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Re: Changing the Natural Sort Order
*To*: mathgroup at smc.vnet.net
*Subject*: [mg49184] Re: Changing the Natural Sort Order
*From*: ab_def at prontomail.com (Maxim)
*Date*: Tue, 6 Jul 2004 03:34:28 -0400 (EDT)
*References*: <ccb5hr$ekg$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
"David Park" <djmp at earthlink.net> wrote in message news:<ccb5hr$ekg$1 at smc.vnet.net>...
> Dear MathGroup,
>
> Is it possible to change the natural sort order of symbols that is used in Sort?
>
> I would like something like the following statement (that does not work).
>
> Assuming[d < b < c, Sort[{a, b, c, d, f}]]
>
> giving the desired output
>
> {a,d,b,c,f}
>
> I won't be sorting simple lists of symbols, but lists of similar, but unspecified, expressions that contain the symbols. For example...
>
> {h[x,g[a]], h[x,g[b]], h[x,g[c]], h[x,g[d]], h[x,g[f]]}
>
> which should give
>
> {h[x,g[a]], h[x,g[d]], h[x,g[b]], h[x,g[c]], h[x,g[f]]}
>
> Is there any way to do this?
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
The first thing to try when dealing with assumptions is to use Refine:
In[1]:=
Assuming[b < a && b > c,
Sort[{a, b, c}, Refine[# < #2]& ]]
Out[1]=
{c, b, a}
In general, this won't work if the test is undefined for some pairs as
in your example:
In[2]:=
Assuming[a > b,
Sort[{a, x, b}, Refine[# < #2]& ]]
Out[2]=
{a, x, b}
If you have h[x,g[a]] instead of a, then you can simply use
#[[2,1]]<#2[[2,1]] instead of #<#2, or you can pull out the occurences
of a,b,c with Cases.
However, it is hard to guarantee anything with Refine; even a
simplification of the form Refine[Sign[expr],expr>0] may fail:
In[3]:=
Refine[Sign[x1*y1+x2*y2+x3*y3], x1*y1+x2*y2+x3*y3 > 0]
Out[3]=
Sign[x1*y1+x2*y2+x3*y3]
So you're probably better off just defining a function which will
return the desired ordering of the elements, without relying on
assumptions.
You can use Reduce for more complicated cases:
In[4]:=
Assuming[1 < E^x < E,
Sort[{x, 0, 1},
Reduce[ForAll[x, $Assumptions, # < #2], Reals]& ]]
Out[4]=
{0, x, 1}
Maxim Rytin
m.r at inbox.ru
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