Constrained optimal discrimination designs for Fourier regression models

In this article, the problem of constructing efficient discrimination designs in a Fourier regression model is considered. We propose designs which maximize the power of the F-test, which discriminates between the two highest order models, subject to the constraints that the tests that discriminate between lower order models have at least some given relative power. A complete solution is presented in terms of the canonical moments of the optimal designs, and for the special case of equal constraints even more specific formulae are available.     

Some new Chebyshev spaces

In this note we see another circumstance where Chebyshev polynomials play a significant role. In particular, we present some new extended Chebyshev spaces that arise in the asymptotic stability of the zero solution of first order linear delay differential equations with m commensurate delays where a_j,j=0,…,m, are constants and τ>0 is constant.     

过抽样四相混沌序列的构成及其扩频应用

结合Chebyshev源映射和通信原理中抽样理论,产生了过抽样混沌映射(OSCM),证明了OSCM也具备混沌特性.并提出利用最大平衡差函数考察序列平衡性.通过和源映射序列广义相关函数的对比,分析了自相关、互相关旁瓣的最大值和平均值,进而分析了四相OSCM序列的相关性.仿真结果表明:针对现行移动通信扩频系统地址码,四相ChebyshevOSCM序列具有良好的自相关和互相关性,可以增强系统的保密性和提升系统容量,是CDMA移动通信扩频系统地址码的优选方案之一.     

Bernstein Operators for Exponential Polynomials

Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ ₀,…,λ _n . Assume that the set U _n of all solutions of the equation Lf=0 is closed under complex conjugation. If the length of the interval [a,b] is smaller than π/M _n , where M _n :=max {|Im λ _j |:j=0,…,n}, then there exists a basis p _n,k , k=0,…,n, of the space U _n with the property that each p _n,k has a zero of order k at a and a zero of order n−k at b, and each p _n,k is positive on the open interval (a,b). Under the additional assumption that λ ₀ and λ ₁ are real and distinct, our first main result states that there exist points a=t ₀<t ₁<⋅⋅⋅<t _n =b and positive numbers α ₀,…,α _n , such that the operator

Chebyshev constants and the inheritance problem

It is proven that any set E consisting of finitely many intervals can be approximated with order 1/n by polynomial inverse images of [-1,1]. This leads to a new proof of the fact that the n-th Chebyshev constant is Kcap(E)ⁿ with some K independent of n. The proof uses properties of monotone systems, in particular, the statement in the so-called inheritance problem.     

基于Chebyshev多项式零点的Lagrange插值多项式逼近的注记

本文通过一个例子说明了文献[3]中定理6.9的不完善之处,并建立了:若f∈C^r_[-1,1],则     

THE CONSTITUTIVE RELATION OF CRACK-WEAKENED ROCK MASSES UNDER AXIAL-DIMENSIONAL UNLOADING

An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.     

Optimal polynomial smoothers for multigrid V-cycles

The idea of using polynomial methods to improve simple smoother iterations within a multigrid method for a symmetric positive definite system is revisited. A two-level bound going back to Hackbusch is optimized by a very simple iteration, a close cousin of the Chebyshev semi-iterative method, but based on the Chebyshev polynomials of the fourth instead of first kind. A full V-cycle bound for general polynomial smoothers is derived using the V-cycle theory of McCormick. The fourth-kind Chebyshev iteration is quasi-optimal for the V-cycle bound. The optimal polynomials for the V-cycle bound can be found numerically, achieving about an 18% lower error contraction factor bound than the fourth-kind Chebyshev iteration, asymptotically as the number of smoothing steps $k\to \infty$. Implementation of the optimized iteration is discussed, and the performance of the polynomial smoothers is illustrated with numerical examples.