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Group Generalized Intervalvalued Intuitionistic Fuzzy Soft Sets and Their Applications in\ Decision Making
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Intervalvalued intuitionistic fuzzy sets (IVIFSs) are widely used to handle uncertainty and imprecision in decision making. However, in more complicated environment, it is difficult to express the uncertain information by an IVIFS with considering the decisionmaking preference. Hence, this paper proposes a group generalized intervalvalued intuitionistic fuzzy soft set (GGIVIFSS) which contains the basic description by intervalvalued intuitionistic fuzzy soft set (IVIFSS) on the alternatives and a group of experts' evaluation of it. It contributes the following threefold: 1) A generalized intervalvalued intuitionistic fuzzy soft set (GIVIFSS) is proposed by introducing an intervalvalued intuitionistic fuzzy parameter, which reflects a new and senior expert's opinion on the basic description. The operations, properties and aggregation operators of GIVIFSS are discussed. 2) Based on GIVIFSS, a GGIVIFSS is then proposed to reduce the impact of decisionmaking preference by introducing more parameters by a group of experts. Its important operations, properties and the weighted averaging operator are also defined. 3) A multiattribute group decision making model based on GGIVIFSS weighted averaging operator is built to solve the group decision making problems in the more universal IVIF environment, and two practical examples are taken to validate the efficiency and effectiveness of the proposed model.
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Hua
Wu
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China and University of Chinese Academy of Sciences, Beijing, China.
Key Laboratory of Ultrafast Photoelectric
China
sunshinesmilewh@gmail.com


