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On Lagrangian and Legendrian Singularities 关于拉格朗日奇点和勒让德奇点
Arnold Mathematical Journal Pub Date : 2020-10-20 DOI: 10.1007/s40598-020-00161-9
Vyacheslav D. Sedykh
{"title":"On Lagrangian and Legendrian Singularities","authors":"Vyacheslav D. Sedykh","doi":"10.1007/s40598-020-00161-9","DOIUrl":"10.1007/s40598-020-00161-9","url":null,"abstract":"<div><p>We describe the topology of stable simple multisingularities of Lagrangian and Legendrian maps. In particular, the tables of adjacency indices of monosingularities to multisingularities are given for generic caustics and wave fronts in spaces of small dimensions. The paper is an extended version of the author’s talk in the International Conference “Contemporary mathematics” in honor of the 80th birthday of V. I. Arnold (Moscow, Russia, 2017).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00161-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46717562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey 点涡动力学的辛约化可积性:综述
Arnold Mathematical Journal Pub Date : 2020-10-15 DOI: 10.1007/s40598-020-00162-8
Klas Modin, Milo Viviani
{"title":"Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey","authors":"Klas Modin,&nbsp;Milo Viviani","doi":"10.1007/s40598-020-00162-8","DOIUrl":"10.1007/s40598-020-00162-8","url":null,"abstract":"<div><p>Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for <span>(N=2)</span>, 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00162-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45350512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Probabilistic Schubert Calculus: Asymptotics 概率舒伯特微积分:渐近性
Arnold Mathematical Journal Pub Date : 2020-09-18 DOI: 10.1007/s40598-020-00160-w
Antonio Lerario, Léo Mathis
{"title":"Probabilistic Schubert Calculus: Asymptotics","authors":"Antonio Lerario,&nbsp;Léo Mathis","doi":"10.1007/s40598-020-00160-w","DOIUrl":"10.1007/s40598-020-00160-w","url":null,"abstract":"<div><p>In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by <span>(delta _{k,n})</span> the average number of projective <i>k</i>-planes in <span>({mathbb {R}}mathrm {P}^n)</span> that intersect <span>((k+1)(n-k))</span> many random, independent and uniformly distributed linear projective subspaces of dimension <span>(n-k-1)</span>. They called <span>(delta _{k,n})</span> the expected degree of the real Grassmannian <span>({mathbb {G}}(k,n))</span> and, in the case <span>(k=1)</span>, they proved that: </p><div><div><span>$$begin{aligned} delta _{1,n}= frac{8}{3pi ^{5/2}} cdot left( frac{pi ^2}{4}right) ^n cdot n^{-1/2} left( 1+{mathcal {O}}left( n^{-1}right) right) . end{aligned}$$</span></div></div><p>Here we generalize this result and prove that for every fixed integer <span>(k&gt;0)</span> and as <span>(nrightarrow infty )</span>, we have </p><div><div><span>$$begin{aligned} delta _{k,n}=a_k cdot left( b_kright) ^ncdot n^{-frac{k(k+1)}{4}}left( 1+{mathcal {O}}(n^{-1})right) end{aligned}$$</span></div></div><p>where <span>(a_k)</span> and <span>(b_k)</span> are some (explicit) constants, and <span>(a_k)</span> involves an interesting integral over the space of polynomials that have all real roots. For instance: </p><div><div><span>$$begin{aligned} delta _{2,n}= frac{9sqrt{3}}{2048sqrt{2pi }} cdot 8^n cdot n^{-3/2} left( 1+{mathcal {O}}left( n^{-1}right) right) . end{aligned}$$</span></div></div><p>Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for <span>(delta _{1,n})</span> involving a one-dimensional integral of certain combination of Elliptic functions.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00160-w","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43125185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Quotients of Torus Endomorphisms and Lattès-Type Maps Torus自同态的商与Lattès型映射
Arnold Mathematical Journal Pub Date : 2020-09-14 DOI: 10.1007/s40598-020-00156-6
Mario Bonk, Daniel Meyer
{"title":"Quotients of Torus Endomorphisms and Lattès-Type Maps","authors":"Mario Bonk,&nbsp;Daniel Meyer","doi":"10.1007/s40598-020-00156-6","DOIUrl":"10.1007/s40598-020-00156-6","url":null,"abstract":"<div><p>We show that if an expanding Thurston map is the quotient of a torus endomorphism, then it has a parabolic orbifold and is a Lattès-type map.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00156-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46984252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
On Nörlund–Voronoi Summability and Instability of Rational Maps 论Nörlund-Voronoi Rational映射的可和性与不稳定性
Arnold Mathematical Journal Pub Date : 2020-09-08 DOI: 10.1007/s40598-020-00158-4
Carlos Cabrera, Peter Makienko, Alfredo Poirier
