1,156 research outputs found
Spreading lengths of Hermite polynomials
The Renyi, Shannon and Fisher spreading lengths of the classical or
hypergeometric orthogonal polynomials, which are quantifiers of their
distribution all over the orthogonality interval, are defined and investigated.
These information-theoretic measures of the associated Rakhmanov probability
density, which are direct measures of the polynomial spreading in the sense of
having the same units as the variable, share interesting properties: invariance
under translations and reflections, linear scaling and vanishing in the limit
that the variable tends towards a given definite value. The expressions of the
Renyi and Fisher lengths for the Hermite polynomials are computed in terms of
the polynomial degree. The combinatorial multivariable Bell polynomials, which
are shown to characterize the finite power of an arbitrary polynomial, play a
relevant role for the computation of these information-theoretic lengths.
Indeed these polynomials allow us to design an error-free computing approach
for the entropic moments (weighted L^q-norms) of Hermite polynomials and
subsequently for the Renyi and Tsallis entropies, as well as for the Renyi
spreading lengths. Sharp bounds for the Shannon length of these polynomials are
also given by means of an information-theoretic-based optimization procedure.
Moreover, it is computationally proved the existence of a linear correlation
between the Shannon length (as well as the second-order Renyi length) and the
standard deviation. Finally, the application to the most popular
quantum-mechanical prototype system, the harmonic oscillator, is discussed and
some relevant asymptotical open issues related to the entropic moments
mentioned previously are posed.Comment: 16 pages, 4 figures. Journal of Computational and Applied Mathematics
(2009), doi:10.1016/j.cam.2009.09.04
Multiple Scattering Formulation of Two Dimensional Acoustic and Electromagnetic Metamaterials
This work presents a multiple scattering formulation of two dimensional
acoustic metamaterials. It is shown that in the low frequency limit multiple
scattering allows us to define frequency-dependent effective acoustic
parameters for arrays of both ordered and disordered cylinders. This
formulation can lead to both positive and negative acoustic parameters, where
the acoustic parameters are the scalar bulk modulus and the tensorial mass
density and, therefore, anisotropic wave propagation is allowed with both
positive or negative refraction index. It is also shown that the surface fields
on the scatterer are the main responsible of the anomalous behavior of the
effective medium, therefore complex scatterers can be used to engineer the
frequency response of the effective medium, and some examples of application to
different scatterers are given. Finally, the theory is extended to
electromagnetic wave propagation, where Mie resonances are found to be the
responsible of the metamaterial behavior. As an application, it is shown that
it is possible to obtain metamaterials with negative permeability and
permittivity tensors by arrays of all-dielectric cylinders and that anisotropic
cylinders can tune the frequency response of these resonances
Shannon Entropy as Characterization Tool in Acoustics
We introduce Shannon's information entropy to characterize the avoided
crossing appearing in the resonant Zener-like phenomenon in ultrasonic
superlattices made of two different fluidlike meta- materials. We show that
Shannon's entropy gives a correct physical insight of the localization effects
taking place and manifest the informational exchange of the involved acoustic
states in the narrow region of parameters where the avoided crossing occurs.
Results for ultrasonic structures consisting of alternating layers of
methyl-metacrylate and water cavities, in which the acoustic Zener effect were
recently demonstrated, are also reported.Comment: 4 pages, 5 figures. Submitted to Phys. Rev. Let
Entropy and complexity properties of the d-dimensional blackbody radiation
Space dimensionality is a crucial variable in the analysis of the structure
and dynamics of natural systems and phenomena. The dimensionality effects of
the blackbody radiation has been the subject of considerable research activity
in recent years. These studies are still somewhat fragmentary, pos- ing
formidable qualitative and quantitative problems for various scientific and
technological areas. In this work we carry out an information-theoretical
analysis of the spectral energy density of a d-dimensional blackbody at
temperature T by means of various entropy-like quantities (disequilibrium,
Shannon entropy, Fisher information) as well as by three (dimensionless)
complexity measures (Cr\'amer-Rao, Fisher-Shannon and LMC). All these
frequency-functional quantities are calculated and discussed in terms of
temperature and dimensionality. It is shown that all three measures of
complexity have an universal character in the sense that they depend neither on
temperature nor on the Planck and Boltzmann constants, but only on the the
space dimensionality d. Moreover, they decrease when d is increasing; in
particular, the values 2.28415, 1.90979 and 1.17685 are found for the
Cr\'amer-Rao, Fisher-Shannon and LMC measures of complexity of the
3-dimensional blackbody radiation, respectively. In addition, beyond the
frequency at which the spectral density is maximum (which follows the
well-known Wien displacement law), three further characteristic frequencies are
defined in terms of the previous entropy quantities; they are shown to obey
Wien-like laws. The potential usefulness of these distinctive features of the
blackbody spectrum is physically discussed.Comment: 10 pages, 2 figure
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