Einige Eigenschaften der Poincaréschen Reihen der Dimension-2

We present a new short and simple proof of the classical theorem that the Poincaré series of dimension-2 for a finitely generated Fuchsian group of the first kind diverges. Our proof is only based on the finiteness of the invariant measure of a fundamental domain of such a group and does not use any detailed information about the geometrical structure of special suitably chosen fundamental domains. The starting point of our argument is a slight generalization of a theorem stated by K. V. RAJESWARA RAO [7] saying that a certain Poincaré series for a Fuchsian group, if it does converge, is a bounded function on the unit disk.     

GL(n,K)中一类准可解子群的结构及其在Fuchsian系统中的应用(英文)

本文研究了单值群的可解性与Fuchsian方程可积性之间的关系.利用可解群及GL(n,K)的性质,获得GL(n,K)中一类准可解子群的结构定理,并将之心用于Fuchsian系统的可积性研究之中.     

Constructing an automorphic form from the orbit of a transformation

We consider the Fuchsian groups of linear-fractional transformations. We propose a new method for presenting automorphic forms as gap series over an appropriate subset of transformations of the group which is not a subgroup. Comparative analysis of the Poincaré theta-series and gap series demonstrates that the use of gap series requires less transformations and parameters that the summands of series depend on.     

关于特征和的混合均值

研究了特征和及其算术性质,并利用解析方法得到了两个关于特征和的混合均值的渐近公式,从而推广了特征和的算术性质。     

Fuchsian convex bodies: basics of Brunn–Minkowski theory

The hyperbolic space ${\mathbb{H}^d}$ can be defined as a pseudo-sphere in the (d + 1) Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that ${\mathbb{H}^d/\Gamma}$ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn–Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere. The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov–Fenchel and Brunn–Minkowski inequalities. Here the inequalities are reversed.     

拟Fuchs群的收敛指数

根据H2上的双曲距离在拟共形变换下的拟不变性,给出了K-拟共形抛物循环Fuchs群的收敛指数的估计．     

SL(n,C)上一类特殊可解子群的结构及其在Fuchs系统中的应用

给出了SL(n,C)中一类特殊的具有有限个生成元的可解子群的结构定理.由单值群的可解性与Fuchs系统的可积性之间的关系,研究对应的单值群是可解的环面上只有一个正则起点的n阶Fuchs方程的解Riemann曲面结构,进而研究其解的大范围性质.     

On the additive energy of the distance set in finite fields

We use character sums to derive new bounds on the additive energy of the set of distances (counted with multiplicities) between two subsets of a vector space over a given finite field. We also give applications to sumsets of distance sets.     

关于广义$k$次高斯和的均值

The main purpose of this paper is using residue system and character sums methods to investigate the mean value properties of general k-th Gauss sums,and two exact calculating formulas are given.