The main goal of this thesis is to characterize the weak-type (resp. strong-type) boundedness of the Hilbert transform H on weighted Lorentz spaces Λpu(w). The characterization is given in terms of some geometric conditions on the weights u and w and the weak-type (resp. strong-type) boundedness of the Hardy-Littlewood maximal operator on the same spaces. Our results extend and unify simultaneously the theory of the boundedness of H on weighted Lebesgue spaces Lp(u) and Muckenhoupt weights Ap, and the theory on classical Lorentz spaces Λp(w) and Ariño-Muckenhoupt weights Bp. / Títol: Acotaciò de l'operador de Hilbert sobre espais de Lorentz amb pesos Resum: L'objectiu principal d'aquesta tesi es caracteritzar l'acotació de l'operador de Hilbert sobre els espais de Lorentz amb pesos Λpu(w). També estudiem la versió dèbil. La caracterització es dona en terminis de condicions geomètriques sobre els pesos u i w, i l'acotació de l'operador maximal de Hardy-Littlewood sobre els mateixos espais. Els nostres resultats unifiquen dues teories conegudes i aparentment no relacionades entre elles, que tracten l'acotació de l'operador de Hilbert sobre els espais de Lebegue amb pesos Lp(u) per una banda i els espais de Lorentz clàssics Λp(w) per altre banda.

Aquesta tesi presenta una contribució a l'estudi de les estructures algebraïques dels sistemes lògics deductius

En aquesta tesi doctoral s'obtenen i estudien les àlgebres d-completes com les àlgebres implicatives associades a uns determinats càlculs proposicionals implicatius, que satisfan un teorema de la deducció feble i contenen als càlculs proposicionals multi-valorats donats per Lukasiewicz. Per altra banda, també s'estudien els sistemes deductius d'aquestes àlgebres.

En aquesta tesi s'estudien les tipologies de l'ordre i els productes d'espais mètrics generalitzats aplicant els resultats al cas del producte numerable d'espais mètrics probabilístics (Sigma i Tau Productes) de Menger, Wald i Serstnev. Aquest apartat ens du a analitzar el comportament de les funcions triangulars "T" amb les sèries de funcions de distribució. Es construeixen immersions en espais seqüencialment convexos que són subespais de productes numerables i es dona un procés de completació que generalitza els processos usuals. / Se estudian las topologías del orden y los productos de espacios métricos generalizados aplicando los resultados al caso del producto numerable de espacios métricos probabilísticos (Sigma y Tau Productos) de Menger, Wald y Serstnev. Dicho apartado lleva a analizar el comportamiento de las funciones triangulares T con las series de funciones de distribución. Se construyen inmersiones en espacios secuencialmente convexos que son subespacios de productos numerables y se da un proceso de completación que generaliza los procesos usuales.

