A combinatorial proof of the Removal Lemma for Groups

Green [B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.     

Finiteness conditions and distributive laws for Boolean algebras

We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

一个与Dedekind和相关的新和式及其在特殊点的值

本文利用正整数模q的正则数的定义以及解析方法研究一类与Dedekind和有关的和式的计算问题,并给出这个和式在一些特殊点上有趣的恒等式.     

Group 6 metal carbonyl complexes of 3′H-spiro[indole-3,2′-[1,3]benzothiazole-2(1H)]-one

The spiro-compound 3′H-spiro[indole-3,2′-[1,3]benzothiazole-2(1H)-one (IBTH₂) was synthesized and its structure was determined using spectroscopic techniques (FTIR, ¹H NMR and mass) and X-ray crystallography. This ligand possesses different centers for coordination. Reactions of [M(CO)₆], M = Cr, Mo or W with IBTH₂ in THF under reduced pressure were studied. For chromium a complex with molecular formula [Cr(ITP)₂] was isolated; where ITP is the opened form of the ligand which occurred through C_spiro–S bond, while [Mo(CO)₅(IBTH₂)] and [W(CO)₅(IBTH₂)] were isolated from the reaction of IBTH₂ with molybdenum and tungsten carbonyls, respectively. All complexes were characterized by elemental analysis, IR, mass and ¹H NMR spectroscopy. The biological activity of the ligand and its complexes were studied and compared with the parent compound isatin.     

银杏叶化学组分群动态FTIR指纹图谱研究

比较了相同产地不同月份与相同月份不同产地银杏叶的FTIR图谱,研究了银杏叶化学组分群随气候、季节、地域等的差异而呈现出的动态分布规律。结果表明,从4月份到10月份,银杏叶内化学组分群呈现动态分布与变化,其变化在6月16日,9月1日,10月4日3个时段。5月份所采集的不同产地银杏叶的化学组分群分布基本一致。这在银杏叶药材种植质量管理规范(GAP)基地的建设、确保银杏叶原料的稳定性以及其后各类相关指纹图谱的研究及制定等方面具有重要意义。     

Homology of Centralizers

Given a group Π, we study the group homology of centralizers Π _g , g ? Π, and of their central quotients Π _g /〈 g〉. This study is motivated by the structure of the Hochschild and the cyclic homology of group algebras, and is based on Quillen's approach to the cyclic homology of algebras via algebra extensions. A method of computing the de Rham complex of a group algebra by means of a Gruenberg resolution is also developed.     

Large subdirect product of modules as direct summand of their direct product

Abstract

The Zappa-Szép product of a pair of groups generalizes the semidirect product in that neither factor is assumed to be normal in the result. We extend the applicability of the Zappa-Szép product to multiplicative structures more general than groups with emphasis on categories and monoids. We also explore the preservation of various properties of the multiplication under the Zappa-Szép product.     

Extending Sets of Idempotents to Ring Extensions

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T, from a corresponding set in the base ring R. Examples and applications are given for rings that occur in functional analysis and group ring theory.