Date: Aug 2, 2013 2:49 AM
Author: Daniel Lichtblau
Subject: Re: Pattern with powers

On Wednesday, July 31, 2013 11:14:07 PM UTC-5, Dr. Wolfgang Hintze wrote:
> Am Mittwoch, 31. Juli 2013 10:49:04 UTC+2 schrieb Alexei Boulbitch:
>

> > I must admit that I am an absolute beginner in patterns, as I cannot cope with a little problem with patterns consisting of powers of variables x and y.
>
> >
> > Specifically, I would like to select from a list all terms of the form

>
> > c x^u y^v (numerical coefficient c times x to the power u times y to the power v)
>
>

> > where u and v are allowed to take the values 0 and 1.
>
> > How can I do this using Cases?
>
> >
>
> > I have already accomplished the first non trivial step using _. (blank followed by a dot) in order to get first powers of the variables:
>
> >
>
> > ls = List@@Expand[5 (x + y)^3]
>
> >
>
> > {5*x^3, 15*x^2*y, 15*x*y^2, 5*y^3}
>
> >
>
> > Example 1
>
> > a = 2; Cases[ls, (_.)*x^(u_.)*y^(v_.) /; u >= a && v < a]
>
> > gives
>
> > {15*x^2*y}
>
> > but misses the term
>
> > 5*x^3
>
> > Example 2: this would be the form I would like most
>
> > Cases[ls, (_.)*x^_?(#1 >= a & )*y^_?(#1 < a & )]
>
> > gives
>
> > {}
>
> > Here even I didn't get the dot behind the blank before the test, so it misses first powers.
>
> > Thanks in advance for any help.
>
> > Best regards,
>
> >
>
> > Wolfgang
>
> >
>
> > Hi, Wolfgang,
>
> > Your explanation is not quite clear. Have a look:
>
> >
>
> > Clear[a, u, v, b];
>
> > a = 2;
>
> > Cases[ls, (Times[_, x, Power[y, v_]] /;
>
> > v <= a) | (Times[_, Power[x, u_]] /; u >= a)]
>
> > {5 x^3, 15 x^2 y, 15 x y^2}
>
> > Is it, what you are after? Or this:
>
> > Clear[a, u, v, b];
>
> > a = 2;
>
> > Cases[ls, (Times[_, x, Power[y, v_]] /;
>
> > v < a) | (Times[_, Power[x, u_]] /; u >= a)]
>
> > {5 x^3, 15 x^2 y}
>
> > ??
>
> > Have fun, Alexei
>
> > Alexei BOULBITCH, Dr., habil.
>
>
> Alexei,
>
>
>
> thanks for your message.
>
> As I tried to explain, I wish to extract from the list ls all terms of the form
>
> c x^u y^v, with c a numerical factor, u and v integers subject to the conditions u>=a and v<a with integer a>0.
>
> I have no problem as long as all terms in the list are "true" powers, i.e. as long as u>=2, v>=2.
>
> Therefore I asked for a solution which also covers the values 0 and 1 for the powers.
>
> Your first solution contains the wrong conditions, and the second one fails for a term x^2 y^3 which is selected by your proposal but it shouldn't be selected because v>2.
>
> Best regards,
>
> Wolfgang


Powers of zero, that is, missing factors, will be difficult to handle via patterns. A more natural choice, in my view, would be to define predicate functions. Here is an example.

In[19]:= s = List @@ Expand[5 (x + y)^3]

Out[19]= {5 x^3, 15 x^2 y, 15 x y^2, 5 y^3}

In[20]:= a = 1;

In[21]:= Select[s, Exponent[#, x] >= a && Exponent[#, y] <= a &]

Out[21]= {5 x^3, 15 x^2 y}

Obviously this can also be done with Cases, but that seems a bit more awkward to me.

In[22]:= Cases[s, aa_ /; Exponent[aa, x] >= a && Exponent[aa, y] <= a]

Out[22]= {5 x^3, 15 x^2 y}

Since "natural" and "awkward" are in the mind of the beholder, one might well hold an opinion the reverse of my own.

Daniel Lichtblau
Wolfram Research