Positive homogeneity is represented as a constraint 0-homogeneity and generalized into z-homogeneity, called also z-end point linearity. Several special z-homogeneous aggregation functions are studied, in particular semicopulas, quasi-copulas, copulas, overlap functions, etc.
{"title":"(2011-6313) End-point linear functions","authors":"A. Stupnanov, Y. Su, R. Mesiar","doi":"10.22111/IJFS.2021.6165","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6165","url":null,"abstract":"Positive homogeneity is represented as a constraint 0-homogeneity and generalized into z-homogeneity, called also z-end point linearity. Several special z-homogeneous aggregation functions are studied, in particular semicopulas, quasi-copulas, copulas, overlap functions, etc.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79328818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper deals with uninorms and nullnorms as basic semi-group operations which are commutative and monotone (increasing). These operations were first introduced on the unit interval and later generalized to bounded lattices. In [Kalina 2019] they were introduced on bounded posets. This contribution is a generalization and extension of the results in [Kalina 2019]. Some necessary and some sufficient conditions for the existence of idempotent uninorms and idempotent nullnorms on bounded posets are studied. Finally, some application examples are provided.
{"title":"(2008-6077) Idempotent uninorms and nullnorms on bounded posets","authors":"M. Kalina, O. Stasová","doi":"10.22111/IJFS.2021.6162","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6162","url":null,"abstract":"The paper deals with uninorms and nullnorms as basic semi-group operations which are commutative and monotone (increasing). These operations were first introduced on the unit interval and later generalized to bounded lattices. In [Kalina 2019] they were introduced on bounded posets. This contribution is a generalization and extension of the results in [Kalina 2019]. Some necessary and some sufficient conditions for the existence of idempotent uninorms and idempotent nullnorms on bounded posets are studied. Finally, some application examples are provided.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72377237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The preference ranking organization method for enrichment of evaluations (PROMETHEE) {constitutes a family of outranking} multiple-attribute decision-making (MADM) methods {that has been adopted by researchers from many areas during} the last years.It provides reliable and clear results {thanks to the} advantages of different types of preference functions.{In this paper, we incorporate the benefits of q-rung orthopair fuzzy set (for short, q-ROFS) in this strategy of solution.} {This model, q-ROFS, is a generalized form of Pythagorean fuzzy set (PFS)}, as it {broadens} the space of acceptable orthopairs and {has} an ability to deal with more elaborate and vague information. The technique of {our extension of the} PROMETHEE method {uses} q-rung orthopair fuzzy numbers to {render} the ratings of alternatives, {which allows us} to express {uncertain and vague information more accurately}.The usual criterion preference function {has} been used to measure the preferences of {the} alternatives.A partial ordering of alternatives is obtained by considering the outgoing and incoming flows of alternatives, which is known as PROMETHEE I.Furthermore, a complete ordering is accomplished by taking into account the procedure of PROMETHEE II.As a numerical {exercise, we consider the selection of a contractor for a construction project}.{A full analysis} is performed {to illustrate the application of the technique that stems from our approach}.{Then we compare the results that we obtain with the results from existing approaches, including q-rung orthopair fuzzy ELECTRE, q-rung orthopair fuzzy TOPSIS, q-rung orthopair fuzzy VIKOR and q-rung orthopair fuzzy aggregation operators.In this way the accuracy and effectiveness of the presented work is conclusively validated
{"title":"(2009-6145) Multi-criteria decision making based on q-rung orthopair fuzzy promethee approach","authors":"M. Akram, D. Shumaiza","doi":"10.22111/IJFS.2021.6163","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6163","url":null,"abstract":"The preference ranking organization method for enrichment of evaluations (PROMETHEE) {constitutes a family of outranking} multiple-attribute decision-making (MADM) methods {that has been adopted by researchers from many areas during} the last years.It provides reliable and clear results {thanks to the} advantages of different types of preference functions.{In this paper, we incorporate the benefits of q-rung orthopair fuzzy set (for short, q-ROFS) in this strategy of solution.} {This model, q-ROFS, is a generalized form of Pythagorean fuzzy set (PFS)}, as it {broadens} the space of acceptable orthopairs and {has} an ability to deal with more elaborate and vague information. The technique of {our extension of the} PROMETHEE method {uses} q-rung orthopair fuzzy numbers to {render} the ratings of alternatives, {which allows us} to express {uncertain and vague information more accurately}.The usual criterion preference function {has} been used to measure the preferences of {the} alternatives.A partial ordering of alternatives is obtained by considering the outgoing and incoming flows of alternatives, which is known as PROMETHEE I.Furthermore, a complete ordering is accomplished by taking into account the procedure of PROMETHEE II.As a numerical {exercise, we consider the selection of a contractor for a construction project}.{A full analysis} is performed {to illustrate the application of the technique that stems from our approach}.