This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case 1$?> , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.
{"title":"Iterated Laurent series over rings and the Contou-Carrère symbol","authors":"S. Gorchinskiy, D. Osipov","doi":"10.1070/RM9975","DOIUrl":"https://doi.org/10.1070/RM9975","url":null,"abstract":"This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case 1$?> , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48732400","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}
K e−ihξ dν(h) is the characteristic function of the measure ν. It was also shown that when (2) fails, there can be an infinite-dimensional kernel in this problem. The problem (1) is a natural generalization of boundary-value problems for elliptic differential-difference equations [5], [6] and functional-differential equations with contracted/extended independent variables [2], [3]. We note a connection between (possibly degenerate) elliptic functional-differential operators and Kato’s well-known problem of the square root of a regular accretive operator [6], [7].
K e−ihξ dν(h)是测度ν的特征函数。还表明，当(2)失效时，该问题可能存在一个无限维核。问题(1)是椭圆型微分-差分方程[5]，[6]和具有收缩/扩展自变量[2]，[3]的泛函-微分方程边值问题的自然推广。我们注意到(可能退化的)椭圆函数微分算子和加托著名的正则加积算子[6]，[7]的平方根问题之间的联系。
{"title":"The spectral radius of a certain parametric family of functional operators","authors":"N. B. Zhuravlev, L. Rossovskii","doi":"10.1070/RM9967","DOIUrl":"https://doi.org/10.1070/RM9967","url":null,"abstract":"K e−ihξ dν(h) is the characteristic function of the measure ν. It was also shown that when (2) fails, there can be an infinite-dimensional kernel in this problem. The problem (1) is a natural generalization of boundary-value problems for elliptic differential-difference equations [5], [6] and functional-differential equations with contracted/extended independent variables [2], [3]. We note a connection between (possibly degenerate) elliptic functional-differential operators and Kato’s well-known problem of the square root of a regular accretive operator [6], [7].","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005123","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":"Ramsey theory in the -space with Chebyshev metric","authors":"A. Kupavskii, A. Sagdeev","doi":"10.1070/RM9958","DOIUrl":"https://doi.org/10.1070/RM9958","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48009887","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 survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter , is analytic in the right-hand complex half-plane , and its zeros in coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed. Bibliography: 83 titles.
{"title":"Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes","authors":"A. Il'ichev","doi":"10.1070/RM9953","DOIUrl":"https://doi.org/10.1070/RM9953","url":null,"abstract":"This survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter , is analytic in the right-hand complex half-plane , and its zeros in coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed. Bibliography: 83 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43539043","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}
M. Isaev, I. Rodionov, Ruizhe Zhang, M. Zhukovskii
The central result of extreme value theory proved by Gnedenko [1] classifies all types of asymptotic distributions that the normalized maximum of a sample of independent identically distributed random variables could possibly have. Does a similar result hold if the variables are not identically distributed or are dependent? We consider a sequence of random vectors Xn = (X1,n, . . . , Xd,n) ∈ R, where d = d(n) ∈ N and n ∈ N. Let [d] := {1, . . . , d}. If, for any fixed x ∈ R, ∣∣∣∣P(max i∈[d] Xi,n ⩽ x)− ∏ i∈[d] P(Xi,n ⩽ x) ∣∣∣∣ → 0, as n →∞, (1)
{"title":"Extreme value theory for triangular arrays of dependent random variables","authors":"M. Isaev, I. Rodionov, Ruizhe Zhang, M. Zhukovskii","doi":"10.1070/RM9964","DOIUrl":"https://doi.org/10.1070/RM9964","url":null,"abstract":"The central result of extreme value theory proved by Gnedenko [1] classifies all types of asymptotic distributions that the normalized maximum of a sample of independent identically distributed random variables could possibly have. Does a similar result hold if the variables are not identically distributed or are dependent? We consider a sequence of random vectors Xn = (X1,n, . . . , Xd,n) ∈ R, where d = d(n) ∈ N and n ∈ N. Let [d] := {1, . . . , d}. If, for any fixed x ∈ R, ∣∣∣∣P(max i∈[d] Xi,n ⩽ x)− ∏ i∈[d] P(Xi,n ⩽ x) ∣∣∣∣ → 0, as n →∞, (1)","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49370695","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}
A. Artem’ev, S. Bolotin, D. Vainchtein, A. Vasiliev, S. Dobrokhotov, L. Zelenyi, V. V. Kozlov, A. Petrukovich, V. V. Sidorenko, D. Treschev, A. I. Shafarevich
