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Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces 有规则的页面上的涂鸦和斑点：曲面上穿越流的凸准包络线
Arnold Mathematical Journal Pub Date : 2024-05-16 DOI: 10.1007/s40598-024-00249-6
Gabriel Katz
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引用次数: 0
Kirillov Polynomials for the Exceptional Lie Algebra (mathfrak g_{2}) 特殊列代数 $$mathfrak g_{2}$ 的基里洛夫多项式
Arnold Mathematical Journal Pub Date : 2024-04-04 DOI: 10.1007/s40598-024-00247-8
Martin T. Luu
{"title":"Kirillov Polynomials for the Exceptional Lie Algebra (mathfrak g_{2})","authors":"Martin T. Luu","doi":"10.1007/s40598-024-00247-8","DOIUrl":"10.1007/s40598-024-00247-8","url":null,"abstract":"<div><p>As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of <i>q</i> elements and fixed Jordan type. One obtains polynomials with respect to <i>q</i> with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra <span>(mathfrak g_2)</span>. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-024-00247-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140744435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Polynomial Superpotential for Grassmannian ({text {Gr}}(k,n)) from a Limit of Vertex Function 从顶点函数极限看格拉斯曼$${text {Gr}}(k,n)$$ 的多项式超势垒
Arnold Mathematical Journal Pub Date : 2024-03-15 DOI: 10.1007/s40598-024-00245-w
Andrey Smirnov, Alexander Varchenko
{"title":"Polynomial Superpotential for Grassmannian ({text {Gr}}(k,n)) from a Limit of Vertex Function","authors":"Andrey Smirnov,&nbsp;Alexander Varchenko","doi":"10.1007/s40598-024-00245-w","DOIUrl":"10.1007/s40598-024-00245-w","url":null,"abstract":"<div><p>In this note, we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, <span>(X=T^{*}{text {Gr}}(k,n))</span>. This integral representation can be used to compute the <span>(hbar rightarrow infty )</span> limit of the vertex function, where <span>(hbar )</span> denotes the equivariant parameter of a torus acting on <i>X</i> by dilating the cotangent fibers. We show that in this limit, the integral turns into the standard mirror integral representation of the <i>A</i>-series of the Grassmannian <span>({text {Gr}}(k,n))</span> with the Laurent polynomial Landau–Ginzburg superpotential of Eguchi, Hori and Xiong.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140237665","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
A Note on Contact Manifolds with Infinite Fillings 关于无限填充接触流形的说明
Arnold Mathematical Journal Pub Date : 2024-03-01 DOI: 10.1007/s40598-024-00244-x
Zhengyi Zhou
{"title":"A Note on Contact Manifolds with Infinite Fillings","authors":"Zhengyi Zhou","doi":"10.1007/s40598-024-00244-x","DOIUrl":"10.1007/s40598-024-00244-x","url":null,"abstract":"<div><p>We use spinal open books to construct contact manifolds with infinitely many different Weinstein fillings in any odd dimension <span>(&gt; 1,)</span> which were previously unknown for dimensions equal to <span>(4n+1.)</span> The argument does not involve understanding factorizations in the symplectic mapping class group.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409338","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
Higher Dimensional Versions of Theorems of Euler and Fuss 欧拉和福斯定理的高维版本
Arnold Mathematical Journal Pub Date : 2024-01-22 DOI: 10.1007/s40598-023-00243-4
Peter Gibson, Nicolau Saldanha, Carlos Tomei
{"title":"Higher Dimensional Versions of Theorems of Euler and Fuss","authors":"Peter Gibson,&nbsp;Nicolau Saldanha,&nbsp;Carlos Tomei","doi":"10.1007/s40598-023-00243-4","DOIUrl":"10.1007/s40598-023-00243-4","url":null,"abstract":"<div><p>We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let <i>A</i> be the positive symmetric matrix taking the outer ellipsoid to the inner one. If <span>({text {tr}}, A = 1)</span>, there exists a bijection between the orthogonal group <i>O</i>(<i>n</i>) and the set of such labeled simplices. Similarly, if <span>({text {tr}}, A^2 = 1)</span>, there are families of parallelotopes and of cross polytopes, also indexed by <i>O</i>(<i>n</i>).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413037","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
Toric Orbit Spaces Which are Manifolds 属于流形的环形轨道空间
Arnold Mathematical Journal Pub Date : 2024-01-03 DOI: 10.1007/s40598-023-00242-5
Anton Ayzenberg, Vladimir Gorchakov
