歷史演講 PAST TALKS

We will discuss the asymptotic behavior of the principal eigenvalue of an elliptic operator with large advection and its applications for advection-mediated coexistence and advection-induced extinction in a competition model.

A 1-factorization of a (regular) graph is a decomposition of the edge set of the graph into disjoint 1-factors (perfect matchings). The problem of determining the existence of a 1-factorization and/or of explicitly nding a 1-factorization (if one exists) is an important problem in discrete mathematics, with several possible applications (e.g. scheduling, job assignment, etc.) The existence version of the problem is known to be NP-complete, even for 3-regular graphs. From the point of view of edge colouring, a 1 factorization of a graph G (if it exists) is nothing else than an optimal edge colouring of G, i.e. one which uses the minimum number of colours. In 1965 Vizing asked whether one can always obtain an optimal edge colouring of any graph starting from a non-optimal edge colouring and using only the technique of colour exchange, i.e. using only colour interchanges along maximal bicoloured paths or cycles until all unnecessary colours disappear. In this talk we shall give an answer to Vizing's question for all the regular graphs of small order (8 or less) by presenting a new technique (based on colour exchange) that allows one to obtain a 1-factorization of any 1-factorizable small graph in the way described above.

Mathematical models and numerical methods for simulating computer devices fabricated by semiconductor materials will be given. The models are based on both classical and quantum mechanics ranging from drift-diffusion, energy-transport to density-gradient equations. Numerical methods include adaptive finite element, monotone iterative and exponential-fitting methods. Simulation results of MOSFET (metal-oxide-semiconductor field-effect transistor) with gate length of 34nm will be presented.

Existence of unique global (in time) solutions to three dimensional incompressible Navier-Stokes equations with periodic boundary in Besov-type function spaces are established by a mix of probabilistic and analytic methods. First, we expose the ideas of probabilitic approach by using two examples. Then, we study the solutions of LANS  equations which includes the limiting  = 0 case of incompressible Navier-Stocks equations with periodic domain. Finally, a rate of convergence in a mixed L1 − L2 time space norm is obtained as  ! 0+.

Plasma, referred to as the fourth state of matter, is a complex system with N-body or many charged particles interacting self-consistently with external as well as internal electromagnetic and/or gravitational fields. Fifty-year exploration of our solar system and the cosmos has revealed that 99% of the matter in the Universe is in ionized or plasma state. Statistical treatment of such a complex system results in a set of partial differential equations in various forms describing temporal and spatial evolutions of a number of physical variables. In this talk the mathematical aspects of the physical laws and governing equations describing the properties of such a statistical system are introduced. Personal journey in mathematical world of plasma physics is shared.

Singular integral equation method is one of the most effective numerical methods solving a (plane) crack problem in fracture mechanics. In this talk we address how the integral equations with hypersingular kernels are applied in solving crack problems. In terms of materials, we consider the examples that include classical linear elastic fracture mechanics (LEFM), functionally graded materials (FGM), and strain-gradient elasticity theory. The corresponding governing partial differential equations are derived for each different type of material. Some semi-analytical results and numerical simulation (including crack displacement profiles and stress singularities in front of the crack-tip) will be presented.

The investigation of thermodynamics formalism is widely studied on finite Markov shifts. Normalized Haar measure is an equilibrium measure, i.e., the measure- theoretic entropy with respect to normalized Haar measure equals the topological entropy. In this talk, we consider convolution sofic systems and demonstrate the thermodynamics formalism on it. We show that, for a certain type of convolution sofic systems, the normalized Haar measure is an equilibrium measure and the Hausdorff dimension is the unique zero of Bowen's equation.

In this talk, we will introduce the concept of multifractal analysis on dynamical systems, which become a popular topic in dimension theory and has wide applications on the researches of turbulence, distribution of galaxies and biology. Next, we prove that C1 averaged conformal repeller admits the multifractal analysis of dimension spectrum by comparing the variational formula of L. Barreira and D. Feng, here, the dynamical system is only needed C1. Finally, we show that the equilibrium measure of variational principle is weak Gibbs, these two results generalize the work of L. Barreira for 2-D case and D. Feng for conformal case.

The three-body problem concerns the motion of three celestial bodies moving in space in accordance with Newton's law of universal gravitation. Except for self-similar solutions, which were completely classified by Euler and Lagrange in the 18th century, classical existence proofs for solutions of the three-body problem are largely relying on a perturbation method due to Poincaré. A vast amount of solutions were discovered by numerical simulations but yet without any analytical proof for their existence. In recent years variational methods have been successfully applied to the n-body problem to construct miscellaneous solutions. In this talk I will briefly describe these recent progresses on the three-body problem. In particular, I will outline a variational proof for the existence of retrograde triple stars and some bizarre planets for double stars.

分析研討會

The Julia sets of the quadratic map $f_\mu(z)=\mu z(1-z)$ for $\mu$ not belonging to the Mandelbrot set form a continuous family as $\mu$ varies. We show that this family can be obtained explicitly by solving a system of infinitely coupled differential equations. We also present numerical approximations to the Julia sets integrated from the anti-integrable limit $\mu=\infty$.

In this paper we consider the nonlinear relativistic wave equation,the modulated Klein-Gordon equation, and we obtain the connection between the nonlinear Schr"odinger equation, compressible Euler equations and the modulated Klein-Gordon equation. Indeed, we study the various singular limits of the modulated Klein-Gordon equations,including nonrelativistic, semiclassical and nonrelativistic-semiclassical limits, from its hydrodynamical structure.

We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we use the Kirchgaessner reduction to obtain a spatial dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. We investigate such a "Bogdanov-Takens bifurcation without parameters'' by blow-up and averaging techniques. In particular we describe the set of all small bounded solutions. In the classical Kolmogorov case, this includes periodic profiles, homoclinic pulses and a heteroclinic front-back pair. This is a consequence of the double symmetry of the external force. In the more general case where only one symmetry is preserved,the set of bounded solutions consist entirely of trivial equilibria and multipulse heteroclinic pairs. The latter form  a very complicated, albeit non-recurrent, set. This is joint work with A. Afendikov and B. Fiedler.

We consider the Cauchy problem for a semilinear parabolic equation with  a supercritical power nonlinearity.  It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space.  In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state.  In this talk we consider solutions whose initial value decays in an anisotropic way.  Then we show that the solution converges to a steady state which is explicitly determined by an average formula.  The proof is given by using our previous results on the global stability and quasi-convergence of solutions, self-similar solutions of the linearized equation around a singular steady state, and a comparison technique.   This is a joint work with Peter Polacik of the University of Minnesota.