Chemically reacting systems exhibiting a repeatable delay period before a visible and sudden change are referred to as clock reactions; they have a long history in education and provide an idealisation of various biochemical and industrial processes. We focus on a purely substrate-depletive clock reaction utilising vitamin C, hydrogen peroxide, iodine and starch. Building on a recent study of a simplified two-reaction model under high hydrogen peroxide concentrations, we develop a more detailed model which breaks the slow reaction into two steps, one of which is rate-limiting unless hydrogen peroxide levels are very high. Through asymptotic analysis, this model enables the effect of hydrogen peroxide concentration to be elucidated in a principled way, resolving an apparent discrepancy with earlier literature regarding the order of the slow reaction kinetics. The model is analysed in moderate- and high-hydrogen peroxide regimes, providing approximate solutions and expressions for the switchover time which take into account hydrogen peroxide concentration. The solutions are validated through simultaneously fitting the same set of parameters to several experimental series, then testing on independent experiments across widely varying hydrogen peroxide concentration. The study thereby presents and further develops a validated mechanistic understanding of a paradigm chemical kinetics system.
{"title":"Mathematical modelling of the vitamin C clock reaction: a study of two kinetic regimes","authors":"Aliya Alsaleh, David J. Smith, Sara Jabbari","doi":"arxiv-2408.04108","DOIUrl":"https://doi.org/arxiv-2408.04108","url":null,"abstract":"Chemically reacting systems exhibiting a repeatable delay period before a\u0000visible and sudden change are referred to as clock reactions; they have a long\u0000history in education and provide an idealisation of various biochemical and\u0000industrial processes. We focus on a purely substrate-depletive clock reaction\u0000utilising vitamin C, hydrogen peroxide, iodine and starch. Building on a recent\u0000study of a simplified two-reaction model under high hydrogen peroxide\u0000concentrations, we develop a more detailed model which breaks the slow reaction\u0000into two steps, one of which is rate-limiting unless hydrogen peroxide levels\u0000are very high. Through asymptotic analysis, this model enables the effect of\u0000hydrogen peroxide concentration to be elucidated in a principled way, resolving\u0000an apparent discrepancy with earlier literature regarding the order of the slow\u0000reaction kinetics. The model is analysed in moderate- and high-hydrogen\u0000peroxide regimes, providing approximate solutions and expressions for the\u0000switchover time which take into account hydrogen peroxide concentration. The\u0000solutions are validated through simultaneously fitting the same set of\u0000parameters to several experimental series, then testing on independent\u0000experiments across widely varying hydrogen peroxide concentration. The study\u0000thereby presents and further develops a validated mechanistic understanding of\u0000a paradigm chemical kinetics system.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937130","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}
A Weierstrass Prym eigenform is an Abelian differential with a single zero on a Riemann surface possessing some special kinds of symmetries. Such surfaces come equipped with an involution, known as a Prym involution. They were originally discovered by McMullen and only arise in genus 2, 3 and 4. Moreover, they are classified by two invariants: discriminant and spin. We study how the fixed points for the Prym involution of Weierstrass Prym eigenforms are permuted. In previous work, the authors computed the permutation group induced by affine transformations in the case of genus 2, showing that they are dihedral groups depending only on the residue class modulo 8 of the discriminant $D$. In this work, we complete this classification by settling the case of genus 3, showing that the permutation group induced by the affine group on the set of its three (regular) fixed points is isomorphic to $mathrm{Sym}_2$ when $D$ is even and a quadratic residue modulo 16, and to $mathrm{Sym}_3$ otherwise. The case of genus 4 is trivial as the Pyrm involution fixes a single (regular) point. In both cases, these same groups arise when considering only parabolic elements of the affine group. By recent work of Freedman, when the Teichm"uller curve induced by Weierstrass Prym eigenform is not arithmetic, the fixed points of the Prym involution coincide with the periodic points of the surface. Hence, in this case, our result also classifies how periodic points are permuted.
