In this work, for the range $frac{n-1}{n} < p leq 1$, we give a molecular reconstruction theorem for $H^p(mathbb{Z}^n)$. As an application of this result and the atomic decomposition developed by S. Boza and M. Carro in [Proc. R. Soc. Edinb., 132 A (1) (2002), 25-43], we prove that the discrete Riesz potential $I_{alpha}$ defined on $mathbb{Z}^n$ is a bounded operator $H^p(mathbb{Z}^n) to H^q(mathbb{Z}^n)$ for $frac{n-1}{n} < p < frac{n}{alpha}$ and $frac{1}{q} = frac{1}{p} - frac{alpha}{n}$, where $0 < alpha < n$.
在这项工作中，对于 $frac{n-1}{n} 的范围< p leq 1$，我们给出了 $H^p(mathbb{Z}^n)$ 的分子重构定理。作为这一结果以及 S. Boza 和 M. Carro 在 [Proc.R. Soc. Edinb、132 A (1) (2002)，25-43]中提出的原子分解[Proc.R. Soc. Edinb, 132 A (1) (2002)，25-43]的应用，我们证明了在 $mathbb{Z}^n$ 上定义的离散李斯势 $I_{alpha}$ 是一个有界算子$H^p(mathbb{Z}^n) to H^q(mathbb{Z}^n)$ for $frac{n-1}{n}.< p < {frac{n}{alpha}$ 并且 $frac{1}{q} = frac{1}{p}- 其中 $0< alpha < n$。
{"title":"A molecular decomposition for $H^p(mathbb{Z}^n)$ and applications","authors":"Pablo Rocha","doi":"arxiv-2408.09528","DOIUrl":"https://doi.org/arxiv-2408.09528","url":null,"abstract":"In this work, for the range $frac{n-1}{n} < p leq 1$, we give a molecular\u0000reconstruction theorem for $H^p(mathbb{Z}^n)$. As an application of this\u0000result and the atomic decomposition developed by S. Boza and M. Carro in [Proc.\u0000R. Soc. Edinb., 132 A (1) (2002), 25-43], we prove that the discrete Riesz\u0000potential $I_{alpha}$ defined on $mathbb{Z}^n$ is a bounded operator\u0000$H^p(mathbb{Z}^n) to H^q(mathbb{Z}^n)$ for $frac{n-1}{n} < p <\u0000frac{n}{alpha}$ and $frac{1}{q} = frac{1}{p} - frac{alpha}{n}$, where $0\u0000< alpha < n$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
María Ángeles García-Ferrero, Joaquim Ortega-Cerdà
In this paper, we study the stability of the concentration inequality for one-dimensional complex polynomials. We provide the stability of the local concentration inequality and a global version using a Wehrl-type entropy.
{"title":"Stability of the concentration inequality on polynomials","authors":"María Ángeles García-Ferrero, Joaquim Ortega-Cerdà","doi":"arxiv-2408.07424","DOIUrl":"https://doi.org/arxiv-2408.07424","url":null,"abstract":"In this paper, we study the stability of the concentration inequality for\u0000one-dimensional complex polynomials. We provide the stability of the local\u0000concentration inequality and a global version using a Wehrl-type entropy.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Esra Güldoğan Lekesiz, Bayram Çekim, Mehmet Ali Özarslan
In this paper, a finite set of biorthogonal polynomials in two variables is produced using Konhauser polynomials. Some properties containing operational and integral representation, Laplace transform, fractional calculus operators of this family are studied. Also, computing Fourier transform for the new set, a new family of biorthogonal functions are derived via Parseval's identity. On the other hand, this finite set is modified by adding two new parameters in order to have semigroup property and construct fractional calculus operators. Further, integral equation and integral operator are also derived for the modified version.
{"title":"The finite bivariate biorthogonal I -- Konhauser polynomials","authors":"Esra Güldoğan Lekesiz, Bayram Çekim, Mehmet Ali Özarslan","doi":"arxiv-2408.07811","DOIUrl":"https://doi.org/arxiv-2408.07811","url":null,"abstract":"In this paper, a finite set of biorthogonal polynomials in two variables is\u0000produced using Konhauser polynomials. Some properties containing operational\u0000and integral representation, Laplace transform, fractional calculus operators\u0000of this family are studied. Also, computing Fourier transform for the new set,\u0000a new family of biorthogonal functions are derived via Parseval's identity. On\u0000the other hand, this finite set is modified by adding two new parameters in\u0000order to have semigroup property and construct fractional calculus operators.\u0000Further, integral equation and integral operator are also derived for the\u0000modified version.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show sharp square function estimates for curves in the plane whose curvature degenerates at a point and estimates sharp up to endpoints for cones over these curves. To this end, for curves of finite type we extend the classical C'ordoba--Fefferman biorthogonality. For cones over degenerate curves, we analyze wave envelope estimates proved via High-Low-decomposition. The arguments are subsequently extended to the cone over the complex parabola.
