Σκοπός της παρούσας εργασίας είναι η ανάλυση προβλημάτων συνοριακών τιμών. Αρχικά αναφέρονται στοιχεία γραμμικής ανάλυσης και συγκεκριμένα εισάγεται η έννοια ενός τελεστή και τα είδη τελεστών που υπάρχουν, καθώς και η σημασία τους στη Φυσική. Επίσης, δίνεται ο ορισμός της διαφορικής εξίσωσης (Σ.Δ.Ε), ο ορισμός ενός προβλήματος αρχικών τιμών και ο ορισμός ενός προβλήματος συνοριακών τιμών. Έπειτα, αναλύεται η θεωρία Sturm-Liouville και περιγράφονται παραδείγματα συνοριακών τιμών τα οποία επιλύονται με αυτή. Ακόμη, μελετώνται οι συναρτήσεις Green και δίνονται παραδείγματα εφαρμογών τους. Στη συνέχεια εξάγεται η κυματική εξίσωση με τη βοήθεια του μοντέλου της ταλαντούμενης χορδής και επιλύεται με τη μέθοδο του χωρισμού των μεταβλητών για διάφορους τύπους αρχικών και συνοριακών τιμών. Κατόπιν, περιγράφονται μέθοδοι για την επίλυση προβλημάτων συνοριακών τιμών που συνδέονται με την εξίσωση της θερμότητας και μετά αναφέρονται εφαρμογές που προκύπτουν από την επίλυση προβλημάτων διάδοσης θερμότητας. Τέλος αναφέρεται η θεωρία Fredholm και η έννοια της κατανομής και δίνονται παραδείγματα λύσεων των διαφορικών εξισώσεων με την έννοια των κατανομών. Η θεωρία Fredholm είναι ιδιαίτερα σημαντική σε προβλήματα διαφορικών εξισώσεων που είναι μη ομογενή. / In the present project, the initial boundary value problems are analyzed. Firstly, elements of linear analysis are introduced. Particularly the concept of an operator and its types are introduced as well as the importance in the physics sector. Also, the definition of a differential equation and the initial boundary value problems are presented. Additionally, the theory of Sturm-Liouville and its example are described. Moreover, Green function and their applications are introduced. Furthermore, the wave equation was elicited with the basis of vibrating spring model and solved with the method of separating variables. Also with this method and by using Fourier series the heat equation was solved. Finally the theory of Fredholm and the concept of distribution are described. The theory of Fredholm is important in problems of not homogeneous differential equation problems.

Orientador: German Jesus Lozada Cruz / Banca: Luiz Augusto Fernandes de Oliveira / Banca: José Marcio Machado / Resumo: O objetivo deste trabalho é estudar problemas de valor de contorno do tipo {ÿ + f(t) =0 y(0)=0˙ y(1)= ky(η), (1) onde η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). Para antingirmos nosso objetivo usamosas funções de Green G(t,s)que nos permitem escrever a solução do problema(1)na seguinte forma: w(t)= ∫ 1 0 G(t,s)f(s)ds. Usando esta solução, investigamos através do ponto fixo de Schauder a solvabilidade do problema não linear { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / Abstract: The main goal of this work is study the following boundary value problems {ÿ + f(t) = 0 =0 y(0)=0˙ y(1)= ky(η), (1), where η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). To achieve our goal we use the Green's function G(t,s) which allow us to write the solution of the problem (2) in the form: w(t)= ∫ 1 0 G(t,s)f(s)ds. Using this solution and the Schauder point theory, also we study the solvability of a nonlinear problem { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / Mestre

Les signaux enregistrés à la surface du globe sont composés de trois types d'ondes : des ondes directes, des ondes réfléchies plus ou moins facilement interprétables, et des ondes multiplement diffusées beaucoup plus complexes à interpréter. Certaines propriétés physiques de ces ondes multiplement diffusées permettent de les assimiler au bruit sismique ambiant. Nous appliquons des techniques de corrélation de bruit sismique développées ces dernières années à la coda sismique, afin de tirer des informations sur le caractère multiplement diffusant du milieu. En particulier, nous montrons que le théorème reliant fonction de Green et champ ambiant peut être utilisé dans la coda. La dynamique temporelle de la reconstruction de la fonction de Green est alors un indicateur de la répartition de l'énergie dans le milieu. En reconstruisant la fonction de Green à l'aide de corrélations pour plusieurs fenêtres de temps dans la coda, nous montrons qu'il est possible de relier la symétrie des parties causales et acausales de la fonction de Green au libre parcours moyen du milieu. Nous développons ensuite des simulations numériques de propagation d'onde acoustique en 2D. Nous observons que l'évolution