HDTenEightyP/NetNews huggingface dataset:
From: hrubin@mentor.cc.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Re: nonstandard analysis
Date: 12 Dec 92 13:33:39 GMT
Organization: Purdue University Statistics Department

In article <1992Dec11.234035.1668@news.Hawaii.Edu> ross@tarski.tmc.edu (David Ross) writes:
>In article <24210@galaxy.ucr.edu> baez@guitar.ucr.edu (john baez) writes:

>>By now, so much has been done standardly that the advantages of nonstandard
>>analysis, if any, are not enough to make many mathematicians want to retool
>>and go nonstandard.

>There are many results in analysis, especially probability theory, for which
>the only known proofs use nonstandard analysis.  

This is at best illusory.  Nonstandard analysis is useful in finding theorems
and simplifying arguments, but with caveats involved in interpretation.

However, since any nonstandard proof of a standard theorem can be mechanically
translated into a standard proof, at most one can say that the standard proofs
have not been written, not that only a nonstandard proof is known.
-- 
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: doudd@sofya.math.byu.edu (Darrin Doud)
Newsgroups: sci.math
Subject: Re:       '-' operation
Date: 15 Dec 1992 22:40:15 GMT
Organization: Brigham Young University
Distribution: world
Reply-To: doudd@sofya.math.byu.edu (Darrin Doud)
NNTP-Posting-Host: sofya.math.byu.edu

>Recently my daughter brought back Problem Set II of Wisconsin Mathematics
>Science and Engineering Talent Search, and Question 4 in it is as following:
>         Operation @ satisfies the conditions that
>                X @ (Y @ Z) = X @ Y + Z and X @ X = 0 for any real numbers
>          X, Y, Z. Show that @ must be subtraction.
>   Since my daughter does not care much whether she has talent so I tried to
> solve it myself:-). It took me more than an hour but still I am not sure I get
> a right answer. The problem I have is: How is operation '-' defined? How about
> other operations like '+', '*', and '/'? Although I do not have Ph.D
> in math but I consider myself one of the best in learning math in those
> math classes. It is sort of ashameful that I never learned or remember
> how the basic operation is defined. For the above question, is only
> thing I have to prove is X @ 0 = X?, which is not very difficult because
> X @ 0 = X @ (X @ X) = X @ X + X = 0 + X = X.


Subtraction is defined to be the inverse operation of addition.  We do need to
prove more than that  X@0=X, although that is a helpful first step.  What we do 
need to do is show that X@Y=X-Y for any pair of numbers X and Y.  We can
do this as follows:

X@Y = (X@Y+Y)-Y = X (Y@Y)-Y = X@0 -Y = X-Y
From: blom@husc15.harvard.edu
Newsgroups: sci.math
Subject: The Mathematics of Secret Santa
Date: 18 Dec 92 03:04:16 GMT
Article-I.D.: husc15.1992Dec17.220416.18649
Organization: Harvard University Science Center

What is the probability that in a Secret Santa arrangement of n people, there
is at least one pair of participants with each other as Secret Santas?

No participant may have himself.

The total number of Secret Santa arrangements turns out to be the subfactorial
of the number of participants.  Here, the subfactorial of the number n is
defined as
                 k
         n   (-1)
!n = n! sum ------
        k=0   k!

For example, !5 = 5! (1/0! - 1/1! + 1/2! - 1/3! + 1/4! - 1/5!)
                = 120 (1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
                = 44.

The subfactorial, I think, should be the denominator of the fraction for the
probability with n people, but I can't figure out the numerator.

        ??
P(n) = ----
        !n

For the record, here are the first few values of P(n)*!n, which should be the
numerator of the fraction:

!2*P(2) = 1
!3*P(3) = 0
!4*P(4) = 3
!5*P(5) = 20
!6*P(6) = 105
!7*P(7) = 504

Eric Blom
Harvard U
Newsgroups: sci.math,sci.physics
From: mkohlhaa@silver.ucs.indiana.edu (mike)
Subject: Help me deal w/ infinity
Organization: Indiana University
Date: Mon, 21 Dec 1992 02:04:31 GMT

Some friends and I have recently been having some discussions/arguments 
about infinity and its meaning in math.

