We show that the difference between the Green energy of a discrete signed measure relative to a circular annulus concentrated at some points on concentric circles and the energy of the signed measure at symmetric points is non-decreasing during the expansion of the annulus. As a corollary, generalizations of the classical Pólya-Schur inequality for complex numbers are obtained. Some open problems are formulated.
{"title":"Green energy of discrete signed measure on concentric circles","authors":"V. Dubinin","doi":"10.4213/im9343e","DOIUrl":"https://doi.org/10.4213/im9343e","url":null,"abstract":"We show that the difference between the Green energy of a discrete signed measure relative to a circular annulus concentrated at some points on concentric circles and the energy of the signed measure at symmetric points is non-decreasing during the expansion of the annulus. As a corollary, generalizations of the classical Pólya-Schur inequality for complex numbers are obtained. Some open problems are formulated.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70327211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a wide class of integer-valued random walks, we obtain exact expressions for the distribution of the first excess over level and the corresponding renewal function as well as for the distribution of the trajectory supremum if it is finite. We discuss possibilities of obtaining explicit expressions for pre-stationary and stationary distributions of a random walk with switchings at the strip boundaries. The research is based on the factorization representations for the double moment generating functions of the distributions under study.
{"title":"Exact formulas in some boundary crossing problems\u0000for integer-valued random walks","authors":"V. I. Lotov","doi":"10.4213/im9323e","DOIUrl":"https://doi.org/10.4213/im9323e","url":null,"abstract":"For a wide class of integer-valued random walks, we obtain exact expressions for the distribution of the first excess over level and the corresponding renewal function as well as for the distribution of the trajectory supremum if it is finite. We discuss possibilities of obtaining explicit expressions for pre-stationary and stationary distributions of a random walk with switchings at the strip boundaries. The research is based on the factorization representations for the double moment generating functions of the distributions under study.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70327201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An answer is given to the question of M. Lohrey and B. Steinberg on decidability of the submonoid membership problem for a finitely generated nilpotent group. Namely, a finitely generated submonoid of a free nilpotent group of class $2$ of sufficiently large rank $r$ is constructed, for which the membership problem is algorithmically undecidable. This implies the existence of a submonoid with similar property in any free nilpotent group of class $l geqslant 2$ of rank $r$. The proof is based on the undecidability of Hilbert's tenth problem.
{"title":"Undecidability of the submonoid membership problem for free nilpotent group of class $lgeqslant 2$ of sufficiently large rank","authors":"Vitalii Anatol'evich Roman'kov","doi":"10.4213/im9342e","DOIUrl":"https://doi.org/10.4213/im9342e","url":null,"abstract":"An answer is given to the question of M. Lohrey and B. Steinberg on decidability of the submonoid membership problem for a finitely generated nilpotent group. Namely, a finitely generated submonoid of a free nilpotent group of class $2$ of sufficiently large rank $r$ is constructed, for which the membership problem is algorithmically undecidable. This implies the existence of a submonoid with similar property in any free nilpotent group of class $l geqslant 2$ of rank $r$. The proof is based on the undecidability of Hilbert's tenth problem.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","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":"135009167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study general settings of the Dirichlet problem, the Neumann problem, and other boundary value problems for equations and systems of the form $mathcal{L}^+ Amathcal{L}u=f$ with general (matrix, generally speaking) differential operation $mathcal{L}$ and some linear or non-linear operator $A$ acting in $L^k_2(Omega)$-spaces. For these boundary value problems, results on well-posedness, existence and uniqueness of a weak solution are obtained. As an operator $A$, we consider Nemytskii and integral operators. The case of operators involving lower-order derivatives is also studied.
{"title":"On weak solutions of boundary value problems for some general differential equations","authors":"Vladimir Petrovich Burskii","doi":"10.4213/im9403e","DOIUrl":"https://doi.org/10.4213/im9403e","url":null,"abstract":"We study general settings of the Dirichlet problem, the Neumann problem, and other boundary value problems for equations and systems of the form $mathcal{L}^+ Amathcal{L}u=f$ with general (matrix, generally speaking) differential operation $mathcal{L}$ and some linear or non-linear operator $A$ acting in $L^k_2(Omega)$-spaces. For these boundary value problems, results on well-posedness, existence and uniqueness of a weak solution are obtained. As an operator $A$, we consider Nemytskii and integral operators. The case of operators involving lower-order derivatives is also studied.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","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":"135661297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we obtain an equivalent relation between the log-plurisubharmonicity of the relative Bergman kernel, the Griffiths and Nakano positivity for the direct image with the natural $L^2$ metric, by finding a converse of Berndtsson's theorem on the direct image. A converse of Berndtsson's generalization of Kiselman minimal principle is also obtained.
{"title":"On the positivity of direct image bundles","authors":"Zhi Li, Xiangyu Zhou","doi":"10.4213/im9336e","DOIUrl":"https://doi.org/10.4213/im9336e","url":null,"abstract":"In the present paper, we obtain an equivalent relation between the log-plurisubharmonicity of the relative Bergman kernel, the Griffiths and Nakano positivity for the direct image with the natural $L^2$ metric, by finding a converse of Berndtsson's theorem on the direct image. A converse of Berndtsson's generalization of Kiselman minimal principle is also obtained.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","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":"135661674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work continues investigations of special Bohr-Sommerfeld geometry for compact symplectic manifolds. By using natural deformation parameters we circumvent the difficulties involved in the definition of moduli spaces of special Bohr-Sommerfeld cycles for compact simply connected algebraic varieties. As a byproduct, we present some ideas of how our constructions can be exploited in the studies of Weinstein structures and Eliashberg conjectures.
