On the exactness of a class of nondifferentiable penalty functions

In this paper, we consider a class of nondifferentiable penalty functions for the solution of nonlinear programming problems without convexity assumptions. Preliminarily, we introduce a notion of exactness which appears to be of relevance in connection with the solution of the constrained problem by means of unconstrained minimization methods. Then, we show that the class of penalty functions considered is exact, according to this notion. This research was partially supported by the National Research Program on “Modelli e Algoritmi per l'Ottimizzazione,” Ministero della Pubblica, Istruzione, Roma, Italy.     

Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the Colombeau simplified model. This generalizes the notion of G^∞-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.     

Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T  : X⇉X ^* relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.     

Topological Complexity of the Relative Closure of a Semi-Pfaffian Couple

Gabrielov introduced the notion of relative closure of a Pfaffian couple as an alternative construction of the o-minimal structure generated by Khovanskii’s Pfaffian functions. In this paper, we use the notion of format (or complexity) of a Pfaffian couple to derive explicit upper bounds for the homology of its relative closure. We consider both the singular and the Borel–Moore homology theories.     

Subcommuting and commuting real homographic functions

If the difference of two real homographic functions is nonnegative, then it is constant. Motivated by this property, we determine all pairs of subcommuting (supercommuting) real homographic functions. Simple modification of subcommuting functions transforms them into commuting ones. Introduced here notion of a generalized iteration group of homographic functions is illustrated by a suitable example.     

Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis

The notion of non-uniform Robust Global Asymptotic Stability(RGAS) presented in this paper generalizes the notion of non-uniformin time RGAS for finite- or infinite-dimensional discrete-timesystems. Lyapunov characterizations for this stability notionare provided. The results are applied to finite-dimensionaldiscrete-time systems obtained by time discretization of continuous-timesystems by the explicit Euler method.     

Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps

We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps is introduced; it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar.     

On roughly transitive amenable graphs and harmonic Dirichlet functions

We introduce the notion of rough transitivity and prove that there exist no non-constant harmonic Dirichlet functions on amenable roughly transitive graphs.

     

Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients

We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmander's definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.     

Global analytic geometry

Motivated by the well-known lack of archimedean information in algebraic geometry, we define, formalizing Ostrowski's classification of seminorms on Z, a new type of valuation of a ring that combines the notion of Krull valuation with that of a multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations artificially introduced in arithmetic geometry by the theory of schemes. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic spaces. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.     

Exceptional crooked functions

Crooked functions are combinatorial objects of great interest. It is already known that the only monomial and binomial crooked functions are quadratic. In this paper, we investigate conditions on the shape of a polynomial to be crooked. Furthermore, the notion of exceptional crooked is introduced, similarly to those of APN or PN exceptional functions. Via a connection with algebraic varieties over finite fields, we provide non-existence results of exceptional crooked functions.     

A new functional calculus for noncommuting operators

In this paper we use the notion of slice monogenic functions [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math., in press] to define a new functional calculus for an n-tuple T of not necessarily commuting operators. This calculus is different from the one discussed in [B. Jefferies, Spectral Properties of Noncommuting Operators, Lecture Notes in Math., vol. 1843, Springer-Verlag, Berlin, 2004] and it allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.     

Strongly internal sets and generalized smooth functions

Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal extension of Colombeau generalized functions. Generalized smooth functions as morphisms between sets of generalized points form a sub-category of the category of topological spaces. In particular, they can be composed unrestrictedly.     

Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds

In this paper, a notion of generalized gradient on Riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented. A characterization of the tangent cone to a nonempty subset S of a Riemannian manifold M at a point x is obtained. Then, these results are applied to characterize epi-Lipschitz subsets of complete Riemannian manifolds.     

BV solutions of a singular diffusion equation

In this paper, we study discontinuous solutions of a partial differential equation of strongly degenerate parabolic type. A notion of weak solutions of BV class is proposed, and existence and uniqueness results are obtained.     

Level Function Method for Quasiconvex Programming

In this paper, we introduce the notion of level function for a continuous real-valued quasiconvex function. The existence, construction, and application of level functions are discussed. Further, we propose a numerical method based on level functions for the solution of quasiconvex minimization problems. Several versions of the algorithms are presented. Also, we apply the idea of the level function method to the solution of a class of variational inequality problems. Finally, the results of numerical experiments on the proposed algorithms are reported.     

On a Theorem of Deutsch and Singer

We introduce the notion of a starshaped set-valued function, and relate it to convex set-valued functions. We prove a theorem which, in the convex case, implies that single-valuedness is exceptional if the function is not everywhere single-valued. A second result shows a remarkable costantness of images for starshaped set-valued functions. Both theorems were inspired from a result of Deutsch and Singer, which they strengthen.     

Irregularities of semilinear hyperbolic equations

We consider local solvability of semilinear hyperbolic Cauchy problems for Gevrey functions. To obtain a general result, we define the notion of irregularities, and we give a criterion for the local solvability. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized domination in closure systems

In the context of extracting maximal item sets and association rules from a binary data base, the graph-theoretic notion of domination was recently used to characterize the neighborhood of a concept in the corresponding lattice.In this paper, we show that the notion of domination can in fact be extended to any closure operator on a finite universe and be efficiently encoded into propositional Horn functions. This generalization enables us to endow notions and algorithms related to Formal Concept Analysis with Horn minimization and minimal covers of functional dependencies in Relational Databases.     

Stable generalization of convex functions

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.