Spring 2016

Inverse problems contain many topics. One of them is to reconstruct unknown inclusions by using a non-invasive method. Enclosure method is one such reconstruction method. In this talk, I will briefly introduce the idea of enclosure method and take two models as examples to show some difficulties encountered. One model is the Lame system, which is an isotropic elastic system; the other is the anisotropic Maxwell system.

When a population grows old, it is always interesting to know what happened to it in the past. The coalescence problem provides a way to understand the ancestry of the individuals in the population.

We consider a rapidly-growing Galton-Watson branching process and pick two individuals in the current generation by simple random sampling without replacement and trace their lines of descent backward in time till they meet for the first time. We call the common ancestor of these chosen individuals at the coalescent time their most recent common ancestor. The coalescence problem is to investigate the limit behaviors of some characteristics of this most recent common ancestor such as its death time and its generation number.

Moreover, in this talk, we will introduce branching random walks by imposing movement structures to the above processes and we will see what happens to the limit distribution of the positions of the particles in the branching random walks by means of the coalescence theory.

Mammalian hearing (including humans) is sensitive to signals as weak as 10e-17 Watt. It was found that "outer hair cells", a special kind of sensory cells located inside the cochlea, are responsible for amplification of acoustic signals. These cells can convert mechanical energy and electrical energy in both directions, thus forming a positive feedback loop that acts as an amplifier for the traveling waves. However, it is still debatable how exactly this electro-mechanical feedback happens at the cell level. My on-going research is to construct dynamic-system models to simulate nonlinear responses of both normal and pathological ears. In this talk, I will first highlight important experimental findings in cochlear mechanics during the past 20 years or so. Then, I will present my work on modeling the micro- and macromechanics of the cochlea, and development of numerical methods thereof, so as to investigate the generation mechanisms for otoacoustic emissions (namely, sounds coming from the ear).

In this talk we study the topics about integrable systems. This area has many applications in shallow and deep water waves and nonlinear special functions (the Painlevé transcendents). First we introduce some basic idea about integrable systems. For example, the inverse scattering transform which is a nonlinear version of Fourier transform. Then we focus on the problems about long-time asymptotics of integrable water wave equations. Although the analytical results of asymptotics were obtained, but how long is the time need to get well approximation between analytical and asymptotic solutions is still an open problem. We construct a specified initial condition of Camassa-Holm equation and then use numerical approaches to compare the finite difference and analytical solutions to answer the above problems.

The backup 2-center problem is a center facility location problem, in which one is asked to deploy two facilities, with a given probability to fail, in a graph. Given that the two facilities do not fail simultaneously, the goal is to find two locations, possibly on edges, that minimize the expected value of the maximum of the distances from all vertices to their closest functioning facility. In this talk, we focus on the algorithmic results of this problem. In particular, we will briefly review the algorithm for computing the center of a tree, and then show how this idea is applied to solve the backup 2-center problem on a tree.

In this talk we will discuss some of the history of fractional differentiation along with some recent advances in the area.

A mathematical model is derived and studied to describe the outbreak of an infectious disease which is not transmissible between humans until a mutation of the virus or bacterium takes place. The feared outbreak of a transmissible form of avian influenza leading to a global epidemic is the paradigm for this study, for which we model the mutation of the virus from a bird-human to a human-human transmissible form. An extension to the SIR approach is applied, leading to a system of ordinary differential equations describing the evolution of two classes of susceptible and infected states and a removed state. The model is analysed to determine in terms of the parameters the necessary conditions and timescales for the onset of the epidemic, the size and duration of the epidemic and the maximum level of the infected individuals at one time. Two biologically reasonable asymptotic limits are considered, 1) small mutation rate and 2) small mutation rate and small infection rate of the bird-human form. A stochastic version of the model is presented to provide alternative estimates when these rate constants are very small to account for the stochasticity. The model can be extended to investigate the effectiveness of a range of quarantine and vaccination programmes giving quantitative estimates on the outcomes of these control measures.

In this talk, we will recall and explain the classical branching rule of the finite dimensional irreducible $\mathfrak{gl}_m$-module $L_m(\lambda)$, which is about how to decompose the $L_m(\lambda)$ into a direct sum of irreducible $\mathfrak{gl}_{m-1}$-modules. Repeating this process, a linear basis for $L_m(\lambda)$ is obtained, called the {\em Gelfand-Tsetlin basis} for $L_m(\lambda)$. We will explain an analogue of the above phenomenon when we consider the $\mathfrak{gl}_{m|n}$-module $L_{m|n}(\lambda)$. It is a well-known fact obtained by Berele and Regev 1987 by careful calculations and combinatorial techniques. We have an alternative proof based on the representation theory of the Lie superalgebra $\mathfrak{gl}_{m|n}$. As a consequence, some interesting combinatorial identities are obtained. This talk is based on a joint work with Sean Clark and Kuang Thamrongpairoj.

