Pub Date : 2018-01-01Epub Date: 2017-11-09DOI: 10.1007/s13373-017-0109-6
Pere Ara, Kang Li, Fernando Lledó, Jianchao Wu
In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Amenability of coarse spaces and <ns0:math><ns0:mi>K</ns0:mi></ns0:math> -algebras.","authors":"Pere Ara, Kang Li, Fernando Lledó, Jianchao Wu","doi":"10.1007/s13373-017-0109-6","DOIUrl":"https://doi.org/10.1007/s13373-017-0109-6","url":null,"abstract":"<p><p>In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.</p>","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s13373-017-0109-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37162254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-12-01DOI: 10.1007/S13373-017-0110-0
F. Schwarz
{"title":"Decomposition of ordinary differential equations","authors":"F. Schwarz","doi":"10.1007/S13373-017-0110-0","DOIUrl":"https://doi.org/10.1007/S13373-017-0110-0","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88615524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-11-13DOI: 10.1007/s13373-018-0130-4
E. Detomi, M. Morigi, P. Shumyatsky
{"title":"Words of Engel type are concise in residually finite groups","authors":"E. Detomi, M. Morigi, P. Shumyatsky","doi":"10.1007/s13373-018-0130-4","DOIUrl":"https://doi.org/10.1007/s13373-018-0130-4","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77202189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-11-01DOI: 10.1007/S13373-018-0118-0
B. Simon
{"title":"Tosio Kato’s work on non-relativistic quantum mechanics: part 1","authors":"B. Simon","doi":"10.1007/S13373-018-0118-0","DOIUrl":"https://doi.org/10.1007/S13373-018-0118-0","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91194549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-16DOI: 10.1007/s13373-018-0123-3
E. Carlen, E. Lieb
{"title":"Some trace inequalities for exponential and logarithmic functions","authors":"E. Carlen, E. Lieb","doi":"10.1007/s13373-018-0123-3","DOIUrl":"https://doi.org/10.1007/s13373-018-0123-3","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87553611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-29DOI: 10.1007/S13373-017-0108-7
V. Lampret
{"title":"Asymptotic inequalities for alternating harmonics","authors":"V. Lampret","doi":"10.1007/S13373-017-0108-7","DOIUrl":"https://doi.org/10.1007/S13373-017-0108-7","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83824960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-01DOI: 10.1007/S13373-016-0093-2
Artem Hulko
{"title":"On the number of eigenvalues of the discrete one-dimensional Schrödinger operator with a complex potential","authors":"Artem Hulko","doi":"10.1007/S13373-016-0093-2","DOIUrl":"https://doi.org/10.1007/S13373-016-0093-2","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87205065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-01DOI: 10.1007/S13373-016-0092-3
E. Korotyaev
{"title":"Lieb–Thirring type inequality for resonances","authors":"E. Korotyaev","doi":"10.1007/S13373-016-0092-3","DOIUrl":"https://doi.org/10.1007/S13373-016-0092-3","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78388302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-01DOI: 10.1007/S13373-016-0087-0
A. Alsaedi, B. Ahmad, Mukhtar Bin Muhammad Kirane, Fatma S. K. Al Musalhi, F. Alzahrani
{"title":"Blowing-up solutions for a nonlinear time-fractional system","authors":"A. Alsaedi, B. Ahmad, Mukhtar Bin Muhammad Kirane, Fatma S. K. Al Musalhi, F. Alzahrani","doi":"10.1007/S13373-016-0087-0","DOIUrl":"https://doi.org/10.1007/S13373-016-0087-0","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73810045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
AbstractThis is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand–Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59–124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164:175–188, 2006; Heckenberger and Yamane in Math Z 259:255–276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in Angiono (J Eur Math Soc 17:2643–2671, 2015; J Reine Angew Math 683:189–251, 2013); the PBW-basis; the dimension or the Gelfand–Kirillov dimension; the associated Lie algebra as in Andruskiewitsch et al. (Bull Belg Math Soc Simon Stevin 24(1):15–34, 2017). Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory.
摘要本文综述了有限维的对角线型尼克尔斯代数，或广义上的算术根代数。这些代数的知识是有限维或有限Gelfand-Kirillov维的点Hopf代数分类程序的基石;它们的结构对于理解有限维点Hopf代数的表示理论、各种上同调的计算以及许多其他方面都是必不可少的。这些Nichols代数在Heckenberger (Adv Math 220:59-124, 2009)中被分类为Weyl群和广义根系统概念的显著应用(Heckenberger in Invent Math 164:175-188, 2006;Heckenberger and Yamane in Math Z 259:255-276, 2008)。在本专著的第一部分，我们给出了对角型尼科尔斯代数的理论概述。本文讨论了广义根的概念及其在对角型尼科尔斯代数和(模)李超代数中的出现。在第二部分和第三部分中，我们描述了Heckenberger(2009)列表中的每个Nichols代数的以下基本信息:广义根系统;它在李氏理论中的标签;数学学报(自然科学版)，2015;数学学报(英文版);2013);PBW-basis;维数或Gelfand-Kirillov维数;相关的李代数如Andruskiewitsch et al.(数学学报，2012,24(1):15 - 34,2017)。实际上，第二部分是关于任意特征的李代数和超代数的尼科尔斯代数，而第三部分是关于小特征的李代数和超代数的尼科尔斯代数的信息，以及李论中尚未确定的少数例子。
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