Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.04
Kuldeep Kaur Shergill, S. S. Billing
In the present paper, we study certain differential inequalities involving meromorphic functions in the open unit disk and obtain certain sufficient conditions for starlikeness and close-to-convexity of meromorphic functions. Mathematics Subject Classification (2010): 30C45, 30C80. Received 06 April 2021; Accepted 11 October 2022
{"title":"Starlikeness and close-to-convexity involving certain differential inequalities","authors":"Kuldeep Kaur Shergill, S. S. Billing","doi":"10.24193/subbmath.2024.1.04","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.04","url":null,"abstract":"In the present paper, we study certain differential inequalities involving meromorphic functions in the open unit disk and obtain certain sufficient conditions for starlikeness and close-to-convexity of meromorphic functions.\u0000Mathematics Subject Classification (2010): 30C45, 30C80.\u0000Received 06 April 2021; Accepted 11 October 2022","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140391617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.03
M. Aouf, A. Moustafa, Fawziah Al-Quhali
In this paper, we obtain coefficient estimates, distortion theorems, radii of close-to-convexity, starlikeness and convexity for functions belonging to the class of analytic starlike and convex functions defined by q−analogue of Ruscheweyh differential operator. Also we find closure theorems, Nk,q,δ (e, g) neighborhood and partial sums for functions in this class. Mathematics Subject Classification (2010): 30C45. Received 07 June 2020; Accepted 06 August 2020
{"title":"Certain class of analytic functions defined by q-analogue of Ruscheweyh differential operator","authors":"M. Aouf, A. Moustafa, Fawziah Al-Quhali","doi":"10.24193/subbmath.2024.1.03","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.03","url":null,"abstract":"In this paper, we obtain coefficient estimates, distortion theorems, radii of close-to-convexity, starlikeness and convexity for functions belonging to the class of analytic starlike and convex functions defined by q−analogue of Ruscheweyh differential operator. Also we find closure theorems, Nk,q,δ (e, g) neighborhood and partial sums for functions in this class.\u0000Mathematics Subject Classification (2010): 30C45.\u0000Received 07 June 2020; Accepted 06 August 2020","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140391376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.14
M. Crășmăreanu, G. Pripoae
We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it Pólya, being the difference between the usual curvature and the inner product of the normal vector field with the Pólya vector field of a given planar function f. We computed it for several examples, since the general problem of vanishing or constant values of this new curvature involves the general expression of f. Mathematics Subject Classification (2010): 53A04, 53A45, 53A55. Received 03 May 2023; Accepted 17 January 2024
{"title":"The Polya f-curvature of plane curves","authors":"M. Crășmăreanu, G. Pripoae","doi":"10.24193/subbmath.2024.1.14","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.14","url":null,"abstract":"We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it Pólya, being the difference between the usual curvature and the inner product of the normal vector field with the Pólya vector field of a given planar function f. We computed it for several examples, since the general problem of vanishing or constant values of this new curvature involves the general expression of f.\u0000Mathematics Subject Classification (2010): 53A04, 53A45, 53A55.\u0000Received 03 May 2023; Accepted 17 January 2024","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140392060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.09
Viet Duoc Trinh, Huy Nguyen Ngoc
In this paper we investigate the homogeneous linear differential equation vi(t) = A(t)v(t) and the semi-linear differential equation vi(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A : R → L(X) is a strongly continuous function, g : R × X → X is continuous and satisfies ϕ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E∞), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C1. Mathematics Subject Classification (2010): 34C45, 34D09, 34D10. Received 14 June 2021; Accepted 09 September 2022
本文研究了巴拿赫空间 X 中的同质线性微分方程 vi(t) = A(t)v(t) 和半线性微分方程 vi(t) = A(t)v(t) + g(t，v(t))，其中 A : R → L(X) 是强连续函数，g ：R × X → X 是连续的，且满足 j-Lipschitz 条件。首先，我们通过空间对（E, E∞）描述了与同质线性微分方程相关的演化族的指数二分法，这是一个 Perron 类型的结果。应用已取得的结果，我们建立了指数二分法的稳健性。接下来，我们证明了半线性微分方程的稳定流形和不稳定流形的存在，并证明这些流形的每个纤维都是C1类的可微分子流形：34C45, 34D09, 34D10.2021 年 6 月 14 日收到；2022 年 9 月 9 日接受
{"title":"Exponential dichotomy and invariant manifolds of semi-linear differential equations on the line","authors":"Viet Duoc Trinh, Huy Nguyen Ngoc","doi":"10.24193/subbmath.2024.1.09","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.09","url":null,"abstract":"In this paper we investigate the homogeneous linear differential equation vi(t) = A(t)v(t) and the semi-linear differential equation vi(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A : R → L(X) is a strongly continuous function, g : R × X → X is continuous and satisfies ϕ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E∞), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C1.\u0000Mathematics Subject Classification (2010): 34C45, 34D09, 34D10.\u0000Received 14 June 2021; Accepted 09 September 2022","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140392251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.13
I. Rus
Let (X, d) be a metric space, f : X → X be a mapping and G(·, f (·)) be an admissible perturbation of f. In this paper we study the following problems: In which conditions imposed on f and G we have the following: (DDE) data dependence estimate for the mapping f perturbation; (UH) Ulam-Hyers stability for the equation, x = f (x); (WP) well-posedness of the fixed-point problem for f; (OP) Ostrowski property of the mapping f. Some research directions are suggested. Mathematics Subject Classification (2010): 47H25, 54H25, 47H09, 65J15, 37N30, 39A30. Received 22 October 2023; Accepted 16 November 2023
假设 (X, d) 是一个度量空间，f : X → X 是一个映射，G(-, f (-)) 是 f 的可允许扰动：在对 f 和 G 施加的条件中，我们有以下条件：（DDE）映射 f 扰动的数据依赖性估计；（UH）方程 x = f (x) 的 Ulam-Hyers 稳定性；（WP）f 的定点问题的好求性；（OP）映射 f 的 Ostrowski 特性。提出了一些研究方向：47H25, 54H25, 47H09, 65J15, 37N30, 39A30.Received 22 October 2023; Accepted 16 November 2023.