This talk studies the entropy of tree shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of binary Markov tree shifts defined on two symbols is either 0 or ln 2. Meanwhile, the realization of entropy of one-dimensional shifts of finite type is elaborated, which indicates that tree shifts are capable of rich phenomena. Considering the influence of three different types of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient condition for the coincidence of entropy with and without boundary condition are addressed.

Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. For the numerical solution of a general partial differential equaition on a domain, Schwarz iterative methods use a decomposition of the domain into two or more (possibly overlapping) subdomains. In essence one is introducing new artificial boundary conditions on the interfaces between these subdomains. In the classical formulation, these artificial boundary conditions are of Dirichlet type. Given an initial approximation, the method progresses by solving for the PDE restricted to each subdomain using as boundary data on the artificial interfaces the values of the solution on the neighboring subdomain from the previous step. This procedure is inherently parallel, since the (approximate) solutions on each subdomain can be performed by a different processor. In the case of optimized Schwarz, the boundary conditions on the artificial interfaces are of Robin or mixed type. In this way one can optimize the Robin parameter(s) and obtain a very fast method.

In this talk, an asynchronous version of the optimized Schwarz method is presented for the solution of differential equations on a parallel computational environment. In a one-way subdivision of the computational domain, with overlap, the method is shown to converge when the optimal artificial interface conditions are used. Convergence is also proved under very mild conditions on the size of the subdomains, when approximate (non-optimal) interface conditions are utilized. Numerical results are presented on large three-dimensional problems illustrating the efficiency of the proposed asynchronous parallel implementation of the method. The main application shown is the calculation of the gravitational potential in the area around the Chicxulub crater, in Yucatan, where an asteroid is believed to have landed 66 million years ago contributing to the extintion of the dinosaurs.

(Joint work with Fr\'ederic Magou\'es and Cedric Venet).

In this talk, we will introduce multiple zeta values in positive characteristic and give an effective algorithm for determining whether a given multiple zeta value is zeta-like, namely those whose ratio with the zeta value of the same weight is rational. On the other hand, we state conjectures on certain families of zeta-like multiple zeta values.

We consider sequences defined as the sum of the output of an automaton. This is a generalization of automatic sequences, the sum-of-digits function and other digital sequences. We asymptotically analyze these sequences when the input of the transducer is a random integer in [0, N).

Depending on properties of the automaton, the sequence is asymptotically normally distributed. We give the expected value and the variance of the sequence, including the main term and the periodic fluctuation in the second order term. We further investigate properties of this periodic fluctuation.

This is joint work with Clemens Heuberger and Helmut Prodinger.

In this talk, we consider p-Laplace operators on complete noncompact manifold M. First, we define the p-parabolic ends and p-hyperbolic ends according to the p-capacity on M, which imply the volume estimates of ends with positive p-spectrum. Next, if Ricci curvature of M has the lower bound depending on the spectrum of M, we show the Liouville properties for p-harmonic functions with finite p-energy, which also infer the topological property of this manifold, that is, M has at most one p-hyperbolic end. Finally, whenever sectional curvature of M has lower bound, we prove the gradient estimate of the positive eigenfunction associated to p-Laplace operator.

Many reconstruction schemes for inverse boundary value problems for heat equation to identify some anomalies in a heat conductor such as unknown cavities and inclusions from the measured data called Neumann to Dirichlet map have been developed. They are for examples, dynamical probe method, linear sampling type method, enclosure method etc..

One of the most attractive and important application of these reconstruction methods is to the active thermography. This is a non-contact and very quick measurement which can be repeated many times for non-destructing testing to identify anomalies inside any heat conductor. More precisely it injects a heat flux by flash lamp and measured the corresponding temperature distribution on the surface of the conductor by an infrared light camera without having any contact to the conductor. The resolution of infrared light camera is very high now a days. Hence the mathematical model of thermography fits very well to the formulation of our inverse problem.

Compared with the other reconstruction schemes, the dynamical probe method and the linear sampling type method are using the most physical and effective input sources to identify the anomalies in details. It can give some possibility to provide a good basis for the active tomography.

In this talk we will introduce some recent development on the dynamical probe method and linear sampling type method for active thermography. The first one and second methods are good when we probe the anomalies from their outside and inside, respectively. Based on this we will propose a true sampling method to identify an anomaly by just one measurement over a relatively short time interval.

There are many applications about eigenvalues of matrices. In this talk, I will first introduce some topics about eigenvalues on graphs, and then show some our results about spectral radius on bipartite graphs.

Computational conformal mapping has been widely applied in animation industries. It provides an angle-preserving parameterization to a surface. In this talk, I will introduce some basics for computing a disk conformal mapping for a simply connected Riemann surface of single boundary, and then demonstrate some applications on 3D animation.

In this talk, I will present a piece of unpublished recent joint work with Professors Chi-Hin Chan and Magdalena Czubak. We study asymptotic properties of stationary Navier-Stokes flows on the exterior domain on hyperbolic plane. During the talk, I first mention some classical results of stationary Navier-Stokes flows passing an obstacle in the 2D-Euclidean setting. Then, I present our work which addresses the far range decay of the velocity and vorticity of the stationary Navier-Stokes flow in the hyperbolic setting.