Pub Date : 2022-07-15DOI: 10.1080/10586458.2022.2158969
M. Deraux
We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.
{"title":"On Subgroups Finite Index in Complex Hyperbolic Lattice Triangle Groups","authors":"M. Deraux","doi":"10.1080/10586458.2022.2158969","DOIUrl":"https://doi.org/10.1080/10586458.2022.2158969","url":null,"abstract":"We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47854555","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 : 2022-06-25DOI: 10.1080/10586458.2022.2088982
Alex Kontorovich
Here at Experimental Mathematics, we like to live up to our moniker. One of the seemingly major innovations in recent times is the rapid expansion of the capabilities of software for formalizing mathematics in interactive theorem provers, to the point that certain papers in top journals can be proved formally about a year or so after the appearance of their “human” versions. The goal of this Special Issue is to record the current state-of-the-art in formalization, and to understand the potential impact such software may have on research mathematics in the near to mid-term future. It certainly seems plausible that 20 years from now, there will exist very highly regarded journals which will only accept formalized proofs. Some of the more committed proselytizers argue that all of the top journals will switch, if not in 20 years, then in 50. (This is not as radical a claim as it may first seem, nor would such a transition be all that unusual in mathematics; e.g., presumably there was a point in the 19th century when it became required for research papers in calculus to include “rigorous” proofs using Cauchy’s ε/δ formalism. From Newton to the Bernoullis to Euler’s nearly thousand publications, no ε’s or δ’s were harmed.) This is particularly interesting from the viewpoint of what it may mean for the publishing process, which currently suffers from a number of inefficiencies. The first of these is error detection and correction. It is common these days to send out first for “quick opinions” of the form: assuming the results are correct, would the main theorems be of sufficient novelty and importance to warrant publication in such-andsuch selective journal. If these reports (often from senior, seasoned experts, and usually returned within a month or two) are positive, then an editor has the much more daunting task of securing referees willing to go through the paper with a fine-toothed comb and check, as best they can, for mathematical errors. (In practice, these are frequently more junior researchers, who may both have fewer other service-type obligations occupying their time, and may also stand to gain valuable experience from reading the submitted paper extremely thoroughly.) Naturally, some of the most important results are also the most difficult to verify, and prone to errors which are not discovered at the refereeing stage. Instances of such abound, so we will not repeat them here. Beyond error correction, submission of formalized mathematics may allow for a much more rapid refereeing process, in which one may need only check that the definitions and theorems have been formalized correctly (which in and of itself is a rather subtle, nontrivial task!), and then let the compiler do the rest.1 Indeed, this is largely how this Special Issue was assembled: referees had plenty of comments on the exposition and quality of results submitted, as well as in some cases correcting the very definitions being formalized, but beyond that, everything was up to
{"title":"Foreword to: Special Issue on Interactive Theorem Provers","authors":"Alex Kontorovich","doi":"10.1080/10586458.2022.2088982","DOIUrl":"https://doi.org/10.1080/10586458.2022.2088982","url":null,"abstract":"Here at Experimental Mathematics, we like to live up to our moniker. One of the seemingly major innovations in recent times is the rapid expansion of the capabilities of software for formalizing mathematics in interactive theorem provers, to the point that certain papers in top journals can be proved formally about a year or so after the appearance of their “human” versions. The goal of this Special Issue is to record the current state-of-the-art in formalization, and to understand the potential impact such software may have on research mathematics in the near to mid-term future. It certainly seems plausible that 20 years from now, there will exist very highly regarded journals which will only accept formalized proofs. Some of the more committed proselytizers argue that all of the top journals will switch, if not in 20 years, then in 50. (This is not as radical a claim as it may first seem, nor would such a transition be all that unusual in mathematics; e.g., presumably there was a point in the 19th century when it became required for research papers in calculus to include “rigorous” proofs using Cauchy’s ε/δ formalism. From Newton to the Bernoullis to Euler’s nearly