51 research outputs found
Fronts and interfaces in bistable extended mappings
We study the interfaces' time evolution in one-dimensional bistable extended
dynamical systems with discrete time. The dynamics is governed by the
competition between a local piece-wise affine bistable mapping and any
couplings given by the convolution with a function of bounded variation. We
prove the existence of travelling wave interfaces, namely fronts, and the
uniqueness of the corresponding selected velocity and shape. This selected
velocity is shown to be the propagating velocity for any interface, to depend
continuously on the couplings and to increase with the symmetry parameter of
the local nonlinearity. We apply the results to several examples including
discrete and continuous couplings, and the planar fronts' dynamics in
multi-dimensional Coupled Map Lattices. We eventually emphasize on the
extension to other kinds of fronts and to a more general class of bistable
extended mappings for which the couplings are allowed to be nonlinear and the
local map to be smooth.Comment: 27 pages, 3 figures, submitted to Nonlinearit
Weak limits of zeros of orthogonal polynomials
Let ÎŒ be a positive unit Borel measure with infinite support on the interval [â1, 1]. Let P n ( x, ÎŒ ) denote the monic orthogonal polynomial of degree n associated with ÎŒ , and let v n ( ÎŒ ) denote the unit measure with mass 1/ n at each zero of P n ( x, ÎŒ ). A carrier is a Borel subset of the support of ÎŒ having unit ÎŒ -measure, and a measure v is carrier related to ÎŒ when it has the same carriers as ÎŒ . We demonstrate that for each carrier B of positive capacity there is a measure v , which is carrier related to ÎŒ , such that the equilibrium measure of the carrier B is the weak limit of the sequence { v n ( v )} n =1/â .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41341/1/365_2005_Article_BF01893436.pd
Quantum mechanical potentials related to the prime numbers and Riemann zeros
Prime numbers are the building blocks of our arithmetic, however, their
distribution still poses fundamental questions. Bernhard Riemann showed that
the distribution of primes could be given explicitly if one knew the
distribution of the non-trivial zeros of the Riemann function.
According to the Hilbert-P{\'o}lya conjecture there exists a Hermitean operator
of which the eigenvalues coincide with the real part of the non-trivial zeros
of . This idea encourages physicists to examine the properties of
such possible operators, and they have found interesting connections between
the distribution of zeros and the distribution of energy eigenvalues of quantum
systems. We apply the Mar{\v{c}}henko approach to construct potentials with
energy eigenvalues equal to the prime numbers and to the zeros of the
function. We demonstrate the multifractal nature of these potentials
by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope
for further analytical progress.Comment: 7 pages, 5 figures, 2 table
Physics of the Riemann Hypothesis
Physicists become acquainted with special functions early in their studies.
Consider our perennial model, the harmonic oscillator, for which we need
Hermite functions, or the Laguerre functions in quantum mechanics. Here we
choose a particular number theoretical function, the Riemann zeta function and
examine its influence in the realm of physics and also how physics may be
suggestive for the resolution of one of mathematics' most famous unconfirmed
conjectures, the Riemann Hypothesis. Does physics hold an essential key to the
solution for this more than hundred-year-old problem? In this work we examine
numerous models from different branches of physics, from classical mechanics to
statistical physics, where this function plays an integral role. We also see
how this function is related to quantum chaos and how its pole-structure
encodes when particles can undergo Bose-Einstein condensation at low
temperature. Throughout these examinations we highlight how physics can perhaps
shed light on the Riemann Hypothesis. Naturally, our aim could not be to be
comprehensive, rather we focus on the major models and aim to give an informed
starting point for the interested Reader.Comment: 27 pages, 9 figure
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