HDTenEightyP/NetNews huggingface dataset:
From: kosowsky@minerva.harvard.edu (Jeffrey J. Kosowsky)
Subject: Re: Undivided distaste for algebra, more fuel for the fire...
Organization: Harvard Robotics Lab, Harvard University
	<1993Jan19.083812@usno06.nor.chevron.com>
	<1993Jan19.214027.21851@schaefer.math.wisc.edu>
	<20JAN199311462556@mary.fordham.edu>
Distribution: usa
Date: Thu, 21 Jan 1993 18:22:53 GMT


>
>   In article <1993Jan19.214027.21851@schaefer.math.wisc.edu>, 
>   mueller@schaefer.math.wisc.edu (Carl Douglas Mueller) writes...
>
>   >>2:  My opinion:  (you may wish to strech your legs here and skip this part)
>   >>    Our schools
>   >>    teach FOIL (first, outer, inner, last) instead of the distributive
>   >>    principle for doing binomial expansion.  
>   > 
>   >    I learned FOIL from my students the first time I was a TA.  Nearly all of
>   >the students I've taught since then have told me that this is how they had 
>   >learned to multiply binomials.  I think that this is a rather TERRIBLE way
>   >to teach this!  It hides what is going on and doesn't allow for the
>   >(rather easy -- if you learn it right) generalization to multiplication of
>   >more general polynomials.
>   > 
>   >Carl Mueller (mueller@math.wisc.edu)
>
>   I agree.  I also learned FOIL for the first time at age 37 when teaching 
>   Precalculus here.  A student complained that FOIL was easier than the 
>   distributive law; when I asked what FOIL was, he was shocked that I had
>   never heard of it.  He was unimpressed by my statement that FOIL was a 
>   special case of a more general law, and that it would be better to under-
>   stand the general law rather than to memorize the special case.  
>   -------------------------------------------------------------------------------
>   Leonard J. Nissim  (nissim@mary.fordham.edu)   
>   Disclaimer:  "I speak only for myself."
>
>   ---------------------------------------------------------------------------


O.K. I give-up. I am 29, a math graduate student, and still don't know
what FOIL is. Should I wait another 8 years or could someone tell me
what this "trick" is? :.)

Thanks,
	Jeff Kosowsky
Organization: Penn State University
Date: Fri, 22 Jan 1993 11:45:12 EST
From: <DCC117@psuvm.psu.edu>
Newsgroups: sci.math
Subject: SOLUTIONS BOOK EXIST? Combinatorics.

  Hi, I just wanted to know if anybody knew of a
 solutions manual ( or something along those lines )
 for the following book :

   INTRODUCTORY COMBINATORICS, 2nd Ed., by Richard A. Brualdi.
   Published by North-Holland.
From: pete@bignode.equinox.gen.nz (Pete Moore)
Subject: Re: why is pi irrational
Newsgroups: sci.math
Organization: Equinox Networks
Reply-To: pete@bignode.equinox.gen.nz
X-Newsreader: TIN [version 1.1 PL8]
Date: 23 Jan 93 15:14:04 +1200
Organization: Equinox Networks

Vaughan R. Pratt (pratt@Sunburn.Stanford.EDU) wrote:
>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:
>>>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.

That's the verse I had in mind.

A similar verse is Proverbs 8:27. The King James says "When he prepared
the heavens, I was there: when he set a compass on the face of the deep."
The RSV translates the original Hebrew more accurately as "...when he drew
a circle on the face of the deep".

Of course, all the Biblical writers were simply following the `cosmology'
of the time, which pictured the Earth as a circular surface, covered by a
tent-like sky to which the sun, planets & stars were attached. Various
other verses speak of `stars falling to Earth' & similar occurrences.

