O objetivo principal deste trabalho é apresentar uma exposição detalhada do Teorema de Poincaré-Andronov-Hopf para uma família de transformações do plano. Apresentaremos também uma aplicação a um sistema dinâmico que modela a evolução do preço e excesso de demanda em um mercado constituído por uma única mercadoria. / The main purpose of this work is to present a detailed exposition of the Poincaré-Andronov-Hopf Theorem for a family of transformations in the plane. We also present an application to a dynamical system modelling the evolution of the price and the excess demand in a single asset market.

O objetivo principal deste trabalho é apresentar uma exposição detalhada do Teorema de Poincaré-Andronov-Hopf para uma família de transformações do plano. Apresentaremos também uma aplicação a um sistema dinâmico que modela a evolução do preço e excesso de demanda em um mercado constituído por uma única mercadoria. / The main purpose of this work is to present a detailed exposition of the Poincaré-Andronov-Hopf Theorem for a family of transformations in the plane. We also present an application to a dynamical system modelling the evolution of the price and the excess demand in a single asset market.

Nodal semimetals host either degenerate points (Dirac/Weyl points) or lines whose band topology in Brillouin zone can be classified either as trivial (normal nodal semimetals) or non trivial (topological nodal semimetals). This thesis investigates the electronic structure of two different categories of topological nodal semimetals probed by angleresolved photoemission spectroscopy (ARPES): The first material is Indium Bismuth (InBi). InBi is a semimetal with simple tetragonal structure with P4/nmm space group. This space group is predicted to host protected nodal lines along the perpendicular momentum direction at the high symmetry lines of the Brillouin zone boundary even under strong spin-orbit coupling (SOC) situation. As a semimetal with two heavy elements, InBi is a suitable candidate to test the prediction. The investigation by ARPES demonstrates not only that InBi hosts the nodal line in the presence of strong SOC, it also shows the signature of type-II Dirac crossing along the perpendicular momentum direction from the center of Brillouin zone. However, as the nodal line observed is trivial in nature, there is no exotic drumhead surface states observed in this material. This finding demonstrates that Dirac crossings can be protected in a non-symmorphic space group. The second material is NbIrTe₄ which is a semimetal that breaks inversion symmetry predicted to host only four Weyl points. This simplest configuration is confirmed by the measurement from the top and bottom surface of NbIrTe₄ showing only a pair of Fermi arcs each. Furthermore, it is found that the Fermi arc connectivity on the bottom surface experiences re-wiring as it evolves from Weyl points energy to the ARPES Fermi energy level. This change is attributed to the hybridisation between the surface and the bulk states as their projection lie within the vicinity of each other. The finding in this work demonstrates that although Fermi arcs are guaranteed in Weyl semimetals, their shape and connectivity are not protected and may be altered accordingly.

The general problem of multi-agent persistent monitoring finds applications in a variety of domains ranging from meter to kilometer-scale systems, such as surveillance or environmental monitoring, down to nano-scale systems such as tracking biological macromolecules for studying basic biology and disease. The problem can be cast as moving the agents between targets, acquiring information from or in some fashion controlling the states of the targets. Under this formulation, at least two questions need to be addressed. The first is the design of motion trajectories for the agents as they move among the spatially distributed targets and jointly optimize a given cost function that describes some desired application. The second is the design of the controller that an agent will use at a target to steer the target's state as desired. The first question can be viewed in at least two ways: first, as an optimal control problem with the domain of the targets described as a continuous space, and second as a discrete scheduling task. In this work we focus on the second approach, which formulates the target dynamics as a hybrid automaton, and the geometry of the targets as a graph. We show how to find solutions by translating the scheduling problem into a search for the optimal route. With a route specifying the visiting sequence in place, we derive the optimal time the agent spends at each target analytically. The second question, namely that of steering the target's state, can be formulated from the perspective of the target, rather than the agent. The mobile nature of the agents leads to intermittencontrol, such that the controller is assumed to be disconnected when no agent is at the target. The design of the visiting schedule of agents to one target can affect the reachability (controllability) of this target's control system and the design of any specific controller. Existing test techniques for reachability are combined with the idea of lifting to provide conditions on systems such that reachability is maintained in the presence of periodic disconnections from the controller. While considering an intermittently connected control with constraints on the control authority and in the presence of a disturbance, the concept of 'degree of controllability' is introduced. The degree is measured by a region of states that can be brought back to the origin in a given finite time. The size of this region is estimated to evaluate the performance of a given sequence.

