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Mención Internacional en el título de doctor SUMMARY, REMARKS AND FUTURE WORK. Culture is one of the classical examples that is normally used to define a complex system. The inherent complexity of humans interacting directly provides enough diversity of phenomena, but if we also consider sociological problems that live in wider timescales, then it becomes boundless. The subject of the interaction is no more the person alone, but the different social constructs that coexist within a society, as, for example, different kind of social norms and expressions. The growing complexity of social systems is also confusing because it has not been traditionally understood in this way, which is logical because we have studied society from the different perspectives that have appeared historically. Humanities have studied the different holistic products that appear from complexity, like literature, philosophy or the different types of artistic studies. As we stated in the introduction, a different perspective appeared when the scientific method started comparative studies of society, bringing new tools, mainly modelling of general effects and statistics. Herbert Spencer popularized the analogy of considering the society as a human being, this is, as a cooperative organization of the different organs and tissues in order to provide a collective structure. And this is the main spirit of complexity sciences, which started to have more importance in the end of the XX century, giving rise to a new wave on how to understand quantitatively social systems. At this moment, we recall the title of this thesis: Behavior and context. The problem in the description of the context is a methodological one. Context is the conjunction of relevant circumstances correlated with the social description of an individual. Traditionally, the social description of a variable has been related with socioeconomic indicators, where the presence of the social structure is only understood with scalar variables. In this work, we show the importance of the network structure in the resemblance of this social profile, and how geometry may act as a proxy of this social variables containing, potentially, more information than just scalar variables. The behavior is correlated with the geometry of this social profile. The importance of the network version of the social profile can be checked by its correlation with the cultural variables of the ego. This is important because it reveals an idiosyncratic manner of making social relationships, and, therefore, the role of social institutions. But the network version is not only a proxy for cultural variables, it is a true identifier of people. A proof of this is the connection between personal profiles and probability of connection, showing that the similarity in the close structural network may be a proxy for social homophily, connected, therefore, to personal and cultural values. Therefore, we have identified a unit component (in terms of measure) of the individual, social and cultural influence on individuals, providing a personal signature around which social homophily appears, even in a complex context where culture and social values play a key role. At the end of the day, this thesis could be summarized by saying that the geometric structure of the close social environment is a signature of the individual and of its social and cultural values. Similarity in this signature is related to social homophily. The other main message that can be extracted from this thesis is related to the importance of ethnic groups in social modelling. Social modelling can not be replicated easily with a markovian model with binary interactions. This is due to the inherent complexity that was pointed out at the beginning of the chapter. Social phenomena exist in several timescales, and because of this they have a lot of implicit information. An ethnic group may exist because of this implicit information, due to, for example, a pre-existing social symbolism; but, in order to arise spontaneously from the social outcome of a binary process (in the way game theory works), some optimization is needed. This is, the existence of a marker does not provide, necessarily, a social separation depending on that variable, if this variable does not provide some kind of improvement in the fitness. This advantage may exist due to an asymmetrical social configuration, for example, in the case of a hierarchy. Previous works have shown the spontaneous appearance of this kind of structures [102], remarking again the importance of the different timescales in this kind of phenomena. An asymmetric social configuration allows the function to be optimized with ethnic markers, and the arena where cultures may compete, reinforcing the roles of their individuals, and building therefore, the different social profiles. With respect to the experimental part, the results that we obtain suggest that ethnic markers experiment should be replicated in contexts with different social configurations, studying, again, as the title suggests, the dependence on the context. It would be interesting the replication of simple ethnic separation altogether with a hierarchy or some reputation system. Once more, following the spirit of all the thesis, connects the importance of the ethnic, personal identifications with the social organization, that regulates its importance in the social equilibrium. In view of these conclusions, future works in this field should go deeper into the importance of the geometric social profile of the ego networks as an individual identifier. In this case, more data needs to be collected, in order to research other sociological variables like different aspects of culture, religion, political views or socioeconomic measures. Note that in the case of the present work, we have focused on the nationality, as the dataset provided us with a diverse sample of nationalities and the control group. In order to study other sociological variables, like wealth, race, gender or a particular social norm, new data is needed that fits those necessities. Moreover, we have focused on networks centered in the ego, where we have a perspective on the close social structure. It would be also interesting to study how much information is needed in order to study different kinds of networks. This is interesting from a practical point of view, as it could help to use different datasets to provide further proofs of the existence of the social signature, but also to check more properties of this measure. It would be interesting, for example, to study if family networks provide us enough information to talk about the different social values and/or material conditions of different families. If the information about the signatures still can be extracted, new datasets could be reused to make a stronger point about the cultural signature, saving the time of collecting new ego networks. Another possible path to follow in the future could be the study of signed networks together with the cultural profile. In our works, we have studied the importance of the social signature in order to determine the cultural profile, and its importance in the outcome of a relationship. It would be interesting to show the connection between these two elements, providing a relationship between cultural or socioeconomic variables and the friends or enemies an individual may have. The final suggestion for future work is related to the measure of properties of this social signature.We have stressed in the introduction (and in several chapters of the thesis) that the cultural values are a holistic product, which may be assumed and created by collectivities. In this work we have focused on a model that is based in the relationship between individual and its environment (in a mean-field like approach), but this is not needed. Considering again the example of the family, perhaps the family could be understood as a whole, and study the problem of its close social environment from a perspective of networks. Given that the problem of culture exists in several scales, it would be interesting to check if new relationships arise from this point of view. At the end of the day, all of these suggestions are basically related to checking the potential of the measure. This poses also an ethical question: if the potential of the social signature is shown in more contexts, and it correlates with personal information, it is important that these studies (public or private) are carried with the utmost concert about their possible implications and misuses. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Jesús Gómez Gardeñes.- Secretario: Alberto Antonioni.- Vocal: Andreia Sofía Monteiro Teixeira

Mención Internacional en el título de doctor This doctoral thesis was supported through the FPI contract BES-2017-079755 associated to the grant FIS2016-78313-P from Ministerio de Ciencia e Innovación (MCIN)/Agencia Estatal de Investigación (AEI)/ 10.13039/501100011033/ and FEDER Una manera de hacer Europa. Additionally the author also acknowledge finantial support from MCIN/AEI/10.13039/501100011033 through grant BADS, no. PID2019-109320GB-100 Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Guillermo Morelli.- Secretaria: María Pilar Guerrero Contreras.- Vocal: Vicente Mariscal Romero

Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: José Navarro Salas.- Secretario: Salvador José Robles Pérez.- Vocal: Iñaki Garay Elizondo

Mención Internacional en el título de doctor Online web services have grown dramatically in size and diversity in the last years, becoming essential components of our daily life and allowing us to conduct elementary tasks like working, getting informed, or keeping in contact with relatives and friends. However, all the changes and evolution experimented on by the online web services had not have been possible without implementing a profitable economic model that sustains it. Despite a suitable percentage of these services being fee-based, they represent a lucrative business that generates billions of dollars, allowing the creation of some of the biggest companies in the world in terms of market capitalization, like Alphabet Inc. or Meta Inc. (Previously known as Facebook Inc.). Being costless and lucrative is possible due to an advertising-based monetization model, which consists of delivering ads to the users in exchange for their services (e.g., Facebook or YouTube). Although online advertising dates back to the middle of the 90s, its popularity has experienced an increase among brands and advertising agencies in the last decade, mainly due to its capacity to reach precise audiences at a low cost. Converting online web services into advertising walls is a double-edged sword for the users. The capacity offered by online advertising to segment their audiences requires a massive collection of personal data from the users, including their web browsing histories or even more invasive data such as age, gender, or location to infer the online profile of the users. This data collection is possible due to implementing a complex tracking ecosystem by online advertising companies from which multiple stakeholders collect, process, and exchange information. The many privacy cases of abuse inflicted by this industry motivated the implementation of new regulatory efforts to protect consumers’ privacy in the last years. Some notable examples are the General Data Protection Regulation (GDPR) [1] in the European Union or the California Consumer Privacy Act Regulations (CCPA) [2] in California, USA. Further, these privacy regulations typically contain specific provisions and strict requirements for websites that provide sensitive material to end users, including sexual, religious, and health services. Implementing new regulatory frameworks, alongside the growth of online web services, forces an endless evolution of current techniques to study and audit online web services. Furthermore, there is a need to emphasize the online advertising ecosystem, as it represents the primary economic support of a high percentage of web services. Also, the activities and abuses conducted by this ecosystem drove the implementation of current privacy regulations to control the use and collection of personal data. This dissertation falls within the topics of Internet measurements, tackling the need for new measurement techniques and methodological approaches to audit and study online web services. These efforts want to increase the limited knowledge about web subsystems offering sensitive material, including their regulatory compliance regarding current privacy regulations. Also, this dissertation tackles the need to study and measure how big ad tech companies create and use the online profiles of their users to distribute tailored ads. Furthermore, the work presented in this dissertation raises the need for a more in-depth understanding of fundamental tools for conducting Internet measurement works, including their limitations and suitability for academic research. Specifically, this dissertation presents three main contributions: The first one corresponds with implementing a novel methodology to audit sensitive web services’ privacy, transparency, and regulatory compliance. We validate our method by looking at pornographic websites concerning the GDPR in the European Union. We focus our analysis on such types of websites for two main reasons: (i) the GDPR establishes specific provisions and strict requirements on sensitive websites, including pornographic ones. (ii) big ad tech companies set strict constraints for porn-related publishers. As a result, it opened new market opportunities for other actors who have specialized in advertising and tracking technologies for adult sites, creating a semi-decoupled ecosystem from the rest of the web. We perform a holistic analysis of over 6,843 pornographic websites, finding a prevalent absence of regulatory compliance and very extended use of tracking techniques, including advanced ones such as fingerprinting. These results stress the importance of studying the World Wide Web subsets that have not been scrutinized by regulators, policymakers, and the research community in depth. Second, we empirically and comprehensively analyze 13 domain classification services to study their labeling strategy and performance. These services have multiple applications, from business applications such as online advertising to academic research works to conduct category-dependent measurements or to identify the purpose of a website or online service. We study each domain classification service’s methodologies, scalability limitations, label constellations, and suitability for academic research studies. In some cases, their findings depend on the results provided by the domain classification services. We find that the limitations and shortcomings of each domain classification service heavily affect their suitability and applicability, both for practical solutions and academic studies. In the third and last contribution, we implement a novel methodology with real users to study the performance and quality of the profiling and ad targeting algorithms from the two most important stakeholders in the online advertising business, Google and Meta (previously Facebook). We find that half of the categories associated with the profiles are incorrectly assigned. We also observe the presence of sensitive categories in Facebook users, posing a privacy risk and potential regulatory noncompliance. In summary, this dissertation brings new methodologies and results to increase our limited knowledge about the web. This work has been supported by IMDEA Networks Institute. Programa de Doctorado en Ingeniería Telemática por la Universidad Carlos III de Madrid Presidente: Nikolaos Laoutaris.- Secretario: Hamed Haddadi.- Vocal: Leyla Bilge

Mención Internacional en el título de doctor Recent technological advances have led to an exponential growth in the volume of data generated. The quest to make sense of these data, some of which are usually complex, has led to recent interest in development of statistical methods for analysing data with complex structures. One such field of interest is functional data analysis (FDA), which deals with the analysis of data that can be considered as functions, curves, or surfaces observed over a domain set. Outlier detection is a challenging but important part of the exploratory analysis process in FDA because functional observations can exhibit outlyingness in various ways compared to the bulk of the data. This thesis addresses the problem of detecting and classifying outliers in functional data with three main contributions. First, the fdaoutlier R package is presented in Chapter 2. The package contains implementations of some of the state-of-the-art functional outlier detection methods in the literature. Some of the methods implemented include directional outlyingness, magnitude-shape plot, sequential transformations, total variation depth, and modified shape similarity index. Detailed illustrations of the functions of the package are provided, using various simulated and real functional datasets curated from the functional outlier detection literature. Overviews of the functional outlier detection methods implemented in the package are also presented in Chapter 2. This chapter therefore, serves as a review of some of the current literature in outlier detection for functional data. Next, two new methods, named ‘Semifast- MUOD’ and ‘Fast-MUOD’, are presented in Chapter 3. These methods work by computing for each curve three indices (magnitude, amplitude and shape index) that measure the outlyingness of that curve in terms of its magnitude, amplitude and shape. ‘Semifast- MUOD’ computes these indices with respect to (w.r.t.) a random sample of the dataset, while ‘Fast-MUOD’ computes these indices w.r.t. to the point-wise or L1 median. The classical boxplot is then used as a cutoff on the three indices to identify curves that are outliers of different types. A byproduct of the methods is an unsupervised classification of the outliers into different types, without the need for visualisation. Performance evaluation of the methods, using various real and simulated datasets, shows that Fast-MUOD is the better of the two new proposed methods for outlier detection, in addition to being very scalable. Comparisons with latest functional outlier detection methods in the literature also show superior or comparable outlier detection performance. In Chapter 4, some theoretical properties of the Fast-MUOD indices are presented. These include some definitions of the indices, as well as convergence proofs of the sample approximations. Some properties of the indices under simple transformations are also presented in this chapter. Finally, three techniques are presented in Chapter 5 for extending the Fast-MUOD indices to outlier detection in multivariate functional data observed on the same domain. These techniques include the use of random projections and identifying outliers on the marginal components of the multivariate functional data. The use of random projections showed the best result in performance evaluations with various real and simulated datasets. Chapter 6 contains some concluding remarks and possible future research work. This work has been supported by IMDEA Networks Institute Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Francisco Javier Prieto Fernández.- Secretario: Alba María Franco Pereira.- Vocal: Fabian Scheipl

This thesis investigates the metric and tetrad formulations of three gravitational field theories in manifolds with timelike boundaries within the covariant phase space program. With the recently developed relative bicomplex framework, we explore the space of solutions and presymplectic structures associated with each action principle and analyse their equivalence. The first action we consider is the Einstein-Hilbert (EH) action with the Gibbons- Hawking-York boundary term. By including the appropriate boundary terms in the variational principles, we show that the metric and tetrad formulations derived from them are equivalent. Furthermore, we show that their solution spaces are the same and that their presymplectic structures and associated charges coincide. The second action we consider is the Palatini action with the Obukhov boundary term, assuming torsion and non-metricity, and we prove the equivalence between its metric and tetrad formulations. Furthermore, we show that the metric and tetradsector of the first-order Palatini formulation are equivalent to the metric and tetrad formulations of the EH action. Lastly, we introduce the Hojman-Mukku-Sayed (HMS) action, a generalisation of the Palatini action plus the Holst term in the presence of boundaries with non-metricity and torsion. We prove that the space of solutions of the HMS and Palatini actions coincided and conclude that HMS’s metric and tetrad sectors are identical to their corresponding versions of the EH action. Additionally, we prove that the Palatini and HMS Lagrangians are not cohomologically equal despite defining the same space of solutions. Consequently, a careful analysis is required for the presymplectic structures and the charges because they may differ. However, we show that the covariant phase spaces of both theories were equivalent. This sheds light on some open problems regarding the equivalence of their associated charges in different formulations. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: José Navarro Salas.- Secretario: Bogar Díaz Jiménez.- Vocal: Mercedes Martín Benito

Mención Internacional en el título de doctor This research has been funded in part by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” through grant PGC2018-098186-B-I00 (BASIC). Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Sandro Meloni.- Secretaria: Francesca Lipari.- Vocal: Giulia Andrighetto

Mención Internacional en el título de doctor Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Arnoldus Bernardus Jacobus Kuijlaa

The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space and Riemannian manifolds of negative sectional curvature. It is remarkable that a simple concept leads to such a rich general theory. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this Ph. D. Thesis we characterize the hyperbolicity constant of interval graphs and circular-arc graphs. Likewise, we provide relationships between dominant sets and the hyperbolicity constant. Finally, we study the invariance of the hyperbolicity constant when the graphs are transformed by several operators. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Lojo

Self-organization processes, in which some form of collective behavior arises from local interactions in a physical system, are promising mechanisms in the context of manufacturing at ultra small scales, where processing techniques are technically challenging. In many cases, the approach is to induce self-organization or self-assembly phenomena on the system surface |which indeed acquires an increased importance at small scales because of the enhanced surface-to-volume ratio| as a result of which a desired surface morphology is achieved, with different properties depending on its intended application. In this thesis, we consider two important examples of self-organization processes which take place at the surfaces of many small non-equilibrium systems. One of them, kinetic roughening, reflects the dominance of fluctuations in the surface morphology, with strong correlations which are quite similar to those of an equilibrium system at a continuous phase transition. The second, opposite self-organization process is the formation of ordered patterns. In the thesis, we will deal with the control of the level and type of various surface properties, like roughness and other, and the conditions for the emergence of a varying degree of spatial order in patterns via spontaneous physical processes. We work in the context of ultrathin fluid films on solid substrates, focusing on the role of the physical effects that become relevant at these very small scales -but not so much at larger scales-, while other loose their relevance. One of these aspects is the thermal noise. The other one is the interaction between the uid surface and the substrate, that play a key role in the two reference physical systems we are going to study: an ultrathin fluid film falling down an inclined plane |where the dynamics of the fluid surface follows the celebrated Kuramoto-Sivashinsky (KS) equation -and a ferrofluid ultrathin film under a magnetic field. Both kinetic roughening and pattern formation usually exhibit some kinds of universal behavior. The universality classes in the kinetic roughening processes that occur in several approximations of the KS equation are widely studied in this thesis. These universality classes are characterized by both how the fluctuations scale with space and time and how these fluctuations are statistically distributed. We will deal with the emergence of symmetries in the fluctuation distributions that are unexpected considering the bare microscopic interactions; the non-trivial relation between the universality class of closely related models; with a novel physical mechanism that induces the transition between different universality (sub)classes as the system temperature and hence the dominant nature of the fluctuations (chaotic or stochastic) changes, and finally with some anomalous kinetic roughening processes in the limit of vanishing viscosity and surface tension. Finally, on the other hand, the formation of highly ordered patterns is assessed in the context of ultrathin ferrofluid films under a magnetic field, due to the spontaneous physical break-up of the film into drops. The conditions under which higher levels of order are achieved will be described. This is intended as a proof-of-concept, previous step that could encourage experiments being performed for this type of systems. This Doctoral Thesis has been supported by Ministerio de Educación, Cultura y Deporte (Spain) through Formación del Profesorado Universitario scolarship No. FPU16/06304. Additionally, research by the author has been partially supported by Ministerio de Economía y Competitividad, Agencia Estatal de Investigación, and Fondo Europeo de Desarrollo Regional (Spain and European Union) through grant Nos. FIS2015-66020-C2-1-P and PGC2018-094763-B-I00. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Miguel Ángel Rodríguez Díaz.- Secretario: Juan Jesús Ruiz Lorenzo.- Vocal: Silvia Noemi Santalla Arribas

Mención Internacional en el título de doctor El principal objetivo de esta tesis es presentar y probar la viabilidad de un método no estándar para controlar el estado de un sistema cuántico, cuya dinámica está gobernada por la ecuación de Schrödinger, modificando sus condiciones de frontera en lugar de interaccionar con el sistema a través de campos externos para dirigir su estado. El control del estado de sistemas cuánticos está volviéndose más y más significativo dado el asombroso avance experimental que está teniendo lugar motivado por la carrera para alcanzar tecnologías cuánticas efectivas. Un requisito natural para cualquier procesador de información cuántico es el control de los estados de un cúbit. Por ejemplo, los cúbits basados en espín pueden controlarse a través de puertas universales basadas en la rotación del espín un cierto ángulo sobre un eje dado [60]. Una demostración de la viabilidad del control y de la lectura de alta fidelidad de un cúbit basado en espín puede encontrarse en [55, 59]. El marco matemático necesario para trabajar con el control de sistemas cuánticos basados en el espín es la Teoría de Control Geométrica (véase [19] para un repaso de este enfoque). También puede mencionarse [15, 56], dónde se realiza un estudio del control de sistemas cuánticos de espín utilizando dicha teoría. Una revisión del estado de la cuestión en control (óptimo) de sistemas cuánticos puede verse en [31]. Sin embargo, la Teoría de Control Geométrica encuentra problemas serios a la hora de tratar sistemas infinito dimensionales provenientes de las dificultades intrínsecas a la geometría en dimensión infinita. Éste constituye un problema estándar que se aborda también, por ejemplo, en Teoría de Campos y puede por lo tanto ser sorteado. Sin embargo, lo cierto es que tan sólo un pequeño número de resultados han aparecido en las fuentes bibliográficas, sobre todo referentes a la controlabilidad de sistemas bilineales (cf. sec. 3.2). Véanse por ejemplo los trabajos de Beauchard et al. [11, 12] y Chambrion et al. [18]. El enfoque habitual del control de sistemas cuánticos se basa en el uso de campos externos para manipular el estado del sistema. Desde un punto de vista tecnológico, aparecen dificultades al intentar controlar un sistema cuántico de este modo que tienen que ver con las complicaciones para manipular un sistema hecho de unas pocas partículas mientras se intenta mantener las correlaciones cuánticas. Como consecuencia, el sistema cuántico debe mantenerse a muy baja temperatura y las interacciones debe producirse muy rápidamente [5], lo cual es inconveniente para las aplicaciones. El paradigma de Control Cuántico en la Frontera aborda el control de sistemas cuánticos de una forma diametralmente opuesta al enfoque estándar. En lugar de buscar el control del sistema cuántico interactuando directamente con el sistema a través de un campo externo, el control se consigue manipulando las condiciones de contorno del sistema. El espectro de un sistema cuántico, por ejemplo un electrón moviéndose en un pozo de potencial, depende de las condiciones de contorno (típicamente Dirichlet o Neumann). Por lo tanto, modificando dichas condiciones de contorno puede modificarse el estado del sistema, lo cual permitiría en última instancia controlarlo [34]. Este tipo de interacción, que es más débil en cierto sentido, hace esperar que sea más sencillo mantener las correlaciones cuánticas. Llamamos sistema (cuántico) de control en la frontera a los sistemas definidos dentro de este paradigma de control. El marco establecido por el Control Cuántico en la Frontera ha sido usado para mostrar cómo generar estados entrelazados en sistemas compuestos modificando las condiciones de frontera del sistema [38]. La relación entre dicho paradigma y la topología del sistema ha sido explorada en [58] y recientemente usada para describir las propiedades físicas de sistemas con paredes móviles [25–28,30]. Sin embargo, a pesar del interés intrínseco que tiene, problemas básicos como la controlabilidad de sistemas sencillos dentro de este paradigma no ha sido aún estudiada. La mayoría de los sistemas cuánticos que aparecen en las aplicaciones para desarrollar ordenadores cuánticos, tales como las trampas de iones o los circuitos superconductores, son infinito dimensionales. Mientras tanto, los modelos usados para describirlos son aproximaciones finito dimensionales [49, 70], lo que introduce errores en dicha descripción. Es por tanto natural buscar mejores modelos matemáticos para estos sistemas. Buenos candidatos para estos modelos pueden construirse a partir de lo que llamamos Circuitos Cuánticos, una generalización de los grafos cuánticos (esto es, grafos métricos equipados con un operador diferencial en sus aristas junto con condiciones de contornos apropiadas que definan una extensión autoadjunta de dicho operador [46,47]). Por un lado, los Circuitos Cuánticos comparten con los dispositivos físicos la estructura tipo grafo y la dinámica en ellos está gobernada por operadores diferenciales (en concreto, consideraremos el operador de Laplace-Beltrami y Laplacianos magnéticos junto con algunos potenciales que representan campos magnéticos y eléctricos en los Circuitos Cuánticos). Por otro lado, los Circuitos Cuánticos son suficientemente sencillos como para encontrar soluciones de la ecuación de evolución numéricamente (o incluso analíticamente). Todo esto hace interesante considerar la dinámica y la controlabilidad de sistemas definidos en Circuitos Cuánticos. Algunos ejemplos concretos de Circuitos Cuánticos pueden encontrarse al final del Capítulo 6, véase la Figura 6.1. Más aún, los grafos cuánticos pueden verse como un caso concreto de Circuito Cuántico, por lo que los resultados presentados en esta tesis pueden aplicarse directamente a ellos. Y, además de el interés matemático de estudiar la dinámica de sistemas definidos en grafos cuánticos, creemos que éstos pueden ser utilizados para modelizar circuitos superconductores. Más concretamente, es sabido que las junturas de Josephson (que son un dispositivo fundamental en los circuitos superconductores, cf. [22, 70, 71]) pueden modelizarse usando condiciones de frontera [48,68]. Nótese que los circuitos superconductores son una de las prometedoras nuevas tecnologías en el desarrollo de ordenadores cuánticos. Para mostrar la viabilidad del método de Control Cuántico en la Frontera, definimos una familia de sistemas de control cuántico en la frontera sobre Circuitos Cuánticos. Para ello, utilizamos la caracterización de las extensiones autoadjuntas del operador de Laplace-Beltrami desarrolladas en [37], que parametriza las extensiones autoadjuntas en función de operadores unitarios actuando en el espacio de datos en la frontera. Estos operadores unitarios son adecuados para representar los grupos de simetría de la variedad subyacente [36], hecho que fue usado en [8] para caracterizar las condiciones de frontera compatibles con las estructuras de grafo que consideramos aquí. Antes de poder abordar el problema de controlabilidad, es necesario afrontar el problema de existencia de soluciones de la ecuación de Schrödinger con condiciones de contorno dependientes del tiempo, problema que tiene su propia importancia. Situaciones similares a la que presentaremos han sido consideradas anteriormente para algunos sistemas cuánticos concretos [21, 23, 58]. La dependencia temporal del dominio del Hamiltoniano del sistema añade dificultades extra a las habituales complicaciones que surgen de la naturaleza discontinua de los operadores no acotados, lo que compromete la existencia de soluciones para la ecuación de Schrödinger. En este sentido, los resultados más generales de que disponemos son debidos a J. Kisyński [44]. Basándose en una idea aproximativa de K. Yosida [72], Kisyński propone condiciones suficientes para la existencia de operadores de green para problemas de Cauchy abstractos en espacios de Banach, incluyendo el caso en el que los operadores que definen la evolución tienen dominios dependientes del tiempo. Estos resultados pueden ser aplicados al caso concreto de la ecuación de Schrödinger, tal y como hace el propio Kisyński, cf. [44, Sec. 8]. Más concretamente, proporciona condiciones suficientes para la existencia de propagadores unitarios para sistemas cuánticos cuyos Hamiltonianos tienen dominio dependiente del tiempo pero no así las formas sesquilineales asociadas a éstos. Este caso también fue estudiado por B. Simon [65], cuyo enfoque se basa en el mismo método aproximativo de Yosida, pero abordado el problema de manera ligeramente distinta. Para simplificar la exposición, en lugar de aplicar los resultados más generales de Kisyński directamente a nuestros sistemas cuánticos de control en la frontera, introduciremos una familia de sistemas de control estándar relacionada que llamamos sistemas de inducción cuántica, cuyos Hamiltonianos tendrán dominio de forma constante. La relación entre estas dos familias de sistemas es usada frecuentemente en la discusión, lo que nos permite probar resultados primero para los sistemas de inducción cuántico y transferirlos a los sistemas de control en la frontera. Siguiendo esta estrategia, usaremos los resultados de Simon y de Kisyński para Hamiltonianos con dominio de forma constante para proporcionar condiciones suficientes para que ambos sistemas tengan dinámica bien definida. En el estudio de las propiedades dinámicas de los Hamiltonianos dependientes del tiempo con dominio de forma constante, la escala de espacios de Hilbert asociada a el Hamiltoniano, H+ ⊂ H ⊂ H−, aparece como una herramienta básica tanto en el desarrollo de Simon como en el de Kisyński. A través de esta tesis, usamos nociones como la existencia de dinámica o la controlabilidad tanto en H como en H−; por ello, usaremos los adjetivos fuerte y débil para referirnos a las nociones en H y en H− respectivamente. Una vez resuelto el problema de la existencia de soluciones de la ecuación de Schrödinger, podremos abordar el problema de controlabilidad. Usaremos como una herramienta útil para este fin el Teorema 4.3.4, un resultado de estabilidad que generaliza los resultados de A.D. Sloan [66] constituyendo uno de los resultados originales principales expuestos e esta tesis. Que sepamos, esta generalización es el resultado de estabilidad más general en el contexto de la ecuación de Schrödinger dependiente del tiempo. La demostración de este teorema se basa en las técnicas usadas por Simon y, en la prueba de dicho resultado, obtenemos una ligera generalización de algunos resultados en [65] (cf. Lemma 4.3.7). La importancia de este resultado de estabilidad no reside tan solo en su papel a la hora de demostrar la controlabilidad de los sistemas de control bajo estudio permitiendo demostrar la viabilidad del Control Cuántico en la Frontera; tiene consecuencias de mayor alcance para la Teoría de Control Cuántico ya que permite obtener cotas a priori del error cometido al controlar el estado del sistema. Demostramos la controlabilidad aproximada de los sistemas de inducción cuántica basándonos en un resultado de T. Chambrion et al. in [18], donde estudian la controlabilidad de una clase de sistemas de control cuántico bilineales con controles constates a trozos. La estructura que tienen los sistemas de inducción cuántica es similar, pero no exactamente igual, a la de los problemas estudiados por Chambrion et al. Por ello, definimos un sistema auxiliar para el cual el teorema de Chambrion et al. asegura controlabilidad aproximada y después, usando el resultado de estabilidad, mostramos que esta controlabilidad aproximada se extiende al sistema de inducción cuántica original. Si embargo, la clase de controles considerada por Chambrion et al. no puede ser llevada automáticamente a sistemas con Hamiltonianos cuyos dominios dependen del tiempo, ya que las soluciones de la ecuación de Schrödinger (en el sentido fuerte) podrían no existir. Por lo tanto, la controlabilidad aproximada obtenida para los sistemas de inducción cuántica usando este procedimiento es en el sentido débil (es decir, en H−). Usando de nuevo el resultado de estabilidad podemos demostrar controlabilidad aproximada en sentido fuerte para los sistemas de inducción cuántica con controles suaves, y ésta puede ser transferida a los sistemas de control cuántico en la frontera usando la relación entre ambos. Estos resultados de controlabilidad, demostrados en el Capítulo 7, junto con los resultados de estabilidad en el Capítulo 4 constituye las contribuciones originales más relevantes de esta tesis. Por último, remarcamos el hecho de que la caracterización de las extensiones autoadjuntas que usamos aquí puede ser aplicada a otros operadores diferenciales como el operador de Dirac [35, 40, 57], por lo que los resultados presentados en esta tesis podrían extenderse a tales casos. Más aún, dicha caractrerización está adaptada al cálculo numérico [39, 51], por lo que podrían desarrollarse algoritmos con aplicaciones al control óptimo de éstos sistemas a partir de los resultados aquí mostrados. This work was partially supported by the “Ministerio de Economía, Industria y Competitividad” research project MTM2017-84098-P, the QUITEMAD project P2018/TCS-4342 funded by “Comunidad Autónoma de Madrid” and by “Universidad Carlos III de Madrid” through its Ph.D. program grant PIPF UC3M 01-1819 and through its mobility grant in 2020. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid. Presidente: Fernando Lledó Macau.- Secretario: Paolo Facchi.- Vocal: Thomas Chambrion

Topological indices have been widely used in different fields associated with scientific research. They are recognized as useful tools in applied research in Chemistry, Ecology, Biology, Physics, among others. For many years, scientists have been trying to improve the predictive power of the famous Randi’c index. This led to the introduction and study of new topological descriptors that correlate or improve the level of prediction of the Randi’c index. Among the most commonly used descriptors are the Inverse index, the first general Zagreb index and the recently introduced Arithmetic- Geometric index. In this work we study the mathematical properties and relationships of the aforementioned topological indices. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Lojo

Mención Internacional en el título de doctor This thesis develops the discrete sub- and supersolutions method and applies it to prove the convergence of the nonlinear finite element method applied to the generalized di usive logistic equation...... Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección de la profesora Marcela Molina Meyer. Se contó con una beca de la UC3M de ayuda al estudio de máster de dos años de duración y posteriormente con un contrato de investigador en formación de la UC3M durante tres años más. Adicionalmente se recibió una ayuda de la UC3M para realizar una estancia de investigación, financiada por el programa "Ayudas para la movilidad de investigadores de la Universidad Carlos III de Madrid en centros de investigación nacionales o extranjeros, Convocatoria 2015. Modalidad B: investigadores en formación predoctoral". La estancia de investigación se realizó en el Departamento de Matematicas de Heriot-Watt University (Edimburgo). Además, esta tesis ha sido respaldada parcialmente por el Ministerio de Economía y Competitividad de España con el proyecto MTM2015-65899-P y por el Ministerio de Ciencia, Innovación y Universidades de España con el proyecto PGC2018-097104-B-100. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: John Werner Peter Volker.- Secretario: Julia María Novo Martín.- Vocal: Sara Pollock

Mención Internacional en el título de doctor This Doctoral Thesis has been supported by the Severo-Ochoa grants SEV-2015- 0554 and BES-2016-077968, from the Spanish Ministry of Economy and Competition (MINECO), via the Instituto de Ciencias Matemáticas (ICMAT) of the Centro Superior de Investigaciones Científicas (CSIC). Additionally, the author has been partially supported by the Ministry of Economy and Competitiveness of Spain under Research Project MTM2017-84098-P. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Enrique Pardo Espino.- Secretario: Luis Alberto Ibort Latre.- Vocal: Nadia S. Larsen

Mención Internacional en el título de doctor In this thesis we present a pseudospectral method in an interval, in the disk and in the circular annulus. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under non-homogeneous Dirichlet, Neumann and Robin boundary conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary di erential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this thesis is applied to estimate Sherwood numbers integrating the mass flux to the disk and we also implement it to solve Lotka-Volterra systems and nonlinear diffusion problems involving chemical reactions. Furthermore, we simulate numerically the classical positive solutions, large solutions and metasolutions of the degenerate diffusive logistic equation with spatial heterogeneities in an interval and in the disk through a collocation spectral method presented here. Some metasolutions of the different branches of metasolutions for the logistic equation introduced in [27] are simulated by the first time. Moreover, we simulate numerically through a collocation spectral method jointly with pathfollowing techniques the classical and non-classical non-negative branches of solutions of Lotka- Volterra competing systems allowing the presence of spatial heterogeneities. Some of these solutions are simulated for the first time. Also we prove the convergence of an innovative Chebyshev-Gauss-Lobatto( (CGL) pseudospectral method applied to fourth order boundary value problems. The proposed method enjoyed all the advantageous of the pseudospectral methods. Moreover, we can select (N - 3) interior CGL collocation points to enforce the equation, meanwhile we use the remaining four collocation points to assure the boundary conditions. In addition, the numerical methods introduced in this thesis are extremely innovative because they can be used to solve non-radially symmetric problems. The models are of a huge interest in Spatial Ecology because they enable us to analyse the effects of the spatial heterogeneity on the evolution of the terrestrial ecosystems. En esta tesis doctoral presentamos un método pseudoespectral en un intervalo, en el disco y en un anillo circular. A diferencia de otros métodos conocidos, el disco no está duplicado. Más aún, resolvemos la ecuación de Laplace bajo condiciones de frontera de tipo Dirichlet, Neumann y Robin, haciendo uso únicamente de los elementos de las matrices de diferenciación correspondientes. Vale la pena mencionar que no utilizamos ningún tipo de cuadratura ni necesitamos resolver ningún sistema desacoplado de ecuaciones diferenciales ordinarias, ni tampoco el uso de una condición de polo, ni siquiera se requiere un ”lifting”. También resolvemos numéricamente diversos ejemplos para mostrar la convergencia espectral. El método pseudoespectral desarrollado en esta tesis es aplicado para estimar el número de Sherwood integrando el flujo de masa en el borde del disco. También es implementado para resolver sistemas Lotka-Volterra de competición y problemas de difusión no lineales. Asimismo, simulamos numéricamente a través de un método espectral de colocación las soluciones clásicas positivas, las soluciones largas y las metasoluciones de la ecuación logística difusiva degenerada con heterogeneidades espaciales en un intervalo y en el disco. Algunas metasoluciones de diferentes ramas de metasoluciones para la ecuación logística introducidas en [27] son simuladas por primera vez. Además, simulamos numéricamente las soluciones clásicas no negativas y las soluciones no clásicas de los sistemas de competición de Lotka-Volterra a través de un método espectral de colocación junto con técnicas de continuación numérica de ramas de soluciones, permitiendo la presencia de heterogeneidades espaciales. Algunas de estas soluciones se simulan por primera vez. También probamos la convergencia de un método pseudoespectral innovador de Chebyshev-Gauss-Lobatto (CGL) aplicado a problemas de valores contorno de cuarto orden. El método propuesto disfruta de todas las ventajas de los métodos pseudoespectrales. Además, podemos seleccionar (N - 3) CGL puntos interiores de colocación donde forzar la ecuación, mientras utilizamos los cuatro puntos de colocación restantes para asegurar las condiciones de frontera. Los métodos numéricos presentados en esta tesis son extremadamente innovadores porque sirven para resolver problemas radialmente no simétricos. Además, los modelos son de un enorme interés en Ecología Espacial porque nos permiten analizar los efectos de las heterogeneidades espaciales en la evolución de los ecosistemas terrestres. Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección de la profesora Marcela Molina Meyer. Se contó con una beca de la UC3M de ayuda al estudio de máster y posteriormente con un contrato predoctoral de la UC3M. Adicionalmente se recibió una ayuda parcial del Ministerio de Ciencias, Tecnología y Universidades de España bajo proyecto de investigación MTM2015-65899-P. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: J. Chris Eilbeck.- Secretario: Luis Francisco López Bonilla.- Vocal: Víctor Pereyra

Mención Internacional en el título de doctor This PhD thesis belongs to the area of Numerical Linear Algebra. Specifically, to the numerical solution of the Rational Eigenvalue Problem (REP). This is a type of eigenvalue problem associated with rational matrices, which are matrices whose entries are rational functions. REPs appear directly from applications or as approx imations to arbitrary Nonlinear Eigenvalue Problems (NLEPs). Rational matrices also appear in linear systems and control theory, among other applications. Nowa days, a competitive method for solving REPs is via linearization. This is due to the fact that there exist backward stable and efficient algorithms to solve the linearized problem, which allows to recover the information of the original rational problem. In particular, linearizations transform the REP into a generalized eigenvalue pro blem in such a way that the pole and zero information of the corresponding rational matrix is preserved. To recover the pole and zero information of rational matrices, it is fundamental the notion of polynomial system matrix, introduced by Rosenbrock in 1970, and the fact that rational matrices can always be seen as transfer functions of polynomial system matrices. This thesis addresses different topics regarding the problem of linearizing REPs. On the one hand, one of the main objectives has been to develop a theory of li nearizations of rational matrices to study the properties of the linearizations that have appeared so far in the literature in a general framework. For this purpose, a definition of local linearization of rational matrix is introduced, by developing as starting point the extension of Rosenbrock’s minimal polynomial system matrices to a local scenario. This new theory of local linearizations captures and explains rigor ously the properties of all the different linearizations that have been used from the 1970’s for computing zeros, poles and eigenvalues of rational matrices. In particu lar, this theory has been applied to a number of pencils that have appeared in some influential papers on solving numerically NLEPs through rational approximation. On the other hand, the work has focused on the construction of linearizations of rational matrices taking into account different aspects. In some cases, we focus on preserving particular structures of the corresponding rational matrix in the li nearization. The structures considered are symmetric (Hermitian), skew-symmetric (skew-Hermitian), among others. In other cases, we focus on the direct construc tion of the linearizations from the original representation of the rational matrix. The representations considered are rational matrices expressed as the sum of their polynomial and strictly proper parts, rational matrices written as general trans fer function matrices, and rational matrices expressed by their Laurent expansion around the point at infinity. In addition, we describe the recovery rules of the information of the original rational matrix from the information of the new lineari zations, including in some cases not just the zero and pole information but also the information about the minimal indices. Finally, in this dissertation we tackle one of the most important open problems related to linearizations of rational matrices. That is the analysis of the backward stability for solving REPs by running a backward stable algorithm on a linearization. On this subject, a global backward error analysis has been developed by considering the linearizations in the family of “block Kronecker linearizations”. An analysis of this type had not been developed before in the literature. Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección del profesor Froilán Martínez Dopico y codirección de la profesora Silvia Marcaida Bengoechea. Se contó durante cuatro años con un contrato predoctoral FPI, referencia BES-2016-076744, asociado al proyecto ALGEBRA LINEAL NUMERICA ESTRUCTURADA PARA MATRICES CONSTANTES, POLINOMIALES Y RACIONALES, referencia MTM2015-65798-P, del Ministerio de Economía y Competitividad, y cuyo investigador principal fue Froilán Martínez Dopico. Asociado a este contrato, se contó con una ayuda para realizar parte de este trabajo durante dos es tancias internacionales de investigación. La primera estancia de investigación se realizó del 30 de enero de 2019 hasta el 1 de marzo de 2019 en el Department of Mathematical En gineering, Université catholique de Louvain (Bélgica), bajo la supervisión del profesor Paul Van Dooren. La segunda estancia de investigación se realizó del 15 de septiembre de 2019 hasta el 19 de noviembre de 2019 en el Department of Mathematical Sciences, University of Montana (EEUU), bajo la supervisión del profesor Javier Pérez Alvaro. Dado que la entidad beneficiaria del contrato predoctoral es la UC3M mientras que el otro codirector de tesis, la profesora Silvia Marcaida Bengoechea, pertenece al Departamento de Matemáticas de la Universidad del País Vasco (UPV/EHU), el trabajo con la profesora Silvia Marcaida se reforzó mediante visitas a la UPV/EHU, financiadas por ayudas de la RED temática de Excelencia ALAMA (Algebra Lineal, Análisis Matricial y Aplicaciones) asociadas al los proyectos MTM2015-68805-REDT y MTM2017-90682-REDT. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Ion Zaballa Tejada.- Secretario: Fernando de Terán Vergara.- Vocal: Vanni Noferini

Mención Internacional en el título de doctor Cancer kills 26.4% of Spanish people. It is the second cause of death, just behind diseases of the circulatory system, 28.3% [1]. The growth of new blood vessels from the existing vasculature in response to chemical signals from a tumor is called tumorinduced angiogenesis and it is closely related to cancer and metastasis. The growth rate of a tumor is considerably increased in its vascular stage compared to its avascular and solid stage, therefore treating cancer turns excessively difficult and the survival rates rapidly decrease [2]. Among diseases that cause disability but not substantial mortality, age-related macular degeneration may cause severe loss of vision or blindness in many people, particularly the elderly. It is projected that 196 million people will be affected by age-related macular degeneration in 2020, increasing to 288 million by 2040 [3], which is likely an underestimation [4]. With age, Bruch’s membrane gets thicker and some damaged cells in the retina become inflamed. The secretion of chemical signals from those cells due to their inflammation induces angiogenesis, but the new blood vessels are disorganized and leaky causing the loss of vision. John Hunter was the pioneer in describing the vessel formation process in 1787 [5], but the first person who coined the word “angiogenesis” was Arthur T. Hertig in 1935 [6]. He was studying the formation of new blood vessel in the primary placenta of the macaque monkey when this word was used for the first time. Years later, in 1971, Judah Folkman hypothesized that tumors emit Tumor Angiogenic Factors (TAF) to attract blood vessels to them [7]. This investigation triggered the research field of angiogenesis in cancer and in 1989 one of the most important angiogenic factors was discovered: the Vascular Endothelial Growth Factor (VEGF). Since then, drugs with antiangiogenic effects have been investigated for cancer, age-related macular degeneration and other diseases, as it is involved in more than seventy different diseases. However, angiogenesis also occurs in normal and vital processes such as wound healing or the growth of a fetus. The difference between physiological and pathological angiogenic processes is a matter of balance. In a healthy process, angiogenesis develops to its proper extent and then stops, while in pathological processes angiogenesis does not stop or it does not develop sufficiently. Angiogenesis keeps the number of blood vessels needed in balance: few blood vessels cause tissue death, while uncontrolled vascular proliferation can lead to cancer, macular degeneration and other diseases. Angiogenesis is a complex, multistep and well regulated process where biochemistry and physics are intertwined. The process entails signaling in vessel cells being driven by both chemical and mechanical mechanisms that result in vascular cell movement, deformation and proliferation. In a later stage of angiogenesis, vessel cells rearrange to form lumen and allow the perfusion of the blood inside the sprout. Depending on what induces the angiogenesis, different environments and cells should be considered, for instance in the retina. A detailed review of the processes involved in angiogenesis from the biological point of view is given in section 1.1. Beyond experimental investigations, mathematical models of angiogenesis try to help in understanding the process and how the relevant mechanisms of angiogenesis interact. The approach of some models focus on a single scale or a single process of those involved to deepen the knowledge about it. Others span multiple scales or the whole process to give an idea about how to prevent or favor angiogenesis. In section 1.2, we briefly review the mathematical models of angiogenesis that have been used to date as well as those when angiogenesis occurs in the retina and models of lumen formation, the late stage of angiogenesis. A crucial question about modeling is how to integrate the multiple scales and mechanisms present in angiogenesis in a mathematical model. A model is expected to be useful to explore methods for promoting and inhibiting angiogenesis. However, answering this question with this expectation is not a simple task. Assembling all the processes involved with their different time and length scales requires to develop a cellular dynamics model combined with models for the continuum fields. We achieved this objective by developing a hybrid cellular Potts model of early stage angiogenesis, given in chapter 2. In contrast to recent models, this mathematical and computational model is able to explore the role of biochemical signaling and tissue mechanics. A exhaustive description of the results of the numerical simulations complete the chapter 2. The advantages of discovering the reasons why angiogenesis starts in the retina or inhibitory mechanisms are innumerable. Unraveling the causes of neovascularization in the retina and giving possible solutions for age-related macular degeneration are our motivation to adapt the angiogenesis model of chapter 2 to the retina. In chapter 3, we present the model and the numerical results. If mathematical models of angiogenesis that incorporate multiple scales and cellular signaling processes are not that common, those that also include lumen formation are almost nonexistent. In chapter 4, we describe two models of lumen formation and their results. The lumen formation in the first model takes place in a already developed sprout. Although some restrictions in the model make its applications and possibilities limited, its study is convenient to establish the basis of the second proposed model. In this second model, the lumenization occurs while the sprout is developing and the pressure of the blood is involved, following recent experiments of lumen formation during angiogenesis. This model is work in progress, but we believe that showing the preliminary results in chapter 4 may be interesting. A critical step in the development of a mathematical and computational model is to analyze the viability of its simulations. The simulations of the model in chapter 2 have been carried out thanks to the parallel computing on Graphics Processing Units (GPUs), as well as simulations of chapters 3 and 4. The large amount of square elements of the grid, nodes, cells and sprouts make this type of computation suitable for these models. The way they have been implemented is explained in chapter 5. Finally, conclusions of this thesis and future work are drawn in the last chapter 6. This chapter highlights and summarizes the research that has been carried out and proposes future extensions and applications of this work. La investigación de esta tesis ha sido financiada por los proyectos de investigación del Ministerio de Economía y Competitividad (ahora FEDER/Ministerio de Ciencia, Innovación y Universidades–Agencia Estatal de Investigación) No. MTM2014-56948-C2-2-P y No. MTM2017-84446-C2-2-R. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Ana María Carpio Rodríguez.- Secretario: Filippo Terragni.- Vocal: Stephen W. Teitsworth

La presente tesis está dedicada al estudio de ciertos operadores de tipo integrodiferencial, de orden diferencial cercano a cero, de interés en aplicaciones y en teoría matemática.(...) El trabajo se divide en dos partes principales. En la primera parte estudiamos operadores lineales definidos por núcleos de Léevy no integrables poniendo el énfasis en núcleos con singularidad muy débil, en el límite de integrabilidad; en la segunda parte estudiamos operadores no lineales de tipo p-laplaciano fraccionario, incluyendo también núcleos débilmente no integrables. The present thesis is dedicated to the study of certain integro-differential operators with differential order close to zero, which are interesting both, from the theoretical point of view and for the applications.(...) This work is divided into two main parts. In the first part we study linear operators defined by non integrable Lévy kernels, placing the emphasis on kernels with a very weak singularity. In the second part we study nonlinear operators of the fractional p-Laplacian type, including also weakly non-integrable kernels. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Salvador Villegas Barranco.- Secretario: Cristina Brandle Cerqueira.- Vocal: Raúl Ferreira de Pablo

In chapter 2 we presented results obtained through experiments on a vibrated monolayer of granular metallic rods, focusing on the effects that introducing a circular obstacle to the centre of the cavity (thus creating an annulus) would have on the steady states of the system. Although topological arguments suggested that defects found on systems with no obstacle would not appear, we observed a separation of the system into regions consisting of smectic and tetratic ordering, the latter containing completely unordered point-like defects. We argue that the topological approach based on a continuum argument is rendered invalid by the size of the elements involved, in particular the ratio of the central obstacle size to particle lengths. The tetratic structures can be viewed as domain walls separating the smectic regions, which were invariably equal to four in our experiments. In chapter 3 we used the SPT to study two different systems of two-dimensional particles. In the first one, composed of particles shaped like hard isosceles triangles, we found, for the one component fluid, 1st and 2nd order Isotropic-Nematic transitions and for symmetric binary mixtures a 1st order Nematic-Nematic transition ending in a critical point and a Landau point where two first-order Nematic-Triatic transitions and a Nematic-Nematic demixing transition coalesce. We also found that for these particular mixtures a Triatic phase can be stabilized even though both species involved do no exhibit a stable Triatic phase on the one-component limit. To the best of our knowledge this is the first example of a Nematic-Nematic transition on one-component hard particle systems. For the second system, of hard rectangular particles, we found that polydispersity has an important effect on the phase diagram, strengthening the 1st order Isotropic-Nematic transition and decreasing the stability of the Tetratic phase. Also we found that polydispersity induces strong fractionation between the coexisting phases. Finally in chapter 4 we studied the effects of nano-confinement on a system of hardboard like particles. Following the restricted orientation (Zwanzig) approach we obtained a DF using the FMT to approximate the excess part of the free-energy of the system by means of numerical minimization. We considered a particular set of particle aspect ratios as well as slightly biaxial particles. We hoped to determine whether homeotropic configurations could be stabilized by the presence of two hard walls. We obtained that, for particular particle aspect ratios, homeotropic phases do stabilize and also that allowing for particle biaxiality greatly enriches the phase diagram. The results obtained can be explained by considering how entropy is maximized as the system approaches close packing. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Mederos Martín.- Secretario: Daniel de las Heras Díaz-Plaza.- Vocal: Giorgo Cinacchi

Mención Internacional en el título de doctor Los polinomios ortogonales de Sobolev han sido estudiados en profundidad a lo largo de las últimas tres décadas como extensión natural de los polinomios ortogonales estándar. Existen tres grandes clasificaciones de los polinomios de Sobolev según las medidas involucradas en el producto interno que los origina: caso continuo, caso discreto, y el caso discreto-continuo. En nuestro trabajo se abordan cada uno de ellos en los Capítulos II, III y IV respectivamente. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: José Mauel Rodríguez García.- Secretario: Ramón Ángel Orive Rodríguez.- Vocal: Ana Pilar Foulquié Moreno

