The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element, g2 HzL = -1 ê z, and an element of infinite order, g• HzL = z + 1. Viewing the action of the group elements on a...

In this paper, we consider hyperbolic rotation (G0), hyperbolic translation (G1) and horocyclic rotation (G2) groups in H3, which is called Minkowski model of hyperbolic space. Then, we investigate extrinsic differential geometry of invariant surfaces under subgroups of G0 in H3. Also, we give explicit parametrization of this invariant surfaces with respect to constant hyperbolic curvature of p...

The artist M.C. Escher was the first artist to create patterns in the hyperbolic plane. He used both the Poincaré disk model and the Poincaré half-plane model of hyperbolic geometry. We discuss some of the theory of hyperbolic patterns and show Escher-inspired designs in both of these models.

We consider a coupled PDE system arising in noise reduction problems. In a two dimensional chamber, the acoustic pressure (unwanted noise) is represented by a hyperbolic wave equation. The floor of the chamber is subject to the action of piezo-ceramic patches (smart materials). The goal is to reduce the acoustic pressure by means of the vibrations of the floor which is modelled by a hyperbolic ...

A Study of Option Pricing Models – Lognormal or Hyperbolic Levy ? By Chen Chen This paper is an investigation into two option pricing models: widely-used Black-Scholes model and one of its augmented extensions – hyperbolic Levy model. Firstly, we have a detailed discussion about the celebrated Black-Scholes model. However, clearly there are many deficiencies in Black-Scholes assumptions. In ord...

T he optimal allocation of natural gas resources to various uses such as final and intermediate consumption, injection into oil fields, and exports can help policymakers to use this kind of resources efficiently. Empirical evidence support using hyperbolic discount rates instead of fixed discount rates in the economic literature. The purpose of this study is to maximize the social we...

We consider a cosmological model consisting of two scalar fields defined in the hyperbolic plane known as inflation. For background space, we homogeneous and isotropic spacetime with nonzero curvature. study asymptotic behaviour solutions search for attractors expanding regime. prove that inflationary stages are stable can solve flatness problem describe acceleration both open closed models, ad...