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

System for detecting and tracking moving objects

This paper considers the construction of a system for detecting and tracking moving objects. It is proposed to pre-process the frame using digital image stabilization algorithms based on optical flow. To detectobjects, it is supposed to use the longest optical flow vectors formed after stabilization, and to implement tracking using several classical algorithms using a prefetch mechanism built on classification neural networks

A new basis for eigenmodes on the Sphere

The usual spherical harmonics $Y_{\ell m}$ form a basis of the vector space ${\cal V} ^{\ell}$ (of dimension $2\ell+1$) of the eigenfunctions of the Laplacian on the sphere, with eigenvalue $\lambda_{\ell} = -\ell ~(\ell +1)$. Here we show the existence of a different basis $\Phi ^{\ell}_j$ for ${\cal V} ^{\ell}$, where $\Phi ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$, the $\ell ^{th}$ power of the scalar product of the current point with a specific null vector $N_j$. We give explicitly the transformation properties between the two bases. The simplicity of calculations in the new basis allows easy manipulations of the harmonic functions. In particular, we express the transformation rules for the new basis, under any isometry of the sphere. The development of the usual harmonics $Y_{\ell m}$ into thee new basis (and back) allows to derive new properties for the $Y_{\ell m}$. In particular, this leads to a new relation for the $Y_{\ell m}$, which is a finite version of the well known integral representation formula. It provides also new development formulae for the Legendre polynomials and for the special Legendre functions.Comment: 6 pages, no figure; new version: shorter demonstrations; new references; as will appear in Journal of Physics A. Journal of Physics A, in pres

A high order qq-difference equation for qq-Hahn multiple orthogonal polynomials

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when $q\to1$ are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in \cite{Lee} is corrected and obtained as a limiting case

Mathematical Structure of Relativistic Coulomb Integrals

We show that the diagonal matrix elements $,$ where $O$ $={1,\beta,i\mathbf{\alpha n}\beta}$ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, may be considered as difference analogs of the radial wave functions. Such structure provides an independent way of obtaining closed forms of these matrix elements by elementary methods of the theory of difference equations without explicit evaluation of the integrals. Three-term recurrence relations for each of these expectation values are derived as a by-product. Transformation formulas for the corresponding generalized hypergeometric series are discussed.Comment: 13 pages, no figure

Arbitrary l-state solutions of the rotating Morse potential by the asymptotic iteration method

For non-zero $\ell$ values, we present an analytical solution of the radial Schr\"{o}dinger equation for the rotating Morse potential using the Pekeris approximation within the framework of the Asymptotic Iteration Method. The bound state energy eigenvalues and corresponding wave functions are obtained for a number of diatomic molecules and the results are compared with the findings of the super-symmetry, the hypervirial perturbation, the Nikiforov-Uvarov, the variational, the shifted 1/N and the modified shifted 1/N expansion methods.Comment: 15 pages with 1 eps figure. accepted for publication in Journal of Physics A: Mathematical and Genera

Motion of vortices in ferromagnetic spin-1 BEC

The paper investigates dynamics of nonsingular vortices in a ferromagnetic spin-1 BEC, where spin and mass superfluidity coexist in the presence of uniaxial anisotropy (linear and quadratic Zeeman effect). The analysis is based on hydrodynamics following from the Gross-Pitaevskii theory. Cores of nonsingular vortices are skyrmions with charge, which is tuned by uniaxial anisotropy and can have any fractal value between 0 and 1. There are circulations of mass and spin currents around these vortices. The results are compared with the equation of vortex motion derived earlier in the Landau-Lifshitz-Gilbert theory for magnetic vortices in easy-plane ferromagnetic insulators. In the both cases the transverse gyrotropic force (analog of the Magnus force in superfluid and classical hydrodynamics) is proportional to the charge of skyrmions in vortex cores.Comment: 19 pages, 2 figures, to be published in the special issue of Fizika Nizkikh Temperatur dedicated to A.M.Kosevich. arXiv admin note: substantial text overlap with arXiv:1801.0109

Thermoconvective flow velocity in a high-speed magnetofluid seal after it has stopped

Convective flow is investigated in the high-speed (linear velocity of the shaft seal is more than 1 m/s) magnetofluid shaft seal after it has been stopped. Magnetic fluid is preliminarily heated due to viscous friction in the moving seal. After the shaft has been stopped, nonuniform heated fluid remains under the action of a high-gradient magnetic field. Numerical analysis has revealed that in this situation, intense thermomagnetic convection is initiated. The velocity of magnetic fluid depends on its viscosity. For the fluid with viscosity of 2 × 10 -4 m 2/s the maximum flow velocity within the volume of magnetic fluid with a characteristic size of 1 mm can attain a value of 10 m/s

Raising and lowering operators, factorization and differential/difference operators of hypergeometric type

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.Comment: LaTeX, 24 pages, iopart style (late submission