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Open Problems on Billiards and Geometric Optics 台球与几何光学的开放问题
Arnold Mathematical Journal Pub Date : 2022-01-17 DOI: 10.1007/s40598-022-00198-y
Misha Bialy, Corentin Fierobe, Alexey Glutsyuk, Mark Levi, Alexander Plakhov, Serge Tabachnikov
{"title":"Open Problems on Billiards and Geometric Optics","authors":"Misha Bialy,&nbsp;Corentin Fierobe,&nbsp;Alexey Glutsyuk,&nbsp;Mark Levi,&nbsp;Alexander Plakhov,&nbsp;Serge Tabachnikov","doi":"10.1007/s40598-022-00198-y","DOIUrl":"10.1007/s40598-022-00198-y","url":null,"abstract":"<div><p>This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 3-4","pages":"411 - 422"},"PeriodicalIF":0.0,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43199280","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
Partial Duality of Hypermaps 超映射的部分对偶性
Arnold Mathematical Journal Pub Date : 2022-01-03 DOI: 10.1007/s40598-021-00194-8
S. Chmutov, F. Vignes-Tourneret
{"title":"Partial Duality of Hypermaps","authors":"S. Chmutov,&nbsp;F. Vignes-Tourneret","doi":"10.1007/s40598-021-00194-8","DOIUrl":"10.1007/s40598-021-00194-8","url":null,"abstract":"<div><p>We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or <span>(tau )</span>-model), or as three permutations on the set of half-edges (rotation system or <span>(sigma )</span>-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 3-4","pages":"445 - 468"},"PeriodicalIF":0.0,"publicationDate":"2022-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444764","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
A Symplectic Dynamics Proof of the Degree–Genus Formula 度-亏格公式的辛动力学证明
Arnold Mathematical Journal Pub Date : 2021-12-20 DOI: 10.1007/s40598-021-00195-7
Peter Albers, Hansjörg Geiges, Kai Zehmisch
{"title":"A Symplectic Dynamics Proof of the Degree–Genus Formula","authors":"Peter Albers,&nbsp;Hansjörg Geiges,&nbsp;Kai Zehmisch","doi":"10.1007/s40598-021-00195-7","DOIUrl":"10.1007/s40598-021-00195-7","url":null,"abstract":"<div><p>We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"9 1","pages":"41 - 68"},"PeriodicalIF":0.0,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-021-00195-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49395259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations 嵌入紧复流形的邻域等价与高余维叶
Arnold Mathematical Journal Pub Date : 2021-10-15 DOI: 10.1007/s40598-021-00192-w
Xianghong Gong, Laurent Stolovitch
{"title":"Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations","authors":"Xianghong Gong,&nbsp;Laurent Stolovitch","doi":"10.1007/s40598-021-00192-w","DOIUrl":"10.1007/s40598-021-00192-w","url":null,"abstract":"<div><p>We consider an embedded <i>n</i>-dimensional compact complex manifold in <span>(n+d)</span> dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of <span>(C_n)</span> in <span>(M_{n+d})</span> is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in <span>(M_{n+d})</span> having <span>(C_n)</span> as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving <i>small divisors condition</i> arising from solutions to their cohomological equations.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 1","pages":"61 - 145"},"PeriodicalIF":0.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41425542","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: 4
On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field 完全非阿基米德域上Schneider的连分数映射
Arnold Mathematical Journal Pub Date : 2021-10-15 DOI: 10.1007/s40598-021-00190-y
A. Haddley, R. Nair
{"title":"On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field","authors":"A. Haddley, R. Nair","doi":"10.1007/s40598-021-00190-y","DOIUrl":"https://doi.org/10.1007/s40598-021-00190-y","url":null,"abstract":"","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 1","pages":"19 - 38"},"PeriodicalIF":0.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52850248","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 Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field 关于完全非阿基米德域上Schneider的连分式映射
Arnold Mathematical Journal Pub Date : 2021-10-15 DOI: 10.1007/s40598-021-00190-y
A. Haddley, R. Nair
{"title":"On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field","authors":"A. Haddley,&nbsp;R. Nair","doi":"10.1007/s40598-021-00190-y","DOIUrl":"10.1007/s40598-021-00190-y","url":null,"abstract":"<div><p>Let <span>({mathcal {M}})</span> denote the maximal ideal of the ring of integers of a non-Archimedean field <i>K</i> with residue class field <i>k</i> whose invertible elements, we denote <span>(k^{times })</span>, and a uniformizer we denote <span>(pi )</span>. In this paper, we consider the map <span>(T_{v}: {mathcal {M}} rightarrow {mathcal {M}})</span> defined by </p><div><div><span>$$begin{aligned} T_v(x) = frac{pi ^{v(x)}}{x} - b(x), end{aligned}$$</span></div></div><p>where <i>b</i>(<i>x</i>) denotes the equivalence class to which <span>(frac{pi ^{v(x)}}{x})</span> belongs in <span>(k^{times })</span>. We show that <span>(T_v)</span> preserves Haar measure <span>(mu )</span> on the compact abelian topological group <span>({mathcal {M}})</span>. Let <span>({mathcal {B}})</span> denote the Haar <span>(sigma )</span>-algebra on <span>({mathcal {M}})</span>. We show the natural extension of the dynamical system <span>(({mathcal {M}}, {mathcal {B}}, mu , T_v))</span> is Bernoulli and has entropy <span>(frac{#( k)}{#( k^{times })}log (#( k)))</span>. The first of these two properties is used to study the average behaviour of the convergents arising from <span>(T_v)</span>. Here for a finite set <i>A</i> its cardinality has been denoted by <span>(# (A))</span>. In the case <span>(K = {mathbb {Q}}_p)</span>, i.e. the field of <i>p</i>-adic numbers, the map <span>(T_v)</span> reduces to the well-studied continued fraction map due to Schneider.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 1","pages":"19 - 38"},"PeriodicalIF":0.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-021-00190-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50485675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Varieties in Cages: A Little Zoo of Algebraic Geometry 笼中的变种：代数几何的小动物园
