Weierstrass points on cyclic covers of the projective line

We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form , where is a polynomial of degree . Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, . We obtain a lower bound for , which we show is exact if and are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula):

where is the genus of the curve. In the case that (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes , the branch points and the non-branch Weierstrass points remain distinct modulo .

     

Some Geometrical Aspects of Efficient Points in Vector Optimization

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phelps theorem. We study also the density of positive proper efficient point sets. This research was supported by a Central Research Grant of Hong Kong Polytechnic University, Grant G-T 507. Research of the first author was also supported by the National Natural Science Foundation of P.R. China, Grant 10361008, and the Natural Science Foundation of Yunnan Province, China, Grant 2003A002M. Research of the second author was also supported by the Natural Science Foundation of Chongqing. Research of the third author was supported by a research grant from Australian Research Counsil.     

正态分布在学生学习成绩评估中的应用

本文将正态分布应用于学生学习成绩评估中，将原始分数转换为标准分数后，各科的分数方可比较或相加，从而改变了将原始分数直接相加比较的那种不够科学也不甚合理的传统做法。     

On the absence of cycles for unimodal mappings of the square

Unimodal 2-endomorphisms of a special kind of the unit square are considered. Conditions for the absence of periodic points are presented. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 370–378, March, 1998.     

General Existence Theorem of Zero Points

Let X be a nonempty, compact, convex set in and let be an upper semicontinuous mapping from X to the collection of nonempty, compact, convex subsets of . It is well known that such a mapping has a stationary point on X; i.e., there exists a point X such that its image under has a nonempty intersection with the normal cone of X at the point. In the case where, for every point in X, it holds that the intersection of the image under with the normal cone of X at the point is either empty or contains the origin 0 ⁿ , then must have a zero point on X; i.e., there exists a point in X such that 0 ⁿ lies in the image of the point. Another well-known condition for the existence of a zero point follows from the Ky Fan coincidence theorem, which says that, if for every point the intersection of the image with the tangent cone of X at the point is nonempty, the mapping must have a zero point. In this paper, we extend all these existence results by giving a general zero-point existence theorem, of which the previous two results are obtained as special cases. We discuss also what kind of solutions may exist when no further conditions are stated on the mapping . Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.     

On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals

A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space of continuous functions, under the supremum norm, from a compact Hausdorff topological space into a Banach space equipped with its weak topology . Strongly extreme points of the unit ball of are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of . It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of . Lastly, some naturally arising and previously known extreme points of the unit ball of are shown to actually be strongly exposed points.

     

Mizoguchi–Takahashi’s type common fixed point theorem

Recently Kamran extended the result of Mizoguchi and Takahashi for closed multi-valued mappings and proved a fixed point theorem. In this paper we further extend the result concluded by Kamran and prove a common fixed point theorem by using the concept of lower semi-continuity.     

On the compactly locally uniformly rotund points of Orlicz spaces

In this paper, locally uniformly rotund points and compactly locally uniformly rotund points are introduced. Moreover, criteria for compactly locally uniformly rotund points in Orlicz spaces are given.     

The fractal structures of the exceptional sets of Lévy processes

In this paper we review some results on the fractal geometry properties of the sets of thick points, thin points, fast points and slow points derived from Lévy processes. This work was supported by National Natural Science Foundation of China (Grant No. 10871200)     

Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces

For vector-valued functions, cone saddle points are defined, and some existence theorems for them are established in infinite-dimensional spaces. Most of our results rely on Condition 2.1, which has been given by Sterna-Karwat. In particular, it shows that the scalarization of vector-valued functions plays an important role for the condition. Some interesting examples in infinite-dimensional spaces are presented. Moreover, necessary conditions for the existence of cone saddle points are investigated.The author would like to thank the referees for their valuable suggestions on the original draft.