HDTenEightyP/NetNews huggingface dataset:
From: johnmpaz@phoenix.princeton.edu (John Manuel Paz)
Subject: nearest points
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Princeton University
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 30 Dec 1992 03:25:10 GMT
Keywords: number theory

Hi,
	In "Number Theory" Hardy presents a result which shows that the
closest rational approximation to an irrational number is:

	| r - p/q | <= 1/q^2,	r - irrational, 0<p<=q, p, q rational

which is the same thing as saying the closest point of an approximating
lattice to a line is O(1/q^2).  I was wondering if anyone could direct
me to any results concerning circles or any curves with curvature.
Thanks in advance.

Newsgroups: sci.math.research
From: robert@capmkt.com
Subject: Re: Variational Problem
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Illinois at Urbana
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Mon, 28 Dec 1992 19:10:44 GMT


A very simple solution is to consider the level curves {P(a,b)=c}
and to choose L to be the level curve corresponding the largest value of
c for which the probability mass interior to the level curve is F. The
area enclosed by L is also the minimum such area for the given probability
mass. The reason this algorithm works is the same as any other "greedy"
algorithm, area is only traded for the maximum amount of probability mass
avalilable at all times.

Newsgroups: sci.math.research
From: cgrinst1@cc.swarthmore.edu (Charles M. Grinstead)
Subject: Edgeworth Expansions and Normal Approximation
Originator: dan@symcom.math.uiuc.edu
Followup-To: poster
X-Submissions-To: sci-math-research@uiuc.edu
Organization: Swarthmore College
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 30 Dec 1992 22:21:24 GMT

Is there anyone out there who is familiar with Edgeworth expansions, and in
particular, familiar with the book "Normal Approximation and Asymptotic
Expansions" by Bhattacharya and Rao?  I have a problem to which I have
applied Corollary 23.2 in this book, and I'm getting an answer that I have
reason to believe is incorrect.  I would love to pick someone's brain using
email about this problem, and I think that someone who understands
asymptotic expansions of distribution functions should be able to answer my
question relatively quickly.  Feel free to respond via email; my address
is: cgrinst1@cc.swarthmore.edu . 
Thanks.

Newsgroups: sci.math.research
From: phillips@bright.uoregon.edu (Chris Phillips)
Subject: Algebraic topology question
Originator: dan@symcom.math.uiuc.edu
Followup-To: poster
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Illinois at Urbana
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Sat, 2 Jan 1993 01:27:47 GMT

Let P_n be the product of n copies of the 2-sphere S^2. In the
singular cohomology group H^2 (P_n, Z/rZ), let g_1, ..., g_n be the
generators corresponding to the factors of P_n. I have a topological
space E, an element x in H^2 (E, Z/rZ), and a continuous map f from
P_n to E such that f*(x) = g_1 + ... + g_n. Here E is an open subset
of a compact manifold, and even probably has finitely generated
cohomology. The object is to get lower bounds on the dimension of the
manifold that contains E, and which go to infinity when n goes to
infinity.

A while ago, I did something similar using Z coefficients instead of Z/rZ.
In this case, f*(x^n) = (g_1 + ... + g_n)^n, which is n! times a generator
of H^2n (P_n, Z) = Z. So H^2n (E, Z) is not zero. In the present case, r
is fixed (and presumably much smaller than n), but (g_1 + ... + g_n)^n
is zero as soon as r divides n!. If r is prime, this means the method only
works for n<r. In that case, if n=r then a cohomology operation called a
generalized Pontrjagin power, when applied to g_1 + ... + g_r, yields a
nonzero element of H^2r (P_r, Z/r^2Z). Therefore H^2r (E, Z/r^2Z) is not
zero. But even this seems not to be helpful for n>r.

What I am really hoping to find is a cohomology operation which one can
apply to g_1 + ... + g_n and get a nonzero element of H^2n (P_n, A) for
some coefficient group A, for n >> r. Of course, it doesn't have to be
exactly H^2n; anything close would help. Any other ideas would also be
appreciated. Of course, what I want may be false. If somebody can produce
for me an open subset U of R^5, say, and a continuous map f from
S^2 x S^2 x S^2 to U such that the restrictions of f to the sets
S^2 x point x point, point x S^2 x point, and point x point x S^2 are
all homotopically nontrivial, then I would probably give up on this
approach entirely.

