Geometrical Solution to the Fermat Problem with Arbitrary Weights

The prime motivation for the present study is a famous problem, allegedly first formulated in 1643 by Fermat, and the so-called Complementary Problem (CP), proposed but incorrectly solved in 1941 by Courant and Robbins. For a given triangle, Fermat asks for a fourth point such that the sum of its Euclidean distances, each weighted by +1, to the three given points is minimized. CP differs from Fermat in that the weight associated with one of these points is –1 instead of +1. The geometrical approach suggested in 1998 by Krarup for solving CP is here extended to cover any combination of positive and negative weights associated with the vertices of a given triangle. Among the by-products are surprisingly simple correctness proofs of the geometrical constructions of Torricelli (around 1645), Cavalieri (1647), Viviani (1659), Simpson (1750), and Martelli (1998). Furthermore, alternative proofs of Ptolemy's theorem (around A.D. 150) and an observation by Heinen (1834) are provided.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

关于TOPSIS法应用中的逆序问题及消除的方法

在多指标决策问题中，TOPSIS法是一种常用的方法，然而传统的TOPSIS法在实际应用中容易产生逆序的现象。本分析了逆序产生的主要原因，并提出了一种改进的方法-RTOPSIS法。应用本的方法不仅能消除逆序的现象，而且还能正确反映指标权重对决策结果的影响。     

Partition identities involving gaps and weights

We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers-Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.

     

中国上市公司经营绩效的综合量化模型研究

根据中国上市公司的特点 ,以经营链条为主线 ,构建了我国上市公司经营绩效量化的指标体系 ,提出了一种新的权重确定方法 .并在此基础上 ,利用模糊综合评价方法建立了中国上市公司经营绩效的综合量化模型 ,从而为我国上市公司经营绩效的评价提供了一个全面、公正、简捷的途径 .     

On weight functions which admit explicit Gauss-Turán quadrature formulas

The main purpose of this paper is the construction of explicit Gauss-Turán quadrature formulas: they are relative to some classes of weight functions, which have the peculiarity that the corresponding -orthogonal polynomials, of the same degree, are independent of . These weights too are introduced and discussed here. Moreover, highest-precision quadratures for evaluating Fourier-Chebyshev coefficients are given.

     

Lamperti-type operators on a weighted space of continuous functions

For a locally convex Hausdorff topological vector space and for a system of weights vanishing at infinity on a locally compact Hausdorff space , let be the weighted space of -valued continuous functions on with the locally convex topology derived from the seminorms which are weighted analogues of the supremum norm. A characterization of the orthogonality preserving (Lamperti-type) operators on is presented in this paper.

     

Weak type bounds for a class of rough operators with power weights

In this note we show that and the fractional integral and maximal operators with rough kernel respectively, are bounded operators from to where and

     

己二酸—三羟甲基丙烷缩聚反应的理论分析

用唐敖庆等曾得到的含内环化A_a-B_b型缩聚反应的固化理论,对配料比为1.3的己二酸-三羟甲基丙烷缩聚反应实验数据进行了分析,得到的溶胶-凝胶分配曲线与实验结果符合的较好,进一步讨论了内环化对凝胶点的影响,算出了凝胶点以上数均、重均和Z均分子量的理论曲线.