48 research outputs found
Solitary and periodic solutions of the generalized Kuramoto-Sivashinsky equation
The generalized Kuramoto-Sivashinsky equation in the case of the power
nonlinearity with arbitrary degree is considered. New exact solutions of this
equation are presented
Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations
Rational solutions and special polynomials associated with the generalized
K_2 hierarchy are studied. This hierarchy is related to the Sawada-Kotera and
Kaup-Kupershmidt equations and some other integrable partial differential
equations including the Fordy-Gibbons equation. Differential-difference
relations and differential equations satisfied by the polynomials are derived.
The relationship between these special polynomials and stationary
configurations of point vortices with circulations Gamma and -2Gamma is
established. Properties of the polynomials are studied. Differential-difference
relations enabling one to construct these polynomials explicitly are derived.
Algebraic relations satisfied by the roots of the polynomials are found.Comment: 23 pages, 8 figure
Expansions of solutions to the equation PâÂČ by algorithms of power geometry
Algorithms of Power Geometry allow to find all power expansions of solutions to ordinary differential equations of a rather general type. Among these, there are PainlevÂŽe equations and their generalizations. In the article we demonstrate how to find by these algorithms all power expansions of solutions to the equation PâÂČ at the points z = 0 and z = â. Two levels of the exponential additions to the expansions of solutions near z = â are computed. We also describe an algorithm of computation of a basis of a minimal lattice containing a given set
Information decomposition of symbolic sequences
We developed a non-parametric method of Information Decomposition (ID) of a
content of any symbolical sequence. The method is based on the calculation of
Shannon mutual information between analyzed and artificial symbolical
sequences, and allows the revealing of latent periodicity in any symbolical
sequence. We show the stability of the ID method in the case of a large number
of random letter changes in an analyzed symbolic sequence. We demonstrate the
possibilities of the method, analyzing both poems, and DNA and protein
sequences. In DNA and protein sequences we show the existence of many DNA and
amino acid sequences with different types and lengths of latent periodicity.
The possible origin of latent periodicity for different symbolical sequences is
discussed.Comment: 18 pages, 8 figure
Universality of a double scaling limit near singular edge points in random matrix models
We consider unitary random matrix ensembles Z_{n,s,t}^{-1}e^{-n tr
V_{s,t}(M)}dM on the space of Hermitian n x n matrices M, where the confining
potential V_{s,t} is such that the limiting mean density of eigenvalues (as
n\to\infty and s,t\to 0) vanishes like a power 5/2 at a (singular) endpoint of
its support. The main purpose of this paper is to prove universality of the
eigenvalue correlation kernel in a double scaling limit. The limiting kernel is
built out of functions associated with a special solution of the P_I^2
equation, which is a fourth order analogue of the Painleve I equation. In order
to prove our result, we use the well-known connection between the eigenvalue
correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal
polynomials, together with the Deift/Zhou steepest descent method to analyze
the RH problem asymptotically. The key step in the asymptotic analysis will be
the construction of a parametrix near the singular endpoint, for which we use
the model RH problem for the special solution of the P_I^2 equation.
In addition, the RH method allows us to determine the asymptotics (in a
double scaling limit) of the recurrence coefficients of the orthogonal
polynomials with respect to the varying weights e^{-nV_{s,t}} on \mathbb{R}.
The special solution of the P_I^2 equation pops up in the n^{-2/7}-term of the
asymptotics.Comment: 32 pages, 3 figure
Point vortices and classical orthogonal polynomials
Stationary equilibria of point vortices with arbitrary choice of circulations
in a background flow are studied. Differential equations satisfied by
generating polynomials of vortex configurations are derived. It is shown that
these equations can be reduced to a single one. It is found that polynomials
that are Wronskians of classical orthogonal polynomials solve the latter
equation. As a consequence vortex equilibria at a certain choice of background
flows can be described with the help of Wronskians of classical orthogonal
polynomials.Comment: 20 pages, 12 figure
Prompt K_short production in pp collisions at sqrt(s)=0.9 TeV
The production of K_short mesons in pp collisions at a centre-of-mass energy
of 0.9 TeV is studied with the LHCb detector at the Large Hadron Collider. The
luminosity of the analysed sample is determined using a novel technique,
involving measurements of the beam currents, sizes and positions, and is found
to be 6.8 +/- 1.0 microbarn^-1. The differential prompt K_short production
cross-section is measured as a function of the K_short transverse momentum and
rapidity in the region 0 < pT < 1.6 GeV/c and 2.5 < y < 4.0. The data are found
to be in reasonable agreement with previous measurements and generator
expectations.Comment: 6+18 pages, 6 figures, updated author lis