{Then we compare the results that we obtain with the results from existing approaches, including q-rung orthopair fuzzy ELECTRE, q-rung orthopair fuzzy TOPSIS, q-rung orthopair fuzzy VIKOR and q-rung orthopair fuzzy aggregation operators.In this way the accuracy and effectiveness of the presented work is conclusively validated","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75115787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the conception of centroid of n-dimensional fuzzy number is introduced viaregarding its membership function as the density function on its support set, and some properties of it are obtained. Compared with the mean of the multi dimensional fuzzy number, the centroid takes into account the overall relationship between the edge membership functions of the membership function of the multi dimensional fuzzy number. Therefore, it can approximate (characterize) the fuzzy number more objectively and reasonably than using the mean of the multi dimensional fuzzy number. The most important work of this paper is that for two special kinds of multi dimensional fuzzy numbers (fuzzy n-cell numbers and fuzzy n-ellipsoid numbers), we respectively give calculation formulas, which can be used conveniently in application since the formulas are based on a definite integral of the level set functions of the multi dimensional fuzzy number on the unit interval [0,1], rather than the multiple integral of the membership function of the multi dimensional fuzzy number itself on its support set. Then, by using the calculation formulas, we obtain another special property of the centroid for fuzzy n-cell number and fuzzy n-ellipsoid number. Finally, as an example of application, by using the centroid of multi dimensional fuzzy number, we define a fuzzy order on n-dimensional fuzzy number space, which can be used to rank uncertain or imprecise multichannel digital information.
{"title":"(2008-6104) Calculation of centroid of high dimensional fuzzy number and application","authors":"G. Wang, Y. Xu","doi":"10.22111/IJFS.2021.6161","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6161","url":null,"abstract":"In this paper, the conception of centroid of n-dimensional fuzzy number is introduced viaregarding its membership function as the density function on its support set, and some properties of it are obtained. Compared with the mean of the multi dimensional fuzzy number, the centroid takes into account the overall relationship between the edge membership functions of the membership function of the multi dimensional fuzzy number. Therefore, it can approximate (characterize) the fuzzy number more objectively and reasonably than using the mean of the multi dimensional fuzzy number. The most important work of this paper is that for two special kinds of multi dimensional fuzzy numbers (fuzzy n-cell numbers and fuzzy n-ellipsoid numbers), we respectively give calculation formulas, which can be used conveniently in application since the formulas are based on a definite integral of the level set functions of the multi dimensional fuzzy number on the unit interval [0,1], rather than the multiple integral of the membership function of the multi dimensional fuzzy number itself on its support set. Then, by using the calculation formulas, we obtain another special property of the centroid for fuzzy n-cell number and fuzzy n-ellipsoid number. Finally, as an example of application, by using the centroid of multi dimensional fuzzy number, we define a fuzzy order on n-dimensional fuzzy number space, which can be used to rank uncertain or imprecise multichannel digital information.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81497989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper further studies topological structures induced by L-fuzzifying approximation operators, where Ldenotes a completely distributive De Morgan algebra. Firstly, the Alexandrov topologies induced by L-fuzzy relations are investigated with respect to L-fuzzifying approximation operators. Especially, the relationships among those Alexandrov topologies are discussed. Secondly, pseudo-similarity sets of L-fuzzy relations are proposed based on the Alexandrov topologies induced by upper and lower L-fuzzifying approximation operators. Meanwhile, the properties of pseudo-similarity sets are discussed, where some examples are presented to show the differences between similarity set of fuzzy relations and pseudo-similarity set of L-fuzzy relations.
{"title":"(1909-5496) Topological structures induced by L-fuzzifying approximation operators","authors":"Chun Yong Wang, Lijuan Wan, B. Zhang","doi":"10.22111/IJFS.2021.6142","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6142","url":null,"abstract":"This paper further studies topological structures induced by L-fuzzifying approximation operators, where Ldenotes a completely distributive De Morgan algebra. Firstly, the Alexandrov topologies induced by L-fuzzy relations are investigated with respect to L-fuzzifying approximation operators. Especially, the relationships among those Alexandrov topologies are discussed. Secondly, pseudo-similarity sets of L-fuzzy relations are proposed based on the Alexandrov topologies induced by upper and lower L-fuzzifying approximation operators. Meanwhile, the properties of pseudo-similarity sets are discussed, where some examples are presented to show the differences between similarity set of fuzzy relations and pseudo-similarity set of L-fuzzy relations.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72800296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the concept of pseudo-continuous semicopula. We show its relationship with continuity in the second variable and provide a characterization of all semicopulas S such that the smallest semicopula-based universal integral is S-homogeneous. This completely solves Open problem 2.29 proposed by J. Borzova-Molnarova et al. in the paper [2].
