Equipped absolutely continuous subspaces and stationary construction of the wave operators in the non-self-adjoint scattering theory

For a pair of non-self-adjoint operators with characteristic operator-functions having boundary values in the strong operator topology almost everywhere on the real axis, (local) wave operators are constructed. The investigation is based on the local stationary approach arising from the self-adjoint scattering theory and a certain interpretation of the absolutely continuous subspace as an equipped Hilbert space. Sufficient conditions for the existence of the wave operators are obtained and some of their properties are described. Bibliography:28 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 144–171.     

Scattering problem for a slowly decreasing potential that is periodically dependent on time

The multidimensional Schrödinger operator with a periodic timedependent potential is considered. The potential may decrease slowly with respect to the space variable and its mean value with respect to time is equal to zero. It is shown that for the pair H₀=?Δ, H, there exist wave operators, and their ranges coincide with the absolutely continuous subspace of the corresponding monodromy operator.     

度量射影的Lipschitz连续选择

It has shown that, for Lipschitz continuous metric projections, it has a Lips-chitz continuous selection for l-Chebyshev subspace and, for each subspace M which is ismorphism to a Hilbert space ,P_M admits a selection s which is locally Lipschitz continuous with index 1/2.     

Absolutely continuous spectrum of fourth order difference equations

We have studied the absolutely continuous spectrum of a selfadjoint subspace extension generated by a three‐term fourth order difference equation with bounded coefficients using subspace theory. In particular, we have shown that the absolutely continuous spectrum exists outside a certain bounded interval. In addition, we have computed the spectral multiplicity as well as the location of absolutely continuous spectrum of selfadjoint subspace extension under certain asymptotic conditions.     

Metric projection onto finite-dimensional subspaces of C and L

In the space C(Q) of real functions that are continuous on the compact set Q, a finite-dimensional subspace P will have a uniformly continuous metric projection if and only if Q is a finite sum of compact sets Qi, and either P is on each Qi a one-dimensional Chebyshev space, or x(t)≡0 for any x belonging to P. The metric projection onto any finite-dimensional subspace of the space L[a, b] of real integrable functions is not uniformly continuous.     

自反算子代数的导子和同构

本文证明了：Ｂａｎａｃｈ空间上完全分配格代数间的导子都是自动连续的；进而证明了套代数的可加导子是内的，套代数间的代数同构是自动连续的、空间的     

Completely continuous and related multilinear operators

Completely continuous multilinear operators are defined and their properties investigated. This class of operators is shown to form a closed multi-ideal. Unlike the linear case, compact multilinear operators need not be completely continuous. The completely continuous maps are shown to be the closure of a subspace of the finite rank operators. Hilbert-Schmidt operators are also considered. An application to finding error bounds for solutions of multipower equations is presented.     

Generic Local Uniqueness and Stability in Polarization Tomography

The problem of polarization tomography is considered on a Riemannian manifold. This problem comes from the physical problem of recovering the anisotropic part of the dielectric permittivity tensor of a quasi-isotropic medium from polarization measurements made around the boundary, but is more general. In greater than three dimensions local uniqueness and stability are established for generic background metrics, and near generic tensor fields through the study of a related linear inverse problem. The same results are established on a natural subspace of tensor fields in dimension three.     

Interaction of Vortical and Acoustic Waves: From General Equations to Integrable Cases

The equations of the (2+1)-dimensional boundary-layer perturbation split into eigenmodes: a vortex wave and two acoustic waves. We assume that the equations of state (Taylor series approximation) are arbitrary. We realize a mode definition via local-relation equations extracted from the linearization of the general system over the boundary-layer flow. Each such link determines an invariant subspace and the corresponding projector. We examine the nonlinear equation for a vortex wave using a special orthogonal coordinate system based on streamlines. The equations for the orthogonal curves are linked to the Laplace equations via Laplace and Moutard transformations. The nonlinearity determines the proper form of the interaction between vortical and acoustic boundary-layer perturbation fields fixed by projecting to a subspace of the Orr-Sommerfeld equation solutions for the Tollmienn-Schlichting (linear vortical) wave and by the corresponding procedure for the acoustic wave. We suggest a new mechanism for controlling the nonlinear resonance of the Tollmienn-Schlichting wave by sound via a four-wave interaction.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 171–181, July, 2005.     

(3+1)维短波方程的不变子空间和精确解

利用不变子空间方法研究了(3+1)维短波方程的不变子空间和精确解.在(2+1)维短波方程增加一维的情形下,构造了更加广泛的精确解,同时也得到了超曲面的爆破解.主要结果不仅推广了不变子空间理论在高维非线性偏微分方程中的应用,而且对研究高维方程的动力系统有重要意义.     