Anatolii Iserovish Neishtadt, a prominent researcher and outstanding expert in the theory of perturbations of dynamical systems and the theory of adiabatic invariants, observed his 70th birthday on 27 July 2020. He was born in Moscow in the family of a chemical engineer and a flight technician (his mother). As a high-school senior he attended lessons at a voluntary evening physicsmathematics school under the auspices of the Bauman Moscow State Technical School (now Technical University), from which he graduated with honours. In 1967 he graduated from Moscow school no. 358 with a gold medal and enrolled in the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University. He was a student in the Department of Theoretical Mechanics, where M. L. Lidov was his scientific advisor. Neishtadt graduated from the university with distinction and, after defending his diploma thesis, was admitted to postgraduate studies in the faculty, where in 1975 he presented, and in 1976 defended, his Ph.D. thesis “Some resonance problems in non-linear systems”, written under the supervision of Lidov. From 1975 to 1987 Neishtadt worked in the All-Union Scientific Research Institute of Medical Instrument Design, where his work involved applied programming. Apart from his duties there, he continued active research in mathematics, working on some problems posed by V. I. Arnold. In 1989 Neishtadt defended his D.Sc. thesis “Questions of perturbation theory for non-linear resonance systems” in the Faculty of Mechanics and Mathematics at Moscow State University. In 1987 he was hired on Arnold’s recommendation by the Institute of Space Research, as a researcher in G. M. Zaslavsky’s laboratory (subsequently, department). Neishtadt then worked there for a long time as head of the Laboratory of Non-Linear and Chaotic Dynamics, and is currently a leading researcher at the institute.
{"title":"Anatolii Iserovish Neishtadt","authors":"A. Artem’ev, S. Bolotin, D. Vainchtein, A. Vasiliev, S. Dobrokhotov, L. Zelenyi, V. V. Kozlov, A. Petrukovich, V. V. Sidorenko, D. Treschev, A. I. Shafarevich","doi":"10.1070/RM9965","DOIUrl":"https://doi.org/10.1070/RM9965","url":null,"abstract":"Anatolii Iserovish Neishtadt, a prominent researcher and outstanding expert in the theory of perturbations of dynamical systems and the theory of adiabatic invariants, observed his 70th birthday on 27 July 2020. He was born in Moscow in the family of a chemical engineer and a flight technician (his mother). As a high-school senior he attended lessons at a voluntary evening physicsmathematics school under the auspices of the Bauman Moscow State Technical School (now Technical University), from which he graduated with honours. In 1967 he graduated from Moscow school no. 358 with a gold medal and enrolled in the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University. He was a student in the Department of Theoretical Mechanics, where M. L. Lidov was his scientific advisor. Neishtadt graduated from the university with distinction and, after defending his diploma thesis, was admitted to postgraduate studies in the faculty, where in 1975 he presented, and in 1976 defended, his Ph.D. thesis “Some resonance problems in non-linear systems”, written under the supervision of Lidov. From 1975 to 1987 Neishtadt worked in the All-Union Scientific Research Institute of Medical Instrument Design, where his work involved applied programming. Apart from his duties there, he continued active research in mathematics, working on some problems posed by V. I. Arnold. In 1989 Neishtadt defended his D.Sc. thesis “Questions of perturbation theory for non-linear resonance systems” in the Faculty of Mechanics and Mathematics at Moscow State University. In 1987 he was hired on Arnold’s recommendation by the Institute of Space Research, as a researcher in G. M. Zaslavsky’s laboratory (subsequently, department). Neishtadt then worked there for a long time as head of the Laboratory of Non-Linear and Chaotic Dynamics, and is currently a leading researcher at the institute.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59004886","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 the problem of online aggregation of experts’ predictions with a quadratic loss function. At the beginning of each round t = 1, 2, . . . , T , experts n = 1, . . . , N provide predictions γ t , . . . , γ t ∈ H (where H is a Hilbert space). The player aggregates the predictions to a single prediction γt ∈ H. Then nature provides the true outcome ω ∈ H. The player and the experts n = 1, . . . , N suffer the losses ht = ∥ω−γt∥ and l t = ∥ω−γ t ∥, respectively, and the next round t + 1 begins. The goal of the player is to minimize the regret, that is, the difference between the total loss of the player and the loss of the best expert: RT = ∑T t=1 ht −minn=1,...,N ∑T t=1 l n t .
