HDTenEightyP/NetNews huggingface dataset:
From: gaylord@ux1.cso.uiuc.edu (Richard J. Gaylord)
Newsgroups: sci.math.symbolic
Subject: select from list with & w/o replacement
Date: 20 Aug 92 13:50:36 GMT
Article-I.D.: news.BtABsD.Cpt
Organization: University of Illinois
X-Xxdate: Thu, 20 Aug 92 08:51:18 GMT
X-Useragent: Nuntius v1.1.1d8
X-Xxmessage-Id: <A6B90D8661029B13@mm-mac19.mse.uiuc.edu>

just waiting to go to a meeting to discuss (1) the teaching of the
required introductory cs course for all engineering students using
Mathematica (2) the teaching of calculus with Mathematica (3) using
Mathematica as an alternative language to FORTRAN, C and Pascal for
engineering students [all of this is actually taking place on the U. of
I. campus right now (not to mention the other 3 or 4 courses being taught
on campus using Mathematica)].

any way, i'd thought i'd finish up the 'choose from a list' problem by
putting both the deal [choose without replacement] and the 'choose with
replacement'  programs in the same message.

chooseWithReplacement[list_, n_Integer] :=
         list[[ Table[Random[Integer,{1,Length[list]}],{n}] ]]

deal[lis_List,n_Integer] :=
    Complement[lis, Nest[Delete[#,Random[Integer,{1,Length[#]}]]&,lis,n]]

these are the most straightforward one i can come up with. 

any alternatives?
Newsgroups: sci.math.symbolic,sci.math.num-analysis,sci.physics
From: jonathan@chpc.utexas.edu (Jonathan Thornburg)
Subject: want refs to latest manuals for Reduce & Macsyma
Summary: want to cite a published description of these systems
Keywords: symbolic algebra manual reference bibliography Reduce Macsyma
Organization: U of Texas at Austin / Physics Dept / Center for Relativity
Date: Sat, 15 Aug 92 23:17:57 GMT

I'm seeking the full bibliographic citations (authors, title, date,
publisher, ISBN if possible) of the current manuals for Reduce and
Macsyma (I already have Maple & Mathematica).  If these manuals are
unpublished, then a recent published description of the systems by
their authors would be a 2nd best alternative.  Can anyone help?

E-mail replies only, please -- I don't think most people will be
interested.  And please, if you could, E-mail anything to me in
duplicate to both addresses given below -- our building ethernet
seems to be broken about 1/3 of the time, so my .forward may not
be working :-( .

Thanks 1.0e6 for any info,

- Jonathan Thornburg
  <jonathan@einstein.ph.utexas.edu> or <jonathan@hermes.chpc.utexas.edu>
  University of Texas at Austin / Physics Dept / Center for Relativity
  and (for a few more months) U of British Columbia / {Astronomy,Physics}
Xref: sparky sci.math.symbolic:2231 sci.math.num-analysis:2485 sci.physics:13060
From: qmdbms@gsusgi2.gsu.edu (Brian Schott)
Subject: Can Theorist prove theorems?
Keywords: symbolic theorem proving
Organization: Georgia State University
Date: Mon, 17 Aug 1992 17:03:38 GMT


Below is a Theorist session which I Copied into the clipboard
and pasted directly into a text file for upload to post here.
If you are working from a Macintosh, I believe you can reverse
this process and see what I have been doing.

(BTW, while I was copying the clipboard into this post, I got several
beeps because some of the characters are invisible.  I hope this does
not mess up the transfer.) 

TV?fnormal POISp0 0 V?fnormal poisp0 0 V?cnormal mp0 0 V?vnormal Bp0 0 V?vnormal xp0 0 V?vnormal np0 0 V?c0 0p0 0 V?ce ep0 0 Ht?s@?n	0*?r^?s/?s!	S(-?s@0	?n*?r^?s/?s!+e^?r,(?n>0);e^?r,(?n<=0))p0 0 
pois((?n,?x))=e^(-?x)*?x^?n/?n!p0 0 dQ14 268 33 22 POIS((?n,?x))=n@0	?n*pois((n,?x))	p0 0 HtPOIS((?n,?x))	n@0	?n*pois((n,?x))p0 0 
POIS((1,0.2))p0 0 
POIS((1,0.2))=n@0	1*pois((n,0.2))p0 1 
POIS((1,0.2))=pois((1,0.2))+pois((0,0.2))p0 2 
POIS((1,0.2))=0.9824769036935782305p0 3 
POIS((n,m))p0 0 
POIS((2,0.2))p0 0 
POIS((2,0.2))=n@0	2*pois((n,0.2))p0 1 
POIS((2,0.2))=0.9988515187551378677p0 2 
POIS((0,0.2))p0 0 tp0 0 
 