Neste trabalho estudamos medidas invariantes para aplicações unimodais. Estamos especialmente interessados em detectar as situações que levam uma aplicação unimodal a não possuir uma medida piac, ou seja, uma medida de probabilidade invariante e absolutamente contínua em relação à medida de Lebesgue. Mostramos que a ordem do ponto crítico e a sua capacidade de recorrência são os fatores mais relevantes nesta questão. Os valores das derivadas da aplicação nos pontos periódicos tem uma infuência menor, mas suficiente para garantir que numa mesma classe de conjuga ção topológica podem existir duas aplicações unimodais com ponto crítico de mesma ordem, sendo que uma delas possui medida piac e a outra não possui. A capacidade de recorrência do ponto crítico, talvez o principal fator nesta questão, depende de aspectos combinatórios bem sofisticados. As ferramentas principais para analisar estes aspectos envolvem os conceitos de tempos de corte e de aplicações kneading. A existência ou não de medidas piac é uma propriedade de natureza métrica, e por isto, é necessário que tenhamos controle de como os iterados da aplicação unimodal distorcem a medida de Lebesgue. Então precisamos usar ferramentas de controle de distorção que incluem principalmente os Princípios de Koebe. Um ponto culminante deste trabalho diz respeito a relação entre existência de mediada piac e existência de atratores selvagens, isto é: atratores métricos que não são atratores topológicos e vice versa. Usamos aqui um argumento probabilístico de rara beleza. / In this work we study invariant measures for unimodal maps. We are especially interested in detecting situations that cause a unimodal map not to have a piac measure, i.e., a measure that is Probability Invariant and Absolutely Continuous with respect to Lebesgue measure. We show that the order of the critical point and its capacity for recurrence are the most relevant factors in this matter. The values of the derivatives of the map at periodic points have a small inuence, but enough to ensure that within a single class of topological conjugacy, there can be two unimodal maps with critical points of the same order, one of which has a piac measure and the other does not. The recurrence capacity of the critical point depends on very sophisticated combinatorial aspects and is probably the main factor in this issue. The main tools to analyze these aspects involve the concepts of cutting times and kneading maps. The existence of piac measures is a property of metric nature, and for this reason we need to have control of how iterations of the unimodal map distort the Lebesgue measure. We therefore need to use distortion control tools, including especially the Principles of Koebe. A culmination of this work concerns the relationship between existence of piac measures and the existence of wild attractors, i.e., metric attractors that not are topological attractors. Here we use a probabilistic argument of rare beauty.

The main topic of this Thesis is a new simplified proof of the Sunyach theorem that provides suffici- ent conditions for existence and uniqueness of an invariant measure for a Markov kernel on a complete separable metric space equipped with its Borel σ-algebra. Weak convergence of measures following from Sunyach's theorem is strengthened to convergence in the total variation norm provided that the Markov kernel is strong Feller. Furthermore, sufficient conditions for geometric ergodicity are stated. Another topic treated is the strong Feller property: its characterization by absolute measurability and uniform integrability and derivation of some other sufficient conditions.

