Upper Semicontinuous Valuations on the Space of Convex Discs

We show that every rigid motion invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, the length, the area, and a suitable curvature integral of the convex disc.     

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.     

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．     

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.     

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.     

Optimally Dense Packings of Hyperbolic Space

In previous work, a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an ergodic theorem of Nevo to provide an appropriate definition of those packings to be considered optimally dense. Examples are given to illustrate various aspects of the density problem, in particular the shift in emphasis from the analysis of individual packings to spaces of packings.     

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 $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.     

Groups Quasi-isometric to Complex Hyperbolic Space

We show that any finitely generated group quasi-isometric to complex hyperbolic space is a finite extension of a properly discontinuous, cocompact subgroup of the isometry group.

     

Compact Homogeneous Hypersurfaces in Hyperbolic Space

In this paper we prove that an n-dimensional compact homogeneous Riemannian manifold isometrically immersed in the hyperbolic space Hⁿ⁺¹ is isometric to a sphere.AMS Subject Classification (1990): 53A07, 53C42, 57S15     

Hyperbolic Harmonic Function in Minkowski Space

The paper discusses the properties of four-dimensional Minkowski space by using Clifford algebra, then gives the concept of hyperbolic harmonic function by constructing a system (P₄) in the four-dimensional Minkowski space, and obtains several properties and a sufficient and necessary condition for the solvability of the system (P₄) .     

Cohomogeneity One Hypersurfaces of the Hyperbolic Space

In this paper we extend to the hyperbolic space results firstly obtainedin Podestà and Spiro (Ann. Global Anal. Geom. 13(2) (1995), 169–184) for compact cohomogeneity, Riemannian manifoldsimmersed as hypersurfaces of the Euclidean space and in Seixas(Ph.D. Thesis, 1996), wherethe complete case was studied. Both works give sufficient conditionsfor the hypersurface to be of revolution. We study cohomogeneity, complete hypersurfaces of the hyperbolic space and prove a similar resultof Seixas (Ph.D. Thesis, 1996), obtaining also a class of nonrotational examples.     

Valuations on lattice polytopes

Let L be a lattice (that is, a Z-module of finite rank), and let L=P(L) denote the family of convex polytopes with vertices in L; here, convexity refers to the underlying rational vector space V=Q⊗L. In this paper it is shown that any valuation on L satisfies the inclusion-exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group L=P(L) for valuations on L can be identified with its characteristic function in V. In fact, the same arguments are shown to apply to P(M), when M is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra.     

HFI-代数的拟Fuzzy-赋值

给出 FI-代数的生成滤子的几个性质 ,然后在 FI-代数上定义了一种拟 Fuzzy-赋值 ,在研究它的一些性质的同时并利用它对 FI-代数的生成滤子予以刻画。     

A Singular Trudinger-Moser Inequality in Hyperbolic Space

In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space $$H^n: sup_{u∈W^{1,n}(H^n),∫_{H^n}|∇_H^nu|^ndμ ≤ 1}∫_{H^n}\frac{e^{α|u|\frac{n}{n-1}}-Σ^{n-2}_{k=0}\frac{α^k|u|^\frac{nk}{n-1}}{k!}}{ρ^β}dμ‹∞ ⇔ \frac{α}{α_n}+\frac{β}{n} ≤ 1,$$ where α>0,α ∈ [0,n), ρ and dμ are the distance function and volume element of $H^n$ respectively.     

Quasi—Einstein Hypersurfaces in a Hyperbolic Space

§1.　IntroductionLetRijbethecomponentsofRiccitensorofann-dimensionalRiemannianmanifoldM.IfRij=Agij Bξiξj,　(i,j=1,2,…,n)(1.1)whereξisanunitvectorfield,thenMiscalledaquasi-EinsteinmanifoldanddenotedbyQE(ξ).Ifξisanisotropicvectorfield,thenMiscalledageneralizedquasi-Einsteinmanifold.Intheequality(1.1),AandBarescalarfunctions.WeknowQE(ξ)manifoldisEinsteinwhenB≡0.Especially,if〈ξ,ξ〉=e=±1,thenQE(ξ)iscalledanormalquasi-Einsteinmani-fold.Itiseasytoknowfrom[1]and[2]:Rij=R-Tn-1…     

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.     

Some Physics Questions in Hyperbolic Complex Space

In hyperbolic complex space, the Clifford algebra is isomorphic to that of a corresponding Minkowski geometry. We define the hyperbolic imaginary unit j (j² = 1, j ≠   ±  1, j*  =   − j) to generate a class of Clifford algebras. We can introduce a class of non-Euclidean spaces and discuss the general form of 4-dimensional Lorentz transformation, and related special relativistic physics.     

Valuations on arithmetic surfaces

In this paper, we give the definition of the height of a valuation and the definition of the big field Cp,G, where p is a prime and GR is an additive subgroup containing 1. We conclude that Cp,G is a field and Cp,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ≤n∈ Z, let Vm,n be an R-vector space of dimension n-m + 1, whose coordinates are indexed from m to n. We generalize the definition of Cp,G, where p i...     

Valuations on a Module

The purpose of this article is to introduce the notion of valuations on a module over a commutative ring. Some basic results on valuations on a module are established. Moreover, the notion of valuation triples of a module is also introduced. In this article, it is shown that the so-called valuation triples correspond to valuations on a module.