HDTenEightyP/NetNews huggingface dataset:
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Pi in beatiful form --lost
Organization: Computer Science Department,  Stanford University.
Date: Wed, 16 Dec 1992 21:54:34 GMT

In article <1992Dec16.144508.19371@nntpd.lkg.dec.com> edp@math.zk3.dec.com (Eric Postpischil) writes:
>In article <1gmvieINN7al@aludra.usc.edu>, rmurphy@aludra.usc.edu (Bob
>Murphy) writes:
>
>>(pi - 3)/4 = 1/(2*3*4) + 1/(4*5*6) + 1/(6*7*8) + ...
>
>The first term is 1/24, which is already greater than (pi-3)/4.  Should
>that have been an alternating series?

Yep.  As an alternating series the first 14 terms sum to 1/4 of 0.14151898.
-- 
Vaughan Pratt	   		Interactive Proofs = Polynomial Space
Newsgroups: sci.math,rec.games.abstract
From: keving@cory.Berkeley.EDU (Kevin Gong)
Subject: Re: Game of pentominos
Organization: University of California, at Berkeley
Date: Thu, 17 Dec 1992 04:20:31 GMT

In article <1992Dec16.215437.8916@ll.mit.edu> nates@ll.mit.edu ( Nate Smith) writes:
>the game of pentominoes, as described by martin gardner, consists
>of the 12 pieces and an 8x8 board.  each of the 2 players takes one
>of the unplayed pieces and positions it over 5 unoccupied squares.
>the last player to move wins.  martin was impressed with its
>complexity.
>...
>a bit as a kid and found it quite intriguing.  i never got to play
>the 8x8 version because by the time i found out about it, nobody
>wanted to have anything to do with me & my pentominoes.  :-)  :-(

Well, now you can play against my program. =)  (see previous post)

>
>the sequence 1,1,2,5,12,35,108,....  is still unsolved, i bet.
>

This refers to the number of polyominoes of n squares.  Thus, for n = 5,
there are 12 such shapes.  The problem is unsolved.  It has been calculated
that there are 654999700403 polyominoes of size 24.  (see The Mathematical
Gardner, edited by David A. Klarner, Wadsworth 1981)

If anyone is interested, I have calculated that there are 9371094
3-dimensional polyominoes of size 12 (18598427 if rotations in 4-space
aren't allowed).  I don't know if anyone has ever calculated such numbers
(I've never seen them anywhere).

 - kevin
Newsgroups: sci.math
From: greil@guug.de (Anton Greil)
Subject: a question to HNN extensions
Keywords: HNN extension, combinatorial group theory
Organization: Technical University of Munich, Germany
Date: Thu, 17 Dec 1992 09:46:25 GMT


I have a conceptual question concerning "HNN extensions".

In the book

  B. Chandler / W. Magnus, "The History of Combinatorial Group Theory" (1982)

on page 111, the definition of the HNN extension is given. It is followed
by the remark: "In particular, t (the "stable letter") is always of infinite
order in H (the extended group)."

Is this remark really correct - and if, then why ?

Thank you!
--Toni
Newsgroups: sci.math
From: ross@tarski.tmc.edu (David Ross)
Subject: Re: nonstandard analysis
Organization: University of Hawaii Mathematics Department
Date: Fri, 11 Dec 1992 23:40:35 GMT

In article <24210@galaxy.ucr.edu> baez@guitar.ucr.edu (john baez) writes:

>By now, so much has been done standardly that the advantages of nonstandard
>analysis, if any, are not enough to make many mathematicians want to retool
>and go nonstandard.

There are many results in analysis, especially probability theory, for which
the only known proofs use nonstandard analysis.  

>Are there are any speedup theorems about nonstandard analysis?  E.g.,
>I'm sure there must be standard proofs that can be shortened by an
>arbitrarily large factor using nonstandard analysis, but has this
>been shown?

Henson and Keisler, 'The strength of nonstandard analysis', J. Symbolic Logic,
1986.  Also, Henson, Kaufmann, and Keisler, JSL, 1985.

