Pub Date : 2024-06-25DOI: 10.3103/s1066369x2470018x
V. B. Levenshtam
Abstract
A multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly time-oscillating right-hand side is considered on a positive time semiaxis. For this problem, which depends on a large parameter (high oscillation frequency), a limiting (averaged) multipoint boundary value problem is constructed and a limiting transition in the Hölder space of bounded vector functions defined on the considered semiaxis is justified. Thus, for normal systems of differential equations in the case of a multipoint boundary value problem, the Krylov–Bogolyubov averaging method on the semiaxis is justified.
{"title":"Averaging of a Normal System of Ordinary Differential Equations of High Frequency with a Multipoint Boundary Value Problem on a Semiaxis","authors":"V. B. Levenshtam","doi":"10.3103/s1066369x2470018x","DOIUrl":"https://doi.org/10.3103/s1066369x2470018x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly time-oscillating right-hand side is considered on a positive time semiaxis. For this problem, which depends on a large parameter (high oscillation frequency), a limiting (averaged) multipoint boundary value problem is constructed and a limiting transition in the Hölder space of bounded vector functions defined on the considered semiaxis is justified. Thus, for normal systems of differential equations in the case of a multipoint boundary value problem, the Krylov–Bogolyubov averaging method on the semiaxis is justified.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"12 7 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508830","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 : 2024-06-25DOI: 10.3103/s1066369x2470021x
T. Kh. Rasulov, D. E. Ismoilova
Abstract
A matrix model (mathcal{A}) is considered related to a system describing two identical fermions and one particle of another nature on a lattice, interacting via annihilation and creation operators. The problem of the study of the spectrum of a block operator matrix (mathcal{A}) is reduced to the investigation of the spectrum of block operator matrices of order three with a discrete variable, and the relations for the spectrum, essential spectrum, and point spectrum are established. Two-particle and three-particle branches of the essential spectrum of the block operator matrix (mathcal{A}) are singled out.
{"title":"Spectral Relations for a Matrix Model in Fermionic Fock Space","authors":"T. Kh. Rasulov, D. E. Ismoilova","doi":"10.3103/s1066369x2470021x","DOIUrl":"https://doi.org/10.3103/s1066369x2470021x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A matrix model <span>(mathcal{A})</span> is considered related to a system describing two identical fermions and one particle of another nature on a lattice, interacting via annihilation and creation operators. The problem of the study of the spectrum of a block operator matrix <span>(mathcal{A})</span> is reduced to the investigation of the spectrum of block operator matrices of order three with a discrete variable, and the relations for the spectrum, essential spectrum, and point spectrum are established. Two-particle and three-particle branches of the essential spectrum of the block operator matrix <span>(mathcal{A})</span> are singled out.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508833","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 : 2024-06-25DOI: 10.3103/s1066369x24700142
N. Yu. Agafonova, S. S. Volosivets
Abstract
We give some necessary and sufficient conditiosn for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray, and Č.V. Stanojević and extend some results of F. Móricz proved for Walsh–Fourier series.
摘要 我们给出了关于乘法系统和相应傅里叶级数的广义导数级数和收敛的一些必要条件和充分条件。这些条件与 S. Sheng、W.O. Bray 和 Č.V. Stanojević 的三角函数结果相对应，并扩展了 F. Móricz 为沃尔什-傅里叶级数证明的一些结果。
{"title":"Integrability of Series with Respect to Multiplicative Systems and Generalized Derivatives","authors":"N. Yu. Agafonova, S. S. Volosivets","doi":"10.3103/s1066369x24700142","DOIUrl":"https://doi.org/10.3103/s1066369x24700142","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We give some necessary and sufficient conditiosn for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray, and Č.V. Stanojević and extend some results of F. Móricz proved for Walsh–Fourier series.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"30 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508826","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 : 2024-06-04DOI: 10.3103/s1066369x24700099
R. Kumar, S. Kaushal, Pragati
Abstract
In the present study, we explored a new mathematical formulation involving modified couple stress thermoelastic diffusion (MCTD) with nonlocal, voids and phase lags. The governing equations are expressed in dimensionless form for the further investigating. The desired equations are expressed in terms of elementary functions by assuming time harmonic variation of the field variables (displacement, temperature field, chemical potential and volume fraction field). The fundamental solutions are constructed for the obtained system of equations for steady oscillation and some basic features of the solutions are established. Also, plane wave vibrations has been examined for two dimensional cases. The characteristic equation yields the attributes of waves like phase velocity, attenuation coefficients, specific loss and penetration depth which are computed numerically and presented in form of distinct graphs. Some unique cases are also deduced. The results provide the motivation for the researcher to investigate thermally conducted modified couple stress elastic material under nonlocal, porosity and phase lags impacts as a new class of applicable materials.