{"title":"Generalized square function estimates for curves and their conical extensions","authors":"Robert Schippa","doi":"arxiv-2408.07248","DOIUrl":"https://doi.org/arxiv-2408.07248","url":null,"abstract":"We show sharp square function estimates for curves in the plane whose\u0000curvature degenerates at a point and estimates sharp up to endpoints for cones\u0000over these curves. To this end, for curves of finite type we extend the\u0000classical C'ordoba--Fefferman biorthogonality. For cones over degenerate\u0000curves, we analyze wave envelope estimates proved via High-Low-decomposition.\u0000The arguments are subsequently extended to the cone over the complex parabola.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the global comparison problem of Gini means with fixed number of variables on a subinterval $I$ of $mathbb{R}_+$, i.e., the following inequality begin{align}tag{$star$}label{ggcabs} G_{r,s}^{[n]}(x_1,dots,x_n) leq G_{p,q}^{[n]}(x_1,dots,x_n), end{align} where $ninmathbb{N},ngeq2$ is fixed, $(p,q),(r,s)inmathbb{R}^2$ and $x_1,dots,x_nin I$. Given a nonempty subinterval $I$ of $mathbb{R}_+$ and $ninmathbb{N}$, we introduce the relations [ Gamma_n(I):={((r,s),(p,q))inmathbb{R}^2timesmathbb{R}^2mid eqref{ggcabs}mbox{ holds for all } x_1,dots,x_nin I},qquad Gamma_infty(I):=bigcap_{n=1}^inftyGamma_n(I). ] In the paper, we investigate the properties of these sets and their dependence on $n$ and on the interval $I$ and we establish a characterizations of these sets via a constrained minimum problem by using a variant of the Lagrange multiplier rule. We also formulate two open problems at the end of the paper.
{"title":"Comparison of Gini means with fixed number of variables","authors":"Richárd Grünwald, Zsolt Páles","doi":"arxiv-2408.07658","DOIUrl":"https://doi.org/arxiv-2408.07658","url":null,"abstract":"In this paper, we consider the global comparison problem of Gini means with\u0000fixed number of variables on a subinterval $I$ of $mathbb{R}_+$, i.e., the\u0000following inequality begin{align}tag{$star$}label{ggcabs} G_{r,s}^{[n]}(x_1,dots,x_n) leq G_{p,q}^{[n]}(x_1,dots,x_n), end{align} where $ninmathbb{N},ngeq2$ is fixed, $(p,q),(r,s)inmathbb{R}^2$ and\u0000$x_1,dots,x_nin I$. Given a nonempty subinterval $I$ of $mathbb{R}_+$ and $ninmathbb{N}$, we\u0000introduce the relations [ Gamma_n(I):={((r,s),(p,q))inmathbb{R}^2timesmathbb{R}^2mid\u0000eqref{ggcabs}mbox{ holds for all } x_1,dots,x_nin I},qquad Gamma_infty(I):=bigcap_{n=1}^inftyGamma_n(I). ] In the paper, we investigate the properties of these sets and their\u0000dependence on $n$ and on the interval $I$ and we establish a characterizations\u0000of these sets via a constrained minimum problem by using a variant of the\u0000Lagrange multiplier rule. We also formulate two open problems at the end of the\u0000paper.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an extension of a criterion of van der Corput on uniform distribution of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions $Delta^2x_nto 0,quad n^2Delta^2x_nto infty $. Our proof is straightforward and uses a Diophantine approximation by rational numbers, while van der Corput's approach is based on some estimates of exponential sums.
