The purpose of this paper is to define an integral for real-valued functions which are defined on a field of sets and to demonstrate several properties of such an integral.

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Dans cette thèse, nous nous intéressons aux propriétés statistiques des systèmes dynamiques aléatoires et non-autonomes. Dans le premier chapitre, consacré aux systèmes aléatoires, nous établissons un cadre fonctionnel abstrait, couvrant une large classe de systèmes dilatants en dimension 1 et supérieure, permettant de démontrer de nombreux théorèmes limites annealed. Nous donnons aussi une condition nécessaire et suffisante pour que la version quenched du théorème de la limite centrale soit valide en dimension 1. Dans le chapitre deux, après avoir introduit la notion de système non-autonome, nous étudions un système composé d'applications en dimension 1 ayant un point fixe neutre commun, et nous montrons que celui-ci admet une vitesse de perte de mémoire polynomiale. Le chapitre trois est consacré aux inégalités de concentration. Nous établissons de telles inégalités pour des systèmes dynamiques aléatoires et non-autonomes, et nous étudions diverses applications. Dans le chapitre quatre, nous nous intéressons aux lemmes dynamiques de Borel-Cantelli pour l'induction de Rauzy-Veech-Zorich, et présentons quelques résultats liés aux statistiques de récurrence pour cette application. / The first chapter, devoted to random systems, we establish an abstract functional framework, including a large class of expanding systems in dimension 1 and higher, under which we can prove annealed limit theorems. We also give a necessary and sufficient condition for the quenched central limit theorem to hold in dimension 1. In chapter 2, after an introduction to the notion of non-autonomous system, we study an example consisting of a family of maps of the unit interval with a common neutral fixed point, and we show that this system admits a polynomial loss of memory. The chapter 3 is devoted to concentration inequalities. We establish such inequalities for random and non-autonomous dynamical systems in dimension 1, and we study some of their applications. In chapter 4, we study dynamical Borel-Cantelli lemmas for the Rauzy-Veech-Zorich induction, and we present some results concerning statistics of recurrence for this map.

In this work we cover some counting techniques used to solve some classic problems in Combinatorics. We also show a link between the so called ârencontre problemâ, the âmÃnage problemâ and the permanent of a square matrix. / Neste trabalho abordamos algumas tÃcnicas de contagem utilizadas para solucionar alguns problemas clÃssicos da AnÃlise CombinatÃria. Mostramos tambÃm uma relaÃÃo entre o problema das cartas mal endereÃadas, o problema de Lucas e os permanentes de uma matriz quadrada.

Orientador: Alice Kimie Miwa Libardi / Banca: Edson de Oliveira / Banca: Thiago de Melo / Resumo: O principal objetivo deste trabalho é apresentar conceitos básicos sobre coincidências e pontos fixos de aplicações contínuas usando como ferramentas os Lemas Combinatórios de Sperner e grau de aplicações. Apresentamos também um cálculo do número de Lefschetz de f; g : T2 ¡! T3, onde Th denota uma superfície de genus h, através da fórmula dada por Gonçalves e Oliveira em [3] / Abstract: The main goal of this work is present basic concepts on coincidences and fixed points of continuous maps with Sperner's Combinatorial Lemmas, and degree maps approaches. We also present a calculation of the Lefschetz number of f; g : T2 ¡! T3, where Th denotes surface of genus h, by using the formula given by Gonçalves and Oliveira in [3] / Mestre

