Weekly Colloquium

The Department hosts a weekly colloquium on Fridays at 2pm, usually in Southern Hall, room 303.

 Date Presenter, Affiliation Title, Abstract 25 Apr 2014 Jiu Ding, USM Infinitely Many Solutions of Some Yang-Baxter Matrix EquationsWe present some recent results on solving the Yang-Baxter matrix equation for some classes of matrices. 11 Apr 2014 Dongsheng Wu, University of Alabama in Huntsville Regularity of Local Times of Fractional Brownian SheetsAs typical anisotropic Gaussian random fields, fractional Brownian sheets have been intensively studied in recent years. In this talk, we study the regularity of local times of fractional Brownian sheets, including the existence, joint continuity and smoothness (in the Meyer-Watanabe sense) of the local times. As applications, we derive regularity results of their collision local times, intersection local times and self-intersection local times. The main tools applied in our derivation are sectorial local nondeterminism of fractional Brownian sheets, Fourier analysis and chaos expansion of the local times. 4 Apr 2014 Chaoyang Zhang, USM Mathematical Modeling of Biological NetworksModeling and reconstruction of biological networks is a challenging inverse problem because of its nonlinearity, high dimensionality, non-uniqueness, sparse and noisy data, and significant computational cost. In this presentation, I will introduce a new Bayesian learning and optimization model (BLOM) for inferring gene regulatory networks from time series data. After extensive testing and validation using both synthetic and yeast cell cycle benchmark datasets, the BLOM successfully identified hub genes and several important gene regulation relationships in the pathways and also addressed the dynamic change of biological networks in the course of the treatment and recovery of earthworm. 21 Mar 2014 Eric Carpenter, NOAA Math in Weather Prediction: The Importance of Numerical Weather Prediction in Forecasting a Severe Weather EventPrimitive equations approximate fundamental motions and processes in the atmosphere responsible for weather.  Given adequate observational data for input, supercomputers can solve these equations and give us high-resolution forecast output on a global scale.  This output is critical to forecasting severe weather events, as was the case last year in Hattiesburg, MS.  After some background information on the National Weather Service and a brief history of numerical weather prediction, we describe the primitive equations and how these equations ultimately helped us predict the Hattiesburg Tornado of 2013.See this link for more details on numerical weather prediction. Mr. Carpenter kindly made a copy of his presentation available, and you can download it by clicking here. 28 Feb 2014 Qixiang Dong, Yangzhou University Some recent results on fractional differential equationsFractional differential equations have attracted a considerable interest in both mathematics and applications, since they are valuable tools in modeling many physical phenomena. In this talk, we first discuss a class of weighted Riemann-Liouville fractional differential equations with infinite delay. Existence and continuous dependence results of solutions are obtained. Then we study the viability of solutions for a class of nonlinear Caputo fractional differential equations with initial conditions at inner points. Existence of solutions to the terminal value problems is obtained. It is proved that the tangency condition is sufficient for a locally closed set to be a viable domain of the Caputo fractional differential equation. 21 Feb 2014 Sungwook Lee, USM Doing Quantum Physics with Split-Complex NumbersThe current quantum mechanics was built upon complex numbers. In this talk, I show that complex numbers are really for light (photons) and assert that split-complex numbers might have been the right choice to describe other particles. I show that quantum mechanics can be completely rebuilt based upon split-complex numbers. The new quantum mechanics exhibits distinct features. Anti-particles show up naturally in this new quantum mechanics. Remarkably, the path integral can be calculated in Minkowski spacetime without turning it into Euclidean path integral via Wick rotation. This new quantum mechanics may also offer an explantion as to why there aren't as many anti-particles as particles in the universe. 14 Feb 2014 Duanmei Zhou Norm inequalities for accretive-dissipative matricesA matrix is accretive-dissipative if both its real and imaginary parts are Hermitian positive semi-definite. In this talk, we will present inequalities between the norm of off-diagonal blocks and the norm of diagonal blocks; and inequalities between the norm of the whole matrix and the norm of its diagonal blocks. These inequalities obtained extend some known results. 7 Feb 2014 D. L. Young, Taiwan University Localized method of approximate particular solutions with Cole–Hopf Transformation for multi-dimensional Burgers equationsBurgers equations depict propagating wave with quadratic nonlinearity. They can describe nonlinear wave and shock wave propagations, where the nonlinear characteristics cause difficulties for numerical analysis.  In this talk, we apply the Cole–Hopf transformation to transform the system of Burgers equations into a linear diffusion equation, and use a combination of finite difference and the localized method of approximate particular solution (FD-LMAPS) for temporal and spatial discretization, respectively. The Burgers equations with behaviors of propagating wave, diffusive $$N$$-wave within multi-dimensional irregular domain will be examined in some experiments. 