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Mathematics & Statistics

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A remark on a theorem of Jech and Shelah

Peter Komjath

Emory University

3:00pm-4:00pm

PHY 109

Vilmos Totik

**Abstract**

Jech and Shelah showed that if \(\aleph_{\omega}\) is a strong limit and \(2^{\aleph_{\omega}}\aleph_{\omega_{1}}\) holds (the consistency of this statement is unknown) then a system of subsets of \(\omega_1\) with certain properties exist. They proved the rather surprising fact that such a system exists anyway.

We give a simpler proof of the latter result.

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Exact Solutions to Integrable Evolution Equations

Tuncay Aktosun

University of Texas at Arlington

3:00pm-4:00pm

PHY 109

Wen-Xiu Ma

**Abstract**

A method is presented to construct certain exact solutions to those nonlinear partial differential equations integrable by the inverse scattering transform with the help of a Marchenko integral equation. Explicit formulas are obtained to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates. The method is applied to some integrable equations such as the Korteweg-de Vries equation on the half line and the nonlinear Schröedinger equation.

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An equivalence of Ward's bound and its applications

Xiaoyu Liu

Wright State University

3:00pm-4:00pm

PHY 109

Xiang-dong Hou

**Abstract**

In 1981, Harold N. Ward introduced the concept of divisibility in linear codes and investigated linear codes from this point of view. Briefly speaking, divisible codes are simply codes whose codewords all have weights divisible by a nontrivial integer, say \(\Delta\) (called a divisor of the code). The reason that these codes are interesting is that people observe nontrivial divisibility in most of the optimal codes, as well as in many other good codes. Ward proved a bound on the dimension of a linear divisible code over a finite field. In this talk, we will show an equivalence of this Ward's bound. Using the equivalence, we generalize Ward's bound to some nonlinear codes. Another application of the result is to generalize the Gleason-Pierce-Ward theorem to additive codes.

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A Short Survey of Extension Theorems

Robb Fry

Thomson Rivers University

3:00pm-4:00pm

PHY 109

LesÅ‚aw Skrzypek

**Abstract**

We shall briefly examine extension theorems in various contexts, as well as a few of their applications. As classical examples we include Tietze's Theorem, Lavrentieff's Theorem, andthe Hahn-Banach Theorem. Recent and less well known results concerning extensions of locally uniformly convex norms and extensions of real-valued, smooth functions on Banach spaces will also be examined.

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Bilinear Forms on the Dirichlet Space

Brett Wick

University of South Carolina

3:00pm-4:00pm

PHY 109

Dmitry Khavinson

**Abstract**

The Dirichlet space is the set of analytic functions on the disc that have a square integrable derivative. In this talk we will discuss necessary and sufficient conditions in order to have a bilinear form on the Dirichlet space be bounded. This condition will be expressed in terms of a Carleson measure condition for the Dirichlet space. One can view this result as the Dirichlet space analogue of Nehari's Theorem for the classical Hardy space on the disc. This talk is based on joint work with N. Arcozzi, R. Rochberg, and E. Sawyer.

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Non-intersecting random paths and multiple orthogonal polynomials

Andrei Martínez-Finkelshtein

Universidad de Almería

Almería, SPAIN

3:00pm-4:00pm

PHY 109

E. Rakhmanov

**Abstract**

Determinantal point processes are of considerable current interest in probability theory and mathematical physics, since they arise naturally in random matrix theory, non-intersecting paths, certain combinatorial and stochastic growth models and representation theory of large groups. We are interested in a situation when the determinantal point process is related to polynomials of multiple orthogonality (that is, when the orthogonality conditions are distributed among several weights). The simplest case is of the non-intersecting paths of particles performing brownian motion, ending at two or more fixed positions (this model is connected also with the eigenvalue distribution of GUE with external source).

We study \(n\) non-intersecting squared Bessel processes in the confluent case: all particles start at time \(t=0\) at the same positive value \(x=a\), remain positive (due to the nature of the squared Bessel process) and are conditioned to end at time \(t=T\) at \(x=0\). In the limit \(n\to\infty\), after appropriate rescaling, the paths fill out a region in the \(tx\)-plane. In particular, the paths initially stay away from the hard edge at \(x=0\), but at a certain critical time the smallest paths hit the hard edge and from then on are stuck to it. The phase transition at the critical time is a new feature of the present model.

A key fact in the analysis is that in the confluent case the biorthogonal ensemble reduces to a multiple orthogonal polynomial ensemble, corresponding to a Nikishin system of two modified Bessel-type weights. This means that there is a \(3\times 3\) RH problem characterizing this model, that can be analyzed in the large \(n\) limit using the Deift-Zhou steepest descent method.

This is a joint work with A. B. J. Kuijlaars and F. Wielonsky.

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TBA

Alfonso Montes Rodríguez

Universidad de Sevilla

Sevilla, SPAIN

4:00pm-5:00pm

PHY 109

Sherwin Kouchekian

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TBA

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Fine asymptotics for Bergman orthogonal polynomials over domains with corners

Nikos Stylianopoulos

University of Cyprus

3:00pm-4:00pm

NES 104

Dmitry Khavinson

**Abstract**

Let \(G\) be a bounded simply-connected domain in the complex plane \(\mathbb{C}\), whose boundary \(\Gamma:=\partial G\) is a Jordan curve, and let \(\{p_n\}^{\infty}_{n=0}\) denote the sequence of Bergman polynomials of \(G\). This is defined as the sequence $$ p_n(z)=\lambda_nz^n+\dotsb,\quad \lambda_n > 0,\quad n=0,1,2,\dotsc, $$ of polynomials that are orthonormal with respect to the inner product $$ \langle f,g\rangle:=\int_G f(z)g(z)\,dm(z), $$ where \(dm\) stands for the area measure.

The purpose of the talk is to report on recent results regarding the fine asymptotic behaviour of the polynomials \(p_n(z)\), in \(\Omega:=\mathbb{C}\setminus(G)\), and the leading coefficients \(\lambda_n\), \(n\in\mathbb{N}\), in cases when the boundary \(\Gamma\) is piecewise analytic without cusps. These results complement an investigation started in the 1920's by T. Carleman, who obtained the fine asymptotics for domains with analytic boundaries and carried over by P. K. Suetin in the 1960's, who established them for domains with smooth boundaries.

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Symmetric quandles and minimal triple point numbers of surface-links

Kanako Oshiro

Hiroshima University

Hiroshima, JAPAN

3:00pm-4:00pm

PHY 109

Masahiko Saito

**Abstract**

For a \(1\)-dimensional link, the crossing number is defined by the minimal number of crossings of any diagram. For \(2\)-dimensional links (closed surfaces embedded in \(4\)-space), minimal triple point numbers correspond to crossing numbers of \(1\)-dimensional links. In this talk, we introduce a method to estimate minimal triple point numbers with symmetric quandles, and we show some results which can be obtained by our method.