A Note on Well-posed Problems

A recent result by Ricceri [Ri] states that a ${C^{1,1}_{loc}}$ function ${f : X \to {\mathbb R}}$ , where X is a Hilbert space, attains its minimum on any small closed ball around a point where its derivative does not vanish, and that the unique minimum point belongs to the boundary of the ball. The proof is based on a saddle-point theorem. We show that the result, which we extend to Banach spaces having a norm with modulus of convexity of power type 2, can be obtained by means of a purely variational argument.     

On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances

We consider the dispersive Degasperis–Procesi equation $u_t?u_{xxt}?\mathtt{c}{u}_{xxx}+4\mathtt{c}{u}_x?u{u}_{xxx}?3u_x{u}_{xx}+4u{u}_x=0$ with $\mathtt{c}\in \mathbb{R}?\{0\}$. In [15] the authors proved that this equation possesses infinitely many conserved quantities. We prove that there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of the Sobolev space $H^s$ with $s\ge 2$, both on $\mathbb{R}$ and $\mathbb{T}$. By the analysis of these conserved quantities we deduce a result of global well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis–Procesi at any order is action-preserving.     

The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces—the local theory

The Helgason Fourier transform on noncompact Riemannian symmetric spaces G/K is generalized to the homogeneous vector bundles over the compact dual spaces U/K. The scalar theory on U/K was considered by Sherman (the local theory for U/K of arbitrary rank, and the global theory for U/K of rank one). In this paper we extend the local theory of Sherman to arbitrary homogeneous vector bundles on U/K. For U/K of rank one we also obtain a generalization of the Cartan-Helgason theorem valid for any K-type.     

Maximum Principles for Infinite Dimensional Diffusion Equations

In this paper we prove the validity of the Maximum Principle for some class of elliptic and parabolic equations of diffusion type in infinite dimension. The main tools are Asplund’s theorem and Preiss’ theorem on differentiability of Lipschitz functions in Banach space.      

Upward planarity testing

Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and single-source digraphs. We also sketch the proof of NP-completeness of upward planarity testing.Research supported in part by the National Science Foundation under grant CCR-9423847.     

A review on probabilistic graphical models in evolutionary computation

Thanks to their inherent properties, probabilistic graphical models are one of the prime candidates for machine learning and decision making tasks especially in uncertain domains. Their capabilities, like representation, inference and learning, if used effectively, can greatly help to build intelligent systems that are able to act accordingly in different problem domains. Evolutionary algorithms is one such discipline that has employed probabilistic graphical models to improve the search for optimal solutions in complex problems. This paper shows how probabilistic graphical models have been used in evolutionary algorithms to improve their performance in solving complex problems. Specifically, we give a survey of probabilistic model building-based evolutionary algorithms, called estimation of distribution algorithms, and compare different methods for probabilistic modeling in these algorithms.     

Modeling and solving the mixed capacitated general routing problem

We tackle the mixed capacitated general routing problem (MCGRP) which generalizes many other routing problems. We propose an integer programming model for the MCGRP and extend some inequalities originally introduced for the capacitated arc routing problem (CARP). Identification procedures for these inequalities and for some relaxed constraints are also discussed. Finally, we describe a branch and cut algorithm including the identification procedures and present computational experiments over instances derived from the CARP.     

Displacement convexity of entropy and related inequalities on graphs

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prékopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal—by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.     

Convergence of relaxation schemes for conservation laws

We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L^∞ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.