Vasileios Chousionis, Sean Li, Vyron Vellis, Scott Zimmerman
The Heisenberg group $mathbb{H}$ equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate which textit{subsets} of $mathbb{H}$ bi-Lipschitz embed into Euclidean spaces. We show that there exists a universal constant $L>0$ such that lines $L$-bi-Lipschitz embed into $mathbb{R}^3$ and planes $L$-bi-Lipschitz embed into $mathbb{R}^4$. Moreover, $C^{1,1}$ $2$-manifolds without characteristic points as well as all $C^{1,1}$ $1$-manifolds locally $L$-bi-Lipschitz embed into $mathbb{R}^4$ where the constant $L$ is again universal. We also consider several examples of compact surfaces with characteristic points and we prove, for example, that Koranyi spheres bi-Lipschitz embed into $mathbb{R}^4$ with a uniform constant. Finally, we show that there exists a compact, porous subset of $mathbb{H}$ which does not admit a bi-Lipschitz embedding into any Euclidean space.
{"title":"Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces","authors":"Vasileios Chousionis, Sean Li, Vyron Vellis, Scott Zimmerman","doi":"10.5186/aasfm.2020.4551","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4551","url":null,"abstract":"The Heisenberg group $mathbb{H}$ equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate which textit{subsets} of $mathbb{H}$ bi-Lipschitz embed into Euclidean spaces. We show that there exists a universal constant $L>0$ such that lines $L$-bi-Lipschitz embed into $mathbb{R}^3$ and planes $L$-bi-Lipschitz embed into $mathbb{R}^4$. Moreover, $C^{1,1}$ $2$-manifolds without characteristic points as well as all $C^{1,1}$ $1$-manifolds locally $L$-bi-Lipschitz embed into $mathbb{R}^4$ where the constant $L$ is again universal. We also consider several examples of compact surfaces with characteristic points and we prove, for example, that Koranyi spheres bi-Lipschitz embed into $mathbb{R}^4$ with a uniform constant. Finally, we show that there exists a compact, porous subset of $mathbb{H}$ which does not admit a bi-Lipschitz embedding into any Euclidean space.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83683405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $R$ be a compact surface and let $Γ$ be a Jordan curve which separates $R$ into two connected components $Σ_1$ and $Σ_2$. A harmonic function $h_1$ on $Σ_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $Γ$. We show that if $Γ$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $Σ_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $Σ_1$ into $Σ_2$ is bounded with respect to the Dirichlet semi-norm.
{"title":"Transmission of harmonic functions through quasicircles on compact Riemann surfaces","authors":"Eric Schippers, W. Staubach","doi":"10.5186/aasfm.2020.4559","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4559","url":null,"abstract":"Let $R$ be a compact surface and let $Γ$ be a Jordan curve which separates $R$ into two connected components $Σ_1$ and $Σ_2$. A harmonic function $h_1$ on $Σ_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $Γ$. We show that if $Γ$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $Σ_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $Σ_1$ into $Σ_2$ is bounded with respect to the Dirichlet semi-norm.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82266428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that every non-compact weighted composition operator $f mapsto ucdot (fcircphi)$ acting on a Hardy space $H^p$ for $1 leq p < infty$ fixes an isomorphic copy of the sequence space $ell^p$ and therefore fails to be strictly singular. We also characterize those weighted composition operators on $H^p$ which fix a copy of the Hilbert space $ell^2$. These results extend earlier ones obtained for unweighted composition operators.
