A note on Arzela–Ascoli's lemma in almost periodic problems

This paper is concerned with a generalized Arzela–Ascoli's lemma, which has been extensively applied in almost periodic problems by the continuation theorem of degree theory. We give a counter example to show that this lemma is incorrect, and there is a gap in the proof of some existing literature, where the addressed generalized Arzela–Ascoli's lemma was used. Moreover, we make some final comments and introduce an open problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Delay‐dependent criteria on the global attractivity of Nicholson's blowflies model with patch structure

In this paper, we investigate the effect of delay on the asymptotic behavior of Nicholson's blowflies model with patch structure and multiple time‐varying delays. By using the fluctuation lemma and some differential inequality technique, delay‐dependent criteria are obtained for the global attractivity of the addressed system. Meanwhile, some numerical examples are given to illustrate the feasibility of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem where 0≤λ < 1,^CD^α is the Caputo's differential operator of order α, and f:[0,1] × [0,∞)→[0,∞) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary α, 1≤n < α≤n + 1: Problem 1: where , 0≤λ < k + 1; Problem 2: where 0≤λ≤α and D^α is the Riemann–Liouville fractional derivative of order α. For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotics of Neumann harmonics when a cavity is close to the exterior boundary of the domain

We construct the asymptotics (as ε→0) of solutions to the Neumann problem for the Laplace equation and of the corresponding Dirichlet integral. The problem concerns a three-dimensional domain having two connected components of the boundary at the distance ε>0. To cite this article: G. Cardone et al., C. R. Mecanique 335 (2007).     

Periodic Solutions of Delay Differential Equations with a Small Parameter: Existence,Stability and Asymptotic Expansion

We consider a scalar delay differential equation with a small parameter, and employ Walthers method to obtain a result on the existence and stability of a slowly oscillatory periodic solution that represents a refinement of the estimate for the Lipschitz constant of a returning map. We also develop a matching method and obtain asymptotic expansions of the slowly oscillatory periodic solution and its minimal period.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthdayAMS subject classifications: 34K15; 34K20; 34C25.     

Application de l'approximation polytropique à la turbulence statistique en moyenne de FavreApplication of the polytropic approximation to turbulence statistics using Favre averaging

The application of the polytropic approximation connecting the quantities of corresponding state, to experimental analysis, is clarified. A method of polytropic determination of the exponent χ (variable but non-fluctuating) in each point of the flow is given. This approximation makes it possible the generation of representative signals of fluctuating quantities, like pressure or density. For heated gases, the problem of measurement of the equations terms written with Favre averaging is thus almost solved. Then, measurement of χ allows the determination by the experiment of crucial terms like turbulent fluxes of mass and momentum, and presso correlation. To cite this article: C. Rey, S. Benjeddou, C. R. Mecanique 332 (2004).     

Asymptotics of eigenfrequencies of an elastic body with a heavy and hard peak-shaped inclusion

A heavy and hard peak-shaped inclusion in an elastic body provokes to concentration of eigenvalues in the low-frequency range of the spectrum and localization of the corresponding eigenmodes near the peak tip. To cite this article: S.A. Nazarov, C. R. Mecanique 335 (2007).     

Variance of the virtual displacement

We show that the affine structure of the 3-dimensional space is deeply enmeshed with the static laws expression. The relevance of tensorial rules for calculus in mechanics is thus enhanced. The virtual work principle is stated precisely but a little twist is given to the usual statements: the tensorial nature of the so-called virtual displacement vector is asserted to be covariant. To cite this article: C. Vallée et al., C. R. Mecanique 332 (2004).

Résumé

Nous montrons que la structure affine de l'espace tridimensionnel est fortement imbriquée avec l'expression des lois de la statique. La pertinence de l'application des règles du calcul tensoriel à la mécanique est ainsi renforcée. Le principe des travaux virtuels est établi précisément avec une seule entorse aux exposés classiques : la nature tensorielle de l'habituel vecteur déplacement virtuel est affirmée covariante. Pour citer cet article : C. Vallée et al., C. R. Mecanique 332 (2004).     

Numerical simulation of corner singularities: a paradox in Maxwell-like problems

This paper sums up some recent studies related to the numerical solution of boundary value problems deriving from Maxwell's equations. These studies bring to light the theoretical origins of the ‘corner paradox’ pointed out by numerical experiments for years: In a domain surrounded by a perfect conductor, a ‘nodal’ discretization can approximate the electromagnetic field only if the domain has no reentrant corners or edges. The explanation lies in a mathematical curiosity: two different interpretations of the same variational equation, which are both well-posed and lead either to the physical or a spurious solution! Two strategies which were recently proposed to remedy this flaw of nodal elements are described. To cite this article: C. Hazard, C. R. Mecanique 330 (2002) 57–68