Pub Date : 2024-01-24DOI: 10.1134/s1063454123040209
A. S. Kuleshov, I. I. Ulyatovskaya
Abstract
In 1986, Ya.V. Tatarinov presented the basis of the theory of weakly nonholonomic systems. Mechanical systems with nonholonomic constraints depending on a small parameter are considered. It is assumed that when the value of this parameter is zero, the constraints of such a system become integrable; i.e., in this case, we have a family of holonomic systems depending on several arbitrary integration constants. We will assume that these holonomic systems are completely integrable Hamiltonian systems. When the small parameter is not zero, the behavior of such systems can be considered with the help of asymptotic methods representing their motion as a combination of the motion of a slightly modified holonomic system with slowly varying previous integration constants (the transgression effect). In this paper, we describe the transgression effect in the problem of motion of an almost holonomic pendulum.
{"title":"The Transgression Effect in the Problem of Motion of an Almost Holonomic Pendulum","authors":"A. S. Kuleshov, I. I. Ulyatovskaya","doi":"10.1134/s1063454123040209","DOIUrl":"https://doi.org/10.1134/s1063454123040209","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In 1986, Ya.V. Tatarinov presented the basis of the theory of weakly nonholonomic systems. Mechanical systems with nonholonomic constraints depending on a small parameter are considered. It is assumed that when the value of this parameter is zero, the constraints of such a system become integrable; i.e., in this case, we have a family of holonomic systems depending on several arbitrary integration constants. We will assume that these holonomic systems are completely integrable Hamiltonian systems. When the small parameter is not zero, the behavior of such systems can be considered with the help of asymptotic methods representing their motion as a combination of the motion of a slightly modified holonomic system with slowly varying previous integration constants (the transgression effect). In this paper, we describe the transgression effect in the problem of motion of an almost holonomic pendulum.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557927","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 : 2024-01-24DOI: 10.1134/s106345412304012x
V. M. Nezhinskij
Abstract
By a diagram we mean a topological space obtained by gluing to a standard circle a finite number of pairwise non-intersecting closed rectangles along their lateral sides, the glued rectangles being pairwise disjoint. Diagrams are not new objects; they have been used in many areas of low-dimensional topology. Our main goal is to develop the theory of diagrams to a level sufficient for application in yet another branch: the theory of tangles. We provide diagrams with simple additional structures: the smoothness of the circles and rectangles that are pairwise consistent with each other, the orientation of the circle, and a point on the circle. We introduce a new equivalence relation (as far as the author knows, not previously encountered in the scientific literature): kindred relation. We define a surjective mapping of the set of classes of kindred diagrams onto the set of classes of diffeomorphic smooth compact connected two-dimensional manifolds with a boundary and note that in the simplest cases this surjection is also a bijection. The application of the constructed theory to the tangle theory requires additional preparation and therefore is not included in this article; the author intends to devote a separate publication to this application.
