Local Influence Under Parameter Constraints

Calculations of local influence curvatures and leverage have been well developed when the parameters are unrestricted. In this article, we discuss the assessment of local influence and leverage under linear equality parameter constraints with extensions to inequality constraints. Using a penalized quadratic function we express the normal curvature of local influence for arbitrary perturbation schemes and the generalized leverage matrix in interpretable forms, which depend on restricted and unrestricted components. The results are quite general and can be applied in various statistical models. In particular, we derive the normal curvature under three useful perturbation schemes for generalized linear models. Four illustrative examples are analyzed by the methodology developed in the article.

Genericity of Nondegenerate Critical Points and Morse Geodesic Functionals

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard-Smale theorem, along the lines of an analogous result of B. White [29]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link (proved in Giambo et al. (2005) [8]) of the multiplicity problem with the famous Seifert conjecture (formulated in Seifert (1948) [1]) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level. (C) 2010 Elsevier Ltd. All rights reserved.

Geodesics and Almost Geodesic Cycles in Random Regular Graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d(G)(u, v) is at least d(C)(u, v) - e(n). Let omega(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= log(d-1)log(d-1) n+omega(n) and vertical bar C vertical bar =2 log(d-1) n+O(omega(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 66: 115-136, 2011