Daktaro disertacijoje sprendžiami trys uždaviniai. Pirmasis nagrinėja Minkovskio “klaustuko” funkcijos Stieltjes’o transformacijos (tai yra, šios funkcijos momentų generuojančios funkcijos, taip vadinamosios diadinės periodo funkcijos), analizines savybes ir jos išraišką uždara ar beveik uždara forma. Pagrindinis rezultatas teigia, kad diadinę periodo funkciją galima išreikšti racionaliųjų funkcijų su racionaliaisiais koeficientais konverguojančia eilute. Įrodyme naudojama kompleksinės dinamikos, analizinės grandininių trupmenų teorijos, kelių kompleksinių kintamųjų funkcijų teorijos technika. Antrasis uždavinys nagrinėja funkcines lygtis, susietas su norminėmis ir kitomis kelių kintamųjų formomis. Yra parodoma, kad šios funkcinės lygtys kartais turi kitų, netrivialiųjų sprendinių. Galiausiai, yra pateikiamas naujas mažosios Fermat teoremos įrodymas. / Doctoral thesis is devoted to investigation of three problems. The first one deals with the analytic properties and representation in closed or almost closed form of the Stieltjes tranform of the Minkowski question mark function (that is, the generating function of moments, the so called dyadic period function). The main result claims that the dyadic period function can be represented as a convergent series of rational functions with rational coefficients. In the proof the techniques from complex dynamics, analytic theory of continued fractions, the theory of several complex variables are being used. The second problem is dealing with functional equations associated with norm and other forms. It is shown that these functional equations sometimes have other solutions apart from the trivial ones. Finally, we present a new proof of Fermat’s little theorem.

Doctoral thesis is devoted to investigation of three problems. The first one deals with the analytic properties and representation in closed or almost closed form of the Stieltjes tranform of the Minkowski question mark function (that is, the generating function of moments, the so called dyadic period function). The main result claims that the dyadic period function can be represented as a convergent series of rational functions with rational coefficients. In the proof the techniques from complex dynamics, analytic theory of continued fractions, the theory of several complex variables are being used. The second problem is dealing with functional equations associated with norm and other forms. It is shown that these functional equations sometimes have other solutions apart from the trivial ones. Finally, we present a new proof of Fermat’s little theorem. / Daktaro disertacijoje sprendžiami trys uždaviniai. Pirmasis nagrinėja Minkovskio “klaustuko” funkcijos Stieltjes’o transformacijos (tai yra, šios funkcijos momentų generuojančios funkcijos, taip vadinamosios diadinės periodo funkcijos), analizines savybes ir jos išraišką uždara ar beveik uždara forma. Pagrindinis rezultatas teigia, kad diadinę periodo funkciją galima išreikšti racionaliųjų funkcijų su racionaliaisiais koeficientais konverguojančia eilute. Įrodyme naudojama kompleksinės dinamikos, analizinės grandininių trupmenų teorijos, kelių kompleksinių kintamųjų funkcijų teorijos technika. Antrasis uždavinys nagrinėja funkcines lygtis, susietas su norminėmis ir kitomis kelių kintamųjų formomis. Yra parodoma, kad šios funkcinės lygtys kartais turi kitų, netrivialiųjų sprendinių. Galiausiai, yra pateikiamas naujas mažosios Fermat teoremos įrodymas.

