This dissertation provides a complete mathematical framework to compute the sensitivities with respect to system parameters for any second order hybrid Ordinary Differential Equation (ODE) and rank 1 and 3 Differential Algebraic Equation (DAE) systems. The hybrid system is characterized by discontinuities in the velocity state variables due to an impulsive forces at the time of event. At the time of event, such system may also exhibit a change in the equations of motion or in the kinematic constraints. The analytical methodology that solves the sensitivities for hybrid systems is structured based on jumping conditions for both, the velocity state variables and the sensitivities matrix. The proposed analytical approach is then benchmarked against a known numerical method. The mathematical framework is extended to compute sensitivities of the states of the model and of the general cost functionals with respect to model parameters for both, unconstrained and constrained, hybrid mechanical systems. This dissertation emphasizes the penalty formulation for modeling constrained mechanical systems since this formalism has the advantage that it incorporates the kinematic constraints inside the equation of motion, thus easing the numerical integration, works well with redundant constraints, and avoids kinematic bifurcations. In addition, this dissertation provides a unified mathematical framework for performing the direct and the adjoint sensitivity analysis for general hybrid systems associated with general cost functions. The mathematical framework computes the jump sensitivity matrix of the direct sensitivities which is found by computing the Jacobian of the jump conditions with respect to sensitivities right before the event. The main idea is then to obtain the transpose of the jump sensitivity matrix to compute the jump conditions for the adjoint sensitivities. Finally, the methodology developed obtains the sensitivity matrix of cost functions with respect to parameters for general hybrid ODE systems. Such matrix is a key result for design analysis as it provides the parameters that affect the given cost functions the most. Such results could be applied to gradient based algorithms, control optimization, implicit time integration methods, deep learning, etc. / Ph. D. / A mechanical system is composed of many different parameters, like the length, weight and inertia of a body or the spring and damping constant of a suspension system. A variation of these constants can modify the motion a mechanical system. This dissertation provides a complete mathematical framework that aims at identifying the parameters that affect at most the motion of a mechanical system. Such system could be hybrid like the human body. Indeed, when walking the foot/ground impact causes an abrupt change of velocity of the foot, while the position of the foot remains the same. Such change makes the velocity of the human body to be discontinuous at such event, which makes the human body when walking a hybrid system. The same can be applied to a vehicle driving over a bump. The main result obtained from the mathematical framework is called the "sensitivity matrix". Such matrix is a key result for design analysis as it identifies the parameters that affect at most the motion of a mechanical system. Such results are very relevant and could be applied to different softwares with prebuilt gradient based algorithms, control optimization, implicit time integration methods, or deep learning, etc.

The estimation approaches examined in this dissertation focus on manipulating system dynamical models to allow the well-known form of the continuous-discrete extended Kalman filter (CDEKF) to accommodate constrained and hybrid systems. This estimation algorithm filters sequential discrete measurements for nonlinear continuous systems modeled with ordinary differential equations. The aim of the research is to broaden the class of systems for which this common tool can be easily applied. Equality constraints, holonomic or nonholonomic, or both, are commonly found in the system dynamics for vehicles, spacecraft, and robotics. These systems are frequently modeled with differential algebraic equations. In this dissertation, three tools for adapting the dynamics of constrained systems for implementation in the CDEKF are presented. These strategies address (1) constrained systems with quasivelocities, (2) kinematically constrained redundant coordinate systems, and (3) systems for which an equality constraint can be broken. The direct linearization work for constrained systems modeled with quasi-velocities is demonstrated to be particularly useful for systems subject to nonholonomic constraints. Concerning redundant coordinate systems, the "constraint force" perspective is shown to be an effective approximation for facilitating implementation of the CDEKF while providing similar performance to that of the fully developed estimation scheme. For systems subject to constraint violation, constraint monitoring methods are presented that allow the CDEKF to autonomously switch between constrained and unconstrained models. The efficacy of each of these approaches is shown through illustrative examples. Hybrid dynamical systems are those modeled with both finite- and infinite-dimensional coordinates. The associated governing equations are integro-partial differential equations. As with constrained systems, these governing equations must be transformed in order to employ the CDEKF. Here, this transformation is accomplished through two finite-dimensional representations of the infinite-dimensional coordinate. The application of these two assumed modes methods to hybrid dynamical systems is outlined, and the performance of the approaches within the CDEKF are compared. Initial simulation results indicate that a quadratic assumed modes approach is more advantageous than a linear assumed modes approach for implementation in the CDEKF. The dissertation concludes with a direct estimation methodology that constructs the Kalman filter directly from the system kinematics, potential energy, and measurement model. This derivation provides a straightforward method for building the CDEKF for discrete systems and relates these direct estimation ideas to the other work presented throughout the dissertation. Together, this collection of estimation strategies provides methods for expanding the class of systems for which a proven, well-known estimation algorithm, the extended Kalman filter, can be applied. The accompanying illustrative examples and simulation results demonstrate the utility of the methods proposed herein.

