Sistemas diferenciais quadráticos planares estão presentes em muitas áreas da matemática aplicada. Embora mais de mil artigos tenham sido publicados sobre os sistemas quadráticos ainda resta muito a se conhecer sobre esses sistemas. Problemas clássicos, e em particular o XVI problema de Hilbert, estão ainda em aberto para essa família. Um dos objetivos dos pesquisadores contemporâneos é obter a classificação topológica completa dos sistemas quadráticos. Devido ao grande número de parâmetros (essa família possui doze parâmetros e, aplicando transformações afins e reescala do tempo, reduzimos esse número a cinco, sendo ainda um número grande para se trabalhar) usualmente subclasses são consideradas nas investigações realizadas. Quando características específicas são levadas em consideração, o número de parâmetros é reduzido e o estudo se torna possível. Nesta tese estudamos principalmente duas subfamílias de sistemas quadráticos: a primeira possuindo um nó triplo semielemental e a segunda possuindo uma selanó semi elemental finita e uma selanó semielemental infinita formada pela colisão de uma sela infinita com um nó infinito. Os diagramas de bifurcação para ambas as famílias são tridimensionais. A família tendo um nó triplo gera 28 retratos de fase topologicamente distintos, enquanto o fecho da família tendo as selasnós dentro do espaço de bifurcação de sua forma normal gera 417. Polinômios invariantes são usados para construir os conjuntos de bifurcação e os retratos de fase topologicamente distintos são representados no disco de Poincaré. Os conjuntos de bifurcação são a união de superfícies algébricas e superfícies cuja presença foi detectada numericamente. Ainda nesta tese, apresentamos todos os retratos de fase de um sistema diferencial conhecido como modelo do tipo SIS (sistema suscetívelinfectadosuscetível, muito comum na matemática aplicada) e a classificação dos sistemas quadráticos possuindo hipérboles invariantes. Ambos sistemas foram investigados usando de polinômios invariantes afins. / Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilberts 16th problem, are still open for this family. One of the goals of recent researchers is the topological classification of quadratic systems. As this attempt is not possible in the whole class due to the large number of parameters (twelve, but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a large number), many subclasses are considered and studied. Specific characteristics are taken into account and this implies a decrease in the number of parameters, which makes possible the study. In this thesis we mainly study two subfamilies of quadratic systems: the first one possessing a finite semielemental triple node and the second one possessing a finite semielemental saddlenode and an infinite semielemental saddlenode formed by the collision of an infinite saddle with an infinite node. The bifurcation diagram for both families are tridimensional. The family having the triple node yields 28 topologically distinct phase portraits, whereas the closure of the family having the saddlenodes within the bifurcation space of its normal form yields 417. Invariant polynomials are used to construct the bifurcation sets and the phase portraits are represented on the Poincaré disk. The bifurcation sets are the union of algebraic surfaces and surfaces whose presence was detected numerically. Moreover, we also present the analysis of a differential system known as SIS model (this kind of systems are easily found in applied mathematics) and the complete classification of quadratic systems possessing invariant hyperbolas.

Cette thèse comporte deux parties indépendantes. Dans la première partie, nous nous intéressons à l’analyse asymptotique de quelques statistiques sur les arbres binaires de recherche (ABR). Dans la deuxième partie, nous nous intéressons à l’étude du problème du collectionneur de coupons impatient. Dans la première partie, en suivant le modèle introduit par Aguech, Lasmar et Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], on définit la profondeur pondérée d’un nœud dans un arbre binaire enraciné étiqueté comme la somme de toutes les clés sur le chemin qui relie ce nœud à la racine. Nous analysons alors dans ABR, les profondeurs pondérées des nœuds avec des clés données, le dernier nœud inséré, les nœuds ordonnés selon le processus de recherche en profondeur, la profondeur pondérée des trajets, l’indice de Wiener pondéré et les profondeurs pondérées des nœuds avec au plus un enfant. Dans la deuxième partie, nous étudions la forme asymptotique de la courbe de la complétion de la collection conditionnée à T_n≤ (1+Λ), Λ>0, où T_n≃n ln⁡n désigne le temps nécessaire pour compléter la collection. Puis, en tant qu’application, nous étudions les automates déterministes et accessibles et nous fournissons une nouvelle dérivation d’une formule due à Korsunov [Kor78, Kor86] / This Phd thesis is divided into two independent parts. In the first part, we provide an asymptotic analysis of some statistics on the binary search tree. In the second part, we study the coupon collector problem with a constraint. In the first part, following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133—141], the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyze the following statistics : the weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search procees, the weighted path length, the weighted Wiener index and the weighted depths of nodes with at most one child in a random binary search tree. In the second part, we study the asymptotic shape of the completion curve of the collection conditioned to T_n≤ (1+Λ), Λ>0, where T_n≃n ln⁡n is the time needed to complete accessible automata, we provide a new derivation of a formula due to Korsunov [Kor78, Kor86]

