Numerical solution of reaction–diffusion equations by compact operators and modified equation methods

A system of reaction–diffusion equations which governs the propagation of an ozone decomposition laminar flame in Lagrangian co-ordinates is analysed by means of compact operators and modified equation methods. It is shown that the use of fourth-order accurate compact operators yields very accurate solutions if sufficient numbers of grid points are located at the flame front, where very steep gradients of temperature and species concentrations exist. Modified equation methods are shown to impose a restriction on the time step under certain conditions. The solutions obtained by means of compact operators and modified equation methods are compared with solutions obtained by other methods; good agreement is obtained.     

A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces

In this paper, we study the existence of solutions and approximation of the solutions by using a Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces. The results presented in this paper generalize the corresponding results of J. Li [J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295 (2004) 115-126].     

Compact operators which factor through subspaces of lp

Let 1 ≤ p ≤ ∞. A subset K of a Banach space X is said to be relatively p ‐compact if there is an 〈x_n 〉 ∈ l^s _p (X) such that for every k ∈ K there is an 〈α_n 〉 ∈ l_{p ′} such that k = σ^∞_n=1 α_n x_n . A linear operator T: X → Y is said to be p ‐compact if T (Ball (X)) is relatively p ‐compact in Y. The set of all p ‐compact operators K_p (X, Y) from X to Y is a Banach space with a suitable factorization norm κ_p and (K_p , κ_p ) is a Banach operator ideal. In this paper we investigate the dual operator ideal (K^d _p , κ^d _p ). It is shown that κ^d _p (T) = π_p (T) for all T ∈ B (X, Y) if either X or Y is finite‐dimensional. As a consequence it is proved that the adjoint ideal of K^d _p is I_{p ′}, the ideal of p ′‐integral operators. Further, a composition/decomposition theorem K^d _p = Π_p K is proved which also yields that (Π^min _p )^inj = K^d _p . Finally, we discuss the density of finite rank operators in K^d _p and give some examples for different values of p in this respect. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Single machine unbounded parallel-batch scheduling with forbidden intervals

In this paper we consider the single machine parallel-batch scheduling with forbidden intervals. There are some forbidden intervals in which the machine cannot be available. The jobs are processed in batches form in the remaining free time-slots without preemption, where the processing time of a batch is defined to be the maximum processing time of the jobs in this batch. We show that, when the objective is bottleneck form, maximum lateness, or makespan with release dates of jobs, the considered problem can be solved in polynomial time.     

Convergence Results of the ERM Method for Nonlinear Stochastic Variational Inequality Problems

This paper considers the expected residual minimization (ERM) method proposed by Luo and Lin (J. Optim. Theory Appl. 140:103–116, 2009) for a class of stochastic variational inequality problems. Different from the work mentioned above, the function involved is assumed to be nonlinear in this paper. We first consider a quasi-Monte Carlo method for the case where the underlying sample space is compact and show that the ERM method is convergent under very mild conditions. Then, we suggest a compact approximation approach for the case where the sample space is noncompact. This work was supported in part by Project 10771025 supported by NSFC and SRFDP 20070141063 of China.     

Binding of a surfactant counterion to low-charge-density poly(acrylic acid) and poly(methacrylic acid)

The binding of a cationic surfactant, dodecylpyridinium (C12Py) chloride, with a low-charge-density poly (methacrylic acid) (PMA) was investigated in buffer solutions under the condition of constant pH. The binding isotherms with PMA consisted of two and three steps at a pH lower and higher than 3.2, respectively. Bindings in the first step were independent of pH and this step was considered to correspond to the solubilization of the hydrocarbon chains of C12Py into the nonpolar region of the compact form of PMA. This is the indication of the compact form from the binding isotherm. At pH higher than 3.2, the second step was discriminated and it depended on the pH. In the third step, a sharp rise in the degree of binding (β) was observed accompanying the solubilization of the precipitates of the PMA–C12Py complex. The binding with poly(acrylic acid) (PAA) and PMA in conventional unbuffered NaCl solutions was also examined and the pH profile of the solution during the binding process was determined. In the case of unbuffered NaCl solutions, the binding with PAA took place cooperatively at the critical association concentration (cac). The binding isotherm consisted of two steps and the pH decreased with the increase in β. The binding isotherm of PMA, on the other hand, consisted of three steps: the pH decreased slightly in the first step and considerably in the second step with the increase in β but it increased with β in the third step, exhibiting a pH minimum around 3.2. The binding in the first step coincided with that obtained in the buffered solutions. Linear relationships between β and the pH were found for both polymers. In the case of PMA, no cac was observed in both buffered and unbuffered NaCl solutions. Received: 24 January 2001 Accepted: 23 May 2001     

基于视场拼接的全视差三维显示系统

提出了一种新的基于视场拼接的全视差三维显示方法及图像生成方案.系统核心部分由一个微投影机阵列、一个反射光路和一个散射屏组构成,图像生成方案采用基于OpenGL的算法实现.为了验证所提出的基于视场拼接理论的正确性,利用液晶显示器、菲涅耳镜组、定向散射屏和背投式反射光路搭建了实验样机.该装置可以在更紧凑的系统结构下显示包括横向视差和纵向视差在内的三维图像,并且通过图像生成算法的配合实现了动态三维显示效果.     

一类函数方程的Cm解的存在性和惟一性

本文讨论了较为广泛的一类函数方程G（x,f(x),f^2(H2(x,f(x)),…,f^n(Hn(x,f(x),…,f^n-1(x))))=0, 对任x∈J.其中n≥,J为实数轴R 的连通闭子集,G∈C^m(J^n 1,R),Hk∈C^m(J^k,R),k=2,…,n.对任一个整数m≥0,本文在较弱的条件下讨论下该方程的C^m解的存在性和惟一性。     

关于双曲空间形式的一个注记

本文主要证明一个具有光滑边界的紧黎曼流形,如果有非平凡解,则就等度量同构与双曲空间形式 会的紧区域,这里Ｄ～２■是■的Ｈｅｓｓｉａｎ与ｇ是Ｍ上的黎曼度量． 还证明关于上述方程的边值问题,只有混合边值问题,而且当ｃ＜－１时有解．     

Some properties for the attractors

For the continuous flows defined on a topological space, we have discussed some properties for the invariant sets and their domains of influence. We have proved the following open problem posed by C. Conley: an attractor in a locally connected compact Hausdorff invariant set has finitely many components. In the meantime, two necessary and sufficient conditions for a set to be an attractor have been given.