The subject of the theory of Dynamical Systems is the evolution of systems with respect to time. Hence, it has many applications to other areas of science, such as Physics, Biology, Economics, etc. and it also has interactions with other parts of Mathematics. The global behavior of a dynamical system is organized by its invariant objects, the simplest ones are equilibria and periodic orbits (and related invariant manifolds). Normally hyperbolic invariant manifolds (NHIM for short) are some of these invariant objects. They have the property to persist under small perturbations of the system. These NHIM are characterized by the fact that the directions on the points of the manifold split into stable, unstable and tangent components. The growth rate of stable directions (for which forward evolution of the system goes to zero) and unstable directions (for which backward evolution goes to zero) dominate the growth rate of the tangent directions. The robustness of normally hyperbolic invariant manifolds makes them very useful to understand the global dynamics. Both the theory and the computation of these objects are important for the general understanding of a dynamical system. The main goal of my thesis is to develop efficient algorithms for the computation of normally hyperbolic invariant manifolds, give a rigorous mathematical theory and implement them to explore new mathematical phenomena. For simplicity, we consider the problem for discrete dynamical systems, since it is known that the discrete case implies the continuous case using time one flow. We consider a diffeomorphism F : Rm → Rm and a d-torus parameterized by K : Td → Rm which is invariant under F. This means that there exists a diffeomorphism f : Td → Td (the internal dynamics) such that it satisfies F ◦ K = K ◦ f, (0.3) called the invariance equation. Our goal is to solve this invariance equation considering two different scenarios: one in which we do not know the internal dynamics of the invariant torus (where K and f are our unknowns), see Chapter 4, and the other in which we impose that the internal dynamics is a rigid rotation with a quasi-periodic frequency (where K is the unknown and f is the rigid rotation), for which we also need to add an adjusting parameter to equation (0.3), see Chapters 2 and 3. Additionally, in both cases we are also interested in computing the invariant tangent and normal bundles. / L’objecte d’estudi dels Sistemes Dinàmics és l’evolució dels sistemes respecte del temps. Per aquesta raó, els Sistemes Dinàmics presenten moltes aplicacions en altres àrees de la Ciència, com ara la Física, Biologia, Economia, etc. i tenen nombroses interaccions amb altres parts de les Matemàtiques. Els objectes invariants organitzen el comportament global d’un sistema dinàmic, els més simples dels quals són els punts fixos i les òrbites periòdiques (així com les seves corresponents varietats invariants). Les Varietats Invariants Normalment Hiperbòliques (NHIM forma abreviada provinent de l’anglès) són alguns d’aquests objectes invariants. Aquests objectes posseeixen la propietat de persistir sota petites pertorbacions del sistema. Les NHIM estan caracteritzades pel fet que les direccions en els punts de la varietat presenten una divisió en components tangent, estable i inestable. L’índex de creixement de les direccions estables (per les quals la iteració endavant del sistema tendeix cap a zero) i inestables (per les quals la iteració enrere del sistema tendeix cap a zero) domina l’índex de creixement de les direccions tangents. La robustesa de les varietats invariants normalment hiperbòliques les fa de gran utilitat a l’hora d’estudiar la dinàmica global. Per aquesta raó, tant la teoria com el càlcul d’aquests objectes sós molt importants per al coneixement general d’un sistema dinàmic. L’objectiu principal d’aquesta tesi és desenvolupar algoritmes eficients pel càlcul de varietats invariants normalment hiperbòliques, donar-ne resultats teòrics rigorosos i implementar-los per a explorar nous fenòmens matemàtics. Per simplicitat, considerarem el problema per a sistemes dinàmics discrets, ja que és ben conegut que el cas discret implica el cas continu usant operadors d’evolució. Considerem així difeomorfismes donats per F : Rm → Rm i un d-tor F-invariant parametritzat per K : Td → Rm. És a dir, existeix un difeomorfisme f : Td → Td (la dinàmica interna) tal que satisfà l’equació F ◦ K = K ◦ f, (0.1) anomenada equació d’invariància. La nostra finalitat és solucionar aquesta equació d’invariància considerant dos possibles escenaris: un en el qual no coneixem quina és la dinàmica interna del tor (on K i f són les nostres incògnites), veure Capítol 4, i un altre en el qual imposem que la dinàmica interna sigui una rotació rígida amb freqüència quasi-periòdica (on K és una incògnita i f és la rotació rígida), pel qual necessitarem, a més a més, afegir un paràmetre ajustador a l’equació (0.1), veure Capítols 2 i 3. En ambdós casos també estarem interessats en el càlcul dels fibrats invariants tangent i normals.

