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Mathematics & Statistics

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Asymptotics for Stieltjes and Van Vleck polynomials

André Martínez-Finkelshtein

University of Almeria, Spain

4:00pm-5:00pm

PHY 013

E. B. Saff

**Abstract**

We are interested in the zeros of the polynomial solutions of a class of second-order linear differential equations with polynomial coefficients. The best-known examples are the hypergeometric and the Heun equations. Heine, Stieltjes, Van Vleck and others, have studied these equations, producing some beautiful results concerning the location of the zeros. In particular, their electrostatic interpretation, due to Stieltjes, is the key to the potential theory approach which allows to answer the question about the asymptotic distribution of the zeros as their number goes to infinity.

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The Poincaré Conjecture; a Million Dollar Problem

Cameron Gordon

University of Texas at Austin

3:00pm-4:00pm

PHY 109

Masahiko Saito

**Abstract**

It has been known since the 19th century that the \(2\)-sphere is the only closed \(2\)-manifold in which every loop contracts, i.e., can be shrunk to a point. In 1904, Henri Poincaré asked whether the analogous statement holds in dimension \(3\): Is the \(3\)-sphere the only closed \(3\)-manifold in which every loop contracts? The assertion that the answer is “yes” has become known as the Poincaré Conjecture. Recently, the Clay Mathematics Institute offered $\(1\) million for the solution of each of seven unsolved mathematical problems; the Poincaré Conjecture is one of them. It is the most famous unsolved problem in the field of topology, and has resisted solution for almost a century. We will explain the terms \(n\)-manifold, \(n\)-sphere, etc., and give some history and background to the conjecture, as well as explaining how non-Euclidean geometry enters into the theory of manifolds in dimensions \(2\) and \(3\).

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On Best Meromorphic Approximation of Markov Functions and \(n\)-widths

Vasiliy Prokhorov

University of South Alabama

3:00pm-4:00pm

TBA

E. B. Saff

**Abstract**

The talk is devoted to some results concerning the best meromorphic approximation on the unit circle in the space \(L_p\) of Markov functions (Cauchy transforms of positive measure \(w\) with support \(\operatorname{supp} w\) in \((-1,1)\)). Connections with \(n\)-widths, the problem of minimal Blaschke products, the theory of Hankel operators will be discussed.

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\(3\)-manifolds that are Seifert union of solid tori

Wolfgang Heil

Florida State University

3:00pm-4:00pm

PHY 109

Mohamed Elhamdadi

**Abstract**

Questions about the Lusternik-Schnirelman category of a \(3\)-manifold \(M\) lead to problems about decompositions of \(M\) into (orientable) handlebodies.

We study the class of \(3\)-manifolds that are *Seifert unions* of solid tori \(V_1,V_2\); i.e. that are obtained from \(V_1,V_2\) by identifying a collection of disjoint surfaces \(\{F'_i\}\) in \(\partial V_1\) with a collection of disjoint surfaces \(\{F''_i\}\) in \(\partial V_2\) under homeomorphisms. (If the collection \(\{F'_i\}\) and \(\{F''_i\}\) consist of essential annuli then such a Seifert union is a Seifert fiber space). In obtaining the classification of Seifert unions of two tori we are lead to consider Seifert unions of any collection of punctured balls and punctured tori. At first one might think that this would include all compact \(3\)-manifolds with boundary; however, it turns out that this collection of Seifert unions yields a very restricted class.

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Frequency Domain Method for Linear Evolutions Equations

Zhuangyi Liu

Univ. of Minnesota at Duluth

3:00pm-4:00pm

PHY 108

Yuncheng You

**Abstract**

Consider a linear control system \begin{gather*} dz(t)/dt=Az(t)+Bu(t),\; z(0)=z_0,\\ y(t)=Cz(t)+Du(t) \end{gather*} where \(z(t)\), \(u(t)\), \(y(t)\) are functions in Hilbert spaces \(Z\), \(U\), \(Y\), respectively; \(A\) generates a \(c_0\)-semigroup of contractions on \(Z\); the operators \(B\), \(C\), \(D\) are in \(L(U,Z)\), \(L(Z,Y)\), \(L(U,Y)\), respectively.

It is known that the properties of the uncontrolled system (\(u(t)=0\)), such as the asymptotic behavior and smoothness of \(z(t)\), provide crutial information to the control design. In this talk, we introduce a Frequency domain method for the study of these properties. This method is based on the relation between the properties of \(z(t)\) and the norm of the resolvent operator, \(||(sI-A)^{-1}||\), on the imaginary axis. We apply this method to several systems governed by partial differential equations with global, local, boundary, or dynamical boundary dissipative terms.

