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Dynamics of metrics in measure spaces and scaling entropy 度量空间中的度量动态和缩放熵
IF 0.9 4区 数学
Russian Mathematical Surveys Pub Date : 2023-11-24 DOI: 10.4213/rm10103e
A. Vershik, Georgii A Veprev, P. Zatitskii
{"title":"Dynamics of metrics in measure spaces and scaling entropy","authors":"A. Vershik, Georgii A Veprev, P. Zatitskii","doi":"10.4213/rm10103e","DOIUrl":"https://doi.org/10.4213/rm10103e","url":null,"abstract":"This survey is dedicated to a new direction in the theory of dynamical systems, the dynamics of metrics in measure spaces and new (catalytic) invariants of transformations with invariant measure. A space equipped with a measure and a metric which are naturally consistent with each other (a metric triple, or an mm-space) defines automatically the notion of its entropy class, thus allowing one to construct a theory of scaling entropy for dynamical systems with invariant measure, which is different and more general in comparison to the Shannon-Kolmogorov theory. This possibility was hinted at by Shannon himself, but the hint went unnoticed. The classification of metric triples in terms of matrix distributions presented in this paper was proposed by Gromov and Vershik. We describe some corollaries obtained by applying this theory. Bibliography: 88 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139238854","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}
引用次数: 0
Derived category of moduli of parabolic bundles on $mathbb{P}^1$ 抛物线束在 $mathbb{P}^1$ 上的模数的衍生类别
IF 0.9 4区 数学
Russian Mathematical Surveys Pub Date : 2023-05-29 DOI: 10.4213/rm10116e
A. Fonarev
{"title":"Derived category of moduli of parabolic bundles on $mathbb{P}^1$","authors":"A. Fonarev","doi":"10.4213/rm10116e","DOIUrl":"https://doi.org/10.4213/rm10116e","url":null,"abstract":"We propose a conjecture on the structure of the bounded derived category of coherent sheaves of the moduli space rank $2$ parabolic bundles on $mathbb{P}^1$.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139372722","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}
引用次数: 0
Igor Moiseevich Krichever (obituary) Igor Moiseevich krichhever(讣告)
IF 0.9 4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10097e
V. Buchstaber, S. Novikov, I. Taimanov
{"title":"Igor Moiseevich Krichever (obituary)","authors":"V. Buchstaber, S. Novikov, I. Taimanov","doi":"10.4213/rm10097e","DOIUrl":"https://doi.org/10.4213/rm10097e","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70328502","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}
引用次数: 0
Strong and weak associativity of weighted Sobolev spaces of the first order 一阶加权Sobolev空间的强、弱结合律
IF 0.9 4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10075e
V. Stepanov, E. Ushakova
{"title":"Strong and weak associativity of weighted Sobolev spaces of the first order","authors":"V. Stepanov, E. Ushakova","doi":"10.4213/rm10075e","DOIUrl":"https://doi.org/10.4213/rm10075e","url":null,"abstract":"A brief overview of the recent results on the problem of characterization of associative and double associative spaces of function classes, including both ideal and non-ideal structures, is presented. The latter include two-weighted Sobolev spaces of the first order on the positive semi- axis. It is shown that, in contrast to the notion of duality, associativity can be ‘strong’ or ‘weak’. In addition, double associative spaces are further divided into three types. In this context it is established that a weighted Sobolev space of functions with compact support possesses weak associative reflexivity, while the strong associative space of a weak associative space is trivial. Weighted classes of Cesàro and Copson type have similar properties; for these classes the problem us fully investigated, and their connections with Sobolev spaces with power weights are established. As an application, the problem of boundedness of the Hilbert transform from a weighted Sobolev space to a weighted Lebesgue space is considered. Bibliography: 49 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70328894","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}
引用次数: 0
Automorphism groups of $mathbb{P}^1$-bundles over a non-uniruled base 非规则基上$mathbb{P}^1$-束的自同构群
4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10093e
Tatiana Bandman, Yuri Gennad'evich Zarhin