我们给出了 van der Corput 关于序列均匀分布准则的扩展。也就是说，我们证明，如果一个序列 $x_n$ 是弱单调的，并且满足条件 $Delta^2x_nto0,quad n^2Delta^2x_nto infty $，那么这个序列就是均匀分布的。 我们的证明简单明了，使用的是有理数的二叉近似，而 van der Corput 的方法是基于指数和的一些估计。
{"title":"On a criterion of uniform distribution","authors":"Grigori Karagulyan, Iren Petrosyan","doi":"arxiv-2408.07061","DOIUrl":"https://doi.org/arxiv-2408.07061","url":null,"abstract":"We give an extension of a criterion of van der Corput on uniform distribution\u0000of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed\u0000modulo 1 if it is weakly monotonic and satisfies the conditions $Delta^2x_nto\u00000,quad n^2Delta^2x_nto infty $. Our proof is straightforward and uses a\u0000Diophantine approximation by rational numbers, while van der Corput's approach\u0000is based on some estimates of exponential sums.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we examine a discrete dynamical system defined by x(n+1) = Ax(n)+ w(n), where x takes values in a Hilbert space H and w is a periodic source with values in a fixed closed subspace W of H. Our goal is to identify conditions on some spatial sampling system G = {gj: j in J} of H that enable stable recovery of the unknown source term w from space-time samples {: n >=0,j in J}. We provide necessary and sufficient conditions on G = {g_j }_{j in J} to ensure stable recovery of any w in W . Additionally, we explicitly construct an operator R, dependent on G, such that R{}_n,j} = w.
本文研究了一个离散动力系统，其定义为 x(n+1) =Ax(n)+ w(n)，其中 x 取值于希尔伯特空间 H，而 w 是一个周期源，其值位于 H 的一个固定闭合子空间 W 中。我们的目标是确定 H 的某个空间采样系统 G = {gj: J 中的 j} 上的条件，以便能够从时空采样{: n >=0,J 中的 j}中恢复未知源项 w。我们提供了 G= {g_j }_{j in J} 的必要条件和充分条件，以确保稳定恢复 W 中的任何 w。此外，我们还明确构建了一个依赖于 G 的算子 R，使得 R{}_n,j} = w。
{"title":"Periodic Source Detection in Discrete Dynamical Systems via space-time sampling","authors":"Akram Aldroubi, Carlos Cabrelli, Ursula Molter","doi":"arxiv-2408.06934","DOIUrl":"https://doi.org/arxiv-2408.06934","url":null,"abstract":"In this paper, we examine a discrete dynamical system defined by x(n+1) =\u0000Ax(n)+ w(n), where x takes values in a Hilbert space H and w is a periodic\u0000source with values in a fixed closed subspace W of H. Our goal is to identify\u0000conditions on some spatial sampling system G = {gj: j in J} of H that enable\u0000stable recovery of the unknown source term w from space-time samples\u0000{<x(n),g_j>: n >=0,j in J}. We provide necessary and sufficient conditions on G\u0000= {g_j }_{j in J} to ensure stable recovery of any w in W . Additionally, we\u0000explicitly construct an operator R, dependent on G, such that\u0000R{<x(n),g_j>}_n,j} = w.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ Big((A(x/varepsilon)+B(x/varepsilon))u'(x)+c(x,u(x))Big)'= d(x,u(x)) mbox{ for } x in (0,1),; u(0)=u(1)=0. $$ Our assumptions are, roughly speaking, as follows: $A in L^infty(mathbb{R};mathbb{M}_n)$ is 1-periodic, $B in L^infty(mathbb{R};mathbb{M}_n))cap L^1(mathbb{R};mathbb{M}_n))$, $A(y)$ and $A(y)+B(y)$ are positive definite uniformly with respect to $y$, $c(x,cdot),d(x,cdot)in C^1(mathbb{R}^n;mathbb{R}^n))$, $c(cdot,u) in C([0,1];mathbb{R}^n)$ and $d(cdot,u) in L^infty((0,1);mathbb{R}^n)$. For small $varepsilon>0$ we show existence of weak solutions $u=u_varepsilon$ as well as their local uniqueness for $|u-u_0|_infty approx 0$, where $u_0$ is a given non-degenerate solution to the homogenized problem, and we prove that $|u_varepsilon-u_0|_inftyto 0$ and, if $c(cdot,u)$ is $C^1$-smooth, that $|u_varepsilon-u_0|_infty=O(varepsilon)$ for $varepsilon to 0$. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singular perturbation as well as to periodic homogenization of nonlinear ODEs and PDEs and, hence, which permits a common approach to existence and local uniqueness results for singularly perturbed problems and for homogenization problems.
