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Title: Factorization of bivariate and multivariate trigonometric polynomials Speaker: Jeff Geronimo, Georgia Tech Time: 3:00pm‐4:00pm Place: PHY 130

An important problem in mathematics is how to write a positive polynomial as a sum of squares of polynomials or rational functions. The simplest case is the Fejer-Riesz Lemma which shows how to write a trigonometric polynomial of a certain degree in one variable as the magnitude square of an algebraic polynomial of the same degree. I will review some applications of this result as well as discuss some recent extensions.

Title: Weierstrass Theorem for homogeneous polynomials on convex bodies and rate of approximation of convex bodies by convex algebraic level surfaces Speaker: Andras Kroo, Georgia Tech Time: 3:00pm‐4:00pm Place: PHY 130

By the classical Weierstrass theorem any function continuous on a compact set can be uniformly approximated by algebraic polynomials. In this talk we shall discuss possible extensions of this basic result of analysis to approximation by homogeneous algebraic polynomials on central symmetric convex bodies.

We shall also consider a related question of approximating convex bodies by convex algebraic level surfaces. It has been known for some time time that any convex body can be approximated arbirarily well by convex algebraic level surfaces. We shall present in this talk some new results specifying rate of convergence.

Title: An efficient algortihm for solving mathematical biology problems Speaker: Ahmet Yildirim, Ege University İzmir, Turkey Time: 3:00pm‐4:00pm Place: PHY 130

The aim of this study is to present a reliable algorithm for solving mathematical biology problems. We will consider the Hantavirus infection model, avian–human influenza epidemic model, fractional order model of HIV infection of \(CD4+\) T Cells, and the fractional-order viral dynamic model. We will apply the multi-stage differential transformation method (MsDTM). The results are compared with the results of Runge–Kutta method (RK-method).

Title: Hilbert problem for a multiply-connected circular domain and applications to fluids and electromagnetics Speaker: Yuri Antipov, Louisiana State University, Baton Rouge Time: 3:00pm‐4:00pm Place: PHY 130

Motivated by the study of supercavitating flow past \(n\) hydrofoils and the Hall effect in a plate with \(n\) circular holes we analyze the Hilbert problem of the theory of analytic functions with discontinuous coefficients for a multiply-connected circular domain. The Hilbert problem maps into the Riemann-Hilbert problem for piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution of this problem is derived in terms of two analogues of the Cauchy kernels, quasi-automorphic and quasi-multiplicative kernels. We show that the use of an automorphic Cauchy kernel requires the solution of an associated real Jacobi inversion problem, while it is bypassed if the quasi-automorphic and quasi-multiplicative kernels are employed. Series representations for both kernels are derived for the Burnside first class groups. In general, the quasi-automorphic kernel is presented in terms of the Schottky-Klein prime function, and the existence of the second, quasi-multiplicative, kernel is proved.

Title: Knotted graphs, embedded foams, and constructing invariants Speaker: J. Scott Carter, University of South Alabama Time: 3:00pm‐4:00pm Place: PHY 130

A general problem in topology is to understand and to classify embeddings of one object into another. When the difference in dimension is \(2\), then there is usually some type of knotting. For example, trivalent graphs can be embedded non-trivially in \(3\)-space. A foam is a \(2\)-dimensional analogue in which \(3\) faces come together along an edge. These can be embedded in \(4\)-space. There is a natural homology theory associated with a family of quandles that is parametrized by a group. Knottings of graphs and foams can be used to represent homology classes.

Title: Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation Speaker: Chunxia Li, Capital Normal University Beijing, PR China Time: 3:00pm‐4:00pm Place: PHY 130

Two families of solutions of a generalized non-Abelian Toda lattice are considered. These solutions are expressed in terms of quasideterminants, constructed by means of Darboux and binary Darboux transformations. As an example of the application of these solutions, we consider the \(2\)-periodic reduction to a matrix sine-Gordon equation. In particular, we investigate the interaction properties of polarized kink solutions.

Title: The Hom-Yang-Baxter Equation Speaker: Donald Yau, Ohio State University at Newark Time: 3:00pm‐4:00pm Place: PHY 130

The Yang-Baxter Equation (YBE) plays an important role in quantum groups and invariants of knots. The Hom-Yang-Baxter Equation (HYBE) is a generalization of the YBE that involves a twisting. In this talk, I will introduce the HYBE and discuss several classes of solutions. Prior knowledge of the YBE is not assumed.