Gravitational waves detection is a challenging scientific objective, faced by scientist in the last 100 years, when Einstein theorized their existence. Despite multiple attempts, it was only in 2016 that the first observation of a gravitational wave was officially announced. The observation, worth a Nobel Prize, was made possible thanks to a worldwide collaboration of three large ground-based detectors. When detecting gravitational waves from ground, the noisy environment limits the frequency bandwidth of the measurement. Thus, the type of cosmic events that are observable is also limited. For this reason, scientists are developing the first gravitational waves detector based in space, which is a much quieter environment, especially in the sub-Hertz bandwidth. The space-based detector is named laser interferometer space antenna (LISA) and its launch is planned for 2034. Due to the extreme complexity of the mission, involving several new technologies, a demonstrator of LISA was launched and operated between 2015 and 2017. The demonstrator mission, called LISA Pathfinder (LPF), had the objective to show the feasibility of the gravitational waves observation directly from space, by characterizing the noise affecting the relative acceleration of two free falling bodies in the milli-Hertz bandwidth. The mission was a success, proving the expected noise level is well below the minimum requirement. The free-falling bodies of LPF, called test masses (TMs), were hosted inside dedicated electrode housings (EH), located approximately 30 cm apart inside the spacecraft. When free falling, each TM stays approximately in the center of the EH, thus having milli-meter wide gaps within the housing walls. Due to the presence of such large gaps, the TMs were mechanically constrained by dedicated mechanisms (named CVM and GPRM) in order to avoid damaging the payload during the launch phase and were released into free fall once the spacecraft was in orbit. Prior to the start of the science phase, the injection procedure of the TMs into free-fall was started. Such a procedure brought each TM from being mechanically constrained to a state where it was electro-statically controlled in the center of the EH. Surprisingly, the mechanical separation of the release mechanism from the TM caused unexpected residual velocities, which were not controllable by the electrostatic control force responsible for capturing the TM once released. Therefore, both the TMs collided with either the surrounding housing walls or the release mechanism end effectors. It was possible to start the science phase by manually controlling the release mechanism adopting non-nominal injection strategies, which should not be applicable in LISA, due to the larger time lag. So, since any release mechanism malfunctioning may preclude the initialization of LISA science phase, the GPRM was extensively tested at the end of LPF, by means of a dedicated campaign of releases, involving several modifications to the nominal injection procedure. The data of the extended campaign are analyzed in this work and the main conclusion is that no optimal automated release strategy is found for the GPRM flight model as-built configuration that works reliably for both the TMs producing a nominal injection procedure. The analysis of the in-flight data is difficult since the gravitational referencesensor of LPF is not designed for such type of analysis. In particular, the low sampling frequency (i.e., 10 Hz) constitutes a limiting factor when detecting instantaneous events such as collisions of the TM. Despite the difficulties of extracting useful information on the TM residual velocity from the in-flight data, it is found that the main cause of the uncontrollable state of the released TM is the collision of the TM with the plunger, i.e., one of the end-effectors of the GPRM. It is shown that the impact is caused by the oscillation of the plunger or by the elastic relaxation of the initial preload force that holds the TM. At the end of the analysis, some improvements to the design of the release mechanism are brie y discussed, aimed at maximizing the probability of performing a successful injection procedure for the six TMs that will be used as sensing bodies in the LISA experiment.

With the roll out of new technologies and the worldbecoming more connected, there is a rising demand for higherbandwidth and new frequency bands. To meet the demand,higher frequencies are used in new communication systems.Higher frequencies come with the need for new antenna designsand one promising type of antenna is the lens antenna. In thispaper, a modulated geodesic Luneburg lens with a novel feedingmethod is proposed for use between 8-10 GHz. Furthermore, themanufacturing of the lens explores the possibility of 3D printingas a method of producing cheap antennas.The paper verifies the viability of using a patch antenna andhorn as a feeding method for a parallel-plate waveguide lens.First the lens is modeled and simulated in CST Microwave Studioand is then 3D-printed in PLA plastic and taped with coppertape. The antenna achieves -5 dB S11-parameter between 8-10GHz. The antenna also achieves 60 scanning in the azimuthplane. The antenna achieves a HPBW of 15. / Med utvecklingen av nya tekniker och envärld som blir allt mer digital är efterfrågan på större bandbreddoch nya frekvensband hög. För att möta efterfrågananvänds högre frekvenser i nya kommunikationssystem. Medanvändningen av högre frekvenser behövs nya antenndesigneroch en lovande typ av antenn är linsantennen. I den härartikeln föreslås en modulerad geodesic Luneburg lins med enny typ av matningsmetod för användning mellan 8-10 GHz. Förtillverkningen av linsen utforskas 3D-printning som en billig ochenkel metod.Artikeln verifierar användningen av en patch-antenn och etthorn som matningsmetod för en lins av parallella metallplattor.Först simuleras linsen i CST Microwave Studio och 3Dprintassedan i PLA-plast och tejpas med koppartejp. Antennenåstakommer -5 dB i S11-parameter mellan 8-10 GHz. Antennenhar en skanning av 60 i azimut-planet och har en HPBW av15. / Kandidatexjobb i elektroteknik 2021, KTH, Stockholm

