QEtaSpecificationCategory¶

qetaspec.spad line 113 [edit on github]

QEtaSpecificationCategory lists the properties that are common for eta-quotients and q-Pochhammer quotients. Specifications can be turned into the purified representation by the purify function.

1: %: from MagmaWithUnit

*: (%, %) -> %: from Magma

/: (%, %) -> %: from Group

<=: (%, %) -> Boolean: from PartialOrder

<: (%, %) -> Boolean: from PartialOrder

=: (%, %) -> Boolean: from BasicType

>=: (%, %) -> Boolean: from PartialOrder

>: (%, %) -> Boolean: from PartialOrder

^: (%, Integer) -> %: from Group
^: (%, NonNegativeInteger) -> %: from MagmaWithUnit
^: (%, PositiveInteger) -> %: from Magma

~=: (%, %) -> Boolean: from BasicType

coerce: % -> OutputForm: from CoercibleTo OutputForm

commutator: (%, %) -> %: from Group

conjugate: (%, %) -> %: from Group

denom: % -> %: denom(x) returns the part of the specification that corresponds to negative exponents. It holds: x=numer(x)/denom(x).

dilate: (%, PositiveInteger) -> %: dilate(x,n) is the respective operation of replacing tau by n*tau. Note that q=exp(2*pi*i*tau).

hash: % -> SingleInteger: from Hashable

hashUpdate!: (HashState, %) -> HashState: from Hashable

inv: % -> %: from Group

latex: % -> String: from SetCategory

leftPower: (%, NonNegativeInteger) -> %: from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %: from Magma

leftRecip: % -> Union(%, failed): from MagmaWithUnit

level: % -> PositiveInteger: level(x) returns the lcm of all the indices of x.

max: (%, %) -> %: from OrderedSet

min: (%, %) -> %: from OrderedSet

monomial: % -> Polynomial Integer: monomial(s) returns monomial(s,e,y) for certain strings e and y that depend on the concrete domain. This function is the inverse of specification.

monomial: (%, String, String) -> Polynomial Integer: monomial(s,u,v) translates the specification to a monomial. Each index i is translated into a variable via indexedSymbol(u,i) and raised to the respective exponent given by the specification if the exponent is positive and via indexedSymbol(v,i) if the exponent is negative. All those variables with their (absolute) value of) the exponent are multiplied together to give the resulting monomial. The vi correspond to 1/ui.

numer: % -> %: numer(x) returns the part of the specification that corresponds to positive exponents.

one?: % -> Boolean: from MagmaWithUnit

purify: % -> %: purify(x) returns an element of this domain that represents the same value, but possibly with other indices. If [2g,g,e] is in parts(x), then it is replaced by the two elements [g,e] and [2g,-e], which corresponds to $(q;q^2)_infty=(q;q))infty/(q^2;q^2)_infty$ and also its equivalent as eta-quotients, i.e. $eta_{2g,g}(tau)=eta(gtau)/eta(2gtau)$.

quotient: % -> Fraction Polynomial Integer: quotient(s) returns quotient(s,e) for a certain string e that depends on the concrete domain. This function is the inverse of specification.

quotient: (%, String) -> Fraction Polynomial Integer: quotient(s,u) returns monomial(numer(s),u)/monomial(denom(s),u).

recip: % -> Union(%, failed): from MagmaWithUnit

rightPower: (%, NonNegativeInteger) -> %: from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %: from Magma

rightRecip: % -> Union(%, failed): from MagmaWithUnit

sample: %: from MagmaWithUnit

smaller?: (%, %) -> Boolean: from Comparable

specification: (Fraction Polynomial Integer, String) -> %: specification(f,e) returns nspec/dspec for nspec:=specification(numer(f),e) and dspec:=specification(denom(f),e).

specification: (Polynomial Integer, String) -> %: specification(p,e) assumes that p is a polynomial with only one monomial. Furthermore, it assumes that p is in the variables ed or ed_g and 0<g<=d and no other variables. The coefficient of this monomial is ignored. The function collects the d's and g's and the corresponding exponent of the variable and creates the respective specification.

specification: Fraction Polynomial Integer -> %: specification(f) returns specification(f,e) for some variable e that depends on the domain. This function is the inverse of quotient.

specification: Polynomial Integer -> %: specification(p) returns specification(p,e) for some variable e that depends on the domain. This function is the inverse of monomial.

BasicType

CoercibleTo OutputForm