歷史演講 PAST TALKS

This work concerns with the asymptotic behavior, monotonicity and uniqueness of traveling wave solutions in a class epidemic model arising from the spread of the epidemic with oral-fecal transmission. Based on the existence results of monostable traveling wave solution, which has been obtained by other authors, we prove the uniqueness and monotonicity provided the traveling wave solutions satisfy some boundness conditions. We also determine the exponential rate that the profile of the traveling wave solution converges to the equilibrium as the moving coordinate tends to infinity.

In this talk, we will introduce some properties of beta-expansions concerning the lengths of the cylinders and the classification of real numbers lager than 1 etc. The asymptotic behaviors of the orbits under beta-transformations are also studied. Finally, we will give some dimensional results of exceptional sets related to the recurrence problem of beta-expansions.

The purpose of this talk is to investigate the concept of a k-stage Euclidean ring, which is a generalization of the concept of a Euclidean ring. As with Euclidean rings, we will give an internal characterization of 2-stage Euclidean rings. Applying this characterization we are capable of providing infinitely many integral domains, which are !-stage Euclidean but not 2-stage Euclidean. Our examples solve finally a fundamental question related to the notion of k-stage Euclidean rings raised by G. E. Cooke [J. reine angew. Math. 282 (1976), 133-156]. The question was stated as follows: “I do not know of an example of an !-stage euclidean ring which is not 2-stage euclidean.” In addition we will study a proposition of P. Samuel, which is related to the concept of Euclidean domains. First we will extend Samuel’s result to the commutative ring with certain property. Also we will derive an analog of Samuel’s proposition for the concept of 2-stage Euclidean rings, and use it to estimate the smallest 2-stage Euclidean algorithm in global fields.

We consider a market model where risky securities are affected by ``hidden'' economic factors, which evolve as a finite-state Markov chain. Our problem is optimizing an expected risk-averse power utility of wealth on finite and infinite time horizons. We will explain that the problem is reduced to a certain stochastic control problem of complete observation. By Bellman's dynamic programming principle, we (formally) derive the Hamilton-Jacobi-Bellman (HJB) equation corresponding to the problem. The HJB equation turns out to be a second order quasilinear degenerate parabolic (elliptic) equation. Under some assumptions, we construct a classical solution of the HJB equation using a probabilistic argument.

For a dynamical system given by a rational map on the projective space, one can define the degrees of the map. The asymptotic growth of the degree under iteration can be measured by the dynamical degrees. They form an interesting family of invariants for the dynamical system. I will first introduce these term with examples. Then I will explain my work of computing the dynamical degrees for monomial maps.

In this talk we will introduce an algebra associated to a cubic curve C defined over a field of characteristic not two or three. This algebra has several interesting properties. In particular, it is an Azumaya algebra of rank nine over its center, which, in turn, is the affine coordinate ring of an elliptic curve, namely the Jacobian of the cubic curve C. These results generalize Haile's work on the Clifford algebra of a binary cubic form.

We propose and analyze two new stabilized FEMs for solving boundary and interior layer problems. The first one is the bubble-stabilized LSFEM which is applied to solve scalar convection-dominated convection-diffusion problems. This bubble-stabilized LSFEM not only inherits the advantages of the primitive LSFEM, such as the resulting linear system being symmetric and positive definite, but also exhibits the characteristics of the residual-free bubble method without involving any stabilization parameters. Numerical experiments are given to demonstrate the effectiveness of the proposed bubble-stabilized LSFEM. The accuracy and computational cost of this method are also compared with those of the primitive LSFEM. We find that for a small diffusivity, the bubble-stabilized LSFEM is much better able to capture the nature of layer structure in the solution than the primitive LSFEM, even if the primitive LSFEM uses a very fine mesh or higher-order elements. The second method is a stabilized FEM in the Galerkin formulation. We analyze the method using continuous piecewise linear elements for reaction-convection-diffusion equations. The equation under consideration is reaction-convection-dominated, involving a small diffusivity and a large reaction coefficient. In addition to giving error estimates of the approximations in L2 and H1 norms, we explicitly establish the dependence of error bounds on the diffusivity, the module of convection field, the reaction coefficient and the mesh size. Several numerical examples exhibiting boundary layers are given to illustrate the high accuracy and stability of the newly proposed stabilized FEM. The results obtained are also compared with those of existing stabilized FEMs.

