Pub Date : 2021-06-20DOI: 10.1080/10586458.2021.1926014
Hang Liu, H. Qin
Abstract In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of L-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these families and special values of L-functions of elliptic curves. We also find identities between the Mahler measures of these nonreciprocal families and tempered polynomials defining genus 1 curves. We will explain these relations by considering the pushforward and pullback of certain elements in K 2 of curves defined by these polynomials and applying Beilinson’s conjecture on K 2 of curves. We show that there are two and three explicit linearly independent elements in K 2 of certain families of genus 2 and genus 3 curves, respectively.
{"title":"Mahler Measure of Families of Polynomials Defining Genus 2 and 3 Curves","authors":"Hang Liu, H. Qin","doi":"10.1080/10586458.2021.1926014","DOIUrl":"https://doi.org/10.1080/10586458.2021.1926014","url":null,"abstract":"Abstract In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of L-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these families and special values of L-functions of elliptic curves. We also find identities between the Mahler measures of these nonreciprocal families and tempered polynomials defining genus 1 curves. We will explain these relations by considering the pushforward and pullback of certain elements in K 2 of curves defined by these polynomials and applying Beilinson’s conjecture on K 2 of curves. We show that there are two and three explicit linearly independent elements in K 2 of certain families of genus 2 and genus 3 curves, respectively.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"32 1","pages":"321 - 336"},"PeriodicalIF":0.5,"publicationDate":"2021-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2021.1926014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41523307","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: 10.1080/10586458.2021.1926016
P. Scholze
Abstract I propose a formalization challenge. The text below is a slightly edited version of the blog post Xena Project – Liquid Tensor Experiment, and I have kept its informal style.
{"title":"Liquid Tensor Experiment","authors":"P. Scholze","doi":"10.1080/10586458.2021.1926016","DOIUrl":"https://doi.org/10.1080/10586458.2021.1926016","url":null,"abstract":"Abstract I propose a formalization challenge. The text below is a slightly edited version of the blog post Xena Project – Liquid Tensor Experiment, and I have kept its informal style.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"31 1","pages":"349 - 354"},"PeriodicalIF":0.5,"publicationDate":"2021-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2021.1926016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48045444","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 : 2021-06-18DOI: 10.1080/10586458.2022.2041135
Greta Fischer, J. Gutt, M. Junger
In this article we explore a symplectic packing problem where the targets and domains are 2n-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to Z, and we require the embeddings to induce isomorphisms between first homology groups. In this case, Maley, Mastrangeli and Traynor [MMT00] showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they determined the simplex packing widths in dimension four for up to k = 12 simplices, along with lower bounds for higher values of k. We present a modified algorithmic approach that allows us to determine the k-simplex packing widths for up to k = 13 simplices in dimension four and up to k = 8 simplices in dimension six. Moreover, our approach determines all simplex-multisets that allow for optimal packings.
{"title":"Algorithmic Symplectic Packing","authors":"Greta Fischer, J. Gutt, M. Junger","doi":"10.1080/10586458.2022.2041135","DOIUrl":"https://doi.org/10.1080/10586458.2022.2041135","url":null,"abstract":"In this article we explore a symplectic packing problem where the targets and domains are 2n-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to Z, and we require the embeddings to induce isomorphisms between first homology groups. In this case, Maley, Mastrangeli and Traynor [MMT00] showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they determined the simplex packing widths in dimension four for up to k = 12 simplices, along with lower bounds for higher values of k. We present a modified algorithmic approach that allows us to determine the k-simplex packing widths for up to k = 13 simplices in dimension four and up to k = 8 simplices in dimension six. Moreover, our approach determines all simplex-multisets that allow for optimal packings.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47942757","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 : 2021-06-11DOI: 10.1080/10586458.2022.2092565
Lior Alon, R. Band, G. Berkolaiko
. An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph’s first Betti number β . We study the distribution of the nodal surplus values in the countably infinite set of the graph’s eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β . We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.
{"title":"Universality of Nodal Count Distribution in Large Metric Graphs","authors":"Lior Alon, R. Band, G. Berkolaiko","doi":"10.1080/10586458.2022.2092565","DOIUrl":"https://doi.org/10.1080/10586458.2022.2092565","url":null,"abstract":". An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph’s first Betti number β . We study the distribution of the nodal surplus values in the countably infinite set of the graph’s eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β . We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48398695","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 : 2021-04-29DOI: 10.1080/10586458.2022.2161676
Benjamin A. Burton, Hsien-Chih Chang, M. Löffler, Clément Maria, Arnaud de Mesmay, S. Schleimer, E. Sedgwick, J. Spreer
We present three"hard"diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $mathbb{S}^2$. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small"standard"example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in $mathbb{S}^2$.
