Colloquia, Spring 2000
Edray Goins, of the
for Advanced Study, School of Mathematics.
On the Multiplicative Properties of the Sums of Squares.
The product of any two integers which are sums of two
squares is again the sum of two squares, and, similarly, the product of
any two integers which are sums of four squares is again the sum of
four squares. We give some thoughts on where these formulas come
from; we also discuss the multiplicative properties of the polynomials
by considering the product of three integers
expressed in such a form. —
This talk will be elementary and accessible to undergraduates.
May 16. Doron
Lubinsky, of the
Johannesburg, South Africa.
A Pade Look at Continued Fractions: Some old, some new.
Every formal power series admits a formal continued fraction
expansion. This is the function theoretic analogue of the continued
fraction expansion of a positive real number. We discuss why they
are of practical and theoretical interest.
— The convergents to the continued fraction are also Pade
approximants, and we examine the convergence of the continued
fraction from a Pade perspective. In particular, we survey old and
new convergence results, and the status of the Baker-Gammel-Wills
Conjecture. Will it be killed by Ramanujan’s continued fraction?
University of Alberta.
The Design of Multiwavelets with Optimum Time-Frequency
In this talk first we will discuss the procedure to design
orthogonal multiwavelets with good time-frequency resolution.
Next, we give formulas to compute the time-durations and
the frequency-bandwidths of scaling functions and multiwavelets.
Later on we discuss the parameterization of symmetric orthogonal
multifilter banks. Finally, we show some results of image
compression with the optimal multiwavelets.
Continuous and discrete models of phenotype evolution in
the ecosystem of red tide plankton.
Heterocapsa circularisquama are vegetative
plankton and form red tides because of active
proliferation in summer. Such red tides cause
mass mortality of bivalves. The coastal ocean is
considered complex multi-component fluid in
which the ecology of Heterocapsa is supported
by other living things and the physical states
depending strongly upon seasonal variations of
air-sea interactions, weather, water temperature,
salinity and water currents. Therefore it is not
straightforward to elucidate the ecosystem of
Heterocapsa in the ocean from the point of view
of a sole field of biology as well as other separate
fields of natural science.
Many problems in mathematics and in applications can be
formulated as constraint satisfaction problems. Among these
problems are the question if a given ordered set has a fixed point
free self-map, the question if a graph has a Hamiltonian Cycle,
the n-queens problem, scheduling problems, etc. In particular,
constraint satisfaction problems (we will give a general definition)
are NP-complete and thus any known solution procedure
potentially requires exponential effort. —
The standard search algorithms that are used to solve such problems
are backtracking and forward checking. In addition to these main
paradigms there are refinements of each algorithm, which may
have shorter search times. The problem now is that if a constraint
satisfaction problem is presented, one would like to know in advance
which algorithm is the fastest to solve the problem. In this fashion
one could automatically choose the “right” algorithm and save
computation time. —
There are many approaches to this idea. The three main approaches
the speaker is aware of are
- Theoretical hierarchies that show one algorithm to be preferable for
certain types of problems,
- Heuristic analysis of the search trees that leads to estimates of the
run time, and
- Probabilistic analysis of the constraint network, which also leads
to estimates of the run time.
In this talk we will focus on the probabilistic analysis of the constraint
network and how it can be used to predict search time. The probabilistic
analysis would also allow us to give an estimate how likely it is that the
problem has a solution at all. —
We will exhibit the ideas behind a probabilistic analysis, the need to
generalize these ideas and combine them with heuristic (or other)
analysis of related problems and some experimental data on randomly
seeded constraint satisfaction problems using different seeding mechanisms.
The talk connects Mathematics and Computer Science and should be
understandable to graduate and advanced undergraduate students in
University of Wisconsin, Madison.
Knot types of Closed Geodesics on surfaces via Curve Shortening.
Closed Geodesics on two dimensional surfaces trace out curves
which look like knot projections. One can ask which particular knot
projections must occur for a given metric, and it turns out that Curve
Shortening (a nonlinear heat equation) allows one to give a partial answer
to this question. In particular one can prove a global version of the
Poincare Birkhoff thoerem for area preserving twistmaps.
Rong-Qing Jia, of
University of Alberta.
Wavelet Frames Generated by Multiresolution Analysis.
A multiresolution of the space of square integrable functions consists
of a family of scaled shift-invariant subspaces. The difference of two
consecutive shift-invariant subspaces is generated by a finite number of
wavelets. Under a very mild condition we show that these wavelets generate
an affine frame for the whole space. Our results are based on a study of
dual wavelet frames with compact support.
