On hyperbolicity of pseudoconvex Reinhardt domains

We give a characterization of Kobayashi hyperbolicity of pseudoconvex Reinhardt domains. All such domains turn out to be biholomorphic to a bounded Reinhardt domain. In particular, any Kobayashi hyperbolic pseudoconvex Reinhardt domain is Kobayashi complete.     

On the hyperbolicity of certain models of polydisperse sedimentation

The sedimentation of a polydisperse suspension of small spherical particles dispersed in a viscous fluid, where particles belong to N species differing in size, can be described by a strongly coupled system of N scalar, nonlinear first‐order conservation laws for the evolution of the volume fractions. The hyperbolicity of this system is a property of theoretical importance because it limits the range of validity of the model and is of practical interest for the implementation of numerical methods. The present work, which extends the results of R. Bürger, R. Donat, P. Mulet, and C.A. Vega (SIAM Journal on Applied Mathematics 2010; 70:2186–2213), is focused on the fluxes corresponding to the models by Batchelor and Wen, Höfler and Schwarzer, and Davis and Gecol, for which the Jacobian of the flux is a rank‐3 or rank‐4 perturbation of a diagonal matrix. Explicit estimates of the regions of hyperbolicity of these models are derived via the approach of the so‐called secular equation (J. Anderson. Linear Algebra and Applications 1996; 246:49–70), which identifies the eigenvalues of the Jacobian with the poles of a particular rational function. Hyperbolicity of the system is guaranteed if the coefficients of this function have the same sign. Sufficient conditions for this condition to be satisfied are established for each of the models considered. Some numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Gromov hyperbolicity of planar graphs

We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ?² with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ?² such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of ?² with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of ?² with tiles which are parallelograms would be non-hyperbolic.     

Gromov hyperbolicity of periodic planar graphs

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it.The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.     

On the classification of certain planar contact structures

We focus on contact structures supported by planar open book decompositions. We study right-veering diffeomorphisms to keep track of overtwistedness property of contact structures under some monodromy changes. As an application we give infinitely many examples of overtwisted contact structures supported by open books whose pages are the four-punctured sphere, and also we prove that a certain family is Stein fillable using lantern relation.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

On certain Hamiltonian cycles in planar graphs

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least five. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than five contains at least distinct Hamiltonian cycles. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 81–96, 1999     

On Hamilton cycles in certain planar graphs

Let G be a 2-connected plane graph with outer cycle X_G such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of X_G. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.     

On the Gromov non-hyperbolicity of certain domains in Cn

In this paper, we prove that if Ω is a bounded convex domain in ${\mathbb{C}}^n$, $n\ge 2$, and S is an affine complex hyperplane such that $\mathrm{\Omega }\cap S$ is not empty, then $\mathrm{\Omega }?S$ is not Gromov hyperbolic with respect to the Kobayashi distance. Next, we show that if X is a bounded convex domain in ${\mathbb{C}}^n$, then $\mathrm{\Omega }=\{(z,w)\in X\times {\mathbb{C}}^{?},\left|w\right|<{\mathrm{e}}^{?\phi (z)}\}$ is not Gromov hyperbolic, where φ is a strictly plurisubaharmonic function on X continuous up to $\stackrel{￣}{X}$.     

On the hyperbolicity of general hypersurfaces

We introduce and study stochastic \(N\)-particle ensembles which are discretizations for general-\(\beta \) log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, \((z,w)\)-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as \(N\to \infty \). The covariance is universal and coincides with its counterpart in random matrix theory.Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.     

On the hyperbolicity constant in graphs

If X is a geodesic metric space and x₁,x₂,x₃∈X, a geodesic triangleT={x₁,x₂,x₃} is the union of the three geodesics [x₁x₂], [x₂x₃] and [x₃x₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if, for every geodesic triangle T in X, every side of T is contained in a δ-neighborhood of the union of the other two sides. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. . In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths , then , and if and only if G is isomorphic to C_m. Moreover, we prove the inequality for every graph, and we use this inequality in order to compute the precise value δ(G) for some common graphs.     

On certain non-linear differential equations in complex domains

We describe meromorphic solutions of certain non-linear (ordinary and partial) differential equations in complex domains. Received: 13 November 2007     

Generic hyperbolicity for reaction diffusion equations on symmetric domains

Summary The problem of generic hyperbolicity for reaction diffusion equations with Dirichlet boundary conditions on a ball inR ⁿ is studied. It is proved that while hyperbolicity is not a generic property, radially symmetric solutions are generically hyperbolic.

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Zusammenfassung Das Problem der generischen Hyperbolizität für Reaktion-Diffusionsgleichungen mit Dirichletschen Randbedingungen in einer Kugel imR ⁿ wird betrachtet. Es wird bewiesen, daß radialsymmetrische Lösungen generisch symmetrische sind, während Hyperbolizität nicht eine generische Eigenschaft ist.

     

On certain domains in cycle spaces of flag manifolds

The affine homogeneous space associated to a real semi-simple Lie group G with maximal compact subgroup K contains a number of naturally defined{\it G}-invariant neighborhoods of its real points which are of interest from various points of view. Here the universal Iwasawa domain is introduced from the point of view of incidence geometry and certain of its properties are derived, e.g., it is Stein, Kobayashi hyperbolic and contains the domain introduced by Akhiezer and Gindikin which is now known to be equivalent to the maximal domain of definition of the adapted complex structure associated to the Killing metric in the tangent bundle . One of the main goals of the paper is to develop methods which lead to a better understanding of the Wolf domain of cycles in an open G-orbit D in a flag manifold . The key is the Schubert domain which is defined by Schubert cycles of complementary dimension to the cycles. These are defined by a Borel subgroup containing an Iwasawa factor AN and consequently and are closely related. Received: 28 May 2001 / Revised version: 19 November 2001 / Published online: 23 May 2002