quotient map R -> R/(f). If the first argument is a field_by_conway(p,n), defined in the CD finfield1, then the same interpretation holds, where R and f are respectively poly_ring_d(GFp(p),1) and conway_polynomial(p,n).
algebraic_cats/
field_negative
This symbol takes one argument which should be a field. It returns a unary function, which is the additive inverse function of the field.
algebraic_cats/
field_one
This symbol takes one argument which should be a field. It returns the multiplicative identity of the field.
algebraic_cats/
field_plus
This symbol takes one argument which should be a field. It returns a binary function, to represent the additive operation of the field.
algebraic_cats/
field_reciprocal
This symbol takes one argument which should be a field. It returns a unary function, which is the multiplicative inverse function of the field.
algebraic_cats/
field_set
This symbol takes one argument which should be a field. It returns the set of the field.
algebraic_cats/
field_times
This symbol takes one argument which should be a field. It returns a binary function, to represent the multiplicative operation of the field.
algebraic_cats/
field_zero
This symbol takes one argument which should be a field. It returns the additive identity of the field.
directives1/
find
This symbol is a binder, whose body should be a clause. When bound to a variable (or list of variables) x with body P(x), it asks for a mathematical expression A such that P(A) holds.
aggregate_cats/
finiteSetType
This symbol represents the type of finite sets.
list2/
first
This symbol represents a function which returns the first elements of its argument, which should be a list.
list2/
first
This symbol represents a function which returns the first elements of its argument, which should be a list.
permutation1/
fix
This symbol is a function with two arguments. The first argument should be a permutation, the second argument a set. When applied to a permutation g and a set X, it represents the subset A of X all points that do not belong to the support of g.
omtypes/
float
The type of floating point numbers
rounding1/
floor
The round down (to -infinity) operation.
meta/
FMP
An optional element which contains an OpenMath Object. This corresponds to a property of the symbol being defined.
mathmltypes/
fn_type
A symbol to be used as the argument of the type symbol to convey the type for a function name.
units_imperial1/
foot
This symbol represents the measure of one foot. This is the standard imperial measure for distance.
units_us1/
foot_us_survey
This symbol represents the measure of one U.S. Survey foot.
prog1/
for
This symbol can be used to encode the for loop. The syntax is for(block1,conditional_block,block3,block4), where block1 is the initialization block, conditional_block is the conditional block that determines the end of the loop, block3 is the incremental block and block4 is the body of the for loop. Each of this blocks should be present (althougth they can be empty).
quant1/
forall
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.
dimensions1/
force
This symbol represents the force physical dimension.
SI_DerivedQuantities1/
force
This symbol represents the physical quantity of force. A variable representing an arbitrary quantity of force is commonly represented with the italic, upper case letter, "F".
mathmlattr/
foreign
A symbol to be used within an OpenMath attribute to specify an attribute of the object. The annotation should be an quadruple of strings constructed via a head foreign_attribute.
mathmlattr/
foreign_attribute
A symbol to be used as the head of the OpenMath application to construct the object used as the value of the foreign attribution. The four arguments of this function should be OpenMath strings representing in order, the Namespace, prefix and local name and value of the MathML attribute.
field3/
fraction_field
This is a unary function. Its argument should be a domain (as in CD ring4). It denotes the fraction field of the domain.
field3/
free_field
This symbol represents a binary function. The first argument should be a natural number p which is zero or a prime number, the second argument a list or a set L. When evaluated on such arguments p and L, the function represents the field of rational functions in L over the rationals if p = 0 and over the field of integers mod p if p is a prime.
group3/
free_group
This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free group generated by the entries of the list or set.
magma3/
free_magma
This symbol represents a binary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free magma generated by the entries of the list or set.
monoid3/
free_monoid
This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free monoid generated by the entries of the list or set.
ring3/
free_ring
This symbol represents a binary function. The first argument should be a ring and the second a list or a set. When evaluated on such arguments R and L, the function represents the free ring over R generated by the elements (or entries) of L. This ring can also be viewed as the ring of non-commutative polynomials over R with variables the elements of L.
semigroup3/
free_semigroup
This symbol represents a binary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free semigroup generated by the entries of the list or set.
SI_DerivedQuantities1/
frequency
This symbol represents the physical quantity of frequency. A variable representing an arbitrary quantity of frequency is commonly represented with the italic, lower case greek variable, "\omega;".
fns3/
function
This symbol denotes a function constructor. When aplied to at least two arguments, which are sets, the first argument is the domain and the second the range of the function. When applied to at least three arguments, the first two of which are stes and the third of which is a lambda expression, the third argument gives the function specification.
prog1/
function_block
The block of code defining the body of the function. The syntax is function_block(local_var,block1), where local_var encodes the local variables (private to the function body) and block1 is the body of the function. Both locar_var and block1 should be present (and of course both can be also empty).
prog1/
function_call
Symbol function_call can be used to "call" already defined functions. The syntax is function_call(name, call_arguments), where name is the encoding of an OpenMath variable (OMV) representing the name of the function and call_arguments are the arguments to pass to the function. Both, name and call_arguments, should be present but call_arguments can be empty.
prog1/
function_definition
The symbol function_definition can be is used to define a function. The syntax is function_definition(name, def_arguments, function_block), where name is the encoding of an OpenMath variable (OMV) representing the name of the funtion, def_arguments is the enconding of the arguments that the function receives and function_block is the body of the function (local variables declarations + body of the function). Functions are completely unaware of the rest of the "world" except for the information they received from the arguments. Functions are only allowed to return values by means of the return construct.
setname3/
function_set
The function_set operator generates the set of functions between the sets specified as its arguments. cf Hom(A,B) in Category theory, and mapsto in the sts CD. For a set of n-argument functions, function_set will take n+1 arguments, specifying the sets of the n arguments and the range.
hypergeo0/
gamma
Euler's gamma function
nums1/
gamma
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.
physical_consts1/
gas_constant
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.
arith1/
gcd
The symbol to represent the n-ary function to return the gcd (greatest common divisor) of its arguments.
poly/
gcd
The n-ary greatest common divisor of its polynomial arguments. This is unique up to units.
polynomial3/
gcd
The n-ary greatest common divisor for univariate polynomials over fields.
polyoperators1/
gcd
The n-ary greatest common divisor for univariate polynomials.
logic3/
Generalisation
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.
gen_hyperbolic1/
generalised_hyperbolic
This symbol represents the generalised hyperbolic function as recorded by Riccati. It is intended to be applied in the curried form, that is, the symbol should be applied to three arguments in order to return a function which should be applied to one argument. The generalised hyperbolic function may be defined as an infinite sum as in the first CMP/FMP .
groupname1/
generalized_quaternion_group
This symbol is a function with one argument, which should be a positive integer. When applied to n it represents the generalized quaternion group of order 4n. This is the group with three generators a, b, and c and relations c = a^2 = b^n, c*a = a*c , b*c = c*b, a*b = b*a*c, and c^2 = 1.
permgp1/
generators
This is a function with one argument, which should be a permutation group. When evaluated with argument G it returns the list of permutations which occur in the definition of G.
relation1/
geq
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.
scscp2/
This symbol is used to find the list of procedures supported by an SCSCP server.
scscp2/
get_service_description
A symbol for the client to ask for some description of a service. Note that this is a very generic description of the service running on a particular port on a particular machine. More details about for example the available symbols there may be obtained with get_allowed_heads, get_signature or get_transient_cd.
scscp2/
get_signature
A symbol for the client to inquire about the signature of a particular function.
scscp2/
get_transient_cd
This symbol is used to get the contents of a transient CD created by a server.
setname2/
GFp
This symbol represents the finite field of integers modulo p, where p is a prime.
setname2/
GFpn
This symbol represents the finite field with p^n elements, where p is a prime.
units_binaryprefix1/
gibi
This symbol represents the fact that the subsequent unit has been effectively multiplied by $2^30$. The full technical name is gigabinary.
units_siprefix1/
giga
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^9$
group3/
GL
This symbol is a function with one argument, which should be a vector space or a module V. When applied to V it represents the group of all invertible linear transformations of V.
group3/
GLn
This symbol is a function with two arguments. The first should be a positive integer n, the second a field F. When applied to n and F it represents the group of all invertible linear transformations of the vector space over F of dimension n.
prog1/
global_var
This symbol, which can have an aribtrary positive number of arguments which must all be variables, can be used to declare global variables as known to functions.
weylalgebra1/
gr
the symbol polynomial of a given differential operator. The inverse of diffop.
veccalc1/
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))
polyd/
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.
polyd2/
Total degree order, graded with the lexicographic ordering.
polyd/
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.
polyd2/
Total degree order, graded with the reverse lexicographic ordering.
SI_NamedDerivedUnits1/
gram
This symbol represents one gram. This unit is implied by the incorporation of the prefix "kilo" in the base unit standard, kilogram. Since SI prefixes may not, by SI standard, be presented by prepending to "kilogram", the gram is introduced for the application of prefixes. By itself, gram should not appear in SI compliant presentation. It has the short symbol form, "g".
units_metric1/
gramme
This symbol represents the measure of one gramme. This is not quite the standard SI measure for mass, which is the kilogramme, but OpenMath chooses to regard the gramme as standard, otherwise one would have to call it the milli-kilogramme.
graph1/
graph
This symbol represents an undirected graph. It takes two arguments: the vertex set of the graph and the edge set. The vertices can be arbitrary OpenMath objects. Each edge should be a set consisting of two vertices.
FundamentalPhysicalConstants1/
gravitational-constant
This symbol represents the constant of proportionality in Newton's law of universal gravitation. By measurement it is found to be approximately equal to 6.6742(10)*10^-11 newton metre^2 per kilogram^2. It is commonly represented with the short, italic symbol, "G".
physical_consts1/
gravitational_constant
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.
SI_NamedDerivedUnits1/
gray
This symbol represents an SI unit of absorbed dose of ionizing, radiation. A gray of absorbed dose represents one joule of energy absorbed per kilogram of mass. It has the short symbol form, "Gy".
polyd/
groebner
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.
polygb1/
groebner
The groebner basis (reduced, minimal) of a set of polynomials, with respect to a given ordering. First argument is a list of variables, the second is an ordering, the third is a list of polynomials. A program that can compute the basis is required to return a "groebner_basis" object.
polygb1/
groebner_basis
The constructor for a Groebner basis (reduced, minimal). The first is a list of variables, the second argument is an ordering, the third is the Groebner Basis itself (with respect to the ordering) that should be represented as a polynomial expression.
polyd/
groebnered
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.
polygb1/
groebnered
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.
polynomial4/
ground_ring_injected
A symbol which denotes that the ground ring of a polynomial algebra is considered to be part of the latter. This is used in the polynomial4.factor symbol to indicate that the factor is part of the factorisation of the common coefficient.
algebraic_cats/
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.
generic_alg_cats/
group
This Symbol represents the generic category of group.
gp1/
group
The n-ary function Group. The group generated by its arguments. The arguments must have a natural group operation associated with them.
group1/
group
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, an element to specify the identity, and a unary operation to specify inverses of group elements. Both the binary and unary operations should act on elements of the set and return an element of the set.
permgp1/
group
This symbol represents an n-ary function. The first argument is a group operation (usually, left_compose or right_compose), the other n-1 arguments represent permutations. When evaluated on such arguments, the function represents the permutation group generated by the last n-1 arguments.
algebraic_cats/
group_identity
This symbol takes one argument which should be a group. It returns the identity of the group.
algebraic_cats/
group_inverse
This symbol takes one argument which should be a group. It returns a unary function, which is the inverse mapping for the group.
algebraic_cats/
group_operation
This symbol takes one argument which should be a group. It returns a binary function, which represents the operation of the group.
algebraic_cats/
group_set
This symbol takes one argument which should be a group. It returns a set, which should be the set of the group.
algebraic_cats/
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.
generic_alg_cats/
groupoid
This Symbol represents the generic category of groupoid.
algebraic_cats/
groupoid_operation
This symbol takes one argument which should be a groupoid. It returns a binary function which should represent the operation of the groupoid.
algebraic_cats/
groupoid_set
This symbol takes one argument which should be a groupoid. It returns the set of the groupoid.
relation1/
gt
This symbol represents the binary greater than function which returns true if the first argument is greater than the second, it returns false otherwise.
setname2/
H
This symbol represents the set of quaternions.
plangeo2/
halfline
The halfline starting at A and going through B. The symbol takes as arguments the points A and B.
hypergeo2/
hankel1
The first Hankel function. This function is one of the famous two solutions of the Bessel differential equation at z=\infty.
hypergeo2/
hankel2
The second Hankel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=\infty.
meta_cats/
has
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.
SI_DerivedQuantities1/
heat
This symbol represents the physical quantity of energy that is transferred from one object to another due to a difference in temperature.
units_siprefix1/
hecto
This symbol represents the fact that the subsequent unit has been effectively multiplied by $100$
SI_NamedDerivedUnits1/
henry
This symbol represents an SI unit of electrical inductance. It has the short symbol form, "H".
linalg5/
Hermitian
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.
linalgsym1/
Hermitian
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.
SI_NamedDerivedUnits1/
hertz
This symbol represents an SI unit of frequency. It has the short symbol form, "Hz".
field4/
homomorphism_by_generators
This is a function with three arguments the first two of which must be fields F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. when applied to F, K, and L, the symbol represents the homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
group5/
homomorphism_by_generators
This is a function with three arguments the first two of which must be groups F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. When applied to F, K, and L, the symbol represents the group homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
ring5/
homomorphism_by_generators
This is a function with three arguments the first two of which must be monoids F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. when applied to F, K, and L, the symbol represents the monoid homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
semigroup4/
homomorphism_by_generators
This is a function with three arguments the first two of which must be semigroups F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. when applied to F, K, and L, the symbol represents the homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
SIUsed_OffSystemUnits1/
hour
This symbol represents the measure of one hour of time. It has the short symbol form, "h".
units_time1/
hour
This symbol represents the measure of one hour of time.
hypergeo1/
hypergeometric0F1
Hypergeometric function {}_0 F_1.
hypergeo1/
hypergeometric1F1
Kummer's confluent hypergeometric function.
hypergeo1/
hypergeometric2F1
The Gauss hypergeometric function. This function has a branch cut on [1,+infinity).
hypergeo1/
hypergeometric_pFq
Generalized hypergeometric function. This function has a branch cut on [1,+infinity).
logic3/
Hypothesis
This symbol represents that a wellformed formula is a hypothesis of a deduction of the propositional or predicate calculus.
nums1/
i
This symbol represents the square root of -1.
plangeo5/
ideal
This symbol is a function in one argument, which should be a coordinatized configuration (that is, each geometric object involved has coordinates). When evaluated at a configuration C it represents a function (given by a lambda binder) mapping the new parameters (arising when the inequality properties in the configuration are being translated into polynomials) to a list of polynomials in the coordinates that are variables which, when equated to zero, represent conditions equivalent to those describing the configuration C. When evaluated at an assertion assertion(C,S) it represents a function (given by a lambda binder) mapping the new parameters (arising when the inequality properties in the configuration are being translated into polynomials) to a list of polynomials in the coordinates that are variables which, when equated to zero, represent conditions equivalent to those describing the configuration C.
ring3/
ideal
This symbol represents a binary function. The first argument is a ring R and the second argument is a list or a set. When evaluated on R and such a second argument, the function represents the ideal in R generated by the entries of the list or set.
field1/
identity
This symbols represents a unary function, whose argument should be a field. It returns the identity element of the field.
fns1/
identity
The identity function, it takes one argument and returns the same value.
group1/
identity
This symbols represents a unary function, whose argument should be a group. It returns the identity element of the group.
linalg4mat/
identity
This symbol denotes a unary function which is used to construct the (nxn) identity matrix where n is the single argument. The argument n must be a natural number.
linalg5/
identity
This symbol denotes a unary function which is used to construct an (nxn) identity matrix where n is the single positive integral argument.
monoid1/
identity
This symbols represents a unary function, whose argument should be a monoid. It returns the identity element of the monoid.
ring1/
identity
This symbols represents a unary function, whose argument should be a ring. It returns the identity element of the ring.
prog1/
if
The symbol can be used to encode the if, then, else construct. The syntax is if(conditional_block,block1,block2), where the conditional_block is the block that determines wich of the block of codes block1 and block2 is going to be executed, block1 is the then block and block2 if the else block. The conditional_block and block1 are required but block2 is optional.
SI_DerivedQuantities1/
illuminance
This symbol represents the physical quantity of illuminance. A variable representing an arbitrary quantity of illuminance is commonly represented with the italic, upper case letter, "E".
fns1/
image
This symbol denotes the image of a given function, which is the set of values the domain of the given function maps to.
complex1/
imaginary
This represents the imaginary part of a complex number
logic1/
implies
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.
list2/
in
This symbol has two arguments, an element and a list. It is used to denote that the element is in the given list.
list3/
in
This symbol represents a boolean function with two arguments, an element and a list. It is used to denote that the element is in the given list.
multiset1/
in
This symbol has two arguments, an element and a multiset. It is used to denote that the element is in the given multiset.
polygb2/
in
This symbol is a function of at least 4 arguments. The first argument is a polynomial p, the second is a list of variables, the third is an ordering the fourth is a list of polynomials B, and, optionally, the fifth is a polynomial_ring. When applied to its arguments, it represents the boolean value of the assertion that p belongs to the ideal generated by B.
set1/
in
This symbol has two arguments, an element and a set. It is used to denote that the element is in the given set.
polygb2/
This symbol is a function of at least 4 arguments. The first argument should be a polynomial p, the second is a list of variables, the third is an ordering the fourth is a list of polynomials B, and optionally: the fifth is a polynomial_ring. When applied to its arguments, it represents the boolean value of the assertion that p belongs to the radical ideal generated by B.
plangeo1/
incident
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.
ThreeDgeo2/
incident
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.
poly1p/
index
index returns the index of a given indexed variable.
poly1p/
indexed_variable
indexed_variable(x,i) returns the variable x_i
indnat/
indNat
Attribution tag to denote the type of inductively defined natural numbers. It is also denoted as setname1:N.
icc/
IndType
Constructor for Inductive Types. Takes arguments the constructor functions for the inhabitants of the type and their signatures.
SI_DerivedQuantities1/
inductance
This symbol represents the physical quantity of electrical inductance. A variable representing an arbitrary quantity of inductance is commonly represented with the italic, upper case letter, "L".
aggregate_cats/
infiniteSetType
This symbol represents the type of infinite sets.
nums1/
infinity
A symbol to represent the notion of infinity.
scscp1/
info_memory
A piece of information from the system, to be used along with a procedure_completed or procedure_terminated message, describing how much memory was spent on the calculation. It should be in bytes, denoted using an OMI.
scscp1/
info_message
A piece of information from the server, to be used along with a procedure_completed or procedure_terminated message, giving some additional information. The client may choose to present this information to its user. The argument is an OMSTR.
scscp1/
info_runtime
A piece of information from the system, to be used along with a procedure_completed or procedure_terminated message, describing how much cputime was spent on the calculation. It should be in milliseconds, denoted using an OMI.
intpath1/
infty
The infty on the Riemann sphere. When the coordinate of the complex plane is z, we call t=1/z the standard coordinate around the infinity of the Riemann sphere.
numerical2/
initial_value
This symbol marks an initial value for a parameter, for example this could be the point from which a newton iteration would start.
polyslp/
inp_node
This constructor takes one argument, which is a variable. The return value is intended to represent an input node.
calculus1/
int
This symbol is used to represent indefinite integration of unary functions. The argument is the unary function.
calculus1/
int
This symbol is used to represent indefinite integration of unary functions. The argument is the unary function.
coercions/
int2flt
The function that converts an integer to a float.
omtypes/
integer
The type of integers
interval1/
integer_interval
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.
mathmltypes/
integer_type
A symbol to be used as the argument of the type symbol to convey the type of an integer.
interval_types/
integerIntervalType
This symbol represents the type of integer intervals.
ring3/
integers
This is a symbol representing the ring of integers.
algebraic_cats/
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).
generic_alg_cats/
integral_domain
This Symbol represents the generic category of integral domain.
algebraic_cats/
integral_domain_negative
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.
algebraic_cats/
integral_domain_one
This symbol takes one argument which should be an integral domain. It returns the multiplicative identity of the integral domain.
algebraic_cats/
integral_domain_plus
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.
algebraic_cats/
integral_domain_set
This symbol takes one argument which should be an integral domain. It returns the set of the integral domain.
algebraic_cats/
integral_domain_times
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.
algebraic_cats/
integral_domain_zero
This symbol takes one argument which should be an integral domain. It returns the additive identity of the integral domain.
calculus2/
integrand
This symbol represents the integrand of the integral.
multiset1/
intersect
This symbol is used to denote the n-ary intersection of multisets. It takes multisets as arguments, and denotes the multiset that contains all the elements that occur in all of them, with multiplicity the minimum of their multiplicities in all multisets.
set1/
intersect
This symbol is used to denote the n-ary intersection of sets. It takes sets as arguments, and denotes the set that contains all the elements that occur in all of them.
interval1/
interval
A symbol to denote a continuous 1-dimensional interval without any information about the character of the end points (used in definite integration). The arguments are the start and the end points of the interval in that order.
sts2/
interval
A constructor for an interval over a set.
interval1/
interval_cc
A symbol to denote a continuous 1-dimensional interval with both end points included in the interval. The arguments are the start and the end points of the interval in that order.
interval1/
interval_co
A symbol to denote a continuous 1-dimensional interval with the first point included in the interval, but the last excluded. The arguments are the start and the end points of the interval in that order.
interval1/
interval_oc
A symbol to denote a continuous 1-dimensional interval with the first point excluded from the interval, but the last included. The arguments are the start and the end points of the interval in that order.
