Pub Date : 2024-03-11DOI: 10.26907/0021-3446-2024-2-22-36
S. Y. Graf
The article is devoted to the definition and properties of the class of diffeomorphisms ofthe unit disk D = { z : | z| < 1} on the complex plane C for which the harmonic measure of theboundary arcs of the slit disk has a limited distortion, i.e. is quasiinvariant. Estimates for derivativemappings of this class are obtained. We prove that such mappings are quasiconformal and are alsoquasiisometries with respect to the pseudohyperbolic metric. An example of a mapping with thespecified property is given. As an application, a generalization of the Hayman–Wu theorem to thisclass of mappings is proved.
文章主要研究复平面 C 上单位圆盘 D = { z : | z| < 1} 的衍射的定义和性质，对于该类衍射，狭缝圆盘边界弧的谐波量具有有限的扭曲，即准不变性。我们得到了该类导数映射的估计值。我们证明了这类映射是准共形的，也是关于伪双曲度量的准等距。我们给出了一个具有上述性质的映射实例。作为应用，我们还证明了海曼-吴（Hayman-Wu）定理对这一类映射的推广。
{"title":"On quasiinvariance of harmonic measure and Hayman-Wu theorem","authors":"S. Y. Graf","doi":"10.26907/0021-3446-2024-2-22-36","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-2-22-36","url":null,"abstract":"The article is devoted to the definition and properties of the class of diffeomorphisms ofthe unit disk D = { z : | z| < 1} on the complex plane C for which the harmonic measure of theboundary arcs of the slit disk has a limited distortion, i.e. is quasiinvariant. Estimates for derivativemappings of this class are obtained. We prove that such mappings are quasiconformal and are alsoquasiisometries with respect to the pseudohyperbolic metric. An example of a mapping with thespecified property is given. As an application, a generalization of the Hayman–Wu theorem to thisclass of mappings is proved.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"82 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140254611","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 : 2024-03-11DOI: 10.26907/0021-3446-2024-2-59-80
A. Trynin
A new method for obtaining a generalized solution of a mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition is proposed. Generalized functions are understood in the sense of a sequential approach. The representative of the class of sequences, which is a generalized function, is obtained using the function interpolation operator, constructed using solutions of the Cauchy problem. The solution is obtained in the form of a series that converges uniformly inside the domain of the solution.
{"title":"On one method for solving a mixed boundary value problem for a parabolic type equation using operators AT ƛ, J","authors":"A. Trynin","doi":"10.26907/0021-3446-2024-2-59-80","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-2-59-80","url":null,"abstract":"A new method for obtaining a generalized solution of a mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition is proposed. Generalized functions are understood in the sense of a sequential approach. The representative of the class of sequences, which is a generalized function, is obtained using the function interpolation operator, constructed using solutions of the Cauchy problem. The solution is obtained in the form of a series that converges uniformly inside the domain of the solution.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"86 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140252020","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 : 2024-03-11DOI: 10.26907/0021-3446-2024-2-81-84
S. V. Berberyan
In this article, we consider a class Ꝝ* consisting of functions, subharmonic in the unit disk and such that their compositions with some families of linear fractional automorphisms of the disk form normal families. We prove a theorem which states that for any function of class Ꝝ* the set of points of the unit circle can be represented as a union of Fatou points, generalized point Plesner, and a set of zero measure.
{"title":"On the classification of points of the unit circle for subharmonic functions of class Ꝝ *","authors":"S. V. Berberyan","doi":"10.26907/0021-3446-2024-2-81-84","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-2-81-84","url":null,"abstract":"In this article, we consider a class Ꝝ* consisting of functions, subharmonic in the unit disk and such that their compositions with some families of linear fractional automorphisms of the disk form normal families. We prove a theorem which states that for any function of class Ꝝ* the set of points of the unit circle can be represented as a union of Fatou points, generalized point Plesner, and a set of zero measure.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"60 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140252959","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 : 2024-02-13DOI: 10.26907/0021-3446-2024-1-69-76
S. Usmanov
Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. We prove the boundedness of these operators and define a critical exponent in the space of summable functions, when hypersurfaces are given by parametric equations.
{"title":"On maximal operators associated with singular hypersurfaces","authors":"S. Usmanov","doi":"10.26907/0021-3446-2024-1-69-76","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-1-69-76","url":null,"abstract":"Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. We prove the boundedness of these operators and define a critical exponent in the space of summable functions, when hypersurfaces are given by parametric equations.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"28 15","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139841090","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 : 2024-02-13DOI: 10.26907/0021-3446-2024-1-77-97
A. M. Shelekhov
We present a much simpler than in [1] solution of the Blaschke problem: Find all regular curvilinear three-webs formed by the pencils of circles.
