NOTES ON ERDOS' CONJECTURE

Let Xn,n 1,be a sequence of independent random variables satisfying P(Xn = O) = 1 － P(Xn = an) = 1 － 1/pn,where an,n 1,is a sequence of real numbers,andpn isthenthprime,setFN(x)=P(N∑Xnn=1 x).The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent,it is both sufficient and necessary that there exist three numbers x0 and x1 ＜ x2 such that limsup(FN(x2) －FN(x1)) ＞ 0 holds,and Lo =N→∞ lim FN(x0) exists.Moreover,the authors point ont that they can also obtain the same N→∞ result in the weakened case of liminfP(Xn = 0) ＞ 0.n→∞     

Summability of Product Jacobi Expansions

Orthogonal expansions in product Jacobi polynomials with respect to the weight function W_α, β(x)=∏^d_j=1 (1−x_j)^α_j (1+x_j)^β_j on [−1, 1]^d are studied. For α_j, β_j>−1 and α_j+β_j−1, the Cesàro (C, δ) means of the product Jacobi expansion converge in the norm of L^p(W_{α, β}, [−1, 1]^d), 1p<∞, and C([−1, 1]^d) if

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Moreover, for α_j, β_j−1/2, the (C, δ) means define a positive linear operator if and only if δ∑^d_i=1 (α_i+β_i)+3d−1.     

An algebraic translation task solved by grade 7–9 students

In this study, the author examined student attempts to translate a verbal problem into an algebraic statement relating two variables, after they had solved an arithmetic question from the same problem. A total of 645 students from New England (U.S.A.) answered the problem on a mathematics assessment administered at the beginning of the school year. Among students who could solve the arithmetic part of the problem, the use of variables in the correct conventional notation appeared from grade 7 and continuously increased through grade 9. These results suggest that there is a relationship between students’ arithmetic understanding and translating verbal problems into algebraic statements relating two variables.     

Confidence Bands for Isotonic Median Curves Using Sign Tests

Suppose that one observes independent random variables (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n) in R² with unknown distributions, except that Median(Y_i | X_i = M(x) for some unknown isotonic function M. We describe an explicit algorithm for the computation of confidence bands for the median function M whose running time is of order O(n²). The bands rely on multiscale sign tests and are shown to have desirable asymptotic properties.     

Nonmonotone Inexact Newton Method for the Extended Linear Complementarity Problem

This paper presents a nonmonotone inexact Newton-type method for the extended linear complementarity problem (ELCP). We first reformulate the optimization system of the ELCP problem into a system of smoothed equations. Then we solve this system by a nonmonotone inexact Newton-type algorithm. The global convergence is obtained and numerical tests for some classes of ELCP include linear complementarity, horizontal linear complementarity, and generalized linear complementarity problems are also given to show the e?ciency of the proposed algorithm.     

独立不同分布随机变盆序列的部分和的增量有多小

关于独立同分布随机变量部分和的增量有多小问题，目前已有了著名的结果.本文通过与独立同分布情形时完全不同的途径，讨论了独立不同分布情形时相应的问题.在矩母函数存在的条件下，获得了与同分布情形相近的结论.     

电感耦合等离子体原子发射光谱(ICP-AES)法测定胡蜂酒中无机元素

采用电感耦合等离子体原子发射光谱法测定胡蜂酒中20种无机元素,建立无机元素对照指纹图谱,对重金属元素进行风险评估。结果表明,胡蜂酒中无机元素种类丰富,其中P、K、Na元素占测定总元素的96.14%。有害元素Pb、Cd、As、Hg、Cu含量符合国家药典标准要求,Cu的靶标危险系数(THQ)为1.00×10^-2,THQ<1说明摄入的重金属对人体健康造成的影响不明显。不同产区胡蜂酒金属元素含量存在差异,根据金属元素图谱得出元素含量按原子序数顺序出现相似的分布态势。结果表明,胡蜂酒中含有丰富的金属元素,很多为营养微量元素,有害元素(Pb、Cd、As、Hg、Cu)含量符合药典要求,从金属元素方面单一来看,饮用胡蜂酒没有明显的健康风险。胡蜂酒无机元素指纹图谱可为胡蜂酒的鉴别提供一定的的参考依据。     

Comparative Analysis of Chemical Constituents in Different Parts of Lotus by UPLC and QToF-MS

Six parts of lotus (seeds, leaves, plumule, stamens, receptacles and rhizome nodes) are herbal medicines that are listed in the Chinese Pharmacopoeia. Their indications and functions have been confirmed by a long history of clinical practice. To fully understand the material basis of clinical applications, UPLC-QToF-MS combined with the UNIFI platform and multivariate statistical analysis was used in this study. As a result, a total of 171 compounds were detected and characterized from the six parts, and 23 robust biomarkers were discovered. The method can be used as a standard protocol for the direct identification and prediction of the six parts of lotus. Meanwhile, these discoveries are valuable for improving the quality control method of herbal medicines. Most importantly, this was the first time that alkaloids were detected in the stamen, and terpenoids were detected in the cored seed. The stamen is a noteworthy part because it contains the greatest diversity of flavonoids and terpenoids, but research on the stamen is rather limited.     

完全未知混合体系中纯组份的红外光谱解析

提出一种新的目标转化因子分析，其初始向量的构造基于实际最大差异光谱，无需分离即可从红外混合体系中解析出纯组份光谱。该算法简单易懂，不必迭代，运算量小，应用于模拟及实际体系，结果令人满意。