Discrete & Continuous Dynamical Systems - S最新文献

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On $ L^1 $ estimates of solutions of compressible viscoelastic system 可压缩粘弹性系统解的$ L^1 $估计
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-06-29 DOI: 10.3934/dcds.2021174
Y. Ishigaki
{"title":"On $ L^1 $ estimates of solutions of compressible viscoelastic system","authors":"Y. Ishigaki","doi":"10.3934/dcds.2021174","DOIUrl":"https://doi.org/10.3934/dcds.2021174","url":null,"abstract":"<p style='text-indent:20px;'>We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to <inline-formula><tex-math id=\"M2\">begin{document}$ W^{2,1} $end{document}</tex-math></inline-formula>, and is sufficiently small in <inline-formula><tex-math id=\"M3\">begin{document}$ H^4cap L^1 $end{document}</tex-math></inline-formula>, the solutions grow in time at the same rate as <inline-formula><tex-math id=\"M4\">begin{document}$ t^{frac{1}{2}} $end{document}</tex-math></inline-formula> in <inline-formula><tex-math id=\"M5\">begin{document}$ L^1 $end{document}</tex-math></inline-formula> due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90783274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local well-posedness for the Maxwell-Dirac system in temporal gauge 时间规范下麦克斯韦-狄拉克系统的局部适定性
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-06-28 DOI: 10.3934/dcds.2022008
H. Pecher
{"title":"Local well-posedness for the Maxwell-Dirac system in temporal gauge","authors":"H. Pecher","doi":"10.3934/dcds.2022008","DOIUrl":"https://doi.org/10.3934/dcds.2022008","url":null,"abstract":"<p style='text-indent:20px;'>We consider the low regularity well-posedness problem for the Maxwell-Dirac system in 3+1 dimensions:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{align*} partial^{mu} F_{mu nu} & = - langle psi, alpha_{nu} psi rangle -i alpha^{mu} partial_{mu} psi & = A_{mu} alpha^{mu} psi , , end{align*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">begin{document}$ F_{mu nu} = partial^{mu} A_{nu} - partial^{nu} A_{mu} $end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id=\"M2\">begin{document}$ alpha^{mu} $end{document}</tex-math></inline-formula> are the 4x4 Dirac matrices. We assume the temporal gauge <inline-formula><tex-math id=\"M3\">begin{document}$ A_0 = 0 $end{document}</tex-math></inline-formula> and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82921530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum 半标准映射泛化的解析线性化:收敛半径和Brjuno和
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-06-25 DOI: 10.3934/dcds.2022009
C. Chavaudret, S. Marmi
{"title":"Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum","authors":"C. Chavaudret, S. Marmi","doi":"10.3934/dcds.2022009","DOIUrl":"https://doi.org/10.3934/dcds.2022009","url":null,"abstract":"<p style='text-indent:20px;'>One considers a system on <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{C}^2 $end{document}</tex-math></inline-formula> close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of <inline-formula><tex-math id=\"M2\">begin{document}$ dalpha $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M3\">begin{document}$ din mathbb{N}^* $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M4\">begin{document}$ alpha $end{document}</tex-math></inline-formula> is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91266052","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}
引用次数: 1
On the structure of $ alpha $-limit sets of backward trajectories for graph maps 图映射的反向轨迹的$ α $极限集的结构
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-06-10 DOI: 10.3934/dcds.2021159
Magdalena Fory's-Krawiec, Jana Hant'akov'a, P. Oprocha
{"title":"On the structure of $ alpha $-limit sets of backward trajectories for graph maps","authors":"Magdalena Fory's-Krawiec, Jana Hant'akov'a, P. Oprocha","doi":"10.3934/dcds.2021159","DOIUrl":"https://doi.org/10.3934/dcds.2021159","url":null,"abstract":"<p style='text-indent:20px;'>In the paper we study what sets can be obtained as <inline-formula><tex-math id=\"M2\">begin{document}$ alpha $end{document}</tex-math></inline-formula>-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those <inline-formula><tex-math id=\"M3\">begin{document}$ alpha $end{document}</tex-math></inline-formula>-limit sets are <inline-formula><tex-math id=\"M4\">begin{document}$ omega $end{document}</tex-math></inline-formula>-limit sets and for all but finitely many points <inline-formula><tex-math