On the existence of infinitely many closed geodesics on orbifolds of revolution

Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S^2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be regarded as a topological two-sphere with metric singularities, we will have extended Bangert's theorem on the existence of infinitely many closed geodesics on any smooth Riemannian two-sphere. In addition, we give an example of a two-sphere cone-manifold of revolution which possesses a single closed geodesic, thus showing that Bangert's result does not hold in the wider class of closed surfaces with cone manifold structures.Comment: 21 pages, 4 figures; for a PDF version see http://www.calpoly.edu/~jborzell/Publications/publications.htm

Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree

Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree we focus upon M(2), the maximum value of the sum of the two largest multiplicities. The corresponding M(1) is already understood. The notion of assignment (of eigenvalues to subtrees) is formalized and applied. Using these ideas, simple upper and lower bounds are given for M(2) (in terms of simple graph theoretic parameters), cases of equality are indicated, and a combinatorial algorithm is given to compute M(2) precisely. In the process, several techniques are developed that likely have more general uses. (C) 2009 Elsevier B.V. All rights reserved

Convex Optimization over Probability Measures

Thesis (Ph.D.)--University of Washington, 2015The thesis studies convex optimization over the Banach space of regular Borel measures on a compact set. The focus is on problems where the variables are constrained to be probability measures. Applications include non-parametric maximum likelihood estimation of mixture densities, optimal experimental design, and distributionally robust stochastic programming. The theoretical study begins by developing the duality theory for optimization problems having non-finite-valued convex objectives over the set of probability measures. It is then shown that the infinite-dimensional problems can be posed as non-convex optimization problems in finite dimensions. The duality theory and constraint qualifications for these finite dimensional problems are derived and applied in each of the applications studied. It is then shown that the non-convex finite-dimensional problems can be decomposed by first optimizing over a subset of the variables, called the "weights", for which the associated optimization problem is convex. Optimization over the remaining variables, called the "support points", is then considered. The objective function in the reduced problem is neither convex nor finite-valued. For these reasons, a smoothing of this function is introduced. The epi-continuity of this smoothing and the convergence of critical points is established. It is shown that all known constraint qualifications fail for this problem, requiring a detailed analysis of the behavior of optimal solutions as the smoothing converges to the original problem. Finally, proof of concept numerical results are given

Closed Geodesics on Orbifolds of Revolution

Using the theory of geodesics on surfaces of revolution, we show that any two-dimensional orbifold of revolution homeomorphic to S2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be regarded as a topological two-sphere with metric singularities, we will have extended Bangert\u27s theorem on the existence of infinitely many closed geodesics on any smooth Riemannian two-sphere. In addition, we give an example of a two-sphere cone-manifold of revolution which possesses a single closed geodesic, thus showing that Bangert\u27s result does not hold in the wider class of closed surfaces with cone manifold structures