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Re: Re: Simplifying and Rearranging Expressions

• To: mathgroup at smc.vnet.net
• Subject: [mg95981] Re: [mg95956] Re: Simplifying and Rearranging Expressions
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 31 Jan 2009 01:15:11 -0500 (EST)
• References: <gls1u8$hjl$1@smc.vnet.net> <200901301047.FAA06653@smc.vnet.net> <78E34FCC-E733-4CAE-AA27-CE95DB401072@mimuw.edu.pl>

Correction, not "Computation Center" but "Calculation Center" which  
has since been renamed CalcCenter and it seems it is still being sold:

http://www.wolfram.com/products/calccenter/

Andrzej Kozlowski


On 30 Jan 2009, at 16:14, Andrzej Kozlowski wrote:

> To my amazement I have found something that I agree here. I do agree  
> that it is largely a pointless waste of time to use computer  
> algebra, which relies on pretty complex algorithms (like Groebner  
> basis) to make this **look** the way you want them to look, for  
> reasons that have no particular relation to these algorithms. Its a  
> waste of computing resources, the effort of mathematicians and  
> programmers and most of all the user, who could much more easily  
> achieve this effect in other ways, one of which is TeX (even better  
> was, in my opinion, was David Bailey's clever idea to use color to  
> manipulate Mathematica expressions almost as one does by hand -  
> unfortunately this appears to have been abandoned due to lack of  
> interest).
> I don't think however there is any chance whatever of WRI  
> incorporating TeX into Mathematica, for two reasons. One is that it  
> would be going against their principal idea of having all  
> Mathematica expressions fully controllable by means of the  
> Mathematica programming language. Clearly this would not be true of  
> TeX strings, if they were meant to be interpreted for display.  
> Secondly, because other CAS systems have essentially tried to do  
> this sort of thing with very little to show for it in terms of  
> market success. You seem to be completely unaware of how tiny the  
> TeX users community is compared with the community of users of  
> programs like Mathematica.
> This reminds me also that a lot of suggestions which you have made  
> about the way Mathematica ought to be (simple, cheap, computation  
> engine, no fancy staff) has already been tried by WRI and clearly  
> failed. It was called something like The Computation Center, and  
> limited version of Mathematica, that Wolfram once sold for a  
> fraction of the price of the full thing. The only problem was that  
> hardly anyone bought it (I suspect you did not either).
> Not surprisingly WRI is likely to be pretty skeptical of bright  
> ideas that remind them of things that they or others have already  
> tired and have been shown not to work.
>
> Andrzej Kozlowski
>
> On 30 Jan 2009, at 11:47, AES wrote:
>
>> In article <gls1u8$hjl$1 at smc.vnet.net>,
>> "David Park" <djmpark at comcast.net> wrote:
>>
>>> I want to start a thread on this because I believe many MathGroup  
>>> people
>>> will have some useful things to say.
>>
>> I'll bite, because I've done a bit of thinking on this.
>>
>>
>>> A common task for Mathematica users is to obtain an expression  
>>> that is in a
>>> particular form. For students and teachers this may often be a  
>>> textbook
>>> form, or there may be other reasons that a particular form is  
>>> desired.
>>
>> Just to add a bit of specificity to this, let's consider expressions
>> that arise in optics and e-m theory, which generally involve a set of
>> physical quantities (velocity of light, propagation constant,
>> permeabilities, index of refraction, frequency, wavelength,
>> characteristics impedance, critical angle of refraction, and multiple
>> others) that are conventionally written as
>>
>>  c, k, (or beta), mu, epsilon, n, f (or omega), lambda,
>>  eta or z_0, thetaCrit, and so on
>>
>> **each of which is directly linked or coupled to (that is, can be
>> calculated from) several others in the same set**.
>>
>> [You have to put up with a side story at this point.  The  
>> distinguished
>> physicist W. K. H. Panofsky, who just recently died, early in his  
>> career
>> co-authored with Melba Phillips a small but excellent text on  
>> classical