Xiuqin
Su
Key Laboratory of Ultrafast Photoelectric Diagnostics Technology,
Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi'an, China
Key Laboratory of Ultrafast Photoelectric
China
suxiuqin@opt.ac.cn
Group decision making
Intervalvalued intuitionistic fuzzy set
Generalized intervalvalued intuitionistic fuzzy soft set
Soft set
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Kar, Group decision making in medical system: an intuitionistic fuzzy soft set##approach, Applied Soft Computing, 24 (2014), 196{211.##[8] B. Dinda and T. Bera and T. Samanta, Generalised intuitionistic fuzzy soft sets and its##application in decision making, arXiv preprint arXiv:1010.2468, 2010.##[9] S. Ebrahimnejad and H. Hashemi and S. Mousavi and B. Vahdani, A new intervalvalued##intuitionistic fuzzy model to group decision making for the selection of outsourcing providers,##Economic Computation and Economic Cybernetics Studies and Research, 49(2) (2015), 269{##[10] F. Feng and C. Li and B. Davvaz and M. Ali, Soft sets combined with fuzzy sets and rough##sets: a tentative approach, Soft Computing, 14(9) (2010), 899{911.##[11] H. Hashemi, J. Bazargan, S. Mousavi and B. Vahdani, An extended compromise ratio model##with an application to reservoir ##ood control operation under an intervalvalued intuitionistic##fuzzy environment, Applied Mathematical Modelling, 38(14) (2014), 3495{3511.##[12] Y. Jiang, Y. Tang,Q. Chen, H. Liu and J. Tang, Intervalvalued intuitionistic fuzzy soft sets##and their properties, Computers & Mathematics with Applications, 60(3) (2010), 906{918.##[13] F. Jin, L. Pei, H. Chen and L. Zhou, Intervalvalued intuitionistic fuzzy continuous weighted##entropy and its application to multicriteria fuzzy group decision making, KnowledgeBased##Systems, 59 (2014), 132{141.##[14] P. Liu, Some hamacher aggregation operators based on the intervalvalued intuitionistic fuzzy##numbers and their application to group decision making, IEEE Transaction on Fuzzy Systems,##22(1) (2014), 83{97.##[15] X. Ma, H. Qin, N. Sulaiman, T. Herawan and J. Abawajy, The parameter reduction of##the intervalvalued fuzzy soft sets and its related algorithms, IEEE trancsactions on Fuzzy##Systems, 22(1) (2014), 57{71.##[16] P. Maji, R. Biswas and A. Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics, 9(3) (2001),##[17] P. Maji, R. Biswas and A. Roy, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics,##9(3) (2001), 677{692.##[18] P. Maji and R. Biswas and A. Roy, Soft set theory, Computers & Mathematics with Appli##cations, 45(4) (2003), 555{562.##[19] P. Majumdar and S. Samanta, Similarity measure of soft sets, New Mathematics and Natural##Computation, 4(1) (2008), 1{12.##[20] P. Majumdar and S. Samanta, Generalised fuzzy soft sets, Computers & Mathematics with##Applications, 59(4) (2010), 1425{1432.##[21] F. Meng, X. Chen and Q. Zhang, Some intervalvalued intuitionistic uncertain linguistic##Choquet operators and their application to multiattribute group decision making, Applied##Mathematical Modelling, 38(9) (2014), 2543{2557.##[22] D. Molodtsov, Soft set theoryrst results, Computers & and Mathematics with Applications,##37(4) (1999), 19{31.##[23] S. Mousavi, B. Vahdani and S. Behzadi, Designing a model of intuitionistic fuzzy VIKOR##in multiattribute group decisionmaking problems, Iranian Journal of Fuzzy Systems, 13(1)##(2016), 45{65.##[24] S. Mousavi and B. Vahdani, Crossdocking location selection in distribution systems: a new##intuitionistic fuzzy hierarchical decision model, International Journal of Computational In##telligence Systems, 9(1) (2016), 91{109.##[25] S. Mousavi and H. Gitinavard and B. Vahdani, Evaluating construction projects by a new##group decisionmaking model based on intuitionistic fuzzy logic concepts, International Jour##nal of EngineeringTransactions C: Aspects, 28(9) (2015), 1312{1319.##[26] A. Roy and P. Maji, A fuzzy soft set theoretic approach to decision making problems, Journal##of Computational and Applied Mathematics, 203(2) (2007), 412{418.##[27] B. Vahdani, S. Mousavi, R. TavakkoliMoghaddam and H. Hashemi, A new design of the##elimination and choice translating reality method for multicriteria group decisionmaking in##an intuitionistic fuzzy environment, Applied Mathematical Modelling, 37(4) (2013), 1781{##[28] W. Wang and X. Liu, The multiattribute decision making method based on intervalvalued##intuitionistic fuzzy Einstein hybrid weighted geometric operator, Computers & Mathematics##with Applications, 66(10) (2013), 1845{1856.##[29] H.Wu and X. Su, Threat assessment of aerial targets based on group generalized intuitionistic##fuzzy soft sets, Control and Decision, 30(8) (2015), 1462{1468.##[30] J. Wu and F. Chiclana, A risk attitudinal ranking method for intervalvalued intuitionistic##fuzzy numbers based on novel attitudinal expected score and accuracy functions, Applied Soft##Computing, 22 (2014), 272{286.##[31] Z. Xu, Method for aggregating intervalvalued intuitionistic fuzzy information and their ap##plication to decision making, Control and Decision, 22(2) (2007), 215{219.##[32] Z. Xu, A method based on distance measure for intervalvalued intuitionistic fuzzy group##decision making, Information Sciences, 180(1) (2010), 181{190.##[33] Z. Xu and J. Chen, An approach to group decision making based on intervalvalued intuition##istic judgment matrices, Systems EngineeringTheory & Practice, 27(4) (2007), 126{133.##[34] Z. Yue, A group decision making approach based on aggregating interval data into interval##valued intuitionistic fuzzy information, Applied Mathematical Modelling, 38(2) (2014), 683{##[35] Z. Yue and Y. Jia, A group decision making model with hybrid intuitionistic fuzzy informa##tion, Computers & Industrial Engineering, 87 (2015), 202{212.##[36] L. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.##[37] X. Zhang and Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS ap##proach to intervalvalued intuitionistic fuzzy group decision making, Applied Soft Computing,##26 (2015), 42{56.##]
Soft Computing Based on a Modified MCDM Approach under Intuitionistic Fuzzy Sets
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The current study set to extend a new VIKOR method as a compromise ranking approach to solve multiple criteria decisionmaking (MCDM) problems through intuitionistic fuzzy analysis. Using compromise method in MCDM problems contributes to the selection of an alternative as close as possible to the positive ideal solution and far away from the negative ideal solution, concurrently. Using Atanassov intuitionistic fuzzy sets (AIFSs) may simultaneously express the degree of membership and nonmembership to decision makers (DMs) to describe uncertain situations in decisionmaking problems. The proposed intuitionistic fuzzy VIKOR indicates the degree of satisfaction and dissatisfaction of each alternative with respect to each criterion and the relative importance of each criterion, respectively, by degrees of membership and nonmembership. Thus, the ratings for the importance of criteria, DMs, and alternatives are in linguistic variables and expressed in intuitionistic fuzzy numbers. Using IFS aggregation operators and with respect to subjective judgment and objective information, the most suitable alternative is indicated among potential alternatives. Moreover, practical examples illustrate the procedure of the proposed method.
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M. R.
Shahriari
Faculty of Management, South Tehran Branch, Islamic Azad University, Tehran, Iran
Faculty of Management, South Tehran Branch,
Iran
Multiple criteria decision making (MCDM)
Decision makers (DMs)
Atanassov intuitionistic fuzzy sets (AIFSs)
Intuitionistic fuzzy numbers
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Vahdani An extended compromise ratio model##with an application to reservoir ##ood control operation under an intervalvalued intuitionistic##fuzzy environment, Applied Mathematical Modelling, 38(14) (2014), 34953511.##[17] A. Jahan, M. Faizal, I. Md Yusof, S. M. Sapuan, and M. Bahraminasab, A comprehensive##VIKOR method for material selection, Materials and Design, 32(3) (2011), 12151221.##[18] L. James JH, C. Y. Tsai, R. H. Lin and and G. H. Tzeng, A modifed VIKOR multiplecriteria##decision method for improving domestic airlines service quality, Journal of Air Transport##Management, 17(2) (2011), 5761.##[19] G. S. Liang and M. J. J. Wang, A fuzzy multicriteria decisionmaking method for facility##site selection, The International Journal of Production Research, 29(11) (1991), 23132330. ##[20] S. M. Mousavi, F. Jolai, R. TavakkoliMoghaddam and B. Vahdani, A fuzzy grey model based##on the compromise ranking for multicriteria group decision making problems in manufac##turing systems, Journal of Intelligent and Fuzzy Systems, 24(4) (2013), 819827.##[21] V. Mohagheghi, S. M. Mousavi and B. Vahdani, A new optimization model for project port##folio selection under intervalvalued fuzzy environment, Arabian Journal for Science and##Engineering, 40(11) (2015), 33513361.##[22] S. M. Mousavi, H. Gitinavard and B. Vahdani, Evaluating construction projects by a new##group decisionmaking model based on intuitionistic fuzzy logic concepts, International Journal##of EngineeringTransactions C: Aspects, 28(9) (2015), 13121319.##[23] S. M. Mousavi, B. Vahdani, M. Amiri, R. TavakkoliMoghaddam, S. Ebrahimnejad and##M. Amiri A multistage decision making process for multiple attributes analysis under an##intervalvalued fuzzy environment, International Journal of Advanced Manufacturing Technology,##64(912) (2013), 12631273.##[24] S. M. Mousavi B. Vahdani and S. Sadigh Behzadi, Designing a model of intuitionistic fuzzy##VIKOR in multiattribute group decisionmaking problems, Iranian Journal of Fuzzy Systems,##13(1) (2016), 4565.##[25] S. M. Mousavi and B. Vahdani Crossdocking location selection in distribution systems: a##new intuitionistic fuzzy hierarchical decision model, International Journal of Computational##Intelligence Systems, 9(1) (2016), 91109.##[26] S. M. Mousavi, B. Vahdani, R. TavakkoliMoghaddam and N. Tajik, Soft computing based##on a fuzzy grey compromise solution approach with an application to the selection problem of##material handling equipment, International Journal of Computer Integrated Manufacturing,##27(6) (2014), 547569.##[27] S. Opcorvic, Multicriteria optimization of Civil engineering systems, Faculty of Civil Engineering,##2(1) (1998), 521.##[28] S. Opricovic and G. H. Tzeng, Multicriteria planning of postearthquake sustainable recon##struction, omputerAided Civil and Infrastructure Engineering, 17(3) (2002), 211220.##[29] S. Opricovic and G. H. Tzeng, The compromise solution by MCDM methods: A comparative##analysis of VIKOR and TOPSIS, European Journal of Operational Research, 156(2) (2004),##[30] S. Opricovic and G. H. Tzeng, Extended VIKOR method in comparison with outranking##methods, European Journal of Operational Research, 178(2) (2007), 514529.##[31] V. Roshanaei, B. Vahdani, S. M. Mousavi, M. Mousakhani and G. Zhang, CAD/CAM system##selection: A multicomponent hybrid fuzzy MCDM model, Arabian Journal for Science and##Engineering, 38(9) (2013), 25792594 .##[32] M. Salimi, B. Vahdani, S. M . Mousavi and R. TavakkoliMoghaddam, A new method based on##multisegment decision matrix for solving decision making problems, Scientia Iranica, 20(6)##(2013), 22592274.##[33] J. R. San Cristobal, Multicriteria decisionmaking in the selection of a renewable energy##project in Spain: The VIKOR method, Renewable Energy, 36(2) (2011), 498502.##[34] B. Vahdani, H. Hadipour, J. S. Sadaghiani and M. Amiri, Extension of VIKOR method##based on intervalvalued fuzzy sets, The International Journal of Advanced Manufacturing##Technology, 47(912) (2010), 12311239.##[35] B. Vahdani and H. Hadipour, Extension of the ELECTRE method based on intervalvalued##fuzzy sets, Soft Computing, 15(3) (2011), 569579.##[36] B. Vahdani, M. Salimi, and M. Charkhchian, A new FMEA method by integrating fuzzy##belief structure and TOPSIS to improve risk evaluation process, The International Journal##of Advanced Manufacturing Technology, 77(14) (2015), 357368.##[37] B. Vahdani, M. Zandieh and A. AlemTabriz, Supplier selection by balancing and ranking##method, Journal of Applied Sciences, 8(19) (2008), 34673472.##[38] B. Vahdani, M. Zandieh, and R. TavakkoliMoghaddam, Two novel FMCDM methods for##alternativefuel buses selection, Applied Mathematical Modelling, 35(3) (2011), 13961412. ##[39] B. Vahdani, S. M. Mousavi and S. Ebrahimnejad, Soft computingbased preference selection##index method for human resource management, Journal of Intelligent and Fuzzy Systems,##26(1) (2014), 393403.##[40] B. Vahdani, H. Hadipour and R. TavakkoliMoghaddam, Soft computing based on interval##valued fuzzy ANPA novel methodology, Journal of Intelligent Manufacturing, 23(5) (2012),##15291544.##[41] B. Vahdani, M. Salimi and S. M. Mousavi, A new compromise decisionmaking model based##on TOPSIS and VIKOR for solving multiobjective largescale programming problems with a##block angular structure under uncertainty, International Journal of Engineering Transactions##B: Applications,27(11) (2014), 16731680.##[42] B. Vahdani, M. Salimi and S. M. Mousavi, A compromise decisionmaking model based on##VIKOR for multiobjective largescale nonlinear programming problems with a block angular##structure under uncertainty, Scientia Iranica. Transaction E, Industrial Engineering,22(6)##(2015), 25712584.##[43] B. Vahdani, S. M. Mousavi and R. TavakkoliMoghaddam, Group decision making based##on novel fuzzy modied TOPSIS method, Applied Mathematical Modelling, 35(9) (2011),##42574269.##[44] B. Vahdani, S. M. Mousavi, R. TavakkoliMoghaddam and H. Hashemi, A new design of the##elimination and choice translating reality method for multicriteria group decision making##in an intuitionistic fuzzy environment, Applied Mathematical Modeling ,37(4) (2013), 1781##[45] B, Vahdani, S. M. Mousavi, H. Hashemi, M. Mousakhani and R. TavakkoliMoghaddam, A##new compromise solution method for fuzzy group decisionmaking problems with an appli##cation to the contractor selection, Engineering Applications of Articial Intelligence, 26(2)##(2012), 779788.##[46] B. Vahdani and M. Zandieh Selecting suppliers using a new fuzzy multiple criteria deci##sion model: the fuzzy balancing and ranking method, International Journal of Production##Research, 48(18) (2010) 53075326.##[47] B. Vahdani, S. M. Mousavi, R. TavakkoliMoghaddam, A. Ghodratnama and M. Mohammadi,##Robot selection by a multiple criteria complex proportional assessment method un##der an intervalvalued fuzzy environment, International Journal of Advanced Manufacturing##Technology, 73(58) (2014), 687697.##[48] B. Vahdani, R. TavakkoliMoghaddam, S. M. Mousavi and A. Ghodratnama, Soft computing##based on new intervalvalued fuzzy modied multicriteria decisionmaking method, Applied##Soft Computing, 13(1) (2013), 165172.##[49] Y. Y. Wu and D. J. Yu, Extended VIKOR for Multicriteria decision making problems under##intuitionistic environment, 18th International Conference on Management Science & Engineering,##[50] Z. Xu, An overview of methods for determining OWA weights, International Journal of Intelligent##Systems,20(8) (2005), 843865.##[51] Z. Xu and R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy##sets, International Journal of General Systems, 35(4) (2006), 417433.##[52] Z. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems, 15(6)##(2007), 11791187.##[53] D. Yong, Plant location selection based on fuzzy TOPSIS, The International Journal of Advanced##Manufacturing Technology ,28(78) (2006), 839844.##[54] L. A. Zadeh, Fuzzy sets, Information and control , 8(3) (1965), 338353.##]
Support vector regression with random output variable and probabilistic constraints
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Support Vector Regression (SVR) solves regression problems based on the concept of Support Vector Machine (SVM). In this paper, a new model of SVR with probabilistic constraints is proposed that any of output data and bias are considered the random variables with uniform probability functions. Using the new proposed method, the optimal hyperplane regression can be obtained by solving a quadratic optimization problem. The proposedmethod is illustrated by several simulated data and real data sets for both models (linear and nonlinear) with probabilistic constraints.
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Maryam
Abaszade
Department of Statistics, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Statistics, Ferdowsi University
Iran


