Lake Erie, one of the Great Lakes in North America, has witnessed recurrent summertime low oxygen dead zones for decades. This is a yearly phenomenon that causes microbial production of the greenhouse gas nitrous oxide from denitrification. Complete denitrification is a microbial process of reduction of nitrate to nitrogen gas via nitrite, nitric oxide, and greenhouse gas nitrous oxide. After scanning the microbial community in Lake Erie, Pseudomonas aeruginosa is decided to be examined, not because it is abundant in Lake Erie, but because it can perform denitrification under anaerobic conditions. This study focuses on a mathematical model of the metabolic network in Pseudomonas aeruginosa under denitrification and testable hypotheses generation using polynomial dynamical systems and stochastic discrete dynamical systems. Analysis of the long-term behavior of the system changing the concentration level of oxygen, nitrate, and phosphate suggests that phosphate highly affects the denitrification performance of the network. / Master of Science

On appelle clinker le matériau obtenu par cuisson de calcaire et d'argile et qui constitue le principal ingrédient du ciment Portland, composant essentiel de la majorité des bétons produits dans le monde. Ce clinker doit être finement broyé avant de pouvoir présenter une réactivité suffisante. La maîtrise des procédés de broyage représente un enjeu considérable pour l'industrie cimentière : il s'agit du premier poste en termes de consommation électrique d'une usine, en partie du fait de l'inefficacité des procédés employés. Les techniques de broyage par compression, apparues au cours des années 80, ont constitué un progrès majeur du point de vue de l'efficacité énergétique, mais la généralisation de leur utilisation a été freinée par des problèmes de maîtrise du procédé, en particulier pour des finesses importantes. L'enjeu de cette thèse est une meilleure compréhension des phénomènes en jeu lors du broyage par compression du clinker, en vue d'un meilleur contrôle des installations industrielles lors de la fabrication de produits fins. Nous nous sommes intéressés en particulier au comportement, du point de vue fondamental, d'un matériau granulaire subissant une fragmentation de ses grains, en nous appuyant sur la simulation numérique d'un Volume Elémentaire Représentatif de matière par les éléments discrets (DEM). Nous avons aussi recherché une loi de comportement permettant de relier contraintes, déformation, et évolution de la taille des particules pour le matériau broyé, en nous appuyant à la fois sur la micromécanique et les techniques d'homogénéisation, et un modèle semi-empirique de bilans de masses. Enfin, un premier pas vers la modélisation du procédé industriel et notamment sa simulation par éléments finis a été esquissé, afin de résorber les difficultés rencontrées en pratique par les industriels / Noindent Clinker is the material obtained by calcination of a mix of clay and limestone, and it is the main component of Portland cement, a crucial ingredient for the majority of concrete used around the world. This clinker must be finely ground to have a sufficient reactivity. Mastering the grinding process is a key issue in the cement industry: it is the first source of expense in terms of electric consumption in a factory, partially because of the overall inefficiency of the process. Compressive grinding techniques, first appeared during the 80's, allow major improvements in terms of energy efficiency, but the general implementation is yet to come, hindered by process control issues, especially for high fineness. The goal of this study is a better understanding of phenomenons occurring during compressive grinding of clinker, in order to provide better process control for industrial installations when dealing with fine products. We particularly choose to study the behaviour, on a fundamental point of view, of a granular material subjected to grain fragmentation, using the numerical simulation of an Elementary Representative Volume of material through Discrete Element Method (DEM). We also looked for a behaviour law able to provide a link between stress, strain, and grain size evolution for the ground material, using at the same time micromechanics and homogenization technique, and a semi-empirical mass balance model. Finally, we made first efforts in the direction of modelling the whole process through numerical simulation by Finite Element Method (FEM), in order to tackle the issue met by the industrials in operations

This thesis deals with solving a real case from one specific part of Operations Research -- Discrete Models. The case can be classified as Vehicle Routing Problem (VRP) which is a subset of classical Travelling Salesman Problem (TSP). The VRP is modified TSP when requirements of customers and capacities of trucks play role. The data needed for calculations were taken from the real situation of Pivovar Svijany a.s. The problem can be defined as VRP with cars with different capacities and split delivery. Even though the mathematic model of the problem is known and described in the thesis, the size of the problem is too big to be optimized. Therefore heuristic was used to solve it. Because of the good computational results in the past the savings algorithm was chosen. Its model was set using Visual Basic for Applications (VBA). The thesis (among others) analyses the sensitivity of the output on the values of the factors that can be chosen by the analyst. At the end of the thesis the best found solution is presented and the initial and the new scheme of the circles are compared.

