Spherical Radon transform and the average of the condition number on certain Schubert subvarieties of a Grassmannian

We study the average complexity of certain numerical algorithms when adapted to solving systems of multivariate polynomial equations whose coefficients belong to some fixed proper real subspace of the space of systems with complex coefficients. A particular motivation is the study of the case of systems of polynomial equations with real coefficients. Along these pages, we accept methods that compute either real or complex solutions of these input systems. This study leads to interesting problems in Integral Geometry: the question of giving estimates on the average of the normalized condition number along great circles that belong to a Schubert subvariety of the Grassmannian of great circles on a sphere. We prove that this average equals a closed formula in terms of the spherical Radon transform of the condition number along a totally geodesic submanifold of the sphere.     

Classes of series identities and associated hypergeometric reduction formulas

The main object of the present paper is to investigate some classes of series identities and their applications and consequences leading naturally to several (known or new) hypergeometric reduction formulas. We also indicate how some of these series identities and reduction formulas would yield several series identities which emerged recently in the context of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order).     

Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind

Recently, the authors introduced some generalizations of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials (see [Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290-302] and [Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917-925]). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol-Genocchi polynomials of higher order. For these generalized Apostol-Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper [Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631-642] and pose two open problems on the subject of our investigation.     

On the operator of exterior derivation on the Riemannian manifolds with cylindrical ends

We formulate some conditions for the normal and compact solvability of the operator of exterior derivation on the cylindrical manifolds equipped with some Riemannian metrics. Some analogous results were obtained in the particular case of warped cylinders [1].     

On multiindices and multivariables presentation of the Voigt functions

This paper aims at presenting multiindices and multivariables study of the unified (or generalized) Voigt functions which play an important rôle in the several diverse field of physics such as astrophysical spectroscopy and the theory of neutron reactions. Some expressions (representations) of these functions are given in terms of familiar special functions of multivariables. Further representations and series expansions involving multidimensional classical polynomials (Laguerre and Hermite) of mathematical physics are established.     

Asymptotic power comparison of the chi-square and likelihood ratio tests

Summary A Hoeffding-type power comparison is made between the likelihood ratio and chi-square tests for a simple hypothesis in a multinomial. The power comparison is based on the fact that under an alternative hypothesis the distribution of the test statistic can be approximated by a normal distribution. The theory of large deviations is used to match the significance levels. This research was partially supported by the National Science Foundation Grant No. GP-33697X2, U.S. Energy Research and Development Agency Grant 7064100, Environmental Protection Agency Grant R805379-01-0 and U.S. Public Health Service Grant USPHS ES01299-15, at the University of California-Berkeley.     

On the convergence of Schröder iteration functions for pth roots of complex numbers

In this work a condition on the starting values that guarantees the convergence of the Schröder iteration functions of any order to a pth root of a complex number is given. Convergence results are derived from the properties of the Taylor series coefficients of a certain function. The theory is illustrated by some computer generated plots of the basins of attraction.     

On the rates of convergence of BBH-Kantorovich operators and their Bézier variant

In the present paper we consider the Bézier variant of BBH-Kantorovich operators J_n,αf for functions f measurable and locally bounded on the interval [0, ∞) with α ? 1. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of J_n,αf(x) at those x > 0 at which the one-sided limits f(x+), f(x−) exist. The very recent result of Chen and Zeng (2009) [L. Chen, X.M. Zeng, Rate of convergence of a new type Kantorovich variant of Bleimann-Butzer-Hahn Operators, J. Inequal. Appl. 2009 (2009) 10. Article ID 852897] is extended to more general classes of functions.     

Transformation of fourier series using power and weakly oscillating sequences

In the present paper, we consider transformations of the Fourier series of functions of several variables by means of the products of power and weakly oscillating sequences. Estimates of the mixed moduli of smoothness of the transformed Fourier series are obtained via the mixed moduli of smoothness of the functions under consideration.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 99–116.Original Russian Text Copyright © 2005 by M. K. Potapov, B. V. Simonov, S. Yu. Tikhonov.This revised version was published online in April 2005 with a corrected issue number.     

On the relationship between differentiability conditions and existence of a strong gradient

It is proved that for any n 2 there exists continuous function f : ⁿ which is differentiable almost everywhere, but has no strong gradient almost everywhere.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 93–98.Original Russian Text Copyright © 2005 by G. G. Oniani.This revised version was published online in April 2005 with a corrected issue number.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration

In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers.     

Quantum Brownian motion and generalizations of Hermite polynomials

We introduce new families of orthogonal polynomials H_D,n, motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The H_D,n’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D) for a non-equilibrium qBp, by means of the Wigner function W, in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function W_eq,D, which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. W_eq,D is chosen to be the weight function which orthogonalizes the H_D,n’s. New results on W_eq,D and on the H_D,n’s are reported. We justify the key role of the H_D,n’s as follows. Using the H_D,n’s, moments W_eq,D,n and W_n are introduced for W_eq,D and W, respectively. At equilibrium, all moments W_eq,D,n except the lowest one (W_eq,D,0) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments W_n (except the lowest moment W₀), will approach zero, while (ii) the lowest non-equilibrium moment W₀ will tend to W_eq,D,0(≠0). To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W₀ for the different models (D) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.