With the rise of 5G and the increasing number ofdevices, novel antenna designs are needed to meet the demandof the future. In this report, the authors present a design andexperimental verification of a 3D-printed Geodesic ModulatedReflective Luneburg lens antenna working at the X-Band, 8-12GHz. The lens profile is calculated from the refractive index of aflat system using transformation optics. Furthermore, the lens ismodulated to minimize the height and chamfers are implementedto reduce reflections. A sliding waveguide connected to a coaxialcable is used to excite the lens while the transmitted signal isradiated from a sinusoidal flare. A copper-lined PLA substrateconstitutes the 3D-printed lens. The authors achieved a S11 below-10 dB across the spectrum and a realized gain exceeding 10 dBacross the sweeping angles at 12 GHz, showcasing the usabilityas a directed antenna. / Med det nya 5G nätverket och den ökandemängden enheter behövs nya antenner för att möta framtidensefterfrågan. I denna rapport presenterar författarna en designoch experimentell verifiering av en 3D-printad geodesisk moduleradreflekterande Luneburg linsantenn i X-bandet, 8-12 GHz.Linsprofilen beräknas från brytningsindexet för ett platt systemmed transformationsoptik. Dessutom är linsen modulerad föratt minimera höjden och kantavfasningar implementeras föratt minska reflektioner. En glidande vågledare ansluten till enkoaxialkabel används för att excitera linsen medan den sända signalenutstrålas från en vågledare med sinusformad avrundning.Ett kopparfodrat PLA-substrat utgör den 3D-printade linsen.Författarna uppnådde en S11 under -10 dB över spektrumet ochen realiserad förstärkning överstigande 10 dB över svepvinklarnavid 12 GHz, vilket visar linsens användbarhet som riktad antenn. / Kandidatexjobb i elektroteknik 2021, KTH, Stockholm

In the present work, we study some aspects of Einstein's theory of gravitation in general spacetimes with an arbitrary number of dimensions. In the first chapter we summarize the foundations of used geometric formalism and we derive the equation of goedesic deviation representing the relation between relative acceleration and the Riemann tensor. Second chapter presents different types of algebraic classification of the Weyl tensor in four and higher dimensions. Third chapter is devoted to a detailed examination of the test particle motions and also to the interpretation of different terms in the general equation of geodesic deviation. The fourth section examines appropriate choice of the interpretation frame and the coordinates. The final fifth chapter contains an analysis of the motion of test particles in the Robinson-Trautman spacetime with an arbitrary higher number of dimensions.

Let M be a possibly non compact smooth manifold. We study genericity in the C^k topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics satisfying suitable endpoints conditions. This extends results of Biliotti, Javaloyes and Piccione for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of MxM that satisfies an admissibility condition. Immediate consequences are generic non conjugacy between two points and non focality between a point and a submanifold (or also between two submanifolds). / Seja M uma variedade suave possivelmente não compacta. Estuda-se a genericidade na topologia C^k (3<=k<=+infty) de propriedades de não degenerescência de fluxos geodésicos semi-Riemannianos em M. A saber, provase uma nova versão do Teorema de Métricas Bumpy para uma tal M e também a genericidade de métricas que não possuem geodésicas degeneradas cujos pontos finais satisfazem certas condições. Isso estende resultados anteriores de Biliotti, Javaloyes and Piccione para geodésicas com extremos fixos para o caso onde os extremos variam em uma subvariedade compacta P de M ×M que satisfaz uma condição de admissibilidade. Consequências imediatas são genericidade de não conjugação entre dois pontos e não focalidade entre um ponto e uma subvariedade (ou também entre duas subvariedades).

