The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite, ordered alphabets is studied using a straightforward Brownian functional approach. Building on the insights gained in both the uniform and non-uniform iid cases, this approach is then applied to iid countable alphabets. Some partial results associated with the extension to independent, growing alphabets are also given. Returning again to the finite setting, and keeping with the same Brownian formalism, a generalization is then made to words arising from irreducible, aperiodic, time-homogeneous Markov chains on a finite, ordered alphabet. At the same time, the probabilistic object, LI_n, is simultaneously generalized to the shape of the associated Young tableau given by the well-known RSK-correspondence. Our results on this limiting shape describe, in detail, precisely when the limiting shape of the Young tableau is (up to scaling) that of the iid case, thereby answering a conjecture of Kuperberg. These results are based heavily on an analysis of the covariance structure of an m-dimensional Brownian motion and the precise form of the Brownian functionals. Finally, in both the iid and more general Markovian cases, connections to the limiting laws of the spectrum of certain random matrices associated with the Gaussian Unitary Ensemble (GUE) are explored.

Spanwise nonuniformity effects are modeled in the cross-sectional analysis of beam theory. This modeling adheres to an established numerical framework on cross-sectional analysis of uniform beams with arbitrary cross-sections. This framework is based on two concepts: decomposition of the rotation tensor and the variational-asymptotic method. Allowance of arbitrary materials and geometries in the cross-section is from discretization of the warping field by finite elements. By this approach, dimensional reduction from three-dimensional elasticity is performed rigorously and the sectional strain energy is derived to be asymptotically-correct. Elastic stiffness matrices are derived for inputs into the global beam analysis. Recovery relations for the displacement, stress, and strain fields are also derived with care to be consistent with the energy. Spanwise nonuniformity effects appear in the form of pointwise and sectionwise derivatives, which are approximated by finite differences. The formulation also accounts for the effects of spanwise variations in initial twist and/or curvature. A linearly tapered isotropic strip is analyzed to demonstrate spanwise nonuniformity effects on the cross-sectional analysis. The analysis is performed analytically by the variational-asymptotic method. Results from beam theory are validated against solutions from plane stress elasticity. These results demonstrate that spanwise nonuniformity effects become significant as the rate at which the cross-sections vary increases. The modeling of transverse shear modes of deformation is accomplished by transforming the strain energy into generalized Timoshenko form. Approximations in this transformation procedure from previous works, when applied to uniform beams, are identified. The approximations are not used in the present work so as to retain more accuracy. Comparison of present results with those previously published shows that these approximations sometimes change the results measurably and thus are inappropriate. Static and dynamic results, from the global beam analysis, are calculated to show the differences between using stiffness constants from previous works and the present work. As a form of validation of the transformation procedure, calculations from the global beam analysis of initially twisted isotropic beams from using curvilinear coordinate axes featuring twist are shown to be equivalent to calculations using Cartesian coordinates.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-06-03Bitstream added on 2014-06-13T18:47:52Z : No. of bitstreams: 1 silvajunior_jb_me_sjrp.pdf: 533913 bytes, checksum: 2ffa5488599336df8a97baf938757756 (MD5) / Neste trabalho estudamos o Método Averaging. Este método é uma ferramenta extremamente útil para quantificar o número de ciclos limites que podem bifurcar de uma singularidade do tipo centro de um sistema de equações diferenciais. A parte inicial do trabalho apresenta a Teoria de Aproximação Assintótica e um primeiro contato com o Averaging. Posteriormente apresentamos uma versão do Averaging via a Teoria do Grau de Brouwer. Finalmente fizemos algumas aplicações do método apresentando uma cota superior para o número de ciclos limites que podem bifurcar a partir das órbitas periódicas de centros de um sistema de equações diferenciais. Além disso, mostramos através de exemplos concretos que esta cota superior pode ser realizada. / In this work we study the Averaging Method. This method is a useful tool in order to give the maximum number of limit cycles that can bifurcate from a center type singularity of a di®erential equation system. In the first part of the work we present the Asymptotic Approximation Theory and a first view of the averaging. After that, we present a version of the averaging via Brouwer Degree Theory. Finally we give some applications of this method presenting an upper bound for the number of limit cycles that can bifurcate from a center type singularity of a di®erential equation system. Moreover, we show by presenting concrete examples that this upper bound can be realized.

