HDTenEightyP/NetNews huggingface dataset:
From: "John C. Baez" <jbaez@BOURBAKI.MIT.EDU>
Subject: Path connectedness
X-Submissions-To: sci-math-research@uiuc.edu
Organization: MIT Department of Mathematics, Cambridge, MA
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 28 Sep 1992 23:36:53 GMT

I am posting this for Allen Knutson, aknaton@math.Princeton.EDU,
who had trouble posting.

I'm curious about the differences between connectedness and path-connectedness.
If X is a connected topological space with two marked points, say that Y is
X-connected if for all y,y' \in Y one can map in X so as to have X's two
marked points hit y,y' respectively.

Now define a pre-partial order on connected spaces: "X connects more
than X'
does" if being X-connected implies being X'-connected. So for instance, all
R^n's (with whatever two marked points you like) are quickly seen to be
equivalent, but the topologist's sine curve connects strictly more than [0,1]
does.

This partial order (once we mod out by equivalence) plainly has a least
element: spaces where the two marked points coincide. Does it have any
maximal elements? Any canonical choice of representative from a given
connectedness class? Should one restrict to a smaller class of spaces
(say, compact Hausdorff) for some reason? How tall is it? (If people are
bugged by the category of connected spaces not being a set, I'm happy
to just take the ones of cardinality \leq 2^{\aleph_0} or whatever.)

References would be good too.                                   Allen K.
From: brnstnd@nyu.edu (D. J. Bernstein)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Date: 2 Oct 92 00:15:57 GMT
Organization: IR

In article <91284@netnews.upenn.edu> weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
> And I for one, never cared again much
> about AC assumptions except when it actually matters. 

All metamathematical issues aside, there is a sound _pedagogical_ reason
to explicitly identify each use of the axiom of choice. Most statements
people prove do not depend on the axiom of choice, and those which do
don't always appear that way at first glance. So the typical reader
won't be thinking of the axiom unless he's given some warning. By
explicitly saying ``if this set is well-ordered'' you make it clear that
the axiom has to be applied, and also _how_ it has to be applied.

For the same reason, it helps the reader to say whether a Noetherian
condition is uniform (any collection of submodules has a maximal
element), or strong (any submodule is finitely generated), or weak (any
ascending chain of submodules stabilizes). By adding one adjective
before the word ``Noetherian'' you make it clear exactly which
assumptions you _really_ want to make.

---Dan
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Infinity
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Fri, 2 Oct 92 03:48:03 GMT

In article <1371.2acabc99@atlas.nafb.trw.com> dsimon@atlas.nafb.trw.com writes:

>I am looking for an infinity of physical things.  There are a finite 
>number of atoms (or electrons or quarks) between Lincoln and Terre Haute. 
>You can say that each atom has an infinite number of "points", but a 
>"point" is a mathematical abstraction - it doesn't really exist.  

>True, there are an infinite number of possible times each rest can last,
>but I'm looking for an infinity of physical things.

>I can actually have "one" of something.  I don't know if it is possible
>for there to be "infinity" of anything.

Not anything that takes up any space, unless they are interpenetrable
or you have an infinite amount of room.  Nobody knows if we have
an infinite amount of room.  Photons are one of the most interpenetrable
sort of things and the "infrared divergences" in quantum field theory
arise from the presence of infinitely many of them.  But really, the
usefulness of infinity has nothing to do with whether you can have
infinitely many physical things.  Can you have -2.79 physical things?
Not really, but -2.79 is a fine concept.
From: maciejw@math.arizona.edu (M P Wojtkowski)
Newsgroups: sci.math
Subject: Re: Poncelet's Theorem, Elliptic Curves & Dynamical Systems
Keywords: conics, geometry, elliptic curves
Date: 27 Sep 92 00:50:18 GMT
Organization: Mathematics Department, University of Arizona


Newsgroups: sci.math
From: epstein@rene.ma.utexas.edu (Paul Epstein)
Subject: Hopf groups
Organization: University of Texas at Austin Mathematics
Distribution: sci
Date: Thu, 1 Oct 1992 03:17:43 GMT

The following question has almost certainly been posed by others and I'm sure 
it can readily be answered by consulting the literature.

I'm asking it because it's an interesting question to which I don't know the
answer.

