Dissertations / Theses on the topic 'Continuum (Mathematics) Dynamics'

In continuum mechanics, the conservation laws for mass, momentum and energy coupled with the constitutive relations of the stress and heat flux could be a powerful solution method for continuum systems. However, it is required that the equations form a closed system, i.e. that the stress and heat flux are formulated as functions of the conserved variables. This thesis studies the constitutive relations of the stress and heat flux in the Fermi-Pasta-Ulam-Tsingou model with Langevin dynamics, by the means of molecular dynamics simulations. In essence, the model consists of a many particle system in the presence of a heat bath, where each particle is chained to its two neighbors by a nonlinear quadratic spring force. A numerical method is implemented to propagate the particle dynamics. Following Hardy [7], formulas relating the macroscopic entities to the particle dynamics are employed in order to study the behavior of the stress and heat flux in relation to the mass, momentum and energy. In fact, the numerical result show that the stress appears as a linear function of the energy.
En värdefull lösningsmetod för kontinuumsystem är att lösa det system av partiella differentialekvationer som utgörs av konserveringslagarna för massa, rörelsemängd och energi tillsammans med två konstitutiva relationer för spänning och värmeflöde. För detta krävs det att ekvationerna bildar ett slutet system, dvs. att spänningen och värmeflödet är funktioner av de konserverade storheterna. Detta examensarbete studerar de konstitutiva relationerna för spänningen och värmeflödet i Fermi-Pasta-Ulam-Tsingou modellen med Langevindynamik, som bestäms genom simulering av molekyldynamiken. Modellen består väsentligen av många partiklar som är kopplade till ett värmebad, där varje partikel är länkad till sina två grannar genom en ickelinjär fjäderkraft. En numerisk metod implementeras för att propagera partikeldynamiken. Baserat på Hardys arbete [7] bestäms först spänningen och värmeflödet, sedan studeras deras beroende av konserveringsvariablerna. De numeriska resultaten visar att spänningen är en linjär funktion av energin.

Thesis (Ph. D.)--University of Alabama at Birmingham, 2009.
Title from PDF title page (viewed Sept. 2, 2009). Additional advisors: Alexander Blokh, Lex G. Oversteegen, Purushotham Bangalore, Vo Thanh Liem, Kyle Siegrist. Degree earned with the cooperation of additional faculty from the University of Alabama and the University of Alabama in Huntsville. Includes bibliographical references.

Doctor of Philosophy
Department of Computing and Information Sciences
William Henry Hsu
Topic models are probabilistic models for discovering topical themes in collections of documents. In real world applications, these models provide us with the means of organizing what would otherwise be unstructured collections. They can help us cluster a huge collection into different topics or find a subset of the collection that resembles the topical theme found in an article at hand. The first wave of topic models developed were able to discover the prevailing topics in a big collection of documents spanning a period of time. It was later realized that these time-invariant models were not capable of modeling 1) the time varying number of topics they discover and 2) the time changing structure of these topics. Few models were developed to address this two deficiencies. The online-hierarchical Dirichlet process models the documents with a time varying number of topics. It varies the structure of the topics over time as well. However, it relies on document order, not timestamps to evolve the model over time. The continuous-time dynamic topic model evolves topic structure in continuous-time. However, it uses a fixed number of topics over time. In this dissertation, I present a model, the continuous-time infinite dynamic topic model, that combines the advantages of these two models 1) the online-hierarchical Dirichlet process, and 2) the continuous-time dynamic topic model. More specifically, the model I present is a probabilistic topic model that does the following: 1) it changes the number of topics over continuous time, and 2) it changes the topic structure over continuous-time. I compared the model I developed with the two other models with different setting values. The results obtained were favorable to my model and showed the need for having a model that has a continuous-time varying number of topics and topic structure.

We address the problem of the consistency between discrete and continuous models for density-dependent size-structured populations. Some earlier works have discussed the consistency of density independent age and size-structured models. Although the issue of consistency between these models has raised interest in recent years, it has not been discussed in depth, perhaps because of the non-linear nature of the equations involved. We construct a numerical scheme of the continuous model and show that the transition matrix of this scheme has the form of the standard discrete model. The construction is based on the theory of Upwind Numerical Schemes for non-Linear Hyperbolic Conservation Laws with one important difference, that we do have a non-linear source at the boundary; interestingly, this case has not been explored in depth from the purely mathematical point of view. We prove the consistency, non-linear stability and hence convergence of the numerical scheme which guarantee that both models yield results that are completely consistent with each other. Several examples are worked out: a simple linear age-structured problem, a density-independent size-structured problem and a non-linear size-structured problem. These examples confirm the convergence just proven theoretically. An ample revision of relevant biological and computational literature is also presented and used to establish realistic restrictions on the objects under consideration and to prepare significant examples to illustrate our points.