interval1/
interval_oo
A symbol to denote a continuous 1-dimensional interval with both end points excluded from the interval. The arguments are the start and the end points of the interval in that order.
interval_types/
intervalType
This symbol represents the type of intervals.
arith2/
inverse
A unary operator which represents the inverse of an element of a set. This symbol could be used to represent additive or multiplicative inverses.
field1/
inverse
This symbol represents a unary function, whose argument should be a field S. It returns the map sending a nonzero element of S to its multiplicative inverse.
fns1/
inverse
This symbol is used to describe the inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the inverse function is ill-defined without further stipulations. No assumptions are made on the semantics of this inverse.
permutation1/
inverse
This symbol is a unary function. Its argument should be a permutation. When applied to argument P, the resulting value is the inverse permutation of P.
group1/
inversion
This symbol represents a unary function, whose argument should be a group G. It returns the map sending an element of G to its inverse.
group3/
invertibles
This symbol is a function with one argument, which should be a monoid M. When applied to M it represents the group of all invertible elements of M.
monoid1/
invertibles
This symbol is a unary function. Its argument should be a monoid M. When applied to M, it denotes the submonoid of M consisting of all invertible elements in M. This is a group.
ring3/
invertibles
This is a unary function, whose argument is a ring R. When applied to R, it denotes the set of invertible elements of R with respect to the multiplication on R.
relation0/
irreflexive
Proposition; the type of irreflexive binary relations.
gp1/
is_abelian
The unary boolean function whose value is true iff the argument is an abelian group
plangeo4/
is_affine
Boolean function testing whether a point or line is affine.
scscp2/
This symbol is used to find whether a particular procedure is supported by an SCSCP server. The reply must be either true or false, described in one of the appropriate symbols from the logic1 content dictionary.
magma1/
is_associative
The unary boolean function whose value is true iff the argument is an associative magma.
field2/
is_automorphism
This symbol is a boolean function with two arguments. The first is a field M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a field automorphism f of M.
graph2/
is_automorphism
This symbol is a boolean function with two arguments. The first is a graph M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a graph automorphism f of M.
group2/
is_automorphism
This symbol is a boolean function with two arguments. The first is a group M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a group automorphism f of M.
magma2/
is_automorphism
This symbol is a boolean function with two arguments. The first is a magma M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a magma automorphism f of M.
monoid2/
is_automorphism
This symbol is a boolean function with two arguments. The first is a monoid M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a monoid automorphism f of M.
ring2/
is_automorphism
This symbol is a boolean function with two arguments. The first is a ring M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a ring automorphism f of M.
semigroup2/
is_automorphism
This symbol is a boolean function with two arguments. The first is a semigroup M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a semigroup automorphism f of M.
permutation1/
is_bijective
This symbol has one argument which should be a endomap p. It returns true if {a_1,...,a_n}={1,...,n} where a_1,...a_n are the arguments of p and false otherwise.
field1/
is_commutative
The unary boolean function whose value is true iff the argument is a commutative field.
group1/
is_commutative
The unary boolean function whose value is true iff the argument is a commutative group.
magma1/
is_commutative
The unary boolean function whose value is true iff the argument is a commutative magma.
monoid1/
is_commutative
The unary boolean function whose value is true iff the argument is a commutative monoid.
ring1/
is_commutative
The unary boolean function whose value is true iff the argument is a commutative ring.
semigroup1/
is_commutative
The unary boolean function whose value is true iff the argument is a commutative semigroup.
plangeo5/
is_coordinatized
This symbol is a boolean valued function of one argument which must be a configuration. When applied to an argument C, it represent the value true if C is a configuration such that each object occurring in C (as well as in its subconfigurations) has coordinates (that is, the set_affine_coordinates field is present as an argument to the object), and value false otherwise. If an object already has coordinates, these are left as they are. If not, then new variables are introduced to coordinatize the object.
order1/
is_Dedekind
This symbol represents a unary boolean function. The argument should be a ring R. When evaluated on R, the function returns true if R is a Dedekind ring and false otherwise. Note that a ring R is a Dedekind ring if it is Noetherian, integrally closed (so integral) and such that every non-zero prime ideal is maximal.
logic3/
is_deduction
This symbol expresses whether or not there is a deduction of the form quoted. Hence for items of type complete_pred_deduction, it is always true
ring4/
is_domain
This symbol represents a boolean unary function. The argument is a ring R. When evaluated on R, the function returns true if R is a domain and false otherwise. A domain is a commutative ring without zero divisors.
linalgeig1/
is_eigenvalue
This symbol represents a Boolean binary function, whose first argument should be a square matrix A over a ring R and whose second argument should be an element of the ring R. Here, the matrix A acts on (row) vectors from the right and the scalar lambda is written to the left of the vector v. When applied to A and lambda, it means that there is an eigenvector vector with eigenvalue lambda.
linalgeig1/
is_eigenvector
This symbol represents a Boolean binary function, whose first argument should be a square matrix A over a ring R and whose second argument should be a vector v of size rowcount(A) over the ring R. When applied to A and v, it means that v is a left eigenvector of A.
permutation1/
is_endomap
This symbol is an n-ary function. Its arguments should be positive integers. When applied to arguments a_1,...,a_n, the resulting value is true if a_i is at most n for all i, otherwise it is false.
field2/
is_endomorphism
This symbol is a boolean function with two arguments. The first argument is a field M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a field endomorphism from M to M.
graph2/
is_endomorphism
This symbol is a boolean function with two arguments. The first argument is a graph M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a graph endomorphism from M to M.
group2/
is_endomorphism
This symbol is a boolean function with two arguments. The first argument is a group M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a group endomorphism from M to M.
magma2/
is_endomorphism
This symbol is a boolean function with two arguments. The first argument is a magma M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a magma endomorphism from M to M.
monoid2/
is_endomorphism
This symbol is a boolean function with two arguments. The first argument is a monoid M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a monoid endomorphism from M to M.
ring2/
is_endomorphism
This symbol is a boolean function with two arguments. The first argument is a ring M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a ring endomorphism from M to M.
semigroup2/
is_endomorphism
This symbol is a boolean function with two arguments. The first argument is a semigroup M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a semigroup endomorphism from M to M.
relation3/
is_equivalence
This symbol represents the boolean binary function which returns true if and only if the second argument is an equivalence relation on the first.
ring4/
is_field
This is unary boolean function whose argument should be a ring R. The value is true if and only if the ring is commutative and every nonzero element has a multiplicative inverse.
field2/
is_homomorphism
This symbol is a boolean function with three arguments. The first and arguments are fields M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a field homomorphism from M to N.
graph2/
is_homomorphism
This symbol is a boolean function with three arguments. The first and arguments are graphs M, N, the third is a map f from the vertex set of M to the vertex set of N. When applied to M, N, and f, it denotes that f is a graph homomorphism from M to N.
group2/
is_homomorphism
This symbol is a boolean function with three arguments. The first two arguments are groups M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a group homomorphism from M to N.
magma2/
is_homomorphism
This symbol is a boolean function with three arguments. The first and arguments are magmas M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a magma homomorphism from M to N.
monoid2/
is_homomorphism
This symbol is a boolean function with three arguments. The first and arguments are monoids M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a monoid homomorphism from M to N.
ring2/
is_homomorphism
This symbol is a boolean function with three arguments. The first and arguments are rings M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a ring homomorphism from M to N.
semigroup2/
is_homomorphism
This symbol is a boolean function with three arguments. The first and arguments are semigroups M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a semigroup homomorphism from M to N.
ring3/
is_ideal
The binary boolean function whose value is true if and only if the second argument is an ideal of the second.
magma1/
is_identity
This symbols represents a binary boolean function, whose arguments should be a magma and an element of the element set of the magma. When applied to the arguments M and x, it returns true if the element x is an identity of the magma M, that is, x*y = y* x = y for all elements y of M.
permgp1/
is_in
This is a Boolean function with two arguments. The first argument should be a permutation, the second a permutation group. When evaluated with first argument x and second argument G, it returns true if and only if x belongs to G.
monoid1/
is_invertible
This symbol represents a binary function, whose first argument is a monoid M and whose second argument is an element x of M. Its value is true iff the argument if x is invertible (that is, has a left and a right inverse) in M.
field2/
is_isomorphism
This symbol is a boolean function with three arguments. The first and arguments are fields M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a field isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.
graph2/
is_isomorphism
This symbol is a boolean function with three arguments. The first and arguments are graphs M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a graph isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.
group2/
is_isomorphism
This symbol is a boolean function with three arguments. The first and arguments are groups M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a group isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.
magma2/
is_isomorphism
This symbol is a boolean function with three arguments. The first and arguments are magmas M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a magma isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.
monoid2/
is_isomorphism
This symbol is a boolean function with three arguments. The first and arguments are monoids M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a monoid isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.
ring2/
is_isomorphism
This symbol is a boolean function with three arguments. The first and arguments are rings M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a ring isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.
semigroup2/
is_isomorphism
This symbol is a boolean function with three arguments. The first and arguments are semigroups M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a semigroup isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M.
permutation1/
is_list_perm
This symbol is an n-ary function. Its arguments should be positive integers. When applied to arguments a_1,...,a_n, the resulting value is true if a_i is at most n for all i and all a_i are distinct, otherwise it is false.
ring4/
is_maximal_ideal
The binary boolean function whose value is true iff the second argument is a maximal ideal of the first.
order1/
is_maximal_order
The unary boolean function whose value is true if and only if the argument is a maximal order.
plangeo3/
is_midpoint
The statement that one point is the midpoint of two others.
ThreeDgeo2/
is_midpoint
The statement that one point is the midpoint of two others.
order1/
is_nonzero_divisor
This symbol represents a boolean binary function. The first argument is a ring R, the second is an element b of R. When evaluated on R and b, the function returns true if b is a nonzero divisor in R.
gp1/
is_normal
If G, H are the group arguments, then IsNormal(G,H) returns true precisely when G is normal in H. That is, g^-1*h*g is defined and contained in H for all h in H and g in G.
group1/
is_normal
If G, H are the group arguments, then IsNormal(G,H) returns true precisely when H is normal in G. That is, inverse(g)*h*g is defined and contained in H for all h in H and g in G.
permutation1/
is_perm
This symbol is a boolean function with one argument. If the argument is not a set of cycles of length at least 2, the boolean value is false. Otherwise it is true if and only if the cycles are disjoint (that is, all entries of all cycles in the argument are mutually distinct.
permutation1/
is_permutation
This symbol is a boolean function with one argument. If the argument is not a set of cycles of length at least 2, the boolean value is false. Otherwise it is true if and only if the cycles are disjoint (that is, all entries of all cycles in the argument are mutually distinct.
ring4/
is_prime_ideal
The binary boolean function whose value is true iff the second argument is a prime ideal of the first.
finfield1/
is_primitive
This symbol represents a binary Boolean function. The first argument should be a finite field, the second a an element of that field. When applied to such arguments, the value represented is true if the second argument is a primitive element of the field, that is, a generator of the multiplicative group of the field.
permgp1/
is_primitive
The unary function with one argument, which should be a permutation group. Its value is true if and only if G acts primitively on the support of G. This means that there is no proper subset B of the support of G with more than one element such that the image of B under an element of G meets B in a proper nonempty subset of B.
permgrp/
is_primitive
The unary function whose value is true iff its permutation group argument acts primitively.
finfield1/
is_primitive_poly
This symbol is a Boolean-valued function with two arguments, the first of which should be a prime number p, and the second of which should be a polynomial with coefficients in GF(p). When applied to p and f, this symbol represents the value true if and only if f is a primitive polynomial, that is, f is irreducible over GF(p), so GF(p)[X]/(f) is a finite field of order p^n, where n is the degree of f, and the image of X in GF(p)[X]/(f) is a generator of the (cyclic) multiplicative group of GF(p)[X]/(f).
order1/
is_principal_ideal_domain
The unary boolean function whose value is true if and only if the argument is a principal ideal domain. R is a principal ideal domain if R is a commutative ring without zero divisors and if every ideal of R is a principal ideal.
relation3/
is_reflexive
This symbol represents the boolean binary function which returns true if and only if the second argument is a reflexive relation on the first.
relation3/
is_relation
This symbol is a boolean function of two arguments, S and R. The first argument should be a set. When applied to S and R, the function returns true if and only if the second argument is a subset of the Cartesian product of S with itself.
field1/
is_subfield
The binary boolean function whose value is true iff the second argument is a subfield of the second.
gp1/
is_subgroup
The binary function whose value is true if the second argument is a subgroup of the first.
group1/
is_subgroup
The binary boolean function whose value is true iff the second argument is a subgroup of the second.
permgp1/
is_subgroup
This is a Boolean function with two arguments, both of which are permutation groups. When evaluated with first argument H and second argument G it returns true if and only if H is a subgroup of G.
magma1/
is_submagma
The binary boolean function whose value is true iff the second argument is a submagma of the first.
monoid1/
is_submonoid
The binary boolean function whose value is true iff the second argument is a submonoid of the second.
ring1/
is_subring
The binary boolean function whose value is true iff the second argument is a subring of the second.
semigroup1/
is_subsemigroup
The binary boolean function whose value is true iff the second argument is a subsemigroup of the second.
relation3/
is_symmetric
This symbol represents the boolean binary function which returns true if and only if the second argument is a symmetric relation on the first.
logic3/
is_theorem
This symbol expresses whether or not there is a theorem of the form quoted. Hence for items of type complete_prop_theorem, it is always true
permgp1/
is_transitive
This is a Boolean function with one argument, which should be a permutation group. When evaluated at a permutation group G, it returns the value true if and only if the permutation group argument acts transitively on the support of G.
permgrp/
is_transitive
The unary function whose value is true iff the permutation group argument acts transitively.
relation3/
is_transitive
This symbol represents the boolean binary function which returns true if and only if the second argument is a transitive relation on the first.
ring4/
is_zero_divisor
This symbol represents a boolean binary function. The first argument is a ring R, the second is an element x of R. When evaluated on R and x, the function returns true if x a zero divisor and nonzero in R.
field2/
isomorphic
This symbol is a Boolean function with n arguments, n at least 2, which are fields. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.
graph2/
isomorphic
This symbol is a Boolean function with n arguments, n at least 2, which are graphs. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.
group2/
isomorphic
This symbol is a Boolean function with n arguments, n at least 2, which are groups. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.
magma2/
isomorphic
This symbol is a Boolean function with n arguments, n at least 2, which are magmas. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.
monoid2/
isomorphic
This symbol is a Boolean function with n arguments, n at least 2, which are monoids. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.
ring2/
isomorphic
This symbol is a Boolean function with n arguments, n at least 2, which are rings. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.
semigroup2/
isomorphic
This symbol is a Boolean function with n arguments, n at least 2, which are semigroups. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.
orthpoly/
jacobiG
The Jacobi polynomial.
SI_NamedDerivedUnits1/
joule
This symbol represents an SI unit of energy. It has the short symbol form, "J".
units_metric1/
Joule
This symbol represents the measure of one Joule. This is the standard SI measure for energy.
set3/
k_subsets
This symbol represents a binary function whose first argument should be a set and whose second argument should be a natural number. When applied to a set X and a number k, it represents the collection of all subsets of X of size k.
SI_NamedDerivedUnits1/
katal
This symbol represents an SI unit of equivalent dose of catalytic activity. A katal of catalytic activity represents the amount of catalyst to effect one mole of catalytic conversion per second. It has the short symbol form, "kat".
SI_BaseUnits1/
kelvin
This symbol represents the measure of one kelvin, the standard SI unit of measure for quantities of thermodynamic temperature. It has the short symbol form, "K", in upright roman font.
fns2/
kernel
This symbol denotes the kernel of a given function. This may be defined as the subset of the range of the given function which maps to the identity element of the image of the given function, however no semantics are assumed. The kernel of a function is also known as the null space of the function.
hypergeon0/
kernel
It returns the kernel of the map defined by a matrix in a specified domain.
linalg1p/
kernel
It returns the kernel of the map defined by a matrix in a specified domain.
ring3/
kernel
This symbol represents a unary function. Its argument is a ring homomorphism f : R -> S. When evaluated on f, the function represents the kernel in R of f, that is, the subset {x in R | f(x) = 0}.
linalgrank1/
kernel_matrix
This symbol represents a unary function whose argument should be a matrix. When applied to a matrix A, it represents a matrix whose rows are a basis of the kernel of A acting on the right.
units_binaryprefix1/
kibi
This symbol represents the fact that the subsequent unit has been effectively multiplied by $2^10$. The full technical name is kilobinary.
units_siprefix1/
kilo
This symbol represents the fact that the subsequent unit has been effectively multiplied by $1000$
SI_BaseUnits1/
kilogram
This symbol represents the measure of one kilogram of mass, the standard SI unit of measure for quantities of mass. It has the short symbol form, "kg", in upright roman font.
SI_functions1/
kind
The symbol to represent the function to return the kind of a quantity. The value of this function is referred to, but not defined in the SI. Its value, kind(Q) for a given quantity, Q, is left to the user to assign.
tensor1/
Kronecker_tensor
This symbol represents the Kronecker tensor or Kronecker delta.
hypergeo2/
kummer
Kummer's hypergeometric function.
norm1/
L_infinity_norm
This symbol signifies the $L_\infty$ norm.
norm1/
L_norm
This symbol signifies the $L_p$ norm for any $p$ (the case of $L_\infty$ is handled specially).
fns1/
lambda
This symbol is used to represent anonymous functions as lambda expansions. It is used in a binder that takes two further arguments, the first of which is a list of variables, and the second of which is an expression, and it forms the function which is the lambda extraction of the expression
lc/
Lambda
The abstraction constructor. It is followed by a list of variables and an OpenMath object.
veccalc1/
Laplacian
This symbol is used to represent the laplacian 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: laplacian(F) = \partial^2(F)/\partial(x_1)^2 + ... + \partial^2(F)/\partial(x_n)^2
altenc/
LaTeX_encoding
A symbol which heads a piece of LaTeX encoding in an attribution.
hypergeon2/
lauricella_FA
Lauricella's hypergeometric series F_A of n variables. In case of one variables, it agrees with the Appel function F_2. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages:
hypergeon2/
lauricella_FC
Lauricella's hypergeometric series F_C of n variables. In case of two variable, it agree with the Appel function F_4. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages:
hypergeon2/
lauricella_FD
Lauricella's hypergeometric series F_D of n variables. In case of two variables, it agree with the Appell function F_1. reference: authors: "Appel, Kampe de Feriet" title: "Les Fonctions Hypergeometriques de Plusieurs Variables et Polynome d'Hermite" pages:
arith1/
lcm
The symbol to represent the n-ary function to return the least common multiple of its arguments.
poly/
lcm
The least common multiple of its polynomial arguments. This is unique up to units, but the choice must be compatible with that made for gcd: see the CMP/FMP.
polynomial3/
lcm
The least common multiple of its polynomial arguments. This is unique up to units, but the choice must be compatible with that made for gcd, so that the product of the gcd and the lcm equals the product of all arguments.
poly/
The leading coefficient with respect to a variable (the second argument). We note that the leading coefficient of 0 is undefined.
polynomial1/
This symbol represents a unary function, whose argument should be univariate polynomial. When applied to a polynomial, it represents the coefficient of the monomial of highest degree. If the polynomial is zero, the value represented is zero.
polynomial1/
This symbol represents a unary function, whose argument should be a nonzero univariate polynomial. When applied to such a polynomial, it represents the highest power of the variable occurring in the polynomial.
polynomial1/
This symbol represents a unary function, whose argument should be univariate polynomial. When applied to a polynomial, it represents its leading term, that is the term that is the product of the highest power of the variable and its coefficient. If the polynomial is zero, the value represented is zero.
fns1/
left_compose
This symbol represents the function which forms the left-composition of its two (function) arguments.
permutation1/
left_compose
This symbol is a binary function. Its arguments should be permutations. When applied to arguments P1 and P2, the resulting value is the permutation which maps x in Support(P1) union Support(P2) to P1(P2(x)).
group4/
left_coset
This symbol represents a ternary function whose first argument is a group G, whose second argument is a subgroup H of G, and whose third argument is an element x of G. Its value on G, H, and x is the left coset of H in G containing x, that is, the set x H.
group4/
left_coset_representative
This symbol represents a quaternary function whose first argument is a group G, whose second argument is a subgroup H of G, whose third argument is left_transversal T of H in G, and whose fourth argument is an element of G. It assigns to G, H, T, g the element of t of T representing the left coset of H containing g, that is, t H = g H .
group4/
left_cosets
The binary function whose value is the set of left cosets of the second argument in the first.
magma1/
left_divides
This symbol is a ternary function. Its first argument should be a magma M and the second and third arguments should be elements of M. When applied to M, a, and b, it denotes the fact that a is a left_divisor of b in M. This means that there is v in M such that av=b.
magma1/
left_expression
This symbol is a binary function. Its first argument should be a magma M, the second argument a list L of elements of M. When applied to M and L, it denotes the left product (L * ( ... (L[n-1] * L[n]) ... )) of all elements in the list L.
fns1/
left_inverse
This symbol is used to describe the left inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the left inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this left inverse.
field2/
left_multiplication
This symbol is a function with two arguments, which should be a field M and an element x of M. When applied to M and x, it denotes left multiplication on M by x.
group2/
left_multiplication
This symbol is a function with two arguments, which should be a group M and an element x of M. When applied to M and x, it denotes left multiplication on M by x.
monoid2/
left_multiplication
This symbol is a function with two arguments, which should be a monoid M and an element x of M. When applied to M and x, it denotes left multiplication on M by x.
ring2/
left_multiplication
This symbol is a function with two arguments, which should be a ring M and an element x of M. When applied to M and x, it denotes left multiplication on M by x.
semigroup2/
left_multiplication
This symbol is a function with two arguments, which should be a semigroup M and an element x of M. When applied to M and x, it denotes left multiplication on M by x.
group5/
left_quotient_map
This symbol is a binary function whose first argument is a group G and whose second argument is an subgroup H of G. When applied to G and H, its value is the natural quotient map from G to the quotient group G/H, sending x to the right coset Hx of G.
polyslp/
left_ref
Takes as argument a node of an slp. Returns the value of the left hand pointer of the node.
monoid3/
left_regular_representation
This is a unary function whose argument must be a monoid M. When applied to M, it represents the map from M to the maps monoid on M that assigns to m left multiplication by m on M.
semigroup3/
left_regular_representation
This is a unary function whose argument must be a semigroup M. When applied to M, it represents the map from M to the maps semigroup on M that assigns to m left multiplication by m on M.
group4/
left_transversal
The binary function whose value is a set of representatives for the left cosets of the second argument as a subgroup of the first.
orthpoly/
legendreP
The first Legendre function. This function is one of the two famous solutions of Legendre differential equation.
orthpoly/
legendreQ
The second Legendre function. This function is the another one of the famous two solutions of Legendre differential equation.
dimensions1/
length
This symbol represents the length physical dimension.
list3/
length
This symbol represents a function whose argument should be a list. It returns the length of its argument.
list3/
length
This symbol represents a function whose argument should be a list. It returns the length of its argument.
permutation1/
length
This symbol is a function with one argument, which must be a cycle. When applied to cycle(a_1,a_2,...,a_n), it returns the number n. This number is referred to as the length of the cycle.
permutation1/
length
This symbol is a function with one argument, which must be a cycle. When applied to cycle(a_1,a_2,...,a_n), it returns the number n. This number is referred to as the length of the cycle.
polyslp/
length
A unary function taking an slp as argument and returning the length of this slp.
