HDTenEightyP/NetNews huggingface dataset:
From: gedgar@magnus.acs.ohio-state.edu (Gerald A Edgar)
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Organization: The Ohio State University, Dept. of Math.
Date: Wed, 16 Dec 1992 13:09:23 GMT

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) ??


-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
From: jrickard@eoe.co.uk (John Rickard)
Newsgroups: sci.math
Subject: Re: A Problem
Date: 16 Dec 92 20:10:25 GMT
Organization: EO Europe Limited, Cambridge, UK
X-Newsreader: TIN [version 1.1 PL6]

Paul Hsieh (hsieh@math.toronto.edu) wrote:
: (Alexander Burshteyn) writes:
: >Here is a nice problem [...]
: >Prove that if, for some non-negative integers A and B, we have that
: >(A^2 + B^2) / ( 1 + AB)  is an integer, then it is a square of an integer.
...
: Actually, this appeared on an IMO question a few years ago.  Just for fun some
: of us at U(T) tried solving the question from scratch (without looking up the
: solution somewhere).  But we cheated by running a computer program for ages to
: find examples.
[Description of how proof was found omitted.]
: Now, I enjoy doing problems as much as the next guy, but the question I have,
: is:  How in the heck is someone supposed to see this under contest conditions?

Well, I could describe how I found it the other evening.  I'd glimpsed
Paul Hsieh's message, and though I tried to ignore it I'm sure it
helped me; nevertheless, I think it's plausible that somebody could
find it this way under contest conditions.

Omitting some approaches I tried that didn't lead anywhere, my
thoughts went roughly:

  We have A^2 + B^2 = C(1+AB).  Let's look at some solutions.  C=1 is
  uninteresting.  Assuming the problem is right, C must be a square,
  so look at C=4.  Then A^2+B^2=4(1+AB).  Taking this mod 4 shows A
  and B are both even; say A=2A', B=2B'.  Then A'^2 - 4A'B' + B'^2 =
  1.  This suggests A'/B' must be close to a root of x^2 - 4x + 1 = 0;
  the larger root of this (let's assume A >= B) is (calculate...)
  2+sqrt(3), which is about 3.732.  We need A'/B' slightly bigger than
  this; the continued fraction approximations are most likely; it's
  obvious that some good approximations are 4/1 and 15/4.
  (Calculate...) Indeed, (8,2) and (30,8) are solutions. So how does
  the continued fraction go?

      2+sqrt(3) = 3 + 1/((1+sqrt(3))/2)
                = 3 + 1/(1 + 1/(1+sqrt(3)))

  so if a/b is one approximation then the next but one (we need
  overestimates so we're only in alternate approximations) is
  3+1/(1+1/(a/b - 1)), which is (4a-b)/a.  Putting a/b = 30/8 gives
  112/30, and (calculate...) (112,30) is indeed a solution; it's
  looking very good.  Now try to prove that all solutions (with C=4
  still) are like this (I'm familiar with similar arguments that show
  that (eg) all solutions of 2x^2 = y^2 + 1 come from the continued
  fraction, which makes the argument easier for me to find); inverting
  the map a/b |-> (4a-b)/a gives a/b |-> b/(4b-a), so we want to check
  that if (a,b) is a solution then (b,4b-a) is also one, which should
  be smaller.  Make a guess at the generalization that for general C, 
  if (a,b) is a solution then so is (b,Cb-a) - (calculate...) and it's
  true!  Since C must obviously be a square if A or B is zero, this
  gives a proof of the theorem, if I write it all down neatly and hide
  all traces of how I found it.  (There are some minor details left,
  such as showing that (b,Cb-a) is indeed smaller in some sense than
  (a,b).)

-- John Rickard
From: rickert@NeXTwork.Rose-Hulman.Edu (John H. Rickert)
Newsgroups: sci.math
Subject: diophantine request
Date: 14 Dec 1992 18:32:33 GMT
Organization: Computer Science Department at Rose-Hulman
Reply-To: rickert@NeXTwork.Rose-Hulman.Edu (John H. Rickert)
NNTP-Posting-Host: g215a-1.nextwork.rose-hulman.edu

Where could I find a good method for determining all solutions 
to a diophantine equation such as  19A^3-18B^3=1?
(I need to solve several of these for a paper I'm working on.)
Is there an efficient program or expert out there somewhere?

