Past OxPDE Special Seminar

In the operation of high frequency resonators in micro electromechanical systems (MEMS)there is a strong need to be able to accurately determine the energy loss rates or alternativelythe quality of the resonance. The resonance quality is directly related to a designer’s abilityto assemble high fidelity system response for signal filtering, for example. This hasimplications on robustness and quality of electronic communication and also stronglyinfluences overall rates of power consumption in such devices – i.e. battery life. Pastdesign work was highly focused on the design of single resonators; this arena of work hasnow given way to active efforts at the design and construction of arrays of coupledresonators. The behavior of such systems in the laboratory shows un-necessarily largespread in operational characteristics, which are thought to be the result of manufacturingvariations. However, such statements are difficult to prove due to a lack of availablemethods for predicting resonator damping – even the single resonator problem is difficult.The physical problem requires the modeling of the behavior of a resonant structure (or setof structures) supported by an elastic half-space. The half-space (chip) serves as a physicalsupport for the structure but also as a path for energy loss. Other loss mechanisms can ofcourse be important but in the regime of interest for us, loss of energy through theanchoring support of the structure to the chip is the dominant effect.

The construction of a basic discretized model of such a system leads to a system ofequations with complex-symmetric (not Hermitian) structure. The complex-symmetryarises from the introduction of a radiation boundary conditions to handle the semi-infinitecharacter of the half-space region. Requirements of physical accuracy dictate rather finediscretization and, thusly, large systems of equations. The core to the extraction of relevantphysical performance parameters is dependent upon the underlying modeling framework.In three dimensional settings of practical interest, such systems are too large to be handleddirectly and must be solved iteratively. In this talk, I will cover the physical background ofthe problem class of interest, how such systems can be modeled, and then solved. Particularinterest will be paid to the radiation boundary conditions (perfectly matched layers versushigher order absorbing boundary conditions), issues associated with frequency domainversus time domain methods, and how these choices interact with iterative solvertechnologies in sometimes unexpected ways. Time permitting I will also touch upon the issue of harmonic inversion methods of this class of problems.

Let $M$ be the Cantor space or an $n$-dimensional manifold with $C(M,M)$ the set of continuous self-maps of $M$. We analyse the behaviour of the generic $f$ in $C(M,M)$ in terms of attractors and some notions of chaos.

We show the solvability of a proposed Generalized Buckley-LeverettSystem, which is related to multidimensional Muskat Problem. More-over, we discuss some important questions concerning singular limitsof the proposed model.

Image segmentation and a number of other problems from image processing and computer vision can be regarded

as interface problems. Recently, diffusive and sharp interface techniques have been used for these problems.

In this talk, we will first briefly explain these models and compare the advantages and disadvantages of these models. Numerically, these models can be solved through some PDEs. In the end, we will show some recent results on how to use graph cut to solve these interface problems. Moreover, the global minimizer can be guaranteed even the problem is nonconex and nonlinear. The use of max-flow in a network setting and also in an infinite dimensional setting will be explained.

• Sufficient conditions for bifurcation from points that are not isolated eigenvalues of the linearisation.

• Odd potential operators.

• Defining min-max critical values using sets of finite genus.

• Formulating some necessary conditions for bifurcation.

• Bifurcation from isolated eigenvalues of finite multiplicity of the linearisation.

• Pseudo-inverses and parametrices for paths of Fredholm operators of index zero.

• Detecting a change of orientation along such a path.

• Lyapunov-Schmidt reduction

• Review of the basic notions concerning bifurcation and asymptotic linearity.

• Review of differentiability in the sense of Gˆateaux, Fréchet, Hadamard.

• Examples which are Hadamard but not Fréchet differentiable.  The Dirichlet problem for a degenerate elliptic equation on a bounded domain. The stationary nonlinear Schrödinger equation on RN

I will discuss two of my papers that develop PDE methods for weak KAM theory, in the context of a singular variational problem that can be interpreted as a regularization of Mather's variational principle by an entropy term. This is, sort of, a statistical mechanics approach to the problem. I will show how the Euler-Lagrange PDE yield approximate changes to action-angle variables for the corresponding Hamiltonian dynamics.

Random matrices were introduced by E. Wigner to model the excitation spectrum of large nuclei. The central idea is based on the hypothesis that the local statistics of the excitation spectrum for a large complicated system is universal. Dyson Brownian motion is the flow of eigenvalues of random matrices when each matrix element performs independent Brownian motions. In this lecture, we will explain the connection between the universality of random matrices and the approach to local equilibrium of Dyson Brownian motion. The main tools in our approach are the logarithmic Sobolev inequality and entropy flow. The method will be applied to the adjacency matrices of Erdos-Renyi graphs.

The problem of isometric embedding of a Riemannian Manifold into

Euclidean space is a classical issue in differential geometry and

nonlinear PDE. In this talk, I will outline recent work my

co-workers and I have done, using ideas from continuum mechanics as a guide,

formulating the problem, and giving (we hope) some new insight

into the role of " entropy".