HDTenEightyP/NetNews huggingface dataset:
From: sal7@midway.uchicago.edu (Rob Salgado)
Newsgroups: sci.math.research
Subject: Looking for info: CAUSAL SETS and/or PARTIALLY-ORDERED SETS
Date: 23 Jan 93 17:52:56 GMT
Article-I.D.: midway.1993Jan23.175256.7389
Reply-To: sal7@midway.uchicago.edu
Followup-To: poster
Organization: University of Chicago Computing Organizations
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
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I am looking for references concerning causal sets (partially-ordered sets)
and other models for a discrete spacetime topology.

Please respond by email.  Thanks.
-- 
Rob Salgado                              sal7@ellis.uchicago.edu
                                         salgado@suhep.phy.syr.edu (preferred)
From: leifj@cyklop.nada.kth.se (Leif Johansson)
Newsgroups: sci.math.research
Subject: A question about extension.
Keywords: extension
Date: 27 Jan 93 13:38:06 GMT
Article-I.D.: kth.1993Jan27.133806.27456
Organization: Royal Institute of Technology, Stockholm, Sweden
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
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Consider the following:

	0->A->E->B->0
        0->A'->E'->B'->0
        0->E->U->E'->0

are three extensions of abelian groups. Is there a way
to express U as a higher ext of something. Are higher
ext's involved in any way ? Any comments and/or hints
would be nice. For various reasons I believe that there
should be an ext^3(something,other) here.

Thank you all in advance.
Leif Johansson
Dept of math. University of Stockholm
leifj@matematik.su.se
Newsgroups: sci.math.research
From: sl25@cus.cam.ac.uk (Steve Linton)
Subject: Re: combinatorics or coding theory question
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: U of Cambridge, England
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 25 Jan 1993 15:56:27 GMT


|> Suppose you have N groups of cells.  Each group contains K cells.
|> Now suppose you make choices consisting of one cell chosen from
|> each of the N groups.  Let each such choice be called a vector.
|> Clearly there are K^N unique vectors.
|> 
|> My question is:  what is the largest set of vectors you can pick
|> such that no two of the vectors overlap (i.e. have the same cell)
|> at more than Q of the groups (where Q < N)?

In more usual language, you want your vectors to differ in at least N-Q places. I
shall call this number D.

You are looking for the largest code with minimum distance D among your K^N
vectors.

Each vector has       (N+D)   neighbours  at diostance less than D, so a 
		      ( D )
		(K-1)^

simple upper bound on the size of your code is  

			  -(N+D/2)	
		K^N (K-1)^ ( D/2 )

(that's K to the N times (K-1) to the N+D/2 choose D/2).

Anyway, you want to look at Codes, Groups and Sphere Packings by J.H.Conway and
N.J.A. Sloane.

		Steve Linton
From: dezan@enstb.enst-bretagne.fr (Catherine Dezan)
Newsgroups: sci.math.research
Subject: looking for an algorithm to separate domains with hyperplanes
Keywords: geometry
Date: 26 Jan 93 11:23:40 GMT
Reply-To: dezan@enstb.enst-bretagne.fr
Organization: Ecole Nationale Superieure des Telecommunications de Bretagne, Brest  FRANCE
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu


I am looking for an algorithm which can separate n convex polyhedral domains
(each one from others) with a minimal number of hyperplans.
Does such an algorithm exist?
What about its complexity?

Thank you in advance for any piece of information.

Catherine Dezan.

email : dezan@enstb.enst-bretagne.fr
Newsgroups: sci.math.research
From: A8131DAL@helios.edvz.univie.ac.at (Peter Schmitt)
Subject: CONFERENCE: Discrete Geometry (Austria)
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Vienna
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 27 Jan 1993 16:19:34 GMT
X-Char-Esc: 29
X-Charset: ASCII

-------------- Announcement ----------------------

May 2-8, 1993.

