HDTenEightyP/NetNews huggingface dataset:
From: ramsay@unixg.ubc.ca (Keith Ramsay)
Subject: Re: Erdos problems
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Fri, 10 Apr 1992 22:54:37 GMT

In article <1992Apr10.144022.10926@husc3.harvard.edu>
fry@zariski.harvard.edu (David Fry) writes:
|There's a famous Erdos problem involving a combinatoric
|problem P for a given integer n. [What is it?]

In article <1992Apr10.201803.29355@sophia.smith.edu> orourke@sophia.smith.edu (Joseph O'Rourke) writes:
|	Determination of the exact value of R(n) = R(n,n), the nth
|Ramsey number?  I'm guessing.  R(n) is the smallest number of
|people such that it must be that either n mutually know one another 
|or n are mutual strangers.  R(3)=6 and R(4)=18, but I think R(5)
|is not known.

That's it. The version of the story I remember is that Erdos commented
if a demon were to threaten the human race with extinction unless we
found R(5), we should use all our computing power to find R(5), but if
the demon asked for R(6), we should devote our efforts to finding a
way to get rid of the demon. He said something like, "if we were smart
enough to find the sixth Ramsey number just by thinking about it, we'd
be so smart we wouldn't have to worry about demons".

I think Erdos is probably as famous for these little amusing stories
and personal quirks as he is for his profuse research. Probably he's
most famous for being a "migrant mathematician", constantly traveling,
without a permanent home.

One of the folk tales which originated with him is that the Supreme
Fascist (an Erdos-ism for God) has a book with all the best proofs in
it. If the SF is being especially benevolent, perhaps he will allow
one a brief glimpse at "the book". Euclid's proof that there are
infinitely many primes is in "the book".

As a youngster I once encountered Erdos. He took the time to talk math
a bit with me, leaving me with the puzzle of finding an uncountable
collection of almost-disjoint subsets of the integers (i.e., the
intersections of pairs of them are all finite).

Our correspondent form Harvard might be amused to know that evidently
Erdos hasn't visited Harvard (at least not in recent memory). When
asked why this might be so, one Harvard professor remarked that it was
a question of style: Erdos steers clear of mathematical questions
which require building up "machinery" first, preferring to deal with
combinatorics, "elementary" number theory and other "elementary"
subjects.
--
Keith Ramsay            Even if a proof from given axioms can, in
ramsay@unixg.ubc.ca     principle, be found mechanically, is it not
                        often found in ways not all that different
                        from finding a new axiom? -Georg Kreisel
Newsgroups: sci.math
From: cyeomans@ms.uky.edu (Charles Yeomans)
Subject: Re: Falling cats and flat SO(3)-bundles
Date: Mon, 6 Apr 1992 13:52:24 GMT
Organization: University Of Kentucky, Dept. of Math Sciences
X-Bytes: 

In article <1992Apr5.011855.28242@midway.uchicago.edu> kwhyte@math.uchicago.edu (Kevin Whyte) writes:
>
>    Some time ago there was a thread on the dynamics of falling
>cats.  Someone gave a reference to some papers which described the
>falling cat more or less by calculating the holonomy of a canonical
>SO(3)-bundle over the space of cat deformations.  They mentioned 
>imagining a nice talk beginning with " Let X be the space of shapes a
>cat can deform itself into, now we construct a natural SO(3)-bundle
>over this space, and compute the holonomy"  Now that I have some free
>time I have been asked to give a lecture to graduate students on anything
>I like, and that seems nice.  Could the person kindly e-mail me the
>references.  Thanks,
>
>Kevin
>kwhyte@math.uchicago.edu

I would be interested to know if this has been generalized to 
other animals.

Charles Yeomans
cyeomans@ms.uky.edu
yeomans@austin.onu.edu

Newsgroups: sci.math,sci.crypt
From: pm@katz.cc.univie.ac.at ()
Subject: Re: New Mersenne Prime
Organization: Vienna University
Date: Wed, 8 Apr 1992 08:39:37 GMT

Bob Silverman writes:

::    I estimate the calculation to take 112 hours. If the Cray people needed
::    19 hours for the single test then it seems my program is reasonably fast:
::    The cycle time of the RS/6000 is six times the cycle time of the Cray 2,
::    and the calculation takes roughly six times nineteen hours.

