The results of bark thickness analysis of pine and spruce trees at the thinning age are presented in the work. The main goal of the work was to investigate regularities of bark thickness variance of pine and spruce logs produced out of trees felled during the thinning cuttings. The main tasks of the work were to find out regularities of bark thickness variance depending on age of the stands, tree species and top diameter of the log, to evaluate adequateness modeled regularities to existing Swedish normatives. The empirical material. 200 pine and 200 spruce felled trees at the Kazlu Ruda state forest enterprise were used for the measurements. The trees were selected at the 4 cutting areas of Nb, Lb, Lc, site types. Diameter and bark thickness of imaginative crosscut at height of 0, 1, 1.3, 3 m and following each second meter were measured for each stem. Methods. The multiple linear and nonlinear regression were used to process empirical data. Calculations were made using software STATISTICA v6.0 and EXCEL. The main results. The differences of bark thickness variation regularities were found while comparing gotten results to Existing Lithuanian and Swedish legalized normatives.

Darbe pateikti beržų, drebulių, juodalksnių žievės storio tyrimai. Tyrimo tikslas: Išanalizuoti beržų, drebulių, juodalksnių stiebų žievės storio kintamumo dėsningumus ir jų pagrindu sudaryti stiebų žievės matematinį modelį. Tyrimo uždaviniai: Nustatyti beržo, drebulės, juodalksnio žievės storio koreliacinius ryšius su stiebų taksacinėmis charakteristikomis. Nustatyti beržų, drebulių, juodalksnių žievės storio kitimo dėsningumus išilgai stiebo ir išreikšti juos regresinėmis lygtimis. Išreikštus beržų, drebulių, juodalksnio žievės storio kitimo dėsningumus palyginti su galiojančiais normatyvais, kitų tyrėjų rezultatais. Tyrimo objektas: Tyrimo bareliuose paimti beržų, drebulių, juodalksnių medžių stiebai iš vidurio Lietuvos ir Žemaitijos regionų. Iš viso tyrimui buvo išmatuota 175 beržo, 136drebulių, 152 juodalksnio stiebų duomenys. Tyrimo metodai: žievės storio priklausomybė nuo jį sąlygojančių veiksnių buvo analizuojama daugianarės tiesinės ir kreivinės regresinės analizės metodais panaudojant EXCEL ir statistika V.6.0 programas. Tyrimo rezultatai: Išanalizuoti beržų, drebulių, juodalksnių kintamumai. Gauti empiriniai duomenys palyginti su kitų autorių matavimo rezultatais, atliekant žievės storio priklausomybės nuo stiebo skersmens 1.3 m aukštyje lyginamąją analizę. Gauti beržų, drebulių juodalksnių žievės storio kintamumo modeliai. / The results of bark thickness analyses of birch, aspen, black alder tree species are presented in the work. The main goal of the work was to investigate regularities of bark thickness variance of birch, aspen, black alder logs felled during all cutting and to make mathematical model of bark of stems. The main tasks of the work were to find out regularities of bark thickness of asp, birch, black – alder and bark thickness dependence on parameters of stems, to make regression analysis of regularities of bark thickness variance along stem and to compare results of work with other similar investigations. The empirical material of asp, birch and black -alder stems were selected from the middle and lowland regions of Lithuania. Overall 465 records of stem were collected, of which 175 records were of aspen, 136 – of birch, 152 – of black-alder. Methods The multiple linear and nonlinear regressions were used to process empirical data. Calculations were made using software STATISTICA v6.0 and MS EXCEL packages. The main results Changes of bark thickness of birch, aspen and black alder were investigated. The acquired empirical data were compared with the data of other authors carrying out comparative analyses of bark thickness dependence on steam diameter at 1.3 m height. Models of bark thickness variance of birch, aspen, black alder were prepared.

