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Radiation and Asymptotics for Spacetimes with Non-Isotropic Mass 具有非各向同性质量的时空的辐射和渐近性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a4
Lydia Bieri
{"title":"Radiation and Asymptotics for Spacetimes with Non-Isotropic Mass","authors":"Lydia Bieri","doi":"10.4310/pamq.2024.v20.n4.a4","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a4","url":null,"abstract":"We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the initial data metric, that is it may include a non-isotropic mass term, we prove new detailed behavior of the radiation field and curvature components at future null infinity. In particular, the limit along the null hypersurface $C_u$ as $t to infty$ of the curvature component $rho =frac{1}{4}{R_{3434}}$ multiplied with $r^3$ tends to a function $P(u, theta, phi)$ on $mathbb{R} times S^2$. When taking the limit $u rightarrow + infty$ (which corresponds to the limit at spacelike infinity), this function tends to a function $P^+(theta, phi)$ on $S^2$. We prove that the latter limit does not have any $l=1$ modes. However, it has all the other modes, $l = 0, l geq 2$. Important derivatives of crucial curvature components do not decay in $u$, which is a special feature of these more general spacetimes We show that peeling of the Weyl curvature components at future null infinity stops at the order $r^{-3}$, that is $(r^{-4}|u|^{+1}$, for large data, and at order $r^{-frac{7}{2}}$ for small data. Despite this fact, we prove that angular momentum at future null infinity is well defined for these spacetimes, due to the good behavior of the $l=1$ modes involved.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743345","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
$H^{frac{11}{4}}(mathbb{R}^2)$ Ill-Posedness for 2D Elastic Wave System $H^{frac{11}{4}}(mathbb{R}^2)$二维弹性波系统的假定性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a11
Xinliang An, Haoyang Chen, Silu Yin
{"title":"$H^{frac{11}{4}}(mathbb{R}^2)$ Ill-Posedness for 2D Elastic Wave System","authors":"Xinliang An, Haoyang Chen, Silu Yin","doi":"10.4310/pamq.2024.v20.n4.a11","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a11","url":null,"abstract":"In this paper, we prove that for the 2D elastic wave equations, a physical system with multiple wave-speeds, its Cauchy problem fails to be locally well-posed in $H frac{11}{4} (mathbb R^2)$. The ill-posedness here is driven by instantaneous shock formation. In 2D Smith-Tataru showed that the Cauchy problem for a single quasilinear wave equation is locally well-posed in $H^s$ with $s gt frac{11}{4}$. Hence our $H ^frac{11}{4}$ ill-posedness obtained here is a desired result. Our proof relies on combining a geometric method and an algebraic wave-decomposition approach, together with detailed analysis of the corresponding hyperbolic system.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743346","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
Initial data on big bang singularities in symmetric settings 对称环境中大爆炸奇点的初始数据
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a2
Hans Ringström
{"title":"Initial data on big bang singularities in symmetric settings","authors":"Hans Ringström","doi":"10.4310/pamq.2024.v20.n4.a2","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a2","url":null,"abstract":"In a recent article, we propose a general geometric notion of initial data on big bang singularities. This notion is of interest in its own right. However, it also serves the purpose of giving a unified perspective on many of the results in the literature. In the present article, we give a partial justification of this statement by rephrasing the results concerning Bianchi class A orthogonal stiff fluid solutions and solutions in the $mathbb{T}^3$-Gowdy symmetric vacuum setting in terms of our general geometric notion of initial data on the big bang singularity.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743073","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
Brief introduction to the nonlinear stability of Kerr 克尔非线性稳定性简介
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI: 10.4310/pamq.2024.v20.n4.a8
Sergiu Klainerman, Jérémie Szeftel
{"title":"Brief introduction to the nonlinear stability of Kerr","authors":"Sergiu Klainerman, Jérémie Szeftel","doi":"10.4310/pamq.2024.v20.n4.a8","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a8","url":null,"abstract":"This a brief introduction to the sequence of works $href{https://doi.org/10.48550/arXiv.2104.11857}{[65]}$, $href{https://doi.org/10.48550/arXiv.2205.14808}{[41]}$, $href{https://doi.org/10.1007/s40818-022-00131-8}{[63]}$, $href{https://doi.org/10.1007/s40818-022-00132-7}{[64]}$, and $href{https://doi.org/10.1007/s40818-023-00152-x}{[85]}$ which establish the nonlinear stability of Kerr black holes with small angular momentum. We are delighted to dedicate this article to Demetrios Christodoulou for whom we both have great admiration. The first author would also like to thank Demetrios for the magic moments of friendship, discussions and collaboration he enjoyed together with him.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743341","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
Topological aspects of Boolean functions 布尔函数的拓扑学方面
