Pub Date : 2025-01-03DOI: 10.1016/j.fss.2024.109260
Hongzhou Wang , Rosana Rodríguez-López
Stopping time problem of first order multidimensional interval-valued dynamic system is discussed. By calculating stopping times of corresponding linear differential equations, we provide some estimation results of stopping times of forward and backward solutions to nonlinear interval-valued differential equations with respect to length or volume constraints. Then, stopping times of some linear and nonlinear interval-valued differential equations models, including predator-prey system, two species mutualism and competition systems, Lorenz equations, are studied as applications.
{"title":"Stopping time estimation of first order multidimensional interval-valued differential equations","authors":"Hongzhou Wang , Rosana Rodríguez-López","doi":"10.1016/j.fss.2024.109260","DOIUrl":"10.1016/j.fss.2024.109260","url":null,"abstract":"<div><div>Stopping time problem of first order multidimensional interval-valued dynamic system is discussed. By calculating stopping times of corresponding linear differential equations, we provide some estimation results of stopping times of forward and backward solutions to nonlinear interval-valued differential equations with respect to length or volume constraints. Then, stopping times of some linear and nonlinear interval-valued differential equations models, including predator-prey system, two species mutualism and competition systems, Lorenz equations, are studied as applications.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"505 ","pages":"Article 109260"},"PeriodicalIF":3.2,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143182379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-03DOI: 10.1016/j.fss.2024.109257
Luis González-De La Fuente , Alicia Nieto-Reyes , Pedro Terán
Statistical depth functions order the elements of a space with respect to their centrality in a probability distribution or dataset. Since many depth functions are maximized in the real line by the median, they provide a natural approach to defining median-like location estimators for more general types of data (in our case, fuzzy data). We analyze the relationships between depth-based medians, medians based on the support function, and some notions of a median for fuzzy data in the literature. We take advantage of specific depth functions for fuzzy data defined in our former papers: adaptations of Tukey depth, simplicial depth, -depth and projection depth.
{"title":"Statistical depth and support medians for fuzzy data","authors":"Luis González-De La Fuente , Alicia Nieto-Reyes , Pedro Terán","doi":"10.1016/j.fss.2024.109257","DOIUrl":"10.1016/j.fss.2024.109257","url":null,"abstract":"<div><div>Statistical depth functions order the elements of a space with respect to their centrality in a probability distribution or dataset. Since many depth functions are maximized in the real line by the median, they provide a natural approach to defining median-like location estimators for more general types of data (in our case, fuzzy data). We analyze the relationships between depth-based medians, medians based on the support function, and some notions of a median for fuzzy data in the literature. We take advantage of specific depth functions for fuzzy data defined in our former papers: adaptations of Tukey depth, simplicial depth, <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-depth and projection depth.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109257"},"PeriodicalIF":3.2,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143158905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-03DOI: 10.1016/j.fss.2024.109258
Enrique Miranda , Arthur Van Camp
The law of iterated expectation tells us how to combine hierarchical pieces of information when our uncertainty is modelled by means of probability measures. It has been extended to the imprecise case through Walley's marginal extension theorem for coherent lower previsions. In this paper, we investigate the extent to which a similar result can be established for other imprecise probability models that are either more general (choice functions) or more particular (possibility measures, belief functions) than coherent lower previsions. By doing this, we also establish links with other results established in the literature in the context of imprecise versions of Jeffrey's rule.
{"title":"The law of iterated expectation and imprecise probabilities","authors":"Enrique Miranda , Arthur Van Camp","doi":"10.1016/j.fss.2024.109258","DOIUrl":"10.1016/j.fss.2024.109258","url":null,"abstract":"<div><div>The law of iterated expectation tells us how to combine hierarchical pieces of information when our uncertainty is modelled by means of probability measures. It has been extended to the imprecise case through Walley's marginal extension theorem for coherent lower previsions. In this paper, we investigate the extent to which a similar result can be established for other imprecise probability models that are either more general (choice functions) or more particular (possibility measures, belief functions) than coherent lower previsions. By doing this, we also establish links with other results established in the literature in the context of imprecise versions of Jeffrey's rule.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109258"},"PeriodicalIF":3.2,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we propose an adaptive event-triggered fuzzy dynamic surface controller for non-strict feedback stochastic nonlinear systems, where the nonlinear stochastic system exhibits input dead-zone characteristics and state constraints. Firstly, in contrast to many existing adaptive backstepping control results, we consider designing a dynamic surface control strategy to simplify complexity and enhance performance in stochastic systems. Then, the unknown nonlinearities within the system are addressed with the help of the approximation capability of the fuzzy logic system. It is worth noting that the design of the event-triggered mechanism effectively reduces resource waste in the data channel and improves communication efficiency. In addition, the adaptive backstepping control is combined with the barrier Lyapunov function in a unified framework to handle the state constraints. Resorting to a novel auxiliary system, the effect of the input dead-zone nonlinearity is countered. The presented fuzzy dynamic surface control scheme not only ensures the semi-globally uniformly ultimately bounded of the controlled system but also constrains the state within a fixed range and gets rid of the Zeno phenomenon. Finally, numerical simulation results validate the effectiveness of the control strategy and illustrate the feasibility of the controller through a single-link manipulator system.