e-mail to
rickert@nextwork.rose-hulman.edu
Newsgroups: sci.math
From: norm@netcom.com (Norman Hardy)
Subject: Re: Philosophy of Pi
Organization: Netcom - Online Communication Services  (408 241-9760 guest) 
Date: Sat, 12 Dec 1992 02:07:52 GMT

In article <1992Dec11.200538.928@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <COLUMBUS.92Dec11123510@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:
>>In article <1992Dec11.155846.10861@ee.ubc.ca> bdow@ee.ubc.ca (DOW BRUCE
>>PAUL) writes: 
>>
>>   I'm looking for some opinions on Pi. Lately I have been thinking about it
>>   a lot. Does anybody know why Pi equals what it does? Why is it an irrational
>>   number? Why is it between 3 and 4 in magnitude? Does it stem from the circle
>>   alone? What is it about the circle that gives Pi its value? I know Pi can
>>   be derived without using the circle (differential equations, infinite series).
>>   What determines the value of Pi? Does it have something to do with time and
>>   space in a three-dimensional universe? Would Pi have a different value in
>>   another universe? Or is it a constant in any universe obeying the laws of 
>>   Euclidean geometry? [...]
>>
>>Boy, are you asking for it in this newsgroup!  But I won't flame; here are
>>three answers.
>>
>>1.  Why does Pi equal what it does?  Well, what do you mean by Pi?  The use
>>    of (quick, count on my fingers!) the 14th letter of the Greek alphabet
>>    to represent the ratio of circumference of the circle to its diameter
>>    is due to Jones, a contemporary of Newton.  (You wouldn't ask, "Why is
>>    a robin a bird?", would you?  Maybe, "Why do English speaking people
>>    call that feathered thing over there a `robin'?"--- but that's
>>    for a different newsgroup.)
>>
>>2.  Pi was first proved irrational by Lambert, I forget when. [...]
>>
>>3.  Would Pi have a different value in a different universe? [...]
>
>I think the last formulation does the most justice to the spirit of
>this question.  So let's imagine a universe where circles are square.
>Then pi used to compute the perimeter would vary between 2.828 (2
>sqrt(2)) and 4 depending on where you measured the diameter, and if
>used to compute the area would vary between 2 and 4.  The arithmetic
>mean of these four quantities is 3.207107 while their geometric mean is
>3.084422.  The arithmetic and geometric means of *those* two quantities
>is in turn 3.145764 and 3.145166, and after one more iteration they
>have converged to 3.145465, within .1% of pi for round circles.
>
>So square wheels should work very well on average.  :-)
>
>Any volunteers to compute pi for octagons or 16-agons?
>-- 
>Vaughan Pratt				All knowledge resides in the going odds

In a 2-D Banach space one can measure the circumference of the 
unit circle (M-body). If the M-body of the space is a hexagon 
then the circumference is 6. If the body is a square then the 
circumference is 8. I think that pi is always between 3 and 4 
inclusively in Banach spaces. There may be no natural 
definition for pi in Banach spaces of more than two dimensions.
Xref: sparky sci.math:16944 misc.education:5277
From: swann@divsun.unige.ch (SWANN Philip)
Newsgroups: sci.math,misc.education
Subject: Re: The Continuum Hypothesis:  Must it be {True or False}, or Not?
Date: 15 Dec 92 09:33:25 GMT
Distribution: usa
Organization: University of Geneva, Switzerland

In article <ARA.92Dec15002026@camelot.ai.mit.edu>, ara@zurich.ai.mit.edu (Allan Adler) writes:
> 
> If there were really such a thing as a set, it might make sense
> to ask whether the continuum hypothesis is tru or false.
> 
> Starting with the early paradoxes and continuing with the independence
> proofs, we have learned that despite our gut feelings to the contrary,
> there really is no such concept as "set". We have some properties of
> sets that we agree on and beyond that we lack the power to make the
> concept more precise, although each individual might have certain
> opinions one way or the other as to which are the common notions.
> 

This is a good point, but not quite correct. The precision of a concept
depends on the language game in which it is used - there are no absolute
concepts, only relative ones - so on the same day you could find a perfectly
precise use of "set" and a frustratingly imprecise usage. Or to put it another
way, a concept is simply a pattern of usage (language + behavior). We can never
have an independent measure of the precision of our concepts - because that would
imply a concept of measurement that was independent (the concept of all concepts...)