{"title":"Kindred Diagrams","authors":"V. M. Nezhinskij","doi":"10.1134/s106345412304012x","DOIUrl":"https://doi.org/10.1134/s106345412304012x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>By a diagram we mean a topological space obtained by gluing to a standard circle a finite number of pairwise non-intersecting closed rectangles along their lateral sides, the glued rectangles being pairwise disjoint. Diagrams are not new objects; they have been used in many areas of low-dimensional topology. Our main goal is to develop the theory of diagrams to a level sufficient for application in yet another branch: the theory of tangles. We provide diagrams with simple additional structures: the smoothness of the circles and rectangles that are pairwise consistent with each other, the orientation of the circle, and a point on the circle. We introduce a new equivalence relation (as far as the author knows, not previously encountered in the scientific literature): kindred relation. We define a surjective mapping of the set of classes of kindred diagrams onto the set of classes of diffeomorphic smooth compact connected two-dimensional manifolds with a boundary and note that in the simplest cases this surjection is also a bijection. The application of the constructed theory to the tangle theory requires additional preparation and therefore is not included in this article; the author intends to devote a separate publication to this application.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557920","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 : 2024-01-24DOI: 10.1134/s1063454123040210
D. Yu. Volkov, K. V. Galunova
Abstract
The paper is devoted to the classical problem of analytical geometry in n-dimensional Euclidean space, namely, finding the canonical equation of a quadric from an initial equation. The canonical equation is determined by the invariants of the second-order surface equation, i.e., by quantities that do not change when the space coordinates are changed affinely. S.L. Pevzner found a convenient system containing the following invariants: q, the rank of the extended matrix of the system for determining the center of symmetry of the surface; the roots of the characteristic polynomial of the quadratic term matrix of the surface equation, i.e., the eigenvalues of this matrix; and Kq, the coefficient of the variable λ to the power n – q in the polynomial that is equal to the determinant of the matrix of order n + 1 that is obtained according to a certain rule from the initial surface equation. The eigenvalues of the matrix of the quadratic terms and coefficient Kq make it possible to write the canonical equation of the surface. In the paper, we propose a new simple proof of Pevzner’s result. In the proof, only elementary properties of the determinants are used. This algorithm for finding the canonical surface equation can find application in computer graphics.
摘要 本文致力于 n 维欧几里得空间解析几何的经典问题，即从初始方程求二次曲面的规范方程。典型方程由二阶曲面方程的不变量决定，即由空间坐标仿射变换时不变的量决定。佩夫兹纳（S.L. Pevzner）发现了一个包含以下不变式的便捷系统：q，用于确定曲面对称中心的系统扩展矩阵的秩；曲面方程二次项矩阵的特征多项式的根，即、该矩阵的特征值；以及 Kq，变量 λ 在多项式中的幂 n - q 的系数，该系数等于从初始曲面方程根据一定规则得到的 n + 1 阶矩阵的行列式。根据二次项矩阵的特征值和系数 Kq，可以写出曲面的典型方程。在本文中，我们对佩夫兹纳的结果提出了一个新的简单证明。在证明中，只使用了行列式的基本性质。这种求典型曲面方程的算法可应用于计算机制图。
{"title":"Metric Invariants of Second-Order Surfaces","authors":"D. Yu. Volkov, K. V. Galunova","doi":"10.1134/s1063454123040210","DOIUrl":"https://doi.org/10.1134/s1063454123040210","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper is devoted to the classical problem of analytical geometry in <i>n</i>-dimensional Euclidean space, namely, finding the canonical equation of a quadric from an initial equation. The canonical equation is determined by the invariants of the second-order surface equation, i.e., by quantities that do not change when the space coordinates are changed affinely. S.L. Pevzner found a convenient system containing the following invariants: <i>q</i>, the rank of the extended matrix of the system for determining the center of symmetry of the surface; the roots of the characteristic polynomial of the quadratic term matrix of the surface equation, i.e., the eigenvalues of this matrix; and <i>K</i><sub><i>q</i></sub>, the coefficient of the variable λ to the power <i>n</i> – <i>q</i> in the polynomial that is equal to the determinant of the matrix of order <i>n</i> + 1 that is obtained according to a certain rule from the initial surface equation. The eigenvalues of the matrix of the quadratic terms and coefficient <i>K</i><sub><i>q</i></sub> make it possible to write the canonical equation of the surface. In the paper, we propose a new simple proof of Pevzner’s result. In the proof, only elementary properties of the determinants are used. This algorithm for finding the canonical surface equation can find application in computer graphics.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"121 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559412","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 : 2024-01-24DOI: 10.1134/s1063454123040192
V. M. Kats, V. A. Morozov, Ya. A. Sevastyanov
Abstract
In this paper, we present an analysis of the dynamic loading of a material carried out by the method of electrical explosion of conductors (EEC). A composite cylinder, which is a thick-walled polymethyl methacrylate cylinder with an axial hole that is pressed into an aluminum shell, is selected as the experimental sample. The sample is loaded by applying an electrical voltage to a conductor in the form of a copper wire placed inside the axial hole of the sample. A state equation for an electrical explosion of the conductor is obtained; the experimental and calculated dependences of pressure in the air channel on the expansion coefficient of the EEC products are plotted. A theoretical model that describes the change in radial stresses arising during loading with allowance for the boundary and initial conditions is developed. The accuracy of the model is substantiated by comparing the obtained results with experimental data. The circumferential stress at which the aluminum shell ruptures directly is estimated.
