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In dieser Arbeit wird das Fraïssé-Hrushowskis Amalgamationsverfahren in Zusammenhang mit nilpotenten graduierten Lie Algebren über einem endlichen Körper untersucht. Die Prädimensionen die in der Konstruktion auftauchen sind mit dem gruppentheoretischen Begriff der Defizienz zu vergleichen, welche auf homologische Methoden zurückgeführt werden kann. Darüber hinaus wird die Magnus-Lazardsche Korrespondenz zwischen den oben genannten Lie Algebren und nilpotenten Gruppen von Primzahl-Exponenten beschrieben. Dabei werden solche Gruppen durch die Baker-Haussdorfsche Formel in den entsprechenden Algebren definierbar interpretiert. Es wird eine omega-stabile Lie Algebra von Nilpotenzklasse 2 und Morleyrang omega + omega erhalten, indem man eine unkollabierte Version der von Baudisch konstruierten "new uncountably categorical group" betrachtet. Diese wird genau analysiert. Unter anderem wird die Unabhängigkeitsrelation des Nicht-Gabelns durch die Konfiguration des freien Amalgams charakterisiert. Mittels eines induktiven Ansatzes werden die Grundlagen entwickelt, um neue Prädimensionen für Lie Algebren der Nilpotenzklassen größer als zwei zu schaffen. Dies erweist sich als wesentlich schwieriger als im Fall 2. Wir konzentrieren uns daher auf die Nilpotenzklasse 3, als Induktionsbasis des oben genannten Prozesses. In diesem Fall wird die Invariante der Defizienz auf endlich erzeugte Lie Algebren adaptiert. Erstes Hauptergebnis der Arbeit ist der Nachweis dass diese Definition zu einem vernüftigen Begriff selbst-genügender Erweiterungen von Lie Algebren führt und sehr nah einer gewünschten Prädimension im Hrushovskischen Sinn ist. Wir zeigen – als zweites Hauptergebnis – ein erstes Amalgamationslemma bezüglich selbst-genügender Einbettungen. / In this work, the so called Fraïssé-Hrushowski amalgamation is applied to nilpotent graded Lie algebras over the p-elements field with p a prime. We are mainly concerned with the uncollapsed version of the original process. The predimension used in the construction is compared with the group theoretical notion of deficiency, arising from group Homology. We also describe in detail the Magnus-Lazard correspondence, to switch between the aforementioned Lie algebras and nilpotent groups of prime exponent. In this context, the Baker-Hausdorff formula allows such groups to be definably interpreted in the corresponding algebras. Starting from the structures which led to Baudisch’ new uncountably categorical group, we obtain an omega-stable Lie algebra of nilpotency class 2, as the countable rich Fraïssé limit of a suitable class of finite Lie algebras. We study the theory of this structure in detail: we show its Morley rank is omega+omega and a complete description of non-forking independence is given, in terms of free amalgams. In a second part, we develop a new framework for the construction of deficiency-predimensions among graded Lie algebras of nilpotency class higher than 2. This turns out to be considerably harder than the previous case. The nil-3 case in particular has been extensively treated, as the starting point of an inductive procedure. In this nilpotency class, our main results concern a suitable deficiency function, which behaves for many aspects like a Hrushovski predimension. A related notion of self-sufficient extension is given. We also prove a first amalgamation lemma with respect to self-sufficient embeddings.

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.

Nesta dissertação, tratamos do estudo das bifurcações de um modelo bi-dimensional de presa-predador, que estende e aperfeiçoa o sistema de Lotka-Volterra. Tal modelo apresenta cinco parâmetros e uma função resposta não monotônica do tipo Holling IV: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ Estudamos as bifurcações do tipo sela-nó, Hopf, transcrítica, Bogdanov-Takens e Bogdanov-Takens degenerada. O método dos centros organizadores é usado para estudar o comportamento qualitativo do diagrama de bifurcação. / In this work are studied the bifurcations of a bi-dimensional predator-prey model, which extends and improves the Volterra-Lotka system. This model has five parameters and a non-monotonic response function of Holling IV type: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ They studied the sadle-node, Hopf, transcritic, Bogdanov-Takens and degenerate Bogdanov-Takens bifurcations. The method of organising centers is used to study the qualitative behavior of the bifurcation diagram.

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.