Sohrab
Effati
Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
Department of Applied Mathematics, Ferdowsi
Iran
seffati.profcms@um.ac.ir
Probabilistic constraints
Support Vector Machine
Support Vector Regression
Quadratic programming
Probability function
Monte Carlo simulation
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Vandewalle, Least squares support vector machine classifiers, Neural##Processing Letters, 9(3) (1999), 293–300.##[36] T. B. Trafalis and S. A. Alwazzi, Support vector regression with noisy data: a second order##cone programming approach, Int. J. General Syst., 36 (2007), 237–250.##[37] T. B. Trafalis and R. C. Gilbert, Robust classification and regression using support vector##machines, Eur. J. Oper. Res., 173(3) (2006), 893–909.##[38] URL http://www.csie.ntu.edu.tw/ cjlin/libsvmtools/datasets/regression.html.##[39] URL http://www.dcc.fc.up.pt/ ltorgo/Regression/DataSets.html.##[40] URL http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml.##[41] V. Vapnik, The nature of statistical learning theory, SpringerVerlag, New York, (1995),##123146, 181186.##[42] V. Vapnik, S. Golowich and A. Smola, Support vector method for multivariate density estimation,##Adv. Neural Inform. Process. Syst., 12 (1999), 659–665.##[43] Y. Xu and L. Wang, A weighted twin support vector regression, KnowledgeBased Syst., 33##(2012), 92–101.##[44] Y. Xu, W. Xi, X. Lv and R. Guo, An improved least squares twin support vector machine,##Journal of information and computational science, 9(4) (2012), 1063–1071.##[45] X. Yang, L. Tan and L. He, A robust least squares support vector machine for regression and##classification with noise, Neurocomputing, 140 (2014), 41–52.##]
A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers
2
2
In this paper, we determine necessary and sufficient Tauberian conditions under which convergence in Pringsheim's sense of a double sequence of fuzzy numbers follows from its $(C,1,1)$ summability. These conditions are satisfied if the double sequence of fuzzy numbers is slowly oscillating in different senses. We also construct some interesting double sequences of fuzzy numbers.
1