According to Zadeh, the term “fuzzy logic” has two different meanings: wide and narrow. In a narrow sense it is a logical system which aims a formalization of approximate reasoning, and so it can be considered an extension of many-valued logic. However, Zadeh also says that the agenda of fuzzy logic is quite different from that of traditional many-valued logic, as it addresses concepts like linguistic variable, fuzzy if-then rule, linguistic quantifiers etc. Hájek, in the preface of his foundational book Metamathematics of Fuzzy Logic, agrees with Zadeh’s distinction, but stressing that formal calculi of many-valued logics are the kernel of the so-called Basic Fuzzy logic (BL), having continuous triangular norms (t-norm) and their residua as semantics for the conjunction and implication respectively, and of its most prominent extensions, namely Lukasiewicz, Gödel and Product fuzzy logics. Taking advantage of the fact that a t-norm has residuum if, and only if, it is left-continuous, the logic of the left-continuous t-norms, called MTL, was soon after introduced. On the other hand, classical modal logic is an active field of mathematical logic, originally introduced at the beginning of the XXth century for philosophical purposes, that more recently has shown to be very successful in many other areas, specially in computer science. That are the most well-known semantics for classical modal logics. Modal expansions of non-classical logics, in particular of many-valued logics, have also been studied in the literature. In this thesis we focus on the study of some modal logics over MTL, using natural generalizations of the classical Kripke relational structures where propositions at possible words can be many-valued, but keeping classical accessibility relations. In more detail, the main goal of this thesis has been to study modal expansions of the logic of a left-continuous t-norm, defined over the language of MTL expanded with rational truth-constants and the Monteiro-Baaz Delta-operator, whose intended (standard) semantics is given by Kripke models with crisp accessibility relations and taking the unit real interval [0, 1] as set of truth-values. To get complete axiomatizations, already known techniques based on the canonical model construction are uses, but this requires to ensure that the underlying (propositional) fuzzy logic is strongly standard complete. This constraint leads us to consider axiomatic systems with infinitary inference rules, already at the propositional level. A second goal of the thesis has been to also develop and automated reasoning software tool to solve satisfiability and logical consequence problems for some of the fuzzy logic modal logics considered. This dissertation is structured in four parts. After a gentle introduction, Part I contains the needed preliminaries for the thesis be as self-contained as possible. Most of the theoretical results are developed in Parts II and III. Part II focuses on solving some problems concerning the strong standard completeness of underlying non-modal expansions. We first present and axiomatic system for the non-nodal propositional logic of a left-continuous t-norm who makes use of a unique infinitary inference rule, the “density rule”, that solves several problems pointed out in the literature. We further expand this axiomatic system in order to also characterize arbitrary operations over [0, 1] satisfying certain regularity conditions. However, since this axiomatic system turn out to be not well-behaved for the modal expansion, we search for alternative axiomatizations with some particular kind of inference rules (that will be called conjunctive). Unfortunately, this kind of axiomatization does not necessarily exist for all left-continuous t-norms (in particular, it does not exist for the Gödel logic case), but we identify a wide class of t-norms for which it works. This “well-behaved” t-norms include all ordinal sums of Lukasiewiczand Product t-norms. Part III focuses on the modal expansion of the logics presented before. We propose axiomatic systems (which are, as expected, modal expansions of the ones given in the previous part) respectively strongly complete with respect to local and global Kripke semantics defined over frames with crisp accessibility relations and worlds evaluated over a “well-behaved” left-continuous t-norm. We also study some properties and extensions of these logics and also show how to use it for axiomatizing the possibilistic logic over the very same t-norm. Later on, we characterize the algebraic companion of these modal logics, provide some algebraic completeness results and study the relation between their Kripke and algebraic semantics. Finally, Part IV of the thesis is devoted to a software application, mNiB-LoS, who uses Satisfability Modulo Theories in order to build an automated reasoning system to reason over modal logics evaluated over BL algebras. The acronym of this applications stands for a modal Nice BL-logics Solver. The use of BL logics along this part is motivated by the fact that continuous t-norms can be represented as ordinal sums of three particular t-norms: Gödel, Lukasiewicz and Product ones. It is then possible to show that these t-norms have alternative characterizations that, although equivalent from the point of view of the logic, have strong differences for what concerns the design, implementation and efficiency of the application. For practical reasons, the modal structures included in the solver are limited to the finite ones (with no bound on the cardinality).

Deep learning has recently been enjoying an increasing popularity due to its success in solving challenging tasks. In particular, deep learning has proven to be effective in a large variety of computer vision tasks, such as image classification, object recognition and image parsing. Contrary to previous research, which required engineered feature representations, designed by experts, in order to succeed, deep learning attempts to learn representation hierarchies automatically from data. More recently, the trend has been to go deeper with representation hierarchies. Learning (very) deep representation hierarchies is a challenging task, which involves the optimization of highly non- convex functions. Therefore, the search for algorithms to ease the learning of (very) deep representation hierarchies from data is extensive and ongoing. In this thesis, we tackle the challenging problem of easing the learning of (very) deep representation hierarchies. We present a hyper-parameter free, off-the-shelf, simple and fast unsupervised algorithm to discover hidden structure from the input data by enforcing a very strong form of sparsity. We study the applicability and potential of the algorithm to learn representations of varying depth in a handful of applications and domains, highlighting the ability of the algorithm to provide discriminative feature representations that are able to achieve top performance. Yet, while emphasizing the great value of unsupervised learning methods when labeled data is scarce, the recent industrial success of deep learning has revolved around supervised learning. Supervised learning is currently the focus of many recent research advances, which have shown to excel at many computer vision tasks. Top performing systems often involve very large and deep models, which are not well suited for applications with time or memory limitations. More in line with the current trends, we engage in making top performing models more efficient, by designing very deep and thin models. Since training such very deep models still appears to be a challenging task, we introduce a novel algorithm that guides the training of very thin and deep models by hinting their intermediate representations. Very deep and thin models trained by the proposed algorithm end up extracting feature representations that are comparable or even better performing than the ones extracted by large state-of-the-art models, while compellingly reducing the time and memory consumption of the model.

