理想的Riesz扩张

本在有单位元e的交换Banach代数B中定义了其闭理想A的Riesz扩张R，并证明了R＝A＋Q的充分条件为A={a,a∈A}在局部凸拓扑{|ρ（x)|:ρ∈Ω}下闭，其中Q为B的根基，x为x（∈B）在商映射θ：B→B/Q下的像，Ω为B的谱空间，ρ(x)=ρ(x),↓Ax∈B,ρ∈Ω。     

随机内积模上的Riesz表示定理及其应用

首先对完备随机内积模上的几乎处处有界的随机线性泛函建立了Riesz表示定理，该定理不仅表明每个完备随机内积模都是随机自共轭的而且也改进了文[1]的主要结果；然后，作为Riesz表示定理的应用，还证明了如下基本定理：设（Ω，σ，u）为任一概率空间，为任一不可分的Hilbert空间，那么任一弱随机元V：Ω→H必弱等价于某个强可测的随机元V：Ω→H     

讲授Riesz表示定理的两点注记

对讲授Riesz表示定理提出了两点可供参考的资料和建议.     

Riesz乘积空间和表示理论

设｛Ｅｉ：ｉ∈Ｉ｝是一族ＡｒｃｈｍｉｄｅａｎＲｉｅｓｚ空间．记Πｉ∈ＩＥｉ为Ｒｉｅｓｚ乘积空间．此文的主要结论是：存在一个完全正则Ｈｏｕｓｄｏｒｆ空间Ｘ使得Πｉ∈ＩＥｉＲｉｅｓｚ同构于Ｃ（Ｘ）的充分必要条件是对每一个ｉ∈Ｉ存在一个完全正则Ｈｏｕｓｄｏｒｆ空间Ｘｉ使得ＥｉＲｉｅｓｚ同构于Ｃ（Ｘｉ）．     

四元数Hilbert空间中Riesz基的刻画*

四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论, 在四元数Hilbert空间中引入了Riesz基的概念, 在此基础上刻画了Riesz基,给出了它们的一些等价条件; 特别地, 得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列; 并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系.     

Riesz空间中的极大不相交系和表示理论

设Ｅ是阿基米德Ｒｉｅｓｚ空间，有弱单位元ｅ和极大不相交系｛ｅｉ：ｉ∈Ｉ｝，其中每一个ｅｉ都是投影元素．由ｅｉ生成的主带记为Ｂ（ｅｉ）．本文考虑如下论述：（ａ）存在完全正则Ｈａｕｓｄｏｒｆ空间Ｘ，使Ｅ是Ｒｉｅｓｚ同构于Ｃ（Ｘ）；（ｂ）对每一个ｉ∈Ｉ，存在一个完全正则Ｈａｕｓｄｏｒｆ空间Ｘｉ使Ｂ（ｅｉ）是Ｒｉｅｓｚ同构于Ｃ（Ｘｉ）．我们证明（ａ）可推出（ｂ）．但其逆在一般情况下不成立．当（ｂ）成立时，我们得到一些与（ａ）等价的论述．     

矩阵的拟直和分解与Craig定理

讨论了矩阵的和与其加项之间,在等价关系及(对称阵的)合同关系上的联系.并进一步讨论了矩阵和的非零特征值与其加项的非零特征值之间的联系,给出了C ra ig定理的一个代数形式和代数证明.     

线性子空间的直和与直积

线性(子)空间的直和是线性代数中的重要概念,本文从泛映射性质的角度刻划线性(子)空间的直和与直积,证明有限个线性(子)空间的直和与直积同构,并将相关结论推广到无穷多个线性(子)空间的直和与直积.     

锥中上调和函数的Riesz 分解定理及其应用

本文给出了锥中上调和函数的Riesz 分解定理. 同时, 得到了它在锥中无穷远点处的增长性质, 并且刻画了其例外集的几何性质. 作为应用, 我们证明了锥内次调和函数的Phragmén-Lindelöf 型定理.     

完备格的关系表示理论及其应用

主要讨论完备格的关系表示问题,分别建立了完全分配格的正则表示定理、超连续格的有限正则表示定理、λ-超连续格的λ-正则关系表示定理、区间拓扑T2完备格的广义有阴正则表示定理,给出了正则关系、完全分配格、超连续格、λ-超连续格、区间拓扑T2完备格的内蕴式刻划;给出了本文所建立的完备格的关系表示理论在Domain理论、格论和拓扑学中的若干应用.     

The Rees Valuations of Complete Ideals in a Regular Local Ring

Let I be a complete m-primary ideal of a regular local ring (R, m) of dimension d ≥ 2. In the case of dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points Lipman proves a unique factorization involving special *-simple complete ideals and possibly negative exponents of the factors. Let T be an infinitely near point to R with dim R = dim T and R/m = T/m _T . We prove that the special *-simple complete ideal P _RT has a unique Rees valuation if and only if either dim R = 2 or there is no change of direction in the unique finite sequence of local quadratic transformations from R to T. We also examine conditions for a complete ideal to be projectively full.     

The Bhattacharya Function of Complete Monomial Ideals in Two Variables

We give explicit formulas for the Bhattacharya function of 𝔪-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals.     

Adjacent Ideals to Simple Complete Ideals in Two-Dimensional Regular Local Rings

Let R be a two-dimensional regular local ring with infinite residue field, and ? be a simple complete residually rational ideal of R of order r which determines R _h . Let 𝒯 be the set of quadratic transforms T of R _h with [T: R _h ] = 1, and 𝒮 the set of simple complete ideals of R of order r which are adjacent to ? from below. If R _h is free respectively a satellite, then there exist T* ∈ 𝒯 respectively T*, T** ∈ 𝒯 and a bijective map between the set 𝒮 and the set 𝒯?{T*} respectively 𝒯?{T*, T**}.     

New Classes of Set-theoretic Complete Intersection Monomial Ideals

Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δ_χ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δ_χ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δ_χ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia.     

Congruences and Ideals of Effect Algebras

This paper is devoted to a general investigation of congruences and ideals in effect algebras. One of our main results is the existence of an order isomorphism between Riesz congruences and Riesz ideals. We also answer an open question of Dvureenskij and Pulmannová by showing that an ideal is a Riesz ideal if and only if it is closed under generalized Sasaki projections.     

关于线性变换的像空间与核空间的直和

讨论数域P上有限维线性空间V上线性变换A的方幂A~k的像空间ImA~k与核空间KerA~k的直和,并将结论推广到无限维线性空间.证明了:V=ImA~k+KerA~k当且仅当ImA~k=ImA~(k+1),以及ImA~k∩KerA~k=0当且仅当KerA~k=KerA~(k+1).     

Monomial Ideals and the Gorenstein Liaison Class of a Complete Intersection

In an earlier work, the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen–Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen–Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d + 1)-times differentiable O-sequence H, there is a nondegenerate arithmetically Cohen–Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen–Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen–Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arithmetically Cohen–Macaulay subschemes of projective space are glicci up to flat deformation.     

The Spectra of General Differential Operators in the Direct Sum Spaces

In this paper, the general ordinary quasi-differential expression M _pof n-th order with complex coefficients and its formal adjoint M _p ⁺ on any finite number of intervals I _p =(a _p ,b _p ),p= 1,...,N, are considered in the setting of the direct sums of L _wp ² (a _p ,b _p )-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new.