Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”-and “ ”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution, the errors are of order O(h ^p + τ ^q ). The estimates hold true even in the hyperbolic case when ɛ = 0.     

Standard position for objects in hyperbolic space

It is shown that, by applying hyperbolic isometries, one may arrangen points in hyperbolic (n–1)-space in a certain standard position. Similar results are developed for complex hyperbolic space, as well as hyperbolic spaces defined over nearly arbitrary fields. The algebraic basis of the paper is the determination of the structure of a double coset space which occurs in representation theory.Partially supported by NSA grant # MDA-904-96-0018.Partially supported by NSA grant # MDA904-95-1-1089.     

A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space

The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in ℝⁿ with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if k>3. The problem is still open for k = 2, 3. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space ℍⁿ.     

Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent

It is proved that commensurable hyperbolic groups are bi-Lipschitz equivalent. Therefore, subgroups of finite index in an arbitrary hyperbolic group also share this property. In addition, it is shown that any two separated nets Γ¹ and Γ² in the hyperbolic space Hⁿ of dimension n≥2 are bi-Lipschitz-equivalent. These results answer the questions posed in [1]. Supported by RFFR grant No. 96-01-01781. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 259–272, May–June, 1997.     

Pointwise fourier inversion: A wave equation approach

Functions of the Laplace operator F(− Δ) can be synthesized from the solution operator to the wave equation. When F is the characteristic function of [0, R ² ], this gives a representation for radial Fourier inversion. A number of topics related to pointwise convergence or divergence of such inversion, as R → ∞, are studied in this article. In some cases, including analysis on Euclidean space, sphers, hyperbolic space, and certain other symmetric spaces, exact formulas for fundamental solutions to wave equations are available. In other cases, parametrices and other tools of microlocal analysis are effective.     

Integer knapsack problems with set-up weights

The Integer Knapsack Problem with Set-up Weights (IKPSW) is a generalization of the classical Integer Knapsack Problem (IKP), where each item type has a set-up weight that is added to the knapsack if any copies of the item type are in the knapsack solution. The k-item IKPSW (kIKPSW) is also considered, where a cardinality constraint imposes a value k on the total number of items in the knapsack solution. IKPSW and kIKPSW have applications in the area of aviation security. This paper provides dynamic programming algorithms for each problem that produce optimal solutions in pseudo-polynomial time. Moreover, four heuristics are presented that provide approximate solutions to IKPSW and kIKPSW. For each problem, a Greedy heuristic is presented that produces solutions within a factor of 1/2 of the optimal solution value, and a fully polynomial time approximation scheme (FPTAS) is presented that produces solutions within a factor of ε of the optimal solution value. The FPTAS for IKPSW has time and space requirements of O(nlog n+n/ε ²+1/ε ³) and O(1/ε ²), respectively, and the FPTAS for kIKPSW has time and space requirements of O(kn ²/ε ³) and O(k/ε ²), respectively.     

The Moduli Space of Points in the Boundary of Complex Hyperbolic Space

We consider the space M(n,m)\mathcal{M}(n,m) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, H_\mathbbCⁿ\mathbf{H}_{\mathbb{C}}^{n}, up to its holomorphic isometry group PU(n,1). An important problem in complex hyperbolic geometry is to construct and describe the moduli space for M(n,m)\mathcal{M}(n,m). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxodromic elements. In the present paper, we give the complete solution to this problem.     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Nonsmooth eigenfunctions in problems of mathematical physics

We study whether V.A. Il’in’s method for proving the uniqueness of the solution of a mixed problem for a hyperbolic equation applies to a problem with transmission conditions in the interior of the interval. We show that the system of eigenfunctions corresponding to this problem is complete in the space L ²(0, l) and is a Riesz basis in this space.     

H m -modified wave operator for nonlinear Hartree equation in space of dimensionn≥2

We consider the scattering problem for the Hartree equation with potential |x|⁻¹ in a space of dimensionn≥2. We prove the existence ofH ^m -modified wave operator for Hartree equation on a dense set of a neighborhood of zero inH ^m (ℝ ⁿ ), meanwhile, we obtain also the global existence for the Cauchy problem of Hartree equation in a space of dimensionn≥2. This project is supported by the National Natural Science Foundation of China, 19601005     

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝⁿ be a bounded convex domain with C ² boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H _k ^p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥_p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H _k ^p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.     

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

On a class of strongly hyperbolic systems

In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold ofT ^* R ⁿ⁺¹-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.     

Isometric immersions of domains of Lobachevski space in Euclidean spaces

Immersions of domains of the n-dimensional Lobachevski space Lⁿ in the (2n−1)-dimensional Euclidean space E²ⁿ⁻¹ are studied. It is shown that the problem of isometric immersion of domains of Lⁿ in E²ⁿ⁻¹ is reduced to the study of a certain system of nonlinear partial differential equations, yielding the sine-Gordon equation as one of the special cases. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 11–16.     

The regularity of local solutions for a generalized Camassa-Holm type equation

The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some prior estimates under certain assumptions on the initial value of the equation. The local existence of its solution in Sobolev space Hs （R） with 1 〈 s ≤ 3/2 is derived.     

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

In this paper we study the topological and metric rigidity of hypersurfaces in ℍ ⁿ⁺¹, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface in ℍ ⁿ⁺¹ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.     

Multipoint problem for hyperbolic equations with variable coefficients

By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.     

Existence and Asymptotic Behavior of Radially Symmetric Solutions to a Semilinear Hyperbolic System in Odd Space Dimensions

This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t→-∞in the energy norm, and to show it has a free profile as t→ ∞. Our approach is based on the work of [11]. Namely we use a weighted L∞norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper.