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.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Nesse trabalho apresentamos as soluÃÃes clÃssicas e quÃnticas de duas classes de osciladores harmÃnicos dependentes de tempo, a saber: (a) o oscilador log-periÃdico 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 trÃs 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 soluÃÃes clÃssicas de cada oscilador resolvemos suas respectivas equaÃÃes 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 soluÃÃes quÃnticas usamos uma transformaÃÃo unitÃria e o mÃtodo dos invariantes quÃnticos de Lewis e Riesenfeld. A funÃÃo de onda obtida à escrita em termos de uma funÃÃo $ ho$, que à soluÃÃo da equaÃÃo de Milne-Pinney. Ainda, para cada sistema resolvemos a respectiva equaÃÃo de Milne-Pinney e discutimos como o produto da incerteza evolui no tempo. / 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.

The purpose of this research is to study the detection of temporal changes between two (or more) multimodal images satellites, i.e., between two different imaging modalities acquired by two heterogeneous sensors, giving for the same scene two images encoded differently and depending on the nature of the sensor used for each acquisition. The two (or multiple) multimodal satellite images are acquired and coregistered at two different dates, usually before and after an event. In this study, we propose new models belonging to different categories of multimodal change detection in remote sensing imagery. As a first contribution, we present a new constraint scenario expressed on every pair of pixels existing in the before and after image change. A second contribution of our work is to propose a spatio-temporal textural gradient operator expressed with complementary norms and also a new filtering strategy of the difference map resulting from this operator. Another contribution consists in constructing an observation field from a pair of pixels and to infer a solution maximum a posteriori sense. A fourth contribution is proposed which consists to build a common feature space for the two heterogeneous images. Our fifth contribution lies in the modeling of patterns of change by anomalies and on the analysis of reconstruction errors which we propose to learn a non-supervised model from a training base consisting only of patterns of no-change in order that the built model reconstruct the normal patterns (non-changes) with a small reconstruction error. In the sixth contribution, we propose a pairwise learning architecture based on a pseudosiamese CNN network that takes as input a pair of data instead of a single data and constitutes two partly uncoupled CNN parallel network streams (descriptors) followed by a decision network that includes fusion layers and a loss layer in the sense of the entropy criterion. The proposed models are enough flexible to be used effectively in the monomodal change detection case. / Cette recherche a pour objet l’étude de la détection de changements temporels entre deux (ou plusieurs) images satellitaires multimodales, i.e., avec deux modalités d’imagerie différentes acquises par deux capteurs hétérogènes donnant pour la même scène deux images encodées différemment suivant la nature du capteur utilisé pour chacune des prises de vues. Les deux (ou multiples) images satellitaires multimodales sont prises et co-enregistrées à deux dates différentes, avant et après un événement. Dans le cadre de cette étude, nous proposons des nouveaux modèles de détection de changement en imagerie satellitaire multimodale semi ou non supervisés. Comme première contribution, nous présentons un nouveau scénario de contraintes exprimé sur chaque paire de pixels existant dans l’image avant et après changement. Une deuxième contribution de notre travail consiste à proposer un opérateur de gradient textural spatio-temporel exprimé avec des normes complémentaires ainsi qu’une nouvelle stratégie de dé-bruitage de la carte de différence issue de cet opérateur. Une autre contribution consiste à construire un champ d’observation à partir d’une modélisation par paires de pixels et proposer une solution au sens du maximum a posteriori. Une quatrième contribution est proposée et consiste à construire un espace commun de caractéristiques pour les deux images hétérogènes. Notre cinquième contribution réside dans la modélisation des zones de changement comme étant des anomalies et sur l’analyse des erreurs de reconstruction dont nous proposons d’apprendre un modèle non-supervisé à partir d’une base d’apprentissage constituée seulement de zones de non-changement afin que le modèle reconstruit les motifs de non-changement avec une faible erreur. Dans la dernière contribution, nous proposons une architecture d’apprentissage par paires de pixels basée sur un réseau CNN pseudo-siamois qui prend en entrée une paire de données au lieu d’une seule donnée et est constituée de deux flux de réseau (descripteur) CNN parallèles et partiellement non-couplés suivis d’un réseau de décision qui comprend de couche de fusion et une couche de classification au sens du critère d’entropie. Les modèles proposés s’avèrent assez flexibles pour être utilisés efficacement dans le cas des données-images mono-modales.