diffalg(deprecated)/equations - Help

diffalg

 equations
 return the defining characteristic set of  a characterizable differential ideal
 inequations
 return the initials and separants of the defining characteristic set of  a characterizable differential ideal
 rewrite_rules
 display the equations of a characterizable differential ideal using a special syntax

 Calling Sequence equations (J) inequations (J) rewrite_rules (J)

Parameters

 J - characterizable differential ideal or a radical differential ideal

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • Characterizable and radical differential ideals are constructed by using the Rosenfeld_Groebner command. They are represented respectively by tables and list of tables.
 • A characterizable differential ideal is defined by a differential characteristic set.
 The differential polynomials forming this characteristic set are accessed by equations. They are sorted by decreasing rank.
 • The inequations of a characterizable differential ideal consist of the factors of the initials and separants of the elements of its characteristic set.
 • If $C$ and $H$ are, respectively, the set of equations and inequations of the characterizable differential ideal J, then J is equal to the saturation differential ideal $\left[C\right]$:${H}^{\mathrm{\infty }}$. It corresponds to the differential system $C=0,H\ne 0$.
 • A differential polynomial $p$  belongs to the characterizable differential ideal J if and only if $p$ is reduced to 0 by $C$ via differential_sprem.
 • The function rewrite_rules displays the equations of a characterizable differential ideal J as rewrite rules with the following the syntax:
 $\mathrm{rank}\left(p\right)=-\frac{q}{\mathrm{initial}\left(p\right)}$, where, of course, $q=p-\mathrm{initial}\left(p\right)\mathrm{rank}\left(p\right)$.
 (see rank, initial)
 The list is sorted decreasingly.
 • If J is  a radical differential ideal given by a characteristic decomposition, that is, as a list of tables representing characterizable differential ideals, then the function is mapped on all its components.
 • The command with(diffalg,equations) allows the use of the abbreviated form of this command.
 • The command with(diffalg,inequations) allows the use of the abbreviated form of this command.
 • The command with(diffalg,rewrite_rules) allows the use of the abbreviated form of this command.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[{\mathrm{lex}}_{u,v}\right]\right):$
 > $\mathrm{p1}≔{v}_{[]}{u}_{x,x}-{u}_{x}:$
 > $\mathrm{p2}≔{u}_{x,y}:$
 > $\mathrm{p3}≔{u}_{y,y}^{2}-1:$
 > $J≔\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{p1},\mathrm{p2},\mathrm{p3}\right],R\right)$
 ${J}{:=}\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (1)
 > $\mathrm{equations}\left(J\right);$$\mathrm{inequations}\left(J\right)$
 $\left[\left[{{u}}_{{x}{,}{x}}{}{{v}}_{{[}{]}}{-}{{u}}_{{x}}{,}{{u}}_{{x}{,}{y}}{,}{{u}}_{{y}{,}{y}}^{{2}}{-}{1}{,}{{v}}_{{y}}\right]{,}\left[{{u}}_{{x}}{,}{{u}}_{{y}{,}{y}}^{{2}}{-}{1}\right]\right]$
 $\left[\left[{{u}}_{{y}{,}{y}}{,}{{v}}_{{[}{]}}\right]{,}\left[{{u}}_{{y}{,}{y}}\right]\right]$ (2)
 > $\mathrm{rewrite_rules}\left(J\right)$
 $\left[\left[{{u}}_{{x}{,}{x}}{=}\frac{{{u}}_{{x}}}{{{v}}_{{[}{]}}}{,}{{u}}_{{x}{,}{y}}{=}{0}{,}{{u}}_{{y}{,}{y}}^{{2}}{=}{1}{,}{{v}}_{{y}}{=}{0}\right]{,}\left[{{u}}_{{x}}{=}{0}{,}{{u}}_{{y}{,}{y}}^{{2}}{=}{1}\right]\right]$ (3)