SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4970-5016, August 2024. Abstract. Optimal transport is a geometrically intuitive, robust, and flexible metric for sample comparison in data analysis and machine learning. Its formal Riemannian structure allows for a local linearization via a tangent space approximation. This in turn leads to a reduction of computational complexity and simplifies combination with other methods that require a linear structure. Recently this approach has been extended to the unbalanced Hellinger–Kantorovich (HK) distance. In this article we further extend the framework in various ways, including measures on manifolds, the spherical HK distance, a study of the consistency of discretization via the barycentric projection, and the continuity properties of the logarithmic map for the HK distance.
SIAM 数学分析期刊》，第 56 卷第 4 期，第 4970-5016 页，2024 年 8 月。 摘要最优传输是一种直观、稳健、灵活的几何度量，可用于数据分析和机器学习中的样本比较。其形式上的黎曼结构允许通过切线空间近似实现局部线性化。这反过来又降低了计算复杂度，简化了与其他需要线性结构的方法的结合。最近，这种方法被扩展到非平衡海灵格-康托洛维奇（HK）距离。在这篇文章中，我们从多方面进一步扩展了这一框架，包括流形上的度量、球形 HK 距离、通过重心投影进行离散化的一致性研究，以及 HK 距离对数映射的连续性特性。
{"title":"Linearized Optimal Transport on Manifolds","authors":"Clément Sarrazin, Bernhard Schmitzer","doi":"10.1137/23m1564535","DOIUrl":"https://doi.org/10.1137/23m1564535","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4970-5016, August 2024. <br/> Abstract. Optimal transport is a geometrically intuitive, robust, and flexible metric for sample comparison in data analysis and machine learning. Its formal Riemannian structure allows for a local linearization via a tangent space approximation. This in turn leads to a reduction of computational complexity and simplifies combination with other methods that require a linear structure. Recently this approach has been extended to the unbalanced Hellinger–Kantorovich (HK) distance. In this article we further extend the framework in various ways, including measures on manifolds, the spherical HK distance, a study of the consistency of discretization via the barycentric projection, and the continuity properties of the logarithmic map for the HK distance.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4870-4927, August 2024. Abstract. In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the [math]-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g., the Allen–Cahn equation) and dissipative systems (e.g., equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka–Volterra equations for [math] and the Brusselator for [math]. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô’s formula and stochastic Gronwall inequalities. Key novelties include the introduction of new [math]-coercivity/dissipativity conditions and the development of an [math]-framework for systems of reaction-diffusion equations, which are needed when treating dimensions [math] in the case of cubic or higher order nonlinearities.
{"title":"Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Global Well-Posedness of Weakly Dissipative Systems","authors":"Antonio Agresti, Mark Veraar","doi":"10.1137/23m1562482","DOIUrl":"https://doi.org/10.1137/23m1562482","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4870-4927, August 2024. <br/> Abstract. In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the [math]-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g., the Allen–Cahn equation) and dissipative systems (e.g., equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka–Volterra equations for [math] and the Brusselator for [math]. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô’s formula and stochastic Gronwall inequalities. Key novelties include the introduction of new [math]-coercivity/dissipativity conditions and the development of an [math]-framework for systems of reaction-diffusion equations, which are needed when treating dimensions [math] in the case of cubic or higher order nonlinearities.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Verena Bögelein, Frank Duzaar, Raffaella Giova, Antonia Passarelli di Napoli
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5017-5078, August 2024. Abstract. In this paper we study the [math] regularity of the weak solutions to the widely degenerate parabolic system [math] where [math], [math] is a parabolic cylinder with a bounded domain [math], and [math]. For the weak solutions [math] with [math], we prove that [math] is continuous in [math] for any continuous function [math] vanishing on the set [math].
{"title":"Gradient Regularity for a Class of Widely Degenerate Parabolic Systems","authors":"Verena Bögelein, Frank Duzaar, Raffaella Giova, Antonia Passarelli di Napoli","doi":"10.1137/23m1589232","DOIUrl":"https://doi.org/10.1137/23m1589232","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5017-5078, August 2024. <br/> Abstract. In this paper we study the [math] regularity of the weak solutions to the widely degenerate parabolic system [math] where [math], [math] is a parabolic cylinder with a bounded domain [math], and [math]. For the weak solutions [math] with [math], we prove that [math] is continuous in [math] for any continuous function [math] vanishing on the set [math].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pierluigi Colli, Gianni Gilardi, Andrea Signori, Jürgen Sprekels
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4928-4969, August 2024. Abstract. This work investigates the well-posedness and optimal control of a sixth-order Cahn–Hilliard equation, a higher-order variant of the celebrated and well-established Cahn–Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improving the understanding of the mathematical properties and control aspects of the sixth-order Cahn–Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.
