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phillies@wpi.wpi.edu (George D. Phillies)
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PostPosted: Thu Aug 20, 1992 3:46 pm    
Subject: Talk.
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Talk.

The following is the title and abstract for a lecture I will be giving
at WPI on wavelet decompositions and their applications. The talk will
be given on Moday, August 31, 1992, at 4PM in the Olin Hall of Physics.

The treatment of wavelets will be strictly introductory, and aimed more
at physicists than mathematicians. The new material will be the use of
wavelet decompositions to study the Ising model of a magnet, and the
sorts of information one can extract from Monte Carlo dynamics
simulations via wavelet decompositions. We have just begun writing up
our results, so preprints are not available.

To reach WPI, take Route 290 to the Route 9-Highland Street exit, go
west on Highland Street approximately 1 mile to West Street (traffic
light), and three blocks north (right turn, if moving away from Route
290). Olin Hall is on the west (left) side of the street, the second
building before West Street comes to an end.

* * * * *

WAVELET DECOMPOSITIONS:

INTRODUCTION AND APPLICATION

TO THE ISING MODEL

George D. J. Phillies
Department of Physics
Worcester Polytechnic Institute


First, I will introduce wavelet expansions, which are a new
mathematical method that may be useful in physics. Second, I will show
how a specific wavelet transform, the Burt-Adelson$^{1}$ decomposition, can
be used to describe statics and dynamics of the Ising model. The
Burt-Adelson decomposition was originally introduced as a tool for
image processing; the Ising model is a highly-simplified statistico-
mechanical description of a spin lattice, nominally similar to the
lattices found in magnets.

Wavelets may formally be viewed as extensions of the orthogonal
functions familiarly used in physics. Isolated examples of wavelet
expansions, such as Haar and Gabor transforms, have long been known.
Recently, Mallat$^{2}$, Meyer$^{3}$, Daubechies$^{4}$ and others have
shown that Haar and Gabor transforms are special cases of an extensive
set of mathematical functions. Wavelets provide complete, sometimes
non-orthogonal, sets of basis functions. Some wavelet expansions can
be generated recursively, suggesting computation efficiencies. Other
wavelet expansions have compact support, so that, e.g., in a wavelet
decomposition of a piece of music, alteration of a single note changes
local coefficients without affecting most of the expansion.

While there is a large mathematical literature on wavelets, concrete
applications of wavelet expansions in the physics literature do not
appear to be common. I will describe Monte Carlo simulations (made by
J. Stott and I last Summer) on static and dynamic properties of the 1-D
Ising model, as revealed by wavelet expansions.

1) P. J. Burt and E. H. Adelson, IEEE Trans. on Communications 31, 532
(1983).

2) S. Mallat, {em A Theory for Multiresolution Signal
Decomposition: The Wavelet Transform } (see IEEE Trans. on Pattern
Analysis and Machine Intelligence}, 11, 674-693 (1989).

3) Y. Meyer, {em Ondelettes et Functions Splines}, Ecole
Polytechnique (Paris) (1986).

4) Ingrid Debauchies, Commun. Pure and Applied Mathematics
41, 909-996 (1988).
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