抽象泛函微分方程：Liapunov泛函与比较原则

本文用比较原则研究一类抽象泛函微分方程解的有界性和稳定性。     

议常微分方程奇解的定义

对比讨论常微分方程教科书中关于奇解的不同定义方式,指出用包络定义奇解的不相容性和用唯一性被破坏定义奇解的合理性.     

一类微分差分方程的周期解

泛函微分方程周期解的存在性问题是重要而困难的。文[1—3]分别用Kaplan—Yorke方法研究了含一个滞量的微分差分方程的周期解问题。文[4]用Kaplan-Yorke方法研究了含二个滞量的微分差分方程周期解的存在性问题。本文研究微分差分方程     

常微分方程奇解的讨论

对常微分方程教科书中采用的不同方式来定义奇解进行了讨论，指出了用包络定义奇解的不相容性和用唯一性被破坏定义奇解的合理性．     

一类二阶非线性常微分方程零解的全局稳定性

本用变量分离法研究了二阶非线性微分方程x φ(x)p(x) g(x)f(x)=0零解的全局稳定性．     

带脉冲的p-Laplacian系统的周期解研究

本文用变分法研究了一类带脉冲的p-Laplacian周期解的存在性问题.其中特别强调了脉冲效应,得到了由脉冲生成的解.特别地,非线性项可以取为零,当p=2时,我们的结果可以用来研究差分方程的周期解的存在性.     

一类高阶混合型方程周期解存在唯一的充要条件

该文利用Fourier级数理论讨论了具有两个偏差变元的常系数线性高阶混合型泛函微分方程的周期解,得到了用系数与时滞表示的周期解存在、唯一的充要条件.     

广义非时齐生灭微分方程解的唯一性条件

本文对广义非时齐生灭微分方程用构造性方法证明了解的存在性以及构造性解的最小非负性并时而获得了使该方程有唯一正规解的一些条件，这些结果在数学和生物学中都有一定的意义     

线性时滞微分方程解的渐近性态

本文用一个简单的方法证明了一类一阶线性时滞微分方程解的有界性帮必有非振动解，分析了振动解的性质。这个方法也被用来讨论一阶时滞方程组和中立型微分方程，所得结果均较简明。     

概率方法解拟线性偏微分方程

本文用随机分析方法证明了拟线性抛物型方程ｕｔ＋ｆ（ｕ）ｕｘ、ｕｘｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）在ｕ０有界可测，ｆ连续且ｆ＞０条件下，其解当→０时收敛于拟线性方程ｕｔ＋ｆ（ｕ）ｕｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）的熵解，即论证了“沾性消失法”解此方程的正确性，１９５７年Ｏｌｅｉｎｉｋ曾用差分方法解决了此问题。这里用概率方法重新获得此结果。     

Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

We study the interlacing property of zeros of Laguerre polynomials of adjacent degree, where the free parameters differ by an integer, and of the same degree, where the free parameter is shifted continuously. Similar interlacing results are proven for the positive zeros of Gegenbauer polynomials. Research by Kathy Driver is supported by the National Research Foundation of South Africa under grant number 2053730. Research by Kerstin Jordaan is partially supported by the National Research Foundation of South Africa under grant number 2054423.     

一类Hamilton系统的Abel积分比值单调性的判别条件

用直接计算的方法对一类Hamilton系统的两个Abel积分比值的单调性进行讨论,指出该单词性条件可由两个判定函数直接确定．     

A graph-theoretic proof of the non-existence of self-orthogonal Latin squares of order 6

The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given, but these proofs generally require extensive prior subject knowledge in order to follow them, or are ‘blind’ proofs in the sense that most of the work is done by computer or by exhaustive enumeration. In this paper we present a graph-theoretic proof of a somewhat weaker result, namely the non-existence of self-orthogonal Latin squares of order six, by introducing the concept of a self-orthogonal Latin square graph. The advantage of this proof is that it is easily verifiable and accessible to discrete mathematicians not intimately familiar with the theory of combinatorial designs. The proof also does not require any significant prior knowledge of graph theory.     

Classification of dicoverings

The dicoverings of a “well pointed” d-space are classified as quotients of the universal dicovering space under congruence relations. We prove that the subcategory of d-spaces generated by the subcategory of directed cubes is equal to the category generated by the interval and the directed interval. Similarly, the category of topological spaces generated by simplices may be generated by the interval.     

Reduction of continuous symmetries of chaotic flows by the method of slices

We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a ‘slice’ defined by minimizing the distance to a single generic ‘template’ intersects the group orbit of every point in the full state space. Global symmetry reduction by a single slice is, however, not natural for a chaotic/ turbulent flow; it is better to cover the reduced state space by a set of slices, one for each dynamically prominent unstable pattern. Judiciously chosen, such tessellation eliminates the singular traversals of the inflection hyperplane that comes along with each slice, an artifact of using the templates local group linearization globally. We compute the jump in the reduced state space induced by crossing the inflection hyperplane. As an illustration of the method, we reduce the SO (2) symmetry of the complex Lorenz equations.     

膨胀土分类的一种新方法

在专家凭经验无法准确地确定评价指标的权重时,用相似系数确定权重,给出了一种新的多指标综合评价方法.将该方法应用于膨胀土的胀缩等级分类,其合理性和有效性得到了验证.     

Generalization of a theorem of Kusmin

A Gauss-Kusmin theorem for the natural extension of the regular continued fraction expansion is given. A generalization of a theorem by D. E. Knuth is obtained by similar techniques.With 3 FiguresResearch of the first author supported by the Netherlands Organization of Scientific Research (NWO).     

非系统风险的回报率及其计量

现代金融理论认为,系统风险无法通过组合投资进行规避,承担系统风险被市场承认从而可以获得风险报酬;非系统风险可以通过组合投资进行有效分散,因而承担非系统风险不应获得风险回报.试图阐述系统风险完全可以规避,指出承担非系统风险也应获得风险报酬,给出计量非系统风险回报率的规划方法,该规划的最优解同样满足两基金分离定理.     

Unimodularity of zeros of self-inversive polynomials

We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang as well as by Lalín and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lalín and Rogers on computational evidence.