自由群的直积的检验元素的注记

本文研究了自由群的直积的检验元素,通过对直积的自同态的分解,得到了直积中的元素为检验元素的充分必要条件,改进了O'neill和Turner的结果.此外,构造了两类具体的检验元素.     

乘方的指数与其幂的位数的关系

为了探究乘方的指数与其幂的位数的关系,定义了几个有关的新概念,并且证明了两个关于乘方以及进制进位的定理,由此建立起关于乘方以及进制进位的理论体系,其中包括进位理论中判定乘方的指数与其幂的位数是否存在周期规律的判别法,以及进位规律的求解法和四条相关的性质.     

Lindelf的工作与Chaplygin的工作的互补性

运用Vakonomic模型导出Lindel f方程 ,表明Lindel f的工作与Vakonomic模型相吻合 ;运用Chetaev模型导出Chaplygin方程 ,表明Chaplygin的工作与Chetaev模型相吻合·　在此基础上 ,通过改进Chaplygin方程和Lindel f方程的表示形式 ,实现了从Lindel f方程向Chaplygin方程的合理过渡和从Chaplygin方程向Lindel f方程的合理的过渡·　最后 ,给出一个典型实例·　结果表明 ,正如Vako nomic模型与Chetaev模型是互补的一样 ,Lindel f的工作与Chaplygin的工作也是互补的·     

一元二次方程根的判别式的应用

<正>一元二次方程根的判别式是初中数学的重要内容,本文以近年中考中所考查的题型为例,归纳整理如下,供同仁们参考.一、求待定字母的取值范围(1)已知方程根的情况,求待定字母的取值范围例1若关于x的方程(k-1)x2+2(k)^(1/2)x+1=0有两个不相等的实数根.求k的獉獉取值范围.析解由题意"方程有两个不相等的实数獉獉根"可知:该方程是一元二次方程,且Δ>0,即     

树的最大特征值的上界的一个注记

设T是一个树,V是T的顶点集．记dv是υ∈V的度,△是T的最大顶点度．设υ∈V且dw=1．记k=ew+1,这里ew是w的excentricity．设δj′= max{dυ:dist(υ,w)=j},j=1,2,…,k-2,我们证明和这里μ1(T)和λ1(T)分别是T的Laplacian矩阵和邻接矩阵的最大特征值．特别地,记δo′=2．     

计算Hamilton矩阵特征值的一个稳定的有效的保结构的算法

提出了一个稳定的有效的保结构的计算Hamilton矩阵特征值和特征不变子空间的算法,该算法是由SR算法改进变形而得到的。在该算法中,提出了两个策略,一个叫做消失稳策略,另一个称为预处理技术。在消失稳策略中,通过求解减比方程和回溯彻底克服了Bunser Gerstner和Mehrmann提出的SR算法的严重失稳和中断现象的发生,两种策略的实施的代价都非常低。数值算例展示了该算法比其它求解Hamilton矩阵特征问题的算法更有效和可靠。     

含有共轭三键的高聚物的掺杂机制的探讨

本文从实验事实出发,对含有共轭三键的共轭高聚物——聚丁二炔的掺杂及电导的特点进行了详细的分析;对聚丁二炔晶体掺杂的机制进行了初步的探讨;从结构相变的角度出发,提出了一个掺杂模型,并从能量的角度估算了由于掺杂引起链段结构改变所需的能量,较好地解释了聚丁二炔晶体掺杂未取得成功的原因.     

多维的一般的BBM-Burgers方程解的逐点估计

研究R~n中一般的BBM-Burgers方程解的渐进行为.运用Green函数法和Fourier分析方法得到了在非零常状态u~*附近小扰动解的逐点估计,作为一个推论,又得到了L~p(R~n)(1≤p∞)空间解的最佳的衰减估计.     

不同的公理不同的几何

一位语文老师在博客上写反思.他在教《自相矛盾》这则寓言时,有学生站起来反驳:有这么傻的人么?太不可信了吧.面对这样大胆质疑的学生,这位老师不知如何处理为好! 现在的学生比以前更敢想敢做了.     

别样的思考 同样的精彩

人们喜欢数学竞赛题是因为它能引发广阔的思考空间,能训练发散思维,引导人们去寻求各种不同的解法;但是我认为它给我的精彩不仅在于有多少种解法,而在于它能引发我从不同的角度来拓展问题,从而享受其中的乐趣.以下以第六届"祖冲之杯"初中数学邀请赛     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本文研究了离散型随机变量次序统计量的分布矩阵的对称性 ,获得了二个定理 .定理 1　服从等概率二点分布或等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵 .定理 2　取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性 .     