:May I ask what variant of FFT you are using for your multiplies???


I am using the FFTs from IBM's Engineering and Scientific Subroutine
Library (ESSL/6000). I understand that these were written - at least in
part - by Jim Cooley himself, and they are reputed to be some of the world's
best FFTs. I know that my program spends nearly half of the time not in the
FFTs but in three lines of code that do nothing but convert from integer to
floating point and vice versa. This could be substantially improved by
manual recoding of the conversion routines, but I have not found the time
to do so.

                          Peter Marksteiner
                          pm@katz.cc.univie.ac.at
                          Vienna University Computer Center


Newsgroups: sci.math
From: mab@wdl39.wdl.loral.com (Mark A Biggar)
Subject: Re: An elementary, "naturally occurring" p
Organization: Loral Western Development Labs
Date: Tue, 7 Apr 1992 18:07:55 GMT

In article <3-cHf19k6@cs.psu.edu> sibley@math.psu.edu writes:
>In article 25777@linus.mitre.org, bs@gauss.mitre.org (Robert D. Silverman) writes:
>>In article <1992Apr1.183035.14916@Princeton.EDU> carabalo@phoenix.Princeton.EDU (David G. Caraballo) writes:
>>:Prove or disprove that there exists a bijection f:[0,1] -> [0,1] that is
>>:not continuous at any point. 
>>Doesn't the following function do it?
>>f(x) = 1-x  if x is irrational.
>>f(x) = x if x is rational.
>Nope. That function is continuous at x = 1/2.

How about

f(x) = x/3      if x is rational and 0 <= x < = .75
f(x) = 3*x-2    if x is rational and .75 <= x <= 1

f(x) = x        if x is algebraic irrational

f(x) = 1-x      if x is trancendential

Each of these mappings is piece-wise linear and preserves the rationalness,
algebric irrationalness or trancendentialness of the mapped value so you
can always tell which reverse mapping to use.  Each is also a bijection over
its domain.  Thus the whole function is a bijection.

All three class of numbers are dense, so any open ball around a given
point will contain members of all three sets.  The three linear mappings
are chosen so that whenever two of the mappings approach each other
the third is always off somewhere else. Thus it is continous nowhere.

--
Mark Biggar
mab@wdl1.wdl.loral.com
From: George.Harris@bbs.oit.unc.edu (George Harris)
Newsgroups: sci.math
Subject: Problem with Covering Space of a Regular Space.
Date: 17 Apr 92 15:42:24 GMT
Organization: Extended Bulletin Board Service


	This is a problem that was assigned in my Algebraic Topology class, 
but now that no one can quite get it, including the teacher, I thought it
would be kosher to ask for help here.

	Given a regular topological space X and a covering space X~ of X.
Let C be a compeact subset of X~.  Show that the set of automorphisms of X~
as a covering space for which the intersection of C with its image is non-
empty, is finite.

	Any help/pointers would be much appreciated.
--
"If you take cranberries and stew them like applesauce, they taste more like
prunes than rhubarb does."		-Groucho Marx.

George.Harris@bbs.oit.unc.edu	or, preferably	gharris@jade.tufts.edu
--
   The opinions expressed are not necessarily those of the University of
     North Carolina at Chapel Hill, the Campus Office for Information
        Technology, or the Experimental Bulletin Board Service.
           internet:  bbs.oit.unc.edu or 152.2.22.80
Newsgroups: sci.math
From: mab@wdl39.wdl.loral.com (Mark A Biggar)
Subject: Re: Tiling problem
Organization: Loral Western Development Labs
Date: Wed, 8 Apr 1992 22:40:56 GMT

In article <2992@cvbnetPrime.COM> dwilson@cvbnet.prime.com (David Wilson x5694 4-1600) writes:
>    Will the pentomino
>        **
>         **
>         *
>    tile the plane minus a quadrant?

