HDTenEightyP/NetNews huggingface dataset:
From: gore@pepper.rutgers.edu (Bittu)
Newsgroups: sci.math
Subject: Re: n doesnt divide 2^n-1
Date: 7 Sep 92 17:02:22 GMT
Organization: Hill Center

In article <1992Sep4.152548.2904@super.org> kedlaya@metropolis.super.org (Kiran Sridhara Kedlaya) writes:

> Mea culpa. I forgot to mention that 2^phi(n) - 1 is divisible by n, and phi(n) 
> is less than n, so d is at most phi(n), so it's less than n.
> 

So we have at least two different proofs that 2^n is not congruent to
1 (mod n) for n > 1. While working on this problem I noticed a few
things that are interesting. Since I still don't have any proofs, my
observations are best summed up in the following conjecture:

Conj: For n >= 1, let 2^n = a.n + b where 0 <= b < n so that a is the
quotient and b the remainder. Then the following hold:

1. b is always even.
2. (even stronger) b is always a power of 2.
3. a is always even.

Note that 1 implies that n doesn't divide 2^n - 1 for n > 1. 2 gives
us a slightly stronger theorem though I don't know how to say it. I
don't know what 3 implies. My question is: Is any of the three
statements true?

--Bittu
Newsgroups: sci.math
From: dy@shire.math.columbia.edu (Deane Yang)
Subject: Re: Freedman-Donaldson Theorem
Organization: Mathematics Department, Columbia University
Keywords: Exotic R^4, differential structures, topology
Date: Thu, 10 Sep 1992 16:12:45 GMT
X-Posted-From: shire.math.columbia.edu
X-Posted-Through: sol.ctr.columbia.edu

In article <4173@seti.UUCP> hussein@bora.inria.fr (Hussein Yahia) writes:
>  I'm interested in understanding the proof of the famous Freedman-Donaldson
>theorem about "fake R^4", that is the existence of exotic diffentiable 
>structures on R^4. (For others R^n , n different of 4, all differentiable
>structures are always diffeomorphic to the standard one). I tried to read
>a book "Connections, Definite Forms, and Four-manifolds" by T. Petrie and 
>J. Randall (Oxford Science Publications) but it is quite a little bit 
>too difficult for me. 
>
>Could someone give me a list of references leading to a proof of that theorem
>for somebody like me who is just at the level of Volumes 1-2 of Spivak's 
>Course on Differential Geometry and Hirsh's Differential Topology (Springer
>Verlag) ?
>

The proof divides into two very different parts,
the topological theory developed by Freedman and the study of the
moduli space of Yang-Mills fields developed by Donaldson. Neither
part is particularly easy to understand, although, as a differential
geometer, I find the latter far more understandable.

My advice would be to focus on one or the other, depending on your
taste. I believe that Freedman has written a few surveys on his
work on 4-manifolds. The Donaldson theory is presented in the book
cited above, a book by Freed-Uhlenbeck, and a set of lecture notes
by Lawson. Any of these references should be used only as a starting
point. You will need to consult other books and papers to fill in the
details.

Given your background, it might be better to pursue more fundamental
material first. For the Freedman stuff, you should learn more topology
(here, I can't help much). For the Donaldson stuff, you should learn
more about analysis and elliptic PDE's.

Deane Yang
Polytechnic University
Newsgroups: sci.math
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: Proof of God's Existence
Organization: U of Cambridge, England
Date: Mon, 7 Sep 1992 23:27:40 GMT

Oo-er. My news program seems to have eaten the end of my last posting
about this. It should have ended:

This makes the idea that people's experiences of God are just the products
of their own minds more tenable than the corresponding assertion about, say,
--
From: cjhs@minster.york.ac.uk
Newsgroups: sci.math
Subject: ZFC etc. (was Re: Report on Philosophies of Physicists)
Date: 14 Sep 92 20:05:45 GMT
Organization: Department of Computer Science, University of York, England
X-Newsreader: Tin 1.1 PL5

I get a bit lost in some of the postings in the "Philosophies..." thread,
but it does seem that some of the particpants might be able to help me
a couple of things straight in my own mind. Maybe the questions have
already been answered, and I just didn't see it...