On continuous restrictions of measurable multilinear mappings

This article deals with measurablemultilinear mappings on Fréchet spaces and analogs of two properties which are equivalent for a measurable (with respect to gaussian measure) linear functional: (i) there exists a sequence of continuous linear functions converging to the functional almost everywhere; (ii) there exists a compactly embedded Banach space X of full measure such that the functional is continuous on it. We show that these properties for multilinear functions defined on a power of the space X are not equivalent; but property (ii) is equivalent to the apparently stronger condition that the compactly embedded subspace is a power of the subspace embedded in X.     

Essential and continuous spectrum of symmetric difference equations

Essential and continuous spectrum of symmetric difference equations have been investigated. It has been shown that the deficiency indices and the existence of these components of the spectrum are determined by the growth conditions of the coefficients. In particular, the deficiency indices are superimposition of those clusters determined by the coefficient growth. Finally, we have proved the neccessary and sufficient conditions for the existence of essential spectrum of selfadjoint subspace extensions using subspace theory and asymtotic summation.     

电磁波在三相土介质中的传播

研究电磁波(EM波)在土中的传播特性,必须获得如介电常数和磁导率等基本电磁参数.在麦克斯韦方程组的基础上,提出了等效模型来计算土体介电常数.利用该模型的计算结果与实测数据符号很好,可以为利用电磁波法测量土体含水量,土体的CT分析以及土工环境监测奠定理论基础.     

On the ergodic limits for unitary periodic propagators

We apply the Wiener theorem and the mean ergodic theorem to establish several results on the ergodic limits for unitary periodic propagator These results can be used to analyze the continuous subspace and pure point subspace of the associated Floquet operator.     

Nonlinear and linear instability of the Rossby-Haurwitz wave

As is known, the large-scale dynamics of barotropic atmosphere can approximately be described by the nonlinear barotropic vorticity equation. It is also well known that the Rossby-Haurwitz (RH) waves, being exact solutions to this equation, represent one of the main features of meteorological fields. Therefore the stability properties of the RH wave are of considerable interest for deeper understanding of the low-frequency variability of the atmosphere. Many works has been devoted to the barotropic instability of flows on a beta-plane and a sphere. However, mathematically, the nonlinear stability problem of the RH wave is still far from its complete solution. Indeed, some of the stability results have been obtained numerically, and hence, contain calculation errors. Severe truncation of perturbations used in the spectral stability analysis, though leads to interesting and useful conclusions, does not allow obtaining comprehensive results. The weak point of some analytical nonlinear instability studies consists in using inappropriate norms for perturbations. It should also be noted that a necessary condition for the linear instability of the RH wave was obtained only recently (Skiba, 2000). In the present work, the nonlinear stability of the RH wave in an ideal incompressible fluid on a rotating sphere is analytically studied. Let H(n) be a subspace of homogeneous spherical polynomials of degree n. Mathematically, a RH wave of degree n is the sum of a super-rotating flow of subspace H(1) and a homogeneous spherical polynomial of subspace H(n). First, we derive a conservation law for arbitrary RH-wave perturbations which asserts that any perturbation evolves in such a way that its kinetic energy E(t) and enstrophy q(t) decrease, remain constant or increase simultaneously. The law is used to divide all the perturbations into three invariant sets depending on the value of their mean spectral number k(t)=q(t)/E(t) introduced by Fjortoft (1953). These sets are denoted as M where k(t)¡n(n+1) (large-scale perturbations), N where k(t)¿n(n+1) (small-scale perturbations), and Z where k(t)=n(n+1) (boundary surface between the sets M and N). Note that Z includes one more invariant set, namely, the subspace H(n). The existence of invariant sets of perturbations allows us to study the RH wave instability in each set separately. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sufficient dimension reduction for the conditional mean with a categorical predictor in multivariate regression

Recent sufficient dimension reduction methodologies in multivariate regression do not have direct application to a categorical predictor. For this, we define the multivariate central partial mean subspace and propose two methodologies to estimate it. The first method uses the ordinary least squares. Chi-squared distributed statistics for dimension tests are constructed, and an estimate of the target subspace is consistent and efficient. Moreover, the effects of continuous predictors can be tested without assuming any model. The second method extends Iterative Hessian Transformation to this context. For dimension estimation, permutation tests are used. Simulated and real data examples for illustrating various properties of the proposed methods are presented.     

On the Stone and Bishop appoximation theorems

Generalizations of the Stone-Weierstrass and Bishop approximation theorems are presented. Given an algebra, a subspace in a continuous function space coinciding with the closure of this algebra is constructed. Analogs of these results are obtained in the case where the set of functions under consideration is not an algebra, but its closure is related to some algebra.     

Hilbert空间的正交分解及其应用

基于Riesz-表示算子,给出了实Hilbert内积空间按某种连续双线性泛函的正交分解的刻划,应用于鞍点变分问题,获得了解的分离及其强制型于问题.     

Continuity properties of Markov semigroups and their restrictions to invariant <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces

We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L ¹-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition.