HDTenEightyP/NetNews huggingface dataset:
From: holmes@opal.idbsu.edu (Randall Holmes)
Newsgroups: sci.math
Subject: Re: Topos theory for mere mortals (?)
Date: 29 Sep 92 16:12:30 GMT
Organization: Boise State University

In article <1992Sep27.232459.15896@husc3.harvard.edu> zeleny@husc9.harvard.edu (Michael Zeleny) writes:
>In article <19qoafINNiqf@usenet.INS.CWRU.Edu> 
>cxm7@po.CWRU.Edu (Colin Mclarty) writes:
>
>>	There is a difference though for Platonists.  Set theoretic
>>realists generally think there is some one universe of all sets, AC 
>>is true in it or is not, and that universe is really the only place 
>>to work.  So the question of what to "postulate" may be hard but
>>it has some right answer--they are not "postulates" at all in the
>>sense of "postulate theorists" like Huntingdon (of the many 
>>axiomatizations of Boolean algebra).  Zeleny has made this point 
>>well in another context.  Category theorists, even if realists, tend 
>>to think it is worth working in all kinds of various ambient 
>>categories for various purposes.  So that you always do need to say 
>>what you postulate about your present ambient.
>
>I have hesitated for a long time before following up on this claim,
>especially since I don't recollect too well just what point it is that I
>made in another context.  However, I wish to go on record as stating that,
>though it probably is a sociological fact that most realists who bother
>thinking about it, think that there is exactly one universe of all sets,
>and that, conversely, some people have gone on record announcing their
>conversion to formalism on the grounds of their loss of belief in the
>unique existence of such a universe, I do not myself believe that a realist
>philosophy of mathematics cannot accomodate a multiplicity of set-theoretic
>(or type-theoretic) universes possessed of different size and structural
>properties.  There is, after all, no good reason to suppose that everything
>in the mathematical universe is a set, or equivalently, that the
>development of a particular set theory is bound to yield privileged
>insights into the nature of all mathematical objects.  So your contrast
>between set theory and category theory may be right, but only for
>accidental reasons.
>
>>Colin McLarty
>
>cordially,
>mikhail zeleny@husc.harvard.edu
>"Un de mes plus grands plaisirs est de jurer Dieu quand je bande."

I first responded to what looks like an iterated reply to this
posting; I think I agree with what Mikhail is saying here.  I am as
rabid a realist as exists, but I think that the one real mathematical
universe can be looked at from a lot of different angles...


-- 
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
Newsgroups: sci.math.research
From: sutin@helios.ucsc.edu (Brian Sutin)
Subject: Bochner's Theorem on S2
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Lick Observatory UCSC
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 1 Oct 1992 01:57:27 GMT

Bochner's Theorem states the optimal condition on the fourier coefficients
for a function to be non-negative (Rudin, Real Analysis, p290).  Has anyone
ever heard of a similar result for orthogonal functions other than e^iwt?
I actually need something for the spherical harmonics on the surface of the
sphere, which are e^imt times Legendre functions, P(l,m).

Brian Sutin				sutin@helios.ucsc.edu
Lick Observatory, UCSC			Santa Cruz, CA 95064
    Fred:   "May I rescue you?"
    Ginger: "No, thank you.  I prefer being in distress."
From: mike@park.bu.edu (Michael Cohen)
Newsgroups: sci.math.research
Subject: Permutation Polytopes
Summary: Polytopes, extreme points
Keywords: Extreme points
Date: 3 Oct 92 01:26:16 GMT
Followup-To: sci.math
Organization: Boston University
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu


Treat the orderings on n letters as permutations.
For each ordering a, define a point in n X n-1/2
space as follows.

U_a(i,j) = 1 if i a j
U_a(i,j) = -1 otherwise.

Consider the convex polytope U which is a convex combination of these
extreme points.  Question:  What is the polar polyhedron.  i.e. the
extreme points of the the of elements <l, u> <= 1. for all u in U.
            -mike



-- 
Boston University (617-353-7857) Email: mike@park.bu.edu
Smail: Michael Cohen                     111 Cummington Street, RM 242
       Center for Adaptive Systems        Boston, Mass 02215
       Boston University
Newsgroups: sci.math.research
From: john@aquarius.unm.edu (John Prentice)
Subject: Coefficients for 12th order symmetric multistep method?
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Dept. of Physics & Astro, University of New Mexico, Albuquerque
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Thu, 1 Oct 1992 00:00:42 GMT