24 Jan 2014 Qixiang Dong, Yangzhou University Introduction to fractional calculusFractional differential equations have attracted considerable interest in both mathematics and applications, since they are valuable tools in modeling many physical phenomena. In this talk, we introduce the concept and some basic results of the Riemann-Liouville fractional integral, the Riemann-Liouville fractional derivative, and the Caputo fractional derivative. 6 Dec 2013 Chenhua Zhang, USM A Characteristic of Long-Tailed Distribution with Application to  Discrete-Time Insurance Risk ModelWe study the tail distribution of weighted sums of form $$\sum_{i=1}^nc_iX_i$$, where the $$X_i$$'s are independent random variables with long-tailed distribution. Using $$h$$-insensitive functions, we establish the uniform asymptotic equivalence of the tail probabilities of $$\sum_{i=1}^nc_iX_i$$, $$\displaystyle{\max_{1\leq k\leq n}}$$$$\left\{\sum_{i=1}^kc_iX_i\right\}$$ and $$\sum_{i=1}^nc_iX_i^+$$ if the $$X_i$$'s follow the long-tailed distribution and the $$c_i$$'s take value in a broad interval. We also prove some uniform asymptotic results for the weighted sums of the $$X_i$$'s with consistently varying tail. Moreover, we present an application to ruin probability in a discrete-time insurance risk model. 15 Nov 2013 Haiyan Tian, USM The method of approximate fundamental solutions for well and ill posed PDEsWe use the method of approximate fundamental solutions for well and ill posed PDEs including elliptic equations and Cauchy problems of elliptic operators. The method allows us to find an approximate fundamental solution of some differential operators when their closed form fundamental solutions are not known. Proper regularization technique is incorporated to obtain a stable solution for an inverse problem. 8 Nov 2013 Joe Brumbleloe, USM Math in music: Explanations and proportionsBefore the 18th c., mathematical descriptions of the materials and structures of music prevailed. A brief accounting of explanations of music will be presented, beginning with Pythagoras (ca. 500 B.C.E.), transmitted by Boethius (De institutione musica, ca. 500), and ending with discovery/formulation of acoustical science by Louis Savuer (Traité de la Théorie de la Musique, 1697). Following this, several mathematically-oriented examples from musical literature will be examined; including Thomas Tallis (1505-1585), Conlon Nancarrow (1912-97), Milton Babbitt (1916-2011), and Béla Bartók (1881-1945). 1 Nov 2013 Lawrence Mead, USM An Axionic Inflationary CosmologyWe have analysed all current data from Supernova type Ia explosions. We find that the data may contain previously unrecognized oscillations in the cosmological evolution factor $$a(t)$$. What is the cause of these oscillations? We have constructed a model of a Big Bang Inflationary cosmology with an axionic field component replacing Dark Matter. We have solved the coupled Axion-Einstein equations, thus extending the Inflationary scenario to the entire 13.7 GYr lifetime of the universe. In doing so, we can (a) explain why the universe is flat, (b) explain that it is the axionic field energy of 23% which causes the (gravitational) effects of the supposed dark matter, (c) provide a perfect fit of the Supernova data both for the brightness-redshift and $$a(t)$$ curves, including the apparent oscillations. 25 Oct 2013 Eowyn Cenek, USM Eulerian and Hamiltonian Cycles, P and NPA graph has a Eulerian cycle if it contains a cycle that includes every edge exactly once.  That same graph has a Hamiltonian path if there exists a path that visits every vertex exactly once.  We will discuss a simple historical algorithm to find a Eulerian cycle in a graph in polynomial time, discuss how we can use this cycle to find the Hamiltonian cycle in the corresponding line graph, and then show that the Hamiltonian cycle problem is, in general, NP-complete. 11 Oct 2013 Songming Hou, Louisiana Tech University Solving Elliptic and Elasticity Interface Problems with Non-body-fitted MeshInterface problems occur in many multi-physics and multi-phase applications in science and engineering. We proposed a non-traditional finite element method for solving elliptic and elasticity interface problems using non-body-fitted mesh. The invertibility of the coefficient matrix for local construction is proved under certain assumptions. The condition number of the large linear system is studied. We presented numerical results in both 2D and 3D. 4 Oct 2013 Jiu Ding, USM The Maximum Entropy Method for the Statistical Study of ChaosThe maximum entropy method has been widely used for the recovery of density functions in physical science and engineering. We introduce the concepts of Shannon entropy and Boltzmann entropy, and describe Jaynes' maximum entropy principle. Then we present a recent work that incorporates the principle with a simple criterion to develop a piecewise linear maximum entropy method for the computation of stationary densities of chaotic dynamical systems. 27 September 2013 Li-Hsuan Shen, National Taiwan University The Local Radial Basis Function Differential Quadrature Method for Convection-Dominated ProblemsWe propose a new technique to solve convection-dominated flow problems. The local support in the formulation of local radial basis function differential quadrature (LRBFDQ) can be chosen using a modified Euclidean distance function, biasing towards the upstream direction to form a comet-like shape. The upwind effect is therefore naturally incorporated when computing the weighting coefficients for LRBF-DQ method. We show remarkable improvement of accuracy and efficiency over the conventional method when solving two-dimensional convection-diffusion equations with various Peclet numbers, as well as magneto-hydrodynamics problems with very high Hartmann numbers. 