Past Talks 歷史演講

We talk about a new C0 finite element method for Maxwell’s equations. Such equations are formulated in terms of curl and div operators. The well-known challenging task for this type of problem is how to use C0 Lagrange elements to numerically solve the singular solution that may be not in H1. The classical C0 finite element method fails to produce a correct convergent solution; meanwhile, even adaptive mesh-refinement technique is useless. Our new C0 finite element method can correctly capture the singular solution and gives an error bound (optimal with respect to the regularity of the exact solution). Our method can deal with any singular solution which may have the regularity from L2 to Hr for any r>0. A numerical test for Maxwell eigenproblem is presented to demonstrate the theoretical error bounds. We also address several other discretization methods, such as edge element method, singular function method, weighted method, H-1 method, discontinuous Galerkin method, finite volume/finite integration/mimetic difference method.

1.講題：A Petrov-Galerkin multiscale finite element method for convection-dominated convection-diffusion problems 演講者：Po-Wen Hsieh (謝博文，國立中央大學數學系博士生) 時間：2010年7 月14日02:00~04:00 pm 地點：M412 研討室 2.講題：A novel least-squares finite element method enriched with residual-free bubbles for solving convection-dominated problems 演講者：Suh-Yuh Yang (楊肅煜，國立中央大學數學系教授) 時間：2010年7 月15日02:00~04:00 pm 地點：M412 研討室 3.講題：Issues related to least-squares finite element methods for incompressible viscous flow problems I, II, III 演講者：Houyuan Duan (段火元，南開大學數學科學學院教授) 時間：2010年7 月16，21，23日02:00~04:00 pm 地點：M412 研討室 4.講題：Issues related to least-squares finite element methods for incompressible viscous flow problems I, II, III 演講者：Houyuan Duan (段火元，南開大學數學科學學院教授) 時間：2010年7 月21日02:00~04:00 pm 地點：M412 研討室 5.講題：Issues related to least-squares finite element methods for incompressible viscous flow problems III 演講者：Houyuan Duan (段火元，南開大學數學科學學院教授) 時間：2010年7 月28，30日及8月4日02:00~04:00 pm 地點：M412 研討室 聯絡人：楊肅煜 教授 Tel: (03) 4227151 ext. 65130, Email: syyang@math.ncu.edu.tw

In this talk, we review mixed finite element method for the Stokes equations in terms of velocity -pressure and several least-squares (LS) finite element methods for the Stokes equations in terms of velocity-pressure-vorticity: 1) mixed method; 2) standard LS method; 3) h2 LS method; 4) H-1 LS method; 5) L2 projected LS method. Advantages and disadvantages of these methods are addressed. Emphasis is placed on L2 projected LS method developed by the author (jointly). In sharp contrast with other LS methods, this method gives optimal error bounds for all variables using unequal order elements and gives optimal error bounds for velocity using equal-order elements. Some numerical tests are presented to confirm theoretical error estimates.

Various neuron models and their coupling behaviors have been studied extensively over the years. In this particular study, we investigate the newly proposed adaptive exponential integrate-and-fire model, because its parameters have a physiological interpretation and are of biophysical relevance to real recordings of pyramidal cells. We show that the stability of phase locked in a pulse-coupled simple network can vary, depending on the type of bifurcation which periodic spiking is produced. We also show that the speed of post-synaptic input or the reset character may play an important role in stablizing the stability as well.

It was conjectured by C.-S. Lin and C.-L.Wang that the number Nk of solutions for the singular mean field equation on tori Δu + peu = pδ0 with coefficient r = 4π(2k + 1) is equal to its topological degree k + 1 for each k ? Z≧0. It turns out that such a non-linear PDE is indeed algebraically integrable and we may reduce the problem to a related one on counting solutions of certain affine polynomial systems. We use projective B´ezout theorem and residual intersection theory to verify the conjecture for k ≦ 5. The structure of the redundant infinity solution for the projectivized system involves some symmetric combinatorics and delicate polyhedron structures. It seems that these structures are related to many other fields of mathematics and deserve further study.

Gibbs sampler has been used exclusively for compatible conditionals that converge to a unique invariant joint distribution. However, conditional models are rarely compatible. In this talk, a Gibbs sampling based approach ---Gibbs ensemble---is proposed to search for a joint distribution that deviates least from a prescribed set of conditional distributions. The algorithm can be easily scalable such that it can handle large data sets of high dimensionality. It is also adaptive to different performance measures. Using simulated data, we show that the proposed approach provides joint distributions that are less discrepant from the incompatible conditionals than those obtained via linear programming and by pure Gibbs sampler. I will explain the central idea of ensemble and discuss its potentially wide applications in statistics. The ensemble approach is applied to a data set regarding geno-polymorphism and response to chemotherapy in patients with metastatic colorectal cancer. Its advantage in selecting conditional models will be illustrated.

Given an elliptic curve E defined over a number field K. Let ³ be a prime in K, F³ be the residue field of RK which is the ring of integers of K and ˜E denote E modulo ³. Denote
E = the discriminant of E, Pt = {³ ` K  prime ideal in K with NK(³) B t and ³ Ñ m
E}, K(t) = lPtl. Let m N. We are interested in the limit In this talk, we will use two methods to compute the limit.