61
75


Ibrahim
Canak
Department of Mathematics, Ege University, 35100, Izmir, Turkey
Department of Mathematics, Ege University,
Turkey
ibrahimcanak@yahoo.com


Umit
Totur
Department of Mathematics, Adnan Menderes University, 09100, Aydin,
Turkey
Department of Mathematics, Adnan Menderes
Turkey
utotur@adu.edu.tr


Zerrin
Onder
Department of Mathematics, Ege University, 35100, Izmir, Turkey
Department of Mathematics, Ege University,
Turkey
zerrin.onder11@gmail.com
Fuzzy numbers
Double sequences
Slow oscillation
Summability $(C
1
1)$
Tauberian theorems
[[1] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, Berlin, 2013.##[2] _I. C.##anak, Tauberian theorems for Cesaro summability of sequences of fuzzy number, J. Intell.##Fuzzy Syst., 27(2) (2014), 937{942.##[3] _I. C.##anak, On Tauberian theorems for Cesaro summability of sequences of fuzzy numbers, J.##Intell. Fuzzy Syst., 30(5) (2016), 2657{2662.##[4] _I. C.##anak, Holder summability method of fuzzy numbers and a Tauberian theorem, Iranian##Journal of Fuzzy Systems, 11(4) (2014), 87{93.##anak, Some conditions under which slow oscillation of a sequence of fuzzy numbers follows##from Cesaro summability of its generator sequence, Iranian Journal of Fuzzy Systems, 11(4)##(2014) 15{22.##[6] D. Dubois and H. Prade, Fuzzy sets and systems: Theory and applications, Academic Press,##New YorkLondon, 1980.##[7] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18(1) (1986)##[8] M. Matlako, Sequences of fuzzy numbers, BUSEFAL, 28 (1986), 28{37.##[9] F. Moricz, Tauberian theorems for Cesaro summable double sequences, Studia Math., 110##(1994), 83{96.##[10] F. Moricz, Necessary and sufficient Tauberian conditions, under which convergence follows##from summability (C; 1), Bull. London Math. Soc., 26 (1994), 288{294.##[11] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst., 33 (1989), 123{126.##[12] E. Savas.##, A note on double sequences of fuzzy numbers, Turkish J. Math., 20 (1996), 175{178.##[13] P. V. Subrahmanyam, Cesaro summability of fuzzy real numbers, J. Anal., 7 (1999), 159{168.##[14] B. C. Tripathy and A. J. Dutta, On fuzzy realvalued double sequence spaces, Soochow J.##Math., 32(4) (2006), 509{520.##[15]O. Talo and F. Bas.ar, On the slowly decreasing sequences of fuzzy numbers, Abstr. Appl.##Anal., Article ID 891986 (2013), doi:10.1155/2013/891986, 17.##[16] O. Talo and C. C.##akan, On the Cesaro convergence of sequences of fuzzy numbers, Appl.##Math. Lett., 25 (2012), 676{681.##[17] O. Talo and C. C.##akan, Tauberian theorems for statistically (C; 1)convergent sequences of##fuzzy numbers, Filomat, 28(4) (2014), 849{858.##[18] B. C. Tripathy and A. Baruah, Norlund and Riesz mean of sequences of fuzzy real numbers,##Appl. Math. Lett., 23 (2010), 651{655. ##[19] B. C. Tripathy and A. J. Dutta, Bounded variation double sequence space of fuzzy real##numbers, Comput. Math. Appl., 59(2) (2010), 1031{1037.##[20] B. C. Tripathy and B. Sarma, Double sequence spaces of fuzzy numbers dened by Orlicz##function, Acta Math. Sci. Ser. B Engl. Ed., 31(1) (2011), 134{140.##[21] B. C. Tripathy and M. Sen, On lacunary strongly almost convergent double sequences of fuzzy##numbers, An. Univ. Craiova Ser. Mat. Inform., 42(2) (2015), 254{259.##[22] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 29{44.##]
Some topological properties of spectrum of fuzzy submodules
2
2
Let $R$ be a commutative ring with identity and $M$ be an$R$module. Let $FSpec(M)$ denotes the collection of all prime fuzzysubmodules of $M$. In this regards some basic properties of Zariskitopology on $FSpec(M)$ are investigated. In particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$FSpec(M)$ is a $T_0$space or Hausdorff.
1