Tesi realitzada a l'IMIM ( Institut Hospital del Mar d'Investigacions Mèdiques) / This work aims to contribute methodologically in the epidemiology of drug use, particularly estimation of incidence. No incidence figures of drug use in Spain had ever been published, prior to those appearing in these articles, and relatively little has been published for other countries. Since around 2000, the European Monitoring Centre for Drugs and Drug Addiction (EMCDDA), which is an agency of the European Union, has been making a concerted effort to promote the determination and publication of drug use incidence figures, given their great importance in designing prevention policies. The approaches used and results obtained by our research have been presented in three EMCDDA meetings (years 2007, 2008 and 2012), at a monographic meeting on incidence promoted by the Norwegian Institute for Alcohol and Drug Research (SIRUS) in 2009, and in the framework of a European project on new methodological tools for policy and programme evaluation (JUST/2010/DPIP/AG/1410) which ran from 2010 to 2012. This work therefore contributes not only by presenting drug use incidence results for Spain, but also by describing the development of methods and sharing ideas that may be adapted for use in other countries.

In this thesis, several methods for the automatic analysis of Intravascular Ultrasound (IVUS) sequences are presented, aimed at assisting physicians in the diagnosis, the assessment of the intervention and the monitoring of the patients with coronary disease. The basis for the developed frameworks are machine learning, pattern recognition and image processing techniques. First, a novel approach for the automatic detection of vascular bifurcations in IVUS is presented. The task is addressed as a binary classification problem (identifying bifurcation and non-bifurcation angular sectors in the sequence images). The multiscale stacked sequential learning algorithm is applied, to take into account the spatial and temporal context in IVUS sequences, and the results are refined using a-priori information about branching dimensions and geometry. The achieved performance is comparable to intra- and inter-observer variability. Then, we propose a novel method for the automatic non-rigid alignment of IVUS sequences of the same patient, acquired at different moments (before and after percutaneous coronary intervention, or at baseline and follow-up examinations). The method is based on the description of the morphological content of the vessel, obtained by extracting temporal morphological profiles from the IVUS acquisitions, by means of methods for segmentation, characterization and detection in IVUS. A technique for non-rigid sequence alignment - the Dynamic Time Warping algorithm - is applied to the profiles and adapted to the specific clinical problem. Two different robust strategies are proposed to address the partial overlapping between frames of corresponding sequences, and a regularization term is introduced to compensate for possible errors in the profile extraction. The benefits of the proposed strategy are demonstrated by extensive validation on synthetic and in-vivo data. The results show the interest of the proposed non-linear alignment and the clinical value of the method. Finally, a novel automatic approach for the extraction of the luminal border in IVUS images is presented. The method applies the multiscale stacked sequential learning algorithm and extends it to 2-D+T, in a first classification phase (the identification of lumen and non-lumen regions of the images), while an active contour model is used in a second phase, to identify the lumen contour. The method is extended to the longitudinal dimension of the sequences and it is validated on a challenging data-set. / En esta tesis, se presentan métodos para el análisis automático de secuencias de Ultrasonido Intravascular (IVUS), destinados a ayudar a los médicos en el diagnóstico, la evaluación de la intervención y el seguimiento de los pacientes con enfermedad coronaria. La base para los métodos desarrollados son técnicas de aprendizaje automático, reconocimiento de patrones y procesamiento de imagen. En primer lugar, se presenta un nuevo método para la detección automática de bifurcaciones vasculares en IVUS. La tarea se aborda como un problema de clasificación binaria (identificando los sectores angulares de bifurcación y de no-bifurcación en las imágenes de la secuencia). Se aplica el algoritmo de multiscale stacked sequential learning, para tener en cuenta el contexto espacial y temporal de las secuencias, y los resultados se refinan utilizando información a-priori acerca de las dimensiones de las ramificaciones y su geometría. El rendimiento obtenido es comparable a la variabilidad intra- e inter-observador. A continuación, se propone un nuevo método para la alineación automática no rígida de secuencias de ecografía intravascular del mismo paciente, adquiridas en diferentes momentos (antes y después de la intervención, o al inicio del estudio y en exámenes de seguimiento). El método se basa en la descripción del contenido morfológico del vaso, que se obtiene mediante la extracción de perfiles temporales morfológicos de las adquisiciones de IVUS. Una técnica para la alineación no rígida de secuencias - Dynamic Time Warping - se aplica a los perfiles y se adapta al problema clínico. Se proponen dos diferentes estrategias para hacer frente a la superposición parcial entre los frame de las secuencias correspondientes. Los beneficios de la estrategia propuesta se demuestran por una amplia validación en datos sintéticos e in vivo. Finalmente, se presenta un enfoque novedoso para la extracción automática de la frontera luminal en imágenes de IVUS. El método aplica el algoritmo de aprendizaje multiscale stacked sequential learning y lo extiende en 2-D+T, en una primera fase de clasificación (la identificación de regiones de lumen y no-lumen de las imágenes), mientras que un modelo de contorno activo se utiliza en una segunda fase, para identificar el contorno luminal.

