In this Ph. D. thesis we discuss four different problems in analysis: (a) sharp inequalities related to the restriction phenomena for the Fourier transform, with emphasis on some Strichartz-type estimates; (b) extremal approximations of exponential type for the Gaussian and for a class of even functions, with applications to analytic number theory; (c) radial symmetrization approach to convolution-like inequalities for the Boltzmann collision operator; (d) regularity of maximal operators with respect to weak derivatives and weak continuity. / text

In this thesis, we focus on the following topics: supereulerian graphs, hamiltonian line graphs, fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees, and several extremal problems on the (minimum and/or maximum) size of graphs under a given graph property. The thesis includes six chapters. The first one is to introduce definitions and summary the main results of the thesis, and in the last chapter we introduce the furture research of the thesis. The main studies in Chapters 2 - 5 are as follows. In Chapter 2, we explore conditions for a graph to be supereulerian.In Section 1 of Chapter 2, we characterize the graphs with minimum degree at least 2 and matching number at most 3. By using the characterization, we strengthen the result in [93] and we also address a conjecture in the paper.In Section 2 of Chapter 2, we prove that if $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$, then $G$ is collapsible except for several special graphs, where $p(n)=0$ for $n$ even and $p(n)=1$ for $n$ odd. As a corollary, a characterization for graphs satisfying $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$ to be supereulerian is obtained. This result extends the result in [21].In Section 3 of Chapter 2, we focus on a conjecture posed by Chen and Lai [Conjecture~8.6 of [33]] that every 3-edge connected and essentially 6-edge connected graph is collapsible. We find a kind of sufficient conditions for a 3-edge connected graph to be collapsible.In Chapter 3, we mainly consider the hamiltonicity of 3-connected line graphs.In the first section of Chapter 3, we give several conditions for a line graph to be hamiltonian, especially we show that every 3-connected, essentially 11-connected line graph is hamilton- connected which strengthens the result in [91].In the second section of Chapter 3, we show that every 3-connected, essentially 10-connected line graph is hamiltonian-connected.In the third section of Chapter 3, we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is hamiltonian. Moreover, if $G$ has 10 vertices of degree 3 and its line graph is not hamiltonian, then $G$ can be contractible to the Petersen graph.In Chapter 4, we consider edge fault-tolerant hamiltonicity of Cayley graphs generated by transposition trees. We first show that for any $F\subseteq E(Cay(B:S_{n}))$, if $|F|\leq n-3$ and $n\geq4$, then there exists a hamiltonian path in $Cay(B:S_{n})-F$ between every pair of vertices which are in different partite sets. Furthermore, we strengthen the above result in the second section by showing that $Cay(S_n,B)-F$ is bipancyclic if $Cay(S_n,B)$ is not a star graph, $n\geq4$ and $|F|\leq n-3$.In Chapter 5, we consider several extremal problems on the size of graphs.In Section 1 of Chapter 5, we bounds the size of the subgraph induced by $m$ vertices of hypercubes. We show that a subgraph induced by $m$ (denote $m$ by $\sum\limits_{i=0}^ {s}2^{t_i}$, $t_0=[\log_2m]$ and $t_i= [\log_2({m-\sum\limits_{r=0}^{i-1}2 ^{t_r}})]$ for $i\geq1$) vertices of an $n$-cube (hypercube) has at most $\sum\limits_{i=0}^{s}t_i2^{t_i-1} +\sum\limits_{i=0}^{s} i\cdot2^{t_i}$ edges. As its applications, we determine the $m$-extra edge-connectivity of hypercubes for $m\leq2^{[\frac{n}2]}$ and $g$-extra edge-connectivity of the folded hypercube for $g\leq n$.In Section 2 of Chapter 5, we partially study the minimum size of graphs with a given minimum degree and a given edge degree. As an application, we characterize some kinds of minimumrestricted edge connected graphs.In Section 3 of Chapter 5, we consider the minimum size of graphs satisfying Ore-condition.

