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May 1st, 2012 (11:15am - 12:05pm)

Seminar: Combinatorics/Partitions Seminar
Title: Analogues of Stanley's Theorem and Elder's Theorem in Odd Partitions and Distinct Partitions.
Speaker: Rebekah Gilbert, PSU
Location: MB106

Held in 115 Osmond Laboratory

May 1st, 2012 (04:00pm - 05:00pm)

Seminar: Applied Analysis Seminar
Title: Multiscale models for network flows and crowd dynamics.
Speaker: Benedetto Piccoli, Rutgers University
Location: MB106

Continuous and discrete models for traffic flow on networks are commonly used. Applications ranges from vehicular traffic to supply chains and data networks. We focus on recent multi scale and mixed models, involving continuous-discrete spaces and ode-pde systems. Finally, we will discuss a measure theoretical framework, particularly efficient for dynamics of large groups.

May 7th, 2012 (09:00am - 11:00am)

Seminar: Ph.D. Thesis Defense
Title: "Analytic methods for Diophantine problems"
Speaker: Jingjing Huang, Advisers: Winnie Li & Bob Vaughan, Penn State
Location: 202 Osmond Laboratory

We are mainly concerned with the Diophantine equation $$\frac{a}{n}=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}$$ and its number of positive integer solutions $R_k(n;a)$. We begin with the binary case $k=2$. Now the distribution of the function $R_2(n;a)$ is well understood. More precisely, by averaging over $n$, the first moment and second moment behaviors of $R_2(n;a)$ have been established. For instance, one of our results is $$\sum_{\substack{n\le N\\(n,a)=1}}R_2(n;a)=N P_2(\log N;a)+O_a(N\log^5 N),$$ where $P_2(\cdot;a)$ is a quadratic function whose coefficients depend on $a$. Furthermore, we have shown that, after normalisation, $R_2(n;a)$ satisfies Gaussian distribution, which is an analog of the classical theorem of Erd\H{o}s and Kac, $$\lim_{N\to\infty}\frac1N\rm{card}\left\{n\le N:\frac{\log R_2(n;a)-(\log 3)\log\log n}{(\log 3)\sqrt{\log\log n}}\le z\right\}=\frac1{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-\frac{t^2}{2}}dt.$$ On the other hand, we change the point of view and study the set of ``exceptional numbers" that do not possess binary representations. Let $E_a(N)$ denote the number of $n\le N$ such that $R_2(n;a)=0$. It is established that when $a\ge3$ we have $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}},$$ with $m$ defined in the talk. I will explain how to prove this theorem. The next project would be to study the ternary case $k=3$. While the conjecture, by Erd\H{o}s, Straus and Schinzel, that for fixed $a\ge 4$, we have $R_3(n;a)>0$ when $n$ is sufficiently large, is still wide open, here I will talk about some partial results on the mean value $\sum_{n\le N}R_3(n;a)$ if time permits.

May 7th, 2012 (09:30am - 11:30am)

Seminar: Ph.D. Oral Comprehensive Examination
Title: "Approximate Solutions to Second Order Parabolic Equations"
Speaker: Chao Liang, Advisers: Xiantao Li, Victor Nistor, Penn State
Location: MB106

We continue the work of expanding the Green's function Gt(x; y) of a uniformly parabolic linear operator @tL with non-constant coecients. Based on dilation and Taylor expansions, we use regular perturbation to get a general computable construction of approximate solutions to parabolic equations which are accurate to arbitrary pre- scribed order in the short-time limit. Our formula is easy to get by programming. We also prove some properties of the approximation se- ries by our formula.

