Z-拟连续domain上的Scott拓扑和Lawson拓扑

对一般子集系统Z,引入了Z-拟连续domain的概念,证明了Z-完备偏序集P是Z-拟连续的当且仅当P上的Z-Scott拓扑σZ(P)在集包含序下是超连续格;Z-拟连续domain P上的Z-Scott拓扑σZ(P)是Sober的当且仅当σZ(P)具有Rudin性质,P贼予Z-Lawson拓扑λZ(P)是pospace,且若P上的Z-Lawson开上集是Z-Scott开的,Z-Lawson开下集是下拓扑开的,则(P,λZ(P))为严格完全正则序空间.     

Z-拟连续domain上的Scott拓扑和Lawson拓扑

对一般子集系统Z,引入了Z-拟连续domain的概念,证明了Z-完备偏序集P是Z-拟连续的当且仅当P上的Z-Scott拓扑σ_z(P)在集包含序下是超连续格;Z-拟连续domain P上的Z-Scott拓扑σ_z(P)是Sober的当且仅当σ_z(P)具有Rudin性质,P赋予Z-Lawson拓扑λ_z(P)是pospace;且若P上的Z-Lawson开上集是Z-Scott开的,Z-Lawson开下集是下拓扑开的,则(P,λ_z(P))为严格完全正则序空间。     

关于紧算子的Orlicz-Pettis型定理

设X是桶空间,Y是序列完备的局部凸空间.本文证明了,由X到Y的紧算子组成的算子级数,其在弱算子拓扑下和一致算子拓扑下的子级数收敛是一致的,当且仅当(X’,β(X’,X))不拓扑同胚地包含CO;同时证明了,N’中σ(X’,X)-子级数收敛级数是β(X’;X)-子级数收敛的,当且仅当(X’,β(X’,X))不拓扑同胚地包含CO.     

半同胚空间类中存在最弱拓扑的条件

若(X,τ)是 S_1-空间,S_τ是它的半开集族[τ]={σ:σ为 X 的拓扑且 S_σ=S_τ)。本文到如下结果:1)若[τ]有最弱拓扑τ(?),则(X,τ(?))是(X,τ)的半正则化空间。2)[τ]中有最弱拓扑的充要条件是(X,τ)的每个非空开集都包含非空的正则开集。因为 T_1一空间是 S_1空间,伪度量空间是 S_1一空间但未必是 T_1一空间。所以,我们的结果推广了[1]中的定理5、推论5和定理6。     

凸幂Domain的极大点的注记

本文讨论了连续Domain D的极大点Max(D)的紧子集Com(Max(D))与凸幂Domain CD的极大点Max(CD)一一对应的条件以及Max(CD)上拓扑的性质, 证明了当X为局部紧Hausdorff空间时,X的上空间UX的凸幂Domain C(UX)的极大点Max(C(UX))与Com(Max(UX))(即X的紧子集)一一对应.X的上空间UX上的Lawson拓扑与X紧子集上的Vietoris拓扑相同,并且与Max(C(UX))带有C(UX)上的相对Scott拓扑同胚.     

双线性Fourier乘子交换子的有界性与紧性

假定T_σ是关于乘子σ的双线性Fourier乘子算子,其中σ满足如下Sobolev正则条件:对某个s∈(n,2n],有sup_(κ∈Z)‖σ_k‖W~s(R~(2m))∞.对于p_1,p_2,p∈(1,∞)且满足1/p=1/p_1+1/p_2和ω=(ω_1,ω_2)∈A_(p/t)(R~(2n)),建立了T_σ及其与函数b=(b_1,b_2)∈(BMO(R~n))~2生成的交换子T_(σ,b)由L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的有界性;同时,在b_1,b_2∈CMO(R~n)(C_c~∞(R~n)在BMO拓扑下的闭包)的条件下,证明交换子T_(σ,b)是L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的紧算子.为了得到主要结果,我们先后建立了几个双(次)线性极大函数在加多权Morrey空间上的有界性以及该空间中准紧集的判定.     

内射拓扑分子格

本文证明了任给T_O拓扑分子格(L,η),以下三条等价:(1)(L,η)为正则内射拓扑分子格;(2)L为完备集环且其完备余素元集ht(L)形成一连续格,余拓扑η为该连续格ht(L)上的Scott闭集格;(3)存在T_O内射拓扑空间(X,Τ),(L,η)同胚于(P(X),Τ~c)在拓扑分子格范畴中的Sober化。此外,还给出了正则内射拓扑分子格、(一般)内射拓扑分子格以及正则内射分子格的一般结构。作为应用,重新证明了有指数元的拓扑分子格的结构。     

带衰退记忆的经典反应扩散方程的强全局吸引子

当任意阶多项式增长的非线性项为耗散,且外力项仅属于L~2(Ω)时,研究了带衰退记忆的经典反应扩散方程的解在强拓扑空间H_0~1(Ω)×L_μ~2(R~+;D(A))的长时间行为.应用抽象函数理论、半群理论以及新的估计技巧,在拓扑空间H_0~1(Ω)×L_μ~2(R~+;D(A))上,验证了强解半群的渐近紧性并且证明了强全局吸引子的存在性.     

