CartanKuranishi(JB, D)

jet.spad line 4569 [edit on github]

CartanKuranishi is a package for the completion of a given differential equation to an involutive equation. Procedures for Cartan characters and Hilbert polynomial are also provided. Based on the Cartan-Kuranishi theorem as it is used in formal theory.

alpha: (NonNegativeInteger, List NonNegativeInteger) -> List NonNegativeInteger

alpha(q, beta) computes the Cartan characters for a differential equation of order q and with characters beta.

alphaHilbert: SparseUnivariatePolynomial Fraction Integer -> List NonNegativeInteger

alphaHilbert(hp) computes the Cartan characters for the Hilbert polynomial hp.

arbFunctions: (NonNegativeInteger, Integer, List NonNegativeInteger) -> List Integer

arbFunctions(q, j, cc) uses the Cartan characters cc to compute the number of arbitrary functions of differentiation order j in the general solution of a differential equation of order q.

bound: (NonNegativeInteger, NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger

bound(n, m, q) computes an upper bound for the number of prolongations needed to make the symbol of an equation of order q with n independent and m dependent variables involutive.

complete2: JetDifferentialEquation(JB, D) -> Record(IDe: JetDifferentialEquation(JB, D), ISys: List D, Order: NonNegativeInteger, NumProj: NonNegativeInteger, Dim: NonNegativeInteger, CarChar: List NonNegativeInteger)

complete2(de) is like complete but returns the involutive equation IDe, a basis ISys for the involutive system without prolongations, its order Order, the number NumProj of needed projections and the Cartan characters CarChar.

complete: JetDifferentialEquation(JB, D) -> Void

complete(de) completes de to an involutive equation. No result is returned; the display depends of the setting of the output flags with setOutput.

gauge: (NonNegativeInteger, Integer, List NonNegativeInteger) -> List Integer

gauge(q, j, gamma) computes the gauge corrections to the number of arbitrary functions of differentiation order j for a system of order q with gamma gauge functions.

gaugeHilbert: (NonNegativeInteger, List NonNegativeInteger) -> SparseUnivariatePolynomial Fraction Integer

gaugeHilbert(q, gamma) computes the gauge correction to the Hilbert polynomial for a system of order q with gamma gauge functions.

hilbert: List NonNegativeInteger -> SparseUnivariatePolynomial Fraction Integer

hilbert(cc) computes the Hilbert polynomial to the Cartan characters cc.

setOutMode: NonNegativeInteger -> NonNegativeInteger

setOutput(i) controls amount of generated output during the completion algorithm: i = 0 –> no display, i = 1 –> result is displayed, i = 2 –> Cartan characters are displayed, i = 3 –> integrability conditions are traced, i = 4 –> intermediate dimensions are traced, i = 5 –> all intermediate systems are traced, i = 6 –> all intermediate systems and symbols are traced, if i > 10 then TeX output is produced. Default is 0. The old value is returned.

setRedMode: NonNegativeInteger -> NonNegativeInteger

setRedMode(i) sets the flag for the reduction mode. Returns old value. Current values are: i = 0 –> No reduction of integrability conditions etc. i = 1 –> Autoreduction of complete system and reduction of all integrability conditions. Default is 0.

setSimpMode: NonNegativeInteger -> NonNegativeInteger

setSimpMode(i) sets the simplification mode used in JetDifferentialEquation. Returns old value.

stirling: (NonNegativeInteger, NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger

stirling(i, k, q) computes the corresponding modified Stirling number.