{"title":"Automorphism groups of $mathbb{P}^1$-bundles over a non-uniruled base","authors":"Tatiana Bandman, Yuri Gennad'evich Zarhin","doi":"10.4213/rm10093e","DOIUrl":"https://doi.org/10.4213/rm10093e","url":null,"abstract":"In this survey we discuss holomorphic $mathbb{P}^1$-bundles $pcolon X to Y$ over a non-uniruled complex compact Kähler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We consider the groups $operatorname{Aut}(X)$ and $operatorname{Bim}(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively, and discuss when these groups are bounded, Jordan, strongly Jordan, or very Jordan. Bibliography: 88 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135103168","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}
引用次数: 0
Stable and historic behavior in replicator equations given by nonlinear mappings 由非线性映射给出的复制方程的稳定和历史行为
4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10084e
Mansoor Khajibaevich Saburov
{"title":"Stable and historic behavior in replicator equations given by nonlinear mappings","authors":"Mansoor Khajibaevich Saburov","doi":"10.4213/rm10084e","DOIUrl":"https://doi.org/10.4213/rm10084e","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136368155","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}
引用次数: 0
Higher-order traps for some strongly degenerate quantum control systems 一些强退化量子控制系统的高阶陷阱
4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10069e
Boris Olegovich Volkov, Alexander Nikolaevich Pechen
{"title":"Higher-order traps for some strongly degenerate quantum control systems","authors":"Boris Olegovich Volkov, Alexander Nikolaevich Pechen","doi":"10.4213/rm10069e","DOIUrl":"https://doi.org/10.4213/rm10069e","url":null,"abstract":"Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor,\"Are there traps in quantum control landscapes?\", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136368141","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}
引用次数: 0
Maps to virtual braids and braid representations 映射到虚拟辫子和辫子表示
4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10087e
Vassily Olegovich Manturov, Igor Mikhailovich Nikonov
{"title":"Maps to virtual braids and braid representations","authors":"Vassily Olegovich Manturov, Igor Mikhailovich Nikonov","doi":"10.4213/rm10087e","DOIUrl":"https://doi.org/10.4213/rm10087e","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136368151","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}
引用次数: 0
On integrability of the deformed Ruijsenaars-Schneider system 变形rujsenaars - schneider系统的可积性
4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10105e
Anton Vladimirovich Zabrodin
{"title":"On integrability of the deformed Ruijsenaars-Schneider system","authors":"Anton Vladimirovich Zabrodin","doi":"10.4213/rm10105e","DOIUrl":"https://doi.org/10.4213/rm10105e","url":null,"abstract":"We find integrals of motion for the recently introduced deformed Ruijsenaars-Schneider many-body system, which is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint of type B. Our method is based on the fact that the equations of motion for this system coincide with those for pairs of Ruijsenaars-Schneider particles which stick together preserving a special fixed distance between the particles. We also obtain Bäcklund transformations and integrable time discretization of the deformed Ruijsenaars-Schneider system, which is shown to be the dynamical system for poles of elliptic solutions to the fully discrete Kadomtsev-Petviashvili equation of type B. In additon, we propose a field analogue of the deformed Ruijsenaars-Schneider system on a space-time lattice. Bibliography: 35 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136368180","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}
引用次数: 1
The number of components of the Pell-Abel equations with primitive solutions of given degree 具有给定阶元解的Pell-Abel方程的分量数
IF 0.9 4区 数学
Russian Mathematical Surveys Pub Date : 2023-01-01 DOI: 10.4213/rm10082e
A. Bogatyrev, Q. Gendron
{"title":"The number of components of the Pell-Abel equations with primitive solutions of given degree","authors":"A. Bogatyrev, Q. Gendron","doi":"10.4213/rm10082e","DOIUrl":"https://doi.org/10.4213/rm10082e","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70328908","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}
引用次数: 0
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