Nesta dissertação, tratamos do estudo das bifurcações de um modelo bi-dimensional de presa-predador, que estende e aperfeiçoa o sistema de Lotka-Volterra. Tal modelo apresenta cinco parâmetros e uma função resposta não monotônica do tipo Holling IV: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ Estudamos as bifurcações do tipo sela-nó, Hopf, transcrítica, Bogdanov-Takens e Bogdanov-Takens degenerada. O método dos centros organizadores é usado para estudar o comportamento qualitativo do diagrama de bifurcação. / In this work are studied the bifurcations of a bi-dimensional predator-prey model, which extends and improves the Volterra-Lotka system. This model has five parameters and a non-monotonic response function of Holling IV type: $$ \\left\\{\\begin \\dot=x(1-\\lambda x-\\frac{\\alpha x^2+\\beta x +1})\\\\ \\dot=y(-\\delta-\\mu y+\\frac{\\alpha x^2+\\beta x +1}) \\end ight. $$ They studied the sadle-node, Hopf, transcritic, Bogdanov-Takens and degenerate Bogdanov-Takens bifurcations. The method of organising centers is used to study the qualitative behavior of the bifurcation diagram.

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

Un entier est dit y-friable si tous ses facteurs premiers n'excèdent pas y. Les valeurs friables de formes binaires interviennent de manière essentielle dans l'algorithme de factorisation du crible algébrique (NFS). Dans cette thèse, nous obtenons des formules asymptotiques pour le nombre de représentations des entiers friables par différentes familles de polynômes. Nous considérons dans la première partie les formes binaires qui se décomposent comme produit d'une forme linéaire et d'une forme quadratique. Nous combinons pour cela le principe d'inclusion-exclusion à des idées issues de travaux sur la distribution multiplicative de certaines suites d'entiers représentés par des formes quadratiques développés par Fouvry et Iwaniec, puis Balog, Blomer, Dartyge et Tenenbaum. Dans un second temps, nous nous concentrons sur les valeurs friables de formes cubiques irréductibles. En adaptant les travaux de Heath-Brown et Moroz sur les nombres premiers représentés par de tels polynômes, nous obtenons des formules asymptotiques valides dans un vaste domaine de friabilité. Notre méthode permet également d'évaluer des moyennes sur les valeurs d'une forme cubique pour d'autres fonctions arithmétiques comprenant en particulier les fonctions de Möbius et de Liouville. Dans le dernier chapitre, nous étudions les corrélations de l'indicatrice des friables avec les nilsuites. En employant la méthode nilpotente de Green et Tao, nous en déduisons une formule pour le nombre de valeurs friables d'un produit de formes affines deux à deux affinement indépendantes / An integer is called y-friable if its largest prime factor does not exceed y. Friable values of binary forms play a central role in the integer factoring algorithm NFS (Number Field Sieve). In this thesis, we obtain some asymptotic formulas for the number of representations of friable integers by various classes of polynomials. In the first part, we focus on binary forms which split as a product of a linear form and a quadratic form. To achieve this, we combine the inclusion-exclusion principle with ideas based on works of Fouvry and Iwaniec and Balog, Blomer, Dartyge and Tenenbaum related to the distribution of some sequences of integers represented by quadratic forms. We then take a closer look at friable values of irreducible cubic forms. Extending some previous works of Heath-Brown and Moroz concerning primes represented by such polynomials, we provide some asymptotic formulas which hold in a large range of friability. With this method, we also evaluate some means over the values of an irreducible cubic form for other multiplicative functions including the Möbius function and the Liouville function. In the last chapter, we investigate the correlations between nilsequences and the characteristic function of friable integers. By using the nilpotent method of Green and Tao, our work provides a formula for the number of friable integers represented by a product of affine forms such that any two forms are affinely independent

Which matrices can be written as sums or products of square-zero matrices? This question is the central premise of this dissertation. Over the past 25 years a signi - cant body of research on products and linear combinations of square-zero matrices has developed, and it is the aim of this study to present this body of research in a consolidated, holistic format, that could serve as a theoretical introduction to the subject. The content of the research is presented in three parts: rst results within the broader context of sums and products of nilpotent matrices are discussed, then products of square-zero matrices, and nally sums of square-zero matrices. / Mathematical Sciences / M. Sc. (Mathematics)

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)