138 research outputs found
Characteristic approximation properties of quadratic irrationals
Some characteristic approximation properties of quadratic irrationals are studied in this paper. It is shown that the limit points of the sequence δn form a subset C(x), and D(x) can be generated from C(x) in a relatively simple way. Another proof of Lekkerkerker's theorem
is given using relations between δn−1, δn, δn+1 which are independent of x and n
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
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
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
Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions
Systems described by an O(n) symmetrical Hamiltonian are considered
in a -dimensional film geometry at their bulk critical points. A detailed
renormalization-group (RG) study of the critical Casimir forces induced between
the film's boundary planes by thermal fluctuations is presented for the case
where the O(n) symmetry remains unbroken by the surfaces. The boundary planes
are assumed to cause short-ranged disturbances of the interactions that can be
modelled by standard surface contributions corresponding
to subcritical or critical enhancement of the surface interactions. This
translates into mesoscopic boundary conditions of the generic
symmetry-preserving Robin type .
RG-improved perturbation theory and Abel-Plana techniques are used to compute
the -dependent part of the reduced excess free energy per
film area to two-loop order. When , it takes the scaling
form as
, where are scaling fields associated with the
surface-enhancement variables , while is a standard
surface crossover exponent. The scaling function
and its analogue for the Casimir force
are determined via expansion in and extrapolated to
dimensions. In the special case , the expansion
becomes fractional. Consistency with the known fractional expansions of D(0,0)
and to order is achieved by appropriate
reorganisation of RG-improved perturbation theory. For appropriate choices of
and , the Casimir forces can have either sign. Furthermore,
crossovers from attraction to repulsion and vice versa may occur as
increases.Comment: Latex source file, 40 pages, 9 figure
Laconicity and redundancy of Toeplitz matrices
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46295/1/209_2005_Article_BF01111000.pd
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 -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
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
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