This thesis studies curves in the symmetric square of a curve, C(2). We characterize them, study their immersion in C(2) and deduce properties of the curve C from the existence of curves in C(2) of specific type. In addition, we give a characterization of C(n), for n general, by the existence of certain subvarieties with particular properties. / En aquesta Tesi s’estudien corbes en el producte simètric d’una corba, C(2). Les caracteritzem, estudiem la seva immersió en C(2) i deduïm propietats de la corba C a partir de l’existència de corbes a C(2) d’un tipus concret. A més, donem una caracterització de C(n), per n general, a partir de l’existència de certes subvarietats amb unes propietats concretes.

This thesis deals with three possible applications of stochastic calculus: modelling prices by supply and demand in a financial market where there is an informed trader, turbulence and financial models using ambit processes and the asymptotic analysis of certain power variation processes. The thesis is organized as follows. Part I contains the basic facts and techniques of mathematics used in the latter parts. Part II deals with the markets with asymmetric information, Chapter 2 presents the basic models by Kyle and Back, and Chapter 3 presents the new results of Kyle’s model with L´evy noise: [Cor14b] and a General Model: [Cor14a], and also a short summary of other related models. Part III is dedicated to ambit processes. Chapter 4 introduces ambit fields and processes and bond markets, summarizes the new results of some applications of ambit processes on energy markets and turbulence: [CFV14], and on a short rate model: [CFSV13]. In Part IV, power variation processes are introduced and new results of [CF10] are summarized in Section 5.3. Finally, the above mentioned articles are included in the appendices.

Wireless Capsule Endoscopy (WCE) is a technique for inner-visualization of the entire small intestine and, thus, offers an interesting perspective on intestinal motility. The two major drawbacks of this technique are: 1) huge amount of data acquired by WCE makes the motility analysis tedious and 2) since the capsule is the first tool that offers complete inner-visualization of the small intestine, the exact importance of the observed events is still an open issue. Therefore, in this thesis, a novel computer-aided system for intestinal motility analysis is presented. The goal of the system is to provide an easily-comprehensible visual description of motility-related intestinal events to a physician. In order to do it, several tools based either on computer vision concepts or on machine learning techniques are presented. A method for transforming 3D video signal to a holistic image of intestinal motility, called motility bar, is proposed. The method calculates the optimal mapping from video into image from the intestinal motility point of view. To characterize intestinal motility, methods for automatic extraction of motility information from WCE are presented. Two of them are based on the motility bar and two of them are based on frame-per-frame analysis. In particular, four algorithms dealing with the problems of intestinal contraction detection, lumen size estimation, intestinal content characterization and wrinkle frame detection are proposed and validated. The results of the algorithms are converted into sequential features using an online statistical test. This test is designed to work with multivariate data streams. To this end, we propose a novel formulation of concentration inequality that is introduced into a robust adaptive windowing algorithm for multivariate data streams. The algorithm is used to obtain robust representation of segments with constant intestinal motility activity. The obtained sequential features are shown to be discriminative in the problem of abnormal motility characterization. Finally, we tackle the problem of efficient labeling. To this end, we incorporate active learning concepts to the problems present in WCE data and propose two approaches. The first one is based the concepts of sequential learning and the second one adapts the partition-based active learning to an error-free labeling scheme. All these steps are sufficient to provide an extensive visual description of intestinal motility that can be used by an expert as decision support system.

