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Several stochastic and sampling processes leading to the logarithmic series distributions and related distributions are studied. An attempt has been made to present a unified picture, using a single notation, of the various derivations and approaches that appear in the literature. These derivations and approaches are compared and some of the relationships among them are brought out. Though the work is mainly interpretative, there are a few results that seem to be new.

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Given the importance placed on probability and statistics in the PK-8 curriculum by the National Council of Teachers of Mathematics (2000) and on teachers by the Interstate New Teacher Assessment and Support Consortium (1995) and the Conference Board of the Mathematical Sciences (2001), it is important to know how well preservice teachers understand topics that are vital to a thorough understanding of the probability and statistics topics emphasized by national standards. It is necessary for a teacher to thoroughly understand the subject matter in order to teach effectively, but that is not sufficient. A teacher must also be able to successfully communicate with the students about that material. Therefore, this study utilized a standards- and literature-based assessment to study 210 preservice teachers with the goal of taking the first step in determining whether current PK-8 preservice teachers are prepared to teach select probability and statistics topics specified in standards documents. The assessment contains 11 probability and statistics items with a total of 23 parts in a variety of shortanswer, multiple-choice, and extended-response formats. It is described in detail in Chapter III and reproduced in Appendix A. A confirmatory factor analysis indicated that for this sample of PK-8 preservice teachers, the assessment measured the underlying constructs on which it was based. Preservice teachers?? ability to answer these items varied greatly. For short-answer and multiple-choice items, the percentage of preservice teachers incorrectly answering an item was as high as 87% and as low as 18%. For extended-response items, incorrect answers were provided by as few as 12% of the participants on one item and by as many as 83% on another. Individual responses were analyzed to illustrate correct conceptions and misconceptions of these preservice teachers. There was not a statistically significant difference between responses based on the grade band the participants were preparing to teach, but students specializing in mathematics and science did perform better than other participants. Although effect sizes were small, the amount of time elapsed since an elementary statistics class was taken and the number of methods courses taken were positively associated with performance on this assessment.

Full-domain multiscale analyses of unidirectional AS4/H3502 open-hole composite tensile specimens were performed to assess the effect of microscale progressive fiber failures in regions with large stress/strain gradients on macroscale composite strengths. The effect of model discretization at the microscale and macroscale on the calculated composite strengths and analysis times was investigated. Multiple sets of microscale analyses of repeating unit cells, each containing varying numbers of fibers with a distinct statistical distribution of fiber strengths and fiber volume fractions, were used to establish the microscale discretization for use in multiscale calculations. In order to improve computational times, multiscale analyses were performed over a reduced domain of the open-hole specimen. The calculated strengths obtained using reduced domain analyses were comparable to those for full-domain analyses, but at a fraction of the computational cost. Such reduced domain analyses likely are an integral part of efficient adaptive multiscale analyses of large all-composite air vehicles.

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Cette thèse porte sur des équations aux dérivées partielles hamiltoniennes à conditions initiales aléatoires. En effet, on étudie ici l'évolution de certaines mesures à travers le flot de telles équations. Cette étude suit deux axes.Premièrement, on considère le caractère globalement bien posé de l'équation d'onde non linéaire quand la donnée initiale est de faible régularité. Cette donnée initiale est une variable aléatoire et on obtient le caractère globalement bien posé de façon presque sûre par rapport à la mesure induite par cette variable. La faible régularité fait référence à l'espace auquel appartient les valeurs de la variable aléatoires et dénote une régularité moins contraignante que celle requise par la théorie déterministe.Dans certaines conditions, des propriétés d'invariance de la loi de la donnée initiale sont nécessaires à la démonstration du caractère bien posé. C'est pourquoi le deuxième axe comprend la question de l'invariance de mesures et leurs stabilités à travers le flot d'EDPs.On donne ainsi une loi invariante à travers le flot de l'équation d'onde cubique et une autre à travers celui de l'équation de Benjamin-Bona-Mahony (BBM). la mesure invariante pour BBM est telle que les amplitudes associées à chaque longueur d'onde de la solution sont des variables aléatoires indépendantes les unes des autres. On considère alors la stabilité de l'invariance pour BBM lorsqu'on ajoute des corrélations entre ces amplitudes.Enfin, en s'inspirant de la littérature physique à propos de la turbulence faible, on s'est demandé ce qu'il advenait de l'indépendance entre les amplitudes dans un contexte plus général. Plus précisément, on a cherché à si les covariances des amplitudes restent petites lorsque celles-ci sont initialement indépendantes et que le terme non quadratique de l'énergie associée à l'équation étudiée est très petit devant l'énergie totale. / This thesis is about Hamiltonian partial differential equations with random initial data. Indeed, the evolution of particular measures are studied here through the flow of such equations. This study is done along two axis.First, the global well-posedness with initial data with low regularity is considered for the non linear wave equation. The initial datum is a random variable and the global well-posedness is obtained almost surely wrt the measure induced by this variable. The low regularity refers to the space which the values of the random initial datum belong to and means a regularity under the one given by deterministic theory.Some properties of invariance of the law of the initial datum are required in the proof of the global well-posedness under certain conditions. Hence, the second axis is the invariance of measures through the flow of PDEs and their stability.An invariant law is given for the cubic non linear wave equation and for the Benjamin-Bona-Mahony equation (BBM). The invariant measure for BBM is such that the amplitudes associated to each wavelength of the solution are random variables independent from each other. The stability of the invariance for BBM is considered when one adds correlations between these amplitudes.Finally, inspired by the Physics literature about wave turbulence, the stability of the independence between the amplitudes is investigated about. Namely, we tried to know if the covariances of the amplitudes remain small when they are initially independent and when the quadratic term of the energy associated to the equation is small compared to the total energy.