Austrian Conference on
Discrete Geometry and related topics

The conference will take place in a hotel near Neuhofen, a small village about
100~km west of Vienna (near Amstetten), situated in a pleasant landscape near
some of the most prominent sights of Austria.  The price of full board will be
approximately AS~4500.-- (about $440) (in a single room),
or AS~3250.-- (about $300) per person in a double room.

Information: Peter Schmitt, Institute of Mathematics, University of Vienna,
             Strudlhofgasse 4, A-1090 Wien, Austria
     e-mail: schmitt@awirap.bitnet
        (or) a8131dal@awiuni11.edvz.univie.ac.at
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math.research
Subject: Re: Set-free measure theory
Date: 27 Jan 93 04:34:54 GMT
Article-I.D.: camelot.ARA.93Jan26233454
Organization: M.I.T. Artificial Intelligence Lab.
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

In article <C1GuEp.M99@sugar.neosoft.com> claird@NeoSoft.com (Cameron Laird) writes:

   Conjecture:  the literature already includes a proof
   that there is nothing novel about "set-free measures",
   that is, they all have realizations as (extended)
   measures.  Does anyone know?

Apparently, you don't require the boolean algebra to be a sigma algebra.
In that case, the Stone representation theorem seems to answer your
question: every boolean algebra is isomorphic to the algebra of
clopen (i.e. simultaneously open and closed) subsets of a totally
disconnected compact Hausdorff space.

If you want to work with sigma algebras and want the representation
to be a sigma homomorphism, the Stone representation doesn't work anymore.
I'm sure a lot of work has been done on the representation of sigma
algebras as sigma algebras of subsets, but I don't know the results
on it.

If your Boolean algebra is atomic, you might get a suitable representation
of it as subsets of the set of its atoms.

If you look at Borel sets modulo sets of measure zero, you don't have
sets any more. This is not an atomic boolean algebra in general.


Allan Adler
ara@altdorf.ai.mit.edu
From: sal7@midway.uchicago.edu (Rob Salgado)
Newsgroups: sci.math.research
Subject: Looking for info: CONTACT MANIFOLDS in physics
Date: 23 Jan 93 18:00:48 GMT
Article-I.D.: midway.1993Jan23.180048.7595
Reply-To: sal7@midway.uchicago.edu
Followup-To: poster
Organization: University of Chicago Computing Organizations
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

I am looking for references on contact manifolds and their applications in
physical problems, particularly in general relativity.  I am already aware
of their use in time-dependent hamiltonian mechanics.  Are there any other
uses?

Please reply by email.  Thanks.
-- 
Rob Salgado                              sal7@ellis.uchicago.edu
                                         salgado@suhep.phy.syr.edu (preferred)
Newsgroups: sci.math.research
From: Allan Adler <ara@zurich.ai.mit.edu>
Subject: Cremona transformations
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: M.I.T. Artificial Intelligence Lab.
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Sun, 24 Jan 1993 20:10:29 GMT



Let X be a variety and f:X-->X a birational automorphism. Can one find
a complete model Y of the function field of X such that f becomes
an actual automorphism of Y (i.e. defined everywhere and its inverse
is defined everywhere)? Can one take Y to be projective?

Allan Adler
ara@altdorf.ai.mit.edu


Newsgroups: sci.math.research
From: rivero@sol.cie.unizar.es (Alejandro Rivero)
Subject: Topos, Foliations and all that
Originator: dan@symcom.math.uiuc.edu
Reply-To: rivero@sol.cie.unizar.es
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Department of Theoretical Physics. University of Zaragoza
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Fri, 22 Jan 1993 21:08:26 GMT

First of all, let me to add to the bibliography a nice book
we have just got:

Sheaves in Geometry and Logic,
By S MacLane and Ieke Moerdijk, Springer.

