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

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)

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

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

A "missing" family of classical orthogonal polynomials

We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.Comment: 20 page

An inverse eigenvalue problem of the theory of optical waveguides

© 2014 IEEE. We present a new method for calculation of permittivities of dielectric materials using optical fiber's propagation constants measurements. Our numerical algorithm is based on approximate solution of a nonlinear nonselfadjoint inverse eigenvalue problem for a system of weakly singular integral equations. We prove that it is enough to measure propagation constants of fundamental eigenmode only at two frequencies for reconstruction of dielectric constants of core and cladding of a waveguide

Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework

In the context of a two-parameter $(\alpha, \beta)$ deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using techniques of supersymmetric quantum mechanics combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters $\alpha$, $\beta$ vanishes, in which case shape invariance under parameter translation occurs. In the special case where $\alpha = \beta \ne 0$, the oscillator Hamiltonian is shown to coincide with that of the q-deformed oscillator with $q > 1$ and its eigenvectors are therefore $n$-$q$-boson states. In the general case where $0 \ne \alpha \ne \beta \ne 0$, the eigenvectors are constructed as linear combinations of $n$-$q$-boson states by resorting to a Bargmann representation of the latter and to $q$-differential calculus. They are finally expressed in terms of a $q$-exponential and little $q$-Jacobi polynomials.Comment: LaTeX, 24 pages, no figure, minor changes, additional references, final version to be published in JP

Projection methods for computation of spectral characteristics of weakly guiding optical waveguides

The original problem on surface and leaky eigen-modes of a weakly guiding step-index optical waveguide is considered. The original problem is reduced to a nonlinear spectral problem for the set of weakly singular boundary integral equations. We approximate the integral operator by collocation and Galerkin methods. Their convergence and quality are proved by numerical experiments. © 2013 IEEE

Occasional derivation as way of era language stylization in V.Aksyonov's novel «Voltairiens and Voltairiennes»

In article the occasional derivation of modern Russian on material of the novel of V. Aksenov "Voltairiens and Voltairiennes" is considered. Applying methods of continuous selection, word-formation, semantic and stylistic analysis, authors make an attempt to systematize the occasional derivatives revealed in the novel, define their structural-semantic features, as well as establish the basic principles of their functioning in work. During research it is revealed that, first, all occasional derivatives can be divided into two types: formed by usual and not usual methods, the suffixation, confixation, preffixation and addition belong to the usual methods used by V. Aksenov in creating nonce words. From not usual the contamination, creation, metathesis from diaeresis are involved. It is established that the most productive are methods of suffixation, confixation, contamination and creation. Dependence of the various method choice of education on author's intention is revealed: for the purpose of stylization the author uses mainly suffixation and addition. For the purpose of combining nominative function and function of language game he mainly addresses methods of contamination and creation. Authors emphasize convention of language material stylization in the novel, refer to examples of deliberately wrong definition of word-formation structure of the word given by the writer, as well as use cases for creating occasional derivatives of the making bases representing deliberate anachronisms as the proof

Symmetry accounting helps solve the Lasing Eigenvalue Problems for optical microcavities

© 2016 IEEE.The microcavity modes are considered accurately using the linear electromagnetic formalism of the Lasing Eigenvalue Problem (LEP) with exact boundary and radiation conditions. We reduce the original problem to a nonlinear eigenvalue problem for the system of Muller boundary integral equations and build a numerical method accounting symmetry of solutions. Symmetry accounting helps split solutions on classes, which contributes to the stability of calculations and reduces the dimension of the nonlinear spectral algebraic problem