I am interested in using a symmetric method to integrate a second
order differential equation in time (we are integrating the
equations of motion of each of the planets in the solar system).
The symmetric method can be written:

  x(n+1) = - sum(j=1 to k) a(k-j) x(n-j+1) +
                   h**2 sum(j=1 to k) b(k-j) f(n-j+1)

where we are solving x''(t) = f(x,t), x(n) is the solution
on the nth time step, and a and b are coefficients (reference:
"A Three Million Year Integration of the Earth's Orbit",
T. R. Quinn, S. Tremaine, and M. Duncan, The Astronomical
Journal, vol 101, no 4, June 1991, pages 2287-2305).
We are interested in reproducing some of the results in
the paper I listed as the reference.  They used k=12,
which means that the coefficients a and b were chosen
to that the method integrates a 13th order polynomial
exactly.  Note, a(j) = a(k-j), b(j) = b(k-j), a(0) = 1,
and b(0) = 1.  So, there are 12 coefficients I need.
Obviously I can work out these coefficients from scratch,
but if anyone happens to have them handy, you could save me
alot of effort if you could kindly send me a list.  Many
thanks!

John
-- 
Dr. John K. Prentice                      
Partner, Quetzal Computational Associates    
3200 Carlisle N.E., Albuquerque, NM   87110-1664
Phone: 505-889-4543  E-mail: john@aquarius.unm.edu -or- jkprent@cs.sandia.gov

Newsgroups: sci.math.research
From: rgep@emu.pmms.cam.ac.uk (Richard Pinch)
Subject: Re: Complete Groups / S_6
Summary: Automorphism of S6
X-Submissions-To: sci-math-research@uiuc.edu
Organization: DPMMS University of Cambridge
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 29 Sep 1992 10:41:54 GMT

According to the "Atlas of finite groups" by Conway et al.
 "The outer automorphism of S_6 interchanges the 15 duads ab (pairs)
  with the 15 synthemes ab.cd.ef (trisections), so interchanges the
  6 points a (which could be specified by 5 duads ab,ac,ad,ae,af)
  with the 6 totals (sets of 5 synthemes containing all duads)."

Richard Pinch, Dept of Pure Mathematics, University of Cambridge
               16 Mill Lane, Cambridge CB2 1SB, U.K.

Newsgroups: sci.math.research
From: "John C. Baez" <jbaez@BOURBAKI.MIT.EDU>
Subject: Re: Complete Groups / S_6
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 23:30:51 GMT

In article <1992Sep28.201204.13907@tamsun.tamu.edu> rlm7638@tamsun.tamu.edu (Jack McKinney) writes:
>     A complete group is one that is isomorphic to its group of
>Automorphisms. The symmetric group on n elements is complete for
>all integers n _except_ 2 and 6. It is also true that the outer
>automorphisms of S_6 form one coset to the set of inner automorphisms.
>Can someone provide an outer automorphism on S_6 ? (This would
>completely specify Aut(S_6), which is isomorphic to S_6 x Z_2)

There was a thread on sci.math where we talked about this.  There is
a nice construction due to Sylvester and Coxeter.  It has some cute
relationships to the icosahedron.  I will send you (and anyone else
who wants) what I have about this.
From: gouvea@zariski.harvard.edu (Fernando Gouvea)
Newsgroups: sci.math.research
Subject: Formal cohomology
Date: 2 Oct 92 16:57:39 GMT
Article-I.D.: husc3.1992Oct2.125739.16026
Organization: Colby College Dept. of Math/CS
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

I've been reading the papers of Washnitzer and Monsky on "formal
cohomology", and I was wondering where the theory has gone since the
foundational papers. Are there current references?

A particular question that occurred to me is the following. In their
papers, Washnitzer and Monsky construct the cohomology of an affine variety
in characteristic p by lifting its coordinate ring (very carefully) to
characteristic zero. Has anyone looked for a "quasi-coherent" version,
where modules over the coordinate ring are lifted to characteristic zero? 


-- 
============================================================================
Fernando Q. Gouvea				fqgouvea@colby.edu
Dept. of Math/CS				(207)-872-3278
Colby College					Mudd 407
Newsgroups: sci.math.research
From: rlm7638@tamsun.tamu.edu (Jack McKinney)
Subject: Hamiltonian Groups
Followup-To: poster
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Texas A&M University, College Station
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 28 Sep 1992 20:09:01 GMT

    Can someone provide me with a list of some Hamiltonian groups?
[A Hamiltonian group is a nonabelian group such that every subgroup
is normal].

