高阶PMD统计特性研究

讨论了三阶偏振模色散(PMD)的统计特性及其对于高阶PMD补偿系统的影响.数值模拟结果表明,随着即时差分群时延(DGD)的增加,二阶PMD单位矢量旋转速率的统计覆盖范围的变化趋势与主偏振态(PSP)旋转斜率的相应变化正好相反.此外,随着即时DGD的增加,二阶PMD模值的统计覆盖范围不断增加,使得高速光通信系统中高阶PMD补偿的实现将更加困难.     

Oscillation and non-oscillation for second-order linear difference equations

We prove new oscillation and non-oscillation theorems for the second-order linear difference equation Δ²x_n−1 + p_nx_n = 0, where is a real sequence with p_n 0. These results are extensions of earlier results of Zhang and Zhou [Comput. Math. Appl. 39 (2000) 1–7].     

Spectral multiplicity of some stochastic processes

In this paper we consider the connection between the canonical and the weak-canonical representations for the given second-order stochastic process in a separable Hilbert space and we extend a well-known theorem of H. Cramer concerning sufficient conditions for a process to be of multiplicity one.

     

A construction of real weight functions for certain orthogonal polynomials in two variables

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are

(x²+y)uxx+2xyuxy+y²uyy+gxux+g(y−1)uy=λu,     

Multiplicity Results for Periodic Solutions to Second-Order Difference Equations

By using the Z _p geometrical index theory, some sufficient conditions on the multiplicity results of periodic solutions to the second-order difference equations

Shape-Constrained Estimation Using Nonnegative Splines

We consider the problem of nonparametric estimation of unknown smooth functions in the presence of restrictions on the shape of the estimator and on its support using polynomial splines. We provide a general computational framework that treats these estimation problems in a unified manner, without the limitations of the existing methods. Applications of our approach include computing optimal spline estimators for regression, density estimation, and arrival rate estimation problems in the presence of various shape constraints. Our approach can also handle multiple simultaneous shape constraints. The approach is based on a characterization of nonnegative polynomials that leads to semidefinite programming (SDP) and second-order cone programming (SOCP) formulations of the problems. These formulations extend and generalize a number of previous approaches in the literature, including those with piecewise linear and B-spline estimators. We also consider a simpler approach in which nonnegative splines are approximated by splines whose pieces are polynomials with nonnegative coefficients in a nonnegative basis. A condition is presented to test whether a given nonnegative basis gives rise to a spline cone that is dense in the space of nonnegative continuous functions. The optimization models formulated in the article are solvable with minimal running time using off-the-shelf software. We provide numerical illustrations for density estimation and regression problems. These examples show that the proposed approach requires minimal computational time, and that the estimators obtained using our approach often match and frequently outperform kernel methods and spline smoothing without shape constraints. Supplementary materials for this article are provided online.     

Three Point Boundary Value Problems on Time Scales

This work formulates existence theorems for solutions to three-point boundary value problems on time scales. The ideas are based on a relationship between the three point boundary conditions, lower and upper solutions and topological degree theory.