Pub Date : 2023-07-14DOI: 10.1515/advgeom-2023-0015
Elsa Ghandour, Sigmundur Gudmundsson
Abstract We use the method of eigenfamilies to construct explicit complex-valued proper p-harmonic functions on the compact real Grassmannians. We also find proper p-harmonic functions on the real flag manifolds which do not descend onto any of the real Grassmannians.
{"title":"Explicit p-harmonic functions on the real Grassmannians","authors":"Elsa Ghandour, Sigmundur Gudmundsson","doi":"10.1515/advgeom-2023-0015","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0015","url":null,"abstract":"Abstract We use the method of eigenfamilies to construct explicit complex-valued proper p-harmonic functions on the compact real Grassmannians. We also find proper p-harmonic functions on the real flag manifolds which do not descend onto any of the real Grassmannians.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41565314","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 : 2023-05-01DOI: 10.1515/advgeom-2023-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/advgeom-2023-frontmatter2","DOIUrl":"https://doi.org/10.1515/advgeom-2023-frontmatter2","url":null,"abstract":"","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134992401","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 : 2023-05-01DOI: 10.1515/advgeom-2023-0004
Y. Oh, Y. Savelyev
Abstract For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S1 = Mid (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form ∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ $$bar{partial}^{pi} w=0, quad w^{*} lambda circ j=f^{*} d theta$$ for the map u = (w, f) : Σ˙→Q×S1$dot{Sigma} rightarrow Q times S^{1}$for a λ-compatible almost complex structure J and a punctured Riemann surface (Σ˙,j).$(dot{Sigma}, j).$In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H1(Σ˙,Z)$H^{1}(dot{Sigma}, mathbb{Z})$and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of (Q, λ) (more generally on arbitrary locally conformal symplectic manifolds).
{"title":"Pseudoholomorphic curves on the LCS-fication of contact manifolds","authors":"Y. Oh, Y. Savelyev","doi":"10.1515/advgeom-2023-0004","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0004","url":null,"abstract":"Abstract For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S1 = Mid (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form ∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ $$bar{partial}^{pi} w=0, quad w^{*} lambda circ j=f^{*} d theta$$ for the map u = (w, f) : Σ˙→Q×S1$dot{Sigma} rightarrow Q times S^{1}$for a λ-compatible almost complex structure J and a punctured Riemann surface (Σ˙,j).$(dot{Sigma}, j).$In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H1(Σ˙,Z)$H^{1}(dot{Sigma}, mathbb{Z})$and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of (Q, λ) (more generally on arbitrary locally conformal symplectic manifolds).","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46111720","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 : 2023-05-01DOI: 10.1515/advgeom-2023-0003
Rikke Eriksen, Markus Kiderlen
Abstract The convex-geometric Minkowski tensors contain information about shape and orientation of the underlying convex body. They therefore yield valuable summary statistics for stationary marked point processes with marks in the family of convex bodies, or, slightly more specialised, for stationary particle processes. We show here that if the distribution of the typical particle is invariant under rotations about a fixed k-plane, then the average volume tensors of the typical particle can be derived from k + 1-dimensional sections. This finding extends the well-known three-dimensional special case to higher dimensions. A corresponding result for the surface tensors is also proven. In the last part of the paper we show how Minkowski tensors can be used to define three ellipsoidal set-valued summary statistics, discuss their estimation and illustrate their construction and use in a simulation example. Two of these, the so-called Miles ellipsoid and the inertia ellipsoid, are based on mean volume tensors of ranks up to 2. The third, based on the mean surface tensor of rank 2, will be called the Blaschke ellipsoid and is only defined when the typical particle has a rotationally symmetric distribution about an axis, as we then can use uniqueness and reconstruction results for centred ellipsoids of revolution from their rank-2 surface tensor. The latter are also established here.
{"title":"Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids","authors":"Rikke Eriksen, Markus Kiderlen","doi":"10.1515/advgeom-2023-0003","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0003","url":null,"abstract":"Abstract The convex-geometric Minkowski tensors contain information about shape and orientation of the underlying convex body. They therefore yield valuable summary statistics for stationary marked point processes with marks in the family of convex bodies, or, slightly more specialised, for stationary particle processes. We show here that if the distribution of the typical particle is invariant under rotations about a fixed k-plane, then the average volume tensors of the typical particle can be derived from k + 1-dimensional sections. This finding extends the well-known three-dimensional special case to higher dimensions. A corresponding result for the surface tensors is also proven. In the last part of the paper we show how Minkowski tensors can be used to define three ellipsoidal set-valued summary statistics, discuss their estimation and illustrate their construction and use in a simulation example. Two of these, the so-called Miles ellipsoid and the inertia ellipsoid, are based on mean volume tensors of ranks up to 2. The third, based on the mean surface tensor of rank 2, will be called the Blaschke ellipsoid and is only defined when the typical particle has a rotationally symmetric distribution about an axis, as we then can use uniqueness and reconstruction results for centred ellipsoids of revolution from their rank-2 surface tensor. The latter are also established here.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48624995","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 : 2023-03-31DOI: 10.1515/advgeom-2023-0013
Sebastian Bischof, B. Mühlherr
Abstract We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affine twin buildings of rank at least 3.
