Some remarks on the class of continuous (Semi-) strictcly quasiconvex functions

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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.

Entradas y salidas de empresas: un contraste de las hipótesis de independencia, simetría y simultaneidad

En este trabajo se analiza una vía alternativa en el debate de la convergencia regional en España: el estudio de la dinámica industrial de sus Comunidades Autónomas. Para ello se analizan los flujos de entrada y salida de establecimientos industriales en las manufacturas delas regiones españolas y se abordan los factores de carácter sectorial o regional que inciden sobre la movilidad industrial. Las specificaciones econométricas adoptan la estructura de un sistema de ecuaciones y cubren las tres hipótesis fundamentales que se han abordado en laliteratura (independencia, simetría y simultaneidad) a partir de un panel de datos construido con informaciones procedentes del Registro de Establecimientos Industriales y la Encuesta Industrial. Palabras clave: demografía industrial, sistemas de ecuaciones, datos de panel

Firm Size and Geographical Aggregation: An Empirical Appraisal in Industrial Location

This paper assesses empirically the importance of size discrimination and disaggregate data for deciding where to locate a start-up concern. We compare three econometric specifications using Catalan data: a multinomial logit with 4 and 41 alternatives (provinces and comarques, respectively) in which firm size is the main covariate; a conditional logit with 4 and 41 alternatives including attributes of the sites as well as size-site interactions; and a Poisson model on the comarques and the full spatial choice set (942 municipalities) with site-specific variables. Our results suggest that if these two issues are ignored, conclusions may be misleading. We provide evidence that large and small firms behave differently and conclude that Catalan firms tend to choose between comarques rather than between municipalities. Moreover, labour-intensive firms seem more likely to be located in the city of Barcelona. Keywords: Catalonia, industrial location, multinomial response model. JEL: C250, E30, R00, R12

A territorial approach to firm entry: The effect of city size

This paper concerns the effects of territorial factors on the processes involved in the creation of manufacturing firms in Spanish cities. Most contributions have focused on regional factors rather than urban ones. Here we assume that it is possible to identify certain urban factors that attract new firms. We use data for the entry of firms in Spanish manufacturing industries between 1994 and 2002. This paper contributes to the existing literature on market entry. Key words: cities, regions, firm entry and Spanish economy. JEL: R0, R12, L60

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.

Educació ambiental en camins tradicionals : una proposta de guia interpretativa pel camí de les Tosques

El primer bloc del treball, que ocupa en extensió la major part del projecte, permet una aproximació a la figura que juga el medi ambient i l’educació ambiental en la societat i en el moment actual. Al mateix temps, s’analitza el context ambiental que envolta el Parc Natural de les Muntanyes de Prades per tal de donar a conèixer l’entorn immediat del camí de les Tosques. Finalment, es descriu l’origen i els diversos camins tradicionals existents, procurant la informació necessària per entendre el valor i el potencial de que disposa aquest viari. El segon bloc, d’altra banda, resulta d’una selecció d’informació del primer i de noves aportacions, en una guia per la realització d’un itinerari autoguiat d’educació ambiental per aquest camí. Aquesta proposta es materialitza en la localització, anàlisi i interpretació de l’itinerari pel camí de les Tosques, i en l’elaboració d’una guia d’educació ambiental per a realitzar-lo.

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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