thousand publications, no ε’s or δ’s were harmed.) This is particularly interesting from the viewpoint of what it may mean for the publishing process, which currently suffers from a number of inefficiencies. The first of these is error detection and correction. It is common these days to send out first for “quick opinions” of the form: assuming the results are correct, would the main theorems be of sufficient novelty and importance to warrant publication in such-andsuch selective journal. If these reports (often from senior, seasoned experts, and usually returned within a month or two) are positive, then an editor has the much more daunting task of securing referees willing to go through the paper with a fine-toothed comb and check, as best they can, for mathematical errors. (In practice, these are frequently more junior researchers, who may both have fewer other service-type obligations occupying their time, and may also stand to gain valuable experience from reading the submitted paper extremely thoroughly.) Naturally, some of the most important results are also the most difficult to verify, and prone to errors which are not discovered at the refereeing stage. Instances of such abound, so we will not repeat them here. Beyond error correction, submission of formalized mathematics may allow for a much more rapid refereeing process, in which one may need only check that the definitions and theorems have been formalized correctly (which in and of itself is a rather subtle, nontrivial task!), and then let the compiler do the rest.1 Indeed, this is largely how this Special Issue was assembled: referees had plenty of comments on the exposition and quality of results submitted, as well as in some cases correcting the very definitions being formalized, but beyond that, everything was up to ","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41405549","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 : 2022-06-08DOI: 10.1080/10586458.2022.2075491
Ernesto Girondo, Cristian Reyes
Abstract
The radius of a packing of metric discs embedded in a compact hyperbolic surface is bounded by an extremal value dependent upon the topology of the surface and the number of discs in the packing. In this paper we discuss the possibility of finding multiple extremal disc-packings within a given surface, determining the combinatorial-arithmetic conditions on the topology of the surface and the number of discs of the packing that allow such a phenomenon to happen. Moreover, we provide explicit examples of surfaces containing multiple extremal packings for each type of packing and each topological type of surface possible. Our construction relies in computer experimentation in two ways: first, by performing numerical computations that suggest certain surfaces as good candidates to contain more than one extremal packing, and second by checking with computer algebra software some lengthy necessary algebraic conditions in certain number fields that prove that the surfaces numerically constructed do indeed contain multiple extremal disc-packings.
{"title":"Multiple Extremal Disc-Packings in Compact Hyperbolic Surfaces","authors":"Ernesto Girondo, Cristian Reyes","doi":"10.1080/10586458.2022.2075491","DOIUrl":"https://doi.org/10.1080/10586458.2022.2075491","url":null,"abstract":"<p><b>Abstract</b></p><p>The radius of a packing of metric discs embedded in a compact hyperbolic surface is bounded by an extremal value dependent upon the topology of the surface and the number of discs in the packing. In this paper we discuss the possibility of finding multiple extremal disc-packings within a given surface, determining the combinatorial-arithmetic conditions on the topology of the surface and the number of discs of the packing that allow such a phenomenon to happen. Moreover, we provide explicit examples of surfaces containing multiple extremal packings for each type of packing and each topological type of surface possible. Our construction relies in computer experimentation in two ways: first, by performing numerical computations that suggest certain surfaces as good candidates to contain more than one extremal packing, and second by checking with computer algebra software some lengthy necessary algebraic conditions in certain number fields that prove that the surfaces numerically constructed do indeed contain multiple extremal disc-packings.</p>","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505178","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 : 2022-05-18DOI: 10.1080/10586458.2023.2174212
Y. Benoist
We prove that on the cyclic groups of odd order d, there exist non zero functions whose convolution square f*f(2t) is proportional to their square f(t)^2 when the proportionality constant is given by an imaginary quadratic integer of norm d which is equal to 1 modulo 2. The proof involves theta functions on elliptic curves with complex multiplication.
