Bivariate Bernstein-Szego weights on the Square
Abstract: Univariate Bernstein-Szego weights, which are weights whose densities are the reciprocal of a positive polynomial, play an important role in the theory of orthogonal polynomials by providing a convenient set of simple approximating weights to a general measure. Recently there has been work on extending these types of measures to the bicircle. I will review this work and describe extensions in the case of the square.
The asymptotic form of the Gauss-Lucas theorem
Abstract: The Gauss-Lucas theorem claims that if the zeros of a polynomial P lie in a convex set K, then the zeros of its derivative also lie in K. This is no longer true if a zero of P may lie outside K, but it was conjectured by Boris Shapiro that then most of the zeros of P' still lie in any fixed (arbitrarily small) neighborhood of K. (The precise conjecture was that if most of the zeros of P lie in K, then most of the zeros of its derivative lie in any fixed neighborhood of K.) In this talk we give a proof and discuss the connection to some old results of Erdos, Niven, de Bruijn and Springer and to their later developments.
Nonsmooth optimization of partial differential equations
Abstract: This talk is concerned with the analysis and numerical solution of optimization problems involving non-differentiable functionals and equality constraints given by partial differential equations. Such problems arise in, e.g. optimal switching controls of distributed systems or parameter identification problems with data corrupted by impulsive noise. Using tools of nonsmooth analysis, explicit pointwise optimality conditions can be derived which are amenable to numerical solution via generalized Newton or hybrid gradient methods.
Algorithmic Principle of Least Revenue for finding market equilibria
Abstract: In analogy to extremal principles in physics, we introduce the Principle of Least Revenue for treating market equilibria. It postulates that equilibrium prices minimize the total excessive revenue of market's participants. As a consequence, the necessary optimality conditions describe the clearance of markets, i.e. at equilibrium prices supply meets demand. It is crucial for our approach that the potential function of total excessive revenue be convex. This facilitates structural and algorithmic analysis of market equilibria by using convex optimization techniques. In particular, results on existence, uniqueness, and efficiency of market equilibria follow easily. The market decentralization fits into our approach by the introduction of trades or auctions. For that, Duality Theory of convex optimization applies. The computability of market equilibria is ensured by applying quasi-monotone subgradient methods for minimizing nonsmooth convex objective - total excessive revenue of the market's participants. We give an explicit implementable algorithm for finding market equilibria which corresponds to real-life activities of market's participants.
• Miguel Jimenez Pozo,
Universidad Autónoma de Puebla, México (Chair of EC and SC).
• Guillermo López Lagomasino,
Universidad Carlos III de Madrid, Spain.
• Jan-J. Rückmann,
Bergen University, Norway.
• Rüdiger Schultz,
Duisburg-Essen University, Germany.