A range space (P, S) consists of a set P of n elements and a collection S = {S1,...,Sm} of subsets of P , referred to as ranges. A hitting set for this range space refers to a subset H of P such that every Si in S contains at least one element of H. The hitting set problem is studied for many geometric range spaces where P is a set of n points in Rd and the ranges are defined by the intersection of geometric objects with P . The algorithmic question of finding the minimum-sized hitting set for a given range space is well studied and is NP-Complete even for geometric range spaces defined by unit disks. The dual of the hitting set problem is the equally well studied set cover problem. A set cover is a sub-collection C⊆S such that every element in P is contained in at least one range in C. A classic problem which is related to the minimum set cover problem is the maximum coverage problem. In this problem, given a range space (P, S) and an integer k, we have to find k ranges from S such that the number of elements in P that are covered by these k ranges are maximized. In this thesis, we study combinatorial questions on a similar variant of hitting set problem for specific geometric range spaces where the size of the hitting set is fixed as a small constant. We study the small hitting set problem mainly for two broad classes of range spaces. We first consider the Dense range space (P, S) where P is a set of n points in Rd and S is defined by “dense” geometric objects i.e, geometric objects that contain more than a constant fraction, say �, of points from P . We fix the size of the hitting set as a small constant k and investigate bounds on the value of � such that all ranges that contain more than �n points from P are hit. Next we consider the Induced range space (P, I) where P isa setof n points in R2 and the ranges are all geometric objects that are induced(spanned) by P i.e., the ranges are defined by geometric objects that have a distinct tuple of points from P on its boundary. For Induced range spaces, we prove bounds on the maximum number of ranges that can be hit using a single point. We also prove combinatorial bounds on the size of the hitting set for various families of induced objects. Now, we describe the problems that we study in this thesis and summarize the results obtained. 1. Strong centerpoints: Here we study the small hitting set question for dense range spaces when the size of hitting set is one. This is related to a classic result in geometry called Centerpoint Theorem. A point x ∈ Rd is said to be the centerpoint of P if x is contained in all convex objects that contain more than dn points from P . Centerpoint Theorem states d+1 that a centerpoint always exists for any point set P . A centerpoint need not be an input point. A natural question to ask is the following: Does there exist a strong centerpoint? i.e., is it true that for any given point set P there exists a point p ∈ P such that p is contained in every convex object that contains more than a constant fraction, say �, of points of P ? It can be easily seen that a strong centerpoint does not exist even for geometric range spaces defined by half spaces. We study the existence and the corresponding bounds for strong centerpoints for some range spaces. In particular, we prove the existence of strong centerpoint and show tight bounds for the following range spaces. Convex polytopes defined by a fixed set of orientations : Geometric range spaces like those induced by axis-parallel boxes, skylines and downward facing equilateral triangles belong to this family of convex polytopes. Hyperplanes in Rd Range spaces with discrete intersection 2. Small Strong Epsilon Nets: This can be considered as an extension of strong centerpoints. This question is related to a well studied area called �-nets. N ⊂ P is called a (strong) �-net of P with respect to S if N ∩ S =�∅ for all objects S ∈S that contain more than �n points of P . We study the following question. Let �S∈ [0, 1] represent the smallest real number such that, for any given point set P , there exists Q ⊂ P of size i which is an �S-net with respect to S. Thus a strong centerpoint will be an �S1 -net. We prove bounds on �Si for small values of i where S is the family of axis-parallel rectangles, halfspaces and disks. 3. Strong First Selection Lemma: Here we consider the hitting question for induced range spaces when the size of the hitting set is one. In other words, given an induced range space, we prove bounds on the maximum number of ranges that can be hit using a single input point. Such questions are referred to as First Selection Lemma and are well studied. We consider the strong version of the First Selection Lemma where the “heavily covered” point is required to be an input point. We study the strong first selection lemma for induced rectangles, induced special rectangles and induced disks. We prove an exact result for the strong variant of the first selection lemma for axis-parallel rectangles. We also prove exact results for the strong variant of the first selection lemma for some subclasses of axis-parallel rectangles like orthants and slabs. We prove strong first selection lemma with almost tight bounds for skylines, another sub-class of axis-parallel rectangles. We prove bounds for first selection lemma for disks in the plane and exact results for a special case when the discs are induced by a centrally symmetric point set. 2 Hitting all Induced Objects: Here we discuss and prove combinatorial bounds on the size of the minimum hitting set for induced range spaces. We prove tight bounds on the hitting set size when induced objects are special rectangles, disks and downward facing equilateral triangles.

