Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2005. / Includes bibliographical references (p. 40).

Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree. Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces.

Thesis (M.A.)--The University of North Carolina at Greensboro, 2008. / Directed by Paul Duvall; submitted to the Dept. of Mathematical Sciences. Title from PDF t.p. (viewed Aug. 26, 2009). Includes bibliographical references (p. 38).

Thesis (M.S.)--Boise State University, 2010. / Includes abstract. Includes bibliographical references (leaf 55).

Thesis (M.S.)--Boise State University, 2010. / Title from t.p. of PDF file (viewed July 30, 2010). Includes abstract. Includes bibliographical references (leaf 55).

The main results of this dissertation show evidence supporting the Successive Surface Scaffolding Conjecture. This is a new conjecture that, if true, guarantees the existence of all the wye-delta-order minimal obstructions of a surface S as subgraphs of the irreducible triangulations of the surface S with a crosscap added. A new data structure, i.e. an augmented rotation system, is presented and used to create an exponential-time algorithm for embedding graphs in any surface with a constant-time check of the change in genus when inserting an edge. A depiction is a new formal definition for representing an embedding graphically, and it is shown that more than one depiction can be given for nonplanar embeddings, and that sometimes two depictions for the same embedding can be drastically different from each other. An algorithm for finding the essential cycles of an embedding is given, and is used to confirm for the projective-plane obstructions, a theorem that shows any embedding of an obstruction must have every edge in an essential cycle. Obstructions of a general surface S that are minor-minimal and not double-wye-delta-minimal are shown to each have an embedding on the surface S with a crosscap added. Finally, open questions for further research are presented. / Graduate

Voltage graph theory can be used to describe embeddings of product graphs if one factor is a Cayley graph. We use voltage graphs to explore embeddings of various products where one factor is a hypercube, describing some minimal and symmetrical embeddings. We then define a graph product, the weak symmetric difference, and illustrate a voltage graph construction useful for obtaining an embedding of the weak symmetric difference of an arbitrary graph with a hypercube.

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

Our world is filled with large-scale engineering systems, which provide various services and conveniences in our daily life. A distinctive trend in the development of today's large-scale engineering systems is the extensive and aggressive adoption of automation and autonomy that enable the significant improvement of systems' robustness, efficiency, and performance, with considerably reduced manning and maintenance costs, and the U.S. Navy's DD(X), the next-generation destroyer program, is considered as an extreme example of such a trend. This thesis pursues a modeling solution for performing simulation-based analysis in the conceptual or preliminary design stage of an intelligent, self-reconfigurable ship fluid system, which is one of the concepts of DD(X) engineering plant development. Through the investigations on the Navy's approach for designing a more survivable ship system, it is found that the current naval simulation-based analysis environment is limited by the capability gaps in damage modeling, dynamic model reconfiguration, and simulation speed of the domain specific models, especially fluid network models. As enablers of filling these gaps, two essential elements were identified in the formulation of the modeling method. The first one is the graph-based topological modeling method, which will be employed for rapid model reconstruction and damage modeling, and the second one is the recurrent neural network-based, component-level surrogate modeling method, which will be used to improve the affordability and efficiency of the modeling and simulation (M&S) computations. The integration of the two methods can deliver computationally efficient, flexible, and automation-friendly M&S which will create an environment for more rigorous damage analysis and exploration of design alternatives. As a demonstration for evaluating the developed method, a simulation model of a notional ship fluid system was created, and a damage analysis was performed. Next, the models representing different design configurations of the fluid system were created, and damage analyses were performed with them in order to find an optimal design configuration for system survivability. Finally, the benefits and drawbacks of the developed method were discussed based on the result of the demonstration.

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