Esta memoria pretende contribuir al estudio de haces reflexivos sobre espacios proyectivos en los dos aspectos siguientes: (A) Caracterización de las clases de Chern de haces reflexivos sobre espacios proyectivos. (B) Estudio de esquemas que parametrizan haces reflexivos sobre espacios proyectivos con clases de Chern prefijadas.

In this thesis, we present a proof of the meromorphic non-integrability for some problems arising from Celestial Mechanics, as well as a new necessary condition for partial integrability in a wider Hamiltonian setting. First of all, a simpler proof is added to those already existing for the Three-Body Problem with arbitrary masses. The N-Body Problem with equal masses is also proven non-integrable. Second of all, a further strengthening of a prior existing result allows us to detect obstructions to the existence of a single additional first integral for classical Hamiltonians with a homogeneous potential. Third of all, using the aforementioned new result, we have proven the non-existence of an additional integral both for the general Three-Body Problem (hence generalizing, in a certain sense, Bruns'Theorem) and for the equal-mass Problem for N=4,5,6. Fourth of all, finally, we have proven the non-integrability of Hill's problem using the most general instance of the Morales-Ramis Theorem.A varying degree of theoretical complexity was involved in these results. Indeed, the proofs involving the given instances of the N-Body Problem required nothing but the exploration of the eigenvalues of a given matrix, with the advantage of knowing four of them explicitly. Thus, not all variational equations were needed but those not corresponding to these four eigenvalues -- this is exactly what transpires from the system reduction and subsequent introduction of normal variational equations, as done by S. Ziglin, J. J. Morales-Ruiz, J.-P. Ramis, and others. Hill's Problem, however, required the whole variational system since only thanks to the special functions introduced in the process of variation of constants was it possible to assure the presence of obstructions to integrability.These results appear to qualify differential Galois theory, and especially a new incipient theory stemming from it, as an amenable setting for the detection of obstructions to total and partial Hamiltonian integrability.

Supply chain formation involves determining the participants and the exchange of goods within a production network. Today’s companies operate autonomously, making local decisions, and coordinating with other companies to buy and sell goods along their supply chains. Decentralized decision making is well suited to this scenario since it better preserves the privacy of the participants, offers better scalability on large-scale scenarios, and is more resilient to failure. Moreover, decentralized supply chain formation can be tackled either by means of peer-to-peer communication between supply chain participants or by introducing local markets that mediate the trading of goods. Unfortunately, current approaches to decentralized supply chain formation, both in the peer- to-peer and the mediated scenario, are unable to provide computationally and economically efficient solutions to the supply chain formation problem. The main goal of this dissertation is to provide computationally and eco- nomically efficient methods for decentralized supply chain formation both in the peer-to-peer and the mediated scenario. This is achieved by means of two optimized max-sum based methods for supply chain formation. On the one hand, we contribute to peer-to-peer supply chain formation via the so-called Reduced Binarized Loopy Belief Propagation (rb-lbp) algorithm. The rb-lbp algorithm is run by a multi-agent system in which each of the participants in the supply chain is represented by a computational agent. Moreover, rb-lbp’s message computation mechanisms allow the efficient computation of max-sum messages. This results in an algorithm that is able to find solutions to the supply chain formation problem of higher value than the state of the art while reducing the memory, bandwidth and computational resources required by several orders of magnitude. On the other hand, we contribute to mediated supply chain formation via the so-called CHaining Agents IN Mediated Environments (chainme) algorithm. The chainme algorithm is run by a multi-agent system in which each of the participants and each of the goods in the supply chain is represented by a computational agent. In chainme participant agents communicate exclusively with the agents representing the goods who act as mediators. Likewise rb-lbp, chainme is also endowed with a message computation mechanism for the efficient computation of max-sum messages. This results in an algorithm that is able to find economically efficient solutions while requiring a fraction of the computa- tional resources needed by the state-of-the-art methods for both peer-to-peer and mediated supply chain formation. Finally, the design and implementation of both of our contributions to decentralized supply chain formation follow the same methodology. That is, we first map the problem at hand into a local term graph over which max-sum can operate. Then, we assign each max-sum local term to a computational agent. Last, we derive computationally efficient expressions to assess the max-sum messages exchanged between these agents. Although our methodology proved to be valid for the design of SCF algorithms, its generality makes it appear as a promising candidate for other multi-agent coordination problems.

