Pub Date : 2021-01-01Epub Date: 2021-07-03DOI: 10.1007/s00605-021-01590-0
A Mukhammadiev, D Tiwari, G Apaaboah, P Giordano
It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers does not generalize classical results. E.g. the sequence and a sequence converges if and only if . 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 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.
众所周知，在Colombeau广义数列的锐拓扑中，极限的概念并不能推广经典的结果。例如，序列1n: o和序列(x n) n∈n收敛当且仅当x n + 1 - x n→0。这在级数、解析广义函数、或广义函数积分中的西格玛加性和经典极限定理的研究中具有深远的影响。这些结果的缺乏也与R ~不一定是完全有序集的事实有关，例如，所有无穷小的集合既没有上限值，也没有上限值。通过引入超自然数、超序、近上和上极值等概念，给出了这些问题的一个解。由此，我们可以推广关于超序列的超极限的所有经典定理。本文探讨了可以应用于其他非阿基米德设置的想法。
{"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}
Pub Date : 2021-01-01Epub Date: 2021-07-19DOI: 10.1007/s00605-021-01605-w
Volker Ziegler
Given a finite set of primes S and an m-tuple of positive, distinct integers we call the m-tuple S-Diophantine, if for each the quantity has prime divisors coming only from the set S. For a given set S we give a practical algorithm to find all S-Diophantine quadruples, provided that .
给定一个素数的有限集合S和一个m元组(a 1，…，a m)的正整数，我们称m元组为S- diophantine，如果对于每一个1≤i≤j≤m，数量a i a j + 1的素数只来自集合S，我们给出一个实用的算法来求出所有S- diophantine四元组，假设| S | = 3。
{"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}
Pub Date : 2020-01-01Epub Date: 2019-09-12DOI: 10.1007/s00605-019-01332-3
Calin Iulian Martin, Ronald Quirchmayr
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.
{"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}
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