>> e-m, colloquially known as "Panofsky and Phillips", from which I and
>> many others studied.  This was long enough ago that cgs and mks units
>> were still fighting it out in the physics community.]
>>
>> [P and P beautifully sidestepped this issue by, as they noted in the
>> Preface to their book, writing every equation in their book using an
>> appropriate (but generally different) subset of the above symbols,  
>> such
>> that every equation was valid in mks units as it stood, **and could  
>> be
>> instantly converted to be exactly valid in cgs units as well,  
>> simply by
>> replacing any factor of epsilon that appeared in any of these  
>> equations
>> by 1/ 4 pi **.]
>>
>>
>>> It might be thought that this should be an easy task but quite  
>>> often it can
>>> be a very difficult task, even involving mathematical derivation  
>>> and many of
>>> the capabilities of Mathematica. Not obtaining a specific form may  
>>> be a
>>> matter of not knowing how to solve the problem in the first place.
>>
>> It may not be just a difficult task;  in fact, **it may be an  
>> impossible
>> task** -- not to mention **an unnecessary and undesirable task**.
>>
>> 1)  As already noted above, you may want to write expressions that
>> contain some subset of a linked set of variables in different ways at
>> different points in an exposition, because these different ways are
>> conventional in the field, and/or make the physical meaning clearer.
>>
>> For example, you may want to write the space-time variation of a  
>> phasor
>> wave amplitude as Exp[ I k z - I omega t] because that's neat,  
>> simple,
>> and conventional.
>>
>> But then, in discussing a waveguide mode where a factor k d (d =
>> waveguide width) appears, you may want to write that factor instead  
>> in
>> the form 2 pi d/lambda to emphasize that it's the width in  
>> wavelengths
>> that's important.
>>
>> But if at some point in your notebook you're going to insert any of  
>> the
>> dependences within this set -- e.g., k := 2 pi / lambda -- then  
>> you're
>> stuck with this from then on.
>>
>> 2)  A second point:  My experience has been that useful identities  
>> that
>> often arise in analyses -- for example, with suitable  
>> qualifications the
>> infinite integral of Exp[- a x^2 + b x] == Exp[b^2/4a] -- sometime  
>> just
>> won't fall out (i.e., won't be explicitly evaluated by Mathematica  
>> if a
>> and b are actually more complicated expressions.
>>
>> 3)  More generally, one very often wants to do the eventual numerical
>> calculations using only one or another form of dimensionless or
>> normalized variables, because that's numerically efficient as well as
>> physically and practically useful in expressing the results.
>>
>> And, if at some point in an exposition you're going to convert your
>> analytical and expositional formulas into dimensionless formulas for
>> numerical calculation purposes -- **at that point you really don't  
>> care
>> how Mathematica arranges the resulting expression**.
>>
>>
>>> Nevertheless, even simple rearrangement can be difficult. I  
>>> sometimes think
>>> of it as doing surgery on expressions. I believe it is generally  
>>> desirable
>>> to use Mathematica to rearrange an expression and not retype the  
>>> expression.
>>> Retyping is too error prone.
>>
>> Last sentence is true; immediately preceding sentence may be true as
>> phrased -- but is a mistaken belief.  Comments on this below.
>>
>>
>>> Simplify and FullSimplify are amazingly useful but it is difficult  
>>> to
>>> control them and obtain a precise result. One will often have to do
>>> additional piecemeal operations. One downside of Simplify and  
>>> FullSimplify
>>> is that they can return different forms with different Mathematica  
>>> versions.
>>> Then any additional operations in an old notebook may no longer  
>>> work. It
>>> would be nice if there was a method of using these commands that  
>>> would be
>>> more version independent.
>>
>>> Various routines such as Together, Apart, Factor, TrigReduce,  
>>> TrigFactor,
>>> TrigExpand, TrigToExp, GroebnerBasis etc., can be useful in  
>>> getting a
>>> specific form. MapAt is very useful for doing surgery on specific  
>>> parts of
>>> an expression. Mathematica often gets two factors that have extra  
>>> minus
