[Axiom-math] Engineering Application

[Raymond E. Rogers]

Hello,
Assuming I get axiom running; would anybody be willing to discuss an
engineering application where I think Algebraic Geometry provides a
mathematically correct answer?
Even though I have just started learning AG, I think I can do all of the
hard work myself.


Ray

[Bill Page]

Post by Raymond E. Rogers
Assuming I get axiom running; would anybody be willing to discuss
an engineering application where I think Algebraic Geometry provides
a mathematically correct answer? Even though I have just started
learning AG, I think I can do all of the hard work myself.

I would be very happy to discuss engineering applications of
algebraic geometry.

Perhaps one place to start with Axiom and algebraic geometry might be
the following tutorial by Donu Arapura (http://www.math.purdue.edu/~dvb):

http://www.math.purdue.edu/~dvb/algeom.html

Introduction to Algebraic Geometry

Arapura includes some computer examples using Maple. We could see how
we might do the same examples using Axiom.

Regards,
Bill Page.

[Raymond E. Rogers]

Thanks, I hadn't seen that page before. It looks great. I will print
out some the materials tomorrow, and practice with axiom when I get it
running.

The problem arises from autozeroing and good design; an error is set
equal to zero during autocalibration, and if the errors are zero the
first derivatives are designed out (more or less). Thus the underlying
equations are made second order (and complicated) by design.

The concrete form of the problem is:
Err(x,a_i,b_k...)=P(x,a_i,b_k...)/Q(x,a_i,b_k...) P&Q polynomials
Q has no zeros in the region of interest; and any poles can probably be
moved to infinity.
The first derivatives of Err(x,0,b_k..) with respect to a_i are all zero
(more or less),
and Err(0,a_i,b_k...)=0
These items are accomplished by autozero/autocalibrate.
The a_i are parameters, the b_k are errors in the parameters (expected
value of 0).
My simple solution is to use a constraint/boundary
c_1*a_1^2+c_2*a_2^2+....=p^2
and intersect the Err and the constraint and look for worst case error.

The p is introduced because to start out with a narrow space around the
center and grow to more reasonable value after the very local problem is
understood.

One way is to find grad(Err)=j*grad(constraint) term by term.
But it seems that this might be avoided if the constraint and Err
intersections are examined for second order roots (?).

Mentally it seems much simpler than it appears in writing. I find the
more general problem interesting, but that seems to be a personal taste.

In any case, any guidance would be appreciated.

Ray

Post by Bill Page

Post by Raymond E. Rogers
Assuming I get axiom running; would anybody be willing to discuss
an engineering application where I think Algebraic Geometry provides
a mathematically correct answer? Even though I have just started
learning AG, I think I can do all of the hard work myself.

I would be very happy to discuss engineering applications of
algebraic geometry.
Perhaps one place to start with Axiom and algebraic geometry might be
http://www.math.purdue.edu/~dvb/algeom.html
Introduction to Algebraic Geometry
Arapura includes some computer examples using Maple. We could see how
we might do the same examples using Axiom.
Regards,
Bill Page.

[Herb Martin]

From: Bill Page

Post by Bill Page

Post by Raymond E. Rogers
Assuming I get axiom running; would anybody be willing to discuss
an engineering application where I think Algebraic Geometry provides
a mathematically correct answer? Even though I have just started
learning AG, I think I can do all of the hard work myself.

I would be very happy to discuss engineering applications of
algebraic geometry.
Perhaps one place to start with Axiom and algebraic geometry might be
the following tutorial by Donu Arapura
http://www.math.purdue.edu/~dvb/algeom.html
Introduction to Algebraic Geometry
Arapura includes some computer examples using Maple. We could see how
we might do the same examples using Axiom.

I too am (newly) interested in Algebraic Geometry
and would prefer to pursue that in Axiom or another
free CAS (rather than Maple) since this is only a
hobby interest for me.

[Bill Page]

Post by Herb Martin
...
I too am (newly) interested in Algebraic Geometry
and would prefer to pursue that in Axiom or another
free CAS (rather than Maple) since this is only a
hobby interest for me.

[Herb Martin]

Although the title of Penrose's book sounds overly grand, it is
none the less a very serious book on differential geometry as
applied to physics.

