Pub Date : 2021-03-06DOI: 10.1142/S0219530521500044
Shouyou Huang, Yunlong Feng, Qiang Wu
Minimum error entropy (MEE) is an information theoretic learning approach that minimizes the information contained in the prediction error, which is measured by entropy. It has been successfully used in various machine learning tasks for its robustness to heavy-tailed distributions and outliers. In this paper, we consider its use in nonparametric regression and analyze its generalization performance from a learning theory perspective by imposing a [Formula: see text]th order moment condition on the noise variable. To this end, we establish a comparison theorem to characterize the relation between the excess generalization error and the prediction error. A relaxed Bernstein condition and concentration inequalities are used to derive error bounds and learning rates. Note that the [Formula: see text]th moment condition is rather weak particularly when [Formula: see text] because the noise variable does not even admit a finite variance in this case. Therefore, our analysis explains the robustness of MEE in the presence of heavy-tailed distributions.
{"title":"Learning theory of minimum error entropy under weak moment conditions","authors":"Shouyou Huang, Yunlong Feng, Qiang Wu","doi":"10.1142/S0219530521500044","DOIUrl":"https://doi.org/10.1142/S0219530521500044","url":null,"abstract":"Minimum error entropy (MEE) is an information theoretic learning approach that minimizes the information contained in the prediction error, which is measured by entropy. It has been successfully used in various machine learning tasks for its robustness to heavy-tailed distributions and outliers. In this paper, we consider its use in nonparametric regression and analyze its generalization performance from a learning theory perspective by imposing a [Formula: see text]th order moment condition on the noise variable. To this end, we establish a comparison theorem to characterize the relation between the excess generalization error and the prediction error. A relaxed Bernstein condition and concentration inequalities are used to derive error bounds and learning rates. Note that the [Formula: see text]th moment condition is rather weak particularly when [Formula: see text] because the noise variable does not even admit a finite variance in this case. Therefore, our analysis explains the robustness of MEE in the presence of heavy-tailed distributions.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46138615","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}
Pub Date : 2021-03-06DOI: 10.1142/S0219530521500032
Xingcai Zhou, Shaogao Lv
This paper considers a class of robust estimation problems for varying coefficient dynamic models via wavelet techniques, which can adapt to local features of the underlying functions and has less restriction to the smoothness of the functions. The convergence rates and asymptotic distributions of the robust wavelet-based estimator are established when the design variables are stationary short-range dependent (SRD) and the errors are long-range dependent (LRD). Particularly, a rate of convergence [Formula: see text] in terms of estimation consistency can be achievable when the true components satisfy certain smoothness for a LRD process. Furthermore, an asymptotic property of the proposed estimator is given to indicate the confidence level of our proposed method for varying coefficient models with LRD.
{"title":"Robust wavelet-based estimation for varying coefficient dynamic models under long-dependent structures","authors":"Xingcai Zhou, Shaogao Lv","doi":"10.1142/S0219530521500032","DOIUrl":"https://doi.org/10.1142/S0219530521500032","url":null,"abstract":"This paper considers a class of robust estimation problems for varying coefficient dynamic models via wavelet techniques, which can adapt to local features of the underlying functions and has less restriction to the smoothness of the functions. The convergence rates and asymptotic distributions of the robust wavelet-based estimator are established when the design variables are stationary short-range dependent (SRD) and the errors are long-range dependent (LRD). Particularly, a rate of convergence [Formula: see text] in terms of estimation consistency can be achievable when the true components satisfy certain smoothness for a LRD process. Furthermore, an asymptotic property of the proposed estimator is given to indicate the confidence level of our proposed method for varying coefficient models with LRD.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45665290","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}
Pub Date : 2021-02-23DOI: 10.1142/s0219530522500129
Lukas Gonon, C. Schwab
Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferental equations (PIDEs) on state spaces of possibly high dimension $d$. Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump L'{e}vy processes. We prove for such PIDEs arising from a class of jump-diffusions on $mathbb{R}^d$, that for any compact $Ksubset mathbb{R}^d$, there exist constants $C,{mathfrak{p}},{mathfrak{q}}>0$ such that for every $varepsilon in (0,1]$ and for every $din mathbb{N}$ the normalized (over $K$) DNN $L^2$-expression error of viscosity solutions of the PIDE is of size $varepsilon$ with DNN size bounded by $Cd^{mathfrak{p}}varepsilon^{-mathfrak{q}}$. In particular, the constant $C>0$ is independent of $din mathbb{N}$ and of $varepsilon in (0,1]$ and depends only on the coefficients in the PIDE and the measure used to quantify the error. This establishes that ReLU DNNs can break the curse of dimensionality (CoD for short) for viscosity solutions of linear, possibly degenerate PIDEs corresponding to Markovian jump-diffusion processes. As a consequence of the employed techniques we also obtain that expectations of a large class of path-dependent functionals of the underlying jump-diffusion processes can be expressed without the CoD.
