In this thesis we analyze lumpability of infinite dimensional dynamical systems. Lumping is a method to project a dynamics by a linear reduction operator onto a smaller state space on which a self-contained dynamical description exists. We consider a well-posed dynamical system defined on a Banach space X and generated by an operator F, together with a linear and bounded map M : X → Y, where Y is another Banach space. The operator M is surjective but not an isomorphism and it represents a reduction of the state space. We investigate whether the variable y = M x also satisfies a well-posed and self-contained dynamics on Y . We work in the context of strongly continuous semigroup theory. We first discuss lumpability of linear systems in Banach spaces. We give conditions for a reduced operator to exist on Y and to describe the evolution of the new variable y . We also study lumpability of nonlinear evolution equations, focusing on dissipative operators, for which some interesting results exist, concerning the existence and uniqueness of solutions, both in the classical sense of smooth solutions and in the weaker sense of strong solutions. We also investigate the regularity properties inherited by the reduced operator from the original operator F . Finally, we describe a particular kind of lumping in the context of C*-algebras. This lumping represents a different interpretation of a restriction operator. We apply this lumping to Feller semigroups, which are important because they can be associated in a unique way to Markov processes. We show that the fundamental properties of Feller semigroups are preserved by this lumping. Using these ideas, we give a short proof of the classical Tietze extension theorem based on C*-algebras and Gelfand theory.

In this thesis we analyze lumpability of infinite dimensional dynamical systems. Lumping is a method to project a dynamics by a linear reduction operator onto a smaller state space on which a self-contained dynamical description exists. We consider a well-posed dynamical system defined on a Banach space X and generated by an operator F, together with a linear and bounded map M : X → Y, where Y is another Banach space. The operator M is surjective but not an isomorphism and it represents a reduction of the state space. We investigate whether the variable y = M x also satisfies a well-posed and self-contained dynamics on Y . We work in the context of strongly continuous semigroup theory. We first discuss lumpability of linear systems in Banach spaces. We give conditions for a reduced operator to exist on Y and to describe the evolution of the new variable y . We also study lumpability of nonlinear evolution equations, focusing on dissipative operators, for which some interesting results exist, concerning the existence and uniqueness of solutions, both in the classical sense of smooth solutions and in the weaker sense of strong solutions. We also investigate the regularity properties inherited by the reduced operator from the original operator F . Finally, we describe a particular kind of lumping in the context of C*-algebras. This lumping represents a different interpretation of a restriction operator. We apply this lumping to Feller semigroups, which are important because they can be associated in a unique way to Markov processes. We show that the fundamental properties of Feller semigroups are preserved by this lumping. Using these ideas, we give a short proof of the classical Tietze extension theorem based on C*-algebras and Gelfand theory.

This thesis is concerned with characterizing the quality of Hidden Markov modeling when learning from limited data. It introduces a new perspective on different sources of errors to describe the impact of undermodeling. Our view is that modeling errors can be decomposed into two primary sources of errors: the approximation error and the estimation error. This thesis takes a first step towards exploring the approximation error of low order HMMs that best approximate the true system of a HMM. We introduce the notion minimality and show that best approximations of the true system with complexity greater or equal to the order of a minimal system are actually equivalent realizations. Understanding this further allows us to explore integer lumping and to present a new way named weighted lumping to find realizations. We also show that best approximations of order strictly less than that of a minimal realization are truly approximations; they are incapable of mimicking the true system exactly. Our work then proves that the resulting approximation error is non-decreasing as the model order decreases, verifying the intuitive idea that increasingly simplified models are less and less descriptive of the true system.

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Ce travail, à l'interface entre les mathématiques appliquées et la physique, s'articule autour de la modélisation numérique des vésicules biologiques, un modéle pour les globules rouges du sang. Pour cela, le modéle de Canham et Helfrich est adopté pour décrire le comportement des vésicules. La modélisation numérique utilise la méthode des fonctions de niveau dans un cadre éléments finis. Un nouvel algorithme de résolution numérique combinant une technique de multiplicateurs de Lagrange avec une adaptation automatique de maillages garantit la conservation exacte des volumes et des surfaces. Cet algorithme permet donc de dépasser une limitation cruciale actuelle de la méthode des fonctions de niveau, à savoir les pertes de masse couramment observées dans ce type de problémes. De plus, les propriétés de convergence de la méthode des fonctions de niveau se trouvent ainsi grandement améliorées, comme l'indiquent de nombreux tests numériques. Ces tests comprennent notamment des problémes d'advection élémentaires, des mouvements par courbure moyenne ainsi que des mouvements par diffusion de surface. Concernant l'équilibre statique des vésicules, une condition générale d'équilibre d'Euler-Lagrange est obtenue à l'aide d'outils de dérivation de forme. En dynamique, le mouvement d'une vésicule sous l'action d'un écoulement de cisaillement est étudié dans le cadre des nombres de Reynolds élevés. L'effet du confinement est considéré, et les régimes classiques de chenille de char et de basculement sont retrouvés. Finalement, pour la premiére fois, l'effet des termes inertiels est étudié et on montre qu'au delà d'une valeur critique du nombre de Reynolds, la vésicule passe d'un mouvement de basculement à un mouvement de chenille de char. / This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that's to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler-Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated in the frames of reference of high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of the number of Reynolds the vesicle passes from a tumbling motion to a tank-treading motion.

This thesis deals with the dividing the dictionary entry into several "sub-meanings", i.e. with the sense division, and with the closely related ordering of these "sub-meanings", i.e. with the sense ordering, with the bilingual passive German-Czech dictionary in the centre of interest. This thesis deals shortly also with the discriminating the senses by different means, i.e. with the sense discrimination. With these three subjects chronologically following the lexicographic decisions the matter of equivalence and the understanding of the term of meaning are correlated. In the theoretical part of this thesis the specifics of the sense division and the sense ordering in the monolingual and in the bilingual lexicography are introduced, in the practical part of this thesis the possibilities of the sense division and the sense ordering are exemplified with ten chosen substantive lemmas prepared for the Large German-Czech Academic Dictionary in progress. For every lemma the most suitable arrangement of the lemma is suggested, which is then compared with the corresponding dictionary entry in the source dictionary Duden - Deutsches Universalwörterbuch. The differences between the arrangements of the microstructure illustrate the necessity of the revision and eventual modification of the adopted structure...

The incompressible Navier-Stokes equations have proven formidable for nearly a century. The present difficulties are mathematical and computational in nature; the computational requirements, in particular, are exponentially exacerbated in the presence of high Reynolds number. The issues are further compounded with the introduction of markers or an immiscible fluid intended to be tracked in an ambient high Reynolds number flow; despite the overwhelming pragmatism of problems in this regime, and increasing computational efficacy, even modest problems remain outside the realm of direct approaches. Herein three approaches are presented which embody direct application to problems of this nature. An LES model based on an entropy-viscosity serves to abet the computational resolution requirements imposed by high Reynolds numbers and a one-stage compressive flux, also utilizing an entropy-viscosity, aids in accurate, efficient, conservative transport, free of low order dispersive error, of an immiscible fluid or tracer. Finally, an integral commutator and the theory of anti-dispersive spaces is introduced as a novel theoretical tool for consistency error analysis; in addition the material engenders the construction of error-correction techniques for mass lumping schemes.