Pub Date : 2023-12-01DOI: 10.1134/s0081543823060056
Abstract
The problem of reachability in a topological space is studied under constraints of asymptotic nature arising from weakening the requirement that the image of a solution belong to a given set. The attraction set that arises in this case in the topological space is a regularization of certain kind for the image of the preimage of the mentioned set (the image and the preimage are defined for generally different mappings). When constructing natural compact extensions of the reachability problem with constraints of asymptotic nature generated by a family of neighborhoods of a fixed set, the case was studied earlier where the topological space in which the results of one or another choice of solution are realized satisfies the axiom (T_{2}). In the present paper, for a number of statements related to compact extensions, it is possible to use for this purpose a (T_{1}) space, which seems to be quite important from a theoretical point of view, since it is possible to find out the exact role of the axiom (T_{2}) in questions related to correct extensions of reachability problems. We study extension models using ultrafilters of a broadly understood measurable space with detailing of the main elements in the case of a reachability problem in the space of functionals with the topology of a Tychonoff power of the real line with the usual (|cdot|)-topology. The general constructions of extension models are illustrated by an example of a nonlinear control problem with state constraints.
{"title":"Closed Mappings and Construction of Extension Models","authors":"","doi":"10.1134/s0081543823060056","DOIUrl":"https://doi.org/10.1134/s0081543823060056","url":null,"abstract":"<h3>Abstract</h3> <p>The problem of reachability in a topological space is studied under constraints of asymptotic nature arising from weakening the requirement that the image of a solution belong to a given set. The attraction set that arises in this case in the topological space is a regularization of certain kind for the image of the preimage of the mentioned set (the image and the preimage are defined for generally different mappings). When constructing natural compact extensions of the reachability problem with constraints of asymptotic nature generated by a family of neighborhoods of a fixed set, the case was studied earlier where the topological space in which the results of one or another choice of solution are realized satisfies the axiom <span> <span>(T_{2})</span> </span>. In the present paper, for a number of statements related to compact extensions, it is possible to use for this purpose a <span> <span>(T_{1})</span> </span> space, which seems to be quite important from a theoretical point of view, since it is possible to find out the exact role of the axiom <span> <span>(T_{2})</span> </span> in questions related to correct extensions of reachability problems. We study extension models using ultrafilters of a broadly understood measurable space with detailing of the main elements in the case of a reachability problem in the space of functionals with the topology of a Tychonoff power of the real line with the usual <span> <span>(|cdot|)</span> </span>-topology. The general constructions of extension models are illustrated by an example of a nonlinear control problem with state constraints. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.1134/s0081543823060081
Abstract
For a number of years, the authors have been working in the sphere of asset tokenization, especially in connection with precious metals. An approach (including algorithms, mathematical models, and software implementation) to the gold reserve management problem has been developed. This approach allows to effectively manage the gold reserve, taking into account the fact that the faster the money is turned over, the less gold is required to ensure the functioning of the financial system. Moreover, the transfer of payments to the online mode dramatically reduces the need for working capital and, hence, the amount of gold required for it. The authors consider the provided algorithms as a very important part of a possible gold-backed settlement system that allows to solve the problem of organizing international payment transactions between countries in their national currencies to avoid the dominance of a single fiat currency.
{"title":"Asset Tokenization and Related Problems","authors":"","doi":"10.1134/s0081543823060081","DOIUrl":"https://doi.org/10.1134/s0081543823060081","url":null,"abstract":"<h3>Abstract</h3> <p>For a number of years, the authors have been working in the sphere of asset tokenization, especially in connection with precious metals. An approach (including algorithms, mathematical models, and software implementation) to the gold reserve management problem has been developed. This approach allows to effectively manage the gold reserve, taking into account the fact that the faster the money is turned over, the less gold is required to ensure the functioning of the financial system. Moreover, the transfer of payments to the online mode dramatically reduces the need for working capital and, hence, the amount of gold required for it. The authors consider the provided algorithms as a very important part of a possible gold-backed settlement system that allows to solve the problem of organizing international payment transactions between countries in their national currencies to avoid the dominance of a single fiat currency. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.1134/s0081543823060123
Abstract
The Weyl derivative (fractional derivative) (f_{n}^{(alpha)}) of real nonnegative order (alpha) is considered on the set (mathscr{T}_{n}) of trigonometric polynomials (f_{n}) of order (n) with complex coefficients. The constant in the Bernstein–Szegő inequality ({|}f_{n}^{(alpha)}costheta+tilde{f}_{n}^{(alpha)}sintheta{| }leq B_{n}(alpha,theta)|f_{n}|) in the uniform norm is studied. This inequality has been well studied for (alphageq 1): G. T. Sokolov proved in 1935 that it holds with the constant (n^{alpha}) for all (thetainmathbb{R}). For (0<alpha<1), there is much less information about (B_{n}(alpha,theta)). In this paper, for (0<alpha<1) and (thetainmathbb{R}), we establish the limit relation (lim_{ntoinfty}B_{n}(alpha,theta)/n^{alpha}=mathcal{B}(alpha,theta)), where (mathcal{B}(alpha,theta)) is the sharp constant in the similar inequality for entire functions of exponential type at most (1) that are bounded on the real line. The value (theta=-pialpha/2) corresponds to the Riesz derivative, which is an important particular case of the Weyl–Szegő operator. In this case, we derive exact asymptotics for the quantity (B_{n}(alpha)=B_{n}(alpha,-pialpha/2)) as (ntoinfty).
