Pub Date : 2025-05-01Epub Date: 2025-05-28DOI: 10.1016/j.rinam.2025.100590
Abdelhafid Younsi
This paper is interested in the existence of singularities for solutions of the Navier–Stokes equations in the whole space. We demonstrate the existence of initial data that leads to the unboundedness of the corresponding strong solution within a finite time. Our approach relies on lower and upper bounds of rates of decay for solutions to the Navier–Stokes equations. This result provides valuable insights into significant open problems in both physics and mathematics.
{"title":"A condition for the finite time blow up of the incompressible Navier–Stokes equations in the whole space","authors":"Abdelhafid Younsi","doi":"10.1016/j.rinam.2025.100590","DOIUrl":"10.1016/j.rinam.2025.100590","url":null,"abstract":"<div><div>This paper is interested in the existence of singularities for solutions of the Navier–Stokes equations in the whole space. We demonstrate the existence of initial data that leads to the unboundedness of the corresponding strong solution within a finite time. Our approach relies on lower and upper bounds of rates of decay for solutions to the Navier–Stokes equations. This result provides valuable insights into significant open problems in both physics and mathematics.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100590"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167461","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 : 2025-05-01Epub Date: 2025-03-22DOI: 10.1016/j.rinam.2025.100566
Junfei Guo , Zhiyuan Huang , Rui Sun , Zhao Yikai
This paper investigates the approximate controllability and approximate null controllability of a class of linear stochastic systems driven by Gaussian random measures. The analysis focuses on controlled systems featuring both deterministic and stochastic components, where the control acts on the drift and jump terms. We establish the equivalence between approximate controllability and approximate null controllability by introducing an invariant subspace , defined by the system’s parameters. The controllability of the system is shown to hinge on whether reduces to the trivial space . These findings provide a unified framework for understanding the controllability properties of stochastic systems with jump and diffusion dynamics.
{"title":"A deterministic criterion for approximate controllability of stochastic differential equations with jumps","authors":"Junfei Guo , Zhiyuan Huang , Rui Sun , Zhao Yikai","doi":"10.1016/j.rinam.2025.100566","DOIUrl":"10.1016/j.rinam.2025.100566","url":null,"abstract":"<div><div>This paper investigates the approximate controllability and approximate null controllability of a class of linear stochastic systems driven by Gaussian random measures. The analysis focuses on controlled systems featuring both deterministic and stochastic components, where the control acts on the drift and jump terms. We establish the equivalence between approximate controllability and approximate null controllability by introducing an invariant subspace <span><math><mi>V</mi></math></span>, defined by the system’s parameters. The controllability of the system is shown to hinge on whether <span><math><mi>V</mi></math></span> reduces to the trivial space <span><math><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></math></span>. These findings provide a unified framework for understanding the controllability properties of stochastic systems with jump and diffusion dynamics.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100566"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-01Epub Date: 2025-05-02DOI: 10.1016/j.rinam.2025.100561
Miao-miao Song , Zui-cha Deng , Xiang Li , Qiu Cui
In this paper, we study the convergence of the inverse drift rate problem of option pricing based on degenerate parabolic equations, aiming to recover the stock price drift rate function by known option market prices. Unlike the classical inverse parabolic equation problem, the article transforms the original problem into an inverse problem with principal coefficients of the degenerate parabolic equation over a bounded region by variable substitution, thus avoiding the error introduced by artificial truncation. Under the optimal control framework, the problem is transformed into an optimization problem, the existence of the minimal solution is proved, and a mathematical proof of the convergence of the optimal solution is given. Finally, the gradient-type iterative method is applied to obtain the numerical solution of the inverse problem, and numerical experiments are conducted to verify it. This study provides an effective theoretical framework and numerical method for inferring the stock price drift rate from the option market price.
