Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps. The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect to vertex-degree sequence, face-degree sequence, and number of edges, and the corresponding generating series for rooted locally orientable maps, can be explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a series defined algebraically in terms of Jack symmetric functions, and the unified theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on rooting, it cannot be directly related to genus. A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant. The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial explanation, for a functional relationship between a generating series for rooted orientable maps and the corresponding generating series for 4-regular rooted orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable maps, and its restriction to undecorated maps is expected to be related to the medial construction. Previous attempts to identify ϕ have suffered from the fact that the existing derivations of the functional relationship involve inherently non-combinatorial steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically.

We study the geometric classi&#64257;cation of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as de&#64257;ned by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (&#937;,B,??).

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (M.S.)--University of Hawaii at Manoa, 2005. / Includes bibliographical references (leaves 47-49). Also available by World Wide Web.

Inaug.-diss.--München. / Lebenslauf. Multigraphed. "Verzeichnis der benutzten literatur und karten": p. 101-106.

Thesis (M.S.)--University of Florida, 2002. / Title from title page of source document. Includes vita. Includes bibliographical references.

Many high value-machining processes such as milling and drilling have been performed by expensive and dedicated (single purpose) machine tools including CNC machine tools. Industrial robots are a good alternative to these conventional dedicated machine tools due to the robots’ many advantages such as lower cost, larger workspace, higher flexibility of motion, and versatility. Despite these advantages, several barriers prevent them from being widely adopted for high value machining purposes. Two of these barriers are low and nonlinear stiffness of the industrial robot arm and the manufacturing end-users not knowing the capabilities and advantages of robots in machining applications. This research sets out to help a typical machining operator who is not an expert in robotics to learn the capability of a given robotic machining system. This study should help the operator plan robotic machining processes by presenting process models and visual maps for a variety of machining processes and workpiece materials. The study shows in particular how the cutting force and the compliance of a robotic machining system affect machining processes. To meet this objective, we present a framework for planning development for any given robotic machining application domain. First, we select primary performance parameters (including joint torque limit and end-effector positional error) and control parameters (including machining parameters, end-effector position, and workpiece position) for robotic machining. Then, we present the system models and visual performance maps for the functional parameters of robotic machining processes. The focus is on cutting forces for the ten selected machining processes and end-effector positional error of a robotic machining system due to the compliance of a robotic system (i.e., robot manipulator and cutting tool) and joint error (due to sensor error and gear backlash). Finally, we present five applications to show how to use visual maps for preliminary planning scnearios of robotic machining processes. The applications present a step-by-step process for selecting from cutting parameters to workpiece position parameters by utilizing performance requirements and visual maps developed in this research. / text

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published_or_final_version / Mathematics / Master / Master of Philosophy

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