Pub Date : 2018-05-18DOI: 10.1007/S13373-018-0125-1
Sheng-Sen Lu, Miaochao Chen, Qilin Liu
{"title":"A nonlinear inverse problem of the Korteweg-de Vries equation","authors":"Sheng-Sen Lu, Miaochao Chen, Qilin Liu","doi":"10.1007/S13373-018-0125-1","DOIUrl":"https://doi.org/10.1007/S13373-018-0125-1","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"35 1","pages":"1-11"},"PeriodicalIF":1.2,"publicationDate":"2018-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79238217","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 : 2018-05-10DOI: 10.1142/s1664360720500058
A. Garreta, A. Miasnikov, D. Ovchinnikov
We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups [Formula: see text]. Then, we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups [Formula: see text].
{"title":"Diophantine problems in solvable groups","authors":"A. Garreta, A. Miasnikov, D. Ovchinnikov","doi":"10.1142/s1664360720500058","DOIUrl":"https://doi.org/10.1142/s1664360720500058","url":null,"abstract":"We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups [Formula: see text]. Then, we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups [Formula: see text].","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"5 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2018-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91236763","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 : 2018-04-01DOI: 10.1007/S13373-016-0094-1
A. Pianzola, A. Stolin
{"title":"Belavin–Drinfeld solutions of the Yang–Baxter equation: Galois cohomology considerations","authors":"A. Pianzola, A. Stolin","doi":"10.1007/S13373-016-0094-1","DOIUrl":"https://doi.org/10.1007/S13373-016-0094-1","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"29 1","pages":"1-14"},"PeriodicalIF":1.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90824966","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 : 2018-04-01DOI: 10.1007/S13373-016-0098-X
Zujin Zhang
{"title":"An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field","authors":"Zujin Zhang","doi":"10.1007/S13373-016-0098-X","DOIUrl":"https://doi.org/10.1007/S13373-016-0098-X","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"73 1","pages":"33-47"},"PeriodicalIF":1.2,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83958420","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 : 2018-02-22DOI: 10.1007/S13373-018-0122-4
Yuan-yuan Chen, J. Toft, P. Wahlberg
{"title":"Correction to: The Weyl product on quasi-Banach modulation spaces","authors":"Yuan-yuan Chen, J. Toft, P. Wahlberg","doi":"10.1007/S13373-018-0122-4","DOIUrl":"https://doi.org/10.1007/S13373-018-0122-4","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"44 1","pages":"1-2"},"PeriodicalIF":1.2,"publicationDate":"2018-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82488526","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 : 2018-02-22DOI: 10.1007/S13373-018-0121-5
B. Simon
{"title":"Tosio Kato’s work on non-relativistic quantum mechanics: part 2","authors":"B. Simon","doi":"10.1007/S13373-018-0121-5","DOIUrl":"https://doi.org/10.1007/S13373-018-0121-5","url":null,"abstract":"","PeriodicalId":9348,"journal":{"name":"Bulletin of Mathematical Sciences","volume":"149 1","pages":"1-99"},"PeriodicalIF":1.2,"publicationDate":"2018-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86583908","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 : 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":"8 2","pages":"257-306"},"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 : 2017-11-13DOI: 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":"53 1","pages":"1-19"},"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}
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