A core of our disagreement has been about .9(r) --that means .9repeating
but I can't make a little bar over the 9.  Anyhow, its been documented that
.9(r) is equal to 1.  Some of my friends think that this is some kind of 
a mathematical trick, given the following equation (keep in mind that (r) 
equals .9repeating):

			N=.9(r)
		      10N=9.9(r)   Multiply both sides by 10
	               9N=9        Subtract N from both sides
		        N=1

My skeptical friends point out the "subract N from both sides" lines as
the problem with the equation.  They think that it's bogus that I am 
litterally subrtracting an N from one side, but subtracting it's equivalent,
namely a .9(r) from the other side.

I appeal to any math gurus out there to comment on the validity of .9(r)
equals one, and anything else which may increase my understand of the
concept of infinity and how putting an "infinite" amount of 9's after
a decimal point can become 1.

Thanks




-- 
					-- Mike
-------------------------------------------------------------------------------
| If the above information is incorrect, then I don't even speak for myself.  |
|      mkohlhaa@silver.ucs.indiana.edu | mkohlhaa@amber.ucs.indiana.edu       |
Newsgroups: sci.math,rec.games.abstract
From: keving@cory.Berkeley.EDU (Kevin Gong)
Subject: Re: Game of pentominos
Organization: University of California, at Berkeley
Date: Wed, 16 Dec 1992 19:57:23 GMT

In article <1992Dec16.182611.10896@progress.com> neil@progress.COM (Neil Galarneau) writes:
>torbenm@diku.dk (Torben AEgidius Mogensen) writes:
>
>>martel@marvin.mr.sintef.no (Paulo Martel) writes:
>
>>>After several tries I gave up a combinatorial analysis of the game of
>>>pentominos. Would someone point me to a reference, or briefly explain
>>>how one could compute the total number of solutions for a grid of a
>>>given size (6x10, 5x12, 4x15, 3x20). 
>
>
>Ah... Good old pentominoes.
>
>Several years ago, motivated by a passage in one of Clarke's novels, several
>of us programmed pentominoes.
>
>It is not very hard to solve it, it is a lot harder to solve it efficiently.
>
>The programming was a lot of fun.

A great new book on polyominoes is "Polyominoes:  A Guide to Puzzles and
Problems in Tiling," by George E. Martin (1991, Mathematical Association of
America).  It says that there are 2339 solutions to 6x10, 1010 solutions to
5x12, 368 to 4x15, and 2 to 3x20.

Also, for those who are interested, I've written a program for the Macintosh
called "Polyominoes."  It has 15 different puzzles for pentominoes, and you
can play Golomb's pentomino game on any of the boards (he originated the
game on the simple 8x8 board).  I intend to later add support for pieces
other than pentominoes, and a puzzle solver.  If you'd like a copy, send
me e-mail (or see if it's still at sumex.stanford.edu, anonymous ftp).

 - kevin
Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: need proof:  (1 + 1/n)^n ==> e
Organization: Computer Science Department,  Stanford University.
Date: Wed, 16 Dec 1992 01:34:18 GMT

In article <1glt0lINNquc@news.aero.org> doner@Aero.org (John Doner) writes:
>In article <1glr2qINN2vg@usenet.INS.CWRU.Edu>, fau@po.CWRU.Edu (Francis A. Uy) writes:
>|> 
>|> I can't believe I forgot the proof of this one.
>|>  Subject line says it all.
>
>First tell us what you want to use for a definition of e.

Indeed.  Easiest would be to just take that limit for the definition.

One can also define e to be exp(1) where exp(x) is that fixpoint of
d/dx satisfying exp(0)=1.  In that case

d/dx (1+x/n)^n = (1+x/n)^{n-1} = n/(n+x) (1+x/n)^n.

So as n -> oo, this function tends to a fixpoint of d/dx, and is 1 at
x=0, hence must be exp(x).  Now set x=1.
-- 
Vaughan Pratt				There's safety in certain numbers.
Newsgroups: sci.math
From: jk87377@cc.tut.fi (Juhana Kouhia)
Subject: Re: Pi in beatiful form --lost
Organization: Tampere University of Technology
Date: Sat, 19 Dec 92 04:09:54 GMT


In article <1992Dec16.201741.24270@CSD-NewsHost.Stanford.EDU>
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>
>And then there is the oldest formula, Vieta's product from 1593:
>
>2/pi = sqrt(1/2) * sqrt((1+sqrt(1/2))/2) * ...


