Hopf Algebras of Dimension 12

We conclude the classification of Hopf algebras of dimension 12 over an algebraically closed field of characteristic zero.     

Representing homology classes by locally flat 2-spheres

We prove an existence result for topologically locally flat embeddings of 2-spheres in simply connected 4-manifolds. This topological result is deduced from a splitting theorem for pointed Hermitian modules over a cyclic group ring. A stability result for such modules is also proved. This applies to the isotopy classification of locally flat embeddings.Partially supported by the NSF.     

Generalized quasiconvexities,cone saddle points,and minimax theorem for vector-valued functions

We establish the following theorems: (i) an existence theorem for weak type generalized saddle points; (ii) an existence theorem for strong type generalized saddle points; (iii) a generalized minimax theorem for a vector-valued function. These theorems are generalizations and extensions of the author's recent results. For such extensions, we propose new concepts of convexity and continuity of vector-valued functions, which are weaker than ordinary ones. Some of the proofs are based on a few key observations and also on the Browder coincidence theorem or the Tychonoff fixed-point theorem. Also, the minimax theorem follows from the existence theorem for weak type generalized saddle points. The main spaces with mathematical structures considered are real locally convex spaces and real ordered topological vector spaces.This paper is dedicated to Professor Kensuke Tanaka on his sixtieth birthday.This paper was written when the author was a visitor at the Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata, Japan. The author is indebted to Prof. K. Tanaka for suggesting this work.The author is very grateful to Prof. P. L. Yu for his useful encouragement and suggestions and to the referees for their valuable suggestions and comments.     

Fuzzy拓扑学—层次结构和点式处理

建设在Ｆｕｚｚｙ集合基础上的Ｆｕｚｚｙ拓扑学，经过２０多年发展，现在已自具特色，日趋成熟，本文力图从点式处理和层次结构角度反映其基本思想、具体构造和主要结果。并且说明Ｆｕｚｚｙ拓扑学研究与经数学的关系。从这里可以看到，Ｆｕｚｚｙ拓扑研究无论是基本思想还是研究方法都有其独到之处。     

Chow groups of the moduli spaces of weighted pointed stable curves of genus zero

The moduli space of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of are generated by the strata of and give all additive relations between them. We also observe that the Chow groups and the homology groups are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data A.     

The Conley index over a base

We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in , and the problem of continuing two periodic orbits which are not homotopic as loops.

     

Classification of semispaces according to their types in infinite-dimensional vector spaces

It is shown that each semispaceC X naturally generates a relation of complete preorder onX with respect to which the pair (X C, C) is a cut ofX. By identifying the type of the semispace with the type of the cut generated by this semispace, the semispaces are classified according to their types. The obtained classification extends the classification of semispaces in finite-dimensional vector spaces due to Martinez-Legaz and Singer to infinite-dimensional spaces.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 191–198, August, 1998.     

不具有关于锥的例外簇的映射

给出Hilbert空间到其自身不具有关于锥的例外族的映射条件，利用Hilbert空间可表为闭凸锥与负对偶锥的特点研究映射关于锥的例外簇的特性，证明了可通过映射在某紧凸子集上的性态判断其例外簇的存在与否，并讨论单调和沿射线单调映射的不具例外簇问题。     

Moduli Stacks of Permutation Classes of Pointed Stable Curves

The notion of m/Γ-pointed stable curves is introduced. It should be viewed as a generalization of the notion of m-pointed stable curves of a given genus, where the labels of the marked points are only determined up to the action of a group of permutations Γ. The classical moduli spaces and moduli stacks are generalized to this wider setting. Finally, an explicit construction of the new moduli stack of m/Γ-pointed stable curves as a quotient stack is given. Received: February 2008