715 research outputs found
A note on the invariant distribution of a quasi-birth-and-death process
The aim of this paper is to give an explicit formula of the invariant
distribution of a quasi-birth-and-death process in terms of the block entries
of the transition probability matrix using a matrix-valued orthogonal
polynomials approach. We will show that the invariant distribution can be
computed using the squared norms of the corresponding matrix-valued orthogonal
polynomials, no matter if they are or not diagonal matrices. We will give an
example where the squared norms are not diagonal matrices, but nevertheless we
can compute its invariant distribution
Absence of magnetic order for the spin-half Heisenberg antiferromagnet on the star lattice
We study the ground-state properties of the spin-half Heisenberg
antiferromagnet on the two-dimensional star lattice by spin-wave theory, exact
diagonalization and a variational mean-field approach. We find evidence that
the star lattice is (besides the \kagome lattice) a second candidate among the
11 uniform Archimedean lattices where quantum fluctuations in combination with
frustration lead to a quantum paramagnetic ground state. Although the classical
ground state of the Heisenberg antiferromagnet on the star exhibits a huge
non-trivial degeneracy like on the \kagome lattice, its quantum ground state is
most likely dimerized with a gap to all excitations. Finally, we find several
candidates for plateaux in the magnetization curve as well as a macroscopic
magnetization jump to saturation due to independent localized magnon states.Comment: new extended version (6 pages, 6 figures) as published in Physical
Review
Defect free global minima in Thomson's problem of charges on a sphere
Given unit points charges on the surface of a unit conducting sphere,
what configuration of charges minimizes the Coulombic energy ? Due to an exponential rise in good local minima, finding global
minima for this problem, or even approaches to do so has proven extremely
difficult. For \hbox{} recent theoretical work based on
elasticity theory, and subsequent numerical work has shown, that for --1000 adding dislocation defects to a symmetric icosadeltahedral lattice
lowers the energy. Here we show that in fact this approach holds for all ,
and we give a complete or near complete catalogue of defect free global minima.Comment: Revisions in Tables and Reference
Askey-Wilson Type Functions, With Bound States
The two linearly independent solutions of the three-term recurrence relation
of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22],
are slightly modified so as to make it transparent that these functions satisfy
a beautiful symmetry property. It essentially means that the geometric and the
spectral parameters are interchangeable in these functions. We call the
resulting functions the Askey-Wilson functions. Then, we show that by adding
bound states (with arbitrary weights) at specific points outside of the
continuous spectrum of some instances of the Askey-Wilson difference operator,
we can generate functions that satisfy a doubly infinite three-term recursion
relation and are also eigenfunctions of -difference operators of arbitrary
orders. Our result provides a discrete analogue of the solutions of the purely
differential version of the bispectral problem that were discovered in the
pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction
Predictions of bond percolation thresholds for the kagom\'e and Archimedean lattices
Here we show how the recent exact determination of the bond percolation
threshold for the martini lattice can be used to provide approximations to the
unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one
of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2)
bond problems, and the other gives estimates of for the homogeneous
kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively
agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure
Heat Kernel Bounds for the Laplacian on Metric Graphs of Polygonal Tilings
We obtain an upper heat kernel bound for the Laplacian on metric graphs
arising as one skeletons of certain polygonal tilings of the plane, which
reflects the one dimensional as well as the two dimensional nature of these
graphs.Comment: 8 page
Exact Site Percolation Thresholds Using the Site-to-Bond and Star-Triangle Transformations
I construct a two-dimensional lattice on which the inhomogeneous site
percolation threshold is exactly calculable and use this result to find two
more lattices on which the site thresholds can be determined. The primary
lattice studied here, the ``martini lattice'', is a hexagonal lattice with
every second site transformed into a triangle. The site threshold of this
lattice is found to be , while the others have and
. This last solution suggests a possible approach to establishing
the bound for the hexagonal site threshold, . To derive these
results, I solve a correlated bond problem on the hexagonal lattice by use of
the star-triangle transformation and then, by a particular choice of
correlations, solve the site problem on the martini lattice.Comment: 12 pages, 10 figures. Submitted to Physical Review
Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization
We give a Riemann-Hilbert approach to the theory of matrix orthogonal
polynomials. We will focus on the algebraic aspects of the problem, obtaining
difference and differential relations satisfied by the corresponding orthogonal
polynomials. We will show that in the matrix case there is some extra freedom
that allows us to obtain a family of ladder operators, some of them of 0-th
order, something that is not possible in the scalar case. The combination of
the ladder operators will lead to a family of second-order differential
equations satisfied by the orthogonal polynomials, some of them of 0-th and
first order, something also impossible in the scalar setting. This shows that
the differential properties in the matrix case are much more complicated than
in the scalar situation. We will study several examples given in the last years
as well as others not considered so far
Simultaneity as an Invariant Equivalence Relation
This paper deals with the concept of simultaneity in classical and
relativistic physics as construed in terms of group-invariant equivalence
relations. A full examination of Newton, Galilei and Poincar\'e invariant
equivalence relations in is presented, which provides alternative
proofs, additions and occasionally corrections of results in the literature,
including Malament's theorem and some of its variants. It is argued that the
interpretation of simultaneity as an invariant equivalence relation, although
interesting for its own sake, does not cut in the debate concerning the
conventionality of simultaneity in special relativity.Comment: Some corrections, mostly of misprints. Keywords: special relativity,
simultaneity, invariant equivalence relations, Malament's theore
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