Lebesgue type decompositions for nonnegative forms

A nonnegative form on a complex linear space is decomposed with respect to another nonnegative form : it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by which is almost dominated by . The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions.     

任意随机序列关于非齐次马氏链的随机和的一类随机偏差定理

In this paper, the notion of limit random logarithmic likelihood ratio of stochastic sequence, as a measure of dissimilarity between the joint distribution on measure P and the Markov distribution on measure Q, is introduced. A class of random approximation theorems for arbitrary stochastic dominated sequence are obtained by using the tools of generating functions and the tailed-probability generating functions.     

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S _n >x) / P(₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(₁ > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.     

Co3O4在多相催化反应中的晶面效应

Co₃O₄是一种重要且性能优异的过渡金属氧化物,可以广泛应用于光、电、磁、热等多个领域,其晶体作为多相催化剂在众多反应中有着不可替代的地位。催化机理研究发现,催化反应过程不但受晶体催化剂颗粒尺寸的影响,而且对晶体的晶面也很敏感。因此,晶面效应的研究对于深入认识多相催化反应过程以及有效设计高活性催化剂都有重要意义。Co₃O₄晶体具有Co³⁺和Co²⁺混合价态的尖晶石结构,其在不同催化反应中均表现出明显的晶面效应。本文综述了近年来Co₃O₄作为多相晶体催化剂在热催化、光催化、电催化以及类过氧化物酶催化中的晶面效应,结合理论计算结果从Co₃O₄晶面的原子结构出发解释了产生晶面效应的原因;最后总结了Co₃O₄在以上反应中晶面效应的一般规律,探讨了目前Co₃O₄晶面效应研究中的不足,并对未来的发展趋势进行了展望。     

用逐点插入法生成Delaunay四面体自适应网格

介绍一种基于Delaunay算法的四面体自适应网格的自动划分方法。该方法用单元尺度场控制生成网格的疏密分布,在不满足尺度场要求的单元面形心处插入新节点,同时计算新节点单元尺寸参数,实现三维实体的Delaunay四面体自动划分。此方法具有几个特点:一是表面网格与体内网格同步划分,无需区分两者;二是结点与单元同时生成;三是生成网格自适应性好,疏密分布任意。另外,还介绍了三维网格划分中两个相关算法:一个是约束面恢复算法,该算法基于约束面不允许有单元边与之相交的性质而提出的;另一个是将二维射线法推广至三维空间,判断一个点是否在一多面体内,实现了凹多面体的划分。最后通过算例对单元质量进行了评价。本文所述方法是一种有效的四面体自适应单元生成算法。     

激光晶体Nd3+:Gd3Ga5O12中的核心和小面生长

Nd3+:Gd3Ga5O12(Nd:GGG)晶体是热容固体激光器的一种重要的工作介质.采用提拉法沿〈111〉方向生长出了Nd:GGG单晶,利用激光器、应力仪和偏光显微镜等仪器和方法,对晶体的小面生长及核心等缺陷进行了观察,分析了小面生长的机理,并提出了消除这些缺陷的办法.通过研究,为改善生长工艺、生长大尺寸优质Nd:GGG晶体提供参考.     

具有主导晶面的BiOIO3/BiOCl异质结制备与光催化性能

分别以乙二醇/去离子水为溶剂,通过溶剂热/水热法分别制备了具有不同主导晶面的BiOIO₃/{110}BiOCl和BiOIO₃/{001}BiOCl异质结。采用X射线衍射、扫描电子显微镜、能量色散谱和紫外可见漫反射光谱对制备的BiOIO₃/BiOCl光催化剂进行了表征。在可见光照射下,通过对罗丹明B和苯酚水溶液的光催化降解,考察了BiOIO₃/BiOCl异质结的光催化活性。结果显示25%BiOIO₃/{110}BiOCl异质结具有最高的光催化效率。BiOIO₃/{110}BiOCl较好的光催化性能是由于其在可见光区较强的光吸收,以及异质结结构和BiOCl所具有的(110)主导晶面有利于光生载流子的分离。超氧自由基(·O₂^-)和空穴(h⁺)是光催化过程中的主要活性物质。此外,根据实验结果探讨了光催化性能增强的机理。     

A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

A modified Crank-Nicolson scheme with incremental unknowns for convection dominated diffusion equations

A modified Crank-Nicolson scheme based on one-sided difference approximations is proposed for solving time-dependent convection dominated diffusion equations in two-dimensional space. The modified scheme is consistent and unconditionally stable. A priori L² error estimate for the fully discrete modified scheme is derived. With the use of the incremental unknowns preconditioner at each time step, a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified Crank-Nicolson scheme.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.