We consider the problem of estimating the failure stresses of bundles (i.e. the tensile forces that destroy the bundles), constructed of several statisti-cally similar fibres, given a particular kind of censored data. Each bundle consists of several fibres which have their own independent identically dis-tributed failure stresses, and where the force applied on a bundle at any moment is distributed equally between the unbroken fibres in the bundle. A bundle with these properties is an example of an equal load-sharing sys-tem, often referred to as the Daniels failure model. The testing of several bundles generates a special kind of censored data, which is complexly struc-tured. Strongly consistent non-parametric estimators of the distribution laws of bundles are obtained by applying the theory of martingales, and by using the observed data. It is proved that random sampling, with replace-ment from the statistical data related to each tested bundle, can be used to obtain asymptotically correct estimators for the distribution functions of deviations of non-parametric estimators from true values. In the case when the failure stresses of the fibres are described by a Weibull distribution, we obtain strongly consistent parametric maximum likelihood estimators of the distribution functions of failure stresses of bundles, by using the complexly structured data. Numerical examples illustrate the behavior of the obtained estimators.

This thesis studies numerical methods for integrating the master equations describing Markov chain models of cardiac ion channels. Such models describe the time evolution of the probability that ion channels are in a particular state. Numerical simulations of such models are often computationally demanding because many solvers require relatively small time steps to ensure numerical stability. The aim of this project is to analyse selected Markov chains and develop more efficient and accurate solvers. We separate a Markov chain model into fast and slow time-scales based on the speed of transitions between states. Eliminating the fast transitions, we find an asymptotic reduction of zeroth-order and first-order in a small parameter describing the time-scales separation. We apply the theory to a Markov chain model of the fast sodium channel INa. We consider several variants for classifying some transitions as fast in order to find reduced systems that yield a good accuracy. However, the time step size is still restricted by numerical instabilities. We adapt the Rush-Larsen technique originally developed for gate models. Assuming that a transition matrix can be considered constant during each time step, we solve the Markov chain model analytically. The solution provides a recipe for a stable exponential solver, which we call "Matrix Rush-Larsen" (MRL). Using operator splitting we design an even more flexible "hybrid" method that combines the MRL with other solvers. The resulting improvement in stability allows a large increase in the time step size. In some models, we obtain reasonably accurate results 27 times faster using a hybrid method than with the forward Euler method, even with the maximal time step allowed by the stability constraint. Finally, we extend the cardiac simulation package BeatBox by the developed exponential solvers. We upgrade a format of "ionic" modules which describe a cardiac cell, in order to allow for a specific definition of Markov chain models. We also modify a particular integrator for ionic modules to include the MRL and the hybrid method. To test the functionality of the code, we have converted a number of cellular models into the ionic format. The documented code is available in the official BeatBox package distribution.

