Clarke critical values of subanalytic Lipschitz continuous functions

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawlucki's extension of the Puiseuxlemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.

The Zariski-Lefschetz principle for higher homotopy groups of nongeneric pencils

We prove a general Zariski-van Kampen-Lefschetz type theorem for higher homotopy groups of generic and nongeneric pencils on singular open complex spaces.

The period function of the generalized Lotka-Volterra centers

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Consenso, constitución y partidos políticos: una reflexión crítica 26 años después

Vint-i-sis desprès de la promulgació de la Constitució espanyola, l’autor fa balanç i arriba a un seguit de conclusions. Entre d’altres, que Espanya viu una etapa de clara partitocràcia que, juntament amb el poder desmesurat dels sindicats, lesiona la vida democràtica; que cal reduir la dualitat suposadament legitimadora entre el Parlament i el carrer; i que malgrat les lloances rebudes, la Constitució de 1976 no ha aconseguit una de les funcions essencials de tota constitució: la tasca integradora.

Increasing quasiconcave production and utility functions with diminishing returns to scale

In microeconomic analysis functions with diminishing returns to scale (DRS) have frequently been employed. Various properties of increasing quasiconcave aggregator functions with DRS are derived. Furthermore duality in the classical sense as well as of a new type is studied for such aggregator functions in production and consumer theory. In particular representation theorems for direct and indirect aggregator functions are obtained. These involve only small sets of generator functions. The study is carried out in the contemporary framework of abstract convexity and abstract concavity.

Weak systems of Gandy, Jensen and Devlin

In Part I, we formulate and examine some systems that have arisen in the study of the constructible hierarchy; we find numerous transitive models for them, among which are supertransitive models containing all ordinals that show that Devlin's system BS lies strictly between Gandy's systems PZ and BST'; and we use our models to show that BS fails to handle even the simplest rudimentary functions, and is thus inadequate for the use intended for it in Devlin's treatise. In Part II we propose and study an enhancement of the underlying logic of these systems, build further models to show where the previous hierarchy of systems is preserved by our enhancement; and consider three systems that might serve for Devlin's purposes: one the enhancement of a version of BS, one a formulation of Gandy-Jensen set theory, and the third a subsystem common to those two. In Part III we give new proofs of results of Boffa by constructing three models in which, respectively, TCo, AxPair and AxSing fail; we give some sufficient conditions for a set not to belong to the rudimentary closure of another set, and thus answer a question of McAloon; and we comment on Gandy's numerals and correct and sharpen other of his observations.

On moduli of smoothness of fractional order

In this paper we consider the properties of moduli of smoothness of fractional order. The main result of the paper describes the equivalence of the modulus of smoothness and a function from some class.

On Boas-Type Problem

R.P. Boas has found necessary and sufficient conditions of belonging of function to Lipschitz class. From his findings it turned out, that the conditions on sine and cosine coefficients for belonging of function to Lip α(0 & α & 1) are the same, but for Lip 1 are different. Later his results were generalized by many authors in the viewpoint of generalization of condition on the majorant of modulus of continuity. The aim of this paper is to obtain Boas-type theorems for generalized Lipschitz classes. To define generalized Lipschitz classes we use the concept of modulus of smoothness of fractional order.

The third order Melnikov function of a quadratic center under quadratic perturbation

We study quadratic perturbations of the integrable system (1+x)dH; where H =(x²+y²)=2: We prove that the first three Melnikov functions associated to the perturbed system give rise at most to three limit cycles.

The mean dehn functions of Abelian groups

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Lipschitz harmonic capacity and Bilipschitz images of cantor sets

For bilipschitz images of Cantor sets in Rd we estimate the Lipschitz harmonic capacity and show this capacity is invariant under bilipschitz homeomorphisms.

Nous perfils professionals de l'actual panorama informatiu audiovisual i multimèdia de Catalunya

Aquest treball forma part de una recerca –finançada pel Consell de l’Audiovisual de Catalunya- dedicada a l’anàlisi de les conseqüències dels processos de digitalització en els mitjans de comunicació de Catalunya (Espanya). El document presenta les transformacions que han afectat al periodista, especialment les noves tasques i funcions. Després d’una introducció dedicada al marc teòric i a la metodologia de la recerca, el document explica les tendències principals observades en les ràdios, emissores de televisió, agències de noticies i mitjans en línia de Catalunya. El document també presenta una sèrie de reflexions sobre la polivalència, la re-qualificació i altres processos que afecten als professionals de la informació. Finalment, es proposa una classificació de les formes que assumeix la polivalència en els mitjans.

CM points and weight 3=2 modular forms

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Stability of periodic solutions obtained via the averaging method for nonsmooth Lipschitz system

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Weighted Hardy spaces for the unit disc: approximation properties

In this paper we study basic properties of the weighted Hardy space for the unit disc with the weight function satisfying Muckenhoupt's (Aq) condition, and study related approximation problems (expansion, moment and interpolation) with respect to two incomplete systems of holomorphic functions in this space.