强向量拟均衡问题解的存在性(英文)

设X是一个实的Hausdorff拓扑向量空间,Y是一个实的局部凸向量空间,C是Y中的闭凸锥,K（?）X是一个紧子集.F:X×X→Y是一个双向量函数,G:K→2K是一个集合值映射.我们考虑下面的强拟均衡问题:存在x∈G（x）,使得对任意的y∈G（x）,成立F（x,y）∈C.本文证明了当F是半连续时,上述问题解的存在性结论.     

点集拓扑学之杨忠道定理的一个机械化证明

本文给出一种用高阶逻辑自动证明语言Isabelle在计算机中表示拓扑空间中开集、闭集、邻域和导集等基本概念的方法,在此基础上证明点集拓扑学中著名的杨忠道定理,即一拓扑空间的任意单点集的导集为闭集,则其任意子集的导集亦为闭集.     

不等式约束的向量优化中值映射的余切上图导数

考虑如下的参数向量优化问题minK{f(w,x)|x∈X,g(w,x)∈C},这里f:W×X→Y是从赋范空间W和X的积到另一个赋范空间Y的Hadamard可微的单值映射,K Y是一个尖闭凸锥,C是Banach空间Z中的一个尖闭凸锥,g:W×X→Z是一个Fréchet可微的映射.借助目标函数的导数、约束映射的余切导数及拉格朗日映射给出了值映射的余切上图导数的两个表示.     

Division and k-th root theorems for Q-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f:K→X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X2 and X×[0,1] are Q-manifolds as well. We construct a countable familyχof spaces with DDP and cd-AP such that no space X∈χis homeomorphic to the Hilbert cube Q whereas the product X×Y of any different spaces X, Y∈χis homeomorphic to Q. We also show that no uncountable familyχwith such properties exists.     

保持谱半径的乘法映射

设X和Y为无限维Banach空间,Φ：B（X）→B（Y）是保持谱半径的满射,且秩为1算子,则Φ具有形式Φ（T）=ATA∧-1,这里A：X→Y或是线性拓扑同构映射或是线性拓扑同构映射的共轭。     

夹心半群S(X,Y,θ)上的α-同余

本文讨论了夹心半群T(X,Y,θ)上的α-同余与集合Y上Tθ-等价关系之间的联系,证明了对于每个Tθ-等价关系E,夹心半群同余格C(T(X,Y,θ))的完全子格γ-1(E)中的最大元是一个α-同余,并判明Symons同余l,d都是α-同余.对于某些拓扑空间X,Y,确定了夹心半群S(X,Y,θ)上的最小(最大)真α-同余.     

Lower Semicontinuity of Kernels of Closed Convex Processes and Local Reachability of Discrete-Time Systems

Let P be a topological space and, for any p ] P, G _p : X M Y be a closed convex process between arbitrary Banach spaces X and Y . This paper shows conditions under which the map p → ker G _p : = {χ ∈ X:0 ∈ G _p (χ)} is lower pseudo-Hausdorff semicontinuous in the sense of (Optimization, 48 (2000) 17). Here, unlike (Optimization, 48 (2000) 17) we do not assume that Y is reflexive. Also, we use an approach quite different from that of (Optimization, 48 (2000) 17). As an application of the above result, we give sufficient conditions for the local reachability of a discrete-time dynamical system which is described by an arbitrary (not necessarily convex) closed set-valued map in Banach spaces.     

Topologies on function spaces

In the present paper we introduce notions of A-splitting and A-jointly continuous topology on the set C(Y,Z) of all continuous maps of a topological space Y into a topological space Z, where A is any family of spaces. These notions satisfy the basic properties of splitting and jointly continuous topologies on C(Y,Z). In particular, for every A, the greatest A-splitting topology on C(Y,Z) (denoted by τ(A) always exists. We indicate some families A of spaces for which the topology τ(A) coincides with the greatest splitting topology on C(X,Y). We give a notion of equivalent families of spaces and try to find a “simple” family which is equivalent to a given family. In particular, we prove that every family is equivalent to a family consisting of one space, and the family of all spaces is equivalent to a family of all T₁-spaces containing at most one nonisolated point. We compare the topologies τ({X}) for distinct compact metrizable spaces X and give some examples. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97. Translated by A. A. Ivanov.     

Banach空间中线性算子方程的约束极值解

设X,Y与Z为Banach空间L∈L(X,Z),T∈L(X,Y)为线性算子.运用线性算子的度量广义逆概念,在L(x)=y的极值解集合中,给出T(x)=h的约束极值解的精确刻画.     