Nous étudions mécanismes généraux qui sont à l'origine de vieillissement de dynamiques en environnements aléatoires, connu sous. Le vieillissement s'observe dans le comportement de certaines fonctions de corrélation, qui ne deviennent jamais indépendantes de l'âge du système. Une approche universelle à ce problème fut développée durant les dernières décennies: le comportement des fonctions de corrélation peut être lié à celui du processus d'horloge, qui est le temps total écoulé le long d'une trajectoire de la dynamique.Une approche élégante fut proposée par Gayrard (2010, 2012) pour étudier le processus d'horloge. Celui-ci est vu comme un processus de sommes partielles à incréments corrélés auquel des critères de convergence, dûs à Durett et Resnick (1978) sont appliqués. Cette méthode fut poussée plus avant par Bovier et Gayrard (2013).Nous étendons les méthodes développées par Gayrard (2012) et Bovier et Gayrard (2013), et étudions vieillissement dans divers modèles. Dans la première partie, nous établissons des critères de convergence vers des processus extrémaux pour des graphes finis et improuver résulats obtenus par Ben Arous et Gun (2012) sur le vieillissement extrémal. La deuxième partie traite de dynamiques sur des graphes infinis. Nous donnons des conditions suffisantes sous lesquelles le processus d'horloge sous-jacent converge vers un subordinateur, et établir l'existence de vieillissement normal dans le modèle assymétrique de pièges de Bouchaud sur $Z^d$ pour $dgeq 2$. La troisième partie concerne le modèle de Bouchaud assymétrique lorsque $dgeq 3$ et sa version symétrique lorsque $d=2$. Nous prouvons l'existence d'un régime de sur-vieillissement. / We study general mechanisms that lead to aging behavior of dynamics in random environments. Aging is observed in the behavior of correlation functions that never become independent of the age of the system. A universal approach to this problem was developed over the past decades: the behavior correlation functions can be linked to the long-time behavior of the clock process, which is the total time elapsed along the trajectory of the random motion. An elegant approach to studying clock processes was proposed by Gayrard (2010,2012). Here, the clock process is viewed as a partial sum process whose increments are dependent random variables and then convergence criteria, due to Durrett and Resnick (1978), are employed. This method was further developed by Bovier and Gayrard (2013).We extend the methods of Gayrard (2012) and Bovier and Gayrard (2013) and use our methods to study the aging behavior of various models. In the first part we establish criteria for the convergence of clock processes on sequences of finite graphs to extremal processes and improve results on extremal aging obtained by Ben Arous and Gun (2012). The second part deals with dynamics that are defined on infinite graphs. We introduce sufficient conditions for the clock process to converge to a subordinator and establish the existence of a normal aging regime in Bouchaud's asymmetric trap model on $Z^d$, for $dgeq 2$. In the third part of this thesis we consider Bouchaud's asymmetric trap model for $dgeq 3$, and its symmetric version for $d=2$. We prove the existence of an super-aging regime.

Este trabalho tem como objetivo calcular intervalos de confiança para altos quantis oriundos de distribuições de caudas pesadas. Para isso, utilizamos os métodos da aproximação pela distribuição normal, razão de verossimilhanças, {\\it data tilting} e gama generalizada. Obtivemos, através de simulações, que os intervalos calculados a partir do método da gama generalizada apresentam probabilidades de cobertura bem próximas do nível de confiança, com amplitudes médias menores do que os outros três métodos, para dados gerados da distribuição Weibull. Todavia, para dados gerados da distribuição Fréchet, o método da razão de verossimilhanças fornece os melhores intervalos. Aplicamos os métodos utilizados neste trabalho a um conjunto de dados reais, referentes aos pagamentos de indenizações, em reais, de seguros de incêndio, de um determinado grupo de seguradoras no Brasil, no ano de 2003 / In this work, confidence intervals for high quantiles from heavy-tailed distributions were computed. More specifically, four methods, namely, normal approximation method, likelihood ratio method, data tilting method and generalised gamma method are used. A simulation study with data generated from Weibull distribution has shown that the generalised gamma method has better coverage probabilities with the smallest average length intervals. However, from data generated from Fréchet distribution, the likelihood ratio method gives the better intervals. Moreover, the methods used in this work are applied on a real data set from 1758 Brazilian fire claims