If G is finitely presented, then can G mod N be isomorphic to G?
What if we insist that N is also finitely presented?
Newsgroups: sci.math
From: prener@watson.ibm.com (Dan Prener)
Subject: Re: A more virus resistant architecture
Date: Mon, 5 Oct 1992 03:44:35 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM T.J. Watson Research Center, Hawthorne, New York

In article <BvMJAo.24F@world.std.com> rjk@world.std.com (robert j kolker) writes:

>While reading a book on the Babbage Analytic Engine the other night, it
>occurred to me that a Babbage machine might be less susceptable to virus
>than a Von Neuman machine. In the Babbage machine, the program is stored
>on an medium external to the data store (punched cards for example). The
>Babbage Analytic Engine was patterned after the Jacquard Loom.

>The question is this: Is a computer whose program space is totally
>disjoint from the the data space less likely to be buggered by viruses
>than a  Von Neuman Machine, where the data and the program are kept in the
>same store.

How disjoint is "totally disjoint"?  You can't really keep program and data
space disjoint for a general-purpose computer.  Somewhere in the sequence

 source-program-creation --> compilation --> linking --> loading --> execution

data is transformed into a program.  Typically, this is in the loading stage.

(If the machine is not to be a general-purpose computer, for example, if it
 is not going to allow the creation of new programs, then the problem of
 virus infection is limited to keeping viruses out of the embedded programs.
 By and large, that seems to be workable.  The world has not yet been troubled
 by viruses in microwave ovens, TVs, automobiles, and such.)
--
                                   Dan Prener (prener@watson.ibm.com)
From: stieger@cco.caltech.edu (Ronald David Stieger)
Newsgroups: sci.math
Subject: Re: HELP! I got a contradiction!
Date: 26 Sep 1992 20:09:58 GMT
Organization: California Institute of Technology, Pasadena
NNTP-Posting-Host: punisher.caltech.edu

cannon@scs.unt.edu (Dustin Cannon (Lab 3460)) writes:


>First off, bear with me.  I'm only a senior math major.
>OK, heres the deal.

>	e^ix  = cos(x) + isin(x)
>	e^-ix = cos(x) - isin(x)

>	Let x = 

>	e^i  = -1
>	e^-i = -1

>	e^i = e^-i
>	i = -i
>	i = -i
>	ii = -ii
>	-1 = 1

>	I had one of my profs tell me the problem is in taking the log of
>e^i, but I didn't quite follow.  Can anyone give me a better explanation.
>Thanks in advance.

OK, I'm taking a class in complex analysis this term, but it doesn't start
till Monday and I am NOT going to start reading the book early, but I think
I understand enough to help you here.  First of all, remember that you're 
dealing with complex numbers here, and the rules for dealing with exponents
and logs are not necessarily the same as with reals, so the prof was 
probably right.  Basically, let c = pi, so e^(ic) = -1, as does e^(-ic) as
you said.  Now, obviously, cos(pi)=cos(-pi)=-1, but you're not going to tell
me that this implies that pi = -pi.  ( At least I hope not. :-)  Taking the
log of e^(ix) is similar to taking the arccos or arcsin of x, except that if
x != k*(pi/2) , e^(ix) has both sin and cos components.  So, as with the
inverse trig functions, I would guess that in taking the log you must somehow
restrict the range, and ic and -ic would not be both included, in fact you
would probably get ic for both.

Well, I hope that helped.  I'm sorry I couldn't be more specific, but hopefully
you can understand my ramblings enough to get a better idea of what happened.

-- 
--Ron Stieger                      **** JUST VOTE NO. ****
  stieger@cco.caltech.edu
"For some, it's simply something you wear around your neck. Just a chain.
It means a lot more than that to me."--Michael W. Smith, "Cross of Gold"
From: hrubin@mentor.cc.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Re: Easy (?) probability question
Date: 2 Oct 92 01:56:21 GMT
Organization: Purdue University Statistics Department

In article <1aflv9INN7p2@agate.berkeley.edu> davidb@triangle.Berkeley.EDU (David T. Blackston) writes:
>Hi all,

>Here's a problem that came up recently in an analysis of a sorting
>algorithm.  It seems really obvious, but I can't come up with a nice
>proof and I was wondering if anyone out there had one.

>Let X be a random variable, and define the random variable Z by
>Z=min{X, k} where k is some constant.  Show that var(Z) <= var(X).

>I have shown this in the case where X is a bounded, integer valued r.v.,
>but the proof is messy and I feel that there has to be a better way.
>Is there?

Here is a simple proof using conditioning and a simple not sufficiently
well known fact.  The fact is that 

*	V(W) = E(V(W|Q)) + V(E(W|Q))

where Q is any partition.  We use the partition into A = [X < k] and ~A.