SI_BaseQuantities/
length
This symbol represents the SI base quantity of length. It has the short symbol form, "L".
relation1/
leq
This symbol represents the binary less than or equal to function which returns true if the first argument is less than or equal to the second, it returns false otherwise.
tensor1/
Levi-Civita
This symbol represents the Levi-Civita alternating pseudo-tensor or permutation symbol. It's definition depends on the number of dimensions, d, of the space: it has as many indexes as there are dimensions in the space. It is totally antisymmetric, its value being: 1 for an even permutation of unequally valued indexes (e.g., (1,2,...,d)); -1 for an odd permutation of unequally valued indexes, and; 0 whenever two indexes take the same value.
polyd/
lexicographic
The lexicographic ordering of terms. 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.
polyd2/
lexicographic
The lexicographic ordering of monomials.
expint/
li
The symbol li defines the basic logarithmic integral as in Abramovitz & Stegun equation 5.1.2. This is a Cauchy principal value integral: $$li(x)=\int_0^x\frac1{\ln t}t dt\qquad(x>1)$$ 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
set2/
lift_binary
This symbol denotes the lift of a binary operator on elements of X to a component-wise operators on subsets of X.
physical_consts1/
light_year
This symbol represents the distant for which a beam of light would take a year to traverse, in a vacuum.
limit1/
limit
This symbol is used to denote the limit of a unary function. It takes 3 arguments: the limiting value of the argument, the method of approach (either null, above, below or both_sides) and the function.
moreerrors/
limitation
This symbol represents the error which is returned when an application reads an error caused by the limitations of an implementation when dealing with OpenMath objects such as limits on the size of objects or on the kind of objects manipulated. This can include limits on the size of a bytearray or integer, a limit on the number of arguments of an application or the inability to deal with Unicode characters outside ISO latin 1. 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.
plangeo1/
line
The symbol is used to indicate a line of planar Euclidean geometry by a variable. The line may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
ThreeDgeo1/
line
The symbol is used to indicate a line of 3-dimensional Euclidean geometry by a variable. The line may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
ThreeDgeo2/
line_parallel
The symbol represents a binary boolean function with input two lines or segments. Its value is true whenever the first argument is parallel to the second.
equations1/
linear
A predicate to indicate that an equation or system of equations is linear, i.e. is expressed in terms of constants and first order terms.
list1/
list
This symbol denotes the list construct which is an n-ary function. The list entries must be given explicitly.
list1/
list
This symbol denotes the list construct which is an n-ary function. The list entries must be given explicitly.
list1/
list
This symbol denotes the list construct which is an n-ary function. The list entries must be given explicitly.
sts2/
list
A constructor for the type of a homogeneous list
list3/
list_of_lengthn
This symbol represents a function with two arguments, the first of which is a natural number and the second of which is a list. The first argument is the length of the list.
list4/
list_of_lengthn
This symbol represents a function with two arguments, the first of which is a natural number and the second of which is a list. The first argument is the length of the list.
permutation1/
list_perm
This symbol is an n-ary constructor. Its arguments should be distinct positive integers in the interval [1,n]. When applied to arguments a_1,...,a_n, the resulting value is the permutation mapping i to a_i for i=1,...,n.
list2/
list_selector
This symbol takes a positive integer n and a list, and represents the n-th element of that list.
list3/
list_selector
This symbol takes a positive integer n and a list, and represents the n-th element of that list.
linalg7/
list_to_matrix
This symbol denotes a binary function. Its first argument must be a ring R, its second argument must be list L of lists of equal lengths whose entries belong to the ring R, up to ring1.expression. When applied to R and L it represents the matrix whose i,j entry consists of the j-th entry from the list L[i]. In particular, the matrix has length(L) rows and length(L) columns.
polyd3/
list_to_poly_d
This symbol is a function with two arguments. The first argument is a ring R and the second argument is a list L. The entries of L are elements of R or can be cast canconically onto elements of R. When applied to R and L, the symbol denotes the distributed (univariate) polynomial over R with terms (L[i-1],i) for i running over the indices of L (i=1, ..., length(L)).
linalg5/
list_to_vector
This symbol denotes a binary function. Its first argument must be a ring R, its second argument must be list L with entries belonging to the ring R, up to ring1.expression. When applied to R and L it represents the vector of the same length as the list L whose i-th coordinate is L[i] (or ring1.expression(L[i])).
linalg7/
list_to_vector
This symbol denotes a binary function. Its first argument must be a ring R, its second argument must be list L with entries belonging to the ring R, up to ring1.expression. When applied to R and L it represents the vector of the same length as the list L whose i-th coordinate is L[i] (or ring1.expression(L[i])).
mathmltypes/
list_type
A symbol to be used as the argument of the type symbol to convey the type for a list.
permutation1/
listendomap
This symbol is a function with one argument which is a permutation whose support consists of positive integers. When applied to such a permutation P, it represents the list of length n whose i-th entry is the image of i under P, where n is the maximum of the support of P.
permutation1/
listperm
This symbol is a function with one argument which is a permutation whose support consists of positive integers. When applied to such a permutation P, it represents the list of length n whose i-th entry is the image of i under P. Here n is at least the maximum of the support of P.
rdf/
literal_lang
A symbol to be used within an OpenMath attribute to specify the language of an RDF literal. The annotation value should be an OpenMath string.
rdf/
literal_type
A symbol to be used within an OpenMath attribute to specify the type of an RDF literal which is represented as an OpenMath object. The annotation value should be an rdf.resource.
SIUsed_OffSystemUnits1/
litre
This symbol represents the volume measure of one litre. It has the short symbol form, "l" or "L".
units_metric1/
litre
This symbol represents the measure of one litre. This is a standard metric measure for physical volume.
units_metric1/
litre_pre1964
This symbol represents the previous (1901-1964) measure of one litre. This used to be a standard metric measure for physical volume.
transc1/
ln
This symbol represents the ln function (natural logarithm) as described in Abramowitz and Stegun, section 4.1. It takes one argument. Note the description in the CMP/FMP of the branch cut. If signed zeros are in use, the inequality needs to be non-strict.
transc3/
ln
This symbol represents the ln function (natural logarithm) as a multivalued function.
prog1/
local_var
This symbol, which can have an aribtrary positive number of arguments which must all be variables, can be used to declare local variables. represents
ThreeDgeo3/
locus
The symbol is used to indicate by a variable the locus set traced by a point T in a 3-dimensional Euclidean geometry configuration C. That is, the set of all points satisfying the conditions imposed on T in the configuration C. The locus may (but need not) be defined by constraints on the point T additional to those in the configuration. The symbol takes the variable as the first argument, the tracer point T as second argument and the additional constraints as further arguments.
transc1/
log
This symbol represents a binary log function; the first argument is the base, to which the second argument is log'ed. It is defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1
transc3/
log
This symbol represents a binary log function; the first argument is the base, to which the second argument is log'ed. It is defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1
directives1/
look_up
This symbol is a function with one argument, which should be a mathematical expression. When applied to a mathematical expression, it asks for mathematical expressions related to the argument. If the argument is a single OpenMath symbol, the service might respond by the list of all properties in the CD containing the argument. Alternatively, the service can provide a list of CDs which use the CD in which the argument occurs. Another service might return all documents in which the symbol occurs. If the argument is a more complicated object, the same services could be called for, but then with all OpenMath symbols occurring in the argument instead of the single one.
physical_consts1/
Loschmidt_constant
This symbol represents the number of particles per unit volume of an ideal gas at standard temperature and pressure. It is approximately 2.686763 * 10^(25) +/- 2.3 * 10^(20) per metre cubed.
linalg5/
lower-Hessenberg
This symbol represents a lower-Hessenberg matrix, it takes one argument, the argument is a vector of vectors representing the non-zero elements. The first element of the argument specifies the value of the first super-diagonal, the subsequent elements specify the value of the diagonal and subsequent subdiagonals, all other elements are zero.
linalg5/
lower-triangular
This symbol represents a lower-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.
matrix1/
lower_band
This symbol is a binary function whose first argument is a non-negative OpenMath integer which denotes the index of the lower band which is specified in the second argument. Hereby the first lower band is the one immediately below the main (generalised) diagonal, its starting coordinates relative to the top-left of the matrix thus are (2, 1).
linalgspec2/
lower_Hessenberg
This symbol represents a lower_Hessenberg matrix, it takes one argument, the argument is a vector of vectors representing the non-zero elements. The first element of the argument specifies the value of the first super-diagonal, the subsequent elements specify the value of the diagonal and subsequent subdiagonals, all other elements are zero.
linalgspec1/
lower_triangular
This symbol represents a lower-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.
relation1/
lt
This symbol represents the binary less than function which returns true if the first argument is less than the second, it returns false otherwise.
SI_NamedDerivedUnits1/
lumen
This symbol represents an SI unit of luminous flux. It has the short symbol form, "lm".
SI_DerivedQuantities1/
luminous-flux
This symbol represents the physical quantity of luminous flux. A variable representing an arbitrary quantity of luminous flux is commonly represented with the italic, upper case letter, "Φv" (\phi; sub V).
SI_BaseQuantities/
luminous-intensity
This symbol represents the SI base quantity of luminous intensity. It has the short symbol form, "J".
SI_NamedDerivedUnits1/
lux
This symbol represents an SI unit of illuminance. It has the short symbol form, "lx".
ring3/
m_poly_ring
This symbol represents a binary function. The first argument should be a ring and the second a list or a set. When evaluated on such arguments R and L, the function represents the free commutative ring over R generated by the elements (or entries) of L. This ring can also be viewed as the ring of polynomials over R with variables the elements of L.
magma1/
magma
This symbol is a constructor for magmas. It takes two arguments in the following order: a set to specify the elements in the magma and a binary operation to specify the magma operation. The binary operation should act on elements of the set and return an element of the set.
semigroup1/
magma
This symbol is a unary function. Its argument should be a semigroup S. When applied to S, it denotes the magma with the same element set and binary operation as S.
SI_DerivedQuantities1/
magnetic-flux
This symbol represents the physical quantity of magnetic flux. A variable representing an arbitrary quantity of magnetic flux is commonly represented with the italic, upper case greek letter, "\Phi;".
SI_DerivedQuantities1/
magnetic-flux-density
This symbol represents the physical quantity of magnetic flux density. A variable representing an arbitrary quantity of magnetic flux density is commonly represented with the italic, upper case letter, "B".
physical_consts1/
magnetic_constant
This symbol represents the ratio of the magnetic flux density in a substance to the external field strength for vacuum. It is equal to 4 pi x 10^(-7) H/m.
semigroup/
make_Semigroup
The contructor for the tuples consisting of a setoid, and an associative binary operation.
setoid/
make_Setoid
The contructor for the tuples consisting of a set, an equivalence relation on the set, and a proof that the relation is actually an equivalence relation.
list1/
map
This symbol represents a mapping function which may be used to construct lists; it takes as arguments a function from X to Y and a list over X in that order. The value that is returned is a list of values in Y. The argument list may be a set or an integer_interval.
list1/
map
This symbol represents a mapping function which may be used to construct lists, it takes as arguments a function from X to Y and a list over X in that order. The value that is returned is a list of values in Y. The argument list may be a set or an integer_interval.
list1/
map
This symbol represents a mapping function which may be used to construct lists; it takes as arguments a function from X to Y and a list over X in that order. The value that is returned is a list of values in Y. The argument list may be a set or an integer_interval.
set1/
map
This symbol represents a mapping function which may be used to construct sets, it takes as arguments a function from X to Y and a set over X in that order. The value that is returned is a set of values in Y. The argument list may be a set or an integer_interval.
set3/
map_with_condition
This symbol represents a function with three arguments. The first argument is a function assignment f (in the form of a lambda binding), the second argument is a set X. The third argument specifies a Boolean function P on X defining the subset Z of X (so Z = {x in X| P(x)}) on which the first argument f is defined, The symbol is used to denote the image {f(x) | x in X and P(x)} of application of the function f on the elements of Z.
set3/
map_with_target
This symbol represents a function with three arguments. The first argument is a function assignment f (in the form of a lambda binding), the second argument is a set X on which the first argument f is defined. The third argument specifies the range Y of the function. The symbol is used to denote the image {f(x) in Y | x in X} of application of the function f on the elements of X (so as to form a subset of Y).
set3/
map_with_target_and_condition
This symbol represents a function with four arguments. The first argument is a function assignment f (in the form of a lambda binding), the second argument is a set X on which the first argument f is defined. The third argument specifies the range Y of the function. The fourth argument specifies a Boolean function P on X defining the subset Z of X (so Z = {x in X| P(x)}) on which the first argument f is defined, The symbol is used to denote the image {f(x) in Y | x in X and P(x)} of application of the function f on the elements of Z.
monoid3/
maps_monoid
This is a unary function whose argument must be a set X or a positive integer. When applied to X, it refers to the monoid of all functions from X to X if X is a set and to {1,...,X} if X is an integer, whose binary operation is composition of maps and whose identity element is the identity map on the set X, respectively {1,...,X}.
semigroup3/
maps_semigroup
This is a unary function whose argument must be a set X or a positive integer. When applied to X, it refers to the semigroup of all functions from X to X if X is a set and to {1,...,X} if X is an integer, whose binary operation is composition of maps and whose identity element is the identity map on the set X, respectively {1,...,X}.
fns4/
maps_to
This symbol denotes a binding constructor. The body of the binder should be a list [A1,A2] of length 2. It is used to represent a function assignment A1 -> A2, where the bound variables occur in A1 and possibly in A2. The expressions A1, A2 should represent objects uniquely determined by given values of the bound variables, wihtin the range of definition of the domain.
lc/
mapsto
The type constructor of non-dependant function space. The first n-1 children denote the types of the arguments, the last denotes the return type. Contrary to sts:mapsto, arguments cannot be variables but have to be OpenMath objects that map to ground OpenMath terms (they contain no variables).
sts/
mapsto
This symbol represents the construction of a function type. The first n-1 children denote the types of the arguments, the last denotes the return type.
dimensions1/
mass
This symbol represents the mass physical dimension.
SI_BaseQuantities/
mass
This symbol represents the SI base quantity of mass. It has the short symbol form, "M".
altenc/
MathML_encoding
A symbol which heads a piece of MathML encoding in an attribution. The MathML encoding is an XML encoding, and the details may be found at: http://www.w3.org/Math/Overview.html
linalg2/
matrix
This symbol is an n-ary matrix constructor which requires matrixrow's as arguments. It is used to represent matrices.
linalg2col/
matrix
This symbol is an n-ary matrix constructor which requires matrixcolumn's as arguments. It is used to represent matrices.
linalg3/
matrix
This symbol is an n-ary matrix constructor which requires matrixcolumn's as arguments. It is used to represent matrices.
matrix1/
matrix
This symbol is a binary function whose first argument must be a matrix algebra constructor and the second argument can be any of the below matrix entry constructors. Additionally it is possible to use the matrix constructors of the linalg2 or linalg3 CDs.
sts2/
matrix
A constructor for the type of a matrix
matrix1/
matrix_domain
This symbol is a ternary function, whose first argument should be a matrix1.entry_domain application. The second and third arguments must be matrix1.row_dimension and matrix1.column_dimension. When applied to these arguments this `creates' the domain of linear mappings between modules of specified dimensions over a common ground domain, conveniently represented by matrices.
polyd/
matrix_ordering
The argument is a matrix with as many columns as indeterminates (= rank). Each row in turm is multiplied by the column vector of exponents to produce a weighting for comparison purposes.
polyd2/
matrix_ordering
The argument is a matrix with as many columns as indeterminates (= rank). Each row in turm is multiplied by the column vector of exponents to produce a weighting for comparison purposes.
ring3/
matrix_ring
This symbol represents a binary function. The first argument is a positive integer n, the second is a ring R. When evaluated on such argument n and R, the function represents the ring of n x n matrices over R.
linalg1/
matrix_selector
This symbol represents the function which allows individual entries to be selected from a matrix. It takes three arguments, the first is the index of the row and the second is the index of the column of the required element, the third argument is the matrix in question. The indexing is one based, i.e. the element in the top left hand corner is indexed by (1,1).
linalg6/
matrix_tensor
This symbol denotes a n-nary function which is used to construct the tensor product matrix of its arguments, which must be matrices.
linalg5/
matrix_to_list
This symbol denotes a unary function. Its argument must be a matrix A. When applied to A it represents the list L whose i-th entry (for i=1,...rowcount(A)) is the lists L[i] whose j-th entry (for j=1,...columncount(A)) is the element A[i,j].
mathmltypes/
matrix_type
A symbol to be used as the argument of the type symbol to convey the type for a matrix (n tuple of rows, where each row is an m tuple for some m, it should be noted that each row must be the same length).
linalg2col/
matrixcolumn
This symbol is an n-ary constructor used to represent columns of matrices. Its arguments should be members of a ring.
linalg3/
matrixcolumn
This symbol is an n-ary constructor used to represent columns of matrices. Its arguments should be members of a ring.
linalg2/
matrixrow
This symbol is an n-ary constructor used to represent rows of matrices. Its arguments should be members of a ring.
minmax1/
max
This symbol denotes the unary maximum function which takes a set as its argument and returns the maximum element in that set.
order1/
maximal_order
This is a binary function, whose first argument is a Dedekind ring R and the second is a polynomial f. When applied to R and f, it returns the maximal order A among the orders of f (over the polynomial ring of R) in the quotient field of A. Note that the result is unique.
s_data1/
mean
This symbol represents an n-ary function denoting the mean of its arguments. That is, their sum divided by their number.
s_dist1/
mean
This symbol represents a unary function denoting the mean of a distribution. The argument is a univariate function to describe the distribution. That is, if f is the function describing the distribution. The mean is the expression integrate(x*f(x)) w.r.t. x over the range (-infinity,infinity).
units_binaryprefix1/
mebi
This symbol represents the fact that the subsequent unit has been effectively multiplied by $2^20$. The full technical name is megabinary.
s_data1/
median
This symbol represents an n-ary function denoting the median of its arguments. That is, if the data were placed in ascending order then it denotes the middle one (in the case of an odd amount of data) or the average of the middle two (in the case of an even amount of data).
units_siprefix1/
mega
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^6$
SI_BaseUnits1/
metre
This symbol represents the measure of one metre of length, the standard SI unit of measure for quantities of length or physical distance. It has the short symbol form, "m", in upright roman font.
units_metric1/
metre
This symbol represents the measure of one metre. This is the standard SI unit measure for physical distance.
units_metric1/
metre_sqrd
This symbol represents the measure of one metre squared. This is the standard SI measure for physical area.
units_metric1/
metres_per_second
This symbol represents the measure of one metre per second. This is the standard SI measure for speed.
units_metric1/
metres_per_second_sqrd
This symbol represents the measure of one metre per second squared. This is the standard SI measure for acceleration.
tensor1/
metric_tensor
This symbol represents the metric tensor, typically depicted using a lower case g. The metric tensor is a nondegenerate, symmetric bilinear form. It defines the ideas of leng th and angle in a metric space, the most common example being the Euclidean metric. The square of a differential length, ds*ds, is given by the bilinear product of the coordinate differentials, dx^i, with the metric tensor.
units_siprefix1/
micro
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^-6$
plangeo3/
midpoint
The midpoint between two points or two endpoints of a segment.
