We show that if a hyperbolic knot manifold MM contains an essential twice-punctured torus FF with boundary slope βbeta and admits a filling with slope αalpha producing a Seifert fibred space, then the distance between the slopes αalpha and βbeta is less than or equal to 55 unless MM is the exterior of the figure eight knot. The result is sharp; the bound of 55 can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the
我们证明，如果一个双曲结流形 M M 包含一个边界斜率为 β β 的本质两次穿刺环 F F，并且允许一个斜率为 α α 的填充，从而产生一个塞弗特纤维空间，那么斜率 α α 和 β β 之间的距离小于等于 5 5，除非 M M 是八字结的外部。这个结果是尖锐的；5 5 的边界可以在无限多的双曲结流形上实现。我们还确定了在α α-填充的基群不包含非阿贝尔自由基的情况下的距离界限。证明分为四种情况：F F 是半纤维、F F 是纤维、F F 是非分离的但不是纤维、F F 是分离的但不是半纤维。
{"title":"Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori","authors":"Steven Boyer, Cameron McA. Gordon, Xingru Zhang","doi":"10.1090/memo/1469","DOIUrl":"https://doi.org/10.1090/memo/1469","url":null,"abstract":"<p>We show that if a hyperbolic knot manifold <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> contains an essential twice-punctured torus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with boundary slope <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta\">\u0000 <mml:semantics>\u0000 <mml:mi>β<!-- β --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">beta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and admits a filling with slope <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\">\u0000 <mml:semantics>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> producing a Seifert fibred space, then the distance between the slopes <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\">\u0000 <mml:semantics>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta\">\u0000 <mml:semantics>\u0000 <mml:mi>β<!-- β --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">beta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is less than or equal to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"5\">\u0000 <mml:semantics>\u0000 <mml:mn>5</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">5</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> unless <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the exterior of the figure eight knot. The result is sharp; the bound of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"5\">\u0000 <mml:semantics>\u0000 <mml:mn>5</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">5</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141216037","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}
{"title":"Subgroup Decomposition in 𝖮𝗎𝗍(𝖥_{𝗇})","authors":"M. Handel, L. Mosher","doi":"10.1090/memo/1280","DOIUrl":"https://doi.org/10.1090/memo/1280","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48171650","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}
We consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t > 0, where f a C1 function. Assuming that 0 and γ > 0 are constant steady states, we investigate the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x, 0) are near γ for x ≈ −∞ and near 0 for x ≈ ∞. If the steady states 0 and γ are both stable, our main theorem shows that at large times, the graph of u(·, t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). We prove this result without requiring monotonicity of u(·, 0) or the nondegeneracy of zeros of f . The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to our theorems, we show that all front-like solutions are quasiconvergent: their ω-limit sets with respect to the locally uniform convergence consist of steady states. In our proofs we employ phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories {(u(x, t), ux(x, t)) : x ∈ R}, t > 0, of the solutions in question.
{"title":"Propagating Terraces and the Dynamics of\u0000 Front-Like Solutions of Reaction-Diffusion Equations\u0000 on ℝ","authors":"P. Polácik","doi":"10.1090/memo/1278","DOIUrl":"https://doi.org/10.1090/memo/1278","url":null,"abstract":"We consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t > 0, where f a C1 function. Assuming that 0 and γ > 0 are constant steady states, we investigate the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x, 0) are near γ for x ≈ −∞ and near 0 for x ≈ ∞. If the steady states 0 and γ are both stable, our main theorem shows that at large times, the graph of u(·, t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). We prove this result without requiring monotonicity of u(·, 0) or the nondegeneracy of zeros of f . The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to our theorems, we show that all front-like solutions are quasiconvergent: their ω-limit sets with respect to the locally uniform convergence consist of steady states. In our proofs we employ phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories {(u(x, t), ux(x, t)) : x ∈ R}, t > 0, of the solutions in question.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47138547","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}
Spiral waves are striking self-organized coherent structures that organize spatio-temporal dynamics in dissipative, spatially extended systems. In this paper, we provide a conceptual approach to various properties of spiral waves. Rather than studying existence in a specific equation, we study properties of spiral waves in general reaction-diffusion systems. We show that many features of spiral waves are robust and to some extent independent of the specific model analyzed. To accomplish this, we present a suitable analytic framework, spatial radial dynamics, that allows us to rigorously characterize features such as the shape of spiral waves and their eigenfunctions, properties of the linearization, and finite-size effects. We believe that our framework can also be used to study spiral waves further and help analyze bifurcations, as well as provide guidance and predictions for experiments and numerical simulations. From a technical point of view, we introduce non-standard function spaces for the well-posedness of the existence problem which allow us to understand properties of spiral waves using dynamical systems techniques, in particular exponential dichotomies. Using these pointwise methods, we are able to bring tools from the analysis of one-dimensional coherent structures such as fronts and pulses to bear on these inherently two-dimensional defects.