Many cells employ cadherin complexes (c-sites) on the cell membrane to attach to neighboring cells, as well as integrin complexes (i-sites) to attach to a substrate in order to accomplish cell migration. This paper analyzes a model for the motion of a group of cells connected by c-sites. We begin with two cells connected by a single c-site and analyze the resultant motion of the system. We find that the system is irrotational. We present a result for reducing the number of c-sites in a system with c-sites between pairs of cells. This greatly simplifies the general system, and provides an exact solution for the motion of a system of two cells and several c-sites.Then a method for analyzing the general cell system is presented. This method involves 0-row-sum, symmetric matrices. A few results are presented as well as conjectures made that we feel will greatly simplify such analyses. The thesis concludes with the proposal of a framework for analyzing a dynamic cell system in which stochastic processes govern the attachment and detachment of c-sites.

Made available in DSpace on 2015-03-04T18:51:12Z (GMT). No. of bitstreams: 1 TeseRosaLuz.pdf: 2324271 bytes, checksum: 38301fe0e11f9cf83a774f763dbb7ec0 (MD5) Previous issue date: 2009-01-01 / Conselho Nacional de Desenvolvimento Cientifico e Tecnologico / In the context of the stochastic perturbations theories we analyze the accuracy of two linear poroelastic models applied to highly heterogeneous porous media subject to uncertainties in the permeability and the elastic constants. The poroelastic models completely and weakly coupled analized arise characterized by the degree of intensity coupling between the hydrodinamics, governor of the percolation of the fluid and poromechanics which governs the deformation of the porous matrix. New equations for the moments of effective solutions using techniques of asymptotic expansion. In light of the perturbation theory are set simplifying assumptions that clarify clearly the domain of validity of weakly coupled model, widely used in simulation of oil reservoirs in the presence of heterogeneities and correlation in poroelastic coefficients. Computational simulations of the primary extraction of oil process are carried out using Monte Carlo techniques in conjunction with finite element methods. Results obtained clearly confirm the conjecture established by the perturbation theorie related with the inaccuracy of the weakly coupled model in the presence of variability in the elastic constants. The methodology used allows to quantify the distance between the two poroelastics models and therefore propose the appropriate model for different conditions of loading and variability of the geological formation. / No contexto das teorias de perturbação estocástica, analisamos a acurácia de dois modelos poroelásticos lineares aplicados a meios porosos altamente heterogêneos sujeitos as incertezas na permeabilidade e nas constantes elásticas. Os modelos poroelásticos completamente e fracamente acoplados analisados surgem caracterizados pelo grau de intensidade de acoplamento entre a hidrodinámica governante da percolação o do fluido e a poromecânica que rege as deformações da matriz porosa. Novas equações efetivas para os momentos das soluções são obtidas fazendo uso de técnicas de expansão assintótica. À luz da teoria de perturbação, são estabelecidas hipóteses simplificadoras que elucidam o domínio de validade do modelo fracamente acoplado, amplamente utilizado nos simuladores de Reservatórios de Petróleo, na presença de heterogeneidades e correlação nos coeficientes poroelásticos. Simulações computacionais do processo de extração primária de petróleo são realizadas utilizando técnicas de Monte Carlo em conjunção com métodos de elementos finitos. Resultados numéricos obtidos confirmam claramente a conjectura estabelecida pela teoria de perturbação relacionada com a inacurácia do modelo fracamente acoplado na presença de variabilidade nas constantes elásticas. A metodologia empregada permite quantificar a distância entre os dois modelos poroelásticos, e consequentemente, propor a escolha do modelo apropriado para diferentes condições de carregamento e variabilidade da formação geológica.