Second, I want to ask if some Topos/Sheaf fanatic have studied
relationship between the generalized spaces etc of Topos theory,
and the non-conmutative "spaces" of Alain Connes. 
Im not expert in the matter, so I could be asking a nonsense. 
Anyway, a motivation for the question is that the Kroneker
foliation of the bidimensional torus, which were the main example
of Connes' theory, is also one principal example of Sheaf/Topos
theory when used to calculate De Rham Homology. At least, it is
used by Moerdijk (althought not in that book) to show all the
constructios of clasifiyng topos, BG etc...

-Alejandro Rivero
Zaragoza Univ, Spain
rivero@cc.unizar.es or rivero@sol.cie.unizar.es
Newsgroups: sci.math.research
From: milanfar@Athena.MIT.EDU
Subject: terminology query
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Illinois at Urbana
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 28 Jan 1993 15:24:29 GMT

Is there a particular name associated with the set of all NxN linear
transformations Q such that |det(Q)|=1? (Note the | | sign)

Thanks.

Newsgroups: sci.math.research
From: claird@NeoSoft.com (Cameron Laird)
Subject: Set-free measure theory
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: NeoSoft Communications Services -- (713) 684-5900
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 26 Jan 1993 14:47:10 GMT

I propose this definition:  a "set-free measure" is
a triple <X, m, T> with
1.  X a lattice (boolean algebra),
2.  m a function defined on the members of X, and
3.  T an equivalence relation on X, for which
4.  m(x + y) = m(x) + m(y) - m(x.y) and
5.  xTy implies m(x) = m(y).

The natural model for such an object is conventional
measure theory, with X a collection of Borel sets, or
measurable sets, taken from an abelian group, and T
translation (that is, read xTy as "x is a translate of
y").  Are there any (non-trivial) others?  I realize I
haven't constrained the range of m (finite, includes
zero, non-negative, ...), or adjoined a countable
version of 4; if doing so makes the problem more
interesting, then I'd want to know.

Conjecture:  the literature already includes a proof
that there is nothing novel about "set-free measures",
that is, they all have realizations as (extended)
measures.  Does anyone know?

I write "set-free" to emphasize the historical attention
directed away from the point-set flavor typical of measure
theory, and toward a categorical attention to the action
of the function m.
-- 

Cameron Laird
claird@Neosoft.com (claird%Neosoft.com@uunet.uu.net)	+1 713 267 7966
claird@litwin.com (claird%litwin.com@uunet.uu.net)  	+1 713 996 8546
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math.research
Subject: Re: Generalisation for Puiseux theorem ?
Date: 26 Jan 93 07:52:58 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu


In article <1k0p4iINN4ti@taloa.unice.fr> guyard@dollar.unice.fr (Frederic Guyard) writes:
>Is there some generalisation for Puiseux Theorem ?

I think you can get one out of the Weierstrass Preparation Theorem.

Later,
Andrew Mullhaupt
From: rinkus@park.bu.edu (Gerard J. Rinkus)
Newsgroups: sci.math.research
Subject: combinatorics or coding theory question
Date: 24 Jan 93 19:27:53 GMT
Organization: Boston University Center for Adaptive Systems
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu




Hi,

I have a combinatorics question that I have been unable to solve.
I have asked a number of mathematics professors, and otherwise
mathematically sophisticated people as well and no one has
been able to show me the general solution to the following
problem.  This question arose to me in connection with a 
neural net model of spatio-temporal memory that I'm working on.

Suppose you have N groups of cells.  Each group contains K cells.
Now suppose you make choices consisting of one cell chosen from
each of the N groups.  Let each such choice be called a vector.
Clearly there are K^N unique vectors.

My question is:  what is the largest set of vectors you can pick
such that no two of the vectors overlap (i.e. have the same cell)
at more than Q of the groups (where Q < N)?

The answer for the special case of Q = N-1 is K^N.  This follows
immediately from the fact that if there are K^N unique vectors
then each vector must be different from any other vector in at
least one group (and therefore no two vectors (in the entire space
of K^N vectors) can overlap at more than N-1 groups).

I've also found for various small special case, that the answer is
K^(Q+1).  