+--------------------------------------------------------+--------------------+
| Martin: Have we done this before?                      | Jack McKinney      |
| Halsey: Are we doing this now?    -from Brain Dead     | jmckinney@tamu.edu |
+--------------------------------------------------------+--------------------+

Newsgroups: sci.math.research
From: werner@pell.anu.edu.au (Werner Nickel)
Subject: Re: Hamiltonian Groups
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Australian National University
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 29 Sep 1992 15:59:10 GMT


>From M.Hall, Jr., The Theory of Groups, p. 190:

Theorem 12.5.4   A Hamiltonian group is the direct product of a
quaternion group with an Abelian group in which every element is
of finite odd order and an Abelian group of exponent 2.

Werner Nickel
Mathematics Research Section
Australian National University

Newsgroups: sci.math.research
From: cjf@maths.uq.oz.au (Chris Fama)
Subject: Request for information about research groups in math.physics
Summary: as the subject says!
Followup-To: poster
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Maths, University of Queensland
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 28 Sep 1992 03:59:35 GMT
Keywords: research,mathematics,physics


I am a graduate student in Mathematics, soon to complete a (first
class, touch wood) honours year.  I would like to hear about active
research groups in such areas as
quantum field theory, quantum gravity, conformal and differential
geometry (and twistor geometry, a subject which I have been looking at
this year and enjoy immensely), in U.K. and Canadian universities.
(OK, you guessed it, I could probably be excited by quite a lot of 
areas in mathematical physics.  Knot theory and lattice models, for
instance.)

   I am about to apply for a Commonwealth Scholarship & Fellowship Plan,
which requires three nominated institutions for study.  I have some idea
of those in the U.K. which would be good (well, any of the big ones!) -
of Canadian institutions though I know nothing.  The application closes
on Wednesday - could some kind souls let me know of largish  research
groups, willing to take on doctoral students?
   (The question of whether I am actually OFFERED the award, of course,
is quite another thing!  Admission to the relvant course is also a
tiny bit uncertain.)

   In view of the deadline I mentioned, could you please E-Mail me
rather than post?  (I doubt this would be of general interest
anyway.)  [If you are reading this AFTER the "deadline" of Wednesday
30/9/92, then I would appreciate a response anyway.  Responses about
U.S. institutions would be welcome too, although that's a little less
urgent.)

Thanks a lot.
Chris.
-- 
+-------------------------------------------------------------------------+
| Chris Fama, maths student.                          cjf@maths.uq.edu.au |
|         "Is this a piece of your brain?" - Basil Fawlty                 |
+-------------------------------------------------------------------------+

Newsgroups: sci.math.research
From: martin@math.rwth-aachen.de (  Martin Schoenert)
Subject: Re: Complete Groups / S_6
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 29 Sep 1992 13:11:47 GMT

In his article of 29-Sep-92 Jack McKinney writes:

         A complete group is one that is isomorphic to its group of
    Automorphisms. The symmetric group on n elements is complete for
    all integers n _except_ 2 and 6. It is also true that the outer
    automorphisms of S_6 form one coset to the set of inner automorphisms.
    Can someone provide an outer automorphism on S_6 ? (This would
    completely specify Aut(S_6), which is isomorphic to S_6 x Z_2)

S_6 operates on 6 points:

    1  2  3  4  5  6

Take the 15 duads (2-sets):

    12  13  14  15  16  23  24  25  26  34  35  36  45  46  56

Then there are 15 synthemes (3-sets of disjoint duads):

    12 34 56   13 24 56   14 23 56   15 23 46   16 23 45
    12 35 46   13 25 46   14 25 36   15 24 36   16 24 35
    12 36 45   13 26 45   14 26 35   15 26 34   16 25 34

Take the 6 pentads  (5-sets of synthemes which contain each duad exactely
once):

    12 34 56  13 25 46  14 26 35  15 24 36  16 23 45   (1')
    12 34 56  13 26 45  14 25 36  15 23 46  16 24 35   (2')
    12 35 46  13 24 56  14 25 36  15 26 34  16 23 45   (3')
    12 35 46  13 26 45  14 23 56  15 24 36  16 25 34   (4')
    12 36 45  13 24 56  14 26 35  15 23 46  16 25 34   (5')
    12 36 45  13 25 46  14 23 56  15 26 34  16 24 35   (6')

S_6  operates on those 6 pentads  and therefore on [1'..6'].  This  gives
the outer automorphism of S_6.

