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$mathbb{Z}$ -graded identities of the Lie algebras $U_1$ in characteristic 2 李代数$U_1$在特征2上的$mathbb{Z}$ -阶恒等式
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-07-22 DOI: 10.1017/S0305004122000123
Claudemir Fidelis, P. Koshlukov
{"title":"$mathbb{Z}$\u0000 -graded identities of the Lie algebras \u0000$U_1$\u0000 in characteristic 2","authors":"Claudemir Fidelis, P. Koshlukov","doi":"10.1017/S0305004122000123","DOIUrl":"https://doi.org/10.1017/S0305004122000123","url":null,"abstract":"Abstract Let K be any field of characteristic two and let \u0000$U_1$\u0000 and \u0000$W_1$\u0000 be the Lie algebras of the derivations of the algebra of Laurent polynomials \u0000$K[t,t^{-1}]$\u0000 and of the polynomial ring K[t], respectively. The algebras \u0000$U_1$\u0000 and \u0000$W_1$\u0000 are equipped with natural \u0000$mathbb{Z}$\u0000 -gradings. In this paper, we provide bases for the graded identities of \u0000$U_1$\u0000 and \u0000$W_1$\u0000 , and we prove that they do not admit any finite basis.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"9 1","pages":"49 - 58"},"PeriodicalIF":0.8,"publicationDate":"2021-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81806569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Non-classical polynomials and the inverse theorem 非经典多项式和反定理
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-07-15 DOI: 10.1017/S0305004121000682
A. Berger, A. Sah, Mehtaab Sawhney, Jonathan Tidor
{"title":"Non-classical polynomials and the inverse theorem","authors":"A. Berger, A. Sah, Mehtaab Sawhney, Jonathan Tidor","doi":"10.1017/S0305004121000682","DOIUrl":"https://doi.org/10.1017/S0305004121000682","url":null,"abstract":"Abstract In this paper we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers \u0000$U^k$\u0000 -norm. We give a brief deduction of the fact that a bounded function on \u0000$mathbb F_p^n$\u0000 with large \u0000$U^k$\u0000 -norm must correlate with a classical polynomial when \u0000$kle p+1$\u0000 . To the best of our knowledge, this result is new for \u0000$k=p+1$\u0000 (when \u0000$p>2$\u0000 ). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers \u0000$U^k$\u0000 -norm over \u0000$mathbb F_p^n$\u0000 for all \u0000$kge p+2$\u0000 , completely characterising when classical polynomials suffice.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"83 1","pages":"525 - 537"},"PeriodicalIF":0.8,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83875875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 3
Random fractals and their intersection with winning sets 随机分形及其与获胜集的交集
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-07-07 DOI: 10.1017/S0305004121000360
Yiftach Dayan
{"title":"Random fractals and their intersection with winning sets","authors":"Yiftach Dayan","doi":"10.1017/S0305004121000360","DOIUrl":"https://doi.org/10.1017/S0305004121000360","url":null,"abstract":"\u0000 We show that fractal percolation sets in \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if \u0000 \u0000 \u0000 $Esubsetmathbb{R}^{d}$\u0000 \u0000 is a realisation of a fractal percolation process, then almost surely (conditioned on \u0000 \u0000 \u0000 $Eneqemptyset$\u0000 \u0000 ), for every countable collection \u0000 \u0000 \u0000 $left(f_{i}right)_{iinmathbb{N}}$\u0000 \u0000 of \u0000 \u0000 \u0000 $C^{1}$\u0000 \u0000 diffeomorphisms of \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 , \u0000 \u0000 \u0000 $dim_{H}left(Ecapleft(bigcap_{iinmathbb{N}}f_{i}left(text{BA}_{d}right)right)right)=dim_{H}left(Eright)$\u0000 \u0000 , where \u0000 \u0000 \u0000 $text{BA}_{d}$\u0000 \u0000 is the set of badly approximable vectors in \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to \u0000 \u0000 \u0000 $dim_{H}left(Eright)$\u0000 \u0000 .\u0000 We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to \u0000 \u0000 \u0000 $mathbb{R}^{d}$\u0000 \u0000 yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"94 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86649696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
PSP volume 171 issue 1 Cover and Front matter PSP第171卷第1期封面和封面问题
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-07-01 DOI: 10.1017/s030500412100044x
{"title":"PSP volume 171 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s030500412100044x","DOIUrl":"https://doi.org/10.1017/s030500412100044x","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"3 1","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84513931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
PSP volume 171 issue 1 Cover and Back matter PSP第171卷第1期封面和封底
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-07-01 DOI: 10.1017/s0305004121000451
{"title":"PSP volume 171 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004121000451","DOIUrl":"https://doi.org/10.1017/s0305004121000451","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"40 1","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85629124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Counting H-free orientations of graphs 计算图的无h方向
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-06-16 DOI: 10.1017/S0305004122000147