But I haven't been able to figure out the general solution, or
prove that it is K^(Q+1) if in fact it is.

Thanks in advance for any help on this.

Rod Rinkus
Dept. of Cognitive and Neural Systems
Boston University
Boston, MA 02215
email: rinkus@park.bu.edu


--
Gerard J. Rinkus
Dept. of Cognitive and Neural Systems
Boston Univ.
rinkus@cns.bu.edu
From: jack@dcs.glasgow.ac.uk (Jack Campin)
Newsgroups: sci.math
Subject: Re: Non-Standard Analysis and philosophy
Date: 22 Jan 93 19:03:50 GMT
Reply-To: jack@dcs.glasgow.ac.uk (Jack Campin)
Organization: COMANDOS Project, Glesga Yoonie

cxm7@po.CWRU.Edu (Colin Mclarty) wrote:
>> Under which philosophies of mathematics is non-standard
>> analysis possible, and to what extent?
> Any philosophy of mathematics that allows the compactness theorem for first
> order logic (and thus for axiomatized higher order logic) has an easy proof
> that standard and non-standard analysis are equally possible.

OK, so just what philosophies of mathematics *do* allow the compactness
theorem?  What is the constructive content of the prime ideal theorem?
(I don't know of any constructive system that accepts it but not AC - is
there one?)

-- 
--  Jack Campin    room G092, Computing Science Department, Glasgow University,
17 Lilybank Gardens, Glasgow G12 8RZ, Scotland   TEL: 041 339 8855 x6854 (work)
INTERNET: jack@dcs.glasgow.ac.uk or via nsfnet-relay.ac.uk    FAX: 041 330 4913
BANG!net: via mcsun and uknet     BITNET: via UKACRL    UUCP: jack@glasgow.uucp
Newsgroups: sci.math
From: stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith)
Subject: Polynomials modulo 2
Keywords: polynomials, division, modular arithmetic
Organization: University of Missouri
Date: 28 Jan 93 15:49:26 GMT

Dear All, 

A friend gave me a polynomial p, whose coefficients come from the field
of order two.  His question was, is there an n such that p divides
1 + x^n, and if so, what is the smallest positive n.  (All division is
modulo 2).

I deduced that the answer is yes --- there is such an n.  To find it, there
is the following algorithm.  The polynomials are represented by binary
numbers:
  a_0 + a_1 x + a_2 x^2 + ... + a_n x^n 
is represented by the binary number a_n ... a_1 a_0.

input(p);
n = 0;
q = 1;
repeat
  if (bit 0 of q = 1) then q = q xor p;
  shift q one place to right;
  n = n+1;
until q = 1;
output(n);

The algorithm is based on synthetic division.

So, here is my question.  Has this been done before (I think yes) and
if so, what is a good reference?  Also, has anyone come up with a better
algorithm than the one I presented here?  

Thanks in advance,

Stephen Montgomery-Smith

Organization: Mathematics, Carnegie Mellon, Pittsburgh, PA
Newsgroups: sci.math
Date: Wed, 27 Jan 1993 16:21:24 -0500 
From: Oswald Wyler <ow0a+@andrew.cmu.edu>
Subject: Puzzle

There seems to be no netnews bb for math puzzles (which I regret), so this
is posted here -- and don't flame me, please.
Here is a sequence of thirty distinct items, separated by dashes:
00 - 02 - 20 - 01 - 22 - 10 - 22 - 14 - 26 - 18 -
06 - 25 - 04 - 23 - 11 - 03 - 15 - 27 - 19 - 07 -
28 - 09 - 17 - 29 - 08 - 16 - 24 - 05 - 13 - 21
Questions:
(1) What does it represent?
(2) From the same thirty items, how many sequences of this kind can be
formed, beginning with 00 - 02 and ending with 21?
Answers and comments to wyler@cs.cmu.edu.
If there's enough interest, I'll post the answer.
From: kimball@stsci.edu (Timothy Kimball)
Subject: Re: Want to learn about FFT's ... where?
Organization: Space Telescope Science Institute
X-Newsreader: TIN [version 1.1 PL6]
Date: Tue, 26 Jan 1993 14:15:39 GMT

_Numerical Recipes_ has the clearest explanation I've seen.

