关于凸曲面的几个定义的关系

本文讨论了凸曲面的几种定义及其关系，发现有的定义是局部凸的定义，有的是整体凸的定义，有的则对于局部凸和整体凸都适合，最后给出了各种定义之间互推的证明，对于局部凸和整体凸定义之间不能推证的，则说明了原因．     

分配格与模格的另一二条件等价定义

针对分配格与模格的格等式定义问题,得知了二条件是定义分配格与模格的最少条件,并进一步证明了Sholander's basis是定义分配格的最短最少变量格等式,最后又从分配格和模格的基本定义出发给出了新的分配格的二条件和三条件等价定义等式及模格的二条件与三条件等价定义等式.     

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．     

Riemann积分的一个等价定义

本文给出了Ｒｉｅｍａｎｎ积分的一个等价定义，用该定义下的达布和的性质证明了Ｒ—可积的必要充分条件定理。并用两种方法证明了定义卫与定义２等价。     

交互作用的两类定义及其关系推导

从正交表的角度定义交互作用和从函数模型角度定义交互作用历来是学者考察交互作用的基本方法,本文引进多边矩阵理论中的分解算子,并用其定义了交互作用,推导了这一定义包含从正交表出发的定义。     

无序和的一个注记

对于任意集合上广义实数取值的非负函数,文献[1]给出了无序和的两种定义,文献[2]给出了无序和的第三种定义,文章证明了这三种定义相互等价.对于任意集合上绝对可和的复值函数,文献[1]和文献[2]分别给出了无序和的两种不同定义,文章证明了这两种定义等价.     

简单随机抽样的一个注记

对简单随机抽样两种定义的关系进行了讨论,澄清了教科书中的一些模糊不清之处;通过反例举证了两种定义在某种意义下的不等价性;给出了变概率抽样与不等概率抽样的定义,指出变概率抽样与不等概率抽样也可以是简单随机抽样;本文最后,给出了简单随机抽样的更具一般性的(广义)定义.     

新的模糊数的模糊距离

文[1]中基于现有的模糊数的距离定义及模糊距离定义的缺陷进行分析并给出了改进的模糊数的距离定义。本文在此基础上给出新的模糊数的模糊距离定义,并给出相应的定理。     

奇异多元正态随机向量的新定义及与其他定义的等价性

通过对奇异协方差阵的行列式重新定义及广义逆矩阵的概念,给出了奇异多元正态随机向量的新定义,并讨论了与其他定义之间的等价关系.     

一个全局最优化问题的填充函数

本文给出了一个非线性全局最优化问题的填充函数定义,此定义不同于以前已有的填充函数定义。根据此定义,本文提出了一簇单参数填充函数和相应的填充函数算法．对几个算例的数据测试表明,该填充函数法是可行和有效的．     

Ramsey Numbers for Partially-Ordered Sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

   Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

Order-free Recursion on the Real Numbers

We investigate operations, computable in the sense of Recursive Analysis, which can be generated recursively from the arithmetic operations and the limit operation without using any tests on the real numbers. These functions, called order-free recursive, can be shown to include a large class of computable functions. As main tools we provide an effective version of the Stone-Weierstraß Approximation Theorem, as well as recursive partitions of unity.     

Multiply Abundant Numbers

A multiply abundant number N of order r is a number for which the sum of the proper divisors, S(N), is greater than r times N. The authors explore abundant, double abundant, and triple abundant numbers as well as abundant numbers of higher orders. They also consider odd abundant numbers of orders greater than 1. They present a proof that abundant numbers of every order r exist. Readers are encouraged to use the included computer code to explore abundant numbers for themselves, look for patterns in the output, and consider further questions about these fascinating numbers.     

李善兰对Stirling数和Euler数的研究

我国清代有影响的数学家李善兰(1811—1882)在他的名著《垛积比类》(1867)中提出并研究了两类系数表(依原著顺序为第二表和第七表),获得了重要成果。它们实际就是现代组合数学中的两种重要计数函数。     

On the Betti Numbers of Chessboard Complexes

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vreica, and ivaljevi in [2]. Our formula also shows that all eigenvalues of the Laplacians of the simplicial complexes are integers, and it gives a formula (involving partitions) for the multiplicities of the eigenvalues.     

New Bounds for the Clique-Chromatic Numbers of Johnson Graphs

Doklady Mathematics - We significantly improve lower bounds on the clique-chromatic numbers for some families of Johnson graphs. A new upper bound on the clique-chromatic numbers for G(n, r, r...     

A Multivalued Strong Law of Large Numbers

We prove a strong law of large numbers for random closed sets in a separable Banach space. It improves upon and unifies the laws of large numbers with convergence in the Wijsman, Mosco and slice topologies, without requiring extra assumptions on either the properties of the space or the kind of sets that can be taken on by the random set as values.     

On the Equivalence of Integral T^k-Cohomology Chern Numbers and T^k-K-Theoretic Chern Numbers

This paper mainly deals with the question of equivalence between equivariant cohomology Chern numbers and equivariant K-theoretic Chern numbers when the transformation group is a torus.By using the equivariant Riemann-Roch relation of AtiyahHirzebruch type,it is proved that the vanishing of equivariant cohomology Chern numbers is equivalent to the vanishing of equivariant K-theoretic Chern numbers.