1) We all know that ZFC is consistent (or maybe just those of us who
are sufficiently naive): I want to know why we have this confidence.

It just the case that lots of very intelligent people have failed
to find an inconsistency? Or are there informal arguments (as there
are for Church's thesis) to suggest consistency.

2) We know that the continuum hypothesis is independent of ZFC. Godel
provided the model (constructible sets) in which the CH is true, and
Cohen provided the model (via forcing) in which the CH is false.

Are the proofs by Cohen and Godel are formal proofs? (I would think
so.) Can we identify the formal system in which the proofs were
performed?

Thanks for any informed comment.

Felicitations -- Chris Ho-Stuart

(PS this is fall-out from the alt.atheism thread, which has evolved to
the point where sci.math is the better location)
From: RVESTERM@vma.cc.nd.edu
Newsgroups: sci.math
Subject: Re: Chess Problem
Date: 12 Sep 92 20:57:13 GMT

by "a win for white," do you mean immediately, or a guaranteed win for white
if the game continues?
From: bill@waikato.ac.nz
Newsgroups: sci.math
Subject: nth prime
Date: 10 Sep 92 19:09:05 +1200
Organization: University of Waikato, Hamilton, New Zealand

Sorry to ask such a FAQ, but what's the quickest algorithm to
find the nth prime? Is it O(nlogn) or something else?

Bill Teahan,
Systems Programmer,
University of Waikato,
Hamilton, New Zealand
Newsgroups: sci.math
Subject: Re: "amslookup" shell script to search AMS database
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 15 Sep 92 22:54:54 GMT

In <BtyswE.87L@watserv1.uwaterloo.ca> shallit@graceland.uwaterloo.ca (Jeffrey Shallit) writes:

>I got a copy of Ken Ribet's shell script below, which he calls
>"amslookup", from Jerry Tunnell.  Ken told me he's posted this 
>to the net before, but I never saw it, so he agreed I could post
>it again.

>Here it is:

>---------------------------------cut here-------------------------
>#
>#From ribet@math.berkeley.edu Tue Sep  1 14:31:59 1992
>#
>#I did indeed write it, and I think that I posted it to sci.math
>#at one point.  You're 100% welcome to spread it around.  The
>#"sed" part of the pipe is to get rid of control-M's that are
>#visible if you use "less" instead of "more".  If you really use
>#"more", there's no need to strip out the control-M's.
>#
>(echo ${1} | /usr/etc/mconnect -p 2050 e-math.ams.com) | sed  's/^M//' | /usr/ucb/more

>--------------------cut here-------------------------------------------

....

but is there a way to send AMS/CML electronic address corrections?  (Only
one of the electronic addresses in my entry is correct.  (Can you guess
which one?)  (Darn, it's in my .signature).
--
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; please mail anything important.
From: kriman@acsu.buffalo.edu (Alfred M. Kriman)
Newsgroups: sci.math
Subject: Re: Origin of Matrices??
Summary: Matrix "origin" controversial; Cayley, Grassmann or Hamilton.
Keywords: quaternion, matrix
Date: 6 Sep 92 22:53:54 GMT
Organization: UB = University at Buffalo = SUNYAB = State U.of NY At Buff.