In this dissertation, we explore the issues related to opportunistic and cooperative communications in a multiuser environment. In the first part of the dissertation, we consider opportunistic scheduling for delay limited systems. Multiuser communication over fading channels is a challenging problem due to fast varying channel conditions. On the other hand, it provides opportunities to exploit the varying nature of the channel and maximize the throughput by scheduling the user (or users) with good channel. This gain is termed as multiuser diversity. The larger the number of users, the greater is the multiuser diversity gain. However, there is an inherent scheduling delay in exploiting multiuser diversity. The objective of this work is to design the scheduling schemes which use multiuser diversity to minimize the system transmit energy. We analyze the schemes in large system limit and characterize the energy--delay tradeoff. We show that delay tolerance in data transmission helps us to exploit multiuser diversity and results in an energy efficient use of the system resources. We assume a general multiuser environment but the proposed scheduling schemes are specifically suitable for the wireless sensor network applications where saving of transmit energyat the cost of delay in transmission is extremely useful to increase the life of battery for the sensor node. In the first part of the thesis, we propose scheduling schemes withthe objective of minimizing transmit energy for a given fixed tolerable transmission delay. The fixed delay is termed as hard deadline. A group of users with channels better than a transmission threshold are scheduled for transmission simultaneously using superposition coding. The transmission thresholds depend onthe fading statistics of the underlying channel and hard deadline of the data to be scheduled. As deadline is approached, the thresholds decrease monotonically to reflect the scheduling priority for theuser. We analyze the proposed schedulers in the large system limit. We compute the optimized transmission thresholds for the proposed scheduling schemes. We analyze the proposed schemes for practically relevant scenarios when the randomly arriving packets have individual, non--identical deadlines. We analyze the case when loss tolerance of the application is exploited to further decrease the system energy. The transmitted energy is not a convex function oftransmission thresholds. Therefore, we propose heuristic optimization procedures to compute the transmission thresholds and evaluate the performance of the schemes. Finally, we study the effect of outer cell interference on the proposed scheduling schemes. The second part of the thesis investigates the problem of cooperative communication between the nodes which relay the data of other sources multiplex with their own data towards a common destination, i.e. a relay node performs as a relay and data source at the same time. This problem setting is very useful in case of some wireless sensor network (WSN) applications where all the nodes relay sensed data towards a common destination sink node. The capacity region of a relay region is still an open problem. We use deterministic network model to study the problem. We characterizethe capacity region for a cooperative deterministic network with single source, multiple relays and single destination. We also characterize the capacity region when communicating nodes have correlated information to be sent to the destination. / Cross Layer Optimization of Wireless Sensor Networks

Legged locomotion has several interesting challenges that need to be addressed, such as the ability of dynamically walk over rough terrain like stairs or stepping stones, as well as the ability to adapt to unexpected changes in the environment and the dynamic model of the robot. This thesis is driven towards solving these challenges and makes contributions on theoretical and experimental aspects to address: dynamic walking, model uncertainty, and rough terrain. On the theoretical front, we introduce and develop a unified robust and adaptive control framework that enables the ability to enforce stability and safety-critical constraints arising from robotic motion tasks under a high level of model uncertainty. We also present a novel method of walking gait optimization and gait library to address the challenge of dynamic robotic walking over stochastically generated stepping stones with significant variations in step length and step height, and where the robot has knowledge about the location of the next discrete foothold only one step ahead. On the experimental front, our proposed methods are successfully validated on ATRIAS, an underactuated, human-scale bipedal robot. In particular, experimental demonstrations illustrate our controller being able to dynamically walk at 0.6 m/s over terrain with step