77
87


R.
Ameri
School of Mathematics, Statistics and Computer Science, College of
Sciences, University of Tehran, Teheran, Iran
School of Mathematics, Statistics and Computer
Iran
rez_ameri@yahoo.com


R.
Mahjoob
Department of Mathematics, Semnan University, Semnan, Iran
Department of Mathematics, Semnan University,
Iran
ra−mahjoob@yahoo.com
Fuzzy prime submodule
Fuzzy prime spectrum
Zariski topology
Irreducible subset
[[1] R. Ameri, Some properties of zariski topology of multiplication modules, Houston Journal of##Mathematics, 36(2) (2009), 337344.##[2] R. Ameri and R. Mahjoob, Prime spectrum of LSubmodules, Fuzzy Sets and Systems, 159(9)##(2008), 11071115.##[3] R. Ameri and R. Mahjoob, Zariski topology on the spectrum of prime Lsubmodules, Soft##Comput., 12(9) (2008), 901908.##[4] S. K. Bhambri, R. Kumar and P. Kumar,Fuzzy prime submodules and radical of a fuzzy##submodules, Bull. Cal. Math. Soc., 87 (1993), 163168.##[5] V. N. Dixit, R. Kummar and N. Ajmal,Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy##Sets and Systems, 44 (1991), 127138.##[6] J. A. Goguen, Lfuzzy sets, Journal Math. Appl., 18 (1967) 145174.##[7] H. HadjiAbadi and M. M. Zahedi, Some results on fuzzy prime spectrum of a ring, Fuzzy##Sets and Systems, 77 (1996), 235240.##[8] R. Kumar, Fuzzy prime spectrum of a ring, Fuzzy Sets and Systems, 46 (1992), 147154.##[9] R. Kumar and J. K. Kohli,Fuzzy prime spectrum of a ring II, Fuzzy Sets and Systems, 59##(1993), 223230.##[10] H. V. Kumbhojkar,Some comments on spectrum of prime fuzzy ideals of a ring, Fuzzy Sets##and Systems, 85 (1997), 109114. ##[11] H. V. Kumbhojkar,Spectrum of prime fuzzy ideals, Fuzzy Sets and Systems, 62 (1994), 101##[12] Chin. Pi. Lu,Prime submodules of modules, Comm. Math. Univ., 33 (1987), 6169.##[13] Chin.Pi. Lu, The zariski topology on the spectrum of a modules, Houston Journal of Mathe##matics, 25(3) (1999), 417432.##[14] Chin.Pi. Lu,Spectra of modules, Comm. in Algebra, 23(10) (1995) 37413752.##[15] R. L. McCasland, M. E. Moore and P. F. Smith,On the Spectrum of Modules Over a Com##mutative Ring, Communications in Algebra, 25(1) (1997), 79103.##[16] John. N. Mordeson and D. S. Malik,Fuzzy Commutative Algebra, World Scientic Publishing##Co. Pet. Ltd, 1998.##[17] T. K. Mukherjee and M. K. Sen,On fuzzy ideals of a ring I; Fuzzy Sets and systems, 21##(1987), 99104.##[18] C. V. Negoita and D. A. Ralescu, Application of fuzzy systems analysis, Basel and Stuttgart,##Birkhauser Verlag; New York, Wiley Halstead, (1975), pp. 191.##[19] F. Z. Pan, Fuzzy nitely generated modules, Fuzzy Sets and Systems, 21 (1987), 105113.##[20] R. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[21] F. I. Sidky, On radical of fuzzy submodules and primary fuzzy submodules, Fuzzy Sets and##Systems, 119 (2001), 419425.##[22] L. A. Zadeh, Fuzzy sets, Inform and Control, 8 (1965), 338353.##]
ON LOCAL HUDETZ gENTROPY
2
2
In this paper, a local approach to the concept of Hudetz $g$entropy is presented. The introduced concept is stated in terms of Hudetz $g$entropy. This representation is based on the concept of $g$ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.
1

