歷史演講 PAST TALKS

The studies on the congruence of combinatorial numbers can be traced back to the famous and also age-old Pascal’s Fractal formed by the triangle of parities (0 and 1 digits) of binomial coefficients . As a pioneer of this area, Kummer formulated the maximum power of dividing . Lucas, another pioneer, developed a very useful tool that , where is a prime, “ ” denotes congruence modulo , and denotes the sequence of digits representing in base . The congruence of combinatorial numbers modulo was considered for a long time, but it is hard and just few results came out. In the first part of this talk, I will present the congruences of Catalan numbers and Motzkin numbers modulo 4 and 8. In particular, this work justifies the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. The units of the modulo ring are the numbers coprime (relatively prime) to . The set of all units forms the multiplicative group of integers modulo n, usually denoted by . It is well known that is isomorphic to , and the group is cyclic for any odd prime . In the second part of this talk, I will present a newest work. By transforming the multiplicative group to its twin as an additive group, we can develop efficient and powerful formulas for enumerating the congruences of many combinatoric numbers, including new proofs for Kummer’s and Lucas’ Theorems.

Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in C. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al on the subject. In addition, the results are used to derive a formula for the maximum dimension of a totally isotropic subspace of a square matrix, and verify the solvability of certain matrix equations. This is joint work with Nung-Sing Sze, University of Connecticut.

We first give necessary and sufficient conditions which ensures validity of the Bedrosian identity for the Hilbert transform of product functions fg. Then we extend the Bedrosian identity for the Hilbert transform of product functions to Calderon-Zygmund singular integral transform. A counterexample shows that sufficient conditions on one-dimensional Euclidean space does not ensure the validity of the Bedrosian for the n-dimensional case.

We will introduce the singular integrals along surfaces on the product spaces. I will discuss some of the ideas to prove the boundedness of those singular integrals with rough kernels.

Abstract The fifth-order KdV-type equation models many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasma. Dual-Petrov-Galerkin approximations to the fifth-order KdV equation is considered. The key idea of this method developed by Jie Shen is to use the trial functions satisfying the underlying boundary conditions of the differential equation and the test functions satisfying the dual boundary conditions. Our theoretical analysis and numerical results indicate the proposed dual-Petrov-Galerkin method is extremely accurate and efficient. This is a joint work with Jie Shen and Jiahong Wu.

Abstract The quantification of synchrony between physiological signals is of crucial importance for the study of functional connectivity. We propose a method to detect the nonlinear interdependence within the coupled chaotic system with the combination of the cluster analysis in one of the coupled system. The simulation from the Rossler-Lorenz system showed our performance of the method is better than that of the conventional one, especially in the coexistence of the 1 : 1 and 2 : 1 phase synchronization between the coupled system. In the self-paced finger movement experiment, the m : 1 cortico-muscular phase synchronization was found between the MEG data in the motor area and the surface EMG on the first dorsal interossei muscle. Therefore, the proposed approach provides a means that may advance the research on both nonlinear dynamics and brain neuroscience.

Abstract Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H^0(M,K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow-Siu surfaces, K\"ahler threefolds given by Deraux and examples of Zheng. If we have time, we will mention a little bit about Mok's conjecture.

Abstract We study the existence, uniqueness, and stability of traveling waves for a system of ordinary differential equations with bistable nonlinearity in discrete periodic media. This system arises from a spatial discrete version of some semilinear parabolic equations with periodic nonlinearity. The main tools to derive the uniqueness and asymptotic stability are comparison principle, spectrum analysis of the linearized operator around a steady state, and the construction of suitable super/subsolutions. To derive the existence of traveling wave, we first convert the system to an integral equation. Then we establish the existence of traveling wave for this system of ordinary differential equations.