Pub Date : 2018-10-22DOI: 10.17323/1609-4514-2019-19-1-37-50
M. Blank
Using ideas borrowed from topological dynamics and ergodic theory we introduce topological and metric versions of the recurrence property for general Markov chains. The main question of interest here is how large is the set of recurrent points. We show that under some mild technical assumptions the set of non recurrent points is of zero reference measure. Necessary and sufficient conditions for a reference measure $m$ (which needs not to be dynamically invariant) to satisfy this property are obtained. These results are new even in the purely deterministic setting.
{"title":"Topological and Metric Recurrence for General Markov Chains","authors":"M. Blank","doi":"10.17323/1609-4514-2019-19-1-37-50","DOIUrl":"https://doi.org/10.17323/1609-4514-2019-19-1-37-50","url":null,"abstract":"Using ideas borrowed from topological dynamics and ergodic theory we introduce topological and metric versions of the recurrence property for general Markov chains. The main question of interest here is how large is the set of recurrent points. We show that under some mild technical assumptions the set of non recurrent points is of zero reference measure. Necessary and sufficient conditions for a reference measure $m$ (which needs not to be dynamically invariant) to satisfy this property are obtained. These results are new even in the purely deterministic setting.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42350129","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}
Pub Date : 2018-10-17DOI: 10.17323/1609-4514-2020-2-277-309
A. Elagin, V. Lunts, Olaf M. Schnurer
We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.
{"title":"Smoothness of Derived Categories of Algebras","authors":"A. Elagin, V. Lunts, Olaf M. Schnurer","doi":"10.17323/1609-4514-2020-2-277-309","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-2-277-309","url":null,"abstract":"We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45020400","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}
Pub Date : 2018-10-16DOI: 10.17323/1609-4514-2019-19-2-275-305
A. Ferreira, J. Rebelo, H. Reis
Starting from some remarkable singularities of holomorphic vector fields, we construct (open) complex surfaces over which the singularities in question are realized by complete vector fields. Our constructions lead to manifolds and vector fields beyond the algebraic setting and provide examples of complete vector fields with some new dynamical phenomena.
{"title":"Palais Leaf-Space Manifolds and Surfaces Carrying Holomorphic Flows","authors":"A. Ferreira, J. Rebelo, H. Reis","doi":"10.17323/1609-4514-2019-19-2-275-305","DOIUrl":"https://doi.org/10.17323/1609-4514-2019-19-2-275-305","url":null,"abstract":"Starting from some remarkable singularities of holomorphic vector fields, we construct (open) complex surfaces over which the singularities in question are realized by complete vector fields. Our constructions lead to manifolds and vector fields beyond the algebraic setting and provide examples of complete vector fields with some new dynamical phenomena.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48846880","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}
Pub Date : 2018-09-25DOI: 10.17323/1609-4514-2019-19-2-307-327
Guan Huang, V. Kaloshin
In this paper, we show that any smooth one-parameter deformations of a strictly convex integrable billiard table $Omega_0$ preserving the integrability near the boundary have to be tangent to a finite dimensional space passing through $Omega_0$.
{"title":"On the Finite Dimensionality of Integrable Deformations of Strictly Convex Integrable Billiard Tables","authors":"Guan Huang, V. Kaloshin","doi":"10.17323/1609-4514-2019-19-2-307-327","DOIUrl":"https://doi.org/10.17323/1609-4514-2019-19-2-307-327","url":null,"abstract":"In this paper, we show that any smooth one-parameter deformations of a strictly convex integrable billiard table $Omega_0$ preserving the integrability near the boundary have to be tangent to a finite dimensional space passing through $Omega_0$.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45466716","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}
Pub Date : 2018-09-20DOI: 10.17323/1609-4514-2020-2-343-374
Richard Griffon
Let $mathbb{F}_q$ be a finite field of odd characteristic and $K= mathbb{F}_q(t)$. For any integer $dgeq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the Mordell--Weil group $E_d(K)$ is unbounded as $d$ varies. The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order of vanishing of its $L$-function at the central point. We provide an explicit expression for the $L$-function of $E_d$, and use it to study this order of vanishing in terms of $d$.
{"title":"A New Family of Elliptic Curves with Unbounded Rank","authors":"Richard Griffon","doi":"10.17323/1609-4514-2020-2-343-374","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-2-343-374","url":null,"abstract":"Let $mathbb{F}_q$ be a finite field of odd characteristic and $K= mathbb{F}_q(t)$. For any integer $dgeq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show that the rank of the Mordell--Weil group $E_d(K)$ is unbounded as $d$ varies. The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order of vanishing of its $L$-function at the central point. We provide an explicit expression for the $L$-function of $E_d$, and use it to study this order of vanishing in terms of $d$.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49356567","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}
Pub Date : 2018-09-19DOI: 10.17323/1609-4514-2022-22-2-295-372
J. East, N. Ruškuc
We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate any congruence; when this number is infinite, it depends on the cofinality of certain limit cardinals.
