In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [32] inverts our integration.
{"title":"Integration of Exact Courant Algebroids","authors":"David Li-Bland, P. Ševera","doi":"10.3934/era.2012.19.58","DOIUrl":"https://doi.org/10.3934/era.2012.19.58","url":null,"abstract":"In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, \u0000and we show that the differentiation procedure from [32] inverts our integration.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"19 1","pages":"58-76"},"PeriodicalIF":0.0,"publicationDate":"2011-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a novel class of functions that can describe the stable and unstable manifolds of the Henon map. We propose an algorithm to construct these functions by using the Borel-Laplace transform. Neither linearization nor perturbation is applied in the construction, and the obtained functions are exact solutions of the Henon map. We also show that it is possible to depict the chaotic attractor of the map by using one of these functions without explicitly using the properties of the attractor.
{"title":"Special functions created by Borel-Laplace transform of Hénon map","authors":"C. Matsuoka, K. Hiraide","doi":"10.3934/ERA.2011.18.1","DOIUrl":"https://doi.org/10.3934/ERA.2011.18.1","url":null,"abstract":"We present a novel class of functions that can describe the stable and unstable manifolds of the Henon map. We propose an algorithm to construct these functions by using the Borel-Laplace transform. Neither linearization nor perturbation is applied in the construction, and the obtained functions are exact solutions of the Henon map. We also show that it is possible to depict the chaotic attractor of the map by using one of these functions without explicitly using the properties of the attractor.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"18 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces a new Parseval frame, based on the 3-D shearlet representation, which is especially designed to capture geometric features such as discontinuous boundaries with very high efficiency. We show that this approach exhibits essentially optimal approximation properties for 3-D functions $f$ which are smooth away from discontinuities along $C^2$ surfaces. In fact, the $N$ term approximation $f_N^S$ obtained by selecting the $N$ largest coefficients from the shearlet expansion of $f$ satisfies the asymptotic estimate ||$f-f_N^S$||$_2^2$ ≍ $N^{-1} (log N)^2, as N to infty.$ Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outperforms wavelet approximations, which only yields a $N^{-1/2}$ rate. Indeed, the wavelet approximation rate was the best published nonadaptive result so far and the result presented in this paper is the first nonadaptive construction which is provably optimal (up to a loglike factor) for this class of 3-D data. Our estimate is consistent with the corresponding 2-D (essentially) optimally sparse approximation results obtained by the authors using 2-D shearlets and by Candes and Donoho using curvelets.
本文介绍了一种新的Parseval框架，该框架基于三维剪切波表示，专门用于高效捕获不连续边界等几何特征。我们表明，这种方法对三维函数$f$具有本质上最优的近似性质，这些函数沿着$C^2$表面平滑地远离不连续。事实上，通过从$f$的shearlet展开中选择$N$最大系数得到的$N$项近似$f_N^S$满足渐近估计|| $f-f_N^S$ || $_2^2$￣$N^{-1} (log N)^2, as N to infty.$直到对数因子，这是该类函数的最优行为，并且显著优于小波近似，小波近似只产生$N^{-1/2}$速率。事实上，小波近似率是迄今为止发表的最好的非自适应结果，而本文提出的结果是第一个可证明对这类三维数据最优(达到loglike factor)的非自适应结构。我们的估计与作者使用二维shearlet和Candes和Donoho使用曲线得到的相应的二维(本质上)最优稀疏近似结果一致。
{"title":"Optimally sparse 3D approximations using shearlet representations","authors":"K. Guo, D. Labate","doi":"10.3934/ERA.2010.17.125","DOIUrl":"https://doi.org/10.3934/ERA.2010.17.125","url":null,"abstract":"This paper introduces a new Parseval frame, based on the 3-D \u0000shearlet representation, which is especially designed to capture \u0000geometric features such as discontinuous boundaries with very high \u0000efficiency. We show that this approach exhibits essentially optimal \u0000approximation properties for 3-D functions $f$ which are smooth \u0000away from discontinuities along $C^2$ surfaces. In fact, the $N$ \u0000term approximation $f_N^S$ obtained by selecting the $N$ largest \u0000coefficients from the shearlet expansion of $f$ satisfies the \u0000asymptotic estimate \u0000 \u0000 ||$f-f_N^S$||$_2^2$ ≍ $N^{-1} (log N)^2, as \u0000N to infty.$ Up to the logarithmic factor, \u0000this is the optimal behavior for functions in this class and \u0000significantly outperforms wavelet approximations, which only yields \u0000a $N^{-1/2}$ rate. Indeed, the wavelet approximation rate was the \u0000best published nonadaptive result so far and the result presented in \u0000this paper is the first nonadaptive construction which is provably \u0000optimal (up to a loglike factor) for this class of 3-D data. \u0000 \u0000 Our estimate is consistent with the corresponding \u00002-D (essentially) optimally sparse approximation results obtained \u0000by the authors using 2-D shearlets and by Candes and Donoho using \u0000curvelets.