HDTenEightyP/NetNews huggingface dataset:
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Mercator Projection
Organization: Computer Science Department,  Stanford University.
Date: Tue, 10 Nov 1992 02:04:08 GMT

In article <israel.721352141@unixg.ubc.ca> israel@unixg.ubc.ca (Robert B. Israel) writes:
>If you conformally map the sphere with the North and South poles removed, 
>1-1 onto a cylinder, don't you have the Mercator projection up to 
>translation and reflection?

This would imply that the only conformations (is that the word?) of the
cylinder onto itself are translations and reflections.  Is that true?

>A physical model: blow up a spherical balloon inside a cylinder, and have
>it stick to the walls of the cylinder when it touches them.  

I bet that turns out not to be conformal.  The poor thing's already
doing its best to stretch horizontally and you're asking it to stretch
the same amount vertically as well!  Not a chance.
-- 
Vaughan Pratt
Xref: sparky sci.math:14816 alt.politics.elections:23530
From: rjk@world.std.com (robert j kolker)
Subject: Re: FREE-ENERGY TECHNOLOGY
Organization: The World Public Access UNIX, Brookline, MA
Date: Thu, 5 Nov 1992 15:28:59 GMT

Factoring the number of capitilized words, we compute a CPI = .762

(CPI = Crack Pot Index)

Bob Kolker
From: columbus@strident.think.com (Michael Weiss)
Newsgroups: sci.math
Subject: What's a knot? (inspired by "What's a manifold?")
Date: 6 Nov 92 10:52:42
Organization: Thinking Machines Corporation, Cambridge MA, USA
Distribution: sci
NNTP-Posting-Host: strident.think.com

I would summarize almost all the discussion in favor of the atlas
definition of a manifold in one phrase:

	proofs drive definitions

in other words, old, "intuitive" definitions are replaced by new "abstract"
definitions when the new definitions turn out to be more convenient for the
logical development of the subject.  If you require your manifolds to be
imbedded in R^n, then you have to come up with an imbedding every time you
construct a manifold, and this is often (to quote John Baez), "a pain in
the butt".

A second theme was the invariant aspect of the (maximal) atlas definition.
One wants to regard two manifolds as "the same" if they have the same
topological (or smooth, or conformal, or piecewise liner, or whatever)
structure, so with an "imbedded" definition, you need to mod out by the
equivalence relation.

This suggests that whenever we have a definition of the form

	"widgets are equivalence classes of wadgets under the boff
    	equivalence relation"

we should look for a definition of a widget that doesn't mention wadgets or
boff.  If this involves generalizing the notion of a widget, fine-- "be
wise, generalize" as I am told Zorn used to say.  If not, we have a
representation theorem.

How would one define a (tame) knot, intrinsically?  Definitions I am
familiar with either involve modding out by ambient isotopy (in fact there
are subtle points here, I believe-- perhaps someone more knowledgeable
would like to post), or by Reidemeister moves.

Is the field just too young to have a suitably slick and (on first
encounter) unintuitive definition?
Newsgroups: sci.math
From: ledwards@leland.Stanford.EDU (Laurence James Edwards)
Subject: Re: definition of topological space
Keywords: Topology; Open sets; Continuity
Organization: DSG, Stanford University, CA 94305, USA
Date: Fri, 6 Nov 92 09:20:37 GMT

In article <1992Nov5.094404.15550@infodev.cam.ac.uk>, rgep@emu.pmms.cam.ac.uk (Richard Pinch) writes:
|> In article <1992Nov5.033835.5180@leland.Stanford.EDU> 
|> ledwards@leland.Stanford.EDU (Laurence James Edwards) writes:
|> [...]
|> Well, here goes.  The sets in the family are called "open" and you should
|> think of them as being rather like the open intervals in the real line.
|> A subset of R is open iff it is the union of a collection of open intervals
|> (a,b); equivalently if it contains an open interval round any of its points.
|> 
|> The epsilon-delta definition of continuity says that f is continuous iff
|> for all x, for all e > 0, there exists d > 0 such that 
|>      |x-x'| < e => |f(x) - f(x')| < d
|> for all x, for all e > 0, there exists d > 0 such that
|>      f*(f(x)-d, f(x)+d) contains (x-e,x+e)
|> where f*(Y) is the set of x such that f(c) is in Y.
|> i.e.
|> Y open => f*(Y) open
|> 
|>  and this last is the topological definition of continuity.

