Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

We develop the asymptotic expansion theory for vector-valued sequences ${\left(F_N\right)}_{N\ge 1}$ of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence $F_N$. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of $F_N$ and we illustrate our results by several examples.     

Classical Backfitting for Smooth-Backfitting Additive Models

Smooth backfitting has been shown to have better theoretical properties than classical backfitting for fitting additive models based on local linear regression. In this article, we show that the smooth backfitting procedure in the local linear case can be alternatively performed as a classical backfitting procedure with a different type of smoother matrices. These smoother matrices are symmetric and shrinking and some established results in the literature are readily applicable. The connections allow the smooth backfitting algorithm to be implemented in a much simplified way, give new insights on the differences between the two approaches in the literature, and provide an extension to local polynomial regression. The connections also give rise to a new estimator at data points. Asymptotic properties of general local polynomial smooth backfitting estimates are investigated, allowing for different orders of local polynomials and different bandwidths. Cases of oracle efficiency are discussed. Computer-generated simulations are conducted to demonstrate finite sample behaviors of the methodology and a real data example is given for illustration. Supplementary materials for this article are available online.     

Finding optimal finite field strengths allowing for a maximum of precision in the calculation of polarizabilities and hyperpolarizabilities

The finite field method, widely used for the calculation of static dipole polarizabilities or the first and second hyperpolarizabilities of molecules and polymers, is thoroughly explored. The application of different field strengths and the impact on the precision of the calculations were investigated. Borders could be defined and characterized, establishing a range of feasible field strengths that guarantee reliable numerical results. The quality of different types of meshes to screen the feasible region is assessed. Extrapolation schemes are presented that reduce the truncation error and allow to increase the precision of finite field calculations by one to three orders of magnitude. © 2013 Wiley Periodicals, Inc.     

A new framework for implicit restarting of the Krylov–Schur algorithm

This paper introduces a new framework for implicit restarting of the Krylov–Schur algorithm. It is shown that restarting with arbitrary polynomial filter is possible by reassigning some of the eigenvalues of the Rayleigh quotient through a rank‐one correction, implemented using only the elementary transformations (translation and similarity) of the Krylov decomposition. This framework includes the implicitly restarted Arnoldi (IRA) algorithm and the Krylov–Schur algorithm with implicit harmonic restart as special cases. Further, it reveals that the IRA algorithm can be turned into an eigenvalue assignment method. Copyright © 2014 John Wiley & Sons, Ltd.     

New q‐Hermite polynomials: characterization,operators algebra and associated coherent states

This paper addresses a construction of new q‐Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three‐term recursive relation as well as the second‐order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self‐adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q‐deformed boson system is studied.     

Multi-variable special polynomials using an operator ordering method

Using an operator ordering method for some commutative superposition operators, we introduce two new multi-variable special polynomials and their generating functions, and present some new operator identities and integral formulas involving the two special polynomials. Instead of calculating complicated partial differential, we use the special polynomials and their generating functions to concisely address the normalization, photocount distributions and Wigner distributions of several quantum states that can be realized physically, the results of which provide real convenience for further investigating the properties and applications of these states.     

An error-free algorithm to solve a linear system of polynomial equations

The paper presents an error-free algorithm to solve a system of linear equations with polynomial coefficients. Modular arithmetic in residual polynomial class and in residual numeric class is employed. The algorithm is iterative and well suited for implementation for computers with vector operations and fast and error-free convolutors.     

Some Inequalities Concerning the Polar Derivative of a Polynomial-II

In this paper, we consider the class of polynomials P（z）= anz^n＋ ∑vn=μan-vz^n-v,1≤μ≤n , having all zeros in ｜z｜≤k, k ≤1 and thereby present an alternative proof, independent of Laguerre＇s theorem, of an inequality concerning the polar derivative of a polynomial.