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Hilbert’s Nullstellensatz for analytic trigonometric polynomials 解析三角多项式的希尔伯特零矩阵
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.09.005
Jie Xiao , Cheng Yuan
{"title":"Hilbert’s Nullstellensatz for analytic trigonometric polynomials","authors":"Jie Xiao ,&nbsp;Cheng Yuan","doi":"10.1016/j.exmath.2022.09.005","DOIUrl":"10.1016/j.exmath.2022.09.005","url":null,"abstract":"<div><p><span>This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if </span><span><math><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msubsup></math></span> are analytic trigonometric polynomials without common zero in the finite complex plane <span><math><mi>ℂ</mi></math></span> then there are analytic trigonometric polynomials <span><math><msubsup><mrow><mrow><mo>{</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msubsup></math></span> obeying <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msubsup><msub><mrow><mi>f</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></math></span> in <span><math><mi>ℂ</mi></math></span>, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on <span><math><mi>ℂ</mi></math></span>.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 910-919"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42459702","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
Hodge–de Rham numbers of almost complex 4-manifolds 几乎复4-流形的Hodge-de Rham数
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.08.005
Joana Cirici , Scott O. Wilson
{"title":"Hodge–de Rham numbers of almost complex 4-manifolds","authors":"Joana Cirici ,&nbsp;Scott O. Wilson","doi":"10.1016/j.exmath.2022.08.005","DOIUrl":"10.1016/j.exmath.2022.08.005","url":null,"abstract":"<div><p>We introduce and study Hodge–de Rham numbers for compact almost complex 4-manifolds, generalizing the Hodge numbers of a complex surface. The main properties of these numbers in the case of complex surfaces are extended to this more general setting, and it is shown that all Hodge–de Rham numbers for compact almost complex 4-manifolds are determined by the topology, except for one (the irregularity). Finally, these numbers are shown to prohibit the existence of complex structures on certain manifolds, without reference to the classification of surfaces.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1244-1260"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086922000548/pdfft?md5=955691cf023e0bdf88e05255afcf52b8&pid=1-s2.0-S0723086922000548-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43715263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Introduction to the combinatorial atlas 组合图集简介
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.08.003
Swee Hong Chan, Igor Pak
{"title":"Introduction to the combinatorial atlas","authors":"Swee Hong Chan,&nbsp;Igor Pak","doi":"10.1016/j.exmath.2022.08.003","DOIUrl":"10.1016/j.exmath.2022.08.003","url":null,"abstract":"<div><p>We give elementary self-contained proofs of the <em>strong Mason conjecture</em> recently proved by Anari et al. (2018) and Brändén and Huh (2020), and of the classical <em>Alexandrov–Fenchel inequality</em>. Both proofs use the <em>combinatorial atlas</em> technology recently introduced by the authors Chan and Pak (2021). We also give a formal relationship between combinatorial atlases and <em>Lorentzian polynomials</em>.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1014-1048"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46473988","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: 8
Complex and tropical counts via positive characteristic 复杂和热带计数通过阳性特征
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.07.003
Marco Pacini , Damiano Testa
{"title":"Complex and tropical counts via positive characteristic","authors":"Marco Pacini ,&nbsp;Damiano Testa","doi":"10.1016/j.exmath.2022.07.003","DOIUrl":"10.1016/j.exmath.2022.07.003","url":null,"abstract":"<div><p>We study two classical families of enumerative problems: inflection lines of plane curves and theta-hyperplanes of canonical curves. In these problems the complex counts and the tropical counts disagree. Each problem suggests a prime with special behavior. On the one hand, we analyze the reduction modulo these special primes, and we prove that the complex solutions coalesce in uniform clusters. On the other hand, we observe that the counts in special characteristic and in tropical geometry match.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1096-1115"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46014874","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 full power of the half-power 半功率的全部功率
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.10.002
P. Amster , J. Ángel Cid
{"title":"The full power of the half-power","authors":"P. Amster ,&nbsp;J. Ángel Cid","doi":"10.1016/j.exmath.2022.10.002","DOIUrl":"10.1016/j.exmath.2022.10.002","url":null,"abstract":"<div><p>We use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully unveiled.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 994-1013"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086922000640/pdfft?md5=104b6946b5aaf4e948ceffa68e902a90&pid=1-s2.0-S0723086922000640-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46786926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Descriptive complexity of subsets of the space of finitely generated groups 有限生成群空间子集的描述复杂度
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.08.001
Mustafa Gökhan Benli̇, Burak Kaya
