Pub Date : 2014-04-09DOI: 10.1090/S0077-1554-2014-00214-4
V. Ryzhikov
This paper is devoted to the disjointness property of powers of a totally ergodic bounded construction of rank 1 and some generalizations of this result. We look at applications to the problem when the Möbius function is independent of the sequence induced by a bounded construction. Interest in the subject matter of this paper is related to the following observation. Bounded constructions of rank 1, under the condition that all their nonzero powers are ergodic, have nontrivial weak limits of powers. This implies that the powers of the constructions are disjoint (in the sense of [1]) and, in view of the results in [2], this results in bounded constructions being independent of the Möbius function. Thus, the problem of disjointness of powers of transformations, which had previously been regarded by specialists as a problem within the framework of self-joining theory, has an interesting application. Sarnak’s well-known conjecture [3] states that a strictly ergodic homeomorphism S : X → X with zero topological entropy has the property
{"title":"Bounded ergodic constructions, disjointness, and weak limits of powers","authors":"V. Ryzhikov","doi":"10.1090/S0077-1554-2014-00214-4","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00214-4","url":null,"abstract":"This paper is devoted to the disjointness property of powers of a totally ergodic bounded construction of rank 1 and some generalizations of this result. We look at applications to the problem when the Möbius function is independent of the sequence induced by a bounded construction. Interest in the subject matter of this paper is related to the following observation. Bounded constructions of rank 1, under the condition that all their nonzero powers are ergodic, have nontrivial weak limits of powers. This implies that the powers of the constructions are disjoint (in the sense of [1]) and, in view of the results in [2], this results in bounded constructions being independent of the Möbius function. Thus, the problem of disjointness of powers of transformations, which had previously been regarded by specialists as a problem within the framework of self-joining theory, has an interesting application. Sarnak’s well-known conjecture [3] states that a strictly ergodic homeomorphism S : X → X with zero topological entropy has the property","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00214-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626884","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}
Pub Date : 2014-04-09DOI: 10.1090/S0077-1554-2014-00215-6
A. Bahri, M. Bendersky, F. Cohen, S. Gitler
. The main goal of this paper is to give a list of problems closely connected to moment-angle complexes, polyhedral products
． 本文的主要目的是给出一系列与矩角配合物、多面体积密切相关的问题
{"title":"On problems concerning moment-angle complexes and polyhedral products","authors":"A. Bahri, M. Bendersky, F. Cohen, S. Gitler","doi":"10.1090/S0077-1554-2014-00215-6","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00215-6","url":null,"abstract":". The main goal of this paper is to give a list of problems closely connected to moment-angle complexes, polyhedral products","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00215-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626922","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}
Pub Date : 2014-04-09DOI: 10.1090/S0077-1554-2014-00210-7
I. Dynnikov, M. Prasolov
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. It is shown that a minimal rectangular diagram maximizes the Thurston-Bennequin number for the corresponding Legendrian links. Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type is proved. A new proof of the monotonic simplification theorem for the unknot is given.
{"title":"Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions","authors":"I. Dynnikov, M. Prasolov","doi":"10.1090/S0077-1554-2014-00210-7","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00210-7","url":null,"abstract":"In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. It is shown that a minimal rectangular diagram maximizes the Thurston-Bennequin number for the corresponding Legendrian links. Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type is proved. A new proof of the monotonic simplification theorem for the unknot is given.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83584961","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}
Pub Date : 2014-04-09DOI: 10.1090/S0077-1554-2014-00213-2
M. Davletshin
In this paper some results of a work by Bolotin and Treshchëv are generalized to the case of g-periodic trajectories of Lagrangian systems. Formulae connecting the characteristic polynomial of the monodromy matrix with the determinant of the Hessian of the action functional are obtained both for the discrete and continuous cases. Applications to the problem of stability of g-periodic trajectories are given. Hill’s formula can be used to study g-periodic orbits obtained by variational methods. §
{"title":"Hill’s formula for -periodic trajectories of Lagrangian systems","authors":"M. Davletshin","doi":"10.1090/S0077-1554-2014-00213-2","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00213-2","url":null,"abstract":"In this paper some results of a work by Bolotin and Treshchëv are generalized to the case of g-periodic trajectories of Lagrangian systems. Formulae connecting the characteristic polynomial of the monodromy matrix with the determinant of the Hessian of the action functional are obtained both for the discrete and continuous cases. Applications to the problem of stability of g-periodic trajectories are given. Hill’s formula can be used to study g-periodic orbits obtained by variational methods. §","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00213-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626867","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}