Well you can tile the whole plain like so:


	aa
	baa
       bbacc
        bbdcc
	 ddcee
	  ddfee
	   ffegg
	    ffhgg
	     hhg
              hh

giving infinte diagonal strips which can be laid side by side.

another way go is like this:

  aabbccddeeff
   aabbccddeeff
   agbhcidjekfl
   gghhiijjkkll
    gghhiijjkkll


Now bring another string like that one at right angles like so:


  aabbccddeeffXXXXXXX
   aabbccddeeffXXXXXX
   agbhcidjekllXXXXXX
   gghhiijjkkllXXXXXX
    gghhiijjkkllXXXXX
                XXXXa
                Xgaaa
                gggab
                ghbbb
                hhhbc
                hiccc
                iiicd
                ijddd
                jjjde
		jkeee
                kkkef
                klfff
                lllf
                l

Now the problem turns into tiling the shape made of X's above.

	XXXXXXX
         XXXXXX
         XXXXXX
         XXXXXX
          XXXXX
          XXXX
          X

This contains 35 x's so 7 may tile it.  But, if we checker board it like so:

	XYXYXYX
         XYXYXY
         YXYXYX
         XYXYXY
          XYXYX
          YXYX
          X

There are 19 X'x and 16 Y's.  combined with the fact that the shape can 
have either 3 x's and 2 Y's or 2 X's and 3 Y's given us the following
over specified set of simultanious equations:

a + b = 7			a pieces of one parity and b of the other
3a + 2b = 19			the a pieces will have 3 x's and the b's 2
2a + 3b = 16			the b pieces will have 3 Y's and the a's 2

There is no solution to this set of equations, so the above shape can not
be tiled with the r pentanimo. 

Note though that if we place two of the L shaped infinte strips above together
(nested and oriented the same way) then to wierd X shapes join into a single
connected shape which may be tilable, but I have not tried it.  If two doesn't
work, then 3 may.

--
Mark Biggar
mab@wdl1.wdl.loral.com
From: somos@ces.cwru.edu (Michael Somos)
Newsgroups: sci.math
Subject: Re: Zero, "Arabic" numerals, etc.
Summary: what is zero?
Date: 8 Apr 92 22:15:29 GMT
Article-I.D.: usenet.1992Apr8.221529.29450
Organization: Computer Engineering and Science, Case Western Reserve University

Does anyone know of a really good source for the study of zero?  I know
that Martin Gardner wrote about "Nothing", but this didn't cover the
topic very well.  The concept of the number zero, the digit zero, and
nothing, are closely related and very subtle.  A glance at history will
demonstrate this.  Even today, I don't think we really understand all
the nuances.  This study would cover the history, the philosophy, and
the mathematics and science of zero.  A highly non-trivial undertaking.
--
Shalom, Michael Somos <somos@ces.cwru.edu>  (No, I don't work for CWRU)
From: bs@gauss.mitre.org (Robert D. Silverman)
Newsgroups: sci.math
Subject: Re: Probabilistic Arguments
Date: 16 Apr 92 16:49:41 GMT
Article-I.D.: linus.1992Apr16.164941.22695
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA

In article <1992Apr16.152831.20128@news.cso.uiuc.edu> chappell@symcom.math.uiuc.edu (Glenn Chappell) writes:
:Lately There have been many references on sci.math to probabilistic or
:heuristic arguments concerning statements whose truth/falsehood is
:unknown. I'm wondering if these are really worthwhile. After all, one of
	      ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Why wonder? Why not check in the library for references??
 
(don't take the next comment personally please. It is not specifically
 aimed at the poster).

It seems that sudents have either forgotten how to use a library, or are
too lazy to be bothered, even to satisfy their own questions.


:the basic ideas behind probability is the idea that the probability
:distribution is what you get as the number of trials approaches infinity.
:
:But in any particular formal system, any particular statement either has
:a proof or it doesn't - period. You get one "trial" and then you're done.
 