Alexander A. Razborov (2007) developed the theory of flag algebras to compute the minimum asymptotic density of triangles in a graph as a function of its edge density. The theory of flag algebras, however, can be used to study the asymptotic density of several combinatorial objects. In this dissertation, we present two original results obtained in the theory of tournaments through application of flag algebra proof techniques. The first result concerns minimization of the asymptotic density of transitive tournaments in a sequence of tournaments, which we prove to occur if and only if the sequence is quasi-random. As a byproduct, we also obtain new quasi-random characterizations and several other flag algebra elements whose density is minimized if and only if the sequence is quasi-random. The second result concerns a class of equivalent properties of a sequence of tournaments that we call quasi-carousel properties and that, in a similar fashion as quasi-random properties, force the sequence to converge to a specific limit homomorphism. Several quasi-carousel properties, when compared to quasi-random properties, suggest that quasi-random sequences and quasi-carousel sequences are the furthest possible from each other within the class of almost balanced sequences. / Alexander A. Razborov (2007) desenvolveu a teoria de álgebras de flags para calcular a densidade assintótica mínima de triângulos em um grafo em função de sua densidade de arestas. A teoria das álgebras de flags, contudo, pode ser usada para estudar densidades assintóticas de diversos objetos combinatórios. Nesta dissertação, apresentamos dois resultados originais obtidos na teoria de torneios através de técnicas de demonstração de álgebras de flags. O primeiro resultado compreende a minimização da densidade assintótica de torneios transitivos em uma sequência de torneios, a qual provamos ocorrer se e somente se a sequência é quase aleatória. Como subprodutos, obtemos também novas caracterizações de quase aleatoriedade e diversos outros elementos da álgebra de flags cuja densidade é minimizada se e somente se a sequência é quase aleatória. O segundo resultado compreende uma classe de propriedades equivalentes sobre uma sequência de torneios que chamamos de propriedades quase carrossel e que, de uma forma similar às propriedades quase aleatórias, forçam que a sequência convirja para um homomorfismo limite específico. Várias propriedades quase carrossel, quando comparadas às propriedades quase aleatórias, sugerem que sequências quase aleatórias e sequências quase carrossel estão o mais distantes possível umas das outras na classe de sequências quase balanceadas.

In the present work we investigate the solution of the univalent twistor equation on an isolated horizon that serves for the definition of the so-called Penrose mass. We start our discussion with the construction of adapted co- ordinates to the isolated horizon and summarizing the main results in this field that are needed for our work. We include a chapter devoted to the extre- mal isolated horizons and prove an important result concerning uniqueness of geometry therein. It is a generalization of the paper by Lewandowski and Pawlowski (Class. Quantum Grav. 31 (17), 2014), which states that the ex- tremal isolated horizons are necessarily isometric to the intrinsic geometry of the Kerr-Newmann black hole. Further we proceed to investigation of the twistor equation on the isolated horizon. We analyze conditions of integra- bility and derive the time dependent solution. Consequently we solve the 2-surface twistor equation and briefly discuss the general approach to the problem of defining the Penrose charge. 1

In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix $W$ as a subset of $V(G)$ such that all vertices in $W$ have been assigned colors from their respective lists. We give a new, simplified proof where there are a small number of precolored vertices on the same face. We also explore cases where $W=\{u,v\}$ and $G$ has a separating $C_3$ or $C_4$ between $u$ and $v$.

The purpose of the dissertation is to prove universal tight bounds for deviation from the mean probability inequalities for functions of random variables. Universal bounds shows that they are uniform with respect to some class of distributions and quantity of variables and other parameters. The bounds are called tight, if we can construct a sequence of random variables, such that the upper bounds are achieved. Such inequalities are useful for example in insurance mathematics, for constructing effective algorithms. We extend the results for Lipschitz functions on general probability metric spaces. / Disertacijos darbo tikslas – įrodyti universalias tiksliąsias nelygybes atsitiktinių dydžių funkcijų nukrypimo nuo vidurkio tikimybėms. Universalios nelygybės pažymi, kad jos yra tolygios pagal tam tikras bendras skirstinių klases ir pagal atsitiktinių dydžių kiekį, kartais ir pagal kitus parametrus. Nelygybės vadinamos tiksliosiomis, jeigu pavyksta sukonstruoti atsitiktinių dydžių seką, kuriai nelygybės virsta lygybėmis. Tokios nelygybės labai naudingos, pavyzdžiui, draudimo matematikoje, konstruojant efektyvius algoritmus. Disertaciją sudaro šeši skyriai. Pirmasis skyrius yra įvadas, kuriame neformaliai pristatomas disertacijoje tiriamas objektas, pateikiamas bendras darbo aprašymas ir motyvacija. Detalesnė kitų autorių rezultatų apžvalga pateikiama atskirai kiekviename skyriuje. Antrasis skyrius skirtas atvejui, kai atsitiktiniai dydžiai yra aprėžti ir simetriniai. Trečiajame skyriuje įrodomos nelygybės atsitiktiniams dydžiams, tenkinantiems dispersijos aprėžtumo sąlygą. Ketvirtajame skyriuje nagrinėjamos sąlyginai aprėžtų atsitiktinių dydžių sumos. Penktajame skyriuje tiriamos atsitiktinių dydžių sekos, sudarančios martingalą arba supermartingalą, ir joms gaunamos universaliosios tikimybinės nelygybės ir sukonstruojama nehomogeninė Markovo grandinė, kuri yra martingalas, ir kuriai minėtos nelygybės virsta lygybėmis. Šeštajame skyriuje rezultatai yra apibendrinami atsitiktinių dydžių sekos Lipšico funkcijoms.

