HDTenEightyP/NetNews huggingface dataset:
From: zeno@athena.mit.edu (Richard Duffy)
Subject: Re: Is Card(R)=Card(R^2)?
Organization: Massachvsetts Institvte of Technology
Date: Sun, 23 Aug 1992 01:17:46 GMT

In article <16g754INNsmk@function.mps.ohio-state.edu> Gerald Edgar writes:
>>How about the unit line minus the end points, and the interior of
>>the unit square?
>
>A continuous bijection from the open interval onto the open square
>is also not possible.  The reason for it is a bit harder, this time.
>But still not beyond a first course in point-set topology.
>

I'm curious to know whether the method Prof. Edgar has in mind is simpler or
just radically different than the following:

Given a continuous bijection  f: I --> I x I  where  I  is an open interval.
I  is a countable union of closed subintervals  K_n , hence  I x I  is the
countable union of the  f(K_n).  But I x I is open in R x R, hence has the
Baire property that it is not a countable union of nowhere-dense subsets.
[If you want to avoid the detail of proving that, just use  I = R  in the
first place, via a homeomorphism, and the fact that  R x R  is a complete
metric space.]

So some particular  f(K_n)  is not nowhere-dense; being also closed (since
it is compact, as the continuous image of the compact set  K_n ),  f(K_n)
therefore contains an open ball  B, which in turn contains a closed sub-ball
C .   Now the inverse image  f^{-1}(C)  is a closed subset of  K_n  and
thus compact, so the restriction of  f  to this set is a homeomorphism
onto  C , i.e.  f^{-1} restricted to  C  is continuous.  But then  f^{-1}(C)
must be a connected subset of  K_n , so it's a closed interval  J.  (We've
essentially reduced the problem to the earlier one).  Now just remove three
points from  C -- you still have a connected set, but its image under
f^{-1},  which is  J \ {three points}  since  f  is bijective, can't
possibly be connected.  [Note that a closed interval with *two* points
removed could still be connected.]  So we've contradicted  f  out of
existence.



--
>> Richard Duffy  -----------\___ ((lambda (x) (list x (list (quote quote) x)))
Internet:  zeno@athena.mit.edu   \____  (quote (lambda (x)
Bitnet:    zeno%athena@MITVMA         \______(list x (list (quote quote) x)))))
Voicenet:  +1 617 253 4045                   \------------------
From: wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys)
Newsgroups: sci.math
Subject: Re: Bell's _Men of Mathematics_ (was: Great and Inspirational ?)
Date: 18 Aug 92 15:28:31 GMT
Reply-To: wsadjw@urc.tue.nl
Organization: Eindhoven University of Technology, The Netherlands

In article <1992Aug13.193848.5184@ariel.ec.usf.edu> mccolm@darwin.math.usf.edu. (Gregory McColm) writes:
>In particular, Bell despises all politicians (he's not 
>alone in this), and has a tendency to think that good 
>mathematics is a mark of good character.  He also 
>repeats gossip incautiously.

You can't trust anything in Bell, unless it's backed up
by somebody else. Better read the somebody else.

>
>I don't know of a conventional, responsible, moderate 
>book containing short biographies of mathematicians.  
>As for histories of mathematics, Carl Boyer is a very 
>good conventional, responsible, and moderate writer 
>of mathematical history;  his book is quite reasonable.  
>Edna Kramer's is more ahistorical.  Other nominations?

D.M. Burton, The history of mathematics. An Introduction.
Allyn and Bacon, Boston etc., 1985.

It's a bit expensive, and not as encyclopedic as Boyer.
But it tends to give more non-mathematical historic background.