{"title":"Deep relu neural networks overcome the curse of dimensionality for partial integrodifferential equations","authors":"Lukas Gonon, C. Schwab","doi":"10.1142/s0219530522500129","DOIUrl":"https://doi.org/10.1142/s0219530522500129","url":null,"abstract":"Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferental equations (PIDEs) on state spaces of possibly high dimension $d$. Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump L'{e}vy processes. We prove for such PIDEs arising from a class of jump-diffusions on $mathbb{R}^d$, that for any compact $Ksubset mathbb{R}^d$, there exist constants $C,{mathfrak{p}},{mathfrak{q}}>0$ such that for every $varepsilon in (0,1]$ and for every $din mathbb{N}$ the normalized (over $K$) DNN $L^2$-expression error of viscosity solutions of the PIDE is of size $varepsilon$ with DNN size bounded by $Cd^{mathfrak{p}}varepsilon^{-mathfrak{q}}$. In particular, the constant $C>0$ is independent of $din mathbb{N}$ and of $varepsilon in (0,1]$ and depends only on the coefficients in the PIDE and the measure used to quantify the error. This establishes that ReLU DNNs can break the curse of dimensionality (CoD for short) for viscosity solutions of linear, possibly degenerate PIDEs corresponding to Markovian jump-diffusion processes. As a consequence of the employed techniques we also obtain that expectations of a large class of path-dependent functionals of the underlying jump-diffusion processes can be expressed without the CoD.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41767821","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}
Pub Date : 2021-02-19DOI: 10.1142/s0219530522500026
C. Schneider, Flóra Orsolya Szemenyei
We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones [Formula: see text] in the specific scale [Formula: see text] of Besov spaces. The regularity of the solution in these spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractional Sobolev scale [Formula: see text] in all cases, which justifies the use of adaptive schemes.
{"title":"Sobolev meets Besov: Regularity for the Poisson equation with Dirichlet, Neumann and mixed boundary values","authors":"C. Schneider, Flóra Orsolya Szemenyei","doi":"10.1142/s0219530522500026","DOIUrl":"https://doi.org/10.1142/s0219530522500026","url":null,"abstract":"We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones [Formula: see text] in the specific scale [Formula: see text] of Besov spaces. The regularity of the solution in these spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractional Sobolev scale [Formula: see text] in all cases, which justifies the use of adaptive schemes.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49652181","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}
Pub Date : 2021-02-05DOI: 10.1142/s0219530522500178
A. Debrouwere, B. Prangoski
We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in [10]. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spacesM w . Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces M w (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.
