In this talk we present a summary of recent and new results on the global character of solutions of the second-order rational difference equation

An n by n nonzero (resp. sign) pattern A is a matrix with entries in {*,0} (resp. {+,-,0}). The inertia of a matrix A is the ordered triple (a₁,a₂,a₃) of nonnegative integers where a₁ (resp. a₂ and a₃) is the number of eigenvalues of A with positive (resp. negative and zero) real part. A is inertially arbitrary if each nonnegative integer triple (a₁,a₂,a₃) with a₁+a₂+a₃=n is the inertia of a matrix with nonzero (resp. sign) pattern A. Some observations regarding which inertias A and B may allow to guarantee A ÅB is inertially arbitrary are presented. It is shown that there exists non-inertially-arbitrary nonzero (resp. sign) patterns A and B such that A ÅB is inertially arbitary.

Let T be a complete hyperbolic surface homeomorphic to a three-holed sphere and let G denote the image under the holonomy representation of its fundamental group. Identifying the group of hyperbolic isometries with an appropriate component of the group of isometries of Minkowski spacetime, we may consider affine deformations of G; we may ask, when does this affine deformation act properly discontinuously on R³? An important invariant for affine isometries with non-elliptic linear part is the Margulis invariant, which is a measure of signed Lorentzian displacement. In this paper, we show that an affine deformation of G acts properly discontinuously if and only if the Margulis invariant is positive for each of the three isometries corresponding to the pant holes of T. More precisely, we show that such an affine deformation admits a fundamental domain.

This is joint work with Drumm and Goldman.

For a given Partial Differential Equation (PDE) or a system of PDEs, with n ³ 2 independent variables, I will describe a systematic (and rather general) framework to find PDE systems nonlocally related to the given one. Such nonlocally related PDE systems have solution sets that are equivalent to the solution set of the given system (i.e., any solution of a nonlocally related system yields a solution of the given system, and, conversely, any solution of the given system yields a solution of the nonlocally related system). Moreover, the solution of any boundary value problem posed for the given PDE system is embedded in the solution of a boundary value problem posed for the nonlocally related (potential) system (and the converse also holds).

Due to nonlocal relations and the equivalence of solution sets, any general method of analysis (symmetry, conservation law, qualitative, perturbation, numerical, etc.), when applied to one of such nonlocally related PDE systems, can yield new results. In particular, new conservation laws, nonlocal symmetries and new exact solutions have been found for many nonlinear PDE systems arising in applications.

I will discuss several illustrative examples: the Nonlinear Wave equation, Planar Gas Dynamics equations, equations of Nonlinear Elastodynamics, and Plasma Equilibrium equations in 3D.

The talk is aimed at the broad audience of applied mathematicians and researchers working with PDE models.

This is a joint work with George Bluman.

We consider the problem of adapting a Genetic Algorithm (GA) to constrained optimization problems, using a dynamic penalty approach as a type of annealing. We present two different methods for ensuring almost sure convergence to a globally optimal (feasible) solution. The first of these involves modifying the GA operators to yield a Boltzmann-type distribution while the second incorporates a dynamic penalty along with a slow annealing of the acceptance probabilities.

Recent precise estimates on the frequency of occurrence of intervals of rapid growth on a Brownian path are discussed. These are then used, together with the theory of Gaussian algebras, to obtain similar estimates for the large fluctuations of complex oscillations, which can be seen as Brownian motion generated by the set of Kolmogorov-Chaitin complex strings. This yields the Hausdorff dimension of the rapid points of such complex oscillations.

In this paper, we study graphs related to knots. We investigate the connection between knot projections and graphs. We use the related graphs to catalog knot projections.