[pt] O Envelope de Retas Médias - ERM consiste da união de três conjuntos invariantes afins: o Affine Envelope Symmetry Set - AESS; o Mid-Parallel Tangents Locus - MPTL; e a Evoluta Afim - EA. O ERM de curvas planas convexas é um assunto que tem sido muito explorado. Porém, não existe na literatura nenhum estudo do ERM para superfícies. Por isso, o objetivo principal desta tese é generalizar o ERM de curvas convexas para superfícies convexas. Para tanto, dividimos a tese em duas partes. A primeira consiste de uma revisão sobre a geometria afim de curvas planas e do estudo do ERM com uma nova abordagem. Na segunda parte realizamos uma breve introdução da geometria afim de hipersuperfícies e a generalização do ERM. Na generalização do ERM, trabalhamos com superfícies, definimos os planos médios e estudamos o que denominamos Envelope de Planos Médios -EPM. Provamos que, o EPM assim como o ERM, é formado por três conjuntos invariantes afins: a Superfície de Centros de 3 mais 3-Cônicas - SC3C; o Mid-Parallel Tangents Surface -MPTS; e a Evoluta de Curvas Médias - ECM. / [en] The Envelope of Mid-Lines - EML consists of the union of three affine invariant sets: the Affine Envelope Symmetry Set - AESS; the Mid-Parallel Tangents Locus - MPTL; and the Affine Evolute. The EML of convex planar curves is a subject that has been very explored. However, there is no study in the literature of the EML for surfaces. Therefore, the main objective of this thesis is to generalize the EML of convex curves to convex surfaces. We divide the writing into two parts. The first part consists of a study of the EML with a new approach. In the second part we consider the EML for surfaces, that we call Envelope of Mid-Planes - EMP. We prove that, the EMP, like the EML, is formed by three affine invariant sets: the Centers of 3 plus 3-Conics Surface - C3CS; the Mid-Parallel Tangents Surface -MPTS; and the Medial Curves Evolute - MCE.

O presente trabalho tem como objetivo desenvolver um processo de fabricação de elementos micro-ópticos utilizando-se litografia por feixe de elétrons, empregando o resiste SU-8, negativo e amplificado quimicamente, sobre substrato de Si. Para tanto, é realizado o estudo dos parâmetros do efeito de proximidade a, b e h para se modelar e controlar os efeitos do espalhamento dos elétrons no resiste e no substrato, e se altera o processamento convencional do SU-8 para se obter um processo com baixo contraste. A determinação dos parâmetros do efeito de proximidade para o sistema de escrita direta e amostra SU-8 / Si é feita experimentalmente e por simulação de Monte Carlo. Particularmente, verifica-se a dependência dos mesmos com a profundidade do resiste. Primeiramente utilizando o software PROXY, obtêm-se a, b e h da observação de padrões de teste revelados. Chega-se a 4m para o parâmetro () que mede o retroespalhamento dos elétrons pelo substrato e 0,7 para a relação (h) entre a intensidade destes com aquela dos elétrons diretamente espalhados pelo resiste (alcance dado por a). Ainda, com esses dados, estima-se o diâmetro do feixe do microscópio eletrônico de varredura a partir da equação de aproximação de espalhamento direto para pequenos ângulos (a = 128nm na superfície do resiste) e se determina a resolução lateral do processo (a = 800nm na interface resiste/ substrato, para um filme de 2,4m). Em seguida, usa-se o software CASINO para se calcular os parâmetros de proximidade a partir da curva de densidade de energia dissipada no resiste obtida pela simulação da trajetória de espalhamento dos elétrons. Confrontam-se, finalmente, os valores obtidos pelos dois métodos. Em relação ao processamento do resiste SU-8, são determinadas as condições experimentais para a fabricação de estruturas tridimensionais por litografia de feixe de elétrons. Especificamente, busca-se desenvolver um processo com características (espessura, contraste, sensibilidade e rugosidade) adequadas para a fabricação de micro-dispositivos ópticos. Inicia-se com o levantamento das curvas de contraste e da sensibilidade do SU-8 para determinadas temperaturas de aquecimento pós-exposição. Obtém-se contraste abaixo de 1 para aquecimento pós-exposição abaixo da temperatura de transição vítrea do resiste, mantendo-se sensibilidade elevada (2C/cm2). Em seguida, mede-se a rugosidade da superfície do filme revelado para diferentes doses de exposição. Para finalizar, submete-se a amostra a um processo de cura e escoamento térmico, para melhorar a dureza e a rugosidade do resiste a ser utilizado como dispositivo final Consegue-se um valor de rugosidade (40nm) inferior a 20 vezes o comprimento de onda de diodo laser de eletrônica de consumo. Por fim, é produzido um dispositivo com perfil discretizado em 16 níveis como prova de conceito. / This work aims at developing an electron-beam lithography process for the fabrication of microoptical elements using the negative tone chemically amplified resist SU-8 on Si substrate. A study of the proximity effect parameters a, b and h is carried out to model and control the electron scattering both in the resist and in the substrate, and the SU-8 standard processing conditions are changed to achieve a low contrast process. The determination of the SU-8 / Si proximity effect parameters and its dependence with resist depth is done employing an experimental method and through Monte Carlo