 
My question is the following.  I would like to demonstrate 
that the Poisson function is a pdf in the sense that it sums 
to 1. Whereas I have been successfull in computing cumulative 
probabilities for the Poisson distribution when I supply 
constants for the variable and the parameter, Theorist seems 
to ignore me when I leave the variable and parameter as names 
and request a Transformation, Expand, Calculate, etc.  
 
Can this be done in Theorist?  BTW, I am working with 
version 1.0. I would like to encourage my students to use 
Theorist if this is possible.
 
(B=)
-- 
Brian Schott/Decision Sciences Dept.             qmdbms@gsusgi2.gsu.edu
College of Business Administration                         404-651-4070
Georgia State University                  interests: approx. reasoning,
Atlanta, Georgia USA 30303-3083     (B=)       decision support systems
From: goodrich@lynx.larc.nasa.gov (Mike Goodrich)
Subject: Re: Can anyone solve this??
Organization: NASA Langley Research Center, Hampton, VA  USA
Date: Thu, 20 Aug 1992 16:38:52 GMT

In article <cbc.714097112@milton>, cbc@milton.u.washington.edu (Charles Cook) writes:
|> As part of my thesis, I came up with this integral that looks as
|> though it should be easy to solve, however, both Mathematica and 
|> Maple choke on it.  I can't seem to find it in *any* integral tables
|> and I've tried every method I can remember to solve it.  Any ideas?
|> 
|> Here it is:
|> 
|>    /Infinity  /X-x0
|>   |          |        (-y*b)      2    2   1/2
|>   |          |       e        ( x  + y   )     dx  dy
|>  /  0       / -x0
|> 
|> X and x0 are arbitrary constants as is b.  X>x0>0 and b = 1/(X*C) (not 
|> that that should affect things.)
|> I'm seeking a symbolic closed form solution to this problem (which may
|> or may not be a pipe dream).  Any help or suggestions would be greatly 
|> appreciated.
|> 
|> Thanks in advance,
|> Charles Cook
|> University of Washington
|> 
|> cbc@u.washington.edu

This looks like a "convert to polar coordinates and integrate over the
first quadrant" candidate. I.e., change variables from

	x,y -> r, theta. (be sure and change limits too)

The tip-off is that r = sqrt( x^2 + y^2 ).


...mike
From: ags@seaman.cc.purdue.edu (Dave Seaman)
Newsgroups: sci.math.symbolic
Subject: Re: Mathematica: defining functions. HELP!
Date: 14 Aug 92 14:30:59 GMT

In article <l8mkk4INN5ll@halcyon.usc.edu> rao@halcyon.usc.edu (Anil Rao)  
writes:
> define a function f(x) = x between x = 2 and x = 5
>                        = x^2/2 for x < 2
>                        = 5 x - 20 for x > 5
> 

At least two methods come to mind. One is to qualify the definitions:

	f[x_] := x /; 2 <= x <= 5
	f[x_] := x^2/2 /; x < 2
	f[x_] := 5 x - 20 /; x > 5

and another way is to use a one-liner with nested conditionals:

	f[x_] := If[ x<2, x^2/2, If[ x<=5, x, 5 x - 20]]

--
Dave Seaman
ags@seaman.cc.purdue.edu
From: rao@halcyon.usc.edu (Anil Rao)
Newsgroups: sci.math.symbolic
Subject: Mathematica: defining functions. HELP!
Date: 13 Aug 1992 23:28:20 -0700
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: halcyon.usc.edu

Hi all,
  I went through the Wolfram book but was still unable to do the
following 'trivial' task:

define a function f(x) = x between x = 2 and x = 5
                       = x^2/2 for x < 2
                       = 5 x - 20 for x > 5

basically, I want a function to behave 'differently' in different regions.
How does one do this in Mma?    