Professor Emeritus at Vanderbilt University.
A Representation of the Riemann Zeta Function in Terms of
Parabolic Cylinder Functions.
April 7. Yurij Kryakin, of Odessa State Polytech
University (I.I. Mechnikov, Institute of Mathematics, Economics and
Mechanics), Ukraine, and
of Wroclaw, Poland.
Local approximation by polynomials.
The remarkable Whitney Theorem states that for each continuous on [0,1]
function f(x) there exists an algebraic polynomial of degree at most
n-1 such that the uniform norm of difference |f(x) – P(x)|
is dominated by the norm of nth divided differences with
some constant W(n) for n=1,2,…. —
Since 1957, a lot of research was done on estimating the values of
W(n). However, at the moment only asymptotic behaviors and exact
values for n<3 are known.
Recent progress in estimating of W(n) is connected with the
special methods of approximations (“approximation on the average”).
The current status of the problem, open questions and the applications in
Approximation Theory, will be discussed.
Palle Jorgensen, of
University of Iowa.
Construction of new wavelet filters, and their classification.
It is well known that wavelet filters satisfy a set of algebraic axioms, but
there aren’t so far systematic ways of constructing examples that are dictated
by applications. Also when examples are constructed, it is often not clear
whether they are minimal. If they are given by a system of numbers, are there
redundancies, and can a given example be further decomposed in elementary
“building blocks”. We first describe all the wavelet filters as a certain
infinite-dimensional unitary group, and we describe how that is effective in
constructing new examples, for example getting high-pass filters consistent
with a given low pass filter. We also provide a harmonic analysis which
resolves the question of equivalence, and elementary building blocks,
making precise the notion of irreducible cases. The theory will apply also
to scaling in several dimensions.
Songmu Zheng, of the
University of Minnesota.
Maximal Attractors for Some Nonlinear PDEs.
In this talk I will report some recent progress in the study of the dynamics
for some nonlinear partial differential equations, including the coupled
Cahn-Hilliard equations, the phase-field equations of Penrose-Fife type arising
from the study of phase transitions, and also the system of nonlinear PDEs
describing the motion of a one-dimensional viscous and heat-conductive ideal
gas. These problems have the following new features regarding their dynamics:
- The metric spaces we work with are incomplete, as can be seen from
the constraint q>0 with q being the absolute temperature, or
u > 0 with u being the specific volume.
- There are some conservation laws for these problems, i.e., conservation
of mass, conservation of energy, etc. It turns out that there is no global
compact attractor for each of these problems when the initial data vary in the
whole space, and we have to restrict ourselves to subspaces characterized
by a sequence of parameters. Of course, the immediate question is how to
define these subspaces for these coupled systems of quasilinear PDEs.
- It is well-known that in order to prove existence of a maximal compact
attractor, one of the major steps is to prove existence of an absorbing set or
an absorbing ball whose size should be independent of the initial data.
To do so, one usually tries to derive the following type of differential
dE/dt + C1E(t) £ C2,
with E being a certain Sobolev norm of the solution, and with
being two positive constants independent of the initial data. If such a
differential inequality is available, then existence of an absorbing ball
immediately follows. However, since all these three systems are quasilinear,
it does not seem feasible to derive such a type of differential inequality.
In this talk I will present a new approach to deal with the above
difficulties. Then we will focus our attention to the coupled
Cahn-Hilliard equations and show that for any given constants b1 > 0,
sequence of closed subspaces
U(b1,b2) is found, and the existence of a compact
maximal attractor in U(b1,b2) is proved.
Factorization in the Composition Algebras.
This broadly-accessible talk will present some arithmetical and geometrical
results concerning the composition algebras, including new results based on a
nonassociative factorization algorithm appropriate for computations involving
the integral octonions. After introducing the composition algebras and some of
their geometrical properties, I will focus on the problem of understanding and
describing the factorizations of elements within certain rings of integers in
these algebras. In particular, if O is a maximal set of integers
a composition algebra and
r Î O
has norm [r] = m.n, I
will present an algorithm that leads to a precise geometrical description of
the set of all factorizations of
as r = ab, where
[a] = m and
[b] = n.
(Part of this work is joint with J. H.
Laci Lovasz, of
Critical graphs and facets of the
stable set polytope.
