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A characterization of differential operators in the ring of complex polynomials 复多项式环中微分算子的表征
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-11-01 DOI: 10.1016/j.indag.2024.07.011
Włodzimierz Fechner , Eszter Gselmann
{"title":"A characterization of differential operators in the ring of complex polynomials","authors":"Włodzimierz Fechner ,&nbsp;Eszter Gselmann","doi":"10.1016/j.indag.2024.07.011","DOIUrl":"10.1016/j.indag.2024.07.011","url":null,"abstract":"<div><div>The paper aims to provide a full characterization of all operators <span><math><mrow><mi>T</mi><mo>:</mo><mi>P</mi><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow><mo>→</mo><mi>P</mi><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> acting on the space of all complex polynomials that satisfy the Leibniz rule <span><span><span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>f</mi><mi>⋅</mi><mi>g</mi><mo>)</mo></mrow><mo>=</mo><mi>T</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mi>⋅</mi><mi>g</mi><mo>+</mo><mi>f</mi><mi>⋅</mi><mi>T</mi><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></math></span></span></span>for all <span><math><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>P</mi><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>. We do not assume the linearity of <span><math><mi>T</mi></math></span>. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1294-1305"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571884","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}
引用次数: 0
Finite odometer factors of rank one group actions 一级群作用的有限里程表因子
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-11-01 DOI: 10.1016/j.indag.2024.04.012
{"title":"Finite odometer factors of rank one group actions","authors":"","doi":"10.1016/j.indag.2024.04.012","DOIUrl":"10.1016/j.indag.2024.04.012","url":null,"abstract":"<div><div><span>In this paper, we give explicit conditions characterizing the Følner rank one group actions that factor onto a finite odometer; those that factor onto an arbitrary, but specified odometer, and those that factor onto an unspecified odometer. We also give explicit conditions describing the Følner rank one actions that are conjugate to a specific odometer, and those that are conjugate to some odometer. These conditions are based on cutting and stacking procedures used to generate the action, and generalize results given by Foreman, Gao, Hill, Silva and Weiss for rank one </span><span><math><mi>Z</mi></math></span>-actions.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1105-1137"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141032515","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}
引用次数: 0
A note on the inverse problem for finite differential Galois groups 关于有限微分伽罗瓦群逆问题的说明
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-11-01 DOI: 10.1016/j.indag.2024.06.005
{"title":"A note on the inverse problem for finite differential Galois groups","authors":"","doi":"10.1016/j.indag.2024.06.005","DOIUrl":"10.1016/j.indag.2024.06.005","url":null,"abstract":"<div><div>In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by van der Put et al. (2020). As an application, we show that there is no hypergeometric equation with projective differential Galois group isomorphic to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>72</mn></mrow></msub></math></span>, thus completing Beukers and Heckman’s answer (Beukers and Heckman, 1989) to the question of which irreducible finite subgroup of <span><math><mrow><mi>P</mi><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> are the projective monodromy of a hypergeometric equation.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1259-1269"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569879","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}
引用次数: 0
An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields 拉斯穆森和玉川对函数域上无性变种猜想的类比
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-11-01 DOI: 10.1016/j.indag.2024.06.007
Mentzelos Melistas
{"title":"An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields","authors":"Mentzelos Melistas","doi":"10.1016/j.indag.2024.06.007","DOIUrl":"10.1016/j.indag.2024.06.007","url":null,"abstract":"<div><div>Let <span><math><mi>L</mi></math></span> be a number field and let <span><math><mi>ℓ</mi></math></span> be a prime number. Rasmussen and Tamagawa conjectured, in a precise sense, that abelian varieties whose field of definition of the <span><math><mi>ℓ</mi></math></span>-power torsion is both a pro-<span><math><mi>ℓ</mi></math></span> extension of <span><math><mrow><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and unramified away from <span><math><mi>ℓ</mi></math></span> are quite rare. In this paper, we formulate an analogue of the Rasmussen–Tamagawa conjecture for non-isotrivial abelian varieties defined over function fields. We provide a proof of our analogue in the case of elliptic curves. In higher dimensions, when the base field is a subfield of the complex numbers, we show that our conjecture is a consequence of the uniform geometric torsion conjecture. Finally, using a theorem of Bakker and Tsimerman we also prove our conjecture unconditionally for abelian varieties with real multiplication.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1270-1281"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141690718","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}
引用次数: 0
The Generalized Riemann Hypothesis from zeros of a single L-function 从单一 L 函数的零点出发的广义黎曼假说
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-11-01 DOI: 10.1016/j.indag.2024.07.009
William Banks
