Proceedings of the American Mathematical Society最新文献

{"title":"Estimation of the eigenvalues and the integral of the eigenfunctions of the Newtonian potential operator","authors":"Abdulaziz Alsenafi, Ahcene Ghandriche, Mourad Sini","doi":"10.1090/proc/16871","DOIUrl":"https://doi.org/10.1090/proc/16871","url":null,"abstract":"<p>We consider the problem of estimating the eigenvalues and the integral of the corresponding eigenfunctions, associated to the Newtonian potential operator, defined in a bounded domain <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega subset-of double-struck upper R Superscript d\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Omega subset mathbb {R}^{d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d equals 2 comma 3\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d=2,3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in terms of the maximum radius of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Ω</mml:mi> <mml:annotation encoding=\"application/x-tex\">Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We first provide these estimations in the particular case of a ball and a disc. Then we extend them to general shapes using a, derived, monotonicity property of the eigenvalues of the Newtonian operator. The derivation of the lower bounds is quite tedious for the 2D-Logarithmic potential operator. Such upper/lower bounds appear naturally while estimating the electric/acoustic fields propagating in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript d\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {R}^{d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the presence of small scaled and highly heterogeneous particles.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Factorization of functions in the Schur-Agler class related to test functions","authors":"Mainak Bhowmik, Poornendu Kumar","doi":"10.1090/proc/16900","DOIUrl":"https://doi.org/10.1090/proc/16900","url":null,"abstract":"<p>We provide necessary and sufficient conditions for operator-valued functions on arbitrary sets associated with a collection of test functions to have factorizations in several situations.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A non-vanishing result on the singularity category","authors":"Xiao-Wu Chen, Zhi-Wei Li, Xiaojin Zhang, Zhibing Zhao","doi":"10.1090/proc/16898","DOIUrl":"https://doi.org/10.1090/proc/16898","url":null,"abstract":"<p>We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomologies of differential graded Leavitt algebras.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Higher residues and canonical pairing on the twisted de Rham cohomology","authors":"Hoil Kim, Taejung Kim","doi":"10.1090/proc/16883","DOIUrl":"https://doi.org/10.1090/proc/16883","url":null,"abstract":"<p>We describe an explicit formula of the canonical pairing on the twisted de Rham cohomology associated with the category of local matrix factorizations and by characterizing its relation to Saito’s higher residue pairings we reprove the conjecture of Shklyarov [Adv. Math. 292 (2016), pp. 181–209].</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Detecting nontrivial products in the stable homotopy ring of spheres via the third Morava stabilizer algebra","authors":"Xiangjun Wang, Jianqiu Wu, Yu Zhang, Linan Zhong","doi":"10.1090/proc/16891","DOIUrl":"https://doi.org/10.1090/proc/16891","url":null,"abstract":"&lt;p&gt;Let &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than-or-equal-to 7\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;mml:mo&gt;≥&lt;/mml:mo&gt; &lt;mml:mn&gt;7&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;p geq 7&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; be a prime number. Let &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S left-parenthesis 3 right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;S(3)&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; denote the third Morava stabilizer algebra. In recent years, Kato-Shimomura and Gu-Wang-Wu found several families of nontrivial products in the stable homotopy ring of spheres &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript asterisk Baseline left-parenthesis upper S right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;π&lt;/mml:mi&gt; &lt;mml:mo&gt;∗&lt;/mml:mo&gt; &lt;/mml:msub&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;pi _* (S)&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; using &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript asterisk comma asterisk Baseline left-parenthesis upper S left-parenthesis 3 right-parenthesis right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;H&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;∗&lt;/mml:mo&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mo&gt;∗&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;H^{*,*} (S(3))&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;. In this paper, we determine all nontrivial products in &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript asterisk Baseline left-parenthesis upper S right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;π&lt;/mml:mi&gt; &lt;mml:mo&gt;∗&lt;/mml:mo&gt; &lt;/mml:msub&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;pi _* (S)&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; of the Greek letter family elements &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha Subscript s Baseline comma beta Subscript s Baseline comma gamma Subscript s Baseline\"&gt; &lt;mml:semantics","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth","authors":"Sebastiano Argenti","doi":"10.1090/proc/16896","DOIUrl":"https://doi.org/10.1090/proc/16896","url":null,"abstract":"<p>We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U upper T 2 left-parenthesis upper W Subscript lamda Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">UT_2(W_lambda )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E n d left-parenthesis upper W Subscript mu Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">End(W_mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some integral dominant weight <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda comma mu\"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">lambda ,mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu not-equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mu neq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the special case <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L equals German s German l Subscript 2\"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"fraktur\">s</mml:mi> <mml:mi mathvariant=\"fraktur\">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L=mathfrak {sl}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove that this is a sufficient condition too.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some maximum principles for parabolic mixed local/nonlocal operators","authors":"Serena Dipierro, Edoardo Proietti Lippi, Enrico Valdinoci","doi":"10.1090/proc/16899","DOIUrl":"https://doi.org/10.1090/proc/16899","url":null,"abstract":"<p>The goal of this paper is to establish new Maximum Principles for parabolic equations in the framework of mixed local/nonlocal operators.</p> <p>In particular, these results apply to the case of mixed local/nonlocal Neumann boundary conditions, as introduced by Dipierro, Proietti Lippi, and Valdinoci [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), pp. 1093–1166].</p> <p>Moreover, they play an important role in the analysis of population dynamics involving the so-called Allee effect, which is performed by Dipierro, Proietti Lippi, and Valdinoci [J. Math. Biol. 89 (2024), Paper No. 19]. This is particularly relevant when studying biological populations, since the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Complex submanifolds of indefinite complex space forms","authors":"Xiaoliang Cheng, Yihong Hao, Yuan Yuan, Xu Zhang","doi":"10.1090/proc/16743","DOIUrl":"https://doi.org/10.1090/proc/16743","url":null,"abstract":"In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al.","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140656804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SAT actions of discrete quantum groups and minimal injective extensions of their von Neumann algebras","authors":"Mehrdad Kalantar, Fatemeh Khosravi, Mohammad Moakhar","doi":"10.1090/proc/16882","DOIUrl":"https://doi.org/10.1090/proc/16882","url":null,"abstract":"<p>We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discrete Schrödinger equations and systems with mixed and concave-convex nonlinearities","authors":"Guanwei Chen, Shiwang Ma","doi":"10.1090/proc/16834","DOIUrl":"https://doi.org/10.1090/proc/16834","url":null,"abstract":"In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models.","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140668665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}