id=\"M5\">begin{document}$ x $end{document}</tex-math></inline-formula>, we can obtain every <inline-formula><tex-math id=\"M6\">begin{document}$ omega $end{document}</tex-math></inline-formula>-limits set as the <inline-formula><tex-math id=\"M7\">begin{document}$ alpha $end{document}</tex-math></inline-formula>-limit set of a backward trajectory starting in <inline-formula><tex-math id=\"M8\">begin{document}$ x $end{document}</tex-math></inline-formula>. For zero entropy maps, every <inline-formula><tex-math id=\"M9\">begin{document}$ alpha $end{document}</tex-math></inline-formula>-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74748144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global propagation of singularities for discounted Hamilton-Jacobi equations 折现Hamilton-Jacobi方程奇异性的全局传播
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-06-03 DOI: 10.3934/dcds.2021179
Cui Chen, Jiahui Hong, K. Zhao
{"title":"Global propagation of singularities for discounted Hamilton-Jacobi equations","authors":"Cui Chen, Jiahui Hong, K. Zhao","doi":"10.3934/dcds.2021179","DOIUrl":"https://doi.org/10.3934/dcds.2021179","url":null,"abstract":"<p style='text-indent:20px;'>The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE333\"> begin{document}$ begin{align} lambda v(x)+H( x, Dv(x) ) = 0 , quad xin mathbb{R}^n. quadquadquad (mathrm{HJ}_{lambda})end{align} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with fixed constant <inline-formula><tex-math id=\"M1\">begin{document}$ lambdain mathbb{R}^+ $end{document}</tex-math></inline-formula>. We reduce the problem for equation <inline-formula><tex-math id=\"M2\">begin{document}$(mathrm{HJ}_{lambda})$end{document}</tex-math></inline-formula> into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of <inline-formula><tex-math id=\"M3\">begin{document}$(mathrm{HJ}_{lambda})$end{document}</tex-math></inline-formula> propagate along locally Lipschitz singular characteristics <inline-formula><tex-math id=\"M4\">begin{document}$ {{bf{x}}}(s):[0,t]to mathbb{R}^n $end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id=\"M5\">begin{document}$ t $end{document}</tex-math></inline-formula> can extend to <inline-formula><tex-math id=\"M6\">begin{document}$ +infty $end{document}</tex-math></inline-formula>. Essentially, we use <inline-formula><tex-math id=\"M7\">begin{document}$ sigma $end{document}</tex-math></inline-formula>-compactness of the Euclidean space which is different from the original construction in [<xref ref-type=\"bibr\" rid=\"b4\">4</xref>]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of <inline-formula><tex-math id=\"M8\">begin{document}$ u $end{document}</tex-math></inline-formula> and the complement of Aubry set using the basic idea from [<xref ref-type=\"bibr\" rid=\"b9\">9</xref>].</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72748514","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}
引用次数: 0
Shadowing as a structural property of the space of dynamical systems 作为动力系统空间结构性质的阴影
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-06-03 DOI: 10.3934/dcds.2021197
J. Meddaugh
{"title":"Shadowing as a structural property of the space of dynamical systems","authors":"J. Meddaugh","doi":"10.3934/dcds.2021197","DOIUrl":"https://doi.org/10.3934/dcds.2021197","url":null,"abstract":"We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate that, for this class of spaces, in order to determine whether a system has shadowing, it is sufficient to check that continuously generated pseudo-orbits can be shadowed.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89616810","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}
引用次数: 2
The nonlocal-interaction equation near attracting manifolds 吸引流形附近的非局部相互作用方程
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-06-03 DOI: 10.3934/dcds.2021142
F. Patacchini, Dejan Slepvcev