{"title":"Calculation of Stresses Initiated by an Electrical Explosion of Conductors in a Composite Thick-Walled Cylinder","authors":"V. M. Kats, V. A. Morozov, Ya. A. Sevastyanov","doi":"10.1134/s1063454123040192","DOIUrl":"https://doi.org/10.1134/s1063454123040192","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, we present an analysis of the dynamic loading of a material carried out by the method of electrical explosion of conductors (EEC). A composite cylinder, which is a thick-walled polymethyl methacrylate cylinder with an axial hole that is pressed into an aluminum shell, is selected as the experimental sample. The sample is loaded by applying an electrical voltage to a conductor in the form of a copper wire placed inside the axial hole of the sample. A state equation for an electrical explosion of the conductor is obtained; the experimental and calculated dependences of pressure in the air channel on the expansion coefficient of the EEC products are plotted. A theoretical model that describes the change in radial stresses arising during loading with allowance for the boundary and initial conditions is developed. The accuracy of the model is substantiated by comparing the obtained results with experimental data. The circumferential stress at which the aluminum shell ruptures directly is estimated.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557994","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 : 2024-01-24DOI: 10.1134/s1063454123040180
B. Yu. Irgashev
Abstract
In previous papers, we obtained some delta-shaped partial solutions of odd-order equations with multiple characteristics and studied some of their properties. In this paper, we first obtain the necessary estimates at infinity for these solutions, and then construct a fundamental solution (FS) of an odd-order equation with multiple characteristics in a rectangular domain as the sum of these particular solutions. We show that the FS is a solution to an inhomogeneous equation with multiple characteristics in a rectangular domain. In turn, knowledge of FS allows us to construct a potential theory for its further use in solving boundary-value problems.
{"title":"Construction of Fundamental Solution for an Odd-Order Equation","authors":"B. Yu. Irgashev","doi":"10.1134/s1063454123040180","DOIUrl":"https://doi.org/10.1134/s1063454123040180","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In previous papers, we obtained some delta-shaped partial solutions of odd-order equations with multiple characteristics and studied some of their properties. In this paper, we first obtain the necessary estimates at infinity for these solutions, and then construct a fundamental solution (FS) of an odd-order equation with multiple characteristics in a rectangular domain as the sum of these particular solutions. We show that the FS is a solution to an inhomogeneous equation with multiple characteristics in a rectangular domain. In turn, knowledge of FS allows us to construct a potential theory for its further use in solving boundary-value problems.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"125 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139557978","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 : 2024-01-24DOI: 10.1134/s1063454123040155
G. V. Shoev
Abstract
The application of hybrid approximate Riemann solvers based on standard HLLC and HLL solvers is discussed. Three different hybrid solvers are considered. The first hybrid solver (rHLLC-HLL) uses a weighted sum of HLLC and HLL so that HLLC is applied in the direction normal to the shock wave, while HLL is applied in the direction along the wave. The second hybrid solver (HLLC-ADC) uses the weighted sum of HLLC and HLL, applying as weights the pressure function at the centers of the left and right cells. The third hybrid solver (HLLC-HLL) computes inviscid fluxes using HLL inside shock waves and HLLC in the other areas of the flow. The faces within the shock waves are determined by a shock-wave indicator based on the reconstructed pressure values to the left and to the right of the face. Several tests have been carried out showing that hybrid solvers prevent the emergence of carbuncle and reduce oscillations on shock waves.