{"title":"Weakly Picard mappings: Retraction-displacement condition, quasicontraction notion and weakly Picard admissible perturbation","authors":"I. Rus","doi":"10.24193/subbmath.2024.1.13","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.13","url":null,"abstract":"Let (X, d) be a metric space, f : X → X be a mapping and G(·, f (·)) be an admissible perturbation of f. In this paper we study the following problems: In which conditions imposed on f and G we have the following:\u0000(DDE) data dependence estimate for the mapping f perturbation; \u0000(UH) Ulam-Hyers stability for the equation, x = f (x);\u0000(WP) well-posedness of the fixed-point problem for f; \u0000(OP) Ostrowski property of the mapping f.\u0000Some research directions are suggested.\u0000Mathematics Subject Classification (2010): 47H25, 54H25, 47H09, 65J15, 37N30, 39A30.\u0000Received 22 October 2023; Accepted 16 November 2023","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140391286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.01
Michael Aristidou, Philip R. Brown, George Chailos
In this paper, we show that the set O/Zp, where p is a prime number, does not form a skew field and discuss idempotent and nilpotent elements in the (finite) ring O/Zp. We provide examples and establish conditions for idempotency and nilpotency. Mathematics Subject Classification (2010): 15A33, 15A30, 20H25, 15A03. Received 27 July 2021; Accepted 14 December 2021
{"title":"Idempotent and nilpotent elements in octonion rings over Z","authors":"Michael Aristidou, Philip R. Brown, George Chailos","doi":"10.24193/subbmath.2024.1.01","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.01","url":null,"abstract":"In this paper, we show that the set O/Zp, where p is a prime number, does not form a skew field and discuss idempotent and nilpotent elements in the (finite) ring O/Zp. We provide examples and establish conditions for idempotency\u0000and nilpotency.\u0000Mathematics Subject Classification (2010): 15A33, 15A30, 20H25, 15A03.\u0000Received 27 July 2021; Accepted 14 December 2021\u0000","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140391639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.08
Karunamurthy Saranya, V. Piramanantham, E. Thandapani, E. Tunç
In this paper, we investigate the oscillatory behavior of solutions to a class of third-order differential equations of the form Lz(t) + f (t)yβ (σ(t)) = 0, where Lz(t) = (p(t)(q(t)zt(t))t)t is a semi-canonical operator and z(t) = y(t) + g(t)y(τ (t)). The main idea is to convert the semi-canonical operator into canonical form and then obtain some new sufficient conditions for the oscillation of all solutions. The obtained results essentially improve and complement to the known results. Examples are provided to illustrate the main results. Mathematics Subject Classification (2010): 34C10, 34K11, 34K40. Received 19 September 2021; Accepted 20 January 2022
{"title":"Oscillation criteria for third-order semi-canonical differential equations with unbounded neutral coefficients","authors":"Karunamurthy Saranya, V. Piramanantham, E. Thandapani, E. Tunç","doi":"10.24193/subbmath.2024.1.08","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.08","url":null,"abstract":"In this paper, we investigate the oscillatory behavior of solutions to a class of third-order differential equations of the form Lz(t) + f (t)yβ (σ(t)) = 0, where Lz(t) = (p(t)(q(t)zt(t))t)t is a semi-canonical operator and z(t) = y(t) + g(t)y(τ (t)). The main idea is to convert the semi-canonical operator into canonical form and then obtain some new sufficient conditions for the oscillation of all solutions. The obtained results essentially improve and complement to the known results. Examples are provided to illustrate the main results.\u0000Mathematics Subject Classification (2010): 34C10, 34K11, 34K40.\u0000Received 19 September 2021; Accepted 20 January 2022","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140391390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.06
Gabriela Apreutesei, Teodor Precupanu
In this paper, we establish some properties of the multivalued mapping (x, d) ⇒ DC (x; d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x; d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC. Mathematics Subject Classification (2010): 32A70, 41A65, 46B20, 46N10. Received 18 May 2023; Accepted 27 November 2023
在本文中，我们建立了多值映射 (x, d) ⇒ DC (x; d) 的一些性质，该映射将线性规范空间 X 的每个元素 x 与规范 d ≥ 0 的线性连续函数集联系起来，并将闭球 B (x; d) 与闭凸集 C ⊂ X 分开。利用这一映射，我们给出了与凸分析中其他重要概念（ε-近似元、距离函数的ε-次微分、对偶映射、极锥）的联系。因此，我们建立了ε-近似元关于非虚闭凸集的对偶表征，这是对加尔卡维已知结果的推广。此外，我们还给出了映射 DC 的单向性和单调性的一些性质：32A70, 41A65, 46B20, 46N10.2023 年 5 月 18 日收到；2023 年 11 月 27 日接受