In this thesis, geometry is used as a basis for conducting experiments aimed at growing and arranging inorganic minerals on curved interfaces. Mineralisation is directed using crystalline and liquid-crystalline metallic soaps and surfactant/water systems as templates.¶ A review of the history, syntheses, structure and liquid crystallinity of metallic soaps and other amphiphiles is presented as a foundation to understanding the interfacial architectures in mesostructured template systems in general.¶ In this study, a range of metallic soaps of varying chain length and cation type are synthesised and characterised to find potentially useful templates for mineral growth. These include alkaline-earth, transition metal, heavy metal and lanthanide soaps. These are systematically characterised using a variety of analytical techniques, including chemical analyses, x-ray diffraction (XRD) infrared spectroscopy (IR) and differential scanning calorimetry (DSC). Their molecular and crystal structures are studied using transmission electron microscopy (TEM), cryo-TEM, electron diffraction (ED), electron paramagnetic spin resonance (EPR), absorption spectroscopy (UV-VIS), high resolution laser spectroscopy, atomic force microscopy (AFM), nuclear magnetic resonance spectroscopy, scanning electron microscopy (SEM), electron dispersive x-ray analysis (EDXA), thermal gravimetric analysis (TGA) and magnetic measurements. Models for the molecular and crystal structures of metallic soaps are proposed. The soaps are predominantly lamellar crystalline or liquid crystalline lamellar rotor phases with tilted and/or untilted molecular constituents. These display evidence of varying degrees of headgroup organisation, including superstructuring and polymerisation. A single crystal structure is presented for a complex of pyridine with cobalt soap. Simple models for their structure are discussed in terms of their swelling properties in water and oils. Experiments are also presented to demonstrate the sorbent properties of aluminium soaps on oil spills.¶ The thermotropic liquid crystallinity of alkaline earth, transition metal, heavy metal and lanthanide soaps is investigated in detail. This is done to assess their suitability as templates, and to document their novel thermotropic behaviour, particularly the relatively unknown lanthanide soaps. Liquid crystalline behaviours are studied using high-temperature XRD (HTXRD), hot-stage optical microscopy and DSC. Models for a liquid crystalline phase progression from crystals to anisotropic liquids are discussed in terms of theories of self-assembly and interfacial curvature. The terminology required for this is drawn from various nomenclature systems for amphiphilic crystals and liquid crystals. General agreement with previous studies is reported for known soaps, while liquid crystallinity is demonstrated in the lanthanide and some non-lanthanide soaps for the first time. A general phase progression of crystalline lamellar through liquid crystalline lamellar to non-lamellar liquid crystalline is discussed in terms of models concerned with the molecular and crystal structures of the soaps and their phase transitions via headgroup and chain re-arrangements.¶ Experiments aimed at guiding growth of metal sulfides using metallic soaps as templates are described, and a model for this growth is discussed. Metal sulfides have been successfully grown by reacting crystalline and liquid crystalline transition metal and heavy metal soaps with H2S gas at room temperature and at elevated temperature. These have been characterised using XRD, TEM, ED and IR. Sulfide growth is demonstrated to be restricted and guided by the reacting soap template architecture. Zinc, cadmium, indium and lead soaps formed confined nanoparticles within the matrix of their reacting soap template. In contrast, curved and flat sheet-like structures, some resembling sponges were found in the products of sulfided iron, cobalt, nickel, copper, tin and bismuth soaps. A model to explain this behaviour is developed in terms of the crystal and liquid crystal structures of the soaps and the crystal structures of the metal sulfide particles.¶ Liquid crystalline iron soaps have been subjected to controlled thermal degradation yielding magnetic iron oxide nanoparticles. Some XRD and TEM evidence has been found for formation of magnetic mesostructures in heat-treated iron soaps. Models for the molecular and liquid crystalline structure of iron soaps, their thermotropic phase progression and eventual conversion to these magnetic products are discussed. Systematic syntheses of mesoporous silicates from sheeted clays are discussed.¶The templates that have been used are cationic surfactants and small, organic molecular salts. Experiments are reported where a cooperative self-assembly of surfactant/water/kanemite plus or minus salt and oils yields 'folded sheet materials' (FSM'S). Templating of kanemite has also been achieved using cobalt cage surfactants. A theoretical prediction of the specific surface areas and specific volumes of homologous sets of FSM's gave excellent agreement with measured values. The geometry and topology of the mesostructures are discussed. A theoretical model is also discussed regarding the curvature found in the sheets of natural clays , and results of templating clays and silica using metallic soaps are presented. Experiments and a model for low temperature nucleation and growth of microporous silicalite-1 are described in terms of silica templating by water clathrates.¶ Finally, the problem of finding minimal surface descriptions of crystal networks is addressed. Combinatoric methods are used to disprove the existence of possible embeddings of type I and II clathrate networks in non-self intersecting periodic minimal surfaces. The crystal network of the clathrate silicate, melanophlogite is successfully embedded in the WI-10 self-intersecting surface. Details of a previously unreported, genus-25 periodic surface with symmetry Im3m are discussed.

