38 research outputs found
The orthogonality of q-classical polynomials of the Hahn class: A geometrical approach
The idea of this review article is to discuss in a unified way the
orthogonality of all positive definite polynomial solutions of the
-hypergeometric difference equation on the -linear lattice by means of a
qualitative analysis of the -Pearson equation. Therefore, our method differs
from the standard ones which are based on the Favard theorem, the three-term
recurrence relation and the difference equation of hypergeometric type. Our
approach enables us to extend the orthogonality relations for some well-known
-polynomials of the Hahn class to a larger set of their parameters. A short
version of this paper appeared in SIGMA 8 (2012), 042, 30 pages
http://dx.doi.org/10.3842/SIGMA.2012.042.Comment: A short version of this paper appeared in SIGMA 8 (2012), 042, 30
pages http://dx.doi.org/10.3842/SIGMA.2012.04
Ratchet effect on a relativistic particle driven by external forces
We study the ratchet effect of a damped relativistic particle driven by both
asymmetric temporal bi-harmonic and time-periodic piecewise constant forces.
This system can be formally solved for any external force, providing the
ratchet velocity as a non-linear functional of the driving force. This allows
us to explicitly illustrate the functional Taylor expansion formalism recently
proposed for this kind of systems. The Taylor expansion reveals particularly
useful to obtain the shape of the current when the force is periodic, piecewise
constant. We also illustrate the somewhat counterintuitive effect that
introducing damping may induce a ratchet effect. When the force is symmetric
under time-reversal and the system is undamped, under symmetry principles no
ratchet effect is possible. In this situation increasing damping generates a
ratchet current which, upon increasing the damping coefficient eventually
reaches a maximum and decreases toward zero. We argue that this effect is not
specific of this example and should appear in any ratchet system with tunable
damping driven by a time-reversible external force.Comment: 1 figur
Symmetries shape the current in ratchets induced by a bi-harmonic force
Equations describing the evolution of particles, solitons, or localized
structures, driven by a zero-average, periodic, external force, and invariant
under time reversal and a half-period time shift, exhibit a ratchet current
when the driving force breaks these symmetries. The bi-harmonic force
does
it for almost any choice of and , provided and are
two co-prime integers such that is odd. It has been widely observed, in
experiments in Josephson-junctions, photonic crystals, etc., as well as in
simulations, that the ratchet current induced by this force has the shape
for
small amplitudes, where depends on the damping ( if
there is no damping, and for overdamped systems). We rigorously
prove that this precise shape can be obtained solely from the broken symmetries
of the system and is independent of the details of the equation describing the
system.Comment: 4 page
Kink topology control by high-frequency external forces in nonlinear Klein-Gordon models
A method of averaging is applied to study the dynamics of a kink in the damped double sine-Gordon equation driven by both external (nonparametric) and parametric periodic forces at high frequencies. This theoretical approach leads to the study of a double sine-Gordon equation with an effective potential and an effective additive force. Direct numerical simulations show how the appearance of two connected π kinks and of an individual π kink can be controlled via the frequency. An anomalous negative mobility phenomenon is also predicted by theory and confirmed by simulations of the original equation.Ministerio de Economía y Competitividad (España) MTM2012-36732-C03-03 FIS2011-24540Junta de Andalucía (España) FQM262 FQM207 FQM-7276 P09-FQM-4643Humboldt Foundation through the Research Fellowship for Experienced Researchers SPA 1146358 ST
Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice
We complete the construction of raising and lowering operators, given in a
previous work, for the orthogonal polynomials of hypergeometric type on
non-homogeneous lattice, and extend these operators to the generalized
orthogonal polynomials, namely, those difference of orthogonal polynomials that
satisfy a similar difference equation of hypergeometric type.Comment: LaTeX, 19 pages, (late submission to arXiv.org
Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports
In this paper we present a survey about analytic properties of polynomials
orthogonal with respect to a weighted Sobolev inner product such that the
vector of measures has an unbounded support. In particular, we are focused in
the study of the asymptotic behaviour of such polynomials as well as in the
distribution of their zeros. Some open problems as well as some new directions
for a future research are formulated.Comment: Changed content; 34 pages, 41 reference
On properties of hypergeometric type-functions
The functions of hypergeometric type are the solutions y = y_(z) of
the differential equation _(z)y′′+_ (z)y′+_y = 0, where _ and _ are polynomials of
degrees not higher than 2 and 1, respectively, and _ is a constant. Here we consider
a class of functions of hypergeometric type: those that satisfy the condition […] 0, where _ is an arbitrary complex (fixed) number. We also assume
that the coefficients of the polynomials _ and _ do not depend on _. To this class
of functions belong Gauss, Kummer and Hermite functions, and also the classical
orthogonal polynomials. In this work, using the constructive approach introduced
by Nikiforov and Uvarov, several structural properties of the hypergeometric type
functions y = y_(z) are obtained. Applications to hypergeometric functions and
classical orthogonal polynomials are also give
Pointwise convergent expansions in q-Fourier-Bessel series
We define q-analogues of Fourier-Bessel series, by means of complete q-
orthogonal systems constructed with the third Jackson q-Bessel function. Sufficient
conditions for pointwise convergence of these series are obtained, in terms of a
general convergence principle valid for other Fourier series on grids defined over
numerable sets. The results are illustrated with specific examples of developments
in q-Fourier-Bessel series