Finite Quasi-Uniform Extensions. I

Finite extensions of quasi-uniformities for prescribed topologies are examined. We present a necessary and sufficient condition for the existence of such an extension. We investigate the set of all compatible extensions and derive that this is a sup-distributive lattice. Then we examine the cardinalities and the lattice theoretic properties of these sets.     

Short Notes on Quasi-Uniform Spaces. V Properties Preserved by Extensions

In §1 we investigate the following problem: are quietness and some of its generalizations preserved by firm extensions, doubly uniformly strict extensions or doubly strict extensions of quasi-uniform spaces. In §2 we investigate similar questions for several symmetry properties of quasi-uniform spaces. We obtain positive as well as negative answers.     

Borderline Case of Traces and Extensions for Weighted Sobolev Spaces

In this paper,we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases.We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces,and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces(on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.     

Elements of Prescribed Order,Prescribed Traces and Systems of Rational Functions Over Finite Fields

Let k 1 and be a system of rational functions forming a strongly linearly independent set over a finite field . Let be arbitrarily prescribed elements. We prove that for all sufficiently large extensions , there is an element of prescribed order such that is the relative trace map from onto We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.classification 11T30, 11G20, 05B15     

Simultaneous Extensions of Topologies Through Traces of Neighbourhood Filters

Let X be a set, X _i X for i I and, for x X, a filter in X _i. The paper gives necessary and sufficient conditions for the existence of a topology on X compatible with , i.e. such that is the trace on X _i of the -neighbourhood filter of x. It is shown that, among these compatible topologies,there are always a coarsest one and a finest one. Some applications are also given.     

Extensions from the Sobolev Spaces H 1 Satisfying Prescribed Dirichlet Boundary Conditions

Extensions from H ¹(_P) into H ¹() (where _P ) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary of . The corresponding extension operator is linear and bounded.     

Completion of Quasi-Uniform Frames

This paper presents the completion of a quasi-uniform frame. When the procedure outlined herein is applied to a uniform frame, the completion yields the (unique) completion of the uniform frame.     

线段被任意折成三段能构成三角形的概率问题的推广

通过线段被任意折成三段能构成三角形的概率问题,推广给出了线段任意折成三段能构成锐角三角形、直角三角形和钝角三角形的概率,任意折成四段能构成四边形、平行四边形和梯形的概率,以及任意折成n段能构成n边形的概率.     

Subsets of the Plane with Small Linear Sections and Invariant Extensions of the Two-Dimensional Lebesgue Measure

We consider some subsets of the Euclidean plane R ², having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on R ².     

On a Problem of Simultaneous Quasi-Uniform Extension

Let T be a topology on a set X and X = 1n Xi, Xi Xj = 0 (i j), Ui a quasi-uniformity on Xi (i = 1,...,n). The paper establishes necessary and sufficient conditions for the existence of a quasi-uniformity U on X with the property that the topology induced by U coincides with T and the restriction of U to Xi is equal to Ui. The rather complicated condition in the general case can be simplified in the special case n = 2.     

Some Remarks on Right K>-Completion of Quasi-Uniform Spaces

We discuss several properties of the so-called right K-completion for certain classes of T ₁ quasi-uniform spaces and derive a result on right K-completeness of the Hausdorff-Bourbaki quasi-uniformity.     

Lidskii-Type Formulae for Dixmier Traces

We establish several analogues of the classical Lidskii Theorem for some special classes of singular traces (Dixmier traces and Connes–Dixmier traces) used in noncommutative geometry.     

Congruences for Traces of Singular moduli

We extend a result of Ahlgren and Ono (Compos. Math. 141, 293–312, 2005) on congruences for traces of singular moduli of level 1 to traces defined in terms of Hauptmodul associated to certain groups of genus 0 of higher levels.      

Extensions for Filter Spaces

A family of extensions for a filter space is defined and its properties, including completeness and total boundedness, are investigated.     

微分算子的对称扩张及Friedrichs扩张的辛几何刻画

本文在加权Hilbert空间L2(I,r(x))(I=(a,6),-∞≤a 0)中,利用辛几何,刻画了n阶对称微分算式的最小算子的对称扩张(含自伴扩张)及 Friedrichs扩张,分别获得了其扩张为对称扩张、Friedrichs扩张的充分必要条件．     

Well-behaved Epireflections for Kan Extensions

Let \(K:\mathbb{B}\rightarrow \mathbb{A}\) be a functor such that the image of the objects in \(\mathbb{B}\) is a cogenerating set of objects for \(\mathbb{A}\). Then, the right Kan extensions adjunction \(\mathbf{Set}^K\dashv Ran_K\) induces necessarily an epireflection with stable units and a monotone-light factorization. This result follows from the one stating that an adjunction produces an epireflection in a canonical way, provided there exists a prefactorization system which factorizes all of its unit morphisms through epimorphisms. The stable units property, for the so obtained epireflections, is thereafter equivalently restated in such a manner it is easily recognizable in the examples. Furthermore, having stable units, there is a strong but quite simple sufficient condition for the existence of an associated monotone-light factorization, which has proved to be fruitful.     

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.