Universal optimality of main effect deletion designs

The universal optimality of certain deletion designs with respect to main effects has been proved. In particular, this settles a recent conjecture by Voss (1986).     

On the optimality of the trigonometric system

We study a new phenomenon of the behaviour of widths with respect to the optimality of trigonometric system. It is shown that the trigonometric system is optimal in the sense of Kolmogorov widths in the case of “super-high” and “super-small” smoothness but is not optimal in the intermediate cases. Bernstein’s widths behave differently when compared with Kolmogorov in the case of “super-small” smoothness. However, in the case of “super-high” smoothness Kolmogorov and Bernstein widths behave similarly, i.e. are realized by trigonometric polynomials.     

Several classes of complete permutation polynomials

In this paper, three classes of monomials and one class of trinomials over finite fields of even characteristic are proposed. They are proved to be complete permutation polynomials.     

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over ${\mathbb{Z}}_b$ by means of the $b$-adic tent transformation, which has recently been introduced by the authors, and employ such folded digital nets as quadrature points. We first analyze the worst-case error of quasi-Monte Carlo rules using folded digital nets in reproducing kernel Hilbert spaces. Here we need to permit digital nets with “infinite digit expansions”, which are beyond the scope of the classical definition of digital nets. We overcome this issue by considering the infinite product of cyclic groups and the characters on it. We then give an explicit means of constructing good folded digital nets as follows: we use higher order polynomial lattice point sets for digital nets and show that the component-by-component construction can find good folded higher order polynomial lattice rules that achieve the optimal convergence rate of the worst-case error in certain Sobolev spaces of smoothness of arbitrarily high order.     

Optimal designs for estimating the coefficients of the lower frequencies in trigonometric regression models

In the common Fourier regression model we determine the optimal designs for estimating the coefficients corresponding to the lower frequencies. An analytical solution is provided which is found by an alternative characterization of c-optimal designs. Several examples are provided and the performance of the D-optimal design with respect to the estimation of the lower order coefficients is investigated. The results give a complete answer to an open question which was recently raised in the literature.     

On the optimality of block designs

Summary A family of highly efficient designs in the sence of theE-criterion is herein described. These designs have strictly betterE-performance than regular graph designs, yet the off-diagonal entries of theirC-matrix differ by as much as two. Some counterexamples to conjectures in experimental design are then supplied. Asymptotic behavior and equivalence of theA- andD-criteria under a certain condition of uniqueness are analyzed as well.     

单个生成元Walsh p-进制平移不变空间伸缩的交与并

p-进制MRA与GMRA是构造L~2(R_+)中小波框架的重要工具. L~2(R+)中嵌套子空间序列交集为{0},并集为L~2(R_+)是其构成p-进制MRA与GMRA的基本要求.本文研究单个生成元Walsh p-进制平移不变子空间伸缩的交与并,证明了:对任意单个生成元Walsh p-进制平移不变子空间,其p-进制伸缩的交是{0};若生成元分为Walsh p-细分函数,则其p-进制伸缩的并是L~2(R_+)中一个Walshp-进制约化子空间.特别地,其伸缩构成L~2(R_+)中p-进制GMRA当且仅当∪_(j∈z)p~j supp(■φ)=R+,其中■为定义在L~2(R_+)上的Walsh p-进制傅里叶变换.值得注意的是:形式上,我们的结果类似于通常L~2(R)的情形,然而其证明不是平凡的.这是因为定义在R_+上的p-进制加法"⊕"不同于定义在R上的通常加法"+".     

The Haar wavelet characterization of weighted Herz spaces and greediness of the Haar wavelet basis

In this paper we consider the Haar wavelet on weighted Herz spaces. Our weight class, whose name is Ap-dyadic local, is the one defined by the first author (2007). We shall investigate the class of Ap-dyadic weights in connection with the maximal inequalities. After obtaining the properties of weights in the first half of the present paper, we consider the Haar wavelet on weighted Herz spaces in the latter half. We shall show that the Haar wavelet basis is an unconditional basis. We also show that the Haar wavelet is not greedy except for the trivial case, that is, the Haar wavelet is greedy if and only if the Herz space under consideration is a weighted Lp space.     

Weighted lattice polynomials of independent random variables

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.     

Lipschitz continuity of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite dimensional Haar subspace of C(X,R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.     

A Note on a Theorem of J. Szabados

In this note, we establish a companion result to the theorem of J. Szabados on the maximum of fundamental functions of Lagrange interpolation based on Chebyshev nodes.     

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.     

A Characterization of MRA Based Wavelet Frames Generated by the Walsh Polynomials

Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly-nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11(6) (2013), 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.     

Chebyshev Spectral Methods and the Lane-Emden Problem

The three-dimensional spherical polytropic Lane-Emden problem is $y_{rr}+(2/r) y_{r} + y^{m}=0, y(0)=1, y_{r}(0)=0$ where $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. We find that a Newton-Kantorovich iteration always converges from an $m$-independent starting point $y^{(0)}(x)=\cos([\pi/2] x), \xi^{(0)}=3$. We apply a Chebyshev pseudospectral method to discretize $x$. The Lane-Emden equation has branch point singularities at the endpoint $x=1$ whenever $m$ is not an integer; we show that the Chebyshev coefficients are $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. However, a Chebyshev truncation of $N=100$ always gives at least ten decimal places of accuracy — much more accuracy when $m$ is an integer. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table.     

Adjoint polynomials of bridge–path and bridge–cycle graphs and Chebyshev polynomials

The chromatic polynomial of a simple graph G with n>0 vertices is a polynomial of degree n, where α_k(G) is the number of k-independent partitions of G for all k. The adjoint polynomial of G is defined to be , where is the complement of G. We find explicit formulas for the adjoint polynomials of the bridge–path and bridge–cycle graphs. Consequence, we find the zeros of the adjoint polynomials of several families of graphs.     

Approximation of Smooth Functions and Constants by Polynomials with Integer and Natural Coefficients

Direct theorems on the approximation of smooth functions by polynomials with integer coefficients are strengthened. First theorems on approximation by polynomials with natural coefficients are obtained.     

Galois groups of polynomials arising from circulant matrices

Circulant matrices are used to construct polynomials, associated with Chebyshev polynomials of the first kind, whose roots are real and made explicit. Then the Galois groups of the polynomials are computed, giving rise to new examples of polynomials with cyclic Galois groups and Galois groups of order p(p−1) that are generated by a cycle of length p and a cycle of length p−1.     

Some inequalities for characteristic functions. II

Recently we established Matysiak and Szablowski's conjecture [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] about a lower bound of real-valued characteristic functions. In this paper, applying an alternative approach we are able to give explicitly the ranges of argument for which the obtained inequalities hold true for general characteristic functions.