The recent popularity of Bayesian inference is due to the practical advantages of the Bayesian approach. The Bayesian analysis makes it possible to reflect ones prior beliefs into the analysis. In this thesis, we explore some asymptotic results in Bayesian nonparametric inference for restricted and unrestricted space of distributions. This thesis is divided into two parts. In the first part, we employ the Dirichlet process in a hypothesis testing framework to propose a Bayesian nonparametric chi-squared goodness-of-fit test. Our suggested method corresponds to Lo's Bayesian bootstrap procedure for chi-squared goodness-of-fit test. Indeed, our bootstrap rectifies some shortcomings of regular bootstrap which only counts number of observations falling in each bin in contingency tables. We consider the Dirichlet process as the prior for the distribution of data and carry out the test based on the Kullback-Leibler distance between the Dirichlet process posterior and the hypothesized distribution. We prove that this distance asymptotically converges to the same chi-squared distribution as the classical frequentist's chi-squared test. Moreover, the results are generalized to the chi-squared test of independence for contingency tables. In the second part, our main focus is on Bayesian nonparametric inference for a restricted group of distributions called spherically symmetric distributions. We describe a Bayesian nonparametric approach to perform an inference for a bivariate spherically symmetric distribution. We place a Dirichlet invariant process prior on the set of all bivariate spherically symmetric distributions and derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of the Dirichlet invariant process for the symmetric distributions on the real line to bivariate spherically symmetric distribution where the underlying distribution is invariant under a finite group of rotations. Further, we obtain the Dirichlet invariant process posterior for the infinite transformation group and we prove that it approaches a certain Dirichlet process. Finally, we develop our approach to obtain the Bayesian nonparametric posterior distribution for functionals of the distribution's support when the support satisfies certain symmetry conditions. When symmetry holds with respect to the parallel lines of axes (for example, in two dimensional space x = a and y = b) we employ our approach to approximate the distribution of certain functionals such as area and perimeter for the support of the distribution. This suggests a Bayesian nonparametric bootstrapping scheme. The estimates can be derived based on posterior averaging. Then, our simulation results demonstrate that our suggested bootstrapping technique improves the accuracy of the estimates.