{"title":"Spiral Waves: Linear and Nonlinear Theory","authors":"Bjorn Sandstede, A. Scheel","doi":"10.1090/memo/1413","DOIUrl":"https://doi.org/10.1090/memo/1413","url":null,"abstract":"Spiral waves are striking self-organized coherent structures that organize spatio-temporal dynamics in dissipative, spatially extended systems. In this paper, we provide a conceptual approach to various properties of spiral waves. Rather than studying existence in a specific equation, we study properties of spiral waves in general reaction-diffusion systems. We show that many features of spiral waves are robust and to some extent independent of the specific model analyzed. To accomplish this, we present a suitable analytic framework, spatial radial dynamics, that allows us to rigorously characterize features such as the shape of spiral waves and their eigenfunctions, properties of the linearization, and finite-size effects. We believe that our framework can also be used to study spiral waves further and help analyze bifurcations, as well as provide guidance and predictions for experiments and numerical simulations. From a technical point of view, we introduce non-standard function spaces for the well-posedness of the existence problem which allow us to understand properties of spiral waves using dynamical systems techniques, in particular exponential dichotomies. Using these pointwise methods, we are able to bring tools from the analysis of one-dimensional coherent structures such as fronts and pulses to bear on these inherently two-dimensional defects.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45564385","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}
Michele D'Adderio, Alessandro Iraci, A. V. Wyngaerd
We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both Haglund (“A proof of the q,tq,t-Schröder conjecture”, 2004) and Aval et al. (“Statistics on parallelogram polyominoes and a q,tq,t-analogue of the Narayana numbers”, 2014). This settles in particular the cases ⟨⋅,en−dhd⟩langle cdot ,e_{n-d}h_drangle and ⟨⋅,hn−dhd⟩langle cdot ,h_{n-d}h_drangle of the Delta conjecture of Haglund, Remmel and Wilson (“The delta conjec
我们讨论了装饰Dyck路径和装饰平行四边形多项式的组合，将Haglund(“q,t q,t -Schröder猜想的证明”，2004)和Aval等人(“平行四边形多项式的统计和Narayana数的q,t q,t模拟”，2014)的主要结果扩展到装饰情况。这特别解决了⟨⋅，e n-d h d⟩langle cdot,e_{n-d}h_drangle和⟨，h n-d h d⟩langle cdot,h_{n-d}h_drangle的Haglund, Remmel和Wilson(“Delta猜想”，2018)的Delta猜想的情况。在此过程中，我们引入了一些新的统计数据，制定了一些新的猜想，证明了对称函数的一些新的恒等式，并回答了文献中的一些开放问题(例如，来自Aval, Bergeron和Garsia [2015]， Haglund, Remmel和Wilson[2018]，以及Zabrocki[2019])。主要的技术工具是麦克唐纳多项式理论中的一个新恒等式，它扩展了哈格伦德在“q,t q,t -Schröder猜想的证明”(2004)中的一个定理。
{"title":"Decorated Dyck paths, polyominoes, and the Delta conjecture","authors":"Michele D'Adderio, Alessandro Iraci, A. V. Wyngaerd","doi":"10.1090/memo/1370","DOIUrl":"https://doi.org/10.1090/memo/1370","url":null,"abstract":"<p>We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both Haglund (“A proof of the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q comma t\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">q,t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-Schröder conjecture”, 2004) and Aval et al. (“Statistics on parallelogram polyominoes and a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q comma t\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">q,t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-analogue of the Narayana numbers”, 2014). This settles in particular the cases <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mathematical left-angle dot comma e Subscript n minus d Baseline h Subscript d Baseline mathematical right-angle\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟨<!-- ⟨ --></mml:mo>\u0000 <mml:mo>⋅<!-- ⋅ --></mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mi>h</mml:mi>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟩<!-- ⟩ --></mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">langle cdot ,e_{n-d}h_drangle</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mathematical left-angle dot comma h Subscript n minus d Baseline h Subscript d Baseline mathematical right-angle\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟨<!-- ⟨ --></mml:mo>\u0000 <mml:mo>⋅<!-- ⋅ --></mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>h</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mi>h</mml:mi>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟩<!-- ⟩ --></mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">langle cdot ,h_{n-d}h_drangle</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of the Delta conjecture of Haglund, Remmel and Wilson (“The delta conjec","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49011543","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}
This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.
