A random walk is called recurrent if it is sure to return to its starting point and transient otherwise. A famous result of Polya is that simple symmetric random walk on the integer lattice is recurrent in dimensions 1 and 2, and transient in higher dimensions. We study random walks that are small perturbations of simple random walk. Our main result is that if the dimension is high enough then these random walks are transient.

We describe some recent developments involving particle methods in
electrostatics and fluid dynamics.

First we describe a Cartesian treecode for charged particle systems
undergoing screened Coulomb interactions in three dimensions. The
treecode reduces the operation count for computing potentials and
forces from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of
particles in the system. The algorithm is especially well suited for
particles distributed on a surface. This is a step towards a treecode-
accelerated boundary integral solver for implicit solvent models in
biomolecular dynamics. (joint work with Peijun Li, Hans Johnston, and
Weihua Geng)

Second we describe a Lagrangian panel method for computing vortex
sheet motion in three dimensional fluid flow. Here too a treecode is
used, to evaluate the regularized Biot-Savart integral for the
sheet's self-induced velocity. The method is applied to compute the
azimuthal instability of a vortex ring. Details of the core dynamics
are clarified by tracking material lines on the sheet surface.
Results show the collapse of the vortex core in each wavelength and
the radial ejection of ringlets. These events are correlated with
local axial flow in the core of the ring. (joint work with Hualong
Feng and Leon Kaganovskiy)

The study of injective envelopes of metric spaces, also known as metric trees (R-trees or T-theory), has its motivation in many sub-disciplines of mathematics as well as biology/medicine and computer science. Its relationship with biology and medicine stems from the construction of phylogenetic trees [5].Concepts of string matching in computer science is closely related with the structure of metric trees [4]. A metric tree is a metric space such that for every in M there is a unique arc between x and y and this arc is isometric to an interval in R. [3],[2]. In this talk, we examine convexity and compact structures in metric trees and show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. We show that a set valued mapping of a metric tree M with convex values has a selection for which for each . Here by we mean the Hausdroff distance [1]. We will mention some applications to k-set contractions as well as an application of the above selection theorem. Furthermore we define n-widths of a subset A of a metric tree M and show that even in the absence of linear structure the limit of n-widths as is equal to the ball measure of noncompactness.

References

1.A. G.Aksoy, M.A. Khamsi A Selection Theorem in Metric Trees, Proc. Amer. Math. Soc. 134, No.10 (2006), 2957-2966.

2.W. B. Johnson, J. Lindenstrauss and D. Preiss, Lipschitz quotients from metric trees and from Banach spaces containing , J. Funct. Anal. 194 (2002), 332-346.

3.A. W. M. Dress, V. Moulton and W. Terhalle, T-Theory, An overview. European J. Combin. 17 (1996), 161-175.

4.I. Bartolini, P. Ciaccia, and M. Patella, String Matching with metric trees using approximate distance. SPIR, Lecture notes in Computer science, Springer Verlag, 2476, (2002), 271-283.

5.C. Semple, and M. Steel, Phylogenetics. Oxford lecture series in mathematics and its applications, 24, (2003).

The processes of adaptation, learning, discovery, and invention are expected to play an increasingly significant role in emerging large-scale computationally intelligent (CI) systems. . The meanings of adaptation and learning are well-known. By discovery we mean the process of creating a new hypothesis based on sufficient new data that does not fit existing hypotheses; while by invention, the process of creating (synthesizing) new prototypes by interpolation or extrapolation of existing ones.
In this lecture we will present a kernel-based mathematical approach to the modeling and design of the processes of adaptation, learning, discovery and invention in nonlinear dynamical systems. The approach is based on our previous work in which uncertainty is handled by mathematical approximation theory methods, specifically, by best approximation of nonlinear functionals in an appropriate Reproducing Kernel Hilbert Space (RKHS). This formulation leads to optimal nonlinear system models in the form of artificial neural networks (ANNs) in a natural way, that is, without imposing, a-priori, a neural structure on the system being modeled. For this reason, while connecting with the mathematical foundations on which they are based, we will use ANNs as generic representations of the nonlinear dynamical systems under discussion.
Computer simulation results, some using real data, will be presented to establish and illustrate the theoretical developments.