This thesis explores the stochastic features of models of ecological systems in discrete and in continuous time. Our interest lies in models formulated at the microscale, from which a mesoscopic description can be derived. The stochasticity present in the models, constructed in this way, is intrinsic to the systems under consideration and stems from their finite size. We start by exploring a susceptible-infectious-recovered model for epidemic spread on a network. We are interested in the case where the connectivity, or degree, of the individuals is characterised by a very broad, or heterogeneous, distribution, and in the effects of stochasticity on the dynamics, which may depart wildly from that of a homogeneous population. The model at the mesoscale corresponds to a system of stochastic differential equations with a very large number of degrees of freedom which can be reduced to a two-dimensional model in its deterministic limit. We show how this reduction can be carried over to the stochastic case by exploiting a time-scale separation in the deterministic system and carrying out a fast-variable elimination. We use simulations to show that the temporal behaviour of the epidemic obtained from the reduced stochastic model yields reasonably good agreement with the microscopic model under the condition that the maximum allowed degree that individuals can have is not too close to the population size. This is illustrated using time series, phase diagrams and the distribution of epidemic sizes. The general mesoscopic theory used in continuous-time models has only very recently been developed for discrete-time systems in one variable. Here, we explore this one-dimensional theory and find that, in contrast to the continuous-time case, large jumps can occur between successive iterates of the process, and this translates at the mesoscale into the need for specifying `boundary' conditions everywhere outside of the system. We discuss these and how to implement them in the stochastic difference equation in order to obtain results which are consistent with the microscopic model. We then extend the theoretical framework to make it applicable to systems containing an arbitrary number of degrees of freedom. In addition, we extend a number of analytical results from the one-dimensional stochastic difference equation to arbitrary dimension, for the distribution of fluctuations around fixed points, cycles and quasi-periodic attractors of the corresponding deterministic map. We also derive new expressions, describing the autocorrelation functions of the fluctuations, as well as their power spectrum. From the latter, we characterise the appearance of noise-induced oscillations in systems of dimension greater than one, which have been previously observed in continuous-time systems and are known as quasi-cycles. Finally, we explore the ability of intrinsic noise to induce chaotic behaviour in the system for parameter values for which the deterministic map presents a non-chaotic attractor; we find that this is possible for periodic, but not for quasi-periodic, states.

In this dissertation we discuss various results for spaces that are wild, i.e. not locally simply connected. We first discuss periodic properties of maps from a given space to itself, similar to Sharkovskii's Theorem for interval maps. We study many non-locally connected spaces and show that some have periodic structure either identical or related to Sharkovskii's result, while others have essentially no restrictions on the periodic structure. We next consider embeddings of solenoids together with their complements in three space. We differentiate solenoid complements via both algebraic and geometric means, and show that every solenoid has an unknotted embedding with Abelian fundamental group, as well as infinitely many inequivalent knotted embeddings with non-Abelian fundamental group. We end by discussing Peano continua, particularly considering subsets where the space is or is not locally simply connected. We present reduced forms for homotopy types of Peano continua, and provide a few applications of these results.

The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts. In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number. In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes). Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients. Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.

The mathematical analysis of dynamical systems covers a wide range of challenging problems related to the time evolution, transient and asymptotic behavior, or regulation and control of physical systems. A large part of my work has been motivated by new mathematical questions arising from biological systems, especially signaling and genetic regulatory networks, where the classical methods usually don't directly apply. Problems include parameter estimation, robustness of the system, model reduction, or model assembly from smaller modules, or control of a system towards a desired state. Although many different formalisms and methodologies can be used to study these problems, in the past decade my work has focused on discrete and hybrid modeling frameworks with the goal of developing intuitive, computationally amenable, and mathematically rigorous, methods of analysis. Discrete (and, in particular, Boolean) models involve a high degree of abstraction and provide a qualitative description of the systems' dynamics. Such models are often suitable to represent the known interactions in gene regulatory networks and their advantage is that a large range of theoretical analysis tools are available using, for instance, graph theoretical concepts. Hybrid (piecewise affine) models have discontinuous vector fields but provide a continuous and more quantitative description of the dynamics. These systems can be analytically studied in each region of an appropriate partition of the state space, and the full solution given as a concatenation of the solutions in each region. Here, I will introduce the two formalisms and then, using several examples, illustrate how a combination of different formalisms permits comparison of results, as well as gaining quantitative knowledge and predictive power on a biological system, through the use of complementary mathematical methods.

There are many types of bioreactors used for producing bacteria populations in commercial, medical and research applications.

This report presents a systematic discussion of some of the most important models corresponding to the well known reproduction kinetics such as the Michaelis-Menten kinetics, competitive substrate inhibition and competitive product inhibition. We propose a modification of a known model, analyze it in the same manner as known models and discuss the most popular types of bioreactors and ways of controlling them.

This work summarises much of the known results and may serve as an aid in attempts to design new models.