Many thanks in advance for your help.

Anil Rao. 
rao@halcyon.usc.edu
From: whitmore@iastate.edu (Kurt D Whitmore)
Subject: Mma (DeclarePackage) Command
Organization: Iowa State University, Ames IA
Date: Tue, 18 Aug 1992 17:19:49 GMT

On page 342 of:

	 Mathematica A System for Doing Mathematics by Computer
	 	Second Edition


it gives the example:

	DeclarePackage["Calculus`VectorAnalysis`",{"Div","Grad","Curl"}]


	This specifies that the symbols Div, Grad and Curl are defined in:
	Calculus`VectorAnalysis`

	Thus by typing the commands Div, Grad or Curl Mma will not
	complain and automatically load the package containing those commands.


The question is:
________________

Has anyone by chance made the proper modifications to (I believe the init.m file) or know of an easy way to tell Mma where all the commmands are in there respective packages so no matter what command a student types it will automatically load that package for them?


Thanks,



Kurt D. Whitmore
whitmore@iastate.edu
From: schwab@ariel.mathematik.uni-kl.de (Hartmut Schwab)
Newsgroups: sci.math.symbolic
Subject: Mma: can't define rule for partial integration
Date: 16 Aug 92 13:21:02 GMT
Reply-To: ricks@berkeley.edu
Organization: UC Berkeley IC CAD Group


I tryed to define a rule for partial Integration in Mathematica.
But I had some Problems doing it. I don't find a solution, how
to put the upper and the lower value in the first part of the 
solution.

In the Book it is written like that:

  b                         b       b
Integral(f*g')  = [ f* g(x)]   - Integral(f' * g)
  a                         a       a

I started the rule like that:
regel = Integrate[Times[a_,b_],c_List] :> a*Integrate[b,c]
	 - Integrate[ D[a, c[[1]] ]* Integrate[b,c], c]

As You see, the upper and lower value are not put in the 
first part of the solution.

To get arround the problem I tryed to do something like:
 a*Integrate[b,c]/. c[[1]] -> c[[3]]

I got an Errormessage because Mathematica evaluated c[[1]] etc.
Part::partd: Part specification c[[1]] is longer than depth of object.

How can I go arround that Problem and put in the upper and lower
values?

Thanks for any help.

Hartmut Schwab
schwab@mathematik.uni-kl.de
Newsgroups: sci.math.symbolic
From: weverka@csn.org (Robert T. Weverka)
Subject: Re: mathematica question- using complex symbols
Organization: University of Colorado, Boulder
Date: Wed, 12 Aug 1992 17:23:15 GMT


Here is another.

MyRealPart=ComplexExpand[Map[Re, MyFunction]]

MyImPart=ComplexExpand[Map[Im, MyFunction]]
From: miller@FS1.cam.nist.gov (Bruce R. Miller)
Newsgroups: sci.math.symbolic
Subject: Macsyma Lisp: Why %SIN
Date: 13 Aug 92 18:30:42 GMT
Organization: NIST - Computing and Applied Mathematics Laboratory


I'm developing a package of special functions for Macsyma and there's
one thing I haven't been able to sort out to my satisfaction.

First a background question:  Why are the standard functions sin, cos,
etc aliased to %sin, %cos, etc?

It seemed like a nice principal for my own package, because it gave a
single, simple hook for cleanup.  That is to say, you can have short &
sweet names for the functions, say $FOO, when the package is loaded.
When you quit macsyma the link between the short names and the internal
implementation, say %%FOO (an extra % so I dont collide w/ any future
built-in %FOO ?!?) (particularly the simplifiers) gets broken. 
So in effect, the functions $FOO become undefined.  On the one hand, 
short names are more natural & less clutter.  On the other, you dont
want to usurp those names permanently when the package is not being
used.

