Thesis (Ph.D.)--University of Western Australia, 2007.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 102-108). Available also in a digital version from Dissertation Abstracts.

by Michael Athans. / Bibliography: leaf [3]. / "August 1983." / Supported by the Office of Naval Research under Grant ONR/N00014-82-K-0582 NR 606-003 NASA Ames and Langley Research Centers under Grant NGL-22-009-124

Local gamma-band (~30-100Hz) oscillations in the brain, produced by feedback inhibition on a characteristic timescale, appear in multiple areas of the brain and are associated with a wide range of cognitive functions. Some regions producing gamma also receive gamma-rhythmic input, and the interaction and coordination of these rhythms has been hypothesized to serve various functional roles. This thesis consists of three stand-alone chapters, each of which considers the response of a gamma-rhythmic neuronal circuit to input in an analytical framework. In the first, we demonstrate that several related models of a gamma-generating circuit under periodic forcing are asymptotically drawn onto an attracting invariant torus due to the convergence of inhibition trajectories at spikes and the convergence of voltage trajectories during sustained inhibition, and therefore display a restricted range of dynamics. In the second, we show that a model of a gamma-generating circuit under forcing by square pulses cannot maintain multiple stably phase-locked solutions. In the third, we show that a separation of time scales of membrane potential dynamics and synaptic decay causes the gamma model to phase align its spiking such that periodic forcing pulses arrive under minimal inhibition. When two of these models are mutually coupled, the same effect causes excitatory pulses from the faster oscillator to arrive at the slower under minimal inhibition, while pulses from the slower to the faster arrive under maximal inhibition. We also show that such a time scale separation allows the model to respond sensitively to input pulse coherence to an extent that is not possible for a simple one-dimensional oscillator. We draw on a wide range of mathematical tools and structures including return maps, saltation matrices, contraction methods, phase response formalism, and singular perturbation theory in order to show that the neuronal mechanism of gamma oscillations is uniquely suited to reliably phase lock across brain regions and facilitate the selective transmission of information.

We review the concept of invariant measure and study conditions under which linear combinations of averages along periodic orbits are dense in the space of invariant measures. / Revisamos el concepto de medida invariante y estudiamos condiciones bajo las cuales combinaciones lineales de promedios a lo largo de órbitas periódicas son densas en el espacio de medidas invariantes.

Differential equations, in particular the nonlinear ones, are commonly used in formulating most of the fundamental laws of nature as well as many technological problems, among others. This makes the need for methods in finding closed form solutions to such equations all-important. In this thesis we study Lie symmetry methods for some nonlinear ordinary differential equations (ODE). The study focuses on identifying and using the underlying symmetries of the given first order nonlinear ordinary differential equation. An extension of the method to higher order ODE is also discussed. Several illustrative examples are presented. / Differentialekvationer, framförallt icke-linjära, används ofta vid formulering av fundamentala naturlagar liksom många tekniska problem. Därmed finns det ett stort behov av metoder där det går att hitta lösningar i sluten form till sådana ekvationer. I det här arbetet studerar vi Lie symmetrimetoder för några icke-linjära ordinära differentialekvationer (ODE). Studien fokuserar på att identifiera och använda de underliggande symmetrierna av den givna första ordningens icke-linjära ordinära differentialekvationen. En utvidgning av metoden till högre ordningens ODE diskuteras också. Ett flertal illustrativa exempel presenteras.

Let $G$ be a second countable, locally compact group which is either compact or abelian, and let $\rho$ be a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_\rho$. We examine frames of the form $\{ \rho(x) f_j \colon x \in G, j \in I\}$ for families $\{f_j\}_{j \in I}$ in $\mathcal{H}_\rho$. In particular, we give necessary and sufficient conditions for the joint orbit of a family of vectors in $\mathcal{H}_\rho$ to form a continuous frame. We pay special attention to this problem in the setting of shift invariance. In other words, we fix a larger second countable locally compact group $\Gamma \supset G$ containing $G$ as a closed subgroup, and we let $\rho$ be the action of $G$ on $L^2(\Gamma)$ by left translation. In both the compact and the abelian settings, we introduce notions of Zak transforms on $L^2(\Gamma)$ which simplify the analysis of group frames. Meanwhile, we run a parallel program that uses the Zak transform to classify closed subspaces of $L^2(\Gamma)$ which are invariant under left translation by $G$. The two projects give compatible outcomes. This dissertation contains previously published material.