On A, Z = X, and so the expected value and variance are equal.  On ~A,
Z = k, which has variance 0, and Z <= X, so E(Z|~A) <= E(X|~A).  Then
a simple calculation shows that both terms of * are reduced for Z.
Only if Z=X or X is constant are the variances equal.
-- 
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)
From: jack@dcs.glasgow.ac.uk (Jack Campin)
Newsgroups: sci.math
Subject: Re: Hyperbolic space?
Date: 29 Sep 92 18:29:41 GMT
Reply-To: jack@dcs.glasgow.ac.uk (Jack Campin)
Organization: COMANDOS Project, Glesga Yoonie


jane@mathematik.uni-Bremen.de (wayne tvedt) writes:
> Allan Adler (ara@zurich.ai.mit.edu) wrote:
>> Maybe it is time someone wrote for hyperbolic space the book analogous to
>> Flatland.
> A French guy, Jean-Pierre Petit, has written some nice cartoon books about
> curved spaces, relativity, and like topics.  I have some German editions
> from Physik-Verlag.
> Unfortunately, Petit's _The Black Hole_ will never be used in American
> primary schools, because it starts off with Anselm and his friend Herr
> Albert toking on a hookah. 

It has been published in America, though:

   Type of Material: Book
     LC Call Number: QC173.59.S65 P4613 1985
             Author: Petit, Jean-Pierre.
      Generic Title: Trou noir. English
              Title: The black hole / Jean-Pierre Petit ; translated by Ian
                      Stewart ; edited by Wendy Campbell.
   Publication Info: Los Altos, Calif. : W. Kaufmann, c1985.
  Phys. Description: 64 p. : ill. ; 28 cm.
        Series Name: The Adventures of Archibald Higgins
              Notes: Translation of: Le trou noir.
              Notes: In his latest adventure, Archibald Higgins investigates
                      concepts of space and time.
           Subjects: Space and time--Juvenile literature.
           Subjects: Black holes (Astronomy)--Juvenile literature.
           Subjects: Space and time--Cartoons and comics.
           Subjects: Black holes (Astronomy)--Cartoons and comics.
           Subjects: Cartoons and comics.
        Other Names: Campbell, Wendy.
 Other Series Names: Petit, Jean-Pierre. Aventures d'Anselme Lanturlu. English.
     LC Card Number:    85025609 /AC
               ISBN: 0-86576-069-1 (pbk.)

-- 
--  Jack Campin    room G092, Computing Science Department, Glasgow University,
17 Lilybank Gardens, Glasgow G12 8RZ, Scotland   TEL: 041 339 8855 x6854 (work)
INTERNET: jack@dcs.glasgow.ac.uk or via nsfnet-relay.ac.uk    FAX: 041 330 4913
BANG!net: via mcsun and uknet     BITNET: via UKACRL    UUCP: jack@glasgow.uucp
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Date: 5 Oct 92 15:46:43 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: holmes@opal.idbsu.edu (Randall Holmes)

In article <1992Oct2.205526.4103@guinness.idbsu.edu>, holmes@opal (Randall Holmes) writes:
>>I certainly don't think you should _not_ mention AC when it's used.  I
>>just think, a la Gunning's comment, the decision on _whether_ to use AC
>>or some other assumption is pretty much a stylistic matter.

>It's not entirely a matter of style.  It is legitimate, for instance
>(although not usually done) to assume that all sets of real numbers
>are Lebesgue measurable.  But then one _cannot_ assume AC in
>unrestricted form...

I assume that one is doing more than set theory, where questions about
whether or not to assume AC are givens.  In which case, merely getting
all sets of reals Lebesgue measurable is the set theoretical result--
what's of interest, a la Gunning, is where you are going with it.

And for _that_, the decision about whether to work inside normal ZFC
or not remains pretty much a stylistic matter, as you will get parallel
results.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Newsgroups: sci.math,rec.puzzles
From: tordm@vana (Tord G.M. Malmgren)
Subject: Re: EASY NUMBER SEQUENCE PUZZLE...
Reply-To: TordM@VanA.PhySto.SE
Organization: Department of Physics, University of Stockholm -- Sweden
Date: Fri, 2 Oct 1992 11:35:05 GMT

In article <1992Oct2.035739.14663@galois.mit.edu>, jbaez@riesz.mit.edu (John C. Baez) writes:

>>>25, 1729, 635318657, ???
>>>what's ??? ?

>> could it be 0 ? is there anybody but me who thinks these kind of
>>"puzzles" are totaly non-interesting?? the "???" could be just ANY number,

>I find them uninteresting too, probably just 'cause I'm not too good
>at them.  But there *are* better and worse answers to these things and
>you know it - the very fact that you wrote 0, an "unlikely" answer, 
>shows you know that there is some pattern to this thing.  

