Transactions of the Moscow Mathematical Society最新文献

{"title":"Asymptotic expansions of solutions of the sixth Painlevé equation","authors":"A. Bruno, I. Goryuchkina","doi":"10.1090/S0077-1554-2010-00186-0","DOIUrl":"https://doi.org/10.1090/S0077-1554-2010-00186-0","url":null,"abstract":"We obtain all asymptotic expansions of solutions of the sixth Painlevé equation near all three singular points x = 0, x = 1, and x = ∞ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable x. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point x = 0 for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones. Introduction In 1884–1885 L. Fuchs [47] and H. Poincaré [70, 71, 72] suggested looking for differential equations whose solutions do not have movable critical points and cannot be expressed in terms of previously known functions. In 1889 S. Kovalevskaya [36] had shown that the absence of movable critical points of solutions allows one to construct solutions analytically. A singular point x = x0 of a function y(x) of a complex variable x is called a critical singular point if the value of the function y(x) changes as x moves along the path surrounding x0. A movable singular point of a solution of a differential equation is a singular point such that its position depends on initial conditions of the problem. For example for the solution y = 1/ √ x− x0, where x0 is an arbitrary constant, the point x = x0 is a movable critical point. By a meromorphic function we mean a function whose only singularities in the finite part of the complex place are poles. In 1887 E. Picard [68] suggested studying the following class of second order equations: (1) y′′ = F (x, y, y′), where the function F is rational in y and y′ and meromorphic in x, and to find, among equations (1), those that have only immovable critical singular points. At the beginning of the 20th century P. Painlevé [65, 66, 67], and his students B. Gambier [49] and R. Garnier [50, 51] solved the problem formulated by Fuchs and Picard. They have found 50 canonical equations of the form (1) whose solutions have no movable critical singular 2010 Mathematics Subject Classification. Primary 34E05; Secondary 34M55. The work was supported by the Russian Foundation of Fundamental Research (Project 08–01–00082) and the Foundation for the Assistance to Russian Science. Editorial Note: The following text incorporates changes and corrections submitted by the authors for the English translation. c ©2010 American Mathematical Society 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 A. D. BRUNO AND I. V. GORYUCHKINA points. Solutions of 44","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"71 1","pages":"1-104"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626206","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 the poles of Picard potentials","authors":"A. Komlov","doi":"10.1090/S0077-1554-2010-00182-3","DOIUrl":"https://doi.org/10.1090/S0077-1554-2010-00182-3","url":null,"abstract":"We study the existence of a global meromorphic fundamental system of solutions for a system of two differential equations Ex = (az + q(x))E, where a is a constant diagonal matrix, and q(x) is an off-diagonal meromorphic function, for each z ∈ C. Following Gesztesy and Weikard (1998), who investigated this property of functions q(x) and its connection to finite-gap solutions of soliton equations, we call such q(x) Picard potentials. We obtain conditions for the Picard property of various potentials q(x).","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"71 1","pages":"241-250"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2010-00182-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626602","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":"Excellent affine spherical homogeneous spaces of semisimple algebraic groups","authors":"R. Avdeev","doi":"10.1090/S0077-1554-2010-00183-5","DOIUrl":"https://doi.org/10.1090/S0077-1554-2010-00183-5","url":null,"abstract":"A spherical homogeneous spaceG/H of a connected semisimple algebraic group G is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group G. All the excellent affine spherical homogeneous spaces are classified up to isomorphism.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"71 1","pages":"209-240"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626142","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 the number of limit cycles of planar quadratic vector fields with a perturbed center","authors":"Trudy Moskov, Matem, Obw, A. Y. Fishkin","doi":"10.1090/S0077-1554-2010-00181-1","DOIUrl":"https://doi.org/10.1090/S0077-1554-2010-00181-1","url":null,"abstract":". We investigate the number of limit cycles of a planar quadratic vector field with a perturbed center-like singular point. An upper bound is obtained on the number of δ -good limit cycles of such a vector field (Theorem 1). Here δ is a parameter characterizing the limit cycles: it shows how far those cycles are from the singular points of the vector field and from the infinite points. The bound also includes another parameter, κ , characterizing the vector field. More precisely, κ gives an estimate on the distance from the vector field to the set consisting of quadratic vector fields with a line of singular points. Earlier, Ilyashenko and Llibre found a bound on the number of δ -good limit cycles of those vector fields which are sufficiently far from the fields with a center-like singular point. Theorem 1 and that bound complement each other and yield a new bound on the number of δ -good limit cycles of a quadratic vector field, regardless of its distance to the vector fields with a center-like singular point (Theorem 2).","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"71 1","pages":"105-139"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2010-00181-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626592","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":"Asymptotic expansion of eigenelements of the Laplace operator in a domain with a large number of ‘light’ concentrated masses sparsely situated on the boundary. Two-dimensional case","authors":"Trudy Moskov, G. Chechkin","doi":"10.1090/S0077-1554-09-00177-0","DOIUrl":"https://doi.org/10.1090/S0077-1554-09-00177-0","url":null,"abstract":"This paper looks at eigenoscillations of a membrane containing a large number of concentrated masses on the boundary. The asymptotic behaviour of the frequencies of eigenoscillations is studied when a small parameter characterizing the diameter and density of the concentrated masses tends to zero. Asymptotic expansions of eigenelements of the corresponding problems are constructed and the expansions are accurately substantiated. The case where the diameter of the masses is much smaller than the distance between them is investigated under the assumption that the limit boundary condition is still a