May 8th, 2012 (01:00pm - 03:00pm)

Seminar: Ph.D. Oral Comprehensive Examination
Title: "Spatial Pattern Dynamics in Evolutionary Games: the Fitness Gradient Flux"
Speaker: Russell deForest, Adviser: Andrew Belmonte, Penn State
Location: MB106

We introduce a non-diffusive spatial coupling into the replicator equation of evolutionary game theory and study the development of patterns for several classes of 2-strategy games in 1D and the rock-paper-scissors game in 2D. Our proposed flux term is based on motion due to local gradients in the relative fitness of each strategy and provides an alternative to diffusive coupling for competitive systems in which the flux depends directly on the game dynamics. We observe asymptotically attracting and wave-generating states in 1D and spiral formation and breakup in 2D. A change of variables is shown to correctly capture the asymptotic behavior of these spatiotemporal systems via a nonlinear diffusion equation.

May 9th, 2012 (04:00pm - 05:00pm)

Seminar: Computational and Applied Mathematics Colloquium
Title: Exponentially growing solutions of the wave equation with compact time-periodic potentials
Speaker: Vesselin Petkov, University Bordeaux 1, Institute Mathematique de Bordeaux
Location: MB114
Abstract: http://www.math.psu.edu/ltz/Petkov_CAM_Talk/abstractPennState.pdf

We study the solutions of the Cauchy problem for the wave equation $\partial_t^2 u - \Delta_x u + V(t, x)u = 0$, where the potential $V(t,x)$ has compact support in $x$ and it is periodic with respect to $t$. If the potential $V(t,x)$ takes negative values, it is possible to construct solutions whose energy is exponentially growing. For positive potentials $V(t, x)$ this problem is more difficult since for time independent potentials $V(x)$ the energy is conserved. This problem was open for more than 30 years. In 1993 G. Majda and M. S. Wei by numerical computations suggested the existence of positive potentials for which it is possible to construct exponentially increasing solutions. In this talk we construct positive time-periodic potentials with compact support for which the wave equation has exponentially growing solutions. Moreover, we prove that the local energy of these solutions is also exponentially growing and we have so called {\it parametric resonance} phenomenon which is well known in the theory of the ordinary differential equations. This is a work in collaboration with F. Colombini and J. Rauch.

May 10th, 2012 (08:00am - 10:00am)

Seminar: Ph.D. Thesis Defense
Title: "Dynamic Modeling of Biological and Physical Systems"
Speaker: Assieh Saadatpour Moghaddam, Advisers: Mark Levi (Math) & Reka Albert (Physics), Penn State
Location: MB106

Given the complexity and interactive nature of many biological and physical systems, constructing informative and coherent network models of these systems and subsequently developing efficient approaches to analyze the models is of utmost importance. The combination of network modeling and dynamic analysis enables one to investigate the behavior of the underlying system as a whole and to make experimentally testable predictions about less-understood aspects of the processes involved. This dissertation reports on a combination of theoretical and computational approaches for network-based dynamic analysis of several highly interactive biological and physical systems. Various dynamic modeling approaches, ranging from Boolean to continuous models, are employed to carry out a systematic analysis of the long-term behavior (attractors) of the respective systems. First, we employ a Boolean dynamic framework to model two biological systems: the abscisic acid (ABA) signal transduction network in plants and the T-LGL leukemia signaling network in humans. Given the relatively large number of components in these networks, we develop a network reduction technique leading to a significant decrease in the computational burden associated with the state space analysis of Boolean models while preserving essential dynamical features. For the ABA system, we utilize a synchronous and three different asynchronous Boolean dynamic methods and compare the attractors of the system and their basins of attraction for both unperturbed and perturbed systems. For the T-LGL signaling network, the best-performing asynchronous Boolean dynamic method identified in our first study is used to identify the disease states of the components of the system and to propose several novel candidate therapeutic targets. Next, we apply a Boolean-continuous hybrid (piecewise linear) dynamic formalism to model a pathogen-immune system interaction network, and present the results of a comparative study of the dynamic characteristics of Boolean and hybrid models. Finally, we rely on continuous dynamic modeling to prove the existence of traveling wave solutions in a better-characterized physical system, namely, a lattice of coupled pendula in the presence of damping and forcing. The theoretical and computational approaches developed in this dissertation provide a bird’s-eye-view of the avenues available for model-driven analysis of complex biological and physical systems.

May 15th, 2012 (02:00pm - 04:00pm)

Seminar: Ph.D. Thesis Defense
Title: "The Generalized Finite Element Method: Numerical Treatment of Singularities, Interfaces, and Boundary Conditions"
Speaker: Qingqin Qu, Advisers: Anna Mazzucato/Victor Nistor, Penn State
Location: MB106

This dissertation is devoted to use the Generalized Finite Element Method to solve partial dierential equations numerically. As an extension of the standard Finite Element Method (FEM), the Generalized Finite Element Method (GFEM) is especially convenient for dealing with complicated do- mains, corner singularities, transmission problems and mixed boundary con- ditions. The GFEM is related to other methods, such as the hp cloud method and the extended nite element method. The GFEM diers from the stan- dard FEM in the construction of the nite-dimensional space in which the approximate solution is sought. Instead of using piecewise polynomials on each element of a triangulation of the domain, we dene the element spaces by using partitions of unity to combine the local approximation spaces, de- ned in each patch of the partition, in order to obtain the global GFEM space. In particular, the GFEM is an example of a meshless method, since the partition of unity need not be subordinated to a particular mesh. The GFEM allows one to include a priori knowledge about the local behavior of the solution, and gives the option of constructing trial spaces of any desired regularity. For transmission (interface) problems on domains with smooth, curved boundaries, we establish quasi-optimal rates of convergence of the numerical solution to the true solution by using a non-conforming Generalized Finite Element Method. A sequence of approximation spaces Sn are constructed that satisfy the following two conditions: (1) nearly zero boundary and inter- face matching, (2) approximability. The numerical solution is then obtained as a Galerkin approximation to the true solution. We prove that the ap- proximation error of order O(dim(Sn) m=2 ), where dim(Sn) is the dimension of the GFEM space Sn, and m is the degree of polynomials used for the local approximation of the solution. Numerical experiments are presented to demonstrate these theoretical results. For the case of singular domains, we study the GFEM approximation to solutions of the Poisson's problem in polygons. It is well-known that the loss of regularity of the exact solution due to domain singularities will deteriorate the convergence rate of the standard FEM if one uses quasi- uniform meshes. To circumvent the loss of regularity, we pose the problem in certain weighted Sobolev spaces, and show that the continuous problem has the expected regularity in these spaces. We construct GFEM approximation spaces using again a partition of unit and local approximation spaces. For the former, we use dilation techniques to deal with corner singularities, while we use standard piecewise polynomial spaces for the latter. We then establish quasi-optimal rate of convergence of the GFEM approximation to the exact solution both in weighted Sobolev spaces and then in Hilbert spaces.

May 22nd, 2012 (10:00am - 12:00pm)

Seminar: Ph.D. Thesis Defense
Title: "Rokhlin Actions on AF algebras and Classifiability"
Speaker: Michael Tseng, Adviser: Nate Brown, Penn State
Location: MB106

The notion of nuclear dimension, introduced by Winter and Zacharias, has provided a new, and very possibly the definitive, point of view for the classification of separable simple nuclear C∗-algebras. In this thesis, the C∗-algebras under consideration are crossed products of the type C oα N where C is an approximately finite dimensional algebra and α is a corner endomorphism with the Rokhlin property. We compute their nuclear dimension and show that they absorb the Jiang-Su algebra tensorially. In light of the new guiding principle that nuclear dimension should imply classifiability, we then revisit previous work, largely due to Rørdam, establishing the classifiability of such crossed products.

May 29th, 2012 (02:30pm - 03:45pm)

Seminar: Logic Seminar
Title: To be announced.
Speaker: Keita Yokoyama, Tokyo Institute of Technology
Location: MB315