[BTW: The cleanup is particularly important in Lisp machine macsyma;
you never _really_ start up a new macsyma session; the lisp is still
there, it just gets disconnected from the macsyma names]

So, what's the problem?  One thing is that you cant use FOO and then
load the package; the pre-alias FOO is different from the aliased FOO.
[this isn't a problem for %SIN, etc, since they're always there]
And once in a while, for reasons unclear, I've had FOO start printing as
%FOO.
And, in general, w/o knowing why its done in other cases, it just seems
screwy :> 

So the question is;  Is this the best way to go about it?  Would it be
better to go ahead and define $FOO, and have the cleanup remove the
OPERATORS property (& if so, hows the best way to do that?).
Any other ideas, suggestions?

bruce
miller@cam.nist.gov

[We now return you to the Mma discussion, in progress ....]
Xref: sparky sci.math.symbolic:2232 sci.math.num-analysis:2490 sci.physics:13076
From: sherod@news.Colorado.EDU (Scott Herod)
Subject: Language Specific Bulletin Boards
Organization: University of Colorado, Boulder
Date: Fri, 14 Aug 1992 19:51:38 GMT

Someone mentioned to me that there existed language specific
BBS.  Can someone send me address and tell me how to access them?

Scott Herod
(sherod@newton.colorado.edu)
From: cbc@milton.u.washington.edu (Charles Cook)
Newsgroups: sci.math.symbolic
Subject: Can anyone solve this??
Summary: Help!!
Date: 18 Aug 92 00:18:32 GMT
Article-I.D.: milton.cbc.714097112
Organization: University of Washington

As part of my thesis, I came up with this integral that looks as
though it should be easy to solve, however, both Mathematica and 
Maple choke on it.  I can't seem to find it in *any* integral tables
and I've tried every method I can remember to solve it.  Any ideas?

Here it is:

   /Infinity  /X-x0
  |          |        (-y*b)      2    2   1/2
  |          |       e        ( x  + y   )     dx  dy
 /  0       / -x0

X and x0 are arbitrary constants as is b.  X>x0>0 and b = 1/(X*C) (not 
that that should affect things.)
I'm seeking a symbolic closed form solution to this problem (which may
or may not be a pipe dream).  Any help or suggestions would be greatly 
appreciated.

Thanks in advance,
Charles Cook
University of Washington

cbc@u.washington.edu
Newsgroups: sci.math.symbolic
From: dvf@kepler.unh.edu (David V Feldman)
Subject: ISETL
Organization: University of New Hampshire  -  Durham, NH
Distribution: worldwide
Date: Fri, 21 Aug 1992 20:51:46 GMT

I would like some basic information about ISETL including:
1)  What is its copyright status (public domain? shareware? copyleft?)?
2)  What platforms are supported?
3)  Where can one obtain it?
4)  What experience have people had using or teaching with it.

David Feldman
From: t68@nikhefh.nikhef.nl (Jos Vermaseren)
Newsgroups: sci.math.symbolic
Subject: Re: Best way to factor out roots
Keywords: factor, polynomials
Date: 19 Aug 92 14:39:14 GMT
Reply-To: t68@nikhefh.nikhef.nl (Jos Vermaseren)
Organization: Nikhef-H, Amsterdam (the Netherlands).

One way to factor out 'known' roots from a large polynomial is by
'substitution'.
Suppose you know that 5 is a root. Then you replace
x-5 by y, or x -> y+5 or whatever the syntax demands.
This operation involves an amount of work which is related to the
sum of all powers in your polynomial (plus some extra for the fact that
the binomial coefficients can become large).
The new polynomial should have a root at y = 0, so it is rather easy
to divide it out. After that you go to the next 'known root'.
Other methods are variations of this. If you divide by (x-5) the
partial fractioning involved is equivalent to the above procedure.

Jos Vermaseren
From: yee@edison.seas.ucla.edu (John Yee)
Newsgroups: sci.math.symbolic
Subject: Convergence for n! x^n ?
Date: 14 Aug 92 23:40:12 GMT
Organization: SEASnet, University of California, Los Angeles

Hi, I was reading in a industry (EE) newspaper that some guy seems to have
used Maple to force the convergence of the summation:
   \sum_{n=0}^\infty n! x^n, x >> 1
and even showing a graph of the function.
Now, maybe I am missing something simple, but I can't see how this
summation can be computed or be "forced to converge".
The article did not go into details, merely hyped the supposed achievement.
Elucidation welcome,
--
jy, yee@seas.ucla.edu
Newsgroups: sci.math.symbolic
From: butscher@helios.physics.utoronto.ca (Adrian Butscher)
Subject:  MMA help on the Mac.
Organization: University of Toronto Physics/Astronomy/CITA
Date: Mon, 17 Aug 1992 13:45:24 GMT

We have MMA on both a workstation (text interface) and on the Mac.  To get
help on a command on the workstation, you just type ?Command or ??Command,
and up comes the usage message and the notes.  On the Mac, however, the input
?Command produces Attributes[Command] = something, even though we have the 
MMA help file installed.  What is the problem here?  