{"title":"Special Bohr-Sommerfeld geometry: variations","authors":"Nikolai Andreevich Tyurin","doi":"10.4213/im9374e","DOIUrl":"https://doi.org/10.4213/im9374e","url":null,"abstract":"This work continues investigations of special Bohr-Sommerfeld geometry for compact symplectic manifolds. By using natural deformation parameters we circumvent the difficulties involved in the definition of moduli spaces of special Bohr-Sommerfeld cycles for compact simply connected algebraic varieties. As a byproduct, we present some ideas of how our constructions can be exploited in the studies of Weinstein structures and Eliashberg conjectures.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","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":"135440255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a presentation of the fundamental group of the complement of a curve $C$ in its "tubular" neighbourhood in a normal surface $S$. The presentation is given in terms of the double weighted dual graph of the resolution of singularities of $C$ (and $S$). This result generalizes the presentation of the fundamental group of the complement of a normal singularity in its neighbourhood given by Mumford in the case, where the dual graph of the resolution is a tree and all exceptional curves of the resolution are rational curves.
{"title":"On the local fundamental group of the complement of a curve in a normal surface","authors":"Victor Stepanovich Kulikov","doi":"10.4213/im9357e","DOIUrl":"https://doi.org/10.4213/im9357e","url":null,"abstract":"We give a presentation of the fundamental group of the complement of a curve $C$ in its \"tubular\" neighbourhood in a normal surface $S$. The presentation is given in terms of the double weighted dual graph of the resolution of singularities of $C$ (and $S$). This result generalizes the presentation of the fundamental group of the complement of a normal singularity in its neighbourhood given by Mumford in the case, where the dual graph of the resolution is a tree and all exceptional curves of the resolution are rational curves.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","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":"135440537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Oleg Nikolaevich German, Nikolai Germanovich Moshchevitin
We consider the problem of simultaneous approximation of real numbers $theta_1, …,theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+theta_1x_1+…+theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German's inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and "one half" of the inequalities by Marnat and Moshchevitin. Moreover, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.
{"title":"On the transference principle and Nesterenko's linear independence criterion","authors":"Oleg Nikolaevich German, Nikolai Germanovich Moshchevitin","doi":"10.4213/im9285e","DOIUrl":"https://doi.org/10.4213/im9285e","url":null,"abstract":"We consider the problem of simultaneous approximation of real numbers $theta_1, …,theta_n$ by rationals and the dual problem of approximating zero by the values of the linear form $x_0+theta_1x_1+…+theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation which proves their result. We also derive several previously unknown corollaries. In particular, we show that, together with German's inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and \"one half\" of the inequalities by Marnat and Moshchevitin. Moreover, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","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":"136079304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider systems with toric configuration space and kinetic energy in the form of a "flat" Riemannian metric on the torus. The potential energy $V$ is a smooth function on the configuration torus. The dynamics of such systems is described by "natural" Hamiltonian systems of differential equations. If $V$ is replaced by $varepsilon V$, where $varepsilon$ is a small parameter, then the study of such Hamiltonian systems for small $varepsilon$ is a part of the "main problem of dynamics" according to Poincaré. We discuss the well-known conjecture on the existence of single-valued momentum-polynomial integrals of motion equations: if there is a momentum-polynomial integral of degree $m$, then there exist a momentum-linear or momentum-quadratic integral. This conjecture was verified in full generality for $m=3$ and $m=4$. We study the cases of "higher" degrees $m=5$ and $m=6$. Similarly to the theory of perturbations of Hamiltonian systems, we introduce resonance lines on the momentum plane. If a system admits a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonance lines are found, from which, in particular, necessary conditions for integrability are derived. Some new criteria for the existence of single-valued polynomial integrals are obtained.
{"title":"Discrete symmetries of equations of dynamics with polynomial integrals of higher degrees","authors":"Valery Vasil'evich Kozlov","doi":"10.4213/im9378e","DOIUrl":"https://doi.org/10.4213/im9378e","url":null,"abstract":"We consider systems with toric configuration space and kinetic energy in the form of a \"flat\" Riemannian metric on the torus. The potential energy $V$ is a smooth function on the configuration torus. The dynamics of such systems is described by \"natural\" Hamiltonian systems of differential equations. If $V$ is replaced by $varepsilon V$, where $varepsilon$ is a small parameter, then the study of such Hamiltonian systems for small $varepsilon$ is a part of the \"main problem of dynamics\" according to Poincaré. We discuss the well-known conjecture on the existence of single-valued momentum-polynomial integrals of motion equations: if there is a momentum-polynomial integral of degree $m$, then there exist a momentum-linear or momentum-quadratic integral. This conjecture was verified in full generality for $m=3$ and $m=4$. We study the cases of \"higher\" degrees $m=5$ and $m=6$. Similarly to the theory of perturbations of Hamiltonian systems, we introduce resonance lines on the momentum plane. If a system admits a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonance lines are found, from which, in particular, necessary conditions for integrability are derived. Some new criteria for the existence of single-valued polynomial integrals are obtained.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","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":"135661673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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