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Monatshefte für Mathematik最新文献

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A central limit theorem for integer partitions into small powers 将整数分割为小幂的中心极限定理
Monatshefte für Mathematik Pub Date : 2023-12-15 DOI: 10.1007/s00605-023-01926-y
Gabriel F. Lipnik, Manfred G. Madritsch, Robert F. Tichy
{"title":"A central limit theorem for integer partitions into small powers","authors":"Gabriel F. Lipnik, Manfred G. Madritsch, Robert F. Tichy","doi":"10.1007/s00605-023-01926-y","DOIUrl":"https://doi.org/10.1007/s00605-023-01926-y","url":null,"abstract":"<p>The study of the well-known partition function <i>p</i>(<i>n</i>) counting the number of solutions to <span>(n = a_{1} + dots + a_{ell })</span> with integers <span>(1 le a_{1} le dots le a_{ell })</span> has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into </p><span>$$begin{aligned} n=leftlfloor a_1^alpha rightrfloor +cdots +leftlfloor a_ell ^alpha rightrfloor end{aligned}$$</span><p>with <span>(1le a_1&lt; cdots &lt; a_ell )</span> and some fixed <span>(0&lt; alpha &lt; 1)</span>. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Solvability of some integro-differential equations with the double scale anomalous diffusion in higher dimensions 一些具有高维双尺度反常扩散的积分微分方程的可解性
Monatshefte für Mathematik Pub Date : 2023-12-13 DOI: 10.1007/s00605-023-01927-x
Vitali Vougalter, Vitaly Volpert
{"title":"Solvability of some integro-differential equations with the double scale anomalous diffusion in higher dimensions","authors":"Vitali Vougalter, Vitaly Volpert","doi":"10.1007/s00605-023-01927-x","DOIUrl":"https://doi.org/10.1007/s00605-023-01927-x","url":null,"abstract":"<p>The article is devoted to the studies of the existence of solutions of an integro-differential equation in the case of the double scale anomalous diffusion with the sum of the two negative Laplacians raised to two distinct fractional powers in <span>({mathbb R}^{d}, d=4, 5)</span>. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for the non-Fredholm elliptic operators in unbounded domains are used.\u0000</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some inequalities for self-mappings of unit ball satisfying the invariant Laplacians 满足不变拉普拉斯算子的单位球自映射的若干不等式
Monatshefte für Mathematik Pub Date : 2023-11-27 DOI: 10.1007/s00605-023-01925-z
Deguang Zhong, Meilan Huang, Dongping Wei
{"title":"Some inequalities for self-mappings of unit ball satisfying the invariant Laplacians","authors":"Deguang Zhong, Meilan Huang, Dongping Wei","doi":"10.1007/s00605-023-01925-z","DOIUrl":"https://doi.org/10.1007/s00605-023-01925-z","url":null,"abstract":"<p>In this paper, we study those mappings in unit ball satisfying the Dirichlet problem of the following differential operators </p><span>$$begin{aligned} Delta _{gamma }=big (1-|x|^{2}big )cdot left[ frac{1-|x|^{2}}{4}cdot sum _{i}frac{partial ^{2}}{partial x_{i}^{2}}+gamma sum _{i}x_{i}cdot frac{partial }{partial x_{i}}+gamma left( frac{n}{2}-1-gamma right) right] . end{aligned}$$</span><p>Our aim is to establish the Schwarz type inequality, Heinz-Schwarz type inequality and boundary Schwarz inequality for those mappings.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Strong tree properties, Kurepa trees, and guessing models 强树属性，Kurepa树和猜测模型
Monatshefte für Mathematik Pub Date : 2023-11-25 DOI: 10.1007/s00605-023-01922-2
Chris Lambie-Hanson, Šárka Stejskalová