Graph labeling has been a rapidly developing subject in the study of graph theory. A graph labelling is an assignment of integers (or other objects) to the vertices or edges, or both, subject to certain conditions. Graph labeling was first introduced in the late 1960s as a tool for resolving graph decomposition problems, since then there have been over one thousand referred research articles in literatures, as appeared in Joseph Gallian's well known dynamic survey article of graph labeling (13th edition, Nov. 2010). In this talk, we will survey and report in particular recent progress on certain graph labeling of magic and antimagic types, along with discussion of open problems and possible applications.

In this talk, we deal with some nonlinear elliptic systems, including sublinear systems, a coupled system in Hamiltonian type, Liouville-type system and simplified Bennett-type system with singular data. Some qualitative properties and complete structures of solutions will be provided by applying the bifurcation theory and linearization approach.

We first consider the semi-linear heat equation u_t=u_{xx}+f(u) whose solution blows up in finite time. Then we consider a finite difference analogue with adaptive time meshes so that the numerical solutions blow up in finite time. The convergence of the blow-up time was also proved. It is interesting that although the convergence is guaranteed, the asymptotic behavior of the numerical solution is different from the solution of the PDE. Next, we consider the semi-linear wave equation u_{tt}=u_{xx}+f(u) and its finite difference analogue. Several significant results have been derived, however, it seems that the stability of a 3-level finite difference scheme is not well-studied. The difficulty and our recent results will be reported.

We consider the thermal transpiration problem in the kinetic theory, which can be modeled by the linearized Boltzmann equation. It is well-known through asymptotic expansions and computations that there is a logarithmic singularity for the fluid velocity around the solid boundary. The goal of this paper is to confirm this basic phenomenon in the kinetic theory through analysis for sufficiently large Knudsen number. We use an iterated scheme, with the "gain" part of the collision operator as a source. The scheme yields an explicit leading term. The remaining converging terms are estimated through a refined pointwise estimate and Maxwellian upper bound for the gain part. Our analysis is motivated by the previous studies of asymptotic and computational analysis. Numerical data supporting the analysis are also provided.

This study studies the dynamic funding policy and investment strategy for the defined benefit pension schemes with the most comprehensive dynamic models of the pension plan to date. We formulate the optimal decisions of pension plans as a stochastic control problem and solve the problem through dynamics programming. The objective function of our model takes into account of the stability and the solvency of the pension. The investable assets in the model include one risk-free asset and two risky assets while the considered liabilities are a three-dimension process. We first investigate the procedures in deriving the optimal strategies under the specified performance measures incorporating asset-liability matching criterions. Then we apply our method to the pension plan of a semi-conductor and electronics company in Taiwan for illustration. Keywords: optimal contribution, asset allocation, dynamic programming, performance measure.

We study an optimal consumption control problem in a jump-diffusion model under the uncertainty. We show a verification result to the existence of a solution of the Hamilton-Jacobi-Bellman equation associated with the stochastic optimization problem, and then give an optimal consumption policy in terms of the solution. An application to the one-sector Ramsey theory in the economic growth is also considered.

In Markov chain Monte Carlo theory, a particular Markov chain is run for a very long time, say T, until its distribution is close enough to the stationarity. In particular, one is interested in the time T to stop the simulation and then choose the random sample. Similar problems arise in many fields including statistical physics, computer science, biology and more. It is well known that T is closely related to the mixing time of Markov chains. In recent years, for models of statistical mechanics and of theoretical computer science, there has been a flourishing of heuristic ideas to rigorously control the time. In this talk, we will introduce the concept of cutoffs and mixing times, and discuss how they are related with each other. For illustration, the classical model, Ehrenfest processes, will be considered and latest results are described in details.

In this talk, we introduce a scheme to get the nonlinear stability of traveling wave solutions of a traffic flow model based on Tong Li’s papers. We shall describe a local existence theorem for a small perturbation combined with a priori estimates. Finally, we derive the global existence and decaying property for this perturbation.

The Primitive equations in the large-scale ocean are derived from the Navier-Stokes equations with Coriolis force, coupled with the thermodynamics equation and salinity diffusion-transport equation under both the Boussinesq and hydrostatic approximations. For decades, this the primitive equations have attracted a lot of attention from both oceanic and mathematical point of view. In this talk, we will briefly introduce the development of the mathematical theory of the primitive equations. In particular, we will focus on the 3-D global well-posedness results obtained by Titi and Cao on 2007.