JWN
>
>-----Greg McColm


Newsgroups: sci.math
From: wjcastre@magnus.acs.ohio-state.edu (W.Jose Castrellon G.)
Subject: Proofs from a finite number of examples (was RE: marrying ages)
Organization: The Ohio State University,Math.Dept.(studnt)
Date: Thu, 13 Aug 1992 01:49:26 GMT

In article <55Z55ZP@math.fu-berlin.de> guckes@math.fu-berlin.de (Sven Guckes) writes:
... 
 [original article deleted]
...
>
>If he uses four examples to prove his statement - tell him to quit math!
>

Maybe not...once I came across a paper by  Chaitin  (I dont have the reference
at hand, sorry) where he writes about the busy beaver function and the
implications that the knowledge of some of its values would bring: if FLT is
true, then it would follow from the proofs for exponents  n, 2 < n < a, for
some fixed a; same for other diophantine equations and various long standing
conjectures. As you might know, busy beaver is non-computable.
Newsgroups: sci.math
From: pedersen@seas.smu.edu (Ted Pedersen)
Subject: Knight's Tour
Organization: SMU School Of Engineering and Applied Science
Date: Wed, 12 Aug 1992 21:17:39 GMT



I realize this might be just a little off the topic (but hopefully not too
much). I would very much like to find some source code for the Knight's Tour
problem. This is the problem where you have a lone knight on a chess board and 
the goal is to have your knight visit all of the squares on the board and 
return to his starting place without visiting any square twice. 

I would especially be interested in C source code that solves our knight's
dilema, either from an ftp site, book, etc.        

If anyone knows of any repositories of source code to solve these sorts of
problems (Knights Tour, N-Queens problem, etc.) I would appreciate any tips. 
I generally enjoy these sorts of computing problems so anyone who knows of 
any interesting variations on the Knight's Tour or N-Queens problem etc. 
I would sure like to hear about them. 

Ted Pedersen

Newsgroups: sci.math
From: dek2@cec1.wustl.edu (Dean Edward Kassmann)
Subject: Re: Continued Fractions
Keywords: Help, Continued Fractions
Organization: Washington University, St. Louis Mo.
Distribution: na
Date: Thu, 20 Aug 1992 22:38:03 GMT

In my own experience I have used continued fractions to sum up divergent
series.  A reference I found useful was G.A. Baker's _Essentials of Pade
Approximants_.
   The second chapter, if I remember correctly, deals with
continued fractions relation to Pade Approximants.  It is nicely written
and fairly easy to understand.
Dean Kassmann
Washington University, ChE Dept.
dek2@wucec1.wustl.edu
kassmann@wuche1.wustl.edu
Newsgroups: sci.math
From: roach@bikini.wri.com (Kelly Roach)
Subject: u(v^n)w prime puzzle
Keywords: Primes, Formal Languages
Organization: Wolfram Research, Inc.
Date: Mon, 17 Aug 1992 15:36:34 GMT




      Here is a little puzzle for sci.math readers.  It is
really a puzzle in formal language theory, but you don't
need to know that to solve the puzzle.  Anyway:

     Observe that the numbers 19, 109, 1009, and 10009
     are all prime.  Unfortunately, 100009 is not.
     {u,v,w}={"1","0","9"} is a "near miss".  The
     question is, are there different u,v,w which
     don't "miss"?

     Prove or disprove:  There are three non-empty
     strings of digits u,v,w such that all the
     numbers in
          L = {u(v^n)w | n is a natural number}
            = {uw, uvw, uvvw, uvvvw, uvvvvw, ...}
     are prime numbers.

I know the solution to this puzzle.  I think the
solution can be understood fairly easily by anyone that
has had a first course in number theory.  See if you
can discover it.

				Kelly
From: sippy+@CS.CMU.EDU (Jay Sipelstein)
Newsgroups: sci.math
Subject: Re: Expansion of set theory
Date: 12 Aug 92 22:01:57 GMT
Article-I.D.: cs.1992Aug12.220157.12782
Reply-To: Jay.Sipelstein@cs.cmu.edu
Organization: School of Computer Science

In article <1992Aug12.164408.10547@thunder.mcrcim.mcgill.edu>, boshuck@triples.math.mcgill.ca (William Boshuck) writes:
>
>[...] there is a natural heuristic which
>is probably how the axioms of set theory got there in the first
>place. Namely, when things exploded after the Russell paradox, 
>the idea to put things together was to begin with a more or
>less concrete model of what was to become set theory and then
>start writing down thigs which just had to be true of this model.
>      [definition of V deleted]
>One way to go beyond ZFC is to look at "axioms" which seem to say
>that V contains very large sets, that is, we might be led to
>believe that anything we do to "push up" the cumulative hierarchy
>ought to be more or less harmless if it does not immediately
>give rise to one of the classical paradoxes. 

