Eléments finis stochastiques en élasticité linéaireA stochastic finite element method in linear mechanics

The stochastic finite element method presented in this Note consists in representing in a probabilistic form the response of a linear mechanical system whose material properties and loading are random. Each input random variable is expanded into a Hermite polynomial series in standard normal random variables. The response (e.g., the nodal displacement vector) is expanded onto the so-called polynomial chaos. The coefficients of the expansion are obtained by a Galerkin-type method in the space of probability. To cite this article: B. Sudret et al., C. R. Mecanique 332 (2004).     

Critical periods of perturbations of reversible rigidly isochronous centers

In this paper we discuss bifurcation of critical periods in an m-th degree time-reversible system, which is a perturbation of an n-th degree homogeneous vector field with a rigidly isochronous center at the origin. We present period-bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function. This procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods. Applying our procedure to the case n=m=2 we determine the maximum number of critical periods and their location; to the case n=m=3 we investigate the bifurcation of critical periods up to the first order in ε and obtain the expression of the second period-bifurcation function when the first one vanishes.     

Effective partitioning method for computing generalized inverses and their gradients

We extend the algorithm for computing {1}, {1, 3}, {1, 4} inverses and their gradients from [11] to the set of multiple-variable rational and polynomial matrices. An improvement of this extension, appropriate to sparse polynomial matrices with relatively small number of nonzero coefficient matrices as well as in the case when the nonzero coefficient matrices are sparse, is introduced. For that purpose, we exploit two effective structures form [6], which make use of only nonzero addends in polynomial matrices, and define their partial derivatives. Symbolic computational package MATHEMATICA is used in the implementation. Several randomly generated test matrices are tested and the CPU times required by two used effective structures are compared and discussed.     

Explicit solution of the finite time L2-norm polynomial approximation problem

The aim of this paper is to present a new approach to the finite time L₂-norm polynomial approximation problem. A new formulation of this problem leads to an equivalent linear system whose solution can be investigated analytically. Such a solution is then specialized for a polynomial expressed in terms of Laguerre and Bernstein basis.     

Asymptotic properties of fractional delay differential equations

In this paper we study the asymptotic properties of d-dimensional linear fractional differential equations with time delay. We present necessary and sufficient conditions for asymptotic stability of equations of this type using the inverse Laplace transform method and prove polynomial decay of stable solutions. Two examples illustrate the obtained analytical results.     

Regenerative cascade homotopies for solving polynomial systems

A key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set. In many problems involving a solution set of a polynomial system, the witness superset contains all the needed information. Sommese and Wampler gave the first numerical method to compute witness supersets, based on dimension-by-dimension slicing of the solution set by generic linear spaces, followed later by the cascade homotopy of Sommese and Verschelde. Recently, the authors of this article introduced a new method, regeneration, to compute solution sets of polynomial systems. Tests showed that combining regeneration with the dimension-by-dimension algorithm was significantly faster than naively combining it with the cascade homotopy. However, in this article, we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets. Computational tests give strong evidence that regenerative cascade is superior in practice to previous methods.     

An efficient linearization technique for mixed 0-1 polynomial problem

This paper addresses a new and efficient linearization technique to solve mixed 0-1 polynomial problems to achieve a global optimal solution. Given a mixed 0-1 polynomial term z=c_tx₁x₂…x_ny, where x₁,x₂,…,x_n are binary (0-1) variables and y is a continuous variable. Also, c_t can be either a positive or a negative parameter. We transform z into a set of auxiliary constraints which are linear and can be solved by exact methods such as branch and bound algorithms. For this purpose, we will introduce a method in which the number of additional constraints is decreased significantly rather than the previous methods proposed in the literature. As is known in any operations research problem decreasing the number of constraints leads to decreasing the mathematical computations, extensively. Thus, research on the reducing number of constraints in mathematical problems in complicated situations have high priority for decision makers. In this method, each n-auxiliary constraints proposed in the last method in the literature for the linearization problem will be replaced by only 3 novel constraints. In other words, previous methods were dependent on the number of 0-1 variables and therefore, one auxiliary constraint was considered per 0-1 variable, but this method is completely independent of the number of 0-1 variables and this illustrates the high performance of this method in computation considerations. The analysis of this method illustrates the efficiency of the proposed algorithm.     

一类融合逼近和插值的曲线细分

采用生成多项式为主的方法对一类融合逼近和插值三重细分格式的支撑区间、多项式生成、连续性、多项式再生及分形性质进行了分析,给出并证明了极限曲线C^k连续的充分条件.通过对融合型细分规则中参数变量的适当选择来实现对极限曲线的形状调整,从而衍生出具有良好性质的新格式,并将这类新格式与现有格式进行比较.数值实例表明这类新格式生成的极限曲线具有较好的保形性.     

Zonal Jet Creation from Secondary Instability of Drift Waves for Plasma Edge Turbulence

A new strategy is presented to explain the creation and persistence of zonal flows widely observed in plasma edge turbulence. The core physics in the edge regime of the magnetic-fusion tokamaks can be described qualitatively by the one-state modified Hasegawa-Mima (MHM for short) model, which creates enhanced zonal flows and more physically relevant features in comparison with the familiar Charney-Hasegawa-Mima (CHM for short) model for both plasma and geophysical flows. The generation mechanism of zonal jets is displayed from the secondary instability analysis via nonlinear interactions with a background base state. Strong exponential growth in the zonal modes is induced due to a non-zonal drift wave base state in the MHM model, while stabilizing damping effect is shown with a zonal flow base state. Together with the selective decay effect from the dissipation, the secondary instability offers a complete characterization of the convergence process to the purely zonal structure. Direct numerical simulations with and without dissipation are carried out to confirm the instability theory. It shows clearly the emergence of a dominant zonal flow from pure non-zonal drift waves with small perturbation in the initial configuration. In comparison, the CHM model does not create instability in the zonal modes and usually converges to homogeneous turbulence.     

Many associated primes of powers of primes

We construct families of prime ideals in polynomial rings for which the number of associated primes of the second power (or higher powers) is exponential in the number of variables in the ring.