In article <43339@skye.dcs.ed.ac.uk> Greg Wilson (gvw@epcc.ed.ac.uk) writes:
>I am trying to find out how matrices (and linear algebra) came to be ---
>who was the first mathematician to use matrix notation, how quickly did
>it spread, etc.
>Please reply directly, as I am not a regular reader of this newsgroup.
>Thanks,
> ========================================================================
>  Gregory V. Wilson                    ||  For in much wisdom is much
>  Technical Coordinator                ||  grief; and he that increaseth
>  Edinburgh Parallel Computing Centre  ||  knowledge, increaseth sorrow.
>  gvw@epcc.edinburgh.ac.uk             ||               - Ecclesiastes
> ========================================================================

   The following summary is gleaned from Section 5.1 of Max Jammer's _The
Conceptual Development of Quantum Mechanics_, and in particular pp. 204-
206 (McGraw-Hill, 1966).  Question of propagation is discussed there 
primarily in context of applications to physics (explosion of use in 1925).

1844: W. R. Hamilton introduces quaternions in article entitled "On
      quaternions; or on a new system of imaginaries in algebra." [1]
1850: J. J. Sylvester used the concept of a matrix as an "arrangement of
      terms" in rows and columns from which determinants can be formed,
      and the word "matrix." [2]  Not clear if this was first such use.
1855: A. Cayley introduced the multiplication rule for matrices. [3]
1858: A. Cayley defined matrix as "an abbreviated notation for a set of
      linear equations."  He is credited with the first rigorous treatment
      of the algebraic theory of matrices. [4]
1884: J. J. Sylvester credited Cayley as exclusive discoverer of matrices.
subsequent years: P. G. Tait (in letter to Cayley), H. Taber (1890), and
(indirectly, while championing H. Grassmann) J. W. Gibbs claim the crown
for Hamilton.  (Ironically, it was Cayley who showed that quaternions
could be represented as 2x2 self-adjoint matrices.)  A scuffle breaks out,
involving W. K. Clifford, A. N.  Whitehead, A. Buchheim and T. J. I'A.
Bromwich.  Humor gets a bad bump on the head.

Refs.: (Much of this is reprinted.  See Jammer for details.)
[1] Phil. Mag. _25_, 10-13, 241-246, 480-495 (1844).  [I guess we would
call this LPU publication today.]
[2] Phil. Mag. _37_, 363-370 (1850).
[3] J. f. d. reine u. angewandte Mathematik (Crelle) _50_, 272-317 (1855).
[4] Phil. Trans. Roy. Soc. (London) _148_, 17-37 (1858).

ec: gvw@epcc.ed.ac.uk (Greg Wilson)

"Beware the man of one book." -- Would someone please inform me of the
source of this quote?
Newsgroups: sci.math
From: victor@watson.ibm.com (Victor Miller)
Subject: Re: Jokes in papers (was Re: Einstein's Erdos number)
Date: Tue, 8 Sep 1992 15:18:50 GMT
Reply-To: victor@watson.ibm.com
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM, T.J. Watson Research Center

>>>>> On 8 Sep 92 09:12:36 GMT, kubo@zariski.harvard.edu (Tal Kubo) said:
Tal> Nntp-Posting-Host: zariski.harvard.edu


Tal> Niall Graham mentioned:

>Source: From Figure 1 of "On properties of a well-known graph or
>what is your ramsey number?"  Annals of the New York Academy of
>Sciences, Vol. 328. The author calls himself Tom Odda of Xanadu University.
Tal>                                              ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Tal> This is reminiscent of one of Andre Weil's early papers
Tal> at the end of which he lists his affiliation as 
Tal> "Univ. of Erewhon, Erewhon, Ut."

Unforunately some good jokes never make it past the editors: One of
the first papers on a practical use of the ideas in Baker's method for
solving Diophantine equations was a paper in the journal of Number
Theory by Don Lewis, and a whole bunch of others.  I was told (on good
authority), that this paper was first submitted under the author name
of "Ann Arbor, University of Michigan".  The editor was not amused,
and insisted that the correct names be put on the paper.