<p>A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK.</p><p>Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics.</p><p>New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance.</p><p>Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK.</p><p>The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.</p>

A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK. Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics. New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance. Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK. The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.

Dans ce mémoire, nous étudions le problème centre-foyer sur un système polynomial. Nous développons ainsi deux mécanismes permettant de conclure qu’un point singulier monodromique dans ce système non-linéaire polynomial est un centre. Le premier mécanisme est la méthode de Darboux. Cette méthode utilise des courbes algébriques invariantes dans la construction d’une intégrale première. La deuxième méthode analyse la réversibilité algébrique ou analytique du système. Un système possédant une singularité monodromique et étant algébriquement ou analytiquement réversible à ce point sera nécessairement un centre. Comme application, dans le dernier chapitre, nous considérons le modèle de Gauss généralisé avec récolte de proies. / In this thesis, we study the center-focus problem in a polynomial system. We describe two mechanisms to conclude that a monodromic singular point in this polynomial system is a center. The first one is the method of Darboux. In this method, one uses invariant algebraic curves to build a first integral. The second method is the algebraic (and analytic) reversibility. A monodromic singularity, which is algebraically or analytically reversible at the singular point, is necessarily a center. As an application, in the last chapter, we consider the generalized Gause model with prey harvesting and a generalized Holling response function of type III.

Le mouvement Brownien réfléchi de manière oblique dans le quadrant, introduit par Harrison, Reiman, Varadhan et Williams dans les années 80, est un objet largement analysé dans la littérature probabiliste. Cette thèse, qui présente l’étude complète de la mesure invariante de ce processus dans tous les cônes du plan, a pour objectif plus global d’étendre au cadre continu une méthode analytique développée initialement pour les marches aléatoires dans le quart de plan par Fayolle, Iasnogorodski et Malyshev dans les années 70. Cette approche est basée sur des équations fonctionnelles, reliant des fonctions génératrices dans le cas discret et des transformées de Laplace dans le cas continu. Ces équations permettent de déterminer et de résoudre des problèmes frontière satisfaits par ces fonctions génératrices. Dans le cas récurrent, cela permet de calculer explicitement la mesure invariante du processus avec rebonds orthogonaux, dans le chapitre 2, et avec rebonds quelconques, dans le chapitre 3. Les transformées de Laplace des mesures invariantes sont prolongées analytiquement sur une surface de Riemann induite par le noyau de l’équation fonctionnelle. L’étude des singularités et l’application de méthodes du point col sur cette surface permettent de déterminer l’asymptotique complète de la mesure invariante selon toutes les directions dans le chapitre 4. / Obliquely reflected Brownian motion in the quadrant, introduced by Harrison, Reiman, Varadhan and Williams in the eighties, has been studied a lot in the probabilistic literature. This thesis, which presents the complete study of the invariant measure of this process in all the cones of the plan, has for overall aim to extend to the continuous framework an analytic method initially developped for random walks in the quarter plane by Fayolle, Iasnogorodski and Malyshev in the seventies. This approach is based on functional equations which link generating functions in the discrete case and Laplace transform in the continuous case. These equations allow to determine and to solve boundary value problems satisfied by these generating functions. In the recurrent case, it permits to compute explicitly the invariant measure of the process with orthogonal reflexions, in the chapter 2, and with any reflexions, in the chapter 3. The Laplace transform of the invariant measure is analytically extended to a Riemann surface induced by the kernel of the functional equation. The study of singularities and the use of saddle point methods on this surface allows to determine the full asymptotics of the invariant measure along every directions in the chapter 4.