Adrian
Newsgroups: sci.math.symbolic
From: king@physics.su.OZ.AU (Peter King)
Subject: Solving Rate Equations
Organization: School of Physics, University of Sydney, Australia
Date: Thu, 20 Aug 1992 23:01:28 GMT

I'm interested in solving the following equations (in Mathematica format)
for a cw optically pumped laser:

eqns = {
	wv N3 - wp (N0 - g0 / g2 N2) == 0
	B21 n (N2 - g2 / g1 N1) - wr N1 == 0
	wp (N0 - g0 / g2 N2) - wr N2 - B21 n (N2 - g2 / g1 N1) == 0
	N0 + N1 + N2 + N3 == N
	B21 n (N2 - g2 / g1 N1) - 2 Pi y n == 0
       }

simultaneously.  In Mathematica I naively tried
	Solve[eqns, {N0, N1, N2, N3, n}]
and ran out of memory on a 40 Mb RAM workstation.  I've tried eliminating n
from the last equation to reduce the system to 4 equations but then I can't
find the solution to n without running out of memory.

	Does anyone know of any tricks/techniques/simplications I can make
to solve this system?  Alternatively does anyone know of simple references
to Groebner bases?

				thanks in advance

				     Peter
--
********************************************************************************
Peter King                               *         The home of TORTUS.
Plasma Physics Department                *   T
University of Sydney, NSW 2006           *   O
AUSTRALIA                                *   Roid of
    voice: 61 2 692 3371                 *   The 
      FAX: 61 2 660 2903                 *   University of
    email: king@physics.su.OZ.AU         *   Sydney
********************************************************************************
Newsgroups: sci.math.symbolic
From: litow@csd4.csd.uwm.edu (bruce e litow)
Subject: mantissa based precision
Originator: litow@proclus.csd.uwm.edu
Keywords: mantissa, digits, modulo 1 
Organization: Computing Services Division, University of Wisconsin - Milwaukee
Date: Wed, 12 Aug 1992 15:17:55 GMT

I often run experiments with a CA package where i am working with
numbers modulo 1. The usual notion of floating precision is not what
I want since digits to the left of the decimal point are irrelevant.
So I have to write little procs to do what I want namely to mean
by k digits precision that I retain k digits in the mantissa. Is there
a feature that does this built into CA packages? I am especially
interested in Maple, Macsyma or Axiom. Please email replies to me.
Thanks.

~
Newsgroups: sci.math.symbolic
From: slaven@iastate.edu (Dave Slaven)
Subject: Schrodinger equation with Mathematica
Keywords: head, wall, beat
Organization: Iowa State University, Ames IA
Date: Wed, 12 Aug 1992 16:15:59 GMT

In preparation for teaching quantum mechanics this fall, i've been
trying to attack the Schrodinger equation with Mathematica.  As this
is my first attempt at using Mathematica, i'm being quite unsuccessful
about it.

Here's the plan.  The starting point is the equation
psi''[x]-(v[x]-e)==0.  For a bound state problem (such as the simple
harmonic oscillator, v[x]=x^2) the condition that the psi vanish at
+-Infinity dictates that only certain values of e are allowed, and the
object is to find them.  So the plan is to start at x=-Infinity, work
our way to Infinity, and find a value of e that gives us 0 there.  Not
that i'm starting out so ambitiously.  I'm just trying now to use a
value of e that i know works (1 for SHO in the units i'm using) and
using NDSolve.

Of course, you can't start NDSolve at -Infinity, so i've changed
variables via x=Tan[u].  The new equation is
Cos[u]^4 (psi''[u]-2 Tan[u] psi'[u])-(v[u]-e) psi[u]==0, where u runs
from -Pi/2 to Pi/2.  I use this and the conditions psi[-Pi/2]==0,
psi'[-Pi/2]==1 (the latter is arbitrary, since i don't care about the
wave function being properly normalized) and use NDSolve from -Pi/2 to
Pi/2.  When i try this, i get "Repeated convergence test failure..."
I've also tried it with values for the minimum and maximum slightly
inside (-Pi/2,Pi/2) and get the same message.  Admittedly, the
potential blows up at the endpoints (v[u]=Tan[u]^2), but is there
nothing that can be done?  Surely someone else has tried this with
more success than i!  Somebody please do my job for me and tell me how
this will work!  Thanks!!