length variation of 23 to 78 cm, as well as simultaneous variation in step length and step height of 35 to 60cm and -22 to 22cm respectively. In addition to that, we also successfully implemented our proposed adaptive controller on the robot, which enables the ability to carry an unknown load up to 68 lb (31 kg) while maintaining very small tracking errors of about 0.01 deg (0.0017 rad) at all joints. To be more specific, we firstly develop robust control Lyapunov function based quadratic program (CLFQP) controller and L1 adaptive control to handle model uncertainty for bipedal robots. An application is dynamic walking while carrying an unknown load. The robust CLF-QP controller can guarantee robustness via a quadratic program that can be extended further to achieve robust safety-critical control. The L1 adaptive control can estimate and adapt to the presence of model uncertainty in the system dynamics. We then present a novel methodology to achieve dynamic walking for underactuated and hybrid dynamcal bipedal robots subject to safety-critical constraints. The proposed controller is based on the combination of control Barrier functions (CBFs) and control Lyapunov functions (CLFs) implemented as a state-based online quadratic program to achieve stability under input and state constraints. The main contribution of this work is the control design to enable stable dynamical bipedal walking subject to strict safety constraints that arise due to walking over a terrain with randomly generated discrete footholds. We next introduce Exponential Control Barrier Functions (ECBFs) as means to enforce high relativedegree safety constraints for nonlinear systems. We also develop a systematic design method that enables creating the Exponential CBFs for nonlinear systems making use of tools from linear control theory. Our method creates a smooth boundary for the safety set via an exponential function, therefore is called Exponential CBFs. Similar to exponential stability and linear control, the exponential boundary of our proposed method helps to have smoother control inputs and guarantee the robustness under model uncertainty. The proposed control design is numerically validated on a relative degree 4 nonlinear system (the two-link pendulum with elastic actuators and experimentally validated on a relative degree 6 linear system (the serial cart-spring system). Thanks to these advantages of Exponential CBFs, we then can apply the method to the problem of 3D dynamic walking with varied step length and step width as well as dynamic walking on time-varying stepping stones. For the work of using CBF for stepping stones, we use only one nominal walking gait. Therefore the range of step length variation is limited ([25 : 60](cm)). In order to improve the performance, we incorporate CBF with gait library and increase the step length range significantly ([10 : 100](cm)). While handling physical constraints and step transition via CBFs appears to work well, these constraints often become active at step switching. In order to resolve this issue, we introduce the approach of 2-step periodic walking. This method not only gives better step transitions but also offers a solution for the problem of changing both step length and step height. Experimental validation on the real robot was also successful for the problem of dynamic walking on stepping stones with step lengths varied within [23 : 78](cm) and average walking speed of 0:6(m=s). In order to address the problems of robust control and safety-critical control in a unified control framework, we present a novel method of optimal robust control through a quadratic program that offers tracking stability while subject to input and state-based constraints as well as safety-critical constraints for nonlinear dynamical robotic systems under significant model uncertainty. The proposed method formulates robust control Lyapunov and barrier functions to provide guarantees of stability and safety in the presence of model uncertainty. We evaluate our proposed control design on different applications ranging from a single-link pendulum to dynamic walking of bipedal robot subject to contact force constraints as well as safety-critical precise foot placements on stepping stones, all while subject to significant model uncertainty. We conduct preliminary experimental validation of the proposed controller on a rectilinear spring-cart system under different types of model uncertainty and perturbations. To solve this problem, we also present another solution of adaptive CBF-CLF controller, that enables the ability to adapt to the effect of model uncertainty to maintain both stability and safety. In comparison with the robust CBF-CLF controller, this method not only can handle a higher level of model uncertainty but is also less aggressive if there is no model uncertainty presented in the system.