Arnold Mathematical Journal Pub Date : 2021-09-30 DOI: 10.1007/s40598-021-00189-5
Gabriel Katz
{"title":"Varieties in Cages: A Little Zoo of Algebraic Geometry","authors":"Gabriel Katz","doi":"10.1007/s40598-021-00189-5","DOIUrl":"10.1007/s40598-021-00189-5","url":null,"abstract":"<div><p>A <span>(d^{{n}})</span>-<span>cage</span> <span>(mathsf K)</span> is the union of <i>n</i> groups of hyperplanes in <span>(mathbb P^n)</span>, each group containing <i>d</i> members. The hyperplanes from the distinct groups are in general position, thus producing <span>(d^n)</span> points where hyperplanes from all groups intersect. These points are called the <span>nodes</span> of <span>(mathsf K)</span>. We study the combinatorics of nodes that impose independent conditions on the varieties <span>(X subset mathbb P^n)</span> containing them. We prove that if <i>X</i>, given by homogeneous polynomials of degrees <span>(le d)</span>, contains the points from such a special set <span>(mathsf A)</span> of nodes, then it contains all the nodes of <span>(mathsf K)</span>. Such a variety <i>X</i> is very special: in particular, <i>X</i> is a complete intersection.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"8 1","pages":"1 - 17"},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-021-00189-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48621450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Extension of the (mathfrak {sl}_2) Weight System to Graphs with (n le 8) Vertices （mathfrak的一个扩展{sl}_2)具有（nle 8）顶点的图的权重系统
Arnold Mathematical Journal Pub Date : 2021-09-06 DOI: 10.1007/s40598-021-00187-7
Evgeny Krasilnikov
{"title":"An Extension of the (mathfrak {sl}_2) Weight System to Graphs with (n le 8) Vertices","authors":"Evgeny Krasilnikov","doi":"10.1007/s40598-021-00187-7","DOIUrl":"10.1007/s40598-021-00187-7","url":null,"abstract":"<div><p>Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that the value of the weight system constructed from the Lie algebra <span>(mathfrak {sl}_2)</span> on a chord diagram depends on the intersection graph of the diagram, rather than the diagram itself. This inspired a question due to Lando about whether it is possible to extend the weight system <span>(mathfrak {sl}_2)</span> to a graph invariant satisfying the four term relations for graphs. We show that for all graphs with up to 8 vertices such an extension exists and is unique, thus answering in affirmative to Lando’s question for small graphs.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"7 4","pages":"609 - 618"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-021-00187-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50457574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Element-Building Games on (mathbb {Z}_n) 元素构建游戏在（mathbb{Z}_n)
Arnold Mathematical Journal Pub Date : 2021-08-18 DOI: 10.1007/s40598-021-00185-9
Bret Benesh, Robert Campbell
{"title":"Element-Building Games on (mathbb {Z}_n)","authors":"Bret Benesh,&nbsp;Robert Campbell","doi":"10.1007/s40598-021-00185-9","DOIUrl":"10.1007/s40598-021-00185-9","url":null,"abstract":"<div><p>We consider a pair of games where two players alternately select previously unselected elements of <span>(mathbb {Z}_n)</span> given a particular starting element. On each turn, the player either adds or multiplies the element they selected to the result of the previous turn. In one game, the first player wins if the final result is 0; in the other, the second player wins if the final result is 0. We determine which player has the winning strategy for both games except for the latter game with nonzero starting element when <span>(n in {2p,4p})</span> for some odd prime <i>p</i>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"7 4","pages":"599 - 608"},"PeriodicalIF":0.0,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-021-00185-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50491925","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
New Invariants of Poncelet–Jacobi Bicentric Polygons Poncelet–Jacobi双中心多边形的新不变量
Arnold Mathematical Journal Pub Date : 2021-08-18 DOI: 10.1007/s40598-021-00188-6
Pedro Roitman, Ronaldo Garcia, Dan Reznik
{"title":"New Invariants of Poncelet–Jacobi Bicentric Polygons","authors":"Pedro Roitman,&nbsp;Ronaldo Garcia,&nbsp;Dan Reznik","doi":"10.1007/s40598-021-00188-6","DOIUrl":"10.1007/s40598-021-00188-6","url":null,"abstract":"<div><p>The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville’s theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family’s limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard <i>N</i>-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard <i>N</i>-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive <i>N</i>-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the <span>(N=4)</span> case).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":"7 4","pages":"619 - 637"},"PeriodicalIF":0.0,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-021-00188-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49216068","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: 8
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