Let (M,g) be a quasi-Sasaki manifold with Reeb vector field xi. Our goal is to understand the structure of M when g is an Einstein metric. Assuming that the S^1 action induced by xi is locally free or assuming a certain non-negativity condition on the transverse curvature, we prove some rigidity results on the structure of (M,g). Naturally associated to a quasi-Sasaki metric g is a transverse Kahler metric g^T. The transverse Kahler-Ricci flow of g^T is the normalized Ricci flow of the transverse metric. Exploiting the transverse Kahler geometry of (M,g), we can extend results in Kahler-Ricci flow to our transverse version. In particular, we show that a deep and beautiful theorem due to Perleman has its counterpart in the quasi-Sasaki setting. We also consider evolving a Sasaki metric g by Ricci flow. Unfortunately, if g(0) is Sasaki then g(t) is not Sasaki for t>0. However, in some instances g(t) is quasi-Sasaki. We examine this and give some qualitative results and examples in the special case that the initial metric is eta-Einstein.

Models are frequently used to represent complex systems in order to test the systems before they are deployed. Some of the most complicated models are those that represent multi-agent systems (MAS), where there are multiple decision makers. Brahms is an agent-oriented language that models MAS. Three major qualities affect the behavior of these MAS models: workframes that change the state of the system, communication activities that coordinate information between agents, and the schedule of workframes. The primary method to test these models that exists is repeated simulation. Simulation is useful insofar as interesting test cases are used that enable the simulation to explore different behaviors of the model, but simulation alone cannot be fully relied upon to adequately cover the test space, especially in the case of non-deterministic concurrent systems. It takes an exponential number of simulation trials to uncover schedules that reveal unexpected behaviors. This thesis defines a coverage metric to make simulation more meaningful before verification of the model. The coverage metric is divided into three different metrics: workframe coverage, communication coverage, and schedule coverage. Each coverage metric is defined through static analysis of the system, resulting in the coverage requirements of that system. These coverage requirements are compared to the logged output of the simulation run to calculate the coverage of the system. The use of the coverage metric is illustrated in several empirical studies and explored in a detailed case study of the SATS concept (Small Aircraft Transportation System). SATS outlines the procedures aircraft follow around runways that do not have communication towers. The coverage metric quantifies the test effort, and can be used as a basis for future automated test generation and active test.

In coding theory we wish to find as many codewords as possible, while simultaneously maintaining high distance between codewords to ease the detection and correction of errors. For linear codes, this translates to finding high-dimensional subspaces of a given metric space, where the induced distance between vectors stays above a specified minimum. In this work I describe the recent advances of this problem in the contexts of lexicodes and Ferrers diagram rank-metric codes. In the first chapter, we study lexicodes. For a ring R, we describe a lexicographic ordering of the left R-module Rn. With this ordering we set up a greedy algorithm which sequentially selects vectors for which all linear combinations satisfy a given property. The resulting output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. We describe a generalization of the algorithm to finite principal ideal rings. In the second chapter, we investigate Ferrers diagram rank-metric codes, which play a role in the construction of subspace codes. A well-known upper bound for dimension of these codes is conjectured to be sharp. We describe several solved cases of the conjecture, and further contribute new ones. In addition, probabilities for maximal Ferrers diagram codes and MRD codes are investigated in a new light. It is shown that for growing field size, the limiting probability depends highly on the Ferrers diagram.

Considerable study has been made concerning the properties of the iterations of a homeomorphism on a metric space. Much of this material is scattered throughout the literature and understood solely by a specialist. The main object of this paper is to put into readable form proofs of theorems found in G.T. Whyburn's "Analytic Topology" pertaining to this topic in topology. Properties of the decomposition space of point-orbits induced by the iterations of a homeomorphism will compose a major part of the study. Some theorems will be established through series of lemmas required to fill in much of the detail lacking in proofs found the book. Although an elementary knowledge of topology is assumed throughout the paper, references are given for basic definitions and theorems used in developing some of the proofs. The following symbols and notation will be used throughout the paper. X will denote a metric space with metric p, S a topological space, I the set of positive integers, A, B, C... sets of points or elements. Small letters, such as a, b, c, x, y, z... will designate elements or points of sets. U and V will denote open sets Sr(x) a spherical neighborhood of x with radius r. A' denotes the set of limit points of A. A- the set of closure points of A/ U, N, C will denote union, intersection, and set inclusion respectively. The symbol E will mean "is an element of". 0 denotes the empty set. S - A is the set of points in S which are not in A.

<p>The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.</p><p>All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.</p><p>Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.</p><p>This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.</p>

Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines and develops discrete homotopy theory for metric spaces, which can be thought of as a discrete analog of classical path-homotopy and covering space theory. Given a metric space, X, this leads to the construction of a collection of covering spaces of X - and corresponding covering groups - parameterized by the positive real numbers, which we call the [epsilon]-covers and the [epsilon]-groups. These covers and groups evolve dynamically as the parameter decreases, changing topological type at specific parameter values which depend on the topology and local geometry of X. This leads to the definition of a critical spectrum for metric spaces, which is the set of all values at which the topological type of the covers change. Several results are proved regarding the critical spectrum and its connections to topology and local geometry, particularly in the context of geodesic spaces, refinable spaces, and Gromov-Hausdorff limits of compact metric spaces. We investigate the relationship between the critical spectrum and covering spectrum in the case when X is geodesic, connections between the geometry of the [epsilon]-groups and the metric and topological structure of the [epsilon]-covers, as well as the behavior of the [epsilon]-covers and critical values under Gromov-Hausdorff convergence.

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This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.

In this thesis, we will present examples of Voronoi diagrams that are not tessellations. Moreover, we will find sufficient conditions on subspaces of E2, S2 and the Poincaré disk and the sets of sites that guarantee that the Voronoi diagrams are pre-triangulations. We will also study g-spaces, which are metric spaces with ‘extendable’ geodesics joining any 2 points and give properties for a set of sites in a g-space that again guarantees that the Voronoi diagram is a pre-triangulation.