[pt] Seja S uma superfície fechada orientável, de gênero > 2 e sem pontos conjugados. Seja F uma folheação no fibrado tangente unitário de S, de codimensão 1, invariante pelo fluxo geodésico e de classe C2. Então, a curvatura de S é constante < 0. A demonstração é conseqüência dos dois seguintes resultados, que têm interesse por si mesmos. O primeiro é que se T1S admite uma folheação contínua de codimensão 1 por folhas C1 invariantes pelo fluxo geodésico então a superfície não tem pontos conjugados e a folheação coincide com a folheação centro-estável ou com a centro-instável. O segundo resultado é o seguinte. Seja S uma superfície fechada orientável, de gênero > 2 e sem pontos conjugados. Então, a folheação centro-estável Fcs de T1S é conjugada à folheação centro-estável da métrica hiperbólica em S. Esta conjugação é da mesma classe de diferenciabilidade de Fcs . Portanto, se Fcs é de classe C2, uma extensão da teoria de Godbillon-Vey implica que a curvatura da superfície é constante negativa. / [en] Lets be a orientable closed surface with no conjugate points. Let F be a foliation in the unitary tangent fiber bundle of S, of codimension 1, invariant by the geodesic flow and of class C2. Then, the curvature of S is constant < 0 . The demonstration is a consequence of the two following results, which are of interest by themselves. The first one is that if T1S admits a continuous foliation of codimension 1 by leaves C1 invariants by the geodesic flow, then the surface is with no conjugate points, and the foliation coincides with either the center stable foliation or the center unstable foliation. The second result is the following. Let S be a orientable closed surface of genus > 2 and with no conjugate points. Then, the center unstable foliation Fcs of T1S is conjugate to the center stable foliation of the hyperbolic metric in S. This conjugation is of the same class of differentiability of Fcs. Therefore, if Fcs is of class C2, an extension of the Godbillon-Vey theory implies that the curvature of the surface is constant negative.

Image segmentation is a process to identify regions of interest from digital images. Image segmentation plays an important role in medical image processing which enables a variety of clinical applications. It is also a tool to facilitate the detection of abnormalities such as cancerous lesions in the brain. Although numerous efforts in recent years have advanced this technique, no single approach solves the problem of segmentation for the large variety of image modalities existing today. Consequently, brain MRI segmentation remains a challenging task. The purpose of this thesis is to demonstrate brain MRI segmentation for delineation of tumors, ventricles and other anatomical structures using Insight Segmentation and Registration Toolkit (ITK) routines as the foundation. ITK is an open-source software system to support the Visible Human Project. Visible Human Project is the creation of complete, anatomically detailed, three-dimensional representations of the normal male and female human bodies. Currently under active development, ITK employs leading-edge segmentation and registration algorithms in two, three, and more dimensions. A goal of this thesis is to implement those algorithms to facilitate brain segmentation for a brain cancer research scientist.

Grâce au théorème d'hyperbolisation, nous savons précisément quand une variété de dimension trois compacte admet une métrique hyperbolique. Par ailleurs, d'après le théorème de rigidité de Mostow, cette structure géométrique est unique. Cependant, trouver des liens pratiques entre la géométrie et la topologie est un problème difficile. La plupart des résultats décrits dans cette thèse visent à concrétiser ces liens. Toute géodésique fermée orientée dans une surface hyperbolique admet un relèvement canonique dans le fibré tangent unitaire de la surface, et on peut donc le voir comme un nœud dans une variété de dimension trois. Les extérieurs des nœuds ainsi construits admettent une structure hyperbolique. Cette thèse a pour objet d'estimer le volume des extérieurs des relèvements canoniques. Pour toute surface hyperbolique on construit une suite de géodésique sur la surface, tel que les extérieurs associées ne sont pas homéomorphes entre elles et dont la suite des volumes respectifs est bornée. Aussi on minore le volume de l'extérieur à l'aide d'un réel explicite qui décrit une relation entre la géodésique et une décomposition en pantalons de la surface. Ceci donne une méthode pour construire une suite de géodésiques dont les volumes des extérieurs associées sont minorées en termes de la longueur de la géodésique correspondant. Dans le cas particulier de la surface modulaire, on obtient des estimations du volume de l'extérieur en termes de la période de la fraction continue associée à la géodésique. / Due to the Hyperbolization Theorem, we know precisely when does a given compact three dimensional manifold admits a hyperbolic metric. Moreover, by the Mostow's Rigidity Theorem this geometric structure is unique. However, finding effective and computable connections between the geometry and topology is a challenging problem. Most of the results on this thesis fit into the theme of making the connections more concrete. To every oriented closed geodesic on a hyperbolic surface has a canonical lift on the unit tangent bundle of the surface, and we can see it as a knot in a three dimensional manifold. The knot complement given in this way has a hyperbolic structure. The objective of this thesis is to estimate the volume of the canonical lift complement. For every hyperbolic surface we give a sequence of geodesics on the surface, such that the knot complements associated are not homeomorphic with each other and the sequence of the corresponding volumes is bounded. We also give a lower bound of the volume of the canonical lift complement by an explicit real number which describes a relation between the geodesic and a pants decomposition of the surface. This give us a method to construct a sequence of geodesics where the volume of the associated knot complements is bounded from below in terms of the length of the corresponding geodesic. For the particular case of the modular surface, we obtain estimations for the volume of the canonical lift complement in terms of the period of the continuous fraction expansion of the corresponding geodesic.