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Fluid Fingering Problems in Hele-Shaw Cells

Jianzhong Su

University of Texas at Arlington

3:00pm-4:00pm

PHY 109

Yuncheng You

**Abstract**

Fluid fingering phenomena arise from various physical problems such as oil recovery and phase transition. The interface between the two different fluids evolves according to the physical laws and generates finger like patterns. Its motion is governed by a partial differential equation with free boundaries. In this talk, we will first provide some physical background of fluid fingering problems, then we will discuss the finger solutions of Hele-Shaw equations. These finger solutions are traveling wave solutions whose finger shaped interfaces are moving along a certain direction at a constant speed. The existence of finger solutions is shown through a fixed point argument of the Hilbert Transformation.

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Meromorphic approximation and inverse boundary problem for the \(2\)-D Laplacian

Laurent Barachart

INRIA — Sophia-Antipolis

France

4:00pm-5:00pm

PHY 013

**Abstract**

We present a new approach to crack detection based on the meromorphic approximation of the complex solution to a Dirichlet Neuman problem.

Such methods are computationally attractive to detect the edges of a sufficiently regular crack. They also raise some conjectures about the behaviour of non-classical meromorphic approximants with real residues in connection with Sobolev-type discrete approximation to Green potentials.

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**Note**

Traveling Wavefronts of Reaction-Diffusion Equations

Kunquan Lan

York University

3:00pm-4:00pm

PHY 108

A. G. Kartsatos

*Speaker is a candidate for Asst. Prof. in PDEs.*

**Abstract**

Traveling waves have been studied extensively for continuous and lattice partial differential equations. These equations arise in a wide variety of models. For example, the reaction-diffusion models of Fisher type were proposed to describe the spread of a favoured gene in a population. Models involving lattice differential equations occur in biology, chemical reaction theory, image processing and pattern recognition, material science and cellular neural networks.

When one considers traveling wave solutions, these partial differential equations are reduced to eigenvalue problems for first or second order ordinary differential equations.

Traveling waves for reaction-diffusion equations correspond to eigenvalue problems for second order ordinary differential equations. They have been widely studied using several theories. For example, the theory of monotone operators together with degree theory and the theory of the phase plane trajectories have been used.

In this talk, I shall review some simple continuous population models for single species and then present our new iterative techniques and show how to apply our theory to obtain the traveling wave solutions for reaction-diffusion equations. This new theory is much simpler than and superior to previous ones. It can be applied not only to obtain the ranges of wave speeds but also to provide powerful numerical schemes to compute the waves.

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Symmetry Constraints of Zero Curvature Equations

Wen-Xiu Ma

Assistant Professor

City University of Hong Kong

3:00pm-4:00pm

PHY 109

Yuncheng You

*Speaker is a candidate for Asst. Prof. in PDEs.*

**Abstract**

Symmetry constraints are proposed to decompose zero curvature equations irrespective of dimensions into specific systems of ODEs, called constrained flows. Functionally independent and involutive systems of functions are generated from stationary zero curvature equations, and used to show the Liouville integrability for the constrained flows. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to zero curvature equations. The resulting constrained flows can be solved by separation of variables, and Jacobi inversion problems of the constrained flows exhibit the integrability by quadratures for zero curvature equations. The theory is illustrated by examples of soliton equations.

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On Nonlinear Wave Propagation in Media With Dispersion and Dissipation

Vladimir Varlamov

Visiting Associate Professor

University of Texas at Austin

3:00pm-4:00pm

PHY 108

Athanassios Kartsatos

*Speaker is a candidate for Asst. Prof. in PDEs.*

**Abstract**

A brief history of the discovery of solitons is given, and the basic semilinear evolution equations describing wave propagation in dispersive media are presented. Typical examples are the Boussinesq equation, the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation and their dissipative counterparts. The issues of wave generation by a moving boundary are discussed. For the damped Boussinesq equation a Cauchy problem is examined and the long-time is asymptotics is presented. As an example of studying multidimensional problems the damped Boussinesq equation in a disc is considered, and the long-time asymptotics is calculated. Series in special functions are used for obtaining long-time asymptotic expansions in bounded domains.

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**Note**

Enumeration of Isomorphism Classes of Extensions of \(p\)-adic Fields and Isomorphism Classes of Finite Commutative Chain Rings

Xiang-Dong Hou

Associate Professor

Wright State University

3:00pm-4:00pm

LIF 260

W. Edwin Clark

*Speaker is a candidate for Asst. Prof. in Alg./Comb.*

**Abstract**

Let \(F\) be a finite extension of \(Q_p\). Given positive integers \(f\) and \(e\), the number of extensions of \(F\) with residue degree \(f\) and ramification index \(e\) in a fixed algebraic closure of \(F\) is finite; Krasner's formulas allow one to compute this number. Our concern is the number \(I(F,f,e)\) of \(F\)-isomorphism classes of the extension of \(F\) with residue degree \(f\) and ramification index \(e\). When \(p^2\) does not divide \(e\), we determine \(I(F,f,e)\) completely; when \(p^2\) divides \(e\) exactly, we determine \(I(F,f,e)\) under some additional assumptions. Our approach is based on results from class field theory and computations with the Galois groups.