Hmm...

 pi = lim_{n \to \infty} 2^{n+1} sqrt(2 - sqrt(2+sqrt(2+sqrt(2+...))))
                                            ^^^^^^^^^^^^^^^^^^^^^^^^^^^
                                                     n items


Juhana Kouhia
From: vxc15931@vax1.mankato.msus.edu
Newsgroups: sci.math
Subject: RE: Jesus loves (?) you
Date: 11 Dec 92 19:49:59 GMT
Organization: Mankato State University

I wonder if Jesus has dealt with quantum mathematics.  I wonder if He started 
it all (together with God).  

If He did, then He sure has a funny way of loving us.


================

Jose' R.C. Cruz
Graduate Assistant
Mankato State University
Mankato, MN

===============

" Let God and Man decree
  Laws for themselves and not for me."
                - A.E. Housman
From: rmurphy@usc.edu (Bob Murphy)
Newsgroups: sci.math
Subject: arithmetic refresher
Date: 12 Dec 1992 19:44:19 -0800
Organization: University of Southern California, Los Angeles, CA
Distribution: world
NNTP-Posting-Host: aludra.usc.edu


I just saw a Dover book entitled "Arithmetic Refresher" by A.A. Klaf.
While skimming through the book I came upon the ridiculous question #530.

Q530  What is the procedure for squaring the mean of two numbers?

The answer is to multiply the two numbers and then add the square of
half the difference.

Well it's simple algebra to see that this is true but what a stupid 
approach!  Why not just take the mean of the two numbers and then
square the result?  I'm sure that books like this are sold mostly
to those whose math skills are poor.  Maybe this is one reason that
they would find math to be just a bunch of cryptic formulas
that one has to memorize.

                         Bob Murphy (rmurphy@aludra.usc.edu)

p.s.  The author also has written a calculus refresher but I didn't
      have time to look through it.

Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Function Terminology
Organization: Computer Science Department,  Stanford University.
Date: Sat, 12 Dec 1992 03:54:15 GMT

In article <Bz48w1.G5C@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>In article <1992Dec11.203802.1770@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>In article <1gaq3tINNg9q@uwm.edu> radcliff@csd4.csd.uwm.edu (David G Radcliffe) writes:
>>>Suppose I have a function f: A --> B, and C is a subset of B which
>>>contains the image set of f.  I define a function g: A --> C by
>>>setting g(a) = f(a) for all a in A.  Usually, f and g can be considered 
>>>as the same function, but sometimes the distinction is important.
>
>>>Is there a standard term or notation for this?
>
>>While I've not heard of one, one could borrow notation used to make the
>>analogous distinction for the integer 2 viewed as the real 2. and write
>>f:A->B as f. and f:A->C as just f without the point.
>
>The function is exactly the same.  A function, in whatever foundational
>system is used, is something which takes arguments in a domain and
>operates on them.  The image set depends only on f and A.  Which
>superset of the range is used does not affect the function.

As the questioner said, "Usually, f and g can be considered as the same
function, but sometimes the distinction is important."  It's a fair
question.  Saying that the choice of codomain makes no difference is
like saying that the integer 2 is identical to the real 2., or that "I
said that" is identical to "I didn't not say that."  In some settings
they are the same, in others not.

As a matter of hygiene, "corestriction" f|/C as subsetting the codomain
doesn't seem quite the right analogy for restriction, since the latter
is always applicable, whereas subsetting the codomain may take you
below its range, unless you don't mind it becoming a partial function.
The dual of restriction should be to quotient the domain.  Supersetting
the codomain should be the dual of lifting f:A->B to f':AxC->B where f'
ignores C.
-- 
Vaughan Pratt				All knowledge resides in the going odds
Newsgroups: sci.math,comp.edu,misc.education
From: prener@watson.ibm.com (Dan Prener)
Subject: Re: Education crisis (was RE: how much math...)
Date: Sat, 12 Dec 1992 06:58:05 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
	<Bz3LvL.L4C@mentor.cc.purdue.edu> <1gac6gINNfeo@agate.berkeley.edu>
Organization: IBM T.J. Watson Research Center, Hawthorne, New York

In article <1gac6gINNfeo@agate.berkeley.edu> bh@anarres.CS.Berkeley.EDU (Brian Harvey) writes:

> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
> > erh0362@tesla.njit.edu (Elliotte Rusty Harold) writes:
> >>If a child wants to learn about dinosaurs, let him or her learn about 
> >>dinosaurs.  If they want learn about football, let them learn about football.
> >
> >This ignores the problem of prerequisites.   The point of a curriculum is
> >so that the background is there when it is needed.  This holds even for
> >graduate students.

> Graduate students, yes.  Little kids, no.  It makes sense that as one gets
> older, more mature, more able to understand the delayed rewards of education,
> one can be expected to accept a more disciplined schooling.

