陽明交通大學應用數學系

Abstract
Semiclassical analysis involves the study of the solutions of singularly-perturbed partial differential equations with given initial data or boundary conditions. The singular perturbation enters via small multipliers on derivatives, and the aim is to describe how the given data taken independent of the small parameter evolves over space and time in the asymptotic limit that the parameter tends to zero. This lecture will introduce the topic, present illustrations from the theory of linear and integrable nonlinear equations, and then embark on a detailed analysis of the Cauchy problem for the defocusing nonlinear Schrödinger equation in the semiclassical limit. The latter problem is solved via an inverse-scattering transform and here we focus on the computation of scattering data in the semiclassical limit, which employs the WKB method.

Abstract
Ice sheet flow laws currently use coefficients that depend on their respective power-law exponents, complicating systematic parameter exploration in models. This study proposes dimensionless formulations for both basal sliding and internal deformation laws, which decouple coefficients from exponents, simplifying sensitivity studies and making independent variation of these parameters feasible in ice-sheet models.

Abstract：
1. Operation method of MOSFET and memory devices
2. Recent development of 3-Dimensional devices (Flash memory & DRAM)
3. Reliability issues of Flash memory
(limitation of number of rewrite cycles and the difficulty to analyze)
4. Short review of my reliability studies

Abstract. We consider nearest-neighbor oriented percolation defined on the product space of a multi-dimensional integer lattice and the set of positive integers.

A point in the product space is described by a vector, where the first component (space component) is a point of the lattice and the second component (time component) is a positive integer.

For a pair of points, where the space components are neighbors and the difference in their time component is 1, we can define a bond, which is independently open with probability p/2d with 0 ≤ p ≤ 2d, regardless of the other bonds. It is well known that oriented percolation exhibits a phase transition as the parameter p varies around a critical point pc which is model-dependent. As the dimension tends to infinity, pc coverges to 1.

However, the best estimate for pc provided by Cox and Durret (Math. Proc. Camb. Phil. Soc. (1983)) give upper and lower bounds, but do not yield an explicit expression for pc.

In this talk, we investigate the explicit expression for pc when d > 4, in a way that pc = 1 + C1d^{-2} + C2d^{-3} + C3d^{-4}+ O(d−5), where C1 to C3 are constants. The proof relies on the lace expansion, which is one of the most powerful tool to analyze the mean-field behavior of statistical-mechanical models in high dimensions. We focus less on the details of the proof and more on the background related to the topic.

Abstract. The study of reaction-diffusion equations on metric graphs has been drawing attention recently. Two different research directions will be introduced. First topic is related to the question of whether excitation waves propagate along the branching of axons of nerve cells or not. Namely, we are going to consider a scaler reaction diffusion equation on a star-shaped metric graph. We can observe propagation blocking depending on the numbers of input and output edges. We also discuss the related problem. Second, we study pattern dynamics on compact metric graphs. We consider systems of reaction-diffusion equations on compact metric graphs with Turing or Wave instability. We construct eigenfunctions of Laplacian on specific metric graphs to see pattern onsets depending on the lengths of the edges. By using the normal form analysis and symmetry arguments we study the local bifurcation structures around the bifurcation points. In both cases, we impose natural boundary conditions, namely, Neumann‒Kirchhoff conditions at the junction.