{"title":"Application of Hybrid Riemann Solvers Based on HLLC and HLL for Simulation of Flows with Gas-Dynamic Discontinuities","authors":"G. V. Shoev","doi":"10.1134/s1063454123040155","DOIUrl":"https://doi.org/10.1134/s1063454123040155","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The application of hybrid approximate Riemann solvers based on standard HLLC and HLL solvers is discussed. Three different hybrid solvers are considered. The first hybrid solver (rHLLC-HLL) uses a weighted sum of HLLC and HLL so that HLLC is applied in the direction normal to the shock wave, while HLL is applied in the direction along the wave. The second hybrid solver (HLLC-ADC) uses the weighted sum of HLLC and HLL, applying as weights the pressure function at the centers of the left and right cells. The third hybrid solver (HLLC-HLL) computes inviscid fluxes using HLL inside shock waves and HLLC in the other areas of the flow. The faces within the shock waves are determined by a shock-wave indicator based on the reconstructed pressure values to the left and to the right of the face. Several tests have been carried out showing that hybrid solvers prevent the emergence of carbuncle and reduce oscillations on shock waves.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139559603","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 : 2024-01-24DOI: 10.1134/s1063454123040064
{"title":"On the Anniversary of Alexander Konstantinovich Belyaev","authors":"","doi":"10.1134/s1063454123040064","DOIUrl":"https://doi.org/10.1134/s1063454123040064","url":null,"abstract":"","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"112 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883421","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 : 2024-01-24DOI: 10.1134/s1063454123040076
A. N. Frolov
Abstract
In this paper, the asymptotic behavior of probabilities of moderate deviations is investigated for combinatorial sums of independent random variables with moments of order p > 2. The zones are found in which these probabilities are equivalent to the tail of the standard normal law. The width of the zones are expressed in terms of the logarithm of the combinatorial variant of the Lyapunov ratio. Previously, similar results have been obtained by the author under the Bernstein and Linnik conditions. The truncation method is used in proving the new results.
摘要 本文研究了具有 p > 2 阶矩的独立随机变量组合和的中等偏差概率的渐近行为。区域的宽度用李亚普诺夫比率的组合变体对数表示。在此之前，作者在伯恩斯坦条件和林尼克条件下也得到过类似的结果。在证明新结果时使用了截断法。
{"title":"On the Asymptotic Behavior of Probabilities of Moderate Deviations for Combinatorial Sums","authors":"A. N. Frolov","doi":"10.1134/s1063454123040076","DOIUrl":"https://doi.org/10.1134/s1063454123040076","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, the asymptotic behavior of probabilities of moderate deviations is investigated for combinatorial sums of independent random variables with moments of order <i>p</i> > 2. The zones are found in which these probabilities are equivalent to the tail of the standard normal law. The width of the zones are expressed in terms of the logarithm of the combinatorial variant of the Lyapunov ratio. Previously, similar results have been obtained by the author under the Bernstein and Linnik conditions. The truncation method is used in proving the new results.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"517 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558064","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 : 2024-01-24DOI: 10.1134/s1063454123040040
T. I. Belyankova, V. V. Kalinchuk
Abstract
An approach to studying the influence of initial mechanical stresses and an electrostatic field on the structure and behavior of Rayleigh waves in piezoelectric media with nonhomogeneous coatings is proposed. This paper considers two-component coatings made of functionally graded piezoelectric material with high-speed (the speed of the shear-wave inclusion is greater than the speed of the shear wave in the substrate) or low-speed (the speed of the shear-wave inclusion is less than the speed of the shear wave in the substrate) inclusions. The initially deformed state of the coating is induced by the separate or combined action of the initial mechanical stresses and external electrostatic field. The influence of the type of non-homogeneity and the nature of the initial mechanical stresses in the presence or absence of an initial electrostatic field on the features of Rayleigh wave propagation for problems with an electrically open or shorted surface is studied. It is established that the presence of a low-intensity initial electrostatic field only slightly affects the action of the initial mechanical stresses depending on its direction. The presence of a high-intensity electrostatic field leads to additional deformation of the material, significant changes in the speeds of the SAW modes, and substantial changes in the structure of the surface-wave field. The obtained results are presented in dimensionless parameters and may be of practical interest in the development, design, and optimization of new materials for micro- and nanoscale devices and devices on Rayleigh surface acoustic waves with high performance characteristics.