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.     

集值度量广义逆的存在性

设X,Y为Banach空间,T∈L(X,Y)为从X到Y的线性算子,D(T),N(T),R(T)分别为T的定义域,核空间与值域,使用算子T的自身性质,给出T具有集值度量广义逆T和R(T)D(T)的充分必要条件.     

Verallgemeinerungen eines Satzes von H. Steinhaus

The following result is due to H. Steinhaus [20]: “If A,B?R are sets of positive inner Lebesgue measure and if the function f: R x R→R is defined by f(x,y):=x+y (x,y?R), then the interior of f(A x B) is non void”. In this note there is proved, that the theorem of H. Steinhaus remains valid, if

1. R is replaced by certain topological measure spaces X, Y and a Hausdorff space Z,
2. f is a continuous function from an open set T?X x Y into Z and satisfies a special local (respectively global) solvability condition in T,
3. A?X is a set of positive outer measure, B?Y contains a set of positive measure and A x B?T.

     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

SUPERCATEGORIES OF THE CATEGORY OF NEARNESS SPACES

Abstract

The framework in which nearness spaces were defined by H. Herrlich [1] and [2], leads one to consider the supercategory Pow of the category Near of nearness spaces, having as objects all pairs (X,ξ), where X is a set and ξ ? P(P(X)) is any subset of the power set of the power set of X, and as morphisms f: (X,ξ) → (Y,n) all functions f: X → Y such that, if A ? ξ then fA □ {f(A) | A ξ A} ? η. In this paper we show that the full subcategories of Pow comprising the objects satisfying subsets of the prenearness space axioms lie in a lattice of bireflections or bicoreflections. This serves as a first step towards the aim of characterizing all bireflective (resp. bicoreflective) and even all initially complete subcategories of Pow.     

The Decomposability for Operator Matrices and Perturbations

Acta Mathematica Sinica, English Series - Let X and Y be Banach spaces. For A ∈ L(X), B ∈ L(Y), C ∈ L(Y, X), let MC be the operator matrix defined on X ⊕ Y by $${M_C} =...     

復式乘積和球面乘積的關係

<正> 1.設X為一拓撲空間,以a_i表示X的n_i次同偷羣∏_(ni)(X)的一個元素.在球面乘積S~(n1)×…×S~(nr)中,最高次元的腔胞記為e~(n1+…+nr).在S~(n1)×…×S~(nr)中除去e~(n1+…+nr)而得一空間Y.設S~(nl)為nl次元球,為了印刷方便起見,把S~(nl)記作S(nl).在S(ni)中取一參考點x_i~o.設i_1相似文献   

序列覆盖的闭映射保持可度量性

设f:X→Y是连续的满映射. f称为序列覆盖映射,若{y})是Y中的收敛序列,则存在X中的收敛序列{xn},使得每一xn∈f-1(yn);f称为1序列覆盖映射,若对于每-y∈Y,存在x∈f-1(y),使得如果{yn}是Y中收敛于点y的序列,则有X中收敛于点x的序列{xn},使得每一xn∈f-1(yn).本文研究度量空间序列覆盖的闭映射之构造,否定地回答了Topology and its Applications上提出的一个问题.     

Positive approximate identities and lattice-ordered dual spaces

It is shown that if E[7] is a locally convex space ordered by a closed, generating positive cone K satisfying certain mild hypotheses relating K and 7, then E′ is lattice-ordered by the dual cone K′ whenever the identity linear transformation on E is the pointwise limit of sums of transformations x→〈x,x′〉z where z∈K and x′∈K′. The converse is true for certain classes of spaces, e.g., Fréchet spaces.     

某类常微系统的一个基本性质

<正> 简介 考虑一n维紧致的C~∞型Riemann流形M~n(n≥2)上,由所有的C~1型常微系统就C~1度量作成的空间.这里为简便,一系统∈将暂称为A_o-系统,如果它只具有有限多个奇点和至多可数多个周期轨道;将暂称为A_1-系统,如果它只具有有限多个奇点和有限多个周期轨道.依据周知的结果,一般绝不是中每一系统都有任意小的邻     

Space of operators acting from one banach lattice to another

In this note we construct a pair of Banach lattices X and Y, which have the following properties:

1. X is not order isomorphic to an AL-space,
2. Y is not order isomorphic to an AM-space,
3. for any continuous linear operator T:X → Y there exists a modulus ¦T¦: X → Y.

This example refutes the conjecture of Cartwright-Lotz, saying that the negation of at least one of the conditions a) or b) is necessary for the validity of c).