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Theorems on Direct and Inverse Approximation by Algebraic Polynomials and Piecewise Polynomials in the Spaces $${{H}^{m}}(a,b)$$ and $$B_{{2,q}}^{s}(a,b)$$ 关于在空间 $${{H}^{m}}(a,b)$$ 和 $$B_{2,q}}^{s}(a,b)$$ 中用代数多项式和分段多项式直接逼近和反逼近的定理
IF 0.4
Russian Mathematics Pub Date : 2024-05-09 DOI: 10.3103/s1066369x24700026
R. Z. Dautov
{"title":"Theorems on Direct and Inverse Approximation by Algebraic Polynomials and Piecewise Polynomials in the Spaces $${{H}^{m}}(a,b)$$ and $$B_{{2,q}}^{s}(a,b)$$","authors":"R. Z. Dautov","doi":"10.3103/s1066369x24700026","DOIUrl":"https://doi.org/10.3103/s1066369x24700026","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926926","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}
引用次数: 0
On Maximal Operators Associated with Singular Hypersurfaces 论与奇异超曲面相关的最大算子
IF 0.4
Russian Mathematics Pub Date : 2024-05-09 DOI: 10.3103/s1066369x24700051
S. E. Usmanov
{"title":"On Maximal Operators Associated with Singular Hypersurfaces","authors":"S. E. Usmanov","doi":"10.3103/s1066369x24700051","DOIUrl":"https://doi.org/10.3103/s1066369x24700051","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Maximal operators associated with singular hypersurfaces in multidimensional Euclidean spaces are considered. These operators have been proven to be bounded, and an exponent of boundedness in the space of integrable functions has been found for the case when hypersurfaces are given by parametric equations.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"126 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926925","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}
引用次数: 0
An Estimate for the Sum of a Dirichlet Series on an Arc of Bounded Slope 有界斜率弧线上狄利克特数列之和的估计值
IF 0.4
Russian Mathematics Pub Date : 2024-05-09 DOI: 10.3103/s1066369x24700014
T. I. Belous, A. M. Gaisin, R. A. Gaisin
{"title":"An Estimate for the Sum of a Dirichlet Series on an Arc of Bounded Slope","authors":"T. I. Belous, A. M. Gaisin, R. A. Gaisin","doi":"10.3103/s1066369x24700014","DOIUrl":"https://doi.org/10.3103/s1066369x24700014","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The article considers the behavior of the sum of the Dirichlet series <span>(F(s) = sumlimits_n {kern 1pt} {{a}_{n}}{{e}^{{{{lambda }_{n}}s}}},)</span> <span>(0 &lt; {{lambda }_{n}} uparrow infty ,)</span> which converges absolutely in the left half-plane <span>({{Pi }_{0}})</span>, 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 <span>(gamma )</span> will the strengthened asymptotic relation the type of Pólya for the sum <i>F</i>(<i>s</i>) of the Dirichlet series be valid in the case when the argument <span>(s)</span> tends to the imaginary axis along <span>(gamma )</span> over a sufficiently massive set.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926908","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}
引用次数: 0
Asymptotic Behavior of Solutions of the Inhomogeneous Schrödinger Equation on Noncompact Riemannian Manifolds 非紧密黎曼曼体上非均质薛定谔方程解的渐近行为
IF 0.4
Russian Mathematics Pub Date : 2024-05-09 DOI: 10.3103/s1066369x24700038
E. A. Mazepa, D. K. Ryaboshlykova
{"title":"Asymptotic Behavior of Solutions of the Inhomogeneous Schrödinger Equation on Noncompact Riemannian Manifolds","authors":"E. A. Mazepa, D. K. Ryaboshlykova","doi":"10.3103/s1066369x24700038","DOIUrl":"https://doi.org/10.3103/s1066369x24700038","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on noncompact 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 noncompact 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.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"28 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926927","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}
引用次数: 0
On Triangulation of the Plane by Pencils of Conics III 论用圆锥曲线的铅笔对平面进行三角测量 III
IF 0.4
Russian Mathematics Pub Date : 2024-05-09 DOI: 10.3103/s1066369x24700063
A. M. Shelekhov
{"title":"On Triangulation of the Plane by Pencils of Conics III","authors":"A. M. Shelekhov","doi":"10.3103/s1066369x24700063","DOIUrl":"https://doi.org/10.3103/s1066369x24700063","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We present a solution of the Blaschke problem much simpler than in [1]: find all regular curvilinear three-webs formed by the pencils of circles.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"64 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926852","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}
引用次数: 0
On the Problem of Solvability of Nonlinear Boundary Value Problems for Shallow Isotropic Shells of Timoshenko Type in Isometric Coordinates 关于等距坐标下各向同性浅壳的非线性边界问题的可解性问题
IF 0.4
Russian Mathematics Pub Date : 2024-05-09 DOI: 10.3103/s1066369x2470004x
S. N. Timergaliev
