Subject: How to find eigenvalues of "bad" matrix
From: Tom Chou
Date: Wed, 08 Jan 1997 13:17:58 +0000
Hello,
I have an infinite, real, nonsymmetric square matrix
of which I want to find the lowest 10 or so eigenvalues.
I am taking larger and larger truncations and seeing if the eigenvalues
converge. I am using balancing, then reduction to Hessenberg form, then
use a QR algorithm as described in Numerical Recipes.
However, for my particular matrix, I find that the eigenvalues don't
quite converge at 40 X 40, where the algorithm uses too
many interations and exits (the lowest eigenval. changes by ~5% in going
from 20 X 20 to 40 X 40). . Looking at the qualitative trends, I figure
I need about a 400 X 400 truncation in the worst cases.
I think the problem is that the off diagonals get very large
numerically. The matrix elements go as n^2*m^3, so numerically
get very large as one goes down the diagonal (~n^5) or, far away from
the diagonals.
My questions are:
(1) Are there analytical bounds on how large a matrix I
need to take for a required accuracy in the lowest few
eigenvalues? Where can I find theories about the convergence of
the eigenvalues as the matrix is taken to be larger and larger?
(2) What codes should I use? Can I simply reset the
number of iterations in the Numerical Recipes routines
without catastrophic consequences? Are there other
routines/packages suited for this kind of matrix?
(3) Now suppose that each matrix element now depends on a parameter,
s. I want to plot the eigenvalues as a function of s. Are there
theorems which can say when or when not any eigenvalues are degenerate?
Or in particular, whether the lowest eigenvalue for one values of
s=s0 can become larger that say the 2nd largest at s=s0
at a different value s=s1? Is it possible to say that the lowest
eigenval. is ALWAYS lower than the second lowest, for all s in
some range?
This problem is related to band structure/floquet matrics.
Any suggestions on where to look for the answers will be greatly
appreciated.
Thx,
Tom
Subject: A question about extending a function to a Borel measure
From: "Alexander E. Gutman"
Date: Thu, 9 Jan 1997 02:29:05 +0600 (NSK)
Hello friends.
Do you know any facts concerning extension
of a real-valued function f: Cl(Q) -> R,
defined on closed subsets of a compact set Q,
to a Borel measure? To a regular Borel measure?
(No additional requirements are imposed on Q.)
If you prefer to deal with open sets, consider
the "dual" question about a function f: Op(Q) -> R
defined on open subsets of Q.
What if f is defined not on all closed/open sets
but on some of them? For instance, on regular ones?
What are the simplest (easily verifiable) known properties of f
that guarantee its extendibility to a (regular) Borel measure?
Could you provide me with a reference?
--
Alexander E. Gutman
Novosibirsk, Russia
root@gutman.nsu.ru
Subject: A question about weakly continuous functions
From: "Alexander E. Gutman"
Date: Thu, 9 Jan 1997 02:26:40 +0600 (NSK)
Hello, friends.
Take a look at the following statement:
For every infinite compact space Q
and every infinite-dimensional Banach space X,
there exists a function from Q into X', the dual of X,
that is weakly* continuous but not norm-continuous.
Is it true?
By the way, the latter statement is equivalent to the following:
For every infinite compact space Q
and every infinite-dimensional Banach space X,
there exists a bounded linear operator from X into C(Q)
that is not compact.
I would be happy to get an answer just for the case Q=N*,
where N* is a one-point compactification of N.
In this case, the statement under consideration
is equivalent to the following:
For every infinite-dimensional Banach space X,
there exists a sequence in X' that is weakly* convergent
but not norm-convergent.
Is it true?
Where can I find a proof or a reference?
--
Alexander E. Gutman
Novosibirsk, Russia
root@gutman.nsu.ru
Subject: Request for Info on Mathematicians
From: Ken Hirano
Date: Wed, 08 Jan 1997 16:01:11 -0800
Dear Members of SCI.MATH.RESEARCH Newsgroup:
The Dean of the Graduate Division of the University of California, Berkeley,
is currently undertaking a research project studying the career outcomes of
Ph.D.'s in selected majors approximately ten years after they have
graduated. 61 U.S. institutions and nearly 6000 Ph.D.'s are involved in this
study. The results of this study will hopefully assist in improving the
effectiveness of doctoral programs in training our future scholars and
scientists.
