Abstract Let S γ , A , B ∗ ( D ) {S}_{gamma ,A,B}^{ast }left({mathbb{D}}) be the usual class of g g -starlike functions of complex order γ gamma in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } {mathbb{D}}=left{zeta in {mathbb{C}}:| zeta | lt 1right} , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) gleft(zeta )=left(1+Azeta )/left(1+Bzeta ) , with γ ∈ C { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D gamma leftin {mathbb{C}}backslash left{0right}right,-1le Alt Ble 1,zeta in {mathbb{D}} . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) fin {S}_{gamma ,A,B}^{ast }left({mathbb{D}}) when ζ = 0 zeta =0 is a zero of order k + 1 k+1 of f ( ζ ) − ζ fleft(zeta )-zeta . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.
{"title":"Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn","authors":"Xiaoying Sima, Z. Tu, L. Xiong","doi":"10.1515/dema-2022-0242","DOIUrl":"https://doi.org/10.1515/dema-2022-0242","url":null,"abstract":"Abstract Let S γ , A , B ∗ ( D ) {S}_{gamma ,A,B}^{ast }left({mathbb{D}}) be the usual class of g g -starlike functions of complex order γ gamma in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } {mathbb{D}}=left{zeta in {mathbb{C}}:| zeta | lt 1right} , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) gleft(zeta )=left(1+Azeta )/left(1+Bzeta ) , with γ ∈ C { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D gamma leftin {mathbb{C}}backslash left{0right}right,-1le Alt Ble 1,zeta in {mathbb{D}} . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) fin {S}_{gamma ,A,B}^{ast }left({mathbb{D}}) when ζ = 0 zeta =0 is a zero of order k + 1 k+1 of f ( ζ ) − ζ fleft(zeta )-zeta . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48565347","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}
Abstract In this article, a new system of Hadamard-type hybrid fractional differential inclusions equipped with Dirichlet boundary conditions was constructed. By virtue of a fixed-point theorem due to B. C. Dhage, (Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Anal. 64 (2006), no. 6, 1290–1306, doi: https://doi.org/10.1016/j.na.2005.06.036), the existence results of solutions for the considered problem are derived in a new norm space for multivalued maps. A numerical example is provided to illustrate our main results.
{"title":"Solvability for a system of Hadamard-type hybrid fractional differential inclusions","authors":"Keyu Zhang, Jiafa Xu","doi":"10.1515/dema-2022-0226","DOIUrl":"https://doi.org/10.1515/dema-2022-0226","url":null,"abstract":"Abstract In this article, a new system of Hadamard-type hybrid fractional differential inclusions equipped with Dirichlet boundary conditions was constructed. By virtue of a fixed-point theorem due to B. C. Dhage, (Existence results for neutral functional differential inclusions in Banach algebras, Nonlinear Anal. 64 (2006), no. 6, 1290–1306, doi: https://doi.org/10.1016/j.na.2005.06.036), the existence results of solutions for the considered problem are derived in a new norm space for multivalued maps. A numerical example is provided to illustrate our main results.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44031533","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}
S. Shah, S. Al-Sa'di, S. Hussain, Asifa Tasleem, A. Rasheed, I. Cheema, M. Darus
Abstract In this article, we study the Fekete-Szegö functional associated with a new class of analytic functions related to the class of bounded turning by using the principle of quasi-subordination. We derived the coefficient estimates including the classical Fekete-Szegö inequality for functions belonging to this class. We also improved some existing results.
{"title":"Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination","authors":"S. Shah, S. Al-Sa'di, S. Hussain, Asifa Tasleem, A. Rasheed, I. Cheema, M. Darus","doi":"10.1515/dema-2022-0232","DOIUrl":"https://doi.org/10.1515/dema-2022-0232","url":null,"abstract":"Abstract In this article, we study the Fekete-Szegö functional associated with a new class of analytic functions related to the class of bounded turning by using the principle of quasi-subordination. We derived the coefficient estimates including the classical Fekete-Szegö inequality for functions belonging to this class. We also improved some existing results.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46121942","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}
M. Bohner, G. Caristi, Ahmad Ghobadi, S. Heidarkhani
Abstract In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class of double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate the applicability of our main theoretical findings.
