Colloquia -- Spring 2005

Friday, April 15, 2005

Abstract

We study problems of invariance of domain and eigenvalues for nonlinear operator equations (*) Tx + Cx = 0, where T, C are from a Banach space X to its dual space X* with T maximal monotone. Invariance of domain refers to the property that the image of a relatively open set in a mapping's domain is an open set in the range space. We show how to extend the famous Schauder invariance of domain theorem, involving injective compact displacements of the identity I + C, to operators T + C, where T is, possibly, densely defined. We also show how to obtain eigenvalues q for operator equations Tx + C(q,x) = 0, where T is maximal monotone and C is demicontinuous, bounded, and of type (S+) w.r.t. the variable x. We use the Leray-Schauder degree theory when C is compact (reducing the problem (*) to a problem of the type (I+C)x = 0), and the Browder degree theory for C demicontinuous, bounded and of type (S+) w.r.t. x.

Friday, April 8, 2005

Abstract

Problems with right-censored data arise frequently in survival analysis and reliability applications, and estimation of the lifetime density function is often of interest. Two inherent problems in kernel density estimation for lifetime data are the “spillover” at the origin and the selection of the smoothing parameter (or bandwidth) values to use in computing the density estimate. To address these issues, we propose the use of asymmetric kernels with a Bayesian approach to bandwidth selection. In particular, the inverse Gaussian density function is used here as the kernel, although other asymmetric densities such as the lognormal can be considered. The (local) Bayes bandwidth obtained is exact for any sample size, only depends on the prior parameters, and can be easily calculated from the censored data. Strong pointwise consistency of the density estimator is proven, and it is also shown that meaningful bandwidths with the same rates of convergence as for the classical asymptotically optimal bandwidths can be obtained for suitable choices of the prior parameters. (Joint work with K.B. Kulasekera, Clemson University.)

Friday, April 1, 2005

Abstract

Monday, March 28, 2005

Abstract

We describe an effective method for locally resolving the zero set of a real-analytic function f(x,y). The method is geometric and involves doing a finite sequence of transformations taking a point (x,y) to a point (x, y - g(x^(1/N))) for appropriate real-analytic functions g, where N is an integer.

After these transformations, a branch of the zero set of f(x,y) will be (locally) given by {(x,y): x > 0, y = 0}or {(x,y): x < 0, y = 0}. This method has applications to oscillatory integral operators, as well as to the determination of the largest e > 0 for which the integral of |f|^(-e) is finite near a given zero of f(x,y).

Friday, March 25, 2005

Abstract

We will review the theory of “fast growing” polynomials and show how to construct them using the exterior conformal mapping. We will expand analytic functions in series of fast growing polynomials and show how to use these expansions to construct the (near) best uniform approximation of these analytic functions on a given curve in the complex plane.
We will show how the near best approximation is used for the purpose of matrix preconditioning. Numerical experiments will then be presented.

Friday, March 25, 2005

Abstract

In 1999, M. Khovanov introduced a graded homology theory for knots, and proved their graded Euler characteristic is the Jones polynomial. These homology groups turn out to be surprisingly strong invariants and have sparked much attention in low dimensional topology. In this talk, we introduce an analogous homology theory for graphs, whose graded Euler characteristic is the chromatic polynomial. Most results are joint work with Laure Helme-Guizon.

Friday, March 11, 2005

Abstract

A directed graph G = (V,E) is called k-admissible if all vertices have the same outdegree k. Let us color the edges using k colors in such a way that in every vertex the outgoing edges have distinct colors. Any sequence w of colors specifies a vertex transformation fw : V -> V where fw(v) is the unique vertex reached from vertex v by following the edges colored by letters of w. Word w is called synchronizing if fw is a constant function, that is, if one reaches the same vertex regardless of the starting position in the graph. The coloring of G is called synchronized if a synchronizing word w exists. We investigate two old open synchronization problems:

1. The road-coloring problem asks which graphs have synchronized colorings. It is conjectured that a synchronized coloring exists for all strongly connected graphs that are not periodic. A graph is periodic if some number m > 1 divides the lengths of all cycles.
2. Suppose we have a synchronized coloring of a graph with n vertices. The Cerny conjecture states that a synchronizing word of length at most (n-1)² must exist.

We prove these conjectures in the special case that the graph is Eulerian, that is, all indegrees of all vertices are also the same constant k. This is an interesting special case as such graphs seem difficult to synchronize due to the lack of vertices with large numbers (>k) of incoming edges.

Friday, March 4, 2005

Friday, March 4, 2005

Abstract

Let for n ∈ N, B_n denotes the Banach-Mazur compactum, i.e., the set of all n-dimensional, real Banach spaces equipped with the Banach-Mazur distance. Let S_n denote a subset of B_n consisting of all symmetric, n-dimensional, real Banach spaces.