{"title":"Convergence analysis of option drift rate inverse problem based on degenerate parabolic equation","authors":"Miao-miao Song , Zui-cha Deng , Xiang Li , Qiu Cui","doi":"10.1016/j.rinam.2025.100561","DOIUrl":"10.1016/j.rinam.2025.100561","url":null,"abstract":"<div><div>In this paper, we study the convergence of the inverse drift rate problem of option pricing based on degenerate parabolic equations, aiming to recover the stock price drift rate function by known option market prices. Unlike the classical inverse parabolic equation problem, the article transforms the original problem into an inverse problem with principal coefficients of the degenerate parabolic equation over a bounded region by variable substitution, thus avoiding the error introduced by artificial truncation. Under the optimal control framework, the problem is transformed into an optimization problem, the existence of the minimal solution is proved, and a mathematical proof of the convergence of the optimal solution is given. Finally, the gradient-type iterative method is applied to obtain the numerical solution of the inverse problem, and numerical experiments are conducted to verify it. This study provides an effective theoretical framework and numerical method for inferring the stock price drift rate from the option market price.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100561"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895722","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 : 2025-05-01Epub Date: 2025-06-09DOI: 10.1016/j.rinam.2025.100597
Soumia EL OMARI, Said Melliani
This study investigates the existence of weak solutions for nonlinear anisotropic elliptic equations characterized by non-local boundary conditions within anisotropic weighted variable exponent Sobolev spaces. By employing variational methods and compact embedding theorems tailored to anisotropic Sobolev spaces, the research focuses on understanding the impact of anisotropy, non-locality, and weighted structures on the solution behavior. We establish sufficient conditions for the existence of solutions under various boundary conditions. These results deepen the understanding of anisotropic elliptic problems by highlighting the role of weighted structures and variable exponents in the interaction between anisotropy and non-locality. The study also explores non-local boundary conditions, which may include integrals of the unknown function over parts of the domain or non-local operators, often encountered in applications such as well modeling in 3D stratified petroleum reservoirs with arbitrary geometries. This work provides a solid theoretical foundation for broader applications in engineering and physics.
{"title":"Study of nonlinear anisotropic elliptic problems with non-local boundary conditions in weighted variable exponent Sobolev spaces","authors":"Soumia EL OMARI, Said Melliani","doi":"10.1016/j.rinam.2025.100597","DOIUrl":"10.1016/j.rinam.2025.100597","url":null,"abstract":"<div><div>This study investigates the existence of weak solutions for nonlinear anisotropic elliptic equations characterized by non-local boundary conditions within anisotropic weighted variable exponent Sobolev spaces. By employing variational methods and compact embedding theorems tailored to anisotropic Sobolev spaces, the research focuses on understanding the impact of anisotropy, non-locality, and weighted structures on the solution behavior. We establish sufficient conditions for the existence of solutions under various boundary conditions. These results deepen the understanding of anisotropic elliptic problems by highlighting the role of weighted structures and variable exponents in the interaction between anisotropy and non-locality. The study also explores non-local boundary conditions, which may include integrals of the unknown function over parts of the domain or non-local operators, often encountered in applications such as well modeling in 3D stratified petroleum reservoirs with arbitrary geometries. This work provides a solid theoretical foundation for broader applications in engineering and physics.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100597"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241790","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 : 2025-05-01Epub Date: 2025-03-18DOI: 10.1016/j.rinam.2025.100564
Nyassoke Titi Gaston Clément , Sadefo Kamdem Jules , Fono Louis Aimé
This paper analyzes the optimal effort for a risk-averse fisherman where the biomass process follows a Hawkes jump–diffusion process with Gilpin–Ayala drift. The main feature of the Hawkes process is to capture the phenomenon of clustering. The price process is of the mean-reverting type. We prove a sufficient maximum principle for the optimal control of a stochastic system consisting of an SDE driven by the Hawkes process and, by the concavity of the Hamiltonian, we obtain the optimal effort of the fisherman for a risk-averse investor.