{"title":"Three solutions for discrete anisotropic Kirchhoff-type problems","authors":"M. Bohner, G. Caristi, Ahmad Ghobadi, S. Heidarkhani","doi":"10.1515/dema-2022-0209","DOIUrl":"https://doi.org/10.1515/dema-2022-0209","url":null,"abstract":"Abstract In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class of double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate the applicability of our main theoretical findings.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45842083","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}
Abstract Cross-border e-commerce platform (CBECP) plays a very important role in the development of a cross-border e-commerce (CBEC). How to select the best CBECP scientifically and reasonably is a very critical multi-attribute group decision-making (MAGDM) issue. With the uncertainty of people’s cognition of the objective world, the decision-making process is full of a lot of fuzzy information. In view of the great advantages of probabilistic dual hesitation fuzzy set (FS) in expressing decision-making information, and in combination with the very extensive use of the Dice similarity measure (DSM), a new MAGDM method is proposed for the optimal CBECP selection (CBECPS) under the probabilistic dual hesitation fuzzy (PDHF) environment. First, on the basis of reviewing a large number of documents on the CBECPS for CBEC, the evaluation index system for the CBECPS is constructed; second, several new DSMs are proposed in the PDHF environment; third, based on the two newly proposed probabilistic dual hesitant weighted generalized Dice similarity measures, two novel MAGDM methods are provided for CBECPS, which are used for CBECPS; finally, the two established MAGDM techniques are compared with the existing decision-making methods, and the parameter analysis is carried out to illustrate the effectiveness and superiority of the two established MAGDM techniques. The two established techniques can not only be used for CBECPS of CBEC, but also be extended to similar related research.
{"title":"The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures","authors":"Baoquan Ning, G. Wei","doi":"10.1515/dema-2022-0239","DOIUrl":"https://doi.org/10.1515/dema-2022-0239","url":null,"abstract":"Abstract Cross-border e-commerce platform (CBECP) plays a very important role in the development of a cross-border e-commerce (CBEC). How to select the best CBECP scientifically and reasonably is a very critical multi-attribute group decision-making (MAGDM) issue. With the uncertainty of people’s cognition of the objective world, the decision-making process is full of a lot of fuzzy information. In view of the great advantages of probabilistic dual hesitation fuzzy set (FS) in expressing decision-making information, and in combination with the very extensive use of the Dice similarity measure (DSM), a new MAGDM method is proposed for the optimal CBECP selection (CBECPS) under the probabilistic dual hesitation fuzzy (PDHF) environment. First, on the basis of reviewing a large number of documents on the CBECPS for CBEC, the evaluation index system for the CBECPS is constructed; second, several new DSMs are proposed in the PDHF environment; third, based on the two newly proposed probabilistic dual hesitant weighted generalized Dice similarity measures, two novel MAGDM methods are provided for CBECPS, which are used for CBECPS; finally, the two established MAGDM techniques are compared with the existing decision-making methods, and the parameter analysis is carried out to illustrate the effectiveness and superiority of the two established MAGDM techniques. The two established techniques can not only be used for CBECPS of CBEC, but also be extended to similar related research.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43632248","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}
W. Lei, Muhammad Ahsan, Waqas Khan, Zaheer Uddin, Masood Ahmad
Abstract In this research work, we proposed a Haar wavelet collocation method (HWCM) for the numerical solution of first- and second-order nonlinear hyperbolic equations. The time derivative in the governing equations is approximated by a finite difference. The nonlinear hyperbolic equation is converted into its full algebraic form once the space derivatives are replaced by the finite Haar series. Convergence analysis is performed both in space and time, where the computational results follow the theoretical statements of convergence. Many test problems with different nonlinear terms are presented to verify the accuracy, capability, and convergence of the proposed method for the first- and second-order nonlinear hyperbolic equations.