Orientadores: Rubens Maciel Filho, Aline Carvalho da Costa
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Química
Made available in DSpace on 2018-08-14T23:39:16Z (GMT). No. of bitstreams: 1 GonzalezQuiroga_Arturo_M.pdf: 2749935 bytes, checksum: c375b22201594f091c1d51f4befd9532 (MD5) Previous issue date: 2009
Resumo: Por mais de um século, a principal fonte de combustível e produtos químicos para a sociedade humana tem vindo a partir de recursos fósseis, os quais são limitados e estão concentrados em poucas regiões do mundo. Biomassa, como a única fonte de carbono renovável, mostra-se promissora para a produção de combustíveis e produtos químicos em grande escala. Na última década, a produção de bioetanol a partir de biomassa lignocelulósica através de hidrólise enzimática tem sido estudada intensamente a nível de bancada. Reatores contínuos, nos quais a fração celulósica e a hemicelulósica de substratos lignocelulósicos são convertidos em açúcares redutores (para serem fermentados a bioetanol), são o tema desta dissertação. As principais questões consideradas aqui são: a cinética, o padrão de contato sólido-líquido, o comportamento fluidodinâmico da lama durante a hidrólise, configurações alternativas de reatores contínuos, e estratégias de operação contínua com relação ao substrato e a enzima. Foram revisados os modelos cinéticos mais recentes para a hidrólise enzimática de substratos lignocelulósicos, dando especial atenção aos modelos úteis para projeto de reatores. Foi proposto um esquema de classificação de modelos cinéticos baseado no número de reações consideradas. Com respeito à utilidade de um modelo cinético na otimização do sistema de reação, é desejável que este inclua a adsorção das enzimas na fração celulósica do substrato e na lignina, a inibição das enzimas por produto final, a reatividade do substrato e a desativação das enzimas. Neste trabalho, um modelo de uma reação foi ajustado a perfis experimentais de glicose e xilose obtidos na hidrólise enzimática de bagaço de cana prétratado com peróxido de hidrogênio alcalino. O modelo cinético apresentado difere de outros modelos em que este inclui a predição do perfil de xilose por meio de uma relação algébrica que faz uso do perfil de glicose, e é especialmente útil na ausência de dados experimentais de adsorção e reatividade do substrato. O comportamento fluidodinâmico da lama de biomassa durante a hidrólise enzimática é complexo devido à ampla distribuição de tamanhos de partícula, às formas incomuns das partículas de biomassa e às mudanças na reología que experimenta o material ao longo da reação. Neste trabalho é apresentado um levantamento detalhado da fluidodinâmica de suspensões de polpas fibrosas e lamas com tendência à sedimentação, devido a que o comportamento fluidodinâmico de lamas de biomassa pode ser entendido a partir dessas duas situações limite. Além disso, foi desenvolvido um modelo fluidodinâmico com balanços microscópicos e solucionado com um software de fluidodinâmica computacional para estudar o escoamento de lamas de biomassa em reatores tubulares com e sem defletores angulares internos. Por outro lado, a micromixtura do material dentro dos reatores foi considerada em duas situações limite: um material que é fracionado em cúmulos discretos que reagem como reatores batch durante o tempo que estejam no sistema de reação, e um material que imediatamente ingressa no sistema de reação entra em contato íntimo com outros elementos de fluido ao nível molecular. As conversões em reatores continuous para as anteriores situações limite de micromistura foram obtidas. Os reatores contínuos considerados foram reatores de tanque agitado em serie, reatores tubulares e combinação entre eles. Devido a que a biomassa adsorve água, no começo da reação de hidrólise a fase móvel pode desaparecer a concentrações de substrato maiores de 10% w/w, aproximadamente. A alimentação distribuída de substrato numa serie de reatores de tanque agitado foi uma alternativa proposta neste trabalho para incrementar a produtividade volumétrica dos reatores. Consequentemente, uma série de reatores de tanque agitado com alimentação distribuída, seguida de um reator tubular, foi a melhor alternativa para incrementar a reatividade volumétrica dos reatores e diminuir o volume de reação. O presente trabalho é em grande parte exploratório sendo que não há procedimentos detalhados de projeto e escalonamento de reatores de hidrólise enzimática de biomassa lignocelulósica na literatura científica atual. Consequentemente, reatores contínuos alternativos assim como procedimentos de modelagem mais detalhados são brevemente discutidos ao final do trabalho. De vital importância são experimentos para elucidar aspectos tais como a reutilização de enzimas por recirculação ou re-adsorção em substrato fresco, adsorção de enzimas a altas concentrações iniciais de substrato (>10% w/w), e a relação entre as propriedades reológicas da lama e a extensão da reação de hidrólise.
Abstract: For ever a century, the main source of fuel and chemicals for human society has come from fossil resources, which are limited and concentrated in a few regions of the world. Biomass, as the only source of renewable carbon, shows great promise for largescale economical production of renewable transportation and fuel chemicals. In the last decade the bioethanol production from lignocellulosic biomass via, enzymatic hydrolysis, has been intensively studied at laboratory level. Continuous reactors in which the cellulose and hemicellulose fractions of lignocellulosic substrates convert to reducing sugars (which are fermented to bioethanol) are the theme of this dissertation. The main issues considered are kinetics, contacting pattern and fluid dynamics, alternative configurations of continuous reactors, and continuous operating strategies with respect to substrate and enzyme. The most recent kinetic models for the enzymatic hydrolysis of lignocellulosic biomass, useful for reactor design, were reviewed and classified based on the number of reaction considered. Regarding to reactor design, the main factors that should be include a kinetic model are adsorption of enzymes on cellulose and lignin, inhibition of enzyme by glucose and cellobiose, substrate reactivity and enzyme deactivation. A kinetic model of a single reaction was fitted to experimental profiles of glucose and xylose obtained by the enzymatic hydrolysis of pretreated of sugarcane bagasse. This kinetic model differs of previous models in that it predicts xylose concentration based on glucose concentration. This kinetic model is a useful in the absence of experimental data on enzyme adsorption and substrate features others than concentration. The fluid dynamic behavior of biomass slurries during enzymatic hydrolysis is very complex due to the wide particle size distribution, the extremes shapes of particles and the significant rheological changes of the slurry with the progress of the enzymatic hydrolysis. This works reviewed the fluid dynamic behavior of fiber pulp suspensions and settling slurries because the fluid dynamic behavior of biomass through continuous reactors can be framed between these two limiting situations. In addition, a computational fluid dynamic model was developed to asses the fluid dynamic behavior of biomass slurries in tubular and baffled tubular reactors, motivated by the benefits of tubular reactors to carry out the enzymatic hydrolysis in terms of lower reaction volume and lower agitation requirements. On the other hand, the micromixing behavior of the flowing material was framed between two limiting situations: an incoming material that is broken up into discrete clumps in which the reaction proceed independently as in a batch reactor, and an incoming material that immediately comes into intimate contact with other fluid elements at molecular level. Conversions in continuous reactors corresponding to the above extreme states of micromixing were obtained. The continuous reactor considered were stirred tanks reactors in series, tubular reactors, and combination between them. As biomass adsorbs water, this may cause the bulk to become unsaturated at initial substrate concentration higher than 10% w/w, approximately. Operating the enzymatic hydrolysis in a distributed feeding mode by adding fresh substrate and enzyme at subsequent stirred tank reactors was proposed as an alternative to increase the volumetric productivity of reactors. A reactors configuration consisting of stirred tank reactors in series with continuous distributed feeding of substrate and enzyme, followed by a tubular reactor allow increasing the volumetric productivity of the reaction system overcoming mixing limitations and lowering the required reaction volume. It should be noted that this work is exploratory and that there are not major reports in open literature about the design and scale-up of continuous reactors for enzymatic hydrolysis. Some alternative continuous reactors, as well as modeling approaches for reactor design, are suggested. Of paramount importance are experiments to elucidate relevant aspects as reutilization of enzymes by recirculation of readsorption, adsorption of enzymes at high substrate concentrations (>10% w/w), and the relation between rheological properties of slurries with the extent of saccharification.
Mestrado
Desenvolvimento de Processos Químicos
Mestre em Engenharia Química