abstract: To establish reliable wireless communication links it is critical to devise schemes to mitigate the effects of the fading channel. In this regard, this dissertation analyzes two types of systems: point-to-point, and multiuser systems. For point-to-point systems with multiple antennas, switch and stay diversity combining offers a substantial complexity reduction for a modest loss in performance as compared to systems that implement selection diversity. For the first time, the design and performance of space-time coded multiple antenna systems that employ switch and stay combining at the receiver is considered. Novel switching algorithms are proposed and upper bounds on the pairwise error probability are derived for different assumptions on channel availability at the receiver. It is proved that full spatial diversity is achieved when the optimal switching threshold is used. Power distribution between training and data codewords is optimized to minimize the loss suffered due to channel estimation error. Further, code design criteria are developed for differential systems. Also, for the special case of two transmit antennas, new codes are designed for the differential scheme. These proposed codes are shown to perform significantly better than existing codes. For multiuser systems, unlike the models analyzed in literature, multiuser diversity is studied when the number of users in the system is random. The error rate is proved to be a completely monotone function of the number of users, while the throughput is shown to have a completely monotone derivative. Using this it is shown that randomization of the number of users always leads to deterioration of performance. Further, using Laplace transform ordering of random variables, a method for comparison of system performance for different user distributions is provided. For Poisson users, the error rates of the fixed and random number of users are shown to asymptotically approach each other for large average number of users. In contrast, for a finite average number of users and high SNR, it is found that randomization of the number of users deteriorates performance significantly. / Dissertation/Thesis / Ph.D. Electrical Engineering 2010

Cette thèse porte sur l’analyse mathématique de différents modèles de la digestion anaérobie. Dans la première partie, nous étudions un modèle à quatre étapes avec dégradation enzymatique du substrat (matière organique) qui peut être sous forme solide. Nous étudions l’effet de l’hydrolyse sur le comportement du processus de la digestion anaérobie et de la production du biogaz (méthane et hydrogène). Nous considèrons, dans un premier modèle, que l’hydrolyse se fait d’une manière enzymatique, alors que dans un second, nous supposons qu’elle est réalisée par un compartiment microbien. Les modèles considérés incluent l’inhibition de croissance des bactéries acétogènes, méthanogènes hydrogénétrophes et acétoclastes par plu- sieurs substrats. Pour étudier l’effet de ces inhibitions en présence de l’étape de l’hydrolyse, nous étudions dans un premier temps un modèle sans inhibition. Nous déterminons les équilibres et nous donnons des conditions nécessaires et suffisantes pour leur stabilité. L’existence et la stabilité des équilibres sont illustrées avec des diagrammes opératoires. Nous montrons que le modèle avec hydrolyse enzymatique change la production du méthane et d’hydrogène. En outre, l’introduction du com- partiment hydrolytique microbien donne de nouveaux équilibres et affecte les régions de stabilité. Nous prouvons que la production de biogaz est maximale en un seul point d’équilibre selon les paramètres opératoires et nous déterminons le taux maxi- mal de biogaz produit, dans chaque cas. Dans la deuxième partie, nous nous sommes intéressés à un modèle à deux étapes décrivant les phases de l’acétogénèse et de la méthanogénèse hydrogénotrophe. Le modèle représente une relation de syntrophie entre deux espèces microbiennes (les bactéries acétogènes et méthanogènes hydro- génotrophes), avec deux substrats à l’entrée (l’acide gras volatile et l’hydrogène), incluant les termes de mortalité et l’inhibition de croissance des bactéries acéto- gènes par un excès d’hydrogène dans le système. L’analyse de l’existence et de la stabilité des équilibres du modèle donne naissance à un nouvel équilibre qui peut être stable selon les paramètres opératoires du système. En utilisant les diagrammes opératoires, on remarque que, quelle que soit la région de l’espace considérée, il existe un seul équilibre localement exponentiellement stable. Cette étude est géné- ralisée dans le cas où la croissance des bactéries méthanogènes hydrogénotrophes est inhibée. Ce modèle donne naissance à deux équilibres strictement positifs et une bistabilité. Nous illustrons, en utilisant les diagrammes opératoires l’effet de cette inhibition sur la réduction des régions de coexistence et l’émergence de régions de bistabilité. / This PhD thesis focuses on the mathematical analysis of different anaerobic digestion (AD) models. In a first part, we study a 4-step model with enzymatic degradation of the substrate (organic matter) that can partly be under a solid form. We investigate the effects of hydrolysis on the behavior of the AD process and the production of biogas (namely, the methane and the hydrogen). We consider, in a first model, that the microbial enzymatic activity is constant, then we take into consideration an explicit hydrolytic microbial compartment for the substrate biodegradation. The considered models include the inhibition of acetogens, hydroge- notrophic methanogens and acetoclastic