- David



--
David Ross, Dept. of Math., Univ. of Hawaii at Manoa, Honolulu HI  96822
Internet: ross@math.hawaii.edu  -or- ross@tarski.math.hawaii.edu
Phone: 808-956-9949
Newsgroups: sci.math,sci.physics,sci.bio,sci.chem,misc.education
From: mkrogers@unix.amherst.edu (MICHAEL K ROGERS)
Subject: Re: Can we have sci.chem back? (was Re: TIME HAS INERTIA abian stuff)
Followup-To: sci.math,sci.physics,sci.bio,sci.chem,misc.education
Organization: Amherst College
X-Newsreader: TIN [version 1.1 PL7]
Date: Tue, 15 Dec 1992 15:38:45 GMT

Dale Atems (atems@igor.physics.wayne.edu) wrote:
: 
: The cross-posting was done either by Abian himself, or by someone who
: mistook his postings for real science. Your suggestion is an excellent
: one. Until this flood subsides, please use killfiles if your
: newsreader supports this feature.

I wish the subject line wouldn't keep changing so that when
I killed the thread, it would remain dead.  There should be
some netiquette about this.

(I shall now kill this thread.)
Newsgroups: sci.math
From: rgep@emu.pmms.cam.ac.uk (Richard Pinch)
Subject: Re: Calculus Puzzle
Organization: Department of Pure Mathematics, University of Cambridge
Date: Fri, 18 Dec 1992 11:59:22 GMT

In article <BzFoLp.EHs@unccsun.uncc.edu> sbardhan@uncc.edu writes:
>X + X + X + X +X.... X times = X^2
>Differtiating both sides w.r.t X we get
>
>1 + 1 + 1 + .....X times = 2X
>or , X = 2X
> as X may not be Zero
>1 = 2.
>
>
>--Soumendu, UNCC
>
No, no, you do it by the product rule:
(1+1+...) X times plus (X+X+...) 1 times = X + X = 2X
So no problem.  

Richard Pinch

Xref: sparky sci.math:17233 comp.edu:2231 misc.education:5452
From: vidynath@function.mps.ohio-state.edu (Vidhyarath K Rao)
Newsgroups: sci.math,comp.edu,misc.education
Subject: Re: Student attitudes
Date: 20 Dec 1992 11:07:09 -0500
Organization: Me? Organized?
Distribution: usa
NNTP-Posting-Host: function.mps.ohio-state.edu

[distribution limited to usa because this is probably irrelevant to others.]

In article <BzH5zo.2uM@mentor.cc.purdue.edu>
  hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>In article <1992Dec17.222321.23782@dsuvax.dsu.edu>
   dbreiden@dsuvax.dsu.edu (Danny Breidenbach) writes:
>>When asked -- they will tell you.  They were asked by people at Purdue ...
>>and guess what they said?    No, I don't have the answers and all the data
>>on hand -- but I doubt you'd have a hard time finding it if you ask the
>>right people.  And hell --- why not talk to a physicist or engineer over
>>lunch and find out what one of them thinks?
>
>At most universities, and I know that this happens at Purdue, people from
>other departments ARE consulted on service courses.  I believe that there
>are, or at least there have been, representatives from engineering on the
>Calculus Committee.  One can find engineers and physicists who would like
>the math department to give such a course, but not too many.

On the other hand here is an anecdote: Somebody involved in one of the
"calculus reform projects" went and asked physicists wha they would like to
see in place of the numerous integral manipulation techniques. One of the
consistent answers was "complex exponentials". But the mathematicians
at the caculus reform meetings wanted to keep partial fractions and
rejected complex exponentials.

Recently, a vice-chair of the Ohio St U Math Department suggested that 
all references to complex numbers be removed from precalculus course(s). This
was in math Dept. meeting, and I do not what physicists/engineers will say.
But this is, to me at least, depressing.

