Fractional Calculus and Applied Analysis最新文献

{"title":"Space–time duality for fractional powers of sectorial operators, with applications to graph laplacians","authors":"Ernesto Estrada, Carlos Lizama","doi":"10.1007/s13540-026-00557-y","DOIUrl":"https://doi.org/10.1007/s13540-026-00557-y","url":null,"abstract":"We establish a space–time duality principle for fractional powers of sectorial operators and for right-sided fractional identities satisfied by their semigroup orbits. Let <inline-formula><alternatives><mml:math><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>α</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$0<alpha <1$$end{document}</tex-math></alternatives></inline-formula>. If <italic>A</italic> is an injective sectorial operator of angle strictly smaller than <inline-formula><alternatives><mml:math><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$pi /2$$end{document}</tex-math></alternatives></inline-formula> and admits a bounded holomorphic <inline-formula><alternatives><mml:math><mml:msup><mml:mi>H</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H^infty $$end{document}</tex-math></alternatives></inline-formula>-calculus, then the semigroup orbit <disp-formula><alternatives><mml:math display=\"block\"><mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>t</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mi>α</mml:mi></mml:msup></mml:mrow></mml:msup><mml:mi>φ</mml:mi></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ u(t)=e^{-tA^alpha }varphi $$end{document}</tex-math></alternatives></disp-formula>which solves the space–fractional problem <inline-formula><alternatives><mml:math><mml:mrow><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>A</mml:mi><mml:mi>α</mml:mi></mml:msup><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$partial _tu+A^alpha u=0$$end{document}</tex-math></alternatives></inline-formula>, also satisfies ","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"93 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148688047","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}

{"title":"Normalized solutions for double critical Schrödinger-Poisson system involving the fractional p-Laplacian in R3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}^3$$end{document}","authors":"Di Xiao, Yueqiang Song, Shaoyun Shi, Sihua Liang","doi":"10.1007/s13540-026-00552-3","DOIUrl":"https://doi.org/10.1007/s13540-026-00552-3","url":null,"abstract":"In this paper, we study the following double critical Schrödinger-Poisson system involving the fractional <italic>p</italic>-Laplacian in <inline-formula><alternatives><mml:math><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}^3$$end{document}</tex-math></alternatives></inline-formula> of the form: <disp-formula><alternatives><mml:math display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msubsup><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mi>s</mml:mi><mml:mo>♯</mml:mo></mml:msubsup><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>λ</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mi>s</mml:mi><mml:mo>∗</mml:mo></mml:msubsup><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"0.333333em\"></mml:mspace><mml:mtext>in</mml:mtext><mml:mspace width=\"0.333333em\"></mml:mspace><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>s</mml:mi></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mi>s</mml:mi><mml:mo>♯</mml:mo></mml:msub","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"57 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687804","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}

{"title":"Fractional Toeplitz operators with radial symbols","authors":"Alexey Karapetyants, Irina Smirnova","doi":"10.1007/s13540-026-00553-2","DOIUrl":"https://doi.org/10.1007/s13540-026-00553-2","url":null,"abstract":"In the present work, we introduce the concept of fractional Toeplitz operators and study boundedness and compactness properties of such operators in the case of radial symbols within the framework of weighted Bergman spaces. We establish sufficient and, in certain cases, necessary conditions for the boundedness, compactness, and membership in Schatten <italic>p</italic>-ideals of fractional Toeplitz operators. These characterizations are primarily derived from the behavior of integral averages of the radial symbol, relating the operator’s structural properties to the behavior of its symbol or an average of the symbol.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"14 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687912","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}

{"title":"Polynomial decay of porous elastic system with fractional damping","authors":"M. L. Santos, O. P. V. Villagrán","doi":"10.1007/s13540-026-00540-7","DOIUrl":"https://doi.org/10.1007/s13540-026-00540-7","url":null,"abstract":"In this work, we prove that the effect of damping given by tempered Caputo derivative does not lead the porous elastic model to exponential stability. Although damping acts on both equations of the system, it is not strong enough to lead the model to exponential stabilization, which is novel for this type of model, where a single damping factor provides exponential stability depending on a relationship between the velocity coefficients. Thus, we prove that the model is well-posed and polynomially stable.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"22 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687833","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}

{"title":"Inverse t-Source problem and a strict positivity property for coupled subdiffusion systems","authors":"Mohamed BenSalah, Yikan Liu","doi":"10.1007/s13540-026-00551-4","DOIUrl":"https://doi.org/10.1007/s13540-026-00551-4","url":null,"abstract":"This article is concerned with the inverse problem on determining the temporal component of the source term in a coupled time-fractional diffusion system by a single point observation. Under a non-degeneracy condition on the known spatial component, we establish the Lipschitz stability by observing all solution components by a series representation of the mild solution. To reduce the observation data, we prove the strict positivity of some fractional integral of the solution to the homogeneous problem by a modified Picard iteration. This, together with a coupled Duhamel’s principle, lead us to the uniqueness of the inverse problem by observing any single solution component under a specific structural constraint on the unknown. Numerically, we propose an iterative regularizing ensemble Kalman method (IREKM) for the simultaneous recovery of the temporal sources. Through extensive numerical tests, we demonstrate its accuracy, robustness against noise, and scalability with respect to the number of components. Our findings highlight the essential roles of the non-degeneracy condition, measurement configuration, and fractional structural constraints in ensuring reliable reconstructions. The proposed framework provides both rigorous theoretical guarantees and a practical algorithmic approach for multi-component source identification in fractional diffusion systems.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"65 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687834","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}