{"title":"Permutations of periodic points of Weierstrass Prym eigenforms","authors":"Rodolfo Gutiérrez-Romo, Angel Pardo","doi":"arxiv-2408.03832","DOIUrl":"https://doi.org/arxiv-2408.03832","url":null,"abstract":"A Weierstrass Prym eigenform is an Abelian differential with a single zero on\u0000a Riemann surface possessing some special kinds of symmetries. Such surfaces\u0000come equipped with an involution, known as a Prym involution. They were\u0000originally discovered by McMullen and only arise in genus 2, 3 and 4. Moreover,\u0000they are classified by two invariants: discriminant and spin. We study how the fixed points for the Prym involution of Weierstrass Prym\u0000eigenforms are permuted. In previous work, the authors computed the permutation\u0000group induced by affine transformations in the case of genus 2, showing that\u0000they are dihedral groups depending only on the residue class modulo 8 of the\u0000discriminant $D$. In this work, we complete this classification by settling the\u0000case of genus 3, showing that the permutation group induced by the affine group\u0000on the set of its three (regular) fixed points is isomorphic to\u0000$mathrm{Sym}_2$ when $D$ is even and a quadratic residue modulo 16, and to\u0000$mathrm{Sym}_3$ otherwise. The case of genus 4 is trivial as the Pyrm\u0000involution fixes a single (regular) point. In both cases, these same groups\u0000arise when considering only parabolic elements of the affine group. By recent work of Freedman, when the Teichm\"uller curve induced by\u0000Weierstrass Prym eigenform is not arithmetic, the fixed points of the Prym\u0000involution coincide with the periodic points of the surface. Hence, in this\u0000case, our result also classifies how periodic points are permuted.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937034","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 construct examples of continuous $mathrm{GL}(2,mathbb{R})$-cocycles which are not uniformly hyperbolic despite having the same non-zero Lyapunov exponents with respect to all invariant measures. The base dynamics can be any non-trivial subshift of finite type. According to a theorem of DeWitt--Gogolev and Guysinsky, such cocycles cannot be H"older-continuous. Our construction uses the nonuniformly hyperbolic cocycles discovered by Walters in 1984.
{"title":"Monochromatic nonuniform hyperbolicity","authors":"Jairo Bochi","doi":"arxiv-2408.03878","DOIUrl":"https://doi.org/arxiv-2408.03878","url":null,"abstract":"We construct examples of continuous $mathrm{GL}(2,mathbb{R})$-cocycles\u0000which are not uniformly hyperbolic despite having the same non-zero Lyapunov\u0000exponents with respect to all invariant measures. The base dynamics can be any\u0000non-trivial subshift of finite type. According to a theorem of DeWitt--Gogolev\u0000and Guysinsky, such cocycles cannot be H\"older-continuous. Our construction\u0000uses the nonuniformly hyperbolic cocycles discovered by Walters in 1984.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"198 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937028","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}
Sagar Ojha, Karl Leodler, Lou Barbieri, TseHuai Wu
This paper presents a unified algorithm for motion and force control for a six degree-of-freedom spatial manipulator. The motion-force controller performs trajectory tracking, maneuvering the manipulator's end-effector through desired position, orientations and rates. When contacting an obstacle or target object, the force module of the controller restricts the manipulator movements with a novel force exertion method, which prevents damage to the manipulator, the end-effector, and the objects during the contact or collision. The core strategy presented in this paper is to design the linear acceleration for the end-effector which ensures both trajectory tracking and restriction of any contact force at the end-effector. The design of the controller is validated through numerical simulations and digital twin validation.
{"title":"Force-Motion Control For A Six Degree-Of-Freedom Robotic Manipulator","authors":"Sagar Ojha, Karl Leodler, Lou Barbieri, TseHuai Wu","doi":"arxiv-2408.04106","DOIUrl":"https://doi.org/arxiv-2408.04106","url":null,"abstract":"This paper presents a unified algorithm for motion and force control for a\u0000six degree-of-freedom spatial manipulator. The motion-force controller performs\u0000trajectory tracking, maneuvering the manipulator's end-effector through desired\u0000position, orientations and rates. When contacting an obstacle or target object,\u0000the force module of the controller restricts the manipulator movements with a\u0000novel force exertion method, which prevents damage to the manipulator, the\u0000end-effector, and the objects during the contact or collision. The core\u0000strategy presented in this paper is to design the linear acceleration for the\u0000end-effector which ensures both trajectory tracking and restriction of any\u0000contact force at the end-effector. The design of the controller is validated\u0000through numerical simulations and digital twin validation.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937132","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 describe the tangent vectors of the stable and unstable manifold of a class of Anosov diffeomorphisms on the torus $mathbb{T}^2$ using the method of formal series and derivative trees. We start with linear automorphism that is hyperbolic and whose eigenvectors are orthogonal. Then we study the perturbation of such maps by trigonometric polynomial. It is known that there exist a (continuous) map $H$ which acts as a change of coordinate between the perturbed and unperturbed system, but such a map is in general, not differentiable. By "re-scaling" the parametrization $H$, we will be able to obtain the explicit formula for the tangent vectors of these maps.
