{"title":"Localization for 𝑇𝐻𝐻(𝑘𝑢) and the Topological\u0000 Hochschild and Cyclic Homology of Waldhausen\u0000 Categories","authors":"A. Blumberg, Michael A. Mandell","doi":"10.1090/memo/1286","DOIUrl":"https://doi.org/10.1090/memo/1286","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"265 1","pages":"0-0"},"PeriodicalIF":1.9,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46602501","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}
Pub Date : 2020-03-01Epub Date: 2019-11-25DOI: 10.1007/s12630-019-01519-0
Alan Hsi-Wen Liao, Shang-Ru Yeoh, Yu-Cih Lin, Fai Lam, Ta-Liang Chen, Chien-Yu Chen
{"title":"In reply: Lidocaine lubricants for intubation-related complications: more details, more significance?","authors":"Alan Hsi-Wen Liao, Shang-Ru Yeoh, Yu-Cih Lin, Fai Lam, Ta-Liang Chen, Chien-Yu Chen","doi":"10.1007/s12630-019-01519-0","DOIUrl":"10.1007/s12630-019-01519-0","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"5 1","pages":"394-395"},"PeriodicalIF":0.0,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12630-019-01519-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82596840","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":" ","pages":"0-0"},"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":"264 1","pages":"0-0"},"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":" ","pages":""},"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
<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"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q comma t"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q,t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schröder conjecture”, 2004) and Aval et al. (“Statistics on parallelogram polyominoes and a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q comma t"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">q,t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-analogue of the Narayana numbers”, 2014). This settles in particular the cases <inline-formula content-type="math/mathml"> <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"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨<!-- ⟨ --></mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩<!-- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">langle cdot ,e_{n-d}h_drangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <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"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨<!-- ⟨ --></mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩<!-- ⟩ --></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">langle cdot ,h_{n-d}h_drangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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":" ","pages":""},"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":" ","pages":""},"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":"263 1","pages":"0-0"},"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}
<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"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a Hilbert space. Our assumption on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is expressed in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α<!-- α --></mml:mi> <mml:annotation encoding="application/x-tex">alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-boundedness for a Euclidean structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α<!-- α --></mml:mi> <mml:annotation encoding="application/x-tex">alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the underlying Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This notion is originally motivated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>- or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi>γ<!-- γ --></mml:mi> <mml:annotation encoding="application/x-tex">gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-boundedness of sets of operators, but for example any operator ideal from the Euclidean space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript n Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">ell ^2_n</mml:annotation> </mml:sema
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":" ","pages":""},"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":" ","pages":""},"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}
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