E~n中n维余弦定理和正弦定理的新证明

利用Grassmann代数的理论与方法,给出了n维欧氏空间En中n维单形第二余弦定理一种简单的证明.然后利用第二余弦定理给出了n维单形正弦定理一种新的简单证明.     

知识规律与规律的属性扰动

By employing the knowledge(R-element equivalence class)in one direction Srough sets and dual of one direction S-rough sets,the concept of knowledge law is given;the generation theorem of knowledge law,the excursion theorem of knowledge law,and the attribute disturbance discernible theorem of knowledge law are proposed.Knowledge law is a new characteristic of S-rough sets.     

Geometric Generalization of the Exponential Law

For the multivariate ℓ₁-norm symmetric distributions, which are generalizations of the n-dimensional exponential distribution with independent marginals, a geometric representation formula is given, together with some of its basic properties. This formula can especially be applied to a new developed and statistically well motivated system of sets. From that the distribution of a t-statistic adapted for the two-parameter exponential distribution and its generalizations is determined. Asymptotic normality of this adapted t-statistic is shown under certain conditions.     

重期望公式的误用

指出[2]中一道习题的错解,并分析出现错误的原因.     

Deviations from the Circular Law

Consider Ginibres ensemble of N×N non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance As N the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane. In this note we describe fluctuations about this Circular Law. First we obtain finite N formulas for the covariance of certain linear statistics of the eigenvalues. Asymptotics of these objects coupled with a theorem of Costin and Lebowitz then result in central limit theorems for a variety of these statistics.This work was supported in part by the grant NSF-9983320.Revised version: 13 February 2004     

Law of the iterated logarithm for the periodogram

We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density function associated with the stationary process. Results for a specified frequency are also given. Our results also lead to the law of the iterated logarithm for the real and imaginary parts of the discrete Fourier transform. The proofs rely on martingale approximations combined with results from harmonic analysis and techniques from ergodic theory. Several applications to linear processes and their functionals, iterated random functions, mixing structures and Markov chains are also presented.     

种群死亡定律的研究

本文详细了研究了种群纯死定律的概率结构，给出了求解转移概率显式的三种方法及一个定理，最后拟合了一个实例。     

On Functional Versions of the Arc-Sine Law

Let X ₁,X ₂,… be a sequence of random variables. Let S _k =X ₁+???+X _k and assume that S _k /b _k converges in distribution for some numerical sequence (b _k ). We study the weak convergence of the random processes {Λ _n (z), z∈?}, where

$\Lambda_{n}(z)=\frac{1}{n}\sum_{k=1}^{n}I\left\{\frac{S_{k}}{b_{k}}\leq z\right\}.$

We consider the same problem when the normalized partial sums S _k /b _k are replaced by other functionals of the sequence (X _n ). In particular, we investigate the case of sample extremes in detail.

     

结合律在数项级数中的巧用

讨论结合律在无穷级数中的运用,并对2015年浙江省高等数学竞赛中一道竞赛题进行了分析和探讨.     

Law of the iterated logarithm for U-statistics

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 1, pp. 97–100, January, 1989.     

Nonexpansivity of the Newton's Cradle Impact Law

The 3-ball Newton's cradle is used as a stepping stone to divulge the structure of impact laws. A continuous cone-wise linear impact law which maps the pre-impact contact velocities to the post-impact contact velocities is proposed for the 3-ball Newton's cradle. The proposed impact law is kinematically, kinetically, and energetically consistent. It reproduces all the classical experimental outcomes. Moreover, the impact law has the mathematical property of being non-expansive. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Compact Law of the Iterated Logarithm for Random Vectors in the Generalized Domain of Attraction of the Multivariate Gaussian Law

For a sequence of independent identically distributed random vectors, we prove that the limiting cluster set of the appropriately operator normed partial sums is, with probability one, the closed unit euclidean ball. The result is proved under the hypotheses that the law of the random vectors belongs to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The two conditions together are still weaker than finite second normed moment and are necessary and sufficient.     

微分中值定理的推广

本文建立了微分中值定理在 n维函数空间的一种推广形式     

Decision Trees and the Second Law

Every system, left unto itself, tends towards disorder. Disorder frequently appears in the form of queues-machines, materials or customers waiting for service or waiting to be served. This article suggests a method of measuring disorder, and shows how this may be used in the analysis of complex decision type of systems.     

Law of the iterated logarithm for random graphs

A milestone in probability theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables with mean 0 and variance 1 In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph H. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random k‐uniform hypergraphs, we obtain the Central Limit Theorem and LIL for the number of Hamilton cycles.     

由ALNQD序列生成的移动平均过程的大数定律及重对数律的精确渐近性

In this paper,we discuss the precise asymptotics of moving-average process Xt =∞∑j=0 ajεt-j under some suitable conditions,where {εt,t ∈ Z} is a sequence of stationary ALNQD randomvariables with mean zeros and finite variances.     

粘聚律的分离功分析以及一种一致关联粘聚律

在粘聚元或者粘聚域模型中,能够预测混合失效模式（mixed mode）的粘聚律是通过关联界面的法向分离和切向分离来建立的．考虑完全剪切失效机制的Xu-Needleman粘聚律是目前文献中应用最多的关联粘聚律之一．基于该文提出的粘聚律一致关联准则,采用界面分离功分析法,证明了考虑剪切失效机制的Xu-Needleman粘聚律在混合失效模式下,属于非一致关联粘聚律．理论分析证明,考虑剪切失效的Xu-Needleman粘聚律仅在界面的法向分离功与切向分离功相等的情况下能够正确刻画混合失效模式．基于粘聚律一致关联准则,在考虑剪切失效的Xu-Needleman粘聚律的基础上,进行修正提出一种一致关联粘聚律（CCC）．分析证明,这种修正的一致关联粘聚律克服了原有Xu-Needleman粘聚律非一致关联的弊端,能够更好刻画混合失效模式下的破坏行为．