For any , we construct ‐energies and the corresponding ‐energy measures on the Sierpiński carpet. A salient feature of our Sobolev space is the self‐similarity of energy. An important motivation for the construction of self‐similar energy and energy measures is to determine whether or not the Ahlfors regular conformal dimension is attained on the Sierpiński carpet. If the Ahlfors regular conformal dimension is attained, we show that any optimal Ahlfors regular measure attaining the Ahlfors regular conformal dimension must necessarily be a bounded perturbation of the ‐energy measure of some function in our Sobolev space, where is the Ahlfors regular conformal dimension. Under the attainment of the Ahlfors regular conformal dimension, the ‐Newtonian Sobolev space corresponding to any optimal Ahlfors regular metric and measure is shown to coincide with our Sobolev space with comparable norms, where is the Ahlfors regular conformal dimension.
{"title":"First‐order Sobolev spaces, self‐similar energies and energy measures on the Sierpiński carpet","authors":"Mathav Murugan, Ryosuke Shimizu","doi":"10.1002/cpa.22247","DOIUrl":"https://doi.org/10.1002/cpa.22247","url":null,"abstract":"For any , we construct ‐energies and the corresponding ‐energy measures on the Sierpiński carpet. A salient feature of our Sobolev space is the self‐similarity of energy. An important motivation for the construction of self‐similar energy and energy measures is to determine whether or not the Ahlfors regular conformal dimension is attained on the Sierpiński carpet. If the Ahlfors regular conformal dimension is attained, we show that any optimal Ahlfors regular measure attaining the Ahlfors regular conformal dimension must necessarily be a bounded perturbation of the ‐energy measure of some function in our Sobolev space, where is the Ahlfors regular conformal dimension. Under the attainment of the Ahlfors regular conformal dimension, the ‐Newtonian Sobolev space corresponding to any optimal Ahlfors regular metric and measure is shown to coincide with our Sobolev space with comparable norms, where is the Ahlfors regular conformal dimension.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"12 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143443282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearized elasticity and an ansatz on the distribution of incompatibilities of the lattice at the interface between two grains. The logarithmic scaling of this formula has been rigorously justified without any ansatz on the geometry of dislocations only recently in an article by Lauteri and Luckhaus. In the present paper, building upon their analysis, we derive a two dimensional sharp interface limiting functional starting from the nonlinear semi‐discrete model introduced in Lauteri and Luckhaus: the line tension we obtain via ‐convergence depends on the rotations of the grains and the relative orientations of the interfaces, and for small angle grain boundaries has the Read and Shockley logarithmic scaling.
{"title":"On the Read‐Shockley energy for grain boundaries in 2D polycrystals","authors":"Martino Fortuna, Adriana Garroni, Emanuele Spadaro","doi":"10.1002/cpa.22245","DOIUrl":"https://doi.org/10.1002/cpa.22245","url":null,"abstract":"In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearized elasticity and an ansatz on the distribution of incompatibilities of the lattice at the interface between two grains. The logarithmic scaling of this formula has been rigorously justified without any ansatz on the geometry of dislocations only recently in an article by Lauteri and Luckhaus. In the present paper, building upon their analysis, we derive a two dimensional sharp interface limiting functional starting from the nonlinear semi‐discrete model introduced in Lauteri and Luckhaus: the line tension we obtain via ‐convergence depends on the rotations of the grains and the relative orientations of the interfaces, and for small angle grain boundaries has the Read and Shockley logarithmic scaling.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"13 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143384980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Perfectly matched layers are extensively used to compute approximate solutions for Maxwell's equations in using a bounded computational domain, usually a rectangular solid. A smaller rectangular domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the boundary of the computational domain, an absorbing boundary condition is imposed that is necessarily imperfect. The method replaces the Maxwell equations by a larger system, and introduces absorption coefficients positive in the layers. Well posedness of the resulting initial boundary value problem is proved here for the first time. The Laplace transform of a resulting Helmholtz system is studied. For positive real values of the transform variable , the Helmholtz system has a unique solution from a variational form that yields limited regularity for rectangular domains. When is not real the complex variational form loses positivity. We smooth the domain and, in spite of this loss, construct solutions with uniform estimates. Using the regularity, we deduce Maxwell from Helmholtz, then remove the smoothing. The boundary condition at the smoothed boundary must be carefully chosen. A method of Jerison‐Kenig‐Mitrea is extended to compensate the nonpositivity of the flux.