There is an excellent non-technical description of this process in 
Rudy Rucker's book Infinity and the Mind.  I'm generally not too
impressed with "pop-math" books, but I was with this one.

Jay Sipelstein
sipelstein@cs.cmu.edu
From: MTURELLA@cmsa.gmr.com
Newsgroups: sci.math
Subject: Re: mathematician's marrying ages.
Date: 13 Aug 92 21:43:37 GMT
Article-I.D.: cmsa.16842EB49.MTURELLA
Organization: GM Research Labs

In article <hubert.713379330@spica>
hubert@spica.ucsb.edu (Chang;Hung-Hsien;H.;;;COMS;Graduate;930630;;;8-402525-199

In my earlier posting, I had placed a P.S. about my marriage
situation at 34.  I was saying that 'early in life' is
a little vague.

It was not my intention to throw my hat in with the Greats.

Sorry, no offense was intended.

Mark
Newsgroups: sci.math
From: sethb@fid.morgan.com (Seth Breidbart)
Subject: Re: You know, the integers (was: Re: Stupid question about FLT)
Organization: Morgan Stanley & Co., New York, NY
Date: Mon, 17 Aug 1992 17:17:16 GMT

In article <1992Aug16.172642.13398@sics.se> torkel@sics.se
(Torkel Franzen) writes:

>   My explanation of the integers is simple: the integers
>are 0, s(0), s(s(0)) and so on. Any obscurity or ambiguity that you
>find in this explanation carries over to any explanation of what is
>meant by "PA".

Q: What do you mean by "and so on"?

A: Any number of s's.

Q: "Any number"?  What about pi s's?

A: I mean any counting number.

Q: What is a "counting number"?

A: A non-negative integer.

Q: What is an integer?


Seth		sethb@fid.morgan.com
Newsgroups: sci.math
From: news@massey.ac.nz (USENET News System)
Subject: Re: Is Card(R)=Card(R^2)?
Organization: Massey University
Date: Thu, 13 Aug 92 00:07:15 GMT

In article <1992Aug12.102140.5231@nntp.hut.fi>, samu@lammio.hut.fi (Samuli Siltanen) writes:
> 
> I have been wondering... It seems first that there are more pairs of
> real numbers than real numbers, but is it really so? And if not, I would
> like to see a bijective mapping f:R->R^2. And furthermore, is there some  
> relation between Card(R) and Card(R^n)?
> 
> Hoping that this is not a trivial one
> 
It is not trivial, but not too hard either. This is the one which slowed
acceptance of set theory - many mathematicians were reluctant to
accept the result and therefore felt that set theory must be wrong.

There is a bijective mapping.

First attempt: use assign alternate digits of the decimal expansion
of z to x and y. (To keep it simple let 0 <= z <= 1). This does not
quite work because 0.52909090... maps to (0.5999..., 0.2000...)
and so does 0.62000... . We have, however, shown that
#([0,1]) >= #([0,1] x [0,1]).
Cantor laboured a great deal to get an explicit bijection, but now
we can just use the Schroeder-Bernstein theorem:
#A >= #B and #A <= #B implies #A = #B.

