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I wrote:
> Of course, it is known that [quadratic programming] is NP-Hard for
> general linear constraints or if x is constrained to a box, even if A
> is positive definite.
That should have been, "even is A is *negative* definite". The point
being: if A is negative definite and x constraned to a box then the
problem is NP-HARD, but if x is constrained to a simplex (and A is
neg. def), the problem is polynomial time solvable.
I would expect the simplex constrained case to be NP-HARD when A is
indefinite, but I have not seen a proof of this special case.
Thanks again,
Chris Stephens.

Does any know of literature which lays a connection between pattern
recognition and coding theory.
Please let me know.
Nico.

Announcing
         The 12'th Annual Geometry Festival
The Festival will be held this year at Duke University (hosted
by both Duke and UNC) in Durham, North Carolina and will take
place during the days 14-16 March 1997.  
The Geometry Festival is sponsored by a consortium of universities 
on the East Coast and supported by a grant from the National Science
Foundation.  It meets once each spring over a weekend, from Friday
afternoon to Sunday noon, and typically has a roster of seven 
featured speakers (one Friday, four Saturday, and two Sunday morning), 
each of whom reports on recent developments in Geometry (broadly 
construed) in which he or she has played a part. 
The attendees of the Festival generally represent a broad range of 
universities in the Eastern half of the US and Canada, though they are 
by no means limited to that area.  For those participants without
grants, support is provided to subsidize lodging costs (to the extent
made possible by NSF support).
For further information, especially concerning the schedule of events,
registration fees, eligibility for support, and other matters,  you may
consult the Festival Web Page  (some
items are under construction) or send e-mail enquiries to either:
    Robert Bryant 
or 
	Angelika Langen ,
the Festival Coordinator.
A second announcement, with the list of speakers, will follow soon.

Greetings!
I'd like to find out about any attempts at generalizing Farey
sequences to R^n (or more general spaces)?
Any pointers will be greatly appreciated!
Regards,
Ajit
====
--
Ajit T. Dingankar
IBM Corporation, Internal Zip 4359
11400 Burnet Road, Austin, TX 78758

Let X, Y, Z be independent random variables with the
same distribution D.
Assuming the moments exist, we have that the variance
of D is the expectation of  (X-Y)^2 / 2.
Similarly, the third central moment of D is the 
expectation of  (2X-Y-Z)^3 / 6.
I guess that higher central moments can be written in a
similar manner as the expectation of some function of
independant variables.  Maybe there is even more than
one way to do it.
Does anyone know a general result of this type?  
Cumulants would do instead of central moments if it
is any easier.
Brendan.

I've been too enthusiastic about the properties of traces at the end of my
proof of the free generation, so here's a simpler one (no series involved):
We must prove that no non-empty sequence of S1 and S2s can be +-1 or +-A.
Well, apply such a sequence to the vector (1 1) transposed: it's a
non-empty sequence, and you're always adding rows to each other, so the
result will have positive components one of which will at least be two.
Hence it can't be (1 1) or (-1 -1) or (1 -1) or (-1 1), which it should be
if the product of Si's were 1, -1, A or -A.
--
Dr. Denis Constales - dcons@world.std.com - http://cage.rug.ac.be/~dc/

In article <32DBD6FE.4D2B@etsu.edu>, Bart Goddard   wrote:
>One of our grad students is looking for the actual
>proof (independently by Julia and Fatou) of the theorem
>"The Julia set of x^2+c is connected iff the orbits of 0
>are bounded."
An accessible modern reference is Carleson and Gamelin's
"Complex Dynamics", Springer 1992, Theorem 4.1 of
Chapter III.
	-Curt McMullen

In article <5bg1l2$mov@news.iastate.edu>,
Fritz Keinert  wrote:
>		 MIDWEST NUMERICAL ANALYSIS DAY 1997
>		       Saturday, April 12, 1997
>		  Iowa State University, Ames, Iowa
...
>  Information concerning the conference is available on the World Wide
>  Web at http://www.math.uwm.edu/Midwest_NA_Day. 
The correct web site for the conference is
	http://www.math.iastate.edu/Midwest_NA_Day
-- 
Fritz Keinert                                   phone:  (515) 294-5223
Department of Mathematics                       fax:    (515) 294-5454
Iowa State University                      e-mail: keinert@iastate.edu
Ames, IA 50011			   http://www.math.iastate.edu/keinert

For AMO stuff, and oscillators in general, check out
Principles of Symmetry, Dynamics, and Spectroscopy bu William Harter, 
Wiley I believe.
Excellent book.....

