# CMU Analysis and Applied Mathematics Seminar

## Spring 2014

### Organizers

If you would like to give a talk, please email any one of us!

### Meeting Times

Fridays, 3:00–4:00pm, in Pearce 223.

### Schedule

 Date Speaker Title 2/14 Xiaoming Zheng Singularity analysis of a reaction-diffusion equation with a solution-dependent Dirac delta source 2/21 Debraj Chakrabarti Boundary Values of Holomorphic Functions 3/7 Yousef Al-Jarrah A Wavelet Based Method for Solving Integral Equations 3/21 Ye Li Comparison Theorems and Geometric Inequalities 4/4 Sivaram Narayan A Gentle Introduction to Composition Operators 4/11 Alex Misiats (Purdue) Long-time behaviour of stochastic reaction-diffusion equaitions. 4/18 Aaron Yip (Purdue) Blow up phenomena for shadow system of Gierer-Meinhardt model 4/25 Yeonhyang Kim Frames with infinite support

### Abstracts

Speaker: Xiaoming Zheng
Title: Singularity analysis of a reaction-diffusion equation with a solution-dependent Dirac delta source
Abstract: We analyze the existence and singularity of a solution to a reaction–diffusion equation, whose reaction term is represented by a Dirac delta function which depends on the solution itself. We prove that there exists a unique analytic solution with a logarithmic singularity at the origin.

Speaker: Debraj Chakrabarti
Title: Boundary Values of Holomorphic Functions
Abstract: We discuss the notion of weak boundary value of a holomorphic function of one or several complex variables. We discuss some recent results regarding weak boundary values on piecewise smooth boundaries and some open questions.

Speaker:Yousef Al-Jarrah
Title: A Wavelet Based Method for Solving Integral Equations
Abstract: Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this talk, we use scaling function interpolation method to solve several different kinds of integral equations such as: 1- Fredholm integral equations of first and second kind 2- Singular Fredolm integral equations 3- Volterra integral equations 4- Fredholm-Volterra integral equations 5- Two-dimensional Fredholm integral equations. Moreover, we prove convergence theorem for the numerical solution of integral equations and present some examples of solving integral equations. Comparisons of the results with other methods will be presented. Finally, an application of the two-dimensional Fredholm integral equation in image denoising will be introduced.

Speaker: Ye Li
Title: Comparison Theorems and Geometric Inequalities
Abstract: We are interested in comparison theorems on manifolds and their applications. We study Hardy type inequality, weighted Hardy inequality and weighted Sobolev inequality via Hessian comparison theorems. We also discuss some Caﬀarelli-Kohn-Nirenberg type inequality on Cartan-Hadamard manifolds.

Speaker: Sivaram Narayan
Title:A Gentle Introduction to Composition Operators
Abstract: Let $\mathbb{D}$ denote the open unit disc of the complex plane and $\varphi$ be a holomorphic self-map of $\mathbb{D}$. The equation $C_{\varphi} f = f\circ \varphi$ defines a composition operator on the space $H(\mathbb{D})$ of holomorphic functions on $\mathbb{D}$. The classical result of Littlewood asserts that any holomorphic self-map $\varphi$ of the unit disc induces a bounded composition operator $C_{\varphi}$ on the Hardy space $H^2$. The goal of the subject is to study how the properties of the analytic function $\varphi$ influence the properties of $C_{\varphi}$ and vice versa. In this talk we will illustrate this connection.

Speaker: Alex Misiats
Title:Long-time behaviour of stochastic reaction-diffusion equaitions.
Abstract: We study the long-time behavior of systems governed by nonlinear reaction-diffusion type equations $du = (Au + f(u))dt + \sigma(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $\sigma$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. This equation is known to have a uniformly bounded (in time) solution ifor $A = \Delta$ provided $f(u)$ possesses certain dissipative properties. The existence of a bounded solution implies, in turn, the existence of an invariant measure for this equation, which is an important step in establishing the ergodic behavior of the underlying physical system. In my presentation I will talk about expanding the existing class of nonlinearities $f$ and $\sigma$, for which the invariant measure exists. We also show that the equation has a unique invariant measure if $A$ is a Shroedinger-type operator $A = 1/\rho({\rm div} \rho \nabla u)$ where $\rho = e^{-|x|^2}$ is the Gaussian weight. In this case the source of dissipation comes from the operator $A$ instead of the nonlinearity $f$. The main idea is to show that the reaction-diffusion equation has a unique bounded solution, defined for all $t \in \mathbb{R}$, i.e. that can be extended backwards in time. This solution is an analog of the trivial solution for the linear heat equation.

Speaker: Aaron N. K. Yip
Title:Blow up phenomena for shadow system of Gierer-Meinhardt model.
Abstract: We will present some results of blow-up phenomena for the shadow system obtained from the Gierer-Meinhardt model. Shadow system is formally derived by letting the diffusion coefficient of one of the components tend to infinity, leading to a coupled system of a diffusion and an ordinary differential equation. There is a huge discrepancy in terms of long time behaviors between the shadow and the original Gierer-Meinhardt systems. We will demonstrate this using integral estimates and a fixed point theorem. This is joint work with Fang Li.

Speaker: Yeonhyang Kim
Title: Frames with infinite support.
Abstract: A \emph{frame} in a Hilbert space $\mathcal{H}$ is a sequence of vectors $\{f_i\}_{i\in I}$ for which there exist constants $0< A\leq B<\infty$ such that for all $f \in\mathcal{H}$, $A\| f \|^2\leq \sum_{i\in {\mathbb Z}}|\langle f,f_i \rangle|^2 \leq B\|f\|^2.$ Due to some desirable properties, frames with infinite support are of interest in applications and have been extensively studied. In this talk, we explore the question of when we can generate a frame with infinite support for $L_2$-space using frame techniques and the Wiener algebra.