Cerebrovascular diseases are by far the largest causes of death in the UK, as well as one of the leading causes of adult disability. The brain's healthy function depends on a steady supply of oxygen, delivered through the microvasculature. Cerebrovascular diseases, such as stroke and dementia, can interrupt the transport of blood (and hence oxygen) rapidly, or over a prolonged period of time. An interruption in flow can lead to ischaemia, with prolonged interruptions leading to tissue death and eventual brain damage. The microvasculature plays a key role in the transport of oxygen and nutrients to brain tissue; however, its role in diseases such as dementia is poorly understood, primarily due to the inability of current clinical imaging techniques to resolve microvessels, and due to the complexity of the underlying microvasculature. Therefore, in order to understand cerebrovascular diseases, it is necessary to be able to resolve and understand the microvasculature. In particular, generating large-scale models of the human microvasculature that can be linked back to contemporary clinical imaging is important in helping plug the current imaging gap that exists. A novel statistical model is proposed here that generates such large-scale models efficiently. Homogenization theory is used to generate a porous continuum capillary bed (characterised by its permeability) that allows for the efficient scaling up of the microvasculature. A novel order-based density-filling algorithm is then developed which generates morphologically accurate penetrating arterioles and venules, also demonstrating that the topology of the vessels only has a minor influence on CBF compared to diameter. Finally, the capillary bed and penetrating vessels are coupled into a large voxel-sized model of the microvasculature from which pressure and flux variations through the voxel can be analysed. A decoupling of the pressure and flux, as well as a layering of flow, was observed within the voxel, driven by the topology of the penetrating vessels. Micro-infarctions were also simulated, demonstrating the large local effects they have on the pressure and flux, whilst only causing a minor drop in CBF within the voxel.

Parameter estimation techniques have been successfully and extensively applied to deterministic models based on ordinary differential equations but are in early development for stochastic models. In this thesis, we first investigate using parameter estimation techniques for a deterministic model to approximate parameters in a corresponding stochastic model. The basis behind this approach lies in the Kurtz limit theorem which implies that for large populations, the realizations of the stochastic model converge to the deterministic model. We show for two example models that this approach often fails to estimate parameters well when the population size is small. We then develop a new method, the MCR method, which is unique to stochastic models and provides significantly better estimates and smaller confidence intervals for parameter values. Initial analysis of the new MCR method indicates that this method might be a viable method for parameter estimation for continuous time Markov chain models.