>>> signs. You can correct that by mapping Minus onto the two factors.  
>>> For
>>> integrals in the wrong form you could cheat by trying to find the  
>>> constant
>>> by which they differ by subtracting and simplifying, and then use  
>>> that in
>>> the derivation.
>>
>> Let's say it like it is:  It's not just "difficult" for ordinary  
>> users
>> to use and control many of these advanced tools:  It's basically
>> **impossible** for the average user to learn what some of these tools
>> do, because they're so complex and the results can depend so  
>> critically
>> on what you put into them; all you end up doing is thrashing around
>> endlessly, trying to get them to produce the results you want.
>>
>> The more powerful they get, the less they're worth trying to learn.
>>
>>
>>> It is very useful to get Mathematica generated expressions into  
>>> the form
>>> that one wants. I believe that this is probably a sticking point  
>>> with many
>>> users. In general it is not a trivial topic. Others may have some  
>>> good
>>> general ideas that I don't know about.
>>
>> My bottom lines are instead:
>>
>> 1)  Accept that "Retyping is...error prone" -- and more generally  
>> that
>> "To err is human..." -- and to the extent that you have to do any  
>> form
>> of retyping, do a _lot_ of checking, rechecking, testing with simple
>> cases, and looking to see that results are physically meaningful.
>>
>> 2)  Nonetheless, in general, "It is GENERALLY NOT very useful to get
>> Mathematica generated expressions into the form that one wants" -- at
>> least, not very often, and not if it involves any significant  
>> amount of
>> effort.  It's wasted energy, and can add its own errors, or divert  
>> one
>> from seeing one's own errors.
>>
>> 3)  Instead, if what you're doing is a complex analysis and/or
>> exposition, tackle the analysis portion initially with paper, pencil,
>> and a good soft eraser, the way God intended analysis to be done.
>>
>> 4)  When and if certain calculations (series expansions, etc.) get
>> messy, run separate Mathematica symbolic calculations in auxiliary
>> notebooks to carry them out.
>>
>> 5)  When it comes time for exposition, do the exposition using a tool
>> that's designed for exposition (e.g., TeX),  while doing the  
>> numerical
>> calculations and graphing using a tool that's good at those things --
>> and while doing this repeat item 1) multiple times.
>>
>>
>>> Someday someone may even write a good tutorial on it.
>>
>> How about instead someone **imbedding real TeX in Mathematica**, as  
>> part
>> of Mathematica's basic capabilities?
>>
>> That is:
>>
>> *  TeX is (I believe) totally open source, free, highly stable, and
>> widely known and studied -- and it's full source code is very  
>> compact.
>>
>> *  So how about building the TeX source code into Mathematica's  
>> already
>> immense repertoire of rules and stuff, and allowing one to include at
>> any point in a "text portion" (I.e., a "non-evaluation portion") of a
>> Mathematica notebook cell the syntax
>>
>>     TeX[ ---any valid TeX syntax---]
>>
>> such as TeX[$\alpha = \beta / \gamma^2$] to get that bit of inline  
>> math
>> into a Text or Header cell, or TeX[$$\alpha = \left( \beta /over
>> \gamma^2 \right)$$] to insert a display equation,
>>
>> and just having Mathematica display the typeset box produced by TeX
>> using that syntax into the Mathematica notebook at that point, with  
>> the
>> stuff inside the [ ] brackets having no other evaluational function  
>> or
>> effect in Mathematica itself, except to be displayed?
>>
>> Is there some reason this would be conceptually impossible?  Would  
>> it be
>> that difficult to accomplish? Could it at least be implemented with  
>> some
>> reasonable subset of TeX syntax and capabilities?
>>
>> If your goal is to have Mathematica notebooks serve simultaneously as
>> "exposition documents" and "calculation performing documents", might
>> this be a lot easier than endless fighting with option-laden and
>> temporally unstable Mathematica expressions like "Together, Apart,
>> Factor, TrigReduce, TrigFactor, TrigExpand, TrigToExp, GroebnerBasis"
>> and all their even more arcane extensions?
>>
>
>
>

• References:
  • Re: Simplifying and Rearranging Expressions
    • From: AES <siegman@stanford.edu>