Estimation of the hyperbolic metric by using the punctured plane

Let denote the density of the hyperbolic metric for a domain Ω in the extended complex plane . We prove the inequality

A characterization of Möbius transformations by use of hyperbolic regular polygons

In this paper we present a new characterization of Möbius transformations by use of hyperbolic regular polygons.     

Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

In this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface.     

双曲型方程具有奇性斜导数的混合问题<英>

本文讨论了2m阶双曲型方程具有奇性斜导数的边值问题。在边界奇点(即不满足Lopatinsky边界条件的点)子流形的一定假设下,证明了所论问题在Sobolev空间H~(s,s)(Q)中解的存在性和唯一性,从而将二阶双曲方程的相应问题的已有的结果(例如[1]、[4—6])推广到了高维的情形。     

Laser diode to single-mode circular core dispersion-shifted/dispersion-flattened fiber excitation via hyperbolic microlens on the fiber tip: Prediction of coupling efficiency by ABCD matrix formalism

The formulated ABCD matrix formalism is employed to prescribe analytical expression of coupling efficiency of a laser diode to single-mode circular core dispersion-shifted as well as dispersion-flattened fiber via hyperbolic microlens on the tip of the fiber. We assume that field distribution in case of both the source and the fiber is one parameter Gaussian type. For maximum excitation efficiency, it is required that the lens transmitted spot size of the source should match with the spot size of the fiber. Further, as regards the spot size of the fiber, we use Petermann II spot size in order to take care of non Gaussian nature of field of such fibers and to make the estimations more realistic thereby. The investigations are carried out for two different wavelengths 1.3 and 1.5 μm. Our simple method predicts the concerned coupling optics excellently and the necessary evaluations require little computations. This simple but accurate technique is expected to benefit the system designers who work in the field of optical technology.     

A discrete eigenfunctions method for computing mixed hyperbolic problems based on an implicit difference scheme

In this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme. This new numerical technique preserves the qualitative properties of the analytic solution due to the Sturm-Liouville structure of the underlying discrete linear boundary-value problem and has computational stability advantages vs other methods. Illustrative examples are included.     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.