A note on hyperbolic transformations

We shall discuss the so‐called hyperbolic Householder and Givens transformations applied to complex matrices, including the case of zero hyperbolic energy of a transformed vector. For each case a numerically stable algorithm is available. Copyright © 2001 John Wiley & Sons, Ltd.     

Very Hyperbolic Polynomials

A real polynomial in one variable is hyperbolic if it has only real roots. A function f is a primitive of order k of a function g if f ^(k) = g. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. In the paper, we prove a property of the domain of very hyperbolic polynomials and describe this domain in the case of degree 4.     

Products of Hyperbolic Metric Spaces

Let (X _i d _i ), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a 'hyperbolic product' X ₁× _h X ₂ which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.     

On the type of triangle groups

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.      

Decomposition of an Updated Correlation Matrix via Hyperbolic Transformations

An algorithm for hyperbolic singular value decomposition of a given complex matrix based on hyperbolic Householder and Givens transformation matrices is described in detail. The main application of this algorithm is the decomposition of an updated correlation matrix.     

Continuity of the hyperbolic zeta function of lattices

The continuity of the hyperbolic zeta function of lattices _H (¦),=+it, >1, as a function on the metric space of lattices is proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 522–526, April, 1998.The authors wish to express their gratitude to Professor N. M. Korobkov and Professor V. I. Nechaev for fruitful discussions.     

Singularities of Hyperbolic Gauss Maps

In this paper we adopt the hyperboloid in Minkowski space asthe model of hyperbolic space. We define the hyperbolic Gaussmap and the hyperbolic Gauss indicatrix of a hypersurface inhyperbolic space. The hyperbolic Gauss map has been introducedby Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135]in the Poincaré ball model, which is very useful forthe study of constant mean curvature surfaces. However, it isvery hard to perform the calculation because it has an intrinsicform. Here, we give an extrinsic definition and we study thesingularities. In the study of the singularities of the hyperbolicGauss map (indicatrix), we find that the hyperbolic Gauss indicatrixis much easier to calculate. We introduce the notion of hyperbolicGauss–Kronecker curvature whose zero sets correspond tothe singular set of the hyperbolic Gauss map (indicatrix). Wealso develop a local differential geometry of hypersurfacesconcerning their contact with hyperhorospheres. 2000 MathematicalSubject Classification: 53A25, 53A05, 58C27.     

带导数边界条件的线性双曲方程的一个二阶L∞收敛格式

In this paper, a difference scheme is derived for the one dimensionallinear hyperbolic differential equation with derivative boundary conditions by themethod of reduction of order. It is proved that the scheme is unconditionally sta-ble, uniquely solvable and second order convergent in L∞ norm. Two numericalexamples are presented.     

双曲矩阵分解修正问题的一些结果

This paper considers the updating problem of the hyperbolic matrix factorizations. The sufficient conditions for the existence of the updated hyperbolic matrix factorizations are first provided. Then, some differential inequalities and first order perturbation expansions for the updated hyperbolic factors are derived. These results generalize the corresponding ones for the updating problem of the classical QR factorization obtained by Jiguang SUN.     

THE FOUR-DIMENSIONAL HYPERBOLIC SPHERICAL HARMONICS

In the four-dimensional hyperbolic space, we can derive Euler formula and establish the hyperbolic spherical polar coordinate and the hyperbolic spherical harmonics.