Since we are using a mail out questionnaire, we have been trying very hard to
locate the most current mailing addresses for these people. We have been
successful in locating approximately 85% of our survey population through
alumni associations and other sources. That still leaves a siginificant
number of persons for whom we have no mailing address.
Therefore, in a last ditch effort, we are utilizing internet resources such
as web pages of academic institutions and field specific mailing discussion
lists (such as yours) in the hope that somebody will know the whereabouts of
some of our lost Ph.D.'s. We are requesting that if anybody recognizes a name
in the following list of math ph.d.'s, that they contact us to provide leads
on where that Ph.D. might be found. Any leads will be greatly appreciated.
We also request that you forward this email to any persons who might know
anything about the whereabouts of these people. Thank you very much.
We apologize for any inconvenience this message may cause you.
Thank you very much.
Sincerely,
Graduate Division
University of California, Berkeley
"Ph.D.'s Ten Years Later" Study
424 Sproul Hall #5900
Berkeley, CA 94720
tel: (510) 643-2791
fax: (510) 642-6366
email: phd10yr@uclink.berkeley.edu
Mathematics PhD's: [Institution/field/last name/first name/middle (if
available)/year of graduation with PhD]
SUNY-Buffalo/Mathematics/Abdullah/Salem/Ali/83
Michigan State University/Mathematics/Attele/Kapila/Rohan/83
Yale University/Mathematics/Bhate/Hemant//83
UC Berkeley/Mathematics/Bolfarine/Heleno//83
University of Chicago/Mathematics/Brandt/Jorgen//83
University of Minnesota/Mathematics/Brierley/Stephen/David/84
Princeton University/Mathematics/Cabrera/Javier/Fernand/83
Columbia University/Mathematics/Chang/Fu//83
SUNY-Buffalo/Mathematics/Chen/Chang-Shan//84
UC Los Angeles/Mathematics/Chou/Jine-Phone//84
SUNY-Buffalo/Mathematics/El-Henawy/Ibrahim/M/83
Stanford University/Mathematics/Fairley/David//83
University of Utah/Mathematics/Faltenbacher/Wolfgang//84
UC Berkeley/Mathematics/Folledo/Manuel//83
Northwestern University/Mathematics/Frank/George/Nelson/85
Tulane University/Mathematics/Franzen/Berthold/Werne/83
University of Iowa/Mathematics/Gallegos-Jarpa/Griceld//83
University of Minnesota/Mathematics/George/Adel/Aziz/83
University of Kentucky/Mathematics/Greenwell/Catherine/El/82
UC Los Angeles/Mathematics/Hsueh/Yuang-Cheh//83
University of Florida/Mathematics/Ireson/Michael/John/83
MIT/Mathematics/Kim/Dong/Yoon/85
University of Michigan/Mathematics/Kim/Hyuk//83
University of Maryland/Mathematics/Kramer/David/Philip/83
University of Pittsburgh/Mathematics/Krishna/Kottekai//83
University of Florida/Mathematics/Kurihara/Eiji//84
MIT/Mathematics/Lasaga/Fernando/Rene/84
University of Minnesota/Mathematics/Lau/Chi-Ping//84
Colorado State University/Mathematics/Leiva/Ricardo/Anibal/83
Temple University/Mathematics/Levitan/Mark/E/85
University of Utah/Mathematics/Luminet/Denis/Laurent/83
University of Washington/Mathematics/Mansfield/Edward/Josep/84
Michigan State University/Mathematics/Merkle/Milan/J/84
Temple University/Mathematics/Moore/Annette/Louise/84
University of New Mexico/Mathematics/Pichardo-Maya/Agustin//85
University of Minnesota/Mathematics/Ricou/Manuel/Oliveira/84
UC Berkeley/Mathematics/Sevilla/Agustin/Ramos/83
University of Utah/Mathematics/Sindler/Frantisek//83
University of Washington/Mathematics/Stedman/Edward//83
University of Florida/Mathematics/Toledo-Manzur/Juan/Ant/82
University of Minnesota/Mathematics/Waksman/Peter//83
SUNY-Binghamton/Mathematics/Wender/Abraham//85
University of Arizona/Mathematics/Xaba/Busa/Abraham/84
Subject: How do K3 manifolds "look" ?
