Thesis (Ph. D.)--Harvard University, 1968. / Typescript. Includes bibliographical references (leaves 1-5 (last group)). Also issued in print.

Thesis (Ph.D.)--University of Central Florida, 2008. / Adviser: Zhihua Qu. Includes bibliographical references (p. 117-121).

Thesis (Ph.D..(Mathematics and Applied Mathematics)) -- University of Pretoria, 2009. / Summary and abstract in English. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 59-61).

Theses (M.A.)--Marshall University, 2006. / Title from document title page. Includes abstract. Document formatted into pages: contains vii, 72 pages. Bibliography: p. 70-72.

Theses (M.A.)--Marshall University, 2005. / Title from document title page. Includes abstract. Document formatted into pages: contains vi, 38 pages. Bibliography: page 38.

The object of this thesis is the study of the decidability properties of linear dynamical systems, which have fundamental ties to theoretical computer science, software verification, linear hybrid systems, and control theory. In particular, we describe a method for deciding the termination of simple linear loops, partly solving a 10-year-old open problem of Tiwari (2004) and Braverman (2006). We also study the membership problem for semigroups of matrix exponentials, which we show to be undecidable in general by reduction from Hilbert's Tenth Problem, and decidable for all instances where the matrices defining the semigroup commute. In turn, this entails the undecidability of the generalised versions of the Continuous Orbit and Skolem Problems to a multi-matrix setting. We also study point-to-point controllability for linear time-invariant systems, which is a central problem in control theory. For discrete-time systems, we show that this problem is undecidable when the set of controls is non-convex, and at least as hard as the Skolem Problem even when it is a convex polytope; for continuous-time systems, we show that this problem reduces to the Continuous Orbit Problem when the set of controls is a linear subspace, which entails decidability. Finally, we show how to decide whether all solutions of a given linear ordinary differential equation starting in a given convex polytope eventually leave it; this problem, which we call the "Polytope Escape Problem'', relates to the liveness of states in linear hybrid automata. Our results rely on a number of theorems from number theory, logic, and algebra, which we introduce in a self-contained way in the preamble to this thesis, together with a few new mathematical results of independent interest.

abstract: Persistence theory provides a mathematically rigorous answer to the question of population survival by establishing an initial-condition- independent positive lower bound for the long-term value of the population size. This study focuses on the persistence of discrete semiflows in infinite-dimensional state spaces that model the year-to-year dynamics of structured populations. The map which encapsulates the population development from one year to the next is approximated at the origin (the extinction state) by a linear or homogeneous map. The (cone) spectral radius of this approximating map is the threshold between extinction and persistence. General persistence results are applied to three particular models: a size-structured plant population model, a diffusion model (with both Neumann and Dirichlet boundary conditions) for a dispersing population of males and females that only mate and reproduce once during a very short season, and a rank-structured model for a population of males and females. / Dissertation/Thesis / Ph.D. Mathematics 2014