 I see the pattern
 x_n=(-1905950809n^3+13341650526n^2-26683285691n)/6+2541264354
 which would yield the above series AND x_4=0

 these series are like having a cross-word that would allow ten different
letters in each tiny square... "fill it in like you wish! if you fill it
in like the author wants it you win the car!"

>I also don't like football, but I don't complain, "What's the point
>of running around with that prolate spheroid?  What difference does
>it make which end of the field it's near?"

 why don't you? it feels good to bitch about nothing every now and then!
you know what they say "supressed feelings lead to frustration and anger"
:-)




---------------+--------------------------------
 Tord Malmgren | InterNet: TordM@VanA.PhySto.SE | These opinions are my OWN,
               | BITNet  : TordM@SESUF51        | and NOT of this department!
---------------+--------------------------------
 Department of Physics, University of Stockholm
Date: Monday, 5 Oct 1992 21:40:16 EST
From: <RVESTERM@vma.cc.nd.edu>
Newsgroups: sci.math
Subject: Re: Irrational?
 <1992Oct1.204515.11508@galois.mit.edu>
 <COLUMBUS.92Oct2172559@strident.think.com> <1aqoe7INN5r7@agate.berkeley.edu>

regarding, "is the word 'irrational' derived from the mathematical term?"

i sincerely doubt it.  mathematicians have long had a history of giving
numbers that they like "good" names, and numbers that they don't like
"bad" names.  for example:

      positive / negative
      rational / irrational
      real / imaginary

then of course there are the "natural" numbers.  though i have never heard
their complement called "unnatural", it would follow the pattern...




Newsgroups: sci.math
From: "peter shirley" <shirley@death.cs.indiana.edu>
Subject: Question about inverting functions
Organization: Computer Science, Indiana University
Date: Sat, 26 Sep 1992 14:06:12 -0500

As part of my computer graphics work, I often have to invert
relatively simply increasing functions, for example:

f(x) = 1 + sin(x)   for 0 < x < pi/2

Sometimes I run into one I can't do analytically, for example:

f(x) = Ax + Bsinx   for 0 < x < pi/2

where A and B are constants.

My question is whether there is a good reference for how to proceed
with such problems.  In my college math courses, I had at least
a semester on integration techniques, but almost nothing on function
inversion.  Any pointers are appreciated!

Pete Shirley
CS Dept
Indiana U
Bloomington
shirley@cs.indiana.edu

Newsgroups: sci.math,rec.puzzles,comp.theory
From: se_karne@rcvie.co.at (Georg Karner)
Subject: Re: more on .999... = 10.0
Organization: Alcatel Austria - Elin Research Center, Vienna
Distribution: inet
Date: Wed, 30 Sep 1992 09:55:02 GMT

Please refer to the sci.math FAQ and do not waste bandwidth!
-- 
+-- Georg.Karner@rcvie.co.at --- or: se_karne@rcvie.co.at ----------+
|   Georg Karner, Alcatel Austria-ELIN Research Centre,             |
|   A-1210 Vienna, Austria                                          |
+-- fax +43 1 391452 --------------------- voice +43 1 391621 263 --+

Xref: sparky sci.math:12798 sci.physics:16025
From: trex.uphwhtt@oscs.montana.edu (Tsunefumi Tanaka)
Newsgroups: sci.math,sci.physics
Subject: Re: Another Cute Math Joke
Date: 6 Oct 92 18:02:42 GMT
Article-I.D.: coe.1992Oct6.180242.17547
Organization: Montana State University, Physics Dept.
X-Xxmessage-Id: <A6F730E18701D739@mph001.oscs.montana.edu>
X-Xxdate: Tue, 6 Oct 92 20:02:41 GMT
X-Useragent: Nuntius v1.1.1d11

Re: Another Cute Math Joke

What do you call the result of three mountain climbers going into a
manufacturing business together?

"Triple Scaler Product"

Bill McHargue
Newsgroups: sci.math,sci.logic
From: frank@Cookie.secapl.com (Frank Adams)
Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
Date: Tue, 29 Sep 1992 14:56:05 GMT
Organization: Security APL, Inc.