Over the past few decades, machine learning (ML) algorithms have become a very useful tool in tasks where designing and programming explicit, rule-based algorithms are infeasible. Some examples of applications where machine learning has been applied successfully are spam filtering, optical character recognition (OCR), search engines and computer vision. One of the most common tasks in ML is supervised learning, where the goal is to learn a general model able to predict the correct label of unseen examples from a set of known labeled input data. In supervised learning often it is assumed that data is independent and identically distributed (i.i.d ). This means that each sample in the data set has the same probability distribution as the others and all are mutually independent. However, classification problems in real world databases can break this i.i.d. assumption. For example, consider the case of object recognition in image understanding. In this case, if one pixel belongs to a certain object category, it is very likely that neighboring pixels also belong to the same object, with the exception of the borders. Another example is the case of a laughter detection application from voice records. A laugh has a clear pattern alternating voice and non-voice segments. Thus, discriminant information comes from the alternating pattern, and not just by the samples on their own. Another example can be found in the case of signature section recognition in an e-mail. In this case, the signature is usually found at the end of the mail, thus important discriminant information is found in the context. Another case is part-of-speech tagging in which each example describes a word that is categorized as noun, verb, adjective, etc. In this case it is very unlikely that patterns such as [verb, verb, adjective, verb] occur. All these applications present a common feature: the sequence/context of the labels matters. Sequential learning (25) breaks the i.i.d. assumption and assumes that samples are not independently drawn from a joint distribution of the data samples X and their labels Y . In sequential learning the training data actually consists of sequences of pairs (x, y), so that neighboring examples exhibit some kind of correlation. Usually sequential learning applications consider one-dimensional relationship support, but these types of relationships appear very frequently in other domains, such as images, or video. Sequential learning should not be confused with time series prediction. The main difference between both problems lays in the fact that sequential learning has access to the whole data set before any prediction is made and the full set of labels is to be provided at the same time. On the other hand, time series prediction has access to real labels up to the current time t and the goal is to predict the label at t + 1. Another related but different problem is sequence classification. In this case, the problem is to predict a single label for an input sequence. If we consider the image domain, the sequential learning goal is to classify the pixels of the image taking into account their context, while sequence classification is equivalent to classify one full image as one class. Sequential learning has been addressed from different perspectives: from the point of view of meta-learning by means of sliding window techniques, recurrent sliding windows or stacked sequential learning where the method is formulated as a combination of classifiers; or from the point of view of graphical models, using for example Hidden Markov Models or Conditional Random Fields. In this thesis, we are concerned with meta-learning strategies. Cohen et al. (17) showed that stacked sequential learning (SSL from now on) performed better than CRF and HMM on a subset of problems called “sequential partitioning problems”. These problems are characterized by long runs of identical labels. Moreover, SSL is computationally very efficient since it only needs to train two classifiers a constant number of times. Considering these benefits, we decided to explore in depth sequential learning using SSL and generalize the Cohen architecture to deal with a wider variety of problems.