This monograph is structured in four chapters. In Chapter 1, we present the context of our problem and the main results proved in this work. We describe the asymptotic behaviour of the reproducing kernel and the construction of new kernels associated to our spaces with a decay away from the diagonal. We shall also explain some tools that will play a fundamental role in the proof of our results. In Chapter 2, we study the problem of a continuous sampling. The role of a discrete family of sampling is played now by a sequence of sets in the manifold called Logvinenko-Sereda sets. We give a complete geometric characterization. A weaker problem is to find a characterization of the Carleson's measures. This question has been also answered in terms of a geometric condition. In Chapter 3, we provide some (qualitative) necessary and sufficient conditions for interpolation and sampling. We define an analog of the Beurling-Landau's density and prove a quantitative necessary condition for sampling and interpolation following the scheme of Landau in the context of the Paley-Wiener spaces. In Chapter 4, we give an application of the density results obtained in Chapter 3 and study the Fekete arrays on compact manifolds with some restriction. Furthermore, we prove from the results of Chapter 3, the equidistribution of the Fekete families on compact manifolds that have a product property (see Definition 4.1 for more details). The results of this monograph are part of the following articles: - J. Ortega-Cerdà, B. Pridhnani. Carleson measures and Logvinenko-Sereda sets on compact manifolds. Forum Mathematicum, to appear ([OCP11b]). - J. Ortega-Cerdà, B. Pridhnani. Beurling-Landau's density on compact manifolds. Preprint ([OCP11a]). / En aquesta tesi, estudiem les famílies d'interpolació i sampling (mostreig) en espais de funcions de banda limitada en varietats compactes. Les nocions de sampling i interpolació juguen un rol fonamental en problemes com ara recuperar un senyal continu a travès de les mostres discretes. Aquestes dues nocions són, en part, de caràcter oposat: un conjunt de sampling ha de ser suficientment dens per tal de poder recuperar la informació i, en un conjunt d'interpolació, els punts han de ser suficientment separats per tal de poder trobar una funció que interpola certs valors. A grans trets, una successió de sampling per a un cert espai de funcions és una successió de punts {lambda(n)}(n) tals que la norma de tota funció “f” de l'espai és equivalent a la norma de la successió que resulta d'avaluar la funció en els punts {lambda(n)}(n). Donada una varietat compacta M de dimensió m>/= 2, considerem el subespai E(L) de L(2)(M) generat per vectors propis del Laplacià de valor propi més petit que L > 0. Aquests espais s'anomenen espais de funcions de banda limitada i són el principal motiu d'estudi de la tesi. Els espais E(L) comparteixen propietats amb els espais clàssics de Paley-Wiener i la tesi explora aquesta connexió. La tesi s'estructura en quatre capítols. En el primer capítol, introduïm el context del nostre problema i els resultats principals provats al llarg d'aquesta tesi. També descrivim el comportament asimptòtic del nucli reproductor i la construcció de nous nuclis associats als nostres espais amb un decaïment fora de la diagonal. A més a més, expliquem algunes eines que jugaran un paper fonamental en les proves dels nostres resultats. En el segon capítol, estudiem el problema del sampling continu. El rol d'una família discreta de sampling el realitza una successió de conjunts en la varietat anomenada successió de Logvinenko-Sereda. Un problema més dèbil és trobar una caracterització de les mesures de Carleson. Aquesta qüestió també s'ha resolt en termes d'una condició geomètrica. En el tercer capítol, provem algunes condicions (qualitatives) necessàries i suficients per a la interpolació i sampling. Definim l'anàleg a la densitat de Beurling-Landau i provem, seguint les idees de Landau en el context dels espais de Paley-Wiener, condicions quantitatives necessàries per a què una família sigui de sampling o d'interpolació. En el quart capítol, donem una aplicació dels resultats de densitat obtinguts en el Capítol 3. Estudiem les famíllies de punts de Fekete en varietats compactes amb certa propietat. Els punts de Fekete són punts que maximitzen un determinant del tipus Vandermond que apareix en la fòrmula d'interpolació del polinomi de Lagrange. Són punts adients per les fòrmules d'interpolació i la integració numèrica. Els punts de Fekete tenen la propietat que són casi d'interpolació i sampling. Per tant, aquest tipus de punts estan ben distribuïts en la varietat ja que contenen informació suficient per recuperar la norma L(2) d'una funció de banda limitada i, són suficientment separats per tal d'interpolar alguns valors fixats. Els resultats d'aquesta tesi són part dels següents articles: - J. Ortega-Cerdà, B. Pridhnani. Carleson measures and Logvinenko-Sereda sets on compact manifolds. Forum Mathematicum 25, no. 1, p. 151-172, 2011. - J. Ortega-Cerdà, B. Pridhnani. Beurling-Landau's density on compact manifolds. Journal of Functional Analysis 263, no. 7, p. 2102-2140, 2012.

