The Hilbert number of a class of differential equations

The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$.     

An Efficient Algorithm for the Knapsack Sharing Problem

The Knapsack Sharing Problem (KSP) is an NP-Hard combinatorial optimization problem, admitted in numerous real world applications. In the KSP, we have a knapsack of capacity c and a set of n objects, namely N, where each object j, j = 1,...,n, is associated with a profit p _j and a weight w _j. The set of objects N is composed of m different classes of objects J _i, i = 1,...,m, and N = ^m _i=1 J _i. The aim is to determine a subset of objects to be included in the knapsack that realizes a max-min value over all classes.In this article, we solve the KSP using an approximate solution method based upon tabu search. First, we describe a simple local search in which a depthparameter and a tabu list are used. Next, we enhance the algorithm by introducing some intensifying and diversifying strategies. The two versions of the algorithm yield satisfactory results within reasonable computational time. Extensive computational testing on problem instances taken from the literature shows the effectiveness of the proposed approach.     

Photosensitive polymers with cinnamate units in the side position of chains: Synthesis, monomer reactivity ratios and photoreactivity

Ethyl α-cyano-4-(methacryloxy)cinnamate was synthesized to prepare new photosensitive polymers. Homopolymerization and copolymerizations were carried out in solution in chloroform at 65 °C using AIBN as an initiator. Methyl methacrylate was chosen as comonomer for the studies of copolymerzation. The structures formed were characterized by IR and ¹H NMR spectroscopies. The values of reactivity ratios of the comonomers calculated according to the Finneman-Ross method indicated that the reactivity of the synthesized photosensitive monomer is lower than that of MMA. Thin films of the obtained polymers were prepared to study their ability to crosslink under UV irradiation. The photodimerization of the cinnamate moieties was characterized.     

Semilinear evolution inclusions with nonlocal conditions

We discuss conditions for the existence of at least one solution of semilinear evolution inclusions with a nonlocal condition and a nonconvex right-hand side. Our technique is based on fixed point theorems for multivalued maps.     

Spectral properties and time asymptotic behaviour of linear transport equations in slab geometry

The goal of this paper is to analyse some spectral properties of time‐dependent monoenergetic linear transport equation with perfect reflecting boundary conditions.It is proved that the streaming operator generates a strongly continuous group and its explicit expression is derived. This enables us to investigate in details the compactness properties of the second‐order remainder term of the Dyson–Phillips expansion for a large class of collision operators. The validity of the obtained results for perfect periodic boundary conditions is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)     

Nonlinear three-point boundary value problems

In the present work, we investigate nonlinear three-point boundary value problems (second-order differential equations with boundary conditions imposed at three consecutive points) in an effort to understand the conditions that must be imposed upon nonlinear perturbations of a linear problem in order to obtain a solution of a three-point problem. Both the noncritical case and the critical case are considered. In the first case, existence and uniqueness results are obtained, and in the second case we deal with local bifurcation problems when the nonlinearities are quadratic and cubic.     

Numerical approximation for a non-linear membrane problem

We present a numerical study of large deformations of non-linearly elastic membranes. We consider the non-linear membrane model obtained by Le Dret and Raoult using Γ-convergence, in the case of a Saint Venant-Kirchhoff bulk material. We consider conforming P₁ and Q₁ finite element approximations of the membrane problem and use a non-linear conjugate gradient algorithm to minimize the discrete energy. We present numerical tests including membranes subjected to live pressure loads.