{"title":"Congruence Lattices of Ideals in Categories and (Partial) Semigroups","authors":"J. East, N. Ruškuc","doi":"10.1090/memo/1408","DOIUrl":"https://doi.org/10.1090/memo/1408","url":null,"abstract":"This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2020-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44460644","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}
Gonzalo Fiz Pontiveros, Simon Griffiths, R. Morris
{"title":"The Triangle-Free Process and the Ramsey\u0000 Number 𝑅(3,𝑘)","authors":"Gonzalo Fiz Pontiveros, Simon Griffiths, R. Morris","doi":"10.1090/memo/1274","DOIUrl":"https://doi.org/10.1090/memo/1274","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60559038","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}
We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators ΓGamma on a Hilbert space. Our assumption on ΓGamma is expressed in terms of αalpha-boundedness for a Euclidean structure αalpha on the underlying Banach space XX. This notion is originally motivated by Rmathcal {R}- or γgamma-boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓn2ell ^2_n
通过表示Hilbert空间上的某些Banach空间算子ΓGamma集，我们提出了一种将Hilbert空间算子的结果推广到Banach空间集的一般方法。我们对ΓGamma的假设是用下面Banach空间X X上欧几里得结构αalpha的αalpha-有界性来表示的。这个概念最初是由算子集的Rmathcal｛R｝-或γgamma-有界性引起的，但例如欧几里得空间中的任何算子理想ℓ n2ell^2_n到X X定义了这样一个结构。因此，我们的方法相当灵活。相反，我们证明了ΓGamma必须是αalpha-有界的，这样一些欧几里得结构αalpha才能在Hilbert空间上表示。通过相应地选择欧几里得结构αalpha，我们得到了Kwapień–Maurey因子分解定理和Maurey、Nikišin和Rubio de Francia的因子分解理论的统一且更通用的方法。这导致了Rubio de Francia的Banach函数空间值扩张定理的改进版本，以及格Hardy–Littlewood极大算子有界性的定量证明。此外，我们使用这些欧几里得结构来构建向量值函数空间。它们具有一个很好的性质，即L2 L^2上的任何有界算子都可以扩展到这些向量值函数空间上的有界算子，这与Bochner空间的扩展问题形成了鲜明的对比。利用这些空间，我们定义了一种插值方法，该方法具有以实数和复数插值方法为模型的公式。利用我们的表示定理，我们证明了Banach空间上扇形算子的转移原理，使我们能够将扇形算子的Hilbert空间结果推广到Banach空间设置。例如，我们扩展和完善了已知的基于联合和算子值H∞H^infty演算的R数学｛R｝-有界性的理论。此外，我们将Hilbert空间上H∞H^infty-演算的有界性的经典刻画用B I P BIP、平方函数和扩张推广到Banach空间设置。此外，我们通过H∞H^infty演算，为一大类扇形算子建立了Littlewood–Paley理论的一个版本和分数光滑的相关空间。我们的抽象设置允许我们减少对X X的几何结构的假设，例如（co）类型和UMD。最后，我们给出了扇形算子的一些复杂反例，重点讨论了在LpL^p的闭子空间上，对于1>p
{"title":"Euclidean Structures and Operator Theory in Banach Spaces","authors":"N. Kalton, E. Lorist, L. Weis","doi":"10.1090/memo/1433","DOIUrl":"https://doi.org/10.1090/memo/1433","url":null,"abstract":"<p>We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Gamma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> on a Hilbert space. Our assumption on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Gamma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is expressed in terms of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\">\u0000 <mml:semantics>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-boundedness for a Euclidean structure <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\">\u0000 <mml:semantics>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> on the underlying Banach space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This notion is originally motivated by <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {R}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>- or <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma\">\u0000 <mml:semantics>\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">gamma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-boundedness of sets of operators, but for example any operator ideal from the Euclidean space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l Subscript n Superscript 2\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">ell ^2_n</mml:annotation>\u0000 </mml:sema","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45838940","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}
We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time T T only at one blow-up point. We also give a sharp description of its profile. In the first part, we formally construct a blow-up solution. In the second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.
{"title":"Construction of Blowup Solutions for the Complex Ginzburg-Landau Equation with Critical Parameters","authors":"G. K. Duong, N. Nouaili, H. Zaag","doi":"10.1090/memo/1411","DOIUrl":"https://doi.org/10.1090/memo/1411","url":null,"abstract":"We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time \u0000\u0000 \u0000 T\u0000 T\u0000 \u0000\u0000 only at one blow-up point. We also give a sharp description of its profile. In the first part, we formally construct a blow-up solution. In the second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43473916","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}
For a finite group GG, we denote by ω(G)omega (G) the number of Aut(G)Aut(G)-orbits on GG, and by o(G)o(G) the number of distinct element orders in GG. In this paper, we are primarily concerned with the two quantities d(G)
对于有限群G G，我们用ω（G）omega（G）表示G G上a u t（G）Aut（G。在本文中，我们主要关注两个量d（G）ω（G）−o（G）mathfrak｛d｝（G）≔omega（G）-o（G）和q（G）（G） ，它们中的每一个都可以被视为G离张意义上的AT群（即ω（G）=o（G）ω（G。我们证明了G的可溶性自由基R a d（G）Rad（G）的指数|G:R a dq（G）mathfrak{q}（G）和o（R a d（G））o（Rad（G）。我们还获得了Fischer Griess Monster群M M的一个奇怪的定量刻画。
{"title":"Automorphism Orbits and Element Orders in Finite Groups: Almost-Solubility and the Monster","authors":"Alexander Bors, Michael Giudici, C. Praeger","doi":"10.1090/memo/1427","DOIUrl":"https://doi.org/10.1090/memo/1427","url":null,"abstract":"<p>For a finite group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we denote by <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"omega left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ω<!-- ω --></mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">omega (G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> the number of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A u t left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Aut(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-orbits on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and by <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"o left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>o</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">o(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> the number of distinct element orders in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this paper, we are primarily concerned with the two quantities <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German d left-parenthesis upper G right-parenthesis colon-equal omega left-parenthesis upper G right-parenthesis minus o left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">d</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 ","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47590446","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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