This thesis investigates the interplay between explanatory issues in special relativity and the theory's metaphysical foundations. Special attention is given to the 'dynamical approach' to relativity, promoted primarily by Harvey Brown and collaborators, according to which the symmetries of dynamical laws are explanatory of relativistic effects, inertial motion, and even the Minkowskian geometrical structure of a specially relativistic world. The thesis begins with a review of Einstein's 1905 introduction to special relativity, after which brief historical introductions are given for the standard 'geometrical' approach to relativity and the unorthodox 'dynamical' approach. After a critical review of recent literature on the topic, the dynamical approach is shown to be in need of a metaphysical package that would undergird the explanatory claims mentioned above. It is argued that the dynamical approach is best understood as a form of relationalism - in particular, as a relativistic form of 'regularity relationalism', promoted recently by Nick Huggett. According to this view, some portion of a world's geometrical structure actually supervenes upon the symmetries of the best-system dynamical laws for a material ontology endowed with a primitive sub-metrical structure. To explore the plausibility of this construal of the dynamical approach, a case study is carried out on solutions to the Klein-Gordon equation. Examples are found for which the field values, when purged of all spatiotemporal structure but their induced topology, are still arguably best-systematized by the Klein-Gordon equation itself. This bolsters the plausibility of the claim that some system of field values, endowed with mere sub-metrical structure, might have as its best-systems dynamical laws a (set of) Lorentz-covariant equation(s), on which Minkowski geometrical structure would supervene. The upshot is that the dynamical approach to special relativity can be defended as what might be called an ontologically and ideologically relationalist approach to Minkowski spacetime structure. The chapters refer regularly to three appendices, which include a brief introduction to topological and differentiable spaces.

Nous étudions certains processus de Lévy à valeurs dans les groupes d'isométries respectifs des espace-temps de Minkowski, de De Sitter et de Anti-De-Sitter. Le groupe d'isométries est vu comme le fibré des repères de l'espace-temps et les processus de Lévy considérés se projettent sur le fibré unitaire en un processus markovien relativiste ; c'est-à-dire que les trajectoires dans l'espace-temps sont de genre temps et que le générateur est invariant par les isométries. Dans la première partie nous adaptons pour les diffusions hypoelliptiques générales un résultat de Ben Arous et Gradinaru concernant la singularité de la fonction de Green hypoelliptique. Nous déduisons de cela un critère d'effilement de Wiener local pour les diffusions relativistes dans le groupe de Poincaré, groupe des isométries de l'espace-temps de Minkowski. Dans les deux dernières parties nous nous intéressons au comportement asymptotique du flot stochastique associé à ces processus de Lévy dans les différents groupes d'isométries. Sous une condition d'intégrabilité de la mesure de Lévy nous calculons explicitement les coefficients de Lyapounov des processus dans le groupe de Poincaré. Nous effectuons un travail similaire pour les espace-temps de De Sitter et Anti-De-Sitter en nous limitant au cas des diffusions. Nous explicitons de plus la frontière de Poisson pour la diffusion dans le groupe d'isométries de l'espace-temps de De Sitter. / We study some Lévy processes with values in the isometry group of Minkowski, De Sitter and Anti-de-Sitter space-times. The isometry group is seen as the frame bundle of the space-time and the Lévy processes we consider are some lift of relativistic markovian processes with values in the unitary tangent bundle of the space-time. Theses processes are relativistic in the sense that theirs trajectories are time-like and their generators are invariant by the isometries of the space-time. In the first part of this work we adapt to the case of a general hypoelliptic diffusion a result of Ben Arous and Gradinaru concerning the singularity of the hypoelliptic Green function. We deduce of this a local Wiener criterion for the relativistic diffusion in the isometry group of Minkowski space-time. In the two last parts we are interested to the asymptotic behavior of the stochastic flow associated to these Lévy processes in the different considered space-times. Under a integrability condition on the Lévy measure we compute explicitly the Lyapunov coefficient for such flows in the isometry group of Minkowski space-time. Then, we do a similar work in the context of de Sitter and Anti-de-Sitter space-times limiting ourselves to the case of diffusions. In fine, we explicit the Poisson boundary of the diffusion in the isometry group of de Sitter space-time.