{"title":"Wave Analysis and Representation of Fundamental Solution in Modified Couple Stress Thermoelastic Diffusion with Voids, Nonlocal and Phase Lags","authors":"R. Kumar, S. Kaushal, Pragati","doi":"10.3103/s1066369x24700099","DOIUrl":"https://doi.org/10.3103/s1066369x24700099","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In the present study, we explored a new mathematical formulation involving modified couple stress thermoelastic diffusion (MCTD) with nonlocal, voids and phase lags. The governing equations are expressed in dimensionless form for the further investigating. The desired equations are expressed in terms of elementary functions by assuming time harmonic variation of the field variables (displacement, temperature field, chemical potential and volume fraction field). The fundamental solutions are constructed for the obtained system of equations for steady oscillation and some basic features of the solutions are established. Also, plane wave vibrations has been examined for two dimensional cases. The characteristic equation yields the attributes of waves like phase velocity, attenuation coefficients, specific loss and penetration depth which are computed numerically and presented in form of distinct graphs. Some unique cases are also deduced. The results provide the motivation for the researcher to investigate thermally conducted modified couple stress elastic material under nonlocal, porosity and phase lags impacts as a new class of applicable materials.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"104 7-8 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254471","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 : 2024-06-04DOI: 10.3103/s1066369x24700129
M. I. Muminov, I. N. Bozorov, T. Kh. Rasulov
Abstract
In this paper, a (2 times 2) block operator matrix (H) is considered as a bounded and self-adjoint operator in a Hilbert space. The location of the essential spectrum ({{sigma }_{{{text{ess}}}}}(H)) of operator matrix (H) is described via the spectrum of the generalized Friedrichs model, i.e., the two- and three-particle branches of the essential spectrum ({{sigma }_{{{text{ess}}}}}(H)) are singled out. We prove that the essential spectrum ({{sigma }_{{{text{ess}}}}}(H)) consists of no more than six segments (components).
