Strong Nearness Frames Having Enough Cauchy Filters

We prove results establishing sufficient conditions for the sum of two nearness frames to have enough Cauchy filters. From these results and the fact that, in the category of strong nearness frames having enough Cauchy filters and uniform frame maps, complete spatial frames form a coreflective subcategory, follow a variety of results where the open-sets contravariant functor from topological spaces to frames transforms products into sums and inverse limits into direct limits.     

Weakly Unconditional Cauchy Series on Locally Convex Spaces

Ｗｅａｋｌｙ　Ｕｎｃｏｎｄｉｔｉｏｎａｌ　Ｃａｕｃｈｙ　Ｓｅｒｉｅｓ　ｏｎ　Ｌｏｃａｌｌｙ　Ｃｏｎｖｅｘ　ＳｐａｃｅｓＬｉＲｏｎｇｌｕ（李容录）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨａｒｂｉｎＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，...     

Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.     

Separation and Epimorphisms in Quasi-Uniform Spaces

We study some categorical aspects of quasi-uniform spaces (mainly separation and epimorphisms) via closure operators in the sense of Dikranjan, Giuli, and Tholen. In order to exploit better the corresponding properties known for topological spaces we describe the behaviour of closure operators under the lifting along the forgetful functor T from quasi-uniform spaces to topological spaces. By means of appropriate closure operators we compute the epimorphisms of many categories of quasi-uniform spaces defined by means of separation axioms and study the preservation (reflection) of epimorphisms under the functor T.     

弱耗散非线性浅水波方程周期柯西问题

建立了弱耗散非线性浅水波方程的局部适定性,对于不同的初始值,我们分别得到了解的整体存在性和解的爆破。     

On the Cauchy Problem for a Class of Linear Weakly Hyperbolic Operators

Abstract

We study a class of linear weakly hyperbolic operators with anisotropic degeneracy on their characteristic manifold and we give sufficient conditions for the well-posedness of the related Cauchy problem.     

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.     

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.     

A Class of Cauchy Problem for Quasilinear Hyperbolic Systems with the Weakly Dissipative Terms

ＡＣｌａｓｓｏｆＣａｕｃｈｙＰｒｏｂｌｅｍｆｏｒＱｕａｓｉｌｉｎｅａｒＨｙｐｅｒｂｏｌｉｃＳｙｓｔｅｍｓｗｉｔｈｔｈｅＷｅａｋｌｙＤｉｓｓｉｐａｔｉｖｅＴｅｒｍｓ￥ＬｉｕＦａｇｕｉ；ＹａｎｇＱｉａｏ（ＮｏｒｔｈＣｈｉｎａＩｎｓｔｉｔｕｔｅｏｆＷａｔｅ...     

一类弱耗散Camassa-Holm 方程Cauchy问题在Besov 空间解的局部适定性

利用Littlewood-Paley 理论和输运方程解的先验估计, 在Besov 空间 中证明了一类弱耗散Camassa-Holm 方程Cauchy 问题解的局部适定性, 同时给出了解的能量估计及爆破准则.     

On Uniformly Locally Compact Quasi-Uniform Hyperspaces

We characterize those Tychonoff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family of nonempty compact subsets of X. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space Xis uniformly locally compact on if and only if Xis paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is -compact if and only if its (lower) semi-continuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on is obtained.     

Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator

The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the sufficient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion. This work was supported by the National Natural Science Foundation of China (Grant No. 10562002), Colleges and Universities Doctoral Subject Research Funds (Grant No. 20070126002) and the Natural Science Foundation of Inner Mongolia (Grant No. 200508010103)     

Quasi-Uniform Extensions for Prescribed Traces of Sections of Entourages

Let (X, U) be a quasi-uniform space and Y X be given. An extension of the given space is a quasi-uniformity V on Y such that U is the trace of V on X and X is dense for the topology T induced by V. The paper looks for extensions V such that the collection of all pairs (V X, V (p)) or all triples (V X, V-1 (p), V(p)) is prescribed, where V runs over a (sub)base for V and p Y - X.     

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.     

Principally Weakly and Weakly Coherent Monoids

We shall call a monoid S principally weakly (weakly) left coherent if direct products of nonempty families of principally weakly (weakly) flat right S-acts are principally weakly (weakly) flat. Such monoids have not been studied in general. However, Bulman-Fleming and McDowell proved that a commutative monoid S is (weakly) coherent if and only if the act S ^I is weakly flat for each nonempty set I. In this article we introduce the notion of finite (principal) weak flatness for characterizing (principally) weakly left coherent monoids. Also we investigate monoids over which direct products of acts transfer an arbitrary flatness property to their components.     

Completeness and fixed-points

In this note the converses of recent fixed-point theorems due toKannan andChatterjea are obtained. An example is constructed to show that a metric space having the fixed-point property for homeomorphisms need not be metrically topologically complete. An example ofConnell is formulated in a more general perspective.     

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

We consider the initial‐value problem for the regularized Boussinesq‐type equation in the class of periodic functions. Validity of the weakly nonlinear solution, given in terms of two counterpropagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the timescales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis.     

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.