methanogens growth bacteria. To examine the effects of these inhibitions in presence of a hydrolysis step, we first study an inhibition-free model. We determine the steady states and give sufficient and neces- sary conditions for their stability. The existence and stability of the steady states are illustrated by operating diagrams. We prove that modeling the hydrolysis phase by a constant enzymatic activity affects the production of methane and hydrogen. Furthermore, introducing the hydrolytic microbial compartment yields new steady states and affects the stability regions. We prove that the biogas production occurs at only one of the steady states according to the operating parameters and state variables and we determine the maximal rate of biogas produced, in each case. In the second part, we are interested in a reduced and simplified model of the AD pro- cess. We focus on the acetogenesis and hydrogenetrophic methanogenesis phases. The model describes a syntrophic relationship between two microbial species (the acetogenic bacteria and the hydrogenetrophic methanogenic bacteria) with two in- put substrates (the fatty acids and the hydrogen) including both decay terms and inhibition of the acetogenic bacteria growth by an excess of hydrogen in the sys- tem. The existence and stability analysis of the steady states of the model points out the existence of a new equilibrium point which can be stable according to the operating parameters of the system. By means of operating diagrams, we show that, whatever the region of space considered, there exists only one locally exponentially stable steady state. This study is generalized to the case where the growth of the hydrogenetrophic methanogens bacteria is inhibited. This model exhibits a rich be- havior with the existence of two positive steady states and bistability. We illustrate by means of operating diagrams the effect of this inhibition on the reduction of the coexistence region and the emergence of a bistability region.

1 Title Helical symmetry and the non-existence of asymptotically flat periodic solutions in general relativity Author Martin Scholtz Department Institute of theoretical physics Faculty of Mathematics and Physics Charles University in Prague Supervisor Prof. RNDr. Jiří Bičák, DrSc., dr. h.c. Abstract. No exact helically symmetric solution in general relativity is known today. There are reasons, however, to expect that such solutions, if they exist, cannot be asymptotically flat. In the thesis presented we investigate a more general question whether there exist periodic asymptotically flat solutions of Einstein's equations. We follow the work of Gibbons and Stewart [3] who have shown that there are no periodic vacuum asymptotically flat solutions an- alytic near null infinity I. We discuss necessary corrections of Gibbons and Stewart proof and generalize their results for the system of Einstein-Maxwell, Einstein-Klein-Gordon and Einstein-conformal-scalar field equations. Thus, we show that there are no asymptotically flat periodic space-times analytic near I if as the source of gravity we take electromagnetic, Klein-Gordon or conformally invariant scalar field. The auxilliary results consist of corresponding confor- mal field equations, the Bondi mass and the Bondi massloss formula for scalar fields. We also...

This thesis concerns the problem of statistical hypothesis testing for mean vector as well as testing for non-normality in a high-dimensional setting which is called the Kolmogorov condition. Since we consider mainly the first and the second moment in testing for mean vector and we utilize the third and the fourth moment in testing for non-normality, this thesis concerns a more general moment testing problem. The research question is related to a data matrix with $p$ rows, which is the number of parameters and $n$ columns which is the sample size, where $p$ can exceed $n$, assuming that the ratio $\frac{p}{n}$ converges when both the number of parameters and the sample size increase.  The first paper reviews the Dempster's non-exact test for mean vector, with a focus on one-sample case. We investigated its size and power properties compared to Hotelling's $\mathit{T}^2$ test as well as Srivastava's test using Monte Carlo simulation.  The second paper concerns the problem of testing for multivariate non-normality in high-dimensional data. We proposed three test statistics which are based on marginal skewness and kurtosis. Simulation studies are carried out for examining the size and power properties of the three test statistics. / Avhandlingen undersöker hypotesprövning i höga dimensioner, under förutsättning att det så kallad Kolmogorovvillkoret (Kolmogorov condition) är uppfyllt. Villkoret innerbär att antalet parametrar ökar tillsammans med storleken på stickprovet med en konstant hastighet. Till kategorin multivariat analys räknas de statistiska metoder som analyserar stickprov från flerdimensionella fördelningar, särskilt multivariat normalfördelning. För högdimensionella data fungerar klassiska skattningar av kovariansmatris inte tillfredställande eftersom komplexiteten med att skatta den inversa kovariansmatrisen ökar när dimensionen ökar. I den första uppsatsen utförs en genomgång av Dempsters (non-exact) test där skattning av den inversa kovariansmatrisen inte behövs. Istället används spåret (trace) av en kovariansmatris. I den andra uppsatsen testas antagandet om normalfördelning med hjälp av tredje och fjärde ordningens moment. Tre olika testvariabler har föreslagits där sumuleringar också presenteras för att jämföra hur väl en icke-normalfördelning identifieras av testet.