Pub Date : 2014-04-09DOI: 10.1090/S0077-1554-2014-00223-5
E. Yu. Netaĭ
. We construct some differential equations describing the geometry of bundles of Jacobians of algebraic curves of genus 1 and 2. For an elliptic curve we produce differential equations on the coefficients of a cometric compatible with the Gauss–Manin connection of the universal bundle of Jacobians of elliptic curves. This cometric is defined in terms of a solution F of the linear system of differential equations
{"title":"Geometric differential equations on bundles of Jacobians of curves of genus 1 and 2","authors":"E. Yu. Netaĭ","doi":"10.1090/S0077-1554-2014-00223-5","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00223-5","url":null,"abstract":". We construct some differential equations describing the geometry of bundles of Jacobians of algebraic curves of genus 1 and 2. For an elliptic curve we produce differential equations on the coefficients of a cometric compatible with the Gauss–Manin connection of the universal bundle of Jacobians of elliptic curves. This cometric is defined in terms of a solution F of the linear system of differential equations","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00223-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627103","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}
Pub Date : 2014-04-09DOI: 10.1090/S0077-1554-2014-00217-X
E. Vinberg
Using the methods of [11], we recover the old result of J. Igusa [3] saying that the algebra of even Siegel modular forms of genus 2 is freely generated by forms of weights 4, 6, 10, 12. We also determine the structure of the algebra of all Siegel modular forms of genus 2.
{"title":"On the algebra of Siegel modular forms of genus 2","authors":"E. Vinberg","doi":"10.1090/S0077-1554-2014-00217-X","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00217-X","url":null,"abstract":"Using the methods of [11], we recover the old result of J. Igusa [3] saying that the algebra of even Siegel modular forms of genus 2 is freely generated by forms of weights 4, 6, 10, 12. We also determine the structure of the algebra of all Siegel modular forms of genus 2.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00217-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626973","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}
Pub Date : 2013-05-14DOI: 10.1090/S0077-1554-2014-00218-1
J. C. Eilbeck, K. Eilers, V. Enolski
The problem of generalisation of classical expressions for periods of second kind elliptic integrals in terms of theta-constants to higher genera is studied. In this context special class of algebraic curves – (n, s)-curves is considered. It is shown that required representations can be obtained by comparison of equivalent expressions for projective connection by Fay-Wirtinger and Klein-Weierstrass. The case of genus two hyperelliptic curve is considered as a principle example and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that the analysis undertaken for genus two curve can be extended to higher genera hyperelliptic curve as well to other classes of (n, s) non-hyperelliptic curves.
{"title":"Periods of second kind differentials of (n,s)-curves","authors":"J. C. Eilbeck, K. Eilers, V. Enolski","doi":"10.1090/S0077-1554-2014-00218-1","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00218-1","url":null,"abstract":"The problem of generalisation of classical expressions for periods of second kind elliptic integrals in terms of theta-constants to higher genera is studied. In this context special class of algebraic curves – (n, s)-curves is considered. It is shown that required representations can be obtained by comparison of equivalent expressions for projective connection by Fay-Wirtinger and Klein-Weierstrass. The case of genus two hyperelliptic curve is considered as a principle example and a number of new Thomae and Rosenhain-type formulae are obtained. We anticipate that the analysis undertaken for genus two curve can be extended to higher genera hyperelliptic curve as well to other classes of (n, s) non-hyperelliptic curves.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00218-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627036","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}
Pub Date : 2013-04-26DOI: 10.1090/S0077-1554-2014-00220-X
F. Santos, G. Ziegler
A seminal result in the theory of toric varieties, due to Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope $P$ there is a positive integer $k$ such that the dilated polytope $kP$ has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that $k=4$ works for every polytope. But this does not imply that every $k>4$ works as well. We here study the values of $k$ for which the result holds showing that: 1. It contains all composite numbers. 2. It is an additive semigroup. These two properties imply that the only values of $k$ that may not work (besides 1 and 2, which are known not to work) are $kin{3,5,7,11}$. With an ad-hoc construction we show that $k=7$ and $k=11$ also work, except in this case the triangulation cannot be guaranteed to be "standard" in the boundary. All in all, the only open cases are $k=3$ and $k=5$.