Philip Swann
From: torbenm@diku.dk (Torben AEgidius Mogensen)
Newsgroups: sci.math
Subject: Re: Tiling problem
Date: 18 Dec 92 11:12:10 GMT
Organization: Department of Computer Science, U of Copenhagen

cet1@cus.cam.ac.uk (C.E. Thompson) writes:

>It occured to me that one could put more constraints on the problem by using
>periodic boundary conditions. This brings the probabilisticly suggested largest
>square down from about 17 to about 11, and so it might be more ammenable to
>a brute-force-and-ignorance computational attack.
>
>This version could be stated as: colour the tiles of the infinite plane square
>lattice black and white, periodically with period N in both directions, such
>that the only (orthogonally oriented) squares with all four corners the same
>colour have both sides divisible by N.

Well, since someone already posted a 13x13 solution, an estimate of 11
is a bit on the low side. But the suggestion sounds interesting.

	Torben Mogensen (torbenm@diku.dk)
Newsgroups: sci.math,sci.physics,sci.bio,sci.chem,misc.education
From: des@helix.nih.gov (David E. Scheim)
Subject: Re: TIME HAS INERTIA - BASS JEALOUS AND INSECURE-ABIAN VALIDATED
Organization: NIH
Date: Mon, 14 Dec 1992 16:32:11 GMT

Whether or not this thread belongs on this particular forum, it is 
frustrating to see all these comments and diatribes without the original 
theory.  I would appreciate a reference to some summary of the original 
hypotheses.

I think some people misunderstand the development of scientific ideas.  
Often a pioneering new theory derives from a succession of hypotheses which 
initially may be wrong but may spark new thoughts that have some validity.  
If a person puts forth an idea with a humble spirit it is almost always 
worthwhile -- it can at least spark new and valuable ideas on the part of 
those who consider and disprove it.  On the other hand, someone who puts 
forth a new idea proclaiming that this is the final solution to the workings 
of the universe is due for criticism.

The notion that a new idea should be ridiculed if it has flaws or doesn't 
jibe with scientific knowledge in 1992 is totally at odds with the 
enlightened scientific method that historically has yielded advances in 
knowledge.  I would go so far to say that a scientist whose ideas are not 
flatly contradicted at least 10% of the time is probably just piddling with 
perterbations of established results.  Knowing nothing of the ideas 
in this particular thread, these are general comments only.

/*********************************************************************/
/*                      --- David E. Scheim ---                      */
/* BITNET: DES@NIHNEI31                                              */
/* INTERNET: desl@helix.nih.gov          PHONE: 301 496-2194         */
/* CompuServe: 73750,3305                  FAX: 301 402-1065         */
/*                                                                   */
/* DISCLAIMER: These comments are offered to share knowledge based   */
/*   upon my personal views.  They do not represent the positions    */
/*   of my employer.                                                 */
/*********************************************************************/
Xref: sparky sci.math:16957 rec.games.abstract:625
From: torbenm@diku.dk (Torben AEgidius Mogensen)
Newsgroups: sci.math,rec.games.abstract
Subject: Re: Game of pentominos
Date: 15 Dec 92 15:47:34 GMT
Organization: Department of Computer Science, U of Copenhagen

martel@marvin.mr.sintef.no (Paulo Martel) writes:

>After several tries I gave up a combinatorial analysis of the game of
>pentominos. Would someone point me to a reference, or briefly explain
>how one could compute the total number of solutions for a grid of a
>given size (6x10, 5x12, 4x15, 3x20). 

I saw a paper once that reported the number of solutions to each of
these rectangle sizes. It used a heavily optimized machine code
program to exhaustively search for all solutions. I remember that for
the 3x20 case there are only two solutions barring reflections and
rotations. These are quite easy to find by hand. The number of
solutions for the 6x10 case was quite large, but I don't recall the
number. I also don't recall the title or author of the paper.

	Torben Mogensen (torbenm@diku.dk)
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Philosophy of Pi
Organization: Computer Science Department,  Stanford University.
Date: Mon, 14 Dec 1992 16:05:52 GMT

In article <1992Dec14.144954.11447@sifon.cc.mcgill.ca> rags@triples.math.mcgill.ca (Robert A. G. Seely) writes:
>In article <1gi1bmINNsou@function.mps.ohio-state.edu> edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:
>>>YESSSSS!!!!!  exp(i*Pi) + 1 = 0
>>>is perhaps the most beautiful identity in mathematics,
>>
>>I disagree.  The beautiful identity is exp(ix) = cos x + i sin x.
>>Unlike the previous one, it is even useful...
>>-- 
>>  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
>>  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
>>  The Ohio State University      telephone: 614-292-0395 (Office)
>>  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
>
>
>But this is just a matter of preferring the general to the particular
>instance - many might prefer the instance as a beautiful representative of 
>the general equation, understanding that the instance also encompasses the 
>general by implication.

