Hyperbolic Case Research Articles

Analytic solutions of an iterative differential equation under Brjuno condition

In this paper, the differential equation involving iterates of the unknown function, $$ x'(z) = [a^2 - x^2 (z)]x^{[m]} (z) $$ with a complex parameter a, is investigated in the complex field ℂ for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the Schroder transformation x(z) = y(αy−1(z)) to the following another functional differential equation without iteration of the unknown function $$ \alpha y'(\alpha z) = [a^2 - y^2 (\alpha z)]y'(z)y(\alpha ^m z), $$ which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y(αy−1(z)) for the original iterative differential equation are obtained. We discuss not only these α given in Schroder transformation in the hyperbolic case 0 < |α| < 1 and resonance, i.e., at a root of the unity, but also those α near resonance (i.e., near a root of the unity) under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.

Squarepants in a tree

We provide efficient constant-factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition , that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.

Symmetric generation of Coxeter groups

We provide involutory symmetric generating sets of finitely generated Coxeter groups, fulfilling a suitable finiteness condition, which in particular is fulfilled in the finite, affine and compact hyperbolic cases.

A note on the global existence of small amplitude solutions to a generalized Davey–Stewartson system

In this paper, we are interested in the Cauchy problem for a generalized Davey–Stewartson (GDS) system. We establish the global time existence of small mass solutions for the GDS system in the elliptic–hyperbolic–hyperbolic case.

Multifractal analysis of the Lyapunov exponent for the backward continued fraction map

AbstractIn this paper we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. To the best of our knowledge this is the first time that conditions of this type have been given. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then becomes equal to zero. We also discuss the existence and uniqueness of equilibrium measures. In order to do so, we introduce a family of countable Markov shifts that can be thought of as a generalization of the renewal shift.

Packing subgroups in relatively hyperbolic groups

We introduce the bounded packing property for a subgroup of a countable discrete group G. This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G. We establish basic properties of bounded packing, and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT(0) cube complexes. Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case.

Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations

We are interested in the large-time behavior of periodic entropy solutions in L ∞ to anisotropic degenerate parabolic-hyperbolic equations of second order. Unlike the pure hyperbolic case, the nonlinear equation is no longer self-similar invariant, and the diffusion term in the equation significantly affects the large-time behavior of solutions; thus the approach developed earlier, based on the self-similar scaling, does not directly apply. In this paper. we develop another approach for establishing the decay of periodic solutions for anisotropic degenerate parabolic-hyperbolic equations. The proof is based on the kinetic formulation of entropy solutions. It involves time translations and a monotonicity-in-time property of entropy solutions and employs the advantages of the precise kinetic equation for the solutions in order to recognize the role of nonlinearity-diffusivity of the equation.

Spectral gap global solutions for degenerate Kirchhoff equations

We consider the second order Cauchy problem u ″ + m ( | A 1 / 2 u | 2 ) A u = 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , where m : [ 0 , + ∞ ) → [ 0 , + ∞ ) is a continuous function, and A is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that u 0 and u 1 are regular enough, depending on the continuity modulus of m , and on the strict/weak hyperbolicity of the equation. We prove that for such initial data ( u 0 , u 1 ) there exist two pairs of initial data ( u ¯ 0 , u ¯ 1 ) , ( u ̂ 0 , u ̂ 1 ) for which the solution is global, and such that u 0 = u ¯ 0 + u ̂ 0 , u 1 = u ¯ 1 + u ̂ 1 . This is a byproduct of a global existence result for initial data with a suitable spectral gap, which extends previous results obtained in the strictly hyperbolic case with a smooth nonlinearity m .

Convergence of Weakly Imposed Boundary Conditions: The One-Dimensional Hyperbolic Case

In this paper we study the convergence of a weakly imposed boundary condition for the one-dimensional Euler equations of gasdynamics. We give a mathematical proof of the convergence in two specific cases, and we analyze two kinds of physically relevant boundary conditions including the nonpenetration condition and the imposed pressure condition. We also show the results of a couple of one-dimensional test cases that confirm our theoretical estimates.

Asymptotics of convex sets in Euclidean and hyperbolic spaces

We study convex sets C of finite (but non-zero) volume in H n and E n . We show that the intersection C ∞ of any such set with the ideal boundary of H n has Minkowski (and thus Hausdorff) dimension of at most ( n − 1 ) / 2 , and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C ∞ is a smooth submanifold of ∂ ∞ H n . In the hyperbolic case, we show that for any k ⩽ ( n − 1 ) / 2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body in E n , and give asymptotic estimates as 1 ≪ k ≪ n .

COMPETITION BETWEEN TWO MICROORGANISMS IN THE CHEMOSTAT WITH GENERAL VARIABLE YIELDS AND GENERAL GROWTH RATES

A model of competition between two organisms in the chemostat is studied in this paper, where it is assumed that the yield coefficients and growth rates are general functions of the nutrient concentration. We give a characterization of the outcome of this competition in terms of the relevant parameters in hyperbolic cases: conditions of the existence and local stability of the rest points are obtained, and the global asymptotic behavior of the solutions is analyzed. In contrast to the corresponding model with constant yields rate, it is demonstrated that the variability of the yield coefficient may lead to oscillatory coexistence of two microorganisms in continuous culture.