本文引入了伪连续半聚的概念。我们在第二个变量中证明了它与连续性的关系，并给出了所有半群S的一个表征，使得最小的基于半群的普适积分是S齐次的。这完全解决了J. Borzova-Molnarova等人在[2]论文中提出的Open problem 2.29。
{"title":"(2006-5941) On homogeneity of the smallest semicopula-based universal integral in the class of pseudo-continuous semicopulas","authors":"T. N. Luan, D. Hoang, T. M. Thuyet","doi":"10.22111/IJFS.2021.6144","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6144","url":null,"abstract":"In this paper, we introduce the concept of pseudo-continuous semicopula. We show its relationship with continuity in the second variable and provide a characterization of all semicopulas S such that the smallest semicopula-based universal integral is S-homogeneous. This completely solves Open problem 2.29 proposed by J. Borzova-Molnarova et al. in the paper [2].","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82165559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The novelty of the type-2 fuzzy variable (T2FV) is its fuzzy secondary possibility distribution. Here, we aim to formulate a new defuzzification model of triangular type-2 fuzzy variables (TT2FVs) based on type reduction. We also show its superiority compared to other existing models by providing some numerical examples. Later, we develop a matrix game problem in the type-2 fuzzy environment to show the applicability of the proposed defuzzification method in a real-world situation. Based on this defuzzification method, two crisp linear programming models are derived, which are subsequently solved by the simplex method using LINGO 17.0 software. Also, the expected pay-off for the maximizing player is calculated as TT2FV, which is desirable. Finally, to show the validity and applicability, the proposed methodology is illustrated with the plastic ban problem.
{"title":"(2005-5916) A novel defuzzification approach of Type-2 fuzzy variable to solving matrix games: An application to plastic ban problem","authors":"M. R. Seikh, S. Karmakar, O. Castillo","doi":"10.22111/IJFS.2021.6143","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6143","url":null,"abstract":"The novelty of the type-2 fuzzy variable (T2FV) is its fuzzy secondary possibility distribution. Here, we aim to formulate a new defuzzification model of triangular type-2 fuzzy variables (TT2FVs) based on type reduction. We also show its superiority compared to other existing models by providing some numerical examples. Later, we develop a matrix game problem in the type-2 fuzzy environment to show the applicability of the proposed defuzzification method in a real-world situation. Based on this defuzzification method, two crisp linear programming models are derived, which are subsequently solved by the simplex method using LINGO 17.0 software. Also, the expected pay-off for the maximizing player is calculated as TT2FV, which is desirable. Finally, to show the validity and applicability, the proposed methodology is illustrated with the plastic ban problem.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89441458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A novel strategy to design optimization is expressedusing the fuzzy preference function concept. This method efficientlyuses the designer's experiences by preference functions and it isalso able to transform a constrained multi-objective optimizationproblem into an unconstrained single-objective optimization problem.These two issues are the most important features of the proposedmethod which using them, you can achieve a more practical solutionin less time. To implement the proposed method, two designoptimizations of an unmanned aerial vehicle are considered whichare: deterministic and non-deterministic optimizations. Theoptimization problem in this paper is a constrained multi-objectiveproblem that with attention to the ability of genetic algorithm,this algorithm is selected as the optimizer. Uncertainties areconsidered and the Monte Carlo simulation (MCS) method is used foruncertainties modeling. The obtained results show a good performanceof this technique in achieving optimal and robust solutions.
{"title":"(2011-6286) A novel method for multi-objective design optimization based on fuzzy systems","authors":"M. R. Setayandeh, A. Babaei","doi":"10.22111/IJFS.2021.6126","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6126","url":null,"abstract":"A novel strategy to design optimization is expressedusing the fuzzy preference function concept. This method efficientlyuses the designer's experiences by preference functions and it isalso able to transform a constrained multi-objective optimizationproblem into an unconstrained single-objective optimization problem.These two issues are the most important features of the proposedmethod which using them, you can achieve a more practical solutionin less time. To implement the proposed method, two designoptimizations of an unmanned aerial vehicle are considered whichare: deterministic and non-deterministic optimizations. Theoptimization problem in this paper is a constrained multi-objectiveproblem that with attention to the ability of genetic algorithm,this algorithm is selected as the optimizer. Uncertainties areconsidered and the Monte Carlo simulation (MCS) method is used foruncertainties modeling. The obtained results show a good performanceof this technique in achieving optimal and robust solutions.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88464511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies the fixed-time synchronization problem of fuzzy stochastic cellular neural networks (FSCNNs) with discrete and distributed delay. Compared with the finite-time synchronization in the existing literature, the fixed-time synchronization of FSCNNs is studied for the first time, and the convergence time obtained does not depend on the upper bound of the initial value of the system. In addition, two kinds of control are designed, one is feedback control and the other is adaptive control. Besides, it is the first time to achieve fixed-time synchronization of FSCNNs via adaptive control. Finally, two numerical examples are also proposed to illustrate the practicability and validity of the results we proposed.
