We study four (a priori) different ways in which an open book decomposition of the 3-sphere can be defined to be braided. These include generalised exchangeability defined by Morton and Rampichini and mutual braiding defined by Rudolph, which were shown to be equivalent by Rampichini, as well as P-fiberedness and a property related to simple branched covers of $S^3$ inspired by work of Montesinos and Morton. We prove that these four notions of a braided open book are actually all equivalent to each other. We show that all open books in the 3-sphere whose binding has a braid index of at most 3 can be braided in this sense.
{"title":"Braided open book decompositions in $S^3$","authors":"Benjamin Bode","doi":"10.4171/rmi/1429","DOIUrl":"https://doi.org/10.4171/rmi/1429","url":null,"abstract":"We study four (a priori) different ways in which an open book decomposition of the 3-sphere can be defined to be braided. These include generalised exchangeability defined by Morton and Rampichini and mutual braiding defined by Rudolph, which were shown to be equivalent by Rampichini, as well as P-fiberedness and a property related to simple branched covers of $S^3$ inspired by work of Montesinos and Morton. We prove that these four notions of a braided open book are actually all equivalent to each other. We show that all open books in the 3-sphere whose binding has a braid index of at most 3 can be braided in this sense.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42454461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend Barker’s weak-strong uniqueness results for the Navier– Stokes equations and consider a criterion involving Besov spaces and weighted Lebesgue spaces.
{"title":"A remark on weak-strong uniqueness for suitable weak solutions of the Navier–Stokes equations","authors":"P. Lemari'e-Rieusset","doi":"10.4171/rmi/1386","DOIUrl":"https://doi.org/10.4171/rmi/1386","url":null,"abstract":"We extend Barker’s weak-strong uniqueness results for the Navier– Stokes equations and consider a criterion involving Besov spaces and weighted Lebesgue spaces.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43410553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the 2d cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s = 2k, k ∈ N, is t2(s−1)/3+ε. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.
{"title":"Growth of Sobolev norms for $2d$ NLS with harmonic potential","authors":"F. Planchon, N. Tzvetkov, N. Visciglia","doi":"10.4171/rmi/1371","DOIUrl":"https://doi.org/10.4171/rmi/1371","url":null,"abstract":"Abstract. We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the 2d cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s = 2k, k ∈ N, is t2(s−1)/3+ε. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44153859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property $PFP_1$ is closed under extensions. In addition, we prove that profinite groups of type $FP_1$ with UBERG are always finitely generated and we characterise UBERG in the class of pro-nilpotent groups. Using infinite products of finite groups, we construct several examples of profinite groups with unexpected properties: (1) an UBERG group which cannot be finitely generated, (2) a group of type $PFP_infty$ which is not UBERG and not finitely generated and (3) a group of type $PFP_infty$ with superexponential subgroup growth.
{"title":"Counting irreducible modules for profinite groups","authors":"Ged Corob Cook, Steffen Kionke, Matteo Vannacci","doi":"10.4171/rmi/1382","DOIUrl":"https://doi.org/10.4171/rmi/1382","url":null,"abstract":"This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property $PFP_1$ is closed under extensions. In addition, we prove that profinite groups of type $FP_1$ with UBERG are always finitely generated and we characterise UBERG in the class of pro-nilpotent groups. Using infinite products of finite groups, we construct several examples of profinite groups with unexpected properties: (1) an UBERG group which cannot be finitely generated, (2) a group of type $PFP_infty$ which is not UBERG and not finitely generated and (3) a group of type $PFP_infty$ with superexponential subgroup growth.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43154860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that constant mean curvature hypersurfaces in $mathbb H^ntimesmathbb R$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces determined by $P$. This is the analogous in $mathbb H^ntimesmathbb R$ of a result in $mathbb R^3$ by A. Ros and H. Rosenberg.
