Pub Date : 2020-06-08DOI: 10.26240/heal.ntua.18569
K. Mamis, Κωνσταντίνος Ι. Μαμής
The topic of this PhD thesis is the derivation of evolution equations for probability density functions (pdfs) describing the non-Markovian response to dynamical systems under Gaussian coloured (smoothly-correlated) noise. These pdf evolution equations are derived from the stochastic Liouville equations (SLEs), which are formulated by representing the pdfs as averaged random delta functions. SLEs are exact yet non-closed, since they contain averaged terms that are expressed via higher-order pdfs. These averaged terms are further evaluated by employing generalizations of the Novikov-Furutsu (NF) theorem. After the NF theorem, SLE averages are expressed equivalently as nonlocal terms depending on the whole history of the response (in some cases, on the history of excitation too). Then, nonlocal terms are approximated by a novel closure scheme, employing the history of appropriate moments of the response (or joint response-excitation moments). Application of this scheme results in a family of novel pdf evolution equations. These equations are nonlinear and retain a tractable amount of the original nonlocality of SLEs, being also in closed form and solvable. Last, the new evolution equations for the one-time response pdf are solved numerically and their results are compared to Monte Carlo (MC) simulations, for the case of a scalar bistable random differential equation under Ornstein-Uhlenbeck excitation. The results show that the novel evolution equations are in very good agreement with the MC simulations, even for high noise intensities and large correlation times of the excitation, that is, away from the white noise limit, where the existing pdf evolution equations found in literature fail. It should be noted that the computational effort for solving the new pdf evolution equations is comparable to the effort required for solving the respective classical Fokker-Planck-Kolmogorov equation.
本博士论文的主题是推导概率密度函数(pdf)的演化方程，该方程描述了高斯彩色(平滑相关)噪声下动力系统的非马尔可夫响应。这些pdf演化方程来源于随机Liouville方程(SLEs)， SLEs通过将pdf表示为平均随机函数来表示。SLEs是精确的但非封闭的，因为它们包含通过高阶pdf表示的平均项。这些平均项通过使用Novikov-Furutsu (NF)定理的推广得到进一步的评估。在NF定理之后，SLE平均根据响应的整个历史(在某些情况下，也取决于激励的历史)等效地表示为非局部项。然后，利用响应的适当矩(或联合响应-激励矩)的历史，用一种新颖的闭包格式逼近非局部项。应用该格式得到了一组新的pdf演化方程。这些方程是非线性的，保留了可处理的SLEs的原始非局域性，也是封闭形式和可解的。最后，对Ornstein-Uhlenbeck激励下的标量双稳随机微分方程进行了数值求解，并与Monte Carlo (MC)模拟结果进行了比较。结果表明，即使在高噪声强度和大相关次数的激励下，即远离白噪声极限的情况下，所建立的演化方程也能很好地与MC模拟吻合，这是现有文献中发现的pdf演化方程所不能达到的。值得注意的是，求解新的pdf进化方程的计算工作量与求解相应的经典Fokker-Planck-Kolmogorov方程所需的计算工作量相当。
{"title":"Probabilistic responses of dynamical systems subjected to gaussian coloured noise excitation: foundations of a non-markovian theory","authors":"K. Mamis, Κωνσταντίνος Ι. Μαμής","doi":"10.26240/heal.ntua.18569","DOIUrl":"https://doi.org/10.26240/heal.ntua.18569","url":null,"abstract":"The topic of this PhD thesis is the derivation of evolution equations for probability density functions (pdfs) describing the non-Markovian response to dynamical systems under Gaussian coloured (smoothly-correlated) noise. These pdf evolution equations are derived from the stochastic Liouville equations (SLEs), which are formulated by representing the pdfs as averaged random delta functions. SLEs are exact yet non-closed, since they contain averaged terms that are expressed via higher-order pdfs. These averaged terms are further evaluated by employing generalizations of the Novikov-Furutsu (NF) theorem. After the NF theorem, SLE averages are expressed equivalently as nonlocal terms depending on the whole history of the response (in some cases, on the history of excitation too). Then, nonlocal terms are approximated by a novel closure scheme, employing the history of appropriate moments of the response (or joint response-excitation moments). Application of this scheme results in a family of novel pdf evolution equations. These equations are nonlinear and retain a tractable amount of the original nonlocality of SLEs, being also in closed form and solvable. Last, the new evolution equations for the one-time response pdf are solved numerically and their results are compared to Monte Carlo (MC) simulations, for the case of a scalar bistable random differential equation under Ornstein-Uhlenbeck excitation. The results show that the novel evolution equations are in very good agreement with the MC simulations, even for high noise intensities and large correlation times of the excitation, that is, away from the white noise limit, where the existing pdf evolution equations found in literature fail. It should be noted that the computational effort for solving the new pdf evolution equations is comparable to the effort required for solving the respective classical Fokker-Planck-Kolmogorov equation.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"83 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90850922","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}
Pub Date : 2020-06-05DOI: 10.1142/S0219887820300020
G. Marmo, Luca Schiavone, A. Zampini
Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem -- connecting symmetries with constants of the motion -- within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism.