Abstract The Weyl derivative (fractional derivative) (f_{n}^{(alpha)}) of real nonnegative order (alpha) is considered on the set (mathscr{T}_{n}) of trigonometric polynomials (f_{n}^{(alpha)}) of order (n) with complex coefficients.研究了伯恩斯坦-塞格（Bernstein-Szegő）不等式 ({|}f_{n}^{(alpha)}costheta+tilde{f}_{n}^{(alpha)}sintheta{| }leq B_{n}(alpha,theta)|f_{n}|) 在统一规范中的常数。这个不等式对于 (alphageq 1) 已经有了很好的研究：G. T. Sokolov 在 1935 年证明，对于所有 (thetainmathbb{R}) 的常数 (n^{alpha}) 它是成立的。对于（0<alpha<1），关于（B_{n}(alpha,theta)）的信息要少得多。在本文中，对于 (0<alpha<1) 和 (thetainmathbb{R}) ，我们建立了极限关系 (lim_{ntoinfty}B_{n}(alpha,theta)/n^{alpha}=mathcal{B}(alpha,theta)) 。其中，(mathcal{B}(alpha,theta))是类似不等式中的尖锐常数，用于在实线上有界的、指数型的整个函数，最多为(1)。(theta=-pialpha/2)的值对应于里兹导数，它是韦尔-塞格ő算子的一个重要特例。在这种情况下，我们推导出 (B_{n}(alpha)=B_{n}(alpha,-pialpha/2)) 的精确渐近量为(ntoinfty) 。
{"title":"On Constants in the Bernstein–Szegő Inequality for the Weyl Derivative of Order Less Than Unity of Trigonometric Polynomials and Entire Functions of Exponential Type in the Uniform Norm","authors":"","doi":"10.1134/s0081543823060123","DOIUrl":"https://doi.org/10.1134/s0081543823060123","url":null,"abstract":"<h3>Abstract</h3> <p>The Weyl derivative (fractional derivative) <span> <span>(f_{n}^{(alpha)})</span> </span> of real nonnegative order <span> <span>(alpha)</span> </span> is considered on the set <span> <span>(mathscr{T}_{n})</span> </span> of trigonometric polynomials <span> <span>(f_{n})</span> </span> of order <span> <span>(n)</span> </span> with complex coefficients. The constant in the Bernstein–Szegő inequality <span> <span>({|}f_{n}^{(alpha)}costheta+tilde{f}_{n}^{(alpha)}sintheta{| }leq B_{n}(alpha,theta)|f_{n}|)</span> </span> in the uniform norm is studied. This inequality has been well studied for <span> <span>(alphageq 1)</span> </span>: G. T. Sokolov proved in 1935 that it holds with the constant <span> <span>(n^{alpha})</span> </span> for all <span> <span>(thetainmathbb{R})</span> </span>. For <span> <span>(0<alpha<1)</span> </span>, there is much less information about <span> <span>(B_{n}(alpha,theta))</span> </span>. In this paper, for <span> <span>(0<alpha<1)</span> </span> and <span> <span>(thetainmathbb{R})</span> </span>, we establish the limit relation <span> <span>(lim_{ntoinfty}B_{n}(alpha,theta)/n^{alpha}=mathcal{B}(alpha,theta))</span> </span>, where <span> <span>(mathcal{B}(alpha,theta))</span> </span> is the sharp constant in the similar inequality for entire functions of exponential type at most <span> <span>(1)</span> </span> that are bounded on the real line. The value <span> <span>(theta=-pialpha/2)</span> </span> corresponds to the Riesz derivative, which is an important particular case of the Weyl–Szegő operator. In this case, we derive exact asymptotics for the quantity <span> <span>(B_{n}(alpha)=B_{n}(alpha,-pialpha/2))</span> </span> as <span> <span>(ntoinfty)</span> </span>. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1134/s008154382304003x
I. B. Bakholdin
{"title":"Periodic and Solitary Waves and Nondissipative Discontinuity Structures in Electromagnetic Hydrodynamics in the Case of Wave Resonance","authors":"I. B. Bakholdin","doi":"10.1134/s008154382304003x","DOIUrl":"https://doi.org/10.1134/s008154382304003x","url":null,"abstract":"","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139344453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1134/s0081543823040107
Abstract
The main result of the paper is a theorem stating that the modulation instability of a carrier periodic wave of small (but finite) amplitude propagating in an arbitrary dispersive medium may only be convective in a reference frame moving at a velocity that differs finitely from the group velocity of this wave. The application of this result to the radiation of a resonant wave by a soliton-like “core” is discussed. Such radiation occurs in media where classical solitary waves are replaced with generalized solitary waves as a result of linear resonance of long and short waves. Generalized solitary waves are traveling waves that form a homoclinic structure doubly asymptotic to a periodic wave.