--

/* tdk -- Opinions are mine, not my employer's. */
From: hougen@focus.csl.uiuc.edu (Darrell Roy Hougen)
Newsgroups: sci.math
Subject: >>godel theorem ...
Date: 22 Jan 1993 21:28:49 GMT
Organization: Center for Reliable and High-Performance Computing, University of Illinois at Urbana-Champaign
NNTP-Posting-Host: focus.csl.uiuc.edu
Summary: godel theorem
Keywords: goedel

cole@alexia.lis.uiuc.edu (Sandra Stewart-Cole) writes:

% For those who have sought to make only a few assumptuions and prove
% various truths that fully represent reality consistently from them, GT
% undermines the attempt.

True.

% It shows that logical deduction is
% insufficient to explain anything complex no matter how many starting
% axioms you take.

Fine.

% For seekers of objective truth backed by logic, this
% is a disturbing concept.

Perhaps, but it is the concept of truth and not the concept of
objectivity that must be modified.  It is still possible obtain
objective knowledge by logical inference from observation.

Darrell R. Hougen
From: Josef Gttgens <jg@techmech.itme.rwth-aachen.de>
Newsgroups: sci.math
Subject: Plotting Graphs
Date: 26 Jan 1993 13:45:50 GMT
Organization: ITM RWTH Aachen
Distribution: world
NNTP-Posting-Host: mac.itm.rwth-aachen.de
X-UserAgent: Nuntius v1.1.1d17
X-XXMessage-ID: <A78AFFC5D401AB03@mac.itm.rwth-aachen.de>
X-XXDate: Tue, 26 Jan 93 14:48:37 GMT

Does anybody know about programs that layout and plot graphs and run
on small machines (e.g. Mac)?

Josef Goettgens
Institut fuer Technische Mechanik   Phone:  +49-241-804626
RWTH Aachen                         Fax:    +49-241-20911
Templergraben 64                    Email:  jg@techmech.itm.rwth-aachen.de
W-5100 Aachen, Germany
Xref: sparky sci.math:18824 sci.physics:23596
Newsgroups: sci.math,sci.physics
Subject: Re: Vectors Question
From: <A54SI@CUNYVM.BITNET>
Date: Tuesday, 26 Jan 1993 20:14:01 EST
Organization: City University of New York/ University Computer Center

Now I see where I was wrong. (-a) X b = -(a X b). I was thinking they were
 actually differing in magnitude but no. I am grateful for all the responses.
-------
CHARLES HOPE   A54SI@CUNYVM   A54SI@CUNYVM.CUNY.EDU
GOVERNMENT BY REPORTERS...MEDIA-OCRACY.
Newsgroups: sci.math
From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Re: request for info on game theory
Organization: M.I.T. Artificial Intelligence Lab.
Date: Wed, 27 Jan 1993 23:40:53 GMT


It might also be instructive to see how some games are actually
implemented. I would look at source code for gnuchess and gnugo,
for example, which are available for free with complete source
code.

Allan Adler
ara@altdorf.ai.mit.edu
From: akay@comlab.ox.ac.uk (Andrew Kay)
Newsgroups: sci.math
Subject: Re: "Cut & Choose" for several players
Date: 21 Jan 93 10:15:33 GMT
Organization: Oxford University Computing Laboratory, UK
Originator: akay@acis.comlab

Anthony Minkoff writes:
>Now I want a method for distributing the pile among n players, so
>that no player feels that any other player received a better
>portion.

and also
>I.e., the statement "each player feels he got a 'fair' share" is not
>a sufficient condition in my formulation of the problem.  Rather, it
>is necessary that *no player feels that _any other player_ received
>a better portion.*

It is not clear to me whether a player is judging (subjective)
values of shares in themselves, or (subjective) values of 
shares to particular players.