L'objectiu principal d'aquesta tesi doctoral és l'estudi de l'efecte de les fluctuacions en poblacions acoblades en sistemes biològics, on cèl·lules en estat latent juguen un paper important. Intentant trobar el significat biològic de la dinàmica dels sistemes. Els punts específics que volem abordar i la organització de la tesi estan explicats a continuació. En el Capítol 2, estudiem el comportament de les poblacions de cèl·lules amb estructura jeràrquica des del punt de vista de les propietats d'estabilitat, En particular: - 1. Divisió simètrica contra asimètrica en el compartiment de les cèl·lules mare. Estudiem la robustesa de les poblacions amb estructura jeràrquica, depenent de si les cèl·lules mare es divideixen simètricament, asimètricament o de les dues maneres. Estudiem com la divisió simètrica afecta a l'estabilitat de la població, ja que això té una gran importància en la progressió del càncer. - 2. La competició entre dues poblacions amb diferents tipus de divisió de les cèl·lules mare. Això és crucial per trobar estratègies òptimes que maximitzin la robustesa (supervivència a llarg termini, resistència a invasions i habilitat per invadir) de poblacions amb estructura jeràrquica. - 3. La influència de paràmetres com son la duplicació i el ritme de mort de cèl·lules mare, el temps de vida mitjà de les cèl·lules completament diferenciades, la longitud de les cadenes de diferenciació i les fluctuacions al compartiment de les cèl·lules mare en la robustesa i arquitectura òptima de les cascades de diferenciació. En el Capítol 3 presentem un model homogeni de combinació de HAART amb teràpies d'activació de les cèl·lules latents del VIH-1 a la sang. Estem interessats en: - 1. L'efecte del ritme d'activació de les cèl·lules latents en el temps mitjà de vida de la infecció. En particular analitzem si les teràpies basades en incrementar aquest ritme són capaces de suprimir la infecció en un temps raonable. - 2. La importància de l'eficiència de les teràpies antiretrovirals, incloent els casos límit en que l'eficàcia és del 100%, en la quantitat de càrrega viral. - 3. La formulació d'una teoria asimptòtica basada en l'aproximació semi-clàssica amb aproximacions quasi estacionàries per descriure la dinàmica del procés. La precisió d'aquest mètode asimptòtic és comparat amb simulacions multi-scale proposades pel Cao et al. En el Capítol 4, estenem el model proposat pel Rong i el Perelson a un model no homogeni de la dinàmica del VIH-1 en el corrent sanguini, considerant que les cèl·lules i els virus no estan distribuïts de manera uniforme en la sang. Els punts específics que volem estudiar són: - 1. El mecanisme que fa que apareguin els episodis de virèmia per sobre els límits de detecció, coneguts com viral blips. En particular volem investigar si són producte de fluctuacions estocàstiques degudes a la inhomogenietat o un altre mecanisme ha de ser considerat. - 2. Si l'aparició dels viral blips està afectada pels procediments duts a terme en el laboratori, com el temps d'espera entre les extraccions i les observacions. - 3. Si la probabilitat, l'amplitud i la freqüència dels viral blips es veu afectada pels diferents possibles tipus de producció viral, és a dir, continua vs burst. En el Capítol 5 presentem i discutim els resultats obtinguts, i comparem, quan és possible, amb altres models o amb resultats experimentals, i discutim el treball que deixem pel futur. Els detalls relatius a qüestions metodològiques, això com una introducció a la modelització estocàstica fent servir equacions mestres es donen en els apèndixs. Per a aquells que no estan familiaritzats amb els models basats en equacions mestres, l'autor recomana llegir primer l'apèndix A que proporciona la base matemàtica per entendre el capítol 2. Els Apèndixs B, C i D juntament amb l'Apèndix A donen la base matemàtica necessària per seguir el capítol 3 i el capítol 4.