20 September 2013 Tài Huy Hà, Tulane University What is a syzygy?In this talk I will describe what algebraists call a syzygy and discuss a few research problems involving syzygies of ideals in polynomial rings. 13 September 2013 John Perry, USM The Zariski TopologyThe Zariski Topology provides an excellent example of Sophie Germain's observation that algebra and geometry are different tongues that describe the same phenomena. In this case, we can view the “geometry” through the lens of topology, and the “algebra” appears in the forms of ideals of a ring. These two tools allow us to give precise notions to algebraic and geometric inuitions in an elegant, compelling way. Students who have taken algebra or topology are encouraged to attend. 6 September 2013 Karen Kohl, USM Identities suggested by the method of bracketsThe method of brackets is a heuristic method for symbolic definite integration.  This method is useful for a large class of integrals including many involving special functions.  Experimentation and testing against a table of integrals suggests several new(?) integration and summation identities. 25 June 2013 Christian Eder, Université Pierre et Marie Curie — Paris 6 Improved Gröbner Basis computation with applications in cryptographyThis talk gives an overview of recent developments in Gröbner Basis theory, illustrating with an application to algebraic cryptanalysis, in particular, the HFE cryptosystem. Recent advances have focused on linear algebra, which can be made more efficient by exploiting known information. This leads to better performance and easier parallelization. By generalizing the linear algebra, signature-based algorithms detect useless rows of the associated matrix in a way that has advantages over traditional criteria for a wide class of examples. 24 June 2013 John Perry, USM Dynamic Gröbner basis computationIn non-linear algebra, we can view the computation of a Gröbner basis as Gaussian reduction of a matrix with infinitely many rows and columns, which miraculously requires only finitely many operations. In linear algebra, it is possible to hasten the result by swapping rows. What about non-linear algebra? This requires more care, but an approach to this was first described in 1993, and in many small systems, the result is a smaller basis, computed more quickly. Unfortunately, the use of the simplex algorithm requires a second matrix, which grows unreasonably as the algorithm progresses. It turns out that two new criteria reduce this growth to a reasonable level, or even eliminate it entirely. 26 Apr 2013 Sungwook Lee, USM Surfaces of Revolution with Constant Mean Curvature in Hyperbolic 3-SpaceI discuss how to construct surfaces of revolution with constant mean curvature $$H=c$$ in hyperbolic 3-space $$\mathbb{H}^3(-c^2)$$ of constant sectional curvature $$-c^2$$. It is intriguing to see that while the hyperbolic 3-space flattens to Euclidean 3-space $$\mathbb{E}^3$$ as $$c\rightarrow0$$, those surfaces approach catenoid, the minimal surface of revolution in $$\mathbb{E}^3$$. I also discuss how to construct minimal surface of revolution in $$\mathbb{H}^3(-c^2)$$. This work was done with Kinsey Zarske as her undergraduate research project. 19 Apr 2013 Louise Perkins, USM Analytic Logical Functions that Provide Weak Interpretations of SatisfyabilityWe introduce a parallel Analytic Logic. We utilize a hyperspace over Boolean disjunctive clauses with a new semantics. The goal of this work is the development of an analytical logical calculus that provides weak solutions to satisfyablility problems in polynomial time. The approach utilizes three novel techniques: a parallel tri-state hyperspace over 2-state Boolean Algebra, a parallel tri-state encoding of disjunctive clauses that induces a metric, and a parallel tri-state concurrent intersection operator that does not require enumeration of all intersectors. 12 Apr 2013 Jiu Ding, USM The mean ergodic theorem of matrices and its application to solving the Yang-Baxter matrix equationWe construct some solutions to the Yang-Baxter matrix equation with the help of the mean ergodic theorem for matrices. 5 Apr 2013 Muge Er, University of Colorado, Colorado Springs Incidence AlgebrasGian-Carlo Rota defined an incidence algebra as a tool for solving combinatorial problems. An incidence algebra is a specific ring of functions defined on the ordered pairs of a partially ordered set to a given ring, and equipped with a module action by this ring. A notable element of an incidence algebra is its Möbius function, which coincides with the well-known Möbius function of number theory. Subsequent work of Rota and other authors investigate incidence algebras as a tool in algebraic topology. After a review of these ideas, I will summarize what I study in the algebraic context of incidence algebras. 22 Mar 2013 John Perry, USM Nim∞While trying to explain my research to students, I invented a game that turned out to be a generalization of the game of Nim, whose mathematical implications are well known. We review some of these implications, and explore how “Nimfinity” distills fundamental ideas of commutative algebra into a challenging game.