Advances in Mathematics最新文献

{"title":"Dual fractional parabolic equations with indefinite nonlinearities","authors":"Wenxiong Chen ,&nbsp;Yahong Guo","doi":"10.1016/j.aim.2024.109891","DOIUrl":"10.1016/j.aim.2024.109891","url":null,"abstract":"<div><p>In this paper, we consider the following indefinite dual fractional parabolic equation involving the Marchaud fractional time derivative<span><span><span><math><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>)</mo><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mi>α</mi><mo>,</mo><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, and the functions <em>a</em> and <em>f</em> are nondecreasing. We prove that there is no positive bounded solutions. To this end, we first show that all positive bounded solutions <span><math><mi>u</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>t</mi><mo>)</mo></math></span> must be strictly monotone increasing along the direction determined by <span><math><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Then by mollifying the first eigenfunction for fractional Laplacian <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> and constructing an appropriate subsolution for the Marchaud fractional operator <span><math><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>−</mo><mn>1</mn></math></span>, we derive a contradiction and thus obtain the non-existence of solutions.</p><p>To overcome the challenges caused by the dual non-locality of the operator <span><math><msubsup><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>, we introduce several new ideas and novel techniques. These novel approaches are not only applicable to the specific problem at hand but can also be extended to address various other fractional problems, be they elliptic or parabolic, including those featuring dual nonlocalities associated with the Marchaud time derivatives.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A classification of complex rank 3 vector bundles on CP5","authors":"Morgan Opie","doi":"10.1016/j.aim.2024.109878","DOIUrl":"10.1016/j.aim.2024.109878","url":null,"abstract":"<div><p>Given integers <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, there is a complex rank 3 topological bundle on <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> with <em>i</em>-th Chern class equal to <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> if and only if <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> satisfy the Schwarzenberger condition. Provided that the Schwarzenberger condition is satisfied, we prove that the number of isomorphism classes of rank 3 bundles <em>V</em> on <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> with <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is equal to 3 if <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are both divisible by 3 and equal to 1 otherwise.</p><p>This shows that Chern classes are incomplete invariants of topological rank 3 bundles on <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span>. To address this problem, we produce a universal class in the <span><math><mrow><mi>tm</mi><msub><mrow><mi>f</mi></mrow><mrow><mi>(</mi><mspace></mspace><mn>3</mn><mi>)</mi></mrow></msub></mrow></math></span>-cohomology of a Thom spectrum related to <span><math><mi>B</mi><mi>U</mi><mspace></mspace><mo>(</mo><mn>3</mn><mo>)</mo></math></span>, where <span><math><mrow><mi>tm</mi><msub><mrow><mi>f</mi></mrow><mrow><mi>(</mi><mspace></mspace><mn>3</mn><mi>)</mi></mrow></msub></mrow></math></span> denotes topological modular forms localized at 3. From this class and orientation data, we construct a <span><math><mi>Z</mi><mo>/</mo><mn>3</mn></math></span>-valued invariant of the bundles of interest and prove that our invariant separates distinct bundles with the same Chern classes.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824003931/pdfft?md5=bab684e3d435c50eb306af6f6b36ae0a&pid=1-s2.0-S0001870824003931-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141984762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonexistence of T4 configurations for hyperbolic systems and the Liu entropy condition","authors":"Sam G. Krupa ,&nbsp;László Székelyhidi Jr.","doi":"10.1016/j.aim.2024.109856","DOIUrl":"10.1016/j.aim.2024.109856","url":null,"abstract":"<div><p>We study the constitutive set <span><math><mi>K</mi></math></span> arising from a <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> system of conservation laws in one space dimension, endowed with one entropy and entropy-flux pair. The convexity properties of the set <span><math><mi>K</mi></math></span> relate to the well-posedness of the underlying system and the ability to construct solutions via convex integration. Relating to the convexity of <span><math><mi>K</mi></math></span>, in the particular case of the <em>p</em>-system, Lorent and Peng (2020) <span><span>[21]</span></span> show that <span><math><mi>K</mi></math></span> does not contain <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> configurations. Recently, Johansson and Tione (2024) <span><span>[14]</span></span> showed that <span><math><mi>K</mi></math></span> does not contain <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> configurations.