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.

Patrimoni Digital de Catalunya, a year and a half experience

E-repositories are part of the e-science, and they are based on the e-infrastructure. The Centre de Supercomputació de Catalunya (CESCA) together with the Consorci de Biblioteques Universitàries de Catalunya (CBUC) started in 1999 a cooperative repository, named TDR, to file, in digital format, the full-text of the read thesis at the universities of our country in order to spread them worldwide in open access, while at the same time, preserving the intellectual copyright of the authors. Since then, four additional cooperative repositories have been created: RECERCAT for research papers; RACO for scientific, cultural and erudite Catalan magazines; MDC for Catalan digital collections of pictures, maps, posters and old magazines; and PADICAT for archiving Catalan digital web content; The main objective of the latter is to archive Catalan web sites. That is, PADICAT collects, processes and provides permanent access to the entire cultural, scientific and general output of Catalonia in digital format. The repository manager is the Biblioteca de Catalunya, as the institution responsible for compiling, processing and distributing the bibliographic heritage of Catalonia, while CESCA is the technology partner. On September 11th, 2006 the repository went into operation for the general public, with some thirty websites archived. After one year and a half, it has 2.720 captures of more than 1.000 websites. This includes 34 million files (HTML, images...) and two terabytes of data. The objective of this paper is to present PADICAT and our experience developing and managing it.We describe the repository briefly, we explain the technology used to implement it and we comment our experiences during its first year and a half.

Formal deformations of Poisson structures in low dimensions

In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson cohomology space, we solve the deformation equations at each step and obtain a large family of formal deformations for each Poisson structure which we consider. With the help of an explicit formula, we show that this family contains, modulo equivalence, all possible formal eformations. We show moreover that, when the Poisson structure is generic, all members of the family are non-equivalent.

Individual versus group strategy-proofness: when do they coincide?

A social choice function is group strategy-proof on a domain if no group of agents can manipulate its final outcome to their own benefit by declaring false preferences on that domain. Group strategy-proofness is a very attractive requirement of incentive compatibility. But in many cases it is hard or impossible to find nontrivial social choice functions satisfying even the weakest condition of individual strategy-proofness. However, there are a number of economically significant domains where interesting rules satisfying individual strategy-proofness can be defined, and for some of them, all these rules turn out to also satisfy the stronger requirement of group strategy-proofness. This is the case, for example, when preferences are single-peaked or single-dipped. In other cases, this equivalence does not hold. We provide sufficient conditions defining domains of preferences guaranteeing that individual and group strategy-proofness are equivalent for all rules defined on the

A Thomason model structure on the category of small n-fold categories

We construct a cofibrantly generated Thomason model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.

On the trace of an endofunctor of a small category

The trace of a square matrix can be defined by a universal property which, appropriately generalized yields the concept of "trace of an endofunctor of a small category". We review the basic definitions of this general concept and give a new construction, the "pretrace category", which allows us to obtain the trace of an endofunctor of a small category as the set of connected components of its pretrace. We show that this pretrace construction determines a finite-product preserving endofunctor of the category of small categories, and we deduce from this that the trace inherits any finite-product algebraic structure that the original category may have. We apply our results to several examples from Representation Theory obtaining a new (indirect) proof of the fact that two finite dimensional linear representations of a finite group are isomorphic if and only if they have the same character.

The simultaneous conjugacy problem in groups of piecewise linear functions

Guba and Sapir asked, in their joint paper [8], if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit. The techniques we develop allow us also to solve the ordinary conjugacy problem as well, and we can compute roots and centralizers. Moreover, these techniques can be generalized to solve the same questions in larger groups of piecewise-linear homeomorphisms.

Condensation of polyhedric structures onto soap films

We study the existence of solutions to general measure-minimization problems over topological classes that are stable under localized Lipschitz homotopy, including the standard Plateau problem without the need for restrictive assumptions such as orientability or even rectifiability of surfaces. In case of problems over an open and bounded domain we establish the existence of a “minimal candidate”, obtained as the limit for the local Hausdorff convergence of a minimizing sequence for which the measure is lower-semicontinuous. Although we do not give a way to control the topological constraint when taking limit yet— except for some examples of topological classes preserving local separation or for periodic two-dimensional sets — we prove that this candidate is an Almgren-minimal set. Thus, using regularity results such as Jean Taylor’s theorem, this could be a way to find solutions to the above minimization problems under a generic setup in arbitrary dimension and codimension.

Pequín, London, Roma: un viatge lingüístic a través de la història de tres llengües: gramàtica comparada entre el xinès, l'anglès i el llatí

Treball de recerca realitzat per un alumne d'ensenyament secundari i guardonat amb un Premi CIRIT per fomentar l'esperit científic del Jovent l'any 2009. En aquest treball es vol trobar un vincle entre les cultures més influents al llarg de la història d’Àsia i Europa, prenent com a element bàsic les seves llengües: el xinès, l’anglès i el llatí. Ara bé, la comparació lingüística entre elles només té veritable interès si es cerca allò comú sobretot semàntica i sintàcticament, posat que d'entrada, visual i fonèticament, qualsevol profà pot observar-ne prou diferències. Aquestes característiques comunes, doncs, tenen valor independentment del temps i l'espai, ja que el que hi ha en joc aquí és, en el fons, el llenguatge verbal com a nervi del fet de ser humà. Els continguts s'estructuren mitjançant l'ús d'un mètode inductiu com a regla d’anàlisi comparativa; no es pot, tanmateix, prescindir d'una introducció teòrica de lingüística, absolutament necessària per a un estudi comparatiu de gramàtica. Una vegada conclòs el treball, es veu que els conceptes que totes tres expressen són comuns i el canvi, quan es dóna, afecta la forma i està relacionat amb les regles de la lògica: aquests casos, doncs, no fan sinó confirmar la tesi, ja que esperar una equivalència total seria no només ingenu, sinó gens natural ni realista. L'anhel del traductor és arribar a expressar, precisament, aquest univers.