simulations. First, a, b and h are obtained comparing exposed patterns calculated by the software PROXY. b, the parameter which measures the backscattering of the electrons by the substrate, is equal to 4m and the value of h, the ratio of the dose contribution of backscattered electrons to that of the forward scattered (related to a), is 0.7. The extrapolation of exposed patterns data is used to estimate the scanning electron microscope beam diameter through the equation for low angle scattering (a = 128nm at the resist surface) and the lateral resolution of the process is determined (a = 800nm at the resist/ substrate interface, for a 2.4m film). With aid of the software CASINO, Monte Carlo simulations of the scattering trajectories of electrons in substrate and resist materials are calculated, recording the energy that they dissipate through collisions along their path. The results obtained representing the profile of the energy dissipated in the resist are used to determine the proximity effect parameters. The experimental method results are compared to that obtained by simulation. Regarding the SU-8 processing, the process parameters for the fabrication of three-dimensional structures by electron-beam lithography are determined. The process is designed to have specifications (thickness, contrast, sensitivity and surface roughness) suitable for microoptical elements fabrication. It begins with the determination of the SU-8 contrast curve and its sensitivity for specific post-exposure bake temperatures. A below the unit contrast process with high sensitivity (2C/cm2) is achieved postannealing the sample below the resist glass transition temperature. The film surface roughness is measured after resist development for different exposure doses, and a controlled hardbake (cure) and reflow is carried to enhance both the mechanical properties and the surface roughness of the structures that will remain as part of the final device. A RMS roughness of 40nm, lower than 20 times the wavelength of consumer electronics laser diode, is obtained. The electron-beam process designed is applied to the fabrication of a microelement with a 16-level profile discretization.

Dans cette thèse nous étudions expérimentalement les propriétés magnétiques de condensats de sodium de spin 1 à l'équilibre. Dans ce système les atomes peuvent occuper chacun des trois états Zeeman caractérisés par la projection de leur spin sur l'axe de quantification m=+1,0,-1. Nous mesurons l'état de spin à N particules du système en fonction du champ magnétique appliqué et et de la magnétisation (différence entre les populations des états m=+1 et m=-1) du nuage atomique. Nos mesures sont en très bon accord avec la prédiction de la théorie de champ moyen, et nous identifions deux phases magnétiques résultant de la compétition entre les interactions de spin antiferromagnétiques et l'effet du champ magnétique. Nous décrivons ces deux phases en terme d'un ordre nématique de spin caractérisant la symétrie de l'état de spin à N particules. Dans une seconde partie nous nous concentrons sur les propriétés du condensat à très faible magnétisation et soumis à un faible champ magnétique. Dans ces conditions, la symétrie du système se manifeste à travers de très grandes fluctuations de spin. Ce phénomène n'est pas explicable par une théorie de champs moyen naïve, et nous développons une approche statistique plus élaborée pour décrire l'état de spin du condensat. Nous mesurons les fluctuations de spin et nous sommes capables de déduire de leur analyse la température caractérisant le degré de liberté de spin du condensat. Nous trouvons que cette température diffère de celle décrivant les atomes thermiques entourant le condensat. Nous interprétons cette différence comme une conséquence du faible couplage entre ces deux systèmes. / In this thesis we study experimentally the magnetic properties of spin-1 Bose-Einstein condensate of Sodium at equilibrium. In this system the atoms can occupy any of the three Zeeman states characterized by their spin projection on the quantization axis m=+1,0,-1. We measure the many-body spin state of the system as a function of the applied magnetic field and of the magnetization (difference between the populations of the spin states m=+1 and m=-1) of the atomic sample. We find that our measurements reproduce very well the mean-field prediction, and we identify two magnetic phases expressing the competition between the antiferromagnetic inter-particle interactions and the effect of the magnetic field. We describe these phases in terms of a spin nematic order characterizing the symmetry of the many-body spin state. In a second part we focus on the properties of condensates of very low magnetization under a weak magnetic field. In these conditions, the symmetry of the system manifests itself in huge spin fluctuations. This phenomenon is not explainable by a naive mean-field theory and we develop a more elaborate statistical approach to describe the spin state of the condensate. We measure the spin fluctuations and are able from their analysis to infer the temperature characterizing the spin degree of freedom of the condensate. We find that this temperature differs from the temperature of the thermal fraction surrounding the condensate. We interpret this difference as a consequence of the weak coupling between these two systems.

