Colloquia in the Mathematics Department
The colloquia are coordinated by Boris Belinskiy. The talks are usually in the EMCS building but the time and location varies. Please browse our prior colloquia.
Wednesday, April 9, 10:00-11:00 am.
- Place: EMCS 239
- Speaker: Dr. Sergei Avdonin, Department of Mathematics and Statistics, University of Alaska, Fairbanks, Alaska, USA
- Title: The Boundary Control Approach to Inverse Problems and Signal Processing
- Abstract: The goal of this talk is to describe connections between non-harmonic Fourier series, control theory, inverse problems of mathematical physics, and signal processing. In particular, we describe an approach to inverse problems (the so-called Boundary Control Method) which is based on deep connections between controllability and identification problems and is applicable to a wide range of linear systems. As an example of the approach, we consider control and inverse problems for differential equations on graphs. We suppose that on each edge of the graph, the wave (or heat, or Schr\"odinger) equation is defined, and that standard compatibility conditions are satisfied at the internal vertices. We prove that the system is exactly controllable if the graph is a tree and the control is applied to all (or to all but one) boundary vertices. Otherwise the system is not generally exactly controllable but may be spectrally controllable. We show how to recover a tree (its connectivity and the lengths of the edges together with coefficients of the equation) from a given response operator or Weyl matrix function. We demonstrate effectiveness of the Boundary Control Method on a classical problem of signal processing --- the spectral estimation problem. The boundary control approach to sampling and interpolation of band-limited and multi-band signals will also be discussed.
- Note: Dr. Avdonin is a candidate for the position of Chair of Excellence in Applied Mathematics.
Monday, April 7, 12:00-1:00
- Place: EMCS 239
- Speaker: Shaobai Kan, Department of Mathematics, Wayne State University, Detroit, Michigan
- Title: Identification of Systems With Regime Switching and Unmodeled Dynamics
- Abstract: This paper is concerned with persistent identification of systems that involve deterministic unmodeled dynamics and stochastic observation disturbances, and whose unknown parameters switch values (possibly large sizes) that can be represented by a Markov chain. Two classes of problems are considered. In the first class, the switching parameters are stochastic processes modeled by irreducible and aperiodic Markov chains with rates of state transitions sufficiently higher than adaptation rates of identification algorithms. We show that an averaged behavior of the parameter process can be derived from the stationary measure of the Markov chain and can be estimated with periodic inputs and least-squares type algorithms. Upper and lower error bounds are established that explicitly show impact of unmodeled dynamics. In contrast, the second class of problems represents systems whose state transitions occur infrequently. An adaptive algorithm with variable step sizes is introduced for tracking the time-varying parameters. Convergence and error bounds are derived. Numerical results are presented to illustrate the performance of the algorithm.
- Note: Shaobai Kan is a candidate for a position in our Department.
Thursday, April 3, 3:00-4:00 pm.
- Place: EMCS 422
- Speaker: Dr. Johnny Henderson, Department of Mathematics, Baylor University, Waco, Texas
- Title: Uniqueness Implies Existence and Uniqueness Criterion for Certain Nonlocal Boundary Value Problems for Third Order Ordinary Differential Equations
- Abstract: For the third order ordinary differential equation, $y'''=f(x,y,y',y'')$, we consider uniqueness implies existence arguments for solutions satisfying the nonlocal 4-point boundary conditions, $y(x_1)=y_1,\;y(x_2)=y_2,\;y(x_3)-y(x_4)=y_3$. Uniqueness of solutions of such boundary value problems are intimately related to solutions of the third order equation satisfying certain 3-point boundary conditions. These relationships will be discussed as well. Finally, in the case when $f$ is Lipschitz, optimal interval lengths will be determined, in terms of the Lipschitz coefficients, on which solutions are unique (hence exist), for the 4-point nonlocal boundary value problems.
- Note: Dr. Henderson is a candidate for the position of Chair of Excellence in Applied Mathematics.
Monday, March 31, 12:00-1:00
- Place: EMCS 239
- Speaker: Dr. Daniela Szatmari-Voicu, Texas Environmental Studies & Analysis, LLC
- Title: Minimax Bias L-estimators of Scale Parameter
- Abstract: We derive the best non-symmetrized and symmetrized L-estimators of scale with respect to their asymptotic bias, under e - contamination neighborhood centered at a known error distribution F0 which is symmetric and unimodal, by first performing a study on the maximum asymptotic bias curves of the non-symmetrized and symmetrized interquantile ranges from which the L-estimators considered arise. For F0 symmetric and unimodal and making use of the generalized method of moment spaces, the solutions to the minimax- bias problems are shown to be convex combinations of at most two interquantile ranges. Sufficient conditions are also found such that the solutions are just one interquantile range.
- Note: Dr. Szatmari-Voicu is a candidate for a tenure-track position in our Department.
Wednesday, March 19, 12:00-1:00 pm.
- Place: EMCS 239
- Speaker: Dr. Ravi Agarwal, Department of Mathematics, Florida Institute of Technology, Melbourne, Florida
- Title: Singular Integral Equations With Real World Applications
- Abstract: We shall provide easily verifiable sufficient conditions which guarantee the existence of solutions to some singular integral equations. The motivation of these problems comes from real world applications, particularly, in communications theory, and the Homann flow.
- Note: Dr. Agarwal is a candidate for the position of the Chair of Excellence in Applied Mathematics.
Tuesday, March 4, 3:05 pm.