In this work, Schr&ouml / dinger and Dirac equations will be examined in geometries that confine the particles to hypersurfaces. For this purpose, two methods will be considered. The first method is the thin layer method which relies on explicit use of geometrical relations and the squeezing of a certain coordinate of space (or spacetime). The second is Dirac&rsquo / s quantization procedure involving the modification of canonical quantization making use of the geometrical constraints. For the Dirac equation, only the first method will be considered. Lastly, the results of the two methods will be compared and some notes on the differences between the results will be included.

Submitted by Anderson Nogueira (andsogueira@hotmail.com) on 2016-03-08T21:06:11Z No. of bitstreams: 1 Tese-Anderson.pdf: 1337697 bytes, checksum: 95996aacf55769a399e7a2c11a2a1e5f (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-03-09T18:28:05Z (GMT) No. of bitstreams: 1 nogueira_aa_dr_ift.pdf: 1337697 bytes, checksum: 95996aacf55769a399e7a2c11a2a1e5f (MD5) / Made available in DSpace on 2016-03-09T18:28:05Z (GMT). No. of bitstreams: 1 nogueira_aa_dr_ift.pdf: 1337697 bytes, checksum: 95996aacf55769a399e7a2c11a2a1e5f (MD5) Previous issue date: 2016-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho tem como objetivo explorar a dinâmica quântica de interação entre partículas escalares e vetoriais e estudar o equilíbrio termodinâmico dessas partículas no ensemble gran-canônico. A dinâmica de interação, escrita em uma linguagem covariante entre o campo de matéria (escalar) e o campo intermediador de interação (vetorial), apresenta uma simetria de calibre local, U(1) no caso quântico e SO(4) no equilíbrio termodinâmico. Sendo assim dividimos o trabalho em dois setores. No primeiro setor analisamos sistematicamente a interação quântica entre partículas escalares (mésons) e partículas vetoriais (fótons) no contexto da eletrodinâmica quântica escalar generalizada de Duffin-Kemmer-Petiau (GSDKP). Para isso quantizamos a teoria, utilizando uma abordagem funcional. Construímos a estrutura Hamiltoniana do sistema seguindo a metodologia de Dirac, o procedimento de Faddeev-Senjanovic para obter a amplitude de transição no calibre de Coulomb generalizado e o método de Faddeev-Popov-DeWitt para escrever a amplitude de transição anterior de maneira covariante na condição de calibre no-mixing. Daí, escrevendo o funcional gerador via Schwinger, as equações de Schwinger-Dyson (SD) e as identidades de Ward-Takahashi (WT) são obtidas. Como introdução à análise das correções radiativas, fizemos um cálculo quantitativo para ver os tipos de divergências superficiais (ultravioleta) que poderiam aparecer na teoria. Depois apresentamos um cálculo explícito das primeiras correções radiativas (1-laço) associadas ao propagador do fóton, propagador do méson, vértice e, estudamos a função de 4 pontos (fóton-fóton) utilizando o método de regularização dimensional, em que a simetria de calibre é manifesta. Como veremos, uma consequência do estudo é que a álgebra de DKP assegura o funcionamento das identidades de WT nas primeiras correções radiativas proibindo certas divergências no ultravioleta. Com o conhecimento das divergências no ultravioleta (UV) e no infravermelho (IV) abordadas nas correções radiativas, estabelecemos o Programa de Renormalização multiplicativo para esta teoria na camada de massa. O fato do propagador do campo escalar possuir uma nova estrutura divergente na massa de Podolsky nos levou a analisar as correções radiativas a 2-laços. Do propagador do fóton definimos o tensor de polarização e com este, de maneira fenomenológica, analisando o comportamento assintótico das funções de Green para altos momentos, abordamos a dependência da constante de estrutura com a escala de energia. No segundo setor estudamos o Formalismo de Matsubara-Fradkin (MF) para descrever campos em equilíbrio termodinâmico. Para isso foi necessário construir as equações em equilíbrio termodinâmico que descrevessem o setor escalar e vetorial e a posteriori extrair a função de partição. Ao construir o setor vetorial, percebemos o surgimento e a importância dos campos fantasmas e sua conexão com a simetria de Bechi-Rouet-Stora-Tyutin (BRST). No caso da escolha de calibre covariante no-mixing, foi necessário contornar o surgimento de uma estrutura pseudo-diferencial. Analisando a função de partição associada aos