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"8 1","pages":"125-137"},"PeriodicalIF":0.0,"publicationDate":"2010-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On reflexive spaces trigonometrically well-bounded operators (abbreviated "twbo's'') have an operator-ergodic-theory characterization as the invertible operators $U$ whose rotates "transfer'' the discrete Hilbert averages $(C,1)$-boundedly. Twbo's permeate many settings of modern analysis, and this note treats advances in their spectral theory, Fourier analysis, and operator ergodic theory made possible by applying classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young to the R.C. James inequalities for super-reflexive spaces. When the James inequalities are combined with spectral integration methods and Young-Stieltjes integration for the spaces $V_{p}(mathbb{T}) $ of functions having bounded $p$-variation, it transpires that every twbo on a super-reflexive space $X$ has a norm-continuous $V_{p}(mathbb{T}) $-functional calculus for a range of values of $p>1$, and we investigate the ways this outcome logically simplifies and simultaneously advances the structure theory of twbo's on $X$. In particular, on a super-reflexive space $X$ (but not on the general reflexive space) Tauberian-type theorems emerge which improve to their $(C,0) $ counterparts the $(C,1) $ averaging and convergence associated with twbo's.
{"title":"Fourier analysis methods in operator ergodic theory onsuper-reflexive Banach spaces","authors":"E. Berkson","doi":"10.3934/ERA.2010.17.90","DOIUrl":"https://doi.org/10.3934/ERA.2010.17.90","url":null,"abstract":"On reflexive spaces trigonometrically well-bounded operators (abbreviated \u0000\"twbo's'') have an operator-ergodic-theory characterization as the \u0000invertible operators $U$ whose rotates \"transfer'' the discrete Hilbert \u0000averages $(C,1)$-boundedly. Twbo's permeate many settings of \u0000modern analysis, and this note treats advances in their spectral theory, \u0000Fourier analysis, and operator ergodic theory made possible by applying \u0000classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young \u0000to the R.C. James inequalities for super-reflexive spaces. When the James \u0000inequalities are combined with spectral integration methods and \u0000Young-Stieltjes integration for the spaces $V_{p}(mathbb{T}) $ \u0000of functions having bounded $p$-variation, it transpires that every twbo on \u0000a super-reflexive space $X$ has a norm-continuous $V_{p}(mathbb{T}) $-functional calculus for a range of values of $p>1$, and we \u0000investigate the ways this outcome logically simplifies and simultaneously \u0000advances the structure theory of twbo's on $X$. In particular, on a \u0000super-reflexive space $X$ (but not on the general reflexive space) \u0000Tauberian-type theorems emerge which improve to their $(C,0) $ \u0000counterparts the $(C,1) $ averaging and convergence associated \u0000with twbo's.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"17 1","pages":"90-103"},"PeriodicalIF":0.0,"publicationDate":"2010-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new characterization of spaces with nonnegative curvature in the sense of Alexandrov.
给出了亚历山德罗夫意义上的非负曲率空间的一个新的刻划。
{"title":"Curvature bounded below: A definition a la Berg--Nikolaev","authors":"N. Lebedeva, A. Petrunin","doi":"10.3934/ERA.2010.17.122","DOIUrl":"https://doi.org/10.3934/ERA.2010.17.122","url":null,"abstract":"We give a new characterization of spaces with nonnegative \u0000curvature in the sense of Alexandrov.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"17 1","pages":"122-124"},"PeriodicalIF":0.0,"publicationDate":"2010-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main results announced in this note are an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and a limit theorem for such flows (Theorem 2). Given an abelian differential on a compact oriented surface, consider the space $mathfrak B^+$ of Holder cocycles over the corresponding vertical flow that are invariant under holonomy by the horizontal flow. Cocycles in $mathfrak B^+$ are closely related to G.Forni's invariant distributions for translation flows [10]. Theorem 1 states that ergodic integrals of Lipschitz functions are approximated by cocycles in $mathfrak B^+$ up to an error that grows more slowly than any power of time. Theorem 2 is obtained using the renormalizing action of the Teichmuller flow on the space $mathfrak B^+$. A symbolic representation of translation flows as suspension flows over Vershik's automorphisms allows one to construct cocycles in $mathfrak B^+$ explicitly. Proofs of Theorems 1, 2 are given in [5].