{"title":"Event-triggered-based fuzzy adaptive tracking control for stochastic nonlinear systems against multiple constraints","authors":"Haina Zhao , Junsheng Zhao , Zong-Yao Sun , Dengxiu Yu","doi":"10.1016/j.fss.2024.109253","DOIUrl":"10.1016/j.fss.2024.109253","url":null,"abstract":"<div><div>In this article, we propose an adaptive event-triggered fuzzy dynamic surface controller for non-strict feedback stochastic nonlinear systems, where the nonlinear stochastic system exhibits input dead-zone characteristics and state constraints. Firstly, in contrast to many existing adaptive backstepping control results, we consider designing a dynamic surface control strategy to simplify complexity and enhance performance in stochastic systems. Then, the unknown nonlinearities within the system are addressed with the help of the approximation capability of the fuzzy logic system. It is worth noting that the design of the event-triggered mechanism effectively reduces resource waste in the data channel and improves communication efficiency. In addition, the adaptive backstepping control is combined with the barrier Lyapunov function in a unified framework to handle the state constraints. Resorting to a novel auxiliary system, the effect of the input dead-zone nonlinearity is countered. The presented fuzzy dynamic surface control scheme not only ensures the semi-globally uniformly ultimately bounded of the controlled system but also constrains the state within a fixed range and gets rid of the Zeno phenomenon. Finally, numerical simulation results validate the effectiveness of the control strategy and illustrate the feasibility of the controller through a single-link manipulator system.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109253"},"PeriodicalIF":3.2,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143158904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-31DOI: 10.1016/j.fss.2024.109256
Huan Huang
This paper discusses the properties of the spaces of fuzzy sets in a metric space with -type metrics, . The metrics are well-defined if and only if the corresponding Haudorff distance functions are measurable. In this paper, we give several fundamental conclusions on the measurability of these Haudorff distance functions. Then we give some characterizations of total boundedness, relative compactness and compactness in fuzzy set spaces with metrics. At last we present the completions of fuzzy set spaces with metrics.
{"title":"Properties of fuzzy set spaces with Lp metrics","authors":"Huan Huang","doi":"10.1016/j.fss.2024.109256","DOIUrl":"10.1016/j.fss.2024.109256","url":null,"abstract":"<div><div>This paper discusses the properties of the spaces of fuzzy sets in a metric space with <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-type <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> metrics, <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>. The <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> metrics are well-defined if and only if the corresponding Haudorff distance functions are measurable. In this paper, we give several fundamental conclusions on the measurability of these Haudorff distance functions. Then we give some characterizations of total boundedness, relative compactness and compactness in fuzzy set spaces with <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> metrics. At last we present the completions of fuzzy set spaces with <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> metrics.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109256"},"PeriodicalIF":3.2,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143158903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1016/j.fss.2024.109255
Yu Xia , Junyang Li , Cheng Wang
This study expands the existing model for microelectromechanical system (MEMS) gyroscopes by transitioning from a deterministic model to a stochastic nonlinear model and proposes an adaptive fuzzy control scheme that is triggered by events and ensures prescribed performance for stochastic MEMS gyroscopes. Unlike current control schemes for MEMS gyroscopes, the scheme guarantees full control over output overshoot and input vibration, while eliminating the need for design parameters to meet feasibility conditions in event-triggered mechanisms. Additionally, it introduces a type-3 fuzzy system with improved modeling capabilities to approximate unknown nonlinear terms. By incorporating command filtering techniques, the scheme effectively addresses issues related to complexity explosion and filtering errors in backstepping designs. Through the application of fixed-time Lyapunov stability theory on stochastic systems, it is demonstrated that all closed-loop signals are fixed-time bounded in probability. Moreover, the tracking error consistently remains within predefined performance boundaries. Simulation experiments validate both the effectiveness and superiority of the scheme.
