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An extremum problem for the power moment of a convex polygon contained in a disc 包含在圆盘中的凸多边形幂矩的极值问题
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-10-01 DOI: 10.1515/advgeom-2021-0021
I. Herburt, S. Sakata
{"title":"An extremum problem for the power moment of a convex polygon contained in a disc","authors":"I. Herburt, S. Sakata","doi":"10.1515/advgeom-2021-0021","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0021","url":null,"abstract":"Abstract In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885854","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}
引用次数: 0
An 𝔽p2-maximal Wiman sextic and its automorphisms 一个𝔽p2-maximal女人的性别及其自同构
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-10-01 DOI: 10.1515/advgeom-2020-0012
M. Giulietti, M. Kawakita, Stefano Lia, M. Montanucci
{"title":"An 𝔽p2-maximal Wiman sextic and its automorphisms","authors":"M. Giulietti, M. Kawakita, Stefano Lia, M. Montanucci","doi":"10.1515/advgeom-2020-0012","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0012","url":null,"abstract":"Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X6+Y6+ℨ6+(X2+Y2+ℨ2)(X4+Y4+ℨ4)−12X2Y2ℨ2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41575118","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}
引用次数: 0
Some remarks on proper actions, proper metric spaces, and buildings 关于适当的行动、适当的度量空间和建筑物的一些评论
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-09-27 DOI: 10.1515/advgeom-2022-0018
L. Kramer
{"title":"Some remarks on proper actions, proper metric spaces, and buildings","authors":"L. Kramer","doi":"10.1515/advgeom-2022-0018","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0018","url":null,"abstract":"Abstract We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49458195","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}
引用次数: 1
A characterization of centrally symmetric convex bodies in terms of visual cones 中心对称凸体的视锥特征
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-08-03 DOI: 10.1515/advgeom-2022-0006
E. Morales-Amaya, J. Jer'onimo-Castro, D. J. Verdusco Hernández
{"title":"A characterization of centrally symmetric convex bodies in terms of visual cones","authors":"E. Morales-Amaya, J. Jer'onimo-Castro, D. J. Verdusco Hernández","doi":"10.1515/advgeom-2022-0006","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0006","url":null,"abstract":"Abstract We prove the following result: Let K be a strictly convex body in the Euclidean space ℝn, n ≥ 3, and let L be a hypersurface which is the image of an embedding of the sphere 𝕊n–1, such that K is contained in the interior of L. Suppose that, for every x ∈ L, there exists y ∈ L such that the support cones of K with apexes at x and y differ by a central symmetry. Then K and L are centrally symmetric and concentric.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49426726","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}
引用次数: 0
Tetrahedral cages for unit discs 单位圆盘用四面体笼
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-07-01 DOI: 10.1515/advgeom-2021-0016
Liping Yuan, T. Zamfirescu, Yanxue Zhang
{"title":"Tetrahedral cages for unit discs","authors":"Liping Yuan, T. Zamfirescu, Yanxue Zhang","doi":"10.1515/advgeom-2021-0016","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0016","url":null,"abstract":"Abstract A cage is the 1-skeleton of a convex polytope in ℝ3. A cage is said to hold a set if the set cannot be continuously moved to a distant location, remaining congruent to itself and disjoint from the cage. In how many positions can (compact 2-dimensional) unit discs be held by a tetrahedral cage? We completely answer this question for all tetrahedra.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41513026","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}
引用次数: 0
Inscribed rectangle coincidences 内接矩形重合
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-07-01 DOI: 10.1515/advgeom-2021-0012
R. Schwartz
{"title":"Inscribed rectangle coincidences","authors":"R. Schwartz","doi":"10.1515/advgeom-2021-0012","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0012","url":null,"abstract":"Abstract We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We use this integral formula to prove the inequality M(γ) ≥ Δ(γ)/2 – 1, where M(γ) denotes the total multiplicity of rectangle coincidences, i.e. pairs, triples, etc. of isometric rectangles inscribed in γ, and Δ(γ) denotes the number of stable diameters of γ, i.e. critical points of the distance function on γ.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49023497","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}
引用次数: 2
A note on large Kakeya sets 关于大型Kakeya电视机的注意事项
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-07-01 DOI: 10.1515/advgeom-2021-0018
M. de Boeck, G. Van de Voorde
{"title":"A note on large Kakeya sets","authors":"M. de Boeck, G. Van de Voorde","doi":"10.1515/advgeom-2021-0018","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0018","url":null,"abstract":"Abstract A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q2 – q q $begin{array}{} displaystyle sqrt{q} end{array}$ + 32 $begin{array}{} displaystyle frac{3}{2} end{array}$q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44795249","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}
引用次数: 0
𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve 𝔽具有许多自同构的p2极大曲线是Hermitian曲线覆盖的Galois
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-06-28 DOI: 10.1515/advgeom-2021-0013
D. Bartoli, M. Montanucci, F. Torres
{"title":"𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve","authors":"D. Bartoli, M. Montanucci, F. Torres","doi":"10.1515/advgeom-2021-0013","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0013","url":null,"abstract":"Abstract Let 𝔽 be the finite field of order q2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+x ${{mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $mathcal{x}$of genus g ≥ 2 with | Aut(𝒳) | > 84(g − 1) is Galois-covered by Hq+1. ${{mathcal{H}}_{q+1}}.$The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $mathcal{x}$for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H72. ${{mathcal{H}}_{72}}.$","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43047106","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}
引用次数: 2
The Beckman–Quarles theorem via the triangle inequality 从三角不等式看Beckman–Quarles定理
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-06-24 DOI: 10.1515/advgeom-2020-0024
V. Totik
{"title":"The Beckman–Quarles theorem via the triangle inequality","authors":"V. Totik","doi":"10.1515/advgeom-2020-0024","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0024","url":null,"abstract":"Abstract We give a short, elementary and non-computational proof for the classical Beckman–Quarles theorem asserting that a map of a Euclidean space into itself that preserves distance 1 must be an isometry.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43029542","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}
引用次数: 0
When is M0,n(ℙ1,1) a Mori dream space? 什么时候M0,n(∈1,1)是Mori梦空间?
IF 0.5 4区 数学
Advances in Geometry Pub Date : 2021-06-24 DOI: 10.1515/advgeom-2021-0019
C. Fontanari
{"title":"When is M0,n(ℙ1,1) a Mori dream space?","authors":"C. Fontanari","doi":"10.1515/advgeom-2021-0019","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0019","url":null,"abstract":"Abstract The moduli space M¯0,n(ℙ1,1) ${{bar{M}}_{0,n}}left( {{mathbb{P}}^{1}},1 right)$of n-pointed stable maps is a Mori dream space whenever the moduli space M¯0,n+3 of (n+3) ${{bar{M}}_{0,n+3}}; text{of} ;(n+3)$pointed rational curves is, and M¯0,n(ℙ1,1) ${{bar{M}}_{0,n}}left( {{mathbb{P}}^{1}},1 right)$is a log Fano variety for n ≤ 5.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44645357","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}
引用次数: 1
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