Mención Internacional en el título de doctor Matrix polynomials arise frequently associated with Polynomial Eigenvalue Problems (PEPs). The standard way to solve PEPs is by means of (strong) linearizations, which are matrix pencils (namely, matrix polynomials of grade 1), that allow to recover the relevant spectral information of the PEP in a relatively simple way, using appropiate algorithms. Any matrix polynomial has infinitely many linearizations. One way to construct such linearizations is by means of symbolic constructions (involving no arithmetic operations) consisting of block-partitioned pencils whose blocks contain the coeficients of the matrix polynomial. (Generalized) companion forms are exactly particular cases of such constructions. In particular, the central notion in this thesis is the notion of (generalized) companion forms (of grade ℓ) of matrix polynomials. To abbreviate, we refer to “(generalized) companion forms" when ℓ = 1 (that is, when they are matrix pencils). One of the properties of (generalized) companion forms (of grade ℓ), we are especially interested in is the sparsity. Sparse (generalized) companion forms (of grade ℓ) are those having the smallest possible number of nonzero entries. This notion is interesting from the point of view of the simplicity in the construction, and it may have some relevance from the numerical point of view. We are also interested in structured (generalized) companion forms (of grade ℓ), that is, companion forms that preserve some of the structures that matrix polynomials frequently present in applications. The most relevant structures are the (skew-)symmetric, (skew)-Hermitian, (anti-)palindromic and alternating structures. These structures correspond to certain symmetries in the entries of the coefficients of the matrix polynomial, and in turn imply some symmetries in their spectral information. To compute this spectral information is the main goal of the PEPs. Therefore, to preserve the symmetries in the block entries of the matrix polynomial is key in the whole process, in order to avoid that, due to numerical errors, the symmetries in the spectrum are lost, which would lead to meaningless results. This is the motivation behind searching for structured (generalized) companion forms, even though we do not deal with numerical issues in this part of the thesis. This dissertation consists of three parts. The first one is devoted to introduce a wide family of quasi-sparse companion forms (which have a small number of nonzero entries, although not necessarily the smallest one) for arbitrary square matrix polynomials, expressed in the monomial basis, which includes all the quasi-sparse companion forms known so far (the Fiedler-like families and the recent block-Kronecker linearizations), and which extends the class of “companion matrix patterns" introduced in [58] for monic scalar polynomials. In addition, a thorough study of the sparsity of companion forms in this new family is carried out. This first part also includes a theoretical analysis of generalized companion forms. In particular, we introduce a new notion of generalized companion forms for matrix polynomials expressed in the monomial basis. This definition is quite general and extends the notions of companion form in [41], generalized companion matrix in [68], and Ma-Zhan companion matrix in [100], as well as the previous class of quasisparse companion forms. Then, we analyze some algebraic properties of generalized companion pencils as well as the sparsity of these constructions (by imposing some natural conditions on their entries). The second part of this thesis is related to structured (generalized) companion forms of matrix polynomials, expressed in the monomial basis. It is well known that is not possible to construct companion forms that preserve any of the structures mentioned before for matrix polynomials of even degree. This motivates the search for more general companion forms, in particular (generalized) companion forms of grade ℓ (for ℓ > 1). These constructions are strong ℓ-ifications and, like (generalized) companion forms, they are constructed in a symbolic way containing the coeficients of the polynomial. The term “structured" here means that these companion forms preserve the structure of the polynomial (that is, the matrix polynomial and its ℓ-ification have the same structure). Then, we present, for the first time, a family of structured (generalized) companion forms of grade ℓ, that preserve any of the structures mentioned before, for matrix polynomials of grade k = sℓ, with s being an odd number, extending recent works (such as [55, 56]), and we give a procedure to obtain sparse ℓ-ifications within this family. Finally, we prove that there are no structured companion quadratifications for quartic matrix polynomials. The third part of this thesis focuses on matrix polynomials expressed in a Lagrange(-like) basis. Matrix polynomials expressed in non-monomial bases are more natural for certain applications than those expressed in the monomial basis and, when we aim to solve the PEP, it is important to avoid reformulating the matrix polynomial into the monomial basis (because this may introduce numerical errors in the computation). Then, this part is focused on new contributions regarding polynomials in non-monomial bases, specifically for the Lagrange basis. In this part, we develop both a numerical and a theoretical analysis. First, regarding the numerical analysis, we perform a first-order backward error analysis of the PEP solved by applying a certain linearization of a matrix polynomial (expressed in the Lagrange basis). More precisely, we obtain that, if the matrix polynomial is suitably scaled, then we get a good backward stability (to first order) from the polynomial point of view when the linearization is combined with a backward stable method for computing the spectral information of the matrix polynomial. Regarding the theoretical analysis, this third part is also devoted to present a family of structured strong linearizations for matrix polynomials of odd grade k, expressed in a Lagrange(-like) basis, which enables the preservation of the spectral symmetries for the matrix polynomial. Se recibió ayuda parcial de los siguientes proyectos de investigación: “Structured Numerical Linear Algebra for Constant, Polynomial and Rational Matrices" (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2015-65798-P, IP: F.M. Dopico) y dos más de la red temática ALAMA (Ministerio de Economía y Competitividad, Números de proyectos: MTM2015-68805-REDT, IP: I. Zaballa, y MTM2017-90682-REDT, IP: J. Mas). Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Froilán César Martínez Dopico.- Secretario: Roberto Canogar Mckenzie.- Vocal: Susana Margar Figueiredo de Sousa

Mención Internacional en el título de doctor In the present dissertation, we are interested in the spectral properties of the magnetic Laplacian. We study the spectrum of the Magnetic Laplacian under some geometrical perturbation on the graph. In particular, we are interested in the existence of spectral gaps, i.e., intervals that do not contain spectrum. We begin this dissertation giving a general context and a brief summary of some studies about the DML and in relation to the work in this thesis. Este trabajo fue apoyado por el Ministerio de Economía y Competitividad mediante los programas DGI MTM2014-54692-P y MTM2017-84098-P. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Carlos Marijuán López.- Secretario: Julio Íñigo de Vicente Majua.- Vocal: Natalia Saburova

Mención Internacional en el título de doctor Collective behavior and, more specifically, flocking are phenomena observed in living systems, from bacterial colonies and spermatozoa, to larger systems such as insects and birds. These organizations exhibit changes from disordered to coherent behavior, which are examples of spontaneous symmetry-breaking out of equilibrium. Collective migrations in these systems can be predicted by simple models such as the Vicsek model (VM) or its variants, in which particles tend to align their velocities to an average of their neighbours’. The change from a disordered state to an ordered state can occur continuously or discontinuously and a variety of resulting patterns are possible. The study of mathematical models of these systems may reveal these changes to be bifurcations in their governing equations. We consider a system of particles moving within a two dimensional box with periodic boundary conditions. In Chapter 2, and following Ihle’s approach, we derive a kinetic equation for a one-particle distribution function in the limit of infinitely many particles by assuming molecular chaos. The kinetic equation is discrete in time and space and it always has a simple uniform solution that corresponds to the disordered state of the system. We have carried out a linear stability analysis of this state and studied the possible bifurcations issuing from it. In the usual case, particles align their velocities to their average velocity with any other particles within a circle of influence plus some angular noise, which has a uniform probability density. The spectrum of the linearized equation has always one multiplier on the unit circumference and there is another one that moves from inside to outside the unit circle as a control parameter crosses a critical value. We use bifurcation methods to derive amplitude equations that describe solutions issuing from the disordered state. The amplitude equations comprise a conservation law for a density disturbance coupled to a two dimensional vector equation for a current density. Analysis and numerical simulations of these equations show that their solutions exhibit an interplay between parabolic and hyperbolic behavior in two different time scales when the distance to the critical value of the bifurcation control parameter goes to zero. In this limit, there appear oscillation frequencies that give rise to resonance phenomena if the alignment rule contains a periodic function of time. Direct simulation of the VM confirms the existence of these resonances. In Chapter 3, we use the same methodology to study the effect of modifying the probability density of the noise in the alignment rule by which VM particles change their velocities. The mechanism of velocity synchronization consists of: active particles may be conformist and align their velocities to the average velocity of their neighbors, or be contrarian and move opposite to the average angle. Depending on the weights of conformist and contrarian or almost contrarian rules, we study the ordered state solutions of the amplitude equations corresponding to period-doubling, Hopf, or pitchfork bifurcations of the disordered state. In Chapter 4, we consider the collective migration of epithelial cell monolayers moving on a surface. This phenomenon is crucial for many relevant processes including wound healing, morphogenesis, and cancer-cell invasion during metastasis. There are many experiments on confluent cellular motion and different mathematical and computational models in the literature. A convenient model based on the physics of foams considers the cells as non-overlapping two dimensional convex polygons. In the active vertex model we study, the cell centers are in a Delaunay triangulation and are subject to forces that constrain them to have target areas and perimeter length, other forces that try to align their velocities to neighboring cells (as in the VM), friction with the substrate, inertia, and stochastic forces. We have simulated numerically this model in two different cases related to wound healing and to invasion of one cell collective by another one: (i) a cellular monolayer spreading on empty space, and (ii) the collision of two different cell populations in an antagonistic migration assay. For (i), we discuss how inertia is necessary to explain the larger size of cells in the boundary with respect to those in the interior of the layer. For (ii), we discuss which parameters of the model produce results that agree with experiments by P. Silberzan’s group. In both cases, the interfaces that separate cells from empty space or cells belonging to different populations are quite rough and may shed and absorb islands as time elapses. To analyze both images from experiments and results of numerical simulations, we use topological data analyses of the interfaces. La investigación de esta tesis ha sido financiada por los proyectos de investigación del Ministerio de Economía y Competitividad (ahora FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación) No. MTM2014-56948-C2-2-P y No. MTM2017-84446-C2-2-R. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Antonio Marquina.- Secretario: Ester Aurora Torrente Orihuela.- Vocal: Björn Birnir

Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Manuel Bello Hernández

Mención Internacional en el título de doctor This thesis set out to contribute to the realm of social physics, with a particular focus on human social networks. Our approach, however, is somewhat di erent from what is typical in disciplines such as complex systems or statistical physics. Rather than simplifying the features of the constituents of our system (people), and stressing their rules of interaction, we focus on better understanding those very same constituents, modelling them as social atoms. Our rationale is that a better understanding of such an atom may shed light on how (and why) it interacts with other atoms to form social collectives. Given its robustness and the evolutionary roots of its premises, we use the Social Brain Hypothesis as our departure point. This theory states that the evolutionary drive behind the development of large brains in humans was the need to process social information and that the limited capacity of our brains imposes a limit to the number of relationships we can manage— the so-called “Dunbar’s number”, roughly 150. Moreover, evidence keeps revealing that these relationships are further organised in a series of hierarchically inclusive layers with decreasing emotional intensity, whose sizes exhibit a more or less constant scaling. Notwithstanding the empirical evidence, neither the presence of scaling in the organisation of personal networks nor its connection with limited cognitive skills had been explained so far. In Chapter 2 we present a mathematical model that solves this puzzle. The assumptions of the model are quite simple, and well founded on empirical evidence. Firstly, the number of relationships we maintain tends to be stable on average. Secondly, these relationships are costly, and our resources are limited. With these two premises, our results show that the hierarchical organisation emerges naturally from the principle of maximum entropy. Not only that, but we also predict a hitherto unnoticed regime of organisation whose existence we prove using several datasets from communities of immigrants. The former model considers that relationships can only belong to a discrete set of categories (layers). In Chapter 3 we extend it so that relationships are classified in a continuum. This modification allows us to test the model with data from very di erent sources such as online communications, face-to-face contacts, and phone calls. Our results show that the two regimes of organisation found in the previous model persist in this variant, and reveal the underlying existence of a (universal) scaling parameter which does not depend on any particular number of layers. To incorporate these ideas into socio-centric models, we build on the so-called Structural Balance Theory. This theory, underpinned by psychological motivations, posits that the structure of social networks of positive and negative relationships are highly interdependent. However, the theory has received little empirical validation, and negative social relationships are poorly understood—both from an ego-centric and a socio-centric perspective. For that reason, we turn to developing an experimental software in order to gather data within a school. In Chapters 4 and 5 we present results from these experiments. In Chapter 4 we analyse the socio-centric networks using machine learning techniques and find that the structure of positive and negative networks is indeed very much connected. Besides, we study the two types of networks separately, showing that they exhibit quite distinct features and that gender e ects in negative social networks are weak and asymmetrical for boys and girls. In Chapter 5, on the other hand, we focus on the structure of negative personal networks. Remarkably, using data from two di erent experimental settings, we show that the structure of personal networks of negative relationships mirrors that of the positive ones and exhibits a similar scaling—albeit their size is significantly smaller. Chapter 6 summarises our results and presents future (and current) lines of investigation. Among them, we outline a model of a social fluid that uses the insights gained with this thesis to build a model of social collectives as ensembles of personal networks. This model is compatible, at the micro-level, with the observations of the social brain hypothesis, and, at the macro-level, with the premises of the structural balance theory. This thesis would not have been possible without the support of Fundación BBVA through its 2016 call project ”Los números de Dunbar y la estructura de las sociedades digitales: modelización y simulación (DUNDIG)”, and we are very thankful for it. Support for early stages of this work through projects IBSEN (European Commission, H2020 FET Open RIA 662725) and VARIANCE (Ministerio de Economía y Competitividad/FEDER, project no. FIS2015-64349-P) is also acknowledged Programa Oficial de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Javier Martín Buldú.- Secretario: José Luis Molina González.- Vocal: Roberta Sinatra

Mención Internacional en el título de doctor This PhD thesis belongs to the areas of Matrix Analysis and Numerical Linear Algebra and has been devoted to solve problems that have arisen as a consequence of the development of new applications during the last two decades. More precisely, these problems belong to two related areas: polynomial eigenvalue problems (PEPs) and rational eigenvalue problems (REPs). Due to the mentioned recent interest into the numerical solution of the PEP, several condition numbers for simple eigenvalues of regular matrix polynomials have appeared in the literature. In order to study the conditioning of all the eigenvalues of a matrix polynomial in a unified framework, some of these eigenvalue condition numbers consider the matrix polynomial to be expressed in homogeneous form. Then, the first problem we address in the field of matrix polynomials is to obtain a relationship between the two main homogeneous eigenvalue condition numbers that have appeared in the literature. Furthermore, we also obtain a relationship between the non-homogeneous and the former homogeneous eigenvalue condition numbers. This latter relationship allows to extend results that have been proved for one of the versions of the eigenvalue condition numbers to the other. We also obtain a relationship between the homogeneous and the non-homogeneous backward errors of approximate eigenpairs that states they are essentially equivalent. Möbius transformations is a classical tool that have been used in the theory of matrix polynomials (and in their applications) since, at least, the 1950s. One of the main use of these transformations is to convert a polynomial eigenvalue problem into another that can be solved more easily. If the numerical solution of a problem is obtained by applying a Möbius transformation, an important question is to study if this transformation worsen the conditioning of the problem and/or the backward errors of the approximate solutions. In this dissertation, we present for the first time, up to our concern, a general study of the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of matrix polynomials. The study of the effect of Möbius transformations on eigenvalue condition numbers and backward errors of approximate eigenpairs is separated in two parts: one for the homogeneous and another for the non-homogeneous version of these magnitudes. Since the homogeneous formulation allows us to obtain two clear and simple results, we perform this analysis first on the homogeneous case. We prove that for any matrix polynomial and any simple eigenvalue, if the matrix that induces the Möbius transformation is well conditioned, then the transformation approximately preserves the homogeneous eigenvalue condition numbers and backward errors of approximate eigenpairs when small perturbations of the matrix polynomial relative to the norm of the whole polynomial are considered in the definition of both the condition number and the backward error. If in the definition of the former magnitudes, perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, well-conditioned matrices induce Möbius transformations that approximately preserve the eigenvalue condition numbers and backward errors of approximate eigenpairs only if a factor that depends on the norm of the coefficients of the polynomial is moderate. For the non-homogeneous versions of the eigenvalue condition number, the conclusions obtained are not so simple. Nevertheless, we present sufficient conditions depending not only on the conditioning of the matrix that induces the Möbius transformation, but also on the eigenvalue and the corresponding eigenvalue of the Möbius transform. In order to perform this analysis, we use the study realized for the homogeneous case and the relationship obtained between the homogeneous and the non-homogeneous eigenvalue condition numbers. Furthermore, the relationship between the homogeneous and the non-homogeneous backward error of approximate eigenpairs allows us to conclude that the effect of Möbius transformations is the same in the homogeneous and the non-homogeneous backward errors. We also address the problems of degree-preserving quasi-triangularization of matrix polynomials and inverse quasi-triangularization. These two problems have already been studied in the literature by Taslaman, Tisseur and Zaballa in 2013, but only for matrix polynomials de_ned over algebraically closed fields or R. In this dissertation we find preserving degree quasi-triangularizations of regular matrix polynomials defined over an arbitrary field, that is, for a given matrix polynomial P(λ), we build a block upper triangular matrix polynomial Q(λ) that is unimodularly equivalent to P(λ) and has the same size and the same degree as P(λ), as well as the same finite and infinite elementary divisors. The maximum size of the blocks of the obtained matrix polynomial Q(λ) is the maximum degree of the irreducible factors of the invariant polynomials of P(λ). Furthermore, we prove that given a list of prescribed data (including possibly infinite elementary divisors), we can find a quasi-triangular matrix polynomial of given degree d and size n×n that realizes that list and such that the maximum size of the blocks does not exceed the maximum degree of the irreducible factors of the prescribed invariant polynomials if and only if dn is equal to the sum of the degrees of the invariant polynomials and the infinite elementary divisors (if any). The rational matrices problem that we study in this PhD thesis (which has been settled recently for matrix polynomials by De Terán, Dopico and Van Dooren in 2015) consists of funding a necessary and sufficient condition for the existence of a rational matrix that realizes a prescribed complete list of structural data (finite and infinite zeros and poles together with their structural indices and the minimal indices of left and right rational nullspaces). This necessary and sufficient condition is that these data satisfy a fundamental relation appearing in Van Dooren's Index Sum Theorem. Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección del profesor Froilán Martínez Dopico. Se contó durante cuatro años con un contrato predoctoral FPI, referencia BES-2013-065688, asociado al proyecto MTM2012- 32542, del Ministerio de Economía y Competitividad, y cuyo investigador principal fue Froilán Martínez Dopico. Gracias a este contrato, se contó con dos ayudas para realizar parte de este trabajo durante sendas estancias de investigación. La primera estancia de investigación se realizó en el Department of Mathematics de la Western Michigan University (EEUU), bajo la supervisión del profesor D. Steven Mackey. Dicha estancia fue financiada por el programa “Ayudas a la movilidad predoctoral para la realización de estancias breves en centros de I+D”, referencia EEBB-I-16-11462, del Ministerio de Economíıa y Competitividad. La segunda estancia de investigación se realizó en el Department of Mathematics de la University of California at Santa Barbara (EEUU), bajo la supervisión de la profesora María Isabel Bueno. Dicha estancia fue financiada por el programa “Ayudas a la movilidad predoctoral para la realización de estancias breves en centros de I+D”, referencia EEBB-I-18-13037, del Ministerio de Economía y Competitividad. Además, se recibióo ayuda parcial de los proyectos de investigación: “Structured Numerical Linear Algebra: Matrix Polynomials, Special Matrices, and Conditioning” (Ministerio de Economía y Competitividad, Número de proyecto: MTM2012-32542, IP: F. M. Dopico), “Structured Numerical Linear Algebra for Constant, Polynomial and Rational Matrices” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2015- 65798-P, IP: F. M. Dopico) y los proyectos de investigación de la red temática ALAMA (Ministerio de Economía y Competitividad, Números de proyectos: MTM2015-68805-REDT, IP: I. Zaballa, y MTM2017-90682-REDT, IP: J. Mas). Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Paul Van Dooren.- Secretario: Fernando de Terán Vergara.- Vocal: Ion Zaballa Tejada

