Simplices of Maximal Volume or Minimal Total Edge Length in Hyperbolic Space

In this paper, we are mainly concerned with n-dimensional simplicesin hyperbolic space Hⁿ. We will also consider simplices withideal vertices, and we suggest that the reader keeps the Poincaréunit ball model of hyperbolic space in mind, in which the sphereat infinity Hⁿ() corresponds to the bounding sphere of radius1. It is known that all hyperbolic simplices (even the idealones) have finite volume. However, explicit calculation of theirvolume is generally a very difficult problem (see, for example,[1] or [16]). Our first theorem states that, amongst all simplicesin a closed geodesic ball, the simplex of maximal volume isregular. We call a simplex regular if every permutation of itsvertices can be realized by an isometry of Hⁿ. A correspondingresult for simplices in the sphere has been proved by Böröczky[4].     

Edge Matrix of Hyperbolic Simplices

In this paper, by using the dual problem which was solved by Feng Luo (Geom. Dedicata 64 (1997), 277–282) and a new method, we give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a hyperbolic n-simplex. By using determinants, we also give necessary and sufficient conditions for given (n(n+1)) /2 positive real numbers to be the edge lengths of a spherical n-simplex.Mathematics Subject Classifications (2000). 51M04, 51M05, 51M20, 51M25, 52A38, 52A37, 52B10.     

On the Volumes of Hyperbolic Simplices

Mathematical Notes - We present an explicit formula for calculating the volume of an arbitrary hyperbolic 4-simplex in terms of the coordinates of its vertices; by this formula, the volume can be...     

Explicit Formula for the Hyperbolic Scattering Determinant

We prove the explicit formula for the hyperbolic scattering determinant in the case of a general subgroup F of PSL （2, R）. The class of test functions involved （not necessarily odd nor continuous） is much broader than that previously known. As an application of the technique, a new representation of the Millson-Shintani zeta function is obtained.     

On the Volume Formula for Hyperbolic Tetrahedra

We present a volume formula for arbitrary hyperbolic tetrahedra and give applications for Coxeter tetrahedra. Received November 6, 1997, and in revised form April 29, 1998.     

The Heat Kernel on Hyperbolic Space

The purpose of this note is to provide a new proof for the explicitformulas of the heat kernel on hyperbolic space. By definition,the hyperbolic space Hⁿ is a (unique) simply connected completen-dimensional Riemannian manifold with a constant negative sectionalcurvature –1. 1991 Mathematics Subject Classification58G32, 58G11.     

广义Fibonacci数的一些求和公式

This note provides the some sum formulas for generalized Fibonacci numbers. The results are proved using clever rearrangements, rather than using induction.     

The Horospherical Geometry of Submanifolds in Hyperbolic Space

Some geometrical properties associated to the contact of submanifoldswith hyperhorospheres in hyperbolic n-space are studied as anapplication of the theory of Legendrian singularities.     

The Homogeneous Geometries of Real Hyperbolic Space

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show that the moduli space of homogeneous structures on real hyperbolic space has two connected components.     

广义中心-细鞍点的若干公式

In this paper, a problem of center-weak focus of a general system of degree n is transformed into a problem of generalized center-weak saddle. It offers: (1) the calculating formula of the saddle values of the former 4 orders in the general system of degree three; (2) some calculating formulae of the saddle values in the general homogeneous system of degree n. The problem of center-focus(saddle) plays an important role in the qualitative theory of ordinary differential equations. However, since the calculations are difficult, the results have not been satisfactory. On the basis of the author's work, Northeast. Math. J., 13(4)(1997), 487-494, the paper offers some calculating formulae of the saddle values.     

A Generalized Face Number for Regular Hyperbolic Honeycombs

A generalized face number, typically taking negative rational values for a regular hyperbolic honeycomb, is constructed in three different ways – via differential geometry, combinatorics, and divergent series.     

Generalized Kantorovich Operators on Bauer Simplices and Their Limit Semigroups

In this paper, we prove an asymptotic formula for generalized Kantorovich operators associated with the canonical Markov projection on a given Bauer simplex K. That formula involves an operator acting on the subalgebra of all products of a?ne functions on K. Moreover, we prove that such an operator is closable and its closure is the generator of a Markov semigroup which, in turn, may be represented in terms of iterates of the above-mentioned generalized Kantorovich operators.     

Cauchy-Like Integral Formula for Functions of a Hyperbolic Variable

It has been recently shown that, working in a plane with the geometry associated with hyperbolic numbers, a complete “Euclidean” formalization of geometry and trigonometry of Minkowski space-time has been obtained.     

关于双曲空间中n维Pedoe不等式及应用

本文应用距离几何的理论与方法,研究了双曲空间中n维单形的几何不等式问题,建立了双曲空间中涉及两个单形棱长的n维Pedoe不等式和彭-常不等式,由此获得双曲空间n维单形的一些新的不等式.     

Cotranchal Bisectors in Complex Hyperbolic Space

Two bisectors in complex hyperbolic space H ⁿ are called cotranchal if they possess a common slice. We define a natural thing called the prespinal angle between two such bisectors and use it to study the intersection of two such bisectors.     

双曲空间上超布朗运动的范围

对双曲空间上的超布朗运动进行了研究,证明了该过程的范围属于某集合的概率可以表示为一个奇异边值问题的解．在得到了这个解的一个极限结果以后,给出了上述概率的一个渐近行为．对于维数d≥2,还证明了从黎曼测度出发的超过程在任何内点非空的有界Borel子集上的占位时是无穷的结论．     

The Krein Formula for Generalized Resolvents in Degenerated Inner Product Spaces

 Let S be a symmetric operator with defect index (1,1) in a Pontryagin space ℋ. The Krein formula establishes a bijective correspondence between the generalized resolvents of S and the set of Nevanlinna functions as parameters. We give an analogue of the Krein formula in the case that ℋ is a degenerated inner product space. The set of parameters is determined by a kernel condition. These results are applied to some classical interpolation problems with singular data. Received 3 February 1997; in revised form 9 June 1997     

Supercritical Elliptic Equation in Hyperbolic Space

In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.     

Extremals for Sobolev and Moser Inequalities in Hyperbolic Space

We review some recent results concernig existence/ non existence/ uniqueness of extremals for Sobolev inequalities in Hyperbolic spaces. We also discuss exponential integrability in the hyperbolic plane and related topics.     

3维双曲空间中曲面的双曲Gauss映照和法Gauss映照

本文导出了3维双曲空间中曲面的双曲Gauss映照和法Gauss映照的关系,发现了一般的曲面由双曲Gauss映照和平均曲率函数唯一确定,并证明了双曲Gauss映照所满足的二阶线性椭圆方程,给出了两种形式的关于双曲Gauss映照的三阶非线性偏微分方程(组)的一个解.