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-18T13:20:13Z No. of bitstreams: 1 arquivototal.pdf: 1301385 bytes, checksum: bef9e67e2f8926afbbf4b9bbad62a632 (MD5) / Made available in DSpace on 2016-03-18T13:20:13Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1301385 bytes, checksum: bef9e67e2f8926afbbf4b9bbad62a632 (MD5) Previous issue date: 2015-08-31 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / A coordinate system has the function to locate the space-time events with respect to a reference system. The construction of the coordinate system depends crucially on the notion of simultaneity associated with the reference. However, there is no natural way, or privileged, set concurrency for non-inertial reference frames, even in Minkowski spacetime. Each procedure leads to different coordinate systems. In this paper, we discuss some methods well known in the literature. We studied the coordinates Rindler, Fermi-Walker radar coordinates and emission coordinates (or GPS). The Rindler coordinate system is one of the prominent systems because it allows to simulate some properties of the geometry of the black hole in a flat space-time. The Rindler coordinates are associated with a family of uniformly accelerated observers who obey the relationship a = (1 / ρ), where is the actual acceleration of the observer and ρ its initial position with respect to some inertial reference system. In this paper, we propose a method for constructing coordinate systems suitable for observers whose acceleration depends on the initial position of the general form a = a (ρ) using this physical principle of locality. The Rindler coordinate system appears as a feature of our generalization. Other particular cases allow us to discuss the relationship between the non-Euclidean geometry of space sections and accelerated reference frames, as was originally proposed by Einstein. Moreover, with the generalization can simulate the behavior of static observers both near the horizon of a black hole, which are subject to a kind of acceleration field (ρ) = 1 / ρ, as in remote areas, for which the (ρ) = 1 / ρ2. In the latter two cases, ρ is from the accelerated observer to the event horizon. With the intention of analyzing the effects of instantaneous acceleration of the rate of atomic clocks, we consider a free massive particle in a box of infinite walls, which is drawn by observers of Rindler. We assume that the particle obeys the Klein-Gordon equation and so we found the frequencies of the stationary states of the system. The transitions between the stationary states are used to set a default frequency for our atomic clock toy. Comparing the accelerated system power spectrum with the energy spectrum of a similar system in an inertial frame, we determined the influence of instantaneous acceleration of the rate of atomic clocks. / Um sistema de coordenadas tem a função de localizar os eventos do espaço-tempo com respeito a um sistema de referência. A construção do sistema de coordenadas depende crucialmente da noção de simultaneidade associada ao referencial. No entanto, não existe uma maneira natural, ou privilegiada, de definir simultaneidade para referenciais não-inerciais, mesmo no espaço-tempo de Minkowski. Cada procedimento conduz a diferentes sistemas de coordenadas. Neste trabalho, discutimos alguns métodos bem conhecidos da literatura especializada. Estudamos as coordenadas de Rindler, de Fermi-Walker, as coordenadas de Radar e as coordenadas de Emissão (ou GPS). O sistema de coordenadas de Rindler é um dos sistemas de grande destaque porque permite simular algumas propriedades da geometria do Buraco Negro num espaço-tempo plano. As coordenadas de Rindler estão associadas a uma família de observadores uniformemente acelerados que obedecem à relação a = (1/ρ), onde a é a aceleração própria do observador e ρ a sua posição inicial com respeito a algum sistema de referência inercial. Neste trabalho, propomos um método para construção de sistemas de coordenadas adaptados a observadores cuja aceleração depende de sua posição inicial da forma geral a=a(ρ), utilizando para isso o princípio físico da localidade. O sistema de coordenadas de Rindler surge como uma particularidade de nossa generalização. Outros casos particulares nos permitem discutir a relação entre a geometria não-Euclidiana das secções espaciais e referenciais acelerados, como originariamente foi proposto por Einstein. Além disso, com a generalização podemos simular o comportamento de observadores estáticos tanto nas proximidades do horizonte de um Buraco Negro, que estão submetidos a um campo de aceleração do tipo a (ρ) = 1/ρ, quanto em regiões afastadas, para as quais, a (ρ) =1/ ρ2. Nestes dois últimos casos, ρ corresponde à distância do observador acelerado até o horizonte de eventos. Com a intenção de analisarmos os efeitos da aceleração instantânea sobre o ritmo de relógios atômicos, consideramos uma partícula livre massiva, dentro de uma caixa de paredes infinitas, que é arrastada por observadores de Rindler. Admitimos que a partícula obedece a equação de Klein-Gordon e assim, encontramos as frequências dos estados estacionários deste sistema. As transições entre os estados estacionários são usadas para definir uma frequência padrão para o nosso relógio atômico de brinquedo. Comparando o espectro de energia do sistema acelerado com o espectro de energia de um sistema idêntico em um referencial inercial, determinamos a influência da aceleração instantânea sobre o ritmo de relógios atômicos.