   [ ... and more ... ]

For two reasons, I arrive at the opposite conclusion.

 (1)  Little kids are less likely to question authority.

 (2)  Little kids learn much faster.

For these two reasons, I think educators have not only the opportunity, but
the obligation to use their authority to cram as much knowledge(*) into
little kids' heads as they reasonably can.  This amounts to a blatantly
paternalistic "Learn this now, because I say so, rather than later, when
you realize it might be useful."  But the fact that little kids' learning
capacity is so much greater than that of older people makes it a kindness.
This argument is in no way specific to mathematics; it applies in many
disciplines.  Think how appropriate it is for languages.

(*)  I do not, of course, mean randomly or capriciously selected knowledge.
     However flawed it may be, the traditional selection for little kids
     is preferable to current fads, since one of the benefits of formal 
     education is its conservative nature. 
--
                                   Dan Prener (prener@watson.ibm.com)
From: andrew@rentec.com (Andrew Mullhaupt)
Newsgroups: sci.math
Subject: Re: Unsolved Integrals
Date: 14 Dec 92 17:17:21 GMT
Organization: Renaissance Technologies Corp., Setauket, NY.

In article <1992Dec10.204653.29154@pellns.alleg.edu> frisinv@alleg.edu writes:
>My hobby is working on solving unsolved integrals (no successes). Lately  
>I've been wondering if I am wasting my time (aside from practice at the  
>method of integration by tricks:)) Is there any way of testing if a  
>particular integral has a closed form solution? Has anyone heard of an  
>unsolved integral being solved recently?


There are some integrals which are impossible to express in closed form,
and this is usually determined by 'Liouville theory'. There is a lot
known about this. When you say _unsolved_ I assume you do not mean
_known to be impossible_. If you work on an _impossible_ integral, then
you do waste your time.

You should consult Zwillinger's _Handbook of Integration_ and then
Hardy and M. Riesz's Cambridge Tract about integration. Should more 
be required, there is the work of J. F. Ritt and more modern stuff
like the Risch algorithm for finding elementary closed forms for
integrals.

Later,
Andrew Mullhaupt


Newsgroups: sci.math
From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Define "Mathematics"
Organization: Computer Science Department,  Stanford University.
Date: Sat, 12 Dec 1992 06:37:25 GMT

In article <1g319dINNa9c@roundup.crhc.uiuc.edu> hougen@focus.csl.uiuc.edu (Darrell Roy Hougen) writes:
>khill@vax1.umkc.edu writes:
>
>% In article <ByMMA7.CA4@undergrad.math.waterloo.edu>, ajknott@descartes.uwaterlo
>% o.ca (Andy Knott) writes:
>%%I am shocked at the misinterpretation of the term "Mathematics" in the media
>%% and the general public.  I have asked several people, including Mathematics
>%% undergraduates, what the definition of Mathematics was. 
>% 	
>% 	Mathematics is undefineable!!!
>%
>% To ask what the definition of Mathematics is, is to as what Mathematics 
>% is about, and Mathematics is about everything and nothing simultaneously.
>
>[ Other stuff about God the Universe and everything deleted ]
>
>Definition (Webster's): Mathematics: (1) the science of numbers and
>their operations, interrelations, combinations, generalizations, and
>abstractions and of space configurations and their structure,
>measurement, transformations, and generalizations.
>
>Definition (my own): Mathematics: (1) a formal language for the
>discussion of logical propositions; (2) the science of formal
>propositions.

All good, except shouldn't you say something about math speeding up
these processes?

Which suggests something along the lines of "Mathematics is rapid
transit for gedanken space."  Or as I can imagine them putting it at a
certain car factory:

	Die Mathematik ist Fahrvergnugen fur Gedankenraum.

This demands to be made into a stanza, but my German muse appears to
have run off with a British soccer fan.  (We were never properly
introduced anyway, c'est la vie.)  Any German speakers care to
volunteer the next line, and/or repair the first?
-- 
Vaughan Pratt				Actions have consequences
Newsgroups: sci.math
Subject: Re: 1992 Putnam problems and unofficial solutions
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 8 Dec 92 17:34:02 GMT
Organization: Beckman Instruments, Inc.

In <1g0bmsINNh22@agate.berkeley.edu> brnstnd@ocf.berkeley.edu (Dan Bernstein) writes:

>As usual, first come the problems, then the problems with solutions.
>Send any followup remarks to the USENET newsgroup sci.math.


