Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.     

Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

We propose, in this paper, a unified method of generating a regularized integral equation in the double layer potential approach for 3D anisotropic elastodynamics. Our regularization preserves the causality in the time-domain. The method is based on a special decomposition of the hypersingular kernel which appears in the integral representation of the stress tensor.     

On the constrained KP hierarchy

An explanation for the so-called constrained hierarchies is presented by linking them with the symmetries of the KP hierarchy. While the existence of ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP hierarchy to the KdV hierarchies, the existence of additional symmetries allows one to reduce the KP to the constrained KP.     

Some Primitive Concepts in Continuum Mechanics Regarded in Terms of Objective Space-Time Molecular Averaging: The Key Rôle Played by Inertial Observers

In Continuum Mechanics the notions of body, material point, and motion, are primitive. Here these concepts are derived for any (possibly time-dependent) material system via mass and momentum densities whose values are local spacetime averages of molecular quantities. The averaging procedure necessary to ensure molecular-based densities can be agreed upon by all observers (that is, are objective) has implications for constitutive relations. Specifically, such relations should first be expressed in terms of Galilean-invariant functions of the motion relative to an inertial frame. Thereafter such relations can be re-phrased for general observers, thereby yielding general-frame constitutive relations compatible with material frame-indifference. Two postulates concerning observer agreement (which together constitute a statement of material frame-indifference) are shown to imply that any stress response function which is assumed to depend upon the motion in an inertial (general) frame must be Galilean-invariant (invariant under superposed rigid body motions). Accordingly, invariance under superposed rigid body motions is not a fundamental tenet of continuum physics, but rather a consequence of material frame-indifference whenever constitutive dependence upon motion in a general observer frame is postulated.     

Growth factor and expected growth factor of some pivoting strategies

Several definitions of growth factors for Gaussian elimination are compared. Some new pivoting strategies, intermediate between partial pivoting and rook pivoting, are introduced. For random matrices, an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed. A stationary behavior of the expected growth factors of the new pivoting strategies is observed. Bounds for the growth factors of these pivoting strategies are provided. It is also shown that partial pivoting by columns produces small growth factors for matrices appearing in practical observations and for which the growth factors produced by partial pivoting are very large.     

Defining equations of modular curves

We obtain defining equations of modular curves X₀(N), X₁(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X₀(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.     

Classical Yang-Mills-Dirac equations: Qualitative analysis of some solutions with a noncompact symmetry group

We consider classical Yang-Mills and Yang-Mills-Dirac equations on Minkowski space, with gauge group SU(2), and look for solutions invariant (up to a gauge transformation) under SO(3)×SO₀(1, 1) and SO₀(2, 1)×SO(2), respectively. In each case, we analyze the qualitative features of the solutions, in particular the asymptotic behavior as the solution approaches its singularities. The method is based on standard theorems from the theory of nonlinear ordinary differential equations.     

基于法布里-珀罗腔反射光相位特性设计的梳状滤波器

提出了一种用于设计密集波分复用全光网络中梳状滤波器的方法，该器件利用法布里-珀罗腔反射光相位的非线性性质，结合光的相位在一定光程差下随频率呈线性周期变化的特点，通过使这样两束相位匹配的光发生干涉，得到所需滤波特性。运用江膜特征矩阵对模型进行了模拟计算，并分析了计算结果，论证了这种结构的可行性。     

High performance solution of the complex symmetric eigenproblem

Complex symmetric matrices often appear in quantum physics in the solution methods of partial differential equations such as the Schrödinger equation. We have recently introduced a new fast and efficient direct eigensolver for this problem in [4], and reported its performance in the eigenvalue calculation in [3]. In this paper, we further report on some benchmark tests for computing the full and partial eigenspectrum on a variety of super computing machines, i.e., the Cray J-932, the DEC Alfa 8400, and the SGI Power Challenge 8000 and 10000. We observe that in all cases the new algorithm is much faster than codes available in standard state of the art eigensolver packages such as LAPACK.     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002