Avian influenza is a contagious viral disease caused by influenza virus type A. Avian influenza can be disastrous (if it occurs), due to the short incubation period (about 1--4 days). Thus it is important to study this disease so that we are more prepared to manage it in the future. A classical system of differential equations (the half-saturated incidence model) and three Filippov models --- an avian-only model with culling of infected birds, an SIIR (Susceptible-Infected-Infected-Recovered) model with quarantine of infected humans and an avian-only model with culling both susceptible and infected birds --- that are governed by ordinary differential equations with discontinuous right-hand sides (i.e., differential inclusion) are proposed to study the transmission of avian influenza. The effect of half-saturated incidence is investigated, and the outcome of this model is compared with the bilinear incidence model. Both models remain endemic whenever their respective basic reproduction numbers are greater than one. The half-saturated incidence model generates more infected individuals than the bilinear incidence model. This may be because the bilinear incidence model is underestimating the number of infected individuals at the outbreak. For the Filippov models, the number of infected individuals is used as a reference in applying control strategies. If this number is greater than a threshold value, a control measure has to be employed immediately to avoid a more severe outbreak. Otherwise, no action is necessary. We perform dynamical system analysis for all models. The existence of sliding modes and the flow on the discontinuity surfaces are determined. In addition, numerical simulations are conducted to illustrate the dynamics of the models. Our results suggest that if appropriate tolerance thresholds are chosen such that all trajectories of the Filippov models are converging to an equilibrium point that lies in the region below the infected tolerance threshold or on the discontinuity surface, then no control strategy is necessary as we consider the outbreak is tolerable. Otherwise, we have to apply control strategies to contain the outbreak. Hence a well-defined threshold policy is crucial for us to combat avian influenza effectively.

This work motivates the study of open dynamical systems corresponding to the doubling map. In particular, the dynamical properties of the attractor of the doubling map when a symmetric, centred open interval is removed are studied. Using the arithmetical properties of the binary expansion of the points on the boundary of the removed interval, we study properties such as topological transitivity, the specification property and intrinsic ergodicity. The properties of the function that associates to each hole $(a,b)$ the topological entropy of the attractor of the considered dynamical system are also shown. For these purposes, a subshift corresponding to an element of the lexicographic world is associated to each attractor and the mentioned properties are studied symbolically.

Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T09:35:17Z (GMT). No. of bitstreams: 1 AchireQuispe_JesusEnrique_D.pdf: 1199686 bytes, checksum: 2c263eb351ad3dfa30b13e0fecc5282b (MD5) Previous issue date: 2014 / Resumo: Recentemente, a Teoria de campos descontínuos (Non-Smooth Dynamic Systems) tem-se desenvolvido rapidamente, motivado principalmente pelas aplicações na física e nas engenharias, e também pela atraente beleza matemática. Neste trabalho, consideraremos campos de vetores suaves por partes, denominados campos de Filippov, e usamos o método convexo de Filippov para definir órbita solução deste tipo de campo. Assim, órbitas soluções passando por um ponto qualquer sempre existem. Há duas principais diferenças com o clássico caso diferenciável: a primeira é que as órbitas neste caso são curvas suaves por partes, enquanto que no caso diferenciável são curvas suaves. A segunda é que as órbitas soluções não tem a propriedade da unicidade, ou seja, podem existir duas ou mais órbitas passando pelo mesmo ponto. São esses fatos que fazem essa teoria um pouco diferente da teoria clássica de campos diferenciáveis. Estamos interessados em estudar qualitativamente os campos de Filippov, especialmente os que são genéricos e estruturalmente estáveis. Assim, nesta tese descrevemos propriedades genéricas necessárias para um campo de Filippov ser estruturalmente estável. Particularmente analisamos estabilidade estrutural local de singularidades tangenciais tais como o rabo de andorinha, a dobradobra,e dobra-cúspide, e adicionalmente pseudoequilíbrios e órbitas fechadas / Abstract: Recently, the Theory of Non-smooth Dynamic Systems has been developed, motivated mostly by their applications in physics and engineering, and also by its attractive mathematical beauty. In this work, we consider piecewise-smooth vector fields, called Filippov's vector fields, and we use the Filippov's convex method to define orbits solutions of this type of vector fields. Thus, orbit solution through any point always exists. But, there are two main differences with the classic differentiable case: the first is that orbits in this case are piecewise smooth curves while that in the differentiable case they are smooth curves. The second is that there is not uniqueness of solutions, this is, it may exist two or more than two orbits passing through a point. We are interested in to study qualitatively the Filippov's vector fields, especially those thatare generic and structurally stable. Thus, in this text we describe generic properties necessaryfor a vector field to be structurally stable. In particular, we analyze local structural stability attangential singularities, such as swallowtail-regular, fold-fold, fold-cusp, and additionally pseudoequilibriumsand closed orbits / Doutorado / Matematica / Doutor em Matemática