Two additional items that might possibly be relevant: E is probably
simply connected, and H_2 (E, Z) is probably isomorphic to Z/rZ. (So
the U above should really be simply connected.)

Please reply to phillips@math.uoregon.edu. Thanks.

Newsgroups: sci.math.research
From: mcintosh@ccwf.cc.utexas.edu (aubrey mcintosh)
Subject: Eigenvectors by orthogonalization of columns.
Posted-Date:  30 Dec 92 19:36:14 GMT
Originator: dan@symcom.math.uiuc.edu
Followup-To: sci.math.num-analysis
X-Submissions-To: sci-math-research@uiuc.edu
Organization: University of Texas, Chemistry
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 30 Dec 1992 19:36:14 GMT


In "Computational Methods of Linear Algebra," by D.K. Faddeev and 
V.N. Faddeeva, translated by Robert C. Williams, W. H. Freeman and 
Company, original publication date 1960, some methods for 
eigenvalue/eigenvector calculations are given.  In particular, the 
method of minimal iterations is given, along with a method of 
orthogonalization of columns method.  I want to pursue the 
calculation of eigenvectors from the orthogonalization of columns 
method, but I haven't worked out the polynomials that would 
correspond to this. 

A similar orthogonalization of rows method is cited as Johann 
Schroeder, Arch. Math., 1953, 4, 322-326.  My German isn't fluent, 
but I think this article should have cited with an earlier chapter.  
I remain unenlightened after viewing it.

Many of the references in this book are in the mid to late 50's, 
e.g. this was an active area of research at publication time.  One 
name that I recognized was J.L.Synge.  So I presume that more work 
is being done.

I'm wondering if there is more recent literature that someone could 
refer me to, or if someone has the polynomials worked out.  I'm 
somewhat excited about this, because the initial orthogonalization 
with pivoting has yielded essentially the correct eigenvalue and 
eigenvector spectrum of a 108x108 correlation matrix from some of my 
lab data, with only a trivial amount of computation time. Although I 
could code an iterative method today to refine these initial values, 
I would like to investigate an exact method.  

I'm looking for real-time with a 650x650 on a PC box...
-- 
Aubrey McIntosh  /  Chemistry  /  University of Texas  /  Austin, TX 78712
..another Gaelic learner...

Newsgroups: sci.math,misc.books.technical
From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Re: High Prices of Math Books. I am pissed.
Organization: M.I.T. Artificial Intelligence Lab.
Distribution: sci
Date: Mon, 28 Dec 1992 11:46:24 GMT


MIR publishers managed to sell bargains in the US. I don't know if they
are still doing that. They distributed their books all over the world,
including (to my personal knowledge) US, France, Belgium and India.
In the days when I could still afford to buy books, I picked up
Differential and Integral Calculus by Ya.S.Bugrov and S.M.Nikolsky
for about $9 in Barnes and Noble after they had marked it up; 
Physique des particules elementaires, by N.Nelipa in France for about
68 francs; Les lasers a semiconducteurs, by O.Bogdankevitch, S.Darznek
and P.Elisseev for about 3 Belgian francs, and many similar bargains.

When I was in India, I picked up some of the East-West bargains of
Addison-Wesley and Van-Nostrand, e.g. I picked up about a dozen fresh copies
of Hick's Notes On Differential Geometry at about 2 rupees each (about
25 cents at the time) thinking that if I ever taught a course on differential
geometry, I could lend the books to the students so they wouldn't have to
pay for expensive books. That never happened and I still have the books,
but maybe someday it will happen.

In the US, I used to get great bargains at Barnes and Noble when they still
sold publishers overstocks. I got Serre's Abelian l-adic representations
for just $2 there, for example.

Instead of comparing the prices at which American technical books sell in 
various countries, it might be useful to compare the typical prices for
comparable books in various countries produced in those countries by 
publishers in those countries for domestic consumption. This means that
we treat as unimportant the fact that the books are not written in 
English. In France, I picked up a text entitled Chimie Pratique
for about 120 francs, which is basically a laboratory manual for
chemistry students (including both inorganic and organic chemistry).
I also picked up Bailly's Dictionnaire Grec-Francais for about
200 francs. It may not be as good as the big Liddell and Scott, but
it is still pretty comprehensive and only a fraction of the price.  In
Germany, I was able to get Ernst Jenni's Lehrbuch der Hebraischen
Sprache des Alten Testaments for 32 DM and I believe I got Max Born's
Optik for a comparable price.

