Pub Date : 2019-10-31DOI: 10.5540/tema.2020.021.03.409
A. D. Báez Sánchez, N. Bobko
We consider an epidemiological SIR model with an infection rate depending on the recovered population. We establish sufficient conditions for existence, uniqueness, and stability (local and global) of endemic equilibria and consider also the stability of the disease-free equilibrium. We show that, in contrast with classical SIR models, a system with a recovery-dependent infection rate can have multiple endemic stable equilibria (multistability) and multiple stable and unstable saddle points of equilibria. We establish conditions for the occurrence of these phenomena and illustrate the results with some examples.
{"title":"On Equilibria Stability in an Epidemiological SIR Model with Recovery-dependent Infection Rate","authors":"A. D. Báez Sánchez, N. Bobko","doi":"10.5540/tema.2020.021.03.409","DOIUrl":"https://doi.org/10.5540/tema.2020.021.03.409","url":null,"abstract":"We consider an epidemiological SIR model with an infection rate depending on the recovered population. We establish sufficient conditions for existence, uniqueness, and stability (local and global) of endemic equilibria and consider also the stability of the disease-free equilibrium. We show that, in contrast with classical SIR models, a system with a recovery-dependent infection rate can have multiple endemic stable equilibria (multistability) and multiple stable and unstable saddle points of equilibria. We establish conditions for the occurrence of these phenomena and illustrate the results with some examples.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74449465","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}
The John-Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal function of Fefferman-Stein, while the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt and Wheeden. Applications to the context of generalized Poincar'e type inequalities and to the context of the $C_p$ class of weights are given. Extensions to the case of polynomial BMO type spaces are also given.
{"title":"Extensions of the John–Nirenberg theorem and applications","authors":"J. Canto, C. P'erez","doi":"10.1090/proc/15302","DOIUrl":"https://doi.org/10.1090/proc/15302","url":null,"abstract":"The John-Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal function of Fefferman-Stein, while the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt and Wheeden. Applications to the context of generalized Poincar'e type inequalities and to the context of the $C_p$ class of weights are given. Extensions to the case of polynomial BMO type spaces are also given.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87338628","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 establish the theory of balayage for the Riesz kernel $|x-y|^{alpha-n}$, $alphain(0,2]$, on $mathbb R^n$, $ngeqslant3$, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan's ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner equilibrium measure $gamma_A$ for $Asubsetmathbb R^n$ arbitrary, in particular given in terms of the total mass of the inner swept measure $mu^A$ with $mu$ suitably chosen. For example, $gamma_A$ exists if and only if $varepsilon^{A^*}nevarepsilon$, where $varepsilon$ is a Dirac measure at $x=0$ and $A^*$ the inverse of $A$ relative to the sphere $|x|=1$, which leads to a Wiener type criterion of inner $alpha$-irregularity. The results obtained are illustrated by examples.
{"title":"A theory of inner Riesz balayage and its applications","authors":"N. Zorii","doi":"10.4064/ba191104-31-1","DOIUrl":"https://doi.org/10.4064/ba191104-31-1","url":null,"abstract":"We establish the theory of balayage for the Riesz kernel $|x-y|^{alpha-n}$, $alphain(0,2]$, on $mathbb R^n$, $ngeqslant3$, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan's ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner equilibrium measure $gamma_A$ for $Asubsetmathbb R^n$ arbitrary, in particular given in terms of the total mass of the inner swept measure $mu^A$ with $mu$ suitably chosen. For example, $gamma_A$ exists if and only if $varepsilon^{A^*}nevarepsilon$, where $varepsilon$ is a Dirac measure at $x=0$ and $A^*$ the inverse of $A$ relative to the sphere $|x|=1$, which leads to a Wiener type criterion of inner $alpha$-irregularity. The results obtained are illustrated by examples.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88806073","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 remarkable theorem of Besicovitch is that an integrable function $f$ on $mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result in the context of ergodic theory that provides a generalization of Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable function on a standard probability space and $T$ is an invertible measure-preserving transformation on that space, then the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated ergodic maximal function $T^*f$ is finite a.e.
{"title":"A theorem of Besicovitch and a generalization of the Birkhoff Ergodic Theorem","authors":"P. Hagelstein, D. Herden, A. Stokolos","doi":"10.1090/BPROC/73","DOIUrl":"https://doi.org/10.1090/BPROC/73","url":null,"abstract":"A remarkable theorem of Besicovitch is that an integrable function $f$ on $mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result in the context of ergodic theory that provides a generalization of Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable function on a standard probability space and $T$ is an invertible measure-preserving transformation on that space, then the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated ergodic maximal function $T^*f$ is finite a.e.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82029106","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 provide the explicit expression of first order $q$-difference system for the Jackson integral of symmetric Selberg type, which is generalized from the $q$-analog of contiguity relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the $q$-KZ equation. Our main result is the explicit expression of the coefficient matrix of the $q$-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called the interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials via Jackson integral representation of symmetric Selberg type, we compute the coefficient matrix.