BESSA, Vagner Henrique Loiola. Osciladores log-periódicos e tipo Caldirola-Kanai. 2012. 66 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2012. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-10-19T18:23:14Z No. of bitstreams: 1 2012_dis_vhlbessa.pdf: 26350485 bytes, checksum: 4eb844c05187fb66d3b274a9f8d1b0ed (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-10-20T20:53:49Z (GMT) No. of bitstreams: 1 2012_dis_vhlbessa.pdf: 26350485 bytes, checksum: 4eb844c05187fb66d3b274a9f8d1b0ed (MD5) / Made available in DSpace on 2015-10-20T20:53:49Z (GMT). No. of bitstreams: 1 2012_dis_vhlbessa.pdf: 26350485 bytes, checksum: 4eb844c05187fb66d3b274a9f8d1b0ed (MD5) Previous issue date: 2012 / In this work we present the classical and quantum solutions of two classes of time-dependent harmonic oscillators, namely: (a) the log-periodic and (b) the Caldirola-Kanai-type oscillators. For class (a) we study the following oscillators: (I) $m(t)=m_0frac{t}{t_0}$, (II) $m(t)=m_0$ and (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$. In all three cases $omega(t)=omega_0frac{t_0}{t}$. For class (b) we study the Caldirola-Kanai oscillator (IV)where $omega(t)=omega_0$ and $m(t)=m_0 ext{exp}ajust{gamma t}$ and the oscillator with $omega(t)=omega_0$ and $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$, for $alpha=2$ (V) and $alpha=4$ (VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of $q(t)$, $p(t)$ as well as the phase diagram $q(t)$ vs $p(t)$. To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time. / Nesse trabalho apresentamos as soluções clássicas e quânticas de duas classes de osciladores harmônicos dependentes de tempo, a saber: (a) o oscilador log-periódico e (b) o oscilador tipo Caldirola-Kanai. Para a classe (a) estudamos os seguintes osciladores: (I) $m(t)=m_0frac{t}{t_0}$, (II) $m(t)=m_0$ e (III) $m(t)=m_0ajust{frac{t}{t_0}}^2$. Nesses três casos $omega(t)=omega_0frac{t_0}{t}$. Para a classe (b) estudamos o oscilador (IV) de Caldirola-Kanai onde $omega(t)=omega_0$ e $m(t)=m_0 ext{Exp}ajust{gamma t}$ e osciladores com $omega(t)=omega_0$ e $m(t)=m_0ajust{1+frac{t}{t_0}}^alpha$, para (V) $alpha=2$ e (VI) $alpha=4$. Para obter as soluções clássicas de cada oscilador resolvemos suas respectivas equações de movimento e analisamos o comportamento de $q(t)$, $p(t)$ assim como do diagrama de fase $q(t)$ vs $p(t)$. Para obter as soluções quânticas usamos uma transformação unitária e o método dos invariantes quânticos de Lewis e Riesenfeld. A função de onda obtida é escrita em termos de uma função $ ho$, que é solução da equação de Milne-Pinney. Ainda, para cada sistema resolvemos a respectiva equação de Milne-Pinney e discutimos como o produto da incerteza evolui no tempo.

Submitted by Guilherme Vituri Fernandes Pinto null (214001018@rc.unesp.br) on 2016-04-11T07:27:24Z No. of bitstreams: 1 Dissertação Guilherme Vituri.pdf: 726432 bytes, checksum: c4db8ed97d1452e129b0f46186ed5a53 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-04-13T14:34:46Z (GMT) No. of bitstreams: 1 pinto_gvf_me_sjrp.pdf: 726432 bytes, checksum: c4db8ed97d1452e129b0f46186ed5a53 (MD5) / Made available in DSpace on 2016-04-13T14:34:46Z (GMT). No. of bitstreams: 1 pinto_gvf_me_sjrp.pdf: 726432 bytes, checksum: c4db8ed97d1452e129b0f46186ed5a53 (MD5) Previous issue date: 2016-03-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é estudar grande parte do artigo [6], no qual Gottlieb define o subgrupo G(X, x_0) de pi_1(X, x_0) (em que X é um CW-complexo conexo por caminhos), posteriormente chamado de grupo de Gottlieb; o calculamos para diversos espaços, como as esferas, o toro, os espaços projetivos, a garrafa de Klein, etc.; posteriormente, estudamos o artigo [22] de Varadarajan, que generalizou o grupo de Gottlieb para um subconjunto G(A, X) de [A, X]_∗ . Por fim, calculamos G(S^n, S^n). / The goal of this work is to study partially the article [6], in which Gottlieb has defined a subgroup G(X, x_0) of pi_1(X, x_0) (where X is a path-connected CW-complex based at x_0), called "Gottlieb group" in the literature. This group is computed in this work for some spaces, namely the spheres, the torus, the projective spaces, and the Klein bottle. Further, a paper by Varadarajan [22] who has generalized Gottlieb group to a subset G(A, X) of [A, X]_* is studied. Finally, the groups G(S^n, S^n) is computed.