In the survival and reliability data analysis, parametric and nonparametric methods are used to estimate the hazard/risk rate and survival functions. A parametric approach is based on the assumption that the underlying survival distribution belongs to some specific family of closed form distributions (normal, Weibull, exponential, etc.). On the other hand, a nonparametric approach is centered around the best-fitting member of a class of survival distribution functions. Moreover, the Kaplan-Meier and Nelson-Aalen type nonparametric approach do not assume either distribution class or closed-form distributions. Historically, well-known time-to-event processes are death of living specie in populations and failure of component in engineering systems. Recently, the human mobility, electronic communications, technological changes, advancements in engineering, medical, and social sciences have further diversified the role and scope of time-to-event processes in cultural, epidemiological, financial, military, and social sciences. To incorporate extensions, generalizations and minimize scope of existing methods, we initiate an innovative alternative modeling approach for time-to-event dynamic processes. The innovative approach is composed of the following basic components: (1) development of continuous-time state of dynamic process, (2) introduction of discrete-time dynamic intervention process, (3) formulation of continuous and discrete-time interconnected dynamic system, (4) utilizing Euler-type discretized schemes, developing theoretical dynamic algorithms, and (5) introduction of conceptual and computational state and parameter estimation procedures. The presented approach is motivated by state and parameter estimation of time-to-event processes in biological, chemical, engineering, epidemiological, medical, military, multiple-markets and social dynamic processes under the influence of discrete-time intervention processes. We initiate (1) a time-to-event process to be a probabilistic dynamic process with unitary state. Action, normal, operational, radical, survival, susceptible, etc. and its complementary states, reaction, abnormal, nonoperational, non-radical, failure, infective and so on (quantitative and qualitative variables), are considered to be illustrations of a unitary state of time-to-event dynamic processes. A unitary state is measured by a probability distribution function. Employing Newtonian dynamic modeling approach and observing the definition of hazard rate as a specific rate, survival or failure probabilistic state dynamic model is developed. This dynamic model is further extended to incorporate internal or external discrete-time dynamic intervention processes acting on unitary state time-to-event processes (2). This further demanded a formulation and development of an interconnected continuous-discrete-time hybrid, and totally discrete-time dynamic models for time-to-event processes (3). Employing the developed hybrid model, Euler-type discretized schemes, a very general fundamental conceptual analytic algorithm is outlined (4). Using the developed theoretical computational procedure in (4), a general conceptual computational data organizational and simulation schemes are presented (5) for state and parameter estimation problems in unitary state time-to-event dynamic processes. The well-known theoretical existing results in the literature are exhibited as special cases in a systematic and unified manner (6). In fact, the Kaplan-Meier and Nelson-Aalen type nonparametric estimation approaches are systematically analyzed by the developed totally discrete-time hybrid dynamic modeling process. The developed approach is applied to two data sets. Moreover, this approach does not require a knowledge of either a closed-form solution distribution or a class of distributions functions. A hazard rate need not be constant. The procedure is dynamic. In the existing literature, the failure and survival distribution functions are treated to be evolving/progressing mutually exclusively with respect to corresponding to two mutually exclusive time varying events. We refer to these two functions (failure and survival) as cumulative distributions of two mutually disjoint state output processes with respect to two mutually exclusive time-varying complementary unitary states of a time-to-event processes in any discipline of interest (7). This kind of time-to-event process can be thought of as a Bernoulli-type of deterministic/stochastic process. Corresponding to these two complementary output processes of the Bernoulli-type of stochastic process, we associate two unitary dynamic states corresponding to a binary choice options/actions (8), namely, ({action, reaction}, {normal, abnormal}, {survival, failure}, {susceptible, infective}, {operational, nonoperational}, {radical, non-radical}, and so on.) Under this consideration, we extend unitary state time-to-event dynamic model to binary state time-to-event dynamic model. Using basic tools in mathematical sciences, we initiate a Newtonian-type dynamic approach for binary state time-to-event processes in the sciences, technologies, and engineering (9). Introducing an innovative concept of “survival state dynamic principle”, an innovative interconnected nonlinear non-stationary large-scale hybrid dynamic model for number of units/species and its unitary survival state corresponding to binary state time-to-event process is formulated (10). The developed model in (10) includes dynamic model (3) as a special case. The developed approach is directly applicable to binary state time-to-event dynamic processes in biological, chemical, engineering, financial, medical, physical, military, and social sciences in a coherent manner. A by-product of this is a transformed interconnected nonlinear hybrid dynamic model with a theoretical discrete-time conceptual computational dynamic process (11). Employing the transformed discrete-time conceptual computational dynamic process, we introduce notions of data coordination, state data decomposition and aggregation, theoretical conceptual iterative processes, conceptual and computational parameter estimation and simulation schemes, conceptual and computational state simulation schemes in a systematic way (12). The usefulness of the developed interconnected algorithm is validated by using three real world data sets (13). We note that the presented algorithm does not need a closed-form representation of distribution/likelihood function. In fact, it is free from any required assumptions of the “Classical Maximum Likelihood Function Approach” in the “Survival and Reliability Analysis.” The rapid electronic communication and human mobility processes have facilitated to transform information, knowledge, and ideas almost instantly around the globe. This indeed generates heterogeneity, and it causes to form nonlinear and non-stationary dynamic processes. Moreover, the heterogeneity, non-linearity, non-stationarity, further generates two types of uncertainties, namely, deterministic, and stochastic. In view of this, it is obvious that nothing is deterministic. In short, the 21st century problems are highly nonlinear, non-stationary and under the influence of internal and external random perturbations. Using tools in stochastic analysis, interconnected deterministic models in (3) and (10) are extended to interconnected stochastic hybrid dynamic model for binary state time-to-event processes (14). The developed model is described by a large-scale nonlinear and non-stationary stochastic differential equations. Moreover, a stochastic version of a survival function is also introduced (15). Analytical, computational, statistical, and simulation algorithms/procedures are also extended and analyzed in a systematic and unified way (16). The presented interconnected stochastic model is motivated to initiate conceptual computational parameter and state estimation schemes for time-to-event statistical data (17). Again, stochastic version of computational algorithms are validated in the context of three real world data sets. The obtained parameter and state estimates show that the algorithm is independent of the choice of nonlinear transformation (18). Utilizing the developed alternative innovative procedure and the recently modified deterministic version of Local Lagged Adapted Generalized Method of Moments (LLGMM) is also extended to stochastic version in a natural way (19). This approach provides a degree of measure of confidence, prediction, and planning assessments (20). In addition, it initiates a conceptual computational parameter and state estimation and simulation schemes that is suitable for the usage of mean square sub-optimal procedure (21). The usefulness and the significance of the approach is illustrated by applying to three data sets (22). The approach provides insight for investigating various type of invariant sets, namely, sustainable/unsustainable, survival/failure, reliable/unreliable (23), and qualitative properties such as sustainability versus unsustainability, reliability versus unreliability, etc. (24) Once again, the presented algorithm is independent of any form of survival distribution functions or data sets. Moreover, it does not require a closed form survival function distribution. We also note that the introduction of intervention processes provides a measure of influence and confidence for the usage of new tools/procedures/approaches in continuous-time binary state time-to-event dynamic process (25). Moreover, the presented dynamic modeling is more feasible for its usage of investigating a more complex time-to-event dynamic process (26). The developed procedure is dynamic and indeed non-parametric (27). The dynamic approach adapts with current changes and updates statistic process (28). The dynamic nature is natural rather than the existing static and single-shot techniques (29).