{"title":"(2012-6329) Adaptive control design for fixed-time synchronization of fuzzy stochastic cellular neural networks with discrete and distributed delay","authors":"Y. Liu, M. Liu, X. Xu","doi":"10.22111/IJFS.2021.6120","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6120","url":null,"abstract":"This paper studies the fixed-time synchronization problem of fuzzy stochastic cellular neural networks (FSCNNs) with discrete and distributed delay. Compared with the finite-time synchronization in the existing literature, the fixed-time synchronization of FSCNNs is studied for the first time, and the convergence time obtained does not depend on the upper bound of the initial value of the system. In addition, two kinds of control are designed, one is feedback control and the other is adaptive control. Besides, it is the first time to achieve fixed-time synchronization of FSCNNs via adaptive control. Finally, two numerical examples are also proposed to illustrate the practicability and validity of the results we proposed.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78599989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The hierarchical Chinese postman problem (HCPP), one of the kinds of Chinese postman problem (CPP), aims to visit at least once the arcs are classified according to their precedence relations and to find the shortest tour or tours. In HCPP, if the travel time values between two nodes constituting the cost are not crisp, this problem type can be said to be the hierarchical Chinese postman problem with fuzzy travel times (HCPP-FTT). In real-life problems, as the travel time between the nodes is not fixed due to various factors, in this study triangular fuzzy numbers are used at objective function coefficients of a small-size sample problem. Different ranking methods are used to defuzzify the fuzzy numbers and the acquired results are compared. The mathematical model formed is resolved using CPLEX solver GAMS 24.2.3 software. Besides, in this study, a large-size real-life problem was discussed. The objective function coefficients representing the travel time between the two nodes in the case study discussed were considered as trapezoidal fuzzy numbers. Due to the NP-hard nature of the problem, the mathematical model cannot find solutions in large size problems in the appropriate time interval. For this reason, in this study, the heuristic method based on the Greedy Search (GS) algorithm and the meta-heuristic method based on ant colony optimization (ACO) is proposed for the solution of the real-life problem. The proposed algorithms are coded using the Matlab-2019b programming language. Using the developed algorithms, the results of different ranking methods and solution times were compared.
{"title":"(2004-5847) Hierarchical Chinese postman problem with fuzzy travel times","authors":"O. Sokmen, M. Yilmaz","doi":"10.22111/IJFS.2021.6100","DOIUrl":"https://doi.org/10.22111/IJFS.2021.6100","url":null,"abstract":"The hierarchical Chinese postman problem (HCPP), one of the kinds of Chinese postman problem (CPP), aims to visit at least once the arcs are classified according to their precedence relations and to find the shortest tour or tours. In HCPP, if the travel time values between two nodes constituting the cost are not crisp, this problem type can be said to be the hierarchical Chinese postman problem with fuzzy travel times (HCPP-FTT). In real-life problems, as the travel time between the nodes is not fixed due to various factors, in this study triangular fuzzy numbers are used at objective function coefficients of a small-size sample problem. Different ranking methods are used to defuzzify the fuzzy numbers and the acquired results are compared. The mathematical model formed is resolved using CPLEX solver GAMS 24.2.3 software. Besides, in this study, a large-size real-life problem was discussed. The objective function coefficients representing the travel time between the two nodes in the case study discussed were considered as trapezoidal fuzzy numbers. Due to the NP-hard nature of the problem, the mathematical model cannot find solutions in large size problems in the appropriate time interval. For this reason, in this study, the heuristic method based on the Greedy Search (GS) algorithm and the meta-heuristic method based on ant colony optimization (ACO) is proposed for the solution of the real-life problem. The proposed algorithms are coded using the Matlab-2019b programming language. Using the developed algorithms, the results of different ranking methods and solution times were compared.","PeriodicalId":54920,"journal":{"name":"Iranian Journal of Fuzzy Systems","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82447723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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