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Re: Semantic information for math representations of physics

From: Roger I Martin PhD <hypernexdev@hypernexinc.com>
Date: Wed, 09 Feb 2005 13:02:26 -0500
Message-ID: <420A5032.3030709@hypernexinc.com>
CC: www-math@w3.org

Do you have an example illustrating the full scope?  Partial 
differentials and solutions?  My example was of only an equation for a 
NURBS curve or surface.  There are many diciplines and equation systems 
that can be accomplished between abstract math in mathml to working 
solutions.  I realize I'm talking from a detail perspective trying to 
understand a more global view of achieving
your "interested in

representing and supporting the validation of
physics-based numerical models generally, from the
conceptual, mathematical statements, encapsulated as a
coherent model, to the translation of that model so as
to incorporate mathematical approximations, and then
to the translation of an encapsulated, approximative
model to a discrete, numerical model."

And I'm trying to gain an understanding of your perspective.  I think I understand the above paragraph:-) When your talking of translation to discrete numerical model I understand it in terms of the difficulties of solving partial differential systems where the abstract level does little to describe the domain and solution spatial resolution.  However mathml does have domain http://www.w3.org/TR/MathML2/chapter4.html#contm.domain and exists http://www.w3.org/TR/MathML2/chapter4.html#contm.elem etc. tags that I use to define where a solution is wanted:

    <apply>
      <exists/>
      <bvar><ci>x</ci></bvar>
      <lowlimit><cn>-1</cn></lowlimit>
      <uplimit><cn>1</cn></uplimit>
      <true/>
    </apply>
    <apply>
      <exists/>
      <bvar><ci>y</ci></bvar>
      <lowlimit><cn>-1</cn></lowlimit>
      <uplimit><cn>1</cn></uplimit>
      <true/>
    </apply>

Working on more complex boundary definitions(hence NURBS and femml.) I think of resolution and say variable/optimized meshes as values which an end user may want to control.  For example a dynamic membrane ending up in a 3D display is greatly effected by spatial and time resolution for a given cpu power; 3GHz can easily be swamped with a practical problem. (An example is modeling cellular locomotion from multiphoton confocal microscopy data)

Another way to put it, I'm looking to take mathml 2.0(especially content) from any Editor that produces it(Mathematica, MathCad, etc.) or from raw data reduction to mathematical modeling and be able to incorporate in a reliable way via open source transforms in systems that then can use the results for the end users purposes.  I use it for example in peak analysis, specifically x-ray diffraction peaks(gaussian etc.), least squares fitting, solving for Avrami equation parameters, etc as peaks change over time.

http://sourceforge.net/projects/mathml-x/ has the stylesheets(not about examples of use yet) but before the site would be useful we would need to define a number of systems that interest you and see whether there is implementation for it.  Mathml-x is  yet a playground to explore mathml and xslt so I'm not suggesting it plugs in and solves everything.  It currently is being applied in dynamic calibration curves for instrumentation, visualization and electro-physiological modeling.  Recently upgraded to XSLT 2.0 applying Michael Kay's Saxon 8.2 and it overcame nearly all of difficulties I was having with xslt 1.0 and custom extensions.

Haven't implemented as much for integrals but for example instead of going numerical immediately I use the pattern matching capability of xslt to go to mathematical solutions/tables.  ie. integral of 1/x goes to ln(x) and depending on the domain readily compute the solution.  Likewise for sin(x)/x where numerical solutions fail at zero, use abstract analytical limits to find solutions.

Part of what started my efforts with mathml-x is getting away from writing for and while loops at the code level(tired of it); codifying what a physicists/scientists does on a tablet by hand is error prone and the connection between the tablet and software gets lost over time and memory fades.  As I talked about before there is a lot of code that does not do what physicists/scientists think it is doing; Seen code that was used for finding a supposedly optimal minimum applied for 25 years where numbers came out but there was no minimum any more than the Peano Surface's http://mathworld.wolfram.com/PeanoSurface.html  minimum:-)  I also know that it is difficult preventing a programmer from inserting what may be an optimization for speed but eventually functions as a bug.  Impossible; part of our genetics I guess:-) And when a non-programmer gets into the code look out!

But I'm writing details again.  If you have the time I would like to hear more of the global perspective of what is needed.  Am also interested in a larger group of collaborators because it is an immense topic.

r,
Roger

JB Collins wrote:

>This sounds interesting and useful, but I'm not sure
>it provides the full scope of what I'd like to do.
>
>Unfortunately, I was unable to investigate by
>accessing the website you gave, i.e.
>http://sourceforge.net/projects/mathml-x/ .
>
>Regards,
>Joe Collins	
>
>  
>
Received on Wednesday, 9 February 2005 18:00:30 GMT

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