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

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