de la reconstruction des fonctions de Green observée dans ce milieu et celle prédite par la théorie est proche. La même approche est ensuite appliquée à des données sismologiques de terrain, sans permettre de retrouver la même dynamique. Nous développons alors une méthode de type MCMC permettant de reconstruire les fonctions de Green du milieu de manière optimale. / Most of the waves recorded by seismometers are hard to interpret because of the complexity of the propagation medium, especially the late part of the seismic coda. These multiply scattered coda waves are close in nature to the ambient noise. We are applying recent noise correlation techniques to coda waves in order to retrieve information about the scattering medium. We show how the relationship between ambient noise and Green's function can be used in the case of the seismic coda. The quality of the Greens function retrieved by cross correlation of time windows in the coda is a proxy indicating the energy partition in the propagation medium. In particular, we establish a link between the symmetry of the causal and acausal parts of the reconstructed Green's function and the mean free path. We validate this theoretical approach with acoustical 2D numerical simulations. The same approach seems inefficient on a seismological dataset, due to the high S/N ratio and the non optimal repartition of receivers. Hence, we develop an MCMC based algorithm in order to optimally reconstruct the green's function in the seismic coda.

abstract: In the traditional setting of quantum mechanics, the Hamiltonian operator does not depend on time. While some Schrödinger equations with time-dependent Hamiltonians have been solved, explicitly solvable cases are typically scarce. This thesis is a collection of papers in which this first author along with Suslov, Suazo, and Lopez, has worked on solving a series of Schrödinger equations with a time-dependent quadratic Hamiltonian that has applications in problems of quantum electrodynamics, lasers, quantum devices such as quantum dots, and external varying fields. In particular the author discusses a new completely integrable case of the time-dependent Schrödinger equation in R^n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. The author also considers several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator. Finally, the author constructs integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrödinger equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2011

Dans cette thèse nous étudions les transitions de phase et leurs propriétés thermodynamiques au sein de couches minces en se basant sur des simulations Monte-Carlo et sur le formalisme de la fonction de Green .Dans le premier chapitre, nous étudions le modèle de Blume-Emery-Griffith pour un film mince sur réseaux triangulaires empilés. Le spin $S_i$ dans ce modèle prend trois valeurs (+/-1,0). Notre travail a été motivé par le désir de vérifier si a nature de la transition de phase se conserve quand on réduit l'épaisseur du film. En utilisant la simulation Monte Carlo, nous montrons qu'il existe une valeur critique d'anisotropie D où la transition change de nature. Nous montrons ainsi que la nature premier ordre ne disparaît pas lorsque nous réduisons l'épaisseur du film contrairement à d'autres systèmes.Dans le deuxième chapitre, nous étudions les propriétés quantiques des couches mince hélimagnétiques. Nous montrons qu'il existe des modes de surface qui affectent la magnétisation de surface, nous montrons également que les fluctuations quantiques provoquent la contraction des spins à T=0 et donnent lieu à un croisement entre les magnétisations des couches à basse température. Nous nous intéressons ensuite à l'effet d'un champ magnétique appliqué perpendiculairement à la surface du film. Nous montrons que les spins réagissent en créant une configuration particulière. En utilisant la simulation Monte Carlo nous étudions la transition de phase en fonction de l'intensité du champ appliqué. Nous montrons que le système subit une transition de phase déclenchée par la destruction des composantes transversales xy des spins de certaines couches. À basse température, nous étudions les effets des fluctuations quantiques en utilisant la méthode des fonctions de Green. Les résultats montrent que la contraction des spins à T=0 est différente d'une couche à l'autre, et que la croisement des magnétisations de couche dépend de l'ampleur des