By the way, the question is not if physicists/engineers >want< to teach math,
but whether they feel that they >have< to.
-- 
Vidhyanath Rao			It is the man, not the method, that solves
nathrao+@osu.edu		the problem. - Henri Poincare
(614)-366-9341			[as paraphrased by E. T. Bell]
Newsgroups: sci.math,rec.puzzles
From: karr@cs.cornell.edu (David Karr)
Subject: Re: Marilyn Vos Savant's error?
Organization: Cornell Univ. CS Dept, Ithaca NY 14853
Date: Tue, 15 Dec 1992 21:10:52 GMT

In article <BzBE0J.A6F@watdragon.uwaterloo.ca> deghare@daisy.uwaterloo.ca (Dave Hare) writes:
>In article <1992Dec15.160000.3714@cs.cornell.edu> karr@cs.cornell.edu (David Karr) writes:
>>It depends entirely on how your probability space is constructed.
>>Specifically, what is the distribution of numbers and kinds of
>>children had by people in the population that you are sampling, and
>>what is the probability, given that a person has a certain set of
>>children, that he or she will give the above response?
>
>This is entirely bogus.  The question is clearly intended to be a particular
>probability problem phrased in familiar terms.  The intent of the problem is
>the basic probability exercise, not a problem in the sociology of people
>meeting on the street.  It is simply more entertaining to phrase it this 
>way rather than in terms of 2-valued iid random variables.

It's not *entirely* bogus.  I think it's bogus to phrase a problem in 
everyday terms, then throw out the assumption of everyday behavior in
favor of some strange set of behaviors known only to the problem poser
and anyone he or she explained it to.

So, all right, my interpretation number 4 is too common-sensical, and
short-circuits the puzzle.  But why is interpretation 1 superior to
interpretation 2 or 3?  In particular, it's my opinion at the moment
(i.e. you may convince me otherwise) that the most naturally "random"
interpretation is number 2, namely the man will randomly pick one
of his two children and tell you the child's sex.

How about the following rephrasing of the problem.  You meet a man
on the street and have the following conversation:
"How many children have you?"
"Two."
"Is one of them a boy?"
"Yes."
What is the probability that the other child is also a boy?

-- David Karr (karr@cs.cornell.edu)



Newsgroups: sci.math,comp.edu,misc.education
From: dbreiden@dsuvax.dsu.edu (Danny Breidenbach)
Subject: Re: Student attitudes
Organization: Dakota State University, Madison, SD
Date: Thu, 17 Dec 1992 22:23:21 GMT

In article <Bz21G2.CHo@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>The math departments are quite willing to teach concept courses to reasonably
>prepared students.  Reasonable preparation is the ability to use symbols to
>formulate problems, to know what a function is, and to understand what is and
>what is not a proof.  

Some members of math departments are willing to teach them to students who 
aren't reasonably prepared with reasonable success.  Of course, a lot of
other people get annoyed that these members first help the students learn
about using symbols, formulating problems, and what a function is.  But once
that is done -- the concepts come along pretty quickly.  It happens all over.
Even at Purdue ...

>When the physicists and engineers tell the math departments that they want
>their students to know what derivatives and integrals ARE, to understand
>their use in word problems, instead of how to calculate them, the
>mathematicians will produce these courses.  I see no signs of this
>happening.  

When asked -- they will tell you.  They were asked by people at Purdue ...
and guess what they said?    No, I don't have the answers and all the data
on hand -- but I doubt you'd have a hard time finding it if you ask the
right people.  And hell --- why not talk to a physicist or engineer over
lunch and find out what one of them thinks?

--Danny
Purdue Alumnus
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Organization: Computer Science Department,  Stanford University.
Date: Wed, 16 Dec 1992 21:34:39 GMT

In article <1992Dec16.130923.1537@magnus.acs.ohio-state.edu> gedgar@magnus.acs.ohio-state.edu (Gerald A Edgar) writes:
>In article <1992Dec16.013418.11120@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>
>>d/dx (1+x/n)^n = (1+x/n)^{n-1} = n/(n+x) (1+x/n)^n.
>>
>>So as n -> oo, this function tends to a fixpoint of d/dx, and is 1 at
>>x=0, hence must be exp(x).  Now set x=1.
>
>What are you claiming?  If  f_n(x) and g_n(x) are two sequences of
>real functions,  d/dx f_n(x) = g_n(x) for each n,
>f_n(x) converges to f(x) for each x, and g_n(x) converges to g(x)
>for each x, then d/dx f(x) = g(x) ??

