On Martingale Approximation of Adapted Processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.     

Stabilization of the stationary motions of non-holonomic mechanical systems

The possible stabilization of the unstable stationary motions of a non-holonomic system is studied from the standpoint of general control theory /1, 2/. As distinct from the case previously considered /3/, when forces of a certain structure are applied with respect to both positional and cyclical coordinates, the stabilization is obtained here by applying control forces only with respect to the cyclical coordinates /4/; the control forces may be applied with respect to some or all of the cyclical coordinates, and depend on the positional coordinates, the velocities, and the corresponding cyclical momenta. It is shown that, just as in the case of holonomic systems /5, 6/, depending on the control properties of the corresponding linear subsystem, the stationary motions, whether stable or unstable, can be stabilized, up to asymptotic stability with respect to all the phase variables, or asymptotic stability with respect to some of the phase variables and stability with respect to the remaining variables. The type of stabilization with respect to the given phase variables depends on the Lyapunov transformations which are needed in order to reduce the critical cases obtained to singular cases /7, 8/.     

Sufficient conditions for stochastic equality of two distributions under some partial orders

We establish some conditions for stochastic equality of two nonnegative random variables which are ordered with respect to variability ordering or with respect to mean residual life ordering or with respect to second order stochastic ordering.     

Conditional orderings and positive dependence

Every univariate random variable is smaller, with respect to the ordinary stochastic order and with respect to the hazard rate order, than a right censored version of it. In this paper we attempt to generalize these facts to the multivariate setting. It turns out that in general such comparisons do not hold in the multivariate case, but they do under some assumptions of positive dependence. First we obtain results that compare the underlying random vectors with respect to the usual multivariate stochastic order. A larger slew of results, that yield comparisons of the underlying random vectors with respect to various multivariate hazard rate orders, is given next. Some comparisons with respect to the orthant orders are also discussed.     

On the properties of orthorecursive expansions with respect to subspaces

In a Hilbert space, for orthorecursive expansions with respect to closed subspaces, we establish a criterion for expansions of elements of a certain finite-dimensional subspace with respect to a finite sequence of subspaces to coincide with the expanded elements. This implies a criterion for an element to be equal to its orthorecursive expansion with respect to a finite sequence of subspaces. We also obtain a number of results related to the best approximations of elements by partial sums of their orthorecursive expansions with respect to a sequence of finite-dimensional subspaces.     

Abelian groups that are small with respect to different classes of groups

In this paper, we study Abelian groups that are small with respect to different classes of groups. Completely decomposable torsion free groups that are small with respect to an arbitrary class of torsion free groups are described completely. Direct products of groups small with respect to the class of slender groups are derived.     

关于非完整力学系统相对部分变量的稳定性*

本文给出研究非完整系统相对部分变量稳定性的一种方法,并得到非完整系统相对部分变量的一些稳定性定理:同时,本文还得到一类非完整系统相对全部变量稳定性与相对部分变量稳定性的关系。     

Vector valued Fourier analysis on unimodular groups

The notion of Fourier type and cotype of linear maps between operator spaces with respect to certain unimodular (possibly nonabelian and noncompact) group is defined here. We develop analogous theory compared to Fourier types with respect to locally compact abelian groups of operators between Banach spaces. We consider the Heisenberg group as an example of nonabelian and noncompact groups and prove that Fourier type and cotype with respect to the Heisenberg group implies Fourier type with respect to classical abelian groups. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ultimate boundedness with respect to part of the variables of solutions of systems of differential equations with partly controlled initial conditions

We introduce the notions of equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables for solutions with partly controlled initial conditions. We obtain sufficient conditions for the equiultimate boundedness and uniform ultimate boundedness with respect to part of the variables of solutions with partly controlled initial conditions. We introduce the notions of equiboundedness and uniform boundedness with respect to part of the variables for solutions of systems with partly controlled initial conditions. We obtain sufficient conditions for the equiboundedness and uniform boundedness with respect to part of the variables of solutions with partly controlled initial conditions.     

具固定脉冲时刻的脉冲微分系统关于部分变元的指数稳定性

研究了具固定脉冲时刻的脉冲微分系统关于部分变元的指数稳定性,得到了保证零解关于部分变元指数稳定的充分条件,并给出了关于部分变元稳定性的一个新的判定准则.最后给出了其相关例子.     

On the local semiconcavity of the solutions of the eikonal equation

We show that if the Hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave. Furthermore, in the 1D case, we show that every viscosity solution of the eikonal equation is semiconcave if and only if the Hamiltonian is Lipschitz continuous with respect to the state variable.     

Temps de vie des solutions d'un problème de Cauchy non linéaire

We solve the global Cauchy problem, with small initial data, in the space of the holomorphic functions with respect to t and Gevrey class with respect to x. We establish the existence and the stability of the solution to Cauchy problem with nul initial data without hyperbolicity hypothesis. In the stationary case, we give estimates of life span of the solutions with respect to size of the initial data.     

Unstable motivic homotopy categories in Nisnevich and cdh-topologies

The motivic homotopy categories can be defined with respect to different topologies and different underlying categories of schemes. For a number of reasons (mainly because of the Gluing Theorem) the motivic homotopy category of smooth schemes with respect to the Nisnevich topology plays a distinguished role but in some cases it is desirable to be able to work with all schemes instead of the smooth ones. In this paper we prove that, under the resolution of singularities assumption, the unstable motivic homotopy category of all schemes over a field with respect to the cdh-topology is almost equivalent to the unstable motivic homotopy category of smooth schemes over the same field with respect to the Nisnevich topology.     

On a Malkin — Massera theorem : PMM vol. 43, no. 6, 1979 pp. 975–979

The results obtained in [1, 2] are complemented by an assertion on asymptotic stability uniform with respect to t₀, x₀, and also are extended to the problems of asymptotic stability with respect to a part of the variables and of optimal stabilization with respect to a part of the variables.     

Dirichlet-Type Problems for Systems of Partial Differential Equations Unresolved with Respect to the Highest Time Derivative

We establish conditions for the correct solvability of problems for systems of partial differential equations unresolved with respect to the highest time derivative with Dirichlet-type conditions with respect to time and periodic conditions with respect to space variables. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of these problems.     

Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients.     

A Multipoint Problem for Pseudodifferential Equations

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,...,x _p for equations unsolved with respect to the leading derivative with respect to t and containing pseudodifferential operators. We establish conditions for the unique solvability of this problem and prove metric assertions related to lower bounds for small denominators appearing in the course of its solution.     

相对于半对偶化模的强Ding-投射模（英文）

This paper is a study of strongly Ding projective modules with respect to a semidualizing module. The class of strongly Ding flat modules with respect to a semidualizing module is also investigated, and the relationship between strongly Ding projective modules and strongly Ding flat modules with respect to a semidualizing module is characterized.Some well-known results on strongly Ding projective modules, n-strongly Ding projective modules and strongly D_C-projective modules are generalized and unified.     

A geometrical construction of inverse points with respect to an ellipse

In this note, we introduce the inverse point respect to an ellipse, which generalizes the classical inversion in a circle. Specifically, we show a geometrical construction of inverse points with respect to an ellipse. For this, we extend the classical method to construct the inverse of a point respect to a circle.