For example, three players A, B, C and shares x, y and z.

A might like x a lot, and be happy for B to get y and C to
get z. However, A knows that y is of particular advantage to
C and z to B, so would not be happy for B to get z and C to
get y. 

Actually, what is to stop B and C covertly swapping y and z
after A has gone home ? I suspect that this question uncovers
a problem with the whole question of A being satisfied about
the way B and C share y and z.

I seem to remember that games with more than two players are
very hard to analyse mathematically, because of the formation
of coalitions which don't fall within the rules of the game.
B may try to help C win a game in return for a share of the
winnings, or to get a date for example !

Interesting puzzle.

Andrew Kay,             
Email: Andrew.Kay@prg.oxford.ac.uk
From: ard@cs.bham.ac.uk (Antoni Diller)
Newsgroups: sci.math
Subject: Non-number theoretcial proofs in number theory
Date: 20 Jan 93 11:24:02 GMT
Organization: School of Computer Science, University of Birmingham, UK

It's years since I've done any proper maths, but what I'm after are
examples of proofs of statements about the natural numbers or integers
that use ideas and methods external to number theory, like for example
results from the theory of complex numbers.  Ideally I'd like an example
which is such that no proof only involving natural numbers or integers
is known.  A reference to a good textbook would be acceptable.
Please reply by email and I'll post any good examples.

Antoni Diller
A.R.Diller@cs.bham.ac.uk
Newsgroups: sci.math
From: dzk@cs.brown.edu (Danny Keren)
Subject: Re: Travelling Salesman Problem
Organization: Brown University Department of Computer Science
Date: Thu, 21 Jan 1993 14:34:45 GMT

If it may be of help, there is a C program for TSP in "Numerical Recipis".
I never used it, though, and can't say how good it is.

-Danny Keren.
From: datta@durban.berkeley.edu (Ruchira Datta)
Newsgroups: sci.math
Subject: Calculus Theorem Discovered By Freshman
Date: 24 Jan 1993 01:58:37 GMT
Organization: U.C. Berkeley Math. Department.
Distribution: world
NNTP-Posting-Host: durban.berkeley.edu

I read in the Contra Costa Times of Saturday, January 23, that in fall of 1991,
Christy Randall, then a freshman at Miami University, discovered a calculus
theorem "by accident".  But the article made no attempt to describe the theorem.
Does anyone else know about this?  Could someone tell me what this theorem is
about, or where I could read about it?  Please e-mail responses, as I don't
read this newsgroup.  (Maybe this has been discussed here before; if so I
apologize.)  Anyway, I'm really curious.

Thanks in advance.

Ruchira Datta
datta@math.berkeley.edu
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: why is pi irrational
Organization: Computer Science Department,  Stanford University.
Date: Fri, 22 Jan 1993 06:29:47 GMT

In article <1993Jan21.133623.20957@infodev.cam.ac.uk> gjm11@cus.cam.ac.uk (G.J. McCaughan) writes:
>In article <pete.03vb@bignode.equinox.gen.nz> pete@bignode.equinox.gen.nz writes:
>
>>>So the Bible is accurate, just not very precise.  :-)
>>
>>When it says the Earth is flat, however, it is inaccurate, rather than
>>imprecise.
>
>Where does the Bible say that the earth is flat? I do not deny that most
>of the writers of the Bible thought the earth to be flat (as did everyone
>else at the time), but did they say so in the writings that now form the
>Bible?

The phrase "circle of the earth" appearing in Isaiah is sometimes cited
as biblical authority for the shape of the earth being circular (hence
flat), rather than spherical:

Isaiah 40:22.  It is he that sitteth upon the circle of the earth, and
the inhabitants thereof are as grasshoppers; that stretcheth out the
heavens as a curtain, and spreadeth them out as a tent to dwell in.
-- 
Vaughan Pratt
Newsgroups: sci.math