HDTenEightyP/NetNews huggingface dataset:
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Newsgroups: sci.math
Subject: Re: Generalizing Prime Numbers
Date: 10 Nov 92 00:26:22 GMT
Organization: U of Cambridge, England

In article <1992Nov8.191948.14975@athena.mit.edu>, frisch1@athena.mit.edu (Jonathan Katz) writes:

> I was wondering if anyone knows about any method of generalizing the concept of
> prime numbers to the complex plane.
> Is there more than one way to generalize it?
> Any help on the would be appreciated.

Yeah. Prime numbers live in the integers, which sit inside the rationals.
If you take any finite-degree field extension of the rationals (translation:
write down finitely many algebraic numbers, like "root(3)" or "the smallest
positive real root of x^5+x+1=0" and consider the set of linear combinations
of them with rational coefficients), then there is a corresponding "ring of
integers" inside it (namely the set of elements of the field which are roots
of polynomials with integral coefficients, whose "top" coefficient is 1).

An element of this ring is a "unit" if it has an inverse in the ring (note
that the units in the usual ring of integers are precisely 1 and -1). We say
that an element of the ring is "irreducible" (which is probably what you mean;
"prime" is in fact a different concept, for nasty technical reasons) if it
can't be written as the product of two elements neither of which is a unit.

Just about the simplest example is in the case of the so-called Gaussian
integers, namely the integers in the smallest field containing the rationals
and i. (i^2=-1). The integers here turn out to be exactly the numbers of the
form a+bi with a,b both integers (warning: things don't always work out in
quite such an obvious way). If you do a bit of work you can prove that the
(usual) primes which are of the form 4m+3 are still irreducible in this ring,
and that the ones of the form 4m+1 factor into two irreducible factors, and
that 2 (the only prime not of form 4m+1 or 4m+3 !) factors as (1+i)(1-i).

I said something about primes and irreducibles being different; I might as
well explain very briefly and handwavingly. It so happens that "irreducible"
isn't really quite the right concept in general; what we really need for
most purposes about prime numbers is not that they don't factor, but that
if a prime divides the product of two numbers then it divides one or other
of the two numbers. By a happy coincidence prime=irreducible in the usual
integers, but this doesn't always hold. (Example: if you throw in root(-5)
instead of root(-1), the integers are again the numbers of form a+bw where
a and b are integers (and w=root(-5)); now 6 is both 2x3 and (1+w)(1-w),
and 2,3,1+w,1-w are irreducible.) So in fact it's no longer the case that
every number factors in a unique way into irreducibles, which is a pain.
The way out involves things called "ideals", which you can think of as
non-existent divisors of numbers, sort of. (Very much sort of.) 

I hope this makes some kind of sense to you...

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: Q: Math in Financial World
Date: 11 Nov 92 15:45:14 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <8400002@acf3.NYU.EDU> wut@acf3.NYU.EDU (Tom Wu) writes:
>
> 	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?

The need is not so much to go beyond the MBA as to avoid it. The MBA
looks like a good idea only when you see people who have one doing well
in technical finance.  Most of these are _extremely_ smart people who
would do twice as well with a math degree.

Yes, if you want to be a top notch analyst, you will need a math Ph.D.
or something pretty close to it, like Physics, EE, etc. But right now
you should make sure you try and skew your undergrad program to look
as much like you are majoring in pure and applied math. You are at a
quite good school for applied math (NYU/Courant) and there are tons of
people there who could help. In fact, there are a couple of guys in the
Econ. department who could be very helpful to you.

Later,
Andrew Mullhaupt

N.B. I am an analyst with a math Ph. D. working in the financial world.
Newsgroups: sci.math,sci.physics,sci.astro,sci.bio,sci.chem,misc.education
From: solman@athena.mit.edu (Jason W Solinsky)
Subject: Re: 2-dimensional grading system (was Re: What can one do for the bright?)
Organization: Massachusetts Institute of Technology
Date: Fri, 13 Nov 1992 18:34:59 GMT