O Reconhecimento de faces consiste em, a partir de uma imagem, identificar ou verificar um ou mais indivíduos através de um banco de dados de faces. O reconhecimento de faces é uma tarefa de grande interesse, principalmente pelo grande número de possíveis aplicações. Dessa forma, existem diversos métodos para lidar com o problema. No entanto, apesar da maioria dos métodos conseguirem bons resultados em ambientes controlados, quando há variações de iluminação, pose ou expressão facial, esse desempenho é reduzido. Buscando lidar com as dificuldades existentes, este trabalho propõe um método para o reconhecimento de faces utilizando os conjuntos-K. Os conjuntos-K são modelos conexionistas baseados em populações de neurônios, concebidos através de estudos e análises do sistema olfativo animal. Estes modelos apresentam estrutura e comportamento biologicamente mais plausíveis que os modelos tradicionais de redes neurais. Os conjuntos-K vêm sendo usados em diversas tarefas de aprendizado de máquina, apresentando bons resultados principalmente na resolução de problemas complexos ou com ruídos. Devido ao grande potencial dos conjuntos-K para reconhecimento de padrões em ambientes complexos e ruidosos, é levantada a hipótese de que um método baseado nos conjuntos-K alcance um melhor desempenho que os métodos existentes na literatura. O método proposto foi avaliado utilizando dois bancos de dados, AT&T e Yale B, o primeiro com pequenas variações em relação a pose e expressão facial e o segundo com grandes variações de iluminação fornecendo um cenário mais complexo. Os resultados mostraram que o método proposto consegue um desempenho equivalente ou um pouco inferior que os outros métodos avaliados para o primeiro banco de dados. Porém, para o segundo banco de dados, que fornece o cenário mais complexo, o método proposto supera os demais métodos. / Face recognition consists of, from a picture, identifying or checking one or more individuals through a face database. Face recognition is an interesting task mainly because of the large number of possible applications. This way, there are various methods to deal with the problem. However, although most methods achieve good results in controlled environments, when there are lighting, pose or facial expression variations, this performance is reduced. Seeking to deal with the existing difficulties, this work proposes a method for recognizing faces using K-sets. The K-sets are connectionist models based on neuron populations, designed from studies and analyses of the animal olfactory system. These models present more biologically plausible structure and behavior than traditional neural network models. K-sets have been used in various machine learning tasks with good results, mainly in the resolution of complex or noisy problems. Due to the great potential of K-sets for pattern recognition in complex and noisy environments, a hypothesis is raised that a method based on K-sets achieves a better performance than existing methods. The proposed method was evaluated using two databases, AT&T and Yale B, the first with small variations of pose and facial expressions and the second with large variations in illumination providing a more complex scenario. The results show that the proposed method achieve an equivalent or slightly lower performance than the other methods evaluated for the first database. However, for the second database, which provides the more complex scenario, the proposed method outperforms the other methods.

There is a long history of interesting connections between topological dynamical systems and C*-algebras. Iterated function systems are an important topic in dynamics, but the diversity of these systems makes it challenging to develop an associated class of C*-algebras. Kajiwara and Watatani were the first to construct a C*-algebra from an iterated function system. They used an algebraic approach involving Cuntz-Pimsner algebras; however, when investigating properties such as ideal structure, they needed to assume that the functions in the system are the inverse branches of a continuous map. This excludes many famous examples, such as the standard functions used to construct the Siérpinski Gasket. In this thesis, we provide a construction of an inductive limit of étale equivalence relations for a broad class of affine iterated function systems, including the Siérpinski Gasket and its relatives, and consider the associated C*-algebras. This approach provides a more dynamical perspective, leading to interesting results that emphasize how properties of the dynamics appear in the C*-algebras. In particular, we show that the C*-algebras are isomorphic for conjugate systems, and find ideals related to the open set condition. In the case of the Siérpinski Gasket, we find explicit isomorphisms to subalgebras of the continuous functions from the attractor to a matrix algebra. Finally, we consider the K-theory of the inductive limit of these algebras. / Graduate