Anyway, this thread has strayed away from pi & no longer belongs in
sci.math 
--
 +-------------------  pete@bignode.equinox.gen.nz  -------------------+
 | The effort to understand the universe is one of the very few things |
 | that lifts human life above the level of farce, and gives it some   |  
 | of the grace of tragedy   -  Steven Weinberg                        |
 +---------------------------------------------------------------------+
Newsgroups: sci.math
From: goldstrn@math.fu-berlin.de (Martin Goldstern)
Subject: Re: Alephs again (two answers)
Summary: Injex does not follow from ZFC and does not imply GCH
Organization: Free University of Berlin, Germany
Date: Thu, 21 Jan 1993 20:39:50 GMT


Daniel Asimov defined 

(Injex) Let A and B be any two cardinal numbers.  
        Then 2^A = 2^B implies that A = B.

and asked if Injex was provable from ZFC or equivalent to GCH. 
The answer to both questions is no.

Injex does not imply the GCH, as for example 

        2^{aleph_n} = aleph_{n+2} for all n < omega
        (and GCH otherwise) 

is known to be consistent with ZFC (by Easton's theorem).  

Injex does not follow from ZFC, as for example

        2^{aleph_0} = 2^{aleph_1} = aleph_2  

is consistent with ZFC (already by Cohen's result). 

These theorems were proved in the 60s. 
They both only assume the consistency of ZF. 

I recommend the books of Kunen and Jech for proofs of these and
related results. 

Martin Goldstern
goldstrn@math.fu-berlin.de
From: matmcinn@nuscc.nus.sg (brett mcinnes)
Subject: Compact quotients of symmetric spaces
Organization: National University of Singapore
X-Newsreader: Tin 1.1 PL4
Date: Thu, 28 Jan 1993 04:45:01 GMT


A beautiful result due to Borel states that given any non-compact
symmetric space, one can always find a discrete torsion-free subgroup of
the identity component of the isometry group such that the quotient is
compact. So for example if you take the Grassmannian GR5,2 =
SO[2,3]/[SO(2)xSO(3)] then there exists a subgroup W of SO[2,3] such that
W\GR5,2 is a compact Kahler-Einstein manifold with negative Ricci curvature.
Question: How much is known about W in general [ie apart from hyperbolic
spaces]? Suppose I want to find an antiholomorphic involution with no
fixed points on W\GR5,2. How do I do it without knowing W? Any advice
would be gratefully received.
From: jeroenr@wsinfo01.win.tue.nl (Jeroen *vi* Rutten)
Newsgroups: sci.math
Subject: Reed-Muller codes: help wanted
Keywords: Coding theory
Date: 22 Jan 93 12:40:34 GMT
Reply-To: jeroenr@win.tue.nl

Let V_m be the m-dimensional vector space over GF(2), so V_m contains
all then (0,1)-vectors of length m. Let n:=2^m and let u_0,...,u_{n-1}
be all elements of V_m in lexicographical order, u_i=(x_1,...,x_m),
i=0,...,n-1.

Then the r-th order Reed-Muller code of length m is defined by

   RM(r,m) = {(f(u_0),f(u_1),...,f(u_{n-1}))|degree(f)<=r}

where f is a polynomial in x_1,...,x_m.

The automorphism group Aut(RM(r,m)) of RM(r,m) contains the group of
invertible affine transformations of V_m, denoted by GA(m,2).
For r<m-1 we even have: Aut(RM(r,m))=GA(m,2).

For the Reed-Muller code RM(3,6) I would like to know the orbits under
GA(m,2). More in particular I would like to know the cardinality of the
orbits, and a code word in each orbit.

Any help or reference is much appreciated.
Thanks in advance,

J. Rutten
Eindhoven University of Technology
The Netherlands
Newsgroups: sci.math
From: psm@mullian.ee.mu.OZ.AU (Phil Malin)
Subject: Need references on tensors and non-Riemannian spaces...
Organization: Computer Science, University of Melbourne, Australia
Date: Thu, 28 Jan 1993 04:43:15 GMT

Hi all.

The subject says it all.  Any help would be greatly appreciated.

Thanks in advance.

Phil Malin
psm@murlibobo.ee.mu.oz.au
Newsgroups: sci.math
From: don@civet.phys.oxy.edu (Don Goldberg)
Subject: Re: Circumference of ellipse
Organization: Occidental College, Los Angeles, CA.  USA
Date: Thu, 28 Jan 1993 22:03:27 GMT

In article <1993Jan28.073503.22736@dde.dk> ct@dde.dk (Claus Tondering)  
writes:
> 
> I have searched in vain in my collection of mathematical formulae
> for a formula for the circomference of an ellipse.
> 
> Can anyone help me? Given the two axes of an ellipse, what is its
> circumference?
> 
> -- 
Using the standard integral for an arc length, and some algebra, the  
value is expressed as "an elliptic integrl of the second kind."  This is  
an integral which is not expressible as an elementary function.

I don't actually have a citation for the last asertion.  Can anyone  
supply it?

--
-------------------------------------------
Don Goldberg
Mathematics Dept
Occidental College
Los Angeles CA 90041
Internet (including NeXT mail): don@oxy.edu
Newsgroups: sci.math
From: harshec@mail.auburn.edu (Eric C. Harshbarger)
Subject: Sorry 
Reply-To: harshec@mail.auburn.edu
Organization: Auburn University
Date: Thu, 21 Jan 1993 21:44:06 GMT

Guys and gals,
I just realized that in my recent post I forgot
to hard return my text, so there may be some
trouble reading the message. I'm sorry if it is a 
mess... I'm still pretty new at this whole newsgroup/
posting game.

E.C. Harshbarger

Newsgroups: sci.math
From: Harry.Cohen@trincoll.edu ()
Subject: Fields medal questions
Followup-To: sci.math
Organization: Trinity College, Hartford, CT.
Date: Tue, 26 Jan 1993 21:39:54 GMT

			I'm a senior doing a little research on fields medals in order to
incorporate into a paper.  However our library has but a page on who won
the last one and that's all.  If you could e-mail me the following.
Where is it given?
Maximum age you can be to get it?
How often it is given?
Max./Min. number of recipients in any year?

Thanks in advance

Harry

------------------------------------------------------------------------
It's a dog-eat-dog world and I'm wearing milkbone shorts.
Women, can't live with them....pass the beer nuts.
                                   Norm Peterson, Cheers
------------------------------------------------------------------------
Newsgroups: sci.math
From: louray@seas.gwu.edu (Michael Panayiotakis)
Subject: Fxy not equal to Fyx
Organization: George Washington University
Date: Thu, 21 Jan 1993 13:36:05 GMT



hey now.

I'm looking for an example where the second partial or a function
F(x,y), taking the partial with respect to x first then with respect to
y, is *not* the same as the partial with respect to y first, then x.

i.e., where d is partial:

  dF          dF
------ =/=  -------
dx dy        dy dx

(where dF : partial F; dx: partial x, etc.
       =/=: NOT equal)

Can anyone please provide me with an example?
Also, the conditions under which Fxy =/= Fyx ?
Please e'mail, since I don't always read the news.

Thanks in advance.
Michael

-- 
pe | ep     MICHAEL PANAYIOTAKIS: louray@seas.gwu.edu
ac | ca     
e / \ e     "I don't believe you, you have the whole damn thing all wrong:
 /   \        He's not the kind you have to wind up on Sundays" (J TULL)
Newsgroups: sci.math
From: kimball@stsci.edu (Timothy Kimball)
Subject: Re: Stationary points on jello
Organization: Space Telescope Science Institute
X-Newsreader: TIN [version 1.1 PL6]
Date: Mon, 25 Jan 1993 15:01:05 GMT

Ramin Zabih (rdz@sail.stanford.edu) wrote:
: Suppose that you are given some surface that deforms totally
: arbitrarily.  Is there any sensible definition for what it would mean
: for a point to stay in the "same place" as the surface deforms?
: --
: 
It seems to my tiny, physics-trained mind that for something to 
have a "place" means it has coordinates.  Where you put the coordinates
would be all-important.  If you attach it to the jello and let it
deform withe the jello, then the points could, I suppose be thought
of as always being in the same place.  If, however, you fix the 
coordinates to the surrounding laboratory, then the points could
move with respect to it.

I know this must be mathematically naive, but it's a shot.
A more mathematical point of view would be welcome.