So? What's your point?
 
:It is quite conceivable for a probabilistic argument to suggest very
:strongly that a statement is true, while the statement is in fact false.

If done naiively, Yes. 

:
:On the other hand, such arguments certainly seem to be worthwhile, from
:an intuitive viewpoint.
:
:Therefore, I propose the following test of the worthiness of
:probabilistic arguments:
 
Additional sophistry deleted............
 
Your test is not needed.


I suggest that those of you who want to learn something
about probabilistic methods get hold of P.D.T. Eliott's 
"Probabilistic Number Theory".

It is a legitimate and rigorous field of study. 

After you've read it, and you still would like to debate its philosophical
implications I'll be glad to.

--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Newsgroups: sci.math
From: sibley@math.psu.edu (David Sibley)
Subject: Re: A simple question about R2 - Q2 (irrationa
Reply-To: sibley@math.psu.edu
Organization: Dept. of Mathematics, Penn State University
Date: Tue, 7 Apr 1992 20:24:05 GMT

In article 174510@masg1.epfl.ch, baumgart@masg1.epfl.ch (Marcel Baumgartner) writes:
>Keywords: In R2 - Q2, does there exist an oblique straight line ?
>


Yes. Let a and s be any reals. There are only countably many numbers of
the form sa-sq+r, where q and r are rational. Choose a number b not
among these. Then the line through (a,b) with slope s contains no
point whose coordinates are both rational, for an easy computation
shows this would be contrary to the way b was chosen.
(Presuming I did the high school algebra right...Something like this
will certainly work, even if I got it wrong.)


David Sibley
sibley@math.psu.edu
From: petry@corona.math.washington.edu (David Petry)
Newsgroups: sci.math
Subject: Re: Series involving primes
Date: 7 Apr 92 04:07:04 GMT
Article-I.D.: u.1992Apr7.040704.5147
Distribution: usa
Organization: University of Washington, Mathematics, Seattle

In article <1992Apr1.152202.15747@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
 
(here, d(n) = p(n+1)-p(n),  p(n) is the n'th prime)

>                 I posted a sketch of the derivation that sum(1 to N) d(i)/p(i)
>= C log N +  O(log N/exp(sqrt(log N))). In fact, one can show that the constant
>C is exp(gamma) ~= 1.78. 

Let t(n) = (p(n+1)-p(n)) / p(n),  then  p(n+1) = p(n) * (1 + t(n) ),

so p(n+1) = prod(1 to n) (1 + t(i)) * p(1).  Take the log of both sides.

log(p(n+1))  =  sum(1 to n) log(1+t(n))  + log(2)
             
             =  sum(1 to n) t(n)  -  (1/2) sum t(n)^2 + 1/3 sum t(n)^3 ...

A heuristic argument (which someone could probably prove rigorously) suggests 
that all the terms in the right hand side other than the first expression add
up to some constant in the limit of large n.

It looks to me like  sum(1 to n) d(i)/p(i) = log(n) + log(log(n)) + O(c) 



David Petry
From: Jason.Breitweg@bbs.oit.unc.edu (Jason Breitweg)
Newsgroups: sci.math
Subject: Interesting problem help!!
Date: 17 Apr 92 14:14:04 GMT
Organization: Extended Bulletin Board Service

Hello,

	A friend challenged me to do a problem.  Surprisingly enough
I am having problems with the problem.  If anyone could help me it
would be much appreciated.  Here goes:
Knowing that exp(i(a+b))=exp(ia)exp(ib) derive the relations for
 
sin(a+b) and cos(a+b).  Hope to hear from somebody soon.  Thanx.