From: schiller@prl.philips.nl (schiller c)
Date: Thu, 9 Jan 1997 09:14:36 GMT
In string theory, the so-called K3 manifolds are a central ingredient.
However, it is difficult to find papers explaining how to visualise them. Is
it possible to get an idea on how they ``look'' - their shape, for example?
Is it possible at all to visualize them in 3d or 4d somehow? Is there a 3d or
4d object - a knot or a link or some other object - which shares at least some
properties - perhaps some topological ones - with these manifolds?
Regards
Christoph Schiller
P.S. The most detailed paper I found on the net about the topic is the one
below, but even that one is poor on visualization.
High Energy Physics - Theory, abstract hep-th/9611137
http://xxx.lanl.gov/abs/hep-th/9611137
K3 Surfaces and String Duality
Author: Paul S. Aspinwall
Comments: 107 pages, LaTeX 2.09, 11 figures, minor corrections
Report-no: RU-96-98
The primary purpose of these lecture notes is to explore the moduli space of
type IIA, type IIB, and heterotic string compactified on a K3 surface. The
main tool which is invoked is that of string duality. K3 surfaces provide a
fascinating arena for string compactification as they are not trivial spaces
but are sufficiently simple for one to be able to analyze most of their
properties in detail. They also make an almost ubiquitous appearance in the
common statements concerning string duality. We review the necessary facts
concerning the classical geometry of K3 surfaces that will be needed and then
we review "old string theory" on K3 surfaces in terms of conformal field
theory. The type IIA string, the type IIB string, the E8 x E8 heterotic
string, and Spin(32)/Z2 heterotic string on a K3 surface are then each
analyzed in turn. The discussion is biased in favour of purely geometric
notions concerning the K3 surface itself. These are an extended form of the
notes from lectures given at TASI 96.
-------------------------
Electronic mail address : schiller@natlab.research.philips.com
Postal address : Dr. Christoph Schiller, Philips Research Laboratories,
Building WY 71, Prof. Holstlaan 4, 5656AA Eindhoven, The Netherlands
tel +31-40-2742896 , fax +31-40-2743859
Everything which can be thought at all, can be thought clearly.
(Wittgenstein, Tractatus Logico-Philosphicus, 4.116)
--
Electronic mail address : schiller@natlab.research.philips.com
Postal address : Dr. Christoph Schiller, Philips Research Laboratories,
Building WY 71, Prof. Holstlaan 4, 5656AA Eindhoven, The Netherlands
Subject: Nagata-Smirnov theorem and axiom of choice
From: Markus Reitenbach
Date: Thu, 09 Jan 1997 16:12:53 -0800
Recently, I studied the proof of the Nagata-Smirnov metrization theorem,
which states that a topological space is metrizable if and only if it is
regular and has a countably locally finite base.
The proof uses the well-ordering theorem, which is equivalent to the
axiom of choice.
I suppose that the Nagata-Smirnov metrization theorem vice versa implies
the axiom of choice (i.e. it is equivalent to the axiom of choice), but
so far I am unable to proof this.
Does anyone know something about this problem?
Markus Reitenbach, University of Ulm
Subject: arc length question
From: lriddle@ness.agnesscott.edu (Larry Riddle)
Date: Wed, 08 Jan 1997 22:04:02 -0500
Let f be a continuous function defined on [0,1]. Define g by
g(x) = 1/2 * (f(x) + f(1-x))
Then g is symmetric about the line x = 1/2. My question is, will g have a
smaller arc length than f, with the lengths being the same only if g = f? I
believe the answer is yes. It is easy to verify if f is a piecewise linear
function over a uniform partition with an even number of subintervals. But
what about the more general case? I'd be satisfied with a proof for the
case when f is a polynomial.
--
Larry Riddle, Chair | lriddle@ness.AgnesScott.edu
Mathematics Dept | 404-638-6222, 404-638-6177 (fax)
Agnes Scott College | http://www.scottlan.edu/academic/
Decatur, GA 30030 | math/welcome.htm