Dirichlet condition. Introduction The behaviour of bodies with singular density has attracted the attention of scientists for many years. They have studied the influence of concentrated masses (singular lumps) on the variation of the frequencies of eigenoscillations of such bodies. This question proved to be rather too difficult for the mathematics available at the end of the 19th century. We must give due credit to a pioneering paper from the beginning of the 20th century [1]; its author investigated the problem of the oscillation of a string loaded with concentrated masses. This paper was far ahead of its time and was forgotten for some years. It was only after asymptotic methods appeared that the interest of researchers returned to problems with concentrated masses and it became possible to investigate problems with singular density adequately. The author of [2] considered the problem for the Laplace operator with Dirichlet boundary conditions in the three-dimensional case where the mass attached to the system is concentrated in an e-neighbourhood of an interior point, e being a small parameter describing the concentration and size of the mass. In that paper the methods of spectral perturbation theory were used. Another approach was proposed in [3, 4, 5, 6, 7]. As has already frequently been pointed out, a new basic parameter for oscillatory systems with locally attached masses was introduced in these papers, namely, the ratio of the attached mass to the mass of the whole system. This approach has long been firmly established in research papers. It turns out that the introduction of this parameter made it possible not only to analyse the 2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B25, 35B27, 35B40.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"70 1","pages":"71-134"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-09-00177-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626492","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 the determinant of an integral lattice generated by rational approximants of the Euler constant","authors":"A. Aptekarev, D. N. Tulyakov","doi":"10.1090/S0077-1554-09-00175-7","DOIUrl":"https://doi.org/10.1090/S0077-1554-09-00175-7","url":null,"abstract":"We investigate rational approximants of the Euler constant constructed using a certain system of jointly orthogonal polynomials and “averaging” such approximants of mediant type. The properties of such approximants are related to the properties of an integral lattice in R3 constructed from recurrently generated sequences. We also obtain estimates on the metric properties of a reduced basis, which imply that the γ-forms with coefficients constructed from basis vectors of the lattice tend to zero. The question of whether the Euler constant is irrational is reduced to a property of the bases of lattices.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"70 1","pages":"237-249"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-09-00175-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626451","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":"A global dimension theorem for quantized Banach algebras","authors":"Trudy Moskov, N. Volosova","doi":"10.1090/S0077-1554-09-00174-5","DOIUrl":"https://doi.org/10.1090/S0077-1554-09-00174-5","url":null,"abstract":". We prove that for a commutative quantized ( h ⊗ and o ⊗ ) algebra with infinite spectrum, the maximum of its left and right global homological dimensions and, as a consequence, its homological bidimension are strictly greater than one. This result is a quantum analog of the global dimension theorem of A. Ya. Helemskii.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"70 1","pages":"207-235"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-09-00174-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626441","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":"Nonlocal problems in the theory of hyperbolic differential equations","authors":"B. Paneah, P. Paneah","doi":"10.1090/S0077-1554-09-00179-4","DOIUrl":"https://doi.org/10.1090/S0077-1554-09-00179-4","url":null,"abstract":". There are relatively few local problems for general hyperbolic differential equations in a bounded domain on the plane, and all these problems are well studied, and, in simple cases, are included in almost any textbook on partial differential equations. On the contrary, nonlocal problems (even more general than boundary problems) remain practically not studied, although a number of problems of this type were successfully studied in connection with elliptic or parabolic equations. In the present paper, we consider two nonlocal quasiboundary problems of sufficiently general type in the characteristic rectangle for equations of the above type. In both cases we find conditions for unique solvability and (for the first time in the theory of hyperbolic equations) the conditions for problems to be Fredholm. Examples show that these conditions are sharp: if they are violated, the resulting problems may fail to have the required solvability properties. The proofs (in their nonanalytic part) are given in the framework of perturbation theory of operators in Banach spaces.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"70 1","pages":"135-170"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-09-00179-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626538","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":"Stochastic and deterministic characteristics of orbits in chaotically looking dynamical systems","authors":"Trudy Moskov, V. Arnold, Cn− C0X","doi":"10.1090/S0077-1554-09-00180-0","DOIUrl":"https://doi.org/10.1090/S0077-1554-09-00180-0","url":null,"abstract":",","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"36 1","pages":"31-69"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-09-00180-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626552","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":"Arens-Michael envelopes, homological epimorphisms, and relatively quasi-free algebras","authors":"A. Pirkovskii","doi":"10.1090/S0077-1554-08-00169-6","DOIUrl":"https://doi.org/10.1090/S0077-1554-08-00169-6","url":null,"abstract":"We describe and investigate Arens–Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some finitely generated algebras, we explicitly describe their Arens–Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens–Michael envelopes are homological epimorphisms (i.e., localizations in the sense of J. Taylor). For that purpose we introduce and study the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum (2 × 2)-matrices, and universal enveloping algebras of some Lie algebras of small dimensions.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"69 1","pages":"27-104"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626281","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}