Foundations of Computational Mathematics最新文献

{"title":"Toward Computational Morse–Floer Homology: Forcing Results for Connecting Orbits by Computing Relative Indices of Critical Points","authors":"Jan Bouwe van den Berg, Marcio Gameiro, Jean-Philippe Lessard, Rob Van der Vorst","doi":"10.1007/s10208-023-09623-w","DOIUrl":"https://doi.org/10.1007/s10208-023-09623-w","url":null,"abstract":"<p>To make progress toward better computability of Morse–Floer homology and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper, we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"30 12","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509660","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}

{"title":"Improved Resolution Estimate for the Two-Dimensional Super-Resolution and a New Algorithm for Direction of Arrival Estimation with Uniform Rectangular Array","authors":"Ping Liu, H. Ammari","doi":"10.1007/s10208-023-09618-7","DOIUrl":"https://doi.org/10.1007/s10208-023-09618-7","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43551054","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}

{"title":"BGG Sequences with Weak Regularity and Applications","authors":"Andreas Čap, Kaibo Hu","doi":"10.1007/s10208-023-09608-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09608-9","url":null,"abstract":"<p>We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in <span>({mathbb {R}}^{n})</span>. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"33 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509772","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}

{"title":"Optimality of Robust Online Learning","authors":"Zheng-Chu Guo, Andreas Christmann, Lei Shi","doi":"10.1007/s10208-023-09616-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09616-9","url":null,"abstract":"<p>In this paper, we study an online learning algorithm with a robust loss function <span>(mathcal {L}_{sigma })</span> for regression over a reproducing kernel Hilbert space (RKHS). The loss function <span>(mathcal {L}_{sigma })</span> involving a scaling parameter <span>(sigma &gt;0)</span> can cover a wide range of commonly used robust losses. The proposed algorithm is then a robust alternative for online least squares regression aiming to estimate the conditional mean function. For properly chosen <span>(sigma )</span> and step size, we show that the last iterate of this online algorithm can achieve optimal capacity independent convergence in the mean square distance. Moreover, if additional information on the underlying function space is known, we also establish optimal capacity-dependent rates for strong convergence in RKHS. To the best of our knowledge, both of the two results are new to the existing literature of online learning.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"31 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534307","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}

{"title":"Further $$exists {mathbb {R}}$$-Complete Problems with PSD Matrix Factorizations","authors":"Y. Shitov","doi":"10.1007/s10208-023-09610-1","DOIUrl":"https://doi.org/10.1007/s10208-023-09610-1","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42845528","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}

{"title":"High-Order Lohner-Type Algorithm for Rigorous Computation of Poincaré Maps in Systems of Delay Differential Equations with Several Delays","authors":"Robert Szczelina, Piotr Zgliczyński","doi":"10.1007/s10208-023-09614-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09614-x","url":null,"abstract":"Abstract We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"379 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135099805","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}

{"title":"Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations","authors":"Nicolas Clozeau, Marc Josien, Felix Otto, Qiang Xu","doi":"10.1007/s10208-023-09613-y","DOIUrl":"https://doi.org/10.1007/s10208-023-09613-y","url":null,"abstract":"Abstract We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior $$a_{textrm{hom}}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mtext&gt;hom&lt;/mml:mtext&gt; &lt;/mml:msub&gt; &lt;/mml:math&gt; of a stationary random medium. The latter is described by a coefficient field a ( x ) generated from a given ensemble $$langle cdot rangle $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;⟨&lt;/mml:mo&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mo&gt;⟩&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; and the corresponding linear elliptic operator $$-nabla cdot anabla $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mi&gt;∇&lt;/mml:mi&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mi&gt;∇&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . In line with the theory of homogenization, the method proceeds by computing $$d=3$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; correctors ( d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a ( x ) from the whole-space ensemble $$langle cdot rangle $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;⟨&lt;/mml:mo&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mo&gt;⟩&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . We make this point by investigating the bias (or systematic error), i.e., the difference between $$a_{textrm{hom}}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mtext&gt;hom&lt;/mml:mtext&gt; &lt;/mml:msub&gt; &lt;/mml:math&gt; and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a ( x ), we heuristically argue that this error is generically $$O(L^{-1})$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . In case of a suitable periodization of $$langle cdot rangle $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;⟨&lt;/mml:mo&gt; &lt;mml:mo&gt;·&lt;/mml:mo&gt; &lt;mml:mo&gt;⟩&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; , we rigorously show that it is $$O(L^{-d})$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mi&gt;d&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . In fact, we give a ","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135692550","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}

{"title":"Multi-index Sequential Monte Carlo Ratio Estimators for Bayesian Inverse problems","authors":"Ajay Jasra, Kody J. H. Law, Neil Walton, Shangda Yang","doi":"10.1007/s10208-023-09612-z","DOIUrl":"https://doi.org/10.1007/s10208-023-09612-z","url":null,"abstract":"Abstract We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of $$hbox {MSE}^{-1}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msup&gt; &lt;mml:mtext&gt;MSE&lt;/mml:mtext&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;/mml:math&gt; , while single-level methods require $$hbox {MSE}^{-xi }$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msup&gt; &lt;mml:mtext&gt;MSE&lt;/mml:mtext&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;-&lt;/mml:mo&gt; &lt;mml:mi&gt;ξ&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;/mml:math&gt; for $$xi &gt;1$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ξ&lt;/mml:mi&gt; &lt;mml:mo&gt;&gt;&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where $$xi =5/4$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ξ&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mn&gt;5&lt;/mml:mn&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mn&gt;4&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; and $$xi =3/2$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ξ&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; , respectively. It is also illustrated on more challenging log-Gaussian process models, where single-level complexity is approximately $$xi =9/4$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ξ&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mn&gt;9&lt;/mml:mn&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mn&gt;4&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $$xi = 5/4 + omega $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ξ&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mn&gt;5&lt;/mml:mn&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mn&gt;4&lt;/mml:mn&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mi&gt;ω&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; , for any $$omega &gt; 0$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;ω&lt;/mml:mi&gt; &lt;mml:mo&gt;&gt;&lt;/mml:mo&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; , whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the s","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135846075","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}

{"title":"A New Approach to Handle Curved Meshes in the Hybrid High-Order Method","authors":"Liam Yemm","doi":"10.1007/s10208-023-09615-w","DOIUrl":"https://doi.org/10.1007/s10208-023-09615-w","url":null,"abstract":"Abstract We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in $$L^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and energy norms. We present numerical examples of the method on a domain with curved boundary and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135932461","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}

{"title":"Correction to: Conormal Spaces and Whitney Stratifications","authors":"M. Helmer, Vidit Nanda","doi":"10.1007/s10208-022-09602-7","DOIUrl":"https://doi.org/10.1007/s10208-022-09602-7","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"1-8"},"PeriodicalIF":3.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45899607","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}