Thesis (Ph. D.)--Carnegie Mellon University, 2003. / Title from PDF title page (viewed Dec. 18, 2009). Includes bibliographical references (p. 49-52).

This dissertation solves variations of three mathematical problems for autonomous vehicle formations: (1) formation shape control in the plane, (2) robust information architecture design, and (3) formation attitude synchronization. An autonomous vehicle formation is a collection of vehicles, each with computation, communication, sensing, and control capabilities, that cooperate to achieve a common objective. Accelerating advancements are making possible a range of science and engineering applications, such as satellite formations for deep-space imaging, teams of unmanned aircraft for military reconnaissance and surveillance missions, and submarine swarms for oceanic exploration. The ubiquitous potential of these applications is driving theoretical work on autonomous vehicle formations across a range of disciplines. A major theoretical question in the field of control theory, and the main focus of this dissertation, is how the properties of the information architecture (i.e. a mapping of the information flow amongst the agents), relate to the stability properties of the desired shape and orientation under certain control laws. A secondary focus is how to design the information flow so that loss of an agent does not destroy the formation's ability to maintain a desired shape. As a motivating example, a solution to a coordinated standoff tracking problem is presented to demonstrate how an instance of a class of information architectures, which are called persistent and related to rigid graph theory, can be used to achieve a formation objective in a practical scenario involving a team of unmanned aircraft. A generalized formation shape control problem is then solved for a class of persistent architectures. This solution gives only local stability results; global stability is analyzed for a four-agent formation and several open problems are identified. The problem of agent loss is addressed by performing a self-repair operation in the event of agent loss and separately by designing robustness into the information architecture a priori. Finally, a rigid body attitude synchronization problem with communication time delays is solved for a class of information architectures based on spectral graph theory. / text

Thesis (Ph.D.)--University of Wollongong. / Typescript. Includes bibliographical references: leaf 169-179.

Thesis (M.S.)--Rochester Institute of Technology, 2008. / Typescript. Includes bibliographical references (leaves 64-66).

Orientadores: Celia Picinin de Mello, Anamaria Gomide / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T19:50:54Z (GMT). No. of bitstreams: 1 Vulcani_RenatadeLacerdaMartins_M.pdf: 2431212 bytes, checksum: 702947f1e783d410ef77eb0234852d6a (MD5) Previous issue date: 2015 / Resumo: Neste trabalho apresentamos uma breve introdução à teoria dos grafos, elucidando alguns conceitos básicos e destacando grafos eulerianos. Usamos o conceito de grafos eulerianos para resolver alguns passatempos e jogos conhecidos. Finalizamos apresentando algumas aplicações que envolvem grafos que não são necessariamente eulerianos / Abstract: In this work we present a brief introduction to graph theory, explaining some basic concepts and highlighting eulerians graphs. We use the concept of eulerians graphs to solve some well known puzzles and games. We finalize by presenting some applications involving graphs that are not necessarily eulerians / Mestrado / Matemática em Rede Nacional / Mestra

We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.

The main contribution of this dissertation proposes a new paradigm, called the parentheses matching paradigm. It claims that this paradigm is well suited for designing efficient parallel algorithms for a broad class of nonnumeric problems. To demonstrate its applicability, we present three cost-optimal parallel algorithms for breadth-first traversal of general trees, sorting a special class of integers, and coloring an interval graph with the minimum number of colors.

Master of Science / Department of Mathematics / Sarah Reznikoff / A (1, ≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1, ≤ 2)-identifiable. As such, we discuss properties that must be satisfied by a valid (1, ≤ 2)-identifying code, characteristics of a graph which preclude the existence of a (1, ≤ 2)-identifying code, and relationships between the maximum degree and order of (1, ≤ 2)-identifiable graphs. Additionally, we show that (1, ≤ 2)-identifiable graphs have no forbidden induced subgraphs and provide a list of (1, ≤ 2)-identifiable graphs with minimum (1, ≤ 2)-identifying codes indicated.

Call number: LD2668 .T4 1984 T56 / Master of Science

Recently graph theory has been successfully applied to magnetic resonance imaging data. However, it remains unclear as to what the nodes and edges in a network should represent. This problem is particularly difficult when extracting morphological networks (i.e., from grey matter segmentations). Existing morphological network studies have used anatomical regions as nodes that are connected by edges when these regions covary in thickness or volume across a sample of subjects. Covariance in cortical thickness or volume has been hypothesised to be caused by anatomical connectivity, experience driven plasticity and/or mutual trophic influences. A limitation of this approach is that it requires magnetic resonance imaging (MRI) scans to be warped into a standard template. These warping processes could filter out subtle structural differences that are of most interest in, for example, clinical studies. The focus of the work in this thesis was to address these limitations by contributing a new method to extract morphological networks from individual cortices. Briefly, this method divides the cortex into small regions of interest that keep the three-dimensional structure intact, and edges are placed between any two regions that have a statistically similar grey matter structure. The method was developed in a sample of 14 healthy individuals, who were scanned at two different time points. For the first time individual grey matter networks based on intracortical similarity were studied. The topological organisation of intracortical similarities was significantly different from random topology. Additionally, the graph theoretical properties were reproducible over time supporting the robustness of the method. All network properties closely resembled those reported in other imaging studies. The second study in this thesis focussed on the question whether extracting networks from individual scans would be more sensitive than traditional methods (that use warping procedures) to subtle grey matter differences in MRI data. In order to investigate this question, the method was applied to the first round of scans from the Edinburgh High Risk study of Schizophrenia (EHRS), before any of the subjects was diagnosed with (symptoms of) the disease. Where traditional methods failed to find differences at the whole brain level between the high risk group and healthy controls, the new method did find subtle disruptions of global network topology between the groups. Finally, the diagnostic value of the networks was studied with exploratory analyses that found that, in comparison to healthy controls, people at high risk of schizophrenia showed more intracortical similarities in the left angular gyrus. Furthermore within the high risk group an increase of intracortical similarities could predict disease outcome up to 74% accuracy. The main conclusion of this thesis was that the new method provides a robust and concise statistical description of the grey matter structure in individual cortices, that is of particular importance for the study of clinical populations when structural disruptions are subtle.