ThreeDgeo1/
midpoint
The symbol is used to indicate the midpoint of a segment in 3-dimensional Euclidean geometry by a variable. The symbol takes the variable as the first argument and the segment as second argument.
units_imperial1/
mile
This symbol represents the measure of one (land, or statute) mile. This is a standard imperial measure for distance, defined in terms of the foot.
units_us1/
mile_us_survey
This symbol represents the measure of one U.S. Survey mile.
units_imperial1/
miles_per_hr
This symbol represents the measure of one mile per hour. This is a standard imperial measure for speed.
units_imperial1/
miles_per_hr_sqrd
This symbol represents the measure of one mile per hour squared. This is a standard imperial measure for acceleration.
units_siprefix1/
milli
This symbol represents the fact that the subsequent unit has been effectively multiplied by $0.001$
minmax1/
min
This symbol denotes the unary minimum function which takes a set as its argument and returns the minimum element in that set.
polygb2/
minimal_groebner_element
This symbol is a function with 3 arguments. First argument is a list of variables, the second is an ordering, the third is a list B of polynomials. [Optionally, the fourth is a polynomial ring.] When applied to its arguments, it represents the polynomial in the Groebner basis of B with respect to the ordering with the least leading monomial.
finfield1/
minimal_polynomial
This symbol represents a function with one or two arguments. Its first argument should be an element x of a finite field F. The second argument should be a subfield K of F. It returns the minimal polynomial of x over K. If there is only one argument, K defaults to the prime subfield of F.
linalgpoly1/
minimum_poly
This symbol represents a binary function. This first argument should be a square matrix A defined over a field F, the second argument a variable X. When applied to A and X, it represents the minimum polynomial of A in the variable X over the field F. (The output should be semantically equivalent to an object obtained by the poly_ring_d_named constructor of the CD polyd1.)
arith1/
minus
The symbol representing a binary minus function. This is equivalent to adding the additive inverse.
field1/
minus
This symbol represents a unary function, whose argument should be a field S. It returns the map sending an element of S to its additive inverse.
opnode/
minus
A constant value, constructs the minus for subtraction nodes.
polyd1/
minus
The sum. The argument is a DMPL. The sum lies within the same "poly_ring_d", i.e., a program implementing this operation should return a DMP with the same "poly_ring_d".
hypergeon0/
minus_part
The argument is a vector. It replaces positive elements in the vector to zero and negative elements to their absolute values.
linalg1p/
minus_part
The argument is a vector. It replaces positive elements in the vector to zero and negative elements to their absolute values.
SIUsed_OffSystemUnits1/
minute
This symbol represents the measure of one minute of time. It has the short symbol form, "min".
units_time1/
minute
This symbol represents the measure of one minute of time.
SIUsed_OffSystemUnits1/
minute-of-arc
This symbol represents the angular measure of one minute of arc. It has the short symbol form, "'".
s_data1/
mode
This symbol represents an n-ary function denoting the mode of its arguments. That is the value which occurs with the greatest frequency.
integer2/
modulo_relation
This symbol represents a univariate function, whose argument should be an integer. When applied to an integer m, it denotes the equivalence relation of being equal modulo m on Z.
polynomial2/
modulo_relation
This symbol represents a univariate function, whose argument should be a polynomial. When applied to a polynomial m, it denotes the equivalence relation of being equal modulo m.
logic3/
ModusPonens
This symbol represents the generation of a line of a proof by application of Modus Ponens. The first argument is the new well-formed formula (B), the second is the line number in the proof for A and the third is the line number in the proof for A implies B.
physical_consts1/
mole
This symbol represents the number of atoms in one gramme of carbon(12).
SI_BaseUnits1/
mole
This symbol represents the measure of one mole, the standard SI unit measure for quantities of amount of substance. It has the short symbol form, "mol", in upright roman font.
s_data1/
moment
This symbol is used to denote the i'th moment of a set of data. The first argument should be the degree of the moment (that is, for the i'th moment the first argument should be i), the second argument should be the point about which the moment is being taken and the rest of the arguments are treated as the data. For n data values x_1, x_2, ..., x_n the i'th moment about c is (1/n) ((x_1-c)^i + (x_2-c)^i + ... + (x_n-c)^i). See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, section 7.7.1.
s_dist1/
moment
This symbol represents a ternary function to denote the i'th moment of a distribution. The first argument should be the degree of the moment (that is, for the i'th moment the first argument should be i), the second argument is the value about which the moment is to be taken and the third argument is a univariate function to describe the distribution. That is, if f is the function which describe the distribution. The i'th moment of f about a is the integral of (x-a)^i*f(x) with respect to x, over the interval (-infinity,infinity).
SI_DerivedQuantities1/
moment-of-force
This symbol represents the physical quantity of force.
dimensions1/
momentum
This symbol represents the momentum physical dimension, it is mass times velocity.
SI_DerivedQuantities1/
momentum
This symbol represents the physical quantity of momentum.
algebraic_cats/
monoid
This is the constructor for monoids. A monoid comprises a set and an operation over elements of the set. The set must contain a unique identity element (relative to the operation). That is an element, I, such that I*a=a*I=a where a represents an arbitrary element of S and * represents the operation. The operation * must be associative over S. The monoid constructor takes three arguments, the set of the monoid, a binary function taking two elements of the set into itself to represent the operation of the monoid and an element of the set to represent the identity of the monoid.
generic_alg_cats/
monoid
This Symbol represents the generic category of monoid.
group1/
monoid
This symbol is a unary function, whose argument should be a group G. When applied to G its value is the monoid underlying G.
monoid1/
monoid
This symbol is a constructor for monoids. It takes three arguments in the following order: a set to specify the elements in the monoid, a binary operation to specify the monoid operation, and an element to specify the identity. The binary operation should act on elements of the set and return an element of the set.
algebraic_cats/
monoid_identity
This symbol takes one argument which should be a monoid, it returns the identity of the monoid.
algebraic_cats/
monoid_operation
This symbol takes one argument which should be a monoid, it returns the operation of the monoid.
algebraic_cats/
monoid_set
This symbol takes one argument which should be a monoid, it returns the set of the monoid.
polyslp/
monte_carlo_eq
This is a Monte-Carlo equality test, it takes three arguments, the first two are slps representing polynomials, the third argument is the maximum probability of incorrectness that is required of the equality test. (Monte-Carlo equality tests are very important for slps as they offer the only tractable method of solving the equality problem in many cases)
hypergeon2/
multi_pochhammer
multi_pochhammer is a product of pochhammer symbols.
hypergeon0/
multi_power
multi_power is for using the multi-index notation.
poly1p/
multi_power
multi_power is for using the multi-index notation.
combinat1/
multinomial
The multinomial coefficient, multinomial(n, n1, ... nk) is the number of ways of choosing ni objects of type i (i from 1 to k) without regard to order, in such a way that the total number of objects chosen is n. multinomial(n, n1, ... nk) is equal to n!/(n1!*n2! ...*nk!).
field1/
multiplication
This symbol represents a unary function, whose argument should be a field S. It returns the multiplication map on the field. We allow for the map to be n-ary.
group1/
multiplication
This symbol represents a unary function, whose argument should be a group G. It returns the multiplication map on G. We allow for the map to be n-ary.
magma1/
multiplication
This symbol represents a unary function, whose argument should be a magma G. It returns the multiplication map on G. We allow for the map to be n-ary.
monoid1/
multiplication
This symbol represents a unary function, whose argument should be a monoid M. It returns the multiplication map on M. We allow for the map to be n-ary.
ring1/
multiplication
This symbol represents a unary function, whose argument should be a ring S. It returns the multiplication map on S. We allow for the map to be n-ary.
semigroup1/
multiplication
This symbol represents a unary function, whose argument should be a semigroup S. It returns the multiplication map on S. We allow for the map to be n-ary.
field1/
multiplicative_group
This symbol is a unary function, whose argument should be a field S. When applied to S its value is the multiplicative group on the nonzero elements of S.
ring3/
multiplicative_group
This is a unary function, whose argument is a ring R. When applied to R, it denotes the group of invertible elements of R with respect to the multiplication on R.
ring1/
multiplicative_monoid
This symbol is a unary function, whose argument should be a ring S. When applied to S its value is the monoid underlying S.
polynomial4/
multiplicity
A symbol which represents the multiplicity of a factor in a factorisation and takes exactly one argument which must be a positive integer.
multiset1/
multiset
This symbol represents the multiset construct. It is either an n-ary function, in which case the multiset entries are given explicitly, or it works on an elements construct. There is no implied ordering to the elements of a multiset.
aggregate_cats/
multisetType
This symbol represents the type of multisets.
setname1/
N
This symbol represents the set of natural numbers (including zero).
meta/
Name
An element containing the string corresponding to the name of the symbol being defined. This must match the syntax for symbol names given in the OpenMath Standard. Here and elsewhere white space occurring at the begining or end of the string will be ignored.
nums1/
NaN
A symbol to convey the notion of not-a-number. The result of an ill-posed floating computation. See IEEE standard for floating point representations.
logic1/
nand
This symbol represents the logical nand function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if all arguments are true or true otherwise.
units_siprefix1/
nano
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^-9$
sts/
nary
Constructs a child of mapsto which denotes an arbitrary number of copies of the argument of nary.
sts/
nassoc
Constructs a child of mapsto which denotes an arbitrary number of copies of the argument of nassoc. The operator is associative on these arguments which means that repeated uses may be flattened/unflattened.
ring1/
negation
This symbol represents a unary function, whose argument should be a ring S. It returns the map sending an element of S to its additive inverse.
SIUsed_OffSystemUnits1/
neper
This symbol represents the dimensionless measure of one neper, the natural unit for representing logarithms of ratios of field amplitudes, such as voltage or pressure. It has the short symbol form, "Np".
relation1/
neq
This symbol represents the binary inequality function.
integer2/
neqmod
This symbol represents a Boolean valued trivariate function, whose arguments should be integers. When applied to integers a, b, m, it denotes the Boolean evalue of the assertion that a and b are not equal modulo m.
polynomial2/
neqmod
This symbol represents a Boolean valued trivariate function, whose arguments should be polynomials. When applied to polynomials a, b, m, it denotes the Boolean evalue of the assertion that a and b are not equal modulo m.
SI_NamedDerivedUnits1/
newton
This symbol represents an SI unit of force. It has the short symbol form, "N".
units_metric1/
Newton
This symbol represents the measure of one Newton. This is the standard SI measure for force.
units_metric1/
Newton_per_sqr_metre
This symbol represents the measure of one Newton per square metre. This is another (deprecated in OpenMath) name for the standard SI measure for pressure, the Pascal.
list2/
nil
The empty list
list2/
nil
The empty list
scscp2/
no_such_transient_cd
Used for errors that arise when the client asks for a transient cd that the server cannot handle.
polyslp/
node_selector
Takes an slp as the first argument, the second argument is the position of the required node. Returns the node of the slp at this position.
algebraic_cats/
non_commutative_ring
This symbol is the constructor for non commutative rings, these are rings over which the * operator is not commutative. the non_commutative_ring constructor takes five arguments: The set of the non-commutative ring. 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 non-commutative ring to specify the additive identity, 0. And a unary function taking the set of the non-commutative ring into itself to represent the additive inverses of the non-commutative ring (i.e. inverses under +, or negatives).
generic_alg_cats/
non_commutative_ring
This Symbol represents the generic category of non-commutative ring.
algebraic_cats/
non_commutative_ring_negative
This symbol takes one argument which should be a non-commutative ring. It returns a unary function, which represents the multiplicative inverse of the non-commutative ring.
algebraic_cats/
non_commutative_ring_plus
This symbol takes one argument which should be a non-commutative ring. It returns a binary function, which represents the additive function of the non-commutative ring.
algebraic_cats/
non_commutative_ring_set
This symbol takes one argument which should be a non-commutative ring. It returns the set of the non-commutative ring.
algebraic_cats/
non_commutative_ring_times
This symbol takes one argument which should be a non-commutative ring. It returns a binary function, which represents the multiplicative function of the non-commutative ring.
algebraic_cats/
non_commutative_ring_zero
This symbol takes one argument which should be a non-commutative ring. It returns the zero of the non-commutative ring.
aggregate_cats/
non_continuousSetType
This symbol represents the type of non-continuous sets.
equations1/
non_linear
A predicate to indicate that an equation or system of equations is non-linear, i.e. contains terms of order greater than 1.
linalg6/
nonZeroRowSparseMatrix
This symbol may be used for representing matrices, it is designed for efficiently representing sparse matrices where every row has at least one non-zero entry. This is an n+1 ary symbol, where n is the number of rows in the matrix. The first argument must be the number of columns in the matrix, every following argument of the symbol must be an application of a sparseMatrixRow symbol which has arguments which are sparseMatrixElement2, one sparseMatrixElement2 element for each row in the matrix, in the order in which they occur in the matrix. Any non-specified entry is implicitly zero.
linalgspars1/
nonZeroRowSparseMatrix
This symbol may be used for representing matrices, it is designed for efficiently representing sparse matrices where every row has at least one non-zero entry. This is an n+1 ary symbol, where n is the number of rows in the matrix. The first argument must be the number of columns in the matrix, every following argument of the symbol must be an application of a sparseMatrixRow symbol which has arguments which are sparseMatrixElement2, one sparseMatrixElement2 element for each row in the matrix, in the order in which they occur in the matrix. Any non-specified entry is implicitly zero.
linalg6/
nonZeroRowSparseMatrix01
This symbol may be used to represent matrices which have no zero rows, and for which every row is in Z_2 efficiently. The first argument is the number of columns in the matrix, the following arguments are sparseMatrixRow elements where the arguments are sparseMatrixElement4 elements. Any non-specified entry is implicitly zero.
linalgspars1/
nonZeroRowSparseMatrix01
This symbol may be used to represent matrices which have no zero rows, and for which every row is in Z_2 efficiently. The first argument is the number of columns in the matrix, the following arguments are sparseMatrixRow elements where the arguments are sparseMatrixElement4 elements. Any non-specified entry is implicitly zero.
logic1/
nor
This symbol represents the logical nor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if any of the arguments are true or true otherwise.
ThreeDgeo2/
normal
The symbol represents a binary boolean function with a line as first argument and a plane as second argument. Its value is true whenever the first argument is normal to the second.
gp1/
normal_closure
The binary function whose value is the set of conjugates of the elements of the second group by elements of the first, where multiplication between them is defined.
group1/
normal_closure
The binary function whose value is the set of conjugates of the elements of the second group by elements of the first, where multiplication between them is defined.
group3/
normalizer
This symbols represents a binary function whose first argument should be a group G and whose second argument should be a set of elements or a subgroup L of the group G. Its value is the subgroup of G of all elements normalizing L.
logic1/
not
This symbol represents the logical not function which takes one boolean argument, and returns the opposite boolean value.
multiset1/
notin
This symbol has two arguments, an element and a multiset. It is used to denote that the element is not in the given multiset.
set1/
notin
This symbol has two arguments, an element and a set. It is used to denote that the element is not in the given set.
multiset1/
notprsubset
This symbol has two (multiset) arguments. It is used to denote that the first multiset is not a proper subset of the second. A proper subset of a multiset is a subset of the multiset but not actually equal to it.
set1/
notprsubset
This symbol has two (set) arguments. It is used to denote that the first set is not a proper subset of the second. A proper subset of a set is a subset of the set but not actually equal to it.
multiset1/
notsubset
This symbol has two (multiset) arguments. It is used to denote that the first multiset is not a subset of the second.
set1/
notsubset
This symbol has two (set) arguments. It is used to denote that the first set is not a subset of the second.
calculus1/
nthdiff
This symbol is used to express the nth-iterated ordinary differentiation of a unary function. The first argument is n, and the second the unary function.
calculus1/
nthdiff
This symbol is used to express the nth-iterated ordinary differentiation of a unary function. The first argument is n, and the second the unary function.
calculus1/
nthpartialdiff
This symbol is used to express the nth-iterated partial differentiation of a function of more than one variable. It has three arguments, the first is a list of positive integers which index the variables of the function, the second is a list of integers which specify the order of differentiation with respect to the corresponding variable, the third argument is the function. Application of the symbol should be taken as meaning the following: differentiation of the third argument with respect to the variables indexed by the first argument. The orders of differentiation are specified by the second argument, in the following manner: The i'th element of the second argument is the order of differentiation of the variable indexed by the i'th element of the first argument.
limit1/
null
This symbol is used within a limit construct to avoid specifying the method of approach to the limit. It takes no arguments.
SI_functions1/
num
The symbol to represent the function to return the numerical value of a quantity in terms of a product of powers of SI base units.
order1/
number_field
This symbol is a constructor for number fields. It takes two arguments in the following order: a ring R and a monic irreducible univariate polynomial f. If the ring R is Z (or Q), it returns the absolute number field. Otherwise it returns the relative number field over the number field whose ring of integers is R. This symbol is intended to be used in upcoming CDs for e.g. describing discriminants of number fields, or Galois groups, unit groups, class groups, regulators, etc.; all useful number theoretical notions.
sts/
NumericalValue
Denotes an OpenMath object that is to be thought of as something that represents a numerical value, or a numerical value.
asymp1/
O
The O symbol represents a unary function which constructs a set of certain functions of type reals to reals. The condition f(n)=O(g(n)) is intended to express an upper bound condition on f.
asymp1/
o
The o symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The condition f(n) = o(g(n)) is intended to express a lower bouund condition on f. Formally we say that f(n) = o(g(n)) if and only if the limit as n tends to infinity of f(n)/g(n) exists and is equal to 0.
sts/
Object
Denotes any OpenMath object.
SI_NamedDerivedUnits1/
ohm
This symbol represents an SI unit of electrical resistance. It has the short symbol form, "\Omega;".
asymp1/
omega
The omega symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The omega symbol represents a set of functions such that for any function in the set omega(g(x)), f(x); it is not true that f(x) is in o(g(x)). Formally we say that f(x) = omega(g(x)) if and only if there is an epsilon > 0 and an infinite sequence x_1, x_2, x_3, ... such that for all j then abs(f(x_j)) > epsilon * g(x_j).
asymp1/
Omega
The Omega symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The Omega symbol represents a set of functions such that for any function in the set Omega(g(x)), f(x); it is not true that f(x) is in O(g(x)).
omtypes/
omtype
The type of symbolic type symtype
alg1/
one
This symbol represents the multiplicative identity element.
alg1/
one
This symbol represents the multiplicative identity element.
SI_BaseQuantities/
one
This symbol represents the proposed SI base quantity of dimension one, or the dimensionless quantity. It has the short symbol form, "1".
SI_BaseUnits1/
one
This symbol represents the dimensionless unit corresponding to the dimensionless quantity dimension. It has the short symbol form, "1".
linalg1p/
ones
It returns a vector of a specifed size of which elements are one. 1-ary function.
polyslp/
op_node
This constructor takes three arguments. The first argument is a symbol from opnode, meant to specify whether the node is a plus, minus times or divide node, the second and third arguments are integers, which are the numbers of the lines which are the arguments of the operation
scscp1/
option_debuglevel
An option, to be given along with a procedure call, describing the amount of debug information the client is interested in. Should be an integer.
scscp1/
option_max_memory
An option, to be given along with a procedure call, describing the maximum amount of memory (in bytes) the system should spend on this call.
scscp1/
option_min_memory
An option, to be given along with a procedure call, describing the minimum amount of memory (in bytes) the system should be able to spend on this call. The idea is that in certain cases we know in advance that we will need a large amount of memory. If the system will never be able to provide that, it would be a waste of time and resources to even start the computation.
scscp1/
An option, to be given along with a procedure call, indicating that the client would like to have a cookie (i.e. a reference to an OpenMath object residing somewhere) as return value.
scscp1/
option_return_nothing
An option, to be given along with a procedure call, indicating that the client expects no return value.
scscp1/
option_return_object
An option, to be given along with a procedure call, indicating that the client would like to have the actual OpenMath object as return value.
scscp1/
option_runtime
An option, to be given along with a procedure call, describing the maximum amount of time (in milliseconds) the system should spend on this call.
logic1/
or
This symbol represents the logical or function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if any of the arguments are true or false otherwise.
permgp1/
orbit
The binary function whose first argument should be a permutation group G. If the second argument is an element of the support of G, the value is the orbit of the second argument under the action of G. Otherwise, it is the singleton consisting of the second argument.
permgrp/
orbit
The binary function whose value is the set of integers which are in the orbit of the second argument under the action of the first argument which is a permutation group.
permgp1/
orbits
This is a function with one argument, which should be a permutation group. When evaluated at a permutation group G, it returns the set of all orbits of G on elements from the support of G.
integer2/
ord
This symbol denotes a binary function. Its first argument shoud be a prime number p, the second an integer n. When applied to p and n, it represents the highest power of p occurring in a factorization of n.
order1/
order
This is a binary function, whose first argument is a Dedekind ring R and the second is a polynomial f. When applied to R and f, it returns an order of f over the polynomial ring R: it is a ring A containing R, which is finitely generated R-module with no nilpotent non-zero ideal and as a R-module it is torsion-free. Note that the result is not unique. Also this function allows to compute an order of a polynomial over another polynomial ring. The idea behind this computation is to coerce f into the polynomial ring of R and then compute the order.
permgp1/
order
This is a function with one argument, which should be a permutation group. When evaluated with argument G it returns the size of the group G.
permutation1/
order
This symbol is a function with one argument which should be a permutation. When applied to a permutation P, it represents the least positive integer n for which composition of n copies of P represents the identity (that is, a permutation with empty support). Note: in this definition of the order, it does not matter whether left_compose or right_compose is being used.
permutation1/
order
This symbol is a function with one argument which should be a permutation. When applied to a permutation P, it represents the least positive integer n for which composition of n copies of P represents the identity (that is, a permutation with empty support). Note: in this definition of the order, it does not matter whether left_compose or right_compose is being used.
relation0/
order
Proposition; the type of order relations, namely relations that are reflexive, antisymmetric and transitive.
algebraic_cats/
ordered_Abelian_group
This symbol is the constructor for ordered Abelian groups, that is an Abelian group on which there is an ordering relation. The ordered_Abelian_group constructor takes five arguments, the set of the ordered Abelian group, a binary function taking two elements of the set into itself to represent the operation of the ordered Abelian group, an element of the set to represent the identity of the ordered Abelian group, a unary function taking the set into itself to specify inverse elements and a binary function taking two elements of the set into the booleans to specify the ordering of the ordered Abelian group.
generic_alg_cats/
ordered_Abelian_group
This Symbol represents the generic category of ordered Abelian group.
