Pub Date : 2024-08-13DOI: 10.1016/j.jfa.2024.110626
The norm of the gradient measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope . In this work we propose an axiomatic definition of descent modulus of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.
梯度的规范‖∇f(x)‖度量实值光滑函数 f 在 x 点的最大下降量。对于(非光滑)凸函数,可用子微分到原点的距离 dist(0,∂f(x)) 表示,而对于定义在度量空间上的一般实值函数,可用度量斜率 |∇f|(x)概念表示。在这项工作中,我们提出了实值函数 f 在一般(不一定是度量)空间上定义的每一点 x 上的下降模 T[f](x) 的公理定义。该定义包括上述所有实例以及定义在概率空间上的函数的平均下降模。我们证明,一大类函数完全由其下降模数和相应的临界值决定。在光滑情况下,这一结果已经令人吃惊:一维信息(梯度的规范)几乎与完整梯度映射的知识一样强大。在非光滑情况下,这一判定结果的关键因素是,根据度量斜率定义的精神,由下坡方向引起的对称性破坏。最后一节研究了定义在有限空间上的函数的特殊情况。在这种情况下,我们得到了在某种意义上具有典型性的下降算子的明确分类。
{"title":"Descent modulus and applications","authors":"","doi":"10.1016/j.jfa.2024.110626","DOIUrl":"10.1016/j.jfa.2024.110626","url":null,"abstract":"<div><p>The norm of the gradient <span><math><mo>‖</mo><mi>∇</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>‖</mo></math></span> measures the maximum descent of a real-valued smooth function <em>f</em> at <em>x</em>. For (nonsmooth) convex functions, this is expressed by the distance <span><math><mrow><mi>dist</mi></mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>∂</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope <span><math><mo>|</mo><mi>∇</mi><mi>f</mi><mo>|</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. In this work we propose an axiomatic definition of descent modulus <span><math><mi>T</mi><mo>[</mo><mi>f</mi><mo>]</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></span> of a real-valued function <em>f</em> at every point <em>x</em>, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003148/pdfft?md5=c3c69cd4d272de9fe328368006f32de8&pid=1-s2.0-S0022123624003148-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142040953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1016/j.jfa.2024.110613
We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results of Burq, Gérard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss -estimates on intervals of length for initial data whose frequencies are comparable to λ, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.
{"title":"Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds","authors":"","doi":"10.1016/j.jfa.2024.110613","DOIUrl":"10.1016/j.jfa.2024.110613","url":null,"abstract":"<div><p>We obtain improved Strichartz estimates for solutions of the Schrödinger equation on negatively curved compact manifolds which improve the classical universal results of Burq, Gérard and Tzvetkov <span><span>[11]</span></span> in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>x</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msubsup></math></span>-estimates on intervals of length <span><math><mi>log</mi><mo></mo><mi>λ</mi><mo>⋅</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> for initial data whose frequencies are comparable to <em>λ</em>, which, given the role of the Ehrenfest time, is the natural analog of the universal results in <span><span>[11]</span></span>. We also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1016/j.jfa.2024.110611
We introduce a relaxation of the Aleksandrov condition for the Gauss Image Problem. This weaker condition turns out to be a necessary condition for two measures to be related by a convex body. We provide several properties of the new condition. A solution to the Gauss Image Problem is obtained for the case when one of the measures is assumed to be discrete and the another measure is assumed to be absolutely continuous, under the new relaxed assumption.
{"title":"The Gauss Image Problem with weak Aleksandrov condition","authors":"","doi":"10.1016/j.jfa.2024.110611","DOIUrl":"10.1016/j.jfa.2024.110611","url":null,"abstract":"<div><p>We introduce a relaxation of the Aleksandrov condition for the Gauss Image Problem. This weaker condition turns out to be a necessary condition for two measures to be related by a convex body. We provide several properties of the new condition. A solution to the Gauss Image Problem is obtained for the case when one of the measures is assumed to be discrete and the another measure is assumed to be absolutely continuous, under the new relaxed assumption.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142048589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1016/j.jfa.2024.110612
We study the inverse problem of determining uniquely and stably quasilinear terms appearing in an elliptic equation from boundary excitations and measurements associated with the solutions of the corresponding equation. More precisely, we consider the determination of quasilinear terms depending simultaneously on the solution and the gradient of the solution of the elliptic equation from measurements of the flux restricted to some fixed and finite number of points located at the boundary of the domain generated by Dirichlet data lying on a finite dimensional space. Our Dirichlet data will be explicitly given by affine functions taking values in . We prove our results by considering a new approach based on explicit asymptotic properties of solutions of this class of nonlinear elliptic equations with respect to a small parameter imposed at the boundary of the domain.