{"title":"A dual mapping associated to a closed convex set and some subdifferential properties","authors":"Gabriela Apreutesei, Teodor Precupanu","doi":"10.24193/subbmath.2024.1.06","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.06","url":null,"abstract":"In this paper, we establish some properties of the multivalued mapping (x, d) ⇒ DC (x; d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x; d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC.\u0000Mathematics Subject Classification (2010): 32A70, 41A65, 46B20, 46N10.\u0000Received 18 May 2023; Accepted 27 November 2023","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140283927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.15
Marius Costandin, B. Gavrea
In this paper, given a fixed reference point and a fixed intersection of finitely many equal radii balls, we consider the problem of finding a point in the said set which is the most distant, under Euclidean distance, to the said reference point. This proble is NP-complete in the general setting. We give sufficient conditions for the existence of an algorithm of polynomial complexity which can solve the problem, in a particular setting. Our algorithm requires that any point in the said intersection to be no closer to the given reference point than the radius of the intersecting balls. Checking this requirement is a convex optimization problem hence one can decide if running the proposed algorithm enjoys the presented theoretical guarantees. We also consider the problem where a fixed initial reference point and a fixed polytope are given and we want to find the farthest point in the polytope to the given reference point. For this problem we give sufficient conditions in which the solution can be found by solving a linear program. Both these problems are known to be NP-complete in the general setup, i.e. the existence of an algorithm which solves any of the above problems without restrictions on the given reference point and search set is undecided so far. Mathematics Subject Classification (2010): 90-08. Received 21 December 2021; Accepted 01 August 2023
{"title":"A polynomial algorithm for some instances of NP-complete problems","authors":"Marius Costandin, B. Gavrea","doi":"10.24193/subbmath.2024.1.15","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.15","url":null,"abstract":"In this paper, given a fixed reference point and a fixed intersection of finitely many equal radii balls, we consider the problem of finding a point in the said set which is the most distant, under Euclidean distance, to the said reference point. This proble is NP-complete in the general setting. We give sufficient conditions for the existence of an algorithm of polynomial complexity which can solve the problem, in a particular setting. Our algorithm requires that any point in the said intersection to be no closer to the given reference point than the radius of the intersecting balls. Checking this requirement is a convex optimization problem hence one can decide if running the proposed algorithm enjoys the presented theoretical guarantees. We also consider the problem where a fixed initial reference point and a fixed polytope are given and we want to find the farthest point in the polytope to the given reference point. For this problem we give sufficient conditions in which the solution can be found by solving a linear program. Both these problems are known to be NP-complete in the general setup, i.e. the existence of an algorithm which solves any of the above problems without restrictions on the given reference point and search set is undecided so far.\u0000Mathematics Subject Classification (2010): 90-08.\u0000Received 21 December 2021; Accepted 01 August 2023","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140283988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.24193/subbmath.2024.1.02
Asif R. Khan, Hira Nabi, J. Pečarić
In this article, we derive the Popoviciu-type inequalities by using the weighted version of the extension of Montgomery’s identity and Green functions. Some results for n-convex functions at a point are also obtained. Besides that, some Ostrowski-type inequalities are also presented, which are interrelated with the obtained inequalities. Mathematics Subject Classification (2010): 26A51, 26D15, 26D20. Received 26 September 2021; Accepted 13 May 2022
{"title":"Popoviciu type inequalities for n-convex functions via extension of weighted Montgomery identity","authors":"Asif R. Khan, Hira Nabi, J. Pečarić","doi":"10.24193/subbmath.2024.1.02","DOIUrl":"https://doi.org/10.24193/subbmath.2024.1.02","url":null,"abstract":"In this article, we derive the Popoviciu-type inequalities by using the weighted version of the extension of Montgomery’s identity and Green functions. Some results for n-convex functions at a point are also obtained. Besides that, some Ostrowski-type inequalities are also presented, which are interrelated with the obtained inequalities.\u0000Mathematics Subject Classification (2010): 26A51, 26D15, 26D20.\u0000Received 26 September 2021; Accepted 13 May 2022","PeriodicalId":517948,"journal":{"name":"Studia Universitatis Babes-Bolyai Matematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140391923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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