abstract:We establish that if $dgeq 2k+6$ and $q$ is odd and sufficiently large with respect to $alphain (0,1)$, then every set $Asubseteq{bf F}_q^d$of size $|A|geqalpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.
{"title":"Spherical configurations over finite fields","authors":"N. Lyall, Á. Magyar, Hans Parshall","doi":"10.1353/ajm.2020.0010","DOIUrl":"https://doi.org/10.1353/ajm.2020.0010","url":null,"abstract":"abstract:We establish that if $dgeq 2k+6$ and $q$ is odd and sufficiently large with respect to $alphain (0,1)$, then every set $Asubseteq{bf F}_q^d$of size $|A|geqalpha q^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"142 1","pages":"373 - 404"},"PeriodicalIF":1.7,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48742730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-06DOI: 10.1353/ajm.2023.a902956
Xinliang An, Hao-Lei Chen, Silu Yin
abstract:In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in $H^3(Bbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.
{"title":"Low regularity ill-posedness for elastic waves driven by shock formation","authors":"Xinliang An, Hao-Lei Chen, Silu Yin","doi":"10.1353/ajm.2023.a902956","DOIUrl":"https://doi.org/10.1353/ajm.2023.a902956","url":null,"abstract":"abstract:In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in $H^3(Bbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"145 1","pages":"1111 - 1181"},"PeriodicalIF":1.7,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44630089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-03DOI: 10.1353/ajm.2023.a902954
Robert Haslhofer, Or Hershkovits, B. White
abstract:In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space.
{"title":"Moving plane method for varifolds and applications","authors":"Robert Haslhofer, Or Hershkovits, B. White","doi":"10.1353/ajm.2023.a902954","DOIUrl":"https://doi.org/10.1353/ajm.2023.a902954","url":null,"abstract":"abstract:In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"145 1","pages":"1051 - 1076"},"PeriodicalIF":1.7,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42276168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-02-16DOI: 10.1353/ajm.2023.a897495
C. Aistleitner, Niclas Technau, Agamemnon Zafeiropoulos
abstract:Given an irrational number $alphain(0,1)$, the Sudler product is defined by $P_N(alpha) = prod_{r=1}^{N}2|sinpi ralpha|$. Answering a question of Grepstad, Kaltenb"ock and Neum"uller we prove an asymptotic formula for distorted Sudler products when $alpha$ is the golden ratio $(sqrt{5}+1)/2$ and establish that in this case $limsup_{N to infty} P_N(alpha)/N < infty$. We obtain similar results for quadratic irrationals $alpha$ with continued fraction expansion $alpha = [a,a,a,ldots]$ for some integer $a geq 1$, and give a full characterisation of the values of $a$ for which $liminf_{N to infty} P_N(alpha)>0$ and $limsup_{N to infty} P_N(alpha) / N < infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first author, Larcher, Pillichshammer, Saad Eddin, and Tichy.
文摘：给定一个无理数$alphain（0,1）$，Sudler乘积定义为$P_N（alpha=prod_｛r=1｝^{N}2|sinpi-ralpha|$。在回答Grepstad、Kaltenb和Neum的一个问题时，我们证明了当$alpha$是黄金比率$（sqrt｛5｝+1）/2$时，失真Sudler乘积的一个渐近公式，并证明了在这种情况下$limsup_｛Ntoinfty｝P_N（alpha）/N0$和$limsup_｛N to infty}P_N（alpha）/N
{"title":"On the order of magnitude of Sudler products","authors":"C. Aistleitner, Niclas Technau, Agamemnon Zafeiropoulos","doi":"10.1353/ajm.2023.a897495","DOIUrl":"https://doi.org/10.1353/ajm.2023.a897495","url":null,"abstract":"abstract:Given an irrational number $alphain(0,1)$, the Sudler product is defined by $P_N(alpha) = prod_{r=1}^{N}2|sinpi ralpha|$. Answering a question of Grepstad, Kaltenb\"ock and Neum\"uller we prove an asymptotic formula for distorted Sudler products when $alpha$ is the golden ratio $(sqrt{5}+1)/2$ and establish that in this case $limsup_{N to infty} P_N(alpha)/N < infty$. We obtain similar results for quadratic irrationals $alpha$ with continued fraction expansion $alpha = [a,a,a,ldots]$ for some integer $a geq 1$, and give a full characterisation of the values of $a$ for which $liminf_{N to infty} P_N(alpha)>0$ and $limsup_{N to infty} P_N(alpha) / N < infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first author, Larcher, Pillichshammer, Saad Eddin, and Tichy.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"145 1","pages":"721 - 764"},"PeriodicalIF":1.7,"publicationDate":"2020-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47208201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:For real quadratic number fields, we consider the order of ideal classes of split prime ideals, $P$, whose norm is a fixed rational prime. We collect fundamental discriminants satisfying a trivial condition for $P$ to be principal, and show that for a positive density of such discriminants, the cyclic subgroup of the ideal class group generated by $P$ does not have a 2-part.