Sistemas diferenciais quadráticos planares estão presentes em muitas áreas da matemática aplicada. Embora mais de mil artigos tenham sido publicados sobre os sistemas quadráticos ainda resta muito a se conhecer sobre esses sistemas. Problemas clássicos, e em particular o XVI problema de Hilbert, estão ainda em aberto para essa família. Um dos objetivos dos pesquisadores contemporâneos é obter a classificação topológica completa dos sistemas quadráticos. Devido ao grande número de parâmetros (essa família possui doze parâmetros e, aplicando transformações afins e reescala do tempo, reduzimos esse número a cinco, sendo ainda um número grande para se trabalhar) usualmente subclasses são consideradas nas investigações realizadas. Quando características específicas são levadas em consideração, o número de parâmetros é reduzido e o estudo se torna possível. Nesta tese estudamos principalmente duas subfamílias de sistemas quadráticos: a primeira possuindo um nó triplo semielemental e a segunda possuindo uma selanó semi elemental finita e uma selanó semielemental infinita formada pela colisão de uma sela infinita com um nó infinito. Os diagramas de bifurcação para ambas as famílias são tridimensionais. A família tendo um nó triplo gera 28 retratos de fase topologicamente distintos, enquanto o fecho da família tendo as selasnós dentro do espaço de bifurcação de sua forma normal gera 417. Polinômios invariantes são usados para construir os conjuntos de bifurcação e os retratos de fase topologicamente distintos são representados no disco de Poincaré. Os conjuntos de bifurcação são a união de superfícies algébricas e superfícies cuja presença foi detectada numericamente. Ainda nesta tese, apresentamos todos os retratos de fase de um sistema diferencial conhecido como modelo do tipo SIS (sistema suscetívelinfectadosuscetível, muito comum na matemática aplicada) e a classificação dos sistemas quadráticos possuindo hipérboles invariantes. Ambos sistemas foram investigados usando de polinômios invariantes afins. / Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilberts 16th problem, are still open for this family. One of the goals of recent researchers is the topological classification of quadratic systems. As this attempt is not possible in the whole class due to the large number of parameters (twelve, but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a large number), many subclasses are considered and studied. Specific characteristics are taken into account and this implies a decrease in the number of parameters, which makes possible the study. In this thesis we mainly study two subfamilies of quadratic systems: the first one possessing a finite semielemental triple node and the second one possessing a finite semielemental saddlenode and an infinite semielemental saddlenode formed by the collision of an infinite saddle with an infinite node. The bifurcation diagram for both families are tridimensional. The family having the triple node yields 28 topologically distinct phase portraits, whereas the closure of the family having the saddlenodes within the bifurcation space of its normal form yields 417. Invariant polynomials are used to construct the bifurcation sets and the phase portraits are represented on the Poincaré disk. The bifurcation sets are the union of algebraic surfaces and surfaces whose presence was detected numerically. Moreover, we also present the analysis of a differential system known as SIS model (this kind of systems are easily found in applied mathematics) and the complete classification of quadratic systems possessing invariant hyperbolas.

The content of the master thesis is processing of project documentation for The Mountain Guesthouse in Trojanovice, first done as a pilot project and then as a feasibility study. It concerns a two-storey detached building with a parking stand, with a basement. This house is designed from the Porotherm building system, ceiling construction - ceramic ceiling panels, saddle roofs with dormer windows. The work also comprises thermal technical assessment chosen constructions, fire protection report, protocol - the energy label of the building envelope, dispositional design and further calculations.

A master thesis „Clinic of holistic medicine“ is processed as a project documentation. The building is projected for the plat no.170/1 in Kladruby u Vlašimy (Benešov district). Is a three-storyed wooden building from large-format wooden panels. Ceilings are ribbed constructions from spruce wood. Stairspaces are made of reinforced concrete. The house is roofed with a saddle roof made by woodentrusses. Stairspaces and machine room of air conditioning are roofed with a one-sheat flat roof. The project is planned as an extension of the Rehabilitation institute complex in Kladruby.