angles hélicoïdaux.Dans le troisième chapitre, nous introduisons l'interaction de Dzyaloshinskii-Moriya. Il a été montré dans divers travaux que l'interaction DM est à l'origine de formation des skyrmions et de nouveau genre de domaines Walls. Nous nous intéressons aux propriétés quantiques d'un système de spins qui interagissent les uns avec les autres via une interaction DM et une interaction ferromagnétique. En utilisant la méthode "steepest descend", nous avons trouvé un état fondamental non-colinéaire qui est dû à la compétition entre l'interaction ferromagnétique et l'interaction asymétrique DM. Utilisant la théorie des fonctions de Green pour calculer le spectre des ondes de spin et la magnétisation des couches à température finie en deux et trois dimensions ainsi que dans un film mince avec des effets de surface. Nous avons constaté que l'excitation des ondes de spin dans les cristaux 2D et 3D est stable à T=0 sans nécessiter d'anisotropie, mais dans le cas d'un film mince nous avons besoin d'une faible anisotropie pour stabiliser le spectre en raison du manque de voisins à la surface. On trouve aussi que l'énergie des ondes de spin est proportionnelle à $K^2$ pour les faibles valeurs de DM et une proportionnalité en $K$ pour les interactions fortes.Dans le quatrième chapitre, nous nous intéressons aux cristaux de skyrmion crées grâce à la compétition entre l'interaction ferromagnétique, le DM et le champ magnétique appliqué. Ces skyrmions s'organisent dans une structure périodique, ils ont été observés expérimentalement dans les composés MnSi, FeCoSi et dans les semiconducteurs dopés. En utilisant la simulation Monte Carlo, nous montrons que les cristaux de skyrmions sont stable à des températures finies et jusqu'à la transition où la structure topologique de chaque skyrmion et la structure périodique sont détruites. Nous étudions également la relaxation des skyrmions dans la phase cristalline et nous constatons que le temps de relaxation suit une loi exponentielle étirée. / In this thesis, we study the phase transition and thermodynamic properties of classical and quantum spin models in thin films using both Green's function and standard Monte Carlo simulation.In chapter 1, we study the Blume-Emery-Griffith model. This model has been introduced to describe the mixing phase of superfluid He$^4$ ($S_i=pm$ 1) and normal fluid He$^3$ ($S_i$= 0) at low temperatures, such system undergoes two kinds of phase transition, first and second-order ones. Using Monte Carlo simulation, we show that there exists a critical value of anisotopy D$below (above) which the transition is of second (first) order, and that the first order nature of transition does not disappear when we reduce the film thickness unlike in other systems where the bulk first-order transition becomes second order with small thickness. In the Helium vocabulary, we show that the film surfaces have a deficit of He$^4$ with respect to interior layers of the film.In chapter 2 we first study quantum properties of a helimagnetic thin film. We show that there exist surface acoustic and optical modes which affect the surface magnetization. We also show that quantum fluctuations cause the spin contraction at $T$=0 and give rise to a cross-over between layer magnetizations at low temperatures. In the second part of chapter 2, we are interested in the effect of an external magnetic field applied. We show that spins react to a moderate applied magnetic field by creating a particular spin configuration along the $c$-axis. Using Monte Carlo simulation we study the phase transition as functions of the magnetic field strength. We show that the system undergoes a phase transition triggered by the destruction of the transverse xy spin-components. At low temperatures, we investigate effects of quantum fluctuations using Green's function method. The results show that the zero-point spin contraction is different from layer to layer. We also find a crossover of layer magnetizations which depends on the magnitude of helical angles.In the third chapter, we introduce the in-plane Dzyaloshinskii-Moriya interaction (DM). It has been showed in various works that the DM interaction is at the origin of topological skyrmions and a new kind of magnetic domain walls. In this chapter, we are interested in the spin-wave properties of a system of spins interacting with each other via a DM interaction. Using the steepest descend method we found a non-collinear