Some geometric inequalities in non-euclidean space

In this paper, we obtained three geometric inequalities for theu-dimensional polar sines and the dihedral angles of ann-dimensional simples. Besides, we obtained an inequality for the dihedral angles of ann-dimensional simplex in then-dimensional hyperbolic spaceH _n.Project Supported by National Natural Foundation P. R. China     

The maximum angle condition is not necessary for convergence of the finite element method

We show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems. This condition is only sufficient. In fact, finite element approximations may converge even though some dihedral angles of simplicial elements tend to π.     

预给二面角的单形在球面型空间Sn,r的嵌入

本文获得预给二面角的单形嵌入球面型空间Ｓ_n,r的一个充分必要条件，并利用它得到关于球面单形二面角的两类几何不等式。     

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.     

Cusp Cross-Sections of Hyperbolic Orbifolds by Heisenberg Nilmanifolds I

Long and Reid [Algebr. Geom. Topol. 2: 285–296, 2002] have shown that the diffeomorphism class of every Riemannian flat manifold of dimension n≥ 3 arises as a cusp cross-section of a complete finite volume real hyperbolic (n+1)-orbifold. For the complex hyperbolic case, McReynolds [Algebr. Geom. Topol. 4: 721–755, 2004] proved that every 3-dimensional infranilmanifold is diffeomorphic to a cusp cross-section of a complete finite volume complex hyperbolic 2-orbifold. Moreover, he gave a necessary and sufficient condition for a Heisenberg infranilmanifold to be realized as a cusp cross-section of finite volume (arithmetically) complex hyperbolic orbifold. We study these realization problems by using Seifert fibrations.     

Sections of Angles and nth Roots of Numbers

It is known since Galois that an algebraic equation can be solved using suitable nth roots whenever the corresponding Galois group is soluble. The object of this note is to construct real numbers with the use of the nth parts of suitable angles and to state necessary and sufficient conditions for this to be possible.     

预给二面角的m面凸多胞形嵌入Rd的充分必要条件

本文给出了预给二面角的ｍ面凸多胞形嵌入Ｒ^d的充分必要条件     

Espace des modules de certains polyèdres projectifs miroirs

A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in (0,\fracp2]{(0,\frac{\pi}{2}]}, and the union of n copies of \mathbbR^d{\mathbb{R}^{d}}, when the polyhedron has the combinatorics of an ecimahedron, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers n and d can be computed explicitly in terms of the combinatorics and the fixed dihedral angles.     

Conformal paracontact curvature and the local flatness theorem

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.     

A necessary and sufficient condition for the nirenberg problem

We seek metrics conformal to the standard ones on Sⁿ having prescribed Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescribed scalar curvature for n ≧ 3 (the Kazdan-Warner problem). There are well-known Kazdan-Warner and Bourguignon-Ezin necessary conditions for a function R(x) to be the scalar curvature of some conformally related metric. Are those necessary conditions also sufficient? This problem has been open for many years. In a previous paper, we answered the question negatively by providing a family of counter examples. In this paper, we obtain much stronger results. We show that, in all dimensions, if R(x) is rotationally symmetric and monotone in the region where it is positive, then the problem has no solution at all. It follows that, on S², for a non-degenerate, rotationally symmetric function R(θ), a necessary and sufficient condition for the problem to have a solution is that Rθ changes signs in the region where it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as will be shown in this paper. We also consider similar necessary conditions for non-symmetric functions. ©1995 John Wiley & Sons, Inc.     

Remarks on the Cauchy problem for first order hyperbolic systems with constant coefficient principal part

Abstract In this article we shall introduce the results obtained in [16], i.e., we shall give a necessary and sufficient condition that the Cauchy problem for first order hyperbolic systems with constant coefficient principal part is C^∞ well-posed under the maximal rank condition (see the condition (R) below). We shall also give a simple sufficient condition without any assumptions on the rank. Keywords: Hyperbolic system, Cauchy problem, Constant coefficient principal part     

Self-dual 2-quasi-cyclic codes and dihedral codes

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if $\mathrm{char}F=2$), or a consta-dihedral code (if $\mathrm{char}F\ne 2$). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. We also obtain necessary and sufficient conditions under which the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) of codes coincide with each other.     

Link Homology in 4-Manifolds

We define link homology in 4-manifolds, and show that it hasa close connection to linking numbers and intersection matricesof 4-manifolds. We also define null-homologous links in 4-manifolds.We give a necessary and sufficient condition for links to benull-homologous in 4-manifolds. This condition implies thatfor any 4-manifold with second Betti number n, there are (n+ 2)-component links which are not nullhomologous in the 4-manifold.     

On a property of the class of n-colorable graphs

An obvious lower bound on the chromatic number of a graph is the largest possible number of points in a complete subgraph. A sufficient condition is presented for these numbers to be equal. From this is derived a necessary and sufficient condition, believed to be new, for a graph to be n-colorable.     

Large hyperbolic lattices

For a fundamental group of a compact orientable manifold, a condition is specified that is sufficient to guarantee the presence of a “virtual” epimorphism onto a free non-Abelian group. A consequence is deriving a strong Tits alternative. An arbitrary noncompact finitely generated discrete subgroup in PO(3, 1) either is large or is virtually Abelian. An application is provided to the problem of uniform exponential growth for lattices in a 3-dimensional hyperbolic space and of growth of Betti numbers for lattices in a hyperbolic n-dimensional space, where n is an odd number. Supported by RFBR (project No. 08-01-00067), by DFG grant Gr 627-11, and by Forschergruppe “Spektrale Analysis, asymptotical Verteilungen und stochastische Dynamiken,” Billfold University. (G. A. Noskov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 174–189, March–April, 2009.     

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.     

Bounded Classes in the Cohomology of Manifolds

We discuss the image of a natural homomorphism from the bounded cohomology to the ordinary cohomology of a manifold. We give a necessary and sufficient condition for a Haken 3-manifold M to have the property that any class in the nth cohomology of M is bounded if n > 1.The first author was partially supported by JSPS.     

Harmonious orbits of linear transformations

We describe necessary and sufficient conditions for orbits of linear transformations onR ⁿ ,n1, and sets arising as sums of elements from orbits, to be harmonious subsets. This is done via a generalization of the notion of Pisot-Vijayaraghavan and Salem numbers.     

Construction of Matrices with Prescribed Singular Values and Eigenvalues

Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m ( n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.     

Tight bounds for the number of edge guards for spiral polygons

In this paper we give tight lower and upper bounds for the number of edge guards required for covering spiral polygons. We have proved that [(n + 2)/5]edge guards are necessary and sufficient to cover a spiral polygon. It has been shown by Aggarwal [2] that [(n + 2)/5]diagonal guards are necessary and sufficient to cover a spiral polygon. Edge guards are more restrictive than diagonal guards. Hence the previous theorem can be got as a corollary using our theorem. The necessary condition of the edge guard problem for spiral polygons has not been investigated although the diagonal guard problem for the same has been solved [2]. The necessary proof of the edge guard problem follows from the necessary condition of the diagonal guard problem but we have given an alternate proof of necessity.