Re: Re: Use of Mathematica with Rule-based Equation Derivations

• To: mathgroup at smc.vnet.net
• Subject: [mg61993] Re: [mg61935] Re: [mg61914] Use of Mathematica with Rule-based Equation Derivations
• From: János <janos.lobb at yale.edu>
• Date: Tue, 8 Nov 2005 03:41:33 -0500 (EST)
• References: <200511041011.FAA14913@smc.vnet.net> <200511050652.BAA01989@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On Nov 5, 2005, at 1:52 AM, Andrzej Kozlowski wrote:

> On 4 Nov 2005, at 19:11, mmorriss at gcn.ou.edu wrote:
>
>> 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
>> about.
>>
>> 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
>>
>>
>
>
> First of all, it is not a good idea to use a protected symbol (E) in
> this way. But concerning your question: there is no reason why you
> shouldn't use NumberQ or NumericQ in the way you indicated.
>
> First define the rule exactly in the way you said you could not:
>
> Expectation[b_?NumericQ* x_]:=b*Expectation[x]
>
> Now set:
>
> NumericQ[a]=True;
>
>
> Now you get
>
> Expectation[a x]
>
> a Expectation[x]
>
> and of course as a bonus you automatically get:
>
> Expectation[3 x]
>
>
> 3 Expectation[x]
>
>
> This gives me a chance to offer a puzzle (for which I offer no prize)
> for those who like this sort of things. Note that
>
>
> NumericQ[a]
>
> True
>
> but where is this information stored by Mathematica? It can't be a
> DownValue for NumericQ, since NumericQ has the Attribute Protected
> and I did not unprotect it. And indeed:
>
>
> DownValues[NumericQ]
>
> {}
>
> Or, we could try to give a an UpValue:
>
> Expectation[b_?NumericQ* x_]:=b*Expectation[x]
>
>
> a/:NumericQ[a]=True;
>
>
> NumericQ[a]
>
> True
>
> Expectation[a x]
>
>
> a Expectation(x)
>
> Everything works fine yet:
>
>
> UpValues[a]
>
> {}
>
> So now again the puzzle: where is this information stored?
>
> Note also that NumberQ behaves quite differently. In fact you can't
> set NumberQ[a]=True without first unprotecting NumberQ. On the other
> hand if you use an UpValue to make a into a "number"
>
>
> a/:NumberQ[a]=True;
>
> then as expected:
>
>
>
> UpValues[a]
>
>
> {HoldPattern[NumberQ[a]] :> True}
>
> Well, any guesses?

According to Trurl the Constructor, the value is stored in kingdom of
Krool, but its dragonic value is set to close to zero.  That is why
neither DownValues or UpValues as junior constructors are able to
reach it.

János

>
>
> Andrzej Kozlowski
>
>
> P.S. All the above with Mathematica 5.1 but I expect it;s the same in
> 5.2
>
>
>
>

```

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