{"title":"Strong tree properties, Kurepa trees, and guessing models","authors":"Chris Lambie-Hanson, Šárka Stejskalová","doi":"10.1007/s00605-023-01922-2","DOIUrl":"https://doi.org/10.1007/s00605-023-01922-2","url":null,"abstract":"<p>We investigate the generalized tree properties and guessing model properties introduced by Weiß and Viale, as well as natural weakenings thereof, studying the relationships among these properties and between these properties and other prominent combinatorial principles. We introduce a weakening of Viale and Weiß’s Guessing Model Property, which we call the Almost Guessing Property, and prove that it provides an alternate formulation of the slender tree property in the same way that the Guessing Model Property provides and alternate formulation of the ineffable slender tree property. We show that instances of the Almost Guessing Property have sufficient strength to imply, for example, failures of square or the nonexistence of weak Kurepa trees. We show that these instances of the Almost Guessing Property hold in the Mitchell model starting from a strongly compact cardinal and prove a number of other consistency results showing that certain implications between the principles under consideration are in general not reversible. In the process, we provide a new answer to a question of Viale by constructing a model in which, for all regular <span>(theta ge omega _2)</span>, there are stationarily many <span>(omega _2)</span>-guessing models <span>(M in {mathscr {P}}_{omega _2} H(theta ))</span> that are not <span>(omega _1)</span>-guessing models.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Infinite products involving the period-doubling sequence 包含倍周期序列的无穷积
Monatshefte für Mathematik Pub Date : 2023-11-24 DOI: 10.1007/s00605-023-01923-1
John M. Campbell
{"title":"Infinite products involving the period-doubling sequence","authors":"John M. Campbell","doi":"10.1007/s00605-023-01923-1","DOIUrl":"https://doi.org/10.1007/s00605-023-01923-1","url":null,"abstract":"<p>We explore the evaluation of infinite products involving the automatic sequence <span>((d_{n}: n in mathbb {N}_{0}))</span> known as the period-doubling sequence, inspired by the work of Allouche, Riasat, and Shallit on the evaluation of infinite products involving the Thue–Morse or Golay–Shapiro sequences. Our methods allow for the application of integral operators that result in new product expansions for expressions involving the dilogarithm function, resulting in new formulas involving Catalan’s constant <i>G</i>, such as the formula </p><span>$$begin{aligned} prod _{n=1}^{infty } left( left( frac{n+2}{n}right) ^{n+1} left( frac{4 n + 3}{4 n+5}right) ^{4 n+4}right) ^{d_{n}} = frac{e^{frac{2 G}{pi }}}{sqrt{2}} end{aligned}$$</span><p>introduced in this article. More generally, the evaluation of infinite products of the form <span>( prod _{n=1}^{infty } e(n)^{d_{n}} )</span> for an elementary function <i>e</i>(<i>n</i>) is the main purpose of our article. Past work on infinite products involving automatic sequences has mainly concerned products of the form <span>( prod _{n=1}^{infty } R(n)^{a(n)} )</span> for an automatic sequence <i>a</i>(<i>n</i>) and a rational function <i>R</i>(<i>n</i>), in contrast to our results as in above displayed product evaluation. Our methods also allow us to obtain new evaluations involving <span>(frac{zeta (3)}{pi ^2})</span> for infinite products involving the period-doubling sequence.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cohomology of quasi-abelianized braid groups 拟阿贝尔化辫群的上同调
Monatshefte für Mathematik Pub Date : 2023-11-24 DOI: 10.1007/s00605-023-01924-0
Filippo Callegaro, Ivan Marin
{"title":"Cohomology of quasi-abelianized braid groups","authors":"Filippo Callegaro, Ivan Marin","doi":"10.1007/s00605-023-01924-0","DOIUrl":"https://doi.org/10.1007/s00605-023-01924-0","url":null,"abstract":"<p>We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincaré dualities for them and prove a stabilization property for the infinite series of reflection groups.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A short note on coproducts of Abelian pro-Lie groups 关于阿贝尔亲李群的副积的简短说明