Orientador: Selene Maria Coelho Loibel / Banca: Maiko Fernandes Buzzi / Banca: Carmen Maria Andreazza / Resumo: Estimar os estados de um sistema é um problema que a cada dia assume maior importância devido ao grande interesse por conhecer com exatidão os resultados dados pelos sistemas dinâmicos em qualquer tempo. Principalmente nos casos onde o sistema é estocástico, o problema da estimação apresenta uma maior complexidade. É nesse contexto que os estudos que Kalman realizou no século XX, sobre a estimação de sistemas dinâmicos estocásticos, ganharam maior relevância. O ltro de Kalman foi o principal resultado desses estudos, pela e cácia demonstrada dentro desse campo de estudo. Este trabalho tem como eixo principal o ltro de Kalman e sua aplicação tendo importância como o melhor estimador para os estados de sistemas dinâmicos lineares estocásticos em tempo discreto / Abstract: Estimating the states of a system is a problem of great importance due to interest in knowing exactly the results given by dynamic systems at any time. Moreover, if the system is stochastic, what causes the estimation problem to have complexity. In this context, Kalman studies in the previous century on the estimation of stochastic dynamical systems, whose result is the lter, which, due to its e ciency, is the most used in this eld. In this work the main focus is the Kalman lter and its application having in view its importance as the best estimator for the states of linear dynamic stochastic systems of discrete time / Mestre

Nous établissons un théorème de convergence locale de l'algorithme classique d'optimisation de système dynamique RTRL, appliqué à un système non linéaire. L'algorithme RTRL est un algorithme en ligne, mais il doit maintenir une grande quantités d'informations, ce qui le rend impropre à entraîner des systèmes d'apprentissage de taille moyenne. L'algorithme NBT y remédie en maintenant une approximation aléatoire non biaisée de faible taille de ces informations. Nous prouvons également la convergence avec probabilité arbitrairement proche de un, de celui-ci vers l'optimum local atteint par l'algorithme RTRL. Nous formalisons également l'algorithme LLR et en effectuons une étude expérimentale, sur des données synthétiques. Cet algorithme met à jour de manière adaptive le pas d'une descente de gradient, par descente de gradient sur celui-ci. Il apporte ainsi une réponse partielle au problème de la fixation numérique du pas de descente, dont le choix influence fortement la procédure de descente et qui doit sinon faire l'objet d'une recherche empirique potentiellement longue par le praticien. / We prove a local convergence theorem for the classical dynamical system optimization algorithm called RTRL, in a nonlinear setting. The rtrl works on line, but maintains a huge amount of information, which makes it unfit to train even moderately big learning models. The NBT algorithm turns it by replacing these informations by a non-biased, low dimension, random approximation. We also prove the convergence with arbitrarily close to one probability, of this algorithm to the local optimum reached by the RTRL algorithm. We also formalize the LLR algorithm and conduct experiments on it, on synthetic data. This algorithm updates in an adaptive fashion the step size of a gradient descent, by conducting a gradient descent on this very step size. It therefore partially solves the issue of the numerical choice of a step size in a gradient descent. This choice influences strongly the descent and must otherwise be hand-picked by the user, following a potentially long research.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22582号 / 情博第719号 / 新制||情||123(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 矢ヶ崎 一幸, 教授 中村 佳正, 教授 梅野 健 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

abstract: Neglected tropical diseases (NTDs) comprise of diverse communicable diseases that affect mostly the developing economies of the world, the “neglected” populations. The NTDs Visceral Leishmaniasis (VL) and Soil-transmitted Helminthiasis (STH) are among the top contributors of global mortality and/or morbidity. They affect resource-limited regions (poor health-care literacy, infrastructure, etc.) and patients’ treatment behavior is irregular due to the social constraints. Through two case studies, VL in India and STH in Ghana, this work aims to: (i) identify the additional and potential hidden high-risk population and its behaviors critical for improving interventions and surveillance; (ii) develop models with those behaviors to study the role of improved control programs on diseases’ dynamics; (iii) optimize resources for treatment-related interventions. Treatment non-adherence is a less focused (so far) but crucial factor for the hindrance in WHO’s past VL elimination goals. Moreover, treatment non-adherers, hidden from surveillance, lead to high case-underreporting. Dynamical models are developed capturing the role of treatment-related human behaviors (patients’ infectivity, treatment access and non-adherence) on VL dynamics. The results suggest that the average duration of treatment adherence must be increased from currently 10 days to 17 days for a 28-day Miltefosine treatment to eliminate VL. For STH, children are considered as a high-risk group due to their hygiene behaviors leading to higher exposure to contamination. Hence, Ghana, a resource-limited country, currently implements a school-based Mass Drug Administration (sMDA) program only among children. School staff (adults), equally exposed to this high environmental contamination of STH, are largely ignored under the current MDA program. Cost-effective MDA policies were modeled and compared using alternative definitions of “high-risk population”. This work optimized and evaluated how MDA along with the treatment for high-risk adults makes a significant improvement in STH control under the same budget. The criticality of risk-structured modeling depends on the infectivity coefficient being substantially different for the two adult risk groups. This dissertation pioneers in highlighting the cruciality of treatment-related risk groups for NTD-control. It provides novel approaches to quantify relevant metrics and impact of population factors. Compliance with the principles and strategies from this study would require a change in political thinking in the neglected regions in order to achieve persistent NTD-control. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2020