关于弱正则环的一些结果

本文第一部分讨论了弱正则环。引进了半平坦模的概念,并证明了一个有单位元的环是弱正则的当且仅当所有右R-模是半平坦的.第二部分讨论了Reduced弱正则环。主要结果有:(1)Reduced弱正则环R是强正则的当且仅当R有有限的素维数;(2)Reduced弱正则环是p. p. 环;(3)如果一个环R是Reduced弱正则的,那么Spec(R)是紧的,Hausdorff的和全不连通的拓扑空间。从而改进了[3]的一些结果。本文中所讨论的环若与其对应的模范畴有关,就自然认为其有单位元。     

Fixed points of meromorphic functions and of their differences,divided differences and shifts

Let f(z) be a finite order meromorphic function and let c∈C\{0} be a constant.If f(z)has a Borel exceptional value a∈C,it is proved that max{τ(f(z)),τ(△_cf(z))}=max{τ(f(z)),τ(f(z+c))}=max{τ(△_cf(z)),τ(f(z+c))}=σ(f(z)).If f(z) has a Borel exceptional value b∈(C\{0})∪{∞},it is proved that max{τ(f(z)),τ(△cf(z)/f(z))}=max{τ(△cf(z)/f(z)),τ(f(z+c))}=σ(f(z)) unless f(z) takes a special form.Here τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z),and σ(g(z)) denotes the order of growth of g(z).     

Borel measures in consonant spaces

A topology on a set X is called consonant if the Scott topology of the lattice is compactly generated; equivalently, if the upper Kuratowski topology and the co-compact topology on closed sets of X coincide. It is proved that every completely regular consonant space is a Prohorov space, and that every first countable regular consonant space is hereditarily Baire. If X is metrizable separable and co-analytic, then X is consonant if and only if X is Polish. Finally, we prove that every pseudocompact topological group which is consonant is compact. Several problems of Dolecki, Greco and Lechicki, of Nogura and Shakmatov, are solved.     

L-Fuzzy Domain上的广义Scott拓扑

本文研究了L-fuzzy domain上的广义Scott拓扑,利用[1]中引入的L-fuzzy domain.获得了其上的广义Scott拓扑,它是Domain上Scott拓扑的推广,证明了一个L-fuzzy单调映射是L-fuzzy Scott连续映射当且仅当它关于L-fuzzy domain上的广义Scott拓扑连续.     

ON THE SPACES OF THE MAXIMAL POINTS

For a continuous domain D, some characterization that the convex powerdomain CD is a domain hull of Max(CD) is given in terms of compact subsets of D. And in this case, it is proved that the set of the maximal points Max(CD) of CD with the relative Scott topology is homeomorphic to the set of all Scott compact subsets of Max(.D) with the topology induced by the Hausdorff metric derived from a metric on Max(D) when Max(D) is metrizable.     

A Poset with Spectral Scott Topology is a Quasialgebraic Domain

We prove that a spectral Scott space (i.e. a poset on which the Scott topology is spectral) is a quasialgebraic domain.     

连续信息基

D.Scott在20世纪60年代引入了domain理论,并给出了两种等价形式——信息系统和邻域系。在型论(typetheory)基础上,G.Sambin提出了形式拓扑(Formal topology)理论,并证明了一元形式拓扑与代数Scott domain等价,从而推出代数信息基的概念,同时作为信息系统与邻域系的推广。本文从信息基的观点出发,提出了连续信息基的概念,证明了它与连续Scott domain的等价性。     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

The bounded-open topology on function spaces

For any regular space Z It is shown, 1) that the bounded-open topology T on C(Y,Z) is splitting and it is also the smallest jointly continuous topology whenever Y is locally bounded, 2) if Y is locally bounded or if X × Y is a boundedly generated space, then there is a natural bijection on C(X × Y,Z) onto C(X,(C(Y,Z),T_eo) which is actually a homeomorphism with respect to the bounded-open topology on both function spaces, 3) The path components of (C(Y,Z),T_eo) are exactly its homotopy classes whenever Y is boundedly generated, 4) The bounded-open topology Teo induces contravariant and covariant Homotopy preserving function-space functors. Further, 5) T_eo reduces to the compact-open topology t_co whenever the domain Y is regular; but in general, T_eo is finer than T_co (assuming the domain is Hausdorff or the range is either Hausdorff or regular).     

Complexity spaces as quantitative domains of computation

We study domain theoretic properties of complexity spaces. Although the so-called complexity space is not a domain for the usual pointwise order, we show that, however, each pointed complexity space is an ω-continuous domain for which the complexity quasi-metric induces the Scott topology, and the supremum metric induces the Lawson topology. Hence, each pointed complexity space is both a quantifiable domain in the sense of M. Schellekens and a quantitative domain in the sense of P. Waszkiewicz, via the partial metric induced by the complexity quasi-metric.     

Strongly continuous posets and the local Scott topology

In this paper, the concept of strongly continuous posets (SC-posets, for short) is introduced. A new intrinsic topology—the local Scott topology is defined and used to characterize SC-posets and weak monotone convergence spaces. Four notions of continuity on posets are compared in detail and some subtle counterexamples are constructed. Main results are: (1) A poset is an SC-poset iff its local Scott topology is equal to its Scott topology and is completely distributive iff it is a continuous precup; (2) For precups, PI-continuity, LC-continuity, SC-continuity and the usual continuity are equal, whereas they are mutually different for general posets; (3) A T₀-space is an SC-poset equipped with the Scott topology iff the space is a weak monotone convergence space with a completely distributive topology contained in the local Scott topology of the specialization order.