[EN] The Ph.D. Thesis "Atomic decompositions and frames in Fréchet spaces and their duals" presented here treats different areas of functional analysis with applications. Schauder frames are used to represent an arbitrary element x of a function space E as a series expansion involving a fixed countable set {xj} of elements in that space such that the coefficients of the expansion of x depend in a linear and continuous way on x. Unlike Schauder bases, the expression of an element x in terms of the sequence {xj}, i.e. the reconstruction formula for x, is not necessarily unique. Atomic decompositions or Schauder frames are a less restrictive structure than bases, because a complemented subspace of a Banach space with basis has always a natural Schauder frame, that is obtained from the basis of the superspace. Even when the complemented subspace has a basis, there is not a systematic way to find it. Atomic decompositions appeared in applications to signal processing and sampling theory among other areas. Very recently, Pilipovic and Stoeva [55] studied series expansions in (countable) projective or inductive limits of Banach spaces. In this thesis we begin a systematic study of Schauder frames in locally convex spaces, but our main interest lies in Fréchet spaces and their duals. The main difference with respect to the concept considered in [55] is that our approach does not depend on a fixed representation of the Fréchet space as a projective limit of Banach spaces. The text is divided into two chapters and appendix that gives the notation, definitions and the basic results we will use throughout the thesis. The first one focuses on the relation between the properties of an existing Schauder frame in a Fréchet space E and the structure of the space. In the second chapter frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. In what follows, we give a brief description of the different chapters: In Chapter 1, we study Schauder frames in Fréchet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete Schauder xviiframes on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional Schauder frame is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete Schauder frames in function spaces are also presented. Most of the results included in this chapter are published by Bonet, Fernández, Galbis and Ribera in [13]. The second chapter of the thesis is devoted to study ¿-Bessel sequences, ¿-frames and frames with respect to ¿ in the dual of a Hausdorff locally convex space E, in particular for Fréchet spaces and complete (LB)-spaces E, with ¿ a sequence space. We investigate the relation of these concepts with representing systems in the sense of Kadets and Korobeinik [34] and with the Schauder frames, that were investigated in Chapter 1. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions. We present several abstract results about ¿-frames in complete (LB)-spaces. Finally, many applications, results and examples concerning sufficient sets for weighted Fréchet spaces of holomorphic functions and weakly sufficient sets for weighted (LB)-spaces of holomorphic functions are collected. Most of the results are submitted for publication in a preprint of Bonet, Fernández, Galbis and Ribera in [12]. / [ES] La presente memoria "Descomposiciones atómicas y frames en espacios de Fréchet y sus duales" trata diferentes áreas del análisis funcional con aplicaciones. Los frames de Schauder se utilizan para representar un elemento arbitrario x de un espacio de funciones E mediante una serie a partir de un conjunto numerable fijado {xj} de elementos de este espacio de manera que los coeficientes de la reconstrucción de x dependen de forma lineal y continua de x. A diferencia de las bases de Schauder, la expresión de un elemento x en términos de la sucesión {xj}, i.e. la fórmula de reconstrucción para x, no es necesariamente única. Las descomposiciones atómicas o los frames de Schauder son un estructura menos restrictiva que las bases, porque un subespacio complementado de un espacio de Banach con base tiene siempre un frame de Schauder natural, que se obtiene a partir de una base del superespacio. Incluso cuando el subespacio complementado tiene una base, no hay una forma sistemática de encontrarla. Las descomposiciones atómicas