{"title":"Isometries of wall-connected twin buildings","authors":"Sebastian Bischof, B. Mühlherr","doi":"10.1515/advgeom-2023-0013","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0013","url":null,"abstract":"Abstract We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affine twin buildings of rank at least 3.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47548686","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 : 2023-03-21DOI: 10.1515/advgeom-2023-0008
Arijit Nath, Kuldeep Saha
Abstract We discuss some embedding results in the category of open books, Lefschetz fibrations, contact manifolds and contact open books. First we prove an open book version of the Haefliger–Hirsch embedding theorem by showing that every k-connected closed n-manifold (n ≥ 7, k < (n − 4)/2) with signature zero admits an open book embedding in the trivial open book of 𝕊2n−k. We then prove that every closed manifold M2n+1 that bounds an achiral Lefschetz fibration admits an open book embedding in the trivial open book of 𝕊2⌊3n/2⌋+3. We also prove that every closed manifold M2n+1 bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on ℝ2n+3. Finally, we give various examples of contact open book embeddings of contact (2n + 1)-manifolds in the trivial supporting open book of the standard contact structure on 𝕊4n+1.
{"title":"Open books and embeddings of smooth and contact manifolds","authors":"Arijit Nath, Kuldeep Saha","doi":"10.1515/advgeom-2023-0008","DOIUrl":"https://doi.org/10.1515/advgeom-2023-0008","url":null,"abstract":"Abstract We discuss some embedding results in the category of open books, Lefschetz fibrations, contact manifolds and contact open books. First we prove an open book version of the Haefliger–Hirsch embedding theorem by showing that every k-connected closed n-manifold (n ≥ 7, k < (n − 4)/2) with signature zero admits an open book embedding in the trivial open book of 𝕊2n−k. We then prove that every closed manifold M2n+1 that bounds an achiral Lefschetz fibration admits an open book embedding in the trivial open book of 𝕊2⌊3n/2⌋+3. We also prove that every closed manifold M2n+1 bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on ℝ2n+3. Finally, we give various examples of contact open book embeddings of contact (2n + 1)-manifolds in the trivial supporting open book of the standard contact structure on 𝕊4n+1.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44226521","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 : 2023-01-01DOI: 10.1515/advgeom-2022-0026
Sihem Mesnager, F. Özbudak
Abstract We obtain the Boomerang Connectivity Table of power permutations F(x)=x2m−1 of F2n $F(x)={{x}^{{{2}^{m}}-1}}text{ }!!~!!text{ of }!!~!!text{ }{{mathbb{F}}_{{{2}^{n}}}}$with m ∈ { 3,n−12,n+12,n−2 }. $left{ 3,frac{n-1}{2},frac{n+1}{2},n-2 right}.$In particular, we obtain the Boomerang uniformity and the Boomerang uniformity set of F(x) at b∈F2n∖F2. $F(x)text{ }!!~!!text{ at }!!~!!text{ }bin {{mathbb{F}}_{{{2}^{n}}}}setminus {{mathbb{F}}_{2}}.$Moreover we determine the complete Boomerang distribution spectrum of F(x) using the number of rational points of certain concrete algebraic curves over F2n. ${{mathbb{F}}_{{{2}^{n}}}}.$We also determine the distribution spectra of Boomerang uniformities explicitly.