{"title":"Congruences on Infinite Partition and Partial Brauer Monoids","authors":"J. East, N. Ruškuc","doi":"10.17323/1609-4514-2022-22-2-295-372","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-2-295-372","url":null,"abstract":"We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate any congruence; when this number is infinite, it depends on the cofinality of certain limit cardinals.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42773258","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}
Pub Date : 2018-08-22DOI: 10.17323/1609-4514-2020-2-323-341
N. Goncharuk, Yury Kudryashov
"Tears of the heart" is a hyperbolic polycycle formed by three separatrix connections of two saddles. It is met in generic 3-parameter families of planar vector fields. In [arXiv:1506.06797], it was discovered that generically, the bifurcation of a vector field with "tears of the heart" is structurally unstable. The authors proved that the classification of such bifurcations has a numerical invariant. In this article, we study the bifurcations of "tears of the heart" in more detail, and find out that the classification of such bifurcation may have arbitrarily many numerical invariants.
{"title":"Bifurcations of the Polycycle “Tears of the Heart”: Multiple Numerical Invariants","authors":"N. Goncharuk, Yury Kudryashov","doi":"10.17323/1609-4514-2020-2-323-341","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-2-323-341","url":null,"abstract":"\"Tears of the heart\" is a hyperbolic polycycle formed by three separatrix connections of two saddles. It is met in generic 3-parameter families of planar vector fields. \u0000In [arXiv:1506.06797], it was discovered that generically, the bifurcation of a vector field with \"tears of the heart\" is structurally unstable. The authors proved that the classification of such bifurcations has a numerical invariant. \u0000In this article, we study the bifurcations of \"tears of the heart\" in more detail, and find out that the classification of such bifurcation may have arbitrarily many numerical invariants.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49333772","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}
Pub Date : 2018-08-20DOI: 10.17323/1609-4514-2019-19-2-329-341
A. Khovanskii
According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of this notes is dedicated to a second step, which deals with pure transcendent extensions and their Galois groups which are connected Lie groups. The idea of the proof goes back to J.Liouville and J.Ritt.
{"title":"Integrability in Finite Terms and Actions of Lie Groups","authors":"A. Khovanskii","doi":"10.17323/1609-4514-2019-19-2-329-341","DOIUrl":"https://doi.org/10.17323/1609-4514-2019-19-2-329-341","url":null,"abstract":"According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of this notes is dedicated to a second step, which deals with pure transcendent extensions and their Galois groups which are connected Lie groups. The idea of the proof goes back to J.Liouville and J.Ritt.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43871964","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}
Pub Date : 2018-08-19DOI: 10.17323/1609-4514-2020-20-4-645-694
Cesar Cuenca, G. Olshanski
The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables. We construct an analogue of the multivariable q-Racah polynomials in the algebra of symmetric functions. Next, we show that our q-Racah symmetric functions can be degenerated into the big q-Jacobi symmetric functions, introduced in a recent paper by the second author. The latter symmetric functions admit further degenerations leading to new symmetric functions, which are analogues of q-Meixner and Al-Salam--Carlitz polynomials. Each of the four families of symmetric functions (q-Racah, big q-Jacobi, q-Meixner, and Al-Salam--Carlitz) forms an orthogonal system of functions with respect to certain measure living on a space of infinite point configurations. The orthogonality measures of the four families are of independent interest. We show that they are linked by limit transitions which are consistent with the degenerations of the corresponding symmetric functions.
{"title":"Elements of the q-Askey Scheme in the Algebra of Symmetric Functions","authors":"Cesar Cuenca, G. Olshanski","doi":"10.17323/1609-4514-2020-20-4-645-694","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-20-4-645-694","url":null,"abstract":"The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables. \u0000We construct an analogue of the multivariable q-Racah polynomials in the algebra of symmetric functions. Next, we show that our q-Racah symmetric functions can be degenerated into the big q-Jacobi symmetric functions, introduced in a recent paper by the second author. The latter symmetric functions admit further degenerations leading to new symmetric functions, which are analogues of q-Meixner and Al-Salam--Carlitz polynomials. \u0000Each of the four families of symmetric functions (q-Racah, big q-Jacobi, q-Meixner, and Al-Salam--Carlitz) forms an orthogonal system of functions with respect to certain measure living on a space of infinite point configurations. The orthogonality measures of the four families are of independent interest. We show that they are linked by limit transitions which are consistent with the degenerations of the corresponding symmetric functions.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49459944","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}
Pub Date : 2018-08-05DOI: 10.17323/1609-4514-2022-22-1-103-120
H. Movasati
We give an example of a one dimensional foliation F of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves X0(d), d ∈ N minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief has been that such formulas do not exist and the emphasis in the literature has been on introducing faster algorithms to compute equations for small values of d.
{"title":"On Elliptic Modular Foliations, II","authors":"H. Movasati","doi":"10.17323/1609-4514-2022-22-1-103-120","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-1-103-120","url":null,"abstract":"We give an example of a one dimensional foliation F of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves X0(d), d ∈ N minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief has been that such formulas do not exist and the emphasis in the literature has been on introducing faster algorithms to compute equations for small values of d.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48691151","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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