A topic closely related to the problem considered above is finite commutative chain rings. A finite commutative chain ring is a finite commutative ring whose ideals form a chain under inclusion. Such rings are useful in finite geometry and combinatorics. Each finite commutative chain ring has a set of invariants \((p,n,f,e,t)\). The number \(I(Q_p,f,e)\) is essentially the isomorphism classes of finite commutative chain rings with invariants \((p,n,f,e,t)\). When \(p\) does not divide \(e\), the number of isomorphism classes of finite commutative chain rings with invariants \((p,n,f,e,t)\) has been determined by Clark and Liang. Our results on \(I(Q_p,f,e)\) settle some open cases about the number of isomorphism classes of finite commutative chain rings.

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Bose-Mesner Algebras

Brian Curtin

NSF Post-Doctoral Fellow

University of California, Berkeley

3:00pm-4:00pm

PHY 109

*Speaker is a candidate for Asst. Prof. in Alg./Comb.*

**Abstract**

We survey some of the rich algebraic combinatorics of Bose-Mesner algebras. A Bose-Mesner algebra is a commutative complex matrix subalgebra which, in addition to ordinary matrix product, is closed under entry-wise product and transposition and which contains the identity and all-ones matrices.

We shall recall the origins of Bose-Mesner algebras in permutation groups and design theory. We then discuss the recent influence of quantum algebras and knot theory on the subject of Bose-Mesner algebras. Indeed, a certain quantum algebra arises in connection with an important family of Bose-Mesner algebras, and it was recently shown that spin models (the basic data for a statistical mechanical construction of link invariants) lies in some Bose-Mesner algebra. We have been working to elaborate on these connections.

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**Note**

A Statistical Model for Latitudinal Correlations of Satellite Data

Dongseok Choi

3:00pm-4:00pm

PHY 130

A. N. V. Rao

*Speaker is a candidate for Asst. Prof. in Statistics.*

**Abstract**

We develop a new class of models for the latitudinal correlation patterns of satellite ozone and temperature data. We employ the monthly average series for each \(5\)-degree latitude zone of the TOMS data from the Nimbus \(7\) satellite over 14 years from November 1978 to November 1992. To each latitude zone, a temporal regression model that includes all known physical effects and time dependency is fitted separately by maximizing likelihood function. From the residuals of all latitude zones, we observe strong contemporaneous spatial correlations, which decay at different rates depending on latitudes and become negative at moderate distances. A three-component model with two associated weight functions, which can accommodate the main features of the concurrent spatial correlations of the residuals is developed. We use low order ARMA models as components, which are widely used in time series analysis. A similar analysis is done on the latitudinal correlations of the residuals from the monthly average series of CPC temperature data at 30mb from January 1979 to December 1993. The new class of models can be applied to study covariance structures that consist of a few dominant factors with varying effectiveness.

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**Note**

Invariant Subspaces of Finite-Dimensional Vector Spaces

Markus Schmidmeier

Post-Doctoral Fellow

Florida Atlantic University

3:00pm-4:00pm

PHY 109

W. Edwin Clark

*Speaker is a candidate for Asst. Prof. in Alg./Comb.*

**Abstract**

Consider a linear operator \(T\) which acts nilpotently on a finite-dimensional vector space \(V\) and a subspace \(U\) of \(V\) that is invariant under the action of \(T\). As a classification of all such systems \((T,V,U)\) is not feasible, we focus on those systems for which \(T\) has nilpotency index \(n\) on \(V\), and nilpotency index \(m\) on \(U\). In joint work with C. M. Ringel, we showed the following:

- There are only finitely many indecomposable systems (\(T\), \(V\), \(U\)), up to equivalence, if \(m<3\), or if \(n<6\), or if \((m,n)\) is the pair \((3,6)\).
- There are infinitely many systems, which can be classified, in case \((m,n)\) is one of the pairs \((6,6)\), \((5,6)\), \((4,6)\), or \((3,7)\).
- Otherwise, a classification is not feasible.

Our study of invariant subspaces is motivated by — and related to — the problem of classifying the \(p^m\)-bounded subgroups of a \(p^n\)-bounded finite abelian group.

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Baker-Hausdorff Theorem

Sam Sakmar

Department of Physics, USF

3:00pm-4:00pm

TBA

Masahiko Saito

**Abstract**

Lie groups play an important role both in mathematics and physics. Because of the non-commutativity of the multiplication rule of the Lie Algebras the combination of the group elements of the Lie Groups is non-trivial and is given by the Baker-Hausdorff theorem. We give the proofs of all the theorems needed for the proof of the Baker-Hausdorff theorem.