Estudaremos uma fam lia de sistemas hamiltonianos no R 4 , H : R 4 R, satisfazendo certas condi c oes, dependendo de um parametro . Iremos ca- racterizar algumas condi c oes sobre n veis de energia desse sistema, que nos permitem concluir existencia e destrui c ao de toros invariantes, em tais n veis de energia. Al em disso, podemos concluir que o fluxo hamiltoniano, restrito a esses n veis de energia, possui entropia topol ogica positiva. / We will study a family of Hamiltonian Systems in R 4 , satisfying certain conditions, H : R 4 R, depending of a parameter . We will characterize some conditions about the energy levels of this system, which allow us to conclude existence and destruction of invariant torus, at such energy levels. Moreover, we can conclude that the hamiltonian flow, restricted to these energy level, has positive topological entropy.

In many engineering applications, including radar, sonar, communications and seismology, the direction of impinging signal wavefronts can be used to discriminate between competing sources. Often these source signals cover a wide bandwidth and conventional narrowband beamforming techniques are ineffective, since spatial resolution varies significantly across the band. In this thesis we consider the problem of beamforming for broadband signals, primarily when the spatial response remains constant as a function of frequency. This is called a frequency invariant beamformer (FIB).¶ Rather than applying the numerical technique of multi-parameter optimisation to solve for the beamformer parameters, we attempt to address the fundamental nature of the FIB problem. The general philosophy is to use a theoretical continuous sensor to derive relationships between a desired FI beampattern and the required signal processing structure. Beamforming using an array of discrete sensors can then be formulated as an approximation problem. This approach reveals a natural structure to the FIB which is otherwise buried in a numerical optimisation procedure.¶ Measured results from a microphone array are presented to verify that the simple FIB structure can be successfully implemented. We then consider imposing broadband pattern nulls in the FI beampattern, and show that (i) it is possible to impose an exact null which is present over all frequencies, and (ii) it is possible to calculate a priori how many constraints are required to achieve a null of a given depth in a FIB. We also show that the FIB can be applied to the problem of broadband direction of arrival (DOA) estimation and provides computational advantages over other broadband DOA estimators.¶ Through the theoretical continuous sensor approach, we show that the FIB theory can be generalised to the problem of designing a general broadband beamformer (GBB) which realizes a broadband angle-versus-frequency beampattern specification. Coupled with a technique for radial beampattern transformation, the GBB can be applied to a wide class of problems covering both nearfield beamforming (in which the shape of the impinging wavefront must be considered and farfield beamforming (which is simplified by the assumption of planar wavefronts) for a broadband beampattern specified over both angle and frequency.