ground state which is due to the competition between the ferromagnetic and the asymmetric DM interaction. We use the Green's function theory to calculate the spin-wave spectrum and the layer magnetization at finite temperatures in two and three dimensions as well as in a thin film with surface effects. We found that the spin-wave excitation in 2D and 3D crystals is stable at $T$=0 without the need of an anisotropy, but in the case of a thin film we need a small anisotropy to stabilize the spin-wave spectrum because of the lack of neighbors at the surface. We find also that the spin-wave energy is proportional to $k^2$ for a small DM interaction and is linear in $k$ for a strong one.Finally, in the fourth chapter we are interested in skyrmion crystals created by the competition between the ferromagnetic interaction and the DM interaction under an applied magnetic field. They arrange themselves in a periodic structure. These skyrmion crystals have been experimentally observed in MnSi compounds and in doped semiconductors. Using Monte Carlo simulation, we show that skyrmion crystals are stable at finite temperatures up to a transition temperature where the topological structure of each skyrmion and the periodic structure of skyrmions are destroyed. We also investigate the relaxation of the skyrmions in the crystalline phase and find that the relaxation time follows a stretched exponential law which is a characteristic of slowly-relaxed systems such as spin glasses.

La technique des corrélations de bruit ambiant est aujourd'hui largement utilisée en sismologie. Elle présente l'avantage essentiel de fournir des données qui ne dépendent pas de l'occurrence des séismes. Cette technique a été utilisée pour imager avec une bonne résolution la croûte dans plusieurs régions du monde. Depuis plus récemment, les corrélations de bruit sont également utilisées pour détecter des variations des propriétés élastiques de la croûte terrestre, et suivre l'évolution temporelle de structures géologiques potentiellement dangereuses. Dans ce contexte, mon travail de thèse a consisté en deux parties principales : 1/ Une étude méthodologique pour améliorer la qualité des mesures sur les corrélations de bruit. La distribution des sources de bruit empêche généralement la reconstruction parfaite de la fonction de Green dans les corrélations de bruit, pouvant introduire des biais dans les mesures basées sur le bruit sismique ambiant. Nous présentons dans ce manuscrit deux approches différentes pour essayer d'éliminer l'influence des sources de bruit dans nos mesures. 2/ Une étude de suivi temporel dans la province du Sichuan (Chine), autour du séisme du Wenchuan (12 mai 2008, Mw = 7.9). Cette application offre des conditions exceptionnelles à une application de suivi temporel, avec l'étude d'un très puissant séisme continental et un très grand nombre de données. Nous nous sommes en particulier intéressés au comportement de la croûte à différentes profondeurs, en réponse au séisme. Nous présentons également une méthode d'inversion pour localiser les variations dans le milieu. / The ambient noise correlation technique is now widely used in seismology. It is especially useful since data provided do not depend on the occurrence of earthquakes. This technique has been used to construct high-resolution tomographic images of the crust in various regions of the world. More recently, ambient noise correlations have been used to detect changes in elastic properties of the Earth's crust, and monitor the temporal evolution of potentially dangerous geological structures. In this scientific context, my research work has consisted of two main parts : 1/A methodological study to improve the quality of noise-based measurements. Usually, the distribution of noise sources prevents the exact reconstruction of the Green's function in noise correlations, that may introduce bias in noise-based measurements. We propose here two different approaches that can help in suppressing noise source influence in measurements. 2/A monitoring case study in Sichuan (China), associated with the Wenchuan earthquake (12 May 2008, Mw = 7.9). This case study provides many opportunities, with the study of a great continental earthquake and a large amount of data available. We especially investigated the behavior of the crust at different depths, in response to the Wenchuan earthquake.We also present an inversion method in order to locate variations in the medium.