algebraic_cats/
ordered_Abelian_group_identity
This symbol takes one argument which should be an ordered Abelian group. It returns the identity of the ordered Abelian group.
algebraic_cats/
ordered_Abelian_group_inverse
This symbol takes one argument which should be an ordered Abelian group. It returns a unary function, which is the inverse function of the ordered Abelian group.
algebraic_cats/
ordered_Abelian_group_operation
This symbol takes one argument which should be an ordered Abelian group. It returns a binary function, which represents the operation of the ordered Abelian group.
algebraic_cats/
ordered_Abelian_group_order
This symbol takes one argument which should be an ordered Abelian group. It returns a binary function, which should represent the ordering of the ordered Abelian group.
algebraic_cats/
ordered_Abelian_group_set
This symbol takes one argument which should be an ordered Abelian group. It returns the set of the ordered Abelian group.
algebraic_cats/
ordered_Abelian_monoid
This symbol is the constructor for ordered Abelian monoids, that is Abelian monoids on which there is an ordering relation. The ordered_Abelian_monoid constructor takes four arguments, the set of the ordered Abelian monoid, a binary function taking two elements of the set into itself to represent the operation of the ordered Abelian monoid, an element of the set to represent the identity of the ordered Abelian monoid and a binary function taking two elements of the set into the booleans to represent the ordering of the ordered Abelian monoid.
generic_alg_cats/
ordered_Abelian_monoid
This Symbol represents the generic category of ordered Abelian monoid.
algebraic_cats/
ordered_Abelian_monoid_identity
This symbol takes one argument which should be an ordered Abelian monoid. It returns an element of the set of the ordered Abelian monoid, which should be the identity of the ordered Abelian monoid.
algebraic_cats/
ordered_Abelian_monoid_operation
This symbol takes one argument which should be an ordered Abelian monoid. It returns a binary function between elements of the set of the ordered Abelian monoid, which should represent the operation of the ordered Abelian monoid.
algebraic_cats/
ordered_Abelian_monoid_order
This symbol takes one argument which should be an ordered Abelian monoid. It returns a binary function between elements of the set of the ordered Abelian monoid, which should represent the ordering relation on the ordered Abelian monoid.
algebraic_cats/
ordered_Abelian_monoid_set
This symbol takes one argument which should be an ordered Abelian monoid. It returns a set which should be the set of the ordered Abelian monoid.
algebraic_cats/
ordered_group
This symbol is the constructor for ordered groups, that is a group on which there is an ordering relation. The ordered_group constructor takes five arguments, the set of the ordered group, a binary function taking two elements of the set into itself to represent the operation of the ordered group, an element of the set to represent the identity of the ordered group, a unary function taking the set into itself to specify inverse elements of the ordered group and a binary function taking two elements of the set into the booleans to specify the ordering of the ordered group.
generic_alg_cats/
ordered_group
This Symbol represents the generic category of ordered group.
algebraic_cats/
ordered_group_identity
This symbol takes one argument which should be an ordered group. It returns the identity of the ordered group.
algebraic_cats/
ordered_group_inverse
This symbol takes one argument which should be an ordered group. It returns a unary function, which is the inverse function of the ordered group.
algebraic_cats/
ordered_group_operation
This symbol takes one argument which should be an ordered group. It returns a binary function, which represents the operation of the ordered group.
algebraic_cats/
ordered_group_order
This symbol takes one argument which should be an ordered group. It returns a binary function, which represents the ordering of the ordered group.
algebraic_cats/
ordered_group_set
This symbol takes one argument which should be an ordered group. It returns the set of the ordered group.
algebraic_cats/
ordered_monoid
This is the constructor for ordered monoids, that is monoids on which there is an ordering relation. The ordered_monoid constructor takes four arguments, the set of the ordered monoid, a binary function taking two elements of the set into itself to represent the operation of the ordered monoid, an element of the set to represent the identity of the ordered monoid and a binary function taking two elements of the set into the booleans to represent the ordering on the ordered monoid.
generic_alg_cats/
ordered_monoid
This Symbol represents the generic category of ordered monoid.
algebraic_cats/
ordered_monoid_identity
This symbol takes one argument which should be an ordered monoid. It returns an element of the set of the ordered monoid, which should be the identity of the ordered monoid.
algebraic_cats/
ordered_monoid_operation
This symbol takes one argument which should be an ordered monoid. It returns a binary function between elements of the set of the ordered monoid, which should represent the operation of the ordered monoid.
algebraic_cats/
ordered_monoid_order
This symbol takes one argument which should be an ordered monoid. It returns a binary function between elements of the set of the ordered monoid, which should represent the ordering relation on the ordered monoid.
algebraic_cats/
ordered_monoid_set
This symbol takes one argument which should be an ordered monoid. It returns a set which should be the set of the ordered monoid.
algebraic_cats/
ordered_ring
This symbol is the constructor for ordered rings, that is a ring on which there is an ordering relation. The ordered_ring constructor takes six arguments, the set of the ordered ring, a binary function from the set into itself to represent the multiplicative operation (*), a binary function from the set into itself to represent the additive operation (+), an element of the set of the ordered ring to represent the additive identity 0, a unary function from the set into itself to represent additive inverses (i.e. inverses under +, or negatives) and a binary function from the set into the booleans to represent the ordering relation.
generic_alg_cats/
ordered_ring
This Symbol represents the generic category of ordered ring.
algebraic_cats/
ordered_ring_negative
This symbol takes one argument which should be an ordered ring. It returns a unary function to represent the additive inverse function of the ordered ring.
algebraic_cats/
ordered_ring_order
This symbol takes one argument which should be an ordered ring. It returns a binary function, which represents the order function on the ordered ring.
algebraic_cats/
ordered_ring_plus
This symbol takes one argument which should be an ordered ring. It returns a binary function, which represents the additive operation of the ordered ring.
algebraic_cats/
ordered_ring_set
This symbol takes one argument which should be an ordered ring. It returns the set of the ordered ring.
algebraic_cats/
ordered_ring_times
This symbol takes one argument which should be an ordered ring. It returns a binary function, which represents the multiplicative operation of the ordered ring.
algebraic_cats/
ordered_ring_zero
This symbol takes one argument which should be an ordered ring. It returns the zero of the ordered ring.
polyd/
ordering
Used as an attribute to indicate an ordering of the terms in a polynomial or list of polynomials. The value of this attribute should be one of the constructors specifying ordering.
polyd2/
ordering
Used as an attribute to indicate an ordering of the monomials in a polynomial or list of polynomials. The value of this attribute should be one of the constructors specifying ordering.
interval1/
oriented_interval
A symbol to denote a continuous 1-dimensional interval without any information about the character of the end points (used in definite integration). The arguments are the start and the end points of the integration, in either order.
mathmlattr/
other
A symbol to be used within an OpenMath attribute to specify the MathML "other" attribute of the object. The annotation should be an OpenMath string representing the value of the other attribute.
piece1/
otherwise
This symbol introduces the 'default' value of a piecewise construct. If none of the previous piece constructs can provide the value, this will. It has a single child, the value.
linalg1/
outerproduct
This symbol represents the outer product function. It takes two vector arguments and returns a matrix. It is defined as follows: if we write the {i,j}'th element of the matrix to be returned as m_{i,j}, then: m_{i,j}=a_i * b_j where a_i,b_j are the i'th and j'th elements of a, b respectively.
setname1/
P
This symbol represents the set of positive prime numbers.
ecc/
Pair
The pairing constructor. It takes two OpenMath objects as first element and second element of the pair, and a third optional OpenMath object that represents the type of the pair.
ecc/
PairProj1
The first projection function that extracts the first component of a Pair. It satisfies the sigma-reduction rule.
ecc/
PairProj2
The second projection function that extracts the second component of a Pair. It satisfies sigma-reduction rule.
plangeo3/
parallel
parallel is a binary boolean function with input two lines, halflines or segments. Its value is true whenever the two inputs are parallel.
relation0/
partial_equivalence
Proposition; the type of partial_equivalence relations, namely relations that are symmetric, and transitive.
calculus1/
partialdiff
This symbol is used to express partial differentiation of a function of more than one variable. It has two arguments, the first is a list of integers which index the variables of the function, the second is the function.
calculus1/
partialdiff
This symbol is used to express partial differentiation of a function of more than one variable. It has two arguments, the first is a list of positive integers which index the variables of the function, the second is the function. Application of the symbol should be taken as meaning the first partial differentiation of the function (the second argument) in each one of the variables indexed by the list of integers (its first argument).
weylalgebra1/
partialdiff
partial differentiation of a given function.
calculus1/
partialdiffdegree
This symbol is used to express partial differentiation of a function of more than one variable. It has three arguments, the first is a list of integers which give the degrees by which the function is differentiated by the corresponding variable. The second is the total degree (which should therefore be the sum of the values in the first list, but may be given symbolically). The third is the function.
poly/
partially_factored
The constructor for a factorization. Its arguments are formal powers (see operator above), where nothing in particular is assumed about the polynomials (they may or may not be irreducible, or relatively prime).
SI_NamedDerivedUnits1/
pascal
This symbol represents an SI unit of pressure. It has the short symbol form, "Pa".
units_metric1/
Pascal
This symbol represents the measure of one Newton per square metre. This is the standard SI measure for pressure.
intpath1/
path_in_sector
The symbol path_in_sector(c,t1,t2) is an outgoing path in a sufficiently small sector with the center c and the angles t1 and t2. The path starts from the point c and it is sufficiently short. When the center is intpath1.infty, the angle should be given in the coordinate t=1/z.
intpath1/
path_in_sector2
The symbol path_in_sector2(c,t1,t2,z0) is an outgoing path in the sector with the center c and the angles t1 and t2. The path is the segment from the point c to the point z0 which lies in the sector.
units_binaryprefix1/
pebi
This symbol represents the fact that the subsequent unit has been effectively multiplied by $2^50$. The full technical name is petabinary.
permutation1/
perm
This symbol is an n-ary function. Its arguments should be positive integers. When applied to arguments a_1,...,a_n, the resulting value is the permutation mapping i to a_i.
permut1/
permutation
The n-ary constructor permutation. The arguments are the integers 1 .. n in some order. permutation(p1, ..., pn) is the function which takes 1 to p1, 2 to p2 and so on.
permutation1/
permutation
This symbol is an n-ary constructor whose arguments are cycles of length at least 2 with the property that all entries of all cycles are mutually distinct. The permutation symbol constructs a bijective map from the set X of entries of the cycles to X. The map is defined as follows: if E occurs as an entry of a cycle, then the permutation maps E to the entry following E in the same cycle if it exists and to the first entry in the same cycle otherwise. When applied to an element y outside X, the constructed permutation is considered to fix y.
permutation1/
permutation
This symbols is an n-ary function whose arguments are cycles of length at least 2 with the property that all entries of all cycles are mutually distinct. The permutation symbol constructs a bijective map from the set X of entries of the cycles to X. The map is defined as follows: if E occurs as an entry of a cycle, then the permutation maps E to the entry following E in the same cycle if it exists and to the first entry in the same cycle otherwise.
permutation1/
permutationsn
This symbol is a unary function. Its argument should be a positive integer. When applied to argument n, the resulting value is the set of all permutations of the set {1,..., n}.
plangeo3/
perpbisector
Given two distinct points A and B, this is the line of all points at equal distance to both A and B.
plangeo3/
perpendicular
perpendicular is a binary boolean function with input two lines, halflines or segments. Its value is true whenever the two inputs are perpendicular to each other.
ThreeDgeo2/
perpendicular
The symbol represents a binary boolean function with input two lines or segments. Its value is true whenever the first argument is perpendicular to the second.
plangeo3/
perpline
Given a point p and a line L, this defines the line through p perpendicular to L.
units_siprefix1/
peta
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^15$
nums1/
pi
A symbol to convey the notion of pi, approximately 3.142. The ratio of the circumference of a circle to its diameter.
units_siprefix1/
pico
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^-12$
piece1/
piece
This introduces an individual component of a piecewise definition. It has precisely two arguments: the first is the value, and the second is a Boolean (meant to be a predicate) which is true if and only if this piece is to provide the value of the piecewise construct.
piece1/
piecewise
This operator heads an expression that is being defined piecewise. Its arguments are n objects built with the piece constructor, optionally followed by one built with otherwise constructor.
fns3/
piecewise_continuous
A predicate to indicate that a function is piecewise continuous everywhere, i.e. continuous at all but a finite number of points in its domain.
fns3/
piecewise_continuous_on
A predicate to indicate that a function is continuous at all but a finite number of points in a region.
aggregate_cats/
piecewiseContinuousSetType
This symbol represents the type of piecewise continuous sets.
units_imperial1/
pint
This symbol represents the measure of one (imperial) pint. This is the standard imperial measure for volume. See units_us1 for the U.S. pint.
units_us1/
pint_us_dry
This symbol represents the measure of one U.S. dry pint.
units_us1/
pint_us_liquid
This symbol represents the measure of one U.S. liquid pint.
lc/
PiType
The type constructor of dependant function space. It binds the (type-attributed) variables in the body, that is an OpenMath object.
FundamentalPhysicalConstants1/
Planck-charge
The Planck charge is defined to be sqrt(h-bar*c*4*pi*eps0). Its value derived from measurement is 1.875545870(47) * 10^−18 coulomb. It is commonly represented with the short, italic symbol, "q", subscripted with an upright capital "P".
FundamentalPhysicalConstants1/
Planck-constant
This symbol represents the fundamental constant equal to the ratio of the energy of a photon to its frequency. By measurement it is found to be approximately equal to 6.62606896(33)*10^(-34) J s [CODATA 2006]. It is commonly represented with the short, italic symbol, "h".
FundamentalPhysicalConstants1/
Planck-length
The Planck length is defined to be sqrt(h-bar*G/(c^3)). Its value derived from measurement is 1.616252(81) * 10^−35 metre. It is commonly represented with the short, italic symbol, "l", subscripted with an upright capital "P".
FundamentalPhysicalConstants1/
Planck-mass
The Planck mass is defined to be sqrt(h-bar*c/G). Its value derived from measurement is 2.17644(11) * 10^−8 kilogram. It is commonly represented with the short, italic symbol, "m", subscripted with an upright capital "P".
FundamentalPhysicalConstants1/
Planck-temperature
The Planck temperature is defined to be sqrt(h-bar*c^5/(G*k^3)). Its value derived from measurement is 1.416785(71) × 10^32 kelvin. It is commonly represented with the short, italic symbol, "T", subscripted with an upright capital "P".
FundamentalPhysicalConstants1/
Planck-time
The Planck time is defined to be sqrt(h-bar*G/(c^5)). Its value derived from measurement is 5.39124(27) * 10^−44 second. It is commonly represented with the short, italic symbol, "t", subscripted with an upright capital "P".
physical_consts1/
Planck_constant
This symbol represents the fundamental constant equal to the ratio of the energy of a quantum of energy to its frequency. It is approximately equal to 6.6260755*10^(-34) +/- 4.0*10^(-40) Joule seconds.
ThreeDgeo1/
plane
The symbol is used to indicate a plane in 3-dimensional Euclidean geometry by a variable. The plane may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
ThreeDgeo2/
plane_parallel
The symbol represents a binary boolean function with input two planes. Its value is true whenever the first argument is parallel to the second.
arith1/
plus
The symbol representing an n-ary commutative function plus.
indnat/
plus
Addition of natural numbers defined recursively by using the successor.
opnode/
plus
A constant value, constructs the plus for addition nodes.
polyd/
plus
The sum. The argument is a DMPL. The sum lies within the same "PolyRingD" i.e. a program implementing this operation should return a DMP with the same "poly_ring_d" (or "poly_ring_d_named").
polyd1/
plus
The sum. The argument is a DMPL. The sum lies within the same "poly_ring_d", i.e., a program implementing this operation should return a DMP with the same "poly_ring_d".
hypergeon0/
plus_part
The argument is a vector. It replaces negative elements in the vector to zero.
linalg1p/
plus_part
The argument is a vector. It replaces negative elements in the vector to zero.
hypergeo0/
pochhammer
Pochhammer symbol
plangeo1/
point
The symbol is used to indicate a point of planar Euclidean geometry by a variable. The point may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
ThreeDgeo1/
point
The symbol is used to indicate a point of 3-dimensional Euclidean geometry by a variable. The point may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
plangeo3/
polarline
Given a point p and a circle C this defines the polar line of p with respect to C.
polyd3/
poly_d_named_to_arith
This symbol is a unary function. Its argument is a DMP with named variables. When applied to R, the symbol denotes the arithmetic expression that is the sum of the terms.
polyd3/
poly_d_to_arith
This symbol is a binary function. The first argument is a DMP and the second argument is a list of objects, typically variables or arithmetic expressions, at least as many as there are variables in the ring to which the DMP belongs. When applied to R and L, the symbol denotes the arithmetic expression that is the sum of the terms with the i-th variable of the ring of the DMP being substituted by the i-th expression or variable of the list L.
polyr/
poly_r_rep
A constructor for the representation of polynomials. The first argument is the polynomial variable, the rest are monomials (in decreasing order of exponent).
ring3/
poly_ring
This symbol represents a binary function. The first argument should be a ring and the second a variable. When evaluated on such arguments R and X, the function represents the free commutative ring over R generated by X. This ring can also be viewed as the ring of polynomials over R with indeterminate X.
polyd/
poly_ring_d
The constructor of polynomial ring. The first argument is a ring (the ring of the coefficients), the second is the number of variables as an integer.
polyd1/
poly_ring_d
The constructor of polynomial ring. The first argument is a ring (the ring of the coefficients), the second is the number of variables as an integer.
polyd/
poly_ring_d_named
The constructor of polynomial ring. The first argument is a ring (the ring of the coefficients), the remaining arguments are the names of the variables. The first variable given is the most important from the point of view of lexicographic ordering, then the second, and so on.
polyd1/
poly_ring_d_named
The constructor of polynomial ring. The first argument is a ring (the ring of the coefficients), the remaining arguments are the names of the variables. The first variable given is the most important from the point of view of lexicographic ordering, then the second, and so on.
polyslp/
poly_ring_SLP
The constructor of the polynomial ring. The first argument is a ring, (the ring of the coefficients), the rest are the variables, in any order.
polyu/
poly_u_rep
A constructor for the representation of polynomials. The first argument is the polynomial variable, the rest are monomials (in decreasing order of exponent).
plangeo5/
polynomial_assertion
This symbol is a function in one argument, which should be an assertion whose configuration is coordinatized (that is, each geometric object involved has coordinates). When evaluated at an assertion assertion(C,T) it represents the assertion that the constant polynomial 1 belongs to the ideal of the polynomial ring over a coefficient ring R containing the rationals and all global (unbound) coordinates of C, in the bound variables of ideal(C) and an external variable t, generated by ideal(C)[bound variables] and 1-f_T t. Here f_T is a polynomial such that f_T=0 is equivalent to the thesis T being true. This means f_T is in the radical ideal of ideal(C)[bound variables]. The interpretation is as follows: There are no parameter choices for the bound variables such that f_T is nonzero. In other words, for all parameter choices of a coordinatization of C, we must have f_T=0. So the truth of the assertion that thesis T holds in configuration C is reflected by the truth of polynomial_assertion(C,T).
polyr/
polynomial_r
The constructor of Recursive Polynomials. The first argument is the polynomial ring containing the polynomial and the second is a "poly_r_rep".
polysts/
polynomial_ring
The type of all polynomial rings, e.g. from polyr or polyd OCDs
polyr/
polynomial_ring_r
The constructor of a recursive polynomial ring. The first argument is a ring (the ring of the coefficients), the rest are the variables (in order).
polyu/
polynomial_ring_u
The constructor of a univariate polynomial ring. The first argument is a ring (the ring of the coefficients), the second is the variable.
polyslp/
polynomial_SLP
The constructor of Polynomials built with Straight Line Program representation. The first argument is the polynomial ring containing the polynomial built with poly_ring_SLP, The second argument is the program body built with prog_body.
polyu/
polynomial_u
The constructor of Recursive Polynomials. The first argument is the polynomial ring containing the polynomial and the second is a "poly_u_rep".
polynomial4/
possibly_reducible
A symbol which denotes that the irreducibility of a factor of the factorisation is not guaranteed.
units_imperial1/
pound_force
This symbol represents the measure of force of one pound.
units_imperial1/
pound_mass
This symbol represents the measure of the mass which weighs one pound under the influence of standard gravity.
arith1/
power
This symbol represents a power function. The first argument is raised to the power of the second argument. When the second argument is not an integer, powering is defined in terms of exponentials and logarithms for the complex and real numbers. This operator can represent general powering.
dimensions1/
power
This symbol represents the power physical dimension, it is energy per time.
field1/
power
This is a symbol with two or three arguments. Its first argument should be an element g of a field and the second argument should be an integer. The optional third argument is the field G containing g. It denotes the element g^k in G.
group1/
power
This is a symbol with three arguments. The first argument is a group G. Its second argument is an element g of G and the third argument is an integer k. It denotes the element g^k in G.
poly/
power
Takes a polynomial and a (non-negative) integer and produces a formal power. Although OpenMath does not specify operational semantics, the idea here is that these powers are not evaluated. We note that the power from arith1 would suggest the expanded form.
polyd/
power
The power. First argument is a DMP, second argument is the integer power. The power lies within the same "PolyRingD" i.e. a program implementing this operation should return a DMP with the same "poly_ring_d" (or "poly_ring_d_named").
polyd1/
power
The power. First argument is a DMP, second argument is the integer power. The power lies within the same "poly_ring_d", i.e., a program implementing this operation should return a DMP with the same "poly_ring_d".
ring1/
power
This is a symbol with two or three arguments. Its first argument should be a an element g of a ring and the second argument should be an integer. The optional third argument is the ring G containing g. It denotes the element g^k in G.
