Die Pseudomomente dienen als Charakteristikum der Annäherung der Komponenten einer Summenverteilungsfunktion gegen die Komponenten der Grenzverteilungsfunktion. In der Terminologie der Pseudomomente werden Abschätzungen der Annäherung der Summenverteilungsfunktion gegen eine unbegrenz teilbare Verteilungsfunktion angegeben. Dabei werden die Aussagen ohne die Voraussetzung der sogenannten Infinitesimalitätsbedingung hergeleitet. Es werden Abschätzungen angegeben sowohl unter der Voraussetzung endlicher Streuungen als auch ohne diese Voraussetzung. Abschließend werden einige Literaturhinweise angegeben.:1. Einleitung S. 2 2. Abschätzungen unter Voraussetzung endlicher Streuungen S. 3 3. Abschätzungen ohne die Voraussetzung über die Existenz der Streuungen S. 6 4. Beweise S. 9 5. Beispiel S. 11 Literatur S. 12 / The pseudo-moments serve as a characteristic of the approach of the components of a cumulative distribution function to the components of the limit distribution function. In the terminology of pseudo-moments estimates of the approximation of the cumulative distribution function by an indefinite divisible distribution function can be specified. The results are derived without the assumption of the so-called condition of infinitesimality. There are given some estimations with or without the assumption of finite variances. Finally some references are given.:1. Einleitung S. 2 2. Abschätzungen unter Voraussetzung endlicher Streuungen S. 3 3. Abschätzungen ohne die Voraussetzung über die Existenz der Streuungen S. 6 4. Beweise S. 9 5. Beispiel S. 11 Literatur S. 12

In diesem Artikel werden notwendige und hinreichende Bedingungen für die Gültigkeit von Grenzwertsätzen für mittlere Abweichungen untersucht. In der Terminilogie von J.V.LINNIK (1971) werden die x-Bereiche für mittlere Abweichungen gewöhnlich als "sehr enge" Zonen der integralen normalen Anziehung bezeichnet. Darüber hinaus werden die Restglieder untersucht, die in den asymptotischen Beziehungen auftreten. Die Ordnung der Konvergenzgeschwindigkeit wird angegeben. Frühere Ergebnisse einiger Autoren werden verallgemeinert. Abschließend werden einige Literaturhinweise angegeben.:1. Einleitung S. 2 2. Allgemeine Grenzwertsätze für mittlere Abweichungen mit Angabe der Ordnung der Konvergenzgeschwindigkeit S. 3 3. Die Existenz von Momenten als notwendige Voraussetzung für die Gültigkeit von Grenzwertsätzen für mittlere Abweichungen S. 7 4. Beweise S. 10 Literatur S. 16 / In this paper we study necessary and sufficient conditions for the validity of limit theorems on moderate deviations. Usually x-zones for moderate deviations are called in the terminilogy by YU.V.LINNIK (1971) "very narrow" zones of integral normal attraction. Moreover we analyse the remainder term appearing in the asymptotic relations. Informations on the order of the rate of convergence are given. Earlier results by several authors are generalized. Finally some references are given.:1. Einleitung S. 2 2. Allgemeine Grenzwertsätze für mittlere Abweichungen mit Angabe der Ordnung der Konvergenzgeschwindigkeit S. 3 3. Die Existenz von Momenten als notwendige Voraussetzung für die Gültigkeit von Grenzwertsätzen für mittlere Abweichungen S. 7 4. Beweise S. 10 Literatur S. 16