{"title":"Toric Orbit Spaces Which are Manifolds","authors":"Anton Ayzenberg,&nbsp;Vladimir Gorchakov","doi":"10.1007/s40598-023-00242-5","DOIUrl":"10.1007/s40598-023-00242-5","url":null,"abstract":"<div><p>We characterize the actions of compact tori on smooth closed manifolds for which the orbit space is a topological manifold (either closed or with boundary). For closed manifolds, the result was originally proved by Styrt in 2009. We give a new proof for closed manifolds which is also applicable to manifolds with boundary. In our arguments, we use the result of Provan and Billera who characterized matroid complexes which are pseudomanifolds. We study the combinatorial structure of torus actions whose orbit spaces are manifolds. In two appendix sections, we give an overview of two theories related to our work. The first one is the combinatorial theory of Leontief substitution systems from mathematical economics. The second one is the topological Kaluza–Klein model of Dirac’s monopole studied by Atiyah. The aim of these sections is to draw some bridges between disciplines and motivate further studies in toric topology.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409577","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 Stratified Resolution of Singularities in the Class of Generalized Power Series 牛顿多面体与广义幂级数类奇点的分层解析
Arnold Mathematical Journal Pub Date : 2023-12-08 DOI: 10.1007/s40598-023-00241-6
Jesús Palma-Márquez
{"title":"Newton Polyhedra and Stratified Resolution of Singularities in the Class of Generalized Power Series","authors":"Jesús Palma-Márquez","doi":"10.1007/s40598-023-00241-6","DOIUrl":"10.1007/s40598-023-00241-6","url":null,"abstract":"<div><p>We generalize the construction of a toric variety associated with an integer convex polyhedron to construct generalized analytic varieties associated with polyhedra with not necessarily rational vertices. For germs of generalized analytic functions with a given Newton polyhedron <span>(Gamma )</span>, the generalized analytic variety associated with <span>(Gamma )</span> provides a stratified resolution of singularities of these functions; also ensuring a full resolution for almost all of them. Thus, this constructive and elementary approach replaces the non-effective previous proof of this result based on consecutive blow-ups.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138587734","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
The Probabilistic Method in Real Singularity Theory 实奇点理论中的概率方法
Arnold Mathematical Journal Pub Date : 2023-11-28 DOI: 10.1007/s40598-023-00240-7
Antonio Lerario, Michele Stecconi
{"title":"The Probabilistic Method in Real Singularity Theory","authors":"Antonio Lerario,&nbsp;Michele Stecconi","doi":"10.1007/s40598-023-00240-7","DOIUrl":"10.1007/s40598-023-00240-7","url":null,"abstract":"<div><p>We explain how to use the probabilistic method to prove the existence of real polynomial singularities with rich topology, i.e., with total Betti number of the maximal possible order. We show how similar ideas can be used to produce real algebraic projective hypersurfaces with a rich structure of umbilical points.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-023-00240-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139353370","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
Cohomology of Spaces of Complex Knots 复结空间同调学
Arnold Mathematical Journal Pub Date : 2023-10-24 DOI: 10.1007/s40598-023-00239-0
V. A. Vassiliev
{"title":"Cohomology of Spaces of Complex Knots","authors":"V. A. Vassiliev","doi":"10.1007/s40598-023-00239-0","DOIUrl":"10.1007/s40598-023-00239-0","url":null,"abstract":"<div><p>We develop a technique for calculating the cohomology groups of spaces of complex parametric knots in <span>({{mathbb {C}}}^k)</span>, <span>(k ge 3)</span>, and obtain these groups of low dimensions.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273712","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
Existence of a Conjugate Point in the Incompressible Euler Flow on a Three-Dimensional Ellipsoid 三维椭球体上不可压缩欧拉流中共轭点的存在性
Arnold Mathematical Journal Pub Date : 2023-10-13 DOI: 10.1007/s40598-023-00238-1
L. A. Lichtenfelz, T. Tauchi, T. Yoneda
{"title":"Existence of a Conjugate Point in the Incompressible Euler Flow on a Three-Dimensional Ellipsoid","authors":"L. A. Lichtenfelz,&nbsp;T. Tauchi,&nbsp;T. Yoneda","doi":"10.1007/s40598-023-00238-1","DOIUrl":"10.1007/s40598-023-00238-1","url":null,"abstract":"<div><p>The existence of a conjugate point on the volume-preserving diffeomorphism group of a compact Riemannian manifold <i>M</i> is related to the Lagrangian stability of a solution of the incompressible Euler equation on <i>M</i>. The Misiołek curvature is a reasonable criterion for the existence of a conjugate point on the volume-preserving diffeomorphism group corresponding to a stationary solution of the incompressible Euler equation. In this article, we introduce a class of stationary solutions on an arbitrary Riemannian manifold whose behavior is nice with respect to the Misiołek curvature and give a positivity result of the Misiołek curvature for solutions belonging to this class. Moreover, we also show the existence of a conjugate point in the three-dimensional ellipsoid case as its corollary.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853005","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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