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On uncommon systems of equations 关于不常见的方程组
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2649-2
Nina Kamčev, Anita Liebenau, Natasha Morrison
{"title":"On uncommon systems of equations","authors":"Nina Kamčev, Anita Liebenau, Natasha Morrison","doi":"10.1007/s11856-024-2649-2","DOIUrl":"https://doi.org/10.1007/s11856-024-2649-2","url":null,"abstract":"<p>A linear system <i>L</i> over <span>(mathbb{F}_{q})</span> is common if the number of monochromatic solutions to <i>L</i> = 0 in any two-colouring of <span>(mathbb{F}_{q}^{n})</span> is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of <span>(mathbb{F}_{q}^{n})</span>. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, Saad and Wolf recently initiated the systematic study of common systems of linear equations.</p><p>Building upon earlier work of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have recently been fully characterised by Fox, Pham and Zhao, who asked about common <i>systems</i> of equations. In this paper we move towards a classification of common systems of two or more linear equations. In particular we prove that any system containing an arithmetic progression of length four is uncommon, resolving a question of Saad and Wolf. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Erdős–Moser and IΣ2 厄尔多斯-莫泽尔和 IΣ2
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2643-8
Henry Towsner, Keita Yokoyama
{"title":"Erdős–Moser and IΣ2","authors":"Henry Towsner, Keita Yokoyama","doi":"10.1007/s11856-024-2643-8","DOIUrl":"https://doi.org/10.1007/s11856-024-2643-8","url":null,"abstract":"<p>The first-order part of Ramsey’s theorem for pairs with an arbitrary number of colors is known to be precisely <i>B</i>Σ<span>\u0000<sup>0</sup><sub>3</sub>\u0000</span>. We compare this to the known division of Ramsey’s theorem for pairs into the weaker principles, EM (the Erdős–Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond <i>I</i>Σ<span>\u0000<sup>0</sup><sub>2</sub>\u0000</span> is entirely due to the arbitrary color analog of ADS.</p><p>Specifically, we show that ADS for an arbitrary number of colors implies <i>B</i>Σ<span>\u0000<sup>0</sup><sub>3</sub>\u0000</span> while EM for an arbitrary number of colors is Π<span>\u0000<sup>1</sup><sub>1</sub>\u0000</span>-conservative over <i>I</i>Σ<span>\u0000<sup>0</sup><sub>2</sub>\u0000</span> and it does not imply <i>I</i>Σ<span>\u0000<sup>0</sup><sub>2</sub>\u0000</span>.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
L2-Quasi-compact and hyperbounded Markov operators L2-准紧密和超界马尔可夫算子
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2648-3
Guy Cohen, Michael Lin
{"title":"L2-Quasi-compact and hyperbounded Markov operators","authors":"Guy Cohen, Michael Lin","doi":"10.1007/s11856-024-2648-3","DOIUrl":"https://doi.org/10.1007/s11856-024-2648-3","url":null,"abstract":"<p>A Markov operator <i>P</i> on a probability space (<i>S, Σ μ</i>) with <i>μ</i> invariant, is called hyperbounded if for some 1 ≤ <i>p</i>≤ <i>q</i> ≤ ∞ it maps (continuously) <i>L</i><sup><i>p</i></sup> into <i>L</i><sup><i>q</i></sup>.</p><p>We deduce from a recent result of Glück that a hyperbounded <i>P</i> is quasi-compact, hence uniformly ergodic, in all <i>L</i><sup><i>r</i></sup>(<i>S, μ</i>), 1 &lt; <i>r</i> &lt; ∞. We prove, using a method similar to Foguel’s, that a hyperbounded Markov operator has periodic behavior similar to that of Harris recurrent operators, and for the ergodic case obtain conditions for aperiodicity.</p><p>Given a probability <i>ν</i> on the unit circle, we prove that if the convolution operator <i>P</i><sub><i>ν</i></sub><i>f</i>:= <i>ν</i> ⋇ <i>f</i> is hyperbounded, then <i>ν</i> is atomless. We show that there is <i>ν</i> absolutely continuous such that <i>P</i><sub><i>ν</i></sub> is not hyperbounded, and there is <i>ν</i> with all powers singular such that <i>P</i><sub><i>ν</i></sub> is hyperbounded. As an application, we prove that if <i>P</i><sub><i>ν</i></sub> is hyperbounded, then for any sequence (<i>n</i><sub><i>k</i></sub>) of distinct positive integers with bounded gaps, (<i>n</i><sub><i>k</i></sub><i>x</i>) is uniformly distributed mod 1 for <i>ν</i> almost every <i>x</i> (even when <i>ν</i> is singular).</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Entropy for actions of free groups under bounded orbit-equivalence 有界轨道等价性条件下自由群作用的熵
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2642-9
Lewis Bowen, Yuqing Frank Lin
{"title":"Entropy for actions of free groups under bounded orbit-equivalence","authors":"Lewis Bowen, Yuqing Frank Lin","doi":"10.1007/s11856-024-2642-9","DOIUrl":"https://doi.org/10.1007/s11856-024-2642-9","url":null,"abstract":"<p>The <i>f</i>-invariant is a notion of entropy for probability-measure-preserving actions of free groups. We show it is invariant under bounded orbit-equivalence.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Borel edge colorings for finite-dimensional groups 有限维群的玻尔边着色
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2640-y
Felix Weilacher
