Mathematics of Computation最新文献_第10页

Construction and analysis of a HDG solution for the total-flux formulation of the convected Helmholtz equation 共轭亥姆霍兹方程全通量公式的HDG解的构造和分析

2区数学

Mathematics of Computation Pub Date : 2023-05-04 DOI: 10.1090/mcom/3850

Hélène Barucq, Nathan Rouxelin, Sébastien Tordeux

引用次数: 1

Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem 奇异摄动对流扩散问题局部不连续Galerkin方法的超逼近性

2区数学

Mathematics of Computation Pub Date : 2023-05-04 DOI: 10.1090/mcom/3844

Yao Cheng, Shan Jiang, Martin Stynes

{"title":"Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem","authors":"Yao Cheng, Shan Jiang, Martin Stynes","doi":"10.1090/mcom/3844","DOIUrl":"https://doi.org/10.1090/mcom/3844","url":null,"abstract":"A singularly perturbed convection-diffusion problem posed on the unit square in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R squared\"> <mml:semantics> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 slash 2 right-parenthesis Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>−</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(N^{-(k+1/2)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> accurate in the energy norm induced by the bilinear form of the weak formulation, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\"application/x-tex\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 right-parenthesis Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mro","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136231610","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}

引用次数: 4

Divisibility conditions on the order of the reductions of algebraic numbers 代数数约化阶上的可整除性条件

2区数学

Mathematics of Computation Pub Date : 2023-05-03 DOI: 10.1090/mcom/3848

Pietro Sgobba

引用次数: 0

Splitting integrators for stochastic Lie–Poisson systems 随机Lie-Poisson系统的分裂积分器

2区数学

Mathematics of Computation Pub Date : 2023-04-27 DOI: 10.1090/mcom/3829

Charles-Edouard Bréhier, David Cohen, Tobias Jahnke

引用次数: 1

Explicit bounds for products of primes in AP AP中素数乘积的显界

2区数学

Mathematics of Computation Pub Date : 2023-04-21 DOI: 10.1090/mcom/3853

Ramachandran Balasubramanian, Olivier Ramaré, Priyamvad Srivastav

{"title":"Explicit bounds for products of primes in AP","authors":"Ramachandran Balasubramanian, Olivier Ramaré, Priyamvad Srivastav","doi":"10.1090/mcom/3853","DOIUrl":"https://doi.org/10.1090/mcom/3853","url":null,"abstract":"For all <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q greater-than-or-equal-to 2\"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">qge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and for all invertible residue classes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a\"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=\"application/x-tex\">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> modulo <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=\"application/x-tex\">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a natural number that is congruent to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"a\"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=\"application/x-tex\">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> modulo <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=\"application/x-tex\">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that is the product of exactly three primes, all of which are below <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 10 Superscript 15 Baseline q right-parenthesis Superscript 5 slash 2\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>15</mml:mn> </mml:mrow> </mml:msup> <mml:mi>q</mml:mi> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>5</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(10^{15}q)^{5/2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135463822","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}

引用次数: 0

Robust a posteriori estimates for the stochastic Cahn-Hilliard equation 随机Cahn-Hilliard方程的稳健后验估计

2区数学

Mathematics of Computation Pub Date : 2023-04-19 DOI: 10.1090/mcom/3836

L’ubomír Baňas, Christian Vieth

引用次数: 0

Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations Stiefel流形上齐次多项式优化的jacobi型算法及其在张量逼近中的应用

2区数学

Mathematics of Computation Pub Date : 2023-04-05 DOI: 10.1090/mcom/3834

Zhou Sheng, Jianze Li, Qin Ni

{"title":"Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations","authors":"Zhou Sheng, Jianze Li, Qin Ni","doi":"10.1090/mcom/3834","DOIUrl":"https://doi.org/10.1090/mcom/3834","url":null,"abstract":"This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135956560","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}

引用次数: 0

Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise 热噪声下薄膜方程的热力学一致性和保正离散化

2区数学

Mathematics of Computation Pub Date : 2023-04-04 DOI: 10.1090/mcom/3830

Benjamin Gess, Rishabh S. Gvalani, Florian Kunick, Felix Otto

{"title":"Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise","authors":"Benjamin Gess, Rishabh S. Gvalani, Florian Kunick, Felix Otto","doi":"10.1090/mcom/3830","DOIUrl":"https://doi.org/10.1090/mcom/3830","url":null,"abstract":"In micro-fluidics, both capillarity and thermal fluctuations play an important role. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=\"application/x-tex\">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> driven by space-time white noise. The (formal) gradient flow structure of its deterministic counterpart, the so-called thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure, i.e., on the level of a discretization of energy functional and dissipative metric tensor. Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Grün and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition. As a consequence, and as opposed to previous discretizations of the thin-film equation with thermal noise, the above discretization is not in need of an artificial condition at the boundary of the configuration space orthant <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace h greater-than 0 right-brace\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{h>0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (which, admittedly, could also be avoided by modelling a disjoining pressure). Thus, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more direct discretizations is that when writing the SDE in Itô’s form, which is the basis for the Euler-Mayurama time discretization, a correction term appears. We perform various numerical experiments to compare the behavior and performance of our discretization to that of a particular finite difference discretization of the equation. Among other things, we study numerically the invariance and entropic repulsion of the invariant ","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136088151","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}

引用次数: 0

Explicit and efficient error estimation for convex minimization problems 凸极小化问题的显式有效误差估计

2区数学

Mathematics of Computation Pub Date : 2023-03-22 DOI: 10.1090/mcom/3821

Sören Bartels, Alex Kaltenbach

{"title":"Explicit and efficient error estimation for convex minimization problems","authors":"Sören Bartels, Alex Kaltenbach","doi":"10.1090/mcom/3821","DOIUrl":"https://doi.org/10.1090/mcom/3821","url":null,"abstract":"We combine a systematic approach for deriving general a posteriori error estimates for convex minimization problems based on convex duality relations with a recently derived generalized Marini formula. The a posteriori error estimates are quasi constant-free and apply to a large class of variational problems including the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Dirichlet problem, as well as degenerate minimization, obstacle and image de-noising problems. In addition, these a posteriori error estimates are based on a comparison to a given non-conforming finite element solution. For the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Dirichlet problem, these a posteriori error bounds are equivalent to residual type a posteriori error bounds and, hence, reliable and efficient.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"204 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136174585","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}

引用次数: 1

Pointwise error estimates and local superconvergence of Jacobi expansions Jacobi展开的点态误差估计和局部超收敛性

2区数学

Mathematics of Computation Pub Date : 2023-03-21 DOI: 10.1090/mcom/3835

Shuhuang Xiang, Desong Kong, Guidong Liu, Li-Lian Wang

{"title":"Pointwise error estimates and local superconvergence of Jacobi expansions","authors":"Shuhuang Xiang, Desong Kong, Guidong Liu, Li-Lian Wang","doi":"10.1090/mcom/3835","DOIUrl":"https://doi.org/10.1090/mcom/3835","url":null,"abstract":"As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue x minus a EndAbsoluteValue\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mi>a</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|x-a|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue a EndAbsoluteValue greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|a|>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"95\"> <mml:semantics> <mml:mn>95</mml:mn> <mml:annotation encoding=\"application/x-tex\">95%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> range of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket negative 1 comma 1 right-bracket\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">[-1,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> except for a small neighbourhood near the singular point <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x equals a period\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">x=a.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> In this paper, we rigorously show that the Jacobi expansion for a more general class of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Phi\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Φ</mml:mi> <mml:annotation encoding=\"application/x-tex\">Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions also enjoys such a local convergence ","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338734","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}

引用次数: 0