aparecen en aplicaciones al procesamiento de señales y la teoría de muestreo, entre otras áreas. Recientemente, Pilipovic y Stoeva [55] han estudiado el desarrollo en serie en límites inductivos y proyectivos (numerables) de espacios de Banach. En esta tesis empezamos un estudio sistemático de los frames de Schauder en espacios localmente convexos aunque nuestro interés principal son los espacios de Fréchet y sus duales. La diferencia principal respecto del concepto considerado en [55] es que nuestra aproximación no depende de una representación fijada del espacio de Fréchet como límite proyectivo de espacios de Banach. El texto queda dividido en dos partes y un apéndice que incluye la notación, las definiciones y los resultados básicos que usaremos a lo largo de la tesis. La primera parte se centra en la relación entre las propiedades de un frame de Schauder en un espacio de Fréchet E y la estructura del espacio. En el segundo capítulo se definen y estudian los frames y las sucesiones de Bessel en espacios de Fréchet y sus duales. A continuación, presentamos una breve descripción de los capítulos: En el Capítulo 1, estudiamos los frames de Schauder en los espacios de Fréchet y sus duales así como los resultados de perturbación. Definimos los frames de Schauder contractivos y acotadamente completos en espacios localmente convexos, estudiamos la dualidad de estos dos conceptos y su relación con la reflexividad del espacio. Caracterizamos cuándo un frame de Schauder incondicional es contractivo o acotadamente completo en términos de las propiedades del espacio. También se presentan varios ejemplos de frames de Schauder en espacios de funciones concretos. La mayoría de los resultados incluidos en este capítulo están publicados por Bonet, Fernández, Galbis y Ribera en [13]. El segundo capítulo de la tesis está centrado en el estudio de las sucesiones de ¿-Bessel, ¿-frames y frames respecto de ¿ en el dual de un espacio localmente convexo de Hausdorff E, en particular, para espacios de Fréchet y espacios (LB) completos E, con ¿ un espacio de sucesiones. Investigamos la relación de estos dos conceptos con los sistemas representantes en el sentido de Kadets y Korobeinik [34] y con los frames de Schauder, considerados en el Capítulo 1. Los resultados abstractos presentados aquí, cuando los aplicamos a espacios de funciones analíticas concretos, nos dan muchos ejemplos y consecuencias sobre los conjuntos de muestreo y los desarrollos en serie de Dirichlet. Presentamos varios resultados abstractos sobre ¿-frames en espacios (LB) completos. Finalmente, recogemos muchas aplicaciones, resultados y ejemplos alrededor de los conjuntos suficientes para espacios de Fréchet de funciones holomorfas y conjuntos débilmente suficientes para espacios pesados (LB) de funciones holomorfas. La mayoría de los resultados incluidos en este capítulo están enviados para publicar e / [CAT] La tesi "Descomposicions atòmiques i frames en espais de Fréchet i els seus duals" presentada ací tracta diferents àrees de l'anàlisi funcional amb aplicacions. Els frames de Schauder s'utilitzen per tal de representar un element arbitrari x d'un espai de funcions E com una reconstrucció en sèrie a partir d'un conjunt numerable fixat {xj} d'elements en aquest espai tal que els coeficients de la reconstrucció de x depenen de forma lineal i continua de x. A diferència de les bases de Schauder, l'expressió d'un element x en termes d'una successió {xj}, i.e. la fórmula de reconstrucció per a x, no és necessàriament única. Les descomposicions atòmiques o els frames de Schauder són una estructura menys restrictiva que les bases, donat que un subespai complementat d'un espai de Banach amb base sempre té un frame de Schauder natural, el qual és obtingut a partir d'una base del superespai. Inclòs quan el subespai complementat disposa de una base, no hi ha una forma sistemàtica per tal de trobar-la. Les descomposicions atòmiques apareixen en aplicacions a processat de senyals i teoría de mostreig entre altres àrees. Recentment, Pilipovic i Stoeva [55] han estudiat els desenvolupaments en sèrie en límits inductius o projectius (numerables) en espais de Banach. En aquesta tesi comencem un estudi sistemàtic dels frames de Schauder en espais localment convexos, tot i que el