{"title":"On the parity of ideal classes over a fixed prime","authors":"Jeongho Park","doi":"10.1353/ajm.2020.0003","DOIUrl":"https://doi.org/10.1353/ajm.2020.0003","url":null,"abstract":"abstract:For real quadratic number fields, we consider the order of ideal classes of split prime ideals, $P$, whose norm is a fixed rational prime. We collect fundamental discriminants satisfying a trivial condition for $P$ to be principal, and show that for a positive density of such discriminants, the cyclic subgroup of the ideal class group generated by $P$ does not have a 2-part.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"142 1","pages":"139 - 176"},"PeriodicalIF":1.7,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41842573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:We prove that any finite set $Fsubset{Bbb Z}^2$ that tiles ${Bbb Z}^2$ by translations also admits a periodic tiling. As a consequence, the problem whether a given finite set $F$ tiles ${Bbb Z}^2$ is decidable.
{"title":"Periodicity and decidability of tilings of ℤ2","authors":"S. Bhattacharya","doi":"10.1353/ajm.2020.0006","DOIUrl":"https://doi.org/10.1353/ajm.2020.0006","url":null,"abstract":"abstract:We prove that any finite set $Fsubset{Bbb Z}^2$ that tiles ${Bbb Z}^2$ by translations also admits a periodic tiling. As a consequence, the problem whether a given finite set $F$ tiles ${Bbb Z}^2$ is decidable.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"142 1","pages":"255 - 266"},"PeriodicalIF":1.7,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49377319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrodinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates. As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate.
{"title":"Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds","authors":"Aynur Bulut","doi":"10.1353/ajm.2023.0013","DOIUrl":"https://doi.org/10.1353/ajm.2023.0013","url":null,"abstract":"We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrodinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates. \u0000As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46129609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:We give a short and elementary proof of the $ell^{2}$ decoupling inequality for the moment curve in $hat{Bbb{R}}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley.
{"title":"A short proof of ℓ2 decoupling for the moment curve","authors":"LI Zanekun, Po-Lam Yung, Pavel Zorin-Kranich","doi":"10.1353/ajm.2021.0048","DOIUrl":"https://doi.org/10.1353/ajm.2021.0048","url":null,"abstract":"abstract:We give a short and elementary proof of the $ell^{2}$ decoupling inequality for the moment curve in $hat{Bbb{R}}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"143 1","pages":"1983 - 1998"},"PeriodicalIF":1.7,"publicationDate":"2019-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46400960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of automorphisms of affine varieties.
{"title":"Families of commuting automorphisms, and a characterization of the affine space","authors":"Serge Cantat, A. Regeta, Junyi Xie","doi":"10.1353/ajm.2023.0009","DOIUrl":"https://doi.org/10.1353/ajm.2023.0009","url":null,"abstract":"In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of automorphisms of affine varieties.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2019-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45817402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.
给出了几何Picard秩-1在任意零特征域上的Fano三倍的唯一性和合理性的充分必要条件。
{"title":"Rationality of Fano threefolds over non-closed fields","authors":"A. Kuznetsov, Yuri Prokhorov","doi":"10.1353/ajm.2023.0008","DOIUrl":"https://doi.org/10.1353/ajm.2023.0008","url":null,"abstract":"We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48893246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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