Abstract In this paper, a (2 times 2) block operator matrix (H) is considered as a bounded and self-adjoint operator in a Hilbert space.通过广义弗里德里希模型的谱来描述算子矩阵 ({{sigma }_{{{text{ess}}}}}(H)) 的本质谱位置，即挑出本质谱 ({{sigma }_{{{text{ess}}}}}(H)) 的两粒子和三粒子分支。我们证明本质谱（{{sigma }_{{text{ess}}}}}(H)）由不超过六个段（成分）组成。
{"title":"On the Number of Components of the Essential Spectrum of One 2 × 2 Operator Matrix","authors":"M. I. Muminov, I. N. Bozorov, T. Kh. Rasulov","doi":"10.3103/s1066369x24700129","DOIUrl":"https://doi.org/10.3103/s1066369x24700129","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, a <span>(2 times 2)</span> block operator matrix <span>(H)</span> is considered as a bounded and self-adjoint operator in a Hilbert space. The location of the essential spectrum <span>({{sigma }_{{{text{ess}}}}}(H))</span> of operator matrix <span>(H)</span> is described via the spectrum of the generalized Friedrichs model, i.e., the two- and three-particle branches of the essential spectrum <span>({{sigma }_{{{text{ess}}}}}(H))</span> are singled out. We prove that the essential spectrum <span>({{sigma }_{{{text{ess}}}}}(H))</span> consists of no more than six segments (components).</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"43 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254602","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 : 2024-06-04DOI: 10.3103/s1066369x24700130
V. N. Paimushin, A. N. Nuriev, S. F. Chumakova
Abstract
A transformation model of the dynamic deformation of an elongated orthotropic composite rod-type plate, consisting of two sections (fastened and free) along its length, is proposed. In the free section, the orthotropic axes of the material do not coincide with the axes of the Cartesian coordinate system chosen for the plate, and in the fastened section, the displacements of points of the contact’s boundary surface (clamping) with the elastic support element are considered to be known. The constructed model is based on the use for the free section of the relations of the refined Timoshenko shear model, compiled for rods in a geometrically nonlinear approximation without taking into account lateral contraction. For the section fastened on the elastic support element, a one-dimensional shear deformation model is built taking into account lateral contraction, which is transformed into another model by satisfying the conditions of kinematic coupling with the elastic support element with given displacements of the interface points with the plate. The conditions for the kinematic coupling of the free and fastened sections of the plate are formulated. Based on the Hamilton–Ostrogradsky variational principle, the corresponding equations of motion and boundary conditions, as well as the static conditions for the matching of sections, are derived. The constructed model is intended to simulate natural processes and structures when solving applied engineering problems aimed at developing innovative oscillatory biomimetic propulsors.
{"title":"Transformation Model of the Dynamic Deformation of an Elongated Cantilever Plate Mounted on an Elastic Support Element","authors":"V. N. Paimushin, A. N. Nuriev, S. F. Chumakova","doi":"10.3103/s1066369x24700130","DOIUrl":"https://doi.org/10.3103/s1066369x24700130","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A transformation model of the dynamic deformation of an elongated orthotropic composite rod-type plate, consisting of two sections (fastened and free) along its length, is proposed. In the free section, the orthotropic axes of the material do not coincide with the axes of the Cartesian coordinate system chosen for the plate, and in the fastened section, the displacements of points of the contact’s boundary surface (clamping) with the elastic support element are considered to be known. The constructed model is based on the use for the free section of the relations of the refined Timoshenko shear model, compiled for rods in a geometrically nonlinear approximation without taking into account lateral contraction. For the section fastened on the elastic support element, a one-dimensional shear deformation model is built taking into account lateral contraction, which is transformed into another model by satisfying the conditions of kinematic coupling with the elastic support element with given displacements of the interface points with the plate. The conditions for the kinematic coupling of the free and fastened sections of the plate are formulated. Based on the Hamilton–Ostrogradsky variational principle, the corresponding equations of motion and boundary conditions, as well as the static conditions for the matching of sections, are derived. The constructed model is intended to simulate natural processes and structures when solving applied engineering problems aimed at developing innovative oscillatory biomimetic propulsors.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"70 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257244","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 : 2024-06-04DOI: 10.3103/s1066369x24700117
S. L. Berberyan
Abstract
A class (mathfrak{A}{kern 1pt} text{*}) consisting of subharmonic functions in the unit disk such that their superpositions with some families of linear fractional automorphisms of the disk form normal families is considered. A theorem stating that for any function of class (mathfrak{A}{kern 1pt} text{*}) the set of points of the unit circle can be represented as a union of a set of Fatou points, a set of generalized Plesner points, and a set of measure zero is proved.