SI_DerivedQuantities1/
power
This symbol represents the physical quantity of power, or energy divided by time. A variable representing an arbitrary quantity of power is commonly represented with the italic, upper case letter, "P".
set3/
powerset
This symbol represents unary function whose argument should be a set. When applied to a set X, it represents the collection of all subsets of X.
relation0/
pre_order
Proposition; the type of preorder relations, namely relations that are reflexive and transitive.
logic3/
pred_deduction
This symbol is used to claim 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).
logic3/
pred_theorem
This symbol is used to claim 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).
fns2/
predicate_on_list
This symbol is used to denote the chains of application or a binary predicate typified by a < b < c. In particular it is used to support the usage in MathML, where transative relations are classed as nary, but the underlying OpenMath symbols are binary. The symbol takes two arguments; the first of which is the binary predicate, the second a list. It is true if every application of the predicate on a pair of elements of the list, taken in order, returns true, otherwise it is false.
rdf/
prefix
A symbol to be used as the head of the OpenMath application to construct a prefix mapping that can be used as a value of the prefixes attribution. The two arguments of this function should be OpenMath strings representing in order, the prefix and the corresponding namespace URI.
units_ops1/
prefix
This symbol represents the fact that the subsequent unit has been effectively multiplied by 1,000 ($10^{3}$)
rdf/
prefixes
A symbol to be used within an OpenMath attribute to specify one or more RDF namespace prefixes. The annotation value should be a set1.set of pairs of strings (prefix name, namespace URI) constructed with the prefix symbol.
dimensions1/
pressure
This symbol represents the pressure physical dimension.
SI_DerivedQuantities1/
pressure
This symbol represents the physical quantity of pressure. A variable representing an arbitrary quantity of pressure is commonly represented with the italic, lower case letter, "p".
finfield1/
primitive_element
This symbol has one or two arguments. If there is only one argument, it must be a prime power q. The optional second argument is a polynomial m which is primitive over the prime subfield of GF(q). This symbol returns a primitive element for GF(q) with minimal polynomial m. If there is only one argument, then the minimal polynomial is assumed to be the conway polynomial for GF(q).
order1/
primitive_element
This is a unary function, whose argument is a number field K. It returns a primitive element of K. Note that the result is not unique.
ring3/
principal_ideal
This symbol represents a binary function. The first argument is a ring R and the second argument is an element of R. When evaluated on R and such a second argument, the function represents the ideal in R generated by the second argument.
prog1/
procedure_block
The block of code defining the body of the procedure. The syntax is procedure_block(local_var, global_var, block1), where local_var encodes the local variables (private to the procedure body), gloval_var are global variables that are know to the procedure and block1 is the body of the procedure. All these elements, locar_var, global_var and block1, should be present (but they can also be empty).
prog1/
procedure_call
Symbol procedure_call can be used to "call" already defined procedures. The syntax is procedure_call(name, call_arguments), where name is the encoding of an OpenMath variable (OMV) representing the name of the function and call_arguments are the arguments to pass to the function. Both, name and call_arguments, should be present but call_arguments can be empty.
scscp1/
procedure_call
The actual procedure call. Its only argument is an OpenMath Application, whose head symbol describes the procedure to be called, and whose arguments are the arguments to the procedure.
scscp1/
procedure_completed
The result of a successful computation. Should come along with a call_id and, possibly, some extra information.
prog1/
procedure_definition
This symbol can be used to define a procedure. The sintax is procedure_definition(name, def_arguments, procedure_block), where name is the encoding of an OpenMath variable representing the name of the procedure, def_arguments encodes the argument the procedure can receive and procedure_block encodes the body of the procedure. Contrary to function procedures can have knowledge about global objects by means of the global_var construct (see procedure block).
scscp1/
procedure_terminated
The result of a failed computation. Should come along with a call_id, an error description, and possibly some extra information.
arith1/
product
An operator taking two arguments, the first being the range of multiplication e.g. an integral interval, the second being the function to be multiplied. Note that the product may be over an infinite interval.
polyslp/
prog_body
The constructor of the body of the straight line program the arguments represent straight line instructions, as constructed by the following three constructors, op_node, inp_node and const_node, possibly wrapped in the return symbol (from the opnode CD). The order is taken to be the order in which they appear.
logic3/
proof
This symbol represents a sequence of justified well-formed formulae (i.e. objects of type ProofLine). The single argument is a List of ProofLine objects.
typesorts/
Prop
The type of propositions
logic3/
prop_deduction
This symbol is used to claim 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).
logic3/
prop_theorem
This symbol is used to claim that a statement 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).
directives1/
prove
This symbol is a function with one argument, which should be a clause. When applied to a clause C, it asks for a proof of C.
directives1/
prove_in_theory
This symbol is a function with two arguments, the first of which should be a clause and the second of which should be a symbol indicating a logic theory. When applied to arguments C, T, it asks for a proof of C in theory T.
multiset1/
prsubset
This symbol has two (multiset) arguments. It is used to denote that the first multiset is a proper subset of the second, that is a subset of the second multiset but not actually equal to it.
set1/
prsubset
This symbol has two (set) arguments. It is used to denote that the first set is a proper subset of the second, that is a subset of the second set but not actually equal to it.
fieldname1/
Q
This is a symbol representing the field of rational numbers.
setname1/
Q
This symbol represents the set of rational numbers.
groupname1/
quaternion_group
This symbol represents the quaternion group of order 8.
permgp2/
quaternion_group
This symbol represents the quaternion group of order 8, viewed as a permutation group by means of the regular representation (multiplication from the right). It is generated by (1,2,3,4)(5,8,6,7) and (1,5,2,6)(3,7,4,8). (In the usual notation, the 8 elements are 1, -1, i, -i, j, -j, k, -k.)
ringname1/
quaternions
This symbol represents a unary function. Its argument is a ring R. When evaluated on R, the function represents the ring of quaternions over R, that is, the ring with basis 1,i,j,k over R such that ij=-ji=k, i^2=j^2=k^2=-1.
integer1/
quotient
The symbol to represent the integer (binary) division operator. That is, for integers a and b, quotient(a,b) denotes q such that a=b*q+r, with |r| less than |b| and a*r positive.
polynomial3/
quotient
This symbol represents the binary division operator on univariate polynomials over fields. That is, for univariate polynomials a and b, quotient(a,b) denotes the polynomial q such that a=b*q+r, with degree(r) less than degree(b).
polynomial4/
quotient
This symbol contains the quotient of polynomial4.divide. Cf. polynomial4.quotient_remainder for an example.
polyslp/
quotient
A quotient function for polynomials represented by slps. It is a requirement that this is an exact division.
ring5/
quotient_by_poly_map
This symbol is a binary function whose first argument is a ring R, and whose second argument is a univariate polynomial f with coefficients from R. So, if the indeterminate is X, when applied to R and f, the function has value the natural quotient map from R[X] to the quotient ring R[X]/(f).
gp1/
quotient_group
The binary function whose value is the factor group of the first argument by the second, assuming the second is normal in the first.
group3/
quotient_group
The binary function whose value is the factor group of the first argument by the second, assuming the second is normal in the first.
ring5/
quotient_map
This symbol is a binary function whose first argument is a ring R and whose second argument is an ideal I of R. When applied to R and I, its value is the natural quotient map from R to the quotient ring R/I.
polynomial4/
quotient_remainder
This symbol is a container for the result of polynomial4.divide. It takes 2 arguments in unspecified order, polynomial4.quotient and polynomial4.remainder. Using the above polynomial4.divide call we may obtain:
ring3/
quotient_ring
This is a binary function, whose first argument is a ring R and whose second argument is an ideal I of R. When applied to R and I, it denotes the quotient ring of R by I.
setname2/
QuotientField
This symbol represents the quotient field of any integral domain.
fieldname1/
R
This is a symbol representing the field of real numbers.
setname1/
R
This symbol represents the set of real numbers.
SI_NamedDerivedUnits1/
This symbol represents one radian, the natural unit of measure for angle. It has the short symbol form, "rad".
SI_DerivedQuantities1/
This symbol represents the physical quantity of radio nuclide activity, or radioactivity. A variable representing an arbitrary quantity of radioactivity is commonly represented with the italic, upper case letter, "A".
plangeo3/
plangeo3/
Gives the radius of a circle.
hypergeon1/
raising_multi_factorial
raising_multi_factorial is a product of pochhammer symbols. 2-ary function. reference: authors: "Saito, Sturmfels, Takayama" title: "Grobner Deformations of Hypergeometric Differential Equations" pages: 127
fns1/
range
This symbol denotes the range of a function, that is a set that the function will map to. The single argument should be the function whos range is being queried. It should be noted that this is not necessarily equal to the image, it is merely required to contain the image.
linalg4/
rank
This symbol represents the function which takes one matrix argument and returns the number of linearly independent rows (or columns) of that matrix.
linalgrank1/
rank
This symbol represents the function which takes one matrix argument and returns the number of linearly independent rows (or columns) of that matrix.
polyd1/
rank
This is a unary function, whose argument can be a DMP, a poly_ring_d, or a poly_ring_d_named. When applied to its argument, it represents the number of variables of the polynomial ring involved.
nums1/
rational
This symbol represents the constructor function for rational numbers. It takes two arguments, the first is an integer p to denote the numerator and the second a nonzero integer q to denote the denominator of the rational p/q.
mathmltypes/
rational_type
A symbol to be used as the argument of the type symbol to convey the type of a rational number.
interval_types/
rationalIntervalType
This symbol represents the type of rational intervals.
complex1/
real
This represents the real part of a complex number
mathmltypes/
real_type
A symbol to be used as the argument of the type symbol to convey the type of a real number.
interval_types/
realIntervalType
This symbol represents the type of real intervals.
polyd/
reduce
The reduction of a polynomial with respect to a Groebner basis. First argument is a DMP, the second argument is a "groebnered" object. i.e. a program implementing this operation should return a DMP which represents the polynomial reduced with respect to the Groebner basis.
polygb1/
reduce
The reduction of a polynomial with respect to a list P of polynomials. First argument is a polynomial expression p, the second argument is the list P of polynomials, the third argument is a list of variables, the fourth argument is a monomial reduction ordering. A program implementing this operation should return a polynomial which represents a polynomial reduced from p with respect to P. This means that p is expressible as the sum of the returned polynomial and a linear combination of the polynomials from P with coefficients being polynomials in the variables given in the third argument, and that no monomial of the returned polynomial is divisible by the leading monomial of an element from P.
FundamentalPhysicalConstants1/
reduced-Planck-constant
This symbol represents the Planck constant divided by 2*pi. It is commonly represented with the short, italic symbol, h with a horizontal bar ("h-bar"), Unicode: U+210F , HTML: ℏ.
relation0/
reflexive
Proposition; the type of reflexive binary relations.
relation3/
reflexive_closure
This symbol represents a binary function whose first argument is a set S, whose second argument is a relation R on S. When applied to S and R, it represents the smallest reflexive relation (with respect to inclusion) on S containing R.
relation0/
relation
Type constructor; returns the type of binary relations on a set.
numerical1/
relative_error_bound
This symbol marks an estimated upper bound for the relative error ( |true-computed|/|computed| ) on a computation.
numerical1/
relative_error_requested
This symbol marks a requirement for the relative error ( |true-computed|/|computed| ) on a computation.
integer1/
remainder
The symbol to represent the integer remainder after (binary) division. For integers a and b, remainder(a,b) denotes r such that a=b*q+r, with |r| less than |b| and a*r positive.
polynomial3/
remainder
The symbol represents a binary function, whose arguments should be univariate polynomials in the same polynomial ring whose coefficient ring is a field. When applied to a and b, it represents the polynomial remainder after division of a by b.
polynomial4/
remainder
This symbol contains the remainder of polynomial4.divide. Cf. polynomial4.quotient_remainder for an example.
list4/
remove
This symbol represents a function with two arguments, both lists. When applied to two lists, it represents a list made up of all the elements of the first list with those elements removed whose entries occur in the second list.
dimensions1/
resistance
This symbol represents the resistance physical dimension, it is the resistance that an electrical circuit has to flow of charge.
SI_DerivedQuantities1/
resistance
This symbol represents the physical quantity of electrical resistance, the resistance that an electrical circuit has to electrical current. A variable representing an arbitrary quantity of electrical resistance is commonly represented with the italic, upper case letter, "R".
rdf/
resource
This symbol represents an unary construction function for representing a specific RDF resource. It takes one string argument denoting an IRI reference as prefixed name in the form "prefix:resourceName" or as a full IRI in the form ">IRI<".
rdf/
resourceset
This symbol represents an unary construction function for constructing a set of RDF resources. It takes one string argument representing a Manchester Syntax description as described by http://www.w3.org/TR/owl2-manchester-syntax/#Descriptions in order to construct a set of RDF resources. Please note that it may also be possible to represent the class description by using OpenMath set operations: rdf.resourceset( set1.intersect( rdf.resourceset("foaf:Person"), set1.suchthat(rdf.resourceset("rdfs:Resource"), fns1.lambda[$r -> set1.size(rdf.valueset("foaf:age",$r) > 0)]) ) )
directives1/
response
This symbol is a function of one argument, which should be a query. When applied to a query, it refers to the response a service might give. It will mainly be used in this CD to express formal mathematical properties of queries.
list2/
rest
This symbol represents a function which returns a list made up of all the elements except the first of its argument, which should be a list.
list2/
rest
This symbol represents a function which returns a list made up of all the elements except the first of its argument, which should be a list.
fns1/
restriction
restriction takes two arguments, a function f, and a set S, which should be a subset of domain(f) and returns the function f restricted to S.
poly/
resultant
Function taking three arguments, it represents the resultant of two polynomials, which are the first two arguments, with respect to the given variable which is the third argument.
scscp2/
retrieve
Using the cookie that was obtained earlier by calling the scscp2.store_session or scscp2.store_persistent procedure or another procedure call, return to the client an OM object representing the object, referred by the cookie.
opnode/
return
A unary function, takes a node of an slp, returns the value of the polynomial which corresponds to this node of the slp.
prog1/
return
This symbol, which can have an aribtrary positive number of arguments, can be used to return values from functions.
polyslp/
return_node
Takes an slp as the argument, and returns the return node of the slp.
list2/
reverse
The reverse of a list
list4/
reverse
The reverse of a list
polyd/
reverse_lexicographic
The reverse lexicographic ordering of terms. 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.
polyd2/
reverse_lexicographic
The reverse lexicographic ordering of monomials
fns2/
right_compose
This symbol represents a function forming the right-composition of its two functional arguments.
permutation1/
right_compose
This symbol is a binary function. Its arguments should be permutations. When applied to arguments P1 and P2, the resulting value is the permutation which maps x in Support(P1) union Support(P2) to P2(P1(x)).
group4/
right_coset
This symbol represents a ternary function whose first argument is a group G, whose second argument is a subgroup H of G, and whose third argument is an element x of G. Its value on G, H, and x is the right coset of H in G containing x, that is, the set H x.
group4/
right_coset_representative
This symbol represents a quaternary function whose first argument is a group G, whose second argument is a subgroup H of G, whose third argument is right_transversal T of H in G, and whose fourth argument is an element of G. It assigns to G, H, T, g the element of t of T representing the right coset of H containing g, that is, H t = H g.
group4/
right_cosets
The binary function whose value is the set of right cosets of the second argument in the first.
magma1/
right_divides
This symbol is a ternary function. Its first argument should be a magma M and the second and third arguments should be elements of M. When applied to M, a, and b, it denotes the fact that a is a right_divisor of b in M. This means that there is v in M such that va = b.
magma1/
right_expression
This symbol is a binary function. Its first argument should be a magma M, the second argument a list L of elements of M When applied to M and L, it denotes the right product (( ... (L * L) * ... ) * L[n]) of all elements in the list L.
fns1/
right_inverse
This symbol is used to describe the right inverse of its argument (a function). This inverse may only be partially defined because the function may not have been surjective. If the function is not surjective the right inverse function is ill-defined without further stipulations. No other assumptions are made on the semantics of this right inverse.
group2/
right_inverse_multiplication
This symbol is a function with two arguments, which should be a group M and an element x of M. When applied to M and x, it denotes right multiplication on M by the inverse of x.
field2/
right_multiplication
This symbol is a function with two arguments, which should be a field M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.
group2/
right_multiplication
This symbol is a function with two arguments, which should be a group M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.
monoid2/
right_multiplication
This symbol is a function with two arguments, which should be a monoid M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.
ring2/
right_multiplication
This symbol is a function with two arguments, which should be a ring M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.
semigroup2/
right_multiplication
This symbol is a function with two arguments, which should be a semigroup M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.
group5/
right_quotient_map
This symbol is a binary function whose first argument is a group G and whose second argument is an subgroup H of G. When applied to G and H, its value is the natural quotient map from G to the quotient group G/H, sending x to the left coset xH of G.
polyslp/
right_ref
Takes as argument a node of an slp. Returns the value of the right hand pointer of the node.
gp1/
right_transversal
The binary function whose value is a set of representatives for the right cosets of the second argument as a subgroup of the first.
group4/
right_transversal
The binary function whose value is a set of representatives for the right cosets of the second argument as a subgroup of the first.
algebraic_cats/
ring
This symbol is the constructor for rings. A ring is a set together with two operations + and *. A ring is an Abelian group under + and a semigroup under *. A ring has a further rule which associates the two operation, that is left and right distributivity. The ring constructor takes five arguments, the set of the ring, 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 additive identity 0 and a unary function from the set into itself to represent additive inverses (i.e. inverses under +, or negatives).
generic_alg_cats/
ring
This Symbol represents the generic category of ring.
ring1/
ring
This symbol is a constructor for rings. It takes six arguments R, a, o, i, m, e,: which are, respectively, a set R to specify the elements in the ring, a binary operation a on R, an element o of R, and a unary operation i on R such that [R,a,o,i] is a commutative group, a binary operation m on R and an element e of R such that [R,m,e] is a monoid.
order1/
ring_integers
This is a unary function, whose argument is a number field K. When applied to K, it returns the ring of integers of K. It is the Dedekind ring of K.
algebraic_cats/
ring_negative
This symbol takes one argument which should be a ring. It returns a unary function which should be the negative function of the ring.
algebraic_cats/
ring_plus
This symbol takes one argument which should be a ring. It returns a binary function which represents the additive operation of the ring.
algebraic_cats/
ring_set
This symbol takes one argument which should be a ring. It returns the set of the ring.
algebraic_cats/
ring_times
This symbol takes one argument which should be a ring. It returns a binary function which represents the multiplicative operation of the ring.
algebraic_cats/
ring_zero
This symbol takes one argument which should be a ring. It returns the additive identity of the ring.
algebraic_cats/
ringoid
This symbol is the constructor for ringoids. A ringoid is a set together with two operations + and *. * is left and right distributive over +. The ringoid constructor takes three arguments, the set of the ringoid, a binary function from the set into itself to represent the * operation and a binary function from the set into itself to represent the + operation.
generic_alg_cats/
ringoid
This symbol represents the generic category of ringoid.
algebraic_cats/
ringoid_plus
This symbol takes one argument which should be a ringoid. It returns a binary function which represents the additive operation (+) of the ringoid.
algebraic_cats/
ringoid_set
This symbol takes one argument which should be a ringoid. It returns a set which represents the set of the ringoid.
algebraic_cats/
ringoid_times
This symbol takes one argument which should be a ringoid. It returns a binary function which represents the multiplicative operation (*) of the ringoid.
meta/
Role
An element containing the string corresponding to the role of the symbol being defined.
arith1/
root
A binary operator which represents its first argument "lowered" to its n'th root where n is the second argument. This is the inverse of the operation represented by the power symbol defined in this CD. Care should be taken as to the precise meaning of this operator, in particular which root is represented, however it is here to represent the general notion of taking n'th roots. As inferred by the signature relevant to this symbol, the function represented by this symbol is the single valued function, the specific root returned is the one indicated by the first CMP. Note also that the converse of the second CMP is not valid in general.
rounding1/
round
The round to nearest operation.
matrix1/
row_dimension
This symbol is a unary function whose first argument must be either a non-negative OpenMath integer or nums1.infinity. When applied this creates an object that denotes the dimension of the codomain of the linear mapping represented by the matrix.
linalg3/
rowcount
This symbol represents the function which takes one matrix argument and returns the number of rows in that matrix.
linalg4/
rowcount
This symbol represents the function which takes one matrix argument and returns the number of rows in that matrix.
linalg4mat/
scalar
This symbol represents a square matrix which is a scalar constant times the identity matrix. It should take two arguments, the first and second specify the number of rows and columns in the matrix, respectively, and the third specifies the scalar multiplier.
linalg5/
scalar
This symbol represents a matrix which is a scalar constant times the identity matrix. It should take two arguments, the first specifes the number of rows and columns in the matrix respectively and the third specifies the scalar multiplier.
linalg1/
scalarproduct
This symbol represents the scalar product function. It takes two vector arguments and returns a scalar value. The scalar product of two vectors a, b is defined as |a| * |b| * cos(\theta), where \theta is the angle between the two vectors and |.| is a euclidean size function. Note that the scalar product is often referred to as the dot product.
permgp1/
schreier_tree
This is a function with two arguments. The first argument should be a permutation group G, the second argument a point x permuted by G. When evaluated at G and x, it returns a list of three lists X,V,B. The first list, X, enumerates the points of the G-orbit of x. The second list and the third list both have the same length as X, say n. The second list represents a map V from [1,...,n] to {-m,...,-1,0,1,...,m}, where m is the number of generators of G, and the third list represents a map B from [1,...,n] to X. These maps satisfy the following properties: X(1) = B(1) = x. Moreover, V(i) = 0 if and only if i = 1. For each index i distinct from 1, the value B(i) is equal to X(j) for some index j smaller than i. If V(i) is positive, then X(i) is the image of B(i) under the V(i)-th generator of G. If V(i) is negative, then B(i) is the image of X(i) under the (-V(i))-th generator of G.
s_data1/
sdev
This symbol represents a function requiring two or more arguments, denoting the sample standard deviation of its arguments. That is, the square root of (the sum of the squares of the deviations from the mean of the arguments, divided by the number of arguments). See CRC Standard Mathematical Tables and Formulae, editor: Dan Zwillinger, CRC Press Inc., 1996, (7.7.11) section 7.7.1.
s_dist1/
sdev
This symbol represents a unary function denoting the standard deviation of a distribution. The argument is a univariate function to describe the distribution. The standard deviation of a distribution is the arithmetical mean of the squares of the deviation of the distribution from the mean.
polyd/
SDMP
The constructor for multivariate polynomials without any indication of variables or domain for the coefficients. Its arguments are just "term"s. No terms should differ only by the coefficient (i.e it is not permitted to have both "2*x*y" and "x*y" as terms in a SDMP). SDMP can be attributed with the "ordering" symbol to indicate a particular ordering of its terms. This attribute shall not be set if the SDMP is part of DMPL that has this attribute set. If the SDMP is ordered, explicitly or implicitly via an outer ordering, the terms must be in decreasing order with respect to this order. The zero polynomial is represented by an SDMP with no terms.
polyd1/
SDMP
The constructor for multivariate polynomials without any indication of variables or domain for the coefficients. Its arguments are just "monomial"s. No monomials should differ only by the coefficient (i.e it is not permitted to have both "2*x*y" and "x*y" as monomials in a SDMP). SDMP can be attributed with the "ordering" symbol to indicate a particular ordering of its monomials. This attribute shall not be set if the SDMP is part of DMPL that has this attribute set.
transc1/
sec
This symbol represents the sec function as described in Abramowitz and Stegun, section 4.3. It takes one argument.
transc1/
sech
This symbol represents the sech function as described in Abramowitz and Stegun, section 4.5. It takes one argument.
