包含—受扭迴转轴的半空间的性质*

本文研究包含有一根部份嵌入的迴转轴的半空间的性质.不用知道一给定的嵌入的轴的扭转问题的精确解,这些性质能指出此半空间的位移或应力场的某些特点并且有时可以用来检查数值解.文中给出嵌入半空间的受扭的刚性圆柱的轴的表面上的正确的应力分布的检查的例子.     

方程a~x=loga~x的解

一个方程的解可以看作两个函数的图象的交点的横坐标。反过来,方程的解又可以反映两个函数之间的某种关系,即它们的图象相交的情况。因此,可以利用函数的性质对方程的解,特别是直接求解很困难的某些超越方程的解的情况作出定性的讨论。也可以利用方程的解对函数的图象间的交点个数作出定量的研究,本文主要通过对函数y=x~(1/x)和y=x~x的性质的分析,就方程a~x=x和a~x=loga~x等的解的情况进行讨论。     

关于二次曲线的切线及奇异点的探讨

利用二次曲线的切线的定义,分别讨论过二次曲线上的一点的切线的求法及过二次曲线外的一点的切线的两种求法,并且得到了存在奇异点的二次曲线的具体类型.     

关于直接揭露本貭特征对于学生掌握几何基本概念的作用的問題

一、研究的任务和方法概念乃是一种反映对象和現象的一般的并且是本质的特征的思维形式。概念的形成是在人类历史发展过程中进行的,概念的掌握是掌握現成的、社会上形成的概念。所以概念的掌握就不需要通过概念形成时人类所走过的复杂的、漫长的道路。虽然如此,但是概念的掌握仍然是一种复杂的过程,它取决于过去的經驗、已有的知识、掌握过程中实现的活动(如教学活动、生活活动等)、掌握借以实現的智力过程的系统。概念的教学方法是多种多样的,主要可分为两种:一种是直接揭露本质特征的方法,另一种是間接揭露本质特征的方法(如变式、对比等)。在平面几何的概念的教学中应該采用直接揭露本质特征的方法还是采     

横看成岭侧成峰善于联想思路清——一类三角问题的解法分析

匈牙利数学家乔治·波利亚致力于解题的研究,为了回答"一个好的解法是如何想出来的"这个令人困惑的问题,他专门研究了解题的思维过程,并把研究所得写成<怎样解题>一书.在波利亚的解题表中,拟定计划是解题的关键环节,拟定计划的过程是在"过去的经验和已有的知识"基础上,探索解题思路的发现过程,是不断变换问题,把复杂的问题向简单的问题转化,陌生的问题向熟悉的问题转化,最终把待解决的问题化归为已解决的或易解决的问题的过程,其中善于联想又是转化的关键.下面通过一道习题的分析,体验这种联想转化的思维过程.     

大学校长为何很多是数学家

1 大学校长多是数学家 经常关注数学教育的人可能会发现一个有趣的现象：国内许多大学的校长（包括现任的、离任的，以及正职、副职），都是数学专业出身，一些还是大名鼎鼎的数学家．比如：老一代的，1890-1920年间出生的就有云南大学的熊庆来、重庆大学（安徽大学）的何鲁、中国科大的华罗庚、复旦大学的苏步青、四川大学的柯召、南开大学的吴大任、上海大学的钱伟长等；     

条件中位数的最近邻估计和它的Bootstrap统计量的渐近性质

本文讨论条件中位数的估计问题,证明了条件中位数的最近邻估计序列的渐近正态性。在讨论估计的误差的分布特性时,作者引用了Efron的Bootstrap方法。证明了在一定条件下,最近邻估计的误差的Bootstrap分布在渐近意义下是可用的。即估计量的误差的渐近分布与它的Bootstrsp统计量的渐近分布是相同的。     

斜三角形的面积向量式及其应用

文[1]介绍了余弦定理的向量式：以同一点为起点的任意两向量的数量积等于这个向量的模的平方和与这两个向量终点的连线段所表示的向量的模的平方的差的一半．如△ABC中，     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅱ):向后一步的Euler全离散化格式

1．引言本文的工作主要是讨论非定常的热传导一对流问题的向后一步的Euler全离散化的非线性Galerkin混合元解的存在性及其误差估计．该工作是对山中的同一问题研究的第二部分．在第一部分［1］，我们已经讨论了此问题的半离散化的情形．由于所研究的目标都是非定常的热传导一对流问题，其背景是相同的，在此将不重复了，请参考［1］．本文的安排如下，52先回顾非定常的热传导一对流问题的混合元解的经典性质．53回顾半离散化的非线性Galerkin混合元解的性质，并导出后续讨论需要的一些关于时间导数的估计．54讨论向后一步的Euler全离散化…     