Wormholes in ACH Einstein manifolds

We give a new construction of Einstein manifolds which are asymptotically complex hyperbolic, inspired by the work of Mazzeo-Pacard in the real hyperbolic case. The idea is to develop a gluing theorem for 1-handle surgery at infinity, which generalizes the Klein construction for the complex hyperbolic metric.

Topology of compact space forms from Platonic solids. I.

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253–263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL 2 ( C ) with compact orbit space, Canad. J. Math. 23 (1971) 451–460], Lorimer [P.J. Lorimer, Four dodecahedral spaces, Pacific J. Math. 156 (2) (1992) 329–335], Prok [I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (2) (1998) 497–515], and Richardson and Rubinstein [J. Richardson, J.H. Rubinstein, Hyperbolic manifolds from a regular polyhedron, Preprint]. In this paper we investigate the topology of closed orientable 3-manifolds from Platonic solids. Here we completely recognize those manifolds in the spherical and Euclidean cases, and state topological properties for many of them in the hyperbolic case. The proofs of the latter will appear in a forthcoming paper.

The Chern-Simons invariants of cone-manifolds with Whitehead link singular set

In the present article, we obtain some explicit integral formulas for the generalized Chern-Simons function I(W(α,β)) for Whitehead link cone-manifolds in the hyperbolic and spherical cases. We also give the Chern-Simons invariant for the Whitehead link orbifolds. We find a formula for the Chern-Simons invariant of n-fold coverings of the three-sphere branched over the Whitehead link.

The Performance Analysis of a Multi-Duct Proton Exchange Membrane Fuel Cell Cathode

A 3-D numerical study was carried out first to analyze flow, heat and mass transfer as well as current distribution in a single half-cell cathode duct of a Proton Exchange Membrane (PEM) fuel cell. From practical point of view, it is necessary to put the appropriate number of cells in a stack. Hence, the above study on a single half-cell is extended to a multiple ducts. Due to multi-duct configuration the assumption of uniform flow distribution would no longer hold true. Therefore, the duct flow maldistribution is considered in the simulation of the multiple ducts. The flow arrangement of the PEM fuel cell resembles that of the plate heat exchangers. Hence, the various types of flow distributions pertinent to the plate heat exchangers are applied to the PEM fuel cell ducts and their effects on heat and mass transfer as well as current distribution are studied in detail. The water formed at the active surface due to the electrochemical reaction diffuses through the porous layer and eventually enters the gas flow duct. A high gas velocity removes the product water at a faster rate from the duct. In nominal-linear and hyperbolic flow distribution cases, the mass fraction of water in the duct is more than that in severe-linear and hyperbolic flow distribution cases. This means that the membrane retains higher water content in nominal distribution cases. This in turn results in higher membrane conductivity and lower ohmic loss. This is the reason for better performance for nominal maldistribution cases over the severe maldistribution cases.

Scattering and inverse scattering on ACH manifolds

We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define the radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.

An Output Controllability Problem for Semilinear Distributed Hyperbolic Systems

An Output Controllability Problem for Semilinear Distributed Hyperbolic SystemsThe paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder's fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.

Viscous boundary value problems for symmetric systems with variable multiplicities

Extending investigations of Métivier and Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to “real” or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the Zumbrun–Serre–Rousset theorem, valid for variable multiplicities, characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system. The new effects of viscosity are seen to be surprisingly subtle; in particular, viscous coupling of crossing hyperbolic modes may induce a destabilizing effect. We illustrate the theory with applications to magnetohydrodynamics.

Wavelet‐based adaptive grids as applied to hydrodynamics

AbstractAn adaptive, wavelet‐based, multiscale finite‐volume scheme is developed and employed to investigate applications in the simulation of water waves. Firstly, two one‐dimensional, strictly hyperbolic cases are investigated: shallow water and Euler equations. These are followed by two investigations using a finite‐volume formulation of Madsen and Sørensen's Boussinesq equations. Converged results were obtained in all cases, which demonstrate that the adaptive grid scheme is significantly faster than that on a uniform grid. In some cases, one‐seventh of the number of cells is required to obtain the same accuracy as that of the uniform grid. Issues of stability are discussed in the context of the particular problems modelled here with the Boussinesq equations, related to discretization of the high‐order spatial derivatives on a non‐uniform grid. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators

We consider second order degenerate hyperbolic Cauchy problems, the degeneracy coming either from low regularity (less than Lipschitz continuity) of the coefficients with respect to time, or from weak hyperbolicity. In the weakly hyperbolic case, we assume an intermediate condition between effective hyperbolicity and the Levi condition. We construct the fundamental solution and study the propagation of singularities using an unified approach to these different kinds of degeneracy.