{"title":"Convective Modulation Instability of the Radiation of the Periodic Component in the Case of Resonance of Long and Short Waves","authors":"","doi":"10.1134/s0081543823040107","DOIUrl":"https://doi.org/10.1134/s0081543823040107","url":null,"abstract":"<span> <h3>Abstract</h3> <p> The main result of the paper is a theorem stating that the modulation instability of a carrier periodic wave of small (but finite) amplitude propagating in an arbitrary dispersive medium may only be convective in a reference frame moving at a velocity that differs finitely from the group velocity of this wave. The application of this result to the radiation of a resonant wave by a soliton-like “core” is discussed. Such radiation occurs in media where classical solitary waves are replaced with generalized solitary waves as a result of linear resonance of long and short waves. Generalized solitary waves are traveling waves that form a homoclinic structure doubly asymptotic to a periodic wave. </p> </span>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1134/s0081543823040016
Editorial Board
{"title":"Andrei Gennad’evich Kulikovskii: On the occasion of his 90th birthday","authors":"Editorial Board","doi":"10.1134/s0081543823040016","DOIUrl":"https://doi.org/10.1134/s0081543823040016","url":null,"abstract":"","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1134/s0081543823040028
Abstract
We consider bending vibrations of a fluid-conveying pipe resting on an elastic foundation with nonuniform elasticity coefficient. Previously A. G. Kulikovskii showed analytically that the elasticity parameters can be distributed in such a way that at every point the system is either locally stable or convectively unstable. In this case, despite the absence of local absolute instability, there exists a global growing mode whose formation is associated with the points of internal reflection of waves. In the present paper, we perform a numerical simulation of the development of the initial perturbation in such a system. In the linear formulation we demonstrate how the perturbation is transformed into a growing eigenmode after a series of reflections and passages through a region of local instability. In the nonlinear formulation, where the nonlinear tension of the pipe is taken into account within the von Kármán model, we show that the perturbation growth is limited; in this case the vibrations acquire a quasi-chaotic character but do not leave the region bounded by the internal reflection points determined by the linearized problem.