The blending of digital technologies, such as artificial intelligence, interactive systems, 3D interfaces and the Internet is enabling new services for users. In particular, Hybrid Structured 3D Virtual Environments (VE) provide users with a collaborative space not only for entertainment and socialization but also for developing “serious” applications such as e-learning, e-government and e-commerce. This thesis focuses on Hybrid Structured 3D VE, which are persistent multi­user systems where participants (both human users and software agents) develop “serious” activities. In these systems the 3D interface graphically represents the system and facilitates human participation, and an Organisation Centred Multi-Agent System (OCMAS) structures participants’ interactions. To do so, the OCMAS specifies the roles that participants can enact, the activities where complex tasks can be accomplished, and communication protocols that enable the prosecution of such tasks. Nevertheless, participating in these systems is not a straight-forward process. Specifically, when the system speci.cation is complex, participants have to perform intricate reasoning processes to understand their applicable regulations at current system state; and they do not have access to information about what happened before they entered the system, neither can further process this information. Moreover, software agents speak a computer-based language, which is usually hard to use by human users. Then, as human users interact with staff agents (software agents devoted to support the system activities) to complete tasks, human-agent interaction style becomes a key issue. In order to overcome these limitations, this work proposes Assisted Hybrid Structured 3D Virtual Environments, where both human users and software agents participation in the system is improved by both assistance and human­agent interaction mechanisms. The system is formalised as a two layered infrastructure. The Organisational Layer structures the interactions of participants, and the Assistance Layer is populated by a set of Personal Assistants in charge of providing with a set of Assistance services to a system participant. There are four types of Assistance services: i) an Information service that processes data about the organisation specification, the participant current state, and the organisational historical execution states; ii) a Justi.cation service that can be triggered once a participant tries to execute a non-valid or prohibited action; iii) an Estimation service that processes whether an action can be performed at current state prior to its execution or not and, if it is actually the case, then it also provides the next system state; and iv) an Advice service, which provides participants with a sequence of actions (i.e. plans) to achieve their goals. Moreover, this work implements and evaluates v-mWater, a virtual market based on trading water, modelled as an Assisted Hybrid Structured 3D Virtual Environment. The usability evaluation results of v-mWater show that it is per­ceived as a useful and powerful application that could facilitate everyday tasks in the future. Users like its learnability, its immersiveness, and how scenario settings facilitate task accomplishment. In general, users completed the proposed task well and they were able to go to the right destination in the scenario. After doing the test, users improved their opinion about 3D virtual environments. In addition, the overall opinion of the human-agent interaction was positive. Nevertheless, those users less familiar with new technologies experimented problems when using a command-based system to interact with staff agents. To support assistance services in the system, this work designs and evaluates an Assistance Architecture where the Information service is implemented for software agents; and the Justi.cation, Estimation and the planning Advice service for human users. Nonetheless, these four services could be offered interchangeably for both humans and software agents, since they all simplify the reasoning process as well as the cognitive load required to participate in these complex structured systems. Speci.cally, the Information service has been extended to help sellers to set the price in their transactions. The tests performed compare the values that different agent satisfaction parameters and system goals take when agents request for different information services, using as a base-line a con.guration without enabling assistance services. The experiments show that system performance and agent satisfaction (and thus, the quality of assistance service) increase with the addition of the information service. Furthermore, individual agents following alternative strategies can request di.erent information as a useful decision sup­port tool. The planning Advice service is the most sophisticated one and makes use of the rest of services to provide a plan that has into account other partici­pants actions and, executed at current system state, will lead to the user’s goal. It is implemented as an extension of A*, namely Plan-eA. Evaluation results indicate that assistance impacts positively in usability measures of efficiency, efficacy and satisfaction. Related to human-agent interaction, this work integrates a new conversational mechanism within VIXEE, an execution infrastructure for Hybrid Struc­tured 3D Virtual Environments. This new mechanism includes a task-oriented conversational system, which allows staff agents to dialogue with human users using natural language conversations. To do so, this work proposes an extension of the well-known AIML language, namely Task-Oriented AIML, for dealing with task-oriented conversations, which are based on activities’ specification and current system state. Test results give good usability measures of efficiency, efficacy and user satisfaction for the conversational approach.