nostre interés està en els espais de Fréchet i els seus duals. La diferència més important amb el concepte estudiat en [55] és que el nostre estudi no depén de una representació fixada del espai de Fréchet com a límit projectiu de espais de Banach. El text està dividit en dos capítols i un apèndix que ens aporta la notació, definicions i els resultats bàsics que utilitzarem al llarg de la tesi. El primer dels capítols està centrat en la relació entre les propietats de un frame de Schauder en un espai de Fréchet E i la estructura del espai. En el segon capítol es defineixen i estudien els frames i les successions de Bessel en espais de Fréchet i els seus duals. En el que segueix, donem una breu descripció dels diferents capítols: En el Capítol 1, estudiem els frames de Schauder en els espais de Fréchet i els seus duals, així com els resultats de pertorbació. Definim els frames de Schauder contractius i fitadament complets en espais localment convexos, estudiem la dualitat d'aquests dos conceptes i la seua relació amb la reflexivitat del espai. Caracteritzem, en quines situacions, un frame de Schauder incondicional és contractiu o fitadament complet en termes de les propietats del espai. També presentem alguns exemples de frames de Schauder concrets en espais de funcions. La majoria dels resultats inclosos en aquest capítol estan publicats per Bonet, Fernández, Galbis i Ribera en [13]. El segon capítol de la tesi està centrat en el estudi de les successions ¿-Bessel, ¿-frames i frames respecte de ¿ en el dual d'un espai localment convex de Hausdorff E, en particular, per a espais de Fréchet i espais (LB) complets E, amb ¿ un espai de successions. Investiguem la relació d'aquests dos conceptes amb sistemes representants en el sentit de Kadets i Korobeinik [34] i amb els frames de Schauder, que han sigut investigats en el Capítol 1. Els resultats abstractes presentats ací, quan els apliquem a espais de funcions analítiques concrets, ens donen molts exemples i conseqüències sobre els conjunts de mostreig i els desenvolupaments en sèrie de Dirichlet. Presentem diversos resultats abstractes sobre ¿-frames en espais (LB) complets. Finalment, recollim moltes aplicacions, resultats i exemples al voltant dels conjunts suficients per a espais de Fréchet de funcions holomorfes i conjunts dèbilment suficients per a espais pesats (LB) de funcions holomorfes. La majoria dels resultats inclosos en aquest capítol estan sotmesos a publicació per Bonet, Fernández, Galbis i Ribera en [12]. / Ribera Puchades, JM. (2015). Atomic decompositions and frames in Fréchet spaces and their duals [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/49987 / TESIS

L’étude des vides avec flux est une étape primordiale afin de mieux comprendre la compactification en théorie des cordes ainsi que ses conséquences phénoménologiques. En présence de flux, l’espace interne ne peut plus être Calabi-Yau, mais admet tout de même une structure SU(3) qui devient un outil privilégié. Après une introduction aux notions géométriques nécessaires, cette thèse examine le rôle des flux dans la compactification supersymétrique sous différents angles. Nous considérons tout d’abord des troncations cohérentes de la supergravité IIA. Nous montrons alors que des condensats fermioniques peuvent aider à supporter des flux et générer une contribution positive à la constante cosmologique. Ces troncations admettent donc des vides de Sitter qu’il serait autrement très difficile d’obtenir, si ce n’est impossible. L’argument est tout d’abord employé avec des condensats de dilatini puis améliorer en suggérant un mécanisme pour générer des condensats de gravitini à partir d’instantons gravitationnels. Ensuite l’attention se tourne sur les branes et leur comportement sous T-dualité non abélienne. Nous calculons les configurations duales à certaines solutions avec D branes de la supergravité de type II, et examinons les flux ainsi que leurs charges afin d’identifier les branes après dualité. La solution supersymétrique avec brane D2 est étudiée plus en détails en vérifiant explicitement les équations sur les spineurs généralisés, puis en discutant de la possibilité d’une déformation massive. Le dernier chapitre fournit une construction systématique de structures SU(3) sur une large