We investigate in this thesis the generic properties of curves in the Minkowski plane R2 1 and of smooth Lorentzian surfaces. The generic properties of curves in R2 1 are obtained by studying the contacts of curves in R2 1 with lines and pseudo-circles. These contacts are captured by the singularities of the families of height and distancesquared functions on the curves. On the other hand, the generic properties of smooth Lorentzian surfaces are obtained by studying certain Binary Differential Equations defined on the surfaces.

The diploma thesis is focused on the design of planar filters combining fractal layouts and defected ground substrates. The diploma thesis can be divided into three main parts. First, basic knowledge about fractals is presented (creation of Minkowski Island and Koch loop, e.g.). Then, the principle of defected ground structure is described, and a combination of fractal motives with a defected ground structure is briefly introduced. Properties of investigated structures are verified by CST Microwave Studio and Ansoft HFSS. Second, different defected ground structures under the 50 transmission line are designed, and conventional equivalent filters are created. Filters are simulated and compared. In final, the investigated filters are recalculated for the substrate Arlon 25N, simulated, manufactured, measured and confronted with a conventional filter on the defected ground substrate.

The objective of this dissertation is the application of Minkowskian cross-section measures (i.e., section and projection measures in finite-dimensional linear normed spaces over the real field) to various topics of geometric convexity in Minkowski spaces, such as bodies of constant Minkowskian width, Minkowskian geometry of simplices, geometric inequalities and the corresponding optimization problems for convex bodies. First we examine one-dimensional Minkowskian cross-section measures deriving (in a unified manner) various properties of these measures. Some of these properties are extensions of the corresponding Euclidean properties, while others are purely Minkowskian. Further on, we discover some new results on the geometry of a simplex in Minkowski spaces, involving descriptions of the so-called tangent Minkowskian balls and of simplices with equal Minkowskian heights. We also give some (characteristic) properties of bodies of constant width in Minkowski planes and in higher dimensional Minkowski spaces. This part of investigation has relations to the well known \emph{Borsuk problem} from the combinatorial geometry and to the widely used monotonicity lemma from the theory of Minkowski spaces. Finally, we study bodies of given Minkowskian thickness ($=$ minimal width) having least possible volume. In the planar case a complete description of this class of bodies is given, while in case of arbitrary dimension sharp estimates for the coefficient in the corresponding geometric inequality are found. / Die Dissertation befasst sich mit Problemen fuer spezielle konvexe Koerper in Minkowski-Raeumen (d.h. in endlich-dimensionalen Banach-Raeumen). Es wurden Klassen der Koerper mit verschiedenen metrischen Eigenschaften betrachtet (z.B., Koerper konstante Breite, reduzierte Koerper, Simplexe mit Inhaltsgleichen Facetten usw.) und einige kennzeichnende und andere Eigenschaften fuer diese Klassen herleitet.