Ah, but Robert, would Giotto's audience have been satisfied with his
drawing just the 9 o'clock point of his famous freehand circle?

"Good general theory does not search for the maximum generality, but
for the right generality."  (Mac Lane, CTWM, bottom of p.103.)
Perhaps he should have said "maximum or minimum".
-- 
Vaughan Pratt				There's safety in certain numbers.
From: koc@rize.ECE.ORST.EDU (Cetin Kaya Koc)
Newsgroups: sci.math
Subject: Re: What is the number
Date: 17 Dec 92 19:46:55 GMT
Article-I.D.: flop.1gqlffINN3fq
Organization: College of Engineering, Oregon State University
NNTP-Posting-Host: rize.ece.orst.edu

>    Whose sum of the factorials of the digits is the number itself

Also 40585 = 4! + 0! + 5! + 8! + 5! = 24 + 1 + 120 + 40320 + 120
                                    = 40585
Newsgroups: sci.math
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Organization: U of Cambridge, England
Date: Wed, 16 Dec 1992 01:10:40 GMT

      First proof
      -----------

(1+1/n)^n = sum{r=0..n} of (n choose r)/n^r
          = sum{r=0..n} of n(n-1)...(n-r+1)/n^r.r!
          = sum{r=0..n} of 1(1-1/n)(1-2/n)...(1-(r-1)/n))/r!

On the one hand, this is less than 1/0!+1/1!+1/2!+...+1/n!.
On the other hand, the "r-term" is less than 1/r! by a factor of less
than (1-(r-1)/n)^r, which we may easily check (um, induction on r or
something) is at least 1-r(r-1)/n. So, writing m for the integer part
of n^(1/3), the sum is at least (1-1/m)(1/0!+...+1/m!).

Both of these expressions tend to e as n tends to infinity. So (1+1/n)^n
is sandwiched between two numbers, both of which are arbitrarily close to
e if n is large enough. So it tends to e,

QED.

      Second proof
      ------------

(Taking the definition log(x)=integral{1..x} of dt/t as fundamental,
so e is defined as the number whose logarithm is 1.)

We may show that log(ab)=log(a)+log(b) by splitting the integral into
the ranges 1..a and a..ab, and using substitution on the latter.
It follows that log(a^n)=n.log(a).
Now, log(1+1/n) is at least 1/n times n/(n+1) [width times least height]
and at most 1/n times 1 [width times most height]. So n times that lies
between n/(n+1) and 1, hence tends to 1. Since the inverse of the log
function is continuous (easy exercise) (1+1/n)^n tends to e,

QED.

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
From: anich@festival.ed.ac.uk (Sandy Nicholson)
Newsgroups: sci.math
Subject: Re: Difference between "show" and "prove"
Keywords: proof  demonstration
Date: 17 Dec 92 21:28:20 GMT
Organization: Edinburgh University (opinions expressed are my own)

In article <1992Dec16.195452.29610@u.washington.edu>, mcfarlan@corona.math.washington.edu (Thomas J. McFarlane) writes:

|> A distinction between demonstration and proof can be made as follows.
|> When one is working within an axiomatic system and derives formal
|> consequences of the axioms, this is called a demonstration.  It's
|> something a computer could verify (in principle).  On the other hand,
|> when we prove something we appeal to insight.  You might say proofs
|> are informal, while demonstrations are formal.  Of course, one may
|> add axioms and definitions to formalize an informal argument, in which
|> case the proof becomes a demonstration.