{"title":"Tangent Space of the Stable And Unstable Manifold of Anosov Diffeomorphism on 2-Torus","authors":"Federico Bonneto, Jack Wang, Vishal Kumar","doi":"arxiv-2408.03607","DOIUrl":"https://doi.org/arxiv-2408.03607","url":null,"abstract":"In this paper we describe the tangent vectors of the stable and unstable\u0000manifold of a class of Anosov diffeomorphisms on the torus $mathbb{T}^2$ using\u0000the method of formal series and derivative trees. We start with linear\u0000automorphism that is hyperbolic and whose eigenvectors are orthogonal. Then we\u0000study the perturbation of such maps by trigonometric polynomial. It is known\u0000that there exist a (continuous) map $H$ which acts as a change of coordinate\u0000between the perturbed and unperturbed system, but such a map is in general, not\u0000differentiable. By \"re-scaling\" the parametrization $H$, we will be able to\u0000obtain the explicit formula for the tangent vectors of these maps.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937030","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}
For symbolic dynamics with some mild conditions, we solve the lowering topological entropy problem for subsystems and determine the Hausdorff dimension of the level set with given complexity, where the complexity is represented by Hausdorff dimension of orbit closure. These results can be applied to some dynamical systems such as $beta$-transformations, conformal expanding repeller, etc. We also determine the dimension of the Furstenberg level set, which is related to a problem of Furstenberg on the orbits under two multiplicatively independent maps.
{"title":"On orbit complexity of dynamical systems: intermediate value property and level set related to a Furstenberg problem","authors":"Yuanyang Chang, Bing Li, Meng Wu","doi":"arxiv-2408.04010","DOIUrl":"https://doi.org/arxiv-2408.04010","url":null,"abstract":"For symbolic dynamics with some mild conditions, we solve the lowering\u0000topological entropy problem for subsystems and determine the Hausdorff\u0000dimension of the level set with given complexity, where the complexity is\u0000represented by Hausdorff dimension of orbit closure. These results can be\u0000applied to some dynamical systems such as $beta$-transformations, conformal\u0000expanding repeller, etc. We also determine the dimension of the Furstenberg\u0000level set, which is related to a problem of Furstenberg on the orbits under two\u0000multiplicatively independent maps.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937027","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 study germs of holomorphic foliations, at the origin of the complex plane, tangent to Pfaffian hypersurfaces - integral hypersurfaces of real analytic 1-forms - satisfying the Rolle-Khovanskii condition. This hypothesis leads us to conclude that such a foliation is defined by a closed meromorphic 1-form, also allowing the classification of the simple models in its reduction of singularities.
{"title":"Holomorphic foliations tangent to Rolle-pfaffian hypersurfaces","authors":"Arturo Fernández-Pérez, Rogério Mol, Rudy Rosas","doi":"arxiv-2408.03914","DOIUrl":"https://doi.org/arxiv-2408.03914","url":null,"abstract":"In this paper we study germs of holomorphic foliations, at the origin of the\u0000complex plane, tangent to Pfaffian hypersurfaces - integral hypersurfaces of\u0000real analytic 1-forms - satisfying the Rolle-Khovanskii condition. This\u0000hypothesis leads us to conclude that such a foliation is defined by a closed\u0000meromorphic 1-form, also allowing the classification of the simple models in\u0000its reduction of singularities.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937117","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}
Adrian S. Wong, Robert S. Martin, Daniel Q. Eckhardt
This work explores the conditions under which global contraction manifests in the leaky continuous time reservoirs, thus guaranteeing Generalized Synchronization. Results on continuous time reservoirs make use of the logarithmic norm of the connectivity matrix. Further analysis yields some simple guidelines on how to better construct the connectivity matrix in these systems. Additionally, we outline how the Universal Approximation Property of discrete time reservoirs is readily satisfied by virtue of the activation function being contracting, and how continuous time reservoirs may inherit a limited form of universal approximation by virtue of them overlapping with Neural Ordinary Differential Equations. The ability of the Reservoir Computing framework to universally approximate topological conjugates, along with their fast training, make them a compelling data-driven, black-box surrogate of dynamical systems, and a potential mechanism for developing digital twins.