{"title":"The nonlocal-interaction equation near attracting manifolds","authors":"F. Patacchini, Dejan Slepvcev","doi":"10.3934/dcds.2021142","DOIUrl":"https://doi.org/10.3934/dcds.2021142","url":null,"abstract":"<p style='text-indent:20px;'>We study the approximation of the nonlocal-interaction equation restricted to a compact manifold <inline-formula><tex-math id=\"M1\">begin{document}$ {mathcal{M}} $end{document}</tex-math></inline-formula> embedded in <inline-formula><tex-math id=\"M2\">begin{document}$ {mathbb{R}}^d $end{document}</tex-math></inline-formula>, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on <inline-formula><tex-math id=\"M3\">begin{document}$ {mathcal{M}} $end{document}</tex-math></inline-formula> can be approximated by the classical nonlocal-interaction equation on <inline-formula><tex-math id=\"M4\">begin{document}$ {mathbb{R}}^d $end{document}</tex-math></inline-formula> by adding an external potential which strongly attracts to <inline-formula><tex-math id=\"M5\">begin{document}$ {mathcal{M}} $end{document}</tex-math></inline-formula>. The proof relies on the Sandier–Serfaty approach [<xref ref-type=\"bibr\" rid=\"b23\">23</xref>,<xref ref-type=\"bibr\" rid=\"b24\">24</xref>] to the <inline-formula><tex-math id=\"M6\">begin{document}$ Gamma $end{document}</tex-math></inline-formula>-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on <inline-formula><tex-math id=\"M7\">begin{document}$ {mathcal{M}} $end{document}</tex-math></inline-formula>, which was shown [<xref ref-type=\"bibr\" rid=\"b10\">10</xref>]. We also provide an another approximation to the interaction equation on <inline-formula><tex-math id=\"M8\">begin{document}$ {mathcal{M}} $end{document}</tex-math></inline-formula>, based on iterating approximately solving an interaction equation on <inline-formula><tex-math id=\"M9\">begin{document}$ {mathbb{R}}^d $end{document}</tex-math></inline-formula> and projecting to <inline-formula><tex-math id=\"M10\">begin{document}$ {mathcal{M}} $end{document}</tex-math></inline-formula>. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77430852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems 离散动力系统的高维拓扑和广义Hopf分岔
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-05-26 DOI: 10.3934/dcds.2021204
H. Barge, J. Sanjurjo
{"title":"Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems","authors":"H. Barge, J. Sanjurjo","doi":"10.3934/dcds.2021204","DOIUrl":"https://doi.org/10.3934/dcds.2021204","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in <inline-formula><tex-math id=\"M1\">begin{document}$ n $end{document}</tex-math></inline-formula>-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75739044","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}
引用次数: 3
Optimal control of an HIV model with a trilinear antibody growth function 具有三线性抗体生长函数的HIV模型的最优控制
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-05-21 DOI: 10.3934/dcdss.2021148
K. Allali, Sanaa Harroudi, Delfim F. M. Torres
{"title":"Optimal control of an HIV model with a trilinear antibody growth function","authors":"K. Allali, Sanaa Harroudi, Delfim F. M. Torres","doi":"10.3934/dcdss.2021148","DOIUrl":"https://doi.org/10.3934/dcdss.2021148","url":null,"abstract":"We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83924238","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}
引用次数: 4
Zero-dimensional and symbolic extensions of topological flows 拓扑流的零维和符号扩展
Discrete & Continuous Dynamical Systems - S Pub Date : 2021-05-20 DOI: 10.3934/dcds.2021148
David Burguet, Ruxi Shi
{"title":"Zero-dimensional and symbolic extensions of topological flows","authors":"David Burguet, Ruxi Shi","doi":"10.3934/dcds.2021148","DOIUrl":"https://doi.org/10.3934/dcds.2021148","url":null,"abstract":"<p style='text-indent:20px;'>A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [<xref ref-type=\"bibr\" rid=\"b6\">6</xref>] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-<inline-formula><tex-math id=\"M1\">begin{document}$ t $end{document}</tex-math></inline-formula> map admits an extension by a subshift for any <inline-formula><tex-math id=\"M2\">begin{document}$ tneq 0 $end{document}</tex-math></inline-formula>. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on <inline-formula><tex-math id=\"M3\">begin{document}$ {0,1}^{mathbb Z} $end{document}</tex-math></inline-formula> with a roof function <inline-formula><tex-math id=\"M4\">begin{document}$ f $end{document}</tex-math></inline-formula> vanishing at the zero sequence <inline-formula><tex-math id=\"M5\">begin{document}$ 0^infty $end{document}</tex-math></inline-formula> admits a principal symbolic extension or not depending on the smoothness of <inline-formula><tex-math id=\"M6\">begin{document}$ f $end{document}</tex-math></inline-formula> at <inline-formula><tex-math id=\"M7\">begin{document}$ 0^infty $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73777446","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}
引用次数: 0
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