In article <1992Nov9.171141.5018@hsr.no>, onar@hsr.no (Onar Aam) writes:
|> 
|> >In article <1992Nov5.070342.6288@hsr.no>, onar@hsr.no (Onar Aam) writes:
|> >
|> >|> What the hell are you guys talking about?(!) This is EXACTLY why I proposed the
|> >|> 2D grading system in the first place. With today's grading system you only get
|> >|> one grade which includes both effort and grasp. With this system the employer
|> >|> wouldn't know whether his employee has brains or not. With a 2D system, however,
|> >|> the employer gets that vital information. 
|> >
|> >What are YOU talking about? The question that was being addressed is WHY WOULD
|> >ANYBODY CARE ABOUT THE EFFORT PUT IN CLASSES? It is additonally unclear whether
|> >A's would go to those people who put in the least effort, or if you would give
|> >A's to the people who are dumb and couldn't learn it in twice the time that other
|> >people could, but gave up a social life so they could get an A on the academic
|> >part.
|> >
|> >Jason W. Solinsky
|> 
|> Are you an employer? I find that very unlikely. Employers are in most cases
|> interested in people who are willing to work, not in some brilliant-but-lazy guy.
|> Someone with a grade: effort=A , understanding=C is probably VERY much wanted
|> by an employer. 

How does work relate to handing in homework assignments?

Your effort C, understanding A student ,ay be using his/her extra time to earn
money for a car. Is the A-A student that did the homework instead a harder
worker? In the case of a student capable of understanding A, work C, an employer
will prefer the the student that goes on to a new project when the original one
is completed. Employers do not want somebody who, when given an job to be
completed within 2 weeks, twidles his/her thumbs from the time it is completed
until two weeks are up. The employer will want somebody who will get started on
the next task.

Jason W. Solinsky
From: dchatterjee@kuhub.cc.ukans.edu
Newsgroups: sci.math
Subject: Presidential candidates and complex analysis :-)
Date: 4 Nov 92 22:26:16 CST
Organization: University of Kansas Academic Computing Services


   I read  some of the most interesting things about the three (although Bill
   has already won the scene). Maybe Bill sure has a rudimentary knowledge of
   complex variables. He was a Rhodes scholar at Oxford, right ? Now, it is a
   fact that E.T.Copson was a math. prof. at Oxford. (Perhaps E.T.C had retired
   at time when Bill was pursuing academics at Oxford.)

   Now, Copson and " Theory of Functions of a Complex Variable " are
   inseparable ! O my people of the complex book ! Canst thou deny that ?

   Thus, my assumption is that Clinton might know about complex variables.

   Bush does not know anything about complex variables, although SIAM news
   had sometime ago published a photograph of Bush presenting a prestigious
   award to George F.Carrier, of Harvard University. Now, as an engineer I
   had to go thru': Carrier, Krook and Pearson, " Functions of a Complex
   Variable: theory and technique ", Hod Books, N.Y., 1988. If there is any
   book so demanding than this, I would be very surprised. (The nature of 
   the writing is to provoke the reader to exercise his/ her intellect.) I love
   the book, though the problems are *r-e-a-l k-i-l-l-e-r-s*. At McGill, I had
   the misfortune of getting some the " easier " (quickies) from the book in an
   open book math. exam !!!!!!!!!!!!!!!!!! 

   However, presenting the award does not qualify the president to know that
   complex variables is basically " good applied mathematics ".

    Someone commented that Ross Perot was an engineer. Thus, it is highly 
    likely that Perot would know complex analysis ? BTW, which book could 
    he have read ? Alfhors ????

    Of all the three Perot seems most likely to have worked with complex
    variables. Clinton is expected to be aware of complex-analysis - at least
    as a general knowledge. Bush would be least interested about such stuff.
     
   - dc
     





Newsgroups: rec.puzzle,sci.math,comp.theory
From: ian@ponder.csci.unt.edu (Ian Parberry)
Subject: the 16-puzzle
Organization: University of North Texas, Denton
Date: Tue, 10 Nov 1992 21:22:24 GMT

The 16-puzzle consists of 15 tiles numbered 1-15, laid out on a 4x4
grid with one hole.  A "move" consists of sliding one of the adjacent tiles
into the hole.  The puzzle is set up by making a sequence of random moves,
and then attempting to return the tiles to the "origin", defined to be
row-major order with the hole in the bottom-right position.

The n^2 puzzle is the obvious generalization to an nxn grid.

I have 2 questions:
1.  What is the number of moves made by the best known algorithm?
2.  What is known about the group theory of the puzzle?