				Sincerely,
					JasoN
                       
--
   The opinions expressed are not necessarily those of the University of
     North Carolina at Chapel Hill, the Campus Office for Information
        Technology, or the Experimental Bulletin Board Service.
           internet:  bbs.oit.unc.edu or 152.2.22.80
From: luzeaux@campanile.berkeley.edu (Dominique Luzeaux)
Newsgroups: sci.math
Subject: Re: the domain of a rational function
Keywords: discontinuity, L'Hospital
Date: 13 Apr 92 19:08:18 GMT
Article-I.D.: pasteur.1992Apr13.190818.26743
Reply-To: luzeaux@campanile.berkeley.edu (Dominique Luzeaux)
Organization: University of California


In article <1992Apr13.063539.13454@clinet.fi>, dix@clinet.fi (Risto
Kaivola) writes:
|>                ax + b
|> Let f:  f(x) = ------
|>                cx + d
|> 
|> The coefficients of this function satisfy c <> 0,  d <> 0,  and  ad = bc.
|> (...)
|> Now, the expression manipulation part is, of course, absolutely correct.
|> What I disagree with is that R \ {-d/c} is given as the domain of the
|> function f. (...)
|> So, I see no reason to say that the domain of f isn't the set of all real
|> numbers.  Comments?

The domain of f is R\ {-d/c}; what you noticed is that there is a continuous
extension of f on the set of all real numbers.
	
	Dominique Luzeaux
Newsgroups: sci.math
From: shag@aero.org (Robert M. Unverzagt)
Subject: Re: On knowing who's who (was: Fermat's Last Theorem)
Organization: The Aerospace Corporation, El Segundo, CA
Distribution: na
Date: Thu, 9 Apr 1992 19:53:37 GMT

In article <22604@dog.ee.lbl.gov> sichase@csa2.lbl.gov writes:
>
>	[much Stuff deleted]
>
> Conclusion:  Lederman is as luminary a figure as Erdos, if not moreso in his
> respective field.   Derek probably didn't know who he is either. :)
> But jeez, guys.  I lament the fact that more people don't know who Erdos, or
> Lederman, or Weinberg, etc., are.  But most people don't even know who
> Euler, Gauss, Maxwell, Faraday, Boltzmann, Bernoulli, Fermat, Laplace,
> Lagrange, and so forth... are.  
> 
> So I think that if someone asks, we who know are obliged to put on the best
> smile possible (no condescending allowed) and teach them.  Don't you want them
> to learn?
> 

Often on camping trips around the campfire we play a stupid game
called (appropriately) The Initial Game.  A person gives the two
initials of a (reasonably) famous person, and the game proceeds
like 20 questions until someone guesses the person.  Example:

Initals:	H.H.
Q1:  Is this person alive?	A1:  No.
Q2:  Was this person American?	A2:  Yes.

Until eventually someone guesses Harry Houdini and wins.  (Houdini
was American, right?)

Anyway, one guy always claims that the people we pick are not
famous -- that we are relying on inside jokes to get the
answers.  The above post reminded me of this, because in the past
we have used "unfamous" people like Euler, Maxwell, Francis
Drake, Herman Hesse, etc.

I freely admit that I don't know who Erdos, or Lederman, or
Weinberg, etc., are.  But I think that someone should know
who Francis Drake is!  Sheesh!  Maybe I cross post to alt.peeves.

Shag

-- 
                          |
      Rob Unverzagt       |     Hey you, Whitehouse,
 shag@aerospace.aero.org  |     Ha ha, charade you are.
                          |                             -- The Pink Floyd
Newsgroups: sci.math
Subject: Re: On knowing who's who (was: Fermat's Last Theorem)
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 9 Apr 92 16:53:03 GMT
Distribution: world
Organization: Beckman Instruments, Inc.

In <COLUMBUS.92Apr9091341@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:

>	<STROMME.92Apr6130830@jeeves.math.utah.edu> <3660@tuegate.tue.nl>
>	<1992Apr9.021905.20917@linus.mitre.org>
>NNTP-Posting-Host: strident.think.com
>In-reply-to: bs@gauss.mitre.org's message of Thu, 9 Apr 1992 02:19:05 GMT

>Our story so far:

>   Derek Abbott: "Who is this Erdos dude?"
>   Robert D. Silverman: "You're kidding, right? How could you not know one
>    	of the foremost mathematicians of this century?"  (approximate
>    	quotation)
>   Various posters: give the guy a break.
>   Other posters: give *us* a break.