{"title":"Gabor frame characterisations of generalised modulation spaces","authors":"A. Debrouwere, B. Prangoski","doi":"10.1142/s0219530522500178","DOIUrl":"https://doi.org/10.1142/s0219530522500178","url":null,"abstract":"We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in [10]. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spacesM w . Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces M w (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48117960","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}
Pub Date : 2021-01-17DOI: 10.1142/s021953052150024x
B. Rubin
We study the spherical slice transform which assigns to a function on the $n$-dimensional unit sphere the integrals of that function over cross-sections of the sphere by $k$-dimensional affine planes passing through the north pole. These transforms are well known when $k=n$. We consider all $1< k < n+1$ and obtain an explicit formula connecting the spherical slice transform with the classical Radon-John transform over $(k-1)$-dimensional planes in the $n$-dimensional Euclidean space. Using this connection, known facts for the Radon-John transform, like inversion formulas, support theorem, representation on zonal functions, and others, can be reformulated for the spherical slice transform.
我们研究了一个函数在n维单位球面上通过k维仿射平面经过北极在球面横截面上的积分的球切片变换。当k=n时，这些变换是众所周知的。我们考虑所有$1< k < n+1$，得到了在$n$维欧几里德空间中$(k-1)$维平面上的球面片变换与经典Radon-John变换之间的显式公式。利用这种联系，Radon-John变换的已知事实，如反演公式、支持定理、区域函数的表示等，可以在球片变换中重新表述。
{"title":"On the Spherical Slice Transform","authors":"B. Rubin","doi":"10.1142/s021953052150024x","DOIUrl":"https://doi.org/10.1142/s021953052150024x","url":null,"abstract":"We study the spherical slice transform which assigns to a function on the $n$-dimensional unit sphere the integrals of that function over cross-sections of the sphere by $k$-dimensional affine planes passing through the north pole. These transforms are well known when $k=n$. We consider all $1< k < n+1$ and obtain an explicit formula connecting the spherical slice transform with the classical Radon-John transform over $(k-1)$-dimensional planes in the $n$-dimensional Euclidean space. Using this connection, known facts for the Radon-John transform, like inversion formulas, support theorem, representation on zonal functions, and others, can be reformulated for the spherical slice transform.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43850063","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}
Pub Date : 2021-01-01DOI: 10.1142/s0219530521500160
Yuan Shen, Yannian Zuo, Liming Sun, Xiayang Zhang
We consider the linearly constrained separable convex optimization problem whose objective function is separable with respect to [Formula: see text] blocks of variables. A bunch of methods have been proposed and extensively studied in the past decade. Specifically, a modified strictly contractive Peaceman–Rachford splitting method (SC-PRCM) [S. H. Jiang and M. Li, A modified strictly contractive Peaceman–Rachford splitting method for multi-block separable convex programming, J. Ind. Manag. Optim. 14(1) (2018) 397-412] has been well studied in the literature for the special case of [Formula: see text]. Based on the modified SC-PRCM, we present modified proximal symmetric ADMMs (MPSADMMs) to solve the multi-block problem. In MPSADMMs, all subproblems but the first one are attached with a simple proximal term, and the multipliers are updated twice. At the end of each iteration, the output is corrected via a simple correction step. Without stringent assumptions, we establish the global convergence result and the [Formula: see text] convergence rate in the ergodic sense for the new algorithms. Preliminary numerical results show that our proposed algorithms are effective for solving the linearly constrained quadratic programming and the robust principal component analysis problems.