{"title":"Symmetries and reduction Part I — Poisson and symplectic picture","authors":"G. Marmo, Luca Schiavone, A. Zampini","doi":"10.1142/S0219887820300020","DOIUrl":"https://doi.org/10.1142/S0219887820300020","url":null,"abstract":"Coherently with the principle of analogy suggested by Dirac, we describe a general setting for reducing a classical dynamics, and the role of the Noether theorem -- connecting symmetries with constants of the motion -- within a reduction. This is the first of two papers, and it focuses on the reduction within the Poisson and the symplectic formalism.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86153492","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}
Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $tau$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $tau$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $tau$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as "colour" and "flavour" indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.
加权星座给出黎曼球的加权分支覆盖的图形表示。它们被引入以提供$2$D Toda $tau$-超几何型函数的组合解释，在多项式权重生成函数的情况下作为加权Hurwitz数的生成函数。给定加权星座的所有顶点和边权的乘积，对所有构型求和，得到$tau$-函数。在目前的工作中，这被推广到星座，其中的加权参数是由一个合理的权重生成函数确定的。相关的$tau$-函数可以表示为双标记加权星座的权重和，每个分支覆盖的等价类都有两种加权参数。分支点的双重标记，被称为“颜色”和“味道”指数，是由于以下事实所要求的:在权重生成函数的泰勒展开中，分母参数中的特定颜色可能出现乘法，而味道标签表明了这种多重性。
{"title":"Constellations and $tau$-functions for rationally weighted Hurwitz numbers","authors":"J. Harnad, B. Runov","doi":"10.4171/AIHPD/104","DOIUrl":"https://doi.org/10.4171/AIHPD/104","url":null,"abstract":"Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $tau$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $tau$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $tau$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as \"colour\" and \"flavour\" indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85034429","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}
Pub Date : 2020-06-01DOI: 10.14311/AP.2021.61.0230
Ilyas Haouam
In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in noncommutative phase-space, as well the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic field are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.
{"title":"ON THE THREE-DIMENSIONAL PAULI EQUATION IN NONCOMMUTATIVE PHASE-SPACE","authors":"Ilyas Haouam","doi":"10.14311/AP.2021.61.0230","DOIUrl":"https://doi.org/10.14311/AP.2021.61.0230","url":null,"abstract":"In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in noncommutative phase-space, as well the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic field are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80614518","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}
Pub Date : 2020-05-25DOI: 10.1142/S0217732320502454
S. Rajeev
Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.
{"title":"Infinitesimal quantum group from helicity in fluid mechanics","authors":"S. Rajeev","doi":"10.1142/S0217732320502454","DOIUrl":"https://doi.org/10.1142/S0217732320502454","url":null,"abstract":"Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82625857","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}
Pub Date : 2020-05-25DOI: 10.13140/RG.2.2.30758.55364
F. Sabetghadam
A generalized Weyl integrable geometry (GWIG) is obtained from simultaneous affine transformations of the tangent and cotangent bundles of a (pseudo)-Riemannian manifold. In comparison with the classical Weyl integrable geometry (CWIG), there are two generalizations here: interactions with an arbitrary dark field, and, anisotropic dilation. It means that CWIG already has interactions with a {it null} dark field. Some classical mathematics and physics problems may be addressed in GWIG. For example, by derivation of Maxwell's equations and its sub-sets, the conservation, hyperbolic, and elliptic equations on GWIG; we imposed interactions with arbitrary dark fields. Moreover, by using a notion analogous to Penrose conformal infinity, one can impose boundary conditions canonically on these equations. As a prime example, we did it for the elliptic equation, where we obtained a singularity-free potential theory. Then we used this potential theory in the construction of a non-singular model for a point charged particle. It solves the difficulty of infinite energy of the classical vacuum state.
