The Berry–Esseen bounds of the weighted estimator in a nonparametric regression model

Consider the following nonparametric model: \(Y_{ni}=g(x_{ni})+ \varepsilon _{ni},1\le i\le n,\) where \(x_{ni}\in {\mathbb {A}}\) are the nonrandom design points and \({\mathbb {A}}\) is a compact set of \({\mathbb {R}}^{m}\) for some \(m\ge 1\), \(g(\cdot )\) is a real valued function defined on \({\mathbb {A}}\), and \(\varepsilon _{n1},\ldots ,\varepsilon _{nn}\) are \(\rho ^{-}\)-mixing random errors with zero mean and finite variance. We obtain the Berry–Esseen bounds of the weighted estimator of \(g(\cdot )\). The rate can achieve nearly \(O(n^{-1/4})\) when the moment condition is appropriate. Moreover, we carry out some simulations to verify the validity of our results.

     

超高交联吸附树脂吸附对硝基苯乙酮的热力学研究

比较两种超高交联聚苯乙烯吸附树脂NJ-8、AM-1与Amberlite XAD-4(以下简称XAD-4)对对硝基苯乙酮的静态吸附行为，根据吸附等温线研究了吸附热力学性质．在298～318K和研究的浓度范围内，NJ-8，AM-1、XAD-4对对硝基苯乙酮的吸附平衡数据符合Freundlich吸附等温方程．结果表明：吸附为放热过程，适当降低温度有利于吸附．计算了对硝基苯乙酮在NJ-8，AM-1、XAD-4树脂上的吸附焓变、自由能变，吸附熵变．对吸附行为作了合理的解释。     

间氨基酚在超高交联树脂上的吸附特征

研究了两种超高交联吸附树脂DA-01和DZH-04对间氨基酚的静态吸附热力学及动力学特征。结果表明,在295K~315K及研究的浓度范围内,Freundlich和Langmuir方程能够对静态吸附等温线进行很好的拟合,DA-01和DZH-04树脂对间氨基酚的吸附符合准一级动力学方程。两种树脂吸附间氨基酚的能力都比较强,用乙醇和2mol/LNaOH混合溶液(体积比1:1)洗脱,温度338K,DZH-04树脂的脱附率为99.50%,DA-01树脂的脱附率为75.36%。     

基于聚合诱导自组装制备二氧化硅/聚合物纳米复合材料

纳米二氧化硅(SiO₂)颗粒以其高硬度、高比表面积、高稳定、价格合理等优势被广泛应用于复合材料的制备中,获得的SiO₂/聚合物复合材料通常具有优良的机械性能、很好的热稳定性以及增强的光学和电性能。近年来,随着聚合诱导自组装(PISA)的提出与发展,研究者们基于PISA发展了多种制备不同形貌聚合物纳米粒子的简便方法,为制备SiO₂/聚合物复合材料提供了新的思路。作者调研了近十年来基于PISA制备SiO₂/聚合物复合材料的相关研究,按照SiO₂与聚合物的结合作用和复合机理的不同,创新性地将SiO₂/聚合物复合材料的制备分为物理包封法、化学接枝法、超分子作用法和原位生长法。本综述重点论述复合材料的合成方法、主要性能及用途,同时分析各种复合方法的优缺点并对制备方法的未来发展做出展望,以期为相关领域科研工作者提供更清晰的脉络和更丰富的启示。     

异方差回归模型近邻型估计的相合性

本文研究异方差回归模型Ｙｉ＾（ｎ）＝ｇ（ｘｉ＾（ｎ））＋εｉ＾（ｎ）,ｉ＝１,…,ｎ,其中ｇ是右实函数,ｘｉ＾（ｎ）是非随机设计点列,εｉ＾（ｎ）是随机误差,文中定义了一类ｇ（ｘ）的近邻型估计ｇｎ（ｘ）＝（ｎ）∑（ｉ＝１）Ｗｍ（ｘ）Ｙｉ＾（ｎ）,得到了ｒ阶平均相全和渐近正态性,特别,在（∞）∑（ｎ＝１）（ｎ）∑（ｉ＝１）Ｅ／εｉ＾（ｎ）／＾ｓ／（ｎｉ）＾ｓ／ｒ〈∞,ｍａｘＥ（１≤ｉ≤ｎ）／εｉ（＾     

Rate of Convergence for the Empirical Distribution Function and the Empirical Characteristic Function of Mixing Processes

Ｒａｔｅ　ｏｆ　Ｃｏｎｖｅｒｇｅｎｃｅ　ｆｏｒ　ｔｈｅ　Ｅｍｐｉｒｉｃａｌ　Ｄｉｓｔｒｉｂｕｔｉｏｎ　Ｆｕｎｃｔｉｏｎ　ａｎｄ　ｔｈｅ　Ｅｍｐｉｒｉｃａｌ　Ｃｈａｒａｃｔｅｒｉｓｔｉｃ　Ｆｕｎｃｔｉｏｎ　ｏｆ　Ｍｉｘｉｎｇ　ＰｒｏｃｅｓｓｅｓＨｕＳｈ...     

综合国力的数学建模

本文考虑社会文明与社会导恶现象的制约机制，建立一个综合国力的非线性数学模型，从数学上讨论了相应的二次微分系统的Ｈｏｐｆ分叉，中心与细焦点的判定，极限环的存在唯一性等问题，并对数学结论予以合理解释，把社会相平面划成社发展区域，社会动荡区域和社会崩溃区域，研讨社会走向，确定纳入社会发展区域的途径。     

Extensions of the Borel–Cantelli lemma in general measure spaces

In this paper, an important bilateral inequality for a sequence of nonnegative measurable functions on a measure space \((S,\mathcal {B}_S,\mu )\) is obtained, and some sufficient conditions for \(\mu \left( \limsup \limits _{n\rightarrow \infty }A_n\right) =\mu (S)\) are given. In addition, a weighted version of the Borel–Cantelli Lemma on the measure space is obtained. Our results generalize the corresponding ones for bounded random sequences to the case of unbounded measurable functions.     

The tail probability of the product of dependent random variables from max-domains of attraction

In this article, we investigate the tail probability of the product of finitely many non-negative dependent random variables. They follow distributions from max-domains of attraction of extreme value distributions and their dependence is modeled via a multivariate Farlie–Gumbel–Morgenstern distribution. For each of the Fréchet, Gumbel and Weibull cases, we obtain an explicit asymptotic formula for the tail probability of the product. Our study extends a few known results in the literature.