{"title":"Borel edge colorings for finite-dimensional groups","authors":"Felix Weilacher","doi":"10.1007/s11856-024-2640-y","DOIUrl":"https://doi.org/10.1007/s11856-024-2640-y","url":null,"abstract":"<p>We study the potential of Borel asymptotic dimension, a tool introduced recently in [2], to help produce Borel edge colorings of Schreier graphs generated by Borel group actions. We find that it allows us to recover the classical bound of Vizing in certain cases, and also use it to exactly determine the Borel edge chromatic number for free actions of abelian groups.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On minimal generating sets for the mapping class group of a punctured surface 关于点状曲面映射类群的最小生成集
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2636-7
Naoyuki Monden
{"title":"On minimal generating sets for the mapping class group of a punctured surface","authors":"Naoyuki Monden","doi":"10.1007/s11856-024-2636-7","DOIUrl":"https://doi.org/10.1007/s11856-024-2636-7","url":null,"abstract":"<p>Let Σ<sub><i>g,p</i></sub> be an oriented surface of genus <i>g</i> with <i>p</i> punctures. We denote by <span>(cal{M}_{g,p})</span> and <span>(cal{M}_{g,p}^{pm})</span> the mapping class group and the extended mapping class group of Σ<sub><i>g,p</i></sub>, respectively. In this paper, we show that <span>(cal{M}_{g,p})</span> and <span>(cal{M}_{g,p}^{pm})</span> are generated by two elements for <i>g</i> ≥ 3 and <i>p</i> ≥ 0.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The discriminant Pfister form of an algebra with involution of capacity four 容量为四的内卷代数的判别式菲斯特形式
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2647-4
Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol
{"title":"The discriminant Pfister form of an algebra with involution of capacity four","authors":"Karim Johannes Becher, Nicolas Grenier-Boley, Jean-Pierre Tignol","doi":"10.1007/s11856-024-2647-4","DOIUrl":"https://doi.org/10.1007/s11856-024-2647-4","url":null,"abstract":"<p>To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with involution. In this way we obtain a unified approach to known decomposability criteria for several cases, and a new result for symplectic involutions on degree-8 algebras in characteristic 2.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Group-theoretical property of some integral non-degenerate fusion categories 某些积分非退化融合范畴的群论性质
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2631-z
Zhiqiang Yu
{"title":"Group-theoretical property of some integral non-degenerate fusion categories","authors":"Zhiqiang Yu","doi":"10.1007/s11856-024-2631-z","DOIUrl":"https://doi.org/10.1007/s11856-024-2631-z","url":null,"abstract":"<p>We show that an integral non-degenerate fusion category <span>({cal C})</span> is group-theoretical if the Frobenius–Perron dimensions of its simple objects are either 1 or powers of a prime <i>p</i>.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An analogue of the Blaschke–Santaló inequality for billiard dynamics 台球动力学的布拉什克-桑塔洛不等式类比
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2634-9
Daniel Tsodikovich
{"title":"An analogue of the Blaschke–Santaló inequality for billiard dynamics","authors":"Daniel Tsodikovich","doi":"10.1007/s11856-024-2634-9","DOIUrl":"https://doi.org/10.1007/s11856-024-2634-9","url":null,"abstract":"<p>The Blaschke–Santaló inequality is a classical inequality in convex geometry concerning the volume of a convex body and that of its dual. In this work we investigate an analogue of this inequality in the context of a billiard dynamical system: we replace the volume with the length of the shortest closed billiard trajectory. We define a quantity called the “billiard product” of a convex body <i>K</i>, which is analogous to the volume product studied in the Blaschke–Santaló inequality. In the planar case, we derive an explicit expression for the billiard product in terms of the diameter of the body. We also investigate upper bounds for this quantity in the class of polygons with a fixed number of vertices.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements 超平面排列的旗希尔伯特-庞加莱数列和易格斯塔函数
IF 1 2区 数学
Israel Journal of Mathematics Pub Date : 2024-08-04 DOI: 10.1007/s11856-024-2646-5
Joshua Maglione, Christopher Voll
{"title":"Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements","authors":"Joshua Maglione, Christopher Voll","doi":"10.1007/s11856-024-2646-5","DOIUrl":"https://doi.org/10.1007/s11856-024-2646-5","url":null,"abstract":"<p>We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert–Poincaré series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert–Poincaré series for irreducible Coxeter arrangements of types A, B, and D in terms of total partitions of the respective types. We show that a different specialization of the flag Hilbert–Poincaré series, which we call the coarse flag Hilbert–Poincaré series, exhibits intriguing nonnegativity features and—in the case of Coxeter arrangements—connections with Eulerian polynomials. For numerous classes and examples of hyperplane arrangements, we determine their (coarse) flag Hilbert–Poincaré series. Some computations were aided by a SageMath package we developed.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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