BM-Algebras and related topics

Some connections between BM-algebras and its related topics are studied. It is proved that the class of medial BH-algebras coincides with the class of BM-algebras. Moreover, the congruence lattice of a BM-algebra is investigated.     

Markov processes and related topics

The ＂Workshop on Markov Processes and Related Topics＂ were held nine times since 2002, organized mainly by the Probability Group in the Research Center for Stochastics, Beijing Normal University. The tenth will be held in the coming August, 2014. This workshop happens almost in each year except 2003 （cancelled because of SARS）, and 2009 （cancelled because too many related conferences were held in that year）. In 2005, the ard workshop is merged into the Joint Meeting of CSPS/IMS. The research group was mainly supported by the National Natural Science Foundation of China （NSFC） in the period of 2002-2010, and by the Ministry of Education （985 project） during 2012-2014. Among the ten workshops, five of them were or will be co-organized with some universities out of Beijing.     

Upper-embeddable graphs and related topics

In this paper, we present different results concerning the structure of upperembeddable graphs. Various characterizations of graphs whose maximum genus satisfies additivity properties will be given.     

等参函数及相关问题研究

著名的Yau 猜想断言单位球面中的紧致嵌入极小超曲面的Laplace 算子的第一特征值等于其维数. 近年来有许多几何学家致力于对Yau 猜想的研究, 但是到目前为止, 已有的结论只是一些关于第一特征值估计的不等式. 作为本文的一个主要结果, 本文证明了对于单位球面中的等参极小超曲面,Yau 猜想是正确的. 进一步地, 对于等参超曲面的焦流形(实际上是球面的极小子流形), 本文还证明了在一定维数条件下, 它的第一特征值也是其维数.
作为本文的第二个主要结果, 以著名的Schoen-Yau-Gromov-Lawson 的关于数量曲率的手术理论为出发点, 本文在一个Riemann 流形的嵌入超曲面处作手术, 构造了一个新的具有丰富几何性质的流形, 称为double 流形. 特别地, 本文在单位球面的极小等参超曲面处实行了这一手术, 发现得到的double 流形不仅有很复杂的拓扑(但其示性类有精确描述), 还存在数量曲率为正的度量, 更重要的是保持了等参叶状结构.
比Willmore 曲面更广泛的定义是Willmore 子流形, 即Willmore 泛函在球面中的的极值子流形.单位球面中的Willmore 子流形的例子在已有文献中是非常罕见的. 作为本文的另外两个主要结果, 通过深入挖掘单位球面上的OT-FKM- 型等参函数的焦流形的性质, 本文发现其极大值对应的焦流形是单位球面的一系列Willmore 子流形; 之后, 本文用几何办法统一证明了单位球面中具有4 个不同主曲率的等参超曲面的焦流形都是单位球面的Willmore 子流形. 这些新的Willmore 子流形是极小的,但一般不是Einstein 的.     

Factorizations of groups and related topics

This is a survey of some recent progress in the theory of groups with factorizations. Some of the methods can be used to obtain information about finite groups in general, nilpotent algebras and nearrings.      

The Muskat problem and related topics

In the paper, we consider the evolution of the free boundary separating two immiscible viscous fluids with different constant densities in an absolutely rigid solid body and in an elastic skeleton. The motion of the liquids is described by the Stokes equations driven by the input pressure and the force of gravity. For flows in a bounded domain, we prove the existence and uniqueness of classical solutions and emphasize the study of the properties of the moving boundary separating the two fluids.     

Singularities of Lagrangian varieties and related topics

We study the classification problem for generic projections of Lagrangian submanifolds. A classification list for symmetric Lagrangian submanifolds is obtained and the generic evolutions of symmetric caustics are illustrated. We show how the singular Lagrangian varieties appear in the invariant theory of binary forms and we introduce the basic concepts of the desingularization procedure. Applications to differential geometry, geometrical optics, and mechanics are presented.     

The topological Tverberg theorem and related topics

The classical Borsuk–Ulam theorem, established some eighty years ago, may now be seen as a consequence of the nonvanishing of the mod 2 cohomology Euler class of a certain vector bundle over a real projective space. A theorem of Kakutani from the 1940s that any continuous real-valued function on the 2–sphere must be constant on some set of three orthogonal vectors may be deduced similarly from the nontriviality of some mod 3 cohomology Euler class. The more recent topological Tverberg theorem of Bárány, Shlosman and Szücs, concerning a prime p, and the extensions of that theorem which have appeared in the last few years in the work of Blagojevi?, Karasev, Matschke, Ziegler and others, may be proved by showing that some mod p Euler class is nonzero. This paper presents a survey of these, and related, results from the viewpoint of topological fibrewise fixed–point theory.     

Multivariate value at risk and related topics

Multivariate Value at Risk, or MVaR, is defined as the quantile set of a multivariate probability distribution. It has already been introduced and used in the literature under the name of p-Level Efficient Points, or pLEP’s, or briefly p-efficient points. Some of the topics connected with it are surveyed: discrete convexity, algorithmic generation, relation to logconcavity. A related notion: Multivariate Conditional Value at Risk, or MCVaR, is also introduced and some of its properties are explored. Finally, optimization problems, based on these notions, are presented and discussed, from the point of view of convexity and algorithmic solution.     

On b-locally closed sets and related topics

The aim of this paper is to introduce the class of b-locally closed sets and two different notions of generalizations of continuous maps namely b-LC-continuous and b-LC-irresoluteness in a topological space. We discuss some of their properties. Several examples are provided to illustrate the behaviour of these new types of sets and functions.     

On the theorem of Napoleon and related topics

The famous theorem of Napoleon is one of the most interesting assertions from elementary geometry of planar figures. We shall give a survey on various aspects, applications and generalizations of this theorem. Received: 22.2.1995 / Accepted: 21.8.1995     

Short review of Yasuda-Shimada theorem and related topics

In this paper, we will present the past and the present situation of a very controversial topic in Finsler geometry: the necessary and sufficient conditions for a Randers space to be of constant curvature. The research on this subject started more than a quarter of century ago, but its final act was played only two years ago. Moreover, the history of Yasuda-Shimada theorem and few words on the life of Professor Yasuda are added. Finally miscellaneous topics studied by the Japanese group of Finsler geometers are reviewed.     

Formal completions and idempotent completions of triangulated categories of singularities

The main goal of this paper is to prove that the idempotent completions of triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic. We also discuss Thomason's theorem on dense subcategories and a relation to the negative K-theory.