classe de variétés toriques compactes. Cette construction définit un fibré en sphère au-dessus d’une variété torique 2d quelconque, mais fonctionne tout aussi bien sur une base Kähler-Einstein / The study of flux vacua is a primordial step in the understanding of string compactifications and their phenomenological properties. In presence of flux the internal manifold ceases to be Calabi-Yau, but still admits an SU(3) structure which becomes thus the preferred framework. After introducing the relevant geometrical notions this thesis explores the role that fluxes play in supersymmetric compactification through several approaches. At first consistent truncations of type IIA supergravity are considered. It is shown that fermionic condensates can help support fluxes and generate a positive contribution to the cosmological constant. These truncations thus admit de Sitter vacua which are otherwise extremely difficult to get, if not impossible. The argument is initially performed with dilatini condensates and then improved by suggesting a mechanism to generate gravitini condensates from gravitational instantons. Then the focus shifts towards branes and their behavior under non abelian T-duality. The duals of several D-brane solutions of type II supergravity are computed and the branes are tracked down by investigating the fluxes and the charges they carry. The supersymmetric D2 brane is further studied by checking explicitly the generalized spinor equations and discussing the possibility of a massive deformation. The last chapter gives a systematic construction of SU(3) structures on a wide class of compact toric varieties. The construction defines a sphere bundle on an arbitrary two-dimensional toric variety but also works when the base is Kähler-Einstein

Nous proposons, par voie théorique, une sonde ayant la capacité de détecter et de caractériser les états de surface d'une chaîne Su-Schrieffer-Heeger (SSH). Cette sonde consiste d’un qubit interagissant avec un environnement, et exploite le phénomène de la décohérence afin de retirer de l’information sur l’environnement. Une approximation de faible couplage permet de démontrer analytiquement que le taux de décohérence du qubit est proportionnel à la densité d’états locale de l’environnement. Dans le cas d’un environnement possédant des états discrets, une mesure de la densité d’états locale peut être équivalente à une mesure de l’amplitude d’un état, permettant donc une caractérisation spatiale des états de l’environnement. Un système tripartite consistant d'un qubit couplé à une chaîne SSH discrète muni de canaux conducteurs aux extrémités est étudié afin de valider l’utilité de la sonde pour inférer et caractériser les états de surface. L’espace des paramètres de la sonde est discuté en détail. En étudiant l’impact du couplage des canaux conducteurs, nous notons l’émergence d’états de type-surface sur des sites interdits ainsi que dans des phases topologiques ne supportant pas d’états de surface dans le modèle SSH isolé. Ces excitations, que nous appelons états fantômes, apparaissent dû à un décalage des frontières de la chaîne SSH. / We propose a novel probe with the ability to detect topological edge states in lowdimensional materials. This probe, consisting of a qubit interacting with a system of interest, utilizes the dynamics of decoherence to study the qubit’s environment. We show analytically that, under a weak-coupling approximation, the decoherence rate of the qubit is proportional to the local density of states of the environment. In studying environments featuring finite subsystems with discrete states, the local density of states mapped by the qubit probe can extract state amplitude profiles, resulting in a full spatial characterization of states. We explicitly study a tripartite system consisting of a qubit coupled to a finite SSH chain with conducting leads attached to each end and demonstrate the probe’s ability to infer the presence of, and characterize, edge states. The parameter space of the probe is studied. Notably, we show the lead coupling strength effectively shifts the SSH chain boundaries resulting in emergent edge-type states, dubbed ghost states, with support on sites which are forbidden in an isolated SSH chain for a given topological phase.