This thesis consists of five articles mainly devoted to problems in dynamical systems and ergodic theory. We consider non-uniformly hyperbolic two dimensional systems and limit distributions of point measures which are absolutely continuous with respect to the Lebesgue measure. Let $f_{a_0}(x)=a_0-x^2$ be a quadratic map where the parameter $a_0\in(1,2)$ is chosen such that the critical point $0$ is pre-periodic (but not periodic). In Papers A and B we study skew-products $(\th,x)\mapsto F(\th,x)=(g(\th),f_{a_0}(x)+\al s(\th))$, $(\th,x)\in S^1\times\real$. The functions $g:S^1\to S^1$ and $s:S^1\to[-1,1]$ are the base dynamics and the coupling functions, respectively, and $\al$ is a small, positive constant. Such quadratic skew-products are also called Viana maps. In Papers A and B we show for several choices of the base dynamics and the coupling function that the map $F$ has two positive Lyapunov exponents and for some cases we further show that $F$ admits also an absolutely continuous invariant probability measure. In Paper C we consider certain Bernoulli convolutions. By showing that a specific transversality property is satisfied, we deduce absolute continuity of the to these Bernoulli convolutions associated distributions. In Papers D and E we consider sequences of real numbers in the unit interval and study how they are distributed. The sequences in Paper D are given by the forward iterations of a point $x\in[0,1]$ under a piecewise expanding map $T_a:[0,1]\to[0,1]$ depending on a parameter $a$ contained in an interval $I$. Under the assumption that each $T_a$ admits a unique absolutely continuous invariant probability measure $\mu_a$ and that some technical conditions are satisfied, we show that the distribution of the forward orbit $T_a^j(x)$, $j\ge1$, is described by the distribution $\mu_a$ for Lebesgue almost every parameter $a\in I$. In Paper E we apply the ideas in Paper D to certain sequences which are equidistributed in the unit interval and give a geometrical proof of an old result by Koksma.

Ce manuscrit décrit analytiquement et numériquement les instabilités d'un fluide magnétique dans une cellule de Hele-Shaw. On considère l'interface entre un fluide magnétique et un autre fluide non magnétique, miscible ou non, soumise à un champ magnétique homogène normal à la cellule ou à l'interface. Le champ démagnétisant est inhomogène à cette interface et génère un mouvement convectif des fluides. Dans la première partie, nous avons utilisé une analyse linéaire de stabilité entre deux liquides miscibles pour une distribution donnée de concentration à l'interface. Les résultats s'appliquent aussi à la stabilité d'un réseau de concentration induit par une expérience de Rayleigh forcé. Nous avons démontré que l'équation de Brinkman décrit mieux la dissipation visqueuse dans une cellule de Hele-Shaw que celle de Darcy. Nous avons trouvé que la viscosité (et non la diffusion massique) donnait à l'écoulement une échelle de longueur de l'ordre de l'épaisseur de la cellule dans le cas des forçages élevés. Dans la seconde partie de notre étude, nous avons modélisé la dynamique non linéaire de l'interface avec une tension superficielle par la méthode des intégrales de frontière. Nous avons décrit la modification des doigts de Saffman–Taylor par les forces magnétostatiques. Nous avons obtenu des structures dendritiques proches de celles observées expérimentalement et analysé quelques aspects de la formation des motifs.

Dans cette thèse, nous étudions l'ensemble des solutions d'équations aux dérivées partielles résultant de modèles d'astrophysique et de biologie. Nous répondons aux questions de l'existence, mais aussi nous essayons de décrire le comportement de certaines familles de solutions lorsque les paramètres varient. Tout d'abord, nous étudions deux problèmes issus de l'astrophysique, pour lesquels nous montrons l'existence d'ensembles particuliers de solutions dépendant d'un paramètre à l'aide de la méthode de réduction de Lyapunov-Schmidt. Ensuite un argument de perturbation et le théorème du Point xe de Banach réduisent le problème original à un problème de dimension finie, et qui peut être résolu, habituellement, par des techniques variationnelles. Le reste de la thèse est consacré à l'étude du modèle Keller-Segel, qui décrit le mouvement d'amibes unicellulaires. Dans sa version plus simple, le modèle de Keller-Segel est un système parabolique-elliptique qui partage avec certains modèles gravitationnels la propriété que l'interaction est calculée au moyen d'une équation de Poisson / Newton attractive. Une différence majeure réside dans le fait que le modèle est défini dans un espace bidimensionnel, qui est expérimentalement consistant, tandis que les modèles de gravitationnels sont ordinairement posés en trois dimensions. Pour ce problème, les questions de l'existence sont bien connues, mais le comportement des solutions au cours de l'évolution dans le temps est encore un domaine actif de recherche. Ici nous étendre les propriétés déjà connues dans des régimes particuliers à un intervalle plus large du paramètre de masse, et nous donnons une estimation précise de la vitesse de convergence de la solution vers un profil donné quand le temps tend vers l'infini. Ce résultat est obtenu à l'aide de divers outils tels que des techniques de symétrisation et des inégalités fonctionnelles optimales. Les derniers chapitres traitent de résultats numériques et de calculs formels liés au modèle Keller-Segel

