Complete Asymptotic Expansions for the Chlodovsky Polynomials

The purpose of this article is to study the local rate of convergence of the Chlodovsky operators (C_nf)(x). As the main results, we investigate their asymptotic behaviour and derive the complete asymptotic expansions of these operators. All the coefficients of n^?k (k = 1, 2,…) are calculated in terms of the Stirling numbers of first and second kind. We mention that analogous results for the Bernstein polynomials can be found in Lorentz [2 G. G. Lorentz ( 1953 ). Bernstein Polynomials . University of Toronto Press , Toronto . [Google Scholar]].     

Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading

The behaviour of a bi-piezoelectric ceramic layer with a centre interfacial crack subjected to anti-plane shear and in-plane electric loading has been studied. The dislocation density functions and the Fourier integral transform method have been employed to eliminate the problem of singular integral equations. The normalized energy release rate, stress and electrical displacement intensity factors, G/G₀,K_III/K_III0 and K_D/K_D0, respectively, were determined for different geometric and property parameters by use of two different crack surface electric boundary conditions, i.e. impermeable and permeable. It has been shown that the effects of the thickness and material constants of the piezoelectric layer on all the three parameters, i.e. G/G₀,K_III/K_III0 and K_D/K_D0 were significant.     

Fast Algorithms for Nonparametric Curve Estimation

Abstract

Naive implementations of local polynomial fits and kernel estimators require almost O(n ²) operations. In this article two fast O(n) algorithms for nonparametric local polynomial fitting are presented. They are based on updating normal equations. Numerical stability is guaranteed by controlling ill-conditioned situations for small bandwidths and data-tuned restarting of the updating procedure. Restarting at every output point results in a moderately fast but highly stable O(n ^7/5) algorithm. Applicability of algorithms is evaluated for estimation of regression curves and their derivatives. The idea is also applied to kernel estimators of regression curves and densities.     

Automating the Partition of Indexes

Abstract

In this article we show how attention to the structure of a particular algebraic calculation can lead to the simple implementation of powerful computer algebra tools. The creation of partitions for a set of indexes is required for the implementation of many theoretical structures. This may be difficult to do by hand even when the number of indexes is only moderately large. These partitions arise through the action of differentiation and so we mimic differentiation in a computer algebra package to create partitions of indexes. The strategies employed are extended to the creation of complementary set partitions, their reduction to equivalence classes, and the implementation of Edgeworth expansions and the exlog relations.     

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.     

On the Location of Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.     

An application of Chen’s theorem to convergence of Fourier multiplier transforms

Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.