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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 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

This thesis is mainly motivated by the analysis, understanding, and prediction of human behaviour by means of the study of their digital fingeprints. Unlike a classical PhD thesis, where you choose a topic and go further on a deep analysis on a research topic, we carried out a breadth analysis on the research topic of complex networks, such as those that humans create themselves with their relationships and interactions. These kinds of digital communities where humans interact and create relationships are commonly called Online Social Networks. Then, (i) we have collected their interactions, as text messages they share among each other, in order to analyze the sentiment and topic of such messages. We have basically applied the state-of-the-art techniques for Natural Language Processing, widely developed and tested on English texts, in a collection of Spanish Tweets and we compare the results. Next, (ii) we focused on Topic Detection, creating our own classifier and applying it to the former Tweets dataset. The breakthroughs are two: our classifier relies on text-graphs from the input text and we achieved a figure of 70% accuracy, outperforming previous results. After that, (iii) we moved to analyze the network structure (or topology) and their data values to detect outliers. We hypothesize that in social networks there is a large mass of users that behaves similarly, while a reduced set of them behave in a different way. However, specially among this last group, we try to separate those with high activity, or low activity, or any other paramater/feature that make them belong to different kind of outliers. We aim to detect influential users in one of these outliers set. We propose a new unsupervised method, Massive Unsupervised Outlier Detection (MUOD), labeling the outliers detected os of shape, magnitude, amplitude or combination of those. We applied this method to a subset of roughly 400 million Google+ users, identifying and discriminating automatically sets of outlier users. Finally, (iv) we find interesting to address the monitorization of real complex networks. We created a framework to dynamically adapt the temporality of large-scale dynamic networks, reducing compute overhead by at least 76%, data volume by 60% and overall cloud costs by at least 54%, while always maintaining accuracy above 88%. Publicado Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Rosa María Benito Zafrilla.- Secretario: Ángel Cuevas Rumín.- Vocal: José Ernesto Jiménez Merino

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 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

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

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

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

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

This work presents a study of (M;N)-coherent pairs of order (m; k) of sequences of orthogonal polynomials of a continuous and discrete variable on the real line and on the unit circle. This concept extends all the generalizations of the notion of, in our terminology, (1; 0)-coherent pair, studied in the literature, which was first introduced as coherent pair by A. Iserles, P. E. Koch, S. P. N rsett, and J. M. Sanz-Serna in 1991 (see [48])....., Este trabajo presenta un estudio de los pares (M;N)-coherentes de orden (m; k) de sucesiones de polinomios ortogonales de una variable continua en la recta real y en la circunferencia unidad, y de una variable discreta. Este concepto extiende todas las generalizaciones de la noción de, en nuestra terminología, par (1,0)-coherente, estudiadas en la literatura, el cual fue introducido por primera vez como par coherente por A. Iserles, P. E. Koch, S. P. N rsett, y J. M. Sanz-Serna en 1991 (véase [48])......

