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.     

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.

     

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.

     

The geometry of PLS1 explained properly: 10 key notes on mathematical properties of and some alternative algorithmic approaches to PLS1 modelling

The insight from, and conclusions of this paper motivate efficient and numerically robust ‘new’ variants of algorithms for solving the single response partial least squares regression (PLS1) problem. Prototype MATLAB code for these variants are included in the Appendix. The analysis of and conclusions regarding PLS1 modelling are based on a rich and nontrivial application of numerous key concepts from elementary linear algebra. The investigation starts with a simple analysis of the nonlinear iterative partial least squares (NIPALS) PLS1 algorithm variant computing orthonormal scores and weights. A rigorous interpretation of the squared P ‐loadings as the variable‐wise explained sum of squares is presented. We show that the orthonormal row‐subspace basis of W ‐weights can be found from a recurrence equation. Consequently, the NIPALS deflation steps of the centered predictor matrix can be replaced by a corresponding sequence of Gram–Schmidt steps that compute the orthonormal column‐subspace basis of T ‐scores from the associated non‐orthogonal scores. The transitions between the non‐orthogonal and orthonormal scores and weights (illustrated by an easy‐to‐grasp commutative diagram), respectively, are both given by QR factorizations of the non‐orthogonal matrices. The properties of singular value decomposition combined with the mappings between the alternative representations of the PLS1 ‘truncated’ X data (including P ^t W ) are taken to justify an invariance principle to distinguish between the PLS1 truncation alternatives. The fundamental orthogonal truncation of PLS1 is illustrated by a Lanczos bidiagonalization type of algorithm where the predictor matrix deflation is required to be different from the standard NIPALS deflation. A mathematical argument concluding the PLS1 inconsistency debate (published in 2009 in this journal) is also presented. Copyright © 2014 John Wiley & Sons, Ltd.     

简并谱的涨落统计特征分析

通过引入权重的方式,使简并谱的积累函数形式上连续递增,并用带权重的多元线性拟合提取涨落谱,由此建立一套适用于分析简并谱的NNS分布、谱刚度、能谱分维函数等涨落统计特征的方法,并具体分析了H2O、NH3及CH4分子的振动能谱进行涨落统计特征分析:发现它们的规则谱和约化非简并谱的涨落统计特征均已非Poigson化;而且约化非简并谱的非Poisson化程度较规则谱明显,即前者的能级斥力较后者大;此外,随着简并度的增加,H2O、NH3、CH4的涨落统计特征由非Poisson型向近Poisson型过渡.     

High Molecular Weight Polystyrene Obtained by Cationic Emulsion Polymerization Catalyzed by Imidazolium‐Based Ionic Liquid

The ionic liquid 1‐n‐butyl‐3‐methylimidazolium heptachlorodiferrate (BMI.Fe₂Cl₇) is efficiently used as catalyst in the cationic emulsion polymerization of styrene. The effect of different reaction temperatures, surfactant, and ionic liquid concentrations on polymer properties as molecular weight distribution and particle size is evaluated. High weight average molecular weights, above 1000 kDa, are achieved at 70% of conversion in 100 nm polystyrene particles formed mainly by micellar nucleation. Particle sizes and molecular weights increase with the decrease of the amount of surfactant. Even at low concentrations, BMI.Fe₂Cl₇/styrene molar ratio equal to 1/1000, the ionic liquid proves to be efficient for the emulsion polymerization of styrene, and lower ionic liquid concentrations lead to the formation of longer polymer chains.     

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.     

一种新的变权向量及其应用

在对常权综合和变权综合进行分析的基础上,结合变权的初衷,构造了一个新的变权向量用于变权综合.构造的变权满足变权向量的公理条件和加型综合函数的定义,运用这个变权公式进行加权综合分析时,综合决策值等于加权均值减去加权方差的一半,具有很好的实际意义.