{"title":"Fixed-time adaptive fuzzy control for stochastic MEME gyroscopes with optimized transient behaviors and limited communication resources","authors":"Yu Xia , Junyang Li , Cheng Wang","doi":"10.1016/j.fss.2024.109255","DOIUrl":"10.1016/j.fss.2024.109255","url":null,"abstract":"<div><div>This study expands the existing model for microelectromechanical system (MEMS) gyroscopes by transitioning from a deterministic model to a stochastic nonlinear model and proposes an adaptive fuzzy control scheme that is triggered by events and ensures prescribed performance for stochastic MEMS gyroscopes. Unlike current control schemes for MEMS gyroscopes, the scheme guarantees full control over output overshoot and input vibration, while eliminating the need for design parameters to meet feasibility conditions in event-triggered mechanisms. Additionally, it introduces a type-3 fuzzy system with improved modeling capabilities to approximate unknown nonlinear terms. By incorporating command filtering techniques, the scheme effectively addresses issues related to complexity explosion and filtering errors in backstepping designs. Through the application of fixed-time Lyapunov stability theory on stochastic systems, it is demonstrated that all closed-loop signals are fixed-time bounded in probability. Moreover, the tracking error consistently remains within predefined performance boundaries. Simulation experiments validate both the effectiveness and superiority of the scheme.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109255"},"PeriodicalIF":3.2,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143158902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-24DOI: 10.1016/j.fss.2024.109254
Zhenyu Jin , Conghua Yan
Motivated by Zhang et al. [28], [38], [39], we aim to make a further discussion of ⁎-betweenness relations. In this paper, we provide the method to construct the ⁎-betweenness by GV fuzzy metrics. After considering the description of ⁎-equivalence with respect to strong fuzzy metrics, we investigate the construction of ⁎-E-betweenness. Some properties of ⁎-betweenness and ⁎-E-betweenness are discussed and some conclusions or conditions in [28] are verified. This paper is also a supplement of [28]. Finally, the relationship between fuzzifying convex structures and ⁎-betweenness relations has been studied.
{"title":"⁎-betweenness induced by fuzzy metrics","authors":"Zhenyu Jin , Conghua Yan","doi":"10.1016/j.fss.2024.109254","DOIUrl":"10.1016/j.fss.2024.109254","url":null,"abstract":"<div><div>Motivated by Zhang et al. <span><span>[28]</span></span>, <span><span>[38]</span></span>, <span><span>[39]</span></span>, we aim to make a further discussion of ⁎-betweenness relations. In this paper, we provide the method to construct the ⁎-betweenness by GV fuzzy metrics. After considering the description of ⁎-equivalence with respect to strong fuzzy metrics, we investigate the construction of ⁎-<em>E</em>-betweenness. Some properties of ⁎-betweenness and ⁎-<em>E</em>-betweenness are discussed and some conclusions or conditions in <span><span>[28]</span></span> are verified. This paper is also a supplement of <span><span>[28]</span></span>. Finally, the relationship between fuzzifying convex structures and ⁎-betweenness relations has been studied.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109254"},"PeriodicalIF":3.2,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-24DOI: 10.1016/j.fss.2024.109251
Tomaž Košir , Elisa Perrone
In this paper, we study discrete quasi-copulas associated with imprecise copulas. We focus on discrete imprecise copulas that are in correspondence with the Alternating Sign Matrices and provide some construction techniques of dual pairs. Additionally, we show how to obtain (full-domain) self-dual imprecise copulas through patchwork techniques. We further investigate the properties of the constructed dual pairs and prove that they are coherent and avoid sure loss.