</p><p>In this paper, we provide a substantial generalization of Lorent-Peng, based on a careful analysis of the shock curves for a large class of <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> systems. We provide several sets of hypotheses on general systems which can be used to rule out the existence of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> configurations in the constitutive set <span><math><mi>K</mi></math></span>. In particular, our results show the nonexistence of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> configurations for every well-known <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> hyperbolic system of conservation laws for which both families of shocks verify the Liu entropy condition.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824003712/pdfft?md5=0c19a0dff471e6ae0545af1366cf0957&pid=1-s2.0-S0001870824003712-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141979758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Translating annuli for mean curvature flow","authors":"David Hoffman ,&nbsp;Francisco Martín ,&nbsp;Brian White","doi":"10.1016/j.aim.2024.109875","DOIUrl":"10.1016/j.aim.2024.109875","url":null,"abstract":"<div><p>We construct a family <span><math><mi>A</mi></math></span> of complete, properly embedded, annular translators <em>M</em> such that <em>M</em> lies in a slab and is invariant under reflections in the vertical coordinate planes. For each <em>M</em> in <span><math><mi>A</mi></math></span>, <em>M</em> is asymptotic as <span><math><mi>z</mi><mo>→</mo><mo>−</mo><mo>∞</mo></math></span> to four vertical planes <span><math><mo>{</mo><mi>y</mi><mo>=</mo><mo>±</mo><mi>b</mi><mo>}</mo></math></span> and <span><math><mo>{</mo><mi>y</mi><mo>=</mo><mo>±</mo><mi>B</mi><mo>}</mo></math></span> where <span><math><mn>0</mn><mo>&lt;</mo><mi>b</mi><mo>≤</mo><mi>B</mi><mo>&lt;</mo><mo>∞</mo></math></span>. We call <em>b</em> and <em>B</em> the <strong>inner width</strong> and the <strong>(outer) width</strong> of <em>M</em>. We show that for each <span><math><mi>b</mi><mo>≥</mo><mi>π</mi><mo>/</mo><mn>2</mn></math></span> and each <span><math><mi>s</mi><mo>&gt;</mo><mn>0</mn></math></span>, there is an <span><math><mi>M</mi><mo>∈</mo><mi>A</mi></math></span> with inner width <em>b</em> and with necksize <em>s</em>. (We also show that there are no translators with inner width <span><math><mo>&lt;</mo><mi>π</mi><mo>/</mo><mn>2</mn></math></span> having the properties of the examples we construct.)</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Zero-free half-planes of the ζ-function via spaces of analytic functions","authors":"Aditya Ghosh,&nbsp;Kobi Kremnizer,&nbsp;S. Waleed Noor,&nbsp;Charles F. Santos","doi":"10.1016/j.aim.2024.109872","DOIUrl":"10.1016/j.aim.2024.109872","url":null,"abstract":"<div><p>In this article we introduce a general approach for deriving zero-free half-planes for the Riemann zeta function <em>ζ</em> by identifying topological vector spaces of analytic functions with specific properties. This approach is applied to weighted <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> spaces and the classical Hardy spaces <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> (<span><math><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>≤</mo><mn>2</mn></math></span>). As a consequence precise conditions are obtained for the existence of zero-free half planes for the <em>ζ</em>-function.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824003876/pdfft?md5=8b5885f06eeefd8862c9157c00298da6&pid=1-s2.0-S0001870824003876-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Soluble skew left braces and soluble solutions of the Yang-Baxter equation","authors":"A. Ballester-Bolinches ,&nbsp;R. Esteban-Romero ,&nbsp;P. Jiménez-Seral ,&nbsp;V. Pérez-Calabuig","doi":"10.1016/j.aim.2024.109880","DOIUrl":"10.1016/j.aim.2024.109880","url":null,"abstract":"<div><p>The study of non-degenerate set-theoretic solutions of the Yang-Baxter equation calls for a deep understanding of the algebraic structure of a skew left brace. In this paper, the skew brace theoretical property of solubility is introduced and studied. It leads naturally to the notion of solubility of solutions of the Yang-Baxter equation. It turns out that soluble non-degenerate set-theoretic solutions are characterised by soluble skew left braces. The rich ideal structure of soluble skew left braces is also shown. A worked example showing the relevance of the brace theoretical property of solubility is also presented.