Orientador: Germán Jesus Lozada Cruz / Banca: Marco Aparecido Queiroz Duarte / Banca: Juliana Conceição Precioso Pereira / Resumo: O objetivo desse trabalho é estudar problemas de valor de contorno com três pontos lineares e não ineares, também conhecidos como problemas não Isto é feito, usando as funções de Green, usadas para resolver problemas de valor de contorno com dois pontos. / Abstract: The aim of this work is to study boundary value problems with three points also known as non-classical problems. This is done using the Green's functions, which are used to solve two-point boundary value problems. / Mestre

Submitted by ANDRE RAMALHO DOS SANTOS null (ramalho_inf@yahoo.com.br) on 2018-02-14T03:20:16Z No. of bitstreams: 1 Dissertação André Ramalho.final.pdf: 2789018 bytes, checksum: d4170ea3aaec8f302b447a0aac5e5986 (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-02-14T16:54:15Z (GMT) No. of bitstreams: 1 santos_ar_me_rcla.pdf: 2620534 bytes, checksum: cb88b11f48c3fc7b2fef3c938febb8e0 (MD5) / Made available in DSpace on 2018-02-14T16:54:15Z (GMT). No. of bitstreams: 1 santos_ar_me_rcla.pdf: 2620534 bytes, checksum: cb88b11f48c3fc7b2fef3c938febb8e0 (MD5) Previous issue date: 2017-11-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nós estudamos teoricamente como o calor e a eletricidade são afetados pela sobreposição de dois férmions de Majorana (MFs, de Majorana fermions em Inglês), os quais estão isolados nas bordas de um fio topológico de Kitaev, em particular, na forma de “ferradura”. É considerado que esse fio está assimetricamente acoplado a um único ponto quântico (QD, de Quantum dot em Inglês) hibridizado com contatos metálicos. Em baixas temperaturas e dependente do nível de energia desse QD, nós mostramos que ao ajustar a assimetria acima, as respostas ressonantes das condutâncias termoelétricas mudam inesperadamente de forma drástica. Assim, propomos como aplicação, um sintonizador termoelétrico em nanoescala assistido por MFs. / We study theoretically in a topological U-shaped Kitaev wire, with Majorana fermions (MFs) on the edges, how heat and electricity are affected by them when found overlapped. The asymmetric regime of their couplings with a single quantum dot (QD) hybridized with metallic leads is considered. At low temperatures and dependent upon the QD energy level, we show that by tuning this asymmetry, the resonance positions of the thermoelectrical conductances change drastically. Thereby, the tuner of heat and electricity here proposed is constituted.