This dissertation develops algebraic theory for discrete models in systems biology. Many discrete model types can be translated into the framework of polynomial dynamical systems (PDS), that is, time- and state-discrete dynamical systems over a finite field where the transition function for each variable is given as a polynomial. This allows for using a range of theoretical and computational tools from computer algebra, which results in a powerful computational engine for model construction, parameter estimation, and analysis methods. Formal definitions and theorems for PDS and the concept of PDS as models of biological systems are introduced in section 1.3. Constructing a model for given time-course data is a challenging problem. Several methods for reverse-engineering, the process of inferring a model solely based on experimental data, are described briefly in section 1.3. If the underlying dependencies of the model components are known in addition to experimental data, inferring a "good" model amounts to parameter estimation. Chapter 2 describes a parameter estimation algorithm that infers a special class of polynomials, so called nested canalyzing functions. Models consisting of nested canalyzing functions have been shown to exhibit desirable biological properties, namely robustness and stability. The algorithm is based on the parametrization of nested canalyzing functions. To demonstrate the feasibility of the method, it is applied to the cell-cycle network of budding yeast. Several discrete model types, such as Boolean networks, logical models, and bounded Petri nets, can be translated into the framework of PDS. Section 3 describes how to translate agent-based models into polynomial dynamical systems. Chapter 4, 5, and 6 are concerned with analysis of complex models. Section 4 proposes a new method to identify steady states and limit cycles. The method relies on the fact that attractors correspond to the solutions of a system of polynomials over a finite field, a long-studied problem in algebraic geometry which can be efficiently solved by computing Gröbner bases. Section 5 introduces a bit-wise implementation of a Gröbner basis algorithm for Boolean polynomials. This implementation has been incorporated into the core engine of Macaulay 2. Chapter 6 discusses bistability for Boolean models formulated as polynomial dynamical systems. / Ph. D.

In this thesis, first a higher order dynamic theory for anisotropic thermoelastic plates is developed. Then, based on this plate theory, two dynamic models, discrete and continuum models (DM and CM), are proposed for layered composites consisting of anisotropic thermoelastic layers. Of the two models, CM is more important, which is established in the study of periodic layered composites using smoothing operations. CM has the properties: it contains inherently the interface and Floquet conditions and facilitates the analysis of the composite, in particular, when the number of laminae in the composite is large / it contains all kinds of deformation modes of the layered composite / its validity range for frequencies and wave numbers may be enlarged by increasing, respectively, the orders of the theory and interface conditions. CM is assessed by comparing its prediction with the exact for the spectra of harmonic waves propagating in various directions of a two-phase periodic layered composite, as well as, for transient dynamic response of a composite slab induced by waves propagating perpendicular to layering. A good comparison is observed in the results and it is found that the model predicts very well the periodic structure of spectra with passing and stopping bands for harmonic waves propagating perpendicular to layering. In view of the results, the physical significance of Floquet wave number is also discussed in the study.

In order to understand biochemical networks we need to know not only how their parts work but also how they interact with each other. The goal of systems biology is to look at biological systems as a whole to understand how interactions of the parts can give rise to complex dynamics. In order to do this efficiently, new techniques have to be developed. This work shows how tools from mathematics are suitable to study problems in systems biology such as modeling, dynamics prediction, reverse engineering and many others. The advantage of using mathematical tools is that there is a large number of theory, algorithms and software available. This work focuses on how algebra can contribute to answer questions arising from systems biology. / Ph. D.

