常微分方程数值方的法稳定性(英)

本文针对常微分方程数值方法稳定性问题，证明了一般方法的绝对稳定性定理，同时也指出了绝对稳定性条件的局限性．为了克服这种局限性，本文绘出了Jordan稳定性的概念，并建立了一个相应的判别定理．     

L_P（Ω）空间上一类C_0－半群的稳定性及应用

本文揭示了ＬＰ－空间中一类Ｃ０半群稳定性的特征，从而获得一个在实际问题中较易检验的稳定性条件．作为应用，本文研究了中子迁移系统的稳定性．     

动力系统中的Lipschitz稳定性

本文讨论动力系统的Ｌｉｐｓｃｈｉｔｚ稳定性与吸引性之间的关系．给出动力系统中弱吸引性、吸引性和强吸引性这三个概念相互等价的条件，在一定的条件下证明了（弱、强）吸引子与全局（弱、强）吸引子是一致的．本文还讨论了度量Ｌｙａｐｕｎｏｖ稳定性和拓扑Ｌｙａｐｕｎｏｖ稳定性及它们与Ｌｉｐｓｃｈｉｔｚ稳定性之间的关系．     

解析非线性差分系统的稳定性

本文研究方程右端具有某种解析性质的非线性差分系统解的稳定性问题。通过引入ｋ－稳定性概念，证明了几个简洁的稳定性充分准则．并用例子说明了本文结果的优越性。     

光滑映射芽的开折的分级稳定性

光滑映射芽各种稳定性的讨论,一直是奇点理论的一个重要部分． Ｔｈｏｍ Ｒ．［１］在创立突变论时,提出了映射芽的,ｒ－开折的稳定性理论．Ｗａｓｓｅｒｍａｎｎ Ｇ．［２］将之发展为开折的（ｒ,ｓ）稳定理论．本文将他们的结论发展为（ｒ１,ｒ２,…,ｒｄ）稳定性,在任意的分级情况下,得到强稳定性、弱稳定性及无穷小稳定性的等价性,并得到了一些基本结果．     

关于变系数线性方程的稳定性

本文给出了变系数线性方程有关稳定性的一些简洁的判据．对周期系数线性方程，给出了较为精确的渐近稳定性判据．从理论上解释了原先“冻结系数法”一般不能成立的原因．     

自治泛函微分方程的不变性原理

本文改进了具无限时滞的自治泛函微分方程的不变性原理．作为所得结果的一种应用，建立了关于渐近稳定性的一个判别准则．并且，给出例子来说明所得的稳定性结果．     

径向轴对称冲击载荷下正交异性固支圆板的Liapunov稳定性

本文用Ｌｉａｐｕｎｏｖ第二方法分析了周边受轴对称径向冲击载荷作用的极正交各向异性固支圆板的稳定性．分析是在小挠度和弹性范围内进行的．冲击载荷被假定具有阶梯脉冲形式．导出了相应的稳定性条件．     

一类双曲方程自由边界问题的稳定性

锅炉炉管中汽水界面的稳定性是一个重要的开问题［１］，本文以此问题为模型，讨论了近似带域上双曲方程自由边界问题，利用小参数法和能量不等式，得到了解的渐近稳定性与平均稳定性．所得的稳定性条件，定性地看与工业试验结果相一致．     

平面Hamilton系统和时逆系统平衡点的稳定性

本文研究了拟周期平面Hamilton系统和时逆系统的平衡点的稳定性．在适当的条件下,证明了平衡点的稳定性以及在平衡点附近存在着大量的拟周期解．     

Creep of polymer concrete in the nonlinear region

Two polyester-based polymer concretes with various volume content of diabase as an extender and aggregate are tested in creep under compression at different stress levels. The phenomenological and structural approaches are both used to analyze the experimental data. Common features of changes in the instantaneous and creep compliances are clarified, and a phenomenological creep model which accounts for the changes in the instantaneous compliance and in the retardation spectrum depending on the stress level is developed. It is shown that the model can be used to describe the experimental results of stress relaxation and creep under repeated loading. Modeling of the composite structure and subsequent solution of the optimization problem confirm the possibility of the existence of an interphase layer more compliant than the binder. A direct correlation between the interphase volume content and the instantaneous compliance of the composite is revealed. It is found that the distinction in nonlinearity of the viscoelastic behavior of the two polymer concretes under investigation can be due to the difference in their porosity. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000.) Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 147–164, 2000.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