In defining continuity what is the advantage of using open sets vs. closed sets,
i.e. why not say:

 |x-x'| <= e => |f(x) - f(x')| <= d

and similarly:

 Y closed => f*(Y) closed


Larry Edwards
Newsgroups: sci.math
Subject: Re: Games with Nonmeasurable Sets
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 6 Nov 92 19:14:21 GMT
Organization: Beckman Instruments, Inc.

In <1992Nov5.045644.21270@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:

>pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>>Now where did this depend on the cardinality of the well-ordered deck?
>>The *only* relevant fact, an elementary measure theoretic one, is that
>>only countably many cards can be assigned a nonzero probability.
>>(Proof:  every such card is in the finite set of cards assigned
>>probability at least 1/n for some postive integer n, and there are only
>>countably many such sets.)

>Vaughan, I think you have completely missed the point of the original
>posting. Let me repeat the key points:

>    1. There is one card for every real between 0 and 1.
>    2. Cards are dealt randomly according to the usual Lebesgue
>       measure on reals.

>So the probability of being dealt any particular card is precisely 0.
>However, for any particular real r, the probability of being dealt
>a card greater than r in the ordering LT is the Lebesgue measure of
>the set

>    { r' | LT(r,r') }

>Since this is the complement of a countable set, it has Lebesgue measure
>1.

It has to do with the set being non-measurable.  Cardinality is the red
herring, except that any mesure space on a countable set with measurable
singletons has no non-measurable sets.
--
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.
Newsgroups: sci.math
From: rbrown1@cc.swarthmore.edu (Randolph Gregory Brown)
Subject: Re: group theory for HS students
Reply-To: rbrown1@cc.swarthmore.edu (Randolph Gregory Brown)
Organization: Swarthmore College
Distribution: sci
Date: Sat, 7 Nov 1992 21:48:31 GMT

hrubin@mentor.cc.purdue.edu (Herman Rubin) writes:
> 
> I raise the question as to whether students learn anything after high
> school algebra, before they take abstract algebra, which is even relevant
> to learning group theory.  The only possibility is a good course in logic,
> including the predicate calculus and a discussion of proof, and such a
> course can be taught at any time after a reasonable attainment of reading
> ability is achieved.  Anyhow, most students taking abstract algebra now
> have not had such a course.
> 
> In the rest of an abstract algebra course, there is a slight use made of
> matrix algebra as an example of rings, and this even occurs as an example
> of groups on vector spaces.  But this is not essential.

I couldn't agree more.  A strong understanding of groups and fields,
if given early, would probably help students 1) firm up their
understanding of algebra, 2) firm up their understanding of functions
(if the right groups were shown).  This would help them in calculus
immensely.  Math teaching suffers from a chronalogical problem -- just
because calculus was invented (discovered if you really want) before
group theory doesn't mean that it should be learned beforehand.

	Randy



Xref: sparky sci.math:14904 sci.physics:18813
From: lry1219@cfdd40.boeing.com (Larry Yeagley)
Newsgroups: sci.math,sci.physics
Subject: Cartan, "debauch of indices", seeking a citation
Keywords: Cartan, "debauch of indices", seeking a citation
Date: 13 Nov 92 01:54:19 GMT
Followup-To: sci.math
Organization: Boeing Computer Services, Seattle - CFD lab


Netlanders:

I am trying to track down a citation for a phrase frequently attributed to Elie
Cartan. The phrase is "debauch of indices". I have heard it several times and
seen it in print once (pg 2-16 of Volume 1 of Spivak's differential geometry
opus). However, I am unable to find anyone who knows where in Cartan's oeuvre
this phrase appears.

The primary reason I am interested is to determine something about the context
and connotations of the phrase. At the present, my only inference is that Cartan
was greatly dissatisfied with the fog of indices in tensor analysis. More than
that I am unable to infer.

If anyone out there in Netland knows, would you please post a citation. The
reason I suggest posting is that I suspect many people would be interested. If
you can't post it, "e-mail" it to me and I'll post it.

Of course, feel free to volunteer your interpretations of the phrase.


Larry Yeagley
lry1219@cfdd42.boeing.com
From: jbarnett@nrtc.northrop.com (Jeff Barnett)
Newsgroups: sci.math
Subject: Boolean Algebras
Date: 12 Nov 92 00:32:30 GMT
Reply-To: jbarnett@nrtc.northrop.com (Jeff Barnett)


Is there a free boolean algebra such that the cardinality of
the algebra (not the cardinality of its generators) is that
of the continuum?

