Non‐Hermitian perturbations of Hermitian matrix‐sequences and applications to the spectral analysis of the numerical approximation of partial differential equations

This article concerns the spectral analysis of matrix‐sequences which can be written as a non‐Hermitian perturbation of a given Hermitian matrix‐sequence. The main result reads as follows. Suppose that for every n there is a Hermitian matrix X_n of size n and that {X_n}_n～_λf, that is, the matrix‐sequence {X_n}_n enjoys an asymptotic spectral distribution, in the Weyl sense, described by a Lebesgue measurable function f; if $\Vert Y_n{\Vert }_2=o(\sqrt{n})$ with ‖·‖₂ being the Schatten 2 norm, then {X_n+Y_n}_n～_λf. In a previous article by Leonid Golinskii and the second author, a similar result was proved, but under the technical restrictive assumption that the involved matrix‐sequences {X_n}_n and {Y_n}_n are uniformly bounded in spectral norm. Nevertheless, the result had a remarkable impact in the analysis of both spectral distribution and clustering of matrix‐sequences arising from various applications, including the numerical approximation of partial differential equations (PDEs) and the preconditioning of PDE discretization matrices. The new result considerably extends the spectral analysis tools provided by the former one, and in fact we are now allowed to analyze linear PDEs with (unbounded) variable coefficients, preconditioned matrix‐sequences, and so forth. A few selected applications are considered, extensive numerical experiments are discussed, and a further conjecture is illustrated at the end of the article.     

Markov-modulated Brownian motions perturbed by catastrophes

ABSTRACT

We consider a one-sided Markov-modulated Brownian motion perturbed by catastrophes that occur at some rates depending on the modulating process. When a catastrophe occurs, the level drops to zero for a random recovery period. Then the process evolves normally until the next catastrophe. We use a semi-regenerative approach to obtain the stationary distribution of this perturbed MMBM. Next, we determine the stationary distribution of two extensions: we consider the case of a temporary change of regime after each recovery period and the case where the catastrophes can only happen above a fixed threshold. We provide some simple numerical illustrations.     

关于平稳高斯过程的上穿过期望次数的几点注记

设｛Ｘ（ｔ），０≤ｔ≤Ｔ＜＋∞｝是平稳高斯过程（可以是均方不可微的，即二阶谱矩可以是无限的），本文着重讨论当ｎ→∞时，过程上穿过ｕ的期望次数的渐近性质．     

An exactly solved model of three-dimensional surface growth in the anisotropic KPZ regime

We generalize the surface growth model of Gates and Westcott to arbitrary inclination. The exact steady growth velocity is of saddle type with principal curvatures of opposite sign. According to Wolf, this implies logarithmic height correlations, which we prove by mapping the steady state of the surface to world lines of free fermions with chiral boundary conditions.     

A method for portfolio choice

This paper shows how one can use the theory of hidden Markov models for portfolio optimization. We illustrate our method by a ball and urn experiment. An application to historical data is examined. Copyright © 2003 John Wiley & Sons, Ltd.     

The conjugate gradient method for computing all the extremal stationary probability vectors of a stochastic matrix

Summary The conjugate gradient method is developed for computing stationary probability vectors of a large sparse stochastic matrixP, which often arises in the analysis of queueing system. When unit vectors are chosen as the initial vectors, the iterative method generates all the extremal probability vectors of the convex set formed by all the stationary probability vectors ofP, which are expressed in terms of the Moore-Penrose inverse of the matrix (P−I). A numerical method is given also for classifying the states of the Markov chain defined byP. One particular advantage of this method is to handle a very large scale problem without resorting to any special form ofP. The Institute of Statistical Mathematics     

非完整系统Hamilton原理的驻值特性

本文论证了非完整系统的Hamilton原理的驻值特性，指出运用Appell－Chetaev条件的Hamilton原理和不应用appell-Chetaev条件的Hamilton原理都是驻值变分原理．并且，讨论了几个有关问题．     

Discrete distributions of orderk on a binary sequence

Summary This paper considers discrete distributions of orderk based on a binary sequence which is defined as an extension of independent trials with a constant success probability and is more practical than the independent trials. Some results on calculation of probabilities and characteristics of the distributions are obtained as well as their formal expressions. Examples and an application are also given. The Institute of Statistical Mathematics     

On a characterization question for symmetric random variables

If X₁, X₂ are independent with common density g symmetric about zero, then $\text{P(X}{}_1\text{+ αX}{}_2\text{> 0) =}\text{1}\text{2}$ for all real α. We provide a counter example to show that the converse is false and thus settle a question posed by Burdick (1972).