--

/* tdk -- Opinions are mine, not my employer's. */
/* except maybe for the part about "tiny mind..." */
From: pursell@cheshire.oxy.edu (William Robert Pursell)
Newsgroups: sci.math
Subject: Re: why is pi irrational
Date: 26 Jan 93 23:27:22 GMT
Organization: Occidental College, Los Angeles, CA 90041 USA.

In article <fX#wH9p@quack.sac.ca.us> dfox@quack.sac.ca.us (David Fox) writes:
>In article <7017@tuegate.tue.nl> wsadjw@urc.tue.nl writes:
>
>
>The Israelites no doubt drew circles from time to time, but their holy
>bible says pi is equal to 3.
>

Is there a reference for this?  (pi = 3 in hebrew scripture)

-Bill Pursell
pursell@cheshire.oxy.edu
From: solan@smaug.uio.no (Svein Olav G. Nyberg)
Newsgroups: sci.math
Subject: Re: Non-Standard Analysis and philosophy
Date: 21 Jan 93 20:30:27 GMT
Reply-To: solan@smaug.uio.no (Svein Olav G. Nyberg)
Organization: University of Oslo, Norway

In article <TORKEL.93Jan21204806@bast.sics.se>, torkel@sics.se (Torkel
Franzen) writes:
|> In article <1993Jan21.180359.21766@ulrik.uio.no> solan@smaug.uio.no 
|> (Svein Olav G. Nyberg) writes:
|> 
|>    >Under which philosophies of mathematics is non-standard
|>    >analysis possible, and to what extent?
|> 
|>   Your question makes no apparent sense. Non-standard analysis is simply a
|> field of mathematics that exists, hence is possible.

I am serious, Torkel. Surely finitist mathematicians will not
allow for infinitely large integers. And rumors have it that
Martin-Lof's attempts to fit non-standard analysis into the
Constructive Mathematics programme (mathematics as possible
under a certain philosophy of mathematics) yields a weaker sort
of non-standard analysis than the standard non-standard analysis.

My question stands.


Solan
Newsgroups: sci.math
From: markb@iplmail.orl.mmc.com (Mark Bower)
Subject: Mackey-Glass solutions
Organization: Martin Marietta
Date: Mon, 25 Jan 1993 17:17:32 GMT

I am looking for papers describing solution techniques and giving
a closed form solution of the Mackey-Glass equation:

dx                 x(t - T)
--  = -b x(t) + a ----------   10
dt                1 + x(t - T)^


Thanks in advance for any information!
From: loeb@greco-prog.fr (Daniel LOEB)
Newsgroups: sci.math
Subject: Re: Which CAS have BA's in math?
Date: 26 Jan 93 09:47:14 GMT
Reply-To: loeb@geocub.UUCP (Daniel LOEB)
Organization: Greco Programmation du CNRS

In article <106190@netnews.upenn.edu> weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>In article <1993Jan19.205434.10234@leland.Stanford.EDU>, ilan@leland (ilan vardi) writes:
>>They write papers though. Shalosh B. Ekhad, Doron's Zeilberger's computer,
>>has written a number of papers (more than I have, I think).
>What is the story about SBE?  The papers I've seen seem to be somewhat
>demented in enthusiasm, a feeling reinforced by the phoney name, which
>gives the impression (being Hebrew for 3-in-1) that the author himself
>is a demented with enthusiasm member of the Jews for Jesus.

Actuaully, the "B." doesn't stand for the Hebrew prefix "beh-" but
rather the English letter "B". It seems the model number of Doron's
computer "3B1".

Doron while terribly enthusiastic about everything he does is not a
member of Jews for Jesus (to my knowlege). Actually, Doron is quite
learned about Judaism, and has often made reference to "Gematriyas"
(Cabbalistic homomorphisms between words and numbers) and to the
discovery of the factorial in the Sefer Yetsira (Ch IV, v 12).
(See for example his "Proof of the Alternating Sign Matrix
Conjecture"---inspired in part by 3B1, or Foata and his "Denert's
statistic is Euler-Mahonian".)