Jeff Barnett
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Mercator Projection
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Tue, 10 Nov 92 03:36:57 GMT

In article <israel.721212129@unixg.ubc.ca> israel@unixg.ubc.ca (Robert
B. Israe) writes:
>In <a34uTB4w165w@netlink.cts.com> kfree@netlink.cts.com (Kenneth
Freeman) writs:
>
>>My Mercator projection goes 'up' to only 84 degrees, ~the northern
>>tip of the classically huge Greenland. I'd like to know three things.
>>1) For a given area, what is its apparent increase in size for a 
>>given latitude? I.e., what is the rate of increase the closer you
>>get a pole (and infinity)?
>
>At latitude t, linear dimensions are multiplied by sec(t), so areas are
>multiplied by sec^2(t).

Hmm, maybe I'm confused.  Let t be the angle from the equator
(latitude) and x the distance from the equator on the Mercator
projection.  (The Mercator projection draws a line from the
center of the earth through the surface of the earth and then to an imaginary
cylinder running north-south into which the earth fits snugly, right?)
Then x = sec t, where I'm assuming the earth has unit radius.
But if you want to know the amount by which linear dimensions 
are multiplied *right at latitude t* you need dx/dt = sec^2 t.  
Areas would then go as sec^4 t.  (Here you need to note that
east-west lengths are getting scaled the same way as north-south
lengths, i.e. that the Mercator projection is conformal.)  

I hope I'm wrong, because if it were really dx/dt = sec t, then we would
have a reason to figure out the integral of sec t -- a notoriously
sneaky basic integral.  And I even seem to recall some remark about it
being the Mercator projection that led people to do this integral.  

So corrections are welcome.  

Xref: sparky sci.math:14605 sci.physics:18433 sci.astro:11781 sci.bio:4084 sci.chem:4477 misc.education:4195
From: kfarmer@wsuaix.csc.wsu.edu (keith farmer;S10000)
Newsgroups: sci.math,sci.physics,sci.astro,sci.bio,sci.chem,misc.education
Subject: Re: 2-dimensional grading system (was Re: What can one do for the bright?)
Date: 8 Nov 92 19:06:39 GMT
Article-I.D.: serval.1992Nov8.190639.12667
Organization: Washington State University

In article <1992Nov8.071756.2228@athena.mit.edu> solman@athena.mit.edu (Jason W Solinsky) writes:
>In article <1992Nov3.091430.4135@hsr.no>, onar@hsr.no (Onar Aam) writes:
>
>|> As a current student (Electrical Engineering if you're curious AND a  
>|> National Merit Finalist (read: Seimifinalist with "formalities" done  
>|> NOT! - but that's a different argument I'm perfectly willing to let  
>|> slide)
>
>Well I'm not. Please DO read National Merit Finalist as National Merit
>Semifinalist with formalities. The difference between the two is that
>semifinalists are chosen objectively and finalists are all semi-finalists
>who fit the stereotypes that the organizers have in mind for the nation's
>brightest individuals. This cut of 5% of all semifinalists is highly
>subjective and effectively meritless.
>
>Jason W. Solinsky
>National Merit Finalist

I turned down my Merit Scholarship (yes, the $2000) to U Chicago in an
attempt to gewt a life. I've no regrets. What Solinsky say is true. Know
a Semi-finalist friend of mine: I count him brighter than me... I just
take tests better.

Keith
From: torkel@sics.se (Torkel Franzen)
Newsgroups: sci.math
Subject: Re: Relatives with Crackpot Proofs
Date: 12 Nov 92 18:51:25 GMT
	<israel.721590265@unixg.ubc.ca>
Organization: Swedish Institute of Computer Science, Kista

In article <israel.721590265@unixg.ubc.ca> israel@unixg.ubc.ca 
(Robert B. Israel) writes:

   >My suggestion would be to persuade him that he needs more mathematical
   >background in order to present his "very interesting" ideas in a more
   >effective way.  Then provide him with the most difficult mathematics
   >text you can find, and tell him to come back when he's mastered it.

  A genuine crackpot will merely observe, in a very reasonable way,
that there is no reason why he should master this stuff, since he
already has a proof of his result. It seems to me that several
responses that have been made here are quite unrealistic. They are
based on the idea that the crackpot is an amateur mathematician, a
person who is open to ordinary mathematical arguments, and is willing
to mull over criticism in an ordinary mathematical way. But people who
do this, people who are in fact amateur mathematicians, are not
anything at all like the genuine crackpot. The genuine crackpot is
sincerely incapable of understanding how any reasonable person could
fail to realize the force of his argument. His conclusion that there is
a conspiracy of silence and suppression is to him compelling.
From: choy@skorpio.usask.ca (I am a terminator.)
Newsgroups: sci.math
Subject: Re: definition of topological space
Date: 7 Nov 92 00:26:22 GMT
Article-I.D.: access.1992Nov7.002622.17213
Organization: University of Saskatchewan, Saskatoon, Canada