{"title":"Stability of perfectly matched layers for Maxwell's equations in rectangular solids","authors":"Laurence Halpern, Jeffrey Rauch","doi":"10.1002/cpa.22249","DOIUrl":"https://doi.org/10.1002/cpa.22249","url":null,"abstract":"Perfectly matched layers are extensively used to compute approximate solutions for Maxwell's equations in using a bounded computational domain, usually a rectangular solid. A smaller rectangular domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the boundary of the computational domain, an absorbing boundary condition is imposed that is necessarily imperfect. The method replaces the Maxwell equations by a larger system, and introduces absorption coefficients positive in the layers. Well posedness of the resulting initial boundary value problem is proved here for the first time. The Laplace transform of a resulting Helmholtz system is studied. For positive real values of the transform variable , the Helmholtz system has a unique solution from a variational form that yields limited regularity for rectangular domains. When is not real the complex variational form loses positivity. We smooth the domain and, in spite of this loss, construct solutions with uniform estimates. Using the regularity, we deduce Maxwell from Helmholtz, then remove the smoothing. The boundary condition at the smoothed boundary must be carefully chosen. A method of Jerison‐Kenig‐Mitrea is extended to compensate the nonpositivity of the flux.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"41 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143393170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eric Cancès, Fabian M. Faulstich, Alfred Kirsch, Eloïse Letournel, Antoine Levitt
This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground‐state density matrix is a fixed‐point of the DMET map for non‐interacting systems, (ii) there exists a unique physical solution in the weakly‐interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific ‐representability problem inherent to DMET.
{"title":"Analysis of density matrix embedding theory around the non‐interacting limit","authors":"Eric Cancès, Fabian M. Faulstich, Alfred Kirsch, Eloïse Letournel, Antoine Levitt","doi":"10.1002/cpa.22244","DOIUrl":"https://doi.org/10.1002/cpa.22244","url":null,"abstract":"This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground‐state density matrix is a fixed‐point of the DMET map for non‐interacting systems, (ii) there exists a unique physical solution in the weakly‐interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific ‐representability problem inherent to DMET.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"55 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143125219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct special Lagrangian pair of pants in general dimensions, inside the cotangent bundle of with the Euclidean structure, building upon earlier topological ideas of Matessi. The construction uses a combination of PDE and geometric measure theory.
{"title":"Special Lagrangian pair of pants","authors":"Yang Li","doi":"10.1002/cpa.22248","DOIUrl":"https://doi.org/10.1002/cpa.22248","url":null,"abstract":"We construct special Lagrangian pair of pants in general dimensions, inside the cotangent bundle of with the Euclidean structure, building upon earlier topological ideas of Matessi. The construction uses a combination of PDE and geometric measure theory.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"7 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143056314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the global approximate controllability of the two‐dimensional incompressible Navier–Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier–Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well‐known open problem due to Agrachev.
{"title":"Localized and degenerate controls for the incompressible Navier–Stokes system","authors":"Vahagn Nersesyan, Manuel Rissel","doi":"10.1002/cpa.22246","DOIUrl":"https://doi.org/10.1002/cpa.22246","url":null,"abstract":"We consider the global approximate controllability of the two‐dimensional incompressible Navier–Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier–Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well‐known open problem due to Agrachev.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"20 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143056364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we study the co‐dimensional one interface limit and geometric motions of parabolic Ginzburg–Landau systems with potentials of high‐dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math. 65 (2012), no. 6, 833–888) to a dynamical case. In particular combining modulated energy methods and weak convergence methods, we derive the limiting harmonic heat flows in the inner and outer bulk regions segregated by the sharp interface, and a non‐standard boundary condition for them. These results are valid provided that the initial datum of the system is well‐prepared under natural energy assumptions.
在这项工作中，我们研究了具有高维井势能的抛物金兹堡-朗道系统的共维一界面极限和几何运动。主要结果概括了 Lin 等人 (Comm. Pure Appl. Math.Pure Appl.65 (2012), no. 6, 833-888）的结果。特别是结合调制能量方法和弱收敛方法，我们推导出了被尖锐界面隔离的内外块体区域的极限谐波热流，以及它们的非标准边界条件。只要系统的初始基准在自然能量假设下准备充分，这些结果就是有效的。
{"title":"Phase transition of parabolic Ginzburg–Landau equation with potentials of high‐dimensional wells","authors":"Yuning Liu","doi":"10.1002/cpa.22242","DOIUrl":"https://doi.org/10.1002/cpa.22242","url":null,"abstract":"In this work, we study the co‐dimensional one interface limit and geometric motions of parabolic Ginzburg–Landau systems with potentials of high‐dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math. 65 (2012), no. 6, 833–888) to a dynamical case. In particular combining modulated energy methods and weak convergence methods, we derive the limiting harmonic heat flows in the inner and outer bulk regions segregated by the sharp interface, and a non‐standard boundary condition for them. These results are valid provided that the initial datum of the system is well‐prepared under natural energy assumptions.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"65 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143055044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we study the critical long‐range percolation (LRP) on , where an edge connects and independently with probability 1 for and with probability for some fixed . Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to and from the interval to (conditioned on no edge joining and ) both have a polynomial lower bound in . Our bound holds for all and thus rules out a potential phase transition (around ) which seemed to be a reasonable possibility.