Le présent mémoire contient deux articles reliés par le formalisme de l'ansatz de Bethe. Dans le premier article, l'opérateur de Heun de type Lie est identifié comme une spécialisation de la matrice de transfert d'un modèle de \(BC\)-Gaudin à un site dans un champ magnétique. Ceci permet de le diagonaliser à l'aide de l'ansatz de Bethe algébrique modifié. La complétude du spectre est démontrée en reliant les racines de Bethe aux zéros des solutions polynomiales d'une équation différentielle de Heun inhomogène. Le deuxième article aborde le sujet des représentations irréductibles de l'algèbre de Lie \(su(3)\) dans la réduction \(su(3) \supset so(3) \supset so(2)\). Cette manière de construire les représentations irréductibles de \(su(3)\) porte une ambiguïté qui empêche de distinguer totalement les vecteurs de base, ce qui mène à un problème d'étiquette manquante. Dans cet esprit, l'algèbre des deux opérateurs fournissant cette étiquette est examinée. L'opérateur de degré 4 dans les générateurs de \(su(3)\) est diagonalisé en se servant des techniques de l'ansatz de Bethe analytique. / This Master’s thesis contains two articles linked by the formalism of the Bethe ansatz. In the first article, the Lie-type Heun operator is identified as a specialization of the transfer matrix of a one-site BC-Gaudin model in a magnetic field. This allows its diagonalization by means of the modified algebraic Bethe ansatz. The completeness of the spectrum is proven by relating the Bethe roots to the zeros of the polynomial solutions of an inhomogeneous differential Heun equation. The second article deals with the subject of irreducible representations of the Lie algebra su(3) in the reduction su(3) ⊃ so(3) ⊃ so(2). This way of constructing the irreducible representations of su(3) carries an ambiguity in distinguishing the basis vectors, also known as a missing label problem. In this spirit, the algebra of the two operators providing the missing label is examined. The operator of degree 4 in the generators of su(3) is diagonalized using the techniques of the analytical Bethe ansatz.

Nous considérons une chaîne de Su-Schrieffer-Heeger (SSH) à laquelle nous attachons une chaîne semi-infinie non dimérisée aux deux extrémités. Nous étudions l’effet d’un tel couplage sur les propriétés du modèle de SSH. En particulier, la représentation d’un tel système infini sous forme de système effectif fini nous permet d’examiner ses états de surface topologiques. Nous montrons que, comme ce à quoi on s’attendrait, les états de surface initiaux évoluent à mesure que le couplage entre les systèmes augmente. Alors que ce couplage augmente, deux phénomènes sont observés: d’un côté, ces états de surface disparaissent progressivement, et de l’autre côté, de nouveaux états de surface émergent. Ces nouveaux états, que nous appelons états fantômes, sont aussi des états de basse énergie. Une particularité surprenante de ceux-ci est qu’ils sont localisés sur une nouvelle interface: celle-ci est passée du premier (et dernier) site au deuxième (et avant-dernier) site, ce qui suggère que la topologie du système est fortement influencée par les chaînes semi-infinies. La topologie du système tripartite peut être classifiée selon trois régimes. Pour le régime de faible couplage, le système est dans une phase topologique bien définie; pour de grands couplages, il est dans sa phase opposée; pour le régime intermédiaire, sa nature topologique n’est pas encore bien comprise. / We consider a Su-Schrieffer-Heeger (SSH) chain to which we attach a semi-infinite undimerized chain (lead) to both ends. We study the effect of the openness of the SSH model on its properties. In particular, an accurate representation of the infinite system using an effective Hamiltonian allows us to examine its topological edge states. We show that, as one would expect, the initial edge states evolve as the coupling between the systems is increased. As this coupling grows, these states slowly vanish, while a new type of edge states emerge. These new states, which we refer to as ghost states, are also low-energy states. A surprising property of these states is that they are localized on a new interface: the interface has moved from the first (and last) site to the second (and second to last) site. This suggests that the topology of the system is strongly affected by the leads, with three regimes of behaviour. For very small coupling the system is in a well-defined topological phase; for very large coupling it is in the opposite phase; in the intermediate region, its topological nature is yet to be understood.