Elements of algebraic geometry and the positive theory of partially commutative groups

The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Inverse problems for invariant algebraic curves: explicit computations

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are nondegenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

A general construction of internal sheaves in algebraic set theory

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothen-dieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

Rational first integrals in the Darboux theory of integrability in Cn

"Vegeu el resum a l'inici del document del fitxer adjunt."

Weighted limits in simplicial homotopy theory

We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor.

Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

The Macroeconomics of the labor market: three fundamental views

We distinguish and assess three fundamental views of the labor market regarding the movements in unempoyment: (i) the frictionless equilibrium view; (ii) the chain reaction theory, or prolonged adjustment view; and (iii) the hysteresis view. While the frictionless view implies a clear compartmentalization between the short- and long-run, the hysteresis view implies that all the short-run fluctuations automatically turn into long-run changes in the unemployment rate. We assert the problems faced by these conceptions in explaining the diversity of labor market experiences across the OECD labor markets. We argue that the prolonged adjustment view can overcome these problems since it implies that the short, medium, and long runs are interrelated, merging with one another along an intertemporal continuum.

On the homotopy theory of spectral categories

"Vegeu el resum a l'inici del document del fitxer adjunt."

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

Migration and trade: Theory with an Application to the Eastern- Western European Integration

The remarkable increase in trade flows and in migratory flows of highly educated people are two important features of globalization of the last decades. This paper extends a two-country model of inter- and intraindustry trade to a rich environment featuring technological differences, skill differences and the possibility of international labor mobility. The model is used to explain the patterns of trade and migration as countries remove barriers to trade and to labor mobility. We parameterize the model to match the features of the Western and Eastern European members of the EU and analyze first the effects of the trade liberalization which occured between 1989 and 2004, and then the gains and losses from migration which are expected to occur if legal barriers to labor mobility are substantially reduced. The lower barriers to migration would result in significant migration of skilled workers from Eastern European countries. Interestingly, this would not only benefit the migrants and most Western European workers but, via trade, it would also benefit the workers remaining in Eastern Europe. Key Words: Skilled Migration, Gains from Variety, Real Wages, Eastern-Western Europe. JEL Codes: F12, F22, J61.

The Commons and anti-commons problems in the tourism economy

Countries specialised in tourism tend to face two problems with contradictory effects: the commons and the anti-commons, which lead to tourism over- and under-production, respectively. This paper develops a two-period model to analyse the joint effects of both problems on a small and remote tourism economy. Congestion and the complementariness between foreign transport and local tourism services are key features in this type of markets. As a result, direct selling and the presence of foreign tour-operators emerge as possible market arrangements with different implications in terms of welfare and public intervention. Four main results are obtained. First, in the direct selling situation the optimal policy depends on the relative importance of the problems. Second, the existence of tour-operators always leads to tourism over-production. Third, the presence of a single tour-operator does not solve the congestion problem. Lastly, the switch from several tour-operators to a single one is welfare reducing.

Some variational problems from image processing

"Vegeu el resum a l'inici del document del fitxer adjunt."

A well-posedness theory in measures for some kinetic models of collective motion

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.

Lp theory for the multidimensional aggregation equation

"Vegeu el resum a l'inici del document del fitxer adjunt."

The analysis of approximate quickselect and related problems

Approximate Quickselect, a simple modification of the well known Quickselect algorithm for selection, can be used to efficiently find an element with rank k in a given range [i..j], out of n given elements. We study basic cost measures of Approximate Quickselect by computing exact and asymptotic results for the expected number of passes, comparisons and data moves during the execution of this algorithm. The key element appearing in the analysis of Approximate Quickselect is a trivariate recurrence that we solve in full generality. The general solution of the recurrence proves to be very useful, as it allows us to tackle several related problems, besides the analysis that originally motivated us. In particular, we have been able to carry out a precise analysis of the expected number of moves of the ith element when selecting the jth smallest element with standard Quickselect, where we are able to give both exact and asymptotic results. Moreover, we can apply our general results to obtain exact and asymptotic results for several parameters in binary search trees, namely the expected number of common ancestors of the nodes with rank i and j, the expected size of the subtree rooted at the least common ancestor of the nodes with rank i and j, and the expected distance between the nodes of ranks i and j.