{"title":"Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates","authors":"Lutz Recke","doi":"arxiv-2408.06705","DOIUrl":"https://doi.org/arxiv-2408.06705","url":null,"abstract":"We consider periodic homogenization with localized defects of boundary value\u0000problems for semilinear ODE systems of the type $$\u0000Big((A(x/varepsilon)+B(x/varepsilon))u'(x)+c(x,u(x))Big)'= d(x,u(x)) mbox{\u0000for } x in (0,1),; u(0)=u(1)=0. $$ Our assumptions are, roughly speaking, as\u0000follows: $A in L^infty(mathbb{R};mathbb{M}_n)$ is 1-periodic, $B in\u0000L^infty(mathbb{R};mathbb{M}_n))cap L^1(mathbb{R};mathbb{M}_n))$, $A(y)$\u0000and $A(y)+B(y)$ are positive definite uniformly with respect to $y$,\u0000$c(x,cdot),d(x,cdot)in C^1(mathbb{R}^n;mathbb{R}^n))$, $c(cdot,u) in\u0000C([0,1];mathbb{R}^n)$ and $d(cdot,u) in L^infty((0,1);mathbb{R}^n)$. For\u0000small $varepsilon>0$ we show existence of weak solutions $u=u_varepsilon$ as\u0000well as their local uniqueness for $|u-u_0|_infty approx 0$, where $u_0$ is\u0000a given non-degenerate solution to the homogenized problem, and we prove that\u0000$|u_varepsilon-u_0|_inftyto 0$ and, if $c(cdot,u)$ is $C^1$-smooth, that\u0000$|u_varepsilon-u_0|_infty=O(varepsilon)$ for $varepsilon to 0$. The main\u0000tool of the proofs is an abstract result of implicit function theorem type\u0000which in the past has been applied to singular perturbation as well as to\u0000periodic homogenization of nonlinear ODEs and PDEs and, hence, which permits a\u0000common approach to existence and local uniqueness results for singularly\u0000perturbed problems and for homogenization problems.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we discuss the problem of interpolation on straight lines by linear combinations of ridge functions with fixed directions. By using some geometry and/or systems of linear equations, we constructively prove that it is impossible to interpolate arbitrary data on any three or more straight lines by sums of ridge functions with two fixed directions. The general case with more straight lines and more directions is reduced to the problem of existence of certain sets in the union of these lines.
{"title":"A note on the problem of straight-line interpolation by ridge functions","authors":"Azer Akhmedov, Vugar Ismailov","doi":"arxiv-2408.06443","DOIUrl":"https://doi.org/arxiv-2408.06443","url":null,"abstract":"In this paper we discuss the problem of interpolation on straight lines by\u0000linear combinations of ridge functions with fixed directions. By using some\u0000geometry and/or systems of linear equations, we constructively prove that it is\u0000impossible to interpolate arbitrary data on any three or more straight lines by\u0000sums of ridge functions with two fixed directions. The general case with more\u0000straight lines and more directions is reduced to the problem of existence of\u0000certain sets in the union of these lines.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"122 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We review recent results on analytical properties (monotonicity and bounds) for ratios of contiguous functions of hypergeometric type. The cases of parabolic cylinder functions and modified Bessel functions have been discussed with considerable detail in the literature, and we give a brief account of these results, completing some aspects in the case of parabolic cylinder functions. Different techniques for obtaining these bounds are considered. They are all based on simple qualitative descriptions of the solutions of associated ODEs (mainly Riccati equations, but not only Riccati). In spite of their simplicity, they provide the most accurate global bounds known so far. We also provide examples of application of these ideas to the more general cases of the Kummer confluent function and the Gauss hypergeometric function. The function ratios described in this paper are important functions appearing in a large number of applications, in which simple approximations are very often required.
{"title":"On bounds for ratios of contiguous hypergeometric functions","authors":"Javier Segura","doi":"arxiv-2408.05573","DOIUrl":"https://doi.org/arxiv-2408.05573","url":null,"abstract":"We review recent results on analytical properties (monotonicity and bounds)\u0000for ratios of contiguous functions of hypergeometric type. The cases of\u0000parabolic cylinder functions and modified Bessel functions have been discussed\u0000with considerable detail in the literature, and we give a brief account of\u0000these results, completing some aspects in the case of parabolic cylinder\u0000functions. Different techniques for obtaining these bounds are considered. They\u0000are all based on simple qualitative descriptions of the solutions of associated\u0000ODEs (mainly Riccati equations, but not only Riccati). In spite of their\u0000simplicity, they provide the most accurate global bounds known so far. We also\u0000provide examples of application of these ideas to the more general cases of the\u0000Kummer confluent function and the Gauss hypergeometric function. The function\u0000ratios described in this paper are important functions appearing in a large\u0000number of applications, in which simple approximations are very often required.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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