The problem of computing eigenvalues, eigenvectors, and invariant subspaces of symplectic matrices plays a major role in many applications, in particular in control theory when the focus is on discrete systems. If standard numerical methods for the solution of the symplectic eigenproblem are applied that do not take into account the special symmetry structure of the problem, then not only the existing symmetry in the spectrum of symplectic matrices may be lost in finite precision arithmetic, but more importantly other relevant intrinsic features or invariants may be ignored although they have a major influence in the corresponding computed eigenvalues. The importance of structure-preservation has been acknowledged in the Numerical Linear Algebra community since several decades, and consequently many algorithms have been developed for the symplectic eigenvalue problem that preserve the given structure at each iteration step. The error analysis for such algorithms requires a corresponding stucture-preserving perturbation theory. This is the general framework in which this dissertation can be placed. In this work, a first order perturbation theory for eigenvalues of real or complex Jsymplectic matrices under structure-preserving perturbations is developed. Since the class of symplectic matrices has an underlying multiplicative structure, Lidskii’s classical formulas for small additive perturbations of the form b A = A+εB cannot be applied directly, so a new multiplicative perturbation theory is first developed: given an arbitrary square matrix A, we obtain the leading terms of the asymptotic expansions in the small, real parameter ε of multiplicative perturbations b A(ε) = (I +εB +· · · )A(I +εC +· · · ) of A for arbitrary matrices B and C. The analysis is separated in two complementary cases, depending on whether the unperturbed eigenvalue is zero or not. It is shown that in either case the leading exponents are obtained from the partial multiplicities of the eigenvalue of interest, and the leading coefficients generically involve only appropriately normalized left and right eigenvectors of A associated with that eigenvalue, with no need of generalized eigenvectors. It should be noted that, although initially motivated by the needs for the symplectic case, this multiplicative (unstructured) perturbation theory is of independent interest and stands on its own. After showing that any small structured perturbation bS of a symplectic matrix S can be written as bS = bS(ε) = (I + εB + · · · ) S with Hamiltonian first-order coefficient B, we apply the previously obtained Lidskii-like formulas for multiplicative perturbations to the symplectic case by exploiting the particular connections that symplectic structure induces in the Jordan form between normalized left and right eigenvectors. Special attention is given to eigenvalues on the unit circle, particularly to the exceptional eigenvalues ±1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under arbitrary ones. Also, several numerical examples are generated Although the approach described above via multiplicative expansions works in most situations, there is a very specific one, the one we call the nongeneric case, which requires a separate, completely different analysis. It corresponds to the case in which, in the absence of structure, the rank of the perturbation would break an odd number of oddsized Jordan blocks corresponding to the eigenvalue either 1 or −1. Since this is not allowed by symplecticity, one among that odd number of Jordan blocks does not break, but increases its size by one becoming an even-sized block. This very special behavior lies outside of the theory developed for what we might call the generic cases, and requires a completely different perturbation analysis, based on Newton diagram techniques like the one performed to obtain the multiplicative expansions. The main difference with the previous expansions is that in this nongeneric case the leading coefficients depend not only on eigenvectors, but also on first generalized Jordan vectors. El problema de calcular autovalores, autovectores y subespacios invariantes de matrices simplécticas juega un papel crucial en muchas aplicaciones, en particular en la Teoría de Control cuando ésta se centra en sistemas discretos. Si para resolver el problema simpléctico de autovalores se emplean métodos numéricos estándar que no tienen en cuenta la simetría especial del problema, entonces no solo se perderá en aritmética finita la simetría natural del espectro de las matrices simplécticas, sino que, aún más importante, podemos estar ignorando otras características o invariantes intrínsecas que tienen una influencia crucial en los correspondientes autovalores calulados. La importancia de preservar la estructura ha sido reconocida por la comunidad del Álgebra Lineal Numérica desde hace varias décadas y, en consecuencia, se han desarrollado diversos algoritmos para el problema simpléctico de autovalores que mantienen la estructura simpléctica en cada paso del proceso iterativo. El análisis de errores de tales algoritmos demanda una teoría de perturbación asociada que también preserve la estructura. Este es el marco general en el que se puede inscribir esta tesis doctoral. En este trabajo se desarrolla una teoría de perturbación de autovalores de matrices J-simplécticas frente a perturbaciones que preservan la simplecticidad de la matriz. Dado que la clase de matrices simplécticas tiene una estructura multiplicativa subyacente, las fórmulas clásicas de Lidskii para perturbaciones aditivas pequeñas de la forma b A = A + εB no se pueden aplicar de manera directa, de modo que desarrollamos una nueva teoría de perturbación multiplicativa: dada cualquier matriz cuadrada A, obtenemos el término director del desarrollo asintótico en el parámetro real (y pequeño) ε de autovalores de perturbaciones multiplicativas b A(ε) = (I + εB + · · · )A(I + εC + · · · ) de A para matrices arbitrarias B y C. El análisis se separa en dos casos complementarios, dependiendo de que el autovalor a perturbar sea nulo o no. Se demuestra que en ambos casos los exponentes directores se obtienen a partir de las multiplicidades parciales del autvalor bajo estudio, y que los coeficientes directores solo involucran genéricamente autovectores derechos e izquierdos adecuadamente normalizados, sin necesidad de autovalor generalizado alguno. Debe señalarse que, aunque inicialmente motivados por la necesidad para el caso simpléctico, esta teoría (no estructurada) de perturbación multiplicativa reviste interés per se independientemente de su aplicación al caso simpléctico. Tras mostrar que cualquier perturbación estructurada peque na bS de una matriz simpléctica S puede escribirse como bS = bS(ε) = (I + εB + · · · ) S con coeficiente de primer orden B Hamiltoniano, aplicamos las fórmulas tipo Lidskii obtenidas para perturbaciones multiplicativas al caso simpléctico, explotando la particular conexión que la estructura simpléctica induce entre los autovectores derechos e izquierdos normalizados por la forma de Jordan. Especial atención se le dedica a los autovalores sobre el círculo unidad, particularmente a los autovalores excepcionales ±1, cuyo comportamiento frente a perturbaciones estructuradas es sabido que difiere muy significativemente del comportamiento frente a perturbaciones arbitrarias. Además, presentamos varios ejemplos numéricos que ilustran (y confirman) los desarrollos asintóticos obtenidos. Aunque el enfoque que acabamos de describir via desarrollos multiplicativos funciona en la mayor parte de las situaciones, hay una muy específica, la que llamamos el caso no-genérico, que requiere de un análisis por separado, completamete distinto del anterior. Corresponde al caso en que, en ausencia de estructura, el rango de la perturbación rompería un número impar de bloques de Jordan de tamaño impar asociados a uno de los autovalores 1 ó −1. Como esto es incompatible con la simplecticidad, uno de entre los bloques de tamaño impar no se rompe, sino que incrementa en uno su dimensión, conviertiéndose en un bloque de Jordn de tamaño par. Este comportamiento tan especial no está explicado por la teoría de lo que podríamos llamar los casos ‘genéricos’ , y requiere de un análisis de perturbación completamente distinto, basado en técnicas del Diagrama de Newton, como el llevado a cabo para obtener los desarrollos multiplicativos. La diferencia principal con los desarrollos anteriores es que en el caso no genérico los coeficientes directores dependen no solo de autovectores, sino también de vectores primeros generalizados de Jordan. Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección del profesor Julio Moro Carreño. Se contó con una beca de la UC3M de ayuda al estudio de máster y posteriormente con un contrato predoctoral de la UC3M. Adicionalmente se recibió ayuda parcial del proyecto de investigación: “Matemáticas e Información Cuántica: de las Álgebras de Operadores al Muestreo Cuántico” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2014-54692-P). Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Alberto Ibort Latre.- Secretario: Rafael Cantó Colomina.- Vocal: Francisco Enrique Velasco Angulo

In this work, we introduce a definition of a local fractional derivative and a fractional integral of order alfa > 0 (where this parameter does not need to be integer), with which we overcome some deficiencies of known local derivatives, conformable or not. This definition allows to compute fractional derivatives of functions defined on any open set on the real line (and not just on the positive half-line). Moreover, we extend some classical results to the context of fractional derivatives. Also, applications of the fractional derivative through the direct and inverse problems are shown, as well as the feasibility of the fractional calculus in real and simulated problems. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana Portilla Ferreira.- Vocal: Ana Granados Sanandrés

After an introductory discussion, in which the main notions and background materials of discrete multiorthogonality are addressed, we focus our attention on some new results, namely Chapters 2, 3, and 4. These three chapters constitute the main core of the Thesis. Indeed, Chapter 2 is focused on the study of four new families of q-multiple orthogonal polynomials, namely q-multiple Charlier, q-multiple Meixner of the first and second kind, respectively, and q-multiple Kravchuk. The raising operators and Rodrigues-type formulas, which provide an explicit expression for these new families, are obtained. Chapter 3 contains a detailed study of some algebraic properties for the aforementioned q-families of multiple orthogonal polynomials. More specifically, the (r + 1) order recurrence relation as well as the (r + 1) order difference equations in the discrete variable on the real line are obtained. Here the letter r is used to denote the dimension of the vector measure µ [vector] . Finally, in Chapter 4 some limit relations between the attained q-families of multiple orthogonal polynomials (when the parameter q approaches 1) and discrete multiple orthogonal polynomials are established. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Francisco José Marcellán Español.- Secretario: Alejandro Zarzo Altarejos.- Vocal: Manuel Enrique Mañas Baena

This PhD Thesis is devoted to the study of some elliptic problems involving powers of the positive Laplace operator This Doctoral Thesis has been supported by a PIPF Grant from Departamento de Matemáticas of Universidad Carlos III de Madrid. Additionally, the author has been partially supported by the Ministry of Economy and Competitiveness of Spain and FEDER under Research Project MTM2016-80618-P. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: José Manuel Rodríguez García.- Secretario: Jesús García Azorero.- Vocal: Antonio Suárez Fernández

Large-scale problems have attracted much attention in the last decades since they arise from different applications in several fields. Moreover, the matrices that are involved in those problems are often sparse, this is, the majority of their entries are zero. Around 40 years ago, the most common problems related to large-scale and sparse matrices consisted in solving linear systems, finding eigenvalues and/or eigenvectors, solving least square problems or computing singular value decompositions. However, in the last years, large-scale and sparse problems of different natures have appeared, motivating and challenging numerical linear algebra to develop effective and efficient algorithms to solve them. Common difficulties that appear during the development of algorithms for solving modern large-scale problems are related to computational costs, storage issues and CPU time, given the large size of the matrices, which indicate that direct methods can not be used. This suggests that projection methods based on Krylov subspaces are a good option to develop procedures for solving large-scale and sparse modern problems. In this PhD Thesis we develop novel and original algorithms for solving two large-scale modern problems in numerical linear algebra: first, we introduce the R-CORK method for solving rational eigenvalue problems and, second, we present projection methods to compute the solution of T-Sylvester matrix equations, both based on Krylov subspaces. The R-CORK method is an extension of the compact rational Krylov method (CORK) [104] introduced to solve a family of nonlinear eigenvalue problems that can be expressed and linearized in certain particular ways and which include arbitrary polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems. The R-CORK method exploits the structure of the linearized problem by representing the Krylov vectors in a compact form in order to reduce the cost of storage, resulting in a method with two levels of orthogonalization. The first level of orthogonalization works with vectors of the same size as the original problem, and the second level works with vectors of size much smaller than the original problem. Since vectors of the size of the linearization are never stored or orthogonalized, R-CORK is more efficient from the point of view of memory and orthogonalization costs than the classical rational Krylov method applied to the linearization. Moreover, since the R-CORK method is based on a classical rational Krylov method, the implementation of implicit restarting is possible and we present an efficient way to do it, that preserves the compact representation of the Krylov vectors. We also introduce in this dissertation projection methods for solving the TSylvester equation, which has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems and other applications. The theory concerning T-Sylvester equations is rather well understood, and before the work in this thesis, there were stable and efficient numerical algorithms to solve these matrix equations for small- to medium- sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations was a completely open problem. In this thesis, we introduce several projection methods based on block Krylov subspaces and extended block Krylov subspaces for solving the T-Sylvester equation when the right-hand side is a low-rank matrix. We also offer an intuition on the expected convergence of the algorithm based on block Krylov subspaces and a clear guidance on which algorithm is the most convenient to use in each situation. All the algorithms presented in this thesis have been extensively tested, and the reported numerical results show that they perform satisfactorily in practice. Adicionalmente se recibió ayuda parcial de los proyectos de investigación: “Structured Numerical Linear Algebra: Matrix Polynomials, Special Matrices, and Conditioning” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2012-32542) y “Structured Numerical Linear Algebra for Constant, Polynomial and Rational Matrices” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2015-65798-P), donde el investigador principal de ambos proyectos fue Froilán Martínez Dopico. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: José Mas Marí.- Secretario: Fernando de Terán Vergara.- Vocal: José Enrique Román Moltó

Mención Internacional en el título de doctor In this Thesis we study the extremal problems of maximazing and minimazing the hyperbolicity constant on several families of graphs. In order to properly raise our research problem, we need to introduce some important definitions and make some remarks on the graphs we study. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Elena Romera Colmenarejo.- Secretario: Ana María Portilla Ferreira.- Vocal: José María Sigarreta Almira

Mención Internacional en el título de doctor We consider online task computing environments such as volunteer computing platforms running on BOINC (e.g., SETI@home) and crowdsourcing platforms such as Amazon Mechanical Turk. We model the computations as an Internet-based task computing system under the masterworker paradigm. A master entity sends tasks across the Internet, to worker entities willing to perform a computational task. Workers execute the tasks, and report back the results, completing the computational round. Unfortunately, workers are untrustworthy and might report an incorrect result. Thus, the first research question we answer in this work is how to design a reliable masterworker task computing system. We capture the workers’ behavior through two realistic models: (1) the “error probability model” which assumes the presence of altruistic workers willing to provide correct results and the presence of troll workers aiming at providing random incorrect results. Both types of workers suffer from an error probability altering their intended response. (2) The “rationality model” which assumes the presence of altruistic workers, always reporting a correct result, the presence of malicious workers always reporting an incorrect result, and the presence of rational workers following a strategy that will maximize their utility (benefit). The rational workers can choose among two strategies: either be honest and report a correct result, or cheat and report an incorrect result. Our two modeling assumptions on the workers’ behavior are supported by an experimental evaluation we have performed on Amazon Mechanical Turk. Given the error probability model, we evaluate two reliability techniques: (1) “voting” and (2) “auditing” in terms of task assignments required and time invested for computing correctly a set of tasks with high probability. Considering the rationality model, we take an evolutionary game theoretic approach and we design mechanisms that eventually achieve a reliable computational platform where the master receives the correct task result with probability one and with minimal auditing cost. The designed mechanisms provide incentives to the rational workers, reinforcing their strategy to a correct behavior, while they are complemented by four reputation schemes that cope with malice. Finally, we also design a mechanism that deals with unresponsive workers by keeping a reputation related to the workers’ response rate. The designed mechanism selects the most reliable and active workers in each computational round. Simulations, among other, depict the trade-off between the master’s cost and the time the system needs to reach a state where the master always receives the correct task result. The second research question we answer in this work concerns the fair and efficient distribution of workers among the masters over multiple computational rounds. Masters with similar tasks are competing for the same set of workers at each computational round. Workers must be assigned to the masters in a fair manner; when the master values a worker’s contribution the most. We consider that a master might have a strategic behavior, declaring a dishonest valuation on a worker in each round, in an attempt to increase its benefit. This strategic behavior from the side of the masters might lead to unfair and inefficient assignments of workers. Applying renown auction mechanisms to solve the problem at hand can be infeasible since monetary payments are required on the side of the masters. Hence, we present an alternative mechanism for fair and efficient distribution of the workers in the presence of strategic masters, without the use of monetary incentives. We show analytically that our designed mechanism guarantees fairness, is socially efficient, and is truthful. Simulations favourably compare our designed mechanism with two benchmark auction mechanisms. This work has been supported by IMDEA Networks Institute and the Spanish Ministry of Education grant FPU2013-03792. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Alberto Tarable.- Secretario: José Antonio Cuesta Ruiz.- Vocal: Juan Julián Merelo Guervós

Mención Internacional en el título de doctor This work is concerned with the analysis of new properties of semiclassical orthogonal polynomials in two variables and establish grounds for further research. Four main contributions were made in this direction. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Guillermo López Lagomasino.- Secretario: Iván Carlos Area Carracedo.- Vocal: Amílcar J. Pinto Lopes Branquinho

Mención Internacional en el título de doctor This thesis set out to contribute to the growing body of knowledge pertaining models of the genotype-phenotype map. In the process, we proposed and studied a new computational model, toyLIFE, and a new metaphor for molecular evolution —adaptive multiscapes. We also studied functional promiscuity and the evolutionary dynamics of shifting environments. The first result of this thesis was the definition of toyLIFE, a simplified model of cellular biology that incorporated toy versions of genes, proteins and regulation as well as metabolic laws. Molecules in toyLIFE interact between each other following the laws of the HP protein folding model, which endows toyLIFE with a simplified chemistry. From these laws, we saw how something reminiscent of cell-like behavior emerged, with complex regulatory and metabolic networks that grew in complexity as the genome increased. toyLIFE is, to our knowledge, the first multi-level model of the genotype- phenotype map, compared to previous models studied in the literature, such as RNA, proteins, gene regulatory networks (GRNs) or metabolic networks. All of these models either disregarded cellular context when assigning phenotype and function (RNA and proteins) or omitted genome dynamics, by defining their genotypes from high-level abstractions (GRNs and metabolic networks). toyLIFE shares the same features exhibited by all genotype-phenotype maps studied so far. There is strong degeneracy in the map, with many genotypes mapping into the same phenotype. This degeneracy translates into the existence of neutral networks, that span genotype space as soon as the genotype contains more than two genes. There is also a strong asymmetry in the size distribution of phenotypes: most phenotypes were rare, while a few of them covered most genotypes. Moreover, most common phenotypes are easily accessed from each other. We also studied the prevalence of functional promiscuity (the ability to perform more than one function) in computational models of the genotypephenotype map. In particular, we studied RNA, Boolean GRNs and toy- LIFE. Our results suggest that promiscuity is the norm, rather than the exception. These results prompt us to rethink our understanding of biology as a neatly functioning machine. One of the most interesting results of this thesis came from studying the evolutionary dynamics of shifting environments in populations showing functional promiscuity: our results show that there is an optimal frequency of change that minimizes the time to extinction of the population. Finally, we presented a new metaphor for molecular evolution: adaptive multiscapes. This framework intends to update the fitness landscape metaphor proposed by Sewall Wright in the 1930s. Adaptive multiscapes include many features that we have learned from computational studies of the genotype-phenotype map, and that have been discussed throughout the thesis. The existence of neutral networks, the asymmetry in phenotype sizes -and the concomitant asymmetry in phenotype accessibility- and the presence of functional promiscuity all alter the original fitness landscape picture. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Joshua Levy Payne.- Secretario: Saúl Arés García.- Vocal: Jacobo Aguirre Araujo