{"title":"Descriptive complexity of subsets of the space of finitely generated groups","authors":"Mustafa Gökhan Benli̇,&nbsp;Burak Kaya","doi":"10.1016/j.exmath.2022.08.001","DOIUrl":"10.1016/j.exmath.2022.08.001","url":null,"abstract":"<div><p><span>In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups, groups of exponential growth and groups with decidable word problem are </span><span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-complete and that the sets of periodic groups and groups of intermediate growth are <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span><span>-complete. We also provide bounds for the descriptive complexity of simplicity, amenability, residually finiteness, Hopficity and co-Hopficity. This paper is intended to serve as a compilation of results on this theme.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1116-1134"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46810940","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: 3
On some characterizations of greedy-type bases 论贪心型碱的若干特征
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.06.003
Pablo M. Berná , Hùng Việt Chu
{"title":"On some characterizations of greedy-type bases","authors":"Pablo M. Berná ,&nbsp;Hùng Việt Chu","doi":"10.1016/j.exmath.2022.06.003","DOIUrl":"10.1016/j.exmath.2022.06.003","url":null,"abstract":"<div><p><span>In 1999, S. V. Konyagin and V. N. Temlyakov introduced the so-called Thresholding </span>Greedy Algorithm<span>. Since then, there have been many interesting and useful characterizations of greedy-type bases in Banach spaces. In this article, we study and extend several characterizations of greedy and almost greedy bases in the literature. Along the way, we give various examples to complement our main results. Furthermore, we propose a new version of the so-called Weak Thresholding Greedy Algorithm (WTGA) and show that the convergence of this new algorithm is equivalent to the convergence of the WTGA.</span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1135-1158"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48601996","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: 5
Property FW and wreath products of groups: A simple approach using Schreier graphs 性质FW和群的环积:使用Schreier图的一种简单方法
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.07.001
Paul-Henry Leemann , Grégoire Schneeberger
{"title":"Property FW and wreath products of groups: A simple approach using Schreier graphs","authors":"Paul-Henry Leemann ,&nbsp;Grégoire Schneeberger","doi":"10.1016/j.exmath.2022.07.001","DOIUrl":"10.1016/j.exmath.2022.07.001","url":null,"abstract":"<div><p>The group property FW stands in-between the celebrated Kazhdan’s property (T) and Serre’s property FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended.</p><p>It follows from the work of Y. Cornulier that a finitely generated wreath product <span><math><mrow><mi>G</mi><msub><mrow><mo>≀</mo></mrow><mrow><mi>X</mi></mrow></msub><mi>H</mi></mrow></math></span> has property FW if and only if both <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span> have property FW and <span><math><mi>X</mi></math></span> is finite. The aim of this paper is to give an elementary, direct and explicit proof of this fact using Schreier graphs.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1261-1270"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086922000391/pdfft?md5=a25144589e5c96c13c5679a9a7e61297&pid=1-s2.0-S0723086922000391-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44811719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Homology and AH conjecture for groupoids on one-dimensional solenoids 一维螺线管上群类群的同调性和AH猜想
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.08.002
Inhyeop Yi
{"title":"Homology and AH conjecture for groupoids on one-dimensional solenoids","authors":"Inhyeop Yi","doi":"10.1016/j.exmath.2022.08.002","DOIUrl":"10.1016/j.exmath.2022.08.002","url":null,"abstract":"<div><p>We show that Matui’s AH conjecture holds for groupoids of the Bratteli–Vershik systems embedded in the unstable equivalence relation on one-dimensional solenoids.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1084-1095"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S072308692200055X/pdfft?md5=66417ea013a83aa83fa6577901ec03cc&pid=1-s2.0-S072308692200055X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45194567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Controlling monotonicity of nonlinear operators 非线性算子的单调性控制
IF 0.7 4区 数学
Expositiones Mathematicae Pub Date : 2022-12-01 DOI: 10.1016/j.exmath.2022.07.002
Michał Borowski, Iwona Chlebicka
{"title":"Controlling monotonicity of nonlinear operators","authors":"Michał Borowski,&nbsp;Iwona Chlebicka","doi":"10.1016/j.exmath.2022.07.002","DOIUrl":"10.1016/j.exmath.2022.07.002","url":null,"abstract":"<div><p>Controlling the monotonicity and growth of Leray–Lions’ operators including the <span><math><mi>p</mi></math></span>-Laplacian plays a fundamental role in the theory of existence and regularity of solutions to second order nonlinear PDEs. We collect, correct, and supply known estimates including the discussion on the constants. Moreover, we provide a comprehensive treatment of related results for operators with Orlicz growth. We pay special attention to exposition of the proofs and the use of elementary arguments.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"40 4","pages":"Pages 1159-1180"},"PeriodicalIF":0.7,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0723086922000408/pdfft?md5=49850269d5e28c150251fbd05421ce5f&pid=1-s2.0-S0723086922000408-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46836549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
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