Geometric properties of rank one asymptotically harmonic manifolds

In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h. We prove the following equivalences for asymptotically harmonic manifolds X under the additional assumption that their curvature tensor together with its covariant derivative are uniformly bounded: (a) X has rank one; (b) X has Anosov geodesic flow; (c) X is Gromov hyperbolic; (d) X has purely exponential volume growth with volume entropy equals h. This generalizes earlier results by G. Knieper for noncompact harmonic manifolds and by A. Zimmer for asymptotically harmonic manifolds admitting compact quotients

Some spectral applications of McMullen's Hausdorff dimension algorithm

Using McMullen's Hausdorff dimension algorithm, we study numerically the dimension of the limit set of groups generated by reflections along three geodesics on the hyperbolic plane. Varying these geodesics, we found four minima in the two-dimensional parameter space, leading to a rigorous result why this must be so. Extending the algorithm to compute the limit measure and its moments, we study orthogonal polynomials on the unit circle associated with this measure. Several numerical observations on certain coefficients related to these moments and on the zeros of the polynomials are discussed. - See more at: http://www.ams.org/journals/ecgd/2012-16-10/S1088-4173-2012-00244-5/home.html#sthash.MXrRFUVZ.dpu

The Twinning Problem

In the field of crystallography, some crystals are not made of a single component but are instead twinned.In these cases, the observed intensities at some points in the lattice will be far larger than predictions. If we find the rotation associated to the twinned component, we can model this twin and improve our agreement with observations. In this report, we explore many routes to improve the process of identifying twins: Generation of fake data for better understanding and accurate testing. The representation of a rotation as defined by an axis and angle. The representation of a rotation as a quaternion. Using lattice points which must be equidistant from the origin to create our viable rotations. An algorithm focused on restricted possibilities. An exploration of 2D lattices for which twinning is mathematically impossible. We find that there is much to be investigated in the field of twinning

Spherical Spectral Synthesis and Two-Radius Theorems on Damek-Ricci Spaces

We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems

Quantum site percolation on amenable graphs

We consider the quantum site percolation model on graphs with an amenable group action. It consists of a random family of Hamiltonians. Basic spectral properties of these operators are derived: non-randomness of the spectrum and its components, existence of an self-averaging integrated density of states and an associated trace-formula.Comment: 10 pages, LaTeX 2e, to appear in "Applied Mathematics and Scientific Computing", Brijuni, June 23-27, 2003. by Kluwer publisher

Laconicity and redundancy of Toeplitz matrices

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46295/1/209_2005_Article_BF01111000.pd

Selected Open Problems in Discrete Geometry and Optimization

A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers

New Strategies in Modeling Electronic Structures and Properties with Applications to Actinides

This chapter discusses contemporary quantum chemical methods and provides general insights into modern electronic structure theory with a focus on heavy-element-containing compounds. We first give a short overview of relativistic Hamiltonians that are frequently applied to account for relativistic effects. Then, we scrutinize various quantum chemistry methods that approximate the $N$-electron wave function. In this respect, we will review the most popular single- and multi-reference approaches that have been developed to model the multi-reference nature of heavy element compounds and their ground- and excited-state electronic structures. Specifically, we introduce various flavors of post-Hartree--Fock methods and optimization schemes like the complete active space self-consistent field method, the configuration interaction approach, the Fock-space coupled cluster model, the pair-coupled cluster doubles ansatz, also known as the antisymmetric product of 1 reference orbital geminal, and the density matrix renormalization group algorithm. Furthermore, we will illustrate how concepts of quantum information theory provide us with a qualitative understanding of complex electronic structures using the picture of interacting orbitals. While modern quantum chemistry facilitates a quantitative description of atoms and molecules as well as their properties, concepts of quantum information theory offer new strategies for a qualitative interpretation that can shed new light onto the chemistry of complex molecular compounds.Comment: 43 pages, 3 figures, Version of Recor