Complete convergence and almost sure convergence of weighted sums of random variables

Letr>1. For eachn1, let {X _nk , –<k<} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee for every >0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods ofiid random variables are also established.     

Asymptotic expansions and Richardson extrapolation of approximate solutions for integro-differential equations by mixed finite element methods

In this paper asymptotic error expansions for mixed finite element approximations of the integro-differential equation are derived, and Richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation post-processing technique. The results of this paper provide new asymptotic expansions. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a-posteriori error estimators for this mixed finite element method. Finally, a numerical example is provided to validate the theoretical results. This project was supported in part by the Special Funds for Major State Basic Research Project (2007CB8149), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical Methods for Convertible Bonds, 06JA630047), the NSERC, Tianjin Natural Science Foundation (07JCYBJC14300), Tianjin Educational Committee, Liu Hui Center for Applied Mathematics of Nankai University and Tianjin University, and Tianjin University of Finance and Economics.     

表面粗糙度的变波矢光散射测量方法的研究

提出了表面粗糙度变波矢光散射测量方法，通过散射光强中心亮点光能量的参数ｌｎ，Ｅ随变化的线性关系来确定粗糙度ｗ，对６块粗糙度不同的硅片背面样品进行了实验测量。     

Extraction of Surface Roughness from the Central δ-peak of Light Scattering Using a Multi-wavelength Ar+ Laser

In this paper, a new light scattering method for measuring the roughness of random surfaces is proposed. The method is based on the relation between the specularly reflected intensity Is, i.e., the central δ-peak intensity and wave vector component kz. The linear fit of the logarithm graph of the normalized central δ-peak intensity Isr versus k2z, whose variation is induced by changing illuminating wavelength, gives the square of the surface roughness w. The roughnesses of 6 silicon backside samples were measured.     

基于耦合算法的三维复杂结构冲击动力学特性

光滑粒子流体动力学-有限元耦合算法（SPH-FEM）较好地结合了SPH和FEM的优势,近年来逐渐被引入冲击动力学相关问题研究中。然而早期的研究对象多为单一材料的简单结构,所取得的研究成果距离实际工程应用仍有一定差距。为此,在总结前人工作的基础上,对SPH-FEM耦合算法进行适当改进,通过引入复合材料损伤模型,对复合材料蒙皮结构飞行器舱段结构进行建模计算,分析其在爆炸冲击激励下的冲击动力学特性。将数值计算结果与试验结果进行对比分析,验证该算法和模型的有效性和准确性,初步实现SPH-FEM的工程实际应用。最后总结了复合材料蒙皮结构飞行器在爆炸冲击激励下的一系列结构动态响应规律,以期为航天飞行器结构设计与防护提供参考。     

Synthesis and electrochemical performance of LiFePO<Subscript>4</Subscript>/C cathode materials from Fe<Subscript>2</Subscript>O<Subscript>3</Subscript> for high-power lithium-ion batteries

Carbon-coated olivine-structured LiFePO₄/C composites are synthesized via an efficient and low-cost carbothermal reduction method using Fe₂O₃ as iron source at a relative low temperature (600 °C). The effects of two kinds of carbon sources, inorganic (acetylene black) and organic (sucrose), on the structures, morphologies, and lithium storage properties of LiFePO₄/C are evaluated in details. The particle size and distribution of the carbon-coated LiFePO₄ from sucrose (LiFePO₄/SUC) are more uniform than that obtained from acetylene black (LiFePO₄/AB). Moreover, the LiFePO₄/SUC nanocomposite shows superior electrochemical properties such as high discharge capacity of 156 mAh g^?1 at 0.1 C, excellent cyclic stability, and rate capability (78 mAh g^?1 at 20 C), as compared to LiFePO₄/AB. Cyclic voltammetric test discloses that the Li-ion diffusion, the reversibility of lithium extraction/insertion, and electrical conductivity are significantly improved in LiFePO₄/SUC composite. It is believed that olivine-structured LiFePO₄ decorated with carbon from organic carbon source (sucrose) using Fe₂O₃ is a promising cathode for high-power lithium-ion batteries.     

Fuzzy Integrals of Set-vatued Function

ＦｕｚｚｙＩｎｔｅｇｒａｌｓｏｆＳｅｔ－ｖａｌｕｅｄＦｕｎｃｔｉｏｎｓＺｈａｎｇＤｅｌｉａｎｄＷａｎｇＺｉｘｉａｏ（Ｄｅｐｔ．ｏｆＭａｔｈ．ＪｉｌｉｎＰｒｏｖ，Ｉｎｓｔ．ｏｆＥｄｕｃａｔｉｏｎＣｈａｎｇｃｈｕｎ，Ｊｉｌｉｎ，１３００２２，Ｐ．Ｒ．Ｃｈ...     

The Model Companion of Width-Two Orders

The class of width-two orders has a model companion. The model companion is complete, decidable, non-finitely axiomatizable, and has continuum many countable models. Generalizations of some results in (Pouzet, M., 1978, J. Combin. Theory Ser. B 25) are presented in the width-n case, for n2.     

Atomic force microscopical and surface plasmon resonance spectroscopical investigation of sub-micrometer metal gratings generated by UV laser-based two-beam interference in Au-Ag bimetallic layers

Metal films containing silver and gold layers having different thicknesses were evaporated on glass substrates. Two-beam interference technique was applied to irradiate the surfaces by the fourth harmonic of a pulsed mode Nd:YAG laser. The atomic force microscopical study showed that surface relief grating having a period of 900 nm corresponding to the interference pattern was developed on the metallic films. The modulation amplitude of the laser-induced gratings was increasable by enhancing the number of laser pulses at constant fluence, and a groove depth commensurable with the film thicknesses was generated at the average fluence of 39.5 mJ/cm² on bimetallic layers. The surface structure was more regular, and the modulation amplitude was larger in case of bimetallic films containing thicker gold layers. The threshold fluences of the phase transitions were determined by numerical temperature model calculations for different metal layer compositions, and a good agreement was found between the calculated and experimentally observed threshold values. The division of the metal stripes into droplets and the development of holes were explained by the melting of the entire metal layers and by the vaporization of silver at higher fluences. The angle-dependent surface plasmon resonance spectroscopy realized in Kretschmann arrangement proved that the laser-induced grating formation was accompanied by the change in the optical thickness and by the modification of the structure of the bimetallic films. Broad side wings appeared on the resonance curves caused by grating-coupling in case of appropriate rotation angle and sufficiently large modulation depth of the grating's grooves, according to our calculations. The coupling on deep gratings developed on bimetallic films containing the thinnest gold layer and on monometallic silver films resulted in separated secondary resonance minimum development. The periodic adherence of native streptavidin on the metallic gratings was detected by tapping mode AFM, and based on the shift of the secondary resonance peak.     

About Some Boundary Value Problems for Fractional PDE and their Numerical Solution

A transmission initial-boundary-value problem in disjoint intervals for fractional in time diffusion equation is studied. Its well posedness in the corresponding Sobolev-like spaces is proved. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)