Search

Similar to the field of real numbers R, which can be constructed as the completion of the rational numbers with respect the usual absolute value | · |, the field of p-adic numbers Q_p is also the completion of the rational numbers with the p-adic absolute value | · |_p. These numbers were first introduced by Kurt Hensel to harness the power of analytic tools in number theory, but nowadays non-archimedean mathematics has become a very rich and active area of research in its own right. This thesis is composed of seven chapters whose main theme is non-archimedean.This dissertation begins with the first chapter in which we introduce some necessary back- ground and concepts that are used throughout this thesis. In particular, we recall the notion of valued fields and review some basic facts from non-archimedean algebra and analysis. We then introduce the Euclidean building associated to the reductive group PGL, which will appear throughout this thesis.In the second chapter we study the entropy map for multivariate Gaussian distributions on a non-archimedean local field. As in the real case, the image of this map lies in the supermodular cone. Moreover, given a multivariate Gaussian measure on a local field, its image under the entropy map determines its pushforward under the valuation map. In general, this entropy map can be defined for non-archimedian valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3 and discuss non-archimedean statistical Gaussian models.In the third chapter we give a method for sampling points from an algebraic manifold (affine or projective) over a local field with a prescribed probability distribution. In the spirit of previous work by Breiding and Marigliano on real algebraic manifolds, our method is based on slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of our sampling method and discuss a few applications. In one application, we sample from algebraic p-adic matrix groups and modular curves.In the fourth chapter, we introduce a novel characterization of Bernoulli polynomials by circular convolution. There is a combinatorial and probabilistic model underlying this characterization which we call The Bernoulli clock. We use this model to give a probabilistic perspective to the work of Horton and Kurn [91] and the more recent work of Clifton et al. [33] on counting permutations of the multiset 1^m · · · n^m with longest continuous and increasing subsequence of length n starting from 1.In the fifth chapter we apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the grad- uated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We also advance the ideal theory of graduated orders by introducing their ideal class polytropes. We then extend our study to bolytropes and bolytrope orders. Bolytropes are bounded subsets of an affine building that consist of all points that have distance at most r from some polytrope. We prove that the points of a bolytrope describe the set of all invariant lattices of a bolytrope order, generalizing the correspondence between polytropes and graduated orders.In the sixth chapter, we study non-archimedean Schur representations and their invariant lattices. More precisely, fix a non-archimedean discretely valued field K and let O_K be its valuation ring. Given a vector space V of dimension n over K and a partition λ of an integer d, we study the problem of determining the invariant lattices in the Schur module S_λ(V ) under the action of the group GL(n,O_K). When K is a non-archimedean local field, our results determine the GL(n, O_K )-invariant Gaussian distributions on S_λ (V ).Finally, in the seventh chapter, we turn to arithmetic geometry. We express the reduction types of Picard curves in terms of tropical invariants associated to binary quintics. These invariants are connected to Picard modular forms using recent work [32] of Cl ́ery and van der Geer. We furthermore give a general framework for tropical invariants associated to group actions on arbitrary varieties. The problem of describing reduction types of curves in terms of their associated invariants fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification \overline{M_{0,n}}.

Tesi per compendi de publicacions, amb diferents seccions retallades pels drets d'editor Premi extraordinari doctorat UPC curs 2017-2018. Àmbit d’Enginyeria Industrial The present work addresses two different applications enabled by a specific and useful property of calcium phosphate cements (CPCs): injectability. On the one hand minimally invasive procedures involving the use of CPCs are based on the injectability of such biomaterials, and on the other hand extrusion-based additive manufacturing processes such as robocasting rely on this property to correctly manufacture personalized 3D-printed scaffolds for the treatment of large bone defects. The present thesis is divided in three different sections. The first one consists in a study of the differences of injectability of aqueous pastes of the two allotropic forms of tricalcium phosphate, namely a- and ß-TCP. The reactivity of the powder was shown to play a significant role in the injectability of TCP pastes. Significant differences were observed between the injection behaviour of non-hardening ß-TCP pastes and that of selfhardening a-TCP pastes. The differences were more marked at low liquid-to-powder ratios, using fine powders and injecting through thin needles. Although, as a general trend, faster-setting pastes were less injectable, some exceptions to this rule were found. For example, whereas in the absence of setting accelerants fine TCP powders were more injectable than the coarse ones, in spite of their shorter setting times, this trend was inverted when setting accelerants were added, and coarse powders were more injectable than