Pub Date : 2022-02-03DOI: 10.1080/10586458.2022.2113576
Riccardo Moschetti, Franco Rota, L. Schaffler
. For an Enriques surface S , the non-degeneracy invariant nd( S ) retains information on the elliptic fibrations of S and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on S together with a configuration of smooth rational curves, and gives a lower bound for nd( S ) . We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying nd( S ) = 10 which are not general and with infinite automorphism group. We obtain lower bounds on nd( S ) for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes–Pardini. Finally, we recover Dolgachev and Kond¯o’s computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.
{"title":"A Computational View on the Non-degeneracy Invariant for Enriques Surfaces","authors":"Riccardo Moschetti, Franco Rota, L. Schaffler","doi":"10.1080/10586458.2022.2113576","DOIUrl":"https://doi.org/10.1080/10586458.2022.2113576","url":null,"abstract":". For an Enriques surface S , the non-degeneracy invariant nd( S ) retains information on the elliptic fibrations of S and its polarizations. In the current paper, we introduce a combinatorial version of the non-degeneracy invariant which depends on S together with a configuration of smooth rational curves, and gives a lower bound for nd( S ) . We provide a SageMath code that computes this combinatorial invariant and we apply it in several examples. First we identify a new family of nodal Enriques surfaces satisfying nd( S ) = 10 which are not general and with infinite automorphism group. We obtain lower bounds on nd( S ) for the Enriques surfaces with eight disjoint smooth rational curves studied by Mendes Lopes–Pardini. Finally, we recover Dolgachev and Kond¯o’s computation of the non-degeneracy invariant of the Enriques surfaces with finite automorphism group and provide additional information on the geometry of their elliptic fibrations.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44113465","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 : 2022-01-01DOI: 10.1080/10586458.2019.1592035
R. V. Bommel
{"title":"Numerical Verification of the Birch and Swinnerton-Dyer Conjecture for Hyperelliptic Curves of Higher Genus over ℚ up to Squares","authors":"R. V. Bommel","doi":"10.1080/10586458.2019.1592035","DOIUrl":"https://doi.org/10.1080/10586458.2019.1592035","url":null,"abstract":"","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2019.1592035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59683150","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 : 2021-12-30DOI: 10.1080/10586458.2021.2011491
Stefano Barbero, Umberto Cerruti, N. Murru
{"title":"Periodic Representations and Approximations of p-adic Numbers Via Continued Fractions","authors":"Stefano Barbero, Umberto Cerruti, N. Murru","doi":"10.1080/10586458.2021.2011491","DOIUrl":"https://doi.org/10.1080/10586458.2021.2011491","url":null,"abstract":"","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41285030","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 : 2021-12-20DOI: 10.1080/10586458.2021.2011806
E. Amzallag, L.-P. Arguin, E. Bailey, K. Huib, R. Rao
Abstract
Selberg’s central limit theorem states that the values of , where τ is a uniform random variable on , are asymptotically distributed like a Gaussian random variable of mean 0 and standard deviation . It was conjectured by Radziwiłł that this distribution breaks down for values of order
{"title":"Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem","authors":"E. Amzallag, L.-P. Arguin, E. Bailey, K. Huib, R. Rao","doi":"10.1080/10586458.2021.2011806","DOIUrl":"https://doi.org/10.1080/10586458.2021.2011806","url":null,"abstract":"<p><b>Abstract</b></p><p>Selberg’s central limit theorem states that the values of <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0001.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0001.gif\"}' src=\"//:0\"/><span></span></span><span><img alt=\"\" data-formula-source='{\"type\" : \"mathjax\"}' src=\"//:0\"/><math display=\"inline\"><mrow><mtext>log</mtext><mo> </mo><mo>|</mo><mi>ζ</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi mathvariant=\"normal\">i</mi><mi>τ</mi><mo stretchy=\"false\">)</mo><mo>|</mo></mrow></math></span>, where <i>τ</i> is a uniform random variable on <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0002.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0002.gif\"}' src=\"//:0\"/><span></span></span><span><img alt=\"\" data-formula-source='{\"type\" : \"mathjax\"}' src=\"//:0\"/><math display=\"inline\"><mrow><mo stretchy=\"false\">[</mo><mi>T</mi><mo>,</mo><mn>2</mn><mi>T</mi><mo stretchy=\"false\">]</mo></mrow></math></span>, are asymptotically distributed like a Gaussian random variable of mean 0 and standard deviation <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0003.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0003.gif\"}' src=\"//:0\"/><span></span></span><span><img alt=\"\" data-formula-source='{\"type\" : \"mathjax\"}' src=\"//:0\"/><math display=\"inline\"><mrow><msqrt><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mtext>log</mtext><mo> </mo><mo> </mo><mtext>log</mtext><mo> </mo><mi>T</mi></mrow></msqrt></mrow></math></span>. It was conjectured by Radziwiłł that this distribution breaks down for values of order <span><noscript><img alt=\"\" src=\"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0004.gif\"/></noscript><img alt=\"\" data-formula-source='{\"type\" : \"image\", \"src\" : \"/na101/home/literatum/publisher/tandf/journals/content/uexm20/0/uexm20.ahead-of-print/10586458.2021.2011806/20211220/images/uexm_a_2011806_ilm0004.gif\"}' src=\"//:0\"/><span></span></span><span><img alt=\"\" data-formula-source='{\"type\" : \"mathjax\"}' src=\"//:0\"/><math displ","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505181","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 : 2021-12-11DOI: 10.1080/10586458.2021.2011807
H. H. Rugh, L. Tan, Fei Yang
{"title":"Schwarzian Versus a Family of Moving Parabolic Points","authors":"H. H. Rugh, L. Tan, Fei Yang","doi":"10.1080/10586458.2021.2011807","DOIUrl":"https://doi.org/10.1080/10586458.2021.2011807","url":null,"abstract":"","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43457509","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 : 2021-10-22DOI: 10.1080/10586458.2021.1982079
Christopher H. Cashen, Charlotte Hoffmann
We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups.
我们为更响亮和威尔顿的猜想提供了实验支持，即非原语的排名大于2的词产生双曲单相关群。
{"title":"Short, Highly Imprimitive Words Yield Hyperbolic One-Relator Groups","authors":"Christopher H. Cashen, Charlotte Hoffmann","doi":"10.1080/10586458.2021.1982079","DOIUrl":"https://doi.org/10.1080/10586458.2021.1982079","url":null,"abstract":"We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138541170","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 : 2021-10-16DOI: 10.1080/10586458.2021.1980751
S. Steinerberger
Let $mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $mu^{boxplus k}$ for any real $k geq 1$. The purpose of this short note is to give an elementary description of $mu^{boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.
{"title":"Free Convolution Powers Via Roots of Polynomials","authors":"S. Steinerberger","doi":"10.1080/10586458.2021.1980751","DOIUrl":"https://doi.org/10.1080/10586458.2021.1980751","url":null,"abstract":"Let $mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $mu^{boxplus k}$ for any real $k geq 1$. The purpose of this short note is to give an elementary description of $mu^{boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44902678","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 : 2021-10-16DOI: 10.1080/10586458.2021.1980462
Joshua Coyston, J. McKee
Abstract We attach Mahler measures to digraphs and find combinatorial realizations of nearly all of the known low-degree ( ) small (< 1.3) one-variable Mahler measures. We find one new such measure not on either of the lists maintained by Mossinghoff and Sac-Épée. Considering limits of sequences of measures attached to families of digraphs, we get combinatorial explanations for 57 of the 61 known irreducible two-variable measures below 1.37.
{"title":"Small Mahler Measures From Digraphs","authors":"Joshua Coyston, J. McKee","doi":"10.1080/10586458.2021.1980462","DOIUrl":"https://doi.org/10.1080/10586458.2021.1980462","url":null,"abstract":"Abstract We attach Mahler measures to digraphs and find combinatorial realizations of nearly all of the known low-degree ( ) small (< 1.3) one-variable Mahler measures. We find one new such measure not on either of the lists maintained by Mossinghoff and Sac-Épée. Considering limits of sequences of measures attached to families of digraphs, we get combinatorial explanations for 57 of the 61 known irreducible two-variable measures below 1.37.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44860920","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 : 2021-10-14DOI: 10.1080/10586458.2021.1982427
Justin Chen, P. Dey, P. Dey
ABSTRACT Orthostochastic matrices are the entrywise squares of orthogonal matrices, and naturally arise in various contexts, including notably definite symmetric determinantal representations of real polynomials. However, defining equations for the real variety were previously known only for 3 × 3 matrices. We study the real variety of 4 × 4 orthostochastic matrices, and find a minimal defining set of equations consisting of 6 quintics and 3 octics. The techniques used here involve a wide range of both symbolic and computational methods, in computer algebra and numerical algebraic geometry.
{"title":"The 4 × 4 Orthostochastic Variety","authors":"Justin Chen, P. Dey, P. Dey","doi":"10.1080/10586458.2021.1982427","DOIUrl":"https://doi.org/10.1080/10586458.2021.1982427","url":null,"abstract":"ABSTRACT Orthostochastic matrices are the entrywise squares of orthogonal matrices, and naturally arise in various contexts, including notably definite symmetric determinantal representations of real polynomials. However, defining equations for the real variety were previously known only for 3 × 3 matrices. We study the real variety of 4 × 4 orthostochastic matrices, and find a minimal defining set of equations consisting of 6 quintics and 3 octics. The techniques used here involve a wide range of both symbolic and computational methods, in computer algebra and numerical algebraic geometry.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45816063","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 : 2021-10-13DOI: 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":null,"pages":null},"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：