{"title":"Rayleigh Waves in an Electroelastic Medium with Prestressed Inhomogeneous Coating","authors":"T. I. Belyankova, V. V. Kalinchuk","doi":"10.1134/s1063454123040040","DOIUrl":"https://doi.org/10.1134/s1063454123040040","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>An approach to studying the influence of initial mechanical stresses and an electrostatic field on the structure and behavior of Rayleigh waves in piezoelectric media with nonhomogeneous coatings is proposed. This paper considers two-component coatings made of functionally graded piezoelectric material with high-speed (the speed of the shear-wave inclusion is greater than the speed of the shear wave in the substrate) or low-speed (the speed of the shear-wave inclusion is less than the speed of the shear wave in the substrate) inclusions. The initially deformed state of the coating is induced by the separate or combined action of the initial mechanical stresses and external electrostatic field. The influence of the type of non-homogeneity and the nature of the initial mechanical stresses in the presence or absence of an initial electrostatic field on the features of Rayleigh wave propagation for problems with an electrically open or shorted surface is studied. It is established that the presence of a low-intensity initial electrostatic field only slightly affects the action of the initial mechanical stresses depending on its direction. The presence of a high-intensity electrostatic field leads to additional deformation of the material, significant changes in the speeds of the SAW modes, and substantial changes in the structure of the surface-wave field. The obtained results are presented in dimensionless parameters and may be of practical interest in the development, design, and optimization of new materials for micro- and nanoscale devices and devices on Rayleigh surface acoustic waves with high performance characteristics.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558066","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 : 2024-01-24DOI: 10.1134/s1063454123040143
S. Yu. Pilyugin, V. S. Shalgin
Abstract
The problem of parametric identification (determining the parameters of a system by observing solutions or functions of them) is one of the main problems in the applied theory of differential equations. When solving this problem, the property of local identifiability plays a crucial role. The presence of this property means that by observing solutions, it is possible to determine unambiguously the value of the system parameters in a neighborhood of the selected parameter. Previously, in the context of this problem, researchers mainly studied the case of a finite-dimensional parameter. The problem of local parametric identifiability in the case of an infinite-dimensional parameter has received much less attention. In this paper, we propose a new method for obtaining sufficient conditions for local parametric identifiability in the case of an infinite-dimensional parameter. When these conditions are met, an infinite-dimensional parameter belonging to certain classes is locally identified by observing the solution at a finite set of points. For systems with a linear dependence on the parameter, the genericity of the specified conditions is established.
{"title":"Conditions for Local Parameter Identifiability for Systems of Differential Equations with an Infinite-Dimensional Parameter","authors":"S. Yu. Pilyugin, V. S. Shalgin","doi":"10.1134/s1063454123040143","DOIUrl":"https://doi.org/10.1134/s1063454123040143","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The problem of parametric identification (determining the parameters of a system by observing solutions or functions of them) is one of the main problems in the applied theory of differential equations. When solving this problem, the property of local identifiability plays a crucial role. The presence of this property means that by observing solutions, it is possible to determine unambiguously the value of the system parameters in a neighborhood of the selected parameter. Previously, in the context of this problem, researchers mainly studied the case of a finite-dimensional parameter. The problem of local parametric identifiability in the case of an infinite-dimensional parameter has received much less attention. In this paper, we propose a new method for obtaining sufficient conditions for local parametric identifiability in the case of an infinite-dimensional parameter. When these conditions are met, an infinite-dimensional parameter belonging to certain classes is locally identified by observing the solution at a finite set of points. For systems with a linear dependence on the parameter, the genericity of the specified conditions is established.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139558121","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号