{"title":"Determination of quasilinear terms from restricted data and point measurements","authors":"","doi":"10.1016/j.jfa.2024.110612","DOIUrl":"10.1016/j.jfa.2024.110612","url":null,"abstract":"<div><p>We study the inverse problem of determining uniquely and stably quasilinear terms appearing in an elliptic equation from boundary excitations and measurements associated with the solutions of the corresponding equation. More precisely, we consider the determination of quasilinear terms depending simultaneously on the solution and the gradient of the solution of the elliptic equation from measurements of the flux restricted to some fixed and finite number of points located at the boundary of the domain generated by Dirichlet data lying on a finite dimensional space. Our Dirichlet data will be explicitly given by affine functions taking values in <span><math><mi>R</mi></math></span>. We prove our results by considering a new approach based on explicit asymptotic properties of solutions of this class of nonlinear elliptic equations with respect to a small parameter imposed at the boundary of the domain.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1016/j.jfa.2024.110615
We introduce the notion of a differential operator on -algebras. This is a noncommutative analogue of a differential operator on a smooth manifold. We show that the common closed domain of all differential operators is closed under smooth functional calculus. As a corollary, we show that Schwartz functions on Connes tangent groupoid are closed under smooth functional calculus.
{"title":"Differential operators on C⁎-algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid","authors":"","doi":"10.1016/j.jfa.2024.110615","DOIUrl":"10.1016/j.jfa.2024.110615","url":null,"abstract":"<div><p>We introduce the notion of a differential operator on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras. This is a noncommutative analogue of a differential operator on a smooth manifold. We show that the common closed domain of all differential operators is closed under smooth functional calculus. As a corollary, we show that Schwartz functions on Connes tangent groupoid are closed under smooth functional calculus.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-31DOI: 10.1016/j.jfa.2024.110614
We give a precise and complete description on the spectrum for a class of non-self-adjoint quasi-periodic operators acting on which contains the Sarnak's model as a special case. As a consequence, one can see various interesting spectral phenomena including symmetric breaking, the non-simply-connected two-dimensional spectrum in this class of operators. Particularly, we provide new examples of non-self-adjoint operator in whose spectra (actually a two-dimensional subset of ) can not be approximated by the spectra of its finite-interval truncations.
{"title":"Non-self-adjoint quasi-periodic operators with complex spectrum","authors":"","doi":"10.1016/j.jfa.2024.110614","DOIUrl":"10.1016/j.jfa.2024.110614","url":null,"abstract":"<div><p>We give a precise and complete description on the spectrum for a class of non-self-adjoint quasi-periodic operators acting on <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> which contains the Sarnak's model as a special case. As a consequence, one can see various interesting spectral phenomena including <span><math><mi>P</mi><mi>T</mi></math></span> symmetric breaking, the non-simply-connected two-dimensional spectrum in this class of operators. Particularly, we provide new examples of non-self-adjoint operator in <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> whose spectra (actually a two-dimensional subset of <span><math><mi>C</mi></math></span>) can not be approximated by the spectra of its finite-interval truncations.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-30DOI: 10.1016/j.jfa.2024.110610
We introduce a class of divergence-free vector fields on obtained after a suitable localization of Beltrami fields. First, we use them as initial data to construct unique global smooth solutions of the three dimensional Navier-Stokes equations. The relevant fact here is that these initial data can be chosen to be large in any critical space for the Navier–Stokes problem, however they satisfy the nonlinear smallness assumption introduced in [10]. As a further application of the method, we use these vector fields to provide analytical example of vortex-reconnection for the three-dimensional Navier-Stokes equations on . To do so, we exploit the ideas developed in [13] but differently from this latter we cannot rely on the non-trivial homotopy of the three-dimensional torus. To overcome this obstacle we use a different topological invariant, i.e. the number of hyperbolic zeros of the vorticity field.
