The Strong Chebyshev Distribution and Orthogonal Laurent Polynomials

Orthogonal Laurent polynomials and strong moment theory: a survey

PC-fractions and orthogonal Laurent polynomials for log-normal distributions

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q q -hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2 Φ 1 ( q − n , q b + 1 ; q − c + b − n ; q , q − c + d − 1 z ) } n = 0 ∞ \{\,_2\Phi _1(q^{-n},q^{b+1};q^{-c+b-n}; q, q^{-c+d-1}z)\}_{n=0}^{\infty } , where 0 > q > 1 0 > q > 1 and the complex parameters b b , c c and d d are such that b ≠ − 1 , − 2 , … b \neq -1, -2, \ldots , c − b + 1 ≠ − 1 , − 2 , … c-b+1 \neq -1, -2, \ldots , R e ( d ) > 0 \mathcal {R}e(d) > 0 and R e ( c − d + 2 ) > 0 \mathcal {R}e(c-d+2) > 0 . Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.

The extended Krylov subspace method and orthogonal Laurent polynomials

If a finite Markov chain (discrete time, discrete states) has a number of absorbing states, one of these will eventually be reached. In this paper are given theoretical formulae for the probability distribution, its generating function and moments of the time taken to first reach an absorbing state, and these formulae are applied to an example taken from genetics. While first passage time problems and their solutions are known for a wide variety of Markov chain processes (e.g., [2], [7], [4]), the theory seems not to have been used in population genetics. Suppose a genetic population consists of a constant number of individuals and the state of the population is defined by the numbers of the various genotypes existing at a given time. Then if mutation is absent, all individuals will eventually become of the same genotype because of random influences such as births, deaths, mating, selection, chromosome breakages and recombinations. The population behavior may in some circumstances be approximated by a Markov chain with absorbing states. In Section 2, two alternative approaches are given for the theoretical determination of absorption time properties, using well known techniques. In Section 3, the consequences of the theoretical results are investigated for a particular population model introduced by Moran [9], [10], and explicit expressions for the distribution of the gene fixation time are obtained in terms of Chebyshev's orthogonal polynomials. The derivation requires finding the pre- and post-eigenvectors of the matrix of transition probabilities, and an incidental by-product is the proof of certain identities for the orthogonal polynomials. The material presented in Section 2 and Section 3 is obtained by exact methods. In Section 4, the Fokker-Planck diffusion equation is used to obtain approximate results, and these are compared with those of the exact theory to ascertain the accuracy of the diffusion approximation.

The paper compares three methods of coding nominal variables in regression analysis: coding of each level as a separate variable, coding with binary code, numbering of factor levels. Although these methods have existed for a long time and even have a theoretical justification (except for encoding with binary code), there were no recommendations and comparisons for their practical application. The features of the application of each method and the existing limitations are analyzed. In the article, there are considered two examples that provide a detailed comparison of these three methods. Comparative analysis has been carried out in the following areas: the presence of restrictions in use; statistical properties of plans; labour intensity and difficulty of obtaining mathematical models and the final result of their building; convenience of semantic analysis and use. Additionally, there have been made comparisons with models based on Chebyshev orthogonal polynomials. It has been established that different methods of coding nominal variables, when used correctly, lead to regression models that are approximately identical in their properties. Moreover, the method of encoding each level as a separate variable is possible only if there are experiments in which there is no nominal variable as an influence effect. The binary coding method is inconvenient to use with a large number of levels of variation of the nominal variable and inconvenient to analyze. When coding by level numbering, it is necessary that the average response values, according to the dispersion diagram of this factor, are sorted by value in accordance with the assigned numbers. With this encoding method, a natural number of factors is preserved. Sharply distinguishable best results are achieved with this coding method using Chebyshev orthogonal polynomials. The highest accuracy and uniformity of approximation are ensured.

Chebyshev collocation method for static intrinsic equations of geometrically exact beams

Fingerprint image segmentation heavily influences the performance of automated fingerprint identification systems. We propose a segmentation technique based on the topographic properties of the image surface. The digital image is treated as a noisy sampling of the underlying continuous surface which is approximated by orthogonal Chebyshev polynomials. Ridge and valley regions are discriminated by the sign of the maximum normal curvature of this continuous surface. Different approximation accuracies are used along the orthogonal and parallel directions to the ridge orientation. This determines a lower sensitivity to small irregularities along the ridge direction then along the orthogonal direction, resulting in a high quality segmentation.

Abstract. Buckling of nonuniform carbon nanotubes are studied with the axial load taken as a combination of concentrated and axially distributed loads. Constitutive modelling of the nanotubes is implemented via nonlocal continuum mechanics. Problem solutions are obtained by employing a weak formulation of the problem and the Rayleigh-Ritz method which is implemented by using orthogonal Chebyshev polynomials. The accuracy of the method of solution is verified against available results. Solutions are obtained for the cases of uniformly distributed and triangularly distributed axial loads. Contour plots are given to assess the effect of nonuniform cross-sections and the small-scale parameter on the buckling load for a combination of simply supported, clamped and free boundary conditions.

Via simulating the contribution of spin-orbit interaction to nuclear masses by orthogonal Chebyshev polynomials, the mass excesses are calculated for a group of atomic nuclei characterized by the isospins of Tz = 51/2, 26, 53/2, 55/2, 28, and 57/2 and by the presence of restored Wigner’s spin-isospin SU(4) symmetry. The root-mean-square deviation of the results of these calculations from their experimental counterparts is σ = 140 keV. The potential of the proposed method is discussed.

Orthogonal Chebyshev Polynomials

Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

Orthogonal polynomials and some q-beta integrals of Ramanujan

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

This paper is concerned with double sequencesC={Cn}n=−∞/∞ of Hermitian matrices with complex entriesCn∈Ms×s) and formal Laurent seriesL0(z)=−Σk=1∞C−kzk andL∞(z)=Σk=0∞Ckz−k. Making use of a Favard-type theorem for certain sequences of matrix Laurent polynomials which was obtained previously in [1] we can establish the relation between the matrix counterpart of the so-calledT-fractions and matrix orthogonal Laurent polynomials. The connection with two-point Pade approximants to the pair (L0,L∞) is also exhibited proving that such approximants are Hermitian too. Finally, error formulas are also given.