Yes, with the additional hypothesis f_n(x)/g_n(x) = 1 + O(x/n).  Do you
have a counterexample?
-- 
Vaughan Pratt	   		Interactive Proofs = Polynomial Space
Newsgroups: sci.math
From: cdeloge@cs.ulowell.edu (Carolyn Deloge)
Subject: Difference between "show" and "prove"
Organization: University of Massachusetts at Lowell Computer Science
Date: Sun, 13 Dec 1992 17:37:32 GMT
Expires: Sun, 31 Jan 1993 05:00:00 GMT

Hi, I am a CS graduate student and I have a question for the math people.

What is the difference between "show" and "prove"?

I had a homework problem that asked to "show" the solution.  I answered by
giving an example and some reasoning.  The teaching assistant marked it 
incorrect, saying that I needed to "prove" it inductively.

Can someone give me his/her definitions on these two words?  I would really
appreciate it.  Thanks.

CD

P.S. Yes I did look in the dictionary, but I want to hear it from the math
experts.
Xref: sparky sci.math:17161 rec.games.abstract:648
From: torbenm@diku.dk (Torben AEgidius Mogensen)
Newsgroups: sci.math,rec.games.abstract
Subject: Re: Game of pentominos
Date: 18 Dec 92 15:36:42 GMT
Organization: Department of Computer Science, U of Copenhagen

umatf071@unibi.hrz.uni-bielefeld.de (sio) writes:

>The 4th Pentacube Contest 92/93

>  Build the figure "volcano" with all 29 pentacubes.

[most deleted]

Does this mean that a company produces a game based on the 29 possible
figures made from 5 cubes? If so, do they have copyright to the actual
shape of the pieces? I find it doubtful that such a claim would hold,
as the shapes are of mathematical nature and thus not copyrightable.
Any rules for the game will be copyrightable though.

I made (about 5 years ago) a puzzle out of the 29 "pentacubes". Does
this predate the game?

	Torben Mogensen (torbenm@diku.dk)
Newsgroups: sci.math
From: guckes@math.fu-berlin.de (Sven Guckes)
Subject: Email of Prof. Halin (Hamburg?)
Originator: guckes@medusa
Organization: Free University of Berlin, Germany
Date: Wed, 16 Dec 1992 23:33:40 GMT

Does anyone know the email address of Prof. Halin ?
I think he should be in Hamburg, Germany.

Please reply by email. Thank you!
Sven :)
Newsgroups: sci.math
From: dennis@cadkey.com (Dennis Paul Himes)
Subject: Re: '-' operation
Organization: cadkey
Date: Thu, 17 Dec 1992 18:56:17 GMT

In article <92350.145501B7D@psuvm.psu.edu> <B7D@psuvm.psu.edu> writes:
>Recently my daughter brought back Problem Set II of Wisconsin Mathematics
>Science and Engineering Talent Search, and Question 4 in it is as following:
>         Operation @ satisfies the conditions that
>                X @ (Y @ Z) = X @ Y + Z and X @ X = 0 for any real numbers
>          X, Y, Z. Show that @ must be subtraction.
>
> ... The problem I have is: How is operation '-' defined? ... For the above 
> question, is only thing I have to prove is X @ 0 = X? ...
>
> Duane

     Subtraction is usually defined as X - Y == X + (-Y), where -Y is the
number such that Y + (-Y) = 0.  (Note that the unary minus sign and the
binary minus sign are really two separate symbols, even though they both
use the same character.)  The above definition of @ can be shown to fit
this definition of subtraction as follows:
     X @ Y = (X @ Y) + 0 = (X @ Y) + (Y + -Y) = ((X @ Y) + Y) + -Y
           = (X @ (Y @ Y)) + -Y = (X @ 0) + -Y = (X @ (X @ X)) + -Y
           = ((X @ X) + X) + -Y = (0 + X) + -Y = X + -Y    Q.E.D.
     Note that none of the known properties of subtraction were used in the
above proof, only the definition of @ given above and the fact that
addition is a group.  The definition of @ could therefore be used as
a(n obscure) definition of subtraction.