{"title":"Unimodular triangulations of dilated 3-polytopes","authors":"F. Santos, G. Ziegler","doi":"10.1090/S0077-1554-2014-00220-X","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00220-X","url":null,"abstract":"A seminal result in the theory of toric varieties, due to Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope $P$ there is a positive integer $k$ such that the dilated polytope $kP$ has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that $k=4$ works for every polytope. But this does not imply that every $k>4$ works as well. We here study the values of $k$ for which the result holds showing that: \u00001. It contains all composite numbers. \u00002. It is an additive semigroup. \u0000These two properties imply that the only values of $k$ that may not work (besides 1 and 2, which are known not to work) are $kin{3,5,7,11}$. With an ad-hoc construction we show that $k=7$ and $k=11$ also work, except in this case the triangulation cannot be guaranteed to be \"standard\" in the boundary. All in all, the only open cases are $k=3$ and $k=5$.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00220-X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627052","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}
Pub Date : 2013-04-23DOI: 10.1090/S0077-1554-2014-00216-8
Christopher Braun, A. Lazarev
We define and study the degeneration property for $ mathrm {BV}_infty $ algebras and show that it implies that the underlying $ L_{infty }$ algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity $ Delta (e^{xi })=e^{xi }Big (Delta (xi )+frac {1}{2}[xi ,xi ]Big )$ which holds in all BV algebras. As an application we show that the higher Koszul brackets on the cohomology of a manifold supplied with a generalised Poisson structure all vanish. - See more at: http://www.ams.org/journals/mosc/2013-74-00/S0077-1554-2014-00216-8/#sthash.pBIIcZKa.dpuf
{"title":"Homotopy BV algebras in Poisson geometry","authors":"Christopher Braun, A. Lazarev","doi":"10.1090/S0077-1554-2014-00216-8","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00216-8","url":null,"abstract":"We define and study the degeneration property for $ mathrm {BV}_infty $ algebras and show that it implies that the underlying $ L_{infty }$ algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity $ Delta (e^{xi })=e^{xi }Big (Delta (xi )+frac {1}{2}[xi ,xi ]Big )$ which holds in all BV algebras. As an application we show that the higher Koszul brackets on the cohomology of a manifold supplied with a generalised Poisson structure all vanish. - See more at: http://www.ams.org/journals/mosc/2013-74-00/S0077-1554-2014-00216-8/#sthash.pBIIcZKa.dpuf","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00216-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626935","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}
Pub Date : 2013-03-21DOI: 10.1090/S0077-1554-2013-00204-6
Trudy Moskov, Matem, Obw, A. Komlov, S. Suetin
. We obtain a strong asymptotic formula for the leading coefficient α n ( n ) of a degree n polynomial q n ( z ; n ) orthonormal on a system of intervals on the real line with respect to a varying weight. The weight depends on n as e − 2 nQ ( x ) , where Q ( x ) is a polynomial and corresponds to the “hard-edge case”. The formula in Theorem 1 is quite similar to Widom’s classical formula for a weight independent of n . In some sense, Widom’s formulas are still true for a varying weight and are thus universal. As a consequence of the asymptotic formula we have that α n ( n ) e − nw Q oscillates as n → ∞ and, in a typical case, fills an interval (here w Q is the equilibrium constant in the external field Q ).
． 得到了n次多项式q n (z)的导系数α n (n)的一个强渐近公式;N)在实数线上的区间系统上关于变权值的标准正交。权重取决于n为e - 2 nQ (x)，其中Q (x)是一个多项式，对应于“硬边情况”。定理1中的公式与Widom的经典公式非常相似，它与n无关。在某种意义上，Widom的公式对于不同的权重仍然是正确的，因此是通用的。作为渐近公式的结果，我们得到α n (n) e - nw Q在n→∞时振荡，并且在典型情况下，填充一个区间(这里w Q是外场Q中的平衡常数)。
{"title":"An asymptotic formula for polynomials orthonormal with respect to a varying weight","authors":"Trudy Moskov, Matem, Obw, A. Komlov, S. Suetin","doi":"10.1090/S0077-1554-2013-00204-6","DOIUrl":"https://doi.org/10.1090/S0077-1554-2013-00204-6","url":null,"abstract":". We obtain a strong asymptotic formula for the leading coefficient α n ( n ) of a degree n polynomial q n ( z ; n ) orthonormal on a system of intervals on the real line with respect to a varying weight. The weight depends on n as e − 2 nQ ( x ) , where Q ( x ) is a polynomial and corresponds to the “hard-edge case”. The formula in Theorem 1 is quite similar to Widom’s classical formula for a weight independent of n . In some sense, Widom’s formulas are still true for a varying weight and are thus universal. As a consequence of the asymptotic formula we have that α n ( n ) e − nw Q oscillates as n → ∞ and, in a typical case, fills an interval (here w Q is the equilibrium constant in the external field Q ).","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60626700","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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