-- 

Yours, Daniel Loeb				loeb@geocub.greco-prog.fr
HOME 	150, cours Victor-Hugo; Appt D45; 33000 Bordeaux France
WORK	LABRI; Universite de Bordeaux I; 33405 Talence Cedex France
From: sitarama@cps.msu.edu (Kocherlakota Sitarama)
Newsgroups: sci.math
Subject: sci.math.* groups
Summary: sci.math.groups
Date: 24 Jan 93 07:02:45 GMT
Organization: Dept. of Computer Science, Michigan State University
NNTP-Posting-Host: indian.cps.msu.edu
Originator: sitarama@indian.cps.msu.edu

Okay, after getting lot of thoughts on sci.math.*, I think, I got
a good solution. Let us take the top, say 6, math journals and give a 
news group which coveres more or less what the journal covers.

In addition, we can still keep sci.math and we can request the 
posters to post a copy in sci.math, so that those who want all at one
place can read from sci.math and people interested in the 
special group can just read the special group?

Thanks!

PS: Posting a message on more than one newsgroup will not cause 
	an additional bandwidth
From: hutchins@acf3.nyu.edu (Carol Hutchins)
Newsgroups: sci.math
Subject: Re: sci.math.* groups
Date: 24 Jan 93 18:04:33 GMT
Organization: New York University

sitarama@cps.msu.edu (Kocherlakota Sitarama) writes:

>Okay, after getting lot of thoughts on sci.math.*, I think, I got
>a good solution. Let us take the top, say 6, math journals and give a 
>news group which coveres more or less what the journal covers.

 	Please let me know once you come to a consensus
on which 6 are the "top math journals".

			Thanks.
			Carol Hutchins
			Library
			Courant Institute
--
       --------------------------------------------------------------
	212-988-3314				Carol Hutchins
	hutchins@acfcluster.nyu.edu		Library
	hutchins@nyuacf.bitnet	                NYU Courant Institute
From: yeomans@austin.onu.edu (Charles Yeomans)
Newsgroups: sci.math
Subject: Re: Smooth manifolds and function extensions
Date: 26 Jan 93 16:14:03 GMT
Article-I.D.: austin.1993Jan26.161403.29059
Organization: Ohio Northern University

In article <2828@eagle.ukc.ac.uk>, mrw@ukc.ac.uk (M.R.Watkins) writes:
> 
> Given a manifold M with a smooth (C-infinity) structure, it is possible to
> define E(M), the linear space of all smooth functions on M.  Now is it
> possible to reconstruct the smooth structure (that is the atlas on M) from
> the knowledge of E(M) alone?
> 
I believe the answer is no, in general.  Consider , in C^2, the two sets
X = unit ball z^2 + w^2 < 1 and Y = X - (0,0).  Let H(X), H(Y) be the spaces of
holomorphic functions on X, Y.  Hartogs' theorem implies that any function
holomorphic on Y extends (uniquely) to a holomorphic function on X.  Thus H(X)
is isomorphic to H(Y).  But X is very diferent from Y.

But for a manifold with a C-infinity structure, what you ask might be possible.
However, I think you need more structure on X, say that of a ring or algebra.
With this, you ought to be able to do the following:  Suppose E(M) has the 
structure of a commutative R-algebra.  Let I be a maximal ideal in E(M). What
you'd like is for I to have the form {f in E(M) | f(p) = 0}, for some p
in M. This would be the hard part.  Given such a fact, you could reconstruct
M using the set of all maximal ideals.

I seem to recall that there are such theorems in Banach algebra.

Charles Yeomans
From: cs161fir@sdcc10.ucsd.edu (Anthony Minkoff)
Newsgroups: sci.math
Subject: Re: "Cut & Choose" for several players
Date: 23 Jan 93 17:57:42 GMT
Organization: University of California, San Diego

In article <1993Jan21.101533.19001@acis.comlab.ox.ac.uk>,
akay@comlab.ox.ac.uk (Andrew Kay) writes:
>Anthony Minkoff writes:
>>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.

Ah.  Pardon me for not being too clear on that one.  Each player
measures the value of each portion by his own utility function.
So, when I say "player A feels that player B got a better portion,"
I mean "player A would rather have player B's portion than his own
portion."


