The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube

E[V₄ (cube)]=[3977/216000] - [(p²)/2160]=0.01384277574?E[V_4 {\rm (cube)}]={3977\over 216000} - {\pi^2 \over 2160}=0.01384277574\ldots     

The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube

We calculate E[V ₄(C)], the expected volume of a tetrahedron whose vertices are chosen randomly (i.e. independently and uniformly) in the interior of C, a cube of unit volume. We find

On Random Sections of the Cube

Let f(j,k,n) denote the expected number of j -faces of a random k -section of the n -cube. A formula for f(0,k,n) is presented, and, for j\geq 1 , a lower bound for f(j,k,n) is derived, which implies a precise asymptotic formula for f(n-m,n-l,n) when 1≤ l<m are fixed integers and n→∈fty . Received August 1, 1998, and in revised form December 15, 1998.     

一类由中心对称原子生成的Hardy型空间

对Ｈａｒｄｙ空间Ｈ￣ｑ（Ｒ￣ｎ），，作者引进了一类新Ｈａｒｄｙ型子空间，１＜ｐ＜∞，并且给出了它在Ｍ，Ｆｒａｚｉｅｒ和Ｂ．ＪａＷｅｒｔｎ意义下的变换特征，应用这个特征，我们得到了空间的一些插值定理。特别，对作者还建立了它的Ｌｕｓｉｎ－面积特征     

The Evolution of Random Subgraphs of the Cube

be a random Qⁿ”-process, that is let Q₀ be the empty spanning subgraph of the cube Qⁿ and, for 1 ? t ? M = nN/2 = n2^n?1, let the graph Q_t be obtained from Q_t?1 by the random addition of an edge of Q_n not present in Q_t?1. When t is about N/2, a typical Q_t undergoes a certain “phase transition'': the component structure changes in a sudden and surprising way. Let t = (1 + ?) N/2 where ? is independent of n. Then all the components of a typical Q_t have o(N) vertices if ? < 0, while if ? > 0 then, as proved by Ajtai, Komlós, and Szemerédi, a typical Q_t has a “giant” component with at least α(?)N vertices, where α(?) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if η > 0 is fixed and t ? (1 ? n^?η) N/2, then all components of a typical Q_t have at most n^β(η) vertices, where β(η) > 0. More importantly, if 60(log n)³/n ? ? = ?n = o(1), then the largest component of a typical Q_t has about 2?N vertices, while the second largest component has order O(n?^?2). Loosely put, the evolution of a typical Qⁿ process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.     

古代筮法产生的随机数研究

随机数在现代科学中有着重要应用,随机数隶属可测函数范畴,因此随机数历.史是数学史的重要组成部分.追朔随机数的历史,可以在《易经》中的《系辞传》中发现筮法.筮法是古代中国产生随机数的方法.筮法过程可以看作为带外来输入的随机过程,也就是说筮法是数学史中的随机过程的古代中国实例.在严格的概率论模型的基础上,给出了筮法所产生的随机数概率,并给出了其信息熵.结果修改了向传三的结果.     

On Codes Identifying Sets of Vertices in Hamming Spaces

A code is called (t, 2)-identifying if for all the words x, y(x y) and the sets (B _t (x) B _t (y)) C and are nonempty and different. Constructions of such codes and a lower bound on the cardinality of these codes are given. The lower bound is shown to be sharp in some cases. We also discuss a more general notion of -identifying codes and introduce weakly identifying codes.     

Limiting Case for Interpolation Spaces Generated by Holomorphic Semigroups

tA )_t≥0 on X. We denote by X_A(θ,p), for 0 < θ < 1, the interpolation spaces of X and D(A) introduced by Berens and Butzer in [1]. These spaces play an important role in the approximation theory ( [2] ) as well as in the study of the abstract parabolic equation u′ (t) = Au(t) + f(t) ( [3] ). It has been proved in [6] that these spaces are meaningful (and still enjoy relevant properties) also in the case θ = 0. In this paper we continue the study of [6] and prove new interesting properties of these spaces.     

Locally Convex Spaces of the Sebastian-e-Silva Type Generated by the Operator of Differentiation

It is established in the paper that the spaces of entire functions of exponential type generated by the operator of differentiation defined in the space of summable functions on the axis belong to the well-known Sebastian-e-Silva class of locally convex spaces. Some properties of the spaces relating to the duality and the open map theorem, as well as properties of convolution, are described.     

