Higher Laplacians on pseudo-Hermitian symmetric spaces

One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians $L_m$ are defined for each positive integer m, where $L_1$ coincides with the Laplace–Beltrami operator. Despite their natural definition, these higher Laplacians have not yet been studied in detail. In this paper, we consider the setting of simple pseudo-Hermitian symmetric spaces, i.e., let $X=G/H$ be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. We show that the higher Laplacians $L_1,L_3,\dots ,L_{2r?1}$ form a set of algebraically independent generators for the algebra ${\mathbb{D}}_G(X)$ of G-invariant differential operators on X, where r denotes the rank of X. For higher rank, this is the first instance of a set of generators for ${\mathbb{D}}_G(X)$ defined explicitly in purely geometric terms, and confirms a conjecture of Engli? and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces.     

Topological Fixed Point Theorems Do Not Hold for Random Dynamical Systems

The aim of this paper is to demonstrate that topological fixed point theorems have no canonical generalization to the case of random dynamical systems. This is done by using tools from algebraic ergodic theory. We give a criterion for the existence of invariant probability measures for group valued cocycles. With that, examples of continuous random dynamical systems on a compact interval without random invariant points, which are an appropriate generalization of fixed points, are constructed.     

Polynomial Asymptotic Representation of Subharmonic Functions in a Half-Plane

Let u(z) be a subharmonic function in a half-plane such that its Riesz measure is concentrated on the finite system of rays.In the paper the connection between the behavior of u(z) and the distribution of its measure (including boundary measure) is investigated in terms of polynomial asymptotic representations.     

Jacobi polynomials and design theory I

In this paper, we introduce the notion of Jacobi polynomials of a code with multiple reference vectors, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold polarization operator. Finally, we describe some facts obtained from Type III and Type IV codes that interpret the relation between the Jacobi polynomials and designs.     

Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure—measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples—classes of homogeneous integral operators on various metric spaces—and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article—various specific cases—illustrate general statements and results given in the paper and at the same time are of interest in their own way.     

基于成本优先级的部件维修策略分析

现实中,系统由于任务、环境等因素,无法实时对故障部件进行维修。因此需要在任务间隔期间或对故障部件进行维修的同时对系统各部件进行预防性机会维修。本文考虑系统期望维修成本,提出了基于部件维修优先级的预防性维修策略。首先把系统期望维修成本分为失效部件维修成本、失效部件导致系统故障的成本和预防性维修其他部件的成本,提出了基于成本的二态和多态系统部件维修优先级度量方法,并在两种场景下分析了如何选择预防性维修部件。其次针对多态系统,研究了基于成本重要度的部件最佳维修水平,并讨论了成本约束下的部件预防性维修策略。最后以某型预警机系统为例进行验证,结果表明,基于成本的预防性维修策略不仅与故障部件位置和相关成本有关,而且还与可用于预防性维修的其他部件重要性有关。