--
		Victor S. Miller
		Bitnet: VICTOR at WATSON
		Internet: victor@watson.ibm.com
		IBM, TJ Watson Research Center
		"Great artists steal; lesser artists borrow" Igor Stravinsky
Newsgroups: sci.math
From: ramsay@unixg.ubc.ca (Keith Ramsay)
Subject: Re: Lebesgue integral (was: Couple of questions
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Tue, 15 Sep 1992 02:04:45 GMT

In article <1992Sep11.130033.26063@watson.ibm.com> 
platt@watson.ibm.com (Daniel E. Platt) writes:
[...taking out some layers of quotes...]
|> In other words, how badly would you miss them if you threw out the
|> non-Riemann-square-integrable functions from L^2(R^3)?
...
|I think you would end up flat on your back.  The problem is that many
|(most) of the techniques revolving around Fourier series and
|integrals, completeness of a basis, etc, ultimately involve being able
|to evaluate 'improper' integrals as a limit of an integral of a
|sequence of functions.

I've heard that much (nearly all, perhaps) of what is done with L^2,
regarded as Lebesgue square-integrable functions, modulo functions
supported on sets of measure zero, is just as nicely done (if not
better) by regarding L^2 as the formal completion (relative to the L^2
norm) of a convenient dense subset (where the Lebesgue measure theory
is not needed). I'm not enough of an analyst to confirm this.

Keith Ramsay
ramsay@unixg.ubc.ca
Organization: Penn State University
Date: Mon, 7 Sep 1992 23:28:42 EDT
From: S. Wukovitz <SXW103@psuvm.psu.edu>
Newsgroups: sci.math
Subject: Partitioning of uncountable sets

Hi, this came up in discussion among a few of my (math) friends.
We were working on the last part of Green Rudin Chapter I,
Problem 6.  This is probably a minor issue in the scheme of
things and I don't think the grader will disagree if I simply
state it as a fact, but even so I would truly love to see a
proof of:
---
Every uncountable set can be partitioned into two uncountable
sets.
---
I mean, it's obvious but we proposed a lot of ways to prove it
and none of them worked.  Maybe we're all just burned out.

This is useful in trying to describe the set of measurable functions
of a certain measurable space which is the subject of the aforementioned
problem in Rudin.  I would appreciate any thoughts you might have.
Thanks!

We're allowed to work on problems together and consult basically
anything we want(I think it's pretty much implied that if we do
we should say so).

Thanks!

-Stephanie Wukovitz
From: rickert@mp.cs.niu.edu (Neil Rickert)
Newsgroups: sci.math
Subject: Re: Set-Theory Doubts: Got some conceptual glitches
Date: 12 Sep 92 16:38:18 GMT
Organization: Northern Illinois University

In article <18s2mtINN2gf@matt.ksu.ksu.edu> bubai@matt.ksu.ksu.edu (P.Chatterjee) writes:
>a) According to my textbook, the ORDERED PAIR of two objects x and y is the set <x,y> = {{x}, {x,y}}.

>What throws me off is the definition; what is the motivation behind it?

Don't take these definitions too seriously.  This definition is there
purely to formalize the definitions in terms of a basic set of set
theoretic axioms.  But most mathematicians, once they are satisfied that
an ordered pair can be formalized, stop paying much attention to this
definition, and think of ordered pairs in intuitive terms.

>b) Let f: X --> Y be an injective mapping. By definition, this means:
>  for all x, x' in X: f(x) = f(x') ==> x=x'.

>My question here is: for the firstmentioned implication why is the CONVERSE not true? (I know it's not but am having trouble finding an intuitive/logical answerfor it).

I assume that by CONVERSE you mean that x=x' ==> f(x) = f(x').  This is
certainly true.  It is implicit in the definition of a mapping, so is
not usually stated explicitly.
Newsgroups: sci.math
From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Partitioning of uncountable sets
Organization: Boise State University Math Dept.
Date: Tue, 8 Sep 1992 16:46:57 GMT

In article <1992Sep8.182706.90039@vaxc.cc.monash.edu.au> kevin@vaxc.cc.monash.edu.au writes:
>A proof that every uncountable set can be partioned into two uncountable
>sets.
>Let X be an uncountable set. Consider the set of ordered pairs, (x,0),(x,1)
>where x is in X. Call this set Y. Then Y is also uncountable, moreover,
>{(x,0) with x in X}=Y(0) and {(x,1) with x in X}=Y(1) are both uncountable.
>But the cardinaltiy of X is the cardinality of Y. Thus, there is a bijection
>f from Y to X. f(Y(0)) and f(Y(1)) form the desired partition of X.
>The proof is of course easier if one assumes A.C, which I have avoided.
>
>Love,
>Kevin Davey.
>

You have not avoided using AC.  k + k = k is not a theorem of ZF.  I
also doubt the truth of the assertion "every uncountable set can be
partitioned into two uncountable sets" in the absence of AC, but I'm
not certain of its status.
-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
From: ct@dde.dk (Claus Tondering)
Newsgroups: sci.math
Subject: Re: Proof of God's Existence
Date: 7 Sep 92 07:41:47 GMT
Organization: Dansk Data Elektronik A/S

It seems to me that all attempts to prove God's existence have to
be postponed until we can define exactly what we mean by existence.

Questions: Does New York exist? Does George Bush exist? Does an apple
exist? etc. etc.

Sure, I can see New York, I can hear George Bush, I can taste the apple.
But in these cases I am only relying on my senses. And my senses can
be deceived. How do I know that George Bush is not a hallucination, a
mere product of my own mind? How do I know that everything I see actually
exists and is not simply a hallucination?

There is only one thing I can know for sure: My mind exists! Even if
everything else is hallucinations, *something* must be seeing those halluci-
nations, and that something is what I call "my mind". This is not necessarily
my body. Hey, my arms, feet, even my head may be a hallucination! The only
thing I know for certain is the existence of the slightly metaphysical
concept of "my mind".

Now for my real question: Bearing the above in mind, why is it considered
unscientific to base a belief in the existence of God on personal
experiences of Him, when a belief in the existence of, say, George Bush is
based also on personal experiences of him?

-- 
Claus Tondering
Dansk Data Elektronik A/S, Herlev, Denmark
E-mail: ct@dde.dk
Newsgroups: sci.math
From: goddard@NeXTwork.Rose-Hulman.Edu (Bart Goddard)
Subject: The Last Number Theory Problem
Organization: Rose-Hulman Institute of Technology
Date: Mon, 14 Sep 1992 13:46:24 GMT



A gushy, heartfelt thank-you note will appear in October to those who
have sent solutions for those last few problems.  (I did have to solve
about half of them myself,  anyway.)  There is just one left, in which
I am beginning to suspect a typo, but I can't even guess where it might
be.  So, I'll post it one last time, and hope someone recognizes it and
can supply the correct statement of the problem, or correct me and  
solve the problem as is.  Here it is:

5.2.11.a (Solved) Show that if n is a pseudoprime to the base a but not
a pseudoprime to the base b, where (a,n)=(b,n)=1, then n is not a 
pseudoprime to the base ab.

5.2.11.b Show that if there is an integer b with (b,n)=1 such that n is
not a pseudoprime to the base b, then n is a pseudoprime to <= \phi(n)
different bases a, with 1<=a<n.  (Hint: Show that the sets a_1, a_2,  
.., a_r, and ba_1, ba_2,...,ba_r have no common elements, where a_1,  
a_2, ..., a_r, are the bases less than n to which n is a pseudoprime.)

Part (a) is easy, and I think part (b) was intended to be easy.  At  
this point, we know what \phi(n) is, but we don't know any properties
about it.