{"title":"On Nörlund–Voronoi Summability and Instability of Rational Maps","authors":"Carlos Cabrera,&nbsp;Peter Makienko,&nbsp;Alfredo Poirier","doi":"10.1007/s40598-020-00158-4","DOIUrl":"10.1007/s40598-020-00158-4","url":null,"abstract":"<div><p>We investigate the connection between the instability of rational maps and summability methods applied to the spectrum of a critical point belonging to the Julia set of a rational map.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00158-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49019372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups 映射类群的Euler类幂的不消失性
Arnold Mathematical Journal Pub Date : 2020-09-07 DOI: 10.1007/s40598-020-00159-3
Solomon Jekel, Rita Jiménez Rolland
{"title":"On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups","authors":"Solomon Jekel,&nbsp;Rita Jiménez Rolland","doi":"10.1007/s40598-020-00159-3","DOIUrl":"10.1007/s40598-020-00159-3","url":null,"abstract":"<div><p>The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this largely expository note, we determine the non-vanishing behavior of the powers of this class. Our argument relies on restricting the cohomology classes to torsion subgroups of the mapping class group.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00159-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50461063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups 关于映射类群的欧拉类幂的不灭性
Arnold Mathematical Journal Pub Date : 2020-09-07 DOI: 10.1007/s40598-020-00159-3
Solomon Jekel, Rita Jiménez Rolland
{"title":"On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups","authors":"Solomon Jekel, Rita Jiménez Rolland","doi":"10.1007/s40598-020-00159-3","DOIUrl":"https://doi.org/10.1007/s40598-020-00159-3","url":null,"abstract":"","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00159-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52850038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Newton Polyhedra and Good Compactification Theorem 牛顿多面体与良紧致化定理
Arnold Mathematical Journal Pub Date : 2020-09-03 DOI: 10.1007/s40598-020-00157-5
Askold Khovanskii
{"title":"Newton Polyhedra and Good Compactification Theorem","authors":"Askold Khovanskii","doi":"10.1007/s40598-020-00157-5","DOIUrl":"10.1007/s40598-020-00157-5","url":null,"abstract":"<div><p>A new transparent proof of the well-known good compactification theorem for the complex torus <span>(({mathbb {C}}^*)^n)</span> is presented. This theorem provides a powerful tool in enumerative geometry for subvarieties in the complex torus. The paper also contains an algorithm constructing a good compactification for a subvariety in <span>(({mathbb {C}}^*)^n)</span> explicitly defined by a system of equations. A new theorem on a toroidal-like compactification is stated. A transparent proof of this generalization of the good compactification theorem which is similar to proofs and constructions from this paper will be presented in a forthcoming publication.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00157-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50446883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Twisted Forms of Differential Lie Algebras over C(t)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {C}}(t)$$ C(t)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amssymb} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {C}}(t)$$
Arnold Mathematical Journal Pub Date : 2020-08-18 DOI: 10.1007/s40598-020-00155-7
A. Masuoka, Yuta Shimada
{"title":"Twisted Forms of Differential Lie Algebras over C(t)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {C}}(t)$$","authors":"A. Masuoka, Yuta Shimada","doi":"10.1007/s40598-020-00155-7","DOIUrl":"https://doi.org/10.1007/s40598-020-00155-7","url":null,"abstract":"","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00155-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52850000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Twisted Forms of Differential Lie Algebras over ({mathbb {C}}(t)) Associated with Complex Simple Lie Algebras 与复单李代数相关的（｛mathbb｛C｝｝（t））上微分李代数的扭曲形式
Arnold Mathematical Journal Pub Date : 2020-08-18 DOI: 10.1007/s40598-020-00155-7
Akira Masuoka, Yuta Shimada
{"title":"Twisted Forms of Differential Lie Algebras over ({mathbb {C}}(t)) Associated with Complex Simple Lie Algebras","authors":"Akira Masuoka,&nbsp;Yuta Shimada","doi":"10.1007/s40598-020-00155-7","DOIUrl":"10.1007/s40598-020-00155-7","url":null,"abstract":"<div><p>Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over <span>({mathbb {C}}(t))</span> associated with complex simple Lie algebras. Hopf–Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-020-00155-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50492093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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