{"title":"q-Difference Systems for the Jackson Integral of Symmetric Selberg Type","authors":"Masahiko Ito","doi":"10.3842/sigma.2020.113","DOIUrl":"https://doi.org/10.3842/sigma.2020.113","url":null,"abstract":"We provide the explicit expression of first order $q$-difference system for the Jackson integral of symmetric Selberg type, which is generalized from the $q$-analog of contiguity relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the $q$-KZ equation. Our main result is the explicit expression of the coefficient matrix of the $q$-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called the interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials via Jackson integral representation of symmetric Selberg type, we compute the coefficient matrix.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87240540","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}
Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family. More precisely, for a mean $mathscr{D}$ like above and a sequence $(lambda_n)$ of positive weights such that $lambda_n/(lambda_1+dots+lambda_n)$ is nondecreasing, we determine the smallest number $H in (1,+infty]$ such that $$ sum_{n=1}^infty lambda_n mathscr{D}big((x_1,dots,x_n),(lambda_1,dots,lambda_n)big) le H cdot sum_{n=1}^infty lambda_n x_n text{ for all }x in ell_1(lambda). $$ It turns out that $H$ depends only on the limit of the sequence $(lambda_n/(lambda_1+dots+lambda_n))$ and the behaviour of the mean $mathscr{D}$ near zero.
利用最近关于加权均值的齐次化和Hardy性质的结果，我们建立了凹拟合均值和单调拟合均值的尖锐Hardy常数，以及这个大族的几个特殊子类。更准确地说，对于像上面这样的平均值$mathscr{D}$和一个正权重的序列$(lambda_n)$，使得$lambda_n/(lambda_1+dots+lambda_n)$是非递减的，我们确定最小的数$H in (1,+infty]$，使得$$ sum_{n=1}^infty lambda_n mathscr{D}big((x_1,dots,x_n),(lambda_1,dots,lambda_n)big) le H cdot sum_{n=1}^infty lambda_n x_n text{ for all }x in ell_1(lambda). $$事实证明，$H$仅取决于序列$(lambda_n/(lambda_1+dots+lambda_n))$的极限和平均值$mathscr{D}$接近零的行为。
{"title":"On Hardy type inequalities for weighted quasideviation means","authors":"Zsolt P'ales, P. Pasteczka","doi":"10.7153/mia-2020-23-75","DOIUrl":"https://doi.org/10.7153/mia-2020-23-75","url":null,"abstract":"Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family. More precisely, for a mean $mathscr{D}$ like above and a sequence $(lambda_n)$ of positive weights such that $lambda_n/(lambda_1+dots+lambda_n)$ is nondecreasing, we determine the smallest number $H in (1,+infty]$ such that \u0000$$ \u0000sum_{n=1}^infty \u0000lambda_n mathscr{D}big((x_1,dots,x_n),(lambda_1,dots,lambda_n)big) le H cdot sum_{n=1}^infty lambda_n x_n text{ for all }x in ell_1(lambda). \u0000$$ It turns out that $H$ depends only on the limit of the sequence $(lambda_n/(lambda_1+dots+lambda_n))$ and the behaviour of the mean $mathscr{D}$ near zero.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87204265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence $Gamma$ from the set of equivalent well-posed two-point boundary conditions to $mathrm{gl}(4,mathbb{C})$. Using $Gamma$, we derive eigenconditions for the integral operator $mathcal{K}_mathbf{M}$ for each well-posed two-point boundary condition represented by $mathbf{M} in mathrm{gl}(4,8,mathbb{C})$. Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition $mathbf{M}$ on $mathrm{Spec},mathcal{K}_mathbf{M}$, (2) they connect $mathrm{Spec},mathcal{K}_mathbf{M}$ to $mathrm{Spec},mathcal{K}_{l,alpha,k}$ whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real $lambda not in mathrm{Spec},mathcal{K}_{l,alpha,k}$, there exists a real well-posed boundary condition $mathbf{M}$ such that $lambda in mathrm{Spec},mathcal{K}_mathbf{M}$. This in particular shows that the integral operators $mathcal{K}_mathbf{M}$ arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to $mathcal{K}_{l,alpha,k}$.