Cette thèse porte sur la modélisation de nano-fils ferromagnétiques. La première par-tie est consacrée à la dérivation par processus asymptotique d'un modèle uni-dimen-sionnel de nano-fil ferromagnétique fini, courbé, torsadé et de section elliptique non constante, soumis à un courant électrique. Nous utilisons ensuite le modèle asympto-tique de jonction de fils pour considérer deux cas :- celui d'un fil infini présentant un coude dans la deuxième partie.-celui un fil rectiligne infini sur lequel on branche perpendiculairement un fil fini dans la troisième partie.Dans chacun des cas précédents, on explicite toutes les solutions stationnaires. Nous étudions ensuite la stabilité de ces solutions, en concluant que le coude et la jonction sont des points attracteurs du mur. Dans la dernière partie, nous introduisons une mé-thode numérique de type différences finis d'ordre 2 en espace adaptée à la simulation des systèmes de réseaux de nano-fils. Après avoir établi numériquement l'ordre de convergence de la méthode, nous validons le schéma en simulant soit des phénomènes décrits dans la littérature, soit des propriétés décrites de manières théoriques dans les parties précédents.Ainsi, nous calculons d'abord le seuil de Walker pour un fil rectiligne. De plus, nous vé-rifions que la configuration du mur est stable dans un fil pincé même en présence d'un petit champ appliqué dans la direction du fil. Par la suite nous vérifions les résultats de stabilité pour les cas d'un fil coudé de longueur finie et d'un jonction de trois fils finis. Enfin, nous étudions la propagation de plusieurs murs dans un réseau de fils sous forme d'un peigne en injectant un courant électrique. Dans cette partie toutes les simulations numériques sont faites en Python avec quelques visualisations en Matlab. / This thesis focuses on the modeling of ferromagnetic nanowires. In the first part, we derive a one-dimensional asymptotic model for the dynamics of the magnetic moment in a twisted ferromagnetic nanowire with variable elliptical cross-section, curvature and torsion, subjected to an electric current. Then, we use the new one-dimensional model to consider two cases: - the case of an infinite ferromagnetic nanowire having a bend in the second part.- the second case is when we connect perpendicularly a finite straight wire on a straight infinite horizontal wire in the third part.In both cases, we prove the existence of static solutions. We study the stability of these solutions, we conclude that the bend and the junction attract the wall profiles. In the last part, we introduce a finite difference of order 2 in space adapted to the si-mulation of nanowire network systems. After having numerically established the order of convergence of the method,we validate the scheme by simulating either phenomena described in the literature, or properties described in theoretical ways in the previous parts.We calculate the Walker field limit, for a straight wire. In addition, we verify that the wall configuration is stable in a pinched wire even in the presence of a small field ap-plied in the direction of the wire. Then we check the stability results for the case of a finite bent wire and a junction of three finite wires. Finally, we study the propagation of several walls in a network of wires in the form of a comb by injecting an electric current. In this part all the numerical simulations are made in Python with some visua-lizations in Matlab.