A similar argument shows that #R = #R^n.
From: "ian tuck" <ian.tuck@canrem.com>
Newsgroups: sci.math
Subject: need help ii!
Date: 23 Aug 92 11:08:42 EST
Reply-To: "ian tuck" <ian.tuck@canrem.com>
Distribution: sci
Organization: Canada Remote Systems

Thanks to all who replied, but unfortunately my description of my
problem was too simplified.  Indeed, using the numbers I gave, it is
relatively trivial to figure out the best combination of boxes to
fill the larger box.  However, the real-world values are not as neat.
So, restating my problem:
     I have a box 20'x8'x8' (This part is true).  I have a number of
smaller boxes of 15 different sizes.  Is there a method by which I can
select any number of these smaller types (say, 5 different types), and
figure out numbers which will fill the larger box with a minimum of left
over space?  There may be certain restrictions on the # of smaller boxes,
(i.e. there must be at least > 10 of each of the 5 types, the # after that
is just as many of each as best solve the problem).  Thanks to all who
replied to my first post.  I hope that someone can point me in the right
direction here.  I imagine using Excel 4's Solver program will not suffice.
On the other hand, if you can show me a method, I program C/C++ for a living,
so can hopefully translate the theory into practice.
Ian
---
  DeLuxe/386 1.25 #8528  
--
Canada Remote Systems  - Toronto, Ontario/Detroit, MI
World's Largest PCBOARD System - 416-629-7000/629-7044
From: vanloon@cwi.nl (Maarten van Loon)
Newsgroups: sci.math
Subject: Ellipsoid Problem
Date: 20 Aug 92 06:26:10 GMT
Organization: CWI, Amsterdam

I am posting this for a friend who has no access to the net.
The problem is the following:

We have two ellipsoids

    ellipsoid 1:   SQR((x-Cx)/a) + SQR((y-Cy)/b) + SQR((z-Cz)/c) = 1

    ellipsoid 2:   SQR((x-Nx)/d) + SQR((y-Ny)/e) + SQR((z-Nz)/f) = 1
    
All values are supposed to be known, except for Nx, Ny, Nz.
x, y and z are of course space variables. We also assume that
    
    d < a
    e < b
    f < c
    
The question here is: what values can Nx, Ny, Nz take so that
ellipsoid 2 is contained in ellipsoid 1?
    
Solutions or suggestions are welcome.
    
Thanks in advance on behalf of my friend,
    
Maarten
--------------------------------------------------------------------
| Maarten van Loon      |  The easiest answers are the most        |
| CWI, P.O.Box 4079     |  difficult to find                       |
| 1009 AB Amsterdam     |                                          |
| The Netherlands       |  ----------------------------------------|
| email: vanloon@cwi.nl |  Not knowing Bach = not really living    |
--------------------------------------------------------------------
    
Newsgroups: sci.math
From: israel@unixg.ubc.ca (Robert B. Israel)
Subject: Re: f: Q -> Q , f analytic
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Sun, 16 Aug 1992 08:00:31 GMT

In <1992Aug14.142252.8804@unibi.uni-bielefeld.de> achim@unibi.uni-bielefeld.de (Achim Flammenkamp) writes:

>A analytic number theoretic question (or ?):

>Given a function with domain and target C excepted a finite number of points.
>It maps Q into Q (Q denotes the set of rational numbers) and
>it is meromorph on C (this implies uniquely extendable from Q to C).

>Must this function be representable as the quotient of two polynomies with
>integer coefficients or are there other functions left (present a counter-
>example, please) ?



Let {q_n} be an enumeration of Q.  Let {P_n} be an enumeration of
polynomials with rational coefficients.  Let f_n be an entire
function (a polynomial will do) satisfying:
  (i)  f_n(q_j) = 0 for j < n
  (ii) f_n(z) is real when z is real.
  (iii) sum_{j=1}^n f_j(q_n) is rational, but is not P_n(q_n).
  (iv) |f_n(z)| < 2^(-n) for |z| <= n
Let F(z) = sum_{n=1}^infinity f_n(z).  The series converges
uniformly on compact sets, so F is an entire function.  For each
rational q_n, F(q_n) = sum_{j=1}^n f_j(q_n) is rational, so F
maps Q into Q.  Moreover, F(q_n) <> P_n(q_n), so F is not a
polynomial with rational coefficients.  Since it's entire, it's
also not a quotient of polynomials with rational coefficients.