Variation of the structure and permeability of tensioned fiber layers during impregnation with a thermoplastic polymer melt

The influence of displacements of tensioned fibers on the impregnation of fibrous layers with a polymer melt and on the final composite structure is studied. Using computer simulation, it is shown that, during impregnation, the structure of tensioned fibrous layers changes considerably depending on the initial arrangement and tensioning of fibers. The consolidated regions formed under the melt front move inside the impregnated layer with the advancing melt front. Displacement of the tensioned fibers as well as the formation of “washouts” favors the impregnation of internal layers, but cause significant inhomogeneity of the polymer structure. The surface (on the side of the melt flow) regions are more saturated with the polymer than the internal ones. A difference in the melt percolation mechanisms at various impregnation regimes is revealed. The effective permeability coefficients of a tensioned fiber layer are not constant but depend on the conditions and regimes of impregnation. Submitted to the 11th Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 259–270, March–April, 2000.     

Asymptotics of the distribution of the integral of the positive part of the Brownian bridge for large arguments

The double Laplace transform of the distribution function of the integral of the positive part of the Brownian bridge was determined by M. Perman and J.A. Wellner, as well as the moments of this distribution. The purpose of the present paper is to determine the asymptotics of this distribution for large values of the argument, and the corresponding asymptotics of the moments.     

The nonlinear and scaled growth of the ottoman and Roman empires

In this work, mathematical models for the growth of the Ottoman and Roman Empires are found. The time interval considered for both cases covers the time from the birth of the empire to the end of the fast expansion period. These empires are assumed to be nonlinearly growing and self-multiplying systems. This approach utilizes the concepts of chaos theory, and scaling. The area governed by the empire is taken as the measure of its growth. It was found that the expansion of each empire on lands, seas, and on both (i.e., lands+seas) can be expressed by power laws. In the Ottoman Empire, the nonlinear growth power of total area is approximately equal to the golden ratio, and the nonlinear growth power of the expansion on lands is approximately equal to the square root of 2. In the case of the Romans, some numbers associated with the golden ratio, or the square root of 2, appear as the power of the nonlinear growth term. The appearance of both the golden ratio and the square root of 2 show that both empires had intention on achieving stability during their growth.     

基于偏序集的PROMETHEE方法优化研究

尽管PROMETHEE是当前最受欢迎的多准则决策方法之一,但在实践应用过程中,模型的应用范围与质量依然受制于指标权重问题。一些常用的赋权方法,不仅没有解决不确定权重问题,反而增加了决策风险。在偏序集相关定理的基础上,给出权重的定性信息即权重次序,由流出矩阵、流入矩阵和净流矩阵等定义,得到了PROMETHEE的偏序集表达形式。当流入和流出之和为常数时,证明了模型存在对偶性质。根据对偶性质,简化了PROMETHEE方法的分析步骤,删减模型冗余信息。应用偏序集表示的PROMETHEE,突破了模型没有具体权重便无法应用的思维定势,解决了模型赋权困难,增强了模型的鲁棒性,拓展了模型处理数据类型的范围。     

Analysis of the Stress State of the Surface Layer of a Polymeric Mass upon Filling of Bulky Compression Molds

The stress state of the surface layer of a polymeric mass during filling of bulky compression molds is analyzed. It is shown that, at particular rheological characteristics of the mass, temperature, and filling rates, cracking of the surface layer occurs, which leads to defects in the finished products. A physical analysis of this process makes it possible to conclude that the cracks arise due to the normal stresses operating in the front region of the moving polymeric mass. It is found that, under certain flow conditions, areas with a pressure lower than the atmospheric one appear on the surface of the polymer. If the tensile stresses arising in these local regions are higher than the tensile strength of the mass, the continuity of the composition is broken in the direction determined by the greatest rate of the normal deformation. To confirm the reliability of the crack-formation mechanism proposed, the distribution of the pressure and normal stresses over the free surface is calculated based on a numerical method. These calculations show that, by comparing the stress level achieved in the front region with the tensile-strength characteristics of the polymeric composition, it is possible to predict, with a sufficient accuracy, the possibility of crack formation in the surface layer of such a mass under given flow conditions and thus to solve the question on flawless manufacturing of products.     