{"title":"Curvature Effects in Pattern Formation: Well-Posedness and Optimal Control of a Sixth-Order Cahn–Hilliard Equation","authors":"Pierluigi Colli, Gianni Gilardi, Andrea Signori, Jürgen Sprekels","doi":"10.1137/24m1630372","DOIUrl":"https://doi.org/10.1137/24m1630372","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4928-4969, August 2024. <br/> Abstract. This work investigates the well-posedness and optimal control of a sixth-order Cahn–Hilliard equation, a higher-order variant of the celebrated and well-established Cahn–Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improving the understanding of the mathematical properties and control aspects of the sixth-order Cahn–Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4854-4869, August 2024. Abstract. In this paper, we prove the global existence of analytical solutions to the compressible Oldroyd-B model without retardation near a nonvacuum equilibrium in [math] [math]. Zero retardation results in zero dissipation in the velocity equation, which is the main difficulty that prevents us from obtaining the long time well-posedness of solutions. Through dedicated analysis, we find that the linearized equations of this model have damping effects, which ensure the global-in-time existence of small data solutions. However, the nonlinear quadratic terms have one more order derivative than the linear part and no good structure is discovered to overcome this derivative loss problem. So we can only build the result in the analytical energy space rather than Sobolev space with finite order derivatives.
{"title":"Globally Analytical Solutions of the Compressible Oldroyd-B Model Without Retardation","authors":"Xinghong Pan","doi":"10.1137/23m1588974","DOIUrl":"https://doi.org/10.1137/23m1588974","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4854-4869, August 2024. <br/> Abstract. In this paper, we prove the global existence of analytical solutions to the compressible Oldroyd-B model without retardation near a nonvacuum equilibrium in [math] [math]. Zero retardation results in zero dissipation in the velocity equation, which is the main difficulty that prevents us from obtaining the long time well-posedness of solutions. Through dedicated analysis, we find that the linearized equations of this model have damping effects, which ensure the global-in-time existence of small data solutions. However, the nonlinear quadratic terms have one more order derivative than the linear part and no good structure is discovered to overcome this derivative loss problem. So we can only build the result in the analytical energy space rather than Sobolev space with finite order derivatives.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4834-4853, August 2024. Abstract. This paper investigates the possible scattering and nonscattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic nonscattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not [math] boundary scatters every incident wave that satisfies a nondegeneracy condition.
{"title":"On Scattering Behavior of Corner Domains with Anisotropic Inhomogeneities","authors":"Pu-Zhao Kow, Mikko Salo, Henrik Shahgholian","doi":"10.1137/23m1603029","DOIUrl":"https://doi.org/10.1137/23m1603029","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4834-4853, August 2024. <br/> Abstract. This paper investigates the possible scattering and nonscattering behavior of an anisotropic and inhomogeneous Lipschitz medium at a fixed wave number and with a single incident field. We connect the anisotropic nonscattering problem to a Bernoulli type free boundary problem. By invoking methods from the theory of free boundaries, we show that an anisotropic medium with Lipschitz but not [math] boundary scatters every incident wave that satisfies a nondegeneracy condition.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4811-4833, August 2024. Abstract. This paper is concerned with the global existence of large solutions to the initial and initial boundary value problems for the radiation hydrodynamics model with constant viscosity coefficient and temperature-dependent heat conductivity coefficient. In particular, the initial data and [math] could be arbitrarily large. Compared with the classical compressible Navier–Stokes equations, the research on the radiation hydrodynamics model is more complicated due to the presence of the radiation effect. To overcome difficulties caused by the radiation, inspired by [J. Li and Z. L. Liang, Arch. Ration. Mech. Anal., 220 (2016), pp. 1195–1208; J. H. Zhang and H. J. Zhao, SIAM J. Math. Anal., 55 (2023), pp. 6229–6261], we construct a pointwise estimate between the radiative heat flux and the temperature; then we can establish the desired basic energy estimate by considering that the temperature has lower and upper bounds separately. And once that is obtained, the main result is proved by employing the elementary energy methods.