89
97


M.
Rahimi
Department of Mathematics, Faculty of Science, University of Qom, Qom,
Iran
Department of Mathematics, Faculty of Science,
Iran
m10.rahimi@gmail.com
$g$entropy
$g$ergodic decomposision
Hudetz correction
[[1] D. Dumitrescu, Measurepreserving transformation and the entropy of a fuzzy partition, 13th##Linz Seminar on Fuzzy Set Theory, (Linz, 1991), 2527.##[2] D. Dumitrescu, Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl.,##176 (1993), 359373.##[3] D. Dumitrescu, Entropy of a fuzzy process, Fuzzy Sets and Systems, 55 (1993), 169177. ##[4] D. Dumitrescu, Entropy of fuzzy dynamical systems, Fuzzy Sets and Systems, 70 (1995),##[5] T. Hudetz, Spacetime dynamical entropy of quantum systems, Lett. Math. Phys., 16 (1988),##[6] T. Hudetz, Algebraic topological entropy, In: Eds., G. A. Leonov et al. Eds., Nonlinear##Dynamics and Quantum Dynamical Systems, (Akademie Verlag, Berlin, 1990), 110124.##[7] D. Markechova, The entropy on Fquantum spaces, Math. Slovaca, 40 (1990), 177190.##[8] D. Markechova, The entropy of fuzzy dynamical systems and generators, Fuzzy Sets and##Systems, 48 (1992), 351363.##[9] D. Markechova, Entropy of complete fuzzy partitions, Math. Slovaca, 43(1) (1993), 110.##[10] D. Markechova, A note to the KolmogorovSinaj entropy of fuzzy dynamical systems, Fuzzy##Sets and Systems, 64 (1994), 8790.##[11] R. Mesiar and J. Rybarik, Entropy of fuzzy partitions: A general model, Fuzzy Sets and##Systems, 99 (1998), 7379.##[12] M. Rahimi and A. Riazi, On local entropy of fuzzy partitions, Fuzzy Sets and Systems, 234##(2014), 97108##[13] M. Rahimi and A. Riazi, Fuzzy entropy of action of semigroups, Math. Slovaca, 66(5)##(2016), 11571168.##[14] M. Rahimi, A local approach to gentropy, Kybernetica, 51(2) (2015), 231245.##[15] B. Riecan, On a type of entropy of dynamical systems, Tatra Mountains Math. Publ., 1##(1992), 135140.##[16] B. Riecan and D. Markechova, The entropy of fuzzy dynamical systems, general scheme and##generators, Fuzzy Sets and Systems, 96 (1998), 191199.##[17] B. Riecan, On the gentropy and its Hudetz correction, Kybernetika, 38(4) (2002), 493500.##[18] J. Rybarik, The entropy of the QFdynamical systems, Busefal, 48 (1991), 2426.##[19] J. Rybarik, The entropy based on pseudoarithmetical operations, Tatra Mountains Math.##Publ., 6 (1995), 157164.##]
Probabilistic Normed Groups
2
2
In this paper, we introduce the probabilistic normed groups. Among other results, we investigate the continuityof inner automorphisms of a group and the continuity of left and right shifts in probabilistic groupnorm. We also study midconvex functions defined on probabilistic normed groups and give some results about locally boundedness of such functions.
1

99
113


Kourosh
Nourouzi
Faculty of Mathematics, K.N.Toosi University of Technology,
P.O.Box 163151618, Tehran, Iran.
Faculty of Mathematics, K.N.Toosi University
Iran
nourouzi@kntu.ac.ir


Alireza
Pourmoslemi
Department of Mathematics, Payame Noor University, P.O.BOX
193953697, Tehran, Iran.
Department of Mathematics, Payame Noor University,
Iran
a_pourmoslemy@pnu.ac.ir
Probabilistic normed groups
Invariant probabilistic metrics
Distributionalslowly varying functions
Midconvex functions
[[1] C. Alsina, B. Schweizer and A. Sklar, On the denition of a probabilistic normed space,##Aequationes Math, 46(2) (1993), 91{98.##[2] N. H. Bingham and A. J. Ostaszewski, Normed versus topological groups: dichotomy and##duality, Dissertationes Math, 472 (2010), 138p.##[3] G. Birkho, A note on topological groups, Compositio Math, 3 (1936), 427{430.##[4] D. R. Farkas, The algebra of norms and expanding maps on groups, J. Algebra, 133(2)##(1990), 386{403. ##[5] M. Frechet, Sur quelques points du calcul fonctionnel, Rendiconti de Circolo Matematico di##Palermo, 22 (1906), 1{74.##[6] S. Kakutani, Uber die Metrisation der topologischen Gruppen, (German) Proc. Imp. Acad,##12(4) (1936), 82{84. (also in Selected Papers, Vol. 1, ed. R. Robert Kallman, Birkhuser,##(1986), 60{62.)##[7] V. L. Klee, Invariant metrics in groups (solution of a problem of Banach), Proc. Amer.##Math. Soc, 3 (1952), 484{487.##[8] E. Klement, R. Mesiar and E. Pap, Triangular norms, Trends in Logica{Studia Logica Library,##Kluwer Academic Publishers, Dordrecht, 8 (2000).##[9] M. Kuczma, An introduction to the theory of functional equations and inequalities, Cauchy's##equation and Jensen's inequality, Second edition, Birkhauser Verlag, Basel, 2009.##[10] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U. S. A, 28 (1942), 535{537.##[11] A. A. Pavlov, Normed groups and their application to noncommutative dierential geometry,##J. Math. Sci., 113(5) (2003), 675{682.##[12] B. J. Pettis, On continuity and openness of homomorphisms in topological groups, Ann. of##Math, 52(2) (1950), 293{308.##[13] B. Schweizer and A. Sklar, Probabilistic metric spaces, NorthHolland Series in Probability##and Applied Mathematics, NorthHolland Publishing Co., New York, 1983.##[14] A. N. Serstnev, On the concept of a stochastic normalized space, (Russian), Dokl. Akad.##Nauk SSSR, 149 (1963), 280{283.##[15] D. A. Sibley, A metric for weak convergence of distribution functions, Rocky Mountain J.##Math, 1(3) (1971), 427{430.##]
Implications, coimplications and left semiuninorms on a complete lattice
2
2
In this paper, we firstly show that the $N$dual operation of the right residual implication, which is induced by a leftconjunctive right arbitrary $vee$distributive left semiuninorm, is the right residual coimplication induced by its $N$dual operation. As a dual result, the $N$dual operation of the right residual coimplication, which is induced by a leftdisjunctive right arbitrary $wedge$distributive left semiuninorm, is the right residual implication induced by its $N$dual operation. Then, we demonstrate that the $N$dual operations of the left semiuninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left semiuninorms. Finally, we reveal the relationships between conjunctive right arbitrary $vee$distributive left semiuninorms induced by implications and disjunctive right arbitrary $wedge$distributive left semiuninorms induced by coimplications, where both implications and coimplications satisfy the neutrality principle.
1