Since the introduction of the Riemann integral in the middle of the nineteenth century, integration theory has been subject to significant breakthroughs on a relatively frequent basis. We have now reached a point where integration theory has been thoroughly researched to a point where one has to delve quite deep into a particular subject in order to encounter open conjectures. In education the Riemann integral has for quite some time been the standard integral in elementary analysis courses and as the complexity of these courses incrementally increase the more general Lebesgue integral eventually becomes the standard integral.  Unfortunately, in the transition from the Riemann integral to the Lebesgue integral there are certain topics of pure theoretical interest which to a certain extent are neglected. This is particularly the case for topics regarding the inverse relationship between differential and integral calculus and the integration of exceedingly complicated functions which for example might be of a highly oscillatory nature. From an applied mathematician's point of view, the partial neglection of these topics in the case of highly problematic functions might be justified in the sense that this theory is unnecessary for modeling most problems that appear in nature. From a theoretician's point of view however this negligence is unacceptable. Consequently, there are alternative integrals which give rise to theories which one can use in an attempt to study these aforementioned topics. An example of such an integral is the Henstock–Kurzweil integral, which can be developed in a rather similar manner to that of the Riemann integral.  In this thesis we will develop the Henstock–Kurzweil integral in order to answer some of the questions which to a certain extent are beyond the scope of the Lebesgue integral while using rather basic proof techniques from complex analysis and measure theory. In addition to that we extended various properties of the Lebesgue integral to the Henstock–Kurzweil integral, in particular when it comes to Lebesgue's fundamental theorem of calculus and the basic convergence theorems of the Lebesgue integral.

L'urne de Polya est l'exemple typique de processus stochastique avec renforcement. La limite presque sûre (p.s.) en temps existe, est aléatoire et non dégénérée. L'urne de Friedman est une généralisation naturelle dont la limite (proportion asymptotique en temps) n'est plus aléatoire. De nombreux modèles aléatoires sont fondés sur des processus de renforcement comme pour la conception d'essais cliniques au design adaptatif, en économie, ou pour des algorithmes stochastiques à des fins d'optimisation ou d'estimation non paramétrique. Dans ce mémoire, inspirés par de nombreux articles récents, nous introduisons une nouvelle famille de systèmes (finis) de processus de renforcement où l'interaction se traduit par un phénomène de renforcement collectif additif, de type champ moyen. Les deux taux de renforcement (l'un spécifique à chaque composante, l'autre collectif et commun à toutes les composantes) sont possiblement différents. Nous prouvons deux types de résultats mathématiques. Différents régimes de paramètres doivent être considérés : type de la règle (brièvement, Polya/Friedman), taux du renforcement. Nous prouvons l'existence d'une limite p.s. coommune à toutes les composantes du système (synchronisation). La nature de la limite (aléatoire/déterministe) est étudiée en fonction du régime de paramètres. Nous étudions également les fluctuations en prouvant des théorèmes centraux de la limite. Les changements d'échelle varient en fonction du régime considéré. Différentes vitesses de convergence sont ainsi établies. / The Polya urn is the paradigmatic example of a reinforced stochastic process. It leads to a random (non degenerated) almost sure (a.s.) time-limit.The Friedman urn is a natural generalization whose a.s. time-limit is not random anymore. Many stochastic models for applications are based on reinforced processes, like urns with their use in adaptive design for clinical trials or economy, stochastic algorithms with their use in non parametric estimation or optimisation. In this work, in the stream of previous recent works, we introduce a new family of (finite) systems of reinforced stochastic processes, interacting through an additional collective reinforcement of mean field type. The two reinforcement rules strengths (one componentwise, one collective) are tuned through (possibly) different rates. In the case the reinforcement rates are like 1/n, these reinforcements are of Polya or Friedman type as in urn contexts and may thus lead to limits which may be random or not. We state two kind of mathematical results. Different parameter regimes needs to be considered: type of reinforcement rule (Polya/Friedman), strength of the reinforcement. We study the time-asymptotics and prove that a.s. convergence always holds. Moreover all the components share the same time-limit (synchronization). The nature of the limit (random/deterministic) according to the parameters' regime is considered. We then study fluctuations by proving central limit theorems. Scaling coefficients vary according to the regime considered. This gives insights into the different rates of convergence.

