Re: Use of Mathematica with Rule-based Equation Derivations

• To: mathgroup at smc.vnet.net
• Subject: [mg61933] Re: [mg61914] Use of Mathematica with Rule-based Equation Derivations
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sat, 5 Nov 2005 01:52:16 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Using rules:

Clear[\[DoubleStruckCapitalE],rules];

rules[x_Symbol:x]:={
\[DoubleStruckCapitalE][expr_?(FreeQ[#,x]&)]:>expr,
\[DoubleStruckCapitalE][a_?(FreeQ[#,x]&) expr_?(!FreeQ[#,x]&)]:>
a*\[DoubleStruckCapitalE][expr],
\[DoubleStruckCapitalE][a_ + b_]:>\[DoubleStruckCapitalE][a] + \
\[DoubleStruckCapitalE][b],
\[DoubleStruckCapitalE][\[DoubleStruckCapitalE][expr_]]:>\
\[DoubleStruckCapitalE][expr]
};

{\[DoubleStruckCapitalE][a*x],\[DoubleStruckCapitalE][
b+f[x]],\[DoubleStruckCapitalE][a+\[DoubleStruckCapitalE][
b x]]}//.rules[]

{a \[DoubleStruckCapitalE](x),b+\[DoubleStruckCapitalE](f(x)),a+
b \[DoubleStruckCapitalE](x)}

{\[DoubleStruckCapitalE][a*y],\[DoubleStruckCapitalE][
b+f[y]],\[DoubleStruckCapitalE][a+\[DoubleStruckCapitalE][
b y]]}//.rules[y]

{a \[DoubleStruckCapitalE](y),b+\[DoubleStruckCapitalE](f(y)),a+
b \[DoubleStruckCapitalE](y)}

Using definitions:

Clear[\[DoubleStruckCapitalE]];

\[DoubleStruckCapitalE][expr_]:=\[DoubleStruckCapitalE][expr,x];
\[DoubleStruckCapitalE][expr_, x_Symbol:x]:=expr/;FreeQ[expr,x];
\[DoubleStruckCapitalE][a_ expr_, x_Symbol:x]:=
a*\[DoubleStruckCapitalE][expr,x]/;
FreeQ[a, x]&&!FreeQ[expr,x];
\[DoubleStruckCapitalE][a_ + b_, x_Symbol:x]:=\[DoubleStruckCapitalE][a,x]+
\
\[DoubleStruckCapitalE][b,x];
\[DoubleStruckCapitalE][\[DoubleStruckCapitalE][expr_,x_Symbol:x],
x_Symbol:
x]:=\[DoubleStruckCapitalE][expr,x];

{\[DoubleStruckCapitalE][a*x],\[DoubleStruckCapitalE][
b+f[x]],\[DoubleStruckCapitalE][a+\[DoubleStruckCapitalE][b x]]}

{a \[DoubleStruckCapitalE](x,x),b+\[DoubleStruckCapitalE](f(x),x),a+
b \[DoubleStruckCapitalE](x,x)}

{\[DoubleStruckCapitalE][a*y,y],\[DoubleStruckCapitalE][b+
f[y],y],\[DoubleStruckCapitalE][a+\[DoubleStruckCapitalE][b y,y],y]}

{a \[DoubleStruckCapitalE](y,y),b+\[DoubleStruckCapitalE](f(y),y),a+
b \[DoubleStruckCapitalE](y,y)}

Bob Hanlon

>
> From: mmorriss at gcn.ou.edu
To: mathgroup at smc.vnet.net
> Date: 2005/11/04 Fri AM 05:11:39 EST
> Subject: [mg61933] [mg61914] Use of Mathematica with Rule-based Equation Derivations
>
> Hi All - I'm a many-year user of mathematica, but have always had one
> particular problem with it that I may have just simply missed reading
>
> Mathematica Version: 5.2
>
> Problem: I would like to develop a set of re-rewite rules to apply to the
> Expected Value operator.  For example:
>
> E[a x] = a E[x]  a -> constant, x -> variable
> E[b + f[x]] = b + E[f[x]] , etc.
>
> The issue is how does one using Mathematica distinguish a 'constant
> variable (i.e. a and b)' from a variable 'variable' (i.e. 'x')? The head
> of a, b and x is 'Symbol' and neither a, b nor x contain a number so I
> can't use a_?NumberQ to identify it as a constant.
>
> This actually goes to the wider question of how does one use Mathematica
> for symbolic derviations where numbers are not actually substituted in the
> derviation?
>
> E.g. E[a + E[b x]] /. Rule2
>    out= a + b E[x] etc.
>
> Thanks all - Mark Morrissey
> University of Oklahoma
>
>
>

```

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