    (1,2)         -> (1',2')(3',4')(5',6')
    (1,2,3,4,5,6) -> (1')(2',3')(4',5',6')

There was a very nice article in 'sci.math' by John  C. Baez called "Some
thoughts on the number 6" about 4  months  ago,  from  which  most of the
above  is taken.  If you are  interested I  will see whether I can  still
find it and send it to you.

Martin.

--
Martin Sch"onert,   Martin.Schoenert@Math.RWTH-Aachen.DE,  +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany
From: bdest+@pitt.edu (Benoit Desjardins)
Newsgroups: sci.math.research
Subject: information request: biomathematics / mathematical biology
Date: 6 Oct 92 05:16:37 GMT
Article-I.D.: blue.6887
Followup-To: poster
Organization: University of Pittsburgh
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu


I would like to know what are the good research centers (for
graduate training and postdocs) in biomathematics (mathematical
biology).  Can someone provide me with this information? Also, does
anyone know about the research activities of Oxford's Center for
Mathematical Biology?

                                       Thank you

                                       B. Desjardins
                                       benoit@med.pitt.edu

P.S. Please reply to "benoit@med.pitt.edu"
From: aap@cs.brown.edu (Andrea A. Pietracaprina)
Newsgroups: sci.math.research
Subject: Field Theory Question
Date: 2 Oct 92 18:40:06 GMT
Article-I.D.: cs.1992Oct2.184006.24417
Organization: Brown Computer Science Dept.
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

I have the following question:

Let F_q be the finite field with q = p^n elements (p being any prime),
and let x be a generator for the multiplicative group of Fq. Assume
that the elements of F_q be represented by polynomials in x over F_p
of degree < n. Is there a way to know, without carrying explicit
calculations, the powers of x that correspond to polynomials of degree 1, 
that is, the integers k such that  x^k = ax+b, a,b in F_p ? Are these
*special* integers?

Thanx.

-andrea
Newsgroups: sci.math.research
From: sl25@emu.pmms.cam.ac.uk (SteveMMS Linton)
Newsgroups: sci.math.research
Subject: Re: Complete Groups / S_6
Date: 30 Sep 92 01:17:23 GMT
Organization: Cambridge University D
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

Apologies for the notational error in my posting. What I called synthemes are
totals, each made up of five synthemes.
Newsgroups: sci.math.research
From: sl25@emu.pmms.cam.ac.uk (Steve Linton)
Subject: Re: Complete Groups / S_6
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Cambridge University DPMMS
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Tue, 29 Sep 1992 07:04:07 GMT


|> 
|>      A complete group is one that is isomorphic to its group of
|> Automorphisms. The symmetric group on n elements is complete for
|> all integers n _except_ 2 and 6. It is also true that the outer
|> automorphisms of S_6 form one coset to the set of inner automorphisms.
|> Can someone provide an outer automorphism on S_6 ? (This would
|> completely specify Aut(S_6), which is isomorphic to S_6 x Z_2)

No it isn't. It is of order 240 and contains a normal subgroup isomorphic to S_6, but it is not isomorphic to S_6 x Z_2.

I can't remember exactly how the outer automorphism acts, but you can reconstruct
it by observing that S_6 contains just six synthemes. A syntheme is a set of five elements of the form (a,b)(c,d)(e,f) which together contain all fifteen possible pairs eg:

(1,2)(3,4)(5,6)  (1,3)(2,5)(4,6) (1,4)(2,6)(3,5) (1,5)(2,4)(3,6) and
(1,6)(2,3)(4,5)

Any element of S_6 permutes these six synthemes by conjugation and the permutation it performs can be taken to be its image under the automorphism.

		
			Steve Linton

Newsgroups: sci.math.research
From: dscott@ccu.umanitoba.ca (Doug Scott)
Subject: band width minimization
Summary: band width minimitation?
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Manitoba, Winnipeg, Canada
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Sun, 27 Sep 1992 21:00:40 GMT
Keywords: band width

I am presently developing a bandwidth/profile minimization
algorithm that will hopefully be superior to the Gibbs-Poole-
Stockmeyer algorithm, faster than B.Armstrongs or Burgess-Lai.

I would like to talk to anyone else doing research in this field.

Also, If you have some good test problems <1000 nodes or elements
I would be interested in using them also.