This book is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. We prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

La méthode des régions-solutions a été développée par Frigon [7] en 2018 pour montrer l'existence de solutions à des équations différentielles ordinaires de premier ordre, dont le graphe d'une solution se trouve à l'intérieur d'une région-solution \(R \subset [0, T] \times \mathbb{R}^{N}\). Cette méthode est en particulier une généralisation des sous et sur-solutions et des tubes-solutions. On présente cette méthode et certains résultats d'existence qui en découlent. D'autre part, la dérivée de Stieltjes, communément appelée \(g\)-dérivée, est le fruit du travail de Pouso et Rodríguez [20] en 2014, permettant l'unification des équations différentielles classiques, des équations aux échelles de temps et des équations différentielles avec impulsions. Elle est en particulier liée au théorème fondamental du calcul pour l'intégrale de Lebesgue-Stieltjes. On présente la base de cette théorie dans un premier temps, puis la façon dont cette \(g\)-dérivée généralise d'autres types d'équations différentielles ou aux échelles de temps. On introduit en particulier la notion de \((g \times I_{\mathbb{R}^{N}})\)-différentiabilité et des résultats qui découlent de cette définition. On présente de plus une fonction exponentielle qui permet de résoudre les équations différentielles de Stieltjes linéaires, introduite par Frigon et Pouso [8]. Le but de ce mémoire est de généraliser la méthode des régions-solutions, dont la généralisation s'appellera \(g\)-région-solution, afin de montrer l'existence de solutions aux équations différentielles de Stieltjes. On présente plusieurs exemples de \(g\)-régions admissibles et de \(g\)-régions-solutions, puis des théorèmes d'existence se basant sur cette méthode. On donne de plus des exemples où on applique ces théorèmes. On termine ce mémoire en présentant deux applications des théorèmes d'existence à l'évolution d'une population de cerfs de Virginie ainsi qu'à l'évolution de la tension générée par une diode à effet tunnel résonnant (DTR) dirigée vers une diode laser (DL). / The method of solution-regions has been developed by Frigon [7] in 2018 to show the existence of solutions for first-order ordinary differential equations, where the graph of a solution is inside a solution-region \(R \subset [0, T] \times \mathbb{R}^{N}\). This method is in particular a generalization of the lower and upper solutions and of the solution-tubes. We show this method and some existence results which follow. On the other hand, the Stieltjes derivative, more commonly called \(g\)-derivative, is the fruit of the work of Pouso and Rodríguez [20] in 2014, which unifies classic differential equations, equations on time scales and differential equations with impulses. In particular, it leads to the fundamental theorem of calculus for the Lebesgue-Stieltjes integral. We start by showing the basis of this theory, and then the way this \(g\)-derivative generalizes other types of differential or time scale equations. We introduce in particular the \((g \times I_{\mathbb{R}^{N}})\)-differentiability and results that follow from this definition. Furthermore, we present an exponential function which solves linear Stieltjes differential equations. The goal of this thesis is to generalize the method of solution-regions, where the generalization will be called \(g\)-solution-region, to show the existence of solutions for Stieltjes differential equations. We present multiple examples of \(g\)-admissible regions and \(g\)-solution-regions, then we establish existence theorems based on this method. We also show examples where we apply these theorems. Finally, we end this thesis by showing two applications of the existence theorems to the evolution of a population of white-tailed deer and to the evolution of the voltage generated by a resonant tunneling diode (RTD) to a laser diode (LD).