{"title":"Discrete imprecise copulas","authors":"Tomaž Košir , Elisa Perrone","doi":"10.1016/j.fss.2024.109251","DOIUrl":"10.1016/j.fss.2024.109251","url":null,"abstract":"<div><div>In this paper, we study discrete quasi-copulas associated with imprecise copulas. We focus on discrete imprecise copulas that are in correspondence with the Alternating Sign Matrices and provide some construction techniques of dual pairs. Additionally, we show how to obtain (full-domain) self-dual imprecise copulas through patchwork techniques. We further investigate the properties of the constructed dual pairs and prove that they are coherent and avoid sure loss.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109251"},"PeriodicalIF":3.2,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-20DOI: 10.1016/j.fss.2024.109252
Liangyao Shi , Feng Li , Hao Shen , Ju H. Park
This paper addresses the bipartite synchronization issue for a class of continuous-time nonlinear cooperation-competition networks, in which the relationships between network systems coexist with coordination and antagonism with the network topology switching obeys a semi-Markov jump process. First, the Takagi-Sugeno fuzzy method is employed to modify the nonlinear function in the networks to deal with this complexity. Then, a fuzzy security control law is devised for the networks to ensure that systems enable bipartite synchronization when communication channels are vulnerable to false data injection attacks. In addition, sufficient conditions are provided to guarantee that the resulting closed-loop system is stochastically stable with a prescribed passive performance index by constructing suitable Lyapunov functions. Lastly, the correctness of the proposed controller is validated by a numerical example and a single-link robot arm model example.
{"title":"Fuzzy-model-based bipartite synchronization for nonlinear cooperation-competition networks with semi-Markov switching topologies under FDI attacks","authors":"Liangyao Shi , Feng Li , Hao Shen , Ju H. Park","doi":"10.1016/j.fss.2024.109252","DOIUrl":"10.1016/j.fss.2024.109252","url":null,"abstract":"<div><div>This paper addresses the bipartite synchronization issue for a class of continuous-time nonlinear cooperation-competition networks, in which the relationships between network systems coexist with coordination and antagonism with the network topology switching obeys a semi-Markov jump process. First, the Takagi-Sugeno fuzzy method is employed to modify the nonlinear function in the networks to deal with this complexity. Then, a fuzzy security control law is devised for the networks to ensure that systems enable bipartite synchronization when communication channels are vulnerable to false data injection attacks. In addition, sufficient conditions are provided to guarantee that the resulting closed-loop system is stochastically stable with a prescribed passive performance index by constructing suitable Lyapunov functions. Lastly, the correctness of the proposed controller is validated by a numerical example and a single-link robot arm model example.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109252"},"PeriodicalIF":3.2,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-19DOI: 10.1016/j.fss.2024.109247
Xianliang Liu , Zhihuan Hu , Weidong Zhang
In a type-2 fuzzy logic system, the centroid computation of a general type-2 fuzzy set (T2 FS) is one of the most important type reduction methods. Nearly all of the existing algorithms are based on the α-plane representation or z-slice representation. Therefore, all of these algorithms can only approximately compute the centroid of a general T2 FS and cannot obtain the analytic expression of the centroid of a general T2 FS. The main objective of this study is to obtain the analytic expression of the centroid of a general T2 FS with monotonically increasing second membership functions based on a discrete universe of discourse. First, assuming that the second membership functions of a general T2 FS is linear and monotonically increasing, a method is proposed to compute its centroid. Second, an approach to calculate the centroid of a general T2 FS is also analyzed, if the second membership functions are only monotonically increasing. Finally, two numerical examples are given to demonstrate the calculation processes of the proposed method in this study. Notably, the proposed method can also be employed to obtain the analytic expression of the centroid of a general T2 FS with monotonically decreasing second membership functions.
{"title":"On the centroid of a general type-2 fuzzy set with monotonically increasing second membership functions","authors":"Xianliang Liu , Zhihuan Hu , Weidong Zhang","doi":"10.1016/j.fss.2024.109247","DOIUrl":"10.1016/j.fss.2024.109247","url":null,"abstract":"<div><div>In a type-2 fuzzy logic system, the centroid computation of a general type-2 fuzzy set (T2 FS) is one of the most important type reduction methods. Nearly all of the existing algorithms are based on the <em>α</em>-plane representation or <em>z</em>-slice representation. Therefore, all of these algorithms can only approximately compute the centroid of a general T2 FS and cannot obtain the analytic expression of the centroid of a general T2 FS. The main objective of this study is to obtain the analytic expression of the centroid of a general T2 FS with monotonically increasing second membership functions based on a discrete universe of discourse. First, assuming that the second membership functions of a general T2 FS is linear and monotonically increasing, a method is proposed to compute its centroid. Second, an approach to calculate the centroid of a general T2 FS is also analyzed, if the second membership functions are only monotonically increasing. Finally, two numerical examples are given to demonstrate the calculation processes of the proposed method in this study. Notably, the proposed method can also be employed to obtain the analytic expression of the centroid of a general T2 FS with monotonically decreasing second membership functions.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109247"},"PeriodicalIF":3.2,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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