Generalized derivatives and nonsmooth optimization, a finite dimensional tour

During the early 1960’s there was a growing realization that a large number of optimization problems which appeared in applications involved minimization of non-differentiable functions. One of the important areas where such problems appeared was optimal control. The subject of nonsmooth analysis arose out of the need to develop a theory to deal with the minimization of nonsmooth functions. The first impetus in this direction came with the publication of Rockafellar’s seminal work titledConvex Analysis which was published by the Princeton University Press in 1970. It would be impossible to overstate the impact of this book on the development of the theory and methods of optimization. It is also important to note that a large part of convex analysis was already developed by Werner Fenchel nearly twenty years earlier and was circulated through his mimeographed lecture notes titledConvex Cones, Sets and Functions, Princeton University, 1951. In this article we trace the dramatic development of nonsmooth analysis and its applications to optimization in finite dimensions. Beginning with the fundamentals of convex optimization we quickly move over to the path breaking work of Clarke which extends the domain of nonsmooth analysis from convex to locally Lipschitz functions. Clarke was the second doctoral student of R.T. Rockafellar. We discuss the notions of Clarke directional derivative and the Clarke generalized gradient and also the relevant calculus rules and applications to optimization. While discussing locally Lipschitz optimization we also try to blend in the computational aspects of the theory wherever possible. This is followed by a discussion of the geometry of sets with nonsmooth boundaries. The approach to develop the notion of the normal cone to an arbitrary set is sequential in nature. This approach does not rely on the standard techniques of convex analysis. The move away from convexity was pioneered by Mordukhovich and later culminated in the monographVariational Analysis by Rockafellar and Wets. The approach of Mordukhovich relied on a nonconvex separation principle called theextremal principle while that of Rockafellar and Wets relied on various convergence notions developed to suit the needs of optimization. We then move on to a parallel development in nonsmooth optimization due to Demyanov and Rubinov called Quasidifferentiable optimization. They study the class of directionally differentiable functions whose directional derivatives can be represented as a difference of two sublinear functions. On other hand the directional derivative of a convex function and also the Clarke directional derivatives are sublinear functions of the directions. Thus it was thought that the most useful generalizations of directional derivatives must be a sublinear function of the directions. Thus Demyanov and Rubinov made a major conceptual change in nonsmooth optimization. In this section we define the notion of a quasidifferential which is a pair of convex compact sets. We study some calculus rules and their applications to optimality conditions. We also study the interesting notion of Demyanov difference between two sets and their applications to optimization. In the last section of this paper we study some second-order tools used in nonsmooth analysis and try to see their relevance in optimization. In fact it is important to note that unlike the classical case, the second-order theory of nonsmoothness is quite complicated in the sense that there are many approaches to it. However we have chosen to describe those approaches which can be developed from the first order nonsmooth tools discussed here. We shall present three different approaches, highlight the second order calculus rules and their applications to optimization.     

Mario Pieri’s address at the University of Catania

The contributions made by the Italian mathematician Mario Pieri (1860-1913) are well known in the field of geometry. Pieri was a member of the School of Peano at the University of Turin. There he became engaged both by the problems of logic and by the philosophical aspects of Peano’s epistemology. This article was motivated by Pieri’s address given at the University of Catania, at the inauguration of the 1906-1907 academic year. My aim is to identify Pieri’s philosophical premises as found in his works and to present them in the general framework of the historical development of the Peano School.     

Formulations and valid inequalities for the node capacitated graph partitioning problem

We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.