The SPAMS Schedule - Spring 1997

In numerical linear algebra, various iteration methods (like those based on contraction mechanism and the Krylov space method ) are essentially processes of polynomial acceleration. In signal processing, polynomial is also the unique soul of digital filters. Chebyshev approximation theory therefore is the key to the convergence analysis for the iteration methods and the design of optimal digital filters. In this 45-minute talk, I will focus mainly on the formulation of problems, rather than the results.

We will discuss the following interesting theorem originating with N.G. de Bruijn: Theorem: Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side. A number of different interesting proofs will be presented (including calculus proofs, combinatorial proofs and number theoretic proofs). We will also discuss some generalizations and open problems.

I will give a brief introduction to some of the applications for nonlinear diffusion models in fluid dynamics and mathematical biology. Similarity solutions are special solutions of these PDE's where the PDE reduces to an ODE problem. These solutions are often representative of generic properties of all solutions of the PDE. I will demonstrate how to use similarity solutions with matched asymptotic expansions to derive solutions of more general initial boundary value problems.

Have you ever wondered why the wake of a boat seems to be at the same angle regardless of its speed? Or if you live downstream from a dam how long it would take for your house to get flooded should the dam break? I will show how these two fairly standard fluids problems can surprisingly be answered using mostly simple geometric arguments and (almost) no PDEs. A little knowledge of the method of characteristics would be helpful for the second problem, but is not necessary. }

The Korteweg and deVries (KdV) equation is a canonical equation in nonlinear waves, but as a practical model for water waves it is limited by being strictly one dimensional (whereas water surface is two-dimensional). We will derive a two-dimensional generalization of the KdV equation called Kadomtsev-Petviashvili (KP) equation. It describes evolution which is weakly nonlinear, weakly dispersive and {\bf weakly} two-dimensional, with all three effects being of the same order.

We will further consider generalized nonlinear evolution equations of the KdV type in two spatial (and one time) dimensions. Looking for solitary wave solutions, we will formulate and analyze the resulting nonlinear elliptic boundary value problems.

We show that if an $n X n$ Jordan block is perturbed by an O($\epsilon$) upper $k$-Hessenberg matrix ($k$ subdiagonals including the main diagonal), then generically the eigenvalues split into $p$ rings of size $k$ and one of size $r$ (if $r \neq 0$), where $n=pk+r$. This generalizes the familiar result $( k=n, p=1, r=0)$ that generically the eigenvalues split into a ring of size $n$. We compute the radii of the rings to first order and the result is generalized in a number of directions involving multiple Jordan blocks of the same size. }

Although quantum mechanics, and non-abelian gauge theory in particular, can often be tedious and mired in algebra, there are theories for which simple topological properties of a system can have exciting physical consequences. The Bohm-Ahrononv experiment describes such a system where the amplitude associated to a particle moving around a magnetic core is determined solely by the winding number of its path. We will see this as an example of anyons, particles found in two dimensional systems which are between bosons and fermions, and whose statistics are determined by arguments from homotopy theory and braid group theory. Last, time permitting, we will discuss topological charge and Yang-Mills instantons, where ideas from Chern-Simons theory may contribute to our understanding of particle cosmology.

Sand in an hourglass, salt piles along the side of a highway, screes at the bottom of a mountain and sugar in a bowl are all examples of granular materials which are ubiquitous in our daily lives. Such materials play an important role in many industrial as well as geological processes. Yet, despite their pervasiveness, granular media behave in surprising ways, often having properties different from those commonly associated with solids, liquids or gases: in a large container of sand, sound will not propagate horizontally; during an avalanche, the "fluid" motion is confined to a thin boundary layer near the surface rather than throughout the bulk as in a liquid; in a gas of sand grains, the particles coalesce into long, thin, tendrils. Another example of exceptional behavior is found in a vibrated container filled with grains, where large particles separate from the smaller ones at the top surface despite the fact that entropy arguments favor complete mixing of the sizes. In such conditions, we also find convection rolls forming spontaneously. This lecture will review a few of these unusual properties and describe some recent research on these fascinating materials.

The long-time behavior of solutions of some problems with strong dissipation approach simple self-similar states. Using symmetries and conservation laws for the system the asymptotic solution can be determined from the initial data. We will focus on the problem of describing equilibrium and entropy for the diffusion of a finite quantity in an infinite domain.

We will discuss a number of mathematical problems that arise in the context of gene recognition. No biology background will be necessary to understand the talk.

( there will, however, be a mininum size requirement on the cerebral cortex of each participant ).