The Stirling numbers of the first and second kind, formerly available in the combinat package, are now top-level commands, Stirling1 and Stirling2.  For compatibility with previous releases, they can also be called as part of the combinat package as combinat[stirling1] and combinat[stirling2].

To view a list of mathematical functions available as top-level commands, see initial functions.

The finance package has been deprecated.  The commands formerly in that package are now part of the superseding Finance package.

The former ChangeRanking command got merged into RosenfeldGroebner.

Example 

with(DifferentialAlgebra):

Define a ranking for a DE system that involves two dependent variables $x,y$ and one independent variable $t$

R := DifferentialRing(derivations = [t], blocks = [[x, y]]):

sys := [diff(x(t),t) = -alpha*x(t) + beta*y(t) - (rho*x(t))/(kappa+x(t)), diff(y(t),t) = alpha*x(t) - beta*y(t)];

$\mathrm{sys}≔\left[\frac{\ⅆ}{\ⅆt}x\left(t\right)=-\mathrm{\alpha }x\left(t\right)+\mathrm{\beta }y\left(t\right)-\frac{\mathrm{\rho }x\left(t\right)}{\mathrm{\kappa }+x\left(t\right)}\,\frac{\ⅆ}{\ⅆt}y\left(t\right)=\mathrm{\alpha }x\left(t\right)-\mathrm{\beta }y\left(t\right)\right]$

This call to RosenfeldGroebner bundles the two ODEs in a regular differential chain.

ideal := RosenfeldGroebner(sys, R);

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\right]$

Equations(ideal, solved);

$\left[\left[\frac{\ⅆ}{\ⅆt}x\left(t\right)=-\frac{\mathrm{\alpha }{x\left(t\right)}^2-\mathrm{\beta }y\left(t\right)x\left(t\right)+\mathrm{\alpha }x\left(t\right)\mathrm{\kappa }+\mathrm{\rho }x\left(t\right)-\mathrm{\beta }y\left(t\right)\mathrm{\kappa }}{\mathrm{\kappa }+x\left(t\right)}\,\frac{\ⅆ}{\ⅆt}y\left(t\right)=\mathrm{\alpha }x\left(t\right)-\mathrm{\beta }y\left(t\right)\right]\right]$

As can be seen in the left-hand sides above, $\mathrm{x\text{'}}$ and $\mathrm{y\text{'}}$ are isolated, so the leading derivatives appear with degree $1$, and hence the differential ideal is prime:

Is(prime, ideal);

$\mathrm{true}$

This ideal can be rewritten in decoupled form, solving for $y$ with respect to $x$, by changing the ranking for the dependent variables, from $\left[\left[x\,y\right]\right]$ to $\left[y\,x\right]$. Bearing in mind that this ideal is prime, this change can be performed directly by RosenfeldGroebner passing the ideal as first argument. You only need to additionally pass to RosenfeldGroebner the piece of information that is changing

newideal := RosenfeldGroebner(ideal, blocks = [y, x]);

$\mathrm{newideal}≔\left[\mathrm{regular\_differential\_chain}\right]$

Check the Equations: they are correspondingly solved for $y\left(t\right)$ with respect to $x\left(t\right)$

Equations(newideal, solved);

$\left[\left[y\left(t\right)=-\frac{-\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)x\left(t\right)-\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\mathrm{\kappa }-\mathrm{\alpha }{x\left(t\right)}^2-\mathrm{\alpha }x\left(t\right)\mathrm{\kappa }-\mathrm{\rho }x\left(t\right)}{x\left(t\right)\mathrm{\beta }+\mathrm{\beta }\mathrm{\kappa }}\,\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)=-\frac{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right){x\left(t\right)}^2\mathrm{\alpha }+\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right){x\left(t\right)}^2\mathrm{\beta }+2\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)x\left(t\right)\mathrm{\alpha }\mathrm{\kappa }+2\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)x\left(t\right)\mathrm{\beta }\mathrm{\kappa }+\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\mathrm{\alpha }{\mathrm{\kappa }}^2+\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\mathrm{\beta }{\mathrm{\kappa }}^2+\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\mathrm{\kappa }\mathrm{\rho }+{x\left(t\right)}^2\mathrm{\beta }\mathrm{\rho }+x\left(t\right)\mathrm{\beta }\mathrm{\kappa }\mathrm{\rho }}{{x\left(t\right)}^2+2x\left(t\right)\mathrm{\kappa }+{\mathrm{\kappa }}^2}\right]\right]$

unwith(DifferentialAlgebra):

The requires command has been deprecated.

A new _ATTRIBUTE structure has been added to certain plot data structures and its purpose is to carry information for internal use. This change does not affect usage of Maple's plotting commands and may be relevant only if you directly manipulate the plot data structures (which is generally not recommended).