M. Buci'c, Oliver Janzer, B. Sudakov
{"title":"Counting H-free orientations of graphs","authors":"M. Buci'c, Oliver Janzer, B. Sudakov","doi":"10.1017/S0305004122000147","DOIUrl":"https://doi.org/10.1017/S0305004122000147","url":null,"abstract":"Abstract In 1974, Erdős posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős’s question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"20 1","pages":"79 - 95"},"PeriodicalIF":0.8,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89296972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 4
Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial 随机对称矩阵:秩分布和特征多项式的不可约性
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-06-08 DOI: 10.1017/S0305004122000226
Asaf Ferber, Vishesh Jain, A. Sah, Mehtaab Sawhney
{"title":"Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial","authors":"Asaf Ferber, Vishesh Jain, A. Sah, Mehtaab Sawhney","doi":"10.1017/S0305004122000226","DOIUrl":"https://doi.org/10.1017/S0305004122000226","url":null,"abstract":"Abstract Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric \u0000${pm 1}$\u0000 -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random \u0000${pm 1}$\u0000 -matrices over \u0000$mathbb{F}_p$\u0000 for primes \u0000$2 < p leq exp(O(n^{1/4}))$\u0000 . Previously, such estimates were available only for \u0000$p = o(n^{1/8})$\u0000 . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in \u0000$mathbb{F}_p^{n}$\u0000 with anticoncentration at least \u0000$1/p + Omega(1/p^2)$\u0000 . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least \u0000$C/p$\u0000 for some large constant \u0000$C > 1$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"3 1","pages":"233 - 246"},"PeriodicalIF":0.8,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80205065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 9
Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties 简单李代数中最小幂零轨道的投影和割线变分
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-05-06 DOI: 10.1017/S0305004123000348
D. Panyushev
{"title":"Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties","authors":"D. Panyushev","doi":"10.1017/S0305004123000348","DOIUrl":"https://doi.org/10.1017/S0305004123000348","url":null,"abstract":"Abstract Let G be a simple algebraic group with \u0000${mathfrak g}={textrm{Lie }} G$\u0000 and \u0000${mathcal O}_{textsf{min}}subset{mathfrak g}$\u0000 the minimal nilpotent orbit. For a \u0000${mathbb Z}_2$\u0000 -grading \u0000${mathfrak g}={mathfrak g}_0oplus{mathfrak g}_1$\u0000 , let \u0000$G_0$\u0000 be a connected subgroup of G with \u0000${textrm{Lie }} G_0={mathfrak g}_0$\u0000 . We study the \u0000$G_0$\u0000 -equivariant projections \u0000$varphi,:,overline{{mathcal O}_{textsf{min}}}to {mathfrak g}_0$\u0000 and \u0000$psi:overline{{mathcal O}_{textsf{min}}}to{mathfrak g}_1$\u0000 . It is shown that the properties of \u0000$overline{varphi({mathcal O}_{textsf{min}})}$\u0000 and \u0000$overline{psi({mathcal O}_{textsf{min}})}$\u0000 essentially depend on whether the intersection \u0000${mathcal O}_{textsf{min}}cap{mathfrak g}_1$\u0000 is empty or not. If \u0000${mathcal O}_{textsf{min}}cap{mathfrak g}_1nevarnothing$\u0000 , then both \u0000$overline{varphi({mathcal O}_{textsf{min}})}$\u0000 and \u0000$overline{psi({mathcal O}_{textsf{min}})}$\u0000 contain a 1-parameter family of closed \u0000$G_0$\u0000 -orbits, while if \u0000${mathcal O}_{textsf{min}}cap{mathfrak g}_1=varnothing$\u0000 , then both are \u0000$G_0$\u0000 -prehomogeneous. We prove that \u0000$overline{G{cdot}varphi({mathcal O}_{textsf{min}})}=overline{G{cdot}psi({mathcal O}_{textsf{min}})}$\u0000 . Moreover, if \u0000${mathcal O}_{textsf{min}}cap{mathfrak g}_1nevarnothing$\u0000 , then this common variety is the affine cone over the secant variety of \u0000${mathbb P}({mathcal O}_{textsf{min}})subset{mathbb P}({mathfrak g})$\u0000 . As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of \u0000${mathfrak g}$\u0000 in place of \u0000${mathfrak g}_0$\u0000 or spherical nilpotent G-orbits in place of \u0000${mathcal O}_{textsf{min}}$\u0000 .","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"3 1","pages":"595 - 624"},"PeriodicalIF":0.8,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81657515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
PSP volume 170 issue 3 Cover and Front matter PSP第170卷第3期封面和封面问题
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-05-01 DOI: 10.1017/s0305004121000347
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引用次数: 0
PSP volume 170 issue 3 Cover and Back matter PSP第170卷第3期封面和封底
IF 0.8 3区 数学
Mathematical Proceedings of the Cambridge Philosophical Society Pub Date : 2021-05-01 DOI: 10.1017/s0305004121000359
{"title":"PSP volume 170 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0305004121000359","DOIUrl":"https://doi.org/10.1017/s0305004121000359","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"28 1","pages":"b1 - b5"},"PeriodicalIF":0.8,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83357286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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