115
130


Yuan
Wang
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
College of Information Engineering, Yancheng
China
yctuwangyuan@163.com


Keming
Tang
College of Information Engineering, Yancheng Teachers University,
Jiangsu 224002, People's Republic of China
College of Information Engineering, Yancheng
China
tkmchina@126.com


Zhudeng
Wang
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, People's Republic of China
School of Mathematics and Statistics, Yancheng
China
zhudengwang2004@163.com
Fuzzy connective
Implication
Coimplication
Left semiuninorm
Neutrality principle
[[1] M. Baczynski and B. Jayaram, Fuzzy implications, Studies in Fuzziness and Soft Computing,##Springer, Berlin, 231 (2008).##[2] S. Burris and H. P. Sankappanavar, A course in universal algebra, World Publishing Corporation,##Beijing 1981.##[3] B. De Baets, Coimplicators, the forgotten connectives, Tatra Mountains Mathematical Publications,##12 (1997), 229{240.##[4] B. De Baets and J. Fodor, Residual operators of uninorms, Soft Computing, 3 (1999), 89{100.##[5] F. Durante, E. P. Klement, R. Mesiar and C. Sempi, Conjunctors and their residual impli##cators: characterizations and construction methods, Mediterranean Journal of Mathematics,##4 (2007), 343{356.##[6] P. Flondor, G. Georgescu and A. Lorgulescu, Pseudotnorms and pseudoBLalgebras, Soft##Computing, 5 (2001), 355{371.##[7] J. Fodor and T. Keresztfalvi, Nonstandard conjunctions and implications in fuzzy logic,##International Journal of Approximate Reasoning, 12 (1995), 69{84.##[8] J. C. Fodor and M. Roubens, Fuzzy preference modelling and multicriteria decision support,##Theory and Decision Library, Series D: System Theory, Knowledge Engineering and Problem##Solving, Kluwer Academic Publishers, Dordrecht, 1994.##[9] J. Fodor, R. R. Yager and A. Rybalov, Structure of uninorms, International Journal of##Uncertainty, Fuzziness and KnowledgeBased Systems, 5 (1997), 411{427.##[10] D. Gabbay and G. Metcalfe, Fuzzy logics based on [0; 1)continuous uninorms, Archive for##Mathematical Logic, 46 (2007), 425{449.##[11] G. Gratzer, Lattice theory: foundation, Birkhauser, Springer Basel AG, 2011.##[12] S. Jenei and F. Montagna, A general method for constructing leftcontinuous tnorms, Fuzzy##Sets and Systems, 136 (2003), 263{282.##[13] H. W. Liu, Semiuninorm and implications on a complete lattice, Fuzzy Sets and Systems,##191 (2012), 72{82.##[14] Z. Ma and W. M.Wu, Logical operators on complete lattices, Information Sicences, 55 (1991),##[15] M. Mas, M. Monserrat and J. Torrens, On left and right uninorms, International Journal of##Uncertainty, Fuzziness and KnowledgeBased Systems, 9 (2001), 491{507.##[16] M. Mas, M. Monserrat and J. Torrens, On left and right uninorms on a nite chain, Fuzzy##Sets and Systems, 146 (2004), 3{17.##[17] M. Mas, M. Monserrat and J. Torrens, Two types of implications derived from uninorms,##Fuzzy Sets and Systems, 158 (2007), 2612{2626. ##[18] D. Ruiz and J. Torrens, Residual implications and coimplications from idempotent uninorms,##Kybernetika, 40 (2004), 21{38.##[19] Y. Su and H. W. Liu, Characterizations of residual coimplications of pseudouninorms on a##complete lattice, Fuzzy Sets and Systems, 261 (2015), 44{59.##[20] Y. Su and Z. D. Wang, Pseudouninorms and coimplications on a complete lattice, Fuzzy##Sets and Systems, 224 (2013), 53{62.##[21] Y. Su and Z. D. Wang, Constructing implications and coimplications on a complete lattice,##Fuzzy Sets and Systems, 247 (2014), 68{80.##[22] Y. Su, Z. D. Wang and K. M. Tang, Left and right semiuninorms on a complete lattice,##Kybernetika, 49 (2013), 948{961.##[23] Z. D. Wang and J. X. Fang, Residual operators of left and right uninorms on a complete##lattice, Fuzzy Sets and Systems, 160 (2009), 22{31.##[24] Z. D. Wang and J. X. Fang, Residual coimplicators of left and right uninorms on a complete##lattice, Fuzzy Sets and Systems, 160 (2009), 2086{2096.##[25] R. R. Yager, Uninorms in fuzzy system modeling, Fuzzy Sets and Systems, 122 (2001),##[26] R. R. Yager, Defending against strategic manipulation in uninormbased multiagent decision##making, European Journal of Operational Research, 141 (2002), 217{232.##[27] R. R. Yager and A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80##(1996), 111{120.##]
Structural properties of fuzzy graphs
2
2
Matroids are important combinatorial structures and connect closelywith graphs. Matroids and graphs were all generalized to fuzzysetting respectively. This paper tries to study connections betweenfuzzy matroids and fuzzy graphs. For a given fuzzy graph, we firstinduce a sequence of matroids from a sequence of crisp graph, i.e.,cuts of the fuzzy graph. A fuzzy matroid, named graph fuzzy matroid,is then constructed by using the sequence of matroids. An equivalentdescription of graphic fuzzy matroids is given and their propertiesof fuzzy bases and fuzzy circuits are studied.
1