得到了F2n $F(x)={{x}^{{{2}^{m}}-1}}text{ }!!~!!text{ of }!!~!!text{ }{{mathbb{F}}_{{{2}^{n}}}}$中m∈{3,n−12,n+12,n−2的幂置换F(x)=x2m−}1的回旋连通性表。$left{ 3,frac{n-1}{2},frac{n+1}{2},n-2 right}.$特别地，我们得到了F(x)在b∈F2n∈F2处的Boomerang均匀性和Boomerang均匀性集。$F(x)text{ }!!~!!text{ at }!!~!!text{ }bin {{mathbb{F}}_{{{2}^{n}}}}setminus {{mathbb{F}}_{2}}.$此外，我们还利用F2n上某些具体代数曲线的有理点数确定了F(x)的完整回旋镖分布谱。${{mathbb{F}}_{{{2}^{n}}}}.$我们还明确地确定了回飞镖均匀性的分布谱。
{"title":"Boomerang uniformity of power permutations and algebraic curves over 𝔽2n","authors":"Sihem Mesnager, F. Özbudak","doi":"10.1515/advgeom-2022-0026","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0026","url":null,"abstract":"Abstract We obtain the Boomerang Connectivity Table of power permutations F(x)=x2m−1 of F2n $F(x)={{x}^{{{2}^{m}}-1}}text{ }!!~!!text{ of }!!~!!text{ }{{mathbb{F}}_{{{2}^{n}}}}$with m ∈ { 3,n−12,n+12,n−2 }. $left{ 3,frac{n-1}{2},frac{n+1}{2},n-2 right}.$In particular, we obtain the Boomerang uniformity and the Boomerang uniformity set of F(x) at b∈F2n∖F2. $F(x)text{ }!!~!!text{ at }!!~!!text{ }bin {{mathbb{F}}_{{{2}^{n}}}}setminus {{mathbb{F}}_{2}}.$Moreover we determine the complete Boomerang distribution spectrum of F(x) using the number of rational points of certain concrete algebraic curves over F2n. ${{mathbb{F}}_{{{2}^{n}}}}.$We also determine the distribution spectra of Boomerang uniformities explicitly.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42778725","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 : 2023-01-01DOI: 10.1515/advgeom-2022-0029
M. Brandt, P. A. Helminck
Abstract We correct two errors in our paper Tropical superelliptic curves published in Advances in Geometry 20 (2020), 527–551. These corrections do not change the main results of the paper.
我们修正了我们发表在《Advances in Geometry》20(2020)，527-551上的论文《热带超椭圆曲线》中的两个错误。这些更正不会改变论文的主要结果。
{"title":"Erratum for \"Tropical superelliptic curves\"","authors":"M. Brandt, P. A. Helminck","doi":"10.1515/advgeom-2022-0029","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0029","url":null,"abstract":"Abstract We correct two errors in our paper Tropical superelliptic curves published in Advances in Geometry 20 (2020), 527–551. These corrections do not change the main results of the paper.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46250085","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 : 2022-11-27DOI: 10.48550/arXiv.2211.14807
Mook Kwon Jung, S. Yoon, Hee-Kap Ahn, T. Tokuyama
Abstract We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently, closed curves of length 2) allowing translations and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translations and discrete rotations of π are allowed. We also give convex coverings of closed curves of length 2 under translations and discrete rotations of multiples of π/2 and of 2π/3. We show that no proper closed subset of that covering is a covering for discrete rotations of multiples of π/2, which is an equilateral triangle of height smaller than 1, and conjecture that such a covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translations and discrete rotations of 2π/k for all integers k=3.
{"title":"Universal convex covering problems under translations and discrete rotations","authors":"Mook Kwon Jung, S. Yoon, Hee-Kap Ahn, T. Tokuyama","doi":"10.48550/arXiv.2211.14807","DOIUrl":"https://doi.org/10.48550/arXiv.2211.14807","url":null,"abstract":"Abstract We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently, closed curves of length 2) allowing translations and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translations and discrete rotations of π are allowed. We also give convex coverings of closed curves of length 2 under translations and discrete rotations of multiples of π/2 and of 2π/3. We show that no proper closed subset of that covering is a covering for discrete rotations of multiples of π/2, which is an equilateral triangle of height smaller than 1, and conjecture that such a covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translations and discrete rotations of 2π/k for all integers k=3.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42055441","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 : 2022-10-01DOI: 10.1515/advgeom-2022-0019
R. Löwen
Abstract Using the Klein correspondence, regular parallelisms of PG(3, ℝ) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3, ℝ). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious.
{"title":"Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case","authors":"R. Löwen","doi":"10.1515/advgeom-2022-0019","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0019","url":null,"abstract":"Abstract Using the Klein correspondence, regular parallelisms of PG(3, ℝ) have been described by Betten and Riesinger in terms of a dual object, called a hyperflock determining (hfd) line set. In the special case where this set has a span of dimension 3, a second dualization leads to a more convenient object, called a generalized star of lines. Both constructions have later been simplified by the author. Here we refine our simplified approach in order to obtain similar results for regular parallelisms of oriented lines. As a consequence, we can demonstrate that for oriented parallelisms, as we call them, there are distinctly more possibilities than in the non-oriented case. The proofs require a thorough analysis of orientation in projective spaces (as manifolds and as lattices) and in projective planes and, in particular, in translation planes. This is used in order to handle continuous families of oriented regular spreads in terms of the Klein model of PG(3, ℝ). This turns out to be quite subtle. Even the definition of suitable classes of dual objects modeling oriented parallelisms is not so obvious.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41288677","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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