Newsgroup sci.math.research 6634

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drm@cauchy.math.duke.edu (David R. Morrison) writes:
> In the late 19th century, plaster models were constructed of a number of 
> interesting surfaces, including the Kummer surface, which is a special type of
> K3 surface.  (More precisely, the model is of the set of real points on 
> the Kummer surface.)  You can find a photograph of this model in a book by
> R.W.H.T. Hudson "Kummer's Quartic Surface", originally published in 1905 but
> reissued in paperback about 5 years ago by Cambridge University Press.
For a virtual version of Hudson's plaster model see:
           http://www.maths.usyd.edu.au:8000/gal/nigel/kum-sea.html
-- one of a number of pics illustrating Sydney Uni's u/g brochure:
         http://www.maths.usyd.edu.au:8000/tch/UgPr/
Nigel O'Brian

hi,
	i have the following question. Let D be a (possibly empty)
subset of the real line. Now define two sets of functions as 
follows:
S_1 = { f \in C^1(R) : f vanishes outside a compact set, and f'(D) \ge 0}
S_2 = { g \in C^1(R) : g(x) and g'(x) go to zero as |x| \to \infty, and
							g'(D) \ge 0}
Here g'(D) describes the first derivative of g evaluated on D. 
Is S_1 dense in S_2 (in the C^1 norm)? If so, any hints on proving it 
would be most welcome. Note that if D is an empty set, this assertion
is (relatively) easy to prove. The harder part, i think, is 
showing this result when D is non-empty.
thanks ...
gt
--
% Gautam Tripathi                Phone (Home):(847)-864-2321 %
% Department of Economics        Phone (Work):(847)-491-8253 %
% Northwestern University        Fax   (Work):(847)-491-7001 %
% Evanston, IL - 60208.          E-mail      : gt@nwu.edu    %

Does anyone know if minimization of an indefinite quadratic on a simplex
is an NP-Hard problem. That is, what is known of the complexity of,
                           T     T
                      min c x + x Ax/2
                       x
    such that        x_i >= 0, for all i=1..n, and
                     x_1+x_2+...+x_n <= 1.
Of course, it is known that this problem is NP-Hard for general linear
constraints or if x is constrained to a box, even if A is positive
definite. However, if x is constrained to a simplex, there are
polynomial time algorithms if A is positive or negative semi-definite
(since there are only n+1 vertices). What about the case where A is
indefinite?
Any references or pointers would be much appreciated.
--
Chris Stephens.

I am looking for informations about MUSIC algorithm (high resolution
algo.). Thanks to this algorithm, you can do spectral analysis on very
noisy signals
O. bougeard
ampc3c , Paris, France

In article <32D39E85.41C6@damtp.cam.ac.uk>,
Tom Chou   wrote:
#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
#
Depending on the routine you use, you may want to present
your matrix with the diagonals in reverse order.  In particular,
if your eigenvalue program begins with a preliminary reduction
to Hessenberg form by Householder transformations, you want the
grading of your matrix to be downward.
Pete Stewart

		 MIDWEST NUMERICAL ANALYSIS DAY 1997
		       Saturday, April 12, 1997
		  Iowa State University, Ames, Iowa
Participants:
  This conference is aimed at faculty members, graduate students and
  visitors from universities is the central US. Ivo Babuska has
  tentatively agreed to give an invited talk. For other featured
  speakers, as well as the contributed talks, check the conference web
  site periodically.
Organizers: 
  Roger Alexander (alex@iastate.edu, (515) 294-7579) 
  Fritz Keinert (keinert@iastate.edu, (515) 294-5223)
Deadline:
  If you are interested in presenting a 20-minute talk, submit a title
  and abstract by March 17, 1997, either through the conference web
  page, via e-mail to naday@iastate.edu, or to one of the organizers.
Information:
  Information concerning the conference is available on the World Wide
  Web at http://www.math.uwm.edu/Midwest_NA_Day. 
Special Note:
  The joint annual meeting of the Iowa sections of MAA/ASA/IMATYC will
  be held in the same building on the same day. There will be
  opportunity to hear talks or socialize with participants from both
  conferences.
-- 
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

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."  Every reference we can find stating the 
theorem says that the proof is "beyond the present scope."
So, if anyone has a real reference to the original (or
a more lately polished) proof, in all its gory detail,
we'd love to know about it.
Thanks in advance,
Bart Goddard
mathbg@etsu.edu