fótons livres de Podolsky via método dos parâmetros fictícios, percebemos o fato da simetria BRST assegurar que a função de partição não depende das escolhas covariantes ao fixarmos o calibre. As condições de Lorenz, no-mixing e Lorenz generalizado são amarradas pela simetria BRST e esse fato está contido em uma afirmação geral em teorias de calibre a temperatura finita, atribuída ao trabalho de Tyutin, de que a física não depende das escolhas de calibre, covariantes ou não, devido a simetria BRST. Por fim, com a funções de partição em mãos, construímos as equações de Schwinger-Dyson-Fradkin (SDF) e as identidades de Ward-Takahashi-Fradkin (WTF) em equilíbrio termodinâmico. / This work has as aim to explore the quantum dynamics of interaction between scalar and vectorial particles and to study the thermodynamic equilibrium of these particles in the gran-canonical ensemble. The dynamics of interaction, written in a covariance language, between the matter field (scalar) and the field that intermediate the interaction (vectorial) exhibit a local gauge symmetry, U(1) in a quantum case and SO(4) in a thermodynamic equilibrium. Therefore we divided the work into two sections. In the first section we analyze systematically the quantum interaction between the scalar particles (mesons) and vectorial particles (photons) in the context of the generalized scalar Duffin-Kemmer-Petiau quantum electrodynamics (GSDKP). For this we use the functional approach to quantize the theory. We built the hamiltonian structure by the Dirac methodology, utilize the Faddeev-Senjanovic procedure to obtain the transition amplitude in the generalized Coulomb gauge and the Faadeev-Popov-DeWitt method to write the covariant form of the previously amplitude in the no-mixing gauge condition. Then writing the functional generator by Schwinger, the Schwinger-Dyson (SD) equations and the Ward-Takahashi (WT) identities are obtained. As an introductory analysis to the first radiative corrections we make a quantitative calculus to see the types of ultraviolet (UV) superficial divergences that appear in the theory. After this we show an explicit calculation of the first radiative corrections (1-loop) associated with the photon propagator, meson propagator, vertex and the 4 point function (photon-photon) utilizing the dimensional regularization method, where the gauge symmetry is manifest. As we will see one of the consequences of the study is that the DKP algebra ensures the functioning of the WT identities in the first radiative corrections prohibiting certain UV divergences. With the knowledge of the UV divergences and de infrared (IR) addressed in the radiative corrections we established the multiplicative renormalization procedure to this theory in the mass shell. The fact that the meson propagator has a new divergence structure in terms of the Podolsky mass took us to analyze the radiative correction at 2-loops. With the photon propagator we define the polarization tensor and in a phenomenological manner, analyzing the asymptotic behavior of Green's functions for higher momentum, we derive the dependence of the structure constant by the scale of energy. In the second section we study the Matsubara-Fradkin (MF) formalism to describe fields in thermodynamical equilibrium. For this it was necessary to construct the equations in thermodynamic equilibrium that describe the scalar sector and vectorial sector and then extract the partition function. When we construct the vectorial sector we realize the emergence and the importance of the ghost fields and their connection to the Bechi-Rouet-Stora-Tyutin (BRST) symmetry. In the case of the no-mixing gauge condition was necessary to contour a pseudo-differential structure. Analyzing the free partition function associated with the free Podolsky photons by the method of fictitious parameters we realize that the BRST symmetry ensures that it does not depend of the covariant choices when we fix de gauge. The Lorenz condition, no-mixing and generalized Lorenz are tied by the BRST symmetry and this fact is contained in a general statement in gauge theories at finite temperature, assigned by Tyutin work, that the physics doesn't depend of the gauge choices, covariant or not, due to BRST symmetry. Lastly, with the partition function in hands, we construct the Schwinger-Dyson-Fradkin (SDF) and the Ward-Takahashi-Fradkin (WTF) in thermodynamic equilibrium.

Orientador: Paulo Ricardo da Silva / Banca: João Carlos da Rocha Medrado / Banca: João Carlos Ferreira Costa / Resumo: Neste trabalho, estudamos as famíılias de sistemas forçados com superfície de impasse regular, as formas normais de seus pontos "típicos"bem como seus retratos de fase. Vemos ainda alguns resultados sobre a genericidade desses pontos e a estabilidade estrutural de um sistema forçado. Abordamos o tema de uma forma simples: apresentamos o que é um sistema forçado e uma família de sistemas forçados para depois estudar as formas normais de seus pontos "típicos" através de dois campos de direções, os quais se tornam fundamentais para o assunto. Utilizamos o Teorema de Peixoto (adaptado para este assunto) como norte para dar as características de um sistema forçado estruturalmente estável. No capítulo 3, damos a estratificação da superfície de impasse e, como resultado final, vemos que esta estratificação é genérica (no conjunto de todas as famílias de sistemas forçados). / Abstract:In this work we study the families of constrained systems with regular impasse surface, the normal forms of its "typical"points and the respectively phase portrait. We see some results about the genericity of these points and the structural stability of a constrained system. We broach the theme in a simple way: we introduce a constrained system and a family of a constrained systems, and so, we study the normal forms of its "typical"points through two line fields, which become essential for the subject. We use the Peixoto's Theorem (adapted for this subject) to characterize a structural stable of constrained systems. In the chapter 3, we make a stratification of the impasse surface and, as a last result, we see that stratification is genericity (in the set of all families of constrained systems). / Mestre

The overarching theme of this thesis is the question of how constraints influence collective behavior. Constraints are crucial in shaping both static and dynamic properties of systems across diverse areas within condensed matter physics and beyond. For example, the simple geometric constraint that hard particles cannot overlap at high density leads to slow dynamics and jamming in glass formers. Constraints also arise effectively at low temperature as a consequence of strong competing interactions in magnetic materials, where they give rise to emergent gauge theories and unconventional magnetic order. Enforcing constraints artificially in turn can be used to protect otherwise fragile quantum information from external noise. This thesis in particular contains progress on the realization of different unconventional phases of matter in constrained systems. The presentation of individual results is organized by the stage of realization of the respective phase. Novel physical phenomena after conceptualization are often exemplified in simple, heuristic models bearing little resemblance of actual matter, but which are interesting enough to motivate efforts with the final goal of realizing them in some way in the lab. One form of progress is then to devise refined models, which retain a degree of simplification while still realizing the same physics and improving the degree of realism in some direction. Finally, direct efforts in realizing either the original models or some refined version in experiment today are mostly two-fold. One route, having grown in importance rapidly during the last two decades, is via the engineering of artificial systems realizing suitable models. The other, more conventional way is to search for realizations of novel phases in materials. The thesis is divided into three parts, where Part I is devoted to the study of two simple models, while artificial systems and real materials are the subject of Part II and Part III respectively. Below, the content of each part is summarized in more detail. After a general introduction to entropic ordering and slow dynamics we present a family of models devised as a lattice analog of hard spheres. These are often studied to explore whether low-dimensional analogues of mean-field glass- and jamming transitions exist, but also serve as the canonical model systems for slow dynamics in granular materials more generally. Arguably the models in this family do not offer a close resemblance of actual granular materials. However, by studying their behavior far from equilibrium, we observe the onset of slow dynamics and a kinetic arrest for which, importantly, we obtain an essentially complete analytical and numerical understanding. Particularly interesting is