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

The print_ranking command prints a message describing the ranking defined on a differential polynomial ring R set up with the differential_ring command.

The ranking of a differential polynomial ring R is a total ordering over the set of all the derivatives of the differential indeterminates of R that is compatible with derivation (see ranking)

The command with(diffalg,print_ranking) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{diffalg}\right)\:$

$p≔u\left[x,y\right]+v\left[x,x\right]\;$$q≔v\left[x\right]+v\left[y,y\right]$

$p≔u_{x,y}+v_{x,x}$

$q≔v_x+v_{y,y}$

$Q≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[\mathrm{grlexA}\left[u,v\right]\right]\right)\:$

$\mathrm{print\_ranking}\left(Q\right)$

In lists, leftmost elements are greater than rightmost ones.
The derivatives of [u, v] are ordered by grlexA:
_U [tau] > _V [phi] when
    |tau| > |phi| or
    |tau| = |phi| and _U > _V w.r.t. the list of indeterminates or
    |tau| = |phi| and _U = _V and tau > phi w.r.t. [x, y]

$\mathrm{leader}\left(p\,Q\right),\mathrm{leader}\left(q\,Q\right)$

$u_{x,y},v_{y,y}$

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[\mathrm{grlexB}\left[u,v\right]\right]\right)$

$R≔\mathrm{PDE\_ring}$

$\mathrm{print\_ranking}\left(R\right)$

In lists, leftmost elements are greater than rightmost ones.
The derivatives of [u, v] are ordered by grlexB:
_U [tau] > _V [phi] when
    |tau| > |phi| or
    |tau| = |phi| and tau > phi w.r.t. [x, y] or
    tau = phi and _U > _V w.r.t. the list of indeterminates

$\mathrm{leader}\left(p\,R\right),\mathrm{leader}\left(q\,R\right)$

$v_{x,x},v_{y,y}$

$S≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[\mathrm{lex}\left[u,v\right]\right]\right)$

$S≔\mathrm{PDE\_ring}$

$\mathrm{print\_ranking}\left(S\right)$

In lists, leftmost elements are greater than rightmost ones.
The derivatives of [u, v] are ordered by lex:
_U [tau] > _V [phi] when
    tau > phi for the lex. order [x, y] or
    tau = phi and _U > _V w.r.t. the list of indeterminates

$\mathrm{leader}\left(p\,S\right),\mathrm{leader}\left(q\,S\right)$

$v_{x,x},v_x$

$T≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{indeterminates}=\left\{u\,v\right\}\,\mathrm{leaders\_of}\left(\left[p\,q\right]\right)=\left[u\left[x,y\right]\,v\left[x\right]\right]\right)$

$T≔\mathrm{PDE\_ring}$

$\mathrm{print\_ranking}\left(T\right)$

In lists, leftmost elements are greater than rightmost ones.
The derivatives of [u, v] are ordered by weights:
Weights are [u = 3, v = 0, x = 3, y = 1]
_U [tau] > _V [phi] when
    weight (_U [tau]) > weight (_V [phi]) or
    weights are equal and _U > _V w.r.t. the list of indeterminates or
    weights and indeterminates are equal and
        tau > phi for the lex. order [x, y]