我们提出了一个比 [1] 简单得多的布拉什克问题解决方案：找出所有由圆的铅笔构成的规则曲线三网。
{"title":"On triangulation of the plane by pencils of conics III","authors":"A. M. Shelekhov","doi":"10.26907/0021-3446-2024-1-77-97","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-1-77-97","url":null,"abstract":"We present a much simpler than in [1] solution of the Blaschke problem: Find all regular curvilinear three-webs formed by the pencils of circles.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"409 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139841645","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 : 2024-02-13DOI: 10.26907/0021-3446-2024-1-77-97
A. M. Shelekhov
We present a much simpler than in [1] solution of the Blaschke problem: Find all regular curvilinear three-webs formed by the pencils of circles.
我们提出了一个比 [1] 简单得多的布拉什克问题解决方案：找出所有由圆的铅笔构成的规则曲线三网。
{"title":"On triangulation of the plane by pencils of conics III","authors":"A. M. Shelekhov","doi":"10.26907/0021-3446-2024-1-77-97","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-1-77-97","url":null,"abstract":"We present a much simpler than in [1] solution of the Blaschke problem: Find all regular curvilinear three-webs formed by the pencils of circles.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"59 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139781565","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 : 2024-02-13DOI: 10.26907/0021-3446-2024-1-69-76
S. Usmanov
Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. We prove the boundedness of these operators and define a critical exponent in the space of summable functions, when hypersurfaces are given by parametric equations.
{"title":"On maximal operators associated with singular hypersurfaces","authors":"S. Usmanov","doi":"10.26907/0021-3446-2024-1-69-76","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-1-69-76","url":null,"abstract":"Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. We prove the boundedness of these operators and define a critical exponent in the space of summable functions, when hypersurfaces are given by parametric equations.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"98 14","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139781255","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 : 2024-02-12DOI: 10.26907/0021-3446-2024-1-35-49
E. Mazepa, D. K. Ryaboshlikova
The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on non-compact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary non-compact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.
{"title":"Symptotic behavior of solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds","authors":"E. Mazepa, D. K. Ryaboshlikova","doi":"10.26907/0021-3446-2024-1-35-49","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-1-35-49","url":null,"abstract":"The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on non-compact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary non-compact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"60 49","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139844769","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 : 2024-02-12DOI: 10.26907/0021-3446-2024-1-3-13
T. Belous, A. M. Gaisin, R. A. Gaisin
The article considers the behavior of the sum of the Dirichlet series F(s) = sum nanelambda ns, 0 < lambda n uparrow infty , which converges absolutely in the left half-plane Pi 0, on a curve arbitrarily approaching the imaginary axis — the boundary of this half-plane. We have obtained a solution to the following problem: Under what additional conditions on gamma will the strengthened asymptotic relation be valid in the case when the argument s tends to the imaginary axis along gamma over a sufficiently massive set.
文章考虑了迪里希勒数列 F(s) = sum nanelambda ns, 0 < lambda n uparrow infty 的和的行为，它在左半平面 Pi 0 中绝对收敛于任意接近虚轴--这个半平面的边界--的曲线上。我们得到了下面问题的一个解：当参数 s 在一个足够大的集合上沿着 gamma 趋向于虚轴时，在 gamma 的哪些附加条件下，加强的渐近关系将有效。
{"title":"An estimate for the sum of a Dirichlet series on an arc of bounded slope","authors":"T. Belous, A. M. Gaisin, R. A. Gaisin","doi":"10.26907/0021-3446-2024-1-3-13","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-1-3-13","url":null,"abstract":"The article considers the behavior of the sum of the Dirichlet series F(s) = sum nanelambda ns, 0 < lambda n uparrow infty , which converges absolutely in the left half-plane Pi 0, on a curve arbitrarily approaching the imaginary axis — the boundary of this half-plane. We have obtained a solution to the following problem: Under what additional conditions on gamma will the strengthened asymptotic relation be valid in the case when the argument s tends to the imaginary axis along gamma over a sufficiently massive set.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"13 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139782306","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 : 2024-02-12DOI: 10.26907/0021-3446-2024-1-50-68
S. Timergaliev
The solvability of a boundary value problem for a system, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with loose edges referred to isometric coordinates in the Timoshenko shear model and consists of five non-linear second-order partial differential equations under given non-linear boundary conditions, is studied. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of this equation is established with the help of the contraction mapping principle.
{"title":"On the problem of solvability of nonlinear boundary value problems for shallow isotropic shells of Timoshenko type in isometric coordinates","authors":"S. Timergaliev","doi":"10.26907/0021-3446-2024-1-50-68","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-1-50-68","url":null,"abstract":"The solvability of a boundary value problem for a system, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with loose edges referred to isometric coordinates in the Timoshenko shear model and consists of five non-linear second-order partial differential equations under given non-linear boundary conditions, is studied. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in Sobolev space, the solvability of this equation is established with the help of the contraction mapping principle.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"118 34","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139785238","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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