环上的拓扑和偏序

在环R上引入了拓扑O[R]和偏序≤R,证明了(R,O[R])是可分的,第一可数的局部紧空间,并得出了如下结论:(1)(R*,O*[R])是T1的当且仅当O*[R]是离散的当且仅当R中的任一元r满足r=r2=-r;(2)若(R,O[R])是T0的,则U∈O[R]当且仅当U=↓U;(3)若R是伪有限的且对任意r都有〈r〉>2,则(R,≤R)是代数Domain;(4)若环R的特征数chR为2,则R是伪有限的当且仅当Rop是代数Domain。     

偏序集的内蕴拓扑连通性北大核心

从序与拓扑的交叉考虑,进一步研究偏序集在多种内蕴拓扑下的连通性和局部连通性.主要结果有:(1)一个偏序集是序连通的当且仅当它赋予Alexandrov拓扑是连通的,也当且仅当它赋予Scott拓扑是连通的;(2)每一偏序集赋予Alexandrov拓扑是局部连通的,每一偏序集赋予Scott拓扑是局部连通的;(3)如果拓扑空间的特殊化偏序集序连通,则该拓扑空间是连通的;(4)构造反例说明了存在偏序集赋予下拓扑后是连通空间,但该偏序集本身不是序连通的.     

偏序集的内蕴拓扑连通性

从序与拓扑的交叉考虑,进一步研究偏序集在多种内蕴拓扑下的连通性和局部连通性.主要结果有:(1)一个偏序集是序连通的当且仅当它赋予Alexandrov拓扑是连通的,也当且仅当它赋予Scott拓扑是连通的;(2)每一偏序集赋予Alexandrov拓扑是局部连通的,每一偏序集赋予Scott拓扑是局部连通的;(3)如果拓扑空间的特殊化偏序集序连通,则该拓扑空间是连通的;(4)构造反例说明了存在偏序集赋予下拓扑后是连通空间,但该偏序集本身不是序连通的.     

H-代数偏序集

本文引入了H-代数偏序集的概念,讨论了它的一些基本性质.得到如下主要结果:(1)举例说明了代数偏序集未必是H-代数偏序集;(2)偏序集是H-代数偏序集当且仅当强紧元是它的强基;(3)偏序集是H-代数偏序集当且仅当它的局部Scott拓扑是强代数格.     

交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

群的一个Domain结构

群可以通过在其上赋予循环群拓扑构造群的Domain模型，从而实现群的可计算性。群上的循环群偏序集及其对偶都是连续的偏序集，这个连续偏序集是代数Domain的充分必要条件是群的单位元是代数元。如果在pseudo-有限的群G及其循环群偏序集的对偶￡^op[G]上分别赋予循环群拓扑和Scott拓扑，那么恒同函数是连续的。     

强代数偏序集

引入了一种新的强代数偏序集的概念,证明了偏序集P为强代数的当且仅当其正规完备化δ(P)为强代数格,给出了强代数偏序集的内蕴式刻画。     

S_1-代数偏序集

本文引入了S_1-代数偏序集的概念,证明了偏序集为S_1-代数的当且仅当其上S_1-拓扑为强代数格,并证明了S_1-连续domain的S_1-连续收缩仍为S_1-连续domain.     

基于理想的半单序超半群

本文给出了序超半群上几类正则序超半群的等价刻画和半单偏序超半群的等价刻画,并研究了几类相关性质。本文的主要结论是:设(S,。,≤)为序超半群,f是S上的模糊子集,S为左拟正则的当且仅当f≤1°f°1°f,类似地我们给出右拟正则的刻画;S是半单的当且仅当f≤1°f°1°f°1。     

Z-拟连续domain上的Scott拓扑和Lawson拓扑

对一般子集系统Z,引入了Z-拟连续domain的概念,证明了Z-完备偏序集P是Z-拟连续的当且仅当P上的Z-Scott拓扑σZ(P)在集包含序下是超连续格;Z-拟连续domain P上的Z-Scott拓扑σZ(P)是Sober的当且仅当σZ(P)具有Rudin性质,P贼予Z-Lawson拓扑λZ(P)是pospace,且若P上的Z-Lawson开上集是Z-Scott开的,Z-Lawson开下集是下拓扑开的,则(P,λZ(P))为严格完全正则序空间.     

Linear birth and death models and associated Laguerre and Meixner polynomials

We study birth and death processes with linear rates λ_n = n + α + c + 1, μ_{n + 1} = n + c, n 0 and μ₀ is either zero or c. The spectral measures of both processes are found using generating functions and the integral transforms of Laplace and Stieltjes. The corresponding orthogonal polynomials generalize Laguerre polynomials and the choice μ₀ = c generates the associated Laguerre polynomials of Askey and Wimp. We investigate the orthogonal polynomials in both cases and give alternate proofs of some of the results of Askey and Wimp on the associated Laguerre polynomials. We also identify the spectra of the associated Charlier and Meixner polynomials as zeros of certain transcendental equations.     

Affine-inscribed and affine-circumscribed polygons and polytopes

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ³, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ³ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298. Translated by B. M. Bekker.     

Partitions and pairwise sums and products

A seven cell partition of N is constructed with the property that no infinite set has all of its pairwise sums and products in any one cell. A related Ramsey Theory question is shown to have different answers for two and three cell partitions.     

Distributive and Multiplication Modules and Rings

We study rings in which every ideal is a finitely generated multiplication right ideal.     

Beliefs and beyond: affect and the teaching and learning of mathematics

The importance of beliefs for the teaching and learning of mathematics is widely recognized among mathematics educators. In this special issue, we explicitly address what we call “beliefs and beyond” to indicate the larger field surrounding beliefs in mathematics education. This is done to broaden the discussion to related concepts (which may not originate in mathematics education) and to consider the interconnectedness of concepts. In particular, we present some new developments at the conceptual level, address different approaches to investigate beliefs, highlight the role of student beliefs in problem-solving activities, and discuss teacher beliefs and their significance for professional development. One specific intention is to consider expertise from colleagues in the fields of educational research and psychology, side by side with perspectives provided by researchers from mathematics education.     

Ritz and harmonic Ritz values and the convergence of FOM and GMRES

The Ritz and harmonic Ritz values are approximate eigenvalues, which can be computed cheaply within the FOM and GMRES Krylov subspace iterative methods for solving non‐symmetric linear systems. They are also the zeros of the residual polynomials of FOM and GMRES, respectively. In this paper we show that the Walker–Zhou interpretation of GMRES enables us to formulate the relation between the harmonic Ritz values and GMRES in the same way as the relation between the Ritz values and FOM. We present an upper bound for the norm of the difference between the matrices from which the Ritz and harmonic Ritz values are computed. The differences between the Ritz and harmonic Ritz values enable us to describe the breakdown of FOM and stagnation of GMRES. Copyright © 1999 John Wiley & Sons, Ltd.