A classificação isomorfa dos espaços de Banach separáveis C(K) é devida a Milutin no caso em que K são não enumeráveis e a Bessaga e Pelczynski no caso em que K são enumeráveis. Neste trabalho apresentamos uma extensão vetorial dessa classificação e tiramos várias consequências, por exemplo, considerando o espaço métrico compacto infinito K e Y um espaço de Banach: &nbsp; &nbsp; 1. Sendo 1 < p < &infin; e &Gamma; um conjunto infinito, classificamos, a menos de isomorfismo, os espaços de Banach C(K, Y &oplus; lp(&Gamma;)), quando o dual de Y contém uma cópia de lq, onde 1/p+ 1/q =1. &nbsp; &nbsp; 2. Classificamos os espaços de Banach C(K, Y &oplus; l&infin;(&Gamma;)), quando a densidade de Y é estritamente menor que 2|&Gamma;|. &nbsp; &nbsp; 3. Classificamos os espaços de Banach C(K &times;(S&oplus; &beta;&Gamma;)) e C(S &oplus; (K&times; &beta;&Gamma;)), onde S é um compacto disperso de Hausdorff arbitrário e &beta;&Gamma; é a compactificação de Stone-Cech de &Gamma;. Obtemos, também, algumas leis de cancelamento para espaços de Banach da forma C(K1,X)&oplus; C(K2,Y), onde K1 e K2 são espaços compactos métricos infinitos de Hausdorff e X, Y espaços de Banach satisfazendo condições adequadas. Estabelecemos também um teorema de quase-dicotomia envolvendo os espaços C(K,X), onde X tem cotipo finito. Finalmente, apresentamos algumas majorações nas distorções de isomorfismos positivos de C([0,&omega;k]) em C([0,&omega;]) e também de C([0,&omega;]) em C([0,&omega;k]), k&isin; N, k &ge; 2. / The isomorphic classification of separable Banach spaces C(K) is due Milutin in the case when K are uncountable and to Bessaga and Pelczynski in the case when K are countable. In this work we prove a vectorial extention of this classification and provide several consequences, for example considering the infinite metric compact space K and Y a Banach space: &nbsp; &nbsp; 1. Let 1 < p < &infin; and &Gamma; a infinite set, we classify, up to an isomorphism, the Banach spaces C(K, Y &oplus; lp(&Gamma;)), in the case where the dual of Y contains no copy of lq, where 1/p+ 1/q =1. &nbsp; &nbsp; 2. We classify the Banach spaces C(K, Y &oplus; l&infin;(&Gamma;)), when the density character of Y is strictly less that 2|&Gamma;|. &nbsp; &nbsp; 3. We classify the Banach spaces C(K &times;(S&oplus; &beta;&Gamma;)) and C(S &oplus; (K&times; &beta;&Gamma;)) where S is an arbitrary dispersed compact and &beta;&Gamma; is the Stone-Cech compactification of &Gamma;. We obtain also some cancellation laws for Banach spaces in the form C(K1,X)&oplus; C(K2,Y), where K1 and K2 are metric compact Hausdorff spaces and X, Y Banach spaces satisfying appropriate conditions. We established also a quasi-dichotomy theorem envolving the C(K,X) spaces, where X is of finite cotype. Finally, we present some upper bounds of distortions of positive isomorphisms of C([0,&omega;k]) on C([0,&omega;]) and also of C([0,&omega;]) on C([0,&omega;k]), k&isin; N, k &ge; 2.

