Consider the matrix

A = (1/(i b + lambda_r - lambda_s))_{r,s}

where b is a non-zero real the lambda_r's are reals such that |
lambda_r - lambda_s|>=1 for all r,s distinct.

What is the norm of A? That is, what is the maximum (or supremum, if r
is allowed to run over all integers)
of the absolute values of all its eigenvalues?

A couple of posters here helpfully showed me how to solve the special
case lambda_r = r;
the norm is then

<= 2 pi/(1 - e^{-2pi b}),

where the bound is tight if r is allowed to run over all integers.

I would guess that the norm is <= 2pi/(1-e^{-2 pi b}) in general, but
the proof I was given works only for lambda_r = c r, c a constant. I
have managed to provide a bound for general lambda_r by modifying one
of the best-known proofs of Hilbert's inequality; the bound I get is
about 5%% larger than
2pi/(1-e^{-2pi b}) for b near 0.5, and much closer to 2pi/(1-e^{-2pi
b}) for b small - still, it is much too ugly to be optimal.

Hi,

Let beta>0. For any square-integrable function g:R\to C, let

F(g) = max_{x>0} e^{-beta x}/(|\hat{g}(x)|^2).

I'd like to know: what is the minimum of F(g) over all functions g
that (a) are supported on the
unit interval [-1/2,1/2] and (b) have l_2 norm equal to 1?

What sort of tools would I have to use? What does this relate to?

Harald

On Thu, 28 Aug 2008 20:00:02 +0000 (UTC), ical12345@btinternet.com
wrote:

Some of your post is unintelligible to many readers - you might want
to repost using nothing but plain ASCII.

[Note from the moderator:
All posters, please make sure that your
postings are in ASCII. I am usually able to catch strange encodings,
but missed this one. Sorry.
Maarten Bergvelt]

On Thu, 28 Aug 2008 20:00:02 +0000 (UTC), ical12345@btinternet.com
wrote:

Some of your post is unintelligible to many readers - you might want
to repost using nothing but plain ASCII.

Hi! I developed an unbreakable encryption method that will replace the
One Time Pad. I know that sounds unlikely, as so many people have
attempted this and failed. I have shown this to DoD architects, and
they agreed with my conclusions. I have a Ph.D. in computer science
and a strong background in math. My name is Jack Bosworth. If you wish
to contact me, I have a web site at http://bosworths.net.

I call the method "Perpetual One Time Pad", or POTP for short. I
considered calling it Perpetual One Use Pad, but I thought POUP wasn't
a good nickname for it:).

The Basic Idea
Take a small number of One Time Pads. These are of course shared
between two computers. Add an array of random numbers that can be used
as indexes in those One Time Pads. Now, every time you use a code from
one of the pads, use the next value in the array to determine another
code in that same pad with which to swap values. That's basically it.
Now you have a small set of perpetual One Time Pads.

I don't really read French, but this is evidently an announcement
of the death of one of the premier mathematicians of the 20th century.

This just off the ALGTOP-L Digest:

1. Deces d'Henri Cartan (fwd) (Carlos Prieto (113))

----------------------------------------------------------------------

Message: 1
Date: Fri, 22 Aug 2008 10:42:18 -0500 (CDT)
From: "Carlos Prieto (113)"
Subject: [ALGTOP-L] Deces d'Henri Cartan (fwd)
To: algtop-l@lists.lehigh.edu
Message-ID:
Content-Type: text/plain; charset="iso-8859-15"

I am forwarding the following bad news. Carlos Prieto

---------- Forwarded message ----------

Date: Thu, 21 Aug 2008 10:17:25 +0200 (CEST)
From: Societe mathematique de France
To: emilia@servidor.unam.mx
Subject: [smf-adherents] Deces d'Henri Cartan

La Soci

Pondering operator algebras and their relation to measure theory,
I've blundered into this question:

Are second-countable locally compact Hausdorff spaces Polish?

A Polish space is a topological space that's homeomorphic to
a separable complete metric space.

This article suggests the answer is YES:

H. E. Vaughan
On locally compact metrisable spaces
Bull. Amer. Math. Soc. Volume 43, Number 8 (1937), 532-535.
http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle...

The author claims to show:

Theorem 2: In order that a Hausdorff space be homeomorphic to a
totally complete metric space it is necessary and sufficient that it be
locally compact and perfectly separable.

He says "a totally complete metric space is a metrisable space in
which the metric is chosen such that every bounded set is compact."
Hmm? *Every* bounded set is compact? That seems rather drastic...
maybe he means they're all relatively compact? In another paper
he cites Kuratowski's "Topologie I", page 196 for this definition.

I had proved recently that between any two consecutive multiples of
any natural number n>1 exists at least one integer k such that (k,
i)=1 for all i in the interval [1,n] (I denoted such numbers k as n-
primes).
This result includes as a particular case the Conjecture P of
Sierpinski (see references 1 and 2). That conjecture states that in
every line of the square array formed by the ordered first n^2 natural
numbers exist at least one prime. Obviously, all n-primes in that
array are also primes (in the first line exist only the n-prime 1, but
we have there always at least the prime 2). I want to know which is
the today status of that conjecture, if it remains unsolved or not,
and particularly if it has some known relationship with the Riemann
hypothesis.
References:
1. Sierpinski, W.,

Do you think everything about Fibonacci numbers is easy? I thought so too
until I came across this:

Write P=(1+sqrt(5))/2. Assume that the interval
[nP -1/n , nP + 1/n]
contains an integer. Prove that n is a Fibonacci number.

(Any integer in the interval is then also Fibonacci, but that's easy.)

I emailed the fellow who posted the problem here:
http://www.mathlinks.ro/Forum/viewtopic.php?t=213412
He says he saw it on the web somewhere, but with no proof attached.

LH

Are there examples of compact connected matrix groups other than the
product of SO(n), Sp(n), U(n) and the exceptional groups.