Discrete Darboux transformation for discrete polynomials of hypergeometric type

Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)

Superconformal indices of three-dimensional theories related by mirror symmetry

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories.Comment: 17 pages; minor change

Reconstruction of dielectric constants of multi-layered optical fibers using propagation constants measurements

We present new method for the numerical reconstruction of the variable refractive index of multi-layered circular weakly guiding dielectric waveguides using the measurements of the propagation constants of their eigenwaves. Our numerical examples show stable reconstruction of the dielectric permittivity function $\varepsilon$ for random noise level using these measurements

On the structure of the top homology group of the Johnson kernel

The Johnson kernel is the subgroup $\mathcal{K}_g$ of the mapping class group ${\rm Mod}(\Sigma_{g})$ of a genus $g$ oriented closed surface $\Sigma_{g}$ generated by all Dehn twists about separating curves. In this paper we study the structure of the top homology group ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$. For any collection of $2g-3$ disjoint separating curves on $\Sigma_{g}$ one can construct the corresponding abelian cycle in the group ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$; such abelian cycles will be called simplest. In this paper we describe the structure of $\mathbb{Z}[{\rm Mod}(\Sigma_{g})/ \mathcal{K}_g]$-module on the subgroup of ${\rm H}_{2g-3}(\mathcal{K}_g, \mathbb{Z})$ generated by all simplest abelian cycles and find all relations between them.Comment: 22 page

The top homology group of the genus 3 Torelli group

The Torelli group of a genus $g$ oriented surface $\Sigma_g$ is the subgroup $\mathcal{I}_g$ of the mapping class group ${\rm Mod}(\Sigma_g)$ consisting of all mapping classes that act trivially on ${\rm H}_1(\Sigma_g, \mathbb{Z})$. The quotient group ${\rm Mod}(\Sigma_g) / \mathcal{I}_g$ is isomorphic to the symplectic group ${\rm Sp}(2g, \mathbb{Z})$. The cohomological dimension of the group $\mathcal{I}_g$ equals to $3g-5$. The main goal of the present paper is to compute the top homology group of the Torelli group in the case $g = 3$ as ${\rm Sp}(6, \mathbb{Z})$-module. We prove an isomorphism $${\rm H}_4(\mathcal{I}_3, \mathbb{Z}) \cong {\rm Ind}^{{\rm Sp}(6, \mathbb{Z})}_{S_3 \ltimes {\rm SL}(2, \mathbb{Z})^{\times 3}} \mathcal{Z},$$ where $\mathcal{Z}$ is the quotient of $\mathbb{Z}^3$ by its diagonal subgroup $\mathbb{Z}$ with the natural action of the permutation group $S_3$ (the action of ${\rm SL}(2, \mathbb{Z})^{\times 3}$ is trivial). We also construct an explicit set of generators and relations for the group ${\rm H}_4(\mathcal{I}_3, \mathbb{Z})$.Comment: 35 pages, minor correction

Modeling of pipe-drawing tool for drawing the multifaceted pipes of nonferrous metals on an immediate arbor

A method of mathematical modeling of a pipe-drawing tool for drawing the multifaceted pipes of nonferrous metals and alloys using the vector-matrix apparatus, which can be applied for the analytical description of the bulk deformation region, is presented. Arbors with various geometries of the reduction zone are considered. As a result of modeling the deformation region, which appears when manufacturing the profiled multifaceted pipes by arbor drawing using all types of considered arbors, it is established that the best result with the smallest rounding radii is attained for arbors with a pyramidal input into the reduction zone. © 2013 Allerton Press, Inc

On a modular property of N=2 superconformal theories in four dimensions

In this note we discuss several properties of the Schur index of N=2 superconformal theories in four dimensions. In particular, we study modular properties of this index under SL(2,Z) transformations of its parameters.Comment: 23 page, 2 figure

Self-Similar Potentials and the q-Oscillator Algebra at Roots of Unity

Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schr\"odinger equation provide bases of representations of the $q$-deformed Heisenberg-Weyl algebra. When the parameter $q$ is a root of unity the functional form of the potentials can be found explicitly. The general $q^3=1$ and the particular $q^4=1$ potentials are given by the equianharmonic and (pseudo)lemniscatic Weierstrass functions respectively.Comment: 15 pp, Latex, to appear in Lett.Math.Phy

q-Ultraspherical polynomials for q a root of unity

Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of $q$-beta integrals of Ramanujan.Comment: 7 pages, LATE