{"title":"Global solutions to semilinear subdiffusion-normal transport equations: well-posedness and Strichartz estimates","authors":"Jia Wei He, Shi Qi Li, Li Peng","doi":"10.1007/s13540-026-00558-x","DOIUrl":"https://doi.org/10.1007/s13540-026-00558-x","url":null,"abstract":"This paper is concerned with a class of time-fractional subdiffusion-normal transport equations that describe the transition from subdiffusion to normal diffusion. Our primary contribution is the establishment of the well-posedness and Strichartz estimates for global solutions on the unbounded domain <inline-formula><alternatives><mml:math><mml:msup><mml:mi mathvariant=\"double-struck\">R</mml:mi><mml:mi>N</mml:mi></mml:msup></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb R^N$$end{document}</tex-math></alternatives></inline-formula>. Furthermore, we extend our analysis to bounded domains with smooth boundaries, proving well-posedness and Strichartz estimates for small initial data and under the condition of a polynomial growth nonlinearity.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"46 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687909","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}

{"title":"A Filippov-Type approximation theorem for caputo fractional stochastic differential inclusions","authors":"Le Xuan Truong","doi":"10.1007/s13540-026-00559-w","DOIUrl":"https://doi.org/10.1007/s13540-026-00559-w","url":null,"abstract":"This work deals with certain class of Caputo-type fractional stochastic differential inclusions driven by Brownian motion. Under suitable measurability and Lipschitz conditions on the multivalued drift and diffusion terms, we establish the existence of solutions first via a fixed-point argument and then via a Filippov-type approximation procedure. The proof relies on stochastic analysis, fractional calculus, and a measurable selection argument. Our result extends the classical Filippov theorem to the fractional stochastic framework and partially generalizes prior work by Da Prato and Frankowska (1994).","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"59 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687832","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}

{"title":"Microscopic process of the rough non-affine Heston model and its application","authors":"Yaoteng Huang, Caibin Zeng","doi":"10.1007/s13540-026-00548-z","DOIUrl":"https://doi.org/10.1007/s13540-026-00548-z","url":null,"abstract":"Recent research in financial volatility modeling demonstrates that rough volatility models, driven by fractional Brownian motion (fBm) with a small Hurst parameter, play a crucial role in understanding the implied volatility surface and improving option pricing. However, the affine structure of the standard Heston model only provides analytical expressions for the characteristic function and density function of the variance process under specific forms of volatility dynamics. To overcome this limitation, this paper introduces a non-affine coefficient, enabling the model to more flexibly capture complex characteristics of asset returns such as excess kurtosis and heavy tails. At the macro level, we replace the Brownian motion in the non-affine Heston model with fBm, constructing the rough non-affine Heston model. At the micro level, we establish a theoretical foundation for the model by linking nearly unstable Hawkes processes to fractional volatility models. Finally, empowered by a robust Lifted Semi-Implicit Euler scheme and the CRN technique, we empirically calibrate the models using S&amp;P 500 index options during the March 2020 crash. The results demonstrate that the rough non-affine model outperforms its affine counterpart by capturing extreme left-tail skewness and drastically reducing downside pricing errors.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"35 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687961","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}

{"title":"On solutions of space-time fractional nonlinear Fokker–Planck equation. II","authors":"Komal Singla, Nikolai Leonenko","doi":"10.1007/s13540-026-00546-1","DOIUrl":"https://doi.org/10.1007/s13540-026-00546-1","url":null,"abstract":"This paper extends our previous study [Singla, K. and Leonenko, N.: Fract. Calc. Appl. Anal. <bold>28</bold>, 1094-1105 (2025)] on the time-fractional nonlinear Fokker-Planck equation to a more general nonlinear space-time fractional formulation involving both temporal and spatial Riemann-Liouville fractional derivatives. The considered model incorporates long-memory effects through the time-fractional derivative and nonlocal spatial transport through the space-fractional derivative, allowing the description of anomalous diffusion processes with both temporal memory and Lévy-type spatial interactions. By applying Lie symmetry analysis for space-time fractional partial differential equations, the governing equation is reduced to a nonlinear fractional ordinary differential equation involving Erdélyi-Kober fractional operators. Exact analytical solutions are then constructed using the power series method together with convergence analysis based on a majorant series approach and the implicit function theorem. The obtained results generalize the corresponding solutions of the previously studied time-fractional model and demonstrate how the interplay between temporal memory and spatial nonlocality influences the qualitative behavior of the solutions. Graphical illustrations are also presented to visualize the effects of different fractional orders on the evolution of the solution profiles.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"64 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687962","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}

{"title":"Green’s function and solution representation for a boundary value problem involving the Prabhakar fractional derivative","authors":"Erkinjon Karimov, Doniyor Usmonov, Maftuna Mirzaeva","doi":"10.1007/s13540-026-00544-3","DOIUrl":"https://doi.org/10.1007/s13540-026-00544-3","url":null,"abstract":"We investigate a first boundary value problem for a second-order partial differential equation involving the Prabhakar fractional derivative in time. Using structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions, we reduce the problem to a Volterra-type integral equation. This reduction enables the explicit construction of the corresponding Green’s function. Based on the obtained Green’s function, we derive a closed-form integral representation of the solution and prove its existence and uniqueness. The results extend classical Green-function techniques to a wider class of fractional order operators and provide analytical tools for further study of boundary and inverse problems associated with Prabhakar-type fractional differential equations.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"19 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148687964","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}