Mención Internacional en el título de doctor El objetivo de esta Tesis es estudiar trasformaciones espectrales para matrices que tiene como entradas funcionales lineales. En particular estudiamos las transformaciones de Christoffel, Geronimus y Geronimus-Uvarov. Con el fin de que esta Tesis sea lo más autocontenida posible, la hemos dividido en siete capítulos. • En el Capítulo 1, introducimos algunos conceptos y fijamos la notación que será usada a lo largo de esta Tesis a la vez que exponemos algunas propiedades básicas acerca de matrices semi-infinitas, módulos y polinomios matriciales junto con su teoría espectral. Hecho esto, pasamos a introducir la definición de matriz de funcionales lineales (y su forma sesquilineal asociada) también como el concepto de bi-ortogonalidad. • En el Capítulo 2 resumimos algunos resultados relativos a formas bilineales para las cuales el operador multiplicación por un polinomio es simétrico y su conexión con las relaciones de recurrencia de orden superior. Con esto en mente, pasamos a explicar cuidadosamente el resultado principal de concerniente a la relación existente entre una sucesión de polinomios satisfaciendo una relación de recurrencia de orden superior y una sucesión de polinomios matriciales que son ortogonales respecto a una medida matricial. Para concluir, mostramos algunos ejemplos de gran interés en la literatura. • El Capítulo 3 está dividido en dos partes. La primera parte se basada en los trabajos de A. J. Durán, A. J. Durán y W. Van Assche, y M. Derevyagin and F. Marcellán siendo este último el referente para esta primera parte del capítulo. Aquí, damos algunos resultados relacionados con la transformación espectral de Geronimus en el contexto de funcionales lineales y con esto en mente, motivamos la definición de este tipo de transformaciones para las formas bilineales simétricas B( . ; . ) definidas en términos de una medida de probabilidad con la propiedad de que el operador multiplicación por un cierto polinomio h(x) es simétrico para la forma bilineal, esto es B(h f ;g) = B( f ;gh) para todo f ;g 2 R[x]. Acto seguido, introducimos la noción de transformación de Geronimus múltiple y generalizamos los resultados dados en, encontrando, por ejemplo, que los productos internos tipo Sobolev discretos obtenidos al aplicarle un transformación de Geronimus múltiple a una forma bilineal nos lleva a una transformación de Geronimus matricial. Estos resultados han sido publicados en. Motivados por el resultado anterior, en la segunda parte de este capítulo, estudiamos la transformación de Geronimus (hermitica) pero ahora, sobre una matriz de medidas definida positiva, es decir, estamos interesados en el análisis de formas sesquilineales ( . ; . )W defindas mediante {P(x)W(x);Q(x)W(x)}W = ∫ P(x)dMQ†(x); donde M es una matriz de medidas definida positiva y W(x) es un polinomio matricial de grado fijo pero arbitrario. Aquí, encontramos condiciones para la existencia de la sucesión de polinomios ortogonales respecto a ( . ; . )W así como la fórmula de conexión entre los polinomios ortogonales originales y perturbados. Los resultados de este capítulo han sido publicados en. • El Capítulo 4 se dedica a la extensión de la fórmula de Christoffel para polinomios biortogonales matriciales. Más precisamente, dada una matriz de funcionales lineales u y W(x) un polinomio matricial con coeficiente principal no-singular, nosotros estudiamos la siguiente transformación matricial û =W(x)u y la relación entre sus familias de polinomios bi-ortogonales. El resultado principal de este capítulo es precisamente el Teorema 4.11, que presenta las fórmulas de conexión entre las sucesiones de polinomios bi-ortogonales matriciales originales y perturbados. Para este propósito usamos toda la riqueza de la teoría espectral disponible para polinomios matriciales, en particular las cadenas de Jordan y los "roots polynomials" a izquierda y derecha que serán extremadamente útiles. Finalmente, veremos que estas transformaciones de Christoffel se pueden extender a la teoría de sistemas integrables. Los resultados de este capítulo han sido publicados en. • El Capítulo 5 estudia la extensión de la transformación de Geronimus para matrices de funcionales lineales soportados en la recta real, es decir, multiplicaremos una matriz de funcionales lineales por un polinomio matricial WG(x) y le adicionaremos una suma de masas adecuadas (que dependen de los "roots polynomials" a izquierda y derecha). Aquí, desarrollamos dos diferentes métodos, el espectral y no espectral con el fin de obtener fórmulas de conexión entre las sucesiones de polinomios bi-ortogonales matriciales asociados al funcional original y el perturbado. • En el Capítulo 6 se desarrolla la extensión de las transformación de Geronimus-Uvarov para matrices de funcionales lineales soportados sobre la recta real. Este tipo de transformaciones pueden considerarse como una composición de una transformación de Geronimus y, acto seguido, de una transformación de Christoffel. En términos de matrices de funcionales esto se escribe como u→uˇ→û, donde uˇWG(x) = u y û = WC(x)uˇ; con WC(x) y WG(x) polinomios matriciales. Como en el capítulo 5, usando el método espectral y no-espectral, obtenemos fórmulas de conexión entre los polinomios bi-ortogonales matriciales originales y los perturbados. En el método espectral encontramos una representación para los polinomios bi-ortogonales matriciales perturbados en términos de la sucesiones de polinomios bi-ortogonales originales y las funciones de segunda especie (ver (1.9)). Aquí asumimos que los coeficientes principals de los polinomiosWC(x) yWG(x) son matrices no-singulares. En el método no-espectral damos una representación para los polinomios bi-ortogonales matriciales perturbados sin asumir ninguna hipótesis sobre el coeficiente principal del polinomio WG(x). Finalmente, como una aplicación, estudiamos la transformación de Uvarov matricial, que consiste en adicionarle al funcional original una suma de masas. Los resultados del capítulo 5 y 6 han sido publicados en. • Finalmente, en el Capítulo 7 damos un resumen de los principales resultados de esta tesis, así como una lista de problemas abiertos. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Antonio José Durán Guardeño.- Secretario: Guillermo López Lagomasino.- Vocal: Ana Pilar Foulquié Moreno

Mención Internacional en el título de doctor La presente tesis doctoral tiene por objeto el estudio de familias de polinomios que son soluciones del siguiente problema con valores iniciales donde tanto f como g son polinomios y L en los capítulos 2 y 3 es el operador derivada m-ésima de f. La diferencia en los dos casos anteriores es que mientras en el capítulo 2 se considera que g es un polinomio ortogonal clásico sobre la recta real, en el capítulo 3 denotamos por g al polinomio ortogonal con respecto a una cierta medida soportada sobre un arco de la circunferencia unidad y [omega]k es constante para cada k = 0 ,..., m. El capítulo 4, se dedica a las aplicaciones al procesamiento digital de imágenes, de las soluciones del problema (1) cuando f = g. Como puede apreciarse más adelante, este último caso corresponde a los conocidos polinomios de Krawtchouk. Acerca de la localización de los puntos críticos de polinomios en términos de sus ceros existe una teoría amplia (vea [72, Part I] y [81]), cuyas bases fundamentales son los teoremas de Rolle, Gauss-Lucas y sus refinamientos. Sin embargo, no existen recíprocos generales de estos resultados. Es obvio, que dado un cero de un polinomio y sus puntos críticos, los restantes ceros están unívocamente determinados. No obstante, solo existen unos pocos resultados sobre localización de ceros en función de sus puntos críticos y uno de sus ceros, la mayoría de los cuales se pueden ver en [72, x4.5]. En general, estos resultados son corolarios del Teorema de Composición de Schur-Szego (vea [72, Th.3.4.1d]. Quizás, los resultados más significativos en este sentido sean los Teoremas de Walsh [72, Th. 4.5.1] y Biernacki [72, Th. 4.5.2]. Hasta donde conocemos, sobre la localización de ceros de integrales iteradas de polinomios, normalizados con la condicin de anularse en el origen, solo existe el trabajo [16]. El mencionado artículo estudia varios casos particulares de familias de polinomios, entre ellos los polinomios de Legendre, y plantea una serie de conjeturas, algunas de las cuales se responden en los Capítulos 2 y 3 de esta memoria. El Capítulo 2 de esta memoria está dedicado a las integrales iteradas de polinomios ortogonales clásicos sobre la recta real, con énfasis en el caso Jacobi. Los trabajos [9, 10] muestran el interés de este tipo de polinomios para las aplicaciones a los métodos numéricos de elementos finitos. Es bien conocido que el polinomio mónico de Hermite Hn+m de grado (n + m) 2 Z+, donde tanto n como m son enteros no negativos. Como se mencionó anteriormente, el tercer capítulo se dedica al estudio del comportamientos asintótico los polinomios obtenidos mediante la integración iterada de los polinomios ortogonales con respecto a medias soportadas en un arco de la circunferencia unidad y el conjunto de acumulación de sus ceros. Se encuentra el comportamiento asintótico relativo entre las familias de polinomios ortogonales y sus respectivas familias de polinomios obtenidos por integración iterada. Se muestra la representación gráfica de regiones cerradas que contienen los ceros de las nuevas familias de polinomios y de curvas donde se acumulan los mismos en varios casos particulares. El tema central del Capítulo 4 es la implementación de un algoritmo eficiente para la detección de bordes de imágenes digitales basado en las propiedades de los polinomios ortogonales de Krawtchouk en dos variables. La primera parte del capítulo se dedica a estudiar las propiedades de esta familia de polinomios ortogonales en dos variables, que son de interés para el algoritmo propuesto. Las novedades de este algoritmo que fundamentan la calificación de eficiente son las siguientes: La aproximación de las diferencias parciales (derivadas parciales discretas) se realiza mediante una combinación lineal de polinomios de Krawtchouk en dos variables, los cuales son ortogonales con respecto a un producto interior discreto que involucra a la distribución binomial. En consecuencia, ya no es necesario suavizar la imagen mediante un filtro Gaussiano en dos dimensiones antes de realizar la diferenciación numérica, con el fin de regularizar la naturaleza mal condicionada de la diferenciación (ver [91]) y por lo tanto mejorar la localización de los bordes. En [11, 36] los autores describen un procedimiento para la detección de bordes utilizando los polinomios discretos de Chebyshev y un único umbral de discriminación de bordes para toda la imagen. Aquí, el algoritmo propuesto utiliza dos niveles de umbrales adaptativos, lo que reduce la presencia de falsos positivos o negativos en la selección de pixels-bordes. El operador gradiente para submatrices bloques de 5x5, en lugar del tradicional 3x3, proporciona una mejor localización de los pixels-bordes, ya que los bordes tienden a ser más gruesos cuando el tamaño del bloque incrementa [36, 69]. Para evitar el efecto de bordes gruesos y mejorar el resultado final en el algoritmo se aplican operaciones morfológicas (estrechar, erosionar y adelgazar) a la imagen de borde obtenida después del segundo paso de procesamiento del algoritmo. Para demostrar la efectividad del algoritmo propuesto se utilizaron imágenes tomadas de dos campos de aplicación muy diferentes: imágenes naturales utilizadas para la detección de objetos, vigilancia, etc; así como mapas de profundidad utilizados actualmente en aplicaciones y servicios multimedia de video 3D. Los contornos de objetos superpuestos, como la identificación de objetos de primer plano en mapas de profundidad, se obtienen con bastante buena precisión. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Francisco José Marcellán Español.- Secretario: Ramón Ángel Orive Rodríguez.- Vocal: Wilfredo Óscar Urbina Romero

Hyperbolic spaces, defined by Gromov in, play an important role in geometric group theory and in the geometry of negatively curved spaces. The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature bounded away from 0, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory. The first works on Gromov hyperbolic spaces deal with finitely generated groups. Initially, Gromov spaces were applied to the study of automatic groups in the science of computation; indeed, hyperbolic groups are strongly geodesically automatic, i.e., there is an automatic structure on the group. The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking. For example, it has been shown empirically in that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension; furthermore, it is evidenced that many real networks are hyperbolic. A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers. Another important application of these spaces is the study of the spread of viruses through the internet. Furthermore, hyperbolic spaces are useful in secure transmission of information on the network. The hyperbolicity has also been used extensively in the context of random graphs. For example, it was shown that several types of small-world networks and networks with given expected degrees are not hyperbolic in some sense. The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; and the references therein. In our study on the hyperbolicity in graphs we use the notations (...) Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana María Portilla Ferreira.- Vocal: Eva Touris Lojo

The first and second Frobenius companion matrices appear frequently in numerical application, but it is well known that they possess many properties that are undesirable numerically, which limit their use in applications. Fiedler companion matrices, or Fiedler matrices for brevity, introduced in 2003, is a family of matrices which includes the two Frobenius matrices. The main goal of this work is to study whether or not Fiedler companion matrices can be used with more reliability than the Frobenius ones in the numerical applications where Frobenius matrices are used. For this reason, in this work we present a thorough study of Fiedler matrices: their structure and numerical properties, where we mean by numerical properties those properties that are interesting for applying these matrices in numerical computations, and some of their applications in the field on numerical linear algebra. The introduction of Fiedler companion matrices is an example of a simple idea that has been very influential in the development of several lines of research in the numerical linear algebra field. This family of matrices has important connections with a number of topics of current interest, including: polynomial root finding algorithms, linearizations of matrix polynomials, unitary Hessenberg matrices, CMV matrices, Green’s matrices, orthogonal polynomials, rank structured matrices, quasiseparable and semiseparable matrices, etc., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Paul Van Dooren.- Secretario: Juan Bernardo Zaballa Tejada.- Vocal: Françoise Tisseur

Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: José María Arrieta Algarra.- Secretario: José Manuel Rodríguez García.- Vocal: Ana Rodríguez Santa María

Esta tesis se enmarca dentro del campo de la Alta Precisión Relativa (HRA) en Álgebra Lineal Numérica (ALN). Sus líneas maestras son dos. Por un lado, el diseño y análisis de algoritmos que permitan resolver problemas de Álgebra Lineal con más precisión de la habitual para matrices con estructura. Y por otro el estudio de la teoría específica de perturbaciones necesaria para tratar los problemas que nos ocupan. En nuestra investigación hemos tratado dos: La obtención de soluciones precisas del problema de mínimos cuadrados para matrices con estructura (Capítulo 3). La obtención de autovalores y autovectores precisos para matrices simétricas graduadas (Capítulo 4)....

Sea X un espacio métrico geodésico y x₁, x₂, x₃ ∈ X. Un triángulo geodésico T = {x₁, x₂, x₃} es la unión de tres geodésicas [x₁x₂], [x₂x₃] y [x₃x₁] de X. El espacio X es δ-hiperbólico (en el sentido de Gromov) si todo lado de T está contenido en la δ-vecindad de la unión de los otros dos lados, para todo triángulo geodésico T de X. Se denota por δ(X) la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ ≥ 0 : X es δ-hiperbólico}. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Uno de los principales objetivos de esta tesis de doctorado es obtener información cuantitativa acerca de la constante de hiperbolicidad de varios productos de grafos. Estas desigualdades permiten obtener un resultado cualitativo importante: la caracterización de la hiperbolicidad de varios productos de grafos en términos de la hiperbolicidad de sus componentes. En este trabajo caracterizamos los productos fuertes de grafos G₁ ⊠ G₂ hiperbólicos, en términos de G₁ y G₂: el producto fuerte G₁ ⊠G₂ es hiperbólico si y sólo si uno de los factores es hiperbólico y el otro está acotado. También probamos algunas relaciones óptimas entre δ(G₁ ⊠ G₂), δ(G₁), δ(G₂) y los diámetros de G₁ y G₂ (y encontramos familias de grafos para los cuales se alcanzan las desigualdades). Obtenemos el valor exacto de la constante de hiperbolicidad para varios productos fuertes de grafos. También caracterizamos los productos lexicográficos de grafos G₁ ◦ G₂ hiperbólicos, en términos de G₁ y G₂: el producto lexicográfico G₁ ◦ G₂ es hiperbólico si y sólo si G₁ es hiperbólico, a menos que G₁ sea un grafo trivial; si G₁ es trivial, entonces G₁ ◦ G₂ es hiperbólico si y sólo si G₂ es hiperbólico. En particular, obtenemos las desigualdades δ(G₁) ≤ δ(G₁ ◦ G₂) ≤ δ(G₁) + 3/2 si G₁ es un grafo no trivial, y encontramos familias de grafos para las cuales se alcanzan estas desigualdades. Además, caracterizamos las sumas cartesianas de grafos G₁ ⊕ G₂ hiperbólicas: G₁ ⊕ G₂ es siempre hiperbólica, a menos que G₁ ó G₂ sea el grafo trivial, y en este último caso G₁ ⊕ G₂ es hiperbólica si y sólo si G₂ ó G₁ es hiperbólico, respectivamente. Obtenemos las desigualdades óptimas 1 ≤ δ(G₁ ⊕ G₂) ≤ 3/2 para todos los grafos G₁, G₂ no triviales. Además, caracterizamos las sumas cartesianas de grafos con δ(G₁⊕G₂) = 1, con δ(G₁⊕G₂) = 5/4 y con δ(G₁ ⊕ G₂) = 3/2. También encontramos el valor exacto de la constante de hiperbolicidad para las sumas cartesianas de diversas familias de grafos. Finalmente, probamos que si el producto directo de grafos G₁×G₂ es hiperbólico, entonces uno de los factores es hiperbólico y el otro factor está acotado. También probamos que esta condición necesaria para la hiperbolicidad es, de hecho, una caracterización en muchos casos. En otros casos, encontramos caracterizaciones que no son tan simples. Además, obtenemos buenas cotas para la constante de hiperbolicidad del producto directo de varias clases de grafos importantes., If X is a geodesic metric space and x₁, x₂, x₃ ∈ X, a geodesic triangle T = {x₁, x₂, x₃} is the union of the three geodesics and [x₁x₂], [x₂x₃] y [x₃x₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in the δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic }. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this PhD Thesis is to obtain quantitative information about the hyperbolicity constant of several products of graphs. These inequalities allow to obtain other main results, which characterize in a qualitative way the hyperbolicity of several products of graphs in terms of the hyperbolicity of their components. In this work we characterize the strong product of two graphs G₁ ⊠ G₂ which are hyper-bolic, in terms of G₁ and G₂: the strong product graph G₁ ⊠ G₂ is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between δ(G₁ ⊠ G₂), δ(G₁), δ(G₂) and the diameters of G₁ and G₂ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the exact values of the hyperbolicity constant for many strong product graphs. Furthermore, we characterize the lexicographic product of two graphs G₁ ◦ G₂ which are hyperbolic, in terms of G₁ and G₂: the lexicographic product graph G₁ ◦ G₂ is hyperbolic if and only if G₁ is hyperbolic, unless if G₁ is a trivial graph; if G₁ is trivial, then G₁ ◦ G₂ is hyperbolic if and only if G₂ is hyperbolic. In particular, we obtain that δ(G₁) ≤ δ(G₁ ◦ G₂) ≤ δ(G₁) +3/2 if G₁ is not a trivial graph, and we find families of graphs for which the inequalities are attained. Besides, we characterize the hyperbolic product graphs for the Cartesian sum G₁ ⊕ G₂: G₁ ⊕ G₂ is always hyperbolic, unless either G₁ or G₂ is the trivial graph; if G₁ or G₂ is the trivial graph, then G₁ ⊕ G₂ is hyperbolic if and only if G₂ or G₁ is hyperbolic, respectively. We also obtain the sharp inequalities 1 ≤ δ(G₁ ⊕ G₂) ≤ 3/2 for every non-trivial graphs G₁, G₂. Besides, we characterize the Cartesian sums with δ(G₁⊕G₂) = 1, with δ((G₁⊕G₂) = 5/4 and with δ(G₁⊕G₂) = 3/2. Furthermore, we obtain the precise value of the hyperbolicity constant of the Cartesian sum of many graphs. Finally, we prove that if the direct product G₁×G₂ is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain good bounds for the hyperbolicity constant of the direct product of some important graphs., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana María Portilla Ferreira.- Vocal: Eva Touris Lojo