==============================================================================

               Dennis Paul Himes    <>    dennis@cadkey.com
 
"In watermelon sugar the deeds were done and done again as my life is done
in watermelon sugar."           - Richard Brautigan
Newsgroups: sci.math
From: bontchev@fbihh.informatik.uni-hamburg.de (Vesselin Bontchev)
Subject: Re: Factor 69 digit integer
Reply-To: bontchev@fbihh.informatik.uni-hamburg.de
Organization: Virus Test Center, University of Hamburg
Date: 14 Dec 92 16:40:36 GMT

Why don't you stop this useless discussion? You are just wasting the
bandwidth... The number has been factored completely and the factoring
has been published in sci.crypt.

-- 
Vesselin Vladimirov Bontchev          Virus Test Center, University of Hamburg
Tel.:+49-40-54715-224, Fax: +49-40-54715-226      Fachbereich Informatik - AGN
< PGP 2.0 public key available on request. > Vogt-Koelln-Strasse 30, rm. 107 C
e-mail: bontchev@fbihh.informatik.uni-hamburg.de    D-2000 Hamburg 54, Germany
From: burshtey@eniac.seas.upenn.edu (Alexander Burshteyn)
Newsgroups: sci.math
Subject: Re:  Haughty quote
Date: 11 Dec 92 19:06:32 GMT
Distribution: usa
Organization: University of Pennsylvania

In article <11DEC199212101828@mary.fordham.edu> nissim@mary.fordham.edu (Leonard J. Nissim) writes:
>I was told by my high school calculus teacher many years ago that 
>Leonard Euler said, "A mathematician is someone to whom e^{i*\pi}=-1
>is just as obvious as 2+2=4 is to you."  But I have not seen this 
>in print anywhere.  

That sentence about the integral from -oo to +oo of e^(-x^2), followed by
"Liouville was a mathematician." appears, as I vaguely recall, in Spivak's
"Calculus on Manifolds" at the end of the problem section in one of the
chapters.
                                            
>Leonard J. Nissim  (nissim@mary.fordham.edu)   

Alex Burshteyn (burshtey@eniac.seas.upenn.edu)
Newsgroups: sci.math,sci.philosophy.tech
From: djoyce@black.clarku.edu (Dave Joyce)
Subject: Centillion etc. (was Re: Negative Zero)
Organization: Clark University (Worcester, MA)
Date: 14 Dec 92 14:35:14 GMT

In <Bz7o7s.9zr@max.physics.sunysb.edu> mantell@beta.mathlab.sunysb.edu (Abraham S. Mantell) writes:

>I thought 1 centillion = 10^600, not 10^303.....????
>
>Abe

It's unfortunate, but it seems number names

     billion, trillion, quadrillion, ..., centillion, ...

do not have universal values.  Depending on the country, a billion can be
either a thousand million or a million million, and so on for the rest.
Only a million has a universal value.

I was thinking the other day about this very problem.  Perhaps a solution
would be to "go metric".  That is, use the metric prefixes for large numbers.
I'm not sure how to inflect them, but kilo, mega, giga, milli, micro, and all
the rest could be used for plain numbers as well as measurements.  I don't
know if the unit (namely, one) would have to be named after some famous
counter or mathematician such as Ahmes, Thales, Pythagoras, or Archimedes.
Imagine using "megapyth" or "megaarch" instead of "million". 


-- 
David E. Joyce				Dept. Math. & Comp. Sci.
Internet:  djoyce@black.clarku.edu	Clark University
BITnet:    djoyce@clarku		Worcester, MA 01610-1477
Newsgroups: sci.math,sci.physics,sci.bio,sci.chem,misc.education
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: TIME HAS INERTIA - NewScientist Article
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Sun, 13 Dec 92 19:57:40 GMT

In article <1992Dec11.024139.12143@dmp.csiro.au> lachlan@dmp.csiro.au (Lachlan Cranswick) writes:

>I just read got this weeks New Scientist.  In the Feedback
>section on the second last page they mention this TIME
>HAS INERTIA thing on the Internet but also mention
>there is now a scale called the CRACKPOT INDEX.
>
>i.e. - you get 40 points for claiming a revolutionary 
>theory that gives no concrete testable predictions, etc etc
>
>It also mentions spoof messages that people are writing 
>spoof?  articles to get
>the maximum possible points in the Crackpot INDEX.

Do they mention that I invented the crackpot index?  Probably not.  Yet
another brilliant idea of mine leading to no tangible fame.  :-)
Maybe I should write a letter to the editor.