In dieser Arbeit wird zuerst das Theorem von Chung, Graham und Wilson über quasi-zufällige Graphen zur sogenannten schwachen Quasi-Zufälligkeit für k-uniforme Hypergraphen verallgemeinert und somit eine Reihe äquivalenter Eigenschaften bestimmt. Basierend auf diesen Resultaten werden nichtbipartite Graphen gefunden, welche die Quasi-Zufälligkeit für Graphen ``forcieren''''. Zuvor waren nur bipartite Graphen mit dieser Eigenschaft bekannt. Desweiteren ist ein konzeptionell einfacher Algorithmus zum Verifizieren nicht erfüllbarer zufälliger k-SAT Formeln angegeben. Dann richtet sich der Fokus auf Anwendungen verschiedener Regularitätslemmata für Hypergraphen. Zuerst wird die Menge aller bezeichneten 3-uniformen Hypergraphen auf n Knoten, die keine Kopie des Hypergraphen der Fano Ebene enthalten, studiert. Es wird gezeigt, dass fast jedes Element aus dieser Menge ein bipartiter Hypergraph ist. Dies führt zu einem Algorithmus, der in polynomiell erwarteter Zeit einen zufälligen Fano-freien (und somit einen zufälligen bipartiten 3-uniformen) Hypergraphen richtig färbt. Schließlich wird die folgende extremale Funktion studiert. Es sind r Farben gegeben sowie ein k-uniformer Hypergraph F. Auf wie viele verschiedene Arten kann man die Kanten eines k-uniformen Hypergraphen H färben, so dass keine monochromatische Kopie von F entsteht? Welche Hypergraphen H maximieren die Anzahl erlaubter Kantenfärbungen? Hier wird ein strukturelles Resultat für eine natürliche Klasse von Hypergraphen bewiesen. Es wird für viele Hypergraphen F, deren extremaler Hypergraph bekannt ist, gezeigt, dass im Falle von zwei oder drei Farben die extremalen Hypergraphen die oben beschriebene Funktion maximieren, während für vier oder mehr Farben andere Hypergraphen mehr Kantenfärbungen zulassen. / This thesis presents first one possible generalization of the result of Chung, Graham and Wilson to k-uniform hypergraphs, and studies the so-called weak quasi-randomness. As applications we obtain a simple strong refutation algorithm for random sparse k-SAT formulas and we identify first non-bipartite forcing pairs for quasi-random graphs. Our focus then shifts from the study of quasi-random objects to applications of different versions of the hypergraph regularity lemmas; all these versions assert decompositions of hypergraphs into constantly many quasi-random parts, where the meaning of ``quasi-random'''' takes different contexts in different situations. We study the family of hypergraphs not containing the hypergraph of the Fano plane as a subhypergraph, and show that almost all members of this family are bipartite. As a consequence an algorithm for coloring bipartite 3-uniform hypergraphs with average polynomial running time is given. Then the following combinatorial extremal problem is considered. Suppose one is given r colors and a fixed hypergraph F. The question is: In at most how many ways can one color the hyperedges of a hypergraph H on n vertices such that no monochromatic copy of F is created? What are the extremal hypergraphs for this function? Here a structural result for a natural family of hypergraphs F is proven. For some special classes of hypergraphs we show that their extremal hypergraphs (for large n) maximize the number of edge colorings for 2 and 3 colors, while for at least 4 colors other hypergraphs are optimal.

