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For a vertex x of G, the eccentricity e (x) is the distance between x and a vertex farthest from x. Then x is a contour vertex if there is no neighbor of x with its eccentricity greater than e (x). The x-y path of length d (x,y) is called a x-y geodesic. The geodetic interval I [x,y] of a graph G is the set of vertices of all x-y geodesics in G. For S ⊆ V , the geodetic closure I [S] of S is the union of all geodetic intervals I [x,y] over all pairs x,y ∈S. A vertex set S is a geodetic set for G if I [S] = V (G). In this thesis, we study the contour sets of product graphs and discuss these sets are geodetic sets for some conditions.

[pt] Lentes eletromagnéticas têm sido usadas desde a década de 50 como forma de se obter uma antena com apontamento eletrônico e alta diretividade, capaz de operar numa vasta faixa de freqüências. Por este motivo, tornaram-se populares, principalmente em aplicações de Guerra Eletrônica. Entretanto, uma de suas desvantagens é que, a cada momento, apenas uma fonte emissora é usada para um certo apontamento. Esta tese propõe um novo tipo de lentes geodésica em que se emprega uma distribuição de fontes uniforme sobre a entrada da lente, sendo que todas elas sempre emitem com a mesma potência, independentemente da direção escolhida para o feixe. Os mesmos objetivos são atingidos - diretividade, ampla faixa de freqüências de operação e apontamento eletrônico. Assim, todas as fontes contribuem para a potência total irradiada, com a mesma amplitude, sendo o apontamento obtido pela adequada escolha da fase de cada uma destas fontes. Acrescenta-se que tal dispositivo possui menor queda de desempenho, numa situação de falha de uma ou mais fontes. Dois modelos teóricos para a análise e projeto da estrutura - um baseado na ótica geométrica e outro em técnicas modais - são apresentados e comparados. / [en] Electromagnetic lenses have been applied since early 50´s as a means of obtaining an electronic steerable antenna with high directivity and broadband range of operation. For this reason, they have became very popular in Electronic warfare applications. Some of their weakness, however, would be the fact that only one feed point (supply) - and all the associated circuitry - can be used at a time, in order to generate a single bean in the radiated field. The purpose of this these is to propose a new kind of geodesic lens in which the same goals are obtained (high directivity, broadband range of frequencies, and electronic steering) but with an uniform distribution of feeding along the lens input, all the supplies used at all times, no matter what the direction chosen for the output bean. Therefore, all the feeding points are radiating the same amount of power (same amplitude), the electronic steering is obtained by properly setting the phases of the several supplies at the input of the lens. Moreover, the device proposed here has less degradation of performance when one or a few supplies go down. Two theoretic models for the analysis and design of such device - one based on the geometric optics and the other based on modal techniques - are also provided and compared.

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A new resistive Geodesic Acoustic Mode (GAM) theory is developed by two-fluids analysis and resistive gyro-kinetic formulation in this thesis. An analytical expression is obtained for the resistive GAM frequency. This theory suggests a large collision frequency will prohibit the parallel current in tokamak, which establishes the cross-field charge neutrality condition ∇·J⊥= 0 for the existence of GAM at the edge plasma of tokamak. Therefore, the resistive GAM theory provides a more plausible explanation to edge GAM phenomena. Various probe arrays are designed and installed in the STOR-M tokamak to search for the poloidal GAM phenomena. A series of experiments were conducted in the L-mode and RMP discharges. The FFT and wavelet analyses indicate the existence of GAM phenomena in STOR-M, and the observed GAM frequencies match the theoretical predication using the resistive GAM model.

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 57-59).

This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds. / Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded. / The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially. / We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.

Thesis (M. S.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Tannenbaum, Allen; Committee Member: Barnes, Christopher F.; Committee Member: Niethammer, Marc; Committee Member: Shamma, Jeff; Committee Member: Vela, Patricio.