Yes, it is (overly grand), much like a "wagon
train scout" of the pioneer days pulling out
a many-times-folded map of the route from St.
Louis to California and spending an evening
showing the settlers the things they need to
know to prepare for the trip.

I think your goal to better understand it is
very laudable.

Or "overly grand". <grin> Thanks for the
encouragement -- it is much appreciated.

I have not yet reach page 200 and I have
already learned an amazing amount.

If there is anything in particular in this book
that you would like to discuss, I would be glad participate.
Maybe we can do some examples using Axiom?

Please. I want to discuss everything, but I
am probably not smart enough yet to figure out
how to ask a good question.

So far, I have been trying to learn Axiom & Maxima
in parallel with reading this book and have only
worked out some simple problems (like imaginary roots
and some differentials.)

I am a bit embarrassed to admit that I don't
really understand how much of the work Axiom
can do, nor really what the full capabilities
of Axiom include.

Even before this book I had entertained the
belief that it might be possible to learn "a
lot of math" by following along through Calculus
and Differential Equations text -- learning to
the IDEAS, and letting the system do the tedious
portions.

This probably wouldn't be sufficient for a
serious mathematician or engineer, but for me
this is just a (serious) hobby.

Perhaps seeing how to use Axiom to graph a
conformal map, or to work through a Fourier
transform would be interesting BUT this is
merely the first thing that came to mind when
searching (desperately) for something to
request -- please substitute ANYTHING your
prefer and I will ask for something else as
soon as I have the prerequisites.

I think it is entirely appropriate to discuss Macaulay here.
In fact if there is continued interest, I would be very happy
to look into providing a web interface for Macaulay at the
Axiom Wiki, e.g.

I know even less of Macaulay, but believe that
would be useful, and it certainly would be very
gracious of you.

Please note, that I am NOT modest about my abilities
nor about my intelligence, and any seeming "false
modesty" above is in fact just reality: There is a
LOT of math covered in the book which I have not studied,
and even my Calculus and Diff Eq is rusty and spotty
in places.

Also: I have no particular preference for Maxima versus
Axiom, and so with your help I will focus more of my
attention on Axiom (and Macaulay2).

Both of these fine programs do more than I can currently
use.

Thanks,
--
Herb Martin

[Raymond E. Rogers]

I would like to say that its great to find theory and tools that really
work. I wrote my optimization/tolerance analysis program in about 5
lines and it works, giving answers that I as an Engineer need. I did it
the crude way, but simplicity is nice.
Small problem, when I ran the real problem I ran out of memory and
haven't been able to find information about the "allocation" command; I
haven't received my book yet. Could somebody point me to the right place.

Ray

[Bill Page]

Post by Raymond E. Rogers
I would like to say that its great to find theory and tools
that really work. I wrote my optimization/tolerance analysis
program in about 5 lines and it works, giving answers that
I as an Engineer need. I did it the crude way, but simplicity
is nice.

:) I am glad you find it useful... we need more tesimonals like
that!

Post by Raymond E. Rogers
Small problem, when I ran the real problem I ran out of memory
and haven't been able to find information about the "allocation"
command; I haven't received my book yet. Could somebody point
me to the right place.

I am afraid you wont find much in any of the Axiom documentation
about memory allocation. :(

But if the error message that you get when you "ran out of
Contiguous blocks exhausted.
Currently, 1354 pages are allocated.
Use ALLOCATE-CONTIGUOUS-PAGES to expand the space.

then maybe you can try for example:

(1) -> )lisp (si::allocate-contiguous-pages 3000 t)

See for example:

http://wiki.axiom-developer.org/SandBoxCategoryOfGraphs

Memory allocation is primarily controlled by the underlying
GCL lisp environment. Unfortunately the specific details of
the memory allocation in GCL are beyond me. Camm Macquire
is the main GCL developer. I have some notes from him back
in September with more details of GCL memory allocation when
I was trying to debug the above problem. If you get deeply into
this, I can copy these to you.

The most significant defaults are set during the compilation
of GCL as part of the overall Axiom build. If you are building
Axiom from source, you might be able to modify the GCLOPTS to
specify larger limits.

Regards,
Bill Page.