>achim
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Newsgroups: sci.math
From: dmurdoch@mast.queensu.ca (Duncan Murdoch)
Subject: Diophantine approximation
Organization: Queen's University
Date: Fri, 14 Aug 1992 18:17:39 GMT

I'm looking for a pair of reals (x_1,x_2), such that rational approximations 
to the pair of the form (q_1, q_2) = (m_1/n, m_2/n) do as poorly as 
possible, in the sense that the distance from the pair of reals to the 
pair of rationals shrinks at the slowest possible rate as n is allowed to 
increase.  

I've looked in Gruber and Lekkerkerker, Geometry of Numbers, and this looks 
like their example I in section 45.1 and example II in section 45.4; in the 
latter, they show that for any choice of (x_1, x_2) there are infinitely 
many pairs (q_1, q_2) as above such that

(x_1 - q_1)^2 + (x_2 - q_2)^2   <=  1/(gamma n^3)     (*)

where gamma = sqrt(23)/2, but the assertion is not true for larger gamma.
In this context, what I think I'm after is a particular pair 
(x_1, x_2) such that there is at best equality in (*), never strict 
inequality; failing that, one where the ratio of the LHS to the RHS is 
asymptotically bounded below by 1.

The reason I have some hope to achieve this is that the corresponding one 
dimensional problem does have solutions, for example x = (sqrt(5) - 1)/2.

The reason I'm asking here is that I can't understand 90% of Gruber and 
Lekkerkerker, so I'm hoping that someone who can read it might know the 
answer.
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: Littlewood's Three Principles
Date: 13 Aug 1992 07:46:52 -0400
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

>In Section 3.6 of his book, Real Analysis (2nd ed., MacMillan, 1968),
>H. L. Royden gives a verbatim quotation from J. E. Littlewood's Lectures on
>the Theory of Functions (Oxford, 1944, p. 26)

Thanks to everyone who helped.  In fact, before I asked the question,
I looked in several likely places, including the index of Royden,
without success.

One of my most frequent complaints about technical books is inadequate
(or even nonexistent) index!
-- 
  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)
Newsgroups: sci.math
From: jeanmarc@ecrc.de (Jean-Marc Andreoli)
Subject: Primes in x_{n+1} = ax_n+b (was Re: u(v^n)w prime puzzle)
Reply-To: jeanmarc@ecrc.de
Organization: European Computer industry Research Centre GmbH.
Date: Fri, 21 Aug 1992 10:31:32 GMT

Let a,b be integers, and (x_n) be a sequence s.t. x_{n+1} = a x_n + b

My question is: does it contain infinitely many primes ?

There are trivial cases, where the answer is no:
1/ if (a x_0 + b = x_0) the sequence is constant.
2/ if (a = -1) the sequence alternates between two values x_0 and x_1.
3/ if ((a,b) > 1 or (x_0,b) > 1) then the sequence clearly contains only composite numbers.

What about the other cases ?

I've seen a "theorem of Dirichlet" in Hardy&Wright which handles the case a=1. Can anyone give me a pointer to a proof of that theorem (Hardy&Wright don't give it and the only reference they give is in German).

---

Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: A Non-Cantorian Set Theory question
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 13 Aug 92 19:13:16 GMT

In article <1992Aug13.173137.10878@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
>In article <1992Aug12.113415.1648@gacvx2.gac.edu> kiran@gacvx2.gac.edu writes:
>>Quite a while ago, I read Martin Gardner write in one of his _Mathematical 
>>Games_ column in _Scientific American_ that on a plane, a letter like _O_ can 
>>be written--allowing smaller O's to be written inside larger O's-- _c_ 
>>times where _c_ is the cardinality of the continuum. On the other hand, he
>>pointed out, letters like _T_ can only be written aleph-nought times.
>>
>>A question that has remained in my mind for a long time since is the following:
>>
>>Since we know that non-Cantorian set theories are possible, is there a
>>one-dimensional shape which can be written some aleph times where that aleph is
>>between aleph-nought and _c_? If so, what would the shape be?
>
>Since the continuum hypothesis is undecideable in ZFC it will be
>impossible to construct such a shape in ZFC.  Thus I don't think you
>will ever get a very clear picture of what such a shape would look like
>even if one "exists".  ("Exists" is in quotes here since it really means
>"can be proved to exist in ZF together with some axioms that contradict
>the continuum hypothesis".)  Nonetheless it's an interesting question.