考虑弹性地基上有限梁在竖向荷载作用下挠度的边值问题。我们构造了一个等价的适定两点边界条件集到$mathrm{gl}(4,mathbb{C})$的一一对应$Gamma$。利用$Gamma$，我们为每个由$mathbf{M} in mathrm{gl}(4,8,mathbb{C})$表示的适定两点边界条件导出了积分算子$mathcal{K}_mathbf{M}$的特征条件。本征条件的特殊特征包括;(1)分离了边界条件$mathbf{M}$对$mathrm{Spec},mathcal{K}_mathbf{M}$的影响;(2)将结构已知的$mathrm{Spec},mathcal{K}_mathbf{M}$与$mathrm{Spec},mathcal{K}_{l,alpha,k}$连接起来。利用本征条件，我们证明了，对于每一个非零实数$lambda not in mathrm{Spec},mathcal{K}_{l,alpha,k}$，存在一个实的适定边界条件$mathbf{M}$，使得$lambda in mathrm{Spec},mathcal{K}_mathbf{M}$。这特别表明，由适定边界条件产生的积分算子$mathcal{K}_mathbf{M}$通常可能不是正的，也不是收缩的，这与$mathcal{K}_{l,alpha,k}$相反。
{"title":"Spectral analysis for the class of integral operators arising from well-posed boundary value problems of finite beam deflection on elastic foundation: characteristic equation","authors":"S. Choi","doi":"10.4134/BKMS.B200041","DOIUrl":"https://doi.org/10.4134/BKMS.B200041","url":null,"abstract":"We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence $Gamma$ from the set of equivalent well-posed two-point boundary conditions to $mathrm{gl}(4,mathbb{C})$. Using $Gamma$, we derive eigenconditions for the integral operator $mathcal{K}_mathbf{M}$ for each well-posed two-point boundary condition represented by $mathbf{M} in mathrm{gl}(4,8,mathbb{C})$. Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition $mathbf{M}$ on $mathrm{Spec},mathcal{K}_mathbf{M}$, (2) they connect $mathrm{Spec},mathcal{K}_mathbf{M}$ to $mathrm{Spec},mathcal{K}_{l,alpha,k}$ whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real $lambda not in mathrm{Spec},mathcal{K}_{l,alpha,k}$, there exists a real well-posed boundary condition $mathbf{M}$ such that $lambda in mathrm{Spec},mathcal{K}_mathbf{M}$. This in particular shows that the integral operators $mathcal{K}_mathbf{M}$ arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to $mathcal{K}_{l,alpha,k}$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83329338","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 a pair of joint conditions on the two functions $b_1,b_2$ strictly weaker than $b_1,b_2in operatorname{BMO}$ that almost characterize the $L^2$ boundedness of the iterated commutator $[b_2,[b_1,T]]$ of these functions and a Calder'on-Zygmund operator $T.$ Namely, we sandwich this boundedness between two bisublinear mean oscillation conditions of which one is a slightly bumped up version of the other.
{"title":"Iterated commutators under a joint condition on the tuple of multiplying functions","authors":"T. Hytonen, Kangwei Li, Tuomas Oikari","doi":"10.1090/proc/15101","DOIUrl":"https://doi.org/10.1090/proc/15101","url":null,"abstract":"We present a pair of joint conditions on the two functions $b_1,b_2$ strictly weaker than $b_1,b_2in operatorname{BMO}$ that almost characterize the $L^2$ boundedness of the iterated commutator $[b_2,[b_1,T]]$ of these functions and a Calder'on-Zygmund operator $T.$ Namely, we sandwich this boundedness between two bisublinear mean oscillation conditions of which one is a slightly bumped up version of the other.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78557070","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 a companion paper [On semiclassical orthogonal polynomials via polynomial mappings, J. Math. Anal. Appl. (2017)] we proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations. In this work we use this fact to derive in an unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind. In particular we derive ordinary differential equations for these polynomials. As an application, we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of these polynomials mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.
{"title":"An electrostatic interpretation of the zeros of sieved ultraspherical polynomials","authors":"K. Castillo, M. N. de Jesus, J. Petronilho","doi":"10.1063/1.5063333","DOIUrl":"https://doi.org/10.1063/1.5063333","url":null,"abstract":"In a companion paper [On semiclassical orthogonal polynomials via polynomial mappings, J. Math. Anal. Appl. (2017)] we proved that the semiclassical class of orthogonal polynomials is stable under polynomial transformations. In this work we use this fact to derive in an unified way old and new properties concerning the sieved ultraspherical polynomials of the first and second kind. In particular we derive ordinary differential equations for these polynomials. As an application, we use the differential equation for sieved ultraspherical polynomials of the first kind to deduce that the zeros of these polynomials mark the locations of a set of particles that are in electrostatic equilibrium with respect to a particular external field.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79105562","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 : 2019-09-24DOI: 10.4007/annals.2020.192.2.6
L. Guth, Hong Wang, Ruixiang Zhang
We prove a sharp square function estimate for the cone in $mathbb{R}^3$ and consequently the local smoothing conjecture for the wave equation in $2+1$ dimensions.
{"title":"A sharp square function estimate for the cone in\u0000 ℝ3","authors":"L. Guth, Hong Wang, Ruixiang Zhang","doi":"10.4007/annals.2020.192.2.6","DOIUrl":"https://doi.org/10.4007/annals.2020.192.2.6","url":null,"abstract":"We prove a sharp square function estimate for the cone in $mathbb{R}^3$ and consequently the local smoothing conjecture for the wave equation in $2+1$ dimensions.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87514813","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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