{"title":"A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation","authors":"W. Lei, Muhammad Ahsan, Waqas Khan, Zaheer Uddin, Masood Ahmad","doi":"10.1515/dema-2022-0203","DOIUrl":"https://doi.org/10.1515/dema-2022-0203","url":null,"abstract":"Abstract In this research work, we proposed a Haar wavelet collocation method (HWCM) for the numerical solution of first- and second-order nonlinear hyperbolic equations. The time derivative in the governing equations is approximated by a finite difference. The nonlinear hyperbolic equation is converted into its full algebraic form once the space derivatives are replaced by the finite Haar series. Convergence analysis is performed both in space and time, where the computational results follow the theoretical statements of convergence. Many test problems with different nonlinear terms are presented to verify the accuracy, capability, and convergence of the proposed method for the first- and second-order nonlinear hyperbolic equations.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44217083","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}
Abstract In this article, one standard and four nonstandard finite difference methods are used to solve a cross-diffusion malignant invasion model. The model consists of a system of nonlinear coupled partial differential equations (PDEs) subject to specified initial and boundary conditions, and no exact solution is known for this problem. It is difficult to obtain theoretically the stability region of the classical finite difference scheme to solve the set of nonlinear coupled PDEs, this is one of the challenges of this class of method in this work. Three nonstandard methods abbreviated as NSFD1, NSFD2, and NSFD3 are considered from the study of Chapwanya et al., and these methods have been constructed by the use of a more general function replacing the denominator of the discrete derivative and nonlocal approximations of nonlocal terms. It is shown that NSFD1, which preserves positivity when used to solve classical reaction-diffusion equations, does not inherit this property when used for the cross-diffusion system of PDEs. NSFD2 and NSFD3 are obtained by appropriate modifications of NSFD1. NSFD2 is positivity-preserving when the functional relationship [ ψ ( h ) ] 2 = 2 ϕ ( k ) {left[psi left(h)]}^{2}=2phi left(k) holds, while NSFD3 is unconditionally dynamically consistent with respect to positivity. First, we show that NSFD2 and NSFD3 are not consistent methods. Second, we tried to modify NSFD2 in order to make it consistent but we were not successful. Third, we extend NSFD3 so that it becomes consistent and still preserves positivity. We denote the extended version of NSFD3 as NSFD5. Finally, we compute the numerical rate of convergence in time for NSFD5 and show that it is close to the theoretical value. NSFD5 is consistent under certain conditions on the step sizes and is unconditionally positivity-preserving.
{"title":"Numerical solution of a malignant invasion model using some finite difference methods","authors":"A. Appadu, Gysbert Nicolaas de Waal","doi":"10.1515/dema-2022-0244","DOIUrl":"https://doi.org/10.1515/dema-2022-0244","url":null,"abstract":"Abstract In this article, one standard and four nonstandard finite difference methods are used to solve a cross-diffusion malignant invasion model. The model consists of a system of nonlinear coupled partial differential equations (PDEs) subject to specified initial and boundary conditions, and no exact solution is known for this problem. It is difficult to obtain theoretically the stability region of the classical finite difference scheme to solve the set of nonlinear coupled PDEs, this is one of the challenges of this class of method in this work. Three nonstandard methods abbreviated as NSFD1, NSFD2, and NSFD3 are considered from the study of Chapwanya et al., and these methods have been constructed by the use of a more general function replacing the denominator of the discrete derivative and nonlocal approximations of nonlocal terms. It is shown that NSFD1, which preserves positivity when used to solve classical reaction-diffusion equations, does not inherit this property when used for the cross-diffusion system of PDEs. NSFD2 and NSFD3 are obtained by appropriate modifications of NSFD1. NSFD2 is positivity-preserving when the functional relationship [ ψ ( h ) ] 2 = 2 ϕ ( k ) {left[psi left(h)]}^{2}=2phi left(k) holds, while NSFD3 is unconditionally dynamically consistent with respect to positivity. First, we show that NSFD2 and NSFD3 are not consistent methods. Second, we tried to modify NSFD2 in order to make it consistent but we were not successful. Third, we extend NSFD3 so that it becomes consistent and still preserves positivity. We denote the extended version of NSFD3 as NSFD5. Finally, we compute the numerical rate of convergence in time for NSFD5 and show that it is close to the theoretical value. NSFD5 is consistent under certain conditions on the step sizes and is unconditionally positivity-preserving.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46209462","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}
Abstract The aim of this article is to introduce a new type of operator similar to those of A. Petruşel and G. Petruşel type (Fixed point results for decreasing convex orbital operators, J. Fixed Point Theory Appl. 23 (2021), no. 35) and prove some fixed-point theorems which generalize and complement several results in the theory of nonlinear operators.