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math
From: g813214@math.nthu.edu.tw (Lin S.Y.)
Subject: Re: 2 questions (GRE Subject test)
Organization: National Tsing Hua University (HsinChu)
Date: Sat, 26 Dec 1992 07:01:08 GMT

 For the second question, the group must be of odd order.
 Because if the number of elments in the group is even, then there must be
 an element with order 2, which means that the element is the inverse of 
 itself!
 This is for the finite group.
 
 --------------------------------------------------------------------------
  Showing Lin at National Tsing-Hua University, Institute of Mathematics, 
  E-mail address : g813214@math.nthu.tw.edu
  Live happily in the company of mathematics !
 --------------------------------------------------------------------------
From: darice@yoyo.cc.monash.edu.au (Fred Rice)
Newsgroups: sci.math
Subject: Re: Bayes' theorem and QM
Keywords: qm
Date: 30 Dec 92 00:34:15 GMT
Organization: Monash University, Melb., Australia.

In <1992Dec28.150215.203430@uctvax.uct.ac.za> naturman@uctvax.uct.ac.za writes:

>In article <TED.92Dec27145845@lole.nmsu.edu>, ted@nmsu.edu (Ted Dunning) writes:
>> 
>> In article <1992Dec26.080126.12238@monu6.cc.monash.edu.au> darice@yoyo.cc.monash.edu.au (Fred Rice) writes:
>> 
>>    Interesting... I have had the same thought of this possibility too.  Does 
>>    anyone know if any work on how chaos and fundamental QM might relate has 
>>    been done?  If so, can anyone give any helpful references or names?
>> 
>> this is called the hidden variable interpretation of quantum
>> mechanics.
>> 
>Wrong. Without going into too much details, hidden variable interpretations are
>interpretations which require QM probabilities to be recoverable by integration
>over a space of "hidden variables".  The hidden variables model possible
>unknown physical quantities whose specified values uniquely determine all
>the values of all physical quantities. Hidden variables interpretations were
>born out of an attempt to show that QM worked like the kinetic theory of gases.
>However, one cannot have a hidden variables interpretation without violating
>locality.

>The attempts to explain QM in terms of chaos theory have nothing to do with
>hidden variable theories.

Does this mean that work on how chaos might relate to fundamental QM has
been done?  

If so, does anyone have any references or names?

(None have been given so far.)

 Fred Rice
 darice@yoyo.cc.monash.edu.au   
Newsgroups: sci.math
From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Bayes' theorem and QM
Organization: ORA Corporation
Date: Wed, 23 Dec 1992 13:03:00 GMT

naturman@uctvax.uct.ac.za writes:

[Types of deterministic models ruled out by Bell's Inequality Violation
deleted]

>Since there are deterministic models which are non-equivalent to these
>models, one cannot conclude that deterministic models are ruled out by
>the Bell Inequality. In fact, a computer program algorithm modelling
>QM provides a a very deterministic model of QM although it is unlikely
>to be a physically relevant model. Research is being done into finding
>a plausible pseudorandom recursive mechanism behind QM. (I have not
>seen any results yet though.)

A computer program modelling quantum mechanics works by cheating.
Since the whole world is in its data space, it is not subject to
speed-of-light limitations on the dissemination of information.

>> As I have mentioned several times before, there *is* a
>> hidden-variables theory for quantum mechanics developed by Pitowsky
>> and Gudder. However, the probabilities associated with the hidden
>> variables are non-classical; in particular, the measurable sets do not
>> form a sigma algebra.
>> 
>> Daryl McCullough
>
>The debate over whether QM uses "classical" probability would be aided if
>someone actually bothered to define what they mean by "classical" or
>"non-classical"

Someone has actually bothered to define the difference. In short,
classical probability theory assigns probabilities to sets of
outcomes. (A set of outcomes is called an "event"). The sets that are
assigned probabilities form a sigma-algebra. That is, they are closed
under set difference and countable unions and intersections. 

This allows us to think of events as statements, in a certain sense,
and to think of the probability assigned as a generalized truth
value. If A and B are events, then so are (A /\ B), and (A \/ B).

Nonclassical probability theory is any departure from this scheme.
There is more than one way to generalize things, of course, but in
particular one may abandon the requirement that the measurable sets
form a sigma-algebra. For example, there may be events A and B such
that (A /\ B) is not an event (it is not assigned a probability).

