Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.

APLICACIÓN DE LAS SOLUCIONES DE LA ECUACIÓN DIFERENCIAL DE BESSEL , EN LA INGENIERÍA DE LOS ALIMENTOS

La expresión de Bessel es una ecuación diferencial ordinaria de segundo orden (EDOPO), cuyas soluciones son esféricas y pueden resolverse por los métodos de Frobenius, series y coeficientes indeterminados; cuyos valores se pueden utilizar muy bien y con facilidad en la ingeniería y tecnología de los alimentos. Sus soluciones se particularizan en cuatro partes, de las cuales las del tipo son las que tienen una aplicación en el presente estudio. La EDOPO de Bessel fue resuelta en sus cuatro posibilidades, donde los valores y comportamientos se proporcionan en tablas y gráficas en detalle. Los valores correspondientes de se usaron en la distribución de temperatura para envases cilíndricos de dimensiones conocidas usando las ecuaciones de transferencia de calor por conducción, preferentemente, proveniente de la teoría de Fourier; facilitando mejorar, modificar y diseñar nuevos procesos. Los resultados de la distribución de temperatura fueron calculados por las ecuaciones respectivas y los resultados obtenidos fueron empleados satisfactoriamente en el desarrollo del trabajo.

Analysis of free vibrations in transradially isotropic spherically symmetric thermoelastic spheres

PurposeThe purpose of this manuscript is to study the vibration characteristics of the spherically symmetric solid and hollow spheres poised of a homogeneous thermoelastic material, based on the three dimensional coupled thermoelasticity.Design/methodology/approachIn this paper, matrix Fröbenius series solution is used to derive the frequency equations, for the field functions. Results have been applied on rigidly fixed boundary conditions.FindingsThe main finding of this paper is that the frequency of vibration of spherically symmetric sphere (structure is independent of theta and phi) increases with the increase of radius, for solid spheres and for hollow spheres with thickness to mean radius ratio. Deformation in the given materials increases with thickness to mean radius ratio of the hollow sphere.Originality/valueA numerical simulation has been done with the help of functional iteration method for solid and hollow thermoelastic spheres made of zinc and poly methyl meth acrylate materials for different boundary conditions. The computer simulated results in contempt of frequency, damping of vibration modes and displacement have been obtained graphically and compared with the existed results.

Transverse waves in coronal flux tubes with thick boundaries: The effect of longitudinal flows

Observations show that transverse magnetohydrodynamic (MHD) waves and flows are often simultaneously present in magnetic loops of the solar corona. The waves are resonantly damped in the Alfvén continuum because of plasma and/or magnetic field nonuniformity across the loop. The resonant damping is relevant in the context of coronal heating, since it provides a mechanism to cascade energy down to the dissipative scales. It has been theoretically shown that the presence of flow affects the waves propagation and damping, but most of the studies rely on the unjustified assumption that the transverse nonuniformity is confined to a boundary layer much thinner than the radius of the loop. Here we present a semi-analytic technique to explore the effect of flow on resonant MHD waves in coronal flux tubes with thick nonuniform boundaries. We extend a published method, which was originally developed for a static plasma, in order to incorporate the effect of flow. We allowed the flow velocity to continuously vary within the nonuniform boundary from the internal velocity to the external velocity. The analytic part of the method is based on expressing the wave perturbations in the thick nonuniform boundary of the loop as a Frobenius series that contains a singular term accounting for the Alfvén resonance, while the numerical part of the method consists of solving iteratively the transcendental dispersion relation together with the equation for the Alfvén resonance position. As an application of this method, we investigated the impact of flow on the phase velocity and resonant damping length of MHD kink waves. With the present method, we consistently recover results in the thin boundary approximation obtained in previous studies. We have extended those results to the case of thick boundaries. We also explored the error associated with the use of the thin boundary approximation beyond its regime of applicability.

Cyclic Sieving, Necklaces, and Branching Rules Related to Thrall's Problem

We show that the cyclic sieving phenomenon of Reiner-Stanton-White together with necklace generating functions arising from work of Klyachko offer a remarkably unified, direct, and largely bijective approach to a series of results due to Kraśkiewicz-Weyman, Stembridge, and Schocker related to the so-called higher Lie modules and branching rules for inclusions $ C_a \wr S_b \hookrightarrow S_{ab} $. Extending the approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem.

Historical Notes

Short review of some earlier work on the classical Markoff theory, by Christoffel, Markoff, Hurwitz, Frobenius, Series, Smith, Bachmann, Remak, Dickson, Cassels, Cusick, Flahive, Heawood, Perron, Cassels, Bombieri, Aigner.

A categorification of the chromatic symmetric polynomial

International audience The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$_*($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology. Le polynôme chromatique symétrique d’un graphe $G$ est une généralisation par une fonction symétrique du polynôme chromatique, et possède des propriétés combinatoires intéressantes. Nous appliquons les techniques de l’homologie de Khovanov pour construire une homologie $H$_*($G$) de modules gradués $S_n$, dont la série bigraduée de Frobeniusse $Frob_G(q,t)$ réduit au polynôme chromatique symétrique à $q=t=1$. Nous obtenons également des analogies pour plusieurs propriétés connues des polynômes chromatiques en termes d’homologie.