{"title":"Hard Diagrams of the Unknot","authors":"Benjamin A. Burton, Hsien-Chih Chang, M. Löffler, Clément Maria, Arnaud de Mesmay, S. Schleimer, E. Sedgwick, J. Spreer","doi":"10.1080/10586458.2022.2161676","DOIUrl":"https://doi.org/10.1080/10586458.2022.2161676","url":null,"abstract":"We present three\"hard\"diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $mathbb{S}^2$. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small\"standard\"example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in $mathbb{S}^2$.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43372802","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 : 2021-04-19DOI: 10.1080/10586458.2022.2062073
Anthony Bordg, Lawrence Charles Paulson, Wenda Li
Abstract Church’s simple type theory is often deemed too simple for elaborate mathematical constructions. In particular, doubts were raised whether schemes could be formalized in this setting and a challenge was issued. Schemes are sophisticated mathematical objects in algebraic geometry introduced by Alexander Grothendieck in 1960. In this article we report on a successful formalization of schemes in the simple type theory of the proof assistant Isabelle/HOL, and we discuss the design choices which make this work possible. We show in the particular case of schemes how the powerful dependent types of Coq or Lean can be traded for a minimalist apparatus called locales.
{"title":"Simple Type Theory is not too Simple: Grothendieck’s Schemes Without Dependent Types","authors":"Anthony Bordg, Lawrence Charles Paulson, Wenda Li","doi":"10.1080/10586458.2022.2062073","DOIUrl":"https://doi.org/10.1080/10586458.2022.2062073","url":null,"abstract":"Abstract Church’s simple type theory is often deemed too simple for elaborate mathematical constructions. In particular, doubts were raised whether schemes could be formalized in this setting and a challenge was issued. Schemes are sophisticated mathematical objects in algebraic geometry introduced by Alexander Grothendieck in 1960. In this article we report on a successful formalization of schemes in the simple type theory of the proof assistant Isabelle/HOL, and we discuss the design choices which make this work possible. We show in the particular case of schemes how the powerful dependent types of Coq or Lean can be traded for a minimalist apparatus called locales.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"31 1","pages":"364 - 382"},"PeriodicalIF":0.5,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48872374","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 : 2021-04-16DOI: 10.1080/10586458.2021.1994488
A. Garber
The Voronoi conjecture on parallelohedra claims that for every convex polytope P that tiles Euclidean d-dimensional space with translations there exists a d-dimensional lattice such that P and the Voronoi polytope of this lattice are affinely equivalent. The Voronoi conjecture is still open for the general case but it is known that some combinatorial restriction for the face structure of P ensure that the Voronoi conjecture holds for P . In this paper we prove that if P is the Voronoi polytope of one of the dual root lattices D∗ d , E∗ 6 , E∗ 7 or E∗ 8 = E8 or their small perturbations, then every parallelohedron combinatorially equivalent to P in strong sense satisfies the Voronoi conjecture.
{"title":"On Combinatorics of Voronoi Polytopes for Perturbations of the Dual Root Lattices","authors":"A. Garber","doi":"10.1080/10586458.2021.1994488","DOIUrl":"https://doi.org/10.1080/10586458.2021.1994488","url":null,"abstract":"The Voronoi conjecture on parallelohedra claims that for every convex polytope P that tiles Euclidean d-dimensional space with translations there exists a d-dimensional lattice such that P and the Voronoi polytope of this lattice are affinely equivalent. The Voronoi conjecture is still open for the general case but it is known that some combinatorial restriction for the face structure of P ensure that the Voronoi conjecture holds for P . In this paper we prove that if P is the Voronoi polytope of one of the dual root lattices D∗ d , E∗ 6 , E∗ 7 or E∗ 8 = E8 or their small perturbations, then every parallelohedron combinatorially equivalent to P in strong sense satisfies the Voronoi conjecture.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48133149","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 : 2021-02-23DOI: 10.1080/10586458.2022.2158968
Brendan Owens, Frank Swenton
We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It successfully finds ribbon disks for slice two-bridge knots and for the connected sum of any alternating knot with its reverse mirror, as well as for 662,903 prime alternating knots of 21 or fewer crossings. We also identify some examples of ribbon alternating knots for which the algorithm fails to find ribbon disks, though a related search identifies all such examples known. Combining these searches with known obstructions, we resolve the sliceness of all but 3,276 of the over 1.2 billion prime alternating knots with 21 or fewer crossings.
