{"title":"Pair correlations of Halton and Niederreiter Sequences are not Poissonian.","authors":"Roswitha Hofer, Lisa Kaltenböck","doi":"10.1007/s00605-021-01531-x","DOIUrl":"https://doi.org/10.1007/s00605-021-01531-x","url":null,"abstract":"<p><p>Niederreiter and Halton sequences are two prominent classes of higher-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper we show that these sequences-even though they are uniformly distributed-fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger who hypothesized that the Halton sequences are not Poissonian. The proofs rely on a general tool which identifies a specific regularity of a sequence to be sufficient for not having Poissonian pair correlations.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"194 4","pages":"789-809"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-021-01531-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"25531808","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":"Diophantine equations in separated variables and polynomial power sums.","authors":"Clemens Fuchs, Sebastian Heintze","doi":"10.1007/s00605-021-01560-6","DOIUrl":"10.1007/s00605-021-01560-6","url":null,"abstract":"<p><p>We consider Diophantine equations of the shape <math><mrow><mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mi>g</mi> <mo>(</mo> <mi>y</mi> <mo>)</mo></mrow> </math> , where the polynomials <i>f</i> and <i>g</i> are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many rational solutions (<i>x</i>, <i>y</i>) with a bounded denominator are only possible in trivial cases.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"196 1","pages":"59-65"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550583/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39622553","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":"196 3","pages":"617-641"},"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":"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":"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":"196 1","pages":"163-190"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550461/pdf/","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":"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":"101 1","pages":"A41-A42"},"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":"192 2","pages":"401-407"},"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}