We present estimates (which are necessarily spectral) of the rate of convergence in the von Neumann ergodic theorem in terms of the singularity at zero of the spectral measure of the function to be averaged with respect to the corresponding dynamical system as well as in terms of the decay rate of the correlations (i.e., the Fourier coefficients of this measure). Estimates of the rate of convergence in the Birkhoff ergodic theorem are given in terms of the rate of convergence in the von Neumann ergodic theorem as well as in terms of the decay rate of the large deviation probabilities. We give estimates of the rate of convergence in both ergodic theorems for some classes of dynamical systems popular in applications, including some well-known billiards and Anosov systems.
{"title":"Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems","authors":"Aleksandr G. Kachurovskii, I. Podvigin","doi":"10.1090/MOSC/256","DOIUrl":"https://doi.org/10.1090/MOSC/256","url":null,"abstract":"We present estimates (which are necessarily spectral) of the rate of convergence in the von Neumann ergodic theorem in terms of the singularity at zero of the spectral measure of the function to be averaged with respect to the corresponding dynamical system as well as in terms of the decay rate of the correlations (i.e., the Fourier coefficients of this measure). Estimates of the rate of convergence in the Birkhoff ergodic theorem are given in terms of the rate of convergence in the von Neumann ergodic theorem as well as in terms of the decay rate of the large deviation probabilities. We give estimates of the rate of convergence in both ergodic theorems for some classes of dynamical systems popular in applications, including some well-known billiards and Anosov systems.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/256","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560382","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}
{"title":"The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research","authors":"I. Sabitov","doi":"10.1090/MOSC/257","DOIUrl":"https://doi.org/10.1090/MOSC/257","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/257","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560418","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 prove global existence theorems for initial–boundary value problems for the modified Navier–Stokes equations used when modeling ocean dynamic pro- cesses. First, the case of distinct vertical and horizontal viscosities for the Navier– Stokes equations is considered. Then a result due to Ladyzhenskaya for the modified Navier–Stokes equations is improved, whereby the elliptic operator is strengthened with respect to the horizontal variables alone and only for the horizontal momentum equations. Finally, the global existence and uniqueness of a solution is proved for the primitive equations describing the large-scale ocean dynamics.
{"title":"On the Existence of a Global Solution of the Modified Navier–Stokes Equations","authors":"G. Kobelkov","doi":"10.1090/MOSC/258","DOIUrl":"https://doi.org/10.1090/MOSC/258","url":null,"abstract":". We prove global existence theorems for initial–boundary value problems for the modified Navier–Stokes equations used when modeling ocean dynamic pro- cesses. First, the case of distinct vertical and horizontal viscosities for the Navier– Stokes equations is considered. Then a result due to Ladyzhenskaya for the modified Navier–Stokes equations is improved, whereby the elliptic operator is strengthened with respect to the horizontal variables alone and only for the horizontal momentum equations. Finally, the global existence and uniqueness of a solution is proved for the primitive equations describing the large-scale ocean dynamics.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/258","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560432","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}
{"title":"Integrable Systems, Shuffle Algebras, and Bethe Equations","authors":"B. Feigin","doi":"10.1090/MOSC/259","DOIUrl":"https://doi.org/10.1090/MOSC/259","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/259","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560471","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}
{"title":"On Salikhov’s integral","authors":"V. Sorokin","doi":"10.1090/MOSC/254","DOIUrl":"https://doi.org/10.1090/MOSC/254","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560313","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 use affine W-algebras to quantize Mishchenko-Fomenko subalgebras for centralizers of nilpotent elements in simple Lie algebras under certain assumptions that are satisfied for all cases in type A and all minimal nilpotent cases outside type $E_8$.