Consider for any n ∈ N a function λ_n : S_n → R defined by
$$
λ_n(X) = λ(X, l_∞),
$$
where λ(X, l_∞) denotes the norm of minimal projection from l_∞ onto X. The aim of this talk is to present a construction of n-dimensional, real, symmetric spaces X_n for which λ_n (X_n) is large.

In particular, we show that
$$
liminf_n λ(X_n)/\sqrt{n} > (2-\sqrt{2/π})^-1,
$$
which disproves a conjecture of H. Koenig.

Also some open problems will be indicated.

Friday, February 25, 2005

Abstract

The Riemann Hypothesis is now left as the most famous unsolved problem in mathematics. Extensive computations of zeros have been used not only to provide evidence for its truth, but also for the truth of deeper conjectures that predict fine scale statistics on the distribution of zeros of various zeta functions.

These conjectures connect number theory with physics, and are regarded by many as the most promising avenue towards a proof of the Riemann Hypothesis. However, as is often true in mathematics, numerical data is subject to a variety of interpretations, and it is possible to argue that the numerical evidence we have gathered so far is misleading. Whatever the truth may be, the computational exploration of zeros of zeta functions is flourishing, and through projects such as the ZetaGrid is drawing many amateurs into contact with higher mathematics.

Friday, February 18, 2005

Abstract

Error-correcting codes are widely used to correct errors in either the transmission or storage of information. A specific error-correcting code provides the high fidelity on compact discs. Because of their demonstrated practical usefulness, electrical engineers started studying these codes about fifty years ago. Now they are studied by engineers, mathematicians and computer scientists and a wide theory has been developed with many connections to mathematical topics.

I will give all the basic definitions with examples and main problems in error-correcting codes. We will discuss syndrome decoding and perfect codes.

We will then use this to determine “what color is my hat.”

Friday, February 11, 2005

Abstract

We present the novel model of “topologized graphs,” in which a (possibly finite) graph is a toplogical space. We discuss the role of topological ideas in extending well-known results about cycle spaces from finite to infinite (topologized) graphs. We show how (non-Hausdorff) graph-theoretic paths and trees, respectively, can be un ified with the (Hausdorff) orderable and dendritic spaces of general topology and how our “orderable” spaces are naturally topologized graphs. We show how an attractive relaxation of the T₁ axiom emerges naturally in different ways. Some results are similar to those of Whyburn (topology, 1968), Ward and others (partial order characterizations, 1970s) and Diestel and Kühn (cycle spaces, 2004) in more general and unified settings. This is joint work with Bruce Richter.

Friday, January 28, 2005

Abstract

The great French mathematician Jean Pierre Serre worked in homotopy theory during the decade 1951-1960 and completely revolutionized the subject. I was fortunate to know J. P. Serre well during that period, and I will reminisce about the unique experience of working with him. At the end of the talk I will describe some of Serre's key ideas.

Wednesday, January 26, 2005

Abstract

In practice, quadrilateral mesh is much flexible in the finite element approximation for a curved domain than rectangular and equally well suitable as triangular. However, the existing results of convergence and superconvergence properties of nonconforming elements over rectangular meshes can hardly be extended directly to quadrilaterals. In this talk, we study some low order nonconforming finite elements. The convergence and superconvergence over general quadrilateral meshes are discussed.

Friday, January 21, 2005

Abstract

In this talk we analyze some recent results on classical families of polynomials (Jacobi, Laguerre) when the parameters take non-classical values. In particular, we study the asymptotics of Jacobi polynomials with varying parameters. To this end, a Riemann-Hilbert approach is used.

Friday, January 14, 2005

Abstract

In 1970's, Eliashberg classified pairs of immersions of the 2-disk to the plane up to the regular homotopy. He used the homotopy principle and proved that there are precisely two regular homotopy classes of such pairs.

In this talk, we classify such pairs up to the regular homotopy by using another method. Comparing to Eliashberg's method, our method is combinatorial.

Monday, January 10, 2005

Abstract

Integrable semi-discretizations of two model equations for shallow water waves are investigated. As a result, one integrable differential-difference version for the AKNS equation and three integrable differential-difference versions for the Hirota-Satsuma equation are found. These four differential-difference versions are transformed into bilinear forms. Baecklund transformations, soliton solutions and Lax pairs for these differential-difference equations are presented.

Please direct questions to mthmaster@nosferatu.cas.usf.edu.
Last updated: 07-Apr-2005.
Copyright 2000, USF Department of Mathematics.