Esta tesis doctoral introduce los aquí llamados lenguajes estricto-débiles, y los analiza desde diversos puntos de vista. Los lenguajes estricto débiles son aquellos lenguajes formales que constan de conjunción, disyunción, "falsum" , "rerum", y adicionalmente de una cantidad indeterminada de conectivas que semánticamente se interpretan en las estructuras de Kripke como diferencias débiles. Estos lenguajes nos dan un fragmento de los habituales lenguajes modales puesto que:Donde es la implicación material y es la diferencia material. Para el caso en que hay una sola implicación estricta y ninguna diferencia débil resulta que la cláusula semántica para la implicación estricta que obtenemos es bien conocida, coincide con la de la lógica proposicional intuicionista (y con la de otros muchos ejemplos).Los lenguajes estricto-débiles se analizan en la tesis doctoral, desde tres puntos de vista diferentes: teoría de modelos (donde se introduce la noción de quasi bisimilaridad), teoría de la prueba (donde se introduce la noción de lógica estricto-débil) y computabilidad (se caracterizan las clases de complejidad para las lógicas estricto-débil).Los resultados obtenidos desde estas tres vertientes sugieren que aunque los lenguajes estricto-débiles son un fragmento de los lenguajes modales, en muchas ocasiones un conocimiento de lo que sucede para dichos fragmentos nos aporta información sobre lo que ocurre en la totalidad de los lenguajes modales (incluso fuera de los fragmentos anteriores).