On Convergence Properties of Algorithms for Unconstrained Minimization

Suppose F is a convex function on R" for which there is a sequenceof points on which the function values are bounded below andthe gradients converge to zero. Is it possible that F is unboundedbelow? The answer, perhaps surprisingly, is yes for n > 1.     

On the Arnold cat map and periodic boundary conditions for planar elongational flow

In this paper we show that the periodic boundary conditions used to simulate planar elongational flow are closely related to the Arnold cat map. In particular the relationship between the Arnold cat map and the periodic boundary conditions devised by Kraynik and Reinelt [1992, Int. J. multiphase Flow, 18, 1045], the so-called K-R map, is demonstrated. It is shown that the family of lattices found by Kraynik and Reinelt corresponds to a subset of hyperbolic toral automorphisms. These lattices were previously found to be sufficient to enable molecular dynamics simulations of steady-state planar elongational flow of unrestricted duration. Within the frame of the cat map we provide a re-derivation for the set of eigenvalues, eigenvectors and orientation angles of the K-R map and find it to be considerably simpler than the original derivation provided by Kraynik and Reinelt.     

PROPER AFFINE DEFORMATIONS OF THE ONE-HOLED TORUS

A Margulis spacetime is a complete at Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface ∑ homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when ∑ is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of ∑. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces.