Monatshefte fur Mathematik最新文献

{"title":"Correction to: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.","authors":"Kostadinka Lapkova","doi":"10.1007/s00605-018-1177-8","DOIUrl":"https://doi.org/10.1007/s00605-018-1177-8","url":null,"abstract":"<p><p>[This corrects the article DOI: 10.1007/s00605-017-1061-y.].</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"186 4","pages":"675-678"},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-018-1177-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37044878","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":"On the length of arcs in labyrinth fractals.","authors":"Ligia L Cristea,&nbsp;Gunther Leobacher","doi":"10.1007/s00605-017-1056-8","DOIUrl":"https://doi.org/10.1007/s00605-017-1056-8","url":null,"abstract":"<p><p>Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length (Cristea and Steinsky in Geom Dedicata 141(1):1-17, 2009; Proc Edinb Math Soc 54(2):329-344, 2011). In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"185 4","pages":"575-590"},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-017-1056-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37377944","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":"Number systems over orders.","authors":"Attila Pethő,&nbsp;Jörg Thuswaldner","doi":"10.1007/s00605-018-1191-x","DOIUrl":"https://doi.org/10.1007/s00605-018-1191-x","url":null,"abstract":"<p><p>Let <math><mi>K</mi></math> be a number field of degree <i>k</i> and let <math><mi>O</mi></math> be an order in <math><mi>K</mi></math> . A <i>generalized number system over</i> <math><mi>O</mi></math> (GNS for short) is a pair <math><mrow><mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo></mrow> </math> where <math><mrow><mi>p</mi> <mo>∈</mo> <mi>O</mi> <mo>[</mo> <mi>x</mi> <mo>]</mo></mrow> </math> is monic and <math><mrow><mi>D</mi> <mo>⊂</mo> <mi>O</mi></mrow> </math> is a complete residue system modulo <i>p</i>(0) containing 0. If each <math><mrow><mi>a</mi> <mo>∈</mo> <mi>O</mi> <mo>[</mo> <mi>x</mi> <mo>]</mo></mrow> </math> admits a representation of the form <math><mrow><mi>a</mi> <mo>≡</mo> <msubsup><mo>∑</mo> <mrow><mi>j</mi> <mo>=</mo> <mn>0</mn></mrow> <mrow><mi>ℓ</mi> <mo>-</mo> <mn>1</mn></mrow> </msubsup> <msub><mi>d</mi> <mi>j</mi></msub> <msup><mi>x</mi> <mi>j</mi></msup> <mspace></mspace> <mrow><mo>(</mo> <mo>mod</mo> <mspace></mspace> <mi>p</mi> <mo>)</mo></mrow> </mrow> </math> with <math><mrow><mi>ℓ</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> and <math> <mrow><msub><mi>d</mi> <mn>0</mn></msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub><mi>d</mi> <mrow><mi>ℓ</mi> <mo>-</mo> <mn>1</mn></mrow> </msub> <mo>∈</mo> <mi>D</mi></mrow> </math> then the GNS <math><mrow><mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>D</mi> <mo>)</mo></mrow> </math> is said to have the <i>finiteness property</i>. To a given fundamental domain <math><mi>F</mi></math> of the action of <math> <msup><mrow><mi>Z</mi></mrow> <mi>k</mi></msup> </math> on <math> <msup><mrow><mi>R</mi></mrow> <mi>k</mi></msup> </math> we associate a class <math> <mrow><msub><mi>G</mi> <mi>F</mi></msub> <mo>:</mo> <mo>=</mo> <mrow><mo>{</mo> <mrow><mo>(</mo> <mi>p</mi> <mo>,</mo> <msub><mi>D</mi> <mi>F</mi></msub> <mo>)</mo></mrow> <mspace></mspace> <mo>:</mo> <mspace></mspace> <mi>p</mi> <mo>∈</mo> <mi>O</mi> <mrow><mo>[</mo> <mi>x</mi> <mo>]</mo></mrow> <mo>}</mo></mrow> </mrow> </math> of GNS whose digit sets <math><msub><mi>D</mi> <mi>F</mi></msub> </math> are defined in terms of <math><mi>F</mi></math> in a natural way. We are able to prove general results on the finiteness property of GNS in <math><msub><mi>G</mi> <mi>F</mi></msub> </math> by giving an abstract version of the well-known \"dominant condition\" on the absolute coefficient <i>p</i>(0) of <i>p</i>. In particular, depending on mild conditions on the topology of <math><mi>F</mi></math> we characterize the finiteness property of <math><mrow><mo>(</mo> <mi>p</mi> <mrow><mo>(</mo> <mi>x</mi> <mo>±</mo> <mi>m</mi> <mo>)</mo></mrow> <mo>,</mo> <msub><mi>D</mi> <mi>F</mi></msub> <mo>)</mo></mrow> </math> for fixed <i>p</i> and large <math><mrow><mi>m</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"187 4","pages":"681-704"},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-018-1191-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36634608","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":"Rings of congruence preserving functions.","authors":"C J Maxson,&nbsp;Frederik Saxinger","doi":"10.1007/s00605-017-1105-3","DOIUrl":"https://doi.org/10.1007/s00605-017-1105-3","url":null,"abstract":"<p><p>Let <math> <mrow><msub><mi>C</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </mrow> </math> denote the near-ring of congruence preserving functions of the group <i>G</i>. We investigate the question \"When is <math> <mrow><msub><mi>C</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <mi>G</mi> <mo>)</mo></mrow> </mrow> </math> a ring?\". We obtain information externally via the lattice structure of the normal subgroups of <i>G</i> and internally via structural properties of the group <i>G</i>.