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

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

One of the open problems in graph theory is the characterization of any graph by a polynomial. Research in this area has been largely driven by the advantages offered by the use of computers which make working with graphs: it is simpler to represent a graph by a polynomial (a vector) that by the adjacency matrix (a matrix). We introduce the alliance polynomial of a graph. The alliance polynomial of a graph G with order n and maximum degree δ_1 is the polynomial A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) 〗, where A{_k}(G) is the number of exact defensive k-alliances in G. Also, we develop and implement an algorithm that computes in an efficient way the alliance polynomial. We obtain some properties of A(G; x) and its coefficients for: • Path, cycle, complete and star graphs. In particular, we prove that they are characterized by their alliance polynomials. • Cubic graphs (graphs with all of their vertices of degree 3), since they are a very interesting class of graphs with many applications. We prove that they verify unimodality. Also, we compute the alliance polynomial for cubic graphs of small order, which satisfy uniqueness. • Regular graphs (graphs with the same degree for all vertices). In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of connected ∆-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not connected ∆-regular. 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. We obtain information about the hyperbolicity constant of cubic graphs. These graphs are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is the hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants. Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K_4 or to K_3,3, the inequalities 5k/4, Uno de los problemas abiertos en la teoría de grafos es la caracterización de cualquier grafo por un polinomio. La investigación en este área ha sido impulsada en gran parte por las ventajas que ofrece el uso de las computadoras que hacen que trabajar con grafos sea más simple. En esta Tesis introducimos el polinomio de alianza de un grafo. El polinomio de alianza de un grafo G con orden n y grado m´aximo δ_1 es el polinomio A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) , donde A{_k}(G) es el n´umero de k alianzas defensivas exactas en G. También desarrollamos e implementamos un algoritmo que calcula de manera eficiente el polinomio de alianza. En este trabajo obtenemos algunas propiedades de A(G; x) y sus coeficientes para: • Grafos caminos, ciclos, completos y estrellas. En particular, hemos demostrado que se caracterizan mediante sus polinomios de alianza. • Grafos cúbicos (grafos con todos sus vértices de grado 3), ya que son una clase muy interesante de grafos con muchas aplicaciones. Hemos demostrado que sus polinomios de alianza verifican unimodalidad. Además, calculamos el polinomio de alianza para grafos cúbicos de orden pequeño, los cuales satisfacen unicidad. • Grafos regulares (grafos con todos sus vértices de igual grado). En particular, se caracteriza el grado de los grafos regulares por el n´umero de coeficientes distintos de cero de su polinomio de alianza. Además, se demuestra que la familia de polinomios de alianza de grafos conexos _-regulares con grado pequeño es muy especial, ya que no contiene polinomios de alianza de grafos conexos que no sean _-regulares. Si X es un espacio métrico geodésico y x1, x2, x3 ∈ X, un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] and [x3x1] de X. El espacio X es δ-hiperbólico (en el sentido de Gromov) si todo lado de todo triángulo geodésico T de X está contenido en la δ-vecindad de la unóon de los otros dos lados. 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. Hemos obtenido información acerca de la constante de hiperbolicidad de los grafos cúbicos; dichos grafos son muy importantes en el estudio de la hiperbolicidad, ya que para cualquier grafo G con grado máximo acotado existe un grafo cúbico G∗ tal que G es hiperbólico si y sólo si G∗ es hiperbólico. En esta memoria conseguimos caracterizar los grafos cúbicos con constante de hiperbolicidad pequeña. Además, se obtienen cotas para la constante de hiperbolicidad del grafo complemento de un grafo cúbico; nuestro principal resultado dice que para cualquier grafo cúbico finito G no isomorfo a K4 o K3,3, se cumple la relación 5k/4, Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid, Presidente: Domingo de Guzmán Pestana Galván; Secretario: Eva Tourís Lojo; Vocal: Sergio Bermudo Navarrete

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.