摘要 我们考虑的是位于弹性系数不均匀的弹性地基上的流体输送管的弯曲振动问题。此前，A. G. Kulikovskii 通过分析表明，弹性参数的分布方式可以使系统在每一点上要么局部稳定，要么对流不稳定。在这种情况下，尽管不存在局部绝对不稳定性，但存在一种全局增长模式，其形成与波的内部反射点有关。在本文中，我们对这种系统中初始扰动的发展进行了数值模拟。在线性模型中，我们演示了扰动如何在经过一系列反射和穿过局部不稳定区域后转化为增长特征模。在非线性公式中，管道的非线性张力在 von Kármán 模型中被考虑在内，我们证明扰动的增长是有限的；在这种情况下，振动具有准混沌特性，但不会离开由线性化问题确定的内部反射点所限定的区域。
{"title":"Linear and Nonlinear Development of Bending Perturbations in a Fluid-Conveying Pipe with Variable Elastic Properties","authors":"","doi":"10.1134/s0081543823040028","DOIUrl":"https://doi.org/10.1134/s0081543823040028","url":null,"abstract":"<span> <h3>Abstract</h3> <p> We consider bending vibrations of a fluid-conveying pipe resting on an elastic foundation with nonuniform elasticity coefficient. Previously A. G. Kulikovskii showed analytically that the elasticity parameters can be distributed in such a way that at every point the system is either locally stable or convectively unstable. In this case, despite the absence of local absolute instability, there exists a global growing mode whose formation is associated with the points of internal reflection of waves. In the present paper, we perform a numerical simulation of the development of the initial perturbation in such a system. In the linear formulation we demonstrate how the perturbation is transformed into a growing eigenmode after a series of reflections and passages through a region of local instability. In the nonlinear formulation, where the nonlinear tension of the pipe is taken into account within the von Kármán model, we show that the perturbation growth is limited; in this case the vibrations acquire a quasi-chaotic character but do not leave the region bounded by the internal reflection points determined by the linearized problem. </p> </span>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1134/s0081543823040132
Abstract
Previously, we have obtained a system of fourth-order hyperbolic equations describing long nonlinear small-amplitude longitudinal–torsional waves propagating along an elastic rod. Waves of two types, fast and slow, propagate in each direction along the rod. In the present paper, based on this system of equations, we derive a second-order hyperbolic system that describes longitudinal–torsional waves propagating in one direction along the rod at close velocities. The waves propagating in the opposite direction along the rod are assumed to have a negligible amplitude. We show that the variation of quantities in simple and shock waves described by the system of second-order equations obtained in this paper exactly coincides with the variation of the same quantities in the corresponding waves described by the original system of fourth-order equations, and the velocities of these waves are close. We also analyze the variation of quantities in simple (Riemann) waves and the overturning conditions for these waves.
{"title":"Longitudinal–Torsional Waves in Nonlinear Elastic Rods","authors":"","doi":"10.1134/s0081543823040132","DOIUrl":"https://doi.org/10.1134/s0081543823040132","url":null,"abstract":"<span> <h3>Abstract</h3> <p> Previously, we have obtained a system of fourth-order hyperbolic equations describing long nonlinear small-amplitude longitudinal–torsional waves propagating along an elastic rod. Waves of two types, fast and slow, propagate in each direction along the rod. In the present paper, based on this system of equations, we derive a second-order hyperbolic system that describes longitudinal–torsional waves propagating in one direction along the rod at close velocities. The waves propagating in the opposite direction along the rod are assumed to have a negligible amplitude. We show that the variation of quantities in simple and shock waves described by the system of second-order equations obtained in this paper exactly coincides with the variation of the same quantities in the corresponding waves described by the original system of fourth-order equations, and the velocities of these waves are close. We also analyze the variation of quantities in simple (Riemann) waves and the overturning conditions for these waves. </p> </span>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1134/s0081543823040053
K. V. Brushlinskii, V. V. Kryuchenkov, E. V. Stepin
{"title":"Mathematical Model of Equilibrium Plasma Configurations in Magnetic Traps and Their Stability Analysis","authors":"K. V. Brushlinskii, V. V. Kryuchenkov, E. V. Stepin","doi":"10.1134/s0081543823040053","DOIUrl":"https://doi.org/10.1134/s0081543823040053","url":null,"abstract":"","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1134/s0081543823040193
Abstract
We consider the propagation of plane waves in an ideal gas in the presence of external sources of energy inflow and dissipation. Using the Whitham criterion, we obtain conditions under which small perturbations of a constant solution are transformed into nonlinear quasiperiodic wave packets of finite amplitude that move in opposite directions. The structure of these wave packets is shown to be similar to roll waves in inclined open channels. We perform numerical calculations of the development of self-oscillations and the nonlinear interaction of waves. The calculations show that under a small harmonic perturbation of the initial equilibrium state, two types of wave structures can develop: roll waves and periodic two-peak standing waves.
{"title":"Wave Structures in Ideal Gas Flows with an External Energy Source","authors":"","doi":"10.1134/s0081543823040193","DOIUrl":"https://doi.org/10.1134/s0081543823040193","url":null,"abstract":"<span> <h3>Abstract</h3> <p> We consider the propagation of plane waves in an ideal gas in the presence of external sources of energy inflow and dissipation. Using the Whitham criterion, we obtain conditions under which small perturbations of a constant solution are transformed into nonlinear quasiperiodic wave packets of finite amplitude that move in opposite directions. The structure of these wave packets is shown to be similar to roll waves in inclined open channels. We perform numerical calculations of the development of self-oscillations and the nonlinear interaction of waves. The calculations show that under a small harmonic perturbation of the initial equilibrium state, two types of wave structures can develop: roll waves and periodic two-peak standing waves. </p> </span>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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