{"title":"Quantizing Mishchenko–Fomenko subalgebras for centralizers via affine -algebras","authors":"T. Arakawa, A. Premet","doi":"10.1090/MOSC/264","DOIUrl":"https://doi.org/10.1090/MOSC/264","url":null,"abstract":"We use affine W-algebras to quantize Mishchenko-Fomenko subalgebras for centralizers of nilpotent elements in simple Lie algebras under certain assumptions that are satisfied for all cases in type A and all minimal nilpotent cases outside type $E_8$.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/264","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560528","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 construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain Ω(ε) = Ω Γ̄ε with a thin singular set Γε lying in the cεneighborhood of a simple smooth closed contour Γ. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on ∂Γε, and also a spectral problem with lumped masses on Γε. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter |ln ε|−1 or ε. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of |ln ε|−1 and obtain an asymptotic expansion with the leading term holomorphically depending on |ln ε|−1 and with the remainder O(εδ), δ ∈ (0, 1). The main role in asymptotic formulas is played by solutions of the Dirichlet problem in Ω Γ with logarithmic singularities distributed along the contour Γ. 1. Statement of the problems. Description of the methods and results 1.1. Domain and boundary value problems. Let Γ be a simple smooth (C∞) closed contour on the plane R. In a neighborhood V of Γ, we introduce the natural curvilinear coordinates (n, s), where s is the arc length parameter and n is the signed distance from Γ positive outside the domain surrounded by Γ (Figure 1). In what follows, we slightly abuse the notation by writing s ∈ Γ to mean the point of Γ with coordinate s and by denoting the set {x ∈ R : (x1, x2) ∈ Γ, x3 = 0} in the space R again by Γ. Let ω be a bounded domain on the plane (Figure 2(a)), let U be a neighborhood of Γ in R where the coordinate system (n, s, x3) is defined, and let (1.1) Γε = {x ∈ U : s ∈ Γ, η = (ε−1n, εx3) ∈ ω}. Here ε > 0 is a small parameter; i.e., Γε is a thin toroidal set (Figure 2(b)). Finally, let Ω be a domain in R containing Γ (and hence containing the set (1.1) for small ε ∈ (0, ε0], ε0 > 0). For simplicity, we assume that the boundaries ∂Ω and ∂ω are smooth and place the origin η = 0 in the interior of the set ω ⊂ R. The aim of this paper is to study asymptotic properties of the spectra of several boundary value problems. First, this is the Dirichlet problem in the singularly perturbed 2010 Mathematics Subject Classification. Primary 35J25; Secondary 35B25, 35B40, 35B45, 35P20, 35S05.
{"title":"Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube","authors":"S. Nazarov","doi":"10.1090/MOSC/243","DOIUrl":"https://doi.org/10.1090/MOSC/243","url":null,"abstract":"We construct and justify asymptotic representations for the eigenvalues and eigenfunctions of boundary value problems for the Laplace operator in a three-dimensional domain Ω(ε) = Ω Γ̄ε with a thin singular set Γε lying in the cεneighborhood of a simple smooth closed contour Γ. We consider the Dirichlet problem, a mixed boundary value problem with the Neumann conditions on ∂Γε, and also a spectral problem with lumped masses on Γε. The asymptotic representations are of diverse character: we find an asymptotic series in powers of the parameter |ln ε|−1 or ε. The most comprehensive and complicated analysis is presented for the lumped mass problem; namely, we sum the series in powers of |ln ε|−1 and obtain an asymptotic expansion with the leading term holomorphically depending on |ln ε|−1 and with the remainder O(εδ), δ ∈ (0, 1). The main role in asymptotic formulas is played by solutions of the Dirichlet problem in Ω Γ with logarithmic singularities distributed along the contour Γ. 1. Statement of the problems. Description of the methods and results 1.1. Domain and boundary value problems. Let Γ be a simple smooth (C∞) closed contour on the plane R. In a neighborhood V of Γ, we introduce the natural curvilinear coordinates (n, s), where s is the arc length parameter and n is the signed distance from Γ positive outside the domain surrounded by Γ (Figure 1). In what follows, we slightly abuse the notation by writing s ∈ Γ to mean the point of Γ with coordinate s and by denoting the set {x ∈ R : (x1, x2) ∈ Γ, x3 = 0} in the space R again by Γ. Let ω be a bounded domain on the plane (Figure 2(a)), let U be a neighborhood of Γ in R where the coordinate system (n, s, x3) is defined, and let (1.1) Γε = {x ∈ U : s ∈ Γ, η = (ε−1n, εx3) ∈ ω}. Here ε > 0 is a small parameter; i.e., Γε is a thin toroidal set (Figure 2(b)). Finally, let Ω be a domain in R containing Γ (and hence containing the set (1.1) for small ε ∈ (0, ε0], ε0 > 0). For simplicity, we assume that the boundaries ∂Ω and ∂ω are smooth and place the origin η = 0 in the interior of the set ω ⊂ R. The aim of this paper is to study asymptotic properties of the spectra of several boundary value problems. First, this is the Dirichlet problem in the singularly perturbed 2010 Mathematics Subject Classification. Primary 35J25; Secondary 35B25, 35B40, 35B45, 35P20, 35S05.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/243","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560097","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 find the algebras of unipotent invariants of Cox rings for all double flag varieties of complexity 0 and 1 for the classical groups; namely, we obtain presentations of these algebras. It is well known that such an algebra is simple in the case of complexity 0. We show that, in the case of complexity 1, the algebra in question is either a free algebra or a hypersurface. Knowing the structure of this algebra permits one to effectively decompose tensor products of irreducible representations into direct sums of irreducible representations.
