A method for calculating invariant subspaces of symmetric hyperbolic equations

An algorithm is constructed for calculating invariant subspaces of symmetric hyperbolic systems arising in electromagnetic, acoustic, and elasticity problems. Discrete approximations are calculated for subspaces that correspond to minimal eigenvalues and smooth eigenfunctions. Difficulties related to the presence of an infinite-dimensional kernel in the differential operator are successfully handled. The efficiency of the algorithm is demonstrated using acoustics equations.     

Krylov subspaces and the analytic grade

Typical behaviour of the solution of a linear system of equations obtained iteratively by Krylov methods can be characterized by three stages. Initially the residual diminishes steadily; this is followed by stagnation and finally rapid convergence near the algebraic grade. This study examines this behaviour in terms of the concepts of approximately invariant subspace and what we have called the analytic grade of a Krylov sequence. It is shown how the small Ritz values play a vital role in the convergence and how this knowledge helps in the construction of an effective preconditioner. Copyright © 2004 John Wiley & Sons, Ltd.     

RICE CONDITION NUMBERS OF CERTAIN CHARACTERISTIC SUBSPACES

This paper proposes the Rice condition numbers for invariant subspace, singular sub-spaces of a matrix and deflating subspaces of a regular matrix pair. The first-order perturbation estimations for these subspaces are derived by applying perturbation expansions of orthogonal projection operators.     

关于算子方程AX=XAX与AX=XA=XAX

用A的不变子空间作参数，给出了算子方程AX＝XAX的全部解。当A是单射或稠值域时，或者当A是正规算子时，给出了算子方程AX＝XA＝XAX的全部解。我们还给出正规算子X是算子方程AX＝XZ＝XAX的解的充分必要条件。     

Point-to-Periodic and Periodic-to-Periodic Connections

In this work we consider computing and continuing connecting orbits in parameter dependent dynamical systems. We give details of algorithms for computing connections between equilibria and periodic orbits, and between periodic orbits. The theoretical foundation for these techniques is given by the seminal work of Beyn in 1994, “On well-posed problems for connecting orbits in dynamical systems”, where a numerical technique is also proposed. Our algorithms consist of splitting the computation of the connection from that of the periodic orbit(s). To set up appropriate boundary conditions, we follow the algorithmic approach used by Demmel, Dieci, and Friedman, for the case of connecting orbits between equilibria, and we construct and exploit the smooth block Schur decomposition of the monodromy matrices associated to the periodic orbits. Numerical examples illustrate the performance of the algorithms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Boundary behavior in Hilbert spaces of vector-valued analytic functions

In this paper we study the boundary behavior of functions in Hilbert spaces of vector-valued analytic functions on the unit disc D. More specifically, we give operator-theoretic conditions on Mz, where Mz denotes the operator of multiplication by the identity function on D, that imply that all functions in the space have non-tangential limits a.e., at least on some subset of the boundary. The main part of the article concerns the extension of a theorem by Aleman, Richter and Sundberg in [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (2007)] to the case of vector-valued functions.     

Fields as Kolmogzrov flows

It is widely recognized that for highly unstable dynamical systems there exists a fundamental limitation on predictability and determinism. An important class of such highly unstable systems is the class of K-flow, which is further characterized by the existence of time-asymmetric objects in the form of K-partition. Our recent approach to the problem of irreversibility has shown that when the implications of the limitation on determinism arising from strong form of instability and those of the existence of K-partition are consistently taken into account, one is naturally led from the physically unrealizable deterministic evolution of phase points to an entropy-increasing stochastic Markovian evolution. Furthermore, this transition is not the result of extraneously imposed coarse graining and/or approximation schemes, but can be brought about by an invertible transformation whose existence and construction are determined by the nature of the instability of the dynamical system itself. After a brief review of this theory which also contains some relatively new remarks, we prove that classical Klein-Gordon field (both massive and massless) possess the structure of K-flow. This seems to provide the first examples of relativistic systems that are K-flows. Some of the implications of this result are briefly discussed. From a mathematical point of view, this seems to be a first step toward an ergodic theory of partial differential equations. In the process, we also provide an independant group-theoretic proof of the existence of incoming and outgoing subspaces of the scattering theory of Lax and Phillips for the wave equation.     

A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces

Bearing in mind the insight into the Hohenberg–Kohn theorem for Coulomb systems provided recently by Kryachko (Int J Quantum Chem 103:818, 2005), we present a re-statement of this theorem through an elaboration on Lieb’s proof as well as an extension of this theorem to finite subspaces. Contribution to the Serafin Fraga Memorial Issue.     

Dirichlet空间上的超循环复合算子

设Bn是复平面C中的单位圆盘(n=1)或复空间Cn中的单位球.众所周知,在Hardy空间上存在丰富的符号在Aut(Bn)中的超循环复合算子.然而,在复平面中单位圆盘上的Dirichlet空间中, 任何复合算子都不能是超循环的.本文则证明,当n>1时,Bn上的Dirichlet空间中确有超循环复合算子.     

投影的约化极小模和空间的夹角

Let M and N be nonzero subspaces of a Hilbert space H, and PM and PN denote the orthogonal projections on M and N, respectively. In this note, an exact representation of the angle and the minimum gap of M and N is obtained. In addition, we study relations between the angle, the minimum gap of two subspaces M and N, and the reduced minimum modulus of （I - PN）PM,