>Problem B6

>Let M be a set of real n by n matrices such that
>(i) I \in M, where I is the n by n identity matrix;
>(ii) if A \in M and B \in M, then either AB \in M or -AB \in M, but not both;
>(iii) if A \in M and B \in M, then either AB = BA or AB = -BA;
>(iv) if A \in M and A \noteq I, there is at least one B \in M such that
>     AB = -BA.

>Prove that M contains at most n^2 matrices.

But can it actually contain n^2 matrices?

I KNOW this is NOT part of the problem as stated, but I am a little
curious.
--
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.research
From: asimov@nas.nasa.gov (Daniel A. Asimov)
Subject: Foliations of the Plane
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: NAS, NASA Ames Research Center, Moffett Field, California
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 16 Dec 1992 20:44:28 GMT

Let F denote a smooth foliation of R^2.  

Does there necessarily exist a C^1 function g: R^2 -> R such that each 
leaf of F is a connected component of a level set of g ?  

If this is always possible, can g always be chosen so as to also be smooth?


Dan Asimov
Mail Stop T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000

asimov@nas.nasa.gov
(415) 604-4799
From: mso@ukc.ac.uk (Michael S. Ody)
Subject: Certain minimal surfaces in R^3
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Physics Lab, The University, Canterbury, Kent, UK
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Fri, 18 Dec 1992 11:23:41 GMT


I am interested in certain surfaces in R^3 defined by the following
metric g_ij and second fundamental form L_ij (i,j = 1,2) :

   g_ij = E(x,y) (dx^2 + dy^2)

   L_ij = c (dx^2 - dy^2).

Apologies for the abuse of notation.  The components are

   g_11 = g_22 = E(x,y)

   L_11 = -L_22 = c 

   g_12 = g_21 = L_12 = L_21 = 0

where E(x,y) is a function of x and y, and c is a constant.
This is a minimal surface because the mean curvature is zero:

   K_m = L_11 g_22 + g_11 L_22 = 0.

I want to know whether surfaces of this form have been studied before,
and, if so, references to the literature.

--
Michael S. Ody,
The Physics Laboratory,
University of Kent at Canterbury,
CT2 7NZ,  United Kingdom.

Newsgroups: sci.math.research
From: twk@emu.pmms.cam.ac.uk (T.W. Korner)
Subject: Cambridge Research Fellowships (revised)
Originator: dan@symcom.math.uiuc.edu
Summary: Revised guide now available
Followup-To: poster
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Department of Pure Mathematics, University of Cambridge
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Fri, 18 Dec 1992 15:43:05 GMT
Keywords: Post-doctoral research, fellowships, Cambridge University

I have just revised my guide to research fellowships to bring it up
to date with a number of recent advertisements.  

It is available by anonymous FTP from /pub/twk/fellow.tex on
can.pmms.cam.ac.uk (131.111.24.28) or hard copy from the address
below.

(From my previous post)
Every year the colleges of Cambridge University offer a number of
research fellowships intended for highly qualified candidates with,
or just about to finish, a Ph.D. (or equivalent).  The terms and
conditions vary from college to college, so I have compiled an
_informal_ and _unauthorised_ guide for applicants unfamiliar
with the Cambridge system (intended for mathematicians but of
interest, I hope, in all subjects).

T.W. K\"orner
Department of Pure Mathematics
University of Cambridge
16 Mill Lane
Cambridge CB2 1SB
U.K.



Newsgroups: sci.math.research
From: das18@cunixa.cc.columbia.edu (Debojyoti A Sarkar)
Subject: Help
Originator: dan@symcom.math.uiuc.edu
Reply-To: das18@cunixa.cc.columbia.edu (Debojyoti A Sarkar)
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Columbia University
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Fri, 11 Dec 1992 18:01:53 GMT


I have the following problem:

min J=summation t=0 to inf 1/2(r* - r(t))**2
s.t. q(t+1) = (1-d)q(t)+c(t)+h(t)
r(t)=f(q(t)), f'<0
h(t)=g(r(t-1))
q0 (initial q) given.

Is it possible to get a closed form solution for (c(t) and h(t)) t=0 to inf?
The control variables are c and h and the state variable is q.  Initially I
am interested in a deterministic setup, but later on I would like to have a
stochastic system where the uncertainty would entre as follows :

r(t)=f(q(t), w(t)) where w would be the source of uncertainty.





Newsgroups: sci.math.research