{"title":"Optimal harvest under a Gilpin–Ayala model driven by the Hawkes process","authors":"Nyassoke Titi Gaston Clément , Sadefo Kamdem Jules , Fono Louis Aimé","doi":"10.1016/j.rinam.2025.100564","DOIUrl":"10.1016/j.rinam.2025.100564","url":null,"abstract":"<div><div>This paper analyzes the optimal effort for a risk-averse fisherman where the biomass process follows a Hawkes jump–diffusion process with Gilpin–Ayala drift. The main feature of the Hawkes process is to capture the phenomenon of clustering. The price process is of the mean-reverting type. We prove a sufficient maximum principle for the optimal control of a stochastic system consisting of an SDE driven by the Hawkes process and, by the concavity of the Hamiltonian, we obtain the optimal effort of the fisherman for a risk-averse investor.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100564"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-01Epub Date: 2025-03-13DOI: 10.1016/j.rinam.2025.100549
Joshua Richland , Alexander Strang
We analyze the correlation between randomly chosen edge weights on neighboring edges in a directed graph. This shared-endpoint correlation controls the expected organization of randomly drawn edge flows, assuming each edge’s flow is conditionally independent of others given its endpoints. We model different relationships between endpoint attributes and flow by varying the kernel associated with a Gaussian process evaluated on every vertex. We then relate the expected flow structure to the smoothness of functions generated by the Gaussian process. We investigate the shared-endpoint correlation for the squared exponential, mixture, and Matèrn kernels while exploring asymptotics in smooth and rough limits.
{"title":"Shared-endpoint correlations and hierarchy in random flows on graphs","authors":"Joshua Richland , Alexander Strang","doi":"10.1016/j.rinam.2025.100549","DOIUrl":"10.1016/j.rinam.2025.100549","url":null,"abstract":"<div><div>We analyze the correlation between randomly chosen edge weights on neighboring edges in a directed graph. This shared-endpoint correlation controls the expected organization of randomly drawn edge flows, assuming each edge’s flow is conditionally independent of others given its endpoints. We model different relationships between endpoint attributes and flow by varying the kernel associated with a Gaussian process evaluated on every vertex. We then relate the expected flow structure to the smoothness of functions generated by the Gaussian process. We investigate the shared-endpoint correlation for the squared exponential, mixture, and Matèrn kernels while exploring asymptotics in smooth and rough limits.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100549"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-01Epub Date: 2025-05-03DOI: 10.1016/j.rinam.2025.100578
Wan-Yi Chiu
The standard mean–variance analysis employs quadratic optimization to determine the optimal portfolio weights and to plot the mean–variance efficient frontier (MVEF). It then indirectly evaluates the mean–variance efficiency test (MVET) by considering the maximum Sharpe ratios of the tangency portfolio within the MVEF framework, which assumes a risk-free rate. This paper integrates these procedures without considering the risk-free rate by transitioning to a regression-based efficient frontier (RBEF). The RBEF estimates the optimal portfolio weights and simultaneously implements the MVET based on an OLS F-test, offering a simpler approach to portfolio optimization.
{"title":"The regression-based efficient frontier","authors":"Wan-Yi Chiu","doi":"10.1016/j.rinam.2025.100578","DOIUrl":"10.1016/j.rinam.2025.100578","url":null,"abstract":"<div><div>The standard mean–variance analysis employs quadratic optimization to determine the optimal portfolio weights and to plot the mean–variance efficient frontier (MVEF). It then indirectly evaluates the mean–variance efficiency test (MVET) by considering the maximum Sharpe ratios of the tangency portfolio within the MVEF framework, which assumes a risk-free rate. This paper integrates these procedures without considering the risk-free rate by transitioning to a regression-based efficient frontier (RBEF). The RBEF estimates the optimal portfolio weights and simultaneously implements the MVET based on an OLS F-test, offering a simpler approach to portfolio optimization.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100578"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143899327","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 : 2025-05-01Epub Date: 2025-06-12DOI: 10.1016/j.rinam.2025.100598
Ronan Dupont
The Virtual Element Method (VEM), as a high-order polytopal method, offers significant advantages over traditional Finite Element Methods (FEM). In particular, it allows the handling of polytopal or non-conforming meshes which greatly simplificates the mesh generation procedure. In this paper, the VEM is used for the discretization of the Helmholtz equations with a Robin-type absorbing boundary condition. This problem is crucial in various fields, including coastal engineering, oceanography and the design of offshore structures. Details of the VEM implementation with Robin boundary condition are given. Numerical results on test cases with analytical solutions show that the methods can provide optimal convergence rates for smooth solutions. Then, as a more realistic test case, the computation of the eigenmodes of the port of Cherbourg is carried out.