{"title":"On the Problem of Solvability of Nonlinear Boundary Value Problems for Shallow Isotropic Shells of Timoshenko Type in Isometric Coordinates","authors":"S. N. Timergaliev","doi":"10.3103/s1066369x2470004x","DOIUrl":"https://doi.org/10.3103/s1066369x2470004x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates, is studied. The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in a Sobolev space, with the solvability of this equation being established using the contraction mapping principle.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"32 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926909","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}
引用次数: 0
Main Properties of the Faddeev Equation for 2 × 2 Operator Matrices 2 × 2 算子矩阵的法德夫方程的主要性质
IF 0.4
Russian Mathematics Pub Date : 2024-02-28 DOI: 10.3103/s1066369x2312006x
T. H. Rasulov, E. B. Dilmurodov
{"title":"Main Properties of the Faddeev Equation for 2 × 2 Operator Matrices","authors":"T. H. Rasulov, E. B. Dilmurodov","doi":"10.3103/s1066369x2312006x","DOIUrl":"https://doi.org/10.3103/s1066369x2312006x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper we consider a <span>(2 times 2)</span> operator matrix <span>(H)</span>. We construct an analog of the well-known Faddeev equation for the eigenvectors of <span>(H)</span> and study some important properties of this equation, related with the number of eigenvalues. In particular, the Birman–Schwinger principle for <span>(H)</span> is proven.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"82 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140003650","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}
引用次数: 0
A Problem in an Unbounded Domain with Combined Tricomi and Frankl Conditions on One Boundary Characteristic for One Class of Mixed-Type Equations 一类混合型方程的一个边界特征上包含特里科米条件和弗兰克尔条件的无界域中的问题
IF 0.4
Russian Mathematics Pub Date : 2024-02-28 DOI: 10.3103/s1066369x23120058
M. Mirsaburov, R. N. Turaev
{"title":"A Problem in an Unbounded Domain with Combined Tricomi and Frankl Conditions on One Boundary Characteristic for One Class of Mixed-Type Equations","authors":"M. Mirsaburov, R. N. Turaev","doi":"10.3103/s1066369x23120058","DOIUrl":"https://doi.org/10.3103/s1066369x23120058","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this work, in an unbounded domain, we prove the correctness of the problem with combined Tricomi and Frankl conditions on one boundary characteristic for one class of mixed-type equations.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"42 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140003361","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}
引用次数: 0
Generalized Integration over Nonrectifiable Flat Curves and Boundary Value Problems 不可修正平曲线上的广义积分和边界值问题
IF 0.4
Russian Mathematics Pub Date : 2024-02-28 DOI: 10.3103/s1066369x23120046
D. B. Katz
{"title":"Generalized Integration over Nonrectifiable Flat Curves and Boundary Value Problems","authors":"D. B. Katz","doi":"10.3103/s1066369x23120046","DOIUrl":"https://doi.org/10.3103/s1066369x23120046","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Two closely related problems are discussed, viz., solving the Riemann boundary value problem for analytic functions and some of their generalizations in the domains of the complex plane with nonrectifiable boundaries and constructing a generalization of the curvilinear integral onto nonrectifiable curves that preserves the properties important for the complex analysis. This review reflects the current state of the topic, with many of the results being quite recent. At the end of the work, a number of unsolved problems are given, each of which can serve as a starting point for scientific research.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140003288","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}
引用次数: 0
Convolution Kernel Determination Problem in the Third Order Moore–Gibson–Thompson Equation 三阶摩尔-吉布森-汤普森方程中的卷积核确定问题
IF 0.4
Russian Mathematics Pub Date : 2024-02-28 DOI: 10.3103/s1066369x23120034
D. K. Durdiev, A. A. Boltaev, A. A. Rahmonov
{"title":"Convolution Kernel Determination Problem in the Third Order Moore–Gibson–Thompson Equation","authors":"D. K. Durdiev, A. A. Boltaev, A. A. Rahmonov","doi":"10.3103/s1066369x23120034","DOIUrl":"https://doi.org/10.3103/s1066369x23120034","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This article is concerned with the study of the inverse problem of determining the difference kernel in a Volterra type integral term function in the third-order Moore–Gibson–Thompson (MGT) equation. First, the initial-boundary value problem is reduced to an equivalent problem. Using the Fourier spectral method, the equivalent problem is reduced to a system of integral equations. The existence and uniqueness of the solution to the integral equations are proved. The obtained solution to the integral equations of Volterra-type is also the unique solution to the equivalent problem. Based on the equivalence of the problems, the theorem of the existence and uniqueness of the classical solutions of the original inverse problem is proved.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"82 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140003371","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}
引用次数: 0
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