>In an interview in Scientific American a few months back, then Chief of
>Staff John Sununu did not know (or professed not to know) who Leon Lederman
>is.  I leave it for the reader to decide which side of the debate (if
>either) this factlet supports.

OK, who IS Leon Lederman?

--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
216-5888@mcimail.com 70707.453@compuserve.com arthur@pnet01.cts.com (personal)
My opinions are my own, and do not represent those of my employer.
Ich bin ein Virus.  Mach' mit und kopiere mich in Deine .signature.
Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Fermat's Last Theorem
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Thu, 9 Apr 1992 14:19:33 GMT

In article <1992Apr7.102442.23606@ms.uky.edu> cyeomans@ms.uky.edu (Charles Yeomans) writes:
:In article <1992Apr7.051902.4889@rata.vuw.ac.nz> harper@kauri.vuw.ac.nz (John Harper) writes:
:I understand that Erdos offered $3000 for a solution to the following
:problem:  let {n_i} be a sequence of positive integers such that
:SUM(1/n_i) diverges.  Then the sequence contains arithmetic progressions
:of arbitrary length.  Try this one in your spare time.
 
A mathematician far more competent than myself told me that he had looked
at this problem and that there were much easier ways to earn $3K.
Working at McDonald's, for example :-)

This really appears to be an extraordinarily difficult problem. Uncle
Paul believes that the tools do not yet exist that can tackle it.

--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
From: dabbott@augean.eleceng.adelaide.edu.AU (Derek Abbott)
Newsgroups: sci.math
Subject: Re: Zero, "Arabic" numerals, etc.
Date: 9 Apr 92 04:46:34 GMT
Organization: Electrical and Electronic Eng., University of Adelaide

In article <1992Apr8.221529.29450@usenet.ins.cwru.edu> somos@ces.cwru.edu (Michael Somos) writes:
>Does anyone know of a really good source for the study of zero?  I know
>that Martin Gardner wrote about "Nothing", but this didn't cover the
>topic very well.  The concept of the number zero, the digit zero, and
>nothing, are closely related and very subtle.  A glance at history will
>demonstrate this.  Even today, I don't think we really understand all
>the nuances.  This study would cover the history, the philosophy, and
>the mathematics and science of zero.  A highly non-trivial undertaking.

Yes the science of zero is of great importance. The repercussions are
immeasurable. Maybe we ought to start an alt.nothing to do the full
justice that this esteemed area of research deserves.

Is there anyone who is prepared to donate vast sums of money to the
research of zero? If so I'll email you my bank account number.

<humour mode off>

<serious mode on>

There is something to nothing.

I have heard a story (maybe apocryphal) that there was/is an Indian
mathematitician who devoted all his life to studying zero and producing
complex proofs for his theorums about the nature of zero.

I don't know what the theorums were or who the guy was or if the
story is true or not. But if anyone has a reference on this Indian
dude, I would be interested to read and learn.
From: ruberman@binah.cc.brandeis.edu
Newsgroups: sci.math
Subject: Re: Knotty Views - references please
Date: 10 Apr 92 14:57:49 GMT
Reply-To: ruberman@binah.cc.brandeis.edu
Organization: Brandeis University

In article <1992Apr10.025235.11538@ryn.mro4.dec.com>, moore@aiag.enet.dec.com writes:
>
>
>In the March 21, 1992 issue of Science News (Vol. 141,No.12) on page 186 there
>was an article - "Knotty Views, Tying together different ways of looking at
>knots." The article was very interesting but short on references.
>
>The article mentions work done by Victor A. Vassiliev of the Russian Academy
of Sciences in Moscow about a radically different way of looking at knots.
> ........
>	Jim Moore
>	moore@aiag.enet.dec.com


You could try reading Vassiliev's paper on the subject, which is in a
collection called the theory of singularities and its applications,
edited by Arnol'd.  It is published by the AMS under the series name
Advances in Soviet Mathematics (or something like that.)
D. Ruberman