I have a naive algorithm for the n^2 puzzle that makes O(n^3) moves.
I plan to use it in class to illustrate algorithm design techniques.
The answer to Question 1 will tell me whether it is worthwhile
putting in a lot of effort to analyze or improve the algorithm.
The answer to Question 2 may help me to prove it correct.  It relies on a
(yet unproven) conjecture about the generators of a certain group.  I'm
too lazy to reinvent this from scratch, since I'd be very surprised if
somebody hasn't already answered all of the "interesting" questions
about the puzzle's group-theoretical foundations.  I remember seeing
a paper that described all of the permutations that can be reached from
the origin, but I've lost track of the reference.  This may contain the
algebraic tools that I need.  Any pointers to this and other references
would be appreciated.  The only papers I have found in the library use AI
techniques and hand-waving analyses.

Please respond by email.  I will summarize to the net if there is
sufficient interest.
____
Ian Parberry (ian@ponder.csci.unt.edu)
Dept. of Computer Sciences, Univ. of North Texas
"Bureaucracy is expanding to meet the needs of an expanding bureaucracy"
Organization: Penn State University
Date: Wed, 11 Nov 1992 20:59:10 EST
From: S. Wukovitz <SXW103@psuvm.psu.edu>
Newsgroups: sci.math
Subject: Re: a first year grad student freaks out.
 <BxK1uD.9Bz@mentor.cc.purdue.edu> <1992Nov11.223722.29808@galois.mit.edu>
 <BxKu9s.BAw@news.udel.edu>

Well, I'm only a second-year grad student, but here are some of the
things that helped me to find questions for my undergrad honors
thesis and also some things that will hopefully be publishable
soon:  Try to read as much as you can about something which interests
you or at least might *possibly* interest you, which seems maybe a
bit challenging but which you can still follow.  Whenever there is
a theorem or example which seems particularly interesting, ask
yourself questions like "Can this be generalized somehow?" and
"Can I make up more examples?"  and "Gee, all of these seem to
have a property in common.  Is that true in general?" and "What
more can I prove about this?"  and so forth.

I don't know how "real" mathematicians do it, but this seems to at
make reading more interesting for *me*, and has sometimes helped
to produce more lengthy investigations.

Hope this helps.  Maybe you will get advice from some more
experienced mathematicians...

-Stephanie Wukovitz
steph@math.cornell.edu
Newsgroups: sci.math
From: mroussel@alchemy.chem.utoronto.ca (Marc Roussel)
Subject: Re: Simple Diff Eq
Organization: Department of Chemistry, University of Toronto
Date: Mon, 9 Nov 1992 15:45:58 GMT

In article <1992Nov8.191725.14856@athena.mit.edu> frisch1@athena.mit.edu
(Jonathan Katz) writes:
>In article <13390031@hpspdla.spd.HP.COM>, ric@hpspdla.spd.HP.COM
>(Ric Peregrino) writes:
>|> df(x)/dx - 1/f(x) = 0
>
>I don't know what kind of information  you're looking for, but the answer is 
>f(x)=(2x)^.5 

     Careful there!  That should be f(x) = sqrt(2x + c) where c is
chosen to satisfy the initial or boundary conditions.

				Marc R. Roussel
                                mroussel@alchemy.chem.utoronto.ca
Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: What is catastrophe theory?
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 12 Nov 92 21:30:29 GMT

In article <1992Nov11.213905.44651@kuhub.cc.ukans.edu> dchatterjee@kuhub.cc.ukans.edu writes:
>In article <1992Nov11.180552.14797@progress.com>, neil@progress.COM (Neil Galarneau) writes:
>> I have heard of something called catastrophe theory.
>> 
>> I am looking for recommendations on articles/books (in English!) readable by a
>> non-math major.

Since someone said to ask me, I'll say what I know about catastrophe
theory books.  The father of catastrophe theory is the great French
topologist Rene Thom, but the great popularizer has been Zeeman.  I
don't know how far one can get into it with no math, but knowing Zeeman
there should be some sort of readable thing in:

Zeeman, E. C.
     Catastrophe theory : selected papers, 1972-1977 / E. C. Zeeman.
Reading,  Mass. : Addison-Wesley Pub. Co., Advanced Book Program, 1977.

Looking through the library catalog I see there's a fair number of books
on "catastrophe theory and X", so if you know about X you may find such
a book interesting.  For example, if X = "ecology" - 

 Jones, Dixon D.
      The application of catastrophe theory to ecological systems /
Dixon D. Jones.  [Laxenburg, Austria : Schloss Laxenburg], 1975.