{"title":"An Algorithm to Find Ribbon Disks for Alternating Knots","authors":"Brendan Owens, Frank Swenton","doi":"10.1080/10586458.2022.2158968","DOIUrl":"https://doi.org/10.1080/10586458.2022.2158968","url":null,"abstract":"We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It successfully finds ribbon disks for slice two-bridge knots and for the connected sum of any alternating knot with its reverse mirror, as well as for 662,903 prime alternating knots of 21 or fewer crossings. We also identify some examples of ribbon alternating knots for which the algorithm fails to find ribbon disks, though a related search identifies all such examples known. Combining these searches with known obstructions, we resolve the sliceness of all but 3,276 of the over 1.2 billion prime alternating knots with 21 or fewer crossings.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43721418","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 : 2021-02-23DOI: 10.1080/10586458.2021.1982080
O. Gorodetsky
Sporadic Ap'ery-like sequences were discovered by Zagier, by Almkvist and Zudilin and by Cooper in their searches for integral solutions for certain families of second- and third-order differential equations. We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic sequences. The new representations in turn lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 8 and powers of 3. We use these to establish the supercongruence $B_{np^k} equiv B_{np^{k-1}} bmod p^{2k}$ for all primes $p ge 3$ and integers $n,k ge 1$, where $B_n$ is a sequence discovered by Zagier and known as Sequence $mathbf{B}$. Additionally, for 14 out of the 15 sequences, the Newton polytopes of the Laurent polynomials used in our representations contain the origin as their only interior integral point. This property allows us to prove that these 14 sporadic sequences satisfy a strong form of the Lucas congruences, extending work of Malik and Straub. Moreover, we obtain lower bounds on the $p$-adic valuation of these 14 sequences via recent work of Delaygue.
{"title":"New Representations for all Sporadic Apéry-Like Sequences, With Applications to Congruences","authors":"O. Gorodetsky","doi":"10.1080/10586458.2021.1982080","DOIUrl":"https://doi.org/10.1080/10586458.2021.1982080","url":null,"abstract":"Sporadic Ap'ery-like sequences were discovered by Zagier, by Almkvist and Zudilin and by Cooper in their searches for integral solutions for certain families of second- and third-order differential equations. We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic sequences. The new representations in turn lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 8 and powers of 3. We use these to establish the supercongruence $B_{np^k} equiv B_{np^{k-1}} bmod p^{2k}$ for all primes $p ge 3$ and integers $n,k ge 1$, where $B_n$ is a sequence discovered by Zagier and known as Sequence $mathbf{B}$. Additionally, for 14 out of the 15 sequences, the Newton polytopes of the Laurent polynomials used in our representations contain the origin as their only interior integral point. This property allows us to prove that these 14 sporadic sequences satisfy a strong form of the Lucas congruences, extending work of Malik and Straub. Moreover, we obtain lower bounds on the $p$-adic valuation of these 14 sequences via recent work of Delaygue.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43232337","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 : 2021-02-09DOI: 10.1080/10586458.2020.1870179
Claudia He Yun
Abstract We compute the Sn -equivariant rational homology of the tropical moduli spaces for using a cellular chain complex for symmetric Δ-complexes in Sage.
摘要:本文利用元胞链配合物计算了对称Δ-complexes的热带模空间的Sn -等变有理同调。
{"title":"The Sn -Equivariant Rational Homology of the Tropical Moduli Spaces Δ2,n","authors":"Claudia He Yun","doi":"10.1080/10586458.2020.1870179","DOIUrl":"https://doi.org/10.1080/10586458.2020.1870179","url":null,"abstract":"Abstract We compute the Sn -equivariant rational homology of the tropical moduli spaces for using a cellular chain complex for symmetric Δ-complexes in Sage.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":"31 1","pages":"1345 - 1357"},"PeriodicalIF":0.5,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2020.1870179","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41498224","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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