{"title":"Invariants of the Cox rings of low-complexity double flag varieties for classical groups","authors":"E. Ponomareva","doi":"10.1090/MOSC/244","DOIUrl":"https://doi.org/10.1090/MOSC/244","url":null,"abstract":"We find the algebras of unipotent invariants of Cox rings for all double flag varieties of complexity 0 and 1 for the classical groups; namely, we obtain presentations of these algebras. It is well known that such an algebra is simple in the case of complexity 0. We show that, in the case of complexity 1, the algebra in question is either a free algebra or a hypersurface. Knowing the structure of this algebra permits one to effectively decompose tensor products of irreducible representations into direct sums of irreducible representations.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/244","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560147","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 consider foliations of compact complex manifolds by analytic curves. It is well known that if the line bundle tangent to the foliation is negative, then, in general position, all leaves are hyperbolic. The manifold of universal coverings over the leaves passing through some transversal has a natural complex structure. We show that in a typical case this structure can be defined as a smooth almost complex structure on the product of the base by the unit disk. We prove that this structure is quasiconformal on the leaves and that the corresponding (1 , 0)-forms and their derivatives with respect to the coordinates on the base and in the leaves admit uniform estimates. The derivatives grow no faster than some negative power of the distance to the boundary of the disk.
{"title":"Almost complex structures on universal coverings of foliations","authors":"A. Shcherbakov","doi":"10.1090/MOSC/250","DOIUrl":"https://doi.org/10.1090/MOSC/250","url":null,"abstract":". We consider foliations of compact complex manifolds by analytic curves. It is well known that if the line bundle tangent to the foliation is negative, then, in general position, all leaves are hyperbolic. The manifold of universal coverings over the leaves passing through some transversal has a natural complex structure. We show that in a typical case this structure can be defined as a smooth almost complex structure on the product of the base by the unit disk. We prove that this structure is quasiconformal on the leaves and that the corresponding (1 , 0)-forms and their derivatives with respect to the coordinates on the base and in the leaves admit uniform estimates. The derivatives grow no faster than some negative power of the distance to the boundary of the disk.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/250","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60559938","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 elementary proofs of some properties of the generalized shift generated by a spherical symmetry. We construct B-kernels for Fourier integrals with respect to Bessel j-functions (Fourier–Bessel transforms). These are designed to play the same role as Dirichlet and de la Vall´ee-Poussin–Nikol’ski˘ı kernels in the theory of trigonometric Fourier integrals and in the theory of function approximation.
． 给出了由球对称产生的广义位移的一些性质的初等证明。我们构造关于贝塞尔j函数(傅里叶-贝塞尔变换)的傅里叶积分的b核。这些被设计成在三角傅立叶积分理论和函数逼近理论中扮演与Dirichlet和de la Vall ' ee-Poussin-Nikol 'ski × ×核相同的角色。
{"title":"The construction of Dirichlet and de la Vallée-Poussin–Nikol’skiĭ kernels for -Bessel Fourier integrals","authors":"L. Lyakhov","doi":"10.1090/MOSC/242","DOIUrl":"https://doi.org/10.1090/MOSC/242","url":null,"abstract":". We give elementary proofs of some properties of the generalized shift generated by a spherical symmetry. We construct B-kernels for Fourier integrals with respect to Bessel j-functions (Fourier–Bessel transforms). These are designed to play the same role as Dirichlet and de la Vall´ee-Poussin–Nikol’ski˘ı kernels in the theory of trigonometric Fourier integrals and in the theory of function approximation.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/242","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60560051","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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