Anyway, in answer to your question, most of the crackpot index stuff
appeared on sci.physics, and all of the postings are apparently archived
somewhere.  I don't know where and couldn't care in the least, since for
the most part the idea of archiving newsgroups is like tape-recording
all the conversations in an insane asylum.  

I'll send you the official crackpot index.
Xref: sparky sci.math:16859 sci.physics:21115 sci.bio:4564 sci.chem:5390 misc.education:5240 talk.bizarre:41550
From: barkdoll@cattell.psych.upenn.edu (Edwin Barkdoll)
Newsgroups: sci.math,sci.physics,sci.bio,sci.chem,misc.education,talk.bizarre
Subject: Re: TIME HAS INERTIA - BASS JEALOUS AND INSECURE-ABIAN VALIDATED
Date: 14 Dec 92 14:09:32 GMT
Followup-To: sci.physics
Organization: University of Pennsylvania

Please take this out of sci.bio!

Edwin Barkdoll
barkdoll@cattell.psych.upenn.edu
eb3@world.std.com
From: rags@triples.math.mcgill.ca (Robert A. G. Seely)
Newsgroups: sci.math
Subject: Re: Philosophy of Pi
Date: 18 Dec 92 20:49:29 GMT
Organization: /etc/organization

In article <DREIER.92Dec18094157@durban.berkeley.edu> dreier@durban.berkeley.edu (Roland Dreier) writes:
ove.
>
> ..., it is a very open question as to whether every integer
>appears in the decimal expansion of Pi.  This is called the Goldbach
>conjecture, and I'm not aware of any real progress toward a solution.
>--
>Roland Dreier                                        dreier@math.berkeley.edu

Goldbach Conjecture: Every even no is the sum of (odd) primes.

How does this relate to the above?  (What am I missing here?)
From: martin@lyra.cis.umassd.edu (Gary Martin)
Newsgroups: sci.math
Subject: Re: Need some help with Topology
Date: 12 Dec 92 20:43:45 GMT
Organization: University of Massachusetts Dartmouth

In article <1gc73iINNl4g@matt.ksu.ksu.edu> bubai@matt.ksu.ksu.edu (P. Chatterjee) writes:

   I had a few questions (none homework, though!) and would appreciate any   
   kind of help from the math-knowledgeables on the net.

Seems that you need to give us more background before we can answer these.

   a) What does it mean to say that a set A is 'infinite'? 'Finiteness', by 
   definition, implies that A is equivalent to a portion of the set of 
   positive integers. Can this definition of 'finiteness' be used to 
   motivate one for 'infiniteness'?

Do you have some precise meaning in mind when you say "portion"?
Let's say it means {1,2,3,...,n} for some  n, ok?  Now you've got 
a definition of "finiteness".  This, or any other definition of
"finite" can be formally negated to get a corresponding definition
of "infinite".  A set is "infinite" if it is not equivalent to any
set {1,2,3,...,n} for a natural number n.  (I suppose we should
mention the empty set, too, but that's just a nit to pick.)

   b) Show that A is OPEN <==> X \ A is closed. 
   Isn't this a definition or can it be proven?

This depends on what definitions you are using.  You can say that the
topology is given by specifying its open sets, and then define
"closed" to mean "open complement".  Then yes, it's a definition.
Or you can (again giving the topology by specifying the open sets)
define A to be closed if every point in  X\A  has an open neighborhood
disjoint from  A.  Then you can prove that "open" is equivalent to
"closed complement". Etc.

   c) If A is infinite and p in A, show that A \ {p} is equivalent to A.

You need to choose a definition of "infinite" before attempting this.
One possible definition, which would make the proof easy, is to
say  A  is infinite iff for some  p  in  A,  A\{p} is equiv. to A.
Another is to use the definition I suggested in (a).  Another
is to say  A  is infinite if there is a one-to-one map of the
natural numbers into  A  (in which case the proof is again easy).
Another is to say  A  is infinite if there is a map from  A  onto
the natural numbers.

   d) Show that (A')' is a subset of A'.

What does A' mean?




   Thanks for all the help.
--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Martin@cis.umassd.edu
Newsgroups: sci.math,rec.puzzles