Our Solar system contains a large amount of dust, containing valuable information about our close cosmic environment. If created in a planet's system, the particles stay predominantly in its vicinity and can form extended dust envelopes, tori or rings around them. A fascinating example of these complexes are Saturnian rings containing a wide range of particles sizes from house-size objects in the main rings up to micron-sized grains constituting the E ring. Other example are ring systems in general, containing a large fraction of dust or also the putative dust-tori surrounding the planet Mars. The dynamical life'' of such circumplanetary dust populations is the main subject of our study. In this thesis a general model of creation, dynamics and death'' of circumplanetary dust is developed. Endogenic and exogenic processes creating dust at atmosphereless bodies are presented. Then, we describe the main forces influencing the particle dynamics and study dynamical responses induced by stochastic fluctuations. In order to estimate the properties of steady-state population of considered dust complex, the grain mean lifetime as a result of a balance of dust creation, life'' and loss mechanisms is determined. The latter strongly depends on the surrounding environment, the particle properties and its dynamical history. The presented model can be readily applied to study any circumplanetary dust complex. As an example we study dynamics of two dust populations in the Solar system. First we explore the dynamics of particles, ejected from Martian moon Deimos by impacts of micrometeoroids, which should form a putative tori along the orbit of the moon. The long-term influence of indirect component of radiation pressure, the Poynting-Robertson drag gives rise in significant change of torus geometry. Furthermore, the action of radiation pressure on rotating non-spherical dust particles results in stochastic dispersion of initially confined ensemble of particles, which causes decrease of particle number densities and corresponding optical depth of the torus. Second, we investigate the dust dynamics in the vicinity of Saturnian moon Enceladus. During three flybys of the Cassini spacecraft with Enceladus, the on-board dust detector registered a micron-sized dust population around the moon. Surprisingly, the peak of the measured impact rate occurred 1 minute before the closest approach of the spacecraft to the moon. This asymmetry of the measured rate can be associated with locally enhanced dust production near Enceladus south pole. Other Cassini instruments also detected evidence of geophysical activity in the south polar region of the moon: high surface temperature and extended plumes of gas and dust leaving the surface. Comparison of our results with this in situ measurements reveals that the south polar ejecta may provide the dominant source of particles sustaining the Saturn's E ring. / In unserem Sonnensystem befindet sich eine große Menge an Staub, der viele Informationen über unseren Kosmos enthält. Wird der Staub im System um den Planeten gebildet, bleibt er vorwiegend in dessen Nähe und bildet Staubhüllen, -tori oder -ringe um ihn. Ein faszinierendes Beispiel eines solchen Komplexes sind die Saturnringe, in denen von mikrometergroßen Partikeln bis zu hausgroßen Körpern alle Partikelgrößen vertreten sind. Weitere Beispiele sind Ringsysteme im Allgemeinen, sowie der vermutete Staubring um Mars. Das dynamische Verhalten einer solchen Staubpopulation ist Hauptthema dieser Dissertation. In dieser Arbeit wurde ein allgemeines Modell zur Erzeugung, Dynamik und Vernichtung von planetarem Staub entwickelt. Endogene und exogene Mechanismen zur Produktion von Staub an atmosphärenlosen Körpern werden vorgestellt. Desweiteren werden die wichtigsten Kräfte welche die Teilchendynamik beeinflussen, sowie die Auswirkung von stochastischen Fluktuationen untersucht. Die Lebenszeiten der Staubkörner als Bilanz zwischen Staubproduktion und -vernichtung werden bestimmt, um den stationären Zustand der Staubkonfiguration abzuschätzen. Die Lebenszeit des Staubes hängt stark von den Eigenschaften der Umgebung und der Teilchen sowie von deren dynamischer Vergangenheit ab. Das vorgestellte Modell kann auf alle planetaren Systeme angewandt werden. Als Beispiel wurden zwei Staubpopulationen in unserem Sonnensystem studiert. Zuerst wurde die Dynamik des Staubes untersucht, welcher durch Mikrometeorideneinschläge auf dem Marsmond Deimos produziert wird und die vermuteten Marstori erzeugt. Der Poynting-Robertson-Effekt, als indirekter Einfluss des Strahlungsdruckes, bewirkt eine signifikante Langzeitänderung der Torusgeometrie. Desweiteren verursacht der Strahlungsdruck eine stochastische Dispersion des nichtsphärischen Staubteilchenensembles, was eine Verringerung der Teilchenzahldichten beziehungsweise der entsprechenden optischen Tiefen im Torus bewirkt. Weiterhin wurde die Staubdynamik in der Umgebung des Saturnmondes Enceladus untersucht. Während des Vorbeifluges der Raumsonde Cassini registrierte der Staubdetektor eine Staubpopulation von mikrometergroßen Teilchen um den Mond. Überraschenderweise wurde die maximal registrierte Staubrate eine Minute vor der größten Annäherung an den Mond gemessen. Diese Asymmetrie der Messung kann, wie in dieser Arbeit demonstriert, mit einer lokalen Staubquelle am Südpol des Mondes erklärt werden. Andere Instrumente der Cassini - Sonde belegen die geophysikalische Aktivität der Südpolregion des Mondes in Form einer erhöhten Oberflächentemperatur und Fontänen von Gas und Staub an der Südpolumgebung. Der Vergleich der numerischen Simulationen mit den in - situ - Messungen zeigt, dass die Südpolquelle die voraussichtlich wichtigste Quelle von E-Ringteilchen ist.