我们证明了作用于Hardy空间$H^p$上的每个非紧加权复合算子$f mapsto ucdot (fcircphi)$对于$1 leq p < infty$固定了序列空间$ell^p$的同构副本，因此不能是严格奇异的。我们还描述了$H^p$上的那些加权复合算子，它们固定了Hilbert空间$ell^2$的一个副本。这些结果扩展了之前对未加权复合运算符的结果。
{"title":"Rigidity of weighted composition operators on H^p","authors":"M. Lindstróm, S. Miihkinen, Pekka J. Nieminen","doi":"10.5186/aasfm.2020.4537","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4537","url":null,"abstract":"We show that every non-compact weighted composition operator $f mapsto ucdot (fcircphi)$ acting on a Hardy space $H^p$ for $1 leq p < infty$ fixes an isomorphic copy of the sequence space $ell^p$ and therefore fails to be strictly singular. We also characterize those weighted composition operators on $H^p$ which fix a copy of the Hilbert space $ell^2$. These results extend earlier ones obtained for unweighted composition operators.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84918396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that if $B subset mathbb{R}^n$ and $E subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $mathbb{R}^n$ such that every $P in E$ intersects $B$ in a set of Hausdorff dimension at least $alpha$ with $k-1 < alpha leq k$, then $dim B geq alpha +dim E/(k+1)$, where $dim$ denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every $alpha$-Furstenberg set in the plane has Hausdorff dimension at least $alpha + 1/2$. More generally, we prove that if $B$ and $E$ are as above with $0 < alpha leq k$, then $dim B geq alpha +(dim E-(k-lceil alpha rceil)(n-k))/(lceil alpha rceil+1)$. We also show that this bound is sharp for some parameters. As a consequence, we prove that for any $1 leq k
我们证明了如果$B subset mathbb{R}^n$和$E subset A(n,k)$是$mathbb{R}^n$的$k$的仿射子空间的非空集合，使得每一个$P in E$都与$B$在至少$alpha$与$k-1 < alpha leq k$的Hausdorff维数集合中相交，则$dim B geq alpha +dim E/(k+1)$，其中$dim$表示Hausdorff维数。这个估计推广了众所周知的furstenberg型估计，即平面上的每个$alpha$ -Furstenberg集至少具有$alpha + 1/2$的Hausdorff维数。更一般地，我们证明如果$B$和$E$与$0 < alpha leq k$相同，则$dim B geq alpha +(dim E-(k-lceil alpha rceil)(n-k))/(lceil alpha rceil+1)$。我们还证明了这个界对于某些参数是尖锐的。由此证明了对于任意$1 leq k
{"title":"Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces","authors":"K. H'era","doi":"10.5186/AASFM.2019.4469","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4469","url":null,"abstract":"We show that if $B subset mathbb{R}^n$ and $E subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $mathbb{R}^n$ such that every $P in E$ intersects $B$ in a set of Hausdorff dimension at least $alpha$ with $k-1 < alpha leq k$, then $dim B geq alpha +dim E/(k+1)$, where $dim$ denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every $alpha$-Furstenberg set in the plane has Hausdorff dimension at least $alpha + 1/2$. More generally, we prove that if $B$ and $E$ are as above with $0 < alpha leq k$, then $dim B geq alpha +(dim E-(k-lceil alpha rceil)(n-k))/(lceil alpha rceil+1)$. We also show that this bound is sharp for some parameters. As a consequence, we prove that for any $1 leq k<n$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of $mathbb{R}^n$ has Hausdorff dimension at least $k+frac{s}{k+1}$.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87078093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The object of this paper is to study the powered Bohr radius $rho_p$, $p in (1,2)$, of analytic functions $f(z)=sum_{k=0}^{infty} a_kz^k$ and such that $|f(z)|<1$ defined on the unit disk $|z|<1$. More precisely, if $M_p^f (r)=sum_{k=0}^infty |a_k|^p r^k$, then we show that $M_p^f (r)leq 1$ for $r leq r_p$ where $r_rho$ is the powered Bohr radius for conformal automorphisms of the unit disk. This answers the open problem posed by Djakov and Ramanujan in 2000. A couple of other consequences of our approach is also stated, including an asymptotically sharp form of one of the results of Djakov and Ramanujan. In addition, we consider a similar problem for sense-preserving harmonic mappings in $|z|<1$. Finally, we conclude by stating the Bohr radius for the class of Bieberbach-Eilenberg functions.
{"title":"On a powered Bohr inequality","authors":"I. Kayumov, S. Ponnusamy","doi":"10.5186/AASFM.2019.4416","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4416","url":null,"abstract":"The object of this paper is to study the powered Bohr radius $rho_p$, $p in (1,2)$, of analytic functions $f(z)=sum_{k=0}^{infty} a_kz^k$ and such that $|f(z)|<1$ defined on the unit disk $|z|<1$. More precisely, if $M_p^f (r)=sum_{k=0}^infty |a_k|^p r^k$, then we show that $M_p^f (r)leq 1$ for $r leq r_p$ where $r_rho$ is the powered Bohr radius for conformal automorphisms of the unit disk. This answers the open problem posed by Djakov and Ramanujan in 2000. A couple of other consequences of our approach is also stated, including an asymptotically sharp form of one of the results of Djakov and Ramanujan. In addition, we consider a similar problem for sense-preserving harmonic mappings in $|z|<1$. Finally, we conclude by stating the Bohr radius for the class of Bieberbach-Eilenberg functions.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91317357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that there exist planar Jordan domains $Omega_1$ and $Omega_2$ with boundaries of Hausdorff dimension $1$ such that any conformal maps $varphi_1 colon mathbb D to Omega_1$ and $varphi_2 colon Omega_2 to mathbb D $ cannot be extended as global homeomorphisms between the Riemann spheres of $W^{1,,1}$ class (or even not in $BV$).