the fact that this understanding hinges on the (in-)ability to anneal topological defects in the presence of a hardcore constraints, which resonates with some previous proposals for an understanding of the glass transition. As another example of anomalous dynamics arising in a magnetic system, we also present a detailed study of a two-dimensional fracton spin liquid. The model is an Ising system with an energy function designed to give rise to an emergent higher-rank gauge theory at low energy. We show explicitly that the number of zero-energy states in the model scales exponentially with the system size, establishing a finite residual entropy. A purpose-built cluster Monte-Carlo algorithm makes it possible to study the behavior of the model as a function of temperature. We show evidence for a first order transition from a high-temperature paramagnet to a low-temperature phase where correlations match predictions of a higher-rank coulomb phase. Turning away from heuristic models, the second part of the thesis begins with an introduction to quantum error correction, a scheme where constraints are artificially imposed in a quantum system through measurement and feedback. This is done in order to preserve quantum information in the presence of external noise, and is widely believed to be necessary in order to one day harness the full power of quantum computers. Given a certain error-correcting code as well as a noise model, a particularly interesting quantity is the threshold of the code, that is the critical amount of external noise below which quantum error correction becomes possible. For the toric code under independent bit- and phase-flip noise for example, the threshold is well known to map to the paramagnet to ferromagnet transition of the two-dimensional random-bond Ising model along the Nishimori line. Here, we present the first generalization of this mapping to a family of codes with finite rate, that is a family where the number of encoded logical qubits grows linearly with the number of physical qubits. In particular, we show that the threshold of hyperbolic surface codes maps to a paramagnet to ferromagnet transition in what we call the 'dual'' random-bond Ising model on regular tessellations of compact hyperbolic manifolds. This model is related to the usual random-bond Ising model by the Kramers-Wannier duality but distinct from it even on self-dual tessellations. As a corollary, we clarify long-standing issues regarding self-duality of the Ising model in hyperbolic space. The final part of the thesis is devoted to the study of material candidates of quantum spin ice, a three-dimensional quantum spin liquid. The work presented here was done in close collaboration with experiment and focuses on a particular family of materials called dipolar-octupolar pyrochlores. This family of materials is particularly interesting because they might realize novel exotic quantum states such as octupolar spin liquids, while at the same time being described by a relatively simple model Hamiltonian. This thesis contains a detailed study of ground state selection in dipolar-octupolar pyrochlore magnets and its signatures as observable in neutron scattering. First, we present evidence that the two compounds Ce2Zr2O7 and Ce2Sn2O7 despite their similar chemical composition realize an exotic quantum spin liquid state and an ordered state respectively. Then, we also study the ground-state selection in dipolar-octupolar pyrochlores in a magnetic field. Most importantly, we show that the well-known effective one-dimensional physics -- arising when the field is applied along a certain crystallographic axis -- is expected to be stable at experimentally relevant temperatures. Finally, we make predictions for neutron scattering in the large-field phase and compare these to measurements on Ce2Zr2O7.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-27T14:37:13Z No. of bitstreams: 1 danieloliveiradesouza.pdf: 460696 bytes, checksum: b38f7b2e07b00443eb2e4bf937e1f462 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T20:53:20Z (GMT) No. of bitstreams: 1 danieloliveiradesouza.pdf: 460696 bytes, checksum: b38f7b2e07b00443eb2e4bf937e1f462 (MD5) / Made available in DSpace on 2017-08-07T20:53:20Z (GMT). No. of bitstreams: 1 danieloliveiradesouza.pdf: 460696 bytes, checksum: b38f7b2e07b00443eb2e4bf937e1f462 (MD5) Previous issue date: 2010-08-24 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / As quatro forças fundamentais da natureza conhecidas são: a força gravitacional, a eletromagnética, a força fraca e a força