Extension of Laguerre polynomials with negative arguments

We consider the irreducibility of polynomial $L_n^{\left(\alpha \right)}\left(x\right)$ where $\alpha$ is a negative integer. We observe that the constant term of $L_n^{\left(\alpha \right)}\left(x\right)$ vanishes if and only if $n\ge \left|\alpha \right|=?\alpha$. Therefore we assume that $\alpha =?n?s?1$ where $s$ is a non-negative integer. Let $g\left(x\right)={(?1)}^n{L}_n^{(?n?s?1)}\left(x\right)=\sum _{j=0}^n{a}_j\frac{x^j}{j!}$ and more general polynomial, let $G\left(x\right)=\sum _{j=0}^n{a}_j{b}_j\frac{x^j}{j!}$ where $b_j$ with $0\le j\le n$ are integers such that $\left|b_0\right|=\left|b_n\right|=1$. Schur was the first to prove the irreducibility of $g\left(x\right)$ for $s=0$. It has been proved that $g\left(x\right)$ is irreducible for $0\le s\le 60$. In this paper, by a different method, we prove: Apart from finitely many explicitly given possibilities, either $G\left(x\right)$ is irreducible or $G\left(x\right)$ is linear factor times irreducible polynomial. This is a consequence of the estimate $s>1.9k$ whenever $G\left(x\right)$ has a factor of degree $k\ge 2$ and $(n,k,s)\ne (10,5,4)$. This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.     

Well-Conditioned Spectral Collocation Methods for Problems in Unbounded Domains

Based on the generalized Laguerre and Hermite functions, we construct two types of Birkhoff-type interpolation basis functions. The explicit expressions are derived, and fast and stable algorithms are provided for computing these basis functions. As applications, some well-conditioned collocation methods are proposed for solving various second-order differential equations in unbounded domains. Numerical experiments illustrate that our collocation methods are more efficient than the standard Laguerre/Hermite collocation approaches.     

关于最小二乘QR分解算法(LSQR)的一个注记

本文从最小多项式出发，通过寻找包含奇异线性系统Ax=b最小范数解的一个解空间，获得了一个更简单的求解广义逆的计算公式.并从理论上对最小二乘QR分解算法（LSQR）收敛性进行了简单分析，分析表明LSQR的收敛性与矩阵A的非零奇异值密切相关，并用A的非零奇异值以及所寻找到的最小范数解空间将最小范数解线性表出.     

SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery(PPR)for Robin boundary elliptic problems on triangulartions.First,we improve the convergence rate between the finite element solution and the linear interpolation under the H1-norm by introducing a class of meshes satisfying the Condition(α,σ,μ).Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator.Finally,numerical tests are provided to verify the theoretical findings.     

Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings

This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory-effect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of $g^{\prime }$ controlled by g.     

Conical Kakeya and Nikodym sets in finite fields

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.)Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.