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-05-15 DOI: 10.4310/pamq.2024.v20.n3.a1
Anders Björner, Mark Goresky, Robert MacPherson
{"title":"Topological aspects of Boolean functions","authors":"Anders Björner, Mark Goresky, Robert MacPherson","doi":"10.4310/pamq.2024.v20.n3.a1","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a1","url":null,"abstract":"We discuss ways in which tools from topology can be used to derive lower bounds for the circuit complexity of Boolean functions.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061727","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
On normal Seshadri stratifications 关于正常的塞沙德里分层
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-05-15 DOI: 10.4310/pamq.2024.v20.n3.a3
Rocco Chirivì, Xin Fang, Peter Littelmann
{"title":"On normal Seshadri stratifications","authors":"Rocco Chirivì, Xin Fang, Peter Littelmann","doi":"10.4310/pamq.2024.v20.n3.a3","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a3","url":null,"abstract":"The existence of a Seshadri stratification on an embedded projective variety provides a flat degeneration of the variety to a union of projective toric varieties, called a semi-toric variety. Such a stratification is said to be normal when each irreducible component of the semi-toric variety is a normal toric variety. In this case, we show that a Gröbner basis of the defining ideal of the semi-toric variety can be lifted to define the embedded projective variety. Applications to Koszul and Gorenstein properties are discussed. Relations between LS‑algebras and certain Seshadri stratifications are studied.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061729","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
Milnor fibre homology complexes 米尔诺纤维同源复合体
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-05-15 DOI: 10.4310/pamq.2024.v20.n3.a9
Gus Lehrer, Yang Zhang
{"title":"Milnor fibre homology complexes","authors":"Gus Lehrer, Yang Zhang","doi":"10.4310/pamq.2024.v20.n3.a9","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a9","url":null,"abstract":"Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as “the non-crossing algebra”, which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $tilde{A}$, which seems to be “dual” to the Fomin–Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection between $tilde{A}$ and the Orlik–Solomon algebra, in analogy with the fact that the Fomin–Kirillov algebra contains the coinvariant algebra of $W$. This analysis is applied to compute the multiplicities ${langle rho, H^k (F, mathbb{C}) rangle}_W$ and ${langle rho, H^k (M, mathbb{C}) rangle}_W$, where $M$ and $F$ are respectively the hyperplane complement and Milnor fibre associated to $W$ and $rho$ is a representation of $W$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061698","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
Reducibility and nonlinear stability for a quasi-periodically forced NLS 准周期强迫 NLS 的可重复性和非线性稳定性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-05-15 DOI: 10.4310/pamq.2024.v20.n3.a8
E. Haus, B. Langella, A. Maspero, M. Procesi
{"title":"Reducibility and nonlinear stability for a quasi-periodically forced NLS","authors":"E. Haus, B. Langella, A. Maspero, M. Procesi","doi":"10.4310/pamq.2024.v20.n3.a8","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a8","url":null,"abstract":"Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus $mathbb{T}^2 := (mathbb{R}/2 pi mathbb{Z})^2$, we consider a quasi-periodically forced NLS equation on $mathbb{T}^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061731","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
Actions of finite group schemes on curves 有限群方案在曲线上的作用
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-05-15 DOI: 10.4310/pamq.2024.v20.n3.a2
Michel Brion
{"title":"Actions of finite group schemes on curves","authors":"Michel Brion","doi":"10.4310/pamq.2024.v20.n3.a2","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a2","url":null,"abstract":"Every action of a finite group scheme $G$ on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of $G$-normalization. In particular, every curve equipped with a $G$-action has a unique projective $G$-normal model, characterized by the invertibility of ideal sheaves of all orbits. Also, $G$-normal curves occur naturally in some questions on surfaces in positive characteristics.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061779","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
Catalan numbers and noncommutative Hilbert schemes 加泰罗尼亚数和非交换希尔伯特方案
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-05-15 DOI: 10.4310/pamq.2024.v20.n3.a10
Valery Lunts, Špela Špenko, Michel Van Den Bergh
{"title":"Catalan numbers and noncommutative Hilbert schemes","authors":"Valery Lunts, Špela Špenko, Michel Van Den Bergh","doi":"10.4310/pamq.2024.v20.n3.a10","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a10","url":null,"abstract":"We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m, n)$-Dyck paths, the number of which is given by the Fuss–Catalan number $A_n (m, 1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061730","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
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