Pub Date : 2021-10-13DOI: 10.1080/10586458.2021.1980456
Shanza Ayub, J. Simoi
Abstract We present numerical evidence for robust spectral rigidity among -symmetric domains of ellipses of eccentricity smaller than 0.30.
摘要本文给出了偏心率小于0.30的椭圆的非对称区域的鲁棒谱刚性的数值证据。
{"title":"Numerical Evidence of Robust Dynamical Spectral Rigidity of Ellipses Among Smooth -Symmetric Domains","authors":"Shanza Ayub, J. Simoi","doi":"10.1080/10586458.2021.1980456","DOIUrl":"https://doi.org/10.1080/10586458.2021.1980456","url":null,"abstract":"Abstract We present numerical evidence for robust spectral rigidity among -symmetric domains of ellipses of eccentricity smaller than 0.30.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"32 1","pages":"467 - 476"},"PeriodicalIF":0.5,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41718026","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}
Pub Date : 2021-10-04DOI: 10.1080/10586458.2021.1980466
Olexandr Konovalov, A. Smoktunowicz, L. Vendramin
ABSTRACT Erratum to the paper [Konovalov, Alexander; Smoktunowicz, Agata; Vendramin, Leandro. On skew braces and their ideals. Exp. Math. 30 (2021), no. 1, 95–104. DOI: 10.1080/10586458.2018.1492476.]
{"title":"Erratum to the Paper “On Skew Braces and Their Ideals”","authors":"Olexandr Konovalov, A. Smoktunowicz, L. Vendramin","doi":"10.1080/10586458.2021.1980466","DOIUrl":"https://doi.org/10.1080/10586458.2021.1980466","url":null,"abstract":"ABSTRACT Erratum to the paper [Konovalov, Alexander; Smoktunowicz, Agata; Vendramin, Leandro. On skew braces and their ideals. Exp. Math. 30 (2021), no. 1, 95–104. DOI: 10.1080/10586458.2018.1492476.]","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"31 1","pages":"346 - 346"},"PeriodicalIF":0.5,"publicationDate":"2021-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46078832","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}
Pub Date : 2021-09-14DOI: 10.1080/10586458.2022.2065555
John M. Machacek, N. Ovenhouse
We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the $2$-dimensional situation.
{"title":"Discrete Dynamical Systems From Real Valued Mutation","authors":"John M. Machacek, N. Ovenhouse","doi":"10.1080/10586458.2022.2065555","DOIUrl":"https://doi.org/10.1080/10586458.2022.2065555","url":null,"abstract":"We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the $2$-dimensional situation.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42389099","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}
Pub Date : 2021-09-09DOI: 10.1080/10586458.2022.2062074
Seungjai Lee, M. Vaughan-Lee
We classify the nilpotent Lie rings of order p with maximal class for p ≥ 5. This also provides a classification of the groups of order p with maximal class for p ≥ 11 via the Lazard correspondence.
{"title":"The Groups and Nilpotent Lie Rings of Order p8 with Maximal Class","authors":"Seungjai Lee, M. Vaughan-Lee","doi":"10.1080/10586458.2022.2062074","DOIUrl":"https://doi.org/10.1080/10586458.2022.2062074","url":null,"abstract":"We classify the nilpotent Lie rings of order p with maximal class for p ≥ 5. This also provides a classification of the groups of order p with maximal class for p ≥ 11 via the Lazard correspondence.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47493702","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}
Pub Date : 2021-08-04DOI: 10.1080/10586458.2022.2093802
E. Falbel, I. Pasquinelli, A. Ucan-Puc
We classify representations of a class of Deligne-Mostow lattices into PGL(3;C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are of the same type as the generators of Deligne-Mostow lattices. We also show local rigidity without constraints on the type of generators for six of them and we show the existence of local deformations for a number of representations in three of them. We use formal computations in SAGE and Maple to obtain the results. The code files are available on GitHub.
{"title":"Representations of Deligne-Mostow lattices into","authors":"E. Falbel, I. Pasquinelli, A. Ucan-Puc","doi":"10.1080/10586458.2022.2093802","DOIUrl":"https://doi.org/10.1080/10586458.2022.2093802","url":null,"abstract":"We classify representations of a class of Deligne-Mostow lattices into PGL(3;C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are of the same type as the generators of Deligne-Mostow lattices. We also show local rigidity without constraints on the type of generators for six of them and we show the existence of local deformations for a number of representations in three of them. We use formal computations in SAGE and Maple to obtain the results. The code files are available on GitHub.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45292101","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}
Pub Date : 2021-07-23DOI: 10.1080/10586458.2021.1986176
Thomas Browning, P. Lutz
ABSTRACT We describe a project to formalize Galois theory using the Lean theorem prover, which is part of a larger effort to formalize all of the standard undergraduate mathematics curriculum in Lean. We discuss some of the challenges we faced and the decisions we made in the course of this project. The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory, and the equivalence of several characterizations of finite degree Galois extensions.