Chaos expansion and asymptotic behavior of the Pareto distribution

We give the chaos expansion of a random variable with Pareto distribution and we analyze, by using the Malliavin calculus, the convergence in the distribution of a sequence of random variable with Pareto distribution toward the standard exponential law.     

On the dependence structure of order statistics

Given a random sample from a continuous variable, it is observed that the copula linking any pair of order statistics is independent of the parent distribution. To compare the degree of association between two such pairs of ordered random variables, a notion of relative monotone regression dependence (or stochastic increasingness) is considered. Using this concept, it is proved that for i<j, the dependence of the jth order statistic on the ith order statistic decreases as i and j draw apart. This extends earlier results of Tukey (Ann. Math. Statist. 29 (1958) 588) and Kim and David (J. Statist. Plann. Inference 24 (1990) 363). The effect of the sample size on this type of dependence is also investigated, and an explicit expression is given for the population value of Kendall's coefficient of concordance between two arbitrary order statistics of a random sample.     

A fuzzy bootstrap test for the mean with <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">p,q</Emphasis></Subscript>-distance

In this paper, we consider the problem of testing a simple hypothesis about the mean of a fuzzy random variable. For this purpose, we take a distance between the sample mean and the mean in the null hypothesis as a test statistic. An asymptotic test about the fuzzy mean is obtained by using a central limit theorem. The asymptotical distribution is ω ₂-distribution. The ω ₂-distribution is only known for special cases, thus we have considered random LR-fuzzy numbers. In the fuzzy concept, in addition to the existence of several versions of the central limit theorem, there is another practical disadvantage: The limit law is, in most cases, difficult to handle. Therefore, the central limit theorem for fuzzy random variable does not seem to be a very useful tool to make inferences on the mean of fuzzy random variable. Thus we use the bootstrap technique. Finally, by means of a simulation study, we show that the bootstrap method is a powerful tool in the statistical hypothesis testing about the mean of fuzzy random variables.     

Testing for equality of means of a Hilbert space valued random variable

We propose a test for equality of means of a random variable valued into a real separable Hilbert space. The test statistic is based on projections of empirical means onto spaces spanned by principal directions obtained from principal component analysis of the random variable. The asymptotic distribution of this test statistic is derived under the null hypothesis and the consistency of the obtained test is proved. An application to the case of functional variables is indicated. To cite this article: J.G. Aghoukeng Jiofack, G.M. Nkiet, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Sufficient conditions for Benford’s law

We present two sufficient conditions for an absolutely continuous random variable to obey Benford’s law for the distribution of the first significant digit. These two sufficient conditions suggest that Benford’s law will not often be observed in everyday sets of numerical data. On the other hand, we recall that there are two processes by way of which a random variable can come close to following Benford’s law. The first of these is the multiplication of independent random variables and the second is the exponentiation of a random variable to a large power. Our working tool is the Poisson sum formula of Fourier analysis. Like the central limit theorem, Benford’s law has an asymptotic nature.     

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.     

Optimal time to change premiums

The claim arrival process to an insurance company is modeled by a compound Poisson process whose intensity and/or jump size distribution changes at an unobservable time with a known distribution. It is in the insurance company’s interest to detect the change time as soon as possible in order to re-evaluate a new fair value for premiums to keep its profit level the same. This is equivalent to a problem in which the intensity and the jump size change at the same time but the intensity changes to a random variable with a know distribution. This problem becomes an optimal stopping problem for a Markovian sufficient statistic. Here, a special case of this problem is solved, in which the rate of the arrivals moves up to one of two possible values, and the Markovian sufficient statistic is two-dimensional. This work was partially supported by the US Army Pantheon Project and National Science Foundation under grant DMS-0604491.     

The laws of large numbers for Pareto-type random variables under sub-linear expectation

In this paper, some laws of large numbers are established for random variables that satisfy the Pareto distribution, so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space. Based on the Pareto distribution, we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.     

Statistical inference for multivariate longitudinal data with irregular auto-correlated error process

Multivariate longitudinal data arise frequently in a variety of applications, where multiple outcomes are measured repeatedly from the same subject. In this paper, we first propose a two-stage weighted least square estimation procedure for the regression coefficients when the random error follows an irregular autoregressive(AR) process, and establish asymptotic normality properties for the resulting estimators. We then apply the smoothly clipped absolute deviation(SCAD) variable selection approach to determine the order of the AR error process. We further propose a test statistic to check whether multiple responses are correlated at the same observation time, and derive the asymptotic distribution of the proposed test statistic. Several simulated examples and real data analysis are presented to illustrate the finite-sample performance of the proposed method.