Daryl McCullough
ORA Corp.
Ithaca, NY
From: bz223@cleveland.Freenet.Edu (Danny Guy Frezza)
Newsgroups: sci.math
Subject: John Kemeny,66,Computer Pioneer and Educator is dead
Date: 27 Dec 92 16:30:39 GMT
Reply-To: bz223@cleveland.Freenet.Edu (Danny Guy Frezza)
Organization: Case Western Reserve University, Cleveland, OH (USA)
NNTP-Posting-Host: hela.ins.cwru.edu


As reported in the New York Times Today, John Kemeny died yesterday
from a heart attack.  Dr. Kemeny was a Mathematican at Dartmouth College
and later served as President of Dartmouth.  He is remembered by computer
programmers as a co-inventor of "Basic", a computer language.
     Full text in the NYT.
-- 

Dan Frezza
Internet: bz223@cleveland.Freenet.Edu
Newsgroups: sci.math
From: darice@yoyo.cc.monash.edu.au (Fred Rice)
Subject: Re: Bayes' theorem and QM
Organization: Monash University, Melb., Australia.
Date: Sat, 26 Dec 1992 08:01:26 GMT

In <449@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:

>The same thing effectively happens in chaotic classical systems.
>In theory we can compute a value for every observable, but such a
>computations is, from a practical standpoint, not possible even with
>perfect information of the initial state. 

My understanding is that, in a classical chaotic system, given perfect
knowledge of the initial observables, one can calculate how the system
will evolve (for want of better terminology), i.e. it is deterministic.
However, in practice, this perfect knowledge of the initial observables
that is required is not obtainable due to limitations of the instruments
(classically, remember, forgetting about QM).

However, perhaps that is what you meant when you said "from a practical
standpoint". 

>It seems likely
>to me that some form of deterministic chaos also underlys the seemingly
>fundamental nature of probability in quantum mechanics.

Interesting... I have had the same thought of this possibility too.  Does 
anyone know if any work on how chaos and fundamental QM might relate has 
been done?  If so, can anyone give any helpful references or names?

 Fred Rice
 darice@yoyo.cc.monash.edu.au   

Xref: sparky sci.math:17528 sci.physics:21922
From: charlie@umnstat.stat.umn.edu (Charles Geyer)
Newsgroups: sci.math,sci.physics
Subject: Re: Bayes' theorem and QM
Date: 30 Dec 92 22:59:55 GMT
Organization: School of Statistics, University of Minnesota

In article <C03Er1.6wz@netnews.jhuapl.edu> mfein@netnews.jhuapl.edu (Feinstein
Matthew R. F1B x6554 ) writes:

> So, what's the catch [about the Banach-Tarski paradox]?  The catch is that
> the pieces are not measurable sets, sets whose existence can only be proved
> if you assume the axiom of choice.  In other words the volume of the
> inidividual pieces cannot be well-defined.
>
> This raises the question of whether the axiom of choice has anything to
> do with the real world, among other things.

Don't see why.  The issue is whether there is any reason to worry about
measurability in integration theory.  If you don't mind the possibility
of non-measurable sets, I don't see why Banach-Tarski should bother you.

If you think that every "real world" set should be measurable, and hence
Lebesgue theory is a waste of time, then you object to more of classical
mathematics than just the axiom of choice.

Not so?

-- 
Charles Geyer
School of Statistics
University of Minnesota
charlie@umnstat.stat.umn.edu
From: cka@aifbdelta.aifb.uni-karlsruhe.de (Claus Kaldeich)
Newsgroups: sci.math
Subject: Re: From a Naughty Category Theorist
Date: 3 Jan 1993 14:34:05 GMT
Organization: AIFB, University of Karlsruhe (Baden / Germany)
Distribution: world
NNTP-Posting-Host: nz11.rz.uni-karlsruhe.de

In article <1992Dec3.171517.7362@guinness.idbsu.edu>, holmes@opal.idbsu.edu 
(Randall Holmes) writes:
> By the way, [...]

Dear Mr. Holmes!

Can You send me these article about Naugthy Categories Theorist ?

