Colloquia — Summer 2006

Friday, August 4, 2006

Title: A New Finite Dimensional Integrable System Associated with AKNS Hierarchy
Speaker: Taixi Xu, Southern Polytechnic State University
Time: 3:00pm‐4:00pm
Place: PHY 130

Sponsor: Wen-Xiu Ma


Given the Hamiltonian operator and the infinite dimensional integrable Hamiltonian equation with its first integrals, we use the general Legendre transformation to show that the infinite dimensional integrable equation can be reduced to a finite domensional integrable Hamiltonian system on an invariant set \(S\).

As a special example, we will discuss the AKNS hierarchy, and we get a new finite dimensional completely integrable system associated with AKNS hierarchy.

Tuesday, July 27, 2006

Title: Sobolev spaces on graphs and their applications
Speaker: Mikhail Ostrovskii, St. John's University
Time: 3:00pm‐4:00pm
Place: PHY 118

Sponsor: Lesław Skrzypek


Let \(G\) be a finite simple graph. By \(V_G\) and \(E_G\) we denote its vertex set and its edge set, respectively. By \(d_v\) we denote the degree of a vertex \(v\in V_G\). Let \(f:V_G\to \mathbf{R}\), and let \(1\le p<\infty\). The Sobolev seminorm of \(f\) corresponding to \(E=E_G\) and \(p\) is defined by $$ ||f||=||f||_{E,p}=\left(\sum_{uv\in E}|f(u)-f(v)|^p\right)^{1/p}. $$

If \(G\) is connected, then the only functions \(f\) satisfying \(||f||_{E,p}=0\) are constant functions, so \(||\cdot||_{E,p}\) is a norm on each linear space of functions on \(V=V_G\) which does not contain constants. We consider the subspace in the space of all functions on \(V_G\) given by \(\sum\limits_{v\in V}f(v)d_v=0\). The obtained normed space is called a Sobolev space on \(G\) and will be denoted by \(S_p(G)\).

In this talk we shall discuss:

  1. Why is it natural to call these spaces “Sobolev spaces”?
  2. Applications of Sobolev spaces on graphs.
  3. Banach-space theoretical properties of Sobolev spaces on graphs.

Monday, May 8, 2006

Title: Discontinuous Dynamical Systems: Progress and Problems
Speaker: Xinchu Fu, Shanghai University
Time: 11:00am‐12:00pm
Place: PHY 130

Sponsor: Yuncheng You


In Part I, we discuss dynamical behaviour of a class of discontinuous maps including piecewise linear maps on the \(2\)-torus and planar piecewise isometries. For piecewise linear parabolic maps on the torus, when the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure. For linear elliptic maps with round-off and quantization discontinuities, two examples arising from digital signal processing are examined, and are shown to have the dynamics of piecewise isometries of a union of convex polygons on the plane by an appropriate transformation of the linearized parts into normal form. Properties of invariant disk packings for invertible piecewise isometries are also discussed. It is shown that tangencies between such disks in the packings are very rare. This support the long-standing conjecture that the exceptional sets possess positive Lebesgue measure. Some results about the topological entropy and dynamical complexity of piecewise isometries are given; and finally, global attractors for planar PWIs are characterized via invariant measures and positive continuous functions on phase space. And some typical open problems in this new research area are presented.

In Part II, we discuss coding and symbolic description of dynamical behavior of a class of \(2\)-D discontinuous maps including piecewise linear maps on the \(2\)-torus and planar piecewise isometries.