Mención Internacional en el título de doctor This work presents a study of orthogonal polynomials from a matrix point of view. We deal with semiclassical and coherent orthogonal polynomials. We also consider the difference and q-difference operators. We also study the coherent pairs of measures supported on the unit circle. In Chapter 1 we present the framework on which the different chapters are developed as well as an historical overview about coherent pairs on the real line and on the unit circle, where it should be noted that there exist only few works and it is a field yet to explore. Besides, in Chapter 2 a new tool introduced by L. Verde-Star in [61] and [62], based on the representation of a sequence of polynomials fPn(x)gn_0 by using an infinite lower semi-matrix, is presented in a very detailed way. This tool is used to characterize sequences of classical orthogonal polynomials and sequences of discrete and classical q-orthogonal polynomials in terms of banded matrices. To do this, the polynomials are represented with respect to the monomial basis (…) and the coeficients ak;j are used as entries of the matrix A = [ak;j ]. Several properties of orthogonal polynomials can be easily obtained with this approach. Afterwards, in Chapter 3, a new characterization of semiclassical orthogonal polynomials is presented by using the matrix approach described in Chapter 2. Semiclassical orthogonal polynomials are usually defined by a structure relation such as (…). This is the so-called first structure relation and it was introduced by P. Maroni. Besides, we give a relation between the Jacobi matrix associated with semiclassical orthonormal polynomials and the matrix whose entries are the coeficients appearing in the previous structure relation. We also use the matrix approach to characterize when a pair of sequences of orthogonal polynomials constitutes a coherent pair. The most general characterization for the coherence on the real line was given by M. N. de Jesús, F. Marcellán, J. Petronilho, and N. C. Pinzón-Cortés. They studied the structure relation (…) where, M;N; m; k are non negative integers and the constants (…) satisfy some natural conditions. By using the matrix approach, the coherence relation is equivalent to the existence of a banded matrix whose size is related to the class of coherence. We have studied the cases of (1,0)-coherence and (M; 0)-coherence which are characterized with a (0,1)-banded matrix and a (0;M)-banded matrix, respectively. In the same way, we have studied the cases of (1,0)-coherence of order m and (M; 0)-coherence of order m, where the order stands for the quantity of derivatives taken over the terms of the sequence of orthogonal polynomials corresponding to the measure (…). In this situation we obtained similar results as in the case of no derivatives. It is worth mentioning that in both cases we carry out numerical experiments to illustrate the results we developed. In Chapter 4, we consider the action of the difference operator and the divided difference operator D in the linear space of polynomials with real coefficients. D_ is either D! or Dq defined by (…). We obtain a new structure relation for the D_-semiclassical orthogonal polynomials that is used later on to deduce the D_-semiclassical character of a given linear functional in terms of banded matrices. We also developed a similar characterization for the D_-coherence. Finally, in Chapter 5 we present a study of polynomials which are orthogonal with respect to measures supported on the unit circle, having as a goal to give a characterization of all the pairs of Borel positive measures (…) supported on the unit circle such that the corresponding sequences of monic orthogonal polynomials (…). This problem is an extension of the one studied in [45] by F. Marcellán and A. Sri Ranga where they consider one term less in the right hand side of the previous equation. This work is contained in Chapter 5. Este trabajo presenta un estudio de familias de polinomios ortogonales desde un punto de vista matricial. Abordamos los polinomios ortogonales semiclásicos y coherentes. También consideramos los operadores de diferencias y de q-diferencias. Además se hace un estudio sobre pares coherentes de medidas con soporte en la circunferencia unidad. En el Capítulo 1 se presenta de manera detallada todo el marco de trabajo sobre el cual se desarrollan los distintos capítulos de esta tesis, así como una reseña histórica sobre los pares coherentes en la recta real y en la circunferencia unidad, donde cabe señalar que existen muy pocos trabajos actualmente y es un área de investigación todavía por explorar. Además, en el Capítulo 2 se presenta de manera detallada una herramienta de estudio introducida por L. Verde-Star para estudiar sucesiones de polinomios ortogonales clásicos, así como sucesiones de polinomios ortogonales discretos y q-ortogonales (…). Después, en el Capítulo 3, se presenta una nueva interpretación de los polinomios ortogonales semiclásicos utilizando el enfoque del Capítulo 2. En el Capítulo 4 se considera la acción del operador en diferencias o en diferencias divididas en el espacio lineal de polinomios con coeficientes reales (…). Finalmente, en el Capítulo 5 se presenta un estudio de familias de polinomios que son ortogonales con respecto a medidas con soporte en la circunferencia unidad, teniendo como objetivo dar una descripción de todos los pares de medidas de Borel positivas (…). Este problema es una extensión del estudiado por F. Marcellán y A. Sri Ranga donde consideran un término menos en la parte de la derecha de la ecuación. Este trabajo está contenido dentro del Capítulo 5. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Guillermo López Lagomasino.- Secretario: Amílcar José Pinto Lopes Branquinh.- Vocales: María Alicia Cachafeiro López

The question of analyzing the predictability of human behavior has been widely studied in literature, to unveil how individuals move, how they can be mobilized and, more philosophically, to understand to what extent our decisions are random or whether we are free to choose. As a consequence of humans relate to each other, we also tend to live in groups at different hierarchies in a social way so it is interesting to analyze how individual features and choices affect the global structure of a society. In this work, we explore the limits of human predictability in terms of shopping behavior, observing that, even when we are constrained to a limited set of possible places where we can make a purchase, predicting where the next purchase will happen is not accurately possible to do by only observing the past. The next question is to study how individual decisions affect emergent phenomena such as the economy or information diffusion across a country. We analyze the contents, temporal and mobility patterns extracted from users’ social media publications to build a profile of the geographical regions that allow to predict the unemployment rate. Finally, we also use a mobile phone call dataset to test whether the dynamics at the urban level, how people create and destroy links within a city, affect the inter-urban diffusion of diseases, virus or rumors. Our results suggest that inter-regional structure is robust and does not vary significantly on time so diffusion processes can be well modeled in terms of static properties of the inter-urban network., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Javier Borge Holthoefer.- Secretario: Rubén Cuevas Rumín.- Vocal: Josep Perelló Palou

Mención Internacional en el título de doctor, Cooperation and coordination is a desirable behavior that is fundamental for the harmonious development of society. People need to rely on cooperation with other individuals in many aspects of everyday life, such as teamwork and economic exchange in anonymous markets. However, cooperation may easily fall prey to exploitation by selfish individuals who only care about short-term gain. For cooperation to evolve, specific conditions and mechanisms are required, such as kinship, direct and indirect reciprocity through repeated interactions, or external interventions such as punishment. In this dissertation we investigate the effect of the network structure of the population on the evolution of cooperation and coordination. We consider several kinds of static and dynamical network topologies, such as Barabási–Albert, social network models and spatial networks. We perform numerical simulations and laboratory experiments using the Prisoner’s Dilemma and coordination games in order to contrast human behavior with theoretical results. We show by numerical simulations that even a moderate amount of random noise on the Barabási–Albert scale-free network links causes a significant loss of cooperation, to the point that cooperation almost vanishes altogether in the Prisoner’s Dilemma when the noise rate is high enough. Moreover, when we consider fixed social-like networks we find that current models of social networks may allow cooperation to emerge and to be robust at least as much as in scale-free networks. In the framework of spatial networks, we investigate whether cooperation can evolve and be stable when agents move randomly or performing L´evy flights in a continuous space. We also consider discrete space adopting purposeful mobility and binary birth-death process to discover emergent cooperative patterns. The fundamental result is that cooperation may be enhanced when this migration is opportunistic or even when agents follow very simple heuristics. In the experimental laboratory, we investigate the issue of social coordination between individuals located on networks of contacts. In contrast to simulations, we find that human players dynamics do not converge to the efficient outcome more often in a social-like network than in a random network. In another experiment, we study the behavior of people who play a pure coordination game in a spatial environment in which they can move around and when changing convention is costly. We find that each convention forms homogeneous clusters and is adopted by approximately half of the individuals. When we provide them with global information, i.e. the number of subjects currently adopting one of the conventions, global consensus is reached in most, but not all, cases. Our results allow us to extract the heuristics used by the participants and to build a numerical simulation model that agrees very well with the experiments. Our findings have important implications for policymakers intending to promote specific, desired behaviors in a mobile population. Furthermore, we carry out an experiment with human subjects playing the Prisoner’s Dilemma game in a diluted grid where people are able to move around. In contrast to previous results on purposeful rewiring in relational networks, we find no noticeable effect of mobility in space on the level of cooperation. Clusters of cooperators form momentarily but in a few rounds they dissolve as cooperators at the boundaries stop tolerating being cheated upon. Our results highlight the difficulties that mobile agents have to establish a cooperative environment in a spatial setting without a device such as reputation or the possibility of retaliation. i.e. punishment. Finally, we test experimentally the evolution of cooperation in social networks taking into account a setting where we allow people to make or break links at their will. In this work we give particular attention to whether information on an individual’s actions is freely available to potential partners or not. Studying the role of information is relevant as information on other people’s actions is often not available for free: a recruiting firm may need to call a job candidate’s references, a bank may need to find out about the credit history of a new client, etc.We find that people cooperate almost fully when information on their actions is freely available to their potential partners. Cooperation is less likely, however, if people have to pay about half of what they gain from cooperating with a cooperator. Cooperation declines even further if people have to pay a cost that is almost equivalent to the gain from cooperating with a cooperator. Thus, costly information on potential neighbors’ actions can undermine the incentive to cooperate in dynamical networks., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: José Antonio Cuesta Ruiz.- Secretario: Maxi San Miguel.- Vocal: Daniele Vilone

En este trabajo, se analiza una teoría que es en cierto modo una extensión natural del campo de las telecomunicaciones clásicas a la mecánica cuántica. Dicha teoría se llama tomografía cuántica y es de hecho una imagen de la mecánica cuántica equivalente a las más habituales que son la imagen de Schrödinger o la de Heisenberg. Esta nueva imagen, difiere de las precedentes en que está muy ligada a la capacidad tecnológica a la hora de medir observables (momento, energía, etc.) en el campo de la óptica cuántica, ya que su objetivo primordial es el de conseguir reconstruir el estado de un sistema cuántico a partir de mediciones en el laboratorio, y dado al avance tecnológico del instrumental de laboratorio disponible como láseres y fotodetectores, cada vez está suscitando mayor interés. En el primer capítulo, veremos cómo nace esta idea de reconstruir estados cuánticos discutiendo brevemente la técnica clásica conocida como Tomografía Axial Computerizada (TAC). Esta técnica está basada en los trabajos de Johann Karl August Radon aplicando la transformada que lleva su nombre. Introduciremos la transformada de Radon de una función de probabilidad definida en el espacio de fases para ver cómo se aplica en el caso del TAC. Para aplicar esta idea para la reconstrucción de estados cuánticos, veremos, en primer lugar, que existe una extensión natural de las técnicas de demodulación de señales moduladas en amplitud (AM) en el campo de la óptica cuántica mostrando que el papel que cumple un mezclador puede ser reemplazado por una combinación de divisores de haz y fotodetectores y mostraremos explícitamente cómo reconstruir el operador densidad que describe un estado cuántico a través de un proceso reminiscente de la transformada de Radon clásica. El segundo capítulo será el corazón de este trabajo. En él, introduciremos de manera formal la descripción tomográfica de la mecánica cuántica. Presentaremos una teoría general, para ello, trataremos a los observables como elementos de un álgebra C_ y los estados serán funcionales lineales positivos que actúen en dicha algebra. Veremos que esta descripción de la mecánica cuántica puede dividirse en dos partes, una primera con el objetivo de obtener una fórmula para reconstruir el estado de un sistema cuántico a partir de una función definida sobre un conjunto llamado conjunto tomográfico, que será un conjunto de observables que tendrá que cumplir una serie de condiciones que expondremos debidamente. A esta primera parte de la teoría, la bautizaremos como Teoría de muestreo generalizada en sistemas cuánticos. La segunda parte estará relacionada con la parte puramente experimental. Lo que hace necesario tener que añadir esta segunda parte es el hecho de que la función definida anteriormente sobre el conjunto tomográfico, que llamaremos función de muestreo, en general, no puede ser medida por medio de los dispositivos con los que contamos en un laboratorio de óptica cuántica, sin embargo, a partir de las mediciones hechas con un fotodetector, podemos obtener distribuciones de probabilidad de cantidades relacionadas con los observables. Entonces, tomando esto último como motivación, esta segunda parte de la teoría consistirá en relacionar esa función de muestreo con una distribución de probabilidad que llamaremos tomograma, que será el resultado directo de un proceso de medida en el laboratorio, y la llamaremos Transformada generalizada positiva por motivos que se expondrán convenientemente. Uno de los problemas más sutiles de esta teoría consiste en hallar un conjunto tomográfico que cumpla las condiciones necesarias para permitir reconstruir el estado a partir de él. Sin embargo, veremos que de manera natural, las representaciones unitarias irreducibles de un grupo finito o de Lie compacto proporcionan conjuntos tomográficos que cumplen las condiciones requeridas, por eso, nos centraremos en el estudio de la reconstrucción de estados a partir de grupos relacionados con el sistema físico dado. Aunque también destacaremos que existen otras representaciones unitarias que nos permiten reconstruir el estado cuántico a partir de ellas, como lo es la representación unitaria irreducible del grupo de Heisenberg Weyl dada por el álgebra de Lie que forman los operadores momento y posición cuánticos que es el ejemplo con el que nace la tomografía cuántica introducido en el primer capítulo. En el tercer capítulo presentaremos un algoritmo numérico que se deriva a partir de unos estados que llamaremos estados adaptados que habremos definido en el capítulo anterior. Este algoritmo nace como un problema inverso, ya que hasta entonces nos habremos centrado en reconstruir estados a partir de un conjunto tomográfico, en especial, cuando el conjunto tomográfico está definido a partir de una representación unitaria de un grupo. Este problema inverso consiste en determinar qué información es posible obtener de una representación unitaria de un grupo si se tiene una familia de estados que describe un sistema físico relacionado con un grupo de simetría. La respuesta a esta pregunta es muy satisfactoria ya que es posible conocer la matriz de transformación de base que nos permita transformar la base, en la que está descrita la representación unitaria, en una base adaptada a los subespacios invariantes bajo la acción de todos los elementos de la representación. Este problema se conoce como descomposición de Clebsh Gordan, ya que dicha transformación aplicada a la representación unitaria la convierte en una matriz diagonal por bloques en la que cada bloque corresponde a una representación unitaria irreducible. La manera en la que resolveremos este problema es con un algoritmo numérico que sólo requiere dos estados adaptados como argumentos de entrada, que pueden ser obtenidos de manera directa si se conoce de forma explícita la representación unitaria que queremos reducir. Este algoritmo lo hemos bautizado con el nombre de SMILY. Hay que destacar que como este algoritmo se ha generado sólo aplicando ciertas transformaciones unitarias sobre matrices que representan un estado cuántico, tiene una extensión natural que podría implementarse en un ordenador cuántico. Para acabar esta tesis, generalizaremos la descripción tomográfica a campos clásicos y cuánticos. Para el caso clásico, primero realizaremos una descripción tomográfico para sistemas con finitos grados de libertad y obtendremos el equivalente tomográfico de la ecuación de Liouville para una densidad de probabilidad y tras esto, haremos el mismo análisis para sistemas con infinitos grados de libertad. Para obtener la descripción tomográfica para campos cuánticos, partiremos del concepto de segunda cuantización y mostraremos el equivalente tomográfico de los axiomas de Wightman Streater para una teoría cuántica de campos. Y para terminar, obtendremos un teorema de reconstrucción para campos escalares y calcularemos el tomograma de ciertos estados de un campo cuántico escalar libre. Comentemos esto último diciendo que es el inicio de una teoría que permitirá una descripción tomográfica de estados, por ejemplo ligados, para teorías con interacción. Para concluir este resumen, quisiera resaltar que el lector especializado, si lo considera conveniente, puede comenzar a leer a partir del segundo capítulo, ya que aunque el primer capítulo sirve como motivación de porqué desarrollar una descripción tomográfica de la mecánica cuántica, el texto puede comprenderse si previamente no se ha leído., Este trabajo ha sido posible gracias al proyecto QUITEMAD (Quantum Information Technologies in Madrid) y la beca PIF del Departamento de Matem aticas de la Universidad Carlos III de Madrid, Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Antonio García García.- Secretario: Miguel Ángel Rodríguez González.- Vocal: Manuel Asorey Carballeira