Mención Internacional en el título de doctor, The study of interfacial phenomena has always constituted an integral part of condensed matter physics and materials science. Indeed, most properties of real materials depend crucially on the presence of imperfections, such as bulk vacancies, dislocations, surface roughness, etc. which derive from the non-equilibrium conditions under which the material has formed [1, 2]. In the last century, the increasing interest in systems with considerable surface to volume ratio, such as for instance devices at the nanoscale, has attracted scientists from different fields, from physics to chemistry, biology, or engineering [3]. This is due to the great amount of technological applications of such systems to a wide variety of situations. Moreover, the improvement of production and characterization techniques for the growth of surfaces at micro and nano-scale, such as Molecular Beam Epitaxy, or for surface etching, such as Ion Beam Sputtering , have unveiled unexpected interesting physical properties of the grown interfaces. From a technological point of view, the ability to control and predict the effect induced by disorder and the mechanisms of self-organization that ensue during the growth dynamics is of great interest [4]. For instance, the possibility to control the surface roughness could improve electric conductivity or the mechanical contact of certain devices, whereas the ability to control the formation of a pattern could change, for instance, the optical properties of the material., Esta tesis se centra en fenómenos de invariancia de escala genérica en modelos de crecimiento de superficies. Concretamente, hemos considerado sistemas fuera de equilibrio que presentaran desorden a grandes escalas espaciales y temporales (rugosidad cinética), y que estuvieran descritos por ecuaciones continuas. El objetivo de la tesis ha sido utilizar métodos propios de la mecánica estadística para estudiar la interacción de este tipo de sistemas con otros dos fenómenos: la aparición de inestabilidades morfológicas que dan lugar a la formación de patrones, y la presencia de anisotropía. En cuanto al primer problema, hemos considerado un ejemplo paradigmático de ecuación que presente invariancia de escala genérica e inestabilidades morfológicas, la ecuación de Kuramoto-Siashinsky. Hasta la fecha, el comportamiento asintótico de este modelo no se había entendido completamente. De hecho, existe una controversia acerca de cual sea la clase de universalidad a la que este modelo pertenece en dos dimensiones. Utilizando simulaciones numéricas de gran escala, hemos comprobado que esta ecuación en dos dimensiones pertenece a la clase de universalidad de Kardar-Parisi-Zhang. Sucesivamente, nos hemos centrado en modelos que presenten invariancia de escala genérica, cuyos exponentes críticos sean distintos cuando se miden en direcciones distintas del sustrato. Cuando eso ocurre, decimos que el sistema presenta anisotropía fuerte. A pesar de la presencia generalizada de anisotropía en sistemas naturales, hasta el momento se había dedicado relativamente poca atención a este fenómeno. Eso se debe probablemente al hecho de que los modelos mas estudiados en este ´ambito no presentan anisotropía fuerte, aunque la forma de la ecuación sea en algunos casos completamente anisótropa. Después de introducir un marco teórico para el estudio de modelos fuertemente anisótropos, hemos comprobado nuestra hipótesis a través de simulaciones numéricas de distintas ecuaciones que presentaran anisotropía fuerte. Sucesivamente, hemos llevado a cabo un estudio numérico y analítico de otros modelos, con el objetivo de encontrar condiciones para la aparición de anisotropía fuerte. Encontramos que este fenómeno ocurre en presencia de una dinámica conservada, siempre que la ecuación tenga una forma bastante especifica. Finalmente, hemos presentado evidencias experimentales de la aparición de anisotropía fuerte, a través de un análisis de datos de experimentos de erosión iónica en distintas condiciones experimentales......., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Mario Castro Ponce.- Secretario: Javier Manuel Muñoz García.- Vocal: Gnnar Pruessner

El objetivo principal de esta disertación es el estudio de la teoría de las álgebras de Lie–Jordan Banach y su papel en el marco de la mecánica clásica y cuántica y sus aplicaciones en diferentes ámbitos de las matemáticas. La investigación algebraica de los fundamentos de la física surgió de la búsqueda de una formulación “excepcional” para la mecánica cuántica, que finalmente culminó en la formulación de las álgebras de Jordan y las posteriores C∗–álgebras..........

This work presents a study of (M;N)-coherent pairs of order (m; k) of sequences of orthogonal polynomials of a continuous and discrete variable on the real line and on the unit circle. This concept extends all the generalizations of the notion of, in our terminology, (1; 0)-coherent pair, studied in the literature, which was first introduced as coherent pair by A. Iserles, P. E. Koch, S. P. N rsett, and J. M. Sanz-Serna in 1991 (see [48])..... ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Este trabajo presenta un estudio de los pares (M;N)-coherentes de orden (m; k) de sucesiones de polinomios ortogonales de una variable continua en la recta real y en la circunferencia unidad, y de una variable discreta. Este concepto extiende todas las generalizaciones de la noción de, en nuestra terminología, par (1,0)-coherente, estudiadas en la literatura, el cual fue introducido por primera vez como par coherente por A. Iserles, P. E. Koch, S. P. N rsett, y J. M. Sanz-Serna en 1991 (véase [48])......