{"title":"An arbitrary-order Virtual Element Method for the Helmholtz equation applied to wave field calculation in port","authors":"Ronan Dupont","doi":"10.1016/j.rinam.2025.100598","DOIUrl":"10.1016/j.rinam.2025.100598","url":null,"abstract":"<div><div>The Virtual Element Method (VEM), as a high-order polytopal method, offers significant advantages over traditional Finite Element Methods (FEM). In particular, it allows the handling of polytopal or non-conforming meshes which greatly simplificates the mesh generation procedure. In this paper, the VEM is used for the discretization of the Helmholtz equations with a Robin-type absorbing boundary condition. This problem is crucial in various fields, including coastal engineering, oceanography and the design of offshore structures. Details of the VEM implementation with Robin boundary condition are given. Numerical results on test cases with analytical solutions show that the methods can provide optimal convergence rates for smooth solutions. Then, as a more realistic test case, the computation of the eigenmodes of the port of Cherbourg is carried out.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100598"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144261916","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 study a predator–prey system with a generalist Leslie–Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey’s population often grows much faster than its predator, allowing us to introduce a small time scale parameter that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens–Bogdanov point.
{"title":"Singular bifurcations in a slow-fast modified Leslie-Gower model","authors":"Roberto Albarran-García , Martha Alvarez-Ramírez , Hildeberto Jardón-Kojakhmetov","doi":"10.1016/j.rinam.2025.100558","DOIUrl":"10.1016/j.rinam.2025.100558","url":null,"abstract":"<div><div>We study a predator–prey system with a generalist Leslie–Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey’s population often grows much faster than its predator, allowing us to introduce a small time scale parameter <span><math><mi>ɛ</mi></math></span> that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens–Bogdanov point.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100558"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-01Epub Date: 2025-04-25DOI: 10.1016/j.rinam.2025.100575
L. El Houari , A. Naji , F. Ghafrani , M. Lamnii
This paper solves the polyharmonic equation for the cases and , using an optimal control approach combined with the cubic quasi-interpolation spline collocation method. Specifically, the biharmonic and triharmonic problems are addressed by decomposing the high-order equation into a system of Poisson equations, which are then transformed into a minimization problem, following the principles of optimal control theory. The objective functional is constructed based on Neumann boundary conditions, while the constraints correspond to the Poisson equations resulting from the decomposition of the original problem. As the biharmonic case has been previously studied in Boudjaj et al. (2019), the main novelty of this work lies in the theoretical and numerical treatment of the triharmonic case. This case is reformulated as an optimal control problem, for which we prove the existence and uniqueness of the solution. Numerical experiments are carried out using the cubic quasi-interpolation spline collocation method. The results are compared with those obtained using the Localized Radial Basis Function (LRBFs) collocation method, highlighting the accuracy and efficiency of the proposed approach.
{"title":"Mixing cubic quasi-interpolation spline collocation method and optimal control techniques to solve polyharmonic (p=2 and p=3) problem","authors":"L. El Houari , A. Naji , F. Ghafrani , M. Lamnii","doi":"10.1016/j.rinam.2025.100575","DOIUrl":"10.1016/j.rinam.2025.100575","url":null,"abstract":"<div><div>This paper solves the polyharmonic equation for the cases <span><math><mi>p = 2</mi></math></span> and <span><math><mi>p = 3</mi></math></span>, using an optimal control approach combined with the cubic quasi-interpolation spline collocation method. Specifically, the biharmonic and triharmonic problems are addressed by decomposing the high-order equation into a system of Poisson equations, which are then transformed into a minimization problem, following the principles of optimal control theory. The objective functional is constructed based on Neumann boundary conditions, while the constraints correspond to the Poisson equations resulting from the decomposition of the original problem. As the biharmonic case has been previously studied in Boudjaj et al. (2019), the main novelty of this work lies in the theoretical and numerical treatment of the triharmonic case. This case is reformulated as an optimal control problem, for which we prove the existence and uniqueness of the solution. Numerical experiments are carried out using the cubic quasi-interpolation spline collocation method. The results are compared with those obtained using the Localized Radial Basis Function (LRBFs) collocation method, highlighting the accuracy and efficiency of the proposed approach.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100575"},"PeriodicalIF":1.4,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870710","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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