Under approval voting, every voter may vote for any number of canditates. To model approval voting, we let a political spectrum be the set of all possible political positions, and let each voter have a subset of the spectrum that they approve, called an approval region. The fraction of all voters who approve the most popular position is the agreement proportion for the society. We consider voting in societies whose political spectrum is modeled by $d$-dimensional space ($\mathbb{R}^d$) with approval regions defined by axis-parallel boxes. For such societies, we first consider a Tur&#aacute;n-type problem, attempting to find the maximum agreement between pairs of voters for a society with a given level of overall agreement. We prove a lower bound on this maximum agreement and find in the literature a proof that the lower bound is optimal. By this result we find that for sufficiently large $n$, any $n$-voter box society in $\mathbb{R}^d$ where at least $\alpha\binom{n}{2}$ pairs of voters agree on some position must have a position contained in $\beta n$ approval regions, where $\alpha = 1-(1-\beta)^2/d$. We also consider an extension of this problem involving projections of approval regions to axes. Finally we consider the question of $(k,m)$-agreeable box societies, where a society is said to be $(k, m)$-agreeable if among every $m$ voters, some $k$ approve a common position. In the $m = 2k - 1$ case, we use methods from graph theory to prove that the agreement proportion is at least $(2d)^{-1}$ for any integer $k \ge 2.$

Extremal Functions for Graph Linkages and Rooted Minors Paul Wollan 137 pages Directed by: Robin Thomas A graph G is k-linked if for any 2k distinct vertices s_1,..., s_k,t_1,..., t_k there exist k vertex disjoint paths P_1,...,P_k such that the endpoints of P_i are s_i and t_i. Determining the existence of graph linkages is a classic problem in graph theory with numerous applications. In this thesis, we examine sufficient conditions that guarantee a graph to be k-linked and give the following theorems. (A) Every 2k-connected graph on n vertices with 5kn edges is k-linked. (B) Every 6-connected graph on n vertices with 5n-14 edges is 3-linked. The proof method for Theorem (A) can also be used to give an elementary proof of the weaker bound that 8kn edges suffice. Theorem (A) improves upon the previously best known bound due to Bollobas and Thomason stating that 11kn edges suffice. The edge bound in Theorem (B) is optimal in that there exist 6-connected graphs on n vertices with 5n-15 edges that are not 3-linked. The methods used prove Theorems (A) and (B) extend to a more general structure than graph linkages called rooted minors. We generalize the proof methods for Theorems (A) and (B) to find edge bounds for general rooted minors, as well as finding the optimal edge bound for a specific family of bipartite rooted minors. We conclude with two graph theoretical applications of graph linkages. The first is to the problem of determining when a small number of vertices can be used to cover all the odd cycles in a graph. The second is a simpler proof of a result of Boehme, Maharry and Mohar on complete minors in huge graphs of bounded tree-width.

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [59]-61).

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 80-81). Available also in a digital version from Dissertation Abstracts.

In this thesis, we consider linear extremal problems in the Hp spaces. For many of these extremal problems, a unique solution can be guaranteed. We will examine some of the classical examples of extremal problems in these spaces. With this framework in place we will then consider a particular problem which does not always have a unique solution.

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics. We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such general purpose bound). We then elaborate on some attempted generalizations/modifications of the Lovász number to undirected hypergraphs that we have tried. It is not currently known whether these attempted generalizations/modifications upper bound the capacity of arbitrary hypergraphs. An important method for proving lower bounds on hypergraph capacity is to exhibit a large independent set in a strong power of the hypergraph. We examine methods for this and show a barrier to attempts to usefully generalize certain of these methods to hypergraphs. We then look at cap sets: independent sets in powers of a certain hypergraph. We examine certain structural properties of them with the hope of finding ones that allow us to prove upper bounds on their size. Finally, we consider two interesting generalizations of capacity and use one of them to formulate several conjectures about connections between cap sets and sunflower-free sets.

In this dissertation, a problem related to Hadwiger's conjecture has been studied. We first proved a conjecture of Jakobsen from 1983 which states that every simple graphs on $n$ vertices and at least (11n-35)/2 edges either has a minor isomorphic to K_8 with one edge deleted or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2} and/or K_7 by identifying cliques of size five. We then studied the extremal functions for complete minors. We proved that every simple graph on nge9 vertices and at least 7n-27 edges either has a minor, or is isomorphic to K_{2, 2, 2, 3, 3}, or is isomorphic to a graph obtained from disjoint copies of K_{1, 2, 2, 2, 2, 2} by identifying cliques of size six. This result extends Mader's theorem on the extremal function for K_p minors, where ple7. We discussed the possibilities of extending our methods to K_{10} and K_{11} minors. We have also found the extremal function for K_7 plus a vertex minor.

The purpose of this thesis is to investigate a class of four left-invariant optimal control problems on the special orthogonal group SO(3). The set of all control-affine left-invariant control systems on SO(3) can, without loss, be reduced to a class of four typical controllable left-invariant control systems on SO(3) . The left-invariant optimal control problem on SO(3) involves finding a trajectory-control pair on SO (3), which minimizes a cost functional, and satisfies the given dynamical constraints and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*SO(3) = SO(3) x so (3)* using the optimal Hamiltonian on so(3)*, where the maximum principle yields the optimal control. In a contribution to the study of this class of optimal control problems on SO(3), the extremal equations on so(3)* (ident ified with JR3) are integrated via elliptic functions to obtain explicit expressions for the solution curves in each typical case. The energy-Casimir method is used to give sufficient conditions for non-linear stability of the equilibrium states. / KMBT_363 / Adobe Acrobat 9.54 Paper Capture Plug-in

The aim of this thesis is to study a class of left-invariant optimal control problems on the matrix Lie group SE(2). We classify, under detached feedback equivalence, all controllable (left-invariant) control affine systems on SE(2). This result produces six types of control affine systems on SE(2). Hence, we study six associated left-invariant optimal control problems on SE(2). A left-invariant optimal control problem consists of minimizing a cost functional over the trajectory-control pairs of a left-invariant control system subject to appropriate boundary conditions. Each control problem is lifted from SE(2) to T*SE(2) ≅ SE(2) x se (2)*and then reduced to a problem on se (2)*. The maximum principle is used to obtain the optimal control and Hamiltonian corresponding to the normal extremals. Then we derive the (reduced) extremal equations on se (2)*. These equations are explicitly integrated by trigonometric and Jacobi elliptic functions. Finally, we fully classify, under Lyapunov stability, the equilibrium states of the normal extremal equations for each of the six types under consideration.

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.