In article <5bjiv8$es2$1@nntp.ucs.ubc.ca>,
Robert Israel  wrote:
>In article , jalotosk@sciborg.uwaterloo.ca (John Alexander Lotoski) writes:
>|> 	Hi everybody.  I have a question regarding transcendential
>|> equations.  As far as I am aware, there is no exact direct algebraic
>|> solution to these types of equations.  To solve, I know that it can be
>|> done: graphically, numerically, or by using approximations.
>
>What exactly do you mean by a "transcendental equation", and what do you
>mean by an "algebraic solution"?
I took a shot at defining this kind of question precisely some years ago
and have posted it periodically to the net, without a really satisfactory
response.
Say that a complex number is "elementary" if it can be expressed using
finitely many iterations of arithmetic operations, exp, and/or ln (some
fixed branch) applied to 0.
Thus 1 = exp(0), 1/3  = exp(0)/(exp(0)+exp(0)+exp(0)), e = exp(exp(0)),
i = exp(ln(-1)/2), pi = ln(-1)/i, and 2 cos(1) = exp(i) + exp(-i) are
all elementary numbers.  In my opinion, elementary numbers come close to
capturing the intuition of an "exact" numerical solution to a problem.
Clearly, the field of elementary numbers is countable, so "most" numbers
are not elementary.
One can then ask, for example, if the (real) solution of x = cos(x) is
elementary.  I don't think this theory has been developed, although it
presumably must have been considered by the computer algebra folks.
-- 
Tim Chow       tchow@umich.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949

	Brendan McKay  wrote:
> Assuming the moments exist, we have that the variance
> of D is the expectation of  (X-Y)^2 / 2.
> 
> Similarly, the third central moment of D is the
> expectation of  (2X-Y-Z)^3 / 6.
> 
> I guess that higher central moments can be written in a
> similar manner as the expectation of some function of
> independant variables.  Maybe there is even more than
> one way to do it.
> 
> Does anyone know a general result of this type?
	I think what you're looking for is in this:
%AUTHOR   = Lee, A. J.
%TITLE    = $U$-statistics. Theory and practice
%PUBLISH  = Marcel Dekker
%PAGES    = 302
%YEAR     = 1990
	Mike Hardy
Michael Hardy
hardy@stat.umn.edu

Typically in a course that covers groups one eventually
encounters a table of the number of nonisomorphic groups
of small order, e.g.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 1 1 2 1 2 1 5 2  2  1  5  1  2  1 14  1
I wanted to make a similar table for rings (with multiplicative
identity) of small order, and I came up with
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 1 1 4 1 1 1 ? ?  1  1  ?  1  1  1  ?  1
It appears that there's only 1 if the order is squarefree.
The number of rings of order 8 appears to be >10.
In particular I think 8 is the size of the smallest
noncommutative ring.  Before I waste time finishing
order 8, has anyone seen such a table before?
Also, what is the size of the smallest ring that
does not have a multiplicative identity?
Thanks in advance.
--
Bradley W. Brock      |"Well my first reaction was: I told you so."--G. Shimura
bbrock@pepperdine.edu |
http://137.159.29.175/| Alan Keyes for President in 2000
fax: 310-456-4785     | Winner of the 2/15/96 NH primary debate

The latest issue of "Scientific American" contains information about a
list of outstanding problems compiled by Weierstrass, Hermite and
Mittag-Lefler. The article is about n-body problem. Does anybody knows
what other problems were on the list?
Siemion Fajtlowicz.

I was wondering if it is possible to show the following:
Given that the ith digit of pi is unknown to a person, is it equally
likely for that digit to be any of {0,1,...,9} ?
Just wondering.
Ujji.
email: ujji@thphys.ox.ac.uk
ps. If pi is too hard a number then what about sqrt(2) ?

Dear all,
In my recent work arises the following problem:
Suppose of a sequence of random variables X_1,X_2,... , with values in
{0,...,k}^d.
For each i we have X_{i,1}+...+X_{i,d}=k and the marginal distributions
converge
to binomial distributions (P(X_{n,j}=l)-> B(k,p_j)({l}) for n to
infinity, p_1+...+p_d=1). 
I would like to know now, whether the distribution of (X_1,...,X_d)
converges to a multinomial distribution with parameters k,p_1,...,p_d.  
-- 
Michael Holst.
holst@math.uni-kiel.de
Christian-Albrechts-Universitaet Kiel, Germany. Dept. of Mathematics.