{"title":"Convolution and Square in Abelian Groups I","authors":"Y. Benoist","doi":"10.1080/10586458.2023.2174212","DOIUrl":"https://doi.org/10.1080/10586458.2023.2174212","url":null,"abstract":"We prove that on the cyclic groups of odd order d, there exist non zero functions whose convolution square f*f(2t) is proportional to their square f(t)^2 when the proportionality constant is given by an imaginary quadratic integer of norm d which is equal to 1 modulo 2. The proof involves theta functions on elliptic curves with complex multiplication.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47093228","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 : 2022-05-09DOI: 10.1080/10586458.2022.2063208
Yves Colin de Verdière, David Vicente
{"title":"Large-Time Series Expansion of the Wave Front Length in the Euclidean Disk","authors":"Yves Colin de Verdière, David Vicente","doi":"10.1080/10586458.2022.2063208","DOIUrl":"https://doi.org/10.1080/10586458.2022.2063208","url":null,"abstract":"","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45606648","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 : 2022-04-25DOI: 10.1080/10586458.2023.2184882
Giovanni Staglianò
We present some applications of the Macaulay2 software package SpecialFanoFourfolds, a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel--Mukai fourfolds. In particular, we show how to construct new examples of such fourfolds, some of which turn out to be rational. We also describe how to calculate K3 surfaces associated with cubic or Gushel-Mukai fourfolds, which relies on an explicit unirationality of some moduli spaces of K3 surfaces.
我们介绍了Macaulay2软件包SpecialFanoFourfolds的一些应用，这是一个用于处理Hodge-special cubic four - fold和Hodge-special Gushel- Mukai four - fold的软件包。特别是，我们展示了如何构建这种四倍的新示例，其中一些被证明是合理的。我们还描述了如何计算与三次或Gushel-Mukai四倍相关的K3曲面，这依赖于K3曲面的一些模空间的显式唯一性。
{"title":"Explicit Computations with Cubic Fourfolds, Gushel–Mukai Fourfolds, and their Associated K3 Surfaces","authors":"Giovanni Staglianò","doi":"10.1080/10586458.2023.2184882","DOIUrl":"https://doi.org/10.1080/10586458.2023.2184882","url":null,"abstract":"We present some applications of the Macaulay2 software package SpecialFanoFourfolds, a package for working with Hodge-special cubic fourfolds and Hodge-special Gushel--Mukai fourfolds. In particular, we show how to construct new examples of such fourfolds, some of which turn out to be rational. We also describe how to calculate K3 surfaces associated with cubic or Gushel-Mukai fourfolds, which relies on an explicit unirationality of some moduli spaces of K3 surfaces.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42059071","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 : 2022-04-06DOI: 10.1080/10586458.2022.2050324
Eryk Kopczynski, Dorota Celinska-Kopczynska
Abstract We present novel methods for real-time native geodesic rendering of anisotropic geometries and similar geometries, Nil, twisted . We also include partial results for the Berger sphere and explain why such real-time rendering of this geometry is difficult. Current approaches are not applicable for rendering complex shapes in these geometries, such as traditional 3D models, because of the computational complexity of ray-based approaches or significant rendering artifacts in older primitive-based approaches. We use tessellations to represent large shapes without numerical precision issues. Our efficient methods for computing the inverse exponential mapping are applicable not only for visualization but for games, physics simulations, and machine learning purposes as well.
{"title":"Real-Time Visualization in Anisotropic Geometries","authors":"Eryk Kopczynski, Dorota Celinska-Kopczynska","doi":"10.1080/10586458.2022.2050324","DOIUrl":"https://doi.org/10.1080/10586458.2022.2050324","url":null,"abstract":"Abstract We present novel methods for real-time native geodesic rendering of anisotropic geometries and similar geometries, Nil, twisted . We also include partial results for the Berger sphere and explain why such real-time rendering of this geometry is difficult. Current approaches are not applicable for rendering complex shapes in these geometries, such as traditional 3D models, because of the computational complexity of ray-based approaches or significant rendering artifacts in older primitive-based approaches. We use tessellations to represent large shapes without numerical precision issues. Our efficient methods for computing the inverse exponential mapping are applicable not only for visualization but for games, physics simulations, and machine learning purposes as well.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48550401","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 : 2022-03-21DOI: 10.1080/10586458.2022.2041134
P. S. Kolesnikov, B. K. Sartayev
Abstract
A Gelfand–Dorfman algebra (GD-algebra) is said to be special if it can be embedded into a differential Poisson algebra. In this paper, we prove that the class of all special GD-algebras is closed with respect to homomorphisms and thus forms a variety. We describe a technique for finding potentially all special identities of GD-algebras and derive two known special identities of degree 4 in this way. By computing the Gröbner basis for the corresponding shuffle operad, we show that these two identities imply all special ones up to degree 5.