We discuss a parallel domain decomposition algorithm for the simulation of blood flow in compliant arteries using a fully-coupled system of nonlinear partial differential equations consisting of a linear elasticity equation and the incompressible Navier-Stokes equations with a resistive outflow boundary condition. The system is discretized with a finite element method on unstructured moving meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping restricted additive Schwarz method. The resistive outflow boundary condition plays an interesting role in the accuracy of the blood flow simulation, and we provide a numerical comparison of its accuracy with the standard pressure type boundary condition. We also discuss the parallel performance of the implicit domain decomposition method for solving the fully coupled nonlinear system on a supercomputer with a large number of processors. This is a joint work with A. Barker and Y. Wu.

Given an acyclic digraph D, the competition graph C(D) of D is the graph with the same vertex set with D where two vertices x and y are adjacent in C(D) if and only if there is a vertex v in D such that (x, v) and (y, v) are arcs of D. It can be shown that any graph G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. For a given graph G, the competition number of G, denoted κ(G), is the smallest number r such that G ∪ Ir is the competition graph of an acyclic digraph. In this talk, I will introduce some previous works about this problem.

In this talk, I will first give a brief review on asymptotic speeds of spread (in short, spreading speeds), traveling waves, and global stability for evolution systems with monostable and bistable nonlinearities. Then I will present the mathematical theory of spreading speeds and traveling waves for monotone semiflows. Finally I will discuss its applications to some deterministic models on biological invasions and disease spread.

While the classical linear homotopy continuation method was developed for solving polynomial systems in 1980’s, the polyhedral homotopies were established in 1995, which yields a drastic improvement over the classical linear homotopies. HOM4PS-2.0 is a software package which implements the polyhedral homotopy continuation method for solving polynomial systems. With several sophisticatedly designed algorithms in mixed volume computation and curve tracing, it surpasses the existing packages in finding all isolated zeros of polynomial systems, such as PHCpack, PHoM, and Bertini, in speed by big margins. The marvelous efficiency of HOM4PS-2.0 makes it possible now to solve some very large systems, such as 5-body problem.

Given a graph G, a function f:V(G)→N is called a coloring of G.Let f be a coloring of G and let H be a subgraph of G, we define , and we let S(f)= fS (G). A coloring f of G is called an IC- coloring of G if for any integer k ∈ [1, fS (G) ] there is a connected subgraph H of G such that fS(H)= k. The IC-index of G, denoted by M(G), is defined to be M(G)= max{S(f): f is an IC- coloring of G}. In this talk, I will introduce some previous works about this problem.

Pooling designs are common tools to efficiently distinguish positive clones from egative clones in clone library screening. In some applications, there is a third type of clones called ``inhibitors'' whose effect is in a sense to obscure the positive lones in pools. Various inhibitor models have been proposed in the literature. This talk addresses the inhibitor problems of designing efficient nonadaptive procedures for both identification and classification problems. Our work improves previous results in three aspects: 1. The algorithm that is used to identify the positive clones works on a more general inhibitor model and has a polynomial-time decoding procedure that recovers the set of positives from the knowledge of the outcomes. 2. The algorithm that is used to classify all clones works in one-stage, i.e., all tests are arranged in advance without knowing the outcomes of other tests, along with a polynomial-time decoding procedure. 3. We extend our results to pooling designs on complexes where the property to be screened is defined on subsets of biological objects, instead of on individual ones.

For a positive integer k, we use [k] to denote the set {1, 2, ... ,k}, and the set of all subsets of [k]. A k-rainbow dominating function of G is a function f : V (G) → such that for every vertex v, either or . The weight of f is defined as.The k-rainbow domination number of G is the minimum weight of a k-rainbow dominating function. We prove that the k-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We give a linear-time algorithm for the k-rainbow domination problem on trees.

The field of biostatistics (生物統計) will be introduced. Overview of a typical biostatistics graduate program in United States will be provided including information on admission requirement and expectation, program curriculum, research focus, and career prospects. Key and commonly used statistical concepts and methods in the biomedical field will be illustrated through several examples in the area of breast cancer research. Examples may include breast cancer risk prediction, etiology, prognostic modeling, genetics association, clinical trial of conventional and targeted therapy, and efficacy of mammography screening programs. Challenges faced by future biostatisticians, including methodological questions and expanded interaction with other disciplines, will be discussed.

We study traveling wave front solutions for a two-component system on a one-dimensional lattice. This system arises in the study of the competition between two species with diffusion (or migration), if we divide the habitat into discrete regions or niches. We consider the case when the nonlinear source terms are of Lotka-Volterra type. For the monostable case, we first show that there is a minimal wave speed such that a traveling wave front exists if and only if its speed is above this minimal wave speed. Next, we characterize the minimal wave speed using the parameters in the system. Then we show that any wave profile is strictly monotone. Moreover, under some conditions, we show that the wave profile is unique (up to translations) for a given wave speed. Finally, for the numerical aspect, we derive the convergence of discretized minimal wave speeds as the mesh size tends to zero.

In this talk we discuss the BPS invariants in physics/string theory and their wallcrossing under the variations of the stability conditions. We will also mention the connection with the generalized Donaldson-Thomas invariants. Some concrete examples like the resolved conifold will be discussed. New results of higher rank DT invariants will be presented if time permits.