In article <TORKEL.92Sep28195555@bast.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <1992Sep28.154237.148977@Cookie.secapl.com> frank@Cookie.secapl.
>com (Frank Adams) writes:
>
>   >I'll go you one better: this is a *better* thing to say.  It's still a very
>   >different thing from a simple "that happens to be false", which disproof of
>   >the other conjecture would have elicited.
>
>  Saying that a disproof of "all formulas have a disjunctive normal
>form" would only have elicited a reaction on the lines of "that
>happens to be false" is not really by itself an intelligible
>description of a possible scenario. Are you stipulating that
>"classical propositional logic" means the same in your hypothetical
>world as it does here and now? And if so, are you not assuming that it
>could still be proved in your hypothetical world that every formula
>does have a disjunctive normal form?

Try substituting Fermat's Last Theorem for your example.  A proof that FLT
was false would be interesting.  By contrast, a proof that ZFC was
inconsistent would be a crisis (not a catastrophe; we depend on ZFC much
less than on propositional logic).

Similarly, in my hypothetical world where the disjunctive normal form
theorem was not proved, its disproof would not be a crisis.  The subsequent
proof, whenever it was found, would be a very serious crisis.  But that is
*because* it is an inconsistency.
Newsgroups: sci.math
From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Subject: Re: Opinions Sought (Why major in math?)
Organization: Purdue University Statistics Department
Date: Mon, 5 Oct 1992 13:23:13 GMT

In article <1992Oct5.113921.14691@st-andrews.ac.uk> ad@tamdhu.st-and.ac.uk (Tony Davie) writes:
>In article <stephen.717902097@mont>, stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) writes:


>> A while ago, I phoned up alumni from our math dept, asking people
>> what career they had persued.  While many said that their math
>> degree helped them to think logically, none said that they actually
>> used their math.

>Surely this is the whole point of an education rather than a training.
>After a good education (majoring in ANY subject) one should come out knowing
>how to find things out for yourself. It's a second level activity. You may
>not know just how to do a certain task, but you sure as hell know where to
>look to find out or who to ask. The only purpose of a university education is
>to learn how to use a library.

This is an oversimplification, and leaves out far too much.  Facts can be
looked up in a library, but understanding of concepts does not yield to this
approach; at most one can study.  But  without the understanding, it is even
difficult to know about the facts.

I personally consider the most important item in mathematics to be the
ability to use symbols.  Someone who uses symbols to formulate word problems
is doing what cannot be looked up in the library, and probably does not think
of it as using mathematics.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)
Newsgroups: sci.math
From: aet@mullian.ee.mu.OZ.AU (bert thompson)
Subject: Re: EASY NUMBER SEQUENCE PUZZLE... (SPOILER)
Organization: Computer Science, University of Melbourne, Australia
Date: Fri, 2 Oct 1992 04:44:08 GMT

sichase@csa3.lbl.gov (SCOTT I CHASE) writes:

>In article <Oct.1.10.53.11.1992.29154@remus.rutgers.edu>, clong@remus.rutgers.edu (Chris Long) writes...
>>In article <aet.717945133@munagin>, Bert Thompson writes:
>> 
>>> 25
>>> 1729
>>> 635318657
>> 
>>The next is the smallest positive integer that can be written as
>>a sum of fifth powers in two distinct ways, but I have no idea what
>>it is.

ok. that's half credit so far.

>I believe that nobody knows what it is.  There is no known solution, to 
>the best of my knowledge, to a^5+b^5 = c^5+d^5, and no proof one way or 
>the other as to the existence of a nontrivial solution.

sorry, you do not get the lollipop. 8^)

gee! and i thought it was easy too!
should i bother jamming it through my computer or will the answer be 42?

cheers!

	bert.

--------------------------------------------------------------------------------
bert thompson	aet@munagin.ee.mu.oz.au.
SITEE		school of information technology and electrical engineering
		university of melbourne
--------------------------------------------------------------------------------
Xref: sparky sci.math:12579 rec.puzzles:6563
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math,rec.puzzles
Subject: Re: EASY NUMBER SEQUENCE PUZZLE...
Date: 1 Oct 1992 15:42:05 -0400
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <1992Oct1.175503.10957@kth.se> TordM@VanA.PhySto.SE writes:
>In article <aet.717945133@munagin>, aet@mullian.ee.mu.OZ.AU (bert thompson) writes:
>
>>25, 1729, 635318657, ???
>>what's ??? ?
>
> could it be 0 ? is there anybody but me who thinks these kind of
>"puzzles" are totaly non-interesting?? the "???" could be just ANY number,
>so what's the point?
>

Those who find them uninteresting may skip them.  In fact, such problems
come up often in mathematical research.  You can compute the first few
items of a sequence, then (somehow) you guess the general form (this is
the step like the problem above), and finally you prove the answer.


-- 
  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