关于随机赋范空间与随机内积空间的某些基本理论(英文)

首先提出随机度量空间定义的另一个提法,这提法不仅等价于原始的定义而且也使随机度量空间自动归入广义度量空间的框架,也考虑了关于拓扑结构的某些新的问题;循着同样的思路,对随机赋范空间的定义也作了新的处理并同时简化了随机赋范模的定义．其次本文也证明了一个Ｅ－范空间的商空间等距同构于一个典型的Ｅ－范空间;进一步,在概率赋范空间的框架下证明了一个概率赋伪范空间是伪内积生成空间的充要条件是它等距同构于一个Ｅ－内积空间,这回答了Ｃ．Ａｌｓｉｎａ与Ｂ．Ｓｃｈｗｅｉｚｅｒ等人新近提出的公开问题．最后,本文转向了它的中心部分──关于随机内积空间的研究,对随机内积空间中的特有且复杂的正交性作较系统的讨论,论证了只有几乎处处正交性才是唯一合理的正交性概念,在此基础上本文尤其将Ｇ．Ｓｔａｍｐａｃｃｈｉａ的在众多学科中都有多种用途的一般投影定理（或称变分不等式解存在性定理）以适当形式推广到完备实随机内积模上．     

The Degree of Weakly Discretely Generated Spaces

A space X is discretely generated at a point \({x \in X}\) if for any \({A \subseteqq X}\) with \({x \in \textsf{cl}(A)}\) , there exists a discrete set \({D \subseteqq A}\) such that \({x \in \textsf{cl}(D)}\) . The space X is discretely generated if it is discretely generated at every point \({x \in X}\) . We say that X is weakly discretely generated if for any non-closed set \({A \subseteqq X}\) , there exists a discrete set \({D \subseteqq A}\) such that \({\textsf{cl}(D) \setminus A \neq \emptyset}\) . New results about these properties in the classes of pseudocompact and ?ech-complete spaces are obtained and a theorem of Ivanov and Osipov concerning the ordinal function idc is generalized to the class of ?ech-complete spaces.     

Moderate Deviations for Linear Processes Generated by Martingale-Like Random Variables

In this paper we study the moderate deviation principle for linear statistics of the type S _n =∑ _i∈Z c _n,i ξ _i , where c _n,i are real numbers, and the variables ξ _i are in turn stationary martingale differences or dependent sequences satisfying projective criteria. As an application, we obtain the moderate deviation principle and its functional form for sums of a class of linear processes with dependent innovations that might exhibit long memory. A new notion of equivalence of the coefficients allows us to study the difficult case where the variance of S _n behaves slower than n. The main tools are: a new type of martingale approximations and moment and maximal inequalities that are important in themselves.     

平削算子生成的B值鞅空间及其原子分解

对于一系列由平削算子生成的Banach空间值鞅空间建立了原子分解定理 ,并以此为工具讨论了它们之间的相互嵌入关系 ,其结果与Banach空间的凸性和光滑性有密切联系 .     

Wavelets with Complementary Boundary Conditions — Function Spaces on the Cube

This paper is concerned with the construction of biorthogonal wavelet bases on n-dimensional cubes which provide Riesz bases for Sobolev and Besov spaces with homogeneous Dirichlet boundary conditions on any desired selection of boundary facets. The essential point is that the primal and dual wavelets satisfy corresponding complementary boundary conditions. These results form the key ingredients of the construction of wavelet bases on manifolds [DS2] that have been developed for the treatment of operator equations of positive and negative order.     

The Number of Seymour Vertices in Random Tournaments and Digraphs

Seymour’s distance two conjecture states that in any digraph there exists a vertex (a “Seymour vertex”) that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements along lines suggested by Seymour’s conjecture, proving that almost surely there are a “large” number of Seymour vertices in random tournaments and “even more” in general random digraphs.     

The Fundamental Function of Spaces Generated by Interpolation Methods Associated to Polygons

We consider the K- and J-spaces generated from an N-tuple of rearrangement-invariant function spaces by using the interpolation methods associated to polygons. We compute their fundamental functions and we characterize their associate spaces. Furthermore, an application is given to interpolation of N-tuples of Marcinkiewicz spaces.