SI_BaseUnits1/
second
This symbol represents the measure of one second of time, the standard SI unit of measure for quantities of time. It has the short symbol form, "s", in upright roman font.
units_metric1/
second
This symbol represents the measure of one second. This is the standard SI measure for time.
units_time1/
second
This symbol represents the measure of one second of time. This is the standard SI unit measure for time.
SIUsed_OffSystemUnits1/
second-of-arc
This symbol represents the angular measure of one second-of-arc. It has the short symbol form, '"'.
intpath1/
segment
The symbol segment(a,b) is the segment from the point a to the point b in the complex plane. If the arguments are sectors given by path_in_sector, it means the segment from a point in the circular border of the sector to a point in the circular border of the another sector.
plangeo2/
segment
The segment of a line between two points of the line. The segment is contained in the affine part of the line. The symbol takes as arguments the two points.
ThreeDgeo1/
segment
The symbol is used to indicate a segment of a line in 3-dimensional Euclidean geometry by a variable. The segment is contained in the affine part of the line. The symbol takes the variable as the first argument and the endpoints as second and third arguments.
list3/
select
This symbol takes two lists as arguments, L and M say. The second argument is a list containing only entries from [1..n], where n is the length of L. The symbol represents the function which returns a list whose length is equal to the length of M, and having at position k the value of L at position M_k.
list3/
select
This symbol takes two lists as arguments, L and M say. The second argument is a list containing only entries from [1..n], where n is the length of L. The symbol represents the function which returns a list whose length is equal to the length of M, and having at position k the value of L at position M_k.
algebraic_cats/
semigroup
This symbol is the constructor for semigroups, that is groupoids for which the operator of the semigroup is associative over the set of the semigroup. The semigroup constructor takes two arguments, the set of the semigroup and a binary function which represents the operation of the semigroup.
generic_alg_cats/
semigroup
This Symbol represents the generic category of semigroup.
monoid1/
semigroup
This symbol is a unary function, whose argument should be a monoid M. When applied to M its value is the semigroup underlying M.
semigroup/
Semigroup
The contructor for the type of semigroups as a Setoid with a binary operation.
semigroup1/
semigroup
This symbol is a constructor for semigroups. It takes two arguments in the following order: a set to specify the elements in the semigroup, and a binary operation to specify the semigroup operation. The binary operation should act on elements of the set and return an element of the set.
algebraic_cats/
semigroup_operation
This symbol takes one argument which should be a semigroup. It returns a binary function which should represent the operation of the semigroup.
algebraic_cats/
semigroup_set
This symbol takes one argument which should be a semigroup. It returns the set of the semigroup.
scscp2/
service_description
The symbol for the server to use in a response to scscp2.get_service_description. It takes three OMSTR arguments: Name, Version, and Description.
set1/
set
This symbol represents the set construct. It is an n-ary function. The set entries are given explicitly. There is no implied ordering to the elements of a set.
plangeo4/
set_affine_coordinates
Defines the affine coordinates of an affine point or line.
ThreeDgeo3/
set_affine_coordinates
Defines the affine coordinates of an a point in 3-dimensional Euclidean space. Takes the point as first argument and the vector with the coordinates as second argument.
plangeo4/
set_coordinates
This symbol defines the coordinates of a point or a line. The coordinates are the projective coordinates and consist of a vector of length 3. Points whose third coordinates are zero are the points at infinity. The line whose first two coordinates are zero is the line at infinity.
mathmltypes/
set_type
A symbol to be used as the argument of the type symbol to convey the type for a set.
multiset1/
setdiff
This symbol is used to denote the multiset difference of two multisets. It takes two multisets as arguments, and denotes the multiset that contains all the elements that occur in the first multiset with strictly greater multiplicity than in the second. The multiplicity in the result is the difference of the two.
set1/
setdiff
This symbol is used to denote the set difference of two sets. It takes two sets as arguments, and denotes the set that contains all the elements that occur in the first set, but not in the second.
sts/
SetNumericalValue
Denotes an OpenMath object that is to be thought of as something that represents a set of numerical values, or a set of numerical values.
setoid/
Setoid
The contructor for the type of set with an equivalence relation on it.
aggregate_cats/
setType
This symbol represents the type of sets.
SI_NamedDerivedUnits1/
siemens
This symbol represents an SI unit of electrical conductance. It is not plural. It has the short symbol form, "S".
SI_NamedDerivedUnits1/
sievert
This symbol represents an SI unit of equivalent dose of ionizing, radiation. A sievert of equivalent dose represents one joule of biologically damaging energy absorbed per kilogram of mass. It has the short symbol form, "Sv".
ecc/
SigmaType
The binder symbol used to construct the type of Cartesian products. The (either plain or attributed) variables might occur in the body \OM\ object.
permutation1/
sign
This symbol is a function with one argument which should be a permutation. When applied to a permutation P, it represents the sign of P, which is equal to -1 if P is an odd permutation and equal to 1 otherwise.
permutation1/
sign
This symbol is a function with one argument which should be a permutation. When applied to a permutation P, it represents the sign of P, which is equal to -1 if P is an odd permutation and equal to 1 otherwise.
metasig/
Signature
This symbol is used to represent the element of a signature file which specifies the signature of a symbol. It should take two string children, the first should be the symbol who's signature is being specified, the second should be an 'OMOBJ' element which specifies the signature. Additionally the second argument should specify an object which must represent a valid type in the type system identified by the XML attribute 'type' corresponding to the element which corresponds to the symbol 'CDSignatures' enclosing this symbol.
scscp2/
signature
The symbol to use for describing the types of arguments of a particular function.
transc1/
sin
This symbol represents the sin function as described in Abramowitz and Stegun, section 4.3. It takes one argument.
transc1/
sinh
This symbol represents the sinh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.
linalg3/
size
This symbol represents the function which takes one vector argument and returns the length of that vector.
linalg4/
size
This symbol represents the function which takes one vector argument and returns the length of that vector.
list2/
size
This symbol is used to denote the number of elements in a list. It is either a non-negative integer.
multiset1/
size
This symbol is used to denote the number of elements in a multiset. It is either a non-negative integer, or an infinite cardinal number. The symbol infinity may be used for an unspecified infinite cardinal.
set1/
size
This symbol is used to denote the number of elements in a set. It is either a non-negative integer, or an infinite cardinal number. The symbol infinity may be used for an unspecified infinite cardinal.
linalg5/
skew-symmetric
This symbol represents a skew-symmetric matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix. For j>i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument. This determines the elements above the diagonal of the matrix, the elements below the diagonal of the matrix must conform to the rule M = - transpose M. This rule implies that the elements on the diagonal must be equal to 0, therefore we do not include these in the argument.
linalgsym1/
skew_symmetric
This symbol represents a skew-symmetric matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix. For j>i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument. This determines the elements above the diagonal of the matrix, the elements below the diagonal of the matrix must conform to the rule M = - transpose M. This rule implies that the elements on the diagonal must be equal to 0, therefore we do not include these in the argument.
group3/
SL
This symbol is a function with one argument, which should be a a module V over a commutative ring. When applied to V it represents the group of all invertible linear transformations of V of determinant 1.
group3/
SLn
This symbol is a function with two arguments. The first should be a positive integer n, the second a field F. When applied to n and F it represents the group of all invertible linear transformations of the vector space over F of dimension n having determinant 1.
polyslp/
slp_degree
A unary function taking an slp as argument and returning the apparent multiplicative degree of the slp, without performing any cancellation.
SI_DerivedQuantities1/
solid-angle
This symbol represents the quantity of a two dimensional, geometric solid angle. A variable representing an arbitrary quantity of solid angle is commonly represented with the italic, upper case greek variable, "\Omega;".
poly1p/
sorted_set_of_indexed_variables
sorted_set_of_indexed_variables(x,s) returns the vector of variables indexed by the sorted set s.
graph1/
source
Given an arrow, this symbol refers to the vertex where the arrow starts. It takes one argument, the arrow.
equations1/
sparse
A predicate to indicate that an equation or system of equations is sparse.
linalg4mat/
sparse
The sparse symbol is a constructor for sparse matrices. It is (n+1)-ary, where the first argument is the rowcount (row dimension) of the matrix, the second argument is the columncount (column dimension) of the matrix and every following argument specifies a possibly non-zero element in the following way. The argument is a list which should have length three. The first element in the list is the row index, the second element is the column index (one based), whilst the third element in the list is the value. Every other element of the matrix is implicitly zero.
linalg4vec/
sparse
The sparse symbol is a constructor for sparse vectors. It is (n+1)-ary, where the first argument is the length (dimension) of the vector, and every following argument specifies a possibly non-zero element in the following way. The argument is a list which should have length two. The first element in the list is the position (one based), whilst the second element in the list is the value. Every other element of the vector is implicitly zero.
linalg7/
sparse
The sparse symbol is a constructor for sparse vectors, it is (n+1)-ary, where the first parameter is the length (dimension) of the vector, and every following parameter specifies a possibly non-zero element in the following way. The parameter is a list which should have length two. The first element in the list is the position (one based), whilst the second element in the list is the value. Every other element of the vector is implicitly zero.
matrix1/
sparse
The constructor for sparse matrices without any indication of dimension or domain for the coefficients. Its arguments are just matrix1.sparse_entrys. Attention: No two matrix1.sparse_entrys must specify the same location.
matrix1/
sparse_entry
This symbol denotes a ternary function whose first two arguments specify the location of an entry inside the matrix, and whose final argument is the entry itself. The entry MUST be either from the specified ground domain directly, or be a diagonal constructor as described below, a block constructor as described below, or a banded constructor as described below. In the block case, the dimensions of the block MUST NOT exceed the total dimensions of the matrix algebra. In the diagonal case, the dimension of the diagonal MUST NOT exceed the total dimensions of the matrix algebra.
linalg6/
sparseMatrix
This symbol may be used for representing matrices, it is designed for efficiently representing sparse matrices. The symbol is n+2 ary, where the first argument is the number of rows in the matrix, the second argument is the number of columns in the matrix and n is the number of non-zero entries. The following arguments must be applications of the symbol sparseMatrixElement1. Any non-specified entry is implicitly zero.
linalgspars1/
sparseMatrix
This symbol may be used for representing matrices, it is designed for efficiently representing sparse matrices. The symbol is n+2 ary, where the first argument is the number of rows in the matrix, the second argument is the number of columns in the matrix and n is the number of non-zero entries. The following arguments must be applications of the symbol sparseMatrixElement1. Any non-specified entry is implicitly zero.
linalg6/
sparseMatrix01
This symbol may be used for representing matrices which have all entries in the modular field Z_2, i.e. 1 or 0. It allows efficient representation of sparse matrices, more so than the 'sparseMatrix' symbol, since the value of the entries with values of 1 need not be stored, only their positions. The symbol is n+2 ary, where the first argument is the number of rows in the matrix, the second argument is the number of columns in the matrix. The following arguments are sparseMatrixElement3 elements described in this content dictionary. Any non-specified entry is implicitly zero.
linalgspars1/
sparseMatrix01
This symbol may be used for representing matrices which have all entries in the modular field Z_2, i.e. 1 or 0. It allows efficient representation of sparse matrices, more so than the 'sparseMatrix' symbol, since the value of the entries with values of 1 need not be stored, only their positions. The symbol is n+2 ary, where the first argument is the number of rows in the matrix, the second argument is the number of columns in the matrix. The following arguments are sparseMatrixElement3 elements described in this content dictionary. Any non-specified entry is implicitly zero.
linalg6/
sparseMatrixElement1
This symbol may be used to represent a non-zero element of a sparse matrix in the following way. It takes three arguments, the first of which represents the column index, the second of which represents the row index and the third represents the value. The indexing is one based; that is an element in the top left position of the matrix will have first and second indices of 1,1 respectively. Applications of this symbol will be expected as arguments of the symbol sparseMatrix in this content dictionary.
linalgspars1/
sparseMatrixElement1
This symbol may be used to represent a non-zero element of a sparse matrix in the following way. It takes three arguments, the first of which represents the column index, the second of which represents the row index and the third represents the value. The indexing is one based; that is an element in the top left position of the matrix will have first and second indices of 1,1 respectively. Applications of this symbol will be expected as arguments of the symbol sparseMatrix in this content dictionary.
linalg6/
sparseMatrixElement2
This symbol may be used to represent a non-zero element of a sparse matrix in the following way. It takes two arguments, the first of which represents the column index, the second of which represents the value of the element. The row index is deduced from the index of the sparseMatrixRow symbols of which applications of this symbol are arguments. Applications of this symbol occur as arguments of arguments of the symbol nonZeroRowSparseMatrix only.
linalgspars1/
sparseMatrixElement2
This symbol may be used to represent a non-zero element of a sparse matrix in the following way. It takes two arguments, the first of which represents the column index, the second of which represents the value of the element. The row index is deduced from the index of the sparseMatrixRow symbols of which applications of this symbol are arguments. Applications of this symbol occur as arguments of arguments of the symbol nonZeroRowSparseMatrix only.
linalg6/
sparseMatrixElement3
This symbol may be used to represent a non-zero element of a sparse matrix over Z_2 in the following way. The first and second arguments are the column and row indices of the non-zero elements respectively i.e. elements with value 1. Applications of this symbol occur as arguments of arguments of the symbol sparseMatrix01 only.
linalgspars1/
sparseMatrixElement3
This symbol may be used to represent a non-zero element of a sparse matrix over Z_2 in the following way. The first and second arguments are the column and row indices of the non-zero elements respectively i.e. elements with value 1. Applications of this symbol occur as arguments of arguments of the symbol sparseMatrix01 only.
linalg6/
sparseMatrixElement4
This symbol may be used to represent a non-zero element of a sparse matrix over Z_2 in the following way. The single argument is the column index of non-zero elements of the matrix, i.e. elements with value 1. Applications of this symbol occur as arguments of arguments of the symbol nonZeroRowSparseMatrix01 only.
linalgspars1/
sparseMatrixElement4
This symbol may be used to represent a non-zero element of a sparse matrix over Z_2 in the following way. The single argument is the column index of non-zero elements of the matrix, i.e. elements with value 1. Applications of this symbol occur as arguments of arguments of the symbol nonZeroRowSparseMatrix01 only.
linalg6/
sparseMatrixRow
This symbol may be used to represent rows of sparse matrices, it is a fairly general symbol in that it may be used to represent rows of any type of sparse matrix from this CD. However the particular type of sparse matrix must have as elements symbols of the corresponding type, as described in that symbols description.
linalgspars1/
sparseMatrixRow
This symbol may be used to represent rows of sparse matrices, it is a fairly general symbol in that it may be used to represent rows of any type of sparse matrix from this CD. However the particular type of sparse matrix must have as elements symbols of the corresponding type, as described in that symbols description.
fns3/
specification
This symbol denotes the specification of a function. It is a unary function. When aplied to its argument, which should be a function applied to three arguments, it returns the third argument of the function, that is, the function specification.
dimensions1/
speed
This symbol represents the speed physical dimension. It is the size of the derivative of distance with respect to time.
SI_DerivedQuantities1/
speed
This symbol represents the physical quantity of speed. It is the size of the derivative of position with respect to time.
FundamentalPhysicalConstants1/
speed-of-light
This symbol represents the speed of light in a vacuum. Its value is implied by the definition of the metre [17th CGPM (1983)]. Consequently, the speed of light is defined to be exactly 299,792,458 metre per second (in the SI). It is commonly represented with the short, italic symbol, "c".
physical_consts1/
speed_of_light
This symbol represents the speed of light in a vacuum. It is approximately 299792458 metres per second.
ThreeDgeo1/
sphere
The symbol is used to indicate a sphere in 3-dimensional Euclidean geometry by a variable. The sphere may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.
ThreeDgeo2/
sphere_center
The statement that a sphere in 3-dimensional Euclidean space has a given point as center. Takes the sphere as first argument and the point as second argument.
sts2/
square_matrix
A constructor for the type of a square matrix
poly/
squarefree
The square-free decomposition of its argument. A program that can compute the factorization is required to return a "squarefreed" object.
poly/
squarefreed
The constructor for a square-free factorization. Its arguments should have the structure of the above "factored", where the polynomials should be square-free. Note that this is not necessarily a minimal square-free decomposition: some exponents can occur more than once. Again, this is a statement that we have a square-free factorisation, rather than a request to compute one.
permgp1/
stabilizer
This is an n-ary function with n at least 2. The first argument is a permutation group G, the other arguments are elements x_2,x_3,...,x_n upon which G acts. The value is the subgroup of G consisting of all permutations which stabilize each of x_2,x_3,...,x_n.
permgrp/
stabilizer
The first argument is a permutation group, the second is some object (point or set) upon which the first argument acts. The value is the subgroup of the first argument which stabilize the second argument.
permgp1/
stabilizer_chain
This function takes one argument which should be a permutation group. When applied to the permutation group G, its value is a list consisting of two lists B, H of equal length. The first list B is a base for G, whereas the i-th entry H[i] of the second list is the stabilizer in G of the elements B, ..., B[i].