Orientador: Euclides de Mesquita Neto / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-21T04:39:32Z (GMT). No. of bitstreams: 1 Labaki_Josue_D.pdf: 3777054 bytes, checksum: 98096129397b78d6f60e69e7d8210b9b (MD5) Previous issue date: 2012 / Resumo: Este trabalho apresenta novos modelos para descrever o comportamento harmônico de placas rígidas e flexíveis incrustadas em meios estratificados. Incrustações profundas das placas são representadas por interfaces bi-materiais de meios viscoelásticos transversalmente isotrópicos. Incrustações rasas são representadas por meios estratificados viscoelásticos transversalmente isotrópicos. Os modelos de meios circundantes são obtidos com base em modelos clássicos disponíveis na literatura, que estão em coordenadas cilíndricas. O método usado para obter esses modelos envolve transformadas de Hankel, com o objetivo de obter as equações de movimento em um domínio no qual elas possam ser manipuladas algebricamente. Algumas funções de interesse são expandidas em séries de Fourier, para decompor grandezas como deslocamentos, tensões e carregamentos em termos axissimétricos e antissimétricos. No modelo de placa rígida, a incrustação da placa é descrita assumindo que uma superfície circular do meio circundante é rígida. Um conjunto de equações integrais acopladas, que relaciona deslocamentos e força de superfície atuantes nesta superfície, é resolvido por meio de discretização da superfície em discos anulares concêntricos. A teoria de Kirchhoff para placas finas é usada como modelo de placa flexível. Uma tentativa de solução para o perfil de deflexão da placa é definido em termos de série de potência. Uma função envolvendo as energias de deformação da placa e do meio é definida, cuja minimização, sob as restrições impostas pelas condições de contorno da placa, resulta no perfil de deflexão da placa incrustada. Os modelos de placa rígida e flexível incrustadas são comparados com resultados da literatura para incrustações em meios isotrópicos. Novos resultados são fornecidos para os casos de incrustações rasas ou profundas em meios viscoelásticos estratificados transversalmente isotrópicos, sob a ação de forças harmônicas verticais e transversais, assim como momentos. Os resultados deste trabalho contribuem para o estudo de fundações e âncoras em solos estratificados / Abstract: This work presents new models to describe the time-harmonic behavior of rigid and flexible circular plates embedded in layered media. Deep embedments of the plates are represented by bimaterial viscoelastic transversely isotropic interfaces. Shallow embedments are represented by multilayered viscoelastic transversely isotropic media. The models of surrounding media are derived according to classical models available in the literature, which are in cylindrical coordinates. The method used to derive these models involves Hankel transforms, in order to bring the equations of motion to a domain in which they can be dealt with algebraically. A few functions of interest are expanded in Fourier series, in order to describe quantities such as displacements, stresses and loadings in terms of axisymmetric or antisymmetric components. In the model of rigid plate, the embedded plate is described by considering that a circular surface within the surrounding medium is rigid. A set of coupled integral equations, which relate the displacements and the unknown tractions across this surface, is solved by discretizing the surface into concentric annular disc elements. Kirchhoff theory of thin elastic plates is used to describe the flexible plate. A trial function to describe the deflection profile of the plate is defined in terms of a power series. An energy functional is established, which involves the strain energy of the plate and of its surrounding medium. The minimization of this functional, under the restriction that is must satisfy the boundary conditions at the plate edge, results in the deflection profile of the embedded plate. The models of embedded plates are compared with previous results from the literature for the case of different embedments in isotropic media. New research results are provided for the case of shallow and deep embedments in layered viscoelastic transversely isotropic media, under time-harmonic vertical and transverse loads and rocking moments. The results of this work contribute to the study of embedded foundations and anchors in layered soil / Doutorado / Mecanica dos Sólidos e Projeto Mecanico / Doutor em Engenharia Mecânica

The main body of this thesis describes how to calculate scanning tunneling microscopy (STM) images from first-principles methods. The theory is based on localized orbital density functional theory (DFT), whose limitations for large-vacuum STM models are resolved by propagating localized-basis wave functions close to the surface into the vacuum region in real space. A finite difference approximation is used to define the vacuum Hamiltonian, from which accurate vacuum wave functions are calculated using equations based on standard single-particle Green’s function techniques, and ultimately used to compute the conductance. By averaging over the lateral reciprocal space, the theory is compared to a series of high-quality experiments in the low- bias limit, concerning copper surfaces with adsorbed carbon monoxide (CO) species and adsorbate atoms, scanned by pure and CO-functionalized copper tips. The theory compares well to the experiments, and allows for further insights into the elastic tunneling regime. A second significant project in this thesis concerns first-principles calculations of a simple chemical reaction of a hydroxyl (oxygen-deuterium) monomer adsorbed on a copper surface. The reaction mechanism is provided by tunneling electrons that, via a finite electron-vibration coupling, trigger the deuterium atom to flip between two nearly identical configurational states along a frustrated rotational motion. The theory suggests that the reaction primarily occurs via nuclear tunneling for the deuterium atom through the estimated reaction barrier, and that over-barrier ladder climbing processes are unlikely.