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"187 3","pages":"531-542"},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-017-1105-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36620256","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":"On Weyl products and uniform distribution modulo one.","authors":"Christoph Aistleitner, Gerhard Larcher, Friedrich Pillichshammer, Sumaia Saad Eddin, Robert F Tichy","doi":"10.1007/s00605-017-1100-8","DOIUrl":"10.1007/s00605-017-1100-8","url":null,"abstract":"<p><p>In the present paper we study the asymptotic behavior of trigonometric products of the form <math> <mrow><msubsup><mo>∏</mo> <mrow><mi>k</mi> <mo>=</mo> <mn>1</mn></mrow> <mi>N</mi></msubsup> <mn>2</mn> <mo>sin</mo> <mrow><mo>(</mo> <mi>π</mi> <msub><mi>x</mi> <mi>k</mi></msub> <mo>)</mo></mrow> </mrow> </math> for <math><mrow><mi>N</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> , where the numbers <math><mrow><mi>ω</mi> <mo>=</mo> <msubsup><mrow><mo>(</mo> <msub><mi>x</mi> <mi>k</mi></msub> <mo>)</mo></mrow> <mrow><mi>k</mi> <mo>=</mo> <mn>1</mn></mrow> <mi>N</mi></msubsup> </mrow> </math> are evenly distributed in the unit interval [0, 1]. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points <math><mi>ω</mi></math> , thereby improving earlier results obtained by Hlawka (Number theory and analysis (Papers in Honor of Edmund Landau, Plenum, New York), 97-118, 1969). Furthermore, we consider the special cases when the points <math><mi>ω</mi></math> are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"185 3","pages":"365-395"},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-017-1100-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37377943","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":"Quadrics and Scherk towers.","authors":"S Fujimori,&nbsp;U Hertrich-Jeromin,&nbsp;M Kokubu,&nbsp;M Umehara,&nbsp;K Yamada","doi":"10.1007/s00605-017-1075-5","DOIUrl":"https://doi.org/10.1007/s00605-017-1075-5","url":null,"abstract":"<p><p>We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"186 2","pages":"249-279"},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-017-1075-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37164208","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":"Theoretische Betrachtungen","authors":"","doi":"10.1007/BF01692220","DOIUrl":"https://doi.org/10.1007/BF01692220","url":null,"abstract":"","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"19 1","pages":"A50"},"PeriodicalIF":0.9,"publicationDate":"2017-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88941801","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":"Oeuvres mathématiques d' Évariste Galois","authors":"Évariste Galois","doi":"10.1007/BF01695090","DOIUrl":"https://doi.org/10.1007/BF01695090","url":null,"abstract":"","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"65 1","pages":"A16"},"PeriodicalIF":0.9,"publicationDate":"2017-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89625311","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":"On semidiscrete constant mean curvature surfaces and their associated families.","authors":"Wolfgang Carl","doi":"10.1007/s00605-016-0929-6","DOIUrl":"https://doi.org/10.1007/s00605-016-0929-6","url":null,"abstract":"<p><p>The present paper studies semidiscrete surfaces in three-dimensional Euclidean space within the framework of integrable systems. In particular, we investigate semidiscrete surfaces with constant mean curvature along with their associated families. The notion of mean curvature introduced in this paper is motivated by a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors at the vertices, and extends previous work on semidiscrete surfaces. In the situation of vanishing mean curvature, the associated families are defined via a Weierstrass representation. For the general cmc case, we introduce a Lax pair representation that directly defines associated families of cmc surfaces, and is connected to a semidiscrete [Formula: see text]-Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"182 3","pages":"537-563"},"PeriodicalIF":0.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-016-0929-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34832703","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":"Unconventional height functions in simultaneous Diophantine approximation.","authors":"Lior Fishman,&nbsp;David Simmons","doi":"10.1007/s00605-016-0983-0","DOIUrl":"https://doi.org/10.1007/s00605-016-0983-0","url":null,"abstract":"<p><p>Simultaneous Diophantine approximation is concerned with the approximation of a point <math><mrow><mi>x</mi> <mo>∈</mo> <msup><mi>R</mi> <mi>d</mi></msup> </mrow> </math> by points <math><mrow><mi>r</mi> <mo>∈</mo> <msup><mi>Q</mi> <mi>d</mi></msup> </mrow> </math> , with a view towards jointly minimizing the quantities <math><mrow><mo>‖</mo> <mi>x</mi> <mo>-</mo> <mi>r</mi> <mo>‖</mo></mrow> </math> and <math><mrow><mi>H</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> </math> . Here <math><mrow><mi>H</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo></mrow> </math> is the so-called \"standard height\" of the rational point <math><mi>r</mi></math> . In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"182 3","pages":"577-618"},"PeriodicalIF":0.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-016-0983-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37816115","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}