Surely this is the wrong way round. I would agree with other posters that in
general no distinction is made between `proof' and `demonstration.' However,
in the context of the theory of formal systems, one *demonstrates* (rigorously
we hope) both model-theoretic and proof-theoretic results, the latter
pertaining to the notion of formal *proof* within a framework of strict rules
of inference. The *demonstration* of results about formal systems is what
involves mathematical insight.
-- 
Sandy Nicholson (anich@festival.ed.ac.uk)
Department of Mathematics and Statistics, Edinburgh University, Scotland
Newsgroups: sci.math
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: Function Terminology
Organization: Purdue University Statistics Department
Date: Fri, 11 Dec 1992 22:24:00 GMT

In article <1992Dec11.203802.1770@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <1gaq3tINNg9q@uwm.edu> radcliff@csd4.csd.uwm.edu (David G Radcliffe) writes:
>>Suppose I have a function f: A --> B, and C is a subset of B which
>>contains the image set of f.  I define a function g: A --> C by
>>setting g(a) = f(a) for all a in A.  Usually, f and g can be considered 
>>as the same function, but sometimes the distinction is important.

>>Is there a standard term or notation for this?

>While I've not heard of one, one could borrow notation used to make the
>analogous distinction for the integer 2 viewed as the real 2. and write
>f:A->B as f. and f:A->C as just f without the point.

The function is exactly the same.  A function, in whatever foundational
system is used, is something which takes arguments in a domain and
operates on them.  The image set depends only on f and A.  Which
superset of the range is used does not affect the function.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Newsgroups: sci.math
From: cet1@cus.cam.ac.uk (C.E. Thompson)
Subject: Re: Tiling problem
Organization: U of Cambridge, England
Date: Fri, 18 Dec 1992 00:23:43 GMT

It occured to me that one could put more constraints on the problem by using
periodic boundary conditions. This brings the probabilisticly suggested largest
square down from about 17 to about 11, and so it might be more ammenable to
a brute-force-and-ignorance computational attack.

This version could be stated as: colour the tiles of the infinite plane square
lattice black and white, periodically with period N in both directions, such
that the only (orthogonally oriented) squares with all four corners the same
colour have both sides divisible by N.

Chris Thompson
JANET:    cet1@uk.ac.cam.phx
Internet: cet1@phx.cam.ac.uk
Newsgroups: sci.math
From: martin@lyra.cis.umassd.edu (Gary Martin)
Subject: Re: GRAVITY-NEUTRALIZING AIR/SPACECRAFT
Organization: University of Massachusetts Dartmouth
Date: Tue, 15 Dec 1992 03:52:33 GMT


Having completely lost "it", I sent away for the plans to the
   >
   >                       Gravy-NEUTRALIZING Air/Spacecraft
   >                         or ZERO/REDUCED-Gravy Chamber
   >
I tried it in the school cafeteria and was immediately arrested by
net.police and charged with wasting bandwidth.  They threatened to

   >  suspend me
   >  by a long rope from a tree branch (or hook in the ceiling, etc.).

but instead gave me a "Post-to-Usenet"-NEUTRALIZING Keyboard.

--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Martin@cis.umassd.edu
From: sjohnson@faulen.asd.sgi.com (Scott Johnson)
Newsgroups: sci.math
Subject: anyone here attend university in manchester england?
Date: 15 Dec 1992 00:59:39 GMT
Organization: Silicon Graphics, Inc.  Mountain View, CA
NNTP-Posting-Host: faulen.asd.sgi.com

i know this probably isn't the place for a personnel search, but i'm
trying to find a couple of guys that are math students and live in
manchester.  i'm assuming they also attend school there, but i'm not
even sure if that is true.

please reply if you are a math major at this school, as i'd like to
get ahold of them if possible.

with much thanks,

sj		"The more people I meet, the more I like my dog" - ts
Newsgroups: sci.math
From: kekamins@zeno30.math.UWaterloo.ca (kekaminsky)
Subject: Re: Diff Eq Books
Organization: University of Waterloo
Date: Sun, 13 Dec 1992 19:18:58 GMT

In article <85218@ut-emx.uucp> tlea@ccwf.cc.utexas.edu (Thomas C. Lea) writes:
>I just finshed my first Diff Eq course and realy liked it.  To bad I had
>such a horrible book, along with a teacher.  So I'm looking for a good
>first book on Diff Eq.  The Book we used was by Guterman & Nitecki.  Any
>suggestions would be greatly appreaciated.
>
>				Thanks

The one used here at Waterloo is Simmons, "Differential Equations".  Its rigourous
enough without being (excuse me Pure mathies on the net) pedantic, and includes
lots of excellent history, and applications.  Also includes a chapter
on Calc. of Variations, and Dynamical Systems.

Kirk