{"title":"Formalizing Galois Theory","authors":"Thomas Browning, P. Lutz","doi":"10.1080/10586458.2021.1986176","DOIUrl":"https://doi.org/10.1080/10586458.2021.1986176","url":null,"abstract":"ABSTRACT We describe a project to formalize Galois theory using the Lean theorem prover, which is part of a larger effort to formalize all of the standard undergraduate mathematics curriculum in Lean. We discuss some of the challenges we faced and the decisions we made in the course of this project. The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory, and the equivalence of several characterizations of finite degree Galois extensions.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"31 1","pages":"413 - 424"},"PeriodicalIF":0.5,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42213596","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}
Pub Date : 2021-07-23DOI: 10.1080/10586458.2021.1926012
Lucien Hennecart
Abstract We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the global behavior of the coefficients of Kac polynomials. All computations were made using SageMath.
{"title":"Asymptotic Behavior of Kac Polynomials","authors":"Lucien Hennecart","doi":"10.1080/10586458.2021.1926012","DOIUrl":"https://doi.org/10.1080/10586458.2021.1926012","url":null,"abstract":"Abstract We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the global behavior of the coefficients of Kac polynomials. All computations were made using SageMath.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"32 1","pages":"294 - 312"},"PeriodicalIF":0.5,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2021.1926012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45493468","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}
Pub Date : 2021-07-01DOI: 10.1080/10586458.2021.1926002
Chantal David, M. Lalín, J. Nam
Abstract We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith in order to formulate conjectures for the -moments of L-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the -moments of characteristic polynomials of unitary matrices to extend our conjecture to such that and . Our conjectures are then numerically tested for two families. A novelty of our conjectures is that cubic twists naturally lead us to consider the possibility .
{"title":"Conjectures for Moments Associated With Cubic Twists of Elliptic Curves","authors":"Chantal David, M. Lalín, J. Nam","doi":"10.1080/10586458.2021.1926002","DOIUrl":"https://doi.org/10.1080/10586458.2021.1926002","url":null,"abstract":"Abstract We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith in order to formulate conjectures for the -moments of L-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the -moments of characteristic polynomials of unitary matrices to extend our conjecture to such that and . Our conjectures are then numerically tested for two families. A novelty of our conjectures is that cubic twists naturally lead us to consider the possibility .","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"32 1","pages":"105 - 132"},"PeriodicalIF":0.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2021.1926002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41435460","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}
Pub Date : 2021-06-21DOI: 10.1080/10586458.2021.1926013
Destine Lee, Iris Rosenblum-Sellers, Jakwanul Safin, Anda Tenie
Abstract Given a finite covering of graphs , it is not always the case that is spanned by lifts of primitive elements of . In this article, we study graphs for which this is not the case, and we give here the simplest known nontrivial examples of covers with this property, with covering degree as small as 128. Our first step is focusing our attention on the special class of graph covers where the deck group is a finite p-group. For such covers, there is a representation-theoretic criterion for identifying deck groups for which there exist covers with the property. We present an algorithm for determining if a finite p-group satisfies this criterion that uses only the character table of the group. Finally, we provide a complete census of all finite p-groups of rank and order < 1000 satisfying this criterion, all of which are new examples.
{"title":"Graph Coverings and (Im)primitive Homology: Examples of Low Degree","authors":"Destine Lee, Iris Rosenblum-Sellers, Jakwanul Safin, Anda Tenie","doi":"10.1080/10586458.2021.1926013","DOIUrl":"https://doi.org/10.1080/10586458.2021.1926013","url":null,"abstract":"Abstract Given a finite covering of graphs , it is not always the case that is spanned by lifts of primitive elements of . In this article, we study graphs for which this is not the case, and we give here the simplest known nontrivial examples of covers with this property, with covering degree as small as 128. Our first step is focusing our attention on the special class of graph covers where the deck group is a finite p-group. For such covers, there is a representation-theoretic criterion for identifying deck groups for which there exist covers with the property. We present an algorithm for determining if a finite p-group satisfies this criterion that uses only the character table of the group. Finally, we provide a complete census of all finite p-groups of rank and order < 1000 satisfying this criterion, all of which are new examples.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"32 1","pages":"313 - 320"},"PeriodicalIF":0.5,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2021.1926013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49529663","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}
Pub Date : 2021-06-20DOI: 10.1080/10586458.2021.1926003
James Williams
Abstract In this article, we investigate the powerful nilpotency class of powerfully nilpotent groups of standard nilpotency class 2. We outline the process of collecting data using the computer algebra system GAP, formulating a conjecture based on the data, and finally we prove the conjecture. In particular, we prove that for a powerfully nilpotent group of nilpotency class 2 and order pn , where p is an odd prime, the powerful nilpotency class of G is at most the integer part of . We also identify and explain what this means in terms of the powerful coclass of the group.
{"title":"Powerfully Nilpotent Groups of Class 2","authors":"James Williams","doi":"10.1080/10586458.2021.1926003","DOIUrl":"https://doi.org/10.1080/10586458.2021.1926003","url":null,"abstract":"Abstract In this article, we investigate the powerful nilpotency class of powerfully nilpotent groups of standard nilpotency class 2. We outline the process of collecting data using the computer algebra system GAP, formulating a conjecture based on the data, and finally we prove the conjecture. In particular, we prove that for a powerfully nilpotent group of nilpotency class 2 and order pn , where p is an odd prime, the powerful nilpotency class of G is at most the integer part of . We also identify and explain what this means in terms of the powerful coclass of the group.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"32 1","pages":"133 - 139"},"PeriodicalIF":0.5,"publicationDate":"2021-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2021.1926003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47396801","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}
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