我们考虑了具有二次损失函数的专家预测的在线聚合问题。在每一轮的开始t=1，2，T，专家n=1，N提供预测γ，γt∈H（其中H是希尔伯特空间）。玩家将预测聚合为一个单独的预测γt∈H。然后自然提供了真实的结果ω∈H.玩家和专家n=1，N分别遭受损失ht=½ω-γt½和l t=½ω-伽玛t½，下一轮t+1开始。玩家的目标是最大限度地减少遗憾，即玩家的总损失与最佳专家的损失之差：RT=∑T T=1 ht−minn=1，。。。，N∑T T=1 l N T。
{"title":"Online algorithm for aggregating experts’ predictions with unbounded quadratic loss","authors":"Alexander Korotin, V. V'yugin, E. Burnaev","doi":"10.1070/RM9961","DOIUrl":"https://doi.org/10.1070/RM9961","url":null,"abstract":"We consider the problem of online aggregation of experts’ predictions with a quadratic loss function. At the beginning of each round t = 1, 2, . . . , T , experts n = 1, . . . , N provide predictions γ t , . . . , γ t ∈ H (where H is a Hilbert space). The player aggregates the predictions to a single prediction γt ∈ H. Then nature provides the true outcome ω ∈ H. The player and the experts n = 1, . . . , N suffer the losses ht = ∥ω−γt∥ and l t = ∥ω−γ t ∥, respectively, and the next round t + 1 begins. The goal of the player is to minimize the regret, that is, the difference between the total loss of the player and the loss of the best expert: RT = ∑T t=1 ht −minn=1,...,N ∑T t=1 l n t .","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43344771","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 dynamical systems on the space of functions taking values in a free associative algebra. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commuting symmetries. In this paper we propose a new approach to the problem of quantisation of dynamical systems, introduce the concept of quantisation ideals and provide meaningful examples.
{"title":"Quantisation ideals of nonabelian integrable systems","authors":"A. Mikhailov","doi":"10.1070/RM9966","DOIUrl":"https://doi.org/10.1070/RM9966","url":null,"abstract":"We consider dynamical systems on the space of functions taking values in a free associative algebra. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commuting symmetries. In this paper we propose a new approach to the problem of quantisation of dynamical systems, introduce the concept of quantisation ideals and provide meaningful examples.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42270560","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}
It is explicitly shown that the Poisson bracket on the set of shear coordinates defined by V. V. Fock in 1997 induces the Fenchel–Nielsen bracket on the set of gluing parameters (length and twist parameters) for pair-of-pants decompositions of Riemann surfaces with holes. These structures are generalized to the case of Riemann surfaces with holes and bordered cusps. Bibliography: 49 titles.
{"title":"Fenchel–Nielsen coordinates and Goldman brackets","authors":"L. Chekhov","doi":"10.1070/RM9972","DOIUrl":"https://doi.org/10.1070/RM9972","url":null,"abstract":"It is explicitly shown that the Poisson bracket on the set of shear coordinates defined by V. V. Fock in 1997 induces the Fenchel–Nielsen bracket on the set of gluing parameters (length and twist parameters) for pair-of-pants decompositions of Riemann surfaces with holes. These structures are generalized to the case of Riemann surfaces with holes and bordered cusps. Bibliography: 49 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47421444","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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