The subject of this monograph can be described as the local properties of geometric Steiner minimal trees in finite-dimensional normed spaces. A Steiner minimal tree of a finite set of points is a shortest connected set interconnecting the points. For a quick introduction to this topic and an overview of all the results presented in this work, see Chapter 1. The relevant mathematical background knowledge needed to understand the results and their proofs are collected in Chapter 2. In Chapter 3 we introduce the Fermat-Torricelli problem, which is that of finding a point that minimizes the sum of distances to a finite set of given points. We only develop that part of the theory of Fermat-Torricelli points that is needed in later chapters. Steiner minimal trees in finite-dimensional normed spaces are introduced in Chapter 4, where the local Steiner problem is given an exact formulation. In Chapter 5 we solve the local Steiner problem for all two-dimensional spaces, and generalize this solution to a certain class of higher-dimensional spaces (CL spaces). The twodimensional solution is then applied to many specific norms in Chapter 6. Chapter 7 contains an abstract solution valid in any dimension, based on the subdifferential calculus. This solution is applied to two specific high-dimensional spaces in Chapter 8. In Chapter 9 we introduce an alternative approach to bounding the maximum degree of Steiner minimal trees from above, based on the illumination problem from combinatorial convexity. Finally, in Chapter 10 we consider the related k-Steiner minimal trees, which are shortest Steiner trees in which the number of Steiner points is restricted to be at most k. / Das Thema dieser Habilitationsschrift kann als die lokalen Eigenschaften der geometrischen minimalen Steiner-Bäume in endlich-dimensionalen normierten Räumen beschrieben werden. Ein minimaler Steiner-Baum einer endlichen Punktmenge ist eine kürzeste zusammenhängende Menge die die Punktmenge verbindet. Kapitel 1 enthält eine kurze Einführung zu diesem Thema und einen Überblick über alle Ergebnisse dieser Arbeit. Die entsprechenden mathematischen Vorkenntnisse mit ihren Beweisen, die erforderlich sind die Ergebnisse zu verstehen, erscheinen in Kapitel 2. In Kapitel 3 führen wir das Fermat-Torricelli-Problem ein, das heißt, die Suche nach einem Punkt, der die Summe der Entfernungen der Punkte einer endlichen Punktmenge minimiert. Wir entwickeln nur den Teil der Theorie der Fermat-Torricelli-Punkte, der in späteren Kapiteln benötigt wird. Minimale Steiner-Bäume in endlich-dimensionalen normierten Räumen werden in Kapitel 4 eingeführt, und eine exakte Formulierung wird für das lokale Steiner-Problem gegeben. In Kapitel 5 lösen wir das lokale Steiner-Problem für alle zwei-dimensionalen Räume, und diese Lösung wird für eine bestimmte Klasse von höher-dimensionalen Räumen (den sog. CL-Räumen) verallgemeinert. Die zweidimensionale Lösung wird dann auf mehrere bestimmte Normen in Kapitel 6 angewandt. Kapitel 7 enthält eine abstrakte Lösung die in jeder Dimension gilt, die auf der Analysis von Subdifferentialen basiert. Diese Lösung wird auf zwei bestimmte höher-dimensionale Räume in Kapitel 8 angewandt. In Kapitel 9 führen wir einen alternativen Ansatz zur oberen Schranke des maximalen Grads eines minimalen Steiner-Baums ein, der auf dem Beleuchtungsproblem der kombinatorischen Konvexität basiert ist. Schließlich betrachten wir in Kapitel 10 die verwandten minimalen k-Steiner-Bäume. Diese sind die kürzesten Steiner-Bäume, in denen die Anzahl der Steiner-Punkte auf höchstens k beschränkt wird.

Local-global principle for quadratic forms This work will be focused on the problems of representation and equivalence for quadratic forms. We will prove the fundamental Hasse-Minkowski theorem, which describes the rational representation and equivalence using properties of the form over the completions of Q: the real and p-adic numbers. We will refer to this procedure as local-global principle. Furthermore, we shall describe the methods for computing the p-adic invariants, and show their relation to the representation problem. Finally, we show how the local-global partially extends to integral forms, in particular to indefinite ones of dimension at least 4. 1