In this thesis, the first-order and higher-order interferences on the total (Coulomb+nuclear), Coulomb and nuclear breakup cross sections in the 15C+208Pb, 11Be+208Pb breakup reactions are first studied at 68 MeV/u incident energy. It is shown that the first-order interference reduces by more than 60% the total breakup cross sections, by less than 3% the Coulomb breakup cross sections and by more than 85% the nuclear breakup cross sections, for both reactions. On the other hand, the high-order interference is found to reduce by less than 9% the total breakup cross section, less than 1% the Coulomb breakup cross section and less than 7% the nuclear breakup cross section for the 15C+208Pb reaction. For the 11Be+208Pb reaction however, the high-order interference reduces by less than 7% the total breakup cross section, by less than 1% the Coulomb breakup cross section and by less than 4% the nuclear breakup cross section. It is finally shown that even at first-order, the incoherent sum of the nuclear breakup cross sections is more important than the incoherent sum of the Coulomb breakup cross sections for the two reactions. The role of the diagonal and off-diagonal continuum-continuum couplings on total, Coulomb and nuclear breakup cross sections is also investigated for the 8B+58Ni, 8B+208Pb and 19C+208Pb at 29.3, 170.3 MeV and 1273 MeV incident energies respectively. Qualitatively, we found that, the diagonal continuum-continuum couplings are responsible for the large reduction of the differential total and nuclear breakup cross sections at backward angles. At forward angles, this reduction is due to the off-diagonal continuum-continuum couplings. In the absence of these couplings, the nuclear breakup is the more dominant process, while when they are included, the Coulomb breakup becomes dominant. This shows that, the nuclear breakup is more affected by the continuum-continuum couplings than its Coulomb counterpart. Quantitatively, we found that, the off-diagonal countinuum-countinuum couplings reduce by 13.39%, 12.71% and 11.11% the total breakup cross sections for the 8B+58Ni, 8B+208Pb and 19C+208Pb reactions, respectively.
Physics
D. Phil. (Physics)

In this dissertation we investigate the following problem in ergodic theory: Let X be a Polish space and T:X→X a homeomorphism. We ask whether there exists a non-atomic Borel probability measure m on X, ergodic and quasi-invariant under T, such that the cohomology class of some preassigned continuous function f:X→S1 is trivial. The cohomology class of f is trivial and f is called a coboundary of T, if the equation hTx=f x˙hx m-almosteverywhere admits a measurable solution h:X→S1 In the presence of a non-periodic recurrent point x0∈X , we establish the existence of such a measure m . Indeed, our approach, which is based on a construction of Katznelson and Weiss, yields a continuum of such measures all of which are of type IIinfinity and pointwise orthogonal. And the same statement is true for measures of type III
acase@tulane.edu