利用(伪)黎曼流形的正切束和余切束的同时仿射变换，得到了广义Weyl可积几何(GWIG)。与经典的Weyl可积几何(CWIG)相比，这里有两个推广:与任意暗场的相互作用，以及各向异性膨胀。这意味着CWIG已经与{ It null}暗域进行了交互。一些经典的数学和物理问题可以在GWIG中解决。例如，通过推导麦克斯韦方程组及其子集，GWIG上的守恒方程、双曲方程和椭圆方程;我们用任意的暗场来施加相互作用。此外，利用类似于彭罗斯共形无穷的概念，可以在这些方程上正则地施加边界条件。作为一个主要的例子，我们对椭圆方程做了这个，我们得到了一个无奇点的势理论。然后我们用这个势理论建立了点带电粒子的非奇异模型。它解决了经典真空态能量无限的难题。
{"title":"Dark Fields do Exist in Weyl Geometry","authors":"F. Sabetghadam","doi":"10.13140/RG.2.2.30758.55364","DOIUrl":"https://doi.org/10.13140/RG.2.2.30758.55364","url":null,"abstract":"A generalized Weyl integrable geometry (GWIG) is obtained from simultaneous affine transformations of the tangent and cotangent bundles of a (pseudo)-Riemannian manifold. In comparison with the classical Weyl integrable geometry (CWIG), there are two generalizations here: interactions with an arbitrary dark field, and, anisotropic dilation. It means that CWIG already has interactions with a {it null} dark field. Some classical mathematics and physics problems may be addressed in GWIG. For example, by derivation of Maxwell's equations and its sub-sets, the conservation, hyperbolic, and elliptic equations on GWIG; we imposed interactions with arbitrary dark fields. Moreover, by using a notion analogous to Penrose conformal infinity, one can impose boundary conditions canonically on these equations. As a prime example, we did it for the elliptic equation, where we obtained a singularity-free potential theory. Then we used this potential theory in the construction of a non-singular model for a point charged particle. It solves the difficulty of infinite energy of the classical vacuum state.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82501175","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}
Pub Date : 2020-05-20DOI: 10.1007/978-3-030-53305-2_14
L. Nieto, M. Gadella, J. Mateos-Guilarte, J. M. Muñoz-Castañeda, C. Romaniega
{"title":"Some Recent Results on Contact or Point Supported Potentials","authors":"L. Nieto, M. Gadella, J. Mateos-Guilarte, J. M. Muñoz-Castañeda, C. Romaniega","doi":"10.1007/978-3-030-53305-2_14","DOIUrl":"https://doi.org/10.1007/978-3-030-53305-2_14","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87300937","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}
Pub Date : 2020-05-20DOI: 10.17323/1609-4514-2020-20-4-695-709
B. Derrida, Zhan Shi
We review recent results and conjectures for a simplified version of the depinning problem in presence of disorder which was introduced by Derrida and Retaux in 2014. For this toy model, the depinning transition has been predicted to be of the Berezinskii--Kosterlitz--Thouless type. Here we discuss under which integrability conditions this prediction can be proved and how it is modified otherwise.
{"title":"Results and Conjectures on a Toy Model of Depinning","authors":"B. Derrida, Zhan Shi","doi":"10.17323/1609-4514-2020-20-4-695-709","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-20-4-695-709","url":null,"abstract":"We review recent results and conjectures for a simplified version of the depinning problem in presence of disorder which was introduced by Derrida and Retaux in 2014. For this toy model, the depinning transition has been predicted to be of the Berezinskii--Kosterlitz--Thouless type. Here we discuss under which integrability conditions this prediction can be proved and how it is modified otherwise.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77593950","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}
Pub Date : 2020-05-19DOI: 10.1142/s0219025721500156
J. Dimock
We consider the quantum mechanics of a charged particle in the presence of Dirac's magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We establish a Feynman-Kac formula expressing solutions of the imaginary time Schrodinger equation as stochastic integrals.
{"title":"A Feynman–Kac formula for magnetic monopoles","authors":"J. Dimock","doi":"10.1142/s0219025721500156","DOIUrl":"https://doi.org/10.1142/s0219025721500156","url":null,"abstract":"We consider the quantum mechanics of a charged particle in the presence of Dirac's magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We establish a Feynman-Kac formula expressing solutions of the imaginary time Schrodinger equation as stochastic integrals.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90771256","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 present the correspondence between Lax integrable systems with spectral parameter on a Riemann surface, and Conformal Field Theories, in quite general set-up suggested earlier by the author. This correspondence turns out to give a prequantization of the integrable systems in question.
{"title":"Quantization of Lax integrable systems and Conformal Field Theory","authors":"O. Sheinman","doi":"10.4064/BC123-4","DOIUrl":"https://doi.org/10.4064/BC123-4","url":null,"abstract":"We present the correspondence between Lax integrable systems with spectral parameter on a Riemann surface, and Conformal Field Theories, in quite general set-up suggested earlier by the author. This correspondence turns out to give a prequantization of the integrable systems in question.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81282838","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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