A graph is called alpha-critical
if deleting any edge
increases the number of independent nodes. Such graphs
have a very nice and tight structure, which was studied
in detail in the 70’s. Another important notion is the stable
set polytope, the convex hull of incidence vectors of
independent sets of nodes. —
The starting point of this talk is the fact that
alpha-critical graphs give rise to facets of the stable set
polytope, and that several of their basic properties
can be generalized to facets. In particular, a known
classification of alpha-critical graphs with fixed “defect”
leads to a classification of facets with fixed
“integrality gap.” —
The talk will survey the theory of alpha-critical graphs,
and then the new results on facets, which are joint work
with László Liptak.
Gulisashvili, of Ohio University.
On the heat equation with a time-dependent potential.
The initial value problem for the heat equation with a
time dependent potential describes the time-evolution of
some quantity, diffusing in Rn in the presence of
time-dependent sources and sinks. We will discuss the
weak solvability problem for the heat equation with a
potential, the smoothing properties of the weak propagator,
and the integral representation formulas for the solutions,
including the celebrated Feynman-Kac formula.
Both hierarchical bases and wavelets have as a prerequisite
a multiresolution analysis on the underlying nested sequence of
triangulations. The spaces of
piecewise polynomials are a natural tool for the construction of the
multiresolution analysis. In particular, if the bases are only required
to be continuous, usual finite-element spaces can be used.
In contrast to this, smooth conforming finite elements, such as Argyris
element, are not suitable due to the fact that the corresponding spaces of
piecewise polynomials are not nested.
The idea to make use of
the basis of the full space of C1-splines
Sd1(D) instead is
due to Dahmen, Oswald & Shi (1994). However, the Morgan-Scott basis
used in their work appears to be instable
if the triangulations have so-called near-degenerate edges and
near-singular vertices. Therefore, the triangulation has to be refined
in such a way (e.g., uniformly) that these geometrical configurations do not
We discuss a recent construction (joint work with L. L. Schumaker)
of a stable local basis for C1 and, more general,
which can be used to build stable multiresolution
analyses on arbitrary nested sequences of triangulations, with the only
restriction that the smallest angle of the triangles is controlled.
In particular, this construction applies to adaptively refined
Qiyu Sun, of the
National University of Singapore.
Finitely Generated Shift-Invariant Spaces.
For any vector-valued compactly supported function
F = (f1,…
define the corresponding semi-convolution
1 < p < ¥, and
V(F) be the images of
the semi-convolution operator
In this talk, we shall discuss the annihilator
of the kernel of the semi-convolution operator
F*¢, and the recovery of functions in
some linear independent, stable or frame conditions on
F. For simplicity, we restrict
ourselves to one dimensional case, all results presented
in the talk can be generalized to higher dimensions.
On the local behavior of solutions of some
degenerate/singular parabolic equations.
Models of flow of multiple immiscible fluids in a
porous matrix and/or phenomena of multiple transitions of
phase, result into quasi-linear parabolic equations, with
measurable coefficients and exhibiting multiple
singularities and/or degeneracies. —
I will discuss the problem of the continuity of the
transition parameters, for example saturation in the
flow of immiscible fluids or temperature in isothermal
phase transitions. —
We review and summarize the main points of the
theory and will present some recent results in this
direction, pointing to the new mathematical tools
generated by these investigations.
Peter Jipsen, of
of Cape Town and Vanderbilt University.
Computer assisted research in universal algebra and logic.
In this talk I will present some of the tools that I have used
in my research and discuss some of the results that have been proved using
them. In the first part I will consider general purpose tools that are
currently available for mathematical research, such as
automated/interactive theorem provers and computer algebra packages. After
a few examples illustrating their use in recent research topics, I will
examine some special purpose algorithms and approaches in universal
algebra and logic that I have used over the past decade (e.g. searching
for finite algebras, completing partial algebras and sandwich structures,
searching for winning strategies in logical games, calculating free
algebras in finitely generated varieties). If time permits, I will
demonstrate a few programs that I have written.
Paul Edelman, of the
School of Mathematics, University of Minnesota.
The Poset of Cellular Strings of a Polytope.
Let P be a polytope and l a linear functional that takes on distinct
values on the vertices of P. A monotone path in P (with respect to l)
is a sequence of vertices such that
- Consecutive vertices lie on an edge of P,
- The first and last vertex in the sequence are minimal and maximal with
respect to l, respectively, and
- The sequence is increasing with respect to l.
Monotone paths arise in many contexts in combinatorial geometry.
In studying the structure of monotone paths one is led to consider a
more general poset, the poset of cellular strings of P, in which
the monotone paths are the minimal elements. This poset was introduced
by Baues in his study of loop spaces, and later studied by Billera,
Kapranov, and Sturmfels. I will discuss a number of structural
properties of this poset, including its homotopy-type and the
connectivity of a graph on the monotone paths derived from the poset.