Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

Multiple zigzag chains Zm,n of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains Zm,n. The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the Zm,n multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size m2×m2 consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zkm,n, i.e., derivatives of Zm,n with a single attached polyacene chain of length k. All presented formulas are accompanied by formal proofs. The developed theoretical machinery is applied for predicting aromaticity distribution patterns in large and infinite multiple zigzag chains Zm,n and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains Zm,n.     

Certain bilateral generating relations for generalized hypergeometric functions

Recently, we introduced a class of generalized hypergeometric functionsI n:(b _q)/α:(a _p) (x, w) by using a difference operator Δ _x,w , where . In this paper an attempt has been made to obtain some bilateral generating relations associated withI _n ^ga (x, w). Each result is followed by its applications to the classical orthogonal polynomials.     

The nearest neighbor random walk on subspaces of a vector space and rate of convergence

LetX be the collection ofk-dimensional subspaces of ann-dimensional vector spaceV _n overGF(q). A metric may be defined onX by letting

Characterizations of generalized Hermite and sieved ultraspherical polynomials

A new characterization of the generalized Hermite polyno-
mials and of the orthogonal polynomials with respect to the measure
is derived which is based on a ``reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.

     

Formal orthogonal polynomials revisited. Applications

From some results concerning the formal orthogonal polynomials, already proved in [5], we develop new properties of generalized adjacent polynomials which correspond to a change in the weight function. A new structure of the singular blocks is given. These results have a direct application to Lanczos methods, theG and -algorithms.     

The zeros of Faber polynomials generated by an -star

It is shown that the zeros of the Faber polynomials generated by a regular -star are located on the -star. This proves a recent conjecture of J. Bartolomeo and M. He. The proof uses the connection between zeros of Faber polynomials and Chebyshev quadrature formulas.

     

Bernstein多项式导数的整体收敛速度

研究Ｂｅｒｎｓｔｅｉｎ多项式的导数对可导函数的整体逼近,利用Ｋ－泛函和光滑模的方法,建立了同时逼近的正定理和逆定理     

Degenerate polynomial patches of degree 11 for almostGC 2 interpolation over triangles

We consider the problem of interpolating scattered data in ³ by analmost geometrically smoothGC ² surface, where almostGC ² meansGC ² except in a finite number of points (the vertices), where the surface isGC ¹. A local method is proposed, based on employing so-called degenerate triangular Bernstein-Bézier patches. We give an analysis of quintic patches forGC ¹ and patches of degree eleven for almostGC ² interpolation.     

Analyzing electrochemical noise with Chebyshev’s discrete polynomials

The principles underlying a novel method intended for analyzing experimental data obtained when studying fluctuation processes are considered. The method in question is Chebyshev’s spectroscopy. The application of this method allows one to determine statistic characteristics of steady-state electrochemical noise against the background of severe deterministic interference without invoking the procedure of the fitting of the initial data. The potentialities of this novel method, which is intended for treating noise experiment, are demonstrated by examining model examples and analyzing the electrochemical noise generated by a lithium electrode placed in an aprotic organic electrolyte.