Numerical optimization is one of the most active research areas and is widely used in science, engineering, economics and industry. In numerical black box optimization, the underlying objective functions, which can be non-differentiable, non-convex, multi-modal and noisy, are unknown to optimization algorithms. At any points in the continuous search space, the first and second order information is not available. Only the objective function values are available by means of function evaluations. The optimization algorithms, which consider these optimization problems as a black box, are designed to find the best solutions in the continuous search space.
This thesis focuses on continuous black box optimization and presents a collection of the novel sampling methods improving the state-of-the-art optimization algorithms. The algorithms are Covariance Matrix Adaptation Evolution Strategies (CMA-ES) and Cooperative Coevolutionary Algorithms (CCEAs). We will also study these algorithms in the dynamic environments where the objective functions change during the course of optimization. It is necessary to understand algorithms’ performances because many real world problems are basically dynamic in nature. Examples of real world problems include but not limited to random arrival of new tasks, machine faults and degradation, climate change, market fluctuation and economic factors.
In the first part of this thesis, two novel sampling methods that improves Evolution Strategies (ES) for continuous black-box optimization will be introduced: halfspace sampling and eigenspace sampling. In Halfspace Sampling, the hyperplane which goes through the current solution separates the search space into two halfspaces: a positive halfspace and a negative halfspace. When a candidate solution is sampled, the sample always lies in the positive halfspace that is estimated by successful steps in the recent iterations. We theoretically derive the log-linear convergence rates of a scale-invariant step size ES when ES are used to optimize spherical functions in finite and infinite dimensions. Halfspace sampling is implemented in a (1+1) CMA-ES, and the resulting algorithm is benchmarked on the Black-Box Optimization Benchmarking (BBOB) testbed. In Eigenspace sampling, the optimization algorithms consider the eigenspace of the underlying objective functions. A candidate solution is always sampled in an eigenspace spanned by eigen-vectors with repeated or clustered eigenvalues. This demonstrates experimentally how eigenspace sampling can improve the CMA-ES for the current benchmark problems, In particular, the CMA-ES that uses eigenspace sampling often performs very well in ill-conditioned problems.
In the second part of this thesis, we will study the CMA-ES, ES and CCEA in dynamic environments. Two new types of individuals that address the dynamic environments will be introduced: 1) random immigrants (RIs) that increase the diversity for the changing environments, and 2) elitist individuals that improve the local convergence to the optima. The resulting algorithms are evaluated on a standard suite of benchmark problems. Superior results are observed when the two types of individuals are used. We also investigate the behavior of three CMA-ES variants, which include an elitist (1+1)-CMA-ES, a non-elitist(μ,λ)-CMA-ES and a sep-CMA-ES. Our experimental results show the simple elitist strategies that include the (1+1)-ES and the (1+1)-CMA-ES generally outperform non-elitist CMA-ES variants. The elitist strategies are robust to dynamic changes with different severites, but performance is worsened when the problem dimensions are increased. In higher dimensions, the performance of elitist and non-elitist variants of CMA-ES are marginally identical.
「連續函數最優化」乃係研究領域中一項重要之議題。其廣泛被應用於科學、工程、經濟及工業等範籌。於黑箱最優化下，「優化算法」往往未能掌握目標函數之基本特性如可微分性，非凸性，多模態性及雜訊，甚至未能完全準確地使用連續函數之基本第一階及第二階導數。它們只能通過目標函數值來評估優化之進度。因此，設計最佳優化算法實為一個既困難且富挑戰性之研究議題。
本論文主要探討「連續函數黑箱最優化」，並提出新穎之抽樣方法，分別為「協方差矩陣適應進化策略」 (CMA-ES) 及「協同進化算法」(CCEAs) ，以改善現時研究領域中最佳之優化算法。本文亦深入了解此算法於動態環境中之優化表現。於應用科學中，如氣候變化，市場波動及經濟因素等等問題於本質上是動態，因此了解它們於動態環境中之優化表現是十分重要。
本論文第一部分提出兩種新抽樣方法，分別為「半空間抽樣法」及「特徵空間抽樣法」， 作為提高「進化策略」 (ES) 之最優化表現。一、於「半空間抽樣法」下，連續函數之搜崇空間會被一片超平面分割成兩個半空間，分別是「正半空間」及「負半空間」。當優化算法尋找一個元素時，它每一次會從正半空間里面抽樣。本文由此推算出進化策略在球形函數收斂速度，並將之應用於最先進之(1 十1)CMA-ES，從而測試它在最新之BBOB平台之表現。二、於「特徵空間抽樣法」下，優化算法首先在擁有重複或集群特徵值之特徵空間內抽出一個元素，然後將特徵空間抽樣法應用於CMA-ES。實驗結果發現使用特徵抽樣法能提升CMA-ES在俗稱「病態」函數中之優化表現。
本論文第二部分探討三個優化算法—「協方差矩陣適應進化策略」、「高進化策略」及「協同進化算法」於動態環境中之表現。本文提出以兩種新元素作處理動態環境—「隨機元素」(Random Immigrants)及「優生元素」(Elitist)。「 隨機元素」用以增加優化過程中元素之多樣性：「優生元素」則作增強局部優化之收斂速度。另外，本文更使用最新之測試平台評估三個CMA-ES優化算法之表現，包括(1+1) CMA-ES、(μ,λ)CMA-ES及sep CMA-ES。實驗結果證明，簡單精英進化策略，如(1+1)ES及(1+1)CMA-ES，普遍比非精英進化策略更能於動態環境作優化表現。精英進化策略較能應付不同程度轉變之動態環境。相反，當該連續函數之維度增加時，進化策略之優化表現則開始下降。於高維度下，精英CMA-ES及非精英CMA-ES之優化表現大致相同。
Au, Chun Kit.
Thesis (Ph.D.)--Chinese University of Hong Kong, 2015.
Includes bibliographical references (leaves 200-230).
Abstracts also in Chinese.
Title from PDF title page (viewed on 26, October, 2016).
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.
Detailed summary in vernacular field only.

In this thesis, we have investigated some interesting dynamical problems in microstructural continuum using Eringen's polar theory. These problems are pertaining to surface waves in a microstretch plate, Stoneley waves at an interface between two different microstretch half-spaces, surface waves in a micropolar cylindrical borehole filled with micropolar fluid, reflection and transmission of elastic waves at a liquid/solid half-space and reflection of elastic waves from a micropolar mixture porous half-space.

The objective of this work was to determine the operating conditions which would maximise the crystallisation performance of continuous vacuum pans used in the sugar industry. The specific application investigated in detail is crystallisation of high grade product sugar (A-sugar) in a South African raw cane sugar factory. The optimisation studies are based on a detailed dynamic mathematical model of a continuous pan. Whilst this model is based on the published work of others, the selection of variables and the formulation of the equations have been structured to produce a modular model of an individual compartment with the minimum number of independent variables. The independent variables have also been selected to meet the requirements of both a state-space control formulation and those necessary for the dynamic programming technique of optimisation. The modular compartment models are linked together to model a multi-compartment pan and the steady state model is derived as a special case of the dynamic model. For the model to simulate the conditions in South African sugar factories adequately requires appropriate descriptions of sucrose solubility and growth kinetics. Given the limited applicability of published data, experiments were undertaken to determine these parameters. Sucrose solubility in impure solutions was determined in laboratory tests designed to approach equilibrium by dissolution at conditions approximating those during pan boiling. The dependence of crystal growth rate on the concentration of impurity present in the mother liquor was investigated in both laboratory scale and pilot scale batch pan boiling experiments. The primary dependence of crystal growth rate on the super-saturation driving force was determined by fitting the steady state model to results of tests on an industrial scale continuous pan. The dynamic programming technique was used in conjunction with the mathematical model to determine the operating conditions which maximise steady state crystallisation performance. Using the crystallisation parameters determined for South African conditions, this approach has shown that the conventional wisdom of running with high crystal contents in all compartments of continuous pans boiling A-massecuite is not optimum. Pans should operate at lower crystal contents in earlier compartments, only increasing to higher crystal contents towards the final compartment. The specific values depend on seed conditions, pan design and the solubility and growth kinetics. To reap the benefits of being able to determine the optimum steady state operating condition for a continuous pan, it is necessary to be able to achieve effective steady state operation under industrial conditions. This requires both a steady loading on the pan and effective control of the crystallisation conditions within the pan. To stabilise loading, a strategy has been developed which uses buffer tanks in an optimal way to damp out flow fluctuations. This strategy accommodates multiple buffer tanks in series without the amplification of disturbances that occurs with some of the simpler published techniques. The dynamic behaviour of absolute pressure control and compartment feed control were investigated in an industrial scale pan. This work has demonstrated the importance of high quality absolute pressure control and developed techniques for effective automatic tuning of pan feed controls. As part of this research, computer control systems were developed as tools to provide the appropriate monitoring and control of the experiments undertaken.
Thesis (Ph.D.)-University of Natal, 2002.