     Topics in Number Theory
      July 30-August 3,1997
There will be eight plenary lecturers including:
     H. Darmon     (McGill Univ.)
     A. Granville  (Univ. of Georgia) 
     C. Pomerance  (Univ. of Georgia)
     C. Skinner    (Inst. for Advanced Study and Princeton Univ.)
     R. Stanley    (MIT) 
     F. Villegas   (Princeton Univ.)
     T. Wooley     (Univ. of Michigan)
     D. Zeilberger (Temple Univ.)
There is a very limited number of slots for 15 minute  contributed
talks. If you would like to be considered for a 15 minute
contributed talk, then please send a proposed abstract and
title to ono@math.ias.edu by March 28, 1997. 
Financial Support:
We anticipate funding from the NSA and the NSF. These
funds will be used to support invited speakers (without
other sources of funding), and junior mathematicians.
To be considered for support, please send an e-mail
to ono@math.ias.edu before March 28,1997.
Registration and other information:
This information is available on the conference web
page (http://www.cde.psu.edu/C&I;/NumTheory/).
Please use the registration materials linked
to this page. The regisration fee is waived
for all invited speakers.
George Andrews and Ken Ono 

U Sharan wrote:
> Given that the ith digit of pi is unknown to a person, is it equally
> likely for that digit to be any of {0,1,...,9} ?
If you put it that way, I'd have to say the answer is 
no.  The ith digit (for any given i) is whatever it is,
whether the person knows it or not.  If it happens to
be 7, then the probability of the digit being 7 is 1.
A better way to put the question that I think you really
want to ask is the following: let N(n,d) be the number of
times the ith digit of pi is d for 1 <= i <= n.  Does
N(n,d)/n -> 1/10 as n -> infinity?
Nobody has any idea how to prove this, but AFAIK the 
statistical evidence, based on the digits that have 
been computed so far (the record is over 6 billion) indicates that it is
true.
> ps. If pi is too hard a number then what about sqrt(2) ?
Also too hard.  The same goes for almost any "naturally
occurring" irrational number, excepting those that might
be specially cooked up for the purpose, such as 
                        infinity
                         -----         2
                          \        (- n )
                           )     10
                          /
                         -----
                         n = 1
Robert Israel                     israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia          fax 822-6074
Vancouver, BC, Canada V6T 1Y4

On Fri, 17 Jan 1997, U Sharan wrote:
> I was wondering if it is possible to show the following:
> 
> Given that the ith digit of pi is unknown to a person, is it equally
> likely for that digit to be any of {0,1,...,9} ?
> 
> Just wondering.
> 
> Ujji.
> 
> email: ujji@thphys.ox.ac.uk
> 
> ps. If pi is too hard a number then what about sqrt(2) ?
> 
> 
> 
This question is not very precise. There are two interpretations:
1) Given an algarism a in {0,...9} let N(a;n) be the number of algarisms 
a in the first n decimals of pi. We can ask whether lim as n -> infinity 
of [N(a;n)/n]=1/10, for every a.
2) For fixed i in {1,2,...} and for fixed a in {0,1,...,9} we can ask 
what is the probability that the i-th decimal of pi is a.
The first possibility is mathematically precise and fits well in this 
newsgroup. However, it's probably an open problem, since as far as I know 
almost every question about the distribution of algarisms in the decimal 
expansion of pi are open problems.
The second possibility depends on your philosophycal position about 
probabilities.
If you're a frequentist (or classic) then the question becomes 
inapropriate for probabilities. The decimals of pi are not random, they 
are well determined numbers, it doesn't matter your knowledge about them. 
In the same way, you can't ask a probability about the average height of 
people in USA (you may not know this average, but it's a well determined 
number, it's not random).
If you're bayesian, then the probability depends on you're knowledge 
about pi. It measures the lack of information about pi you have. If you 
do know the i-th decimal of pi, the probability is 0 or 1. If you don't 
have any information about it, it's 1/10, by definition. If you have some 
information about it, then you must calculate the probability based on 
that. For example (a stupid example), if you know that the i-th decimal 
is even, then the probability decomes 1/5. It depends on the person and 
how many information the person has about the thing in question, 
probability is not a "universal" number.
I really do not know much about all that, but I usually consider myself a 
bayesian. And I think that if the discussion takes this course it should 
move to another newsgroup (maybe sci.math).
Daniel Victor Tausk
Instituto de Matematica da Universidade de Sao Paulo - Brasil
e-mail: tausk@ime.usp.br
home page: http://www.ime.usp.br/~tausk