# How can I specify a numerical order for variables?

I need to specify that a variable is less than another, and greater than yet another, in a Mathematica program, because I will later apply a test to them that will determine what functional form to use. I also need Sign[] to give the correct answer. I tried using TagSet like this:

h /: (h>=d0) = True
h /: (h<=d1) = True


but this does not work with Sign[h-d0], for example, and

h /: Sign[h - d0] = 1;


fails because it says h is too deep.

An example of how I will use this:

f[x_] := Exp[ c Sign[h-x] ]


where x can take the values d0 or d1. (This will be used to set up boundary relations to solve for a set of coefficients to solve a PDE.) I suppose I could use some If statements to set the whole thing up, but thought it would be nicer to have a specification of the properties of h.

I'm sure this should be easy to do - any help is appreciated.

• Maybe you could just rescale $x$ to $\xi =x/h$, so you can work on ${\mathrm{sgn}}(1-\xi)$. Aug 18 '12 at 16:05
• is it for numeric or analytic solution of PDE? Aug 18 '12 at 16:23
• Unprotect[Sign]; Sign[h - d0] = 1; Aug 18 '12 at 18:09

In:= \$Assumptions = x < y < z;

• Thanks, Daniel, that's what I had in mind. I want the variable to acquire the property, so I can rely on it being evaluated properly in different contexts (If[h > d0, ...], or Sign[h-d0], etc). What I'm doing is having Mathematica solve a set of equations that are set up for a layered medium, where the behavior of some functions depends on whether the layer is above or below the source. I'll vote you up when I get the points! Aug 19 '12 at 2:59
• @TomDickens Sometimes it is better to use assumptions locally, e.g. Refine[{Sign[x - y], Sign[z - x]}, Assumptions -> x < y < z]. You might find useful answers to these questions : mathematica.stackexchange.com/questions/2404/… and mathematica.stackexchange.com/questions/4973/… Aug 19 '12 at 19:36
• @TomDickens Pay attention to the difference between the second argument between Refine[expr,Assumptions ->assum], and Refine[expr, assum] e.g. see the documentatio of Refine. Aug 19 '12 at 19:47