In article <1dbm2eINN7gr@function.mps.ohio-state.edu>, edgar@function.mps.ohio-state.edu (Gerald Edgar) writes:
|> In article <1992Nov5.033835.5180@leland.Stanford.EDU> ledwards@leland.Stanford.EDU (Laurence James Edwards) writes:
|> >The definition of a topological space is:
|> >
|> >    :a set with a collection of subsets satisfying the conditions that 
|> >    both the empty set and the set itself belong to the collection, the 
|> >    union of any number of the subsets is also an element of the collection, 
|> >    and the intersection of a finite number of the subsets is an element 
|> >    of the collection

What elements of the power set are excluded from the collection?

|> The definition above should come only after many, many examples have been
|> studied.  At that point, the purpose of the definition may be more
|> apparent.

Can you give some examples to help me appreciate the definition?

Henry Choy
choy@cs.usask.ca
Newsgroups: sci.math
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Relatives with Crackpot Proofs
Date: 10 Nov 92 14:23:52 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: keith@celia.UUCP (Keith Goldfarb)

In article <2105@celia.UUCP>, keith@celia (Keith Goldfarb) writes:
>Does anybody have any good suggestions about how to deal with
>crackpot "proof"s from relatives and friends?

Tell him he's a crackpot, and you don't want to see it.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Newsgroups: sci.math,misc.education
From: wut@acf3.NYU.EDU (Tom Wu)
Newsgroups: sci.math
Subject: Q: Math in Financial World
Date: 5 Nov 92 17:37:00 GMT
Organization: New York University


 	Hi. This is my first posting on this net, and it looks
like all of you here have a very high mathematical background.
I am currently a Junior in college, majoring in economics, and 
minoring in math.
	I am wondering if there is someone on this net who is
working on the financial market( equity, bonds, f/x). I've heard
that in order to be a top notch analyst, one has to obtain a
Phd degree in Math, probabily in statistics. Is this true? Could
someone tell me what kinds of mathematical subjects are useful?
And is there really a need to go beyond a MBA or a MA?

	Thanks in advance.


			---Tom
Newsgroups: sci.math,misc.education
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Newsgroups: sci.math
Subject: Re: definition of topological space
Date: 9 Nov 92 02:07:43 GMT
Organization: U of Cambridge, England

In article <1992Nov8.211002.20516@massey.ac.nz>, news@massey.ac.nz (USENET News System) writes:

> Not quite, you need
> 
> 4. Any nbhd N of x contains another nbhd M of x which is a nhbd of
> each of its points.

Oh dear. And someone else pointed out that I forgot to specify that each
point has at least one neighbourhood. Why do I never engage my brain before
posting to sci.math?

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Newsgroups: sci.math
Subject: Re: Assorted questions and problems
Date: 10 Nov 92 00:37:02 GMT
Organization: U of Cambridge, England

In article <96778@netnews.upenn.edu>, weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:

> Of course not.  It's almost embarrassing to mention the counterexamples,
> but here goes: (0,1) and [0,1].  The question you meant to ask, I assume,
> was if the orderings were dense.  In that case, a back and forth argument
> shows the two are isomorphic.  I'm pretty certain you need CH for this--I
> think Shelah has the contrary model.

Er, orderings being dense isn't enough. Consider R and R without irrationals
between 0 and 1. Both dense, but the latter has two things with only a countable
amount of stuff between them. I suspect that the usual back-and-forth thing will
work if any two points have continuum-many points between them, though.

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
Newsgroups: sci.math,misc.education
From: burtt@comet.nscl.msu.edu (BRIAN BURTT)
Subject: Re: Is Math Hard?
News-Software: VAX/VMS VNEWS 1.41    
Organization: NSCL Michigan State University
Date:  9 Nov 1992 06:21 EDT  

In article <1dku2bINNdar@darksttar.UCSC.EDU>, ask@ucscb.UCSC.EDU (Andrew Stanford Klingler) writes...
>  P.S.  Burtt does seem perilously close to whining here.  Actually, we
>   all do, on this and related threads.  It was my impression that this
>   newsgroup represented a slice of the mathematical "elite".  It would
>   be very surprising if, given normal childhoods, most of us were *not*
>   bored and, in retrospect, "underutilized".  Sure, it's a problem -
>   just not a very shocking one.