-- 
Dave Slaven! 

"I'm not downhearted.  I am not downhearted.  
        I'm not downhearted, but I'm gettin' there."  --Mose Allison
Newsgroups: sci.math.symbolic
From: Tobin C. Anthony <tca712@rs710.gsfc.nasa.gov>
Subject: Control Examples in Mathematica
Organization: NASA-Goddard Space Flight Center
X-Useragent: Nuntius v1.1
Date: Fri, 14 Aug 1992 15:24:22 GMT

I am interested in any packages, functions, or plain old algorithms that
people
have used to solve controls problems in Mathematica.  I am mostly
interested in 
aerospace applications but I would like to see what tools people have
used to solve classical or modern controls problems in any discipline.  I
would like to write my own packages but I am interested in seeing what's
out there.  I would appreciate it if any replies could cc:
amk712@rs710.gsfc.nasa.gov.  Thanks in advance. 

Tobin

----------------------->>>>><<<<<-----------------------
Tobin C. Anthony                   tca712@rs710.gsfc.nasa.gov
Code 712                               301-286-1273 (w)
Goddard Space Flight Center
Greenbelt MD 20771
USA

"Marge, you're just a tool of the dog-house makers" - H. Simpson
----------------------->>>>><<<<<-----------------------
From: Randy.Baer@f310.n3603.z1.FIDONET.ORG (Randy Baer)
Newsgroups: sci.math.symbolic
Subject: derive
Date: Thu, 13 Aug 92 22:05:57 EDT
Organization: FidoNet node 1:3603/310 - Tetragrammatton, St Petersburg FL

 
  In Derive 2.07 aftering loading Ode1.mth

  To solve the linear DE of the form:
  y'+p(x)y=q(x)    when y(1)=0  or whatever init condits 
 
  I type:
 
  Linear1(p,q,x,y,x0,y0)
 
   I want to  be able to solve a DE in just the general form, without
the initial conditions of x0 and y0
 What should I put in for x0 and y0 to get the answer. 

--  
Internet: Randy.Baer@f310.n3603.z1.FIDONET.ORG
UUCP: ...!myrddin!tct!psycho!310!Randy.Baer
Note:psycho is a free gateway between Usenet & Fidonet. For info write to
     root@psycho.fidonet.org.
Xref: sparky sci.math:10345 sci.physics:12961
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math,sci.physics
Subject: Re: math and physics degrees
Date: 18 Aug 92 02:26:29 GMT
	<1992Aug17.191102.18697@galois.mit.edu>
Organization: M.I.T. Artificial Intelligence Lab.

In article <1992Aug17.191102.18697@galois.mit.edu> jbaez@euclid.mit.edu (John C. Baez) writes:

   Doing a PhD *simultaneously* in math and physics sounds almost
   impossible to me.  Maybe there are some places where it can be done, or
   maybe you have twice as much energy and time than other grad
   students.  But I suggest that you get a PhD in one or the other, which
   needn't prevent you from taking courses in BOTH subjects, as I did.


People do get PhD's in more than one subject. For example, I believe that
Lila Chichilniski (an economist who is or was at COlumbia University)
has a PhD in math and a PhD in economics. Also, my friend Jeffrey Katz
who since I knew him as an undergraduate had a passion for certain statistical
models of psychology, obtained a PhD in math and a PhD in psychology.
I think his thesis is about 900 pages long...

For a perhaps more celebrated example, Peter Galiston, author of the book
How Experiments End (which I found quite informative), obtained a PhD
in History of Science but in order to be certain of competence in the
particle physics whose history he was studying he also obtained a
PhD in physics.

So I don't think you necessarily have to compromise your dreams of doing
both mathematics and physics. On the other hand, as John Baez points out,
majoring in math doesn't necessarily preclude learning physics, particularly
if you make sure you take enough lab courses so that you understand that
physics is not a branch of mathematics. Max Born was a mathematics major and
then went into physics. 

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math