Wir untersuchen das Verhalten von stochastischen bistabilen und erregbaren Systemen auf der Basis einer Modellierung mit diskreten Zuständen. In Ergänzung zum bekannten Markovschen Zwei-Zustandsmodell bistabiler stochastischer Dynamik stellen wir ein nicht Markovsches Drei-Zustandsmodell für erregbare Systeme vor. Seine relative Einfachheit, verglichen mit stochastischen Modellen erregbarer Dynamik mit kontinuierlichem Phasenraum, ermöglicht eine teilweise analytische Auswertung in verschiedenen Zusammenhängen. Zunächst untersuchen wir den gemeinsamen Einfluß eines periodischen Treibens und Rauschens. Dieser wird entweder mit Hilfe spektraler Größen oder durch Synchronisation des Systems mit dem treibenden Signal charakterisiert. Wir leiten analytische Ausdrücke für die spektrale Leistungsverstärkung und das Signal-zu-Rauschen Verhältnis für periodisch getriebene Renewal-Prozesse her und wenden diese auf das diskrete Modell für erregbare Dynamik an. Stochastische Synchronization des Systems mit dem treibenden Signal wird auf der Basis der Diffusionseigenschaften der Übergangsereignisse zwischen den diskreten Zuständen untersucht. Wir leiten allgemeine Formeln her, um die mittlere Häufigkeit dieser Ereignisse sowie deren effektiven Diffusionskoeffizienten zu berechnen. Über die konkrete Anwendung auf die untersuchten diskreten Modelle hinaus stellen diese Ergebnisse ein neues Werkzeug für die Untersuchung periodischer Renewal-Prozesse dar. Schließlich betrachten wir noch das Verhalten global gekoppelter bistabiler und erregbarer Systeme. Im Gegensatz zu bistabilen System können erregbare Systeme synchronisiert werden und zeigen kohärente Oszillationen. Alle Untersuchungen des nicht Markovschen Drei-Zustandsmodells werden mit dem prototypischen Modell für erregbare Dynamik, dem FitzHugh-Nagumo System, verglichen und zeigen eine gute Übereinstimmung. / We investigate the behavior of stochastic bistable and excitable dynamics based on a discrete state modeling. In addition to the well known Markovian two state model for bistable dynamics we introduce a non Markovian three state model for excitable systems. Its relative simplicity compared to stochastic models of excitable dynamics with continuous phase space allows to obtain analytical results in different contexts. First, we study the joint influence of periodic signals and noise, both based on a characterization in terms of spectral quantities and in terms of synchronization with the periodic driving. We present expressions for the spectral power amplification and signal to noise ratio for renewal processes driven by periodic signals and apply these results to the discrete model for excitable systems. Stochastic synchronization of the system to the driving signal is investigated based on diffusion properties of the transition events between the discrete states. We derive general results for the mean frequency and effective diffusion coefficient which, beyond the application to the discrete models considered in this work, provide a new tool in the study of periodically driven renewal processes. Finally the behavior of globally coupled excitable and bistable units is investigated based on the discrete state description. In contrast to the bistable systems, the excitable system exhibits synchronization and thus coherent oscillations. All investigations of the non Markovian three state model are compared with the prototypical continuous model for excitable dynamics, the FitzHugh-Nagumo system, revealing a good agreement between both models.

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.