The demand for higher quality steel and higher production rates in the production of steel slabs is ever increasing. These slabs are produced using a continuous casting process. The molten steel flow patterns inside the components of the caster play an important role in the quality of these products. A simple yet effective design method that yields optimum designs is required to design the systems influencing the flow patterns in the caster. The tundish is one of these systems. Traditionally, experimental methods were used in the design of these tundishes, making use of plant trials or water modelling. These methods are both costly and time consuming. More recently, Computational Fluid Dynamics (CFD) has established itself as a viable alternative to reduce the number of experimentation required, resulting in a reduction in the time scales and cost of the design process. Furthermore, CFD provides more insight into the flow process that is not available through experimentation only. The CFD process is usually based on a trial-and-error basis and relies heavily on the insight and experience of the designer to improve designs. Even an experienced designer will only be able to improve the design and does not necessarily guarantee optimum results. In this thesis, a more efficient design methodology is proposed. This methodology involves the combination of a mathematical optimiser with CFD to automate the design process. The methodology is tested on a four different industrial test cases. The first case involves the optimisation of a simple dam-weir configuration of a single strand caster. The position of the dam and weir relative to inlet region is optimised to reduce the dead volume and increase the inclusion removal. The second case involves the optimisation of a pouring box and baffle of a two-strand caster. In this case, the pouring box and baffle geometry is optimised to maximise the minimum residence time at operating level and a typical transition level. The third case deals with the geometry optimisation of an impact pad to reduce the surface turbulence that should result in a reduction in the particle entrainment from the slag layer. The last case continues from the third case where a dam position and height is optimised in conjunction with the optimised impact pad to maximise the inclusion removal on the slag layer. The cases studies show that a mathematical optimiser combined with CFD is a superior alternative compared to traditional design methods, in that it yields optimum designs for a tundish in a continuous casting system.
Thesis (PhD (Mechanical Engineering))--University of Pretoria, 2006.
Mechanical and Aeronautical Engineering
unrestricted

In this thesis we investigate single and multi-player stochastic dynamic optimization prob-lems. We consider both discrete and continuous time processes. In the multi-player setup we investigate zero-sum games with both complete and partial information. We study partially observable stochastic games with average cost criterion and the state process be-ing discrete time controlled Markov chain. The idea involved in studying this problem is to replace the original unobservable state variable with a suitable completely observable state variable. We establish the existence of the value of the game and also obtain optimal strategies for both players. We also study a continuous time zero-sum stochastic game with complete observation. In this case the state is a pure jump Markov process. We investigate the nite horizon total cost criterion. We characterise the value function via appropriate Isaacs equations. This also yields optimal Markov strategies for both players. In the single player setup we investigate risk-sensitive control of continuous time Markov chains. We consider both nite and in nite horizon problems. For the nite horizon total cost problem and the in nite horizon discounted cost problem we characterise the value function as the unique solution of appropriate Hamilton Jacobi Bellman equations. We also derive optimal Markov controls in both the cases. For the in nite horizon average cost case we shown the existence of an optimal stationary control. we also give a value iteration scheme for computing the optimal control in the case of nite state and action spaces. Further we introduce a new class of stochastic processes which we call stochastic processes with \age-dependent transition rates". We give a rigorous construction of the process. We prove that under certain assunptions the process is Feller. We also compute the limiting probabilities for our process. We then study the controlled version of the above process. In this case we take the risk-neutral cost criterion. We solve the in nite horizon discounted cost problem and the average cost problem for this process. The crucial step in analysing these problems is to prove that the original control problem is equivalent to an appropriate semi-Markov decision problem. Then the value functions and optimal controls are characterised using this equivalence and the theory of semi-Markov decision processes (SMDP). The analysis of nite horizon problems becomes di erent from that of in nite horizon problems because of the fact that in this case the idea of converting into an equivalent SMDP does not seem to work. So we deal with the nite horizon total cost problem by showing that our problem is equivalent to another appropriately de ned discrete time Markov decision problem. This allows us to characterise the value function and to nd an optimal Markov control.