Orientadores: Jose Wilson Magalhães Bassani, Antonio Carlos Guimarães de Almeida / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-12T22:48:22Z (GMT). No. of bitstreams: 1 Azevedo_GersonFlorenceCarvalheirade_D.pdf: 2274193 bytes, checksum: 9c8a1fee0426b5335f1a55b9dfcdd33d (MD5) Previous issue date: 2009 / Resumo: Durante o burst epileptiforme e a depressão alastrante (DA), são observados um aumento da [K+]o (concentração extracelular de potássio) e uma diminuição da [Ca2+]o (concentração extracelular de cálcio), evidenciando a participação deste mecanismo não-sináptico nestes padrões oscilatórios anormais. Essas variações nas [K+]o e [Ca2+]o elevam a excitabilidade neuronal. No entanto, não está claro se a alta [K+]o é um fator primário na geração destas atividades neuronais ou se apenas desempenha um papel secundário neste processo. Para melhor compreender a dinâmica não-linear destes padrões, as condições experimentais de alta [K+]o e zero [Ca2+]o foram replicadas em um modelo ampliado de Golomb, referente à região CA1 da formação hipocampal. Importantes mecanismos regulatórios de concentração iônica, como a bomba Na+/K+, a difusão iônica e o sistema de buffer da glia, foram acrescentados ao modelo de Golomb. Dentro destas condições, foi possível simular atividades elétricas neuronais tipicamente apresentadas no burst epileptiforme em sua fase ictal. A DA foi iniciada pela interrupção da atividade da bomba Na+/K+. O bloqueio da bomba Na+/K+ por meio da hipóxia celular é uma manobra experimental para se obter a DA, conhecida também como depressão alastrante hipóxica - DAH. A teoria de bifurcação e o método fast-slow analysis foram utilizados para estudar a interferência do K+ extracelular na excitabilidade celular. Este estudo indicou que o sistema perde a sua estabilidade com o aumento da [K+]o, transitando para um elevado estado de excitabilidade. Este crescimento da [K+]o provoca bifurcações no comportamento dinâmico neuronal, que determinam transições entre diferentes estágios dessas atividades elétricas. No primeiro estágio, o aumento da [K+]o propicia a deflagração do burst epileptiforme e da DA via bifurcações sela-nó e de Hopf supercrítica, respectivamente. Ao longo da atividade neuronal, o nível de excitabilidade é mantido por meio de um crescimento contínuo da [K+]o, que deprime as correntes de K+ em um processo de realimentação positiva. Neste estágio, em relação ao burst epileptiforme, a amplitude e a freqüência dos disparos dos potenciais de ação são alteradas via bifurcação de Hopf supercrítica. No último estágio, com a depressão das correntes de K+, a bomba de Na+/K+ tem uma participação importante no término da atividade neuronal. O burst epileptiforme e a DA são finalizados por meio das bifurcações sela-órbita homoclínica e sela-nó, respectivamente. Portanto, este trabalho sugere que o K+ extracelular pode desempenhar um papel fundamental na dinâmica não-linear do burst epileptiforme e da sua transição para a DA. / Abstract: During the epileptiform burst and the spreading depression (SD), it is observed an increase of [K+]o (extracellular potassium concentration) and a decrease of [Ca2+]o (extracellular calcium concentration), pointing out the participation of this nonsynaptic mechanism in these abnormal oscillatory patterns. These ionic variations raise the neuronal excitability. However, whether the high [K+]o is a primary factor in the beginning of these neuronal activities or just plays a secondary role into this process is unclear. To better understand the nonlinear dynamics of these patterns, the experimental conditions of high [K+]o and zero [Ca2+]o were replicated in an extended Golomb model in which we added important regulatory mechanisms of ion concentration as Na+/K+ pump, ion diffusion and glial buffering. Within these conditions, it was possible to simulate epileptiform burst within the ictal phase. The SD was elicited by the interruption of the Na+/K+ pump activity. The blockage of Na+/K+ pump by cellular hypoxia is an experimental procedure to elicit SD, known as hypoxic spreading depression - HSD. The bifurcation theory and the method of fast-slow analysis were used to study the interference of extracellular K+ in the cellular excitability. This analysis indicates that the system loses its stability at a high [K+]o, transiting to an elevated state of neuronal excitability. This raise of [K+]o provokes bifurcations in the neuronal dynamic behavior, that determine transitions between different stages of these electrical activities. In the initial stage, the increase of [K+]o creates favorable conditions to trigger the epileptiform burst and the SD by saddle-node and supercritical Hopf bifurcations, respectively. During the neuronal activity, the level of excitability is maintained by a continuous growth of [K+]o that depresses K+ currents in a positive feedback way. At this stage, concerning epileptiform burst, the amplitude and frequency of action potentials are changed by supercritical Hopf bifurcation. At the last stage, with the depression of K+ currents, the Na+/K+ pump plays an important role in the end of neuronal activity. The epileptiform burst and SD activities terminate by saddle-homoclinic orbit and saddle-node bifurcations, respectively. Thus, this work suggests that [K+]o may play a fundamental role in the nonlinear dynamics of the epileptiform burst and the transition to SD. / Doutorado / Engenharia Biomedica / Doutor em Engenharia Elétrica