Dans cette thèse, on s'intéresse aux plus grands facteur premiers d'entiers consécutifs. Désignons par $P^+(n)$ (resp. $P^-(n)$) le plus grand (resp. plus petit) facteur premier d'un entier générique $n\geq 1$ avec la convention que $P^+(1)=1$ (resp. $P^-(1)=\infty$). Dans le premier chapitre, nous étudions les plus grands facteurs premiers d'entiers consécutifs dans les petits intervalles. Nous démontrons qu'il existe une proportion positive d'entiers $n$ tels que $P^+(n)<P^+(n+1)$ pour $n\in\, ]x,\, x+y]$ avec $y=x^{\theta}, \tfrac{7}{12}<\theta\leq 1$. Nous obtenons un résultat similaire pour la condition $P^+(n)>P^+(n+1)$. Dans le deuxième chapitre, nous nous intéressons à la fonction $P_y^+(n)$, où $P_y^+(n)=\max\{p|n:\, p\leq y\}$ et $2\leq y\leq x.$ Nous montrons qu'il existe une proportion positive d'entiers $n$ tels que $P_y^+(n)<P_y^+(n+1)$. En particulier, la proportion d'entiers $n$ avec $P^+(n)<P^+(n+1)$ est plus grande que $0,1356$ en prenant $y=x.$ Les outils principaux sont le crible et un système de poids bien adapté. Dans le troisième chapitre, nous démontrons que les deux configurations $P^+(n-1)>P^+(n)<P^+(n+1)$ et $P^+(n-1)<P^+(n)>P^+(n+1)$ ont lieu pour une proportion positive d'entiers $n$, en utilisant le système de poids bien adapté que l'on a introduit dans le Chapitre 2. De façon similaire, on peut obtenir un résultat plus général pour $k$ entiers consécutifs, $k\in \mathbb{Z}, k\geq3$. Dans le quatrième chapitre, on étudie les plus grands facteurs premiers d'entiers consécutifs voisins d'un entier criblé. Sous la conjecture d'Elliott-Halberstam, nous montrons d'abord que la proportion de la configuration $P^+(p-1)<P^+(p+1)$ est plus grande que $0,1779$. Puis, nous démontrons qu'il existe une proportion positive d'entiers $n$ tels que $P^+(n)<P^+(n+2), P^-(n)>x^{\beta}$ avec $0<\beta<\frac{1}{3}$ / In this thesis, we study the largest prime factors of consecutive integers. Denote by $P^+(n)$ (resp. $P^-(n)$) the largest (resp. the smallest) prime factors of the integer $n\geq 1$ with the convention $P^+(1)=1$ (resp. $P^-(1)=\infty$). In the first chapter, we consider the largest prime factors of consecutive integers in short intervals. We prove that there exists a positive proportion of integers $n$ for $n\in\, (x,\, x+y]$ with $y=x^{\theta}, \tfrac{7}{12}<\theta\leq 1$ such that $P^+(n)<P^+(n+1)$. A similar result holds for the condition $P^+(n)>P^+(n+1)$. In the second chapter, we consider the function $P_y^+(n)$, where $P_y^+(n)=\max\{p|n:\, p\leq y\}$ and $2\leq y\leq x$. We prove that there exists a positive proportion of integers $n$ such that $P_y^+(n)<P_y^+(n+1)$. In particular, the proportion of the pattern $P^+(n)<P^+(n+1)$ is larger than $0.1356$ by taking $y=x.$ The main tools are sieve methods and a well adapted system of weights. In the third chapter, we prove that the two patterns $P^+(n-1)>P^+(n)<P^+(n+1)$ and $P^+(n-1)<P^+(n)>P^+(n+1)$ occur for a positive proportion of integers $n$ respectively, by the well adapted system of weights that we have developed in the second chapter. With the same method, we derive a more general result for $k$ consecutive integers, $k\in \mathbb{Z}, k\geq 3$. In the fourth chapter, we study the largest prime factors of consecutive integers with one of which without small prime factor. Firstly we show that under the Elliott-Halberstam conjecture, the proportion of the pattern $P^+(p-1)<P^+(p+1)$ is larger than $0.1779$. Then, we prove that there exists a positive proportion of integers $n$ such that $P^+(n)<P^+(n+2), P^-(n)>x^{\beta}$ with $0<\beta<\frac{1}{3}$

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]

Algebraic Quantum Field Theory (AQFT) is a mathematically rigorous framework that was developed to model the interaction of quantum mechanics and relativity. In AQFT, quantum mechanics is modelled by C*-algebras of observables and relativity is usually modelled in Minkowski space. In this thesis we will consider a generalization of AQFT which was inspired by the work of Abramsky and Coecke on abstract quantum mechanics [1, 2]. In their work, Abramsky and Coecke develop a categorical framework that captures many of the essential features of finite-dimensional quantum mechanics. In our setting we develop a categorified version of AQFT, which we call premonoidal C*-quantum field theory, and in the process we establish many analogues of classical results from AQFT. Along the way we also exhibit a number of new concepts, such as a von Neumann category, and prove several properties they possess. We also establish some results that could lead to proving a premonoidal version of the classical Doplicher-Roberts theorem, and conjecture a possible solution to constructing a fibre-functor. Lastly we look at two variations on AQFT in which a causal order on double cones in Minkowski space is considered.