It is trivial to show that with the other interpretation, there may not
be an envy-free division.  E.g, suppose UA is A's utility function,
UB is B's u.f., 0 is the "null portion," and P is the original pile.  If
UB(0) > UA(P)-- unusual, but conceivable-- then it is clear that no
envy-free division exists.

Even if you impose further constraints on the utility functions so
that an envy-free division must exist-- e.g., U(0) is a constant
across all players-- I doubt there could be an envy-free division
algorithm, because there is no way to test for whether UA(x) > UB(y).
It is, however, possible to test whether UA(x) > UA(y), by forcing
A to chose between portion x and portion y.

Tony
-- 
Tony Minkoff                            aminkoff@sdcc13.ucsd.edu
cs161fir@sdcc10.ucsd.edu                cs163wav@sdcc8.ucsd.edu
"This quote is in this .sig for no particular reason."
From: labute@galois.math.mcgill.ca (John LABUTE)
Newsgroups: sci.math
Subject: Re: Can Anyone Solve this?????
Date: 26 Jan 93 17:43:33 GMT
Organization: McGill University


The most elementary way to solve the differential equation y''-2y+y=0 is
to introduce the differential operator D = d/dx. The equation then 
becomes (D^2 - 2D + I)(y) = 0 where I is the identity operator. Factoring
the polynomial in D the equation becomes (D - I)^2(y) = 0 which is
equivalent to the equation D^2(e^{-x}y) = 0 from which follows
e^{-x}y = a+bx and hence y = ae_x + bxe^x. These techniques only require
linear algebra (algebra of linear operators) and elementary calculus and
can easily be generalised to solve any constant coefficient n-th order
linear differential equation.

-- 
John Labute                                   Phone: (514) 393-3819
Mathematics Department                          FAX: (514) 398-3899
McGill University                            e-mail: labute@math.mcgill.ca
From: goddard@NeXTwork.Rose-Hulman.Edu (Bart E. Goddard)
Newsgroups: sci.math
Subject: Re: pyramid volume
Date: 21 Jan 1993 18:12:57 GMT
Organization: Computer Science Department at Rose-Hulman
Reply-To: goddard@NeXTwork.Rose-Hulman.Edu (Bart E. Goddard)
NNTP-Posting-Host: g214-1.nextwork.rose-hulman.edu

In article <1993Jan21.173423.11339@sophia.smith.edu> orourke@sophia.smith.edu  
(Joseph O'Rourke) writes:
> In article <1993Jan21.140402.25519@mr.med.ge.com> carl@crazyman.med.ge.com  
(Carl Crawford) writes:
> >
> >how do show that the volume of a pyramid is 
> >
> >	1/3 * area of base * altitude 
> >	
> >without using calculus?
> 
> Nice question!  Although I don't have an answer, I would like to
> make an observation:  three identical tetrahedra pack half a cube.
Along the same lines, 6 identical (4-sided) pyramids pack a cube: each face of  
the cube is a base of a pyramid with top at the center of the cube.

Thus, for this pyramid, V = Bh/3.   Now, are affine transformations allowed?
--
Bart Goddard                            |Behold, thy King cometh unto thee,
Math. Dept. Rose-Hulman Inst. of Tech.  |meek, and sitting upon an ass, and
goddard@nextwork.rose-hulman.edu        |a colt, the foal of an ass.
From: ferenbaugh-charles@math.yale.edu (Charles Ferenbaugh)
Newsgroups: sci.math
Subject: Sorry (was Re: THE DIVINE MASTERS OF THE UNIVERSE)
Date: 26 Jan 93 14:59:40 GMT
Organization: Yale University Math Dept.
NNTP-Posting-Host: cauchy.math.yale.edu

In article <1k3i81INNqor@CAUCHY.MATH.YALE.EDU> ferenbaugh-charles@math.yale.edu (Charles Ferenbaugh) writes:
> [long article deleted]

I apologize for reprinting the long article.  I hit a wrong key in the
process of responding, and then the machine wouldn't let me cancel
the posting.

- Charles Ferenbaugh
  crf@math.yale.edu
Newsgroups: sci.math