Made available in DSpace on 2016-06-02T20:05:59Z (GMT). No. of bitstreams: 1 DissLEBS.pdf: 1626270 bytes, checksum: c18112f89ed0a1eea09a198885cf2c2c (MD5) Previous issue date: 2007-06-14 / Financiadora de Estudos e Projetos / Abstract In this dissertation we will study how extended Poisson process can be applied to construct discrete probabilistic models. An Extended Poisson Process is a continuous time stochastic process with the state space being the natural numbers, it is obtained as a generalization of homogeneous Poisson process where transition rates depend on the current state of the process. From its transition rates and Chapman-Kolmogorov di¤erential equations, we can determine the probability distribution at any …xed time of the process. Conversely, given any probability distribution on the natural numbers, it is possible to determine uniquely a sequence of transition rates of an extended Poisson process such that, for some instant, the unidimensional probability distribution coincides with the provided probability distribution. Therefore, we can conclude that extended Poisson process is as a very ‡exible framework on the analysis of discrete data, since it generalizes all probabilistic discrete models. We will present transition rates of extended Poisson process which generate Poisson, Binomial and Negative Binomial distributions and determine maximum likelihood estima- tors, con…dence intervals, and hypothesis tests for parameters of the proposed models. We will also perform a bayesian analysis of such models with informative and noninformative prioris, presenting posteriori summaries and comparing these results to those obtained by means of classic inference. / Nesta dissertação veremos como o proceso de Poisson estendido pode ser aplicado à construção de modelos probabilísticos discretos. Um processo de Poisson estendido é um processo estocástico a tempo contínuo com espaço de estados igual ao conjunto dos números naturais, obtido a partir de uma generalização do processo de Poisson homogê- neo onde as taxas de transição dependem do estado atual do processo. A partir das taxas de transição e das equações diferenciais de Chapman-Kolmogorov pode-se determinar a distribuição de probabilidades para qualquer tempo …xado do processo. Reciprocamente, dada qualquer distribuição de probabilidades sobre o conjunto dos números naturais é pos- sível determinar, de maneira única, uma seqüência de taxas de transição de um processo de Poisson estendido tal que, para algum instante, a distribução unidimensional do processo coincide com a dada distribuição de probabilidades. Portanto, o processo de Poisson es- tendido se apresenta como uma ferramenta bastante ‡exível na análise de dados discretos, pois generaliza todos os modelos probabilísticos discretos. Apresentaremos as taxas de transição dos processos de Poisson estendido que ori- ginam as distribuições de Poisson, Binomial e Binomial Negativa e determinaremos os estimadores de máxima verossimilhança, intervalos de con…ança e testes de hipóteses dos parâmetros dos modelos propostos. Faremos também uma análise bayesiana destes mod- elos com prioris informativas e não informativas, apresentando os resumos a posteriori e comparando estes resultados com aqueles obtidos via inferência clássica.

Cette thèse s'intéresse à l'étude des stratégies d'introduction de populations dans l'environnement. Les deux principaux contextes présentés sont la lutte biologique et la réintroduction d'espèces. Si ces deux types d'introduction diffèrent, des processus biotiques et abiotiques les influencent de manière similaire. En particulier les populations introduites, souvent de petite taille, peuvent être sensibles à diverses formes de stochasticité, voire subir une baisse de leur taux de croissance à faible effectif, ce qu'on appelle « effet Allee ». Ces processus peuvent interagir avec les stratégies d'introduction des organismes et moduler leur efficacité. Dans un premier temps, nous modélisons le processus d'introduction à l'aide de systèmes dynamiques impulsionnels : la dynamique de la population est décrite par des équations différentielles ordinaires qui, à des instants donnés, sont perturbées par des augmentations soudaines de la taille de la population. Cette approche se concentre sur l'influence des effets Allee sur les populations isolées (réintroduction) ou dans un cadre proie-prédateur (lutte biologique). Dans un second temps, en nous concentrant sur l'aspect réintroduction, nous étendons ce cadre de modélisation pour prendre en compte des aspects stochastiques liés à l'environnement ou aux introductions elles-mêmes. Finalement, nous considérons un modèle individu centré pour étudier l'effet de la stochasticité démographique inhérente aux petites populations. Ces différentes approches permettent d'analyser l'influence de la distribution temporelle des introductions et ainsi déterminer les stratégies qui maximisent les chances de succès des introductions. / This thesis investigates introduction strategies of populations in the environment. Two main situations are considered: biological control and species reintroduction. Although these two kinds of introductions are different, many biotic and abiotic processes influence them in a similar way. Introduced populations are often small and may be sensitive to various stochastic factors. Further, small populations may suffer from a decrease of their growth rate when the population is small, a feature called "Allee effect". These processes may interact with introduction strategies and modulate their efficiency. First, we represent the introduction process using impulsive dynamical systems: population dynamics are described by ordinary differential equations that are disrupted at some instants by instantaneous increases of the population size. This approach focuses on the influence of Allee effects on single-species (reintroduction) or predator-prey interactions (biological control). Then, we concentrate on the reintroduction approach and extend the previous deterministic framework to take into consideration stochastic factors arising from the environment or from introductions themselves. Finally, we consider an individual-based model to study the effects of demographic stochasticity which is inherent to small populations. These different approaches allow to investigate the temporal distribution of introductions and determine which introduction strategies maximize the probability of success of introductions.