关于多项式映射自同构恒等集的一个注记

本文给出了判定一个仿射代数集是否是一个自同构恒等集的充分条件.作为一个推论,我们给出了Mckay-Wang的一个问题的一个新证明.我们也给出了一些具体的例子来说明主要的定理.     

一个不可定向曲面的极小禁用子图的构造

曲面S的一个极小禁用子图是这样的一个图,它的任何一个顶点的度都不小于3,它不能嵌入在S上,但是删去任何一条边后得到的图能嵌入在S上.本文给出了四种构造一个不可定向曲面的极小禁用子图的方式,即粘合一个顶点,一个图的边被其它的图替换,粘合两个顶点,将一个图放在另一个图的一个曲面嵌入的面内.     

基于身份的门限密码体制

描述一个公钥密码体制,其中参与者的公钥是一个公开值,例如他的身份,这个体制由很多可信中心联合产生一个大合数N=pq,p,q为素数且p≡q≡3(mod 4),任意其中一个可信中心都不知道N的分解.另外,每一个可信中心拥有一个秘密指数的一个分享,这样产生一个门限解密.本文将讨论所提出的方案的安全性,并证明它与解决二次剩余问题的困难性有关.     

Dedekind 群的一个刻画

本文给出了Dedekind群的一个刻画.即如果一个群G有一个无不动点的弱幂同构,则G是一个Dedekind群.     

浅谈中学代数中代数运算概念的教学

代数运算是近代算学中一个很重要的概念。下面是它的一个定义(不是函数观点的定义): 令M是一个集合,这个集合至少含有一个元素。若是有一个确定的法则,通过它对于集合M中一对有次序的元素a与b,能够得到集合M的一个确定的元素c与之对应,那么这个法则叫做集合M的一个代数运算。     

关于k—复盖图的几个条件

设Ｇ是一个图，ｋ为正整数，图Ｇ的一个ｋ－正则支撑子图Ｆ称做图Ｇ的一个ｋ－因子，若图Ｇ的每一条边ｅ都属于Ｇ的一个ｋ－因子，则称Ｇ是一个ｋ－复盖图，本文给出了一个图Ｇ是ｋ的复盖图的几个充分条件。     

一个带约束的拟线性椭圆特征问题

通过构造伪梯度向量场和下降流的方法, 证明了一个带约束的拟线性椭圆特征值问题至少有一个正解、一个负解和一个变号解.     

q-等度连续点及-敏感点

该文介绍了q-等度连续点与-敏感点的概念.证明了:一个点是一个动力系统的平均等度连续点当且及当对任意q∈[0,1),使得它是该系统的一个q-等度连续点;一个点是一个动力系统的平均敏感点当且仅当存在q∈(0,1],使得它是该系统的一个-敏感点.     

关于k－复盖图的几个条件

设Ｇ是一个图，ｋ为正整数．图Ｇ的一个ｋ－正则支撑子图Ｆ称做图Ｇ的一个ｋ－因子．若图Ｇ的每一条边ｅ都属于Ｇ的一个ｋ－因子，则称Ｇ是一个ｋ－复盖图．本文给出了一个图Ｇ是ｋ－复盖图的几个充分条件．     

三对角矩阵求逆问题的思考——从一道课本习题谈起

矩阵求逆是矩阵代数中的一个重要运算,逆矩阵的获得却困难重重.文中介绍一个三对角形矩阵求逆的新思路,它可以作为求解逆矩阵问题的一个"解题模块",学习线性代数课程的一个"认知结构",也可以作为教育数学的一个案例.     

二次曲面和平面位置关系的判式

在解析几何中有二次曲线与直线位置关系的讨论、二次曲面与直线位置关系的讨论,而二次曲面与平面相关位置关系的探讨较少.本文给出二次曲面a11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz+2a14x+2a24y+2a34z+a44=0(1)和平面Ax+By+Cz+D=0(2)的相对位置的判别式Δ=a11a12a13a14Aa21a22a23a24Ba31a32a33a34Ca41a42a43a44DA B C D0(aij=aji).(3)并证明了:若Δ>0,则二次曲面(1)与平面(2)相交;若Δ=0,则(1)和(2)相切;若Δ<0,则(1)和(2)相离.     

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

Staircase Parameterization in Dynamical Selection

This paper deals with a variant of a dynamical selection scheme introduced by Attouch and Cominetti for ill-posed convex minimization which combines approximation with the steepest descent method by mean of a suitable parameterization of the approximation parameter as a function of the time. This variant applies to a general inclusion with a maximal monotone operator by mean of a staircase parameterization. A discrete analogue is also considered. Applications to selecting a particular zero of a maximal monotone operator or a particular fixed point of a nonexpansive mapping via regularization techniques are presented. Finally, the alternative use of well-posedness by perturbations is discussed.     

Box-Rectangular Drawings of Plane Graphs

In this paper we introduce a new drawing style of a plane graph G called a box-rectangular drawing. It is defined to be a drawing of G on an integer grid such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal line segment or a vertical line segment, and the contour of each face is drawn as a rectangle. We establish a necessary and sufficient condition for the existence of a box-rectangular drawing of G. We also give a linear-time algorithm to find a box-rectangular drawing of G if it exists.     

On arithmetic Macaulayfication of Noetherian rings

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

     

Mean-variance portfolio selection for a non-life insurance company

We consider a collective insurance risk model with a compound Cox claim process, in which the evolution of a claim intensity is described by a stochastic differential equation driven by a Brownian motion. The insurer operates in a financial market consisting of a risk-free asset with a constant force of interest and a risky asset which price is driven by a Lévy noise. We investigate two optimization problems. The first one is the classical mean-variance portfolio selection. In this case the efficient frontier is derived. The second optimization problem, except the mean-variance terminal objective, includes also a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target which is a random process. In order to find optimal strategies we apply techniques from the stochastic control theory.     

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.     

The KdV soliton crosses a dissipative and dispersive border

We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a into a number of them. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapor saturation/condensation area. Some rough estimations for a prediction of an output are given using the relative decay (or accumulation) of the KdV conserved quantities in a dissipative area; in particular for a restriction for a number of solitons in the transmitted signal.