	If I seemed to be whining, I apologize profusely.  I consider 
myself very lucky because I survived the system, with my goals and dreams
intact.  I've had many friend, however, who were not as lucky.  For them,
let me make the point, students should be able to realize their full
potential with the full assistance and support of the educational system,
rather than having to fight it tooth-and-nail all the way.  The word
"graduate" should sound a lot less like "battle-worn veteran."
--Brian Burtt

Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Assorted questions and problems
Organization: Computer Science Department,  Stanford University.
Date: Mon, 9 Nov 1992 23:23:00 GMT

In article <96778@netnews.upenn.edu> weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>In article <1992Nov8.181631.13298@Princeton.EDU>, tao@fine (Terry Tao) writes:
>>(3) Assume the axiom of choice and the axiom of the continuum.  Is it
>>true that two chains (totally ordered sets) which both have the
>>cardinality of the continuum have a one-to-one and onto order
>>preserving mapping betweem them?
>
>Of course not.  It's almost embarrassing to mention the counterexamples,
>but here goes: (0,1) and [0,1].  The question you meant to ask, I assume,
>was if the orderings were dense.  In that case, a back and forth argument
>shows the two are isomorphic.  I'm pretty certain you need CH for this--I
>think Shelah has the contrary model.

Actually they are not isomorphic, since only the latter has a least
element and a greatest element.

The back-and-forth argument shows that there are exactly four
*countable* dense chains, depending on whether they have a top and
independently a bottom.  For a counterexample to the generalization of
this to *uncountable* dense chains, remove the irrationals from the
middle third of (0,1).  In any isomorphism of that chain with (0,1),
the middle third of the former will be paired with an *interval* of the
latter of measure 0, which can consist only of one point, so these are
two nonisomorphic dense uncountable chains.  (One of these days I'll
strike it lucky and get a measure-theoretic argument correct.  :-)
-- 
Vaughan Pratt
From: spencer@panix.com (David Spencer)
Newsgroups: sci.math
Subject: open prof exam (was: feynman story probably not true)
Date: 6 Nov 92 14:01:10 GMT
Article-I.D.: panix.1992Nov6.140110.5264
Organization: PANIX Public Access Unix, NYC

In <92309.155316RVESTERM@vma.cc.nd.edu> <RVESTERM@vma.cc.nd.edu> writes:

>a little while ago, somebody posted a story to this effect: for a
>certain physics exam at caltech, students were allowed to use any
>reference material that they could carry in to the room.  one
>enterprising guy supposedly carried richard feynman in.

Is it urban folklore if it's true and happened to you?

When I took differential geometry, I was the only undergrad in the
course. Grad students didn't have to take final exams, so I was the
only person taking the final. It was the only exam being given in the
exam room. About 5 minutes into the exam, the prof (can't remember who
it was, but obviously not Feynman ...) came in, sat down next to me,
and started discussing the exam with me.

Naturally enough, the proctor freaked out. The prof pointed out that
he had specified that it was an "open professor" exam. The admin
office had changed that to "open book" in the instructions to the
proctor. The prof had a hell of a time convincing the proctor that he
was the prof and it wasn't a joke.

It was probably the best exam I ever took, in terms of testing my
understanding of the material. Also one of the hardest.

dhs   spencer@panix.com
From: epstein@rene.ma.utexas.edu (Paul Epstein)
Newsgroups: sci.math
Subject: Re: Well-known people who majored in math (other than mathematicians)
Date: 5 Nov 92 20:21:09 GMT
Distribution: na
Organization: University of Texas at Austin Mathematics

shackle@csd4.csd.uwm.edu (David Henry Fetter) writes:

>Raymond Smullyan, who writes popular puzzle books and books on philosophy.
>By the way, he's also a meta-mathematician of some renown.

He might be insulted to learn you do not consider him a mathematician.
From: choy@skorpio.usask.ca (I am a terminator.)
Newsgroups: sci.math
Subject: Re: RFI: Naming of 16 & 32 sided regular polyons
Date: 7 Nov 92 00:28:17 GMT
Article-I.D.: access.1992Nov7.002817.17320
Organization: University of Saskatchewan, Saskatoon, Canada

In article <1992Oct28.183047.9150@math.utexas.edu>, akessler@bernard.ma.utexas.edu (Adam Kessler) writes:
|> In article <1992Oct28.100704.1@sscvx1.ssc.gov> schludermann@sscvx1.ssc.gov writes:
|> >  What are the names for 16 and 32 sided regular convex polygons and the
|> >recipocals. Thanxs in advance.
|> >
|> 
|> For 16-sided regular polygons with convex interiors I vote for Hexidecagons
|> and for 32-sided ones Sesquiduodecagons.  :-)

I call them all sexy polygons.

Henry Choy