Dans ce mémoire, nous étudions l'entropie des systèmes dynamiques différentiables définis sur des variétés riemanniennes non compactes. Dans un premier temps, nous éclaircissons les liens entre différentes notions d'entropie dans ce cadre non compact. Ensuite, nous utilisons ces premiers résultats pour y étudier la validité de l'inégalité de Ruelle. Rappelons ici que cette inégalité, pour des difféomorphismes de variétés riemanniennes compactes, nous dit que l'entropie est majorée par la somme des exposants de Lyapounov positifs. Nous montrons que, lorsque nous enlevons l'hypothèse de compacité, l'inégalité de Ruelle n'est pas toujours satisfaite. Nous obtenons ce résultat en construisant une famille explicite de contre-exemples. En revanche, nous montrons, dans le cas d'un difféomorphisme de comportement asymptotique linéaire, ou du flot géodésique sur le fibré unitaire tangent d'une variété riemannienne à courbure négative, que l'inégalité de Ruelle est toujours satisfaite. Pour finir, nous nous intéressons au problème de la perte possible de masse d'une suite de mesures de probabilité d'une variété riemannienne non compacte. Dans le cas du flot géodésique, nous montrons que l'entropie permet de contrôler la masse d'une limite vague de mesures de probabilité invariantes par le flot pour une classe particulière de variétés géométriquement finies. Plus précisément, nous montrons qu'une suite de mesures d'entropie assez grande ne peut pas perdre la totalité de sa masse. De plus, le minorant optimal de l'entropie dans ce résultat est lié à la géométrie de la partie non compacte de la variété: c'est l'exposant critique maximal des sous-groupes paraboliques du groupe fondamental. / In this work, we study the entropy of smooth dynamical systems defined on non compact Riemannian manifolds. First, we clarify some relations between different notions of entropy in this setting. Second, we use these first results in order to study the validity of Ruelle's inequality. This inequality, for diffeomorphisms defined on compact Riemannian manifolds, says that the measure-theoretic entropy is bounded from above by the sum of the positive Lyapunov exponents. We show that without the compactness assumption, Ruelle's inequality is not always satisfied. We obtain this result by constructing an explicit family of counterexamples. On the other hand, we prove, in the case of diffeomorphisms with linear asymptotic behavior, or that one of the geodesic flow on the unit tangent bundle of a Riemannian manifold with negative curvature, that Ruelle's inequality is always satisfied. Finally, we are interested in the problem of the possible escape of mass of a sequence of probability measures on a non compact Riemannian manifold. In the case of the geodesic flow, we show that the entropy allows to control the mass of a weak$^\ast$-limit of a sequence of probability measures, on the unit tangent bundle of a particular class of geometrically finite manifolds, which are also invariant by the flow. More precisely, we show that a sequence of measures with large enough entropy cannot lose the whole mass. Moreover, the optimal lower bound of the entropy in this result is related to the geometry of the non compact part of the manifold: it is the maximal critical exponent of the parabolic subgroups of the fundamental group.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Neste trabalho, provaremos alguns resultados sobre espectro essencial de uma classe de variedades Riemannianas, nÃo necessariamente completas, com condiÃÃes de curvatura na vizinhanÃa de um raio. Sobre essas condiÃÃes obtemos que o espectro essencial do operador de Laplace contÃm um intervalo. Como aplicaÃÃo, obteremos o espectro do operador de Laplace de regiÃes ilimitadas dos espaÃos formas, tais como a horobola do espaÃo hiperbÃlico e cones do espaÃo Euclidiano. Construiremos tambÃm um exemplo que indica a necessidade das condiÃÃes globais sobre o supremo das curvaturas seccionais fora de uma bola para que a variedade nÃo tenha espectro essencial. / In this thesis we consider a family of Riemannian manifolds, not necessarily complete, with curvature conditions in a neighborhood of a ray. Under these conditions we obtain that the essential spectrum of the Laplace operator contains an interval. The results presented in this thesis allow to determine the spectrum of the Laplace operator on unlimited regions of space forms, such as horoball in hyperbolic space and cones in Euclidean space. Also construct an example that shows the need of global conditions on the supreme sectional curvature outside a ball, so that the variety has no essential spectrum.