Or if X = "cussing a blue streak" - 

 Berecz, John M. (John Michael)
     Understanding Tourette syndrome, obsessive compulsive disorder, and
related problems : a developmental and catastrophe theory perspective
John M.  Berecz.  New York : Springer Pub. Co., c1992.

There seem to be plenty of introductions like:

Saunders, P. T. (Peter Timothy), 1939-
      An introduction to catastrophe theory / P. T. Saunders.  Cambridge
[Eng.]  ; New York : Cambridge University Press, 1980.



but I have no idea which is best, or if there are any decent "pop"
introductions.  For mathematicians I would recommend Arnold, and for
anyone with an intellectual streak I would recommend going straight to
the horse's mouth and reading the incredibly fascinating and mysterious
book by Thom:

 Thom, Rene, 1923-
     Structural stability and morphogenesis; an outline of a general
theory of models. Translated from the French ed., as updated by the author, by
D. H. Fowler. With a foreword by C. H. Waddington.  [1st English ed.].
Reading, Mass., W. A. Benjamin, 1975.

This is likely to be rather difficult unless you are an expert in BOTH
category theory and developmental biology, but for that very reason it
is quite fun.
Newsgroups: sci.math
From: awpaeth@watpix.uwaterloo.ca (Alan Wm Paeth)
Subject: Caltech, Re: Feynman story
Organization: University of Waterloo
Date: Mon, 9 Nov 1992 21:36:55 GMT

>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.

I heard the story at Caltech in Spring 1975. Feynman was never mentioned --
it concerned a geology final with a grad student lugged in. The story is
somewhat plausable, given that (a) Caltech students are famous for pranks and
(b) open-book exams are quite common there.

If anyone wants to trace this back in hopes of finding an actual event, I can
provide the name of the math TA we called "Father Captain Doctor". (A Novitiate
in ROTC completing his PhD in Math).

   /Alan Paeth
   Computer Graphics Laboratory
   University of Waterloo
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Newsgroups: sci.math
Subject: Re: Trivial! (was: Re: Help X^2 == Y mod N)
Date: 5 Nov 92 03:13:26 GMT
Organization: Computer Science Department,  Stanford University.

In article <1992Nov5.001930.24516@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
>Some of you may remember the post in which I described a system of logic
>with a predicate T such that T(P) means "P is trivial".  T(P) -> P
>but not vice-versa.  (Well, in real life T(P) does not imply P, but
>we're just joking around here.)  T(P) -> T(T(P)), though.

Isn't T->TT contradicted by the example of Norbert Wiener?  After being
challenged about T(P) for some P that probably no one remembers any
more, Wiener wandered away in deep thought and returned 20 minutes
later to announce triumphantly that it was indeed trivial.

Or did you mean Trivial(P) -> Triumphantly(Trivial(P))?

(MIT is the official repository of Wiener stories, so I fully expect a
suitable counterstory from John.)
-- 
Vaughan Pratt				There's no truth in logic, son.
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Assorted questions and problems
Date: 9 Nov 92 16:14:40 GMT
Article-I.D.: netnews.96778
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: tao@fine.princeton.edu (Terry Tao)

In article <1992Nov8.181631.13298@Princeton.EDU>, tao@fine (Terry Tao) writes:
>I have three questions that I can't do.  I hope you can see from the
>diversity of them that they are not homework.

Not just the diversity.  They are on the deep side of things.

>(1) what is the current status of the Bieberbach conjecture, that any
>univalent holomorphic function f on the unit disk such that f(0) = 0 and
>f'(0) = 1 satisfies the fact that the taylor expansion f(x) = \sum a_n x^n
>has the property |a_n|  \leq n? The last I heard, it was proved for n up to
>7 only, and also for all n sufficiently large |a_n| \leq 1.08 n.

It was proven in the mid-80s, amidst a good deal of confusion and some
controversy.  It's already in book form--see, eg, the AMS symposium.

>(2) Suppose X and Y are Banach spaces.  Can one construct a linear mapping
>from X to Y which is NOT continous?e.g. a map from L^2 to L^2 which is not
>bounded.  Is it possible to construct one without AC?

L^2 is a Hilbert space even--not a good example.