Monatshefte für Mathematik Pub Date : 2023-11-19 DOI: 10.1007/s00605-023-01915-1
Wolfgang Herfort, Karl H. Hofmann, Francesco G. Russo
{"title":"A short note on coproducts of Abelian pro-Lie groups","authors":"Wolfgang Herfort, Karl H. Hofmann, Francesco G. Russo","doi":"10.1007/s00605-023-01915-1","DOIUrl":"https://doi.org/10.1007/s00605-023-01915-1","url":null,"abstract":"<p>The notion of <i>conditional coproduct</i> of a family of abelian pro-Lie groups in the category of abelian pro-Lie groups is introduced. It is shown that the cartesian product of an arbitrary family of abelian pro-Lie groups can be characterized by the universal property of the <i>conditional coproduct</i>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An extension of Aigner’s theorem 艾格纳定理的推广
Monatshefte für Mathematik Pub Date : 2023-11-18 DOI: 10.1007/s00605-023-01913-3
Nguyen Xuan Tho
{"title":"An extension of Aigner’s theorem","authors":"Nguyen Xuan Tho","doi":"10.1007/s00605-023-01913-3","DOIUrl":"https://doi.org/10.1007/s00605-023-01913-3","url":null,"abstract":"<p>In 1957, Aigner (Monatsh Math 61:147–150, 1957) showed that the equations <span>(x^6+y^6=z^6)</span> and <span>(x^9+y^9=z^9)</span> have no solutions in any quadratic number field with <span>(xyzne 0)</span>. We show that Aigner’s result holds for all equations <span>(x^{3n}+y^{3n}=z^{3n})</span>, where <span>(nge 2)</span> is a positive integer. The proof combines Aigner’s idea with deep results on Fermat’s equation and its variants.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Voronoi summation formula for non-holomorphic Maass forms of half-integral weight 半积分权的非全纯质量形式的Voronoi求和公式
Monatshefte für Mathematik Pub Date : 2023-11-18 DOI: 10.1007/s00605-023-01921-3
Olga Balkanova, Dmitry Frolenkov
{"title":"A Voronoi summation formula for non-holomorphic Maass forms of half-integral weight","authors":"Olga Balkanova, Dmitry Frolenkov","doi":"10.1007/s00605-023-01921-3","DOIUrl":"https://doi.org/10.1007/s00605-023-01921-3","url":null,"abstract":"<p>We prove a Voronoi summation formula for non-holomorphic half-integral weight Maass forms on <span>(Gamma _0(4))</span> without any restrictions on the denominator of a fraction in the exponential function. As an application we obtain a Voronoi summation formula for the values of Zagier <i>L</i>-series.\u0000</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Eigenvalues of truncated unitary matrices: disk counting statistics 截断酉矩阵的特征值:磁盘计数统计
Monatshefte für Mathematik Pub Date : 2023-11-16 DOI: 10.1007/s00605-023-01920-4
Yacin Ameur, Christophe Charlier, Philippe Moreillon
{"title":"Eigenvalues of truncated unitary matrices: disk counting statistics","authors":"Yacin Ameur, Christophe Charlier, Philippe Moreillon","doi":"10.1007/s00605-023-01920-4","DOIUrl":"https://doi.org/10.1007/s00605-023-01920-4","url":null,"abstract":"<p>Let <i>T</i> be an <span>(ntimes n)</span> truncation of an <span>((n+alpha )times (n+alpha ))</span> Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of <i>T</i>. We prove that as <span>(nrightarrow + infty )</span> with <span>(alpha )</span> fixed, the associated moment generating function enjoys asymptotics of the form </p><span>$$begin{aligned} exp big ( C_{1} n + C_{2} + o(1) big ), end{aligned}$$</span><p>where the constants <span>(C_{1})</span> and <span>(C_{2})</span> are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.\u0000</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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