{"title":"Contraction and Synchronization in Reservoir Systems","authors":"Adrian S. Wong, Robert S. Martin, Daniel Q. Eckhardt","doi":"arxiv-2408.04058","DOIUrl":"https://doi.org/arxiv-2408.04058","url":null,"abstract":"This work explores the conditions under which global contraction manifests in\u0000the leaky continuous time reservoirs, thus guaranteeing Generalized\u0000Synchronization. Results on continuous time reservoirs make use of the\u0000logarithmic norm of the connectivity matrix. Further analysis yields some\u0000simple guidelines on how to better construct the connectivity matrix in these\u0000systems. Additionally, we outline how the Universal Approximation Property of\u0000discrete time reservoirs is readily satisfied by virtue of the activation\u0000function being contracting, and how continuous time reservoirs may inherit a\u0000limited form of universal approximation by virtue of them overlapping with\u0000Neural Ordinary Differential Equations. The ability of the Reservoir Computing\u0000framework to universally approximate topological conjugates, along with their\u0000fast training, make them a compelling data-driven, black-box surrogate of\u0000dynamical systems, and a potential mechanism for developing digital twins.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937026","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}
This paper studies the asymptotic behaviour of the solution of a differential equation perturbed by a fast flow preserving an infinite measure. This question is related with limit theorems for non-stationary Birkhoff integrals. We distinguish two settings with different behaviour: the integrable setting (no averaging phenomenon) and the case of an additive "centered" perturbation term (averaging phenomenon). The paper is motivated by the case where the perturbation comes from the Z-periodic Lorentz gas flow or from the geodesic flow over a Z-cover of a negatively curved compact surface. We establish limit theorems in more general contexts.
本文研究的是微分方程解在保持无限度量的快速流扰动下的渐近行为。这个问题与非稳态伯克霍夫积分的极限定理有关。我们区分了具有不同行为的两种情况：可积分情况（无平均现象）和有加法 "居中 "扰动项的情况（平均现象）。本文以扰动来自 Z 周期洛伦兹气体流或来自负弯曲紧凑曲面的 Z 覆盖面上的大地流的情况为出发点。我们建立了更一般情况下的极限定理。
{"title":"Slow-fast systems in infinite measure, with or without averaging","authors":"Maxence Phalempin","doi":"arxiv-2408.03009","DOIUrl":"https://doi.org/arxiv-2408.03009","url":null,"abstract":"This paper studies the asymptotic behaviour of the solution of a differential\u0000equation perturbed by a fast flow preserving an infinite measure. This question\u0000is related with limit theorems for non-stationary Birkhoff integrals. We\u0000distinguish two settings with different behaviour: the integrable setting (no\u0000averaging phenomenon) and the case of an additive \"centered\" perturbation term\u0000(averaging phenomenon). The paper is motivated by the case where the\u0000perturbation comes from the Z-periodic Lorentz gas flow or from the geodesic\u0000flow over a Z-cover of a negatively curved compact surface. We establish limit\u0000theorems in more general contexts.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937033","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}
Closure models are widely used in simulating complex multiscale dynamical systems such as turbulence and the earth system, for which direct numerical simulation that resolves all scales is often too expensive. For those systems without a clear scale separation, deterministic and local closure models often lack enough generalization capability, which limits their performance in many real-world applications. In this work, we propose a data-driven modeling framework for constructing stochastic and non-local closure models via conditional diffusion model and neural operator. Specifically, the Fourier neural operator is incorporated into a score-based diffusion model, which serves as a data-driven stochastic closure model for complex dynamical systems governed by partial differential equations (PDEs). We also demonstrate how accelerated sampling methods can improve the efficiency of the data-driven stochastic closure model. The results show that the proposed methodology provides a systematic approach via generative machine learning techniques to construct data-driven stochastic closure models for multiscale dynamical systems with continuous spatiotemporal fields.
{"title":"Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator","authors":"Xinghao Dong, Chuanqi Chen, Jin-Long Wu","doi":"arxiv-2408.02965","DOIUrl":"https://doi.org/arxiv-2408.02965","url":null,"abstract":"Closure models are widely used in simulating complex multiscale dynamical\u0000systems such as turbulence and the earth system, for which direct numerical\u0000simulation that resolves all scales is often too expensive. For those systems\u0000without a clear scale separation, deterministic and local closure models often\u0000lack enough generalization capability, which limits their performance in many\u0000real-world applications. In this work, we propose a data-driven modeling\u0000framework for constructing stochastic and non-local closure models via\u0000conditional diffusion model and neural operator. Specifically, the Fourier\u0000neural operator is incorporated into a score-based diffusion model, which\u0000serves as a data-driven stochastic closure model for complex dynamical systems\u0000governed by partial differential equations (PDEs). We also demonstrate how\u0000accelerated sampling methods can improve the efficiency of the data-driven\u0000stochastic closure model. The results show that the proposed methodology\u0000provides a systematic approach via generative machine learning techniques to\u0000construct data-driven stochastic closure models for multiscale dynamical\u0000systems with continuous spatiotemporal fields.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937047","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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