This subject goes under the name of "automatic continuity".  I have no
idea of what happens without AC.  The strongest you can prove in ZFC is
the work of Bade--you can isolate a finite number of bad points.  The
work of Dales and Esterle gave counterexamples under CH.  Woodin and
Solovay showed that CH was necessary.  See the book by Dales and Woodin.

>(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.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Newsgroups: sci.math
From: rscott@csws17.ic.sunysb.edu (Robert Scott)
Subject: Re: Lie group terminology
Organization: State University of New York at Stony Brook
Date: Fri, 6 Nov 1992 05:50:48 GMT

In article <1992Nov5.003021.24615@galois.mit.edu> jbaez@riesz.mit.edu (John C.
Baez) writes:

>In article <1992Nov4.085734.3572@leland.Stanford.EDU>
ledwards@leland.Stanford.EDU (Laurence James Edwards) writes:
>>
>>I just happened across this paper circa 1968 which talks about "Lie group
>>germs". Can anybody tell me what these are? I took a look in a couple of
>>math dictionaries and books on group theory but I didn't find anything.
>>I know very little about group theory so explanations with a minimum
>>of jargon would be most helpful.
>
>Hmm, the terminology is not familiar to me, though I know what a Lie


I THINK THAT "LIE GROUP GERM" IS VERY CLOSE TO A TERMINOLOGY THAT IS USED
IN BOURBAKI'S ENGLISH-LANGUAGE VOLUME ON LIE GROUPS AND LIE ALGEBRAS.  IT
MEANS ABOUT WHAT JOHN EXPECTS IT TO MEAN, OF COURSE, BUT IT'S HANDLED IN A
VERY TECHNICAL WAY FOR SOME REASON.  (PERHAPS SO AS TO BE APPLICABLE TO THE
P-ADIC CASE, OR SOMETHING LIKE THAT.)



-JAMES DOLAN
Newsgroups: sci.math
From: torkel@sics.se (Torkel Franzen)
Subject: Re: Relatives with Crackpot Proofs
Organization: Swedish Institute of Computer Science, Kista
Date: Mon, 9 Nov 1992 22:15:01 GMT

In article <2105@celia.UUCP> keith@celia.UUCP (Keith Goldfarb) writes:

   >When I next see this guy,
   >he's going to ask me what I thought, and I really don't know how
   >I should deal with it.

  "Your proof made me realize that I am not nearly as clever as I thought
I was. I simply did not understand it. I can only suggest that you submit
it for the consideration of people who understand these things better
than I do. Please excuse me; I must retire to reconsider my chosen
career."
From: astein@nysernet.org (Alan Stein)
Newsgroups: sci.math
Subject: Re: DERIVATIVE OF X^X
Date: 6 Nov 92 01:25:30 GMT
Organization: NYSERNet, Inc.

nas_ps@jsc.nasa.gov (Pradip R. Shah) writes:

>This was the problem posed to us in my undergrad math class.  I do not remember
>now, how I solved it then (almost 15 years ago).  Here it goes:

>Using the definition of the derivative (i.e. df/dx = lim   {f(x+h) - f(x)}/h ),
>                                                     h->0  
>find the derivative of y w.r.t. x , given


>          y = x^x   (assume x is a real number and .ne. 0)

>Thanks in advance for your help.

Doing it straight from the definition would not be all that easy;
however, using either logarithmic differentiation or writing x^x as
e^{x \ln x} makes it routine.
-- 
Alan H. Stein   -----  AJIN   -----  astein@israel.nysernet.org
Newsgroups: sci.math
From: alex@cbnewsl.cb.att.com (D.Alex.Damman)
Subject: tile
Organization: AT&T Bell Laboratories
Distribution: na
Date: Wed, 11 Nov 1992 19:44:19 GMT
Keywords: software

I am writing a tiling editor which allows the user to assign colors to a
fundamental region which is automatically replicated.  Then it rotates the
color table:  whee!  Has anyone here written, purchased or used a similar
program?  A friend suggested that design or architectural folks might have
such a tool.  Bit replication a la Scott Kim's _Inversions_ would be helpful
also.

I have the Grunbaum & Shephard book on tilings.  Is this the ultimate or what?

Please reply by e-mail as I do not normally read this entire newsgroup.
-- 
Alex	AMA 526543
Newsgroups: sci.math