No description available.

abstract: Let T be a tournament with edges colored with any number of colors. A rainbow triangle is a 3-colored 3-cycle. A monochromatic sink of T is a vertex which can be reached along a monochromatic path by every other vertex of T. In 1982, Sands, Sauer, and Woodrow asked if T has no rainbow triangles, then does T have a monochromatic sink? I answer yes in the following five scenarios: when all 4-cycles are monochromatic, all 4-semi-cycles are near-monochromatic, all 5-semi-cycles are near-monochromatic, all back-paths of an ordering of the vertices are vertex disjoint, and for any vertex in an ordering of the vertices, its back edges are all colored the same. I provide conjectures related to these results that ask if the result is also true for larger cycles and semi-cycles. A ruling class is a set of vertices in T so that every other vertex of T can reach a vertex of the ruling class along a monochromatic path. Every tournament contains a ruling class, although the ruling class may have a trivial size of the order of T. Sands, Sauer, and Woodrow asked (again in 1982) about the minimum size of ruling classes in T. In particular, in a 3-colored tournament, must there be a ruling class of size 3? I answer yes when it is required that all 2-colored cycles have an edge xy so that y has a monochromatic path to x. I conjecture that there is a ruling class of size 3 if there are no rainbow triangles in T. Finally, I present the new topic of alpha-step-chromatic sinks along with related results. I show that for certain values of alpha, a tournament is not guaranteed to have an alpha-step-chromatic sink. In fact, similar to the previous results in this thesis, alpha-step-chromatic sinks can only be demonstrated when additional restrictions are put on the coloring of the tournament's edges, such as excluding rainbow triangles. However, when proving the existence of alpha-step-chromatic sinks, it is only necessary to exclude special types of rainbow triangles. / Dissertation/Thesis / Ph.D. Mathematics 2011

Thesis--Freie Universität Berlin. / Includes bibliographical references (p. 315-340) and index.

Thesis--Freie Universität Berlin. / Includes bibliographical references (p. 315-340) and index.

A 2-dipath k-colouring of an oriented graph G is an assignment of k colours, 1,2, . . . , k, to the vertices of G such that vertices joined by a directed path of length two are assigned different colours. The 2-dipath chromatic number is the minimum number of colours needed in such a colouring. There are two possible models, depending on whether adjacent vertices must also be assigned different colours. For both models of 2-dipath colouring we develop the basic theory, including characterizing the oriented graphs that can be 2-dipath coloured using a small number of colours, finding bounds on the 2-dipath chromatic number, determining the complexity of deciding the existence of a 2-dipath k-colouring, describing a homomorphism model, and showing how to determine the 2-dipath chromatic number of tournaments and bipartite tournaments. / Graduate

Wimbledon is one of the most popular annual sports tournament. In the Gentlemen's Single 2009 the top seeded and defending champion Rafael Nadal withdrew from the tournament due to injury days prior to the tournament. Here, we try to analyze the effects of Nadal's withdrawal especially on the ability/strength of the main competitor Roger Federer by using bookmakers expectancies to estimate the unknown abilities of the players and compare them for two different odds sets. The comparison shows that the bookmakers did not incorporate Nadal's withdrawal adequately, assigning too high expected winning probabilities to Federer and Murray. / Series: Research Report Series / Department of Statistics and Mathematics

Un graphe est un ensemble de noeuds, ensemble de liens reliant des paires de noeuds. Avec la quantité accumulée de données collectées, il existe un intérêt croissant pour la compréhension des structures et du comportement de très grands graphes. Néanmoins, l’augmentation rapide de la taille des grands graphes rend l’étude de tous les graphes de moins en moins efficace. Ainsi, il existe une demande impérieuse pour des méthodes plus efficaces pour étudier de grands graphes sans nécessiter la connaissance de tous les graphes. Une méthode prometteuse pour comprendre le comportement de grands graphes consiste à exploiter des propriétés spécifiques de structures locales, telles que la taille des grappes ou la présence locale d’un motif spécifique, c’est-à-dire un graphe donné (généralement petit). Un exemple classique de la théorie des graphes (cas avérés de la conjecture d'Erdos-Hajnal) est que, si un graphe de grande taille ne contient pas de motif spécifique, il doit alors avoir un ensemble de noeuds liés par paires ou non liés, de taille exponentiellement plus grande que prévue. Cette thèse abordera certains aspects de deux questions fondamentales de la théorie des graphes concernant la présence, en abondance ou à peine, d’un motif donné dans un grand graphe : - Le grand graphe peut-il être partitionné en copies du motif ? - Le grand graphe contient-il une copie du motif ? Nous discuterons de certaines des conjectures les plus connues de la théorie des graphes sur ce sujet: les conjectures de Tutte sur les flots dans les graphes et la conjecture d'Erdos-Hajnal mentionnée ci-dessus, et présenterons des preuves pour plusieurs conjectures connexes - y compris la conjecture de Barát-Thomassen, une conjecture de Haggkvist et Krissell, un cas particulier de la conjecture de Jaeger-Linial-Payan-Tarsi, une conjecture de Berger et al, et une autre d'Albouker et al. / A graph is a set of nodes, together links connecting pairs of nodes. With the accumulating amount of data collected, there is a growing interest in understanding the structures and behavior of very large graphs. Nevertheless, the rapid increasing in size of large graphs makes studying the entire graphs becomes less and less efficient. Thus, there is a compelling demand for more effective methods to study large graphs without requiring the knowledge of the graphs in whole. One promising method to understand the behavior of large graphs is via exploiting specific properties of local structures, such as the size of clusters or the presence locally of some specific pattern, i.e. a given (usually small) graph. A classical example from Graph Theory (proven cases of the Erdos-Hajnal conjecture) is that if a large graph does not contain some specific pattern, then it must have a set of nodes pairwise linked or not linked of size exponentially larger than expected. This thesis will address some aspects of two fundamental questions in Graph Theory about the presence, abundantly or scarcely, of a given pattern in some large graph: - Can the large graph be partitioned into copies of the pattern? - Does the large graph contain any copy of the pattern?We will discuss some of the most well-known conjectures in Graph Theory on this topic: the Tutte's flow conjectures on flows in graphs and the Erdos-Hajnal conjecture mentioned above, and present proofs for several related conjectures -- including the Barát-Thomassen conjecture, a conjecture of Haggkvist and Krissell, a special case of Jaeger-Linial-Payan-Tarsi's conjecture, a conjecture of Berger et al, and another one by Albouker et al.

A graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices there are at least k other vertices, each of which is adjacent to the first m vertices but not adjacent to any of the latter n vertices. The class of graphs having property P(m.n,k) is denoted by ζ(m,n,k). The problem that arises is that of characterizing the class ζ(m,n,k). One particularly interesting problem that arises concerns the functionP(m,n,k) = min{υ(G) : G є ζ(m,n,k) }.In Chapter 2, we establish some important properties of graphs in the class ζ(m,n,k) and a lower bound on p(m,n,k). In particular, we prove thatp(n,n,k) ≥ 4n-1 (2(n+k) + ½ (3 = (-1)n+k+1} + 1/3 l 1/3One of the results in Chapter 2 is that almost all graphs have property P(m,n,k). However, few members of ζ(m,n,k) have been exhibited. In Chapter 3. we construct classes of graphs having property P(l,n,k) . These classes include the cubes, "generalized" Petersen graphs and "generalized" Hoffman-Singleton graphs.An important graph in the study of the class ζ(m,n,k) is the Paley graph Gq defined as follows. Let q = l(mod 4) be a prime power. The vertices of Gq are the elements of the finite field IFq. Two vertices a and b are joined by an edge if and only if their difference is a quadratic residue, that is a - b = y2 for some y є IFq. In chapter 4, we prove that for a prime p = l(mod 4), all sufficiently large Paley graphs GP satisfy property P(m.n,k). This is established by making use of results from prime number theory.In Chapter 5 , we establish, by making use of results from finite fields, the adjacency properties of Paley graphs of order q = pd , with p a prime.For directed graphs, there is an analogue of the above adjacency property concerning tournaments. A tournament Tq of order q is said to have property Q(n,k) if every subset of n vertices of Tq is dominated (if there is an arc directed from ++ / a vertex u to a vertex v, we say that u dominates v and that v is dominated by u) by at least k other vertices.Let q = 3(mod 4) is a prime power. The Paley tournament Dq is defined as follows. The vertices of Dq are the elements of the finite field IFq. Vertex a is ioined to vertex b by an arc if and only if a - b is a quadratic residue in Fq. In Chapter 6, we prove that the Paley tournament Dq has property Q(n,k) wheneverq > {(n - 3)2n-1 + Z}G + kZn - 1. A graph G is said to have property P*(rn,n,k) if for any set of rn + n distinct vertices of G there are exactly k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The class of graphs having property P*(m.n,k) is denoted by S*(m,n.k). The class S*(m,n,k) has been studied when one of m or n is zero. In Chapter 7, we show that, for m = n = 1, graphs with this property (k + t)' + 1, are the strongly regular graphs with parameters ( k + t,t - 1,t) for some positive integer t. For rn 2 1, n 2 1, and m + n 2 3, we show that, there is no graph having property P*(m.n,k), for any positive integer k. The first Chapter of this thesis provides the motivation, terminology. general concepts and the problems concerning the adjacency properties of graphs. In Chapter 8 . we present some open problems.

The investment management industry has proven to be a fertile ground for theoretical and empirical research over the past forty years, particularly in relation to the nature and quantification of risk. However, the dominance of the U.S. industry has meant that much of the academic research has focused on the U.S. market. This thesis investigates aspects of investment risk using alternative data to that used in much of the prior published research. This thesis contains an extensive analysis of aspects of risk related to both the demand side and the supply side of the managed funds market in Australia. Among the demand side characteristics, attitudes towards risk and their impact on asset allocation decisions will be an important determinant of investors' financial well-being, particularly in retirement. Accordingly, the first part of the thesis examines the financial risk tolerance of investors, exploring the relationship between subjective financial risk tolerance and a range of demographic characteristics that are widely used as a basis for heuristically derived estimates of investors' attitudes towards financial risk. The second part of the thesis contains an analysis of the supply side of the industry, focusing on risk-shifting behavior by investment fund managers. Since the time when performance and risk-shifting behavior of fund managers was first put under the spotlight 40 years ago, it is possible to identify an evolving strand in the research where performance assessment is examined within the framework of the principal-agent literature. One focus that has emerged in this literature is the adaption of the tournament model to the analysis of investment manager behavior, wherein it is hypothesized that fund managers who were interim losers were likely to increase fund volatility in the latter part of the assessment period to a greater extent than interim winners. Against this background, the second part of the thesis examines risk-shifting behavior by Australian fund managers. Both the ability of fund managers to time the market and the applicability of the tournament model of funds management to a segment of the Australian

No description available.