SI_NamedDerivedUnits1/
This symbol represents one steradian, the natural unit of measure for solid angle. It has the short symbol form, "sr".
combinat1/
Stirling1
The Stirling numbers of the first kind. (-1)^(n-m)*Stirling1(n,m) is the number of permutations of n symbols which have exactly m cycles. Note that there are a few slightly different definitions of these numbers.
combinat1/
Stirling2
The Stirling numbers of the second kind. Stirling2(n, m) is the number of partitions of a set with n elements into m non empty subsets. Note that there are a few slightly different definitions of these numbers.
scscp2/
store_persistent
This indicates the request to store an object on the server side (possibly after computing or simplifying it), returning only a cookie (actually, OM reference) pointing to an object that is usable (using OMR) in the foreseeable future, possibly from different sessions, to get access to the actual object. The server is encouraged to describe the expected lifetime of this object and whether references to this object from different SCSCP sessions are allowed in the response to a scscp2.get_signature request on this symbol. However, at this time we provide no automated or machine-readable mechanism for handling these lifetimes.
scscp2/
store_session
This indicates the request to store an object on the server side (possibly after computing or simplifying it), returning only a cookie (actually, OM reference) pointing to an object that is usable (using an OMR) in the remainder of the current SCSCP session to get access to the actual object.
relation0/
strict_order
Proposition; the type of strict order relations, namely relations that are irreflexive, antisymmetric and transitive.
omtypes/
string
The type of character strings
monoid3/
strings
This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the set of all strings whose characters are entries of the list or set.
sts/
structure
The structure element is used to represent a structure of a particular (algebraic) type.
mathmlattr/
style
A symbol to be used within an OpenMath attribute to specify the style attribute of the object. The annotation should be an OpenMath string representing the value of the style attribute.
field1/
subfield
This symbol is a constructor symbol with one or two arguments. The first argument is a list or set, D, of field elements. The optional second argument is the field G containing D. It denotes the subfield of G generated by D.
group1/
subgroup
This symbol is a constructor symbol with one or two arguments. The first argument is a list or set, D, of group elements. The optional second argument is the group G containing D. It denotes the subgroup of G generated by D.
magma1/
submagma
This symbol is a constructor symbol with two arguments. The first argument is a magma M, the second a list or set, D, of elements of M. When applied to M and D, it denotes the submagma of M generated by D.
monoid1/
submonoid
This symbol is a constructor symbol with two arguments. The first argument is a monoid M, the second a list or set, D, of elements of M. When applied to M and D, it denotes the submonoid of M generated by D.
ring1/
subring
This symbol is a constructor symbol with one or two arguments. The first argument is a list or set, D, of ring elements. The optional second argument is the ring G containing D. It denotes the subring of G generated by D.
semigroup1/
subsemigroup
This symbol is a constructor symbol with two arguments. The first argument is a semigroup S, the second a list or set, D, of elements of S. When applied to S and D, it denotes the subsemigroup of S generated by D.
multiset1/
subset
This symbol has two (multiset) arguments. It is used to denote that the first set is a subset of the second, i.e. every element of the first occurs with multiplicity at least as much in the second.
set1/
subset
This symbol has two (set) arguments. It is used to denote that the first set is a subset of the second.
linalgpoly1/
substitute
This symbol represents a binary function. This first argument should be a polynomial f in a single variable X, the second should be a square matrix A defined over a field F. When applied to f and A, it represents the matrix obtained by replacing X by A and the constant term by the corresponding scalar matrix.
field1/
subtraction
This symbols represents a unary function, whose argument should be a field. It returns the binary operation of subtraction on the field.
ring1/
subtraction
This symbols represents a unary function, whose argument should be a ring. It returns the binary operation of subtraction on the ring.
indnat/
succ
Successor function on the natural number. Constructor for the inductively defined natural numbers. Takes argument a a natural number and returns a natural number.
list1/
suchthat
This symbol represents the suchthat function which may be used to construct lists; it takes two arguments. The first argument should be a set X which contains the elements of the list, the second argument should be a predicate, that is a function from the set X to the booleans which describes if an element is to be in the list returned.
list1/
suchthat
This symbol represents the suchthat function which may be used to construct lists, it takes two arguments. The first argument should be the set which contains the elements of the list, the second argument should be a predicate, that is a function from the set to the booleans which describes if an element is to be in the list returned.
list1/
suchthat
This symbol represents the suchthat function which may be used to construct lists; it takes two arguments. The first argument should be a set X which contains the elements of the list, the second argument should be a predicate, that is a function from the set X to the booleans which describes if an element is to be in the list returned.
set1/
suchthat
This symbol represents the suchthat function which may be used to construct sets, it takes two arguments. The first argument should be the set which contains the elements of the set we wish to represent, the second argument should be a predicate, that is a function from the set to the booleans which describes if an element is to be in the set returned.
arith1/
sum
An operator taking two arguments, the first being the range of summation, e.g. an integral interval, the second being the function to be summed. Note that the sum may be over an infinite interval.
permgp1/
support
This represents a unary function whose argument should be a permutation group. When evaluated at a permutation group G, it is the set of points which are moved a member of G.
permutation1/
support
This symbol is a function with one argument which is a permutation. When applied to a permutation whose arguments are the cycles A1,...,An, it represents the set A which is the union of the entries of all Ai for i=1,...,n.
permutation1/
support
This symbol is a function with one argument which is a permutation. When applied to a permutation P whose arguments are the cycles A1,...,An, it represents the set A which is the union of the entries of all Ai for i=1,...,n.
gp1/
sylow_subgroup
The largest p-subgroup of the argument (up to conjugacy).
group3/
sylow_subgroup
This symbol represents a binary function with two arguments, the first is a group G and the second a prime number p. When applied to G and p, it represents a Sylow p-subgroup of G (which is unique up to conjugacy in G).
scscp2/
symbol_set
This symbol is used in the reply to a scscp2.get_allowed_heads call. It should be the head of an OM Application, the contents of the OMA being arbitrarily many OM Symbols (meaning that a particular symbol is supported), OMA's with head meta.CDName (meaning that all symbols of a particular CD are supported) or OMA's with head meta.CDGroupName (meaning that all symbols of all CDs of a particular CD group are supported). See the example at scscp2.get_allowed_heads.
scscp2/
symbol_set_all
This symbol is used in the reply to a scscp2.get_signature message. It means that this particular service takes any OpenMath object as argument.
linalg5/
symmetric
This symbol represents a symmetric matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix. For j>=i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument. This determines the upper triangle of the matrix, the lower triangle is specified by the rule M = transpose M.
linalgsym1/
symmetric
This symbol represents a symmetric matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix. For j>=i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument. This determines the upper triangle of the matrix, the lower triangle is specified by the rule M = transpose M.
relation0/
symmetric
Proposition; the type of symmetric binary relations.
relation3/
symmetric_closure
This symbol represents a binary function whose first argument is a set S, whose second argument is a relation R on S. When applied to S and R, it represents the smallest symmetric relation (with respect to inclusion) on S containing R.
group3/
symmetric_group
This symbol is a function with one argument, which should be a set X. When applied to a set X it represents the group of all permutations on X .
permgp2/
symmetric_group
This symbol represents a unary function. Its argument is either a positive integer or a set. When evaluated on a set, it represents the permutation group of all permutations of that set. When evaluated on a positive integer n, it represents the permutation group of all permutations of the set {1,..., n}.
group3/
symmetric_groupn
This symbol is a function with one argument, which should be a natural number n. When applied to n it represents the group of all permutations on the set {1,2,... ,n}.
omtypes/
symtype
The type of symbolic types introduced in other CDs
transc1/
tan
This symbol represents the tan function as described in Abramowitz and Stegun, section 4.3. It takes one argument.
plangeo3/
tangent
Given a line L and a circle C this boolean checks whether L is a tangent line to C.
transc1/
tanh
This symbol represents the tanh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.
graph1/
target
Given an arrow, this symbol refers to the vertex the arrow points to. It takes one argument, the arrow.
units_binaryprefix1/
tebi
This symbol represents the fact that the subsequent unit has been effectively multiplied by $2^40$. The full technical name is terabinary.
dimensions1/
temperature
This symbol represents the temperature physical dimension.
SI_BaseQuantities/
temperature
This symbol represents the SI base quantity of thermodynamic temperature. It has the short symbol form, "\Theta;".
tensor1/
tensor_selector
This symbol takes 3 arguments: a tensor, a basis, and a tuple of contravariant and/or covariant indexes. It returns the indexed tensor component in the given basis.
units_siprefix1/
tera
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^12$
polyd/
term
The constructor of terms. Valid applications are of the form Term(coeff, exp1, exp2, ... expn) which represents the term coeff * var1^exp1*...varn^expn where n is the number of variables, expi are non-negative integers. coeff should be non-zero.
polyd1/
term
The constructor of monomials. Valid applications are of the form Term(coeff, exp1, exp2, ... expn) which represents the monomial coeff * var1^exp1*...varn^expn where n is the number of variables, expi are non-negative integers.
polyr/
term
A constructor for monomials, that is products of powers and elements of the base ring. First argument is from N (the exponent of the variable implied by an outer poly_r_rep) second argument is a coefficient (from the ground field, or a polynomial in lesser variables).
polyu/
term
A constructor for monomials, that is products of powers and elements of the base ring. First argument is from N (the exponent of the variable implied by an outer poly_u_rep) second argument is a coefficient (from the ground field)
SI_NamedDerivedUnits1/
tesla
This symbol represents an SI unit of magnetic flux density. It has the short symbol form, "T".
asymp1/
theta
The theta symbol represents a unary function which constructs a set of certain functions of type reals to positive reals. The theta symbol represents a set of functions which all have the same 'rate of growth'. Formally we say that f(x) = theta(g(x)) if and only if there are constants c_1 not= 0 and c_2 not= 0 and x_0 such that for all x > x_0 it is true that c_1*g(x) < f(x) < c_2*g(x).
dimensions1/
time
This symbol represents the time physical dimension. Note that the main units for time are defined in the units_time1 CD.
SI_BaseQuantities/
time
This symbol represents the SI base quantity of time. It has the short symbol form, "T".
arith1/
times
The symbol representing an n-ary multiplication function.
arith2/
times
The symbol representing an n-ary multiplication function inheriting from the times in arith1, but with the extra property that here it must be commutative.
freealg1/
times
Multiplication in the free algebra.
indnat/
times
Multiplication of natural numbers defined recursively by using the successor and plus.
opnode/
times
A constant value, constructs the times for multiplication nodes.
polyd/
times
The product. The argument is a DMPL. The product lies within the same "PolyRingD" i.e. a program implementing this operation should return a DMP with the same "poly_ring_d" (or "poly_ring_d_named").
polyd1/
times
The product. The argument is a DMPL. The product lies within the same "poly_ring_d", i.e., a program implementing this operation should return a DMP with the same "poly_ring_d".
weylalgebra1/
times
multiplication in D
SIUsed_OffSystemUnits1/
tonne
This symbol represents the mass measure of one tonne. It has the short symbol form, "t".
relation0/
transitive
Proposition; the type of transitive binary relations.
relation3/
transitive_closure
This symbol represents a binary function whose first argument is a set S, whose second argument is a relation R on S. When applied to S and R, it represents the smallest transitive relation (with respect to inclusion) on S containing R.
linalg1/
transpose
This symbol represents a unary function that denotes the transpose of the given matrix or vector
linalg5/
tridiagonal
This symbol represents a tridiagonal matrix, it takes one argument which should be a vector of vectors which should have three elements. These should be vectors representing the sub-diagonal, the diagonal and the super-diagonal in that order.
linalgspec1/
tridiagonal
This symbol represents a tridiagonal matrix, it takes one argument which should be a vector of vectors which should have three elements. These should be vectors representing the sub-diagonal, the diagonal and the super-diagonal in that order.
logic1/
true
This symbol represents the boolean value true.
rounding1/
trunc
The round to zero operation.
ecc/
Tuple
The n-ary tupling constructor when n>2. The arguments are the element of the tuple. Tuple objects can also be constructed by successive nesting of Pair.
tensor1/
tuple
This symbol is an n-ary symbol, returning an n-tuple of the arguments. The number of arguments, n, is a non-negative integer. The elements of the n-tuple are ordered as the arguments are ordered. Elements of a tuple may have the same type and value as each other, or not. An n-tuple, unlike a list, is generally not mutable.
tensor1/
tuple_selector
This symbol takes 2 arguments, a tuple and a natural number index, and returns the tuple component indicated by the index value.
cc/
type
ecc/
type
icc/
type
lc/
type
mathmltypes/
type
A symbol to be used within an OpenMath attribute to specify the type of the object.
meta_cats/
type
This symbol is unary and returns the type of its argument.
plangeo1/
type
The symbol represents the type of the basic geometric objects: points, lines, configuration.
sts/
type
A symbol to be used within an OpenMath attribute to specify the type of the object.
typesorts/
Type
The cumulative type of the type of sets in a hierarchy of types.
typesorts/
Type0
The type of sets in a hierarchy of types.
cc/
typecoerce
Attribution tag to denote type-judgement with coercion
ecc/
typecoerce
Attribution tag to denote type-judgement with coercion
icc/
typecoerce
Attribution tag to denote type-judgement with coercion
lc/
typecoerce
Attribution tag to denote type-judgement with coercion
arith1/
unary_minus
This symbol denotes unary minus, i.e. the additive inverse.
scscp2/
unbind
This indicates the request to remove the object, referred by the cookie, from the server.
moreerrors/
unexpected
This symbol represents the error which is returned when an application reads an error caused by an unexpected problem. 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.
error/
unexpected_symbol
This symbol represents the error which is raised when an application reads a symbol which is not present in the mentioned content dictionary. When receiving such a symbol, the application should act as if it had received the OpenMath error object constructed from unexpected_symbol and the unexpected symbol as in the example below.
error/
unhandled_symbol
This symbol represents the error which is raised when an application reads a symbol which is present in the mentioned content dictionary, but which it has not implemented. When receiving such a symbol, the application should act as if it had received the OpenMath error object constructed from unhandled_symbol and the unhandled symbol as in the example below.
multiset1/
union
This symbol is used to denote the n-ary union of multisets. It takes multisets as arguments, and denotes the multiset that contains all the elements that occur in any of them, with multiplicity the sum of all the multiplicities in the multiset arguments.
set1/
union
This symbol is used to denote the n-ary union of sets. It takes sets as arguments, and denotes the set that contains all the elements that occur in any of them.
SI_functions1/
unit
The symbol to represent the function that returns the units of its argument in terms of a product of powers of SI base units.
tensor1/
unit_Cartesian
This symbol takes one argument, a natural number, and returns the Cartesian basis element, of a right handed Cartesian coordinate frame, corresponding to the value of the argument. The unit_Cartesian basis elements are each constant with respect to position in the space and define an orthonormal vector space basis.
units_sts/
unit_prefix
The type of all unit prefixes, such as "kilo".
error/
unsupported_CD
This symbol represents the error which is raised when an application reads a symbol where the mentioned content dictionary is not present. When receiving such a symbol, the application should act as if it had received the OpenMath error object constructed from unsupported_CD and the symbol from the unsupported Content Dictionary as in the example below.
transc2/
unwind
The unwinding number denotes the extent to which $z=\ln\exp z$ is not true. It was orignally defined in Corless,R.M. & Jeffrey,D.J., The Unwinding Number. SIGSAM Bulletin 30(1996) 2, pp. 28-35. However, we take the definition (which has a change of sign) from Corless,R.M., Davenport,J.H., Jeffrey,D.J. & Watt,S.M., According to Abramowitz and Stegun. SIGSAM Bulletin 34(2000) 2, pp. 58--65. Note that the symbol is normally denoted by ${\cal K}$.
linalg5/
upper-Hessenberg
This symbol represents an upper-Hessenberg matrix, it takes one argument, the argument is a vector of vectors representing the non-zero elements. The first element of the argument specifies the value of the first subdiagonal, the subsequent elements specify the value of the diagonal and subsequent super-diagonals, all other elements are zero.
linalg5/
upper-triangular
This symbol represents an upper-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.
matrix1/
upper_band
This symbol is a binary function whose first argument is a non-negative OpenMath integer which denotes the index of the upper band which is specified in the second argument. Hereby the first upper band is the one immediately above the main (generalised) diagonal, its starting coordinates relative to the top-left of the matrix thus are (1, 2).
linalgspec2/
upper_Hessenberg
This symbol represents an upper_Hessenberg matrix, it takes one argument, the argument is a vector of vectors representing the non-zero elements. The first element of the argument specifies the value of the first subdiagonal, the subsequent elements specify the value of the diagonal and subsequent super-diagonals, all other elements are zero.
linalgspec1/
upper_triangular
This symbol represents an upper-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.
rdf/
value
This symbol represents a function for accessing the value of an RDF property. It takes two arguments, a string denoting the property and an object denoting the RDF resource whose property value should be retrieved.
rdf/
valueset
This symbol represents a function for accessing all values of multivalued RDF property. It takes two arguments, a string denoting the property and an object denoting the RDF resource whose property values should be retrieved.
calculus2/
variable_of_integration
This symbol represents the variable with respect to which an integral is calculated.
polyd1/
variables
This is a unary function, whose argument is a poly_ring_d_named. When applied to its argument, it represents the list of variables of the polynomial ring.
s_data1/
variance
This symbol represents a function requiring two or more arguments, denoting the variance of its arguments. That is, the square of the standard deviation.
s_dist1/
variance
This symbol represents a unary function denoting the variance of a distribution. The argument is a function to describe the distribution. That is if f is the function which describes the distribution. The variance of a distribution is the square of the standard deviation of the distribution.
linalg2/
vector
This symbol represents an n-ary function used to construct (or describe) vectors. Vectors in this CD are considered to be row vectors and must therefore be transposed to be considered as column vectors.
linalg2col/
vector
This symbol represents an n-ary function used to construct (or describe) vectors. Vectors in this CD are considered to be column vectors, and must therefore be transposed to be considered as row vectors.
linalg3/
vector
This symbol represents an n-ary function used to construct (or describe) vectors. Vectors in this CD are considered to be column vectors, and must therefore be transposed to be considered as row vectors.
sts2/
vector
A constructor for the type of a vector
sts2/
vector_n
A constructor for the type of a vector of size n
poly1p/
vector_of_indexed_variables
vector_of_indexed_variables(x,n) returns the vector of variables (x_1, ..., x_n). vector_of_indexed_variables(x,[m,n]) returns the vector of variables (x_{1,1}, ..., x_{m,n}). Any vector of numbers can be given as an argument.
linalg1/
vector_selector
This symbol represents the function which allows individual entries to be selected from a vector, or a matrixrow. It takes two arguments. The first argument is the position in the vector (or matrixrow) of the required entry, the second argument is the vector (or matrixrow) in question. The indexing is one based, i.e. the first element is indexed by one.
linalg6/
vector_tensor
This symbol denotes a n-nary function which is used to construct the tensor product vector of its arguments, which must be vectors.
linalg5/
vector_to_list
This symbol denotes a unary function. Its argument must be a vector v. When applied to v it represents the list whose entries are the coordinates of v (with the same indexing).
mathmltypes/
vector_type
A symbol to be used as the argument of the type symbol to convey the type of a (column) vector, an n-tuple of entries.
linalg1/
vectorproduct
This symbol represents the vector product function. It takes two three dimensional vector arguments and returns a three dimensional vector. It is defined as follows: if we write a as [a_1,a_2,a_3] and b as [b_1,b_2,b_3] then the vector product denoted a x b = [a_2b_3 - a_3b_2 , a_3b_1 - a_1b_3 , a_1b_2 - a_2b_1]. Note that the vector product is often referred to as the cross product.
dimensions1/
velocity
This symbol represents the velocity physical dimension. It is the derivative of distance with respect to time.
graph1/
vertexset
This symbol represents the vertex set of a (directed or undirected) graph. It takes one argument, the graph.
permgp2/
vierer_group
This symbol represents the Klein Vierer group of order 4, viewed as a permutation group of degree 4. It consists of the identity, (1,2)(3,4), (1,3)(2,4), and (1,4)(2,3).
SI_NamedDerivedUnits1/
volt
This symbol represents an SI unit of voltage or electric tension. It has the short symbol form, "V".
units_metric1/
volt
This symbol represents the measure of one volt. This is the standard SI measure for voltage.
dimensions1/
voltage
This symbol represents the voltage physical dimension.
SI_DerivedQuantities1/
voltage
This symbol represents the physical quantity of voltage or electric tension. A variable representing an arbitrary quantity of voltage is commonly represented with the italic, upper case letter, "V".
dimensions1/
volume
This symbol represents the volume physical dimension.
SI_DerivedQuantities1/
volume
This symbol represents the physical quantity of volume. It has the short symbol form, "V".
SI_NamedDerivedUnits1/
watt
This symbol represents an SI unit of power, a joule per second. It has the short symbol form, "W".
units_metric1/
Watt
This symbol represents the measure of one Watt. This is the standard SI measure for power.
SI_NamedDerivedUnits1/
weber
This symbol represents an SI unit of magnetic flux. It has the short symbol form, "Wb".
units_time1/
week
This symbol represents the measure of one week of time.
polyd/
weighted
The first argument is a list of integers to act as variable weights, and the second is an ordering. The result is an ordering.
polyd2/
weighted
The first argument is a list of integers to act as variable weights, and the second is an ordering. The result is an ordering.
polyd/
weighted_degree
The total degree of its argument, taking into account any weights declared. The value returned is an integer: non-negative if the weights are. We note that the degree of 0 is undefined.
polyd2/
weighted_degree
The total degree of its argument, taking into account any weights declared. The value returned is an integer: non-negative if the weights are. We note that the degree of 0 is undefined.
hypergeon0/
where
The word "where" is often used in mathematical expressions to set variables or to say side conditions. CDname logic1.implies can be used for these purposes, but "where" will be more intuitive and more friendly expression for authors.
logic1p/
where
The word "where" is often used in mathematical expressions to set variables or to say side conditions. CDname logic1.implies can be used for these purposes, but "where" will be more intuitive and more friendly expression for formula book writers.
prog1/
while
This symbol represents the while loop. The syntax is while(conditional_block, block1), where conditional_block is the block that determines when to stop the while loop and block1 is the body of the while loop.
logic1/
xnor
This symbol represents the logical xnor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if there are an odd number of true arguments or true otherwise.
logic1/
xor
This symbol represents the logical xor function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if there are an odd number of true arguments or false otherwise.
units_imperial1/
yard
This symbol represents the measure of one yard. This is a standard imperial measure for distance, defined in terms of the foot.
units_us1/
yard_us_survey
This symbol represents the measure of one U.S. Survey yard.
units_siprefix1/
yocto
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^-24$
units_siprefix1/
yotta
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^24$
ringname1/
Z
This symbol represents the ring of integers.
setname1/
Z
This symbol represents the set of integers, positive, negative and zero.
units_siprefix1/
zepto
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^-21$
alg1/
zero
This symbol represents the additive identity element.
alg1/
zero
This symbol represents the additive identity element.
field1/
zero
This symbols represents a unary function, whose argument should be a field. It returns the zero element of the field.
indnat/
zero
The natural number 0, also constant base function for the inductive definition of the type of natural numbers
linalg4mat/
zero
This symbol denotes a function with two arguments, m and n, which should be natural numbers. When applied to m and n, it represents the m x n zero matrix.
linalg4vec/
zero
This symbol represents a function with one argument, which should be a natural number n. When applied to n, it represents the zero vector of size n (in the terminology of linalg3; dimension n in some terminology).
linalg5/
zero
This symbol denotes a function with two integral arguments m,n which is used to construct an (mxn) zero matrix.
linalg7/
zero
The zero symbol represents the zero vector, it takes one parameter which should be the length (dimension in some terminology) of the vector.
ring1/
zero
This symbols represents a unary function, whose argument should be a ring. It returns the zero element of the ring.
physical_consts1/
zero_Celsius
This symbol represents the zero of the Celsius temperature scale.
physical_consts1/
zero_Fahrenheit
This symbol represents the zero of the Fahrenheit temperature scale.
units_siprefix1/
zetta
This symbol represents the fact that the subsequent unit has been effectively multiplied by $10^21$
ringname1/
Zm
This symbol represents the ring of integers modulo m, where m is not necessarily a prime. It takes one argument, the integer m.
setname2/
Zm
This symbol represents the set of integers modulo m, where m is not necessarily a prime. It takes one argument, the integer m.