This thesis deals with the mathematical modeling of evolutionary processes that take place in heterogeneous populations. Its leitmotif is the response of complex ensembles of replicating entities to multiple (and often opposite) selection pressures. Even though the specific problems addressed in different chapters belong to different organizational levels—genome, population, and community—all of them can be conceptualized as the evolution of a heterogeneous population—let it be a population of genomic elements, viruses, or prokaryotic hosts and phages—facing a complex environment. As a result, the mathematical tools required for their study are quite similar. In contrast, the strategies that each population has discovered to perpetuate vary according to the different evolutionary challenges and environmental constraints that the population experiences. Along this thesis, there has been a special interest on connecting theoretical models with experimental results. To that end, most of the work presented here has been motivated either by laboratory findings or by the bioinformatic analysis of sequenced genomes. We strongly believe that such a multidisciplinary approach is necessary in order to improve our knowledge on how evolution works. Moreover, experiments are a must when it comes to propose antiviral strategies based on theoretical predictions, as we do in Chapter 3. This thesis is structured in two main blocks. The first one focuses on studying instances of viral evolution under the action of mutagenic drugs, paying particular attention to their possible application to the development of novel antiviral therapies. This block comprises chapters 2 and 3; the former dicussing the phenomenon of lethal defection and stochastic viral extinction; the latter dealing with the optimal way to combine mutagens and inhibitors in multidrug antiviral treatments. The second block is devoted to the study of the evolutionary forces underlying genome structure. In chapter 4, we propose a mechanism through which multipartite viruses could have originated. Interestingly, the pathway leading to genome segmentation shares some steps with lethal defection, but each outcome is reached at specific environmental conditions. Chapter 5 analyses the abundance distributions of transposable elements in prokaryotic genomes, with the aim of determining the key processes involved in their spreading. We explicitly explore the hypothesis that transposable elements follow a neutral dynamics, with a negligible fitness cost for their host genomes. A higher level of organization is studied in Chapter 6, where an agent based coevolutionary model based on Lotka-Volterra interactions is used to investigate the evolutionary dynamics of the prokaryotic antiviral immunity system CRISPR-Cas. This chapter also examines the environmental factors that are responsible for its maintenance or loss. Finally, Chapter 7 summarizes the main results obtained along the thesis and sketches possible lines of work based on them.

Esta tesis incluye cinco capítulos, y está dedicada al estudio de familias de polinomios ortogonales de tipo estándar y no estándar. A continuación resumimos brevemente las conclusiones aportadas por la presente memoria. • Se realiza por primera vez un estudio completo del comportamiento de los zeros de familias de polinomios ortogonales con respecto a una medidas modificadas mediante perturbaciones canónicas de tipo Uvarov y Christoffel . El comportamiento de estos ceros se da en términos del parámetro M, el cual determina cómo es la intensidad de la perturbación sobre la medida clásica. Hasta el momento, se habían realizado progresos significativos en esta dirección a través de aproximaciones semiclásicas, y solamente tratando el comportamiento de propiedades promedio de los ceros, usando el método de WKB. • Se obtienen resultados asintóticos de las secuencias de polinomios ortogonales mónicos con respecto a la perturbación de Uvarov de la medida clásica de Laguerre, como ejemplo canónico de perturbación fuera del soporte de una medida clásica con soporte no acotado. Hasta la fecha, los puntos de masa se localizaban en la frontera (o fronteras) del soporte de la medida perturbada. • Se proporciona un modelo electrostático de los ceros de la familia de polinomios ortogonales con respecto a una medida de Laguerre perturbada, con una cantidad numerable m de puntos de masa, fuera del soporte de la medida clásica de Laguerre. Hasta el momento, el único trabajo similar consideraba un solo punto de masa en el origen. Igualmente, describimos el comportamiento de los ceros de los polinomios ortogonales tipo-Krall en términos de los ceros de cierto polinomio de grado 2m (siendo m el número de masas de Dirac que aparecen en la medida), y que son las fuentes de un potencial logarítmico de corto alcance que afecta a la localización de los ceros de las secuencias de polinomios ortogonales Krall-Laguerre, considerados como puntos críticos de un problema de equilibrio. • Igualmente se obtienen propiedades asintóticas de secuencias de polinomios ortogonales mónicos de tipo Laguerre-Sobolev, cuando los puntos de masa están situados fuera del soporte de la medida clásica de Laguerre. Hasta la fecha, los puntos de masa se localizaban en la frontera (o fronteras) de los soportes de las medidas perturbadas.