{"title":"Polynomial lower bound on the effective resistance for the one‐dimensional critical long‐range percolation","authors":"Jian Ding, Zherui Fan, Lu‐Jing Huang","doi":"10.1002/cpa.22243","DOIUrl":"https://doi.org/10.1002/cpa.22243","url":null,"abstract":"In this work, we study the critical long‐range percolation (LRP) on , where an edge connects and independently with probability 1 for and with probability for some fixed . Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to and from the interval to (conditioned on no edge joining and ) both have a polynomial lower bound in . Our bound holds for all and thus rules out a potential phase transition (around ) which seemed to be a reasonable possibility.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"66 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143049912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a stochastic Laplacian growth model, where a set grows according to a reflecting Brownian motion in stopped at level sets of its boundary local time. We derive a scaling limit for the leading‐order behavior of the growing boundary (i.e., “interface”). It is given by a geometric flow‐type pde. It is obtained by an averaging principle for the reflecting Brownian motion. We also show that this geometric flow‐type pde is locally well‐posed, and its blow‐up times correspond to changes in the diffeomorphism class of the growth model. Our results extend those of Dembo et al., which restricts to star‐shaped growth domains and radially outwards growth, so that in polar coordinates, the geometric flow transforms into a simple ode with infinite lifetime. Also, we remove the “separation of scales” assumption that was taken in Dembo et al.; this forces us to understand the local geometry of the growing interface.
{"title":"A flow‐type scaling limit for random growth with memory","authors":"Amir Dembo, Kevin Yang","doi":"10.1002/cpa.22241","DOIUrl":"https://doi.org/10.1002/cpa.22241","url":null,"abstract":"We study a stochastic Laplacian growth model, where a set grows according to a reflecting Brownian motion in stopped at level sets of its boundary local time. We derive a scaling limit for the leading‐order behavior of the growing boundary (i.e., “interface”). It is given by a geometric flow‐type <jats:sc>pde</jats:sc>. It is obtained by an averaging principle for the reflecting Brownian motion. We also show that this geometric flow‐type <jats:sc>pde</jats:sc> is locally well‐posed, and its blow‐up times correspond to changes in the diffeomorphism class of the growth model. Our results extend those of Dembo et al., which restricts to star‐shaped growth domains and radially outwards growth, so that in polar coordinates, the geometric flow transforms into a simple <jats:sc>ode</jats:sc> with infinite lifetime. Also, we remove the “separation of scales” assumption that was taken in Dembo et al.; this forces us to understand the local geometry of the growing interface.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"72 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142905024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new class of multilevel, adaptive, dual‐space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual‐space multilevel kernel‐splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without relying on the FFT. This requires careful attention to the discretization of the Fourier transform at each spatial scale. Like multilevel summation, we make use of a recursive (telescoping) decomposition of the original kernel into the sum of a smooth far‐field kernel, a sequence of difference kernels, and a residual kernel, which plays a role only in leaf boxes in the adaptive tree. At all higher levels in the grid hierarchy, the interaction kernels are designed to be smooth in both physical and Fourier space, admitting efficient Fourier spectral approximations. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians (SOG) with a highly localized remainder. The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.
{"title":"A dual‐space multilevel kernel‐splitting framework for discrete and continuous convolution","authors":"Shidong Jiang, Leslie Greengard","doi":"10.1002/cpa.22240","DOIUrl":"https://doi.org/10.1002/cpa.22240","url":null,"abstract":"We introduce a new class of multilevel, adaptive, dual‐space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (<jats:italic>dual‐space multilevel kernel‐splitting</jats:italic>) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without relying on the FFT. This requires careful attention to the discretization of the Fourier transform at each spatial scale. Like multilevel summation, we make use of a recursive (telescoping) decomposition of the original kernel into the sum of a smooth far‐field kernel, a sequence of difference kernels, and a residual kernel, which plays a role only in leaf boxes in the adaptive tree. At all higher levels in the grid hierarchy, the interaction kernels are designed to be smooth in both physical and Fourier space, admitting efficient Fourier spectral approximations. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians (SOG) with a highly localized remainder. The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"42 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142815825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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