On bifurcations in degenerate resonance zones

Bifurcations in degenerate resonance zones for Hamitonian systems with 3/2 degrees of freedom close to nonlinear integrable ones and for symplectic maps of a cylinder are discussed.     

Asymptotics of instability zones of Hill operators with a two term potential

We give a sharp asymptotics of the instability zones of the Hill operator $Ly=?y^{″}+(acos2x+bcos4x)y$ for arbitrary real $a\text{,}b\ne 0$. To cite this article: P. Djakov, B. Mityagin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On the Rayleigh-Plateau instability

In this paper,we study the surface instability of a cylindrical pore in the absence of stress. This instability is called the Rayleigh-Plateau instabilty. We consider the model developed by Spencer et ...     

On the width of hybrid zones

Hybrid zones occur when two species are found in close proximity and interbreeding occurs, but the species’ characteristics remain distinct. These systems have been treated in the biology literature using partial differential equations models. Here we investigate a stochastic spatial model and prove the existence of a stationary distribution that represents the hybrid zone in equilibrium. We calculate the width of the hybrid zone, which agrees with the PDE formula only in dimensions d≥3$d\ge 3$. Our results also give insight into properties of hybrid zones in patchy environments.     

Asymptotics of instability zones of the Hill operator with a two term potential

Let γn denote the length of the nth zone of instability of the Hill operator Ly=−y^″−[4tαcos2x+2α²cos4x]y, where α≠0, and either both α, t are real, or both are pure imaginary numbers. For even n we prove: if t, n are fixed, then for α→0

     

The geometric structures and instability of entropic dynamical models

In this paper, we characterize two entropic dynamical (ED) models from the viewpoint of information geometry and give the geometric structures of the associated statistical manifolds of the models. The scalar curvatures and the geodesics are obtained. Also the instability of entropic dynamical models is studied from the behavior of the geodesics lengths, statistical volume elements and Jacobi vector fields.     

On local inflexional instability in boundary-layer flows

Summary This paper describes a linearized theory of the two- and three-dimensional incompressible viscous flows ensuing from locally unstable velocity profiles. The theory is used to propose a hypothesis for the mechanisms governing formation of the turbulence wedge behind a fixed roughness element.

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Résumé On a présenté une théorie linearizée pour l'évolution des écoulements visqueux, incompressibles, bi-et tri-dimensionels, suivant l'instabilité locale des profils de vitesse de la couche limite laminaire. Utilisant les résultats de la théorie, on a proposé une hypothèse sur les mécanismes qui régissent la formation du coin de turbulence en aval d'un élément de rugosité fixée.

     

On Weyl structures

We characterize the relation between the geometrical properties of Weyl manifolds and the algebraic properties of the Weyl algebras (§1) and the deformation algebras associated to two conformal Weyl connections (§2). The last section is devoted to the study of the Weyl-Lyra algebras associated to a conformal Weyl connection and a conformal semisymmetric connection.     

On the instability of resonances in parallel-plane waveguides

It has been observed¹³ that the propagation of acoustic waves in the region Ω₀= ?² × (0, 1), which are generated by a time-harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown⁹ in the case of Dirichlet's boundary condition U = 0 on ?Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω₀. By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω₀ ?B, where B is a smooth bounded domain with B??Ω₀. In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n · x ′ ? 0 on ? B, where x ′ = (x₁, x₂, 0) and n is the normal unit vector pointing into the interior B of ? B. In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω₀ will be discussed in a subsequent paper.     

On local Görtler instability

Résumé On a présenté une modification simple de la critère de Görtler pour l'instabilité de la couche limite laminaire sur une paroi de courbature concave, ce qui met en évidence la dépendence de la critère sur la longueur finie de la région de courbature. L'effet de la longueur finie n'est signifiant que pour les tourbillons excités ayant les plus grandes possibles longueurs d'onde. On a suggéré que ce sont les derniers qui peuvent être les plus importants parce qu'ils s'amortissent le moins rapidement avec la distance en aval de la région de courbature.     

On incomplete preference structures

Incomplete preference structures are composed of three relations: preference, indifference and incomparability. We survey some very recent works which model such structures, using interval orders or semi orders. Three approaches are proposed: first, in relation to comparability graph characterization; second, in relation to order dimension theory; and third, representation of the structures on the real line.     

On some incidence structures

In this paper,suggested by André's papers ([2), [3]), we construct geometrical structures (X,?,//}) where X is a finite set of points, ? is a set of lines, and // is an equivalence relation on ?. These constructions are made starting with a finite and not empty set X and a permutation group G which is 2-transitive on X and such that the stabilizer of two distinct points of X is different from the identical subgroup. We look for conditions such that the structure (X, ?) is a (3,q)-Steiner system. We remember that a (3,q)-Steiner system is a pair (X,B), where X is a set of elements (called points), B is a system of subsets of X (called blocks), such that:

1. every block contains q points exactly;
2. given three distinct points x,y,z of X, there is exactly one subset of X belonging to B and containing x,y,z.

At the end we construct such a system with the help of a nearskewfield (according to Zassenhaus [7], [8]).