Finite Differences and Orthogonal Polynomials

By combining finite differences with symmetric functions, we present an elementary demonstration for the limit relation from Laguerre to Hermite polynomials, proposed by Richard Askey. Another limit relation between these two polynomials will also be established.     

Coloring complexes and arrangements

Steingrimsson’s coloring complex and Jonsson’s unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ℬ _n arrangements. The first author was supported by NSF grant DMS-0500638. The second author was supported by NSF grant DMS-0245623.     

Robust designs for models with possible bias and correlated errors

This paper studies the model-robust design problem for general models with an unknown bias or contamination and the correlated errors. The true response function is assumed to be from a reproducing kernel Hilbert space and the errors are fitted by the qth order moving average process MA（q）, especially the MA（1） errors and the MA（2） errors. In both situations, design criteria are derived in terms of the average expected quadratic loss for the least squares estimation by using a minimax method. A case is studied and the orthogonality of the criteria is proved for this special response. The robustness of the design criteria is discussed through several numerical examples.     

具有十个大振幅极限环的五次多项式系统

本文研究一类五次平面多项式系统赤道极限环分支问题.运用奇点量方法,首次证明了五次多项式系统可在赤道分支出十个极限环.     

Partial duality and Bollobás and Riordan’s ribbon graph polynomial

Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan’s ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation which used the homfly polynomial of a knot.     

证明微分中值问题的辅助多项式法

通过几个典型实例,介绍辅助多项式函数在微分中值证明问题中的应用.     

Exact values of relative widths of classes of differentialbe functions

We study relative widths in the spacesC andL of classes of periodic differentiable functionsW ^r,r=1,2,…, when in contrast to the Kolmogorov widths it is additionally required that the approximating functions belong to the classMW ^r with a given majorantM of the norm of the derivative of orderr. It is proved that ifM satisfies the estimate

某些三维流形不变量的计算

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