摘要本文的目的是引入一种类似于a.Petruşel和G.Petruşe l型算子的新型算子（递减凸轨道算子的不动点结果，J.Fixed point Theory Appl.23（2021），no.35），并证明了一些不动点定理，这些定理推广和补充了非线性算子理论中的几个结果。
{"title":"Fixed-point results for convex orbital operators","authors":"O. Popescu","doi":"10.1515/dema-2022-0184","DOIUrl":"https://doi.org/10.1515/dema-2022-0184","url":null,"abstract":"Abstract The aim of this article is to introduce a new type of operator similar to those of A. Petruşel and G. Petruşel type (Fixed point results for decreasing convex orbital operators, J. Fixed Point Theory Appl. 23 (2021), no. 35) and prove some fixed-point theorems which generalize and complement several results in the theory of nonlinear operators.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49348940","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}
Abstract It is known that the Stirling numbers of the second kind are related to normal ordering in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. Recently, Kim-Kim introduced a λ lambda -analogue of the unsigned Stirling numbers of the first kind and that of the r r -Stirling numbers of the first kind. In this article, we introduce a λ lambda -analogue of the shift algebra (called λ lambda -shift algebra) and investigate normal ordering in the λ lambda -shift algebra. From the normal ordering in the λ lambda -shift algebra, we derive some identities about the λ lambda -analogue of the unsigned Stirling numbers of the first kind.
{"title":"Normal ordering associated with λ-Stirling numbers in λ-shift algebra","authors":"Taekyun Kim, Dae San Kim, H. Kim","doi":"10.1515/dema-2022-0250","DOIUrl":"https://doi.org/10.1515/dema-2022-0250","url":null,"abstract":"Abstract It is known that the Stirling numbers of the second kind are related to normal ordering in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. Recently, Kim-Kim introduced a λ lambda -analogue of the unsigned Stirling numbers of the first kind and that of the r r -Stirling numbers of the first kind. In this article, we introduce a λ lambda -analogue of the shift algebra (called λ lambda -shift algebra) and investigate normal ordering in the λ lambda -shift algebra. From the normal ordering in the λ lambda -shift algebra, we derive some identities about the λ lambda -analogue of the unsigned Stirling numbers of the first kind.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47413036","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}
A. Alsaedi, B. Ahmad, H. Al-Hutami, Boshra Alharbi
Abstract In this article, we introduce and study a new class of hybrid fractional q q -integro-difference equations involving Riemann-Liouville q q -derivatives, supplemented with nonlocal boundary conditions containing Riemann-Liouville q q -integrals of different orders. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present the existing criteria for solutions to the problem at hand by applying Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results. Some new results follow as special cases of this work.
{"title":"Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions","authors":"A. Alsaedi, B. Ahmad, H. Al-Hutami, Boshra Alharbi","doi":"10.1515/dema-2022-0222","DOIUrl":"https://doi.org/10.1515/dema-2022-0222","url":null,"abstract":"Abstract In this article, we introduce and study a new class of hybrid fractional q q -integro-difference equations involving Riemann-Liouville q q -derivatives, supplemented with nonlocal boundary conditions containing Riemann-Liouville q q -integrals of different orders. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present the existing criteria for solutions to the problem at hand by applying Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results. Some new results follow as special cases of this work.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44672739","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}
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