The geometric mean and the function (det(·))1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t → p(x+ td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and ...

We discuss non existence of analytic first integrals for analytic diffeomorphisms possessing a hyperbolic fixed point with a homoclinic connection.

Using a completely analytic procedure – based on a suitable extension of a classical method – we discuss an approach to the Poincaré-Mel'nikov theory, which can be conveniently applied also to the case of non-hyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case, and precisely on problems described b...

We prove that a Hamiltonian p ∈ C(T R) is locally integrable near a nondegenerate critical point ρ0 of the energy, provided that the fundamental matrix at ρ0 has no purely imaginary eigenvalues. This is done by using Birkhoff normal forms, which turn out to be convergent in the C sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transfo...

We derive explicit forms of the two–point correlation functions of the O(N) non-linear sigma model at the critical point, in the large N limit, on various three dimensional manifolds of constant curvature. The two–point correlation function, G(x, y), is the only n-point correlation function which survives in this limit. We analyze the short distance and long distance behaviour of G(x, y). It is...

It is shown that a slender elastic fiber moving in a Stokesian fluid can be susceptible to a buckling instability--termed the "stretch-coil" instability--when moving in the neighborhood of a hyperbolic stagnation point of the flow. When the stagnation point is part of an extended cellular flow, it is found that immersed fibers can move as random walkers across time-independent closed-streamline...

A fixed point theorem is proved for inverse transducers, leading to an automatatheoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regular fixed points in the hyperbolic boundary has finitely many orbits under the a...

We investigate a new basic model for asset pricing, the hyperbolic model, which allows an almost perfect statistical t of stock return data. After a brief introduction into the theory supported by an appendix we use also secondary market data to compare the hyperbolic model to the classical Black-Scholes model. We study implicit volatilities, the smile e ect and the pricing performance. Exploit...

Hyperbolic structures are obtained by tiling a hyperbolic surface with negative Gaussian curvature. These structures generally exhibit two percolation transitions: a system-wide connection can be established at a certain occupation probability p = pc1, and there emerges a unique giant cluster at pc2 > pc1. There have been debates about locating the upper transition point of a prototypical hyper...