Mención Internacional en el título de doctor, In this thesis we have studied the effect of fluctuations in two important paradigms of soft condensed matter physics, namely, Fisher fronts and thin fluid films. Our study of stochastic FKPP traveling waves broadens current knowledge on the influence of thermal noise onto the dynamics of these waves in two space dimensions. The FKPP equation describes the evolution of a stable state into an unstable one by creating a surface (a line in one dimension) that delimits the occupancy of the stable state. This 2d surface contains equipotential lines (points, in 1d) perpendicular to the velocity direction. Due to the pulled nature of the FKPP front, the noise decreases the velocity in such way that, on average, the decrease depends logarithmically on the total number of particles. Hence, even for a macroscopic number of particles, the effect of fluctuations can be observed. In two space dimensions, the stochastic front surface roughens in time with the scale-invariance properties. Indeed, we have shown through extensive simulations of the algorithm based on a special stochastic PDE solver, that the 2d sFKPP traveling wave solutions define a surface that roughens with the same scaling exponents as the KPZ equation in 1d. To further support this claim, we have calculated the distribution function of the fluctuations in surface position, and obtained very good agreement with the distribution of the KPZ surface, which is the Tracy-Widom-GOE distribution. The fact that dimensionality of the universality class is smaller than the one of the sFKPP equation lies in the weak-noise mechanism, which only affects the microscopic region of the front. In the macroscopic region, the weak noise does not produce any additional effect. Namely, above certain length-scale, the effect that fluctuations produce in the microscopic region, simply propagates back to the macroscopic one. We show that the morphology of the microscopic surface defines that of the macroscopic surface after a time related to the total number of particles, providing the same universality class for all equipotential lines….., La investigación en física estadística se ha centrado en estudiar los estados en equilibrio que alcanza un sistema físico, mientras que la dinámica fuera del equilibrio, descrita por ejemplo por las ecuaciones diferenciales, solo es conocida en los casos más simples. El desarrollo de los ordenadores ha permitido a los investigadores estudiar numéricamente esta dinámica fuera del equilibrio. Los estudios basados en simulaciones numéricas no sólo han hecho posible observar la evolución dinámica sino también, en muchos casos han contribuido en que se encontraran o mejoraran las predicciones analíticas. Sin embargo, las simulaciones han hecho posible estudiar una gran variedad de modelos, como por ejemplo, modelos de partículas, dadas las relaciones de interacción entre sí mismas, modelos explícitos, implícitos o semi-implícitos, basados en soluciones numéricas de las ecuaciones parciales diferenciales, modelos espectrales o pseudo-espectrales, etc. Estos modelos matemáticos, aparte de modelar los sistemas físicos tienen muchas aplicaciones en otros campos, especialmente en biología, química y economía. Además, la necesidad industrial de entender cuantitativamente los procesos en escalas muy pequeñas (micro y nano) requiere estudios numéricos para ver qué modelo describe mejor dichos procesos. Los modelos en los que se centran los trabajos presentados en esta tesis describen sistemas microfluídicos, como por ejemplo la evolución de una superficie o intercara entre fluido y vapor sobre un sustrato plano. Normalmente, en esas escalas (micro y nanoscópicas) es imposible desarrollar modelos teniendo en cuenta todas las interacciones conocidas, mientras los modelos fenomenológicos a veces son demasiado simples. Por esta razón, hemos estudiado la evolución de dos modelos simples con un ruido específico que trata de agrupar los efectos que se pierden simplificando la naturaleza de las interacciones, creando modelos más realistas. El primer modelo es un proceso reactivo-difusivo dado por la ecuación de Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP), utilizada recientemente en bastantes sistemas biológicos. El segundo modelo representa los sistemas de películas delgadas y la evolución de las películas bajo diferentes situaciones físicas de la interacción fluido-sustrato…., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Aurora Hernández Machado.- Secretario: Olivier Pierre Louis.- Vocal: Javier Manuel Muñoz García

Roughly speaking sampling theory deals with determining whether we can or can not recover a continuous function from some discrete set of its values. The most important result and main pillar of this theory is the well-known Shannon’s sampling theorem which states that: If a signal f(t) contains no frequencies higher than 1/2 cycles per second, it is completely determined by giving its ordinates at a sequence of points spaced one second apart…., A grandes rasgos la teoría de muestreo estudia el problema de recuperar una función continua a partir de un conjunto discreto de sus valores. El resultado más importante y pilar fundamental de esta teoría es el conocido teorema de muestreo de Shannon que afirma que: Si una señal f(t) no contiene frecuencias mayores que 1/2 ciclos por segundo entonces está completamente determinada por sus ordenadas en una sucesión de puntos espaciados en un segundo…., Proyecto de investigación MTM2009–08345 del Ministerio de Ciencia e Innovación de España., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Luis Alberto Ibort Latre.- Secretario: Eugenio Hernández Rodríguez.- Vocal: Ole Christensen

Mención Internacional en el título de doctor, In the last 30 years, the study of orthogonal polynomials in Sobolev spaces has obtained an increasing attention from the research community. The first work on Sobolev orthogonal polynomials was published in 1962 by Althammer, who studied the Legendre-Sobolev polynomials orthogonal with respect to the inner product. The study of this family of orthogonal polynomials is not only interesting for a comparison with the standard theory of orthogonal polynomials, but these polynomials also arise in a natural way in a variety of contexts. In this thesis, we analyze the properties of polynomials orthogonal with respect to a discrete Sobolev inner product. More precisely, we will focus our attention on the study of connection formulas relating Sobolev orthogonal polynomials with the corresponding ordinary ones. Indeed, we deal with some problems on asymptotic behavior of Sobolev orthogonal polynomials as well as we obtain some results on convergence of Fourier-Sobolev series. The present Thesis is organized as follows: In Chapter 1 we introduce the theory of Sobolev orthogonal polynomials and the notation that we will use along this Thesis. We summarize two main differences between the standard orthogonal polynomials and the Sobolev case: recurrence relations and the location of zeros of orthogonal polynomials. Here, we also include a thorough study about the known connection formulas. Finally, for a better understanding of our work, we give the state of the art about asymptotics and Fourier series of orthogonal polynomials, analyzing both the cases of measures with bounded and unbounded support, respectively. In Chapter 2 we study some algebraic and analytic aspects of certain family of Sobolev polynomials orthogonal with respect to a measure with a bounded support on the real line. In Section 2.1 we present an alternative proof for a known result about Outer Relative Asymptotics of Sobolev orthogonal polynomials. In Section 2.2 we also include a new matrix connection relating the matrix associated to the higher order recurrence relation for Sobolev polynomials and the corresponding Jacobi matrix associated to the standard ones. In Section 2.3 we show a result about pointwise convergence of Fourier-Sobolev series in the case of measures with bounded support. In Chapter 3 we summarize some known properties of polynomials orthogonal with respect to a modification of the Laguerre measure, the k-iterated Christoffel one. Later on, we obtain estimates for the norm of such polynomials as well as a generalized Christoffel formula for them. Finally, we present a detailed study about the diagonal Christoffel kernels associated to the Gamma distribution. In particular, we obtain the asymptotic behavior of these kernel polynomials both inside and outside the support of the measure. In Chapter 4 we deal with the Outer and Inner Relative Asymptotics of Sobolevtype orthogonal polynomials when the mass points are located inside the support of the measure, the oscillatory region for such polynomials. Finally, we obtain the asymptotic behavior of the coefficients appearing in the higher order recurrence relation that Sobolev polynomials satisfy. In Chapter 5 we show the divergence of a certain Fourier-Sobolev series. The main tool for this purpose will be a Cohen type inequality. This problem is dealing for the first time for a Sobolev-type inner product with a mass point outside the support of the measure., El desarrollo del estudio de polinomios ortogonales en espacios de Sobolev ha tenido lugar a lo largo de los últimos 30 años. El primer artículo sobre polinomios ortogonales de Sobolev, fue publicado en 1962 por Althammer, quien estudió los polinomios de Sobolev-Legendre ortogonales respecto al producto interno. El estudio de estas nuevas familias de polinomios ortogonales es interesante, no sólo por la comparación entre las propiedades y comportamiento de estos polinomios frente a los polinomios ortogonales, sino por las múltiples aplicaciones que tienen en diferentes contextos. En esta memoria analizaremos el comportamiento y las propiedades de polinomios ortogonales respecto a productos internos de Sobolev discretos. Más concretamente, nuestro interés será el estudio de fórmulas de conexión entre polinomios ortogonales estándar y polinomios ortogonales de Sobolev. De esta forma, podremos abordar algunos problemas de asintótica de polinomios ortogonales de Sobolev, así como obtener resultados de convergencia de series de Fourier asociadas a tales polinomios. Estos contenidos se dividen en los siguientes capítulos: En el capítulo 1 presentamos una introducción a la teoría de polinomios de Sobolev, introduciendo la notación que se utilizaría a lo largo de esta tesis. Se resumirán las principales diferencias entre el caso estándar y el caso Sobolev. En concreto nos centraremos en el estudio de relaciones de recurrencia y localización de ceros de dichos polinomios ortogonales. Se incluirá un estudio bastante completo de los diferentes tipos de fórmulas de conexión existentes. Finalmente, daremos una panorámica de los resultados conocidos en asintótica y desarrollos en series de Fourier de polinomios ortogonales de Sobolev tanto en el caso de soporte acotado como en el no acotado, que permitirá una mejor comprensión de nuestro trabajo. En el capítulo 2 se estudian aspectos analíticos y algebraicos de cierta familia de polinomios de Sobolev ortogonales respecto a una medida de soporte acotado. En la sección 2.1 se presenta una demostración alternativa de un resultado conocido sobre asintótica relativa exterior de ciertos polinomios de Sobolev. Demostraremos una nueva relación matricial entre la matriz asociada a la relación de recurrencia que satisfacen los polinomios de Sobolev y la matriz de Jacobi de los correspondientes polinomios ordinarios en la sección 2.2. En la última sección presentaremos un resultado sobre convergencia puntual de series de Fourier asociadas a ciertos polinomios de Sobolev. En el capítulo 3 resumiremos algunas propiedades conocidas de polinomios ortogonales respecto a una medida de Laguerre modificada, una k-iteración de Christoffel de la medida de Laguerre. A continuación, obtendremos estimaciones para la norma de estos polinomios y proporcionaremos una fórmula generalizada de Christoffel para tal familia de polinomios. Finalmente, presentaremos un estudio completo y detallado de los núcleos de Christoffel diagonales asociados a la distribución Gamma, obteniendo la asintótica de los mismos tanto dentro como fuera del soporte de la medida. En el capítulo 4 se hace un estudio de asintótica relativo de polinomios ortogonales de Sobolev discreto cuando las masas en la parte discreta del producto interno están situadas dentro el soporte de la medida, la región de oscilación de dichos polinomios. El comportamiento de dichos polinomios será estudiado tanto dentro como fuera del soporte de la medida. Finalmente, obtendremos el comportamiento asintótico de los coeficientes que aparecen en la relación de recurrencia que satisfacen los polinomios de Sobolev. En el capítulo 5 abordaremos el problema de convergencia de series de Fourier-Sobolev. Mostraremos la divergencia de la serie de Fourier asociada a cierta familia de polinomios ortogonales de Sobolev y la principal herramienta para ello serán las desigualdades de tipo Cohen. Este problema es tratado por primera vez para un producto de Sobolev discreto con una masa fuera del soporte de la medida., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Guillermo López Lagomasino.- Vocal: M. Alicia Chachafeiro López.- Secretario: José Carlos Soares Petronilho

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in the δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic } . The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this PhD Thesis is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G \ e obtained from the graph G by deleting an arbitrary edge e from it. These inequalities allow to obtain other main result, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity. In this work we also obtain information about the hyperbolicity constant of the line graph L(G) in terms of properties of the graph G. In particular, we prove qualitative results as the following: a graph G is hyperbolic if and only if L(G) is hyperbolic; if {Gn} is a T-decomposition of G ({Gn} are simple subgraphs of G), the line graph L(G) is hyperbolic if and only if supn δ(L(Gn)) is finite. Besides, we obtain quantitative results when k is the length of the edges of G and L(G). Two of them are quantitative versions of our qualitative results. We also prove that g(G)/4 ≤ δ(L(G)) ≤ c(G)/4 + 2k, where g(G) is the girth of G and c(G) is its circumference. We show that δ(L(G)) ≥ sup{L(g) : g is an isometric cycle in G}/4. Besides, we obtain bounds for δ(G) + δ(L(G)). Also, we characterize the graphs G with δ(L(G)) < k. Furthermore, we consider G with edges of arbitrary lengths, and L(G) with edges of non-constant lengths. In particular, we prove that a cycle of the graph G is transformed isometrically into a cycle of the graph L(G) with the same length. We also prove that δ(G) ≤ δ(L(G)) ≤ 5δ(G) + 3lmax, where lmax := supe∈E(G) L(e). This result implies the monotony of the hyperbolicity constant under a non-trivial transformation (the line graph of a graph). Also, we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the “boundary” (the 1-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the 1-skeleton of a general CW 2-complex is hyperbolic if and only if its dual graph is hyperbolic. One of the main problems on this subject is to relate the hyperbolicity with other properties on graph theory. We extend in two ways (edge-chordality and path-chordality) the classical definition of chordal graphs in order to relate this property with Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph (with small path-chordality constant) is hyperbolic. Some previous works characterize the hyperbolic product graphs (for the Cartesian product, strong product and lexicographic product) in terms of properties of the factor graphs. Finally, we characterize the hyperbolic product graphs for two important kinds of products: the graph join G1 ⊎G2 and the corona G1 ⋄G2. The graph join G1 ⊎G2 is always hyperbolic, and G1 ⋄ G2 is hyperbolic if and only if G1 is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join G1 ⊎ G2 and the corona G1 ⋄ G2., Sea X un espacio métrico geodésico y x1, x2, x3 ∈ X. Un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] y [x3x1] de X. El espacio X es δ- hiperbólico (en el sentido de Gromov) si todo lado de T está contenido en la δ-vecindad de la unión de los otros dos lados, para todo triángulo geodésico T de X. Se denota por δ(X) a la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ ≥ 0 : X es δ-hiperbólico }. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Uno de los principales objetivos de esta Tesis Doctoral es obtener información cuantitativa sobre la variación de la constante de hiperbolicidad del grafo G \ e que se obtiene del grafo G mediante la eliminación de una arista arbitraria e de él. Estas desigualdades permiten caracterizar, de forma cuantitativa, la hiperbolicidad de cualquier grafo en términos de su hiperbolicidad local. En esta memoria se obtene información acerca de la constante de hiperbolicidad del grafo línea L(G) en términos de propiedades del grafo G. En particular, se obtienen los siguientes resultados cualitativos: un grafo G es hiperbólico si y sólo si L(G) es hiperbólico; si {Gn} es una T-descomposición de G, el grafo línea L(G) es hiperbólico si y sólo si supn δ(L(Gn)) es finito. Además, se obtienen resultados cuantitativos cuando las aristas de G y L(G) tienen longitud k. Se demuestra que g(G)/4 ≤ δ(L(G)) ≤ c(G)/4 + 2k, donde g(G) es el cuello de G y c(G) es su circunferencia. También se prueba que δ(L(G)) ≥ sup{L(g) : g es un ciclo isométrico de G}/4. Igualmente, se obtienen cotas para δ(G) + δ(L(G)) y se caracterizan los grafos G tales que δ(L(G)) < k. Por otra parte, se consideran grafos G con aristas de longitudes arbitrarias, y L(G) con aristas de longitudes no constante. En particular, se demuestra que los ciclos de G se transforman isométricamente en ciclos de L(G) con la misma longitud. También se obtiene la relación δ(G) ≤ δ(L(G)) ≤ 5δ(G) + 3lmax, donde lmax := supe∈E(G) L(e). Este resultado prueba la monotonía de la constante de hiperbolicidad bajo una transformación no trivial (el grafo línea de un grafo). También, se obtienen criterios que permiten decidir, para una clase amplia de grafos, cuando son estos hiperbólicos o no. Se presta especial atención en los grafos planares que son la “frontera” (el 1-esqueleto) de una teselación del plano euclídeo. Además, se prueba que los grafos que se obtienen como 1-esqueleto de un CW 2-complejo general son hiperbólicos si y sólo si su grafo dual es hiperbólico. Uno de los principales problemas en este área es relacionar la hiperbolicidad de un grafo con otras propiedades de la teoría de grafos. En este trabajo se extiende de dos maneras (arista-cordalidad y camino-cordalidad) la definición clásica de cordalidad con el fin de relacionar esta propiedad con la hiperbolicidad. De hecho, se demuestra que todo grafo arista-cordal es hiperbólico y que todo grafo hiperbólico es camino-cordal. También, se demuestra que todo grafo cúbico camino-cordal (con constante pequeña de camino-cordalidad) es hiperbólico. Algunos trabajos previos caracterizan la hiperbolicidad de productos de grafos (para el producto cartesiano, el producto fuerte y el producto lexicográfico) en términos de propiedades de los grafos factores. En el último capítulo, se caracteriza la hiperbolicidad de dos productos de grafos: el grafo join G1 ⊎ G2 y el corona G1 ⋄ G2. El grafo join G1 ⊎ G2 siempre es hiperbólico, y el corona G1 ⋄ G2 es hiperbólico si y sólo si G1 es hiperbólico. Además, obtenemos fórmulas sencillas para la constante de hiperbolicidad del grafo join G1 ⊎ G2 y del corona G1 ⋄ G2.