{"title":"Modified proximal symmetric ADMMs for multi-block separable convex optimization with linear constraints","authors":"Yuan Shen, Yannian Zuo, Liming Sun, Xiayang Zhang","doi":"10.1142/s0219530521500160","DOIUrl":"https://doi.org/10.1142/s0219530521500160","url":null,"abstract":"We consider the linearly constrained separable convex optimization problem whose objective function is separable with respect to [Formula: see text] blocks of variables. A bunch of methods have been proposed and extensively studied in the past decade. Specifically, a modified strictly contractive Peaceman–Rachford splitting method (SC-PRCM) [S. H. Jiang and M. Li, A modified strictly contractive Peaceman–Rachford splitting method for multi-block separable convex programming, J. Ind. Manag. Optim. 14(1) (2018) 397-412] has been well studied in the literature for the special case of [Formula: see text]. Based on the modified SC-PRCM, we present modified proximal symmetric ADMMs (MPSADMMs) to solve the multi-block problem. In MPSADMMs, all subproblems but the first one are attached with a simple proximal term, and the multipliers are updated twice. At the end of each iteration, the output is corrected via a simple correction step. Without stringent assumptions, we establish the global convergence result and the [Formula: see text] convergence rate in the ergodic sense for the new algorithms. Preliminary numerical results show that our proposed algorithms are effective for solving the linearly constrained quadratic programming and the robust principal component analysis problems.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63880944","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}
Pub Date : 2020-12-08DOI: 10.1142/s0219530520500207
Kui Li, R. Wong
Wiener–Hopf Equations are of the form [Formula: see text] These equations arise in many physical problems such as radiative transport theory, reflection of an electromagnetive plane wave, sound wave transmission from a tube, and in material science. They are also known as the renewal equations on the half-line in Probability Theory. In this paper, we present a method of deriving asymptotic expansions for the solutions to these equations. Our method makes use of the Wiener–Hopf technique as well as the asymptotic expansions of Stieltjes and Hilbert transforms.
{"title":"Asymptotic expansions for Wiener–Hopf equations","authors":"Kui Li, R. Wong","doi":"10.1142/s0219530520500207","DOIUrl":"https://doi.org/10.1142/s0219530520500207","url":null,"abstract":"Wiener–Hopf Equations are of the form [Formula: see text] These equations arise in many physical problems such as radiative transport theory, reflection of an electromagnetive plane wave, sound wave transmission from a tube, and in material science. They are also known as the renewal equations on the half-line in Probability Theory. In this paper, we present a method of deriving asymptotic expansions for the solutions to these equations. Our method makes use of the Wiener–Hopf technique as well as the asymptotic expansions of Stieltjes and Hilbert transforms.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45292396","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}
Pub Date : 2020-11-30DOI: 10.1142/S0219530520500165
N. Antonić, Marko Erceg, M. Mišur
We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from [Formula: see text] to [Formula: see text] by kernels, which we prove to be distributions of order [Formula: see text] in [Formula: see text], but higher, although still finite, order in [Formula: see text]. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in [Formula: see text], and of finite order in [Formula: see text]. This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations.
{"title":"Distributions of anisotropic order and applications to H-distributions","authors":"N. Antonić, Marko Erceg, M. Mišur","doi":"10.1142/S0219530520500165","DOIUrl":"https://doi.org/10.1142/S0219530520500165","url":null,"abstract":"We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from [Formula: see text] to [Formula: see text] by kernels, which we prove to be distributions of order [Formula: see text] in [Formula: see text], but higher, although still finite, order in [Formula: see text]. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in [Formula: see text], and of finite order in [Formula: see text]. This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42829424","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}
Pub Date : 2020-10-19DOI: 10.1142/s0219530520400102
F. Bao, Yanzhao Cao, J. Yong
Forward-backward stochastic differential equation (FBSDE) systems were introduced as a probabilistic description for parabolic type partial differential equations. Although the probabilistic behavior of the FBSDE system makes it a natural mathematical model in many applications, the stochastic integrals contained in the system generate uncertainties in the solutions which makes the solution estimation a challenging task. In this paper, we assume that we could receive partial noisy observations on the solutions and introduce an optimal filtering method to make a data informed solution estimation for FBSDEs.
{"title":"Data informed solution estimation for forward-backward stochastic differential equations","authors":"F. Bao, Yanzhao Cao, J. Yong","doi":"10.1142/s0219530520400102","DOIUrl":"https://doi.org/10.1142/s0219530520400102","url":null,"abstract":"Forward-backward stochastic differential equation (FBSDE) systems were introduced as a probabilistic description for parabolic type partial differential equations. Although the probabilistic behavior of the FBSDE system makes it a natural mathematical model in many applications, the stochastic integrals contained in the system generate uncertainties in the solutions which makes the solution estimation a challenging task. In this paper, we assume that we could receive partial noisy observations on the solutions and introduce an optimal filtering method to make a data informed solution estimation for FBSDEs.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45476973","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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