The main purpose of the work presented here is to study transformations of sequences of orthogonal polynomials associated with a hermitian linear functional L, using spectral transformations of the corresponding C-function F. We show that a rational spectral transformation of F is given by a finite composition of four canonical spectral transformations. In addition to the canonical spectral transformations, we deal with two new examples of linear spectral transformations. First, we analyze a spectral transformation of L such that the corresponding moment matrix is the result of the addition of a constant on the main diagonal or on two symmetric sub-diagonals of the initial moment matrix. Next, we introduce a spectral transformation of L by the addition of the first derivative of a complex Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. In this case, outer relative asymptotics for the new sequences of orthogonal polynomials in terms of the original ones are obtained. Necessary and su cient conditions for the quasi-definiteness of the new linear functionals are given. The relation between the corresponding sequence of orthogonal polynomials in terms of the original one is presented. We also consider polynomials which satisfy the same recurrence relation as the polynomials orthogonal with respect to the linear functional L , with the restriction that the Verblunsky coe cients are in modulus greater than one. With positive or alternating positive-negative values for Verblunsky coe cients, zeros, quadrature rules, integral representation, and associated moment problem are analyzed. We also investigate the location, monotonicity, and asymptotics of the zeros of polynomials orthogonal with respect to a discrete Sobolev inner product for measures supported on the real line and on the unit circle., El objetivo principal de este trabajo es el estudio de las sucesiones de polinomios ortogonales con respecto a transformaciones de un funcional lineal hermitiano L , usando para ello las transformaciones de la correspondiente C-función F . Un primer resultado es que las transformaciones espectrales racionales de F están dadas por una composición finita de cuatro transformaciones espectrales canónicas. Además de estas transformaciones canónicas se estudian dos ejemplos de transformaciones espectrales lineales que son novedosos en la literatura. El primero de estos ejemplos está dado por una modificación del funcional lineal L, de modo que la correspondiente matriz de momentos es el resultado de la adición de una constante en la diagonal principal o en dos subdiagonales simétricas de la matriz de momentos original. El segundo ejemplo es una transformación de L mediante la adición de la primera derivada de una delta de Dirac compleja cuando su soporte es un punto sobre la circunferencia unidad o dos puntos simétricos respecto a la circunferencia unidad. En este caso se obtiene la asintótica relativa exterior de la nueva sucesión de polinomios ortogonales en términos de la original. Se dan condiciones necesarias y suficientes para que los funcionales derivados de las perturbaciones estudiadas sean cuasi-definidos, y se obtiene la relación entre las correspondientes sucesiones de polinomios ortogonales. Se consideran además polinomios que satisfacen las mismas ecuaciones de recurrencia que los polinomios ortogonales con respecto al funcional lineal L, agregando la restricción de que sus coeficientes de Verblunsky son en valor absoluto mayores que 1. Cuando estos coeficientes son positivos o alternan signo, se estudian los ceros, las fórmulas de cuadratura, la representación integral y el problema de momentos asociado. Asimismo, se estudia la localización, monotonicidad y comportamiento asintótico de los ceros asociados a polinomios discretos ortogonales de Sobolev para medidas soportadas tanto en la recta real como en la circunferencia unidad., This work was supported by FPU Research Fellowships, Re. AP2008-00471 and Dirección General de Investigación, Ministerio de Ciencia e Inovación of Spain, grant MTM2009-12740-C03-01.

In the first part of this thesis, I address the classical problem of asset price dynamics based on a new theoretical framework developed for nonequilibrium physical systems. This problem is mainly relevant for two reasons. First, because understanding the true distribution of returns is important for asset allocation, risk management, and option pricing. Second, because in spite of all the effort in determining the origin of non-Gaussian returns no conclusive result has been achieved yet. The most important result of this part is the demostration that the non-Gaussian shape and stable scaling of the returns distribution are due to slow, but significant, fluctuations in volatility. Futhermore, this result suggests that stock price fluctuations are universal, and that return distributions can be described by one functional form. In the second part, I present an empirical study about the execution of large orders in two stock exchanges: the London Stock Exchange, and the Spanish Stock Exchange. This type of orders can cause a tremendous impact because they are larger than the available liquidity in the order book at a time. For this reason, they are split to minimize transaction costs. Market price impact is the basic factor of these costs, so an accurate description of its functional form is necessary to any optimal execution. The most important result in this part is the empirical determination of this functional form in two markets and the finding of a common behavior in both markets that can be summarized into a concave temporary impact, roughly described by a square root function of the hidden order size, and a price reversion after the completion of the hidden order making permanent impact equal to roughly half of the temporary impact.