131
144


Xiaonan
Li
School of Mathematics and Statistics, Xidian University, Xi'an, 710071,
Shaanxi, China
School of Mathematics and Statistics, Xidian
China
xnli@xidian.edu.cn


Huangjian
Yi
School of Information and Technology, Northwest University, Xi'an,
710069, Shaanxi, China
School of Information and Technology, Northwest
China
yhj255@163.com
Fuzzy graph
Partial fuzzy subgraph
Cycle
Fuzzy matroid
[[1] P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letters, 6 (1987), 297##[2] K. R. Bhutani and A. Rosenfeld, Strong arcs in fuzzy graphs, Information Sciences, 152##(2003), 319322.##[3] K. R. Bhutani and A. Rosenfeld, Fuzzy end nodes in fuzzy graphs, Information Sciences, 152##(2003), 323326.##[4] K. R. Bhutani and A. Rosenfeld, On Mstrong fuzzy graphs, Information Sciences, 155 (2003),##[5] M. Blue, B. Bush and J. Puckett, Unied approach to fuzzy graph problems, Fuzzy Sets and##Systems, 125 (2002), 355368.##[6] R. Goetschel and W. Voxman, Fuzzy matroids, Fuzzy Sets and Systems, 27 (1988), 291302.##[7] R. Goetschel and W. Voxman, Bases of fuzzy matroids, Fuzzy Sets and Systems, 31 (1989),##[8] R. Goetschel and W. Voxman, Fuzzy circuits, Fuzzy Sets and Systems, 32 (1989), 3543.##[9] R. Goetschel and W. Voxman, Fuzzy matroids and a greedy algorithm, Fuzzy Sets and Sys##tems, 37 (1990), 201213.##[10] R. Goetschel and W. Voxman, Fuzzy rank functions, Fuzzy Sets and Systems, 42 (1991),##[11] C. E. Huang, Graphic and representable fuzzifying matroids, Proyecciones Journal of Math##ematics, 29 (2010), 1730.##[12] X. N. Li, S. Y. Liu and S. G. Li, Connecttedness of rened GVfuzzy matroids, Fuzzy Sets##and Systems, 161 (2010), 27092723.##[13] X. N. Li and H. J. Yi, Axioms for fuzzy bases of H fuzzy matroids, Journal of Intelligent and##Fuzzy Systems, 29 (2015), 19952001.##[14] S.G. Li, X. Xin, Y. L. Li, Closure axioms for a class of fuzzy matroids and cotowers of##matroids, Fuzzy Sets and Systems, 158 (2007), 12461257.##[15] L. X. Lu and W. W. Zheng, Categorical relations among matroids, fuzzy matroids and fuzzi##fying matroids, Iranian Journal of Fuzzy Systems, 7(1) (2010), 8189. ##[16] J. N. Mordeson and P. S. Nair, Fuzzy graphs and fuzzy hypergraphs, PhysicaVerlag, 2000.##[17] J. N. Mordeson and J. N. Peng, Operators on fuzzy graphs, Information Sciences, 79 (1994),##[18] J. G. Oxley, Matroid Theory, Oxford University Press, New York, 1992.##[19] A. Rosenfeld, Fuzzy graphs, In: L. A. Zadeh, K. S. Fu, M. Shimura(Eds.), Fuzzy sets and##Their Applications to Cognitive and Decision Processes, Academic Press, New York, (1975),##[20] F. G. Shi, A new approach to the fuzzication of matroids, Fuzzy Sets and Systems, 160##(2009), 696705.##[21] M. S. Sunitha and A. Vijayakumar, A characterization of fuzzy trees, Information Sciences,##113 (1999), 293300.##[22] H. Whitney, On the abstract properties of linear dependence, American Journal of Mathe##matics, 57 (1935), 509533.##[23] W. Yao, Basis axioms and circuits axioms for fuzzifying matroids, Fuzzy Sets and Systems,##161 (2010), 31553165.##]
MFUZZIFYING INTERVAL SPACES
2
2
In this paper, we introduce the notion of $M$fuzzifying interval spaces, and discuss the relationship between $M$fuzzifying interval spaces and $M$fuzzifying convex structures.It is proved that the category {bf MYCSA2} can be embedded in the category {bf MYIS} as a reflective subcategory, where {bf MYCSA2} and {bf MYIS} denote the category of $M$fuzzifying convex structures of $M$fuzzifying arity $leq 2$ and the category of $M$fuzzifying interval spaces, respectively. Under the framework of $M$fuzzifying interval spaces, subspaces and product spaces are presented and some of their fundamental properties are obtained.
1

145
162


ZhenYu
Xiu
College of Applied Mathematics, Chengdu University of Information
Technology, Chengdu 610000, P.R. China
College of Applied Mathematics, Chengdu University
China
xyz198202@163.com


FuGui
Shi
chool of Mathematics and Statistics, Beijing Institute of Technology,
Beijing 100081, P.R. China
chool of Mathematics and Statistics, Beijing
China
fugushi@bit.edu.cn
$M$fuzzifying interval spaces
$M$fuzzifying convex structures
$M$fuzzifying interval preserving functions
Subspaces
Product spaces
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COUNTING DISTINCT FUZZY SUBGROUPS OF SOME RANK3 ABELIAN GROUPS
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In this paper we classify fuzzy subgroups of a rank3 abelian group $G = mathbb{Z}_{p^n} + mathbb{Z}_p + mathbb{Z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) nonisomorphic maximal chains of subgroups and (v) classes of isomorphic fuzzy subgroups of $G$. Illustrative examples are provided.
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Isaac K.
Appiah
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
Department of Mathematics, University of
South Africa


B. B.
Makamba
Department of Mathematics, University of Fort Hare, ALICE, 5700,
South Africa
Department of Mathematics, University of
South Africa
bbmakamba@ufh.ac.za
Equivalence
Fuzzy subgroup
Maximal chain
Keychain
Distinguishing factor
Isomorphism
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Persiantranslation vol. 14, no. 1, February 2017
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