{"title":"Hölder cocycles and ergodic integrals for translation flows on flat surfaces","authors":"A. Bufetov","doi":"10.3934/ERA.2010.17.34","DOIUrl":"https://doi.org/10.3934/ERA.2010.17.34","url":null,"abstract":"The main results announced in this note are an asymptotic expansion for ergodic integrals of \u0000translation flows on flat surfaces of higher genus (Theorem 1) \u0000and a limit theorem for such flows (Theorem 2). \u0000Given an abelian differential on a compact oriented surface, \u0000consider the space $mathfrak B^+$ of Holder cocycles over the corresponding vertical flow that are \u0000invariant under holonomy by the horizontal flow. \u0000Cocycles in $mathfrak B^+$ are closely related to G.Forni's invariant distributions for \u0000translation flows [10]. Theorem 1 states that ergodic integrals of Lipschitz functions are approximated \u0000by cocycles in $mathfrak B^+$ up to an error that grows more slowly than any power of time. Theorem 2 is obtained using the renormalizing action of the Teichmuller flow on the space $mathfrak B^+$. \u0000A symbolic representation of translation flows as suspension flows over Vershik's automorphisms allows one to construct cocycles in $mathfrak B^+$ explicitly. \u0000Proofs of Theorems 1, 2 are given in [5].","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"17 1","pages":"34-42"},"PeriodicalIF":0.0,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a two-parameter family of one-dimensional maps and the related (a, b)-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.
{"title":"THEORY OF (a, b)-CONTINUED FRACTION TRANSFORMATIONS AND APPLICATIONS","authors":"S. Katok, Ilie Ugarcovici","doi":"10.3934/ERA.2010.17.20","DOIUrl":"https://doi.org/10.3934/ERA.2010.17.20","url":null,"abstract":"We study a two-parameter family of one-dimensional maps and the related (a, b)-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"17 1","pages":"20-33"},"PeriodicalIF":0.0,"publicationDate":"2010-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T$: $ RRto RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. Applications include most previously known results, as well as some new ones.
{"title":"Multifractal formalism derived from thermodynamics for general dynamical systems","authors":"V. Climenhaga","doi":"10.3934/ERA.2010.17.1","DOIUrl":"https://doi.org/10.3934/ERA.2010.17.1","url":null,"abstract":"We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T$: $ RRto RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. Applications include most previously known results, as well as some new ones.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"14 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2010-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study Gauss curvature for random Riemannian metrics on a compact surface,lying in a fixed conformal class; our questions are motivated by comparisongeometry. We next consider analogous questions for the scalar curvature indimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.
{"title":"Scalar curvature and Q-curvature of random metrics","authors":"Y. Canzani, D. Jakobson, I. Wigman","doi":"10.3934/ERA.2010.17.43","DOIUrl":"https://doi.org/10.3934/ERA.2010.17.43","url":null,"abstract":"We study Gauss curvature for random Riemannian metrics on a compact surface,lying in a fixed conformal class; our questions are motivated by comparisongeometry. We next consider analogous questions for the scalar curvature indimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"17 1","pages":"43-56"},"PeriodicalIF":0.0,"publicationDate":"2010-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a method that associates to a singular space a CW complex whose ordinary rational homology satisfies Poincare duality across complementary perversities as in intersection homology. The method is based on a homotopy theoretic process of spatial homology truncation, whose functoriality properties are investigated in detail. The resulting homology theory is not isomorphic to intersection homology and addresses certain questions in type II string theory related to massless D-branes. The two theories satisfy an interchange of third and second plus fourth Betti number for mirror symmetric conifold transitions. Further applications of the new theory to K-theory and symmetric L-theory are indicated.
{"title":"Singular spaces and generalized Poincaré complexes","authors":"Markus Banagl","doi":"10.3934/ERA.2009.16.63","DOIUrl":"https://doi.org/10.3934/ERA.2009.16.63","url":null,"abstract":"We introduce a method that associates to a singular space a \u0000CW complex whose ordinary rational homology satisfies \u0000Poincare duality across complementary perversities as in intersection \u0000homology. The method is based on a homotopy theoretic \u0000process of spatial homology truncation, whose functoriality properties \u0000are investigated in detail. The resulting homology theory is not \u0000isomorphic to intersection homology and addresses certain questions \u0000in type II string theory related to massless D-branes. \u0000The two theories satisfy an interchange of third and second plus fourth \u0000Betti number for mirror symmetric conifold transitions. \u0000Further applications of the new theory to K-theory and symmetric L-theory \u0000are indicated.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"16 1","pages":"63-73"},"PeriodicalIF":0.0,"publicationDate":"2009-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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