103 research outputs found
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
Solution of a Generalized Stieltjes Problem
We present the exact solution for a set of nonlinear algebraic equations
. These
were encountered by us in a recent study of the low energy spectrum of the
Heisenberg ferromagnetic chain \cite{dhar}. These equations are low
(density) ``degenerations'' of more complicated transcendental equation of
Bethe's Ansatz for a ferromagnet, but are interesting in themselves. They
generalize, through a single parameter, the equations of Stieltjes,
, familiar from Random Matrix theory.
It is shown that the solutions of these set of equations is given by the
zeros of generalized associated Laguerre polynomials. These zeros are
interesting, since they provide one of the few known cases where the location
is along a nontrivial curve in the complex plane that is determined in this
work.
Using a ``Green's function'' and a saddle point technique we determine the
asymptotic distribution of zeros.Comment: 19 pages, 4 figure
The q-harmonic oscillator and an analog of the Charlier polynomials
A model of a q-harmonic oscillator based on q-Charlier polynomials of
Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation
and q-annihilation operators, q-coherent states and an analog of the Fourier
transformation are found. A connection of the kernel of this transform with
biorthogonal rational functions is observed
Approximate volume and integration for basic semi-algebraic sets
Given a basic compact semi-algebraic set \K\subset\R^n, we introduce a
methodology that generates a sequence converging to the volume of \K. This
sequence is obtained from optimal values of a hierarchy of either semidefinite
or linear programs. Not only the volume but also every finite vector of moments
of the probability measure that is uniformly distributed on \K can be
approximated as closely as desired, and so permits to approximate the integral
on \K of any given polynomial; extension to integration against some weight
functions is also provided. Finally, some numerical issues associated with the
algorithms involved are briefly discussed
Finite-Size Corrections to Anomalous Dimensions in N=4 SYM Theory
The scaling dimensions of large operators in N=4 supersymmetric Yang-Mills
theory are dual to energies of semiclassical strings in AdS(5)xS(5). At one
loop, the dimensions of large operators can be computed with the help of Bethe
ansatz and can be directly compared to the string energies. We study
finite-size corrections for Bethe states which should describe quantum
corrections to energies of extended semiclassical strings.Comment: 10 page
The Coulomb phase shift revisited
We investigate the Coulomb phase shift, and derive and analyze new and more
precise analytical formulae. We consider next to leading order terms to the
Stirling approximation, and show that they are important at small values of the
angular momentum and other regimes. We employ the uniform approximation.
The use of our expressions in low energy scattering of charged particles is
discussed and some comparisons are made with other approximation methods.Comment: 13 pages, 5 figures, 1 tabl
A Physicist's Proof of the Lagrange-Good Multivariable Inversion Formula
We provide yet another proof of the classical Lagrange-Good multivariable
inversion formula using techniques of quantum field theory.Comment: 9 pages, 3 diagram
The Bivariate Normal Copula
We collect well known and less known facts about the bivariate normal
distribution and translate them into copula language. In addition, we prove a
very general formula for the bivariate normal copula, we compute Gini's gamma,
and we provide improved bounds and approximations on the diagonal.Comment: 24 page
Toward Forecasting Volcanic Eruptions using Seismic Noise
During inter-eruption periods, magma pressurization yields subtle changes of
the elastic properties of volcanic edifices. We use the reproducibility
properties of the ambient seismic noise recorded on the Piton de la Fournaise
volcano to measure relative seismic velocity variations of less than 0.1 % with
a temporal resolution of one day. Our results show that five studied volcanic
eruptions were preceded by clearly detectable seismic velocity decreases within
the zone of magma injection. These precursors reflect the edifice dilatation
induced by magma pressurization and can be useful indicators to improve the
forecasting of volcanic eruptions.Comment: Supplementary information:
http://www-lgit.obs.ujf-grenoble.fr/~fbrengui/brenguier_SI.pdf Supplementary
video:
http://www-lgit.obs.ujf-grenoble.fr/~fbrengui/brenguierMovieVolcano.av
Long time behaviour for a class of low-regularity solutions of the Camassa-Holm equation
In this paper, we investigate the long time behaviour for a class of
low-regularity solutions of the Camassa-Holm equation given by the
superposition of infinitely many interacting traveling waves with corners at
their peaks.Comment: 30 page
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