Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Gangbo, Wilfrid; Committee Member: Loss, Michael; Committee Member: Pan, Ronghua; Committee Member: Swiech, Andrzej; Committee Member: Tannenbaum, Allen. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 115-121). Available also in a digital version from Dissertation Abstracts.

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The Hamiltonian cycle problem (HCP) can be succinctly stated as: "Given a graph, find a cycle that passes through every single vertex exactly once, or determine that this cannot be achieved". Such a cycle is called a Hamiltonian cycle. The HCP is a special case of the better known Travelling salesman problem, both of which are computationally difficult to solve. An efficient solution to the HCP would help solving the TSP effectively, and therefore would have a great impact in various fields such as computer science and operations research. In this thesis, we obtain novel theoretical results that approach this discrete, deterministic, problem using tools from stochastic processes, matrix analysis, and graph theory. / PhD Doctorate

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

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This thesis consists of two parts. In the first part, we study stability properties of Hamiltonian systems on the Wasserstein space. Let H be a Hamiltonian satisfying conditions imposed in the work of Ambrosio and Gangbo. We regularize H via Moreau-Yosida approximation to get H[subscript Tau] and denote by μ[subscript Tau] a solution of system with the new Hamiltonian H[subscript Tau] . Suppose H[subscript Tau] converges to H as τ tends to zero. We show μ[subscript Tau] converges to μ and μ is a solution of a Hamiltonian system which is corresponding to the Hamiltonian H. At the end of first part, we give a sufficient condition for the uniqueness of Hamiltonian systems. In the second part, we develop a general theory of differential forms on the Wasserstein space. Our main result is to prove an analogue of Green's theorem for 1-forms and show that every closed 1-form on the Wasserstein space is exact. If the Wasserstein space were a manifold in the classical sense, this result wouldn't be worthy of mention. Hence, the first cohomology group, in the sense of de Rham, vanishes.