Bart Goddard
goddard@nextwork.rose-hulman.edu
From: sec@otter.hpl.hp.com (Simon Crouch)
Date: Fri, 11 Sep 1992 09:24:26 GMT
Subject: Re: Freedman-Donaldson Theorem
Organization: Hewlett-Packard Laboratories, Bristol, UK.
Newsgroups: sci.math

In sci.math, ara@zurich.ai.mit.edu (Allan Adler) writes:


> Several people have suggested various books aimed at making it easier to
> prepare one to read the Freedman-Donaldson proof (which I haven't read).
> I usually find it somewhat discouraging to be given a list of several
> books to read, each of which will in turn present its own difficulties,
> and to resolve the variations in notation, conventions and points of
> view of various authors, when all I really want is to read a particular
> paper or accomplish a particular task. It is sort of like saying, "You can't
> just DO that. You first have to become someone who DOES that."

Hi Allan,
	 What you say is very true - I remember the frustrations
	 as a beginning graduate student trying to read the latest
	 and greatest and being directed to weighty tomes!
	 Unfortunately, there is no "Freedman-Donaldson proof" There
	 is a Freedman proof (dealing with topological four-manifolds)
	 and a Donaldson proof (dealing with differentiable four-manifolds).
	 Both appeared
	 in the Journal of Differential Geometry (I seem to remember)
 	in the early to mid 80's. They are both very hard. (For me 
	anyway!)

> Maybe the Freedman-Donaldson paper is too hard just to read directly.
> But why not determine that based on an attempt to read it first hand
> and to decide for oneself exactly what one needs to know to read it
> and how to learn what one needs to know?

Both Freedman and Donaldson have written books that "improve"
on the earlier proofs and give indications to the reader about what
is required as prerequisite to the proofs.
The introductions to these books give
excellent summaries as to what is "going on". I gave these references, since
they genuinely offer a less thorny path towards understanding the theorems.
The other books I suggested (Lawsons "Spin Geometry" in particular) are
good references to look up the stuff you'll need to understand.
I hope somebody does the same for Witten's stuff :-)

> Allan Adler
> ara@altdorf.ai.mit.edu

Simon Crouch.
From: columbus@rachmaninoff.think.com (Michael Weiss)
Newsgroups: sci.math
Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
Date: 16 Sep 92 00:14:06 GMT
	<1992Sep15.184930.10080@guinness.idbsu.edu>
Organization: Thinking Machines Corporation, Cambridge MA, USA
NNTP-Posting-Host: rachmaninoff.think.com
In-reply-to: holmes@opal.idbsu.edu's message of 15 Sep 92 18:49:30 GMT

In article <1992Sep15.184930.10080@guinness.idbsu.edu>
holmes@opal.idbsu.edu (Randall Holmes) answers the question:
   >
   >Are the proofs by Cohen and Godel are formal proofs? (I would think
   >so.) Can we identify the formal system in which the proofs were
   >performed?

thus:

   They are rigorous arguments.  No one ever does proofs in formal
   systems, in practice.  The Godel proof is a construction in ZF (normal
   set theory without choice).  The Cohen proof is a construction in ZFC
   (Choice is used, I believe).

The relative consistency results of Godel and Cohen can be regarded as
purely combinatorial statements about pushing symbols around,
and as such can be expressed in the language of Peano arithmetic.

In principle, the proofs of these relative consistency statements could be
carried out in Peano arithmetic.  Cohen discusses this briefly in the last
chapter of his book "Set Theory and the Continuum Hypothesis".
I believe the treatment in Shoenfield's book "Mathematical Logic" makes the
same point.

In practice, the proofs become far more intuitive if we adopt a less
puritan attitude, and talk about models of ZF.  Cohen makes free use of
axiom SM (="there exists a model of ZF whose universe is a set and whose
element-of relation is the standard element-of relation".)  Cohen proves
with ZF+V=L+SM that there is (for example) a model of ZFC + not CH.
Naturally this implies the relative consistency result, Con(ZF) -->
Con(ZFC+not CH).  But if all you want is the relative consistency result,
then a much weaker set of axioms (such as Peano arithmetic) will do.