The asymptotic average-shadowing property and transitivity for flows

The asymptotic average-shadowing property is introduced for flows and the relationships between this property and transitivity for flows are investigated. It is shown that a flow on a compact metric space is chain transitive if it has positively (or negatively) asymptotic average-shadowing property and a positively (resp. negatively) Lyapunov stable flow is positively (resp. negatively) topologically transitive provided it has positively (resp. negatively) asymptotic average-shadowing property. Furthermore, two conditions for which a flow is a minimal flow are obtained.     

The average-shadowing property and strong ergodicity

Let X be a compact metric space and f:X→X be a continuous map. In this paper, we prove that if f has the average-shadowing property and the minimal points of f are dense in X, then f is weakly mixing and totally strongly ergodic. As applications we obtain that if f is a distal or Lyapunov stable map having the average-shadowing property, then X is consisting of one point. Moreover, we illustrate that the full shift has the average-shadowing property.     

The average-shadowing property and topological ergodicity

In this note we prove that a Lyapunov stable map having the average-shadowing property from a compact metric space onto itself is topologically ergodic, but it is not topologically weakly mixing.     

提升和投射具有伪轨跟踪性质或可扩性的连续流

本文证明了紧度量空间上的连续流经提升或投射后，其伪轨跟踪性质及可扩性是不变的；做为应用给出了不定向闭曲面上具有伪轨跟踪性质的Ｃｒ流的特征。     

流的Birkhoff中心及Ω稳定性条件

首先对紧度量空间上的连续流论证了滤子的存在性与无环性的关系,并给出了Birkhoff中心是非游荡集的一个充分条件;然后对流形上的C1流证明了:Birkhoff中心双曲+无环条件公理A+无环条件,因而它是Ω稳定的.     

Runge-kutta schemes for Hamiltonian systems

We study the application of Runge-Kutta schemes to Hamiltonian systems of ordinary differential equations. We investigate which schemes possess the canonical property of the Hamiltonian flow. We also consider the issue of exact conservation in the time-discretization of the continuous invariants of motion.     

Complementation in spaces of continuous functions on compact lines

We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton.     

Subsets of the square with the continuous and order preserving fixed point property

This study characterizes the convex sets whose complements in the unit square exhibit the fixed point property for mappings which are jointly continuous and order preserving. Hence, one can readily construct simple sets with this fixed point property, but which neither have the fixed point property individually for continuous mappings nor for order preserving mappings. This is the first characterization of any non-trivial set with this property.     

General spectral flow formula for fixed maximal domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory. This work was supported in part by The Danish Science Research Council, SNF grant 21-02-0446. The second author is partially supported by FANEDD 200215, 973, Program of MOST, Fok Ying Tung Edu. Funds 91002, LPMC of MOE of China, and Nankai University.     

On shadowing: Ordinary and ergodic

The aim of this paper is to introduce the notion of ergodic shadowing for a continuous onto map which is equivalent to the map being topologically mixing and has the ordinary shadowing property. In particular, we deduce the chaotic behavior of a map with ergodic shadowing property. Moreover, we define some kind of specification property and investigate its relation to the ergodic shadowing property.     

关于weierstrass逼近定理的几点注记

Weierstrass逼近定理是函数逼近论中的重要定理之一,定理阐述了闭区间上的连续函数可以用一多项式去逼近.将该定理进行推广:即使一个函数是几乎处处连续的,也不一定具有与连续函数相类似的逼近性质,但是一个处处不连续的函数却有可能具有这样的性质.证明了定义在闭区间上且与连续函数几乎处处相等的函数具有类似的逼近性质,并给出了weierstrass逼近定理的一个推广应用.     

Remarkable Classes of Unital AM-Spaces

We define and investigate two classes of unital Banach AM-spaces, the elements of which are the sums of continuous functions and discrete functions. Neither class is almost Dedekind σ-complete, although one has the Cantor property. One class has the rather rare property of having a sequentially order continuous norm and we deduce that any C(K) space can be embedded as a sublattice of a C(X) space with a sequentially order continuous norm. Finally we identify the order continuous and sequentially order continuous duals of spaces in these classes, which promise to be a rich source of further examples.     

Obstacle problem for nonlinear parabolic equations

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Hölder continuous, we prove that the solution inherits the same property.     

广义近似空间的粗糙同胚与拓扑同胚

对广义近似空间之间的映射引入并刻画了粗糙连续性和拓扑连续性,探讨了他们的性质及相互关系.证明了两个粗糙连续映射的复合还是粗糙连续的,每个粗糙连续的映射都是拓扑连续的.在此基础上引入了粗糙同胚性质和拓扑同胚性质的概念,证明了拓扑同胚性质均为粗糙同胚性质并考察了广义近似空间的诸如分离性、连通性、紧性等的粗糙同胚不变性和拓扑同胚不变性.     

On the class of disjoint limited completely continuous operators

We introduce and study new class of sets (almost L-limited sets). Also, we introduce new concept of property in Banach lattice (almost Gelfand–Phillips property) and we characterize this property using almost L-limited sets. On the other hand, we introduce the class of disjoint limited completely continuous operators which is a largest class than that of limited completely continuous operators, we characterize this class of operators and we study some of its properties.     

A seminorm with square property on a complex associative algebra is submultiplicative

The result stated in the title is proved as a consequence of an appropriate generalization replacing the square property of a seminorm with a similar weaker property which implies an equivalence to the supnorm of all continuous functions on a compact Hausdorff space also.

     

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ? is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ? such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.     

Conjugate Functions for Convex and Nonconvex Duality

We study conjugate duality with arbitrary coupling functions. Our only tool is a certain support property, which is automatically fulfilled in the two most widely used special cases, namely the case where the underlying space is a topological vector space and the coupling functions are the continuous linear ones, and the case where the underlying space is a metric space and the coupling functions are the continuous ones. We obtain thereby a simultaneous axiomatic extension of these two classical models. Also included is a condition for global optimality, which requires only the mentioned support property.     

具有跟踪性质的连续半流

在这篇论文中，我们给出了连续半流的跟踪性质与其逆极限的跟踪性质之间的一些等价条件，并且做为应用，我们证明了具有强跟踪性质的连续半流在其游荡集上的限制也具有强跟踪性质以及连续半流的谱分解定理。