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diffalg[equations] - return the defining characteristic set of  a characterizable differential ideal

diffalg[inequations] - return the initials and separants of the defining characteristic set of  a characterizable differential ideal

diffalg[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 C:H. It corresponds to the differential system C=0,H0.

• 

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:

  

rankp=qinitialp, where, of course, q=pinitialprankp.

  

(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.

withdiffalg:

R:=differential_ringderivations=x,y,ranking=lexu,v:

p1:=v[]ux,xux:

p2:=ux,y:

p3:=uy,y21:

J:=Rosenfeld_Groebnerp1,p2,p3,R

J:=characterizable,characterizable

(1)

equationsJ;inequationsJ

ux,xv[]ux,ux,y,uy,y21,vy,ux,uy,y21

uy,y,v[],uy,y

(2)

rewrite_rulesJ

ux,x=uxv[],ux,y=0,uy,y2=1,vy=0,ux=0,uy,y2=1

(3)

See Also

diffalg(deprecated), diffalg(deprecated)/differential_algebra, diffalg(deprecated)[differential_sprem], diffalg(deprecated)[Rosenfeld_Groebner], DifferentialAlgebra[Equations]


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