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824003955/pdfft?md5=e9a68f529e5789dd9884b0404587ef51&pid=1-s2.0-S0001870824003955-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141964487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Integrality of the higher Rademacher symbols","authors":"Cormac O'Sullivan","doi":"10.1016/j.aim.2024.109876","DOIUrl":"10.1016/j.aim.2024.109876","url":null,"abstract":"<div><p>Rademacher symbols may be defined in terms of Dedekind sums, and give the value at zero of the zeta function associated to a narrow ideal class of a real quadratic field. Duke extended these symbols to give the zeta function values at all negative integers. Here we prove Duke's conjecture that these higher Rademacher symbols are integer valued, making the above zeta value denominators as simple as the corresponding Riemann zeta value denominators. The proof uses detailed properties of Bernoulli numbers, including a generalization of the Kummer congruences.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141953843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Classical Whittaker modules for the affine Kac-Moody algebras AN(1)","authors":"Hongjia Chen ,&nbsp;Lin Ge ,&nbsp;Zheng Li ,&nbsp;Longhui Wang","doi":"10.1016/j.aim.2024.109874","DOIUrl":"10.1016/j.aim.2024.109874","url":null,"abstract":"<div><p>Inspired by Sugawara operators, we introduce quasi Sugawara operators to construct several important operators on the universal non-degenerate Whittaker module of level <em>κ</em> over the affine Kac-Moody algebra of type <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>N</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup></math></span>. As a result, we classify simple non-degenerate Whittaker modules for the affine algebras <span><math><mover><mrow><msub><mrow><mi>sl</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> and <span><math><mover><mrow><msub><mrow><mi>sl</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mrow><mo>˜</mo></mrow></mover></math></span> whether at the noncritical or critical level. In addition, we also give an explicit description on the structure of arbitrary non-degenerate Whittaker modules over these algebras. In particular, we recover the results on the classification of simple non-degenerate Whittaker <span><math><mover><mrow><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-modules (<span><math><mover><mrow><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mo>˜</mo></mrow></mover></math></span>-modules) obtained by Adamović, Lü and Zhao.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141962943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Koszul operads governing props and wheeled props","authors":"Kurt Stoeckl","doi":"10.1016/j.aim.2024.109869","DOIUrl":"10.1016/j.aim.2024.109869","url":null,"abstract":"<div><p>In this paper, we construct groupoid coloured operads governing props and wheeled props, and show they are Koszul. This is accomplished by new biased definitions for (wheeled) props, and an extension of the theory of Groebner bases for operads to apply to groupoid coloured operads. Using the Koszul machine, we define homotopy (wheeled) props, and show they are not formed by polytope based models. Finally, using homotopy transfer theory, we construct Massey products for (wheeled) props, show these products characterise the formality of these structures, and re-obtain a theorem of Mac Lane on the existence of higher homotopies of (co)commutative Hopf algebras.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824003840/pdfft?md5=a276c43f8754cc2d38ff043f652c7163&pid=1-s2.0-S0001870824003840-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141960890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a fully nonlinear elliptic equation with differential forms","authors":"Hao Fang,&nbsp;Biao Ma","doi":"10.1016/j.aim.2024.109867","DOIUrl":"10.1016/j.aim.2024.109867","url":null,"abstract":"<div><p>We introduce a fully nonlinear PDE with a differential form, which unifies several important equations in Kähler geometry including Monge-Ampère equations, <em>J</em>-equations, inverse <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> equations, and deformed Hermitian Yang-Mills (dHYM) equations. We pose some natural positivity conditions on Λ, and prove analytical and algebraic criterion for the solvability of the equation. Our results generalize previous works of G. Chen, J. Song, Datar-Pingali and others. As an application, we prove a conjecture of Collins-Jacob-Yau for dHYM equations with small global phase.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141960974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}