SIAM 数学分析期刊》，第 56 卷第 4 期，第 4811-4833 页，2024 年 8 月。 摘要本文关注粘性系数恒定、导热系数随温度变化的辐射流体力学模型的初始值和初始边界值问题的全局大解存在性。特别是，初始数据和[math]可以任意大。与经典的可压缩 Navier-Stokes 方程相比，由于辐射效应的存在，辐射流体力学模型的研究更为复杂。为了克服辐射带来的困难，受[J. Li 和 Z. L. Liang]的启发，我们提出了辐射流体力学模型。Li and Z. L. Liang, Arch.Ration.Mech.Anal., 220 (2016), pp.Anal.，55 (2023)，pp. 6229-6261]，我们构建了辐射热通量与温度之间的点估计；然后，我们可以通过考虑温度分别具有下限和上限来建立所需的基本能量估计。得到这些估计值后，我们就可以利用基本能量方法证明主要结果了。
{"title":"Global Classical Large Solutions for the Radiation Hydrodynamics Model in Unbounded Domains","authors":"Jing Wei, Minyi Zhang, Changjiang Zhu","doi":"10.1137/23m1598581","DOIUrl":"https://doi.org/10.1137/23m1598581","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4811-4833, August 2024. <br/> Abstract. This paper is concerned with the global existence of large solutions to the initial and initial boundary value problems for the radiation hydrodynamics model with constant viscosity coefficient and temperature-dependent heat conductivity coefficient. In particular, the initial data and [math] could be arbitrarily large. Compared with the classical compressible Navier–Stokes equations, the research on the radiation hydrodynamics model is more complicated due to the presence of the radiation effect. To overcome difficulties caused by the radiation, inspired by [J. Li and Z. L. Liang, Arch. Ration. Mech. Anal., 220 (2016), pp. 1195–1208; J. H. Zhang and H. J. Zhao, SIAM J. Math. Anal., 55 (2023), pp. 6229–6261], we construct a pointwise estimate between the radiative heat flux and the temperature; then we can establish the desired basic energy estimate by considering that the temperature has lower and upper bounds separately. And once that is obtained, the main result is proved by employing the elementary energy methods.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4759-4810, August 2024. Abstract. We study an inverse problem for variable coefficient fractional parabolic operators of the form [math] for [math] and show the unique recovery of [math] from exterior measured data. Similar to the fractional elliptic case, we use a Runge-type approximation argument, which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.
{"title":"The Calderón Problem for Space-Time Fractional Parabolic Operators with Variable Coefficients","authors":"Agnid Banerjee, Soumen Senapati","doi":"10.1137/23m1584137","DOIUrl":"https://doi.org/10.1137/23m1584137","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4759-4810, August 2024. <br/> Abstract. We study an inverse problem for variable coefficient fractional parabolic operators of the form [math] for [math] and show the unique recovery of [math] from exterior measured data. Similar to the fractional elliptic case, we use a Runge-type approximation argument, which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4623-4661, August 2024. Abstract. We study nonlocal elliptic and parabolic equations on [math] open sets in weighted Sobolev spaces, where [math]. The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable.
{"title":"Nonlocal Elliptic and Parabolic Equations with General Stable Operators in Weighted Sobolev Spaces","authors":"Hongjie Dong, Junhee Ryu","doi":"10.1137/23m160061x","DOIUrl":"https://doi.org/10.1137/23m160061x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4623-4661, August 2024. <br/> Abstract. We study nonlocal elliptic and parabolic equations on [math] open sets in weighted Sobolev spaces, where [math]. The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4687-4711, August 2024. Abstract. This paper performs a stability analysis of a class of moment closure systems derived with an extended quadrature method of moments (EQMOM) for the one-dimensional Bhatnagar–Gross–Krook equation. The class is characterized with a kernel function. A sufficient condition on the kernel is identified for the EQMOM-derived moment systems to be strictly hyperbolic. We also investigate the realizability of the moment method. Moreover, sufficient and necessary conditions are established for the two-node systems to be well-defined and strictly hyperbolic and to preserve the dissipation property of the kinetic equation.
{"title":"Stability Analysis of an Extended Quadrature Method of Moments for Kinetic Equations","authors":"Ruixi Zhang, Qian Huang, Wen-An Yong","doi":"10.1137/23m157911x","DOIUrl":"https://doi.org/10.1137/23m157911x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4687-4711, August 2024. <br/> Abstract. This paper performs a stability analysis of a class of moment closure systems derived with an extended quadrature method of moments (EQMOM) for the one-dimensional Bhatnagar–Gross–Krook equation. The class is characterized with a kernel function. A sufficient condition on the kernel is identified for the EQMOM-derived moment systems to be strictly hyperbolic. We also investigate the realizability of the moment method. Moreover, sufficient and necessary conditions are established for the two-node systems to be well-defined and strictly hyperbolic and to preserve the dissipation property of the kinetic equation.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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