Weekly Colloquium
The Department hosts a weekly colloquium on Fridays at 2pm, usually in Southern Hall, room 303.
Date  Presenter, Affiliation  Title, Abstract 
13 Feb 2015  D. L. Young, Taiwan University  The Localized Method of Approximated Particular Solutions for Solving Incompressible NavierStokes Equations This talk will focus on demonstrating that the Localized Method of Approximated Particular Solutions (LMAPS) is a stable, accurate tool for simulating multidimensional, incompressible viscous flow fields governed by the NavierStokes equations. The LMAPS is tested by nonuniform point distribution, extremely narrow rectangular domain, internal flow, velocity or pressure driven flow and high velocity or pressure gradient, etc. All results are similar with results of FEM or other existing literature, and it is concluded that the LMAPS has high potential to be applied to more complicated engineering applications. A further attempt to solve threedimensional NavierStokes equations will be addressed and discussed. 
30 Jan 2015  Bernd Schroeder, USM  How not to prove a certain inequality from applied mathematics. This talk will present an inequality that arose in a colleague's analysis of the behavior of microair vehicles. We will discuss how a theorist who had stopped doing analysis a decade ago can become involved in such activities. We will specifically focus on some approaches that turned out to not work, because these indicate the challenges that lie ahead for attempts to generalize the inequality or to simplify that presenter's (rather tedious) proof. 
23 Jan 2015  Drew Lewis, University of Alabama in Tuscaloosa  When is a polynomial a coordinate? For a ring \(R\), a polynomial \(f \in R[x_1,\ldots,x_n]\) is called a coordinate if there exist \(n1\) other polynomials \(g_2,\ldots,g_n\) such that \(R[f,g_2,\ldots,g_n]=R[x_1,\ldots,x_n]\) (so \((f,g_2,\ldots,g_n)\) forms a system of coordinates for the polynomial ring in the usual sense). A natural question to ask is: given a polynomial, how can one tell if it is a coordinate or not? We will discuss several candidate criteria such as pseudocoordinates, stable coordinates, and hyperplanes, with an emphasis on some concrete examples. 
25 Apr 2014  Jiu Ding, USM  Infinitely Many Solutions of Some YangBaxter Matrix Equations We present some recent results on solving the YangBaxter matrix equation for some classes of matrices.

11 Apr 2014  Dongsheng Wu, University of Alabama in Huntsville  Regularity of Local Times of Fractional Brownian Sheets As 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 MeyerWatanabe sense) of the local times. As applications, we derive regularity results of their collision local times, intersection local times and selfintersection 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 Networks Modeling and reconstruction of biological networks is a challenging inverse problem because of its nonlinearity, high dimensionality, nonuniqueness, 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 Event Primitive 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 highresolution 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 equations Fractional 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 RiemannLiouville 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 SplitComplex Numbers The current quantum mechanics was built upon complex numbers. In this talk, I show that complex numbers are really for light (photons) and assert that splitcomplex numbers might have been the right choice to describe other particles. I show that quantum mechanics can be completely rebuilt based upon splitcomplex numbers. The new quantum mechanics exhibits distinct features. Antiparticles 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 antiparticles as particles in the universe. 
14 Feb 2014  Duanmei Zhou  Norm inequalities for accretivedissipative matrices A matrix is accretivedissipative if both its real and imaginary parts are Hermitian positive semidefinite. In this talk, we will present inequalities between the norm of offdiagonal 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 multidimensional Burgers equations Burgers 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 (FDLMAPS) for temporal and spatial discretization, respectively. The Burgers equations with behaviors of propagating wave, diffusive \(N\)wave within multidimensional irregular domain will be examined in some experiments. 
24 Jan 2014  Qixiang Dong, Yangzhou University  Introduction to fractional calculus Fractional 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 RiemannLiouville fractional integral, the RiemannLiouville fractional derivative, and the Caputo fractional derivative. 
6 Dec 2013  Chenhua Zhang, USM  A Characteristic of LongTailed Distribution with Application to DiscreteTime Insurance Risk Model We 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 longtailed 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 longtailed 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 discretetime insurance risk model. 
15 Nov 2013  Haiyan Tian, USM  The method of approximate fundamental solutions for well and ill posed PDEs We 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 proportions Before the 18^{th} 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 mathematicallyoriented examples from musical literature will be examined; including Thomas Tallis (15051585), Conlon Nancarrow (191297), Milton Babbitt (19162011), and Béla Bartók (18811945). 
1 Nov 2013  Lawrence Mead, USM  An Axionic Inflationary Cosmology We 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 AxionEinstein 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 brightnessredshift and \(a(t)\) curves, including the apparent oscillations. 
25 Oct 2013  Eowyn Cenek, USM  Eulerian and Hamiltonian Cycles, P and NP A 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, NPcomplete. 
11 Oct 2013  Songming Hou, Louisiana Tech University  Solving Elliptic and Elasticity Interface Problems with Nonbodyfitted Mesh Interface problems occur in many multiphysics and multiphase applications in science and engineering. We proposed a nontraditional finite element method for solving elliptic and elasticity interface problems using nonbodyfitted 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 Chaos The 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  LiHsuan Shen, National Taiwan University  The Local Radial Basis Function Differential Quadrature Method for ConvectionDominated Problems We propose a new technique to solve convectiondominated 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 cometlike shape. The upwind effect is therefore naturally incorporated when computing the weighting coefficients for LRBFDQ method. We show remarkable improvement of accuracy and efficiency over the conventional method when solving twodimensional convectiondiffusion equations with various Peclet numbers, as well as magnetohydrodynamics 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 Topology The 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 brackets The 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 cryptography This 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, signaturebased 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 computation In nonlinear 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 nonlinear 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 3Space I discuss how to construct surfaces of revolution with constant mean curvature \(H=c\) in hyperbolic 3space \(\mathbb{H}^3(c^2)\) of constant sectional curvature \(c^2\). It is intriguing to see that while the hyperbolic 3space flattens to Euclidean 3space \(\mathbb{E}^3\) 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 Satisfyability We 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 tristate hyperspace over 2state Boolean Algebra, a parallel tristate encoding of disjunctive clauses that induces a metric, and a parallel tristate 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 YangBaxter matrix equation We construct some solutions to the YangBaxter matrix equation with the help of the mean ergodic theorem for matrices. 
5 Apr 2013  Muge Er, University of Colorado, Colorado Springs  Incidence Algebras GianCarlo 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 wellknown 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. 