群与图的对称性

首先对平面图形的对称进行分析和利用其对称群进行量化,进而将此推广到考察一般图的广义"对称性"与图自同构群的关系,最后刻画了无平方因子阶局部本原弧传递图的自同构群结构.     

超立方体Q_n的传递剖分

一个图的传递剖分是它的边集的一个划分,且满足图的一个自同构群在其划分后的各个部分组成的集合上作用是传递的.决定了超立方体Q_n的所有G-传递剖分,其中G为Q_n的全自同构群.     

容许二维线性群作用点本原的二弧传递图

一个图称为本原的如果它的自同构群作用在点集上是本原的.在这篇文章里,我们不但完全分类了容许一类二维线性群作用点本原的2-弧传递图,而且决定了他们的自同构群,并在同构意义下决定了它们的个数.     

非交换单群上三度点传递双凯莱图

如果一个图Γ含有一个自同构群G使得它在顶点集V(Γ)上作用半正则且恰好有两个轨道,则称图r是群G上的双凯莱图.进一步的,如果G在全自同构群Aut(Γ)中正规,我们就称这个双凯莱图是群G上的正规双凯莱图.本文中,我们证明了绝大多数非交换单群G上的三度点传递双凯莱图都是该群上的正规双凯莱图.     

第二小阶数的双本原半对称图

如果一个正则图是边传递但不是点传递的,那么我们称它是半对称的.每一个半对称图X必定是两部分点数相等的二部图,并且它的自同构群Aut(X)在每一部分上是传递的.如果一个半对称图的自同构群在每一部分上作用是本原的,那么我们称它是双本原的.本文决定了第二小阶数的双本原半对称图.     

Bubble-sort图和Modified Bubble-sort图的自同构群

Bubble-Sort图和Modified Bubble-Sort图是两类特殊的Cayley图,由于其在网络构建中的应用而受到广泛关注.本文完全确定了这两类图的自同构群.     

开关图的几个代数性质

首先对开关图的自同构群进行了研究,随即讨论了它的点传递性,并得到Calyley图的开关图依然是Cayley图.     

关于特殊的实单Lie代数D4的内共轭分类

在[2]中,我们讨论了实单Lie代数的内共轭分类问题,但对于稍为困难的特殊实单Lie代数D作为例外,没有讨论.在[3]中,我们提到了可以利用定理2[3]直接证明内共轭的分类定理,但因为篇幅关系,没有给予详细的证明.在本文中,我们将讨论D_4的内共轭分类问题,并详细证明关于Satake图解的内共轭分类定理. 设of是实单Lie代数,g~c是f的复化,Autg~c,Intg~c,分别是g~c的自同构群和内自同构群;Aut(g),Int(g)Int(g)分别是g~c的自同构群拟内自同构群和内自同构群,其他符号参看[1].     

Petersen图的点传递正则覆盖图

运用基图自同构能被提升的线性准则 ,对满足 :1覆叠变换群 K =Znp,2覆盖图的保簇变换群是点传递的 Petersen图的连通正则覆盖图进行了完全分类 .这种图共有 1 2种类型 .     

3p阶亚循环群的连通4度Cayley图

研究3p阶(p是大于3的素数)亚循环群的连通4度Cayley图.主要决定了其全自同构群的结构,并由此得到这类图的CI性、正规性和弧传递性.用到单群分类定理.     

Estimation of the measure of the image of the ball

Under study is the class of ring Q-homeomorphisms with respect to the p-module. We establish a criterion for a function to belong to the class and solve a problem that stems from M. A. Lavrentiev [1] on the estimation of the measure of the image of the ball under these mappings. We also address the asymptotic behavior of these mappings at a point.     

On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.     

Refinement of the asymptotics of the power of the quantile test

One investigates the asymptotic properties of the quantile test, similar to the properties of the Pearson's chi-square test of fit.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 5–15, 1986.The author is grateful to D. M. Chibisov for useful remarks.     

On the approximation of positive operators and the behaviour of the spectra of the approximants

LetT be a positive linear operator on the Banach latticeE and let (S _n ) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS _n andT the peripheral spectra (S _n ) ofS _n converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.