{"title":"Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers.","authors":"A Mukhammadiev, D Tiwari, G Apaaboah, P Giordano","doi":"10.1007/s00605-021-01590-0","DOIUrl":"https://doi.org/10.1007/s00605-021-01590-0","url":null,"abstract":"<p><p>It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers <math><mover><mi>R</mi> <mo>~</mo></mover> </math> does not generalize classical results. E.g. the sequence <math> <mrow><mfrac><mn>1</mn> <mi>n</mi></mfrac> <mo>↛</mo> <mn>0</mn></mrow> </math> and a sequence <math> <msub><mrow><mo>(</mo> <msub><mi>x</mi> <mi>n</mi></msub> <mo>)</mo></mrow> <mrow><mi>n</mi> <mo>∈</mo> <mi>N</mi></mrow> </msub> </math> converges <i>if</i> and only if <math> <mrow><msub><mi>x</mi> <mrow><mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> <mo>-</mo> <msub><mi>x</mi> <mi>n</mi></msub> <mo>→</mo> <mn>0</mn></mrow> </math> . This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that <math><mover><mi>R</mi> <mo>~</mo></mover> </math> is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-021-01590-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39578382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Finding all <i>S</i>-Diophantine quadruples for a fixed set of primes <i>S</i>.","authors":"Volker Ziegler","doi":"10.1007/s00605-021-01605-w","DOIUrl":"https://doi.org/10.1007/s00605-021-01605-w","url":null,"abstract":"<p><p>Given a finite set of primes <i>S</i> and an <i>m</i>-tuple <math><mrow><mo>(</mo> <msub><mi>a</mi> <mn>1</mn></msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub><mi>a</mi> <mi>m</mi></msub> <mo>)</mo></mrow> </math> of positive, distinct integers we call the <i>m</i>-tuple <i>S</i>-Diophantine, if for each <math><mrow><mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>m</mi></mrow> </math> the quantity <math> <mrow><msub><mi>a</mi> <mi>i</mi></msub> <msub><mi>a</mi> <mi>j</mi></msub> <mo>+</mo> <mn>1</mn></mrow> </math> has prime divisors coming only from the set <i>S</i>. For a given set <i>S</i> we give a practical algorithm to find all <i>S</i>-Diophantine quadruples, provided that <math><mrow><mo>|</mo> <mi>S</mi> <mo>|</mo> <mo>=</mo> <mn>3</mn></mrow> </math> .</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-021-01605-w","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39622554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Le rayonnement des corps noirs","authors":"O. Lummer, R. Dongier, M. Lamotte","doi":"10.1007/BF01706970","DOIUrl":"https://doi.org/10.1007/BF01706970","url":null,"abstract":"","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88963702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A steady stratified purely azimuthal flow representing the Antarctic Circumpolar Current.","authors":"Calin Iulian Martin, Ronald Quirchmayr","doi":"10.1007/s00605-019-01332-3","DOIUrl":"10.1007/s00605-019-01332-3","url":null,"abstract":"<p><p>We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7220887/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37947749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}