Sincerly

Your, Anatol Reibold

-- 
 ----------------------------------------------------------------------------- 
|                  9 9 9                        Anatol Reibold                |
|        4 4       9 9 9                        TH Darmstadt                  |
|  1     4 4       9 9 9                        FB-4 AG-1                     |
| reibold@mathematik.th-darmastadt.de           Schlossgartenstrasse 7        |
| Voice +49 (0)6151 317926                                                    |
| Fax   +49 (0)6151 164011                      GERMANY - W-6100 DARMSTADT    |
 ----------------------------------------------------------------------------- 
Xref: sparky sci.math:17255 rec.puzzles:8076
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math,rec.puzzles
Subject: Re: Integral Puzzle (Cute, not Evil)
Date: 21 Dec 1992 08:44:30 -0500
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <1992Dec19.011244.2780@Csli.Stanford.EDU> hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:
>
>Let f be any function such that:
>	* for 0<=x<=1; 0<=f(x)<= a
>
>	  /1
>	* |  f(x)dx = A
>	  /0
>where a and A are given constants (obviously, A<=a).
>
>What are the maximum and minimum values that
>
>	/1       2
>	|  {f(x)}  dx
>	/0
>
>can take?

In more flowery language: given the L_infinity and L_1 norms of f, estimate
the L_2 norm of f.


-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Newsgroups: sci.math
From: tao@fine.princeton.edu (Terry Tao)
Subject: Re: need help in calculus problem.
Originator: news@nimaster
Organization: Princeton University
Date: Sun, 3 Jan 1993 14:06:10 GMT

In article <C09t6w.796@usenet.ucs.indiana.edu> droth@silver.ucs.indiana.edu (David Roth) writes:
>Tangent line?  In 3-space, there are infinitely many tangent lines to a
>surface at a point..
>
>David

it's not a surface, it's a curve.  And the original poster wanted the
tangent space at each point, which is one dimensional (if the equations are
not degenerate).

I think there is no easy formula; the best one can do is a differential
equation.  As t is the parameter, we denote x = x(t), y = y(t), z = z(t), and we have

F(x(t), y(t), t) = 0, etc. for all t.

our job is to find x'(t), y'(t), and z'(t).  So we differentiate the three
equations with respect to t, and we get equations that look like

x'(t) F_1(x(t), y(t), t) + ... = 0

where F_1 is the deriv of F wrt the first variable.  Solving the three
equations in terms of the three unknowns x', y' and z' we get, basically

(x'(t), y'(t), z'(t)) = A(x(t), y(t), z(t), t)

where A is some monster function depending on F, G, and H.  That's about
all one can do - get a DE.  If you try to solve this DE by seperation of
variables, guess what - you're going to end up with F(x(t), y(t), t) = 0
and all the other equations again.  I think this is the best one can do.

Terry
From: garyb@mpd.tandem.com (Gary Bjerke)
Newsgroups: sci.math
Subject: Lagrange successor rule
Date: 23 Dec 92 00:12:20 GMT
Reply-To: garyb@anasazi.UUCP (Gary Bjerke)
Organization: /etc/organization


I was thumbing through an old statistics textbook when I came across the 
Lagrange successor rule. The example given was that of a collection of coins
for which the probability of flipping a head is uniformly distributed over
the set of values {1/N, 2/N, ..., N/N} for N some arbitrary integer. The rule
states that the probability of getting a head on the (n+1)th flip given that
the first n flips were heads, is n/(n+1).

Note that N is arbitrary and not a parameter is the final result. The textbook
said Lagrange had used this to prove that the sun was extremely likely to rise 
tomorrow, given that it had risen every day for the past 5000 years (the 
probability is [5000*365]/[5000*365+1], to be precise). 

I followed the proof, but I have absolutely no intuition for this result at
all. I even fail to see how it applies to the rising of the sun (in what sense
does the unconditional probability of its rising meet the uniform-distribution
requirements?) Can somebody help me get a gut feel for what this result means?
From: hubert@mike.ucsb.edu (Hung-Hsien Chang)
Newsgroups: sci.math
Subject: Re: Names and pictures
Date: 22 Dec 92 04:18:40 GMT

prener@watson.ibm.com (Dan Prener) writes:

>In article <7126@ucsbcsl.ucsb.edu> hubert@ucsb.edu (Hung-Hsien Chang) writes:

>> I think it would be nice if the mathematical papers  published on 
>> journals contain the pictures of the authors as well. So people 
>> will know the authors not just by name.

>Why would that be good?  It seems to me that the predominant effect would
>be to provide an opportunity for invidiousness.
>--
>                                   Dan Prener (prener@watson.ibm.com)

Sometimes it would be admiration and respect. Also it is much more friendly
( if the author's picture is not angry-looking. ) when name is associated 
with faces. 

Newsgroups: sci.math,sci.physics