Furstenberg族与处处混沌及等度连续(英文)

本文引进并研究了Furstenberg族意义下的处处混沌与等度连续的概念.如果一个动力系统是F_1-敏感和F_2-可达的,则称之为(F_1,F_2)-处处混沌的,其中F_1与F_2是Furstenberg族.一个动力系统(X,f)被称为F_1-敏感的,是指存在7>0使得对任意x∈X及x的任意开邻域存在y∈U,有{n∈Z_+:d(f~n(x),f~n(y))>τ}∈F_1成立.一个动力系统(X,f)被称为F_2-可达的,是指对任意的s>O及X的任意非空开集U,V,存在x∈U,y∈V使得{n∈Z_+:d(f~n(x),f~n(y))<ε}∈F_1成立.一个动力系统被称为F-等度连续的,是指对任意的ε>0,存在δ>0,当d(x,y)<δ时有{n∈Z_+:d(f~n(x),f~n(y))<ε}∈F成立,其中F是一个Furstenberg族.     

Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

单子理论对拓扑等度连续与均匀连续的刻画及其应用

在扩大模型下,应用单子理论给出了拓扑等度连续和均匀连续的非标准刻画,并应用均匀连续的非标准特征证明了网收敛与均匀连续之间的关系,最后应用拓扑等度连续和均匀连续的非标准特征证明了拓扑等度连续和均匀连续之间的关系.     

C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module*

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C^*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C^*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of L(H), where L(H) denotes the set of all adjointable operators on H. The C^*-subalgebras of L(H) generated by elements in {P - P ∧ Q, Q - P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π, X e) of i(P, Q, H) such that e π(P - P ∧ Q) = π(P) - π(P) ∧ π(Q),eπ(Q - P ∧ Q) = π(Q) - π(P) ∧ π(Q).When (P, Q) is semi-harmonious, that is, R(P + Q) and R(2I - P - Q) are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I - Q, I - P, H) are unitarily equivalent via a unitary operator in L(H). A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I - Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C^*-modules are also provided.     

Survey on Path-Dependent PDEs*

In this paper,the authors provide a brief introduction of the path-dependent partial differential equations(PDEs for short) in the space of continuous paths,where the path derivatives are in the Dupire(rather than Fréchet) sense.They present the connections between Wiener expectation,backward stochastic differential equations(BSDEs for short) and path-dependent PDEs.They also consider the well-posedness of path-dependent PDEs,including classical solutions,Sobolev solutions and viscosity solution...     

Dual-holomorphic Functions and Problems of Lifts*

The main purpose of this paper is to study the differential geometrical objects on tangent bundle corresponding to dual-holomorphic objects of dual-holomorphic manifold.As a result of this approach, the authors find a new class of lifts(deformed complete lifts)in the tangent bundle.     

A Theory of Orbit Braids*

In this paper, the authors systematically discuss orbit braids in M × I with regards to orbit configuration space FG(M, n), where M is a connected topological manifold of dimension at least 2 with an effective action of a finite group G. These orbit braids form a group, named orbit braid group, which enriches the theory of ordinary braids.The authors analyze the substantial relations among various braid groups associated to those configuration spaces FG(M, n), F(M/G, n) and F(M, n). They also co...     

Bochner-Martinelli Formula for Higher Spin Operators of Several R6 Variables*

The higher spin operator of several R⁶ variables is an analogue of the ■-operator in theory of several complex variables.The higher spin representation of so₆(C)is ⊙^kC⁴ and the higher spin operator D_k acts on ⊙^kC⁴-valued functions.In this paper,the authors establish the Bochner-Martinelli formula for higher spin operator D_k of several R⁶variables.The embedding of R⁶ⁿ into the space of co...     

具有重结点的B-样条曲面G1的连续条件

本文得到了关于两个双三次内部重结点B-样条曲面片G¹连续的充分必要条件和在公共边界线上控制向量的本征条件.这些条件直接由两个B-样条曲面的控制向量表示.文[10]证明了使用内部单结点的双三次B-样条曲面来构造G¹光滑曲面,局部格式不存在.使用本文的这些条件就可以构造出具有局部各式的G¹光滑造型.     

高维Hausdorff算子在Hp(Rn)上的有界性

本文主要研究以下形式的Hausdorff算子H_Φf(x)=∫_RⁿΦ(u₁….,u_n)f(u₁x₁,…,u_nx_n)du₁…du_n,其中Φ是Rⁿ上的缓增分布.当n≥2,0Φ在H^p(Rⁿ)上有界当且仅当Φ≡0.进一步,当n≥2,n/n+1Φ有合适定义,那么H_Φ在H^p(Rⁿ)上有界当且仅当Φ是常数.这些结果都表明Hausdorff算子H_Φ在H^p(Rⁿ)上的有界性很复杂.此外,我们将H_Φ转化成卷积型算子,得到H_Φ在Lebesgue空间上有界的一些新的结果.     

Lattice-Based Cryptography: A Survey*

Most of current public key cryptosystems would be vulnerable to the attacks of the future quantum computers.Post-quantum cryptography offers mathematical methods to secure information and communications against such attacks,and therefore has been receiving a significant amount of attention in recent years.Lattice-based cryptography,built on the mathematical hard problems in(high-dimensional) lattice theory,is a promising post-quantum cryptography family due to its excellent efficiency,moderate s...     

Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.     

Regularity and Compactness of Stationary Map-Varifold Pairs*

The authors introduce the conception of stationary map-varifold pairs and prove a compactness result.As applications,they analyse the asymptotic structure of the pseudo tangent map of stationary harmonic maps.For stationary pair,they also get a strong convergence criterion about the map part and introduce the stratification of the singular set.     

On Descriptions of Products of Simplices*

The authors give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology. In addition, they give a shorter proof of a well known criterion on this subject.     

Reduced Crossed Products of Pro-Banach Algebras*

This paper gives the concept of the reduced pro-Banach algebra crossed product associated with inversely pro-Banach algebra dynamical system, and shows that the reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products. Also, the authors show that if the locally compact group is amenable, then the crossed product and the reduced crossed product are isometrically isomorphic.     

On Fridman Invariants and Generalized Squeezing Functions*

In this paper, the authors introduce the notion of generalized squeezing function and study the basic properties of generalized squeezing functions and Fridman invariants.They also study the comparison of these two invariants, in terms of the so-called quotient invariant.     

Range-Renewal Processes: SLLNs and Power Laws*

Given n samples(viewed as an n-tuple) of a γ-regular discrete distribution π,in this article the authors concern with the weighted and unweighted graphs induced by the n samples. They first prove a series of SLLN results(of Dvoretzky-Erd¨os’ type). Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1 + γ. They also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distribu...     

Haken 3-Manifolds in Small Covers*

The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple polytope is a flag polytope. In addition, they find that M being Haken is also equivalent to the existence of a Riemannian metric with non-positive sectional curvature on M.