{"title":"On the Special Identities of Gelfand–Dorfman Algebras","authors":"P. S. Kolesnikov, B. K. Sartayev","doi":"10.1080/10586458.2022.2041134","DOIUrl":"https://doi.org/10.1080/10586458.2022.2041134","url":null,"abstract":"<p><b>Abstract</b></p><p>A Gelfand–Dorfman algebra (GD-algebra) is said to be special if it can be embedded into a differential Poisson algebra. In this paper, we prove that the class of all special GD-algebras is closed with respect to homomorphisms and thus forms a variety. We describe a technique for finding potentially all special identities of GD-algebras and derive two known special identities of degree 4 in this way. By computing the Gröbner basis for the corresponding shuffle operad, we show that these two identities imply all special ones up to degree 5.</p>","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505180","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 : 2022-03-10DOI: 10.1080/10586458.2022.2041133
David Lehavi
Abstract
We effectively reconstruct the set of enveloping quadrics of a generic curve C of genus 5 from its theta hyperplanes; for a generic genus 5 curve C this data suffices to effectively reconstruct C. As a consequence we get a complete description of the Schottky locus in genus 5 in terms of theta hyperplanes. The computational part of the proof is a certified numerical argument.
{"title":"Effective Reconstruction of Generic Genus 5 Curves from their Theta Hyperplanes","authors":"David Lehavi","doi":"10.1080/10586458.2022.2041133","DOIUrl":"https://doi.org/10.1080/10586458.2022.2041133","url":null,"abstract":"<p><b>Abstract</b></p><p>We effectively reconstruct the set of enveloping quadrics of a generic curve <i>C</i> of genus 5 from its theta hyperplanes; for a generic genus 5 curve <i>C</i> this data suffices to effectively reconstruct <i>C</i>. As a consequence we get a complete description of the Schottky locus in genus 5 in terms of theta hyperplanes. The computational part of the proof is a certified numerical argument.</p>","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505179","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 : 2022-02-20DOI: 10.1080/10586458.2022.2102095
M. Arnold, R. Schwartz, S. Tabachnikov
The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is 2-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map x (cid:55)→ − 4 x mod 1. We also present the results of an experimental study in the case of hexagons.
曲线的渐屈线是其法线的包络线。在本注释中，我们考虑了这种构造的投影自然离散模拟：我们定义了投影平面中多边形边的投影垂直平分线，并研究了将多边形发送到由其边的投影正交平分线形成的新多边形的映射。我们考虑这个映射作用在投影多边形的模空间上。我们分析了五边形的情况；在这种情况下，模量空间是二维的。映射的第二次迭代有一个积分，其水平曲线是三次曲线，并且这些水平曲线上的变换与映射x共轭（cid:55）→ − 4 x mod 1。我们还介绍了六边形情况下的实验研究结果。
{"title":"On Projective Evolutes of Polygons","authors":"M. Arnold, R. Schwartz, S. Tabachnikov","doi":"10.1080/10586458.2022.2102095","DOIUrl":"https://doi.org/10.1080/10586458.2022.2102095","url":null,"abstract":"The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is 2-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map x (cid:55)→ − 4 x mod 1. We also present the results of an experimental study in the case of hexagons.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45417476","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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