形式背景的AE-仿紧性

利用拓扑学的思想定义了形式背景的AE-仿紧性,给出了AE-仿紧背景的充分条件,研究了AE-仿紧背景的若干性质.证明了AE-仿紧性被适当的信息态射所保持,对一类闭嵌入子背景是遗传的.在以形式背景为对象,信息态射为态射的范畴FCC中,给出了两个形式背景乘积对象的表示,证明了两个AE-仿紧背景的乘积对象还是AE-仿紧的.     

紧L-X半群上不同分布的组合乘积的极限性质

本文讨论一类紧拓扑半群上不同分布概率测度组合乘积的极限性质,利用“局部群化”的方法将紧群上,紧交换半群上的一些结果以相应的形式建立到紧L-X半群上.     

动力系统中紧的指数吸引集的存在性

本文研究了一般形式的无穷维动力系统.利用非紧性测度理论,当非紧性测度指数衰减时,获得了紧的指数吸引集的存在性条件.同时给出了易于验证非紧性测度指数衰减的方法.     

二元可分正交紧支集小波基的刻划

本文讨论了当特征函数具有某种特殊形式时,不可分二元紧支集正交小波基的存在性问题．结论为,当特征函数为所给的形式时,不可分二元紧文集正交小波基是不存在的．     

L-Fuzzy拓扑空间中的F紧性

由于L-Fuzzy拓扑空间拥有丰富的层次结构,其中的紧性概念就有各种不同的定义形式。在[2]中,王国俊提出了被广泛采用的良紧性概念并利用α-网的工具给出了改进的F紧性定义,但没有对F紧性进行几何刻划。本文利用α-远域族的工具,在一般LF拓扑空间中引入F紧性,解决了F紧性的几何刻划问题,同时较系统地研究了F紧性的性质。     

具有紧支撑的非张量积形式二维小波有限元

分析论述了构造非张量积形式二维Daubechies小波的几条定理,在此基础上着重构造了具有紧支撑的非张量积形式二维小波,随后用具有紧支撑的非张量积二维小波有限元去解弹性薄板挠度问题,给出了误差阶,最后列举了一个数值例子．     

关于Ortho-紧性的一些注记

B.M.Scott在[1]中提出如下问题:遗传性集态正规是否蕴含ortho-紧性?本文证明了即使是单调正规(蕴含遗传性集态正规性)也不蕴含ortho-紧性,关于各种弱形式的ortho-紧性间的关系,我们证明了点星ortho-紧空闻是离散ortho-紧的,于是,由Junnila的结果我们得到:θ-加细的点星ortho-紧空间是ortho-紧的,改进了D.K.Burke的相应结果。     

紧交换半群上概率测度卷积序列的极限性质

本文用“部分群化”的方法研究紧交换半群上概率测度的极限性质.§3讨论 i.i.d 的情形,将紧群上的 Kawada-It(?)型结果以相应的形式建立到紧交换半群上.§4讨论独立非同分布的情形,建立了紧交换半群上的强 Kloss 收敛准则,它曾由苏联学者 Maksimov先后在有限群([1])与紧群上([2])得到.     

一类紧Fuzzy测度序列的淡收敛

经典测度的弱收敛和淡收敛在经典概率论等随机数学理论中有着很重要的作用。在局部紧Hausdorff空间中首次引入了一类紧Fuzzy 测度序列的淡收敛的概念,得到了它的等价形式及一些性质,推广了相应的经典结果。     

一种利用平衡向量构造$M$-带紧支对称尺度滤波器的方法

本文研究了M-带紧支对称尺度滤波器的表示形式,利用平衡向量,得到一种代数构造方法.利用这种方法可以通过选择半正交矩阵来构造M-带紧支对称尺度滤波器.作为应用,参数化了一类结构优美的4-带紧支对称小波框架系统.     

Locale同伦理论

本文引入了locale连续映射同伦的概念,建立了locale同伦范畴,构造性地证明了任一locale连续映射都同伦等价于一个locale包含映射。通过引入locale H群的概念(它是locale群概念的自然推广),建立了locale同伦范畴到群同态范畴的一个反变函子。特别地,我们建立了locale同伦群范畴上的基本群函子,证明了locale L上以p为基点的基本群同构于L的谱空间pt(L)上以p为基点的基本群。因此,基本群函子是locale范畴中的一个同伦不变量。     

弱空间式Locale

空间式locale范畴SLoc是locale范畴Loc的余反射满子范畴,但对locale乘积不封闭．本文引入弱空间式locale,证明弱空间式locale范畴WSloc为范畴Loc的余反射满子范畴,且对locale秉积封闭．还证明了一个locale A是空间式的当且仅当它的枝映射localeN(A)是弱空间式的;一个空问式locale的每一个子locale都是空间式的当且仅当它的每一个子locale是弱空间式的．最后,证明了弱空间式性在定向函子下保持不变．     

Separating families of locale maps and localic embeddings

We establish the notion of a separating family of locale maps, which is the localic analogue of the topological concept of separating points from closed sets by continuous maps. We then present a localic version of the topological embedding (or diagonal) theorem. Applications to arbitrary locales, zero-dimensional locales, and completely regular locales are given. Using the axiom of choice, we are able to control the number of factors of the target localic products so that it does not exceed the weight of the embeddable locale. Apart from the proofs of results involving the weights of locales, the remaining proofs are valid in topos logic.     