{"title":"The Generalized Riemann Hypothesis from zeros of a single L-function","authors":"William Banks","doi":"10.1016/j.indag.2024.07.009","DOIUrl":"10.1016/j.indag.2024.07.009","url":null,"abstract":"<div><div>For each primitive Dirichlet character <span><math><mi>χ</mi></math></span>, a hypothesis <span><math><mrow><msup><mrow><mi>GRH</mi></mrow><mrow><mi>†</mi></mrow></msup><mrow><mo>[</mo><mi>χ</mi><mo>]</mo></mrow></mrow></math></span> is formulated in terms of zeros of the associated <span><math><mi>L</mi></math></span>-function <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>χ</mi><mo>)</mo></mrow></mrow></math></span>. It is shown that for any such character, <span><math><mrow><msup><mrow><mi>GRH</mi></mrow><mrow><mi>†</mi></mrow></msup><mrow><mo>[</mo><mi>χ</mi><mo>]</mo></mrow></mrow></math></span> is equivalent to the Generalized Riemann Hypothesis.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1282-1293"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141845529","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}
引用次数: 0
Non-Hausdorff germinal groupoids for actions of countable groups 可数群作用的非豪斯多夫萌芽群形
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-11-01 DOI: 10.1016/j.indag.2024.05.005
{"title":"Non-Hausdorff germinal groupoids for actions of countable groups","authors":"","doi":"10.1016/j.indag.2024.05.005","DOIUrl":"10.1016/j.indag.2024.05.005","url":null,"abstract":"<div><div>We study conditions under which the germinal groupoid associated to a minimal equicontinuous action of a countable group on a Cantor set has non-Hausdorff topology. We develop a new criterion, which serves as an obstruction for the étale topology on the groupoid to be non-Hausdorff. We use this and other criteria to study the topology of germinal groupoids for a few classes of actions. In particular, we give examples of families of contracting and non-contracting self-similar groups, which are amenable and whose actions have associated germinal groupoids with non-Hausdorff topology.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1149-1184"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569880","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}
引用次数: 0
Critical curves of rotations 旋转的临界曲线
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2024.02.001
{"title":"Critical curves of rotations","authors":"","doi":"10.1016/j.indag.2024.02.001","DOIUrl":"10.1016/j.indag.2024.02.001","url":null,"abstract":"<div><p>In rotations with a binary symbolic dynamics, a critical curve is the locus of parameters for which the boundaries of the partition that defines the symbolic dynamics are connected via a prescribed number of iterations and symbolic itinerary. We study the arithmetical and geometrical properties of these curves in parameter space.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 989-1008"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000089/pdfft?md5=2f9d5a6610f18fbac9a8c979bf5335b4&pid=1-s2.0-S0019357724000089-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139679261","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}
引用次数: 0
Semicocycle discontinuities for substitutions and reverse-reading automata 替换和反读自动机的半循环不连续
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.05.003
{"title":"Semicocycle discontinuities for substitutions and reverse-reading automata","authors":"","doi":"10.1016/j.indag.2023.05.003","DOIUrl":"10.1016/j.indag.2023.05.003","url":null,"abstract":"<div><p>In this article we define the semigroup associated to a primitive substitution. We use it to construct a minimal automaton which generates a substitution sequence <span><math><mi>u</mi></math></span> in reverse reading. We show, in the case where the substitution has a coincidence, that this automaton completely describes the <em>semicocycle discontinuities</em> of <span><math><mi>u</mi></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 796-812"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000496/pdfft?md5=0a7daca4bb4987075ce1de3e272a8290&pid=1-s2.0-S0019357723000496-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46621925","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}
引用次数: 0
Automorphism groups of random substitution subshifts 随机替换子移的自同构群
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.08.006
{"title":"Automorphism groups of random substitution subshifts","authors":"","doi":"10.1016/j.indag.2023.08.006","DOIUrl":"10.1016/j.indag.2023.08.006","url":null,"abstract":"<div><p>We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 931-958"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000836/pdfft?md5=3e4882926d47d4fbc572c02f04ab7895&pid=1-s2.0-S0019357723000836-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44793068","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}
引用次数: 0
Inter-model sets in Rd are model sets Rd中的模型间集是模型集
IF 0.5 4区 数学
Indagationes Mathematicae-New Series Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.06.007
{"title":"Inter-model sets in Rd are model sets","authors":"","doi":"10.1016/j.indag.2023.06.007","DOIUrl":"10.1016/j.indag.2023.06.007","url":null,"abstract":"<div><p>We show that any union of finitely many shifted model sets from a given cut-and-project scheme is a model set in some modified cut-and-project scheme. Restricting to direct space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we show that any inter-model set is a model set in some modified cut-and-project scheme with second countable internal space. In both cases, the window in the modified cut-and-project scheme inherits the topological and measure-theoretic properties of the original windows.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 865-889"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44478350","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}
引用次数: 0
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