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.
Monday, October 6, 2008, 4:00-5:00 pm.
- Place: EMCS 422
- Speaker: Lingju Kong, Department of Mathematics, University of Tennessee at Chattanooga, Tennessee, USA
- Speaker: Qingkai Kong, Department of Mathematics, Northern Illinois University, DeKalb, Illinois, USA
- Title: On Nodal Solutions for Second Order Boundary Value Problems
- Abstract: We study the nonlinear boundary value problem (BVP) consisting of the equation: $$-(p(t)y')'+q(t)y= w(t)f(y),{\rm on}\;[a,b]$$ and a general separated boundary condition (BC). By comparing it with a linear Sturm-Liouville problem (SLP) we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let $\lambda_n, n=0,1,2,\dots$, be the $n$-th eigenvalue of the corresponding linear SLP. Then the BVP has a pair of solutions with exactly $n$ zeros in $(a,b)$ if $\lambda_n$ is in the interior of the range of $f(y)/y$; and does not have any solution with exactly $n$ zeros in $(a,b)$ if $\lambda_n$ is outside this range. These conditions become necessary and sufficient when $f(y)/y$ is monotone on $(-\infty, 0)$ and on $(0,\infty)$. We also discuss the changes of the number of different types of nodal solutions as the equation or the BC changes. Our results are obtained without assuming the global existence and uniqueness of solutions of the corresponding initial value problems. The proofs are mainly based on a bifurcation theorem by Rabinowitz and some recent results on the dependence of the eigenvalues of SLPs on the problems.
Monday, September 29, 2008, 4:00-4:50 pm.
- Place: EMCS 422
- Speaker: Boris Belinskiy, Department of Mathematics, University of Tennessee at Chattanooga, Tennessee, USA
- Title: My High School Education: Some Thoughts about Two Different Teaching Philosophies
- Abstract: This is a story of my high school education at the fSU (former Soviet Union.) It is based on my own experience and does not lay claim to any generalization. My goal is to describe the mixture of the social, cultural, and political circumstances under which I was lucky to get, I think, a rather good high school education. I will try to describe both advantages and disadvantages of the fSU educational system in comparison with our system.
Monday, September 15, 2008, 4:00-4:50 pm.
- Place: EMCS 422
- Speaker: Francesco Barioli, Department of Mathematics, University of Tennessee at Chattanooga, Tennessee, USA
- Title: Minimum Rank for Symmetric Matrices with Qualitative Constraints
- Abstract: Over the last several years increasing attention has been paid to Inverse Eigenvalue Problems, mainly for the important applications to several fields, like Dynamical Systems, Image Reconstruction, Signal Reconstruction, and many others. In fact, an Inverse Eigenvalue Problem (IEP) consists on the reconstruction of a matrix from partial data, like eigenvalues and/or eigenvectors. Depending on the kind of information available (partial or complete, qualitative or quantitative) and depending on the constraints such a matrix is required to satisfy (nonnegativity, symmetry, zero/nonzero pattern), the solution of an IEP may be hard to obtain. As a matter of fact, for most IEPs a solution is far to be known. For a given class of matrices, a first step toward the solution of an IEP, is the determination of the minimum possible rank among all the matrices in that class. Of particular interest is the Minimum Rank Problem for a simple graph, namely, the minimum rank among real symmetric matrices whose zero-nonzero pattern of off-diagonal entries is described by a given simple graph G. The increasing interest in the problem has been brought to the attention of AIM (American Institute of Mathematics). In October 2006, an AIM workshop held in Palo Alto, CA, on Spectra of families of matrices, brought together people interested in Combinatorial Matrix Theory and Spectral Graph Theory. Since then, a group of 8-10 people, known as the AIM Minimum Rank Special Graphs Work Group is collaborating in a fruitful research. In this talk I will present most of this Work Group research. In particular, the Zero Forcing technique for the determination of the minimum rank of several classes of graphs, and results on several conjectures on minimum rank in terms of complements of graphs and dual graphs.
Friday, August 22, 2008, 2:00-2:50 pm.
- Place: EMCS 422
- Speaker: Linshan Wang, Department of Mathematics, Ocean University of China, Qingdao, P.R. China
- Title: Stability Analysis of Reaction-diffusion; Neural Networks with Delays
- Abstract: In this paper, we survey and utilize results from stability analysis of the reaction-diffusion equations with delays in order to develop a stable theory for the neural networks with delays.
Friday, August 22, 2008, 3:00-3:50 pm.
- Place: EMCS 422
- Speaker: Liping Wang, Department of Mathematics, Ocean University of China, Qingdao, P.R. China
- Title: Multivariate Compound Distribution and its Engineering Applications
- Abstract: Compound extreme value distribution (CEVD) was derived in 1980 as Poisson-Gumbel CEVD for typhoon induced extreme events in China Sea and Poisson-Weibull model for hurricane characteristics along Atlantic coasts and Gulf of Mexico at 1982. After their publication CEVD have aroused some interests of scientists and engineers. According to incomplete statistics for about forty coastal structures the CEVD was successfully used to predict design wave height. Especially, the catastrophe in New Orleans induced by Hurricane Katrina 2005 shows, that the design basis of New Orleans protection structures is ‘Standard Project Hurricane’ proposed by NOAA, which only corresponded to 38 years return period of hurricane central pressure and Hurricane Katrina corresponds to 60 years return period predicted by Poisson-Weibull model. Bivariate compound extreme value distribution was derived in different forms and applied in some engineering aspects, such as platform deck clearance design, disaster prevention design criteria for estuarine city Shanghai, coastal city Qingdao and et al. In this paper, the Poisson-Nested Logistic Trivariate Compound Extreme Value Distribution (PNLTCED) is derived by compounding the discrete distribution of number of data sampling over certain threshold level per year (Poisson distribution) into the multivariate continuous distribution (Nested logistic trivariate distribution). The distinguish of this model from our previous study lies in the data sampling of discrete distribution. In this paper, the number of data sampling over threshold level is taken account in the CEVD instead of typhoon occurrence frequency. Comparison between the predicted results based on the long term observed data and short term data shows its stability in prediction.
