Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type

MSC 2010: 33C47, 42C05, 41A55, 65D30, 65D32In the first part of this survey paper we present a short account on some important properties of orthogonal polynomials on the real line, including computational methods for constructing coefficients in the fundamental three-term recurrence relation for orthogonal polynomials, and mention some basic facts on Gaussian quadrature rules. In the second part we discuss our Mathematica package Orthogonal Polynomials (see [2]) and show some applications to problems with strong nonclassical weights on (0;+1), including a conjecture for an oscillatory weight on [¡1; 1]. Finally, we give some new results on orthogonal polynomials on radial rays in the complex plane

On the geometrical description of fractional Chern insulators based on static structure factor calculations

We study the static structure factor of the fractional Chern insulator Laughlin-like state and provide analytical forms for this quantity in the long-distance limit. In the course of this we identify averaged over Brillouin zone Fubini Study metric as the relevant metric in the long-distance limit. We discuss under which conditions the static structure factor will assume the usual behavior of Laughlin-like fractional quantum Hall system i.e. the scenario of Girvin, MacDonald, and Platzman [Phys. Rev. B 33, 2481 (1986)]. We study the influence of the departure of the averaged over Brillouin zone Fubini Study metric from its fractional quantum Hall value which appears in the long-distance analysis as an effective change of the filling factor. According to our exact diagonalization results on the Haldane model and analytical considerations we find persistence of fractional Chern insulator state even in this region of the parameter space.Comment: 13 pages, 7 figures, published versio

On polynomials orthogonal on the semicircle and applications

Polynomials {TV,) orthogonal on the semicircle F = (z E C: z = eis, 0 < 0 < P) with respect to the inner product (f, g) = /rf(z)g(z)w(z>(it)-' dz, where z * W(Z) is a complex weight function, have been introduced in 1986-1987 by Gautschi, Landau and the author. In this paper we introduce the functions of the second kind, as well as the corresponding associated polynomials, and prove some recurrence relations. For Gauss-Gegenbauer quadrature formulae over the semicircle, applied to analytic functions, we develop error bounds from contour integral representations of the remainder term and give some numerical results

Coexistence of Composite-Bosons and Composite-Fermions in nu=1/2 + 1/2 Quantum Hall Bilayers

In bilayer quantum Hall systems at filling fractions near nu=1/2+1/2, as the spacing d between the layers is continuously decreased, intra-layer correlations must be replaced by inter-layer correlations, and the composite fermion (CF) Fermi seas at large d must eventually be replaced by a composite boson (CB) condensate or "111 state" at small d. We propose a scenario where CBs and CFs coexist in two interpenetrating fluids in the transition. Trial wavefunctions describing these mixed CB-CF states compare very favorably with exact diagonalization results. A Chern-Simons transport theory is constructed that is compatible with experiment.Comment: 4 pages, 2 figure

Color measurement of animal source foods

Rapid and objective assessment of food color is necessary in quality control. The color evaluation of animal source foods using a computer vision system (CVS) and a traditional colorimeter is examined. With the same measurement conditions, color results deviated between these two approaches. The color returned by the CVS had a close resemblance to the perceived color of the animal source foods, whereas the colorimeter returned not typical colors. The effectiveness of the CVS is confirmed by the study results. Considering these data, it could be concluded that the colorimeter is not representative method for color analysis of animal source foods, therefore, the color read by the CVS seemed to be more similar to the real ones