Made available in DSpace on 2014-06-11T19:27:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-12-09Bitstream added on 2014-06-13T20:47:43Z : No. of bitstreams: 1 cobra_tt_me_rcla.pdf: 485593 bytes, checksum: 107d36859b5a9c932411b3a54094c4ac (MD5) / O principal objetivo deste trabalho é apresentar conceitos básicos sobre coincidências e pontos fixos de aplicações contínuas usando como ferramentas os Lemas Combinatórios de Sperner e grau de aplicações. Apresentamos também um cálculo do número de Lefschetz de f; g : T2 ¡! T3, onde Th denota uma superfície de genus h, através da fórmula dada por Gonçalves e Oliveira em [3] / The main goal of this work is present basic concepts on coincidences and fixed points of continuous maps with Sperner’s Combinatorial Lemmas, and degree maps approaches. We also present a calculation of the Lefschetz number of f; g : T2 ¡! T3, where Th denotes surface of genus h, by using the formula given by Gonçalves and Oliveira in [3]

Die Regularitätsmethode für Graphen wurde vor über 30 Jahren von Szemerédi, für den Beweis seines Dichteresultates über Teilmengen der natürlichen Zahlen, welche keine arithmetischen Progressionen enthalten, entwickelt. Grob gesprochen besagt das Regularitätslemma, dass die Knotenmenge eines beliebigen Graphen in konstant viele Klassen so zerlegt werden kann, dass fast alle induzierten bipartiten Graphen quasi-zufällig sind, d.h. sie verhalten sich wie zufällige bipartite Graphen mit derselben Dichte. Das Regularitätslemma hatte viele weitere Anwendungen, vor allem in der extremalen Graphentheorie, aber auch in der theoretischen Informatik und der kombinatorischen Zahlentheorie, und gilt mittlerweile als eines der zentralen Hilfsmittel in der modernen Graphentheorie. Vor wenigen Jahren wurden Regularitätslemmata für andere diskrete Strukturen entwickelt. Insbesondere wurde die Regularitätsmethode für uniforme Hypergraphen und dünne Graphen verallgemeinert. Ziel der vorliegenden Arbeit ist die Weiterentwicklung der Regularitätsmethode und deren Anwendung auf Probleme der theoretischen Informatik. Im Besonderen wird gezeigt, dass vererbbare (entscheidbare) Hypergrapheneigenschaften, das sind Familien von Hypergraphen, welche unter Isomorphie und induzierten Untergraphen abgeschlossen sind, testbar sind. D.h. es existiert ein randomisierter Algorithmus, der in konstanter Laufzeit mit hoher Wahrscheinlichkeit zwischen Hypergraphen, welche solche Eigenschaften haben und solchen die „weit“ davon entfernt sind, unterscheidet. / About 30 years ago Szemerédi developed the regularity method for graphs, which was a key ingredient in the proof of his famous density result concerning the upper density of subsets of the integers which contain no arithmetic progression of fixed length. Roughly speaking, the regularity lemma asserts, that the vertex set of every graph can be partitioned into a constant number of classes such that almost all of the induced bipartite graphs are quasi-random, i.e., they mimic the behavior of random bipartite graphs of the same density. The regularity lemma had have many applications mainly in extremal graph theory, but also in theoretical computer science and additive number theory, and it is considered one of the central tools in modern graph theory. A few years ago the regularity method was extended to other discrete structures. In particular extensions for uniform hypergraphs and sparse graphs were obtained. The main goal of this thesis is the further development of the regularity method and its application to problems in theoretical computer science. In particular, we will show that hereditary, decidable properties of hypergraphs, that are properties closed under isomorphism and vertex removal, are testable. I.e., there exists a randomised algorithm with constant running time, which distinguishes between Hypergraphs displaying the property and those which are “far” from it.

[pt] Esse trabalho visa demonstrar os lemas de Sperner e aplicá-los nasdemonstrações do teorema de Monsky em Q2 e do teorema do ponto fixo deBrouwer em R2. Além disso, relatamos como esses lemas foram abordados com alunos da educação básica tendo como ferramenta educacional jogos de tabuleiro. / [en] This work aims to prove the Sperner s Lemmas and to apply them in proving the Monsky s Theorem in Q2 and the Brouwer fixed point Theorem in R2. Moreover, we report how these lemmas were addressed with students in basic education using board games as educational tools.