我们证明存在平面Jordan域$Omega_1$和$Omega_2$，其边界为Hausdorff维数$1$，使得任何共形映射$varphi_1 colon mathbb D to Omega_1$和$varphi_2 colon Omega_2 to mathbb D $都不能被扩展为$W^{1,,1}$类的Riemann球之间的全局同态(甚至不能在$BV$中)。
{"title":"Schoenflies solutions of conformal boundary values may fail to be Sobolev","authors":"Y. Zhang","doi":"10.5186/AASFM.2019.4441","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4441","url":null,"abstract":"We show that there exist planar Jordan domains $Omega_1$ and $Omega_2$ with boundaries of Hausdorff dimension $1$ such that any conformal maps $varphi_1 colon mathbb D to Omega_1$ and $varphi_2 colon Omega_2 to mathbb D $ cannot be extended as global homeomorphisms between the Riemann spheres of $W^{1,,1}$ class (or even not in $BV$).","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90254018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose $ngeq 2$ and $mathcal{A}_{i}subset {0,1,cdots ,(n-1)}$ for $ i=1,cdots ,l,$ let $K_{i}=bigcupnolimits_{ain mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},cdots ,m_{l}in mathbb{Z}$ with $prodnolimits_{i}m_{i}neq 0,$ we let begin{equation*} S_{x}=left{ mathbf{(}y_{1},cdots ,y_{l}mathbf{)}:m_{1}y_{1}+cdots +m_{l}y_{l}=xtext{ with }y_{i}in K_{i}text{ }forall iright} . end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set begin{equation*} U_{r}={x:mathbf{Card}(S_{x})=r}, end{equation*} where $mathbf{Card}(S_{x})$ denotes the cardinality of $S_{x}$, and $rin mathbb{N}^{+}$. We prove under the so-called covering condition that the Hausdorff dimension of $U_{1}$ can be calculated in terms of some matrix. Moreover, if $rgeq 2$, we also give some sufficient conditions such that the Hausdorff dimension of $U_{r}$ takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of $U_{r}$ is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.
{"title":"Arithmetic representations of real numbers in terms of self-similar sets","authors":"Kan Jiang, Lifeng Xi","doi":"10.5186/aasfm.2019.4463","DOIUrl":"https://doi.org/10.5186/aasfm.2019.4463","url":null,"abstract":"Suppose $ngeq 2$ and $mathcal{A}_{i}subset {0,1,cdots ,(n-1)}$ for $ i=1,cdots ,l,$ let $K_{i}=bigcupnolimits_{ain mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},cdots ,m_{l}in mathbb{Z}$ with $prodnolimits_{i}m_{i}neq 0,$ we let begin{equation*} S_{x}=left{ mathbf{(}y_{1},cdots ,y_{l}mathbf{)}:m_{1}y_{1}+cdots +m_{l}y_{l}=xtext{ with }y_{i}in K_{i}text{ }forall iright} . end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set begin{equation*} U_{r}={x:mathbf{Card}(S_{x})=r}, end{equation*} where $mathbf{Card}(S_{x})$ denotes the cardinality of $S_{x}$, and $rin mathbb{N}^{+}$. We prove under the so-called covering condition that the Hausdorff dimension of $U_{1}$ can be calculated in terms of some matrix. Moreover, if $rgeq 2$, we also give some sufficient conditions such that the Hausdorff dimension of $U_{r}$ takes only finite values, and these values can be calculated explicitly. Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of $U_{r}$ is infinity. Various examples are provided. Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88208078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove an inequality which quantifies the idea that a holomorphic self-map of the disc that perturbs two points is close to the identity function.
我们证明了一个不等式，它量化了扰动两点的圆盘的全纯自映射接近恒等函数的思想。
{"title":"A hyperbolic-distance inequality for holomorphic maps","authors":"Argyrios Christodoulou, I. Short","doi":"10.5186/AASFM.2019.4425","DOIUrl":"https://doi.org/10.5186/AASFM.2019.4425","url":null,"abstract":"We prove an inequality which quantifies the idea that a holomorphic self-map of the disc that perturbs two points is close to the identity function.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88555644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.
{"title":"Differentiating Orlicz spaces with rare bases of rectangles","authors":"E. D. Aniello, L. Moonens, J. Rosenblatt","doi":"10.5186/aasfm.2020.4523","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4523","url":null,"abstract":"In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75109478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider one-dimensional Calder'on's problem for the variable exponent $p(cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(cdot)$ measurable.
{"title":"Variable exponent Calderón's problem in one dimension","authors":"Tommi Brander, D. Winterrose","doi":"10.5186/aasfm.2019.4459","DOIUrl":"https://doi.org/10.5186/aasfm.2019.4459","url":null,"abstract":"We consider one-dimensional Calder'on's problem for the variable exponent $p(cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(cdot)$ measurable.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76696133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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