Directed inverse limits of spatial locales

In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.

     

Universal categories of uniform and metric locales

Recall that a category is called universal if it contains an isomorphic copy of any concrete category as a full subcategory. In particular, if is universal then every monoid can be represented as the endomorphism monoid of an object in . A major obstacle to universality in categories of topological nature are the constant maps (which prevent, for instance, representing nontrivial groups as endomorphism monoids). Thus, to obtain, say, a universal category of uniform spaces, the constants have to be prohibited by artificial additional conditions (for instance, conditions of an openness type). Since in generalized spaces (locales) we do not necessarily have points, the question naturally arises as to whether we can get rid of surplus conditions in search of universality there. In this paper we prove that the category of uniform locales with all uniform morphisms is universal. Indeed we establish the universality already for the subcategory of very special uniform locales, namely Boolean metric ones. Moreover, universality is also obtained for more general morphisms, such as Cauchy morphisms, as well as for special metric choices of morphisms (contractive, Lipschitz). The question whether one can avoid uniformities remains in general open: we do not know whether the category of all locales with all localic morphisms is universal. However, the answer is final for the Boolean case: by a result of McKenzie and Monk ([10], see Section 4) one cannot represent groups by endomorphisms of Boolean algebras without restriction by an additional structure.We use only basic categorical terminology, say, that from the introductory chapters of [9]. All the necesasary facts concerning generalized spaces (frames, locales) and universality are explicitly stated. More detail on frames (locales) can be found in [8] and on universality and embeddings of categories in [11].Presented by E. Fried.     

一致Locale的乘积

本文利用Ｌｏｃａｌｅ上的收敛结构研究一致Ｌｏｃａｌｅ的乘积结构，证明了一致Ｌｏ ｃａｌｅ的完备性关于Ｌｏｃａｌｅ积和弱积封闭．     

Isocompactness in the Category of Locales

We study isocompactness in Loc defined, exactly as in Top, by requiring that every countably compact closed sublocale be compact. This is a genuine extension of the same-named topological concept since every Boolean (or, even more emphatically, every paracompact) locale is isocompact. A slightly stronger variant is defined by decreeing that the closure of every complemented countably compact sublocale be compact. Dropping the adjective “complemented” yields a formally even stronger property, which we show to be preserved by finite products. Metrizable locales (or, more generally, perfectly normal locales) do not distinguish between the three variants of isocompactness. Each of the stronger variants of isocompactness travels across a proper map of locales, and in the opposite direction if the map is a surjection in Loc.     

Locale的弱拓扑表达

本文引入了弱拓扑空间的概念,证明了locale范畴与弱拓扑空间范畴的关系类似于拓扑空间范畴与locale范畴的关系。locale范畴严格包含于弱拓扑空间范畴并且与Sober的弱拓扑空间范畴等价。     

Locale范畴中的零维性

本文讨论ｌｏｃａｌｅ的零维性质，主要结果有：（１）给出ｌｏｃａｌｅＡ的核映射（ｎｕｃｌｅｕｓ）构成的ｌｏｃａｌｅＮ（Ａ）中上确界的点式刻划，并得到了Ｎ（Ａ）的紧性与Ａ的紧性之间的关系；（２）给出零维ｌｏｃａｌｅ与ｃｏｈｅｒｅｎｔｌｏｃａｌｅ之间的关系，以及零维ｌｏｃａｌｅ的紧零维反射；（３）给出零维ｌｏｃａｌｅ范畴在ｌｏｃａｌｅ范畴中的刻划．     

T D property and spatial sublocales

As one of main backgrounds of locale theory, topologies have close connections with locales. But locales have other backgrounds such as algebra, mathematical logic, etc. So there are many differences between locales and topologies. Spatiality is an important localic property to investigate the connections between locales and topologies. TheT _D property is a special separation property which plays an important role in this kind of investigations. Just as it will be proved in this paper, theT _D property often appears as the lowest requirement for many topological spaces such that they can be described with localic properties and vice versa. In this paper, we show these special properties of theT _D axiom and investigate some other interesting and important problems ofT _D -spatiality of locales.Supported by the National Natural Science Foundation of China and the Science Foundation of the State Education Commision of China.Supported by the Fund for Excellent Young University Teachers of the State Educational Commission of China and theE _x -Oversea-Scholars Fund of the Educational Commission of China.