My reply was bogus.  One might have a construction in ZFC of some shape
and a proof that "IF there is a cardinal X between aleph-0 and the
continuum, one can only put this shape X times in the plane."  (Or
perhaps more hypotheses.)
Newsgroups: sci.math
From: mw@ki.fht-mannheim.de (Marc Wachowitz)
Subject: Re: mathematician's marrying ages.
Date: Thu, 20 Aug 92 09:49:44 GMT
Organization: Fachhochschule fuer Technik Mannheim
X-Newsreader: Tin 1.1 PL4

Juhana Kouhia <jk87377@cc.tut.fi> wrote:
: I have also read about how some elder people die soon after the dead of their
: wife/husband.
Two points about this:
1. "Some" doesn't yet indicate a general corrleation.
2. Even if it were "many", that might still be interpreted as strong attachment
   and inability to let the past pass away.

Be careful with concepts of "humans are meant to ..." - the presence of a fence
doesn't imply you have to stay enclosed. You might as well build a gate,
jump over it or remove it.

: Having many friends is not the same than having a caring living partner.
Right, quality and quantity need not be related - but quality doesn't imply
"partner" either (assuming you mean it in the sense of living together for a
longer time).

-------------------------------------------------------------------------------
Marc Wachowitz, mw@gandalf.ki.fht-mannheim.de, 75742@novell1.rz.fht-mannheim.de
  *  wonder everyday    *    nothing in particular    *    all is special  *
Newsgroups: sci.math
Subject: Re: int(x*log(x)*((1-x)^n), x=0..1)
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 20 Aug 92 17:37:20 GMT
Keywords: Hairy integral, Beta function

In <dld.714296391@bruce.cs.monash.edu.au> dld@cs.monash.edu.au (David L Dowe) writes:

>Let  I(n) = int(x*log(x)*((1-x)^n), x=0..1);
>to use Maple's notation.        n is a positive integer.     log is ln  .

>I seek as simple an expression as possible or, that failing, as rapidly
>converging an expression as possible for I(n).

Futher simplification shows:

           n^2+n-1       (1 + 1/2 + ... + 1/n)
I(n) = --------------- - ---------------------,
       (n+1)^2 (n+2)^2    (n+1) (n+2)


which can readily be proved by induction, noting the the integral for

(n+1) I(n) - (n+3) I(n+1) can be written in closed form.

--
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.
My interaction with our news system is unstable; if you want to be sure I see a post, mail it.
Xref: sparky sci.math:10493 sci.physics:13283
Newsgroups: sci.math,sci.physics
Subject: Re: tensors: How about 3rd, 4th rank?
From: mcirvin@husc8.harvard.edu (Mcirvin)
Date: 21 Aug 92 21:43:01 GMT

pierct@rpi.edu (Tom Pierce) writes:

>I know all about 1st and second order tensor, but what's a third order
>tensor look like?  

If you write it in terms of components, it's got three indices.  It
would look like a cube of numbers.  But:

>Also, aren't these the same mathematical critters as
>"matrices", witha the added "square" criterion?

Not really, because the important thing about tensors is the way
they *transform* under rotations (and Lorentz boosts, in
relativity).  A vector in space is not just three numbers; it's a 
three-component object whose components change in a very
specific way under rotations.  Likewise a tensor is characterized
by its transformation properties.  The relation between a tensor
and a matrix is like the relation between a vector and a list of
its components.

>What would you use a third or highr order tensor for?  Any PHYSICAL
>applications?

The Riemann curvature tensor is fourth-order.  It has 4x4x4x4
components in GR, though most of them are related by symmetries so
it has many fewer independent components.  This describes the
curvature of four-dimensional spacetime in general relativity.

-- 
Matt McIrvin, Cambridge, Massachusetts, USA 
Newsgroups: sci.math