我們將介紹現代調和分析中最重要的幾個算子，如Hardy-Littlewood極大算子、奇異積分算子和Littlewood-Paley算子產生的背景及其應用。

In this talk I introduce risk sensitive portfolio optimization problem and optimization problems related to this. Moreover, in particular, using dynamic programming approach, we discuss risk sensitive portfolio optimization problem with linear Gaussian stochastic factor model.

Let G=(V,E) be a graph, N=|V|, and V={v(i)| i=1, 2, 3, …, N}. G is Hamiltoninan if G contains a cycle that visits v(i) exactly once for all 1<=i<=N. We say “G contains m mutually independent hamiltonian cycles” if G contains m Hamiltonian cycles such that for some vertex , and . Recently, there are numerous studies about the existence of mutually independent Hamiltonian cycles in interconnection networks. In this talk, we will review and compare the different construction schemes of mutually independent Hamiltonian cycles in various networks, derive a degree-based theoretical properties, and introduce our current work.

The refined analytic torsion was introduced by M.Braverman and T. Kappeler as a canonical refinement of analytic torsion on odddimensional closed manifolds. In this talk I will describe two newboundary conditions by using the Hodge decomposition ofthe space of the complex flat bundle valued differential forms on the boundaryof an odd dimensional compact manifold and its application to the constructionof the refined analytic torsion on odd dimensional compact manifolds withboundary. Then Iwill discuss the gluing formula for the refined analytic torsion for an acyclicHermitian connection. (Joint work with Yoonweon Lee)

Recently max-plus or min-plus (more generally idempotent) methods are effectively used for deterministic control problems. As a notable advantage of these methods, we can obtain curse-of-dimensionality-free numerical methods for dynamic programming PDEs in deterministic controls. In this talk, we present our attempt to use idempotent methods for continuous-time stochastic control problems. Starting with basic and dynamic programming ideas in stochastic control, we give a new algorithm to compute time-discretized value functions of the stochastic control, which we call idempotent distributed dynamic programming principle (IDDPP). The material of this talk is based on collaboration with W.M. McEneaney (UC San Diego).

In this talk, a coupling interface method (CIM) is proposed for solving complex interface problems, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. The method uses adaptive-order strategy in one dimension and is extended to high dimensions through a dimension-by-dimension approach. To connect information from each dimension, a coupled equation for the principal derivatives is derived through the jump conditions in each coordinate direction. The cross derivatives are approximated by one-side interpolation. The method is easy to implement and flexible to integrate with other approaches. Two applications, elliptic interface problems and wave-guide modes of surface plasmon, are shown. Several examples are tested to show the efficiency, robustness and accuracy of the method.

In this talk, I will present new estimates about the smoothing property of gain term of Boltzmann collision operator for the hard sphere model. These estimates are useful in studying propagation of singularity of solution, we will also discuss this issue if time is allowed.

We consider the search problems in which the goal is definite and the search methods are given. The most general search problem is to find unknown items in a given finite set by asking questions relating to the contents of subsets. We generally look for search strategies which are more economical in the number of queries and less time consuming in determining the unknown items from queries outcomes. Besides dealing with search problems from real applications, we further set up and study search problems that have high potential of being applied. Group testing is probably the most well-known search problem; besides, graph learning problem, a search problem motivated by applications in genome sequencing, have also been studied with much attention. In this talk, I will introduce my recent work on combinatorial group testing on inhibitor complex model, threshold group testing problems and graph learning problems.

This is a study seminar to learn the uses of stochastic control in insurance. We plan to study the materials based on a book by Hanspeter Schmidli. The seminar gives a chance to learn the uses of stochastic control method in solving the optimization problems. The book in particular discusses some problems from insurance. They include: the optimal dividend in risk theory, minimizing the ruin probabilities, and the optimal investment problem. We plan to meet on Wednesday in the morning for every two weeks. The book by Schmidli consists of 4 chapters. The main issues discussed in this seminar are organized as follows. First, we will start with the concept of stochastic control in discrete time and introduce the discrete-time dynamic programming. The stochastic control in continuous time will be illustrated in the second part. Furthermore, several applications in insurance will be discussed after presenting Hamilton-Jacobi-Bellman approach. Then, the problems in life insurance with the continuous time case will be discussed, and the last is the asymptotic properties of the value function obtained from Hamilton-Jacobi-Bellman equation.