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Thesis (doctoral)--Litterarum Universitate Friderica Guilelma, 1858. / Vita.

No description available.

Note:

Thesis (Ph. D.)--Lehigh University, 2003. / Includes vita. Includes bibliographical references (leaves 36-38).

We define the notion of relative sectional curvature for 2-complexes, and prove that a compact angled 2-complex that has negative sectional curvature relative to planar sections has coherent fundamental group. We analyze a certain type of 1-complex that we call flattenable graphs Gamma→ X for an compact angled 2-complex X, and show that if X has nonpositive sectional curvature, and if for every flattenable graph pi1 (Gamma) → pi1( X) is finitely presented, then X has coherent fundamental group. Finally we show that if X is a compact angled 2-complex with negative sectional curvature relative to pi-gons and planar sections then pi1(X) is coherent. Some results are provided which are useful for creating examples of 2-complexes with these properties, or to test a 2-complex for these properties.

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Characterizing a manifold up to isometry is a challenging task. A manifold is a topological space. One may equip a manifold with a metric, and generally speaking, this metric determines how the manifold “looks". An example of this would be the unit sphere in R3. While we typically envision the standard metric on this sphere to give it its familiar shape, one could define a different metric on this set of points, distorting distances within this set to make it seem perhaps more ellipsoidal, something not isometric to the standard round sphere. In an effort to distinguish manifolds up to isometry, we wish to compute meaningful invariants. For example, the Riemann curvature tensor and its surrogates are examples of invariants one could construct. Since these objects are generally too complicated to compare and are not real valued, we construct scalar invariants from these objects instead. This thesis will explore these invariants and exhibit a special family of manifolds that are not flat on which all of these invariants vanish. We will go on to properly define, and gives examples of, manifolds, metrics, tangent vector fields, and connections. We will show how to compute the Christoffel symbols that define the Levi-Civita connection, how to compute curvature, and how to raise and lower indices so that we can produce scalar invariants. In order to construct the curvature operator and curvature tensor, we use the miracle of pseudo-Riemannian geometry, i.e., the Levi-Civita connection, the unique torsion free and metric compatible connection on a manifold. Finally, we examine Generalized Plane Wave Manifolds, and show that all scalar invariants of Weyl type on these manifolds vanish, despite the fact that many of these manifolds are not flat.

We consider the problem of finding a bounded-curvature path in the plane from one configuration αs to another configuration αt that avoids the interior of a set of polygonal obstacles Ε. We call any such path from αs to αt a feasible path. In this thesis, we develop algorithms to find feasible paths that have explicit guarantees on when they will return a feasible path. We phrase our guarantees and run time analysis in terms of the complexity of the desired solution (see k and λ below). In a sense, our algorithms are output sensitive, which is particularly desirable because there are no known bounds on the solution complexity amid arbitrary polygonal environments. Our first major result is an algorithm that given Ε, αs, αt, and a positive integer k either (i) verifies that every feasible path has a descriptive complexity greater than k or (ii) outputs a feasible path. The run time of this algorithm is bounded by a polynomial in n (the total number of obstacle vertices in Ε), m (the bit precision of the input), and k. This result complements earlier work by Fortune and Wilfong: their algorithm considers paths of arbitrary descriptive complexity (it has no dependence on k), but it never outputs a path, just whether or not a feasible path exists. Our second major result is an algorithm that given E, αs, αt, a length λ, and an approximation factor Ε, either (i) verifies that every feasible path has length greater than λ or (ii) constructs a feasible path that is at most (1+ Ε) times longer than the shortest feasible path. The run time of this algorithm is bounded by a polynomial in n, m, Ε-1, and λ. This algorithm is the result of applying the techniques developed earlier in our thesis to the previous approximation approaches. A shortcoming of these prior approximation algorithms is that they only search a special class of feasible paths. This restriction implies that the path that they return may be arbitrarily longer than the shortest path. Our algorithm returns a true approximation because we search for arbitrary shortest paths.

We consider the problem of finding a bounded-curvature path in the plane from one configuration αs to another configuration αt that avoids the interior of a set of polygonal obstacles Ε. We call any such path from αs to αt a feasible path. In this thesis, we develop algorithms to find feasible paths that have explicit guarantees on when they will return a feasible path. We phrase our guarantees and run time analysis in terms of the complexity of the desired solution (see k and λ below). In a sense, our algorithms are output sensitive, which is particularly desirable because there are no known bounds on the solution complexity amid arbitrary polygonal environments. Our first major result is an algorithm that given Ε, αs, αt, and a positive integer k either (i) verifies that every feasible path has a descriptive complexity greater than k or (ii) outputs a feasible path. The run time of this algorithm is bounded by a polynomial in n (the total number of obstacle vertices in Ε), m (the bit precision of the input), and k. This result complements earlier work by Fortune and Wilfong: their algorithm considers paths of arbitrary descriptive complexity (it has no dependence on k), but it never outputs a path, just whether or not a feasible path exists. Our second major result is an algorithm that given E, αs, αt, a length λ, and an approximation factor Ε, either (i) verifies that every feasible path has length greater than λ or (ii) constructs a feasible path that is at most (1+ Ε) times longer than the shortest feasible path. The run time of this algorithm is bounded by a polynomial in n, m, Ε-1, and λ. This algorithm is the result of applying the techniques developed earlier in our thesis to the previous approximation approaches. A shortcoming of these prior approximation algorithms is that they only search a special class of feasible paths. This restriction implies that the path that they return may be arbitrarily longer than the shortest path. Our algorithm returns a true approximation because we search for arbitrary shortest paths.