- Place: EMCS 422
- Speaker: Boris Belinskiy*, Department of Mathematics, University of Tennessee at Chattanooga
- Title: Stochastic Wave Equation Driven by a Fractional Brownian Motion
- Abstract: We consider a linear stochastic wave equation driven by fractional-in-time noise. We prove the existence and uniqueness of the weak solution. We also study the expected energy associated with wave equation and improve our previous results on that matter. Specifically, we find the iff condition of the convergence of the series representing the expected energy using physically natural objects. We discuss the smoothness of the solution. We consider both cases $H>1/2$ and $H<1/2$ for the Hurst parameter.
- Note: The presentation would be of interest to Math students who are majoring in any area of Applied Math.
- Note: * Research of this author was supported in part by the University of Tennessee at Chattanooga Faculty Research Grant.
Tuesday, February 19, EMCS 422, 3:05 pm.
- Place: EMCS 422
- Speaker: Meg Kiessling, Department of Mathematics, University of Tennessee at Chattanooga
- Title: The Use of Projects in Math 136 (Calculus for Management, Life, and Social Sciences), Math 215 (Math for Elementary School Teachers), and Math 123 (Mathematics in Our Modern World)
- Abstract: Over the past three years, I have incorporated projects in all of these classes. I will present information on the types of projects I have used, the expectations I had for these projects, the students' response and performance in completing the projects, sources for project ideas and methods used for grading the projects.
- Note: All are welcome to attend this talk, especially those considering incorporating projects into these or similar courses.
Thursday, January 24, 3:05-4:00 pm.
- Place: EMCS 422
- Speaker: Aniekan Ebiefung , Department of Mathematics, University of Tennessee at Chattanooga
- Title: Disjunctive Programming and Generalized Leontief Input-Output Model
- Abstract: This paper considers a generalization of the Leontief input-output model that is useful in modeling the concept of choice of technology. It is shown that a disjunctive programming problem, together with its dual problem, may be used to effectively solve the new input-output model.
- Note: This is a modeling paper that will be of interest to students and the general public.
Thursday, January 10, 3:05 pm.
- Place: EMCS 422
- Speaker: Marat Akhmet, Department of Mathematics, Middle East Technical Univ., Ankara, Turkey
- Title: Differential Equations with Piecewise Argument of Generalized Type: a New Theory for 25-year-old Equations
- Abstract: Differential equations with an argument of a solution as the greatest integer function were introduced at the beginning of 1980s by K. Cooke and co-authors. Significant theoretical results concerning oscillations, boundary value problems, positive solutions have been obtained. The only method of investigation for these systems has been the reduction of the equations to systems of discrete equations, which involves only the values of solutions at integers or their multiples. Thus, many common questions for all types of differential equations: existence and uniqueness of solutions, dependence of solutions on parameters, stability, existence of integral manifolds and almost periodic solutions, etc., have not been investigated fully. In a few last papers, we have proposed a new approach, which allows us to create a theory of these systems very similar to that of ODE despite the equations considered by us have deviating arguments. In the report, we discuss the results obtained last 2-3 years. The phenomenon of bifurcation and chaos for these equations and applications to certain problems of population dynamics will also be considered.
Thursday, November 8, 3:30 pm.
- Place: EMCS 423
- Speaker: Matt Mathews, Department of Mathematics, University of Tennessee at Chattanooga
- Title: Granular Flows Around Hopper Inserts
- Abstract: It is common to use corrective inserts to encourage better flow patterns in granular material moving through a converging hopper. One model of a steady state granular flow around such an insert leads to a set of hyperbolic conservation laws for the components of the stress tensor and two components of velocity. A Runge-Kutta discontinuous Galerkin (RKDG) method is applied and the resulting flow fields are computed.
- Note: Students who have completed a differential equations course, with linear algebra, will find much of the mathematics accessible. Students who have completed a numerical analysis course will see many familiar concepts.
Thursday, October 11, 3:00 pm.
- Place: EMCS 422
- Speaker: Lingju Kong, Department of Mathematics, University of Tennessee at Chattanooga
- Title: New Existence Results for Higher Order Multi-point Boundary Value Problems
- Abstract: We consider a certain higher order multi-point boundary value problem. We provide sufficient conditions for the existence of a solution of this problem based on the existence of higher order lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has nonlinear dependence on all lower order derivatives of the unknown function and the boundary condition covers many multi-point boundary conditions studied in the literature. Schauder's fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are included to illustrate the results.
- Note: These results are joint work with John R. Graef and Qingkai Kong.
- Note: Students who have had an introduction to ordinary differential equations will find the subject matter of interest.
Thursday, September 13, 3:00 pm.
- Place: EMCS 422
- Speaker: Qingkai Kong, Department of Mathematical Science, Northern Illinois University
- Title: Use Time Scales to Study Impulsive Functional Differential Equations
- Abstract: We explore connections between certain impulsive differential equations and corresponding dynamic equations on time scales. More specifically, for a given impulsive differential equation, we construct a ``counterpart'' as an equation defined on a time scale which has the same qualitative behavior. By doing so, we can simply ``translate'' any results for the equation on time scale to the impulsive differential equations. As applications of this result, we obtain oscillation criteria for two linear impulsive differential equations with delays by applying existing results for equations on time scales. Our work shows that this method provides a new approach for the general impulsive differential equations and can be used to some equations which are difficult to deal with by the traditional ways.
- Note: Students who have had an introduction to ordinary differential equations will find the subject matter of interest.