forte. As três últimas são regidas por aspectos da chamada teoria quântica de campos. O grande interesse da teoria de cordas consiste no fato de que acredita-se que a mesma pode unificar a força gravitacional com a mecância quântica. Sendo portanto, uma candidata a formar a famigerada teoria da uni cação dessas forças fundamentais. Com essa motivação em mente, nessa dissertação abordamos as ações de cordas de Nambu-Goto, Born-Infeld-Nambu-Goto e Dirac-Born-Infeld-Nambu-Goto de cordas bosônicas como sistemas vinculados. O nosso principal caminho para estudar esse assunto fascinante da física teórica é o conhecido método de Dirac para a análise de vínculos. / The four fundamental forces of nature are known: the force gravitational, the electromagnetic, the weak force and strong force. The three latter aspects are governed by the so-called quantum field theory. The great interest in string theory is the fact that it is believed that it can unify the gravitational force with quantum mechanics. Being thus a candidate to form the notorious theory of uni cation these fundamental forces. With this motivation in mind, this dissertation discusses the actions of the Nambu-Goto strings, Born-Infeld-Nambu-Goto and Dirac-Born-Infeld-Nambu-Goto bosonic string as bound systems. Our main way to study this fascinating subject in physics is the known theoretical method for analysis of Dirac links.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-28T14:31:27Z No. of bitstreams: 1 lucianamirandavieiraxavier.pdf: 351536 bytes, checksum: f476ea35c3ced7fc7f8315f85d076f6e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T21:18:56Z (GMT) No. of bitstreams: 1 lucianamirandavieiraxavier.pdf: 351536 bytes, checksum: f476ea35c3ced7fc7f8315f85d076f6e (MD5) / Made available in DSpace on 2017-08-07T21:18:56Z (GMT). No. of bitstreams: 1 lucianamirandavieiraxavier.pdf: 351536 bytes, checksum: f476ea35c3ced7fc7f8315f85d076f6e (MD5) Previous issue date: 2009-02-27 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / Nesta tese, revisa-se os principais métodos de quantização de sistemas vinculados a partir das técnicas Hamiltoniana de Dirac e Lagrangeana de Faddev-Jackiw ( sem vínculos) e sua extenção a de Barcelos Neto- Wotzasek (com vínculos), estes denominados simplesmente por Formalismo Simplético (FS). Em vista da correspondência entre os formalismos, eles serão aplicados ao Modelo de Skyrme SU(2) e ao Eletromagnetimo de Maxwell. Apresenta-se uma técnica contemporânea, que mergulha uma teoria de segunda classe em uma dual com invariância de calibre, a saber, o Formalismo Simplético de Imersão (FSI). Esse método baseia-se no FS e estende-se o espaço de configuração por meio das variáveis de Wess-Zumino. Para ilustrar esse FSI, constroi-se a eletrodinâmica de Maxwell como uma teoria de calibre, na qual as divergências clássicas não estejam presentes. Uma generalização relativística é a eletrodinâmica de Proca e de Chern-Simons, que consideram a possibilidade de existência de um fóton massivo e de um campo com alcance finito. A descrição dual reproduz o mesmo resultado encontrado na literatura através de outros métodos. Apesar da arbitrariedade dos geradores da simetria de calibre, os modos-zeros, mostram uma família de representações dinâmicas duais para o sistema em questão. / In this thesis, it will be revised the main quantization methods of constrained systems using the Dirac Hamiltonian method and Faddev-Jackiw Lagrangian techniques (without constrained), and its extension to the Barcelos Neto- Wotzasek Lagrangian method (with constrained), these known as Symplectic Formalism. Because of the correspondence among the formalisms, they will be applied of the Skyrme SU(2) model and Electromagnetism of Maxwell. It will be presented a contemporary technique that it embed a second class theory in a dual with gauge invariance, the Embedding Symplectic Formalism . This method is based on the Symplectic Formalism, it is extended the configuration space through Wess-Zumino variables. In order to illustrate this Embedding Symplectic Formalism, the Maxwell electrodynamics is built as a gauge theory, without the classic differences. A relativistic generalization is the Proca and Chern-Simons electrodynamics that consider the possibility of existence of a massive photon and a field with finite reach. The dual description reproduce the identic result reported in the literature using other methods. Although, the arbitrariness of the gauge symmetry generator, zero-mode, it reveals a family of dynamic dual representations to this system.