Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

Simultaneous approximation by universal series

We show that, if individual universal series exist, then we can choose a sequence of universal series performing simultaneous universal approximation with the same sequence of indices. As an application we derive the existence of universal Laurent Series on an annulus using only the existence of universal Taylor Series on discs. Our results are generic (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.      

Characterization of Stability for Cone Increasing Constraint Mappings

We investigate stability (in terms of metric regularity) for the specific class of cone increasing constraint mappings. This class is of interest in problems with additional knowledge on some nondecreasing behavior of the constraints (e.g. in chance constraints, where the occurring distribution function of some probability measure is automatically nondecreasing). It is demonstrated, how this extra information may lead to sharper characterizations. In the first part, general cone increasing constraint mappings are studied by exploiting criteria for metric regularity, as recently developed by Mordukhovich. The second part focusses on genericity investigations for global metric regularity (i.e. metric regularity at all feasible points) of nondecreasing constraints in finite dimensions. Applications to chance constraints are given.     

Adaptation of the Generic PDE’s Results to the Notion of Prevalence

Many generic results have been proved, especially concerning the qualitative behaviour of solutions of partial differential equations. Recently, a new notion of “almost always”, the prevalence, has been developed for vectorial spaces. This notion is interesting since, for example, prevalence sets are equivalent to the full Lebesgue measure sets in finite dimensional spaces. The purpose of this article is to adapt the generic PDE’s results to the notion of prevalence. In particular, we consider the cases where Sard–Smale theorems or arguments of analytic perturbations of the parameters are used. Dedicated to Professor Pavol Brunovsky in honor of his 70th birthday.     

On the typical rank of real binary forms

We determine the rank of a general real binary form of degree d?=?4 or d?=?5. In the case d?=?5, the possible values of the rank of such general forms are 3, 4, and 5. This is the first reported case, to our knowledge, where more than two typical ranks have been found. We prove that a real binary form of degree d with d real roots has rank?d.     

Embedding methods for solving variational inequalities

Variational inequality problems (VIP) are an important class of mathematical problems that appear in many practical situations. So, it is important to find efficient and robust numerical solution methods. An appealing idea is to embed the VIP into a one-parametric problem which, then, can be solved numerically by a path-following method. In this article, we study two different types of embeddings and we analyse their generic properties. The non-linear complementarity problem and box-constrained VIP are discussed as special cases.