L’analyse statique de programmes est un domaine crucial en compilation, en optimisation, et en validation de logiciels. Les structures de données complexes (tableaux, listes, graphes...), omniprésentes dans les programmes, posent des problèmes difficiles, du fait qu’elles représentent des ensembles de données de taille importante ou inconnue, et que l’adressage des données dans ces ensembles est calculé (indexation, indirection). La plupart des travaux sur l’analyse des structures de données concernent la vérification de la correction des accès aux données (vérification que les indices d’un tableau sont dans les bornes, que les pointeurs ne sont pas nuls, “shape analysis”). L’analyse du contenu des structures de données est encore peu abordée. A Verimag, ce domaine a été abordé récemment, et a donné lieu à de premiers résultats sur l’analyse de tableaux unidimensionnels. Une méthode d’analyse de programmes simples a été proposée [1], qui permet de découvrir des propriétés des contenus de tableaux, comme par exemple que le résultat d’un programme de tri est bien un tableau trié. Un autre type de propriétés, dites “non positionnelles” a aussi été considéré [2], qui concerne le contenu global d’un tableau, indépendamment de son rangement: par exemple, on montre que le résultat d’un tri est une permutation du tableau initial. Ces premiers résultats sont très encourageants, mais encore embryonnaires. L’objectif du travail de thèse proposé est de les étendre dans plusieurs directions. Notre analyse de propriétés positionnelles est capable de découvrir des relations point- à-point entre des “tranches” de tableaux (ensembles de cellules consécutives). Les extensions envisagées concernent les tableaux multidimensionnels, les ensembles de cellules non nécessairement consécutives, et les structures de données plus générales. Concernant les propriétés non positionnelles, les premiers résultats sont limités aux égalités de contenus de tableaux. Ils doivent être étendus à des relations plus complexes (inclusions, sommes disjointes...) et à d’autres structures de données. Ce travail prend place dans le projet ASOPT (“Analyse statique et optimisation”), accepté dans le programme Arpège de l’ANR en 2008. Références : [1] N. Halbwachs, M. Péron. Discovering properties about arrays in simple programs. ACM Conference on Programming Language Design and Implementation, PLDI 2008. Tucson (Az.), juin 2008. [2] V. Perrelle. Analyse statique du contenu de tableaux, propriétés non positionnelles. Rapport de M2R, Master Parisien de Recherche en Informatique, septembre 2008. / Static analysis is key area in compilation, optimization and software validation. The complex data structures (arrays, dynamic lists, graphs...) are ubiquitous in programs, and can be challenging, because they can be large or of unbounded size and accesses are computed. (through indexing or indirections). Whereas the verification of the validity of the array accesses was one of the initial motivations of abstract interpretation, the discovery of properties about array contents was only adressed recently. Most of the analyses of array contents are based on a partitioning of the arrays. Then, they try to discover properties about each fragment of this partition. The choice of this partition is a difficult problem and each method have its flaw. Moreover, classical representations of array partitions induce an exponential complexity for these analyzes. In this thesis, we generalize the concept of array partitioning into the concept of "fragmentation" which allow overlapping fragments, handling potentially empty fragments and selecting specialized relations. On the other hand, we propose an abstraction of these fragmentations in terms of graphs called "slices diagrams" as well as the operations to manipulate them and ensuring a polynomial complexity. Finally, we propose a new criterion to compute a semantic fragmentation inspired by the existing ones which attempt to correct their flaws. These methods have been implemented in a static analyzer. Experimentations shows that the analyzer can efficiently and precisly prove some challenging exemples in the field of static analysis of programs manipulating arrays.