L-模糊正规子群之间的对应定理

群G的一个L-模糊正规子群A的陪集做成的群G／A与群G的一个商群是自然同构的。如果f：G→G’是群的满同态，则G’的L-模糊正规子群做成的群与G的在f的核上取定值的L-模糊正规子群做成的群之间存在一个保序的双射。     

The mathematical theory of growing bodies. Finite deformations

The fundamentals of the mathematical theory of accreting bodies for finite deformations are explained using the concept of the bundle of a differentiable manifold that enables one to construct a clear classification of the accretion processes. One of the possible types of accretion, as due to the continuous addition of stressed material surfaces to a three-dimensional body, is considered. The complete system of equations of the mechanics of accreting bodies is presented. Unlike in problems for bodies of constant composition, the tensor field of the incompatible distortion, which can be found from the equilibrium condition for the boundary of growth, that is, a material surface in contact with a deformable three-dimensional body, enters into these equations. Generally speaking, a growing body does not have a stress-free configuration in three-dimensional Euclidean space. However, there is such a configuration on a certain three-dimensional manifold with a non-Euclidean affine connectedness caused by a non-zero torsion tensor that is a measure of the incompatibility of the deformation of the growing body. Mathematical models of the stress-strain state of a growing body are therefore found to be equivalent to the models of bodies with a continuous distribution of the dislocations.     

A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization

This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.

     

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.     

Novel method of identifying time series based on network graphs

In this article, we propose a novel method for transforming a time series into a complex network graph. The proposed algorithm is based on the spatial distribution of a time series. The characteristics of geometric parameters of a network represent the dynamic characteristics of a time series. Our algorithm transforms, respectively, a constant series into a fully connected graph, periodic time series into a regular graph, linear divergent time series into a tree, and chaotic time series into an approximately power law distribution network graph. We find that when the dimension of reconstructed phase space increases, the corresponding graph for a random time series quickly turns into a completely unconnected graph, while that for a chaotic time series maintains a certain level of connectivity. The characteristics of the generated network, including the total edges, the degree distribution, and the clustering coefficient, reflect the characteristics of the time series, including diverging speed, level of certainty, and level of randomness. This observation allows a chaotic time series to be easily identified from a random time series. The method may be useful for analysis of complex nonlinear systems such as chaos and random systems, by perceiving the differences in the outcomes of the systems—the time series—in the identification of the systemic levels of certainty or randomness. © 2011 Wiley Periodicals, Inc. Complexity, 2011     

A New Pareto Optimal Solution in a Lagrange Decomposable Multi-Objective Optimization Problem

In this paper a committee decision-making process of a convex Lagrange decomposable multi-objective optimization problem, which has been decomposed into various subproblems, is studied. Each member of the committee controls only one subproblem and attempts to select the optimal solution of this subproblem most desirable to him, under the assumption that all the constraints of the total problem are satisfied. This procedure leads to a new solution concept of a Lagrange decomposable multi-objective optimization problem, called a preferred equilibrium set. A preferred equilibrium point of a problem, for a committee, may or may not be a Pareto optimal point of this problem. In some cases, a non-Pareto optimal preferred equilibrium point of a problem, for a committee, can be considered as a special type of Pareto optimal point of this problem. This fact leads to a generalization of the Pareto optimality concept in a problem.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

Harmonic and pulse excitations of multiply connected cylindrical bodies

The three-dimensional problem of the theory of elasticity of the harmonic oscillations of cylindrical bodies (a layer with several tunnel cavities on a cylinder of finite length) is considered for uniform mixed boundary conditions on its bases. Using the Φ-solutions constructed, the boundary-value problems are reduced to a system of well-known one-dimensional singular integral equations. The solution of the problem of the pulse excitation of a layer on the surface of a cavity is “assembled” from a packet of corresponding harmonic oscillations using an integral Fourier transformation with respect to time. The results of calculations of the dynamic stress concentration in a layer (a plate) weakened by one and two openings of different configuration are given, as well as the amplitude-frequency characteristics for a cylinder of finite length with a transverse cross section in the form of a square with rounded corners, and data of calculations for a trapeziform pulse, acting on the surface of a circular cavity, are presented.     

Modeling of a discrete dynamic system using a modified Petri net

Together with the proposed modification connected with the introduction of a time-dependent net, we define a criterion for optimality of a sequence of initiations of transitions, and study a method of obtaining such a sequence. We given the construction of a model of a discrete dynamic system having finite parameters. to describe it we introduce a modification of the Petri net using a determination of the state of the positions, time delay of the transitions, and variation of the conditions and rules for initiating transitions. Translated fromDinamicheskie Sistemy, No. 13, 1994. pp. 39–50.     

A Priori and A Posteriori: A Bootstrapping Relationship

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.