{"title":"Localization of Beltrami fields: Global smooth solutions and vortex reconnection for the Navier-Stokes equations","authors":"","doi":"10.1016/j.jfa.2024.110610","DOIUrl":"10.1016/j.jfa.2024.110610","url":null,"abstract":"<div><p>We introduce a class of divergence-free vector fields on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> obtained after a suitable localization of <em>Beltrami fields</em>. First, we use them as initial data to construct unique global smooth solutions of the three dimensional Navier-Stokes equations. The relevant fact here is that these initial data can be chosen to be large in any critical space for the Navier–Stokes problem, however they satisfy the nonlinear smallness assumption introduced in <span><span>[10]</span></span>. As a further application of the method, we use these vector fields to provide analytical example of vortex-reconnection for the three-dimensional Navier-Stokes equations on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. To do so, we exploit the ideas developed in <span><span>[13]</span></span> but differently from this latter we cannot rely on the non-trivial homotopy of the three-dimensional torus. To overcome this obstacle we use a different topological invariant, i.e. the number of hyperbolic zeros of the vorticity field.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624002982/pdfft?md5=bd4ea9dcf91041c5a984937fea1ce849&pid=1-s2.0-S0022123624002982-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-30DOI: 10.1016/j.jfa.2024.110608
We study the well-known asymptotic formulas for fractional Sobolev functions à la Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova, in a geometric approach. We show that the key to these asymptotic formulas are Rademacher's theorem and volume growth at infinity respectively. Examples fitting our framework includes Euclidean spaces, Riemannian manifolds, Alexandrov spaces, finite dimensional Banach spaces, and some ideal sub-Riemannian manifolds.
{"title":"On the asymptotic behaviour of the fractional Sobolev seminorms: A geometric approach","authors":"","doi":"10.1016/j.jfa.2024.110608","DOIUrl":"10.1016/j.jfa.2024.110608","url":null,"abstract":"<div><p>We study the well-known asymptotic formulas for fractional Sobolev functions à la Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova, in a geometric approach. We show that the key to these asymptotic formulas are Rademacher's theorem and volume growth at infinity respectively. Examples fitting our framework includes Euclidean spaces, Riemannian manifolds, Alexandrov spaces, finite dimensional Banach spaces, and some ideal sub-Riemannian manifolds.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-30DOI: 10.1016/j.jfa.2024.110609
We consider the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. In the moderate deviation regime, we establish that the quenched density of the motion of the particle (after appropriate centering and scaling) converges weakly to the dimensional stochastic heat equation driven by multiplicative space-time white noise. Our result confirms physics predictions and computations in [66], [7] and is the first rigorous instance of such weak convergence in the moderate deviation regime. Our proof relies on a certain Girsanov transform and works for all Howitt-Warren flows with finite and nonzero characteristic measures. Our results capture universality in the sense that the limiting distribution depends on the flow only via the total mass of the characteristic measure. As a corollary of our results, we prove that the fluctuations of the maximum of an N-point sticky Brownian motion are given by the KPZ equation plus an independent Gumbel on timescales of order .
{"title":"KPZ equation limit of sticky Brownian motion","authors":"","doi":"10.1016/j.jfa.2024.110609","DOIUrl":"10.1016/j.jfa.2024.110609","url":null,"abstract":"<div><p>We consider the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. In the moderate deviation regime, we establish that the quenched density of the motion of the particle (after appropriate centering and scaling) converges weakly to the <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>)</mo></math></span> dimensional stochastic heat equation driven by multiplicative space-time white noise. Our result confirms physics predictions and computations in <span><span>[66]</span></span>, <span><span>[7]</span></span> and is the first rigorous instance of such weak convergence in the moderate deviation regime. Our proof relies on a certain Girsanov transform and works for all Howitt-Warren flows with finite and nonzero characteristic measures. Our results capture universality in the sense that the limiting distribution depends on the flow only via the total mass of the characteristic measure. As a corollary of our results, we prove that the fluctuations of the maximum of an <em>N</em>-point sticky Brownian motion are given by the KPZ equation plus an independent Gumbel on timescales of order <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>N</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-24DOI: 10.1016/j.jfa.2024.110595
We study local-in-time and global-in-time bilinear Strichartz estimates for the Schrödinger equation on waveguides. As applications, we apply those estimates to study global well-posedness of nonlinear Schrödinger equations on these waveguides.
{"title":"On bilinear Strichartz estimates on waveguides with applications","authors":"","doi":"10.1016/j.jfa.2024.110595","DOIUrl":"10.1016/j.jfa.2024.110595","url":null,"abstract":"<div><p>We study local-in-time and global-in-time bilinear Strichartz estimates for the Schrödinger equation on waveguides. As applications, we apply those estimates to study global well-posedness of nonlinear Schrödinger equations on these waveguides.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}