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.     

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.     

A Triangle Inequality for Measurement

We introduce the triangle inequality for measurement. This is a property that when satisfied by a measurement enables one to construct a metric on the set of elements with measure zero that yields the relative Scott topology. The naturality of this construction permits a categorical solution to the model problem in domain theory for locally compact metric spaces. The first time such a solution has been achieved.     

Operators that are points of spectral continuity,II

This paper continues an earlier study of those bounded operators on a Hilbert space at which the spectrum is continuous, where the spectrum is considered as a function whose domain is the set of all bounded operators furnished with the norm topology and whose range is the collection of compact subsets of the complex plane furnished with the Hausdorff metric. In this paper the points of continuity of the essential spectrum, the approximate point spectrum, and certain related subsets of the spectrum are characterized.The first author was supported by National Science Foundation Grant MCS 77-28396.     

Operators that are points of spectral continuity

In this paper a characterization is obtained of those bounded operators on a Hilbert space at which the spectrum is continuous, where the spectrum is considered as a function whose domain is the set of all operators with the norm topology and whose range is the set of compact subsets of the plane with the Hausdorff metric. Similar characterizations of the points of continuity of the Weyl spectrum, the spectral radius, and the essential spectral radius are also obtained.The first author was supported by National Science Foundation Grant MCS 77-28396.     

Spaces of Densely Continuous Forms

The set C(X,Y) of continuous functions from a topological space X into a topological space Y is extended to the set D(X,Y) of densely continuous forms from X to Y, such form being a kind of multifunction from X to Y. The topologies of pointwise convergence, uniform convergence, and uniform convergence on compact sets are defined for D(X,Y), for locally compact spaces X and metric spaces Y having a metric satisfying the Heine–Borel property. Under these assumptions, D(X,Y) with the uniform topology is shown to be completely metrizable. In addition, if X is compact, D(X,Y) is completely metrizable under the topology of uniform convergence on compact sets. For this latter topology, an Ascoli theorem is established giving necessary and sufficient conditions for a subset of D(X,Y) to be compact.     

拟连续Domain的若干拓扑性质

对拟连续Domain D证明了:(1)双拓扑空间(D,σ(D),(D))为两两完全正则空间;(2)若D有可数基,则(max(D),σ(D)max(D))为正则空间当且仅当它为Polish空间;(3)拓扑空间(D,σb(D))为零维Tychonoff空间,其中σb(D)为D上Scott拓扑的b-拓扑。     

Closure of the set of classical Riemannian spaces

The basic result of the paper is a theorem asserting that the closure of the set of compact Riemannian spaces in the set of all compact metric spaces with inner metric consists precisely of the set of compact metric spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov. Here the convergence of a sequence of Riemannian spaces in the topology we consider means its Lipschitz convergence to a limit metric space and the uniform bilateral boundedness of the sectional curvatures of the spaces of the sequence. The results obtained are considered in application to the compactness theorem of M. Gromov.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 21, pp. 43–66, 1989.     

Approximative compactness of the algebraic sum of sets

Let X be a group with an invariant metric, A and B nonempty subsets of X with B compact. It is proved that if A is an existence set [1] (approximatively compact [2]) then A + B and B + A are existence sets (approximatively compact). An example is given of a one-dimensional linear metric space (with an invariant metric) in which there exist an approximatively compact set A and an element v such that A + v is not an existence set.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 55–60, January, 1978.     

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.     

Coupled-expanding maps and one-sided symbolic dynamical systems

This paper studies relationships between coupled-expanding maps and one-sided symbolic dynamical systems. The concept of coupled-expanding map is extended to a more general one: coupled-expansion for a transitive matrix. It is found that the subshift for a transitive matrix is strictly coupled-expanding for the matrix in certain disjoint compact subsets; the topological conjugacy of a continuous map in its compact invariant set of a metric space to a subshift for a transitive matrix has a close relationship with that the map is strictly coupled-expanding for the matrix in some disjoint compact subsets. A certain relationship between strictly coupled-expanding maps for a transitive matrix in disjoint bounded and closed subsets of a complete metric space and their topological conjugacy to the subshift for the matrix is also obtained. Dynamical behaviors of subshifts for irreducible matrices are then studied and several equivalent statements to chaos are obtained; especially, chaos in the sense of Li–Yorke is equivalent to chaos in the sense of Devaney for the subshift, and is also equivalent to that the domain of the subshift is infinite. Based on these results, several new criteria of chaos for maps are finally established via strict coupled-expansions for irreducible transitive matrices in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, respectively, where their conditions are weaker than those existing in the literature.     

The ANR-property of hyperspaces with the Attouch-Wets topology

We characterize metric spaces whose hyperspaces of non-empty closed, bounded, compact or finite subsets, endowed with the Attouch-Wets topology, are absolute (neighborhood) retracts.      

QFS-Domains and their Lawson Compactness

In this paper, concepts of quasi-finitely separating maps and quasi-approximate identities are introduced. Based on these concepts, QFS-domains and quasicontinuous maps are defined. Properties and characterizations of QFS-domains are explored. Main results are: (1) finite products, nonempty Scott closed subsets and quasicontinuous projection images of QFS-domains, as well as FS-domains, are all QFS-domains; (2) QFS-domains are compact in the Lawson topology; (3) An L-domain is a QFS-domain iff it is an FS-domain, iff it is compact in the Lawson topology; (4) Bounded complete quasicontinuous domains, in particular quasicontinuous lattices, are all QFS-domains.     

A note on the representability of compact sets by real sequences

Consider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with the Hausdorff metric. In this paper, we prove that K is isometric to a subset of l_∞( ). An approximation result is also proved.     

A viability theorem for set-valued states in a Hilbert space

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H.     

Logarithms of moduli of entire functions are nowhere dense in the space of plurisubharmonic functions

We prove that the set of logarithms of moduli of entire functions of several complex variables is nowhere dense in the space of plurisubharmonic functions equipped with a topology that is a generalization of the topology of uniform convergence on compact sets. This topology is generated by a metric in which plurisubharmonic functions form a complete metric space. Thus, the logarithms of moduli of entire functions form a set of the first Baire category. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1602 – 1609, December, 2008.     

Banach空间中关于有界集的同时远达问题的适定性

本文研究Banach空间中关于有界集的同时远达问题的适定性,在集合的Hausdorff距离下,证明了：对自反局部一致凸Banach空间中的闭有界集K,使所有关于K的同时远达问题是适定的紧凸子集A全体在紧凸子集全体中是Gδ型集．     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Metric spaces in synthetic topology

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.     

Banach空间中远达和同时远达问题的适定性

本文研究Ｂａｎａｃｈ空间Ｘ中远达和同时远达问题的适定性,在集合的Ｈａｕｓ－ ｄｏｒｆｆ距离下,对Ｘ中的闭凸子集Ｄ和相对弱紧的有界闭子集Ｋ,证明了下述结果： 若Ｄ关于Ｋ严格凸和有Ｋａｄｅｃ性质,则Ｄ中所有使远达问题 ｍａｘ｛ｘ,Ｋ｝是适定的 点ｘ全体在Ｄ中是Ｇδ型集．作为应用,得到了同时远达问题适定性的类似结果．