Casolà o industrial?: estudi organolèptic i microbiològic de la vida útil del pernil cuit

Treball de recerca realitzat per un alumne d'ensenyament secundari i guardonat amb un Premi CIRIT per fomentar l'esperit científic del Jovent l'any 2009. Aquest treball tracta sobre el temps de garantia o de vida útil d’un producte alimentari i està basat en esbrinar quin és el millor mètode o envàs per conservar el pernil cuit llescat durant la seva vida útil, per tal que es mantinguin les propietats organolèptiques i microbiològiques inicials de l’aliment durant el màxim de temps possible. Per comparar envasos he utilitzat els industrials (envàs al buit i envàs en atmosfera) i els envasos que podem trobar quan comprem un producte llescat en el mostrador d’una botiga o supermercat. Per a fer l’estudi doncs s’ha utilitzat el pernil cuit, conegut com a pernil dolç, un dels productes més consumits per la societat actual. El pernil ha estat tractat en tres situacions diferents: llescat en una màquina de carnisseria i envasat en mètodes de conservació domèstics, llescat en màquines industrials i envasat en diferents envasos (al buit i en atmosfera modificada) utilitzats per les grans empreses i llescat en màquines industrials i envasat al buit sotmès a un tractament amb altes pressions. Després d’aquests processos s’ha realitzat un estudi de vida útil, és a dir, s’ha analitzat el producte al laboratori, amb l’objectiu de trobar quin és el millor envàs per a la conservació del pernil cuit.

Complex contact manifolds and S1 actions

We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun and Salamon on Fano contact manifolds but under a symmetry assumption instead of a curvature condition.

Anàlisi i resolució de puzzles

Aquest projecte és una anàlisi d'algunes tècniques de reconeixement d'objectes en 2D, la seva comparació i equivalència. La solució presentada en aquesta memòria ha estat implementada en C++ per a poder avaluar els resultats així com les diferents problemàtiques aparegudes durant tot el procés. La resolució de puzzles és tan sols un exemple dels múltiples casos d'objectes bidimensionals que ens podem trobar que necessiten tècniques de matching per obtenir un objecte global.

Generalized descriptive set theory and classification theory

Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.

Teaching foreign languages through translation: considering multiple intelligences

Aquest treball de recerca tracta de la relació existent entre pedagogia, traducció, llengües estrangeres i intel•ligències múltiples. El debat sobre si la traducció és una eina útil a la classe de llengües estrangeres és un tema actual sobre el qual molts investigadors encara indaguen. Estudis recents, però, han demostrat que qualsevol tasca de traducció -en la qual s’hi poden incloure treballs amb les diferents habilitats- és profitosa si la considerem un mitjà, no una finalitat en ella mateixa. Evidentment, l’ús de la traducció dins l’aula és avantatjosa, però també hem de tenir presents certs desavantatges d’aquesta aplicació. Un possible desavantatge podria ser la creença que, al principi, molta gent té referent a l’equivalència, paraula per paraula, d’una llengua vers una altra. Però després de presentar vàries tasques de traducció als estudiants, aquests poden arribar a controlar, fins i tot, les traduccions inconscients i poden assolir un cert nivell de precisió i flexibilitat que val la pena mencionar. Però l’avantatge principal és que s’enfronten a una activitat molt estesa dins la societat actual que combina dues llengües, la llengua materna i la llengua objecte d’estudi, per exemple. De tot això en podem deduir que utilitzar la llengua materna a la classe no s’ha de considerar un crim, com fins ara, sinó una virtut, evidentment si és emprada correctament. En aquest treball de recerca s’hi pot trobar una síntesi tant de les principals teories d’adquisició i aprenentatge de llengües com de les teories de traducció. A la pregunta de si les teories, tant de traducció com de llengües estrangeres, s’haurien d’ensenyar implícita o explícitament, es pot inferir que segons el nivell d’estudis on estiguin els aprenents els convindrà aprendre les teories explícitament o les aprendran, de totes maneres, implícitament. Com que qualsevol grup d’estudiants és heterogeni -és a dir que cada individu té un ritme i un nivell d’aprenentatge concret i sobretot cadascú té diferents estils de percepció (visual, auditiu, gustatiu, olfactiu, de moviment) i per tant diferents intel•ligències-, els professors ho han de tenir en compte a l’hora de planificar qualsevol programa d’actuació vers els alumnes. Per tant, podem concloure que les tasques o projectes de traducció poden ajudar als alumnes a aprendre millor, més eficaçment i a aconseguir un aprenentatge més significatiu.

Two necessary conditions for strategy-proofness: on what domains are they also sufficient?

A social choice function may or may not satisfy a desirable property depending on its domain of definition. For the same reason, different conditions may be equivalent for functions defined on some domains, while different in other cases. Understanding the role of domains is therefore a crucial issue in mechanism design. We illustrate this point by analyzing the role of different conditions that are always related, but not always equivalent to strategy-proofness. We define two very natural conditions that are necessary for strategy-proofness: monotonicity and reshuffling invariance. We remark that they are not always sufficient. Then, we identify a domain condition, called intertwinedness, that ensures the equivalence between our two conditions and that of strategy-proofness. We prove that some important domains are intertwined: those of single-peaked preferences, both with public and private goods, and also those arising in simple models of house allocation. We prove that other necessary conditions for strategy-proofness also become equivalent to ours when applied to functions defined on intertwined domains, even if they are not equivalent in general. We also study the relationship between our domain restrictions and others that appear in the literature, proving that we are indeed introducing a novel proposal.