the fine ones. In the second section thermoresponsive pastes are developed through the combination of CPCs with an inversethermoresponsive hydrogel. Although calcium phosphate cements (CPCs) are used for bone regeneration in a wide range of clinical applications, various physicochemical phenomena are known to hinder their potential use in minimally invasive surgery or in highly vascularized surgical sites, mainly because of their lack of injectability or their low washout resistance. The proposed strategy allowed to finely tune the cohesive and rheological properties of CPCs to achieve clinical acceptable injectability. It avoided phase separation during implantation and improved cohesion, avoiding washout of the paste. Using the knowledge acquired about the injectability behaviour of TCP pastes, the additive manufacturing of 3D printed scaffolds is studied in the last section. More precisely, this study dealt with the robocasting of alpha-tricalcium phosphate/gelatine reactive slurries as a bioinspired self-setting ink for the production of biomimetic hydroxyapatite/gelatine scaffolds. A controlled and totally interconnected pore network of approximately 300 µm was obtained after ink printing and setting, with the struts consisting of a micro/nanoporous matrix of needle-shaped calcium deficient hydroxyapatite crystals, with a high specific surface area. Gelatine was effectively retained by chemical crosslinking. The setting reaction of the ink resulted in a significant increase of both the elastic modulus and the compressive strength of the scaffolds, which were within the range of the human trabecular bone. In addition to delaying the onset of the setting reaction, thus providing enough time for printing, gelatine provided the viscoelastic properties to the strands to support their own weight, and additionally enhanced mesenchymal stem cell adhesion and proliferation on the surface of the scaffold. Altogether this new processing approach opens good perspectives for the design of hydroxyapatite scaffolds for bone tissue engineering with enhanced reactivity and resorption rate. El presente trabajo contempla dos aplicaciones permitidas por una propiedad específica y útil de los cementos de fosfato de calcio (CPCs): la inyectabilidad. Por un lado, los procedimientos quirúrgicos mínimamente invasivos que implican el uso de los cementos de fosfatos de calcio se basan en su inyectabilidad y por otro los procesos de fabricación aditiva basados en la microextrusión como por ejemplo el robocasting se fundamentan en esta propiedad para fabricar andamios impresos en 3D para el tratamiento de defectos óseos grandes. Esta tesis se divide en tres secciones diferentes. La primera consiste en un estudio de las diferencias en la inyectabilidad de pastas cuyas fases solidas esta compuestas por dos formas alotrópicas del fosfato tricálcico (TCP), respectivamente a- y ß-TCP. La reactividad de los polvos ha sido identificada como teniendo un rol significativo en la inyectabilidad de pastas de TCP. Las diferencias fueron más marcadas a una relación liquido-polvo baja, usando polvos finos e inyectando las pastas por cánulas finas. Sin embargo, como tendencia general, las pastas de fraguado más rápido fueron las menos inyectables, algunas excepciones a esta regla fueron encontradas. Por ejemplo, en ausencia de acelerante del fraguado las pastas de polvos finos de TCP fueron más inyectables que las de polvos gruesos a pesar de sus tiempos de fraguado más corto, esta tendencia fue invertida cuando se empleó soluciones de acelerante y los polvos gruesos resultaron más inyectables que los finos. En la segunda sección de esta tesis, cementos termosensibles fueron desarrollados mediante la combinación de CPCs con un hídrogel termosensible. Aunque los CPCs se usen para la regeneración ósea en una variedad importante de indicaciones clínicas, varios fenómenos fisicoquímicos pueden comprometer su uso para procedimientos quirúrgicos mínimamente invasivos o en sitios quirúrgicos altamente vascularizados, principalmente debido a su falta de inyectabilidad o su baja resistencia cohesiva al lavado por líquidos. La estrategia propuesta ha permitido ajustar la cohesión y las propiedades reológicas del cemento alcanzando una inyectabilidad clínicamente aceptable. Permitió evitar la separación de fases durante la inyección y mejoró la cohesión, evitando el lavado y desintegración de la pasta. Usando el conocimiento adquirido sobre la inyectabilidad de pastas de TCP, se estudió la fabricación aditiva de andamios mediante impresión 3D. Más precisamente, este estudio trató con la fabricación por robocasting empleando una mezcla reactiva de gelatina/a-TCP como tinta bioinspirada de andamios de hidroxiapatita/gelatina con una composición biocinética. Se obtuvo una red de poros totalmente interconectados de unos 300 µm, con los hilos compuestos de una matriz micro/nanoporosa de cristales aciculares de hidroxiapatita deficiente en calcio presentando una elevada superficie especifica. La gelatina fue correctamente retenida gracias a un procedimiento de entrecruzado químico. La reacción de fraguado de la tinta resultó en un aumento significativo del módulo elástico y de la resistencia a compresión, que se situó en el rango de la del hueso trabecular humano. Adicionalmente al retraso en el comienzo de la reacción de fraguado, permitiendo mientras la impresión de la tinta, la gelatina confirió propiedades viscoelasticas a los hilos extruidos para soportar su propio peso, y adicionalmente mejoró la adhesión y proliferación de células mesénquimales sobre el andamio. En su conjunto este trabajo abre perspectivas nuevas para el diseño de andamios de hidroxiapatita para ingeniería de tejido óseo con reactividad y reabsorción mejoradas. Award-winning