Disertacijos darbo tikslas – įrodyti universalias tiksliąsias nelygybes atsitiktinių dydžių funkcijų nukrypimo nuo vidurkio tikimybėms. Universalios nelygybės pažymi, kad jos yra tolygios pagal tam tikras bendras skirstinių klases ir pagal atsitiktinių dydžių kiekį, kartais ir pagal kitus parametrus. Nelygybės vadinamos tiksliosiomis, jeigu pavyksta sukonstruoti atsitiktinių dydžių seką, kuriai nelygybės virsta lygybėmis. Tokios nelygybės labai naudingos, pavyzdžiui, draudimo matematikoje, konstruojant efektyvius algoritmus. Disertaciją sudaro šeši skyriai. Pirmasis skyrius yra įvadas, kuriame neformaliai pristatomas disertacijoje tiriamas objektas, pateikiamas bendras darbo aprašymas ir motyvacija. Detalesnė kitų autorių rezultatų apžvalga pateikiama atskirai kiekviename skyriuje. Antrasis skyrius skirtas atvejui, kai atsitiktiniai dydžiai yra aprėžti ir simetriniai. Trečiajame skyriuje įrodomos nelygybės atsitiktiniams dydžiams, tenkinantiems dispersijos aprėžtumo sąlygą. Ketvirtajame skyriuje nagrinėjamos sąlyginai aprėžtų atsitiktinių dydžių sumos. Penktajame skyriuje tiriamos atsitiktinių dydžių sekos, sudarančios martingalą arba supermartingalą, ir joms gaunamos universaliosios tikimybinės nelygybės ir sukonstruojama nehomogeninė Markovo grandinė, kuri yra martingalas, ir kuriai minėtos nelygybės virsta lygybėmis. Šeštajame skyriuje rezultatai yra apibendrinami atsitiktinių dydžių sekos Lipšico funkcijoms. / The purpose of the dissertation is to prove universal tight bounds for deviation from the mean probability inequalities for functions of random variables. Universal bounds shows that they are uniform with respect to some class of distributions and quantity of variables and other parameters. The bounds are called tight, if we can construct a sequence of random variables, such that the upper bounds are achieved. Such inequalities are useful for example in insurance mathematics, for constructing effective algorithms. We extend the results for Lipschitz functions on general probability metric spaces.

Alexander A. Razborov (2007) developed the theory of flag algebras to compute the minimum asymptotic density of triangles in a graph as a function of its edge density. The theory of flag algebras, however, can be used to study the asymptotic density of several combinatorial objects. In this dissertation, we present two original results obtained in the theory of tournaments through application of flag algebra proof techniques. The first result concerns minimization of the asymptotic density of transitive tournaments in a sequence of tournaments, which we prove to occur if and only if the sequence is quasi-random. As a byproduct, we also obtain new quasi-random characterizations and several other flag algebra elements whose density is minimized if and only if the sequence is quasi-random. The second result concerns a class of equivalent properties of a sequence of tournaments that we call quasi-carousel properties and that, in a similar fashion as quasi-random properties, force the sequence to converge to a specific limit homomorphism. Several quasi-carousel properties, when compared to quasi-random properties, suggest that quasi-random sequences and quasi-carousel sequences are the furthest possible from each other within the class of almost balanced sequences. / Alexander A. Razborov (2007) desenvolveu a teoria de álgebras de flags para calcular a densidade assintótica mínima de triângulos em um grafo em função de sua densidade de arestas. A teoria das álgebras de flags, contudo, pode ser usada para estudar densidades assintóticas de diversos objetos combinatórios. Nesta dissertação, apresentamos dois resultados originais obtidos na teoria de torneios através de técnicas de demonstração de álgebras de flags. O primeiro resultado compreende a minimização da densidade assintótica de torneios transitivos em uma sequência de torneios, a qual provamos ocorrer se e somente se a sequência é quase aleatória. Como subprodutos, obtemos também novas caracterizações de quase aleatoriedade e diversos outros elementos da álgebra de flags cuja densidade é minimizada se e somente se a sequência é quase aleatória. O segundo resultado compreende uma classe de propriedades equivalentes sobre uma sequência de torneios que chamamos de propriedades quase carrossel e que, de uma forma similar às propriedades quase aleatórias, forçam que a sequência convirja para um homomorfismo limite específico. Várias propriedades quase carrossel, quando comparadas às propriedades quase aleatórias, sugerem que sequências quase aleatórias e sequências quase carrossel estão o mais distantes possível umas das outras na classe de sequências quase balanceadas.