{"title":"On the Classification of Points of a Unit Circle for Subharmonic Functions of the Class $$mathfrak{A}{kern 1pt} text{*}$$","authors":"S. L. Berberyan","doi":"10.3103/s1066369x24700117","DOIUrl":"https://doi.org/10.3103/s1066369x24700117","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A class <span>(mathfrak{A}{kern 1pt} text{*})</span> consisting of subharmonic functions in the unit disk such that their superpositions with some families of linear fractional automorphisms of the disk form normal families is considered. A theorem stating that for any function of class <span>(mathfrak{A}{kern 1pt} text{*})</span> the set of points of the unit circle can be represented as a union of a set of Fatou points, a set of generalized Plesner points, and a set of measure zero is proved.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"49 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254666","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 : 2024-06-04DOI: 10.3103/s1066369x24700105
A. Yu. Trynin
Abstract
The paper proposes a new method for obtaining a generalized solution to the mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition. Generalized functions are understood in the sense of the sequential approach. The representative of the class of sequences, which is a generalized function, is obtained using the function interpolation operator, constructed using solutions to 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 $$mathbb{A}{{mathbb{T}}_{{lambda ,j}}}$$","authors":"A. Yu. Trynin","doi":"10.3103/s1066369x24700105","DOIUrl":"https://doi.org/10.3103/s1066369x24700105","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper proposes a new method for obtaining a generalized solution to the mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition. Generalized functions are understood in the sense of the sequential approach. The representative of the class of sequences, which is a generalized function, is obtained using the function interpolation operator, constructed using solutions to the Cauchy problem. The solution is obtained in the form of a series that converges uniformly inside the domain of the solution.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254550","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 : 2024-06-04DOI: 10.3103/s1066369x24700075
O. V. Glazyrina, R. Z. Dautov, D. A. Gubaidullina
Abstract
In order to solve a parabolic variational inequality with a nonlocal spatial operator and a one-sided constraint on the solution, a numerical method based on the penalty method, finite elements, and the implicit Euler scheme is proposed and studied. Optimal estimates for the accuracy of the approximate solution in the energy norm are obtained.
{"title":"Accuracy of an Implicit Scheme for the Finite Element Method with a Penalty for a Nonlocal Parabolic Obstacle Problem","authors":"O. V. Glazyrina, R. Z. Dautov, D. A. Gubaidullina","doi":"10.3103/s1066369x24700075","DOIUrl":"https://doi.org/10.3103/s1066369x24700075","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In order to solve a parabolic variational inequality with a nonlocal spatial operator and a one-sided constraint on the solution, a numerical method based on the penalty method, finite elements, and the implicit Euler scheme is proposed and studied. Optimal estimates for the accuracy of the approximate solution in the energy norm are obtained.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254663","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 : 2024-06-04DOI: 10.3103/s1066369x24700087
S. Yu. Graf
Abstract
This study defines and describes the properties of the class of diffeomorphisms of the unit disk (mathbb{D} = { z,:;|{kern 1pt} z{kern 1pt} |; < 1} ) on the complex plane (mathbb{C}) for which the harmonic measure of the boundary arcs of the slit disk has a limited distortion (i.e., is quasi-invariant). Estimates for derivative mappings of this class are obtained. We prove that such mappings are quasiconformal and quasi-isometries with respect to the pseudohyperbolic metric. An example of a mapping with the specified property is given. As an application, a generalization of the Hayman–Wu theorem to this class of such mappings is proved.
{"title":"On Quasi-Invariance of Harmonic Measure and Hayman–Wu Theorem","authors":"S. Yu. Graf","doi":"10.3103/s1066369x24700087","DOIUrl":"https://doi.org/10.3103/s1066369x24700087","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This study defines and describes the properties of the class of diffeomorphisms of the unit disk <span>(mathbb{D} = { z,:;|{kern 1pt} z{kern 1pt} |; < 1} )</span> on the complex plane <span>(mathbb{C})</span> for which the harmonic measure of the boundary arcs of the slit disk has a limited distortion (i.e., is quasi-invariant). Estimates for derivative mappings of this class are obtained. We prove that such mappings are quasiconformal and quasi-isometries with respect to the pseudohyperbolic metric. An example of a mapping with the specified property is given. As an application, a generalization of the Hayman–Wu theorem to this class of such mappings is proved.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259724","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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