{"title":"Constant mean curvature hypersurfaces in $mathbb{H}^ntimesmathbb{R}$ with small planar boundary","authors":"B. Nelli, Giuseppe Pipoli","doi":"10.4171/rmi/1420","DOIUrl":"https://doi.org/10.4171/rmi/1420","url":null,"abstract":"We show that constant mean curvature hypersurfaces in $mathbb H^ntimesmathbb R$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces determined by $P$. This is the analogous in $mathbb H^ntimesmathbb R$ of a result in $mathbb R^3$ by A. Ros and H. Rosenberg.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42823626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let p be a prime number and ξ an irrational p-adic number. Its multiplicative irrationality exponent μ(ξ) is the supremum of the real numbers μ for which the inequality |bξ − a|p ≤ |ab| /2 has infinitely many solutions in nonzero integers a, b. We show that μ(ξ) can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of ξ. We establish that μ×(ξt,p) = 3, where ξt,p is the p-adic number 1 − p − p + p − p + . . ., whose sequence of digits is given by the Thue–Morse sequence over {−1, 1}.
{"title":"On the rational approximation to $p$-adic Thue–Morse numbers","authors":"Y. Bugeaud","doi":"10.4171/rmi/1384","DOIUrl":"https://doi.org/10.4171/rmi/1384","url":null,"abstract":"Let p be a prime number and ξ an irrational p-adic number. Its multiplicative irrationality exponent μ(ξ) is the supremum of the real numbers μ for which the inequality |bξ − a|p ≤ |ab| /2 has infinitely many solutions in nonzero integers a, b. We show that μ(ξ) can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of ξ. We establish that μ×(ξt,p) = 3, where ξt,p is the p-adic number 1 − p − p + p − p + . . ., whose sequence of digits is given by the Thue–Morse sequence over {−1, 1}.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48754687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that there exists a bounded subset of R such that no system of exponentials can be a Riesz basis for the corresponding Hilbert space. An additional result gives a lower bound for the Riesz constant of any putative Riesz basis of the two dimensional disk.
{"title":"A set with no Riesz basis of exponentials","authors":"G. Kozma, S. Nitzan, A. Olevskiǐ","doi":"10.4171/rmi/1411","DOIUrl":"https://doi.org/10.4171/rmi/1411","url":null,"abstract":"We show that there exists a bounded subset of R such that no system of exponentials can be a Riesz basis for the corresponding Hilbert space. An additional result gives a lower bound for the Riesz constant of any putative Riesz basis of the two dimensional disk.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46862860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are concerned with solutions to the linear wave equation. We give an asymptotic formula for large time, valid in the energy space, via an operator related to the Radon transform. This allows us to show that the energy is concentrated near the light cone. This allows to derive further expressions the exterior energy (outside a shifted light cone). We in particular generalize the formulas of [CKS14] obtained in the radial setting. In odd dimension, we study the discrepancy of the exterior energy regarding initial energy, and prove in the general case the results of [KLLS15] (which were restricted to radial data).
{"title":"Concentration close to the cone for linear waves","authors":"R. Cote, C. Laurent","doi":"10.4171/rmi/1399","DOIUrl":"https://doi.org/10.4171/rmi/1399","url":null,"abstract":"We are concerned with solutions to the linear wave equation. We give an asymptotic formula for large time, valid in the energy space, via an operator related to the Radon transform. This allows us to show that the energy is concentrated near the light cone. This allows to derive further expressions the exterior energy (outside a shifted light cone). We in particular generalize the formulas of [CKS14] obtained in the radial setting. In odd dimension, we study the discrepancy of the exterior energy regarding initial energy, and prove in the general case the results of [KLLS15] (which were restricted to radial data).","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45756990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Carruth, Maximilian F. Eggl, C. Fefferman, C. Rowley, Melanie Weber
We consider a simple control problem in which the underlying dynamics depend on a parameter that is unknown and must be learned. We exhibit a control strategy which is optimal to within a multiplicative constant. While most authors find strategies which are successful as the time horizon tends to infinity, our strategy achieves lowest expected cost up to a constant factor for a fixed time horizon.
{"title":"Controlling unknown linear dynamics with bounded multiplicative regret","authors":"J. Carruth, Maximilian F. Eggl, C. Fefferman, C. Rowley, Melanie Weber","doi":"10.4171/rmi/1377","DOIUrl":"https://doi.org/10.4171/rmi/1377","url":null,"abstract":"We consider a simple control problem in which the underlying dynamics depend on a parameter that is unknown and must be learned. We exhibit a control strategy which is optimal to within a multiplicative constant. While most authors find strategies which are successful as the time horizon tends to infinity, our strategy achieves lowest expected cost up to a constant factor for a fixed time horizon.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43821641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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