## WESTERN MICHIGAN UNIVERSITY ANALYSIS SEMINAR   FALL 2006

### Wednesday at  11 a.m. - 11:50 a.m.Alavi Commons Room, Everett Tower

Comments, questions?  Contact   Yuri Ledyaev ,  phone (269)  387-4557

SEMINAR'S ARCHIVE:  Spring 2006 ,

 Date Speaker, Title and Abstract December 4 Dr.  D. Steven  Mackey Department of Mathematics Western Michigan University  ITERATIVE METHODS FOR MATRIX FACTORIZATIONS (continuation of November 15th talk) The decomposition of matrices into simpler, more structured factors is central to numerical linear algebra, and hence of great importance for applications.  In this talk we discuss the design and analysis of  iterative algorithms of the form  $X_{n+1} = f(X_n)$  for several such matrix factorizations.  In addition we consider the issue of structure preservation  ---  if  $X_0$  is already structured in some way, can we arrange that every iterate  $X_n$  retains this structure? November 29 Dr. Giuseppe Grassi Dipartimento di Ingegneria dell'Innovazione Universita di Lecce, Italy and Department of Electrical and Computer Engineering Western Michigan University GENERATING AND SHAPING CHAOTIC ATTRACTORS In recent years chaotic phenomena have gradually moved from being an academic curiosity to a subject of practical significance, due to potential applications in engineering systems. Since chaos can be useful, designing chaos (or generating/shaping chaos) becomes an important issue in many technological applications. This lecture presents a systematic technique to generate chaotic attractors characterized by rich dynamic behaviours, that is, multi-scroll attractors with an arbitrarily large number of scrolls located around specific unstable equilibriums points. Additionally, the lecture introduces a novel observer-based technique to shape chaotic attractors, that is, to arbitrarily scale all the chaotic drive system states using a scalar synchronizing signal only. November 15 Dr.  D. Steven  Mackey Department of Mathematics Western Michigan University  ITERATIVE METHODS FOR MATRIX FACTORIZATIONS The decomposition of matrices into simpler, more structured factors is central to numerical linear algebra, and hence of great importance for applications.  In this talk we discuss the design and analysis of  iterative algorithms of the form  $X_{n+1} = f(X_n)$  for several such matrix factorizations.  In addition we consider the issue of structure preservation  ---  if  $X_0$  is already structured in some way, can we arrange that every iterate  $X_n$  retains this structure? November 8 Department of Mathematics Western Michigan University ON THE EXTENDED EIGENVALUES OF SOME VOLTERRA OPERATORS For a given operator $A$ it is often of interest to determine which operators satisfy the equation $A X = \lambda X A$ for some  complex number $\lambda$. In this situation $\lambda$ is said to be an extended eigenvalue of $A$. Although the case $\lambda = 1$ is much better understood, even when  $\lambda \neq 1$, $X$ shares many properties with $A$. In particular, when $A$ is compact, $X$ has an invariant subspace. In this work we focus on the integral operators of the Volterra type. We will exhibit some of the possible solutions of theoperator equation above and discuss their significance. November 1 Anirban Dutta Department of Mathematics Western Michigan University BASICS OF STOCHASTIC CALCULUS We introduce  two types of Stochastic Integrals, namely, Stratanovich and Ito Integral. Then we focus on results involving Ito Integral, for example, the Ito Formula. If time permits we'll talk about Stroock-Varadhan Diffusion Theory, thus revealing a deeper connection between Brownian Motion and PDE's. October 25 Anirban Dutta Department of Mathematics Western Michigan University BASICS OF BROWNIAN MOTION We discuss a formalism of Brownian Motion in Euclidean Spaces starting with  definitions and  existence theorems. Then, we  study the properties of brownian paths in single dimension along with occasional generalizations to higher dimensions. If time permits, we will look at some connection between Brownian Motion and classical analysis issues, such as, Harmonic measure, PDEs (Heat Equation in particular) etc. October 18 Ovidiu Furdui Department of Mathematics Western Michigan University LACUNARY SERIES ON FOCK SPACES We give necessary and sufficient conditions for a holomorphic function with gaps to belong to the Fock spaces F. Some applications of these results will discussed. October 11 Dr.Melinda Koelling Department of Mathematics Western Michigan University REPRESENTATIONAL LEARNING One approach to explain neural function is to describe how the circuitry of the brain connects to produce observed properties of the brain. Another approach is to consider what representations might be developed on the basis of statistics of the input and what goals might be served by those representations.  In this talk, I will consider some examples of this second approach. October  4 Dr. Sanath Boralugoda Department of Mathematics University of Sri Jayewardenepura  Colombo, Sri Lanka EVOLUTION PROBLEMS ASSOCIATED WITH NON-SMOOTH FUNCTIONS Recent results show that the Moreau envelope functions associated with some classes of non-smooth functions (eg. Prox-Regular, Primal-Lower-Nice) have C1 smoothness. In this talk we discuss the local existence and uniqueness of absolutely continuous solutions of some differential inclusions determined by subdifferentials  of  related  envelope functions. September 27 Department of Mathematics Western Michigan University   BLACK-SCHOLES OPTION PRICING MODEL In 1997 Scholes and Merton were awarded the Nobel Prize in economics for their contribution to the Black-Scholes option pricing model (unfortunately Black had passed away). What is this model and why is it so important? In this talk I will give a brief introduction following the original Black-Scholes seminal paper published in 1973 and mention some later  developments if time permits. September 20 Dr. Qiji Zhu Department of Mathematics Western Michigan University  OPTION PRICING INTERVAL   . An option is a right without obligation. Options are widely used in the financial industry to mitigate risks or to leverage investment opportunities. If you believe that there is no free lunch then an option always come with a price. How to price an option is a highly active area of research in mathematical finance with an extensive literature. We will try to give an elementary introduction to the widely used classical idea of pricing options by replicating portfolios stemming from Black, Scholes and Merton. We then indicate the difficulties of this method in investment practice and propose an alternative based on information theoretical techniques. This talk is accessible to undergraduate students. September 13 Yuri S.Ledyaev Department of Mathematics  Western Michigan University  CONVEX SETS ON MANIFOLDS OF NONPOSITIVE CURVATURE Manifolds of nonpositive curvature ( Cartan-Hadamard manifolds) have been studied intensively during last three decades, in particular, in the framework of a "geometrization" program for solving some difficult algebraic problems. Notion of convexity plays a significant role in these studies. In this talk we discuss generalization of Junge theorem on a radius of circumscribed ball and new proximal criteria for convexity of sets on Cartan-Hadamard manifolds by using new analytical techniques.