Doug G. Scott
Dept. Mechanical Engr.
U. of Manitoba
Winnipeg Manitoba

dscott@ccu.umanitoba.ca



Newsgroups: sci.math.research
From: rlm7638@tamsun.tamu.edu (Jack McKinney)
Subject: Complete Groups / S_6
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Texas A&M University, College Station
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 28 Sep 1992 20:12:57 GMT

     A complete group is one that is isomorphic to its group of
Automorphisms. The symmetric group on n elements is complete for
all integers n _except_ 2 and 6. It is also true that the outer
automorphisms of S_6 form one coset to the set of inner automorphisms.
Can someone provide an outer automorphism on S_6 ? (This would
completely specify Aut(S_6), which is isomorphic to S_6 x Z_2)

+-----------------------------------------------+-------------------------+
| "A container of ashes might one day be thrown | Jack McKinney           |
|   from the sky, which could burn the land and | jmckinney@tamu.edu      |
|   boil the oceans."                           +-------------------------+
|                -From the Hopi prophecy        |      Koyaanisqatsi      |
+-----------------------------------------------+-------------------------+
From: rgep@emu.pmms.cam.ac.uk (Richard Pinch)
Newsgroups: sci.math.research
Subject: Carmichael numbers up to 10^16
Keywords: Carmichael numbers
Date: 5 Oct 92 15:00:53 GMT
Article-I.D.: infodev.1992Oct5.150053.15133
Organization: DPMMS University of Cambridge
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu

CARMICHAEL NUMBERS UP TO 10^16 (preliminary announcement)

There are 246683 Carmichael numbers up to 10^16.
I hope this list is complete, but it has not yet
been completely checked.

The list is avilable by anonymous FTP from 131.111.24.28,
can.pmms.cam.ac.uk, in file /pub/Carmichael/carmichael-16.

Richard Pinch
Dept of Pure Mathematics
University of Cambridge
16 Mill Lane
Cambridge  CB2 1SB
U.K.

5 October 1992 15:50

The number of Carmichael numbers less than M with k prime factors:

  k   10^12   10^13   10^14   10^15   10^16
  -   -----   -----   -----   -----   -----
  3    1000    1858    3284    6083   10816
  4    2102    3639    6042    9938   16202
  5    3156    7082   14938   29282   55012
  6    1714    5270   14401   36907   86696
  7     262    1340    5359   19210   60150
  8       7      89     655    3622   16348
  9       0       1      27     170    1436
 10       0       0       0       0      23
 11       0       0       0       0       0
 12       0       0       0       0       0
 13       0       0       0       0       0
      -----   -----   -----   -----   -----
       8241   19279   44706  105212  246683


The ratio C(10^n)/C(10^n-1)

  n   C(10^n) C(10^n-1)  ratio
  -    ------    ------  -----
  4         7         1  7.000
  5        16         7  2.286
  6        43        16  2.688
  7       105        43  2.441
  8       255       105  2.429
  9       646       255  2.533
 10      1547       646  2.395
 11      3605      1547  2.330
 12      8241      3605  2.286
 13     19279      8241  2.339
 14     44706     19279  2.319
 15    105212     44706  2.353
 16    246683    105212  2.345

The function k(x) of Pomerance/Selfridge/Wagstaff

  n    k(x)
  -  -------
  3  2.93319
  4  2.19547
  5  2.07632
  6  1.97946
  7  1.93388
  8  1.90495
  9  1.87989
 10  1.86870
 11  1.86421
 12  1.86377
 13  1.86240
 14  1.86293
 15  1.86301
 16  1.86406
From: guy@casbah.acns.nwu.edu (Guy Metcalfe)
Newsgroups: sci.math.research
Subject: splines in multiply connected regions
Date: 29 Sep 92 19:07:09 GMT
Article-I.D.: news.1992Sep29.190709.2393
Organization: Northwestern University, Evanston Illinois.
X-Submissions-To: sci-math-research@uiuc.edu
X-Administrivia-To: sci-math-research-request@uiuc.edu


Is it known how to construct a spline in a multiply connected region, for
